PERRY’S CHEMICAL ENGINEERS’ HANDBOOK SEVENTH EDITION McGraw-Hill New York San Francisco Washington, D.C. Auckland Bogotá Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo Toronto
Prepared by a staff of specialists under the editorial direction of Late Editor Robert H. Perry
Editor Don W. Green Deane E. Ackers Professor of Chemical and Petroleum Engineering, University of Kansas Associate Editor James O. Maloney Professor Emeritus of Chemical Engineering, University of Kansas
Library of Congress Cataloging-in-Publication Data Perry’s chemical engineers’ handbook. — 7th ed. / prepared by a staff of specialists under the editorial direction of late editor Robert H. Perry : editor, Don W. Green : associate editor, James O’Hara Maloney. p. cm. Includes index. ISBN 0-07-049841-5 (alk. paper) 1. Chemical engineering—Handbooks, manuals, etc. I. Perry, Robert H., date. II. Green, Don W. III. Maloney, James O. TP151.P45 1997 660—dc21 96-51648 CIP
Copyright © 1997, 1984, 1973, 1963, 1950, 1941, 1934 by The McGrawHill Companies, Inc. Copyright renewed 1962, 1969 by Robert H. Perry. All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. 1 2 3 4 5 6 7 8 9 0
DOW/DOW 9 0 2 1 0 9 8 7
ISBN 0-07-049841-5
INTERNATIONAL EDITION Copyright © 1997. Exclusive rights by The McGraw-Hill Companies, Inc., for manufacture and export. This book cannot be re-exported from the country to which it is consigned by McGraw-Hill. The International Edition is not available in North America. When ordering this title, use ISBN 0-07-115448-5.
The sponsoring editors for this book were Zoe Foundotos and Robert Esposito, the editing supervisor was Marc Campbell, and the production supervisor was Pamela A. Pelton. It was set in Caledonia by North Market Street Graphics. Printed and bound by R. R. Donnelley & Sons Company. This book was printed on acid-free paper. Information contained in this work has been obtained by The McGrawHill Companies, Inc. (“McGraw-Hill”) from sources believed to be reliable. However, neither McGraw-Hill nor its authors guarantee the accuracy or completeness of any information published herein, and neither McGraw-Hill nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information. This work is published with the understanding that McGraw-Hill and its authors are supplying information but are not attempting to render engineering or other professional services. If such services are required, the assistance of an appropriate professional should be sought.
Michael M. Ahbott, Ph.D., Howard P. Isermann Department of Chemical Engineering, Rensselaer Polytechnic Institute; Member, American Institute of Chemical Engineers (Section 4, Thermodynamics) Terry Allen, Ph.D., Senior Research Associate (retired), Du~ont Central Research and Development \Section 20, Size Reduction and Size Enlargement) John D. Dacha, Ph.D., Consulting Scientist, Chevron Products Company; Member, ASTM (American Society for TestiI1cgand Materials), Committee D02 on Petroleufll Products and Lubri"cants; American Chemical Society; International Association for Stability and Handling of Liquid Fuels, Steering Committee (Section 27, Energy Resources, Conversion, and Utilization) Glenn w. Baldwin, M.S., P.E., Staff Engineer, Union Carbide Corporatio~; Membe!, American Institute of Chemical Engineers (Section 12, Psychrometry, Evaporative Cooling, and Solids Drying) Scott D. Barnicki,
Ph.D., Senior Resear~h Chemical Engineer, Eastman Chemical Company
(Section 13, Distillation) Kenneth J. Bell, Ph.D., P.E., Regents Professor Emeritus, School of Chemical Engineering, Oklahoma State University; Member, American Institute of Chemical Engineers (Section 11, Heat- Transfer Equipment) Richard C. Bennett, B.S., Ch.E., Registered Professional Engineer, Illinois; Member, American Institute of Chemical Engineers (AIChE); President of Crystallization Technology, Inc.; FoJmer Presideptof Swenson Process Equipme~t, Inc. (Section 13, Liquid-Solid Operations and Equipment) Charles E. Benson, M.Eng., M.E., Director, Combustion Technology, Arthur D. Little, lIic.; Member, American Society of Mechanical Engineers; Comb~tion Institute (Section 27, Energy Resources, Conversion, and Utilization) Patrick M. Bernhagen, P.E., B.S.M.E., Senior Mechanical Engineer, Foster wheeler USA Corporation, American Society of Mechanical EngineeFs ( Section 11, Heat -Transfer Equipment) c
ix
x
LIST OF CONTRIBUTORS
Heinz P. Bloch, P.E., B.S.M.E., M.S.M.E., Consulting Engineer, Process Machinery Consulting; American Society of Mechanical Engineers, Vibration Institute; Registered Professional Engineer (New Jersey, Texas) (Section 29, Process Machinery Drives) Frank T. Bodurtha, Sc.D., E.I. DuPont de Nemours and Co., Inc. (retired), Wilmington, Delaware (retired); Consultant, Frank T. Bodurtha, Inc. (Section 26, Process Safety) Meherwan P. Boyce, P.E., Ph.D., President, Boyce Engineering International; ASME Fellow; Registered Professional Engineer (Texas, Oklahoma) (Section 10, Transport and Storage of Fluids; Section 29, Process Mochinery Drives), Laurence G. Britton, Process Safety)
Ph.D.,
Research Scientist, Union Carbide Corporation
(Section 26,
Evan Buck, M.S.Ch.E., Manager, Thermophysical Property Skill Center, Central Technology, Union Carbide Corporation (Section 2, Physical and Chemical Data) Henry R. Bungay, P.E., Ph.D., Professor of Chemical and Environmental Engineering, Rensselaer Polytechnic Institute; Member, American Institute of Chemical Engineers, American Chemical Society, American Society for Microbiology, American Society for Engineering Education, Society for General Microbiology (Section 24, Biochemical Engineering) Anthony I. Buonicore, M.Ch.E., P.E., Diplomate AAEE, CEO, Environmental Data Resources, Inc.; Member, American Institute of Chemical Engineers, Air and Waste Management Association (Section 25, Waste Management) Michael M. Calistrat, B.S.M.E., M.S.M.E., Owner, Michael Calistrat and Associates; Member, American Society of Mechanical Engineers (Section 29, Process Machinery Drives) , Giorgio Carta, Ph.D., Professor, Department of Chemical Engineering, U niversity of Virginia; Member, American Institute of Chemical Engineers, American Chemical Society, International Adsorption Society (Section 16, Adsorption and Ion Exchange) Vincent Conrad, Ph.D., Group Leader, Technical Services Development Laboratory,CONSOL, Inc.; Member, Spectroscopy Society of Pittsburgh, Society for Analytical Chemistry of Pittsburgh, Society for Applied Spectroscopy (Section 27, Energy Resources, Conversion, and Utilization) Harrison Cooper, ph.D., Harrison R. Cooper Systems, Inc., Salt take City, Utah (Section 19, Solid-Solid Operations and Equipment) B. B. Crocker, S.M., P.E., Consulting Chemical Engineer; Fellow, American Institute of chemical Engineers; Member, Air Pollution Control Association (Section 14, Gas Absorption and GasLiquid System Design) Daniel A. Crowl, Ph.D., Professor of Chemical Engineering, Chemical Engineering Department, Michigan Technological University; Member, American Institute of Chemical Engineers, American Chemical Society (Section 26, Process Safety) Roger W. Cusack, Vice President, Glitsch Process Systems, Inc.; Member, American Insi:itute of Chemical Engineers (Section 15, Liquid-Liquid Extraction Operations and Equipment) Donald A. Dahlstrom, Ph.D., Research Professor, Chemical and Fuels Engineering Department and Metallurgical Engineering Department, University of Utah; Member, National Academy of Engineering, American Institute of Chemical Engineers (AIChE), American Chemical Society (ACS), Society of Mining, Metallurgic Exploration (SME) of the American Institute of Mining, Metallurgical and Petrol~um Engine()r~ (AIME), AmericaI) Society of Engineering Education (Section 18, Liquid-Solid °?erations and Equipment) Thomas E. Daubert, Ph.D., Professor, Department of Chemical Engineering, The Pennsylvania State University (Section 2, Physic~ and Chemical Data) R. H. Daugherty, Ph.D., Consulting Engineer, Research Center~ Reliance Electric Company; Member, Institute of Electrical and Electronics Engineers (Section 29, Process Machinery Drives)
LIST OF CONTRIBUTORS
xi
James F. DaVis, Ph.D., Professor of Chemical Engineering, Ohio State University (Section 3, Mathematics) James B. Dunson, B.S., Principal Consultant, E. I. duPont de Nemours & Co.; Member Americanlnstitute of Chemical Engineers; Registered Professional Engineer \Delaware) (Section 17, Gas-Solid Operation and Equipment) Thomas F. Edgar, Ph.p., Professor of Chemical Engineering, y niversity o£Texas, Austin, Texas (Section8, Process Control) Robert C. Emmet, Jr., B,.5., Ch.E., Senior Process Consultant, EIMCO Process Equipment Co.; Member, American Institute of Chemical Engineers (AIChE), American Institute of Mining, Metallurgical and Petroleum Engineers (AIME), Society of Mining, Metallurgical and Exploration Engineers (SME) (Section 18, Liquid-Solid Operations and Equipment) Stanley M. Englund, M.S., Ch.E., Fellow, American Institute of Chemical Engineers; Process Consultant, The Dow Chemical Company (retired) (Section 26, Process Safety) BryanJ. Ennis, Ph.D., President, E&G Associates, and Adjunct Professor of Chemical Engineering; Vanderbilt University; Member and Chair of Powder Technology Programming Group of the Particle Technology Forum, American Institute of Chemical Engineers (Section 20, Size Reduction and Size Enlargement) William Eyka~p, Ph.D., Adjunc~ Professor of Chemical Engineering, Tufts University; Formerly President, Koch Membrane Systems; Member, American Institute of Chemical Engineers, American Chemical Society, American Association for the Advancement of Science, North American Membrane Society, European Society of Membrane Science and Technology (Section 22, Altemative Separation Processes) James R. Fair, Ph.D., P.E., Professor of Chemical Engineering, University of Texas; National Academy of Engineering; Fello'Y, American Institute of Chemical Engineers; Member, American Chemical Society, American S,ociety for Engineering Education, N atio~al Society of Professional Engineers
(Section
14, Gas Ab~orption
and
., Gas-Liquid
System
Design)
, Bruce A. Finlayson, Ph.D., Rehnberg Professor and Chair, Department of Chemical Engineering, University ofWashington; Member, National Academy of Engineering (Section 3, Mathematics) Thomas M. FI~n, Ph.D" P.E., Cryogenic Engineer, President CRYOCO, Louisville, CoJorado; Member, American Institute of Chemical Engineers (Section 11, Heat-Trans£er Equipment) Anthony G.,Fonseca, Ph,P., Director, Coal Utiltzation, CONSOL, Inc,; Member, American Chemical Society, Society for Mining, Metallurgy, and Extraction ( Section 27, Energy Resources, Conversion, and Utilization) D. G. Friend,
National Institutes of Standards and Technology, Boulder, Colorado (Section 2,
Physical and Chemical Data) George W. Gassman, B.S.M.E., Senior Research Specialist, Final Control Systems, Fisher Controls International, Inc., Marshalltown, Iowa (Section 8, Process Control) Fred K. Geitner,
P.Eng., B.S.M.E.,
M.S.M.E.,
Consulting Engineer; Registered Professional
Engineer (Ontario, Canada) (Section 29, Process Machinery Drives) Victor M. Goldschmidt,
Ph.D., P.E., Professor of Mechanical Engineering, Purdue University,
West Lafayette, Indiana (Section 11, Heat-Trallsfer Equipment) Stanley Grossel, President, Process Safety & Design, Inc.; Fellow, American Institute of chemical Engineers; Member, American Chemical Society; Member, The Combustion Institute; Member,Explosion Protection Systems Committee of NFPA (Section 26, Process Safety) Peter Harriott, Ph.D., Professor, School of Chemical Engineering, Cornell University; Member, American Institute of Chemical Engineering, American Chemical Society (ACS) (Section 18; Liquid-Solid Operations and Equipment)
xii
LIST OF CONTRIBUTORS
T. Alan Hatton, Ph.D., Ralph Landau Professor and Director of the DavidH. Koch School of Chemical Engineering Practice, Massachusetts Institute of Technology; Founding Fellow, American Institute of Medical and Biological Engineering; Member. American Institute of Chemical Engineers, American Chemical Society, International Association of Colloid and Interface scientists, American Association for the Advancement of Science, N eutt:on Scattering Society of America (Section 22, Alternative Separation Processes) Joseph D. Henry, Jr., Ph.D., P.E., Senior Fellow, Department of Engineering and Public Policy, Carnegie Mellon University; Member, American InStitute of Chemical Engineers, American Society for Engineering Education (Section 22, Alternative Separation Processes) W. G. High, C.Eng., B.Sc., F.I.Mech.E., tion 26, Process Safety)
Burgoyne Consultants Ltd., W. Yorks, England (Sec-
Richard Hogg, Ph.D., Professor, Department of Mineral Engineering, The Pennsylvania State University, University Park, PA (Section 19, Solid-Solid Operations and Equipment) F. A. Holland, D.Sc., Ph.D., Consultant in Heat Energy Recycling; Research Professor, University of Salford, England; Fellow; Institution of Chemical Engineers, London (Section 9, Process Economics) Hoyt C. Hottel,
S.M., Professor Emeritus of Chemical Engineering, Massachusetts Institute of
Technology; Member, National Academy of Sciences, American Academy of Arts and Sciences, American Institute of Chemical Engineers, American Chemical Society, Combustion Institute (Section 5, Heat and Mass Transfer) Colin S.Howat, Ph.D., P.E., John E. & Winfred E. Sharp Professor, Department of Chemical and Petroleum Engineering, University of Kansas; Member, American Institute of Chemical Engineers; Member, American Society of Engineering Education (Section 30, Analysis ofPlant Performance) Predrag S. Hrnjak, Ph.D., V.Res., Assistant Professor, University of Illinois at Urbana champaign and Principal Investigator-U. of I. Air Conditioning and Refrigeration Center, Assistant Professor, University of Belgrade; Member, International Institute of Refrigeration, American Society of Heating, Refrigeration and Air Conditioning (Section 11, Heat-Transfer Equipment) Arthur
E. Humphrey,
Ph.D., Retired, Professor of Chemical Engineering, Pennsylvania State
University; Member, U.S. National Academy of Engineering, American Institute of Chemical Engineers, American Chemical Society, American Society for Microbiology (Section 24, Biochemical Engineering) Eric Jenett, M.S.Ch.E.,
Manager, Process Engineering, Brown & Root, Inc.; Associate Member,
AIChE, Project Management Institute; Registered Professional Engineer (Texas) (Section 29, Process Machinery Drives) John S. Jeris, Sc.D., P.E., Professor of Environmental Engineering, Manhattan College; Environmental Consultant; Member, American Water Works Association, Water Environment Federation Section Director (Section 25, Waste Management) T. L. P. Jespen, M.S., MiD. Proc;, Metallurgical Engineer, Basic, Inc;; Gabbs, Nevada (Section 19, Solid-Solid Operations and Equipment) Keith P. Johnston, Ph.D., P.E., Professor of Chemical Engineering, University of Texas (Austin); Member, American Institute of Chemical Engineers, American Chemical Society, U niversity of Texas Separations Research Program (Section 22, Alternative Separation Processes) Trevor A. Kletz, D.Sc., Senior Visiting Research Fellow, Department of Chemical Engineering, Loughborough University, U.K.; Fellow, American Institute of Chemical Engineers, Royal Academy of Engineers (U .K.), Institution of Chemical Enginee!s (U .K.), and Royal Society of chemistry (U.K.) (Section 26, Process Safety) Edgar B. Klunder, Ph.D., Project Manager, Energy Technol
U,~.
LIST OF CONTRIBUTORS
xiii
Kent S. Knaebel, Ph.D., President, Adsorption Research, Inc.; Member, American Il;lstitute of Chemical Engineers, American Chemical Society, International Adsorption Society. Professional Engineer (Ohio) (Section 5, Heat and Mass Transfer) Frank Knoll, M.S., Min. Proc., President, Carpco, Inc., Jacksonville, Florida (Section 19, SolidSolid Operations and Equipment) James G.Knudsen, Ph.D.,Professor Emeritus of Chemical Engineering, Oregon State university; Member, American Institute of Chemical Engineers, American Chemical Society; Registered Professional Engineer (Oregon) (Section 5, Heat and Mass Transfer) Michael Krumpelt, Ph.D., Manager, Fuel Cell TechnolQgy, Argonne National Laboratory; Member, American Institute of Chemical Engineers, American Chemical Society, Electrochemical Society (Section 27, Energy Resources, Conversion, and Utilization) Irwin J. Kugelman, Sc.D., Professor of Civil Engineering, Lehigh University; Member, American Society of Civil Engineering, Water Environmental Federation (Section 25, Waste Management) rim Laros, M.S. Mineral Processing, Senior Process Consultant, EIMCO Process Equipment Co,; Member, Society of Mining, Metallurgy and Exploration (SME of AIME) (Section 18, Liquid-Solid Operations and Equipment) Richard M. Lemert, Ph.D., P.E., Assistant Professor of Chemical Engineering, University of Toledo; Member, American Institute of Chemical Engineers, American Chemical Society, Society qf Mining Engineers, American Society for Engineering Education (Section 22, Alternative Separation Processes) Robert Lemlich, Ph.D., P.E., Professor of Chemical Engineering Emeritus, University of Cincinnati; Fellow, American Institute of Chemical Engineers; Member, American Chemical Society, American Society for Engineering Education, American Chemical Society (Sectiqn 22, Alternative Separation Processes) Wallace Leung, Sc.D., Director, Process Technology, Bird Machine Company; Member, American Filtration and Separation Society (Director) (Section 18, Liquid-Solid Operations and Equipment) M. Douglas LeVan, Ph.D., Professor, Department of Chemical Engineering, University of Virginia; Member, American Institute of Chemical Engineers, American Chemical Society, Intemational Adsorption Society (Section 16, Adsorption and Ion Exchange) Peter E. Liley, Ph.D., D.I.C., School of Mechanical Engineering, Purdue University (Section 2, Physical and Chemical Data) James D. Litster, Ph.D., Associate Professor, Department of Chemical Engineering, University of Queensland; Member, Institute of Chemical Engiiieers-Australia (Section 20, Size Reduction and Size Enlargement) Peter J. Lofuis, D. Phil., Arthur D. Little, Inc.; Member, American Society of Mechanical Engineers (Section 27, Energy Resources, Conversion, and Utilization) Hsue-peng Loh, Ph.D., P.E., Federal Energy Technology Center (Morgantown), U .S. Department of Energy; Member, American Institute of Chemical Engineers, American Society of Information Sciences (Section 27, Energy Resources, Conversion, and Utilization) Douglas E. Lowenhaupt, M.S., Group Leader, Coke Laboratory, CONSOL, Inc.; Member, American Society for Testing and Materials, Iron and Steel Making Society, International Committee for Coal Petrology (Section 27, Energy Resources, Conversion, and Utilization) James 0. Maloney, Ph.D., P.E., Emeritus Professor of Chemical Engineering, University of Kansas; Fellow, American Institute of Chemical Engineering; Fellow, American Association for the Advancement of Science; Member, American Chemical Society, American Society for Engineering Education (Section 1, Conversion Factors and Mathematical Symbols)
Thomas J. McAvoy, Ph.D., Professor of Chemical Engineering, University of Maryland, College Park, Maryland (Section 8, Process Control) Chad McCleary, EIMCO Process Equipment Company, Process Consultant (Section 18, Liquid-Solid Operations and Equipment) Thomas F. McGowan, P.E., Senior Consultant, RMT/Four Nines; Member, American Institute of Chemical Engineers, American Society of Mechanical Engineers, Air and Waste Management Association (Section 25, Waste Management) Howard G. Mcllvried, III, Ph.D., Senior Engineer, Burns and Roe Services Corporation, Federal Energy Technology Center (Pittsburgh), Member, American Chemical Society, American Institute of Chemical Engineers (Section 27, Energy Resources, Conversion, and Utilization) John D. McKenna, Ph.D., President and Chairman, ETS International, Inc., Member, American Institute of Chemical Engineers, Air and Waste Management Association (Section 25, Waste Management) Shelhy A. Miller, Ph.D., P.E., Resident Retired Senior Engineer; Argonne National Laboratory; American Association for the Advancement of Science (Fellow), American Chemical Society, American Institute of Chemical Engineers (Fellow), American Institutes of Chemists (Fellow), Filtration Society, New York Academy of Sciences, Society of Chemical Industry (Section 18; Liqmd.; Solid Operations and Equipment; Section 27, Energy Resources, Conversion, and Utilization) Booker Morey, Ph.D., Senior Consultant, SRI International; Member; Society of Mining, Metallurgy and Exploration (SME of AIME), The Filtration Society, Air and Waste Management Association; Registered Professional Engineer (California and Massachusetts) (Section 18, LiquidSolid Operations and Equipment) Charles G. Moyers, Ph.D., P.E., Principal Engineer, Union Carbide Corporation; Fellow, American Institute of Chemical Engineers (Section 12, Psychrometry, Evaporative Cooling, and Solids Drying; Section 22, Alternative Separation Processes) John Newman, Ph.D., Professor of Chemical Engineering, University of California, Berkeley; Principle Investigator; Inorganic Materials Research Division, Lawrence Berkeley Laboratory (Section 22, Alternative Separation Processes) James Y. Oldshue, Ph.D., President, Oldshue Technologies International, Inc.; Member, National Academy of Engineering; Adjunct Professor of Chemical Engineering atBeijing Institute of Chemical Technology, Beijing, China; Member; American Chemical Society (ACE), American Institute of Chemical Engineering (AIChE), Traveler Century Club, Executive Committee on the Transfer of Appropriate Technology for the World Federation of Engineering Organizations (Section 18, Liquid-Solid Operations and Equipment) Robert W. Ormsby, M.S., Ch.E. P.E., Manager of Safety, Chemical Group, Air Products and Chemicals, Inc.; Air Products Corp.; Fellow, American Institute of Chemical Engineers (Section 26, Process Safety) John E. Owens, B.E.E., Electrostatic Consultant, Condux, Inc.; Member, Institute of Electrical and Electronics Engineers, Electrostatics Society of America (Section 26, Process Safety) Bhupendra Parekb, Ph.D., Associate Director, Center for Applied Energy Research, University of Kentucky, Lexington, Kentucky (Section 19, Solid-Solid Operations and Equipment) Mel Pen, Ph.D., Senior Consultant, E. I. duPont de Nemours & Co.; Fellow, American Institute of Chemical Engineers; Registered Professional Engineer (Delaware) (Section 17, Gas-Solid Operations and Equipment) W. R. Penney, Ph.D., P.E., Professor of Chemical Engineering, University of Arkansas; Member, American Institute of Chemical Engineers (Section 14, Gas Absorption and Gas-Liquid System Design)
LIST OF CONTRIBUTORS
xv
WalterF. Podolski, Ph.D., Chemical Engineer, Electrochemical Technology Program, Argonne National Laboratory; Member, American Institute of Chemical Engineers (Section 27, Energy Resources, Conversion, and Utilization) Herbert A. Pohl, Ph.D. (deceased), Professor of Physics, Oklahoma State University (Section 22, Alternative Separation Processes) Kent Pollock, Ph.D., Member of Technical Staff, Group 91, Space Surveillance Techniques, MIT Lincoln Laboratory (Section 22, Alternative Separation Processes) George Priday,B.S., Ch.E., EIMCO Process Equipment Company; Member, American Institute of Chemical Engineering {AIChE), Instrument Society of America (ISA) (Section 18, LiquidSolid Operations and Equipment) Michael E. Prudich, Ph.D" Professor and Chair of Chemical Engineering, Ohio U~iversity; Member, American Institute of Chemical Engineers, American Chemical Society, Society of Mining Engineers, American Society for Engineering Education (Section 22, Alternative Skparation Processes) Raj K. Rajamani, Ph.D., Professor, Department of Metallurgy and Metallurgical Engineering, University of Utah, Salt Lake City, Utah (Section 19, Solid-Solid Operations and Equipment) Lawrence K. Rath, B.S., P.E., Federal Energy Technology Center (Morgantown), U .S. Department of Energy; Member, American Institute of Chemical Engineers (Section 27, Energy Resources, Conversion, and Utilization) Grantges J. Raymus, M.E., M.S., President, Raymus Associates, Incorporated, Packaging consultants; Adjunct Professor and Program Coordinator, Center for Packaging Science and Engineering, College of Engineering, Rutgers, The State University of New Jersey; formerly Manager of Packaging Engineering, Union Carbide Corporation; Registered Professional Engineer, california; Member, Institute of Packaging Professionals, ASME (Section 21, Handling of Bulk Solids and Packagi?g of Solids and Liquids) Lanny A. Robbins, Ph.D., Research Fellow, Dow Chemical Company; Member, American Institute of Chemical Engineers (Section 15, Liquid-Liquid Extraction Operations and Equipment) Joseph J. Santoleri, P.E.,Senior Consultant, RMT/Four Nines; Member, American Institute of Chemical Engineers, American Society of Mechanical Engineers, Air and Waste Management Association (Section 25, Waste Management) Adel F. Sarofim, Sc.D., Lammot DuPont Professor of Chemical Engineering and Assistant Director, Fuels Research Laboratory, Massachusetts Institute of Technology; Member, American Institute of Chemical Engineers, American Chemical Society, Combustion Institute (Section 5, Heat and Mass Transfer) Kalanadh v. S. Sastry, Ph.D., Professor, Department of Materials Science and Mineral Engineering, University of California, Berkeley, CA; Member, American Institute of Chemical Engineers, Society for Mining, Metallurgy and Exploration (Section 19, Solid-Solid Operations and Equipment) Paul J. Schafbuch, Ph.D" Senior Research Specialist, ~inal Control Systems, Fisher Controls International, Inc., Marshalltown, Iowa (Section 8, Process Control) Carl A. Schiappa, B.S., Ch.E., Process Engineering Associate, Michigan Division Engineering, The Dow Chemical Company; Member, AIChE and CCPS (Section 26, Process Safety) David K. Schmalzer, Ph.D., P.E., Fossil Energy Program Manager, Argonne National Laboratory; Member, American Chemical Society, Americanlnstitute of Chemical Engineers (Section 27, Energy Resources, Conversion, and Utilization) J. D. Seader, Ph.D., Professor of Chemical Engineering, University of Utah, Salt Lake City, Utah; Fellow, American Institute of Chemical Engineers; Member, American Chemical Society; Member. American SocietY for Enl!ineerin!! Education (Section 13. Distillation)
xvi
LIST OF CONTRIBUTORS
Dale E. Seborg, Ph.D., Professor of Chemical Engineering, University of California, Santa Barbara, California (Section 8, Process Control) Richard L. Shilling, Fintube Company-a
P.E., B.S.M., B.E.M.E., Manager of Engineering Development, Brown Koch Engineering Company; Member, American Society of Mechanical
Engineers (Section 11, Heat-Transfer Equipment) F. Greg Shinskey, B.S.Ch.E.,
Consultant (retired from Foxboro Co.), North Sandwich, New
Hampshire (Section 8, Process Control) Oliver w. Siebert, P.E., B.S.M.E., Washington University, Graduate Metallurgical Engineering, Sever Institute of Technology; Professor, Department of Chemical Engineering, Washington University, St. Louis, Missouri; President, Siebert Materials Engineering, Inc., St, Louis, Missouri; Senior Engineering Fellow (retired), Monsanto Co.; Mechanical Designer, Sverdrup Corp.; Metallurgist, Carondelet Foundry; United Nations Consultant to the People's Republic of China; Fellow, American Institute of Chemical Engineers; Life Fellow, American Society of Mechanical Engineers; Past Elected Director and Fellow, N ational Association of Corrosion Engineers, Int'I; American Society for Metals, Int'I; American Welding Society; Pi Tau Sigma, Sigma Xi, and Tau Beta Pi (Section 28, Materials of Construction) Jeffrey J. Siirola, Ph.D., Research Fellow, Eastman Chemical Company; Member, National Academy of Engineering; Fellow, American Institute ofChemical Engineers, American Chemical Society, American Association for Artificial Intelligence, American Society for Engineering Education (Section 13, Distillation) Charles E. Silverblatt, M.S., Ch.E., Peregrine International Associates, Inc.; Consultant to WesTech Engineering, Inc., American Institute of Mining, Metallurgical and Petroleum Engines (AIME) (Section 18, Liquid-Solid Operations and Equipment) Richard
Siwek, M.S., Explosion Protection Manager, Corporate Unit Safety and Environment,
Ciba-Geigy Ltd., Basel, Switzerland (Section 26, Process Safety) J. Stephen
Slottee, M.S., Ch.E., Manager, Technology and Development, EIMCO
Equipment
Co.; Member, American Institute of Chemical Engineers (AIChE)
Process
(Section 18,
Liquid-Solid Operations and Equipment) Cecil L. Smith,
Ph.D.,
Principal, Cecil L. Smith Inc., Baton Rouge, Louisiana (Section 8,
Process Control) Julian C. Smith, B. Chem., Ch.E., Professor Emeritus Chemical Engineering, Cornell University; Member, American Chemical Society (ACS), American Institute of Chemical Engineers (AIChE) (Section 18, Liquid-Solid Operations and Equipment) Richard H. Snow, Ph.D., Engineering Advisor, lIT Research Institute; Member, American Chemical Society, Sigma Xi; Fellow, American Institute of Chemical Engineers (Section 20, Size Reduction and Size Enlargement) Thomas Sorenson,
M.B.A.,
MiD. Eng., President, Galigher Ash (Canada) Ltd. (Section 19,
Solid-Solid Operations and Equipment) Rameshwar D. Srivastava, Ph.D., Fuels Group Manager, Burns and Roe Services Corporation, Federal Energy Technology Center (Pittsburgh) (Section 27, Energy Resources, Conversion, and Utilization ) F. C. Standiford,
M.S., P.E., Member, American Institute of Chemical Engineers, American
Chemical Society (Section 11, Heat-Transfer Equipment) D. E. SteiDmeyer, M.A., M.S., P.E., Distinguished Fellow, Monsanto Company; Fellow; American Institute of Chemical Engineers; Member, American Chemical Society (Section 14, Gas Absorption and Gas-Liquid System Design) Gary J. Stiegel, M.S., P.E., Program Coordinator, Federal Energy Technology Center (Pittsburgh), U .S. Department of Energy (Section 27, Energy Resources, Conversion, and Utilization)
LIST OF CONTRIBUTORS
xvii
John G. Stoecker II, B.S.M.E., University of Missouri School of Mines and Metallurgy; principal Consultant, Stoecker & Associates, St. Louis, Missouri; Principal Materials Engineering Specialist (retired), Monsanto Co.; High-Temperature Design/Application Engineer, Abex Corporation; Member, NACE International, ASM International (Section 27, Energy Resources, Conversion, and Utilization) Judson S. Swearingen,
Ph.D., Retired President, Rotoflow Corporation (Section 29, Process
Machinery Drives) Louis Theodore, Sc.D., Professor of Chemical Engineering, Manhattan College; Member, Air and Waste Management Association (Section 25, Waste Management) Michael P. Thien, Sc.D., Senior Research Fellow, Merck & Co., Inc.; Member, American Institute of Chemical Engineers, American Chemical Society, International Society for Pharmaceutical Engineers (Section 22, Alternative Separation Processes) George H. Thomson, and Chemical Data)
AIChE Design Institute for Physical Property Data (Section 2, Physical
James N. Tilton, Ph.D., P.E., Senior Consultant, Process Engineering, E. I. duPont de Nemours & Co.; Member, American Institute of Chemical Engineers; Registered Professional Engineer (Delaware) (Section 6, Fluid and Particle Dynamics) Klaus D. Timmerhaus, Ph.D., P.E., Professor and President's Teaching Scholar, University of Colorado, Boulder, Colorado; Fellow, American Institute of Chemical Engineers, American Society for Engineering Education, American Association for the Advancement of Science; Member, American Astronautical Society, N ational Academy of Engineering, Austrian Academy of Science, International Institute of Refrigeration, American Society of Heating, Refrigerating and Air Conditioning Engineers, American Society of Environmental Engineers, Engineering Society for Advancing Mobility on Land, Sea, Air, and Space, Sigma Xi, The Research Society (Section 11, Heat-Transfer Equipment) David B. Todd, Ph.D., President, Todd Engineering; Member, American Association for the Advancement of Science (AAAS), American Chemical Society (ACS), American Institute of Chemical Engineering (AIChE), American Oil Chemists Society (AOCS), Society of Plastics Engineers (SPE), and Society of the Plastics Industry (SPI); Registered Professional Engineer, Michigan (Section 18, Liquid-Solid Operations and Equipment) George T. Tsao, Ph.D., Director, Laboratory for Renewable Resource Engineering, Purdue University; Member, American Institute of Chemical Engineers, American Chemical Society, American Society for Microbiology (Section 24, Biochemical Engineering) Hendrick c. Van Ness, D.Eng., Howard P. Isermann Department of Chemical Engineering, Rensselaer Polytechnic Institute; Fellow, American Institute of Chemical Engineers; Member, American Chemical Society (Section 4, Thermodynamics) Stanley M. Walas, Ph.D., Professor Emeritus, Department of Chemical and Petroleum Engineering, University of Kansas; Fellow, American Institute of Chemical Engineers (Section 7, Reaction Kinetics; Section 23, Chemical Reactors) Phillip C. Wankat, Ph.D., Professor of Chemical Engineering, Purdue University; Member, American Institute of Chemical Engineers, American Chemical Society, International Adsorption Society (Section 5, Heat and Mass Transfer) Ionel Wechsler, M.S., MiD. and Met., Vice President, Sala Magnetics, Inc., Cambridge, Massachusetts (Section 19, Solid-Solid Operations and Equipment) Arthur W. Westerberg, Ph.D., Swearingen University Professor of Chemical Engineering, Carnegie Mellon University; Member, National Academy of Engineering (Section 3, Mathematics) John M. Wheeldon, Ph.D., Electric Power Research Institute (Section 27, Energy Resources, Conversion. and Utilization)
xviii
LIST OF CONTRIBUTORS
Robert E. White, Ph.D., Principal Engineer; Chemistry and Chemical Engineering Division, Southwest Research Institute (Section 26, Process Safety) J. K. Wilkinson, M.Sc., Consultant Chemical Engineer; Fellow, Institution of Chemical Engineers, London (Section 9, Process Economics) David Winegarder, Ph.D., Engineering Associate, Michigan Division Engineering, The Dow Chemical Company; Member AIChE and CCPS (Section 26, Process Safety) John L. Woodward,
Ph.D., Principal, DNV Technica, Inc. (Section 26, Process Safety)
Yoshiyuki Yamashita, Ph.D., Associate Professor of Chemical Engineering, Tohoku University, Sendai, Japan (Section 3, Mathematics) Cannen M. Yon, M.S., DevelopmentAssociate, UOP, Des Plaines, Illinois; Member, American Institute of Chemical Engineers (Section 16, Adsorption and Ion Exchange)
Preface to the Seventh Edition
Perry’s has been an important source for chemical engineering information since 1934. The significant contributions of the editors who have guided preparation of the previous editions is acknowledged. These include John H. Perry (first to third editions), Robert H. Perry (fourth to sixth editions), Cecil H. Chilton (fourth and fifth editions), and Sidney D. Kirkpatrick (fourth edition). Ray Genereaux (DuPont) contributed to each of the first six editions, and Shelby Miller (Argonne National Lab) worked on the second through the seventh. The current editors directed both the sixth and seventh editions. Advances in the technology of chemical engineering have continued as we have moved toward the twenty-first century, and this edition will carry us into that century. The Handbook has been reorganized. The first group of sections focuses on chemical and physical property data and the fundamentals of chemical engineering. The second and largest group of sections deals with processes, generally divided as heat transfer operations, distillation, kinetics, liquid-liquid, liquid-solid, and so on. The last group treats auxiliary information such as materials of construction, process machinery drives, waste management, and process safety. All sections have been revised and updated, and several sections are entirely new or have been extensively revised. Examples of these sections are mathematics, mass transfer, reaction kinetics, process control, transport and storage of fluids, alternative separation processes, heat-transfer equipment, chemical reactions, liquid-solid operations and equipment, process safety, and analysis of plant performance. Significant new information has also been included in the physical and chemical data sections. Several section editors and contributors worked on this seventh edition, and these persons and their affiliations are listed as a part of the front material. Approximately one-half of the section editors are fellows of the AIChE. In addition, the following chemical engineering students at the University of Kansas assisted in the preparation of the index: Jason Canter, Pau Ying Chong, Mei Ling Chuah, Li Phoon Hor, Siew Pouy Ng, Francis J. Orzulak, Scott C. Renze, Page B. Surbaugh, and Stephen F. Weller. Shari L. Gladman and Sarah Smith provided extensive secretarial assistance. Much of Bob Perry’s work carries over into this edition and his influence is both recognized and remembered. DON W. GREEN JAMES O. MALONEY University of Kansas April, 1997 xix
For the detailed contents of any section, consult the title page of that section. See also the alphabetical index in the back of the handbook.
Section
Conversion Factors and Mathematical Symbols
Jam£s 0. Maloney
1
Physical and Chemical Data Peter E. Liley, George H. Thomson, D. G. Friend, Thomas E. Daubert, Evan Buck
2
Mathematics Bruce A. Finlayson, Jam£s F Davis, Arthur W Westerberg, Yoshiyuki Yamashita
3
Thermodynamics
4
Hendrick C. Van Ness, Michael M. Abbott
Heat and Mass Transfer Jam£s G. Knudsen, Hoyt C. Hottel, Adel F Sarofim, Phillip c. Wankat, Kent S. Knaebel
5
Fluid and Particle Dynamics
6
Reaction Kinetics
Jam£s N. Tilton
Stanley M. Walas
7
Process Control Thomas F Edgar; Cecil L. Smith, F Greg Shinskey, George W Gassman, Paul J. Schajbuch, Thomas J. McAvoy, Dale E. Seborg
8
Process Economics
9
F A. Holland, J. K Wilkinson
Transport and Storage of Fluids
Meherwan 1'. Royce
10
Heat- Transfer Equipment Richard L. Shilling, Kenneth J. Bell, Patrick M. Bemhagen, Thomas M. Flynn, Victor M. Goldschmidt, Predrag S. Hrnjak, F C. Standiford, Klaus D. Timmerhaus
11
Psychrometry, Evaporative Cooling, and Solids Drying Glenn W Baldwin
12
Distillation
Charles G. Moyers,
J. D. Seader;Jeffrey J. Siirola, Scott D. Bamicki
13
vii
viii
CONTENTS
Gas Absorption and Gas-Liquid System Design W R. Penny, B. B. Crocker
james R. Fair; D. E. Steinmeyer;
Liquid-Liquid Extraction Operations and Equipment Roger W Cusack Adsorption and Ion Exchange
14 Lanny A. Robbins,
M. Douglas LeVan, Giorgio Carta, Carmen M. Yon
Gas-Solid Operations and Equipment
Mel Pell, james B. Dunson
15 16 17
Liquid-Solid Operations and Equipment Donald A. Dahlstrom, Richard C. Bennett, Robert G, Emmet, Peter Harriott, Tim Laros, Wallace Leung, Shelby A. Miller; Brooker Morey, james Y. Oldshue, George Priday, Charles E. Silverblatt, I. Stephen Slottee, Iulian c. Smith
18
Solid-Solid Operations and Equipment Kalanadh V S. Sastry, Harrison Cooper; Richard Hogg, T L. p Iespen, Frank Knoll, Bhupendra Parekh, Raj K Rajamani, Thomas Sorenson, Ionel Wechsler; Chad McCleary, David B. Todd
19
Size Reduction and Size Enlargement Bryan I. Ennis, james D. Litster
Richard L. Snow, Terry Allen,
Handling of Bulk Solids and Packaging of Solids and Liquids
Grantges I. Raymus
Alternative Separation Processes joseph D. Henry, Ir:, Michael E. Prudich, William Eykamp, T Alan Hatton, Keith p johnston, Richard M. Lemert, Robert Lemlich, Charles G. Moyers, john Newman, Herbert A. Pohl, Kent Pollock, Michael p Thien Chemical Reactors
Stanley M. Walas
Biochemical Engineering
Henry R. Bungay, Arthur E. Humphrey, George T Tsao
20 21
22 23 24
Waste Management Louis Theodore, Anthony I. Buonicore, John D. McKenna, Irwin I. Kugelman, john s. Ieris, joseph I. Santoleri, Thomas R McGowan
25
Process Safety Stanley M. Englund, Frank T Bodurtha, Laurence G. Britton, Daniel A. Crowl, Stanley Grossel, W G. High, Trevor A. Kletz, Robert W Ormsby, john E. Owens, Carl A. Schiappa, Richard Siwek, Robert E. White, David Winegardner; john L. Woodward
26
Energy Resources, Conversion, and Utilization Walter R Podolski, Shelby A. Miller; David K Schnialzer; Anth9ny G. Fonseca, Vincent Conrad, Douglas E. Lowenhaupt, john Bacha, Lawrence K Rath, Hsue-peng Loh, EdgarB. Klunder; Howard G. McIlvried, III, Gary I. Stiegel, Rameshwar D. Srivastava, Peter I. Loftus, Charles E. Benson, john M. Wheeldon, Michael Krumpelt
27
Materials of Construction
28
Oliver W Siebert, john G. Stoecker
Process Machinery Drives Heinz p Bloch, R. H. Daugherty, Fred K Geitner; Meherwan p Boyce, judson S. Swearingen, Eric jennet, Michael M. Calistrat Analysis of Plant Performance Index follows Section 30.
Colin S. Howat
29 30
Section 1
Conversion Factors and Mathematical Symbols*
James O. Maloney, Ph.D., P.E., Emeritus Professor of Chemical Engineering, University of Kansas; Fellow, American Institute of Chemical Engineering; Fellow, American Association for the Advancement of Science; Member, American Chemical Society, American Society for Engineering Education
Fig. 1-1 Table 1-1 Table 1-2a Table 1-2b Table 1-3 Table 1-4 Table 1-5 Table 1-6 Table 1-7 Table 1-8 Table 1-9
CONVERSION FACTORS Graphic Relationships of SI Units with Names . . . . . . . . . SI Base and Supplementary Quantities and Units. . . . . . . Derived Units of SI that Have Special Names. . . . . . . . . . Additional Common Derived Units of SI . . . . . . . . . . . . . SI Prefixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conversion Factors: U.S. Customary and Commonly Used Units to SI Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metric Conversion Factors as Exact Numerical Multiples of SI Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alphabetical Listing of Common Conversions . . . . . . . . . Common Units and Conversion Factors . . . . . . . . . . . . . . Kinematic-Viscosity Conversion Formulas . . . . . . . . . . . . Values of the Gas-Law Constant. . . . . . . . . . . . . . . . . . . . .
1-2 1-3 1-3 1-3 1-3 1-4 1-13 1-15 1-18 1-18 1-18
Table 1-10 United States Customary System of Weights and Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 1-11 Temperature Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . Table 1-12 Specific Gravity, Degrees Baumé, Degrees API, Degrees Twaddell, Pounds per Gallon, Pounds per Cubic Foot . . . Table 1-13 Wire and Sheet-Metal Gauges . . . . . . . . . . . . . . . . . . . . . . Table 1-14 Fundamental Physical Constants . . . . . . . . . . . . . . . . . . . .
1-19 1-19 1-20 1-21 1-22
CONVERSION OF VALUES FROM U.S. CUSTOMARY UNITS TO SI UNITS Table 1-15 Table 1-16
MATHEMATICAL SYMBOLS Mathematical Signs, Symbols, and Abbreviations . . . . . . . Greek Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-24 1-24
* Much of the material was taken from Sec. 1. of the fifth edition. The contribution of Cecil H. Chilton in developing that material is acknowledged. 1-1
FIG. 1-1
1-2
Graphic relationships of SI units with names (U.S. National Bureau of Standards, LC 1078, December 1976.)
TABLE 1-1
SI Base and Supplementary Quantities and Units
Quantity or “dimension” Base quantity or “dimension” length mass time electric current thermodynamic temperature amount of substance luminous intensity Supplementary quantity or “dimension” plane angle solid angle
SI unit
SI unit symbol (“abbreviation”); Use roman (upright) type
meter kilogram second ampere kelvin mole* candela
m kg s A K mol cd
radian steradian
rad sr
* When the mole is used, the elementary entities must be specified; they may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.
TABLE 1-2a Derived Units of SI that Have Special Names Quantity
Unit
Symbol
Formula
frequency (of a periodic phenomenon) force pressure, stress energy, work, quantity of heat power, radiant flux quantity of electricity, electric charge electric potential, potential difference, electromotive force capacitance electric resistance conductance magnetic flux magnetic-flux density inductance luminous flux illuminance activity (of radionuclides) absorbed dose
hertz newton pascal joule watt coulomb volt
Hz N Pa J W C V
l/s (kg⋅m)/s2 N/m2 N⋅m J/s A⋅s W/A
farad ohm siemens weber tesla henry lumen lux becquerel gray
F Ω S Wb T H lm lx Bq Gy
C/V V/A A/V V⋅s Wb/m2 Wb/A cd⋅sr lm/m2 l/s J/kg
TABLE 1-2b Additional Common Derived Units of SI Quantity
Unit
acceleration angular acceleration angular velocity area concentration (of amount of substance) current density density, mass electric-charge density electric-field strength electric-flux density energy density entropy heat capacity heat-flux density, irradiance luminance magnetic-field strength molar energy molar entropy molar-heat capacity moment of force permeability permittivity radiance
meter per second squared radian per second squared radian per second square meter mole per cubic meter
m/s2 rad/s2 rad/s m2 mol/m3
ampere per square meter kilogram per cubic meter coulomb per cubic meter volt per meter coulomb per square meter joule per cubic meter joule per kelvin joule per kelvin watt per square meter
A/m2 kg/m3 C/m3 V/m C/m2 J/m3 J/K J/K W/m2
candela per square meter ampere per meter joule per mole joule per mole-kelvin joule per mole-kelvin newton-meter henry per meter farad per meter watt per square-metersteradian watt per steradian joule per kilogram-kelvin joule per kilogram joule per kilogram-kelvin cubic meter per kilogram newton per meter watt per meter-kelvin meter per second pascal-second square meter per second cubic meter 1 per meter
cd/m2 A/m J/mol J/(mol⋅K) J/(mol⋅K) N⋅m H/m F/m W/(m2⋅sr)
radiant intensity specific-heat capacity specific energy specific entropy specific volume surface tension thermal conductivity velocity viscosity, dynamic viscosity, kinematic volume wave number
TABLE 1-3
Symbol
W/sr J/(kg⋅K) J/kg J/(kg⋅K) m3/kg N/m W/(m⋅K) m/s Pa⋅s m2/s m3 1/m
SI Prefixes
Multiplication factor
Prefix
Symbol
000 = 1018 000 = 1015 000 = 1012 000 = 109 000 = 106 000 = 103 100 = 102 10 = 101 0.1 = 10−1 0.01 = 10−2 0.001 = 10−3 0.000 001 = 10−6 0.000 000 001 = 10−9 0.000 000 000 001 = 10−12 0.000 000 000 000 001 = 10−15 0.000 000 000 000 000 001 = 10−18
exa peta tera giga mega kilo hecto* deka* deci* centi milli micro nano pico femto atto
E P T G M k h da d c m µ n p f a
1 000 000 000 1 000 000 1 000 1
000 000 000 000 1
000 000 000 000 000 1
*Generally to be avoided.
1-3
TABLE 1-4
Conversion Factors: U.S. Customary and Commonly Used Units to SI Units Customary or commonly used unit
Quantity
SI unit
Alternate SI unit
Conversion factor; multiply customary unit by factor to obtain SI unit
Space,† time Length
naut mi mi chain link fathom yd ft
1.852* E + 00 1.609 344* E + 00 2.011 68* E + 01 2.011 68* E − 01 1.828 8* E + 00 9.144* E − 01 3.048* E − 01 3.048* E + 01 2.54* E + 01 2.54 E + 00 2.54* E + 01
in in mil
km km m m m m m cm mm cm µm
Length/length
ft/mi
m/km
1.893 939 E − 01
Length/volume
ft/U.S. gal ft/ft3 ft/bbl
m/m3 m/m3 m/m3
8.051 964 E + 01 1.076 391 E + 01 1.917 134 E + 00
Area
mi2 section acre ha yd2 ft2 in2
km2 ha ha m2 m2 m2 mm2 cm2
2.589 988 E + 00 2.589 988 E + 02 4.046 856 E − 01 1.000 000* E + 04 8.361 274 E − 01 9.290 304* E − 02 6.451 6* E + 02 6.451 6* E + 00
Area/volume
ft2/in3 ft2/ft3
m2/cm3 m2/m3
5.699 291 E − 03 3.280 840 E + 00
Volume
cubem acre⋅ft
km3 m3 ha⋅m m3 m3 m3 dm3 m3 dm3 m3 dm3 dm3 dm3 dm3 cm3 cm3 cm3
4.168 182 1.233 482 1.233 482 7.645 549 1.589 873 2.831 685 2.831 685 4.546 092 4.546 092 3.785 412 3.785 412 1.136 523 9.463 529 4.731 765 2.841 307 2.957 353 1.638 706
yd3 bbl (42 U.S. gal) ft3 U.K. gal U.S. gal U.K. qt U.S. qt U.S. pt U.K. fl oz U.S. fl oz in3
L L L L L L
E + 00 E + 03 E − 01 E − 01 E − 01 E − 02 E + 01 E − 03 E + 00 E − 03 E + 00 E + 00 E − 01 E − 01 E + 01 E + 01 E + 01
Volume/length (linear displacement)
bbl/in bbl/ft ft3/ft U.S. gal/ft
m3m m3/m m3/m m3/m L/m
6.259 342 E + 00 5.216 119 E − 01 9.290 304* E − 02 1.241 933 E − 02 1.241 933 E + 01
Plane angle
rad deg (°) min (′) sec (″)
rad rad rad rad
1 1.745 329 E − 02 2.908 882 E − 04 4.848 137 E − 06
Solid angle
sr
sr
1
Time
year week h
a d s min s h ns
1 7.0* 3.6* 6.0* 6.0* 1.666 667 1
E + 00 E + 03 E + 01 E + 01 E − 02
1.016 047 9.071 847 5.080 234 4.535 924 4.535 924 3.110 348 2.834 952 6.479 891
E + 00 E − 01 E + 01 E + 01 E − 01 E + 01 E + 01 E + 01
min mµs Mass, amount of substance Mass
1-4
U.K. ton U.S. ton U.K. cwt U.S. cwt lbm oz (troy) oz (av) gr
Mg Mg kg kg kg g g mg
t t
TABLE 1-4
Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Continued ) Customary or commonly used unit
Quantity Amount of substance
lbm⋅mol std m3(0°C, 1 atm) std ft3 (60°F, 1 atm)
SI unit
Alternate SI unit
Conversion factor; multiply customary unit by factor to obtain SI unit 4.535 924 E − 01 4.461 58 E − 02 1.195 30 E − 03
kmol kmol kmol
Enthalpy, calorific value, heat, entropy, heat capacity Calorific value, enthalpy (mass basis)
Btu/lbm cal/g cal/lbm
MJ/kg kJ/kg kWh/kg kJ/kg J/kg
Caloric value, enthalpy (mole basis)
kcal/(g⋅mol) Btu/(lb⋅mol)
kJ/kmol kJ/kmol
Calorific value (volume basis—solids and liquids)
Btu/U.S. gal
MJ/m3 kJ/m3 kWh/m3 MJ/m3 kJ/m3 kWh/m3 MJ/m3 kJ/m3 kWh/m3 MJ/m3 kJ/m3
kJ/dm3
Btu/U.K. gal Btu/ft3
cal/mL (ft⋅lbf)/U.S. gal
J/g J/g
2.326 000 2.326 000 6.461 112 4.184* 9.224 141
E − 03 E + 00 E − 04 E + 00 E + 00
4.184* E + 03 2.326 000 E + 00
kJ/dm3 kJ/dm3
2.787 163 2.787 163 7.742 119 2.320 800 2.320 800 6.446 667 3.725 895 3.725 895 1.034 971 4.184* 3.581 692
E − 01 E + 02 E − 02 E − 01 E + 02 E − 02 E − 02 E + 01 E − 02 E + 00 E − 01
Calorific value (volume basis—gases)
cal/mL kcal/m3 Btu/ft3
kJ/m3 kJ/m3 kJ/m3 kWh/m3
J/dm3 J/dm3 J/dm3
4.184* 4.184* 3.725 895 1.034 971
E + 03 E + 00 E + 01 E − 02
Specific entropy
Btu/(lbm⋅°R) cal/(g⋅K) kcal/(kg⋅°C)
kJ/(kg⋅K) kJ/(kg⋅K) kJ/(kg⋅K)
J/(g⋅K) J/(g⋅K) J/(g⋅K)
4.186 8* 4.184* 4.184*
E + 00 E + 00 E + 00
Specific-heat capacity (mass basis)
kWh/(kg⋅°C) Btu/(lbm⋅°F) kcal/(kg⋅°C)
kJ/(kg⋅K) kJ/(kg⋅K) kJ/(kg⋅K)
J/(g⋅K) J/(g⋅K) J/(g⋅K)
3.6* 4.186 8* 4.184*
E + 03 E + 00 E + 00
Specific-heat capacity (mole basis)
Btu/(lb⋅mol⋅°F) cal/(g⋅mol⋅°C)
kJ/(kmol⋅K) kJ/(kmol⋅K)
4.186 8* 4.184*
E + 00 E + 00
Temperature (absolute)
°R K
K K
5/9 1
Temperature (traditional)
°F
°C
5/9(°F − 32)
Temperature (difference)
°F
K, °C
5/9
Pressure
atm (760 mmHg at 0°C or 14,696 psi)
µmHg (0°C) µ bar mmHg = torr (0°C) cmH2O (4°C) lbf/ft2 (psf) mHg (0°C) bar dyn/cm2
MPa kPa bar MPa kPa MPa kPa bar kPa kPa kPa kPa kPa Pa Pa Pa
1.013 250* E − 01 1.013 250* E + 02 1.013 250* E + 00 1.0* E − 01 1.0* E + 02 6.894 757 E − 03 6.894 757 E + 00 6.894 757 E − 02 3.376 85 E + 00 2.488 4 E − 01 1.333 224 E − 01 9.806 38 E − 02 4.788 026 E − 02 1.333 224 E − 01 1.0* E + 05 1.0* E − 01
Vacuum, draft
inHg (60°F) inH2O (39.2°F) inH2O (60°F) mmHg (0°C) = torr cmH2O (4°C)
kPa kPa kPa kPa kPa
3.376 85 2.490 82 2.488 4 1.333 224 9.806 38
E + 00 E − 01 E − 01 E − 01 E − 02
Liquid head
ft in
m mm cm
3.048* 2.54* 2.54*
E − 01 E + 01 E + 00
Pressure drop/length
psi/ft
kPa/m
2.262 059 E + 01
Temperature, pressure, vacuum
bar mmHg (0°C) = torr
1-5
TABLE 1-4
Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Continued ) Customary or commonly used unit
Quantity
SI unit
Alternate SI unit
Conversion factor; multiply customary unit by factor to obtain SI unit
Density, specific volume, concentration, dosage kg/m3 g/m3 kg/m3 g/cm3 kg/m3 kg/m3 g/cm3 kg/m3 kg/m3
1.601 846 1.601 846 1.198 264 1.198 264 9.977 633 1.601 846 1.601 846 1.0* 1.601 846
E + 01 E + 04 E + 02 E − 01 E + 01 E + 01 E − 02 E + 03 E + 01
ft /lbm U.K. gal/lbm U.S. gal/lbm
m3/kg m3/g dm3/kg dm3/kg dm3/kg
6.242 796 6.242 796 6.242 796 1.002 242 8.345 404
E − 02 E − 05 E + 01 E + 01 E + 00
Specific volume (mole basis)
L/(g⋅mol) ft3/(lb⋅mol)
m3/kmol m3/kmol
1 6.242 796 E − 02
Specific volume
bbl/U.S. ton bbl/U.K. ton
m3/t m3/t
1.752 535 E − 01 1.564 763 E − 01
Yield
bbl/U.S. ton bbl/U.K. ton U.S. gal/U.S. ton U.S. gal/U.K. ton
dm3/t dm3/t dm3/t dm3/t
Concentration (mass/mass)
wt % wt ppm
kg/kg g/kg mg/kg
lbm/bbl g/U.S. gal g/U.K. gal lbm/1000 U.S. gal lbm/1000 U.K. gal gr/U.S. gal gr/ft3 lbm/1000 bbl mg/U.S. gal gr/100 ft3
kg/m3 kg/m3 kg/m3 g/m3 g/m3 g/m3 mg/m3 g/m3 g/m3 mg/m3
ft3/ft3 bbl/(acre⋅ft) vol% U.K. gal/ft3 U.S. gal/ft3 mL/U.S. gal mL/U.K. gal vol ppm U.K. gal/1000 bbl U.S. gal/1000 bbl U.K. pt/1000 bbl
m3/m3 m3/m3 m3/m3 dm3/m3 dm3/m3 dm3/m3 dm3/m3 cm3/m3 dm3/m3 cm3/m3 cm3/m3 cm3/m3
Concentration (mole/volume)
(lb⋅mol)/U.S. gal (lb⋅mol)/U.K. gal (lb⋅mol)/ft3 std ft3 (60°F, 1 atm)/bbl
kmol/m3 kmol/m3 kmol/m3 kmol/m3
Concentration (volume/mole)
U.S. gal/1000 std ft3 (60°F/60°F) bbl/million std ft3 (60°F/60°F)
dm3/kmol dm3/kmol
Throughput (mass basis)
U.K. ton/year U.S. ton/year U.K. ton/day
Density
lbm/ft3 lbm/U.S. gal lbm/U.K. gal lbm/ft3 g/cm3 lbm/ft3
Specific volume
ft3/lbm 3
Concentration (mass/volume)
Concentration (volume/volume)
cm3/g cm3/g
L/t L/t L/t L/t
g/dm3 g/L mg/dm3 mg/dm3 mg/dm3 mg/dm3 mg/dm3
L/m3 L/m3 L/m3 L/m3 L/m3
L/kmol L/kmol
1.752 535 1.564 763 4.172 702 3.725 627
E + 02 E + 02 E + 00 E + 00
1.0* 1.0* 1
E − 02 E + 01
2.853 010 2.641 720 2.199 692 1.198 264 9.977 633 1.711 806 2.288 351 2.853 010 2.641 720 2.288 351
E + 00 E − 01 E − 01 E + 02 E + 01 E + 01 E + 03 E + 00 E − 01 E + 01
1 1.288 931 1.0* 1.605 437 1.336 806 2.641 720 2.199 692 1 1.0* 2.859 403 2.380 952 3.574 253
E − 04 E − 02 E + 02 E + 02 E − 01 E − 01 E − 03 E + 01 E + 01 E + 00
1.198 264 9.977 644 1.601 846 7.518 21
E + 02 E + 01 E + 01 E − 03
3.166 91 1.330 10
E + 00 E − 01
1.016 047 9.071 847 1.016 047 4.233 529 9.071 847 3.779 936 1.016 047 9.071 847 4.535 924
E + 00 E − 01 E + 00 E − 02 E − 01 E − 02 E + 00 E − 01 E − 01
Facility throughput, capacity
U.S. ton/day U.K. ton/h U.S. ton/h lbm/h
1-6
t/a t/a t/d t/h t/d t/h t/h t/h kg/h
TABLE 1-4
Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Continued ) Customary or commonly used unit
Quantity Throughput (volume basis)
bbl/day 3
ft /day bbl/h ft3/h U.K. gal/h U.S. gal/h U.K. gal/min U.S. gal/min
SI unit
Alternate SI unit
Conversion factor; multiply customary unit by factor to obtain SI unit E + 01 E − 01 E − 03 E − 01 E − 02 E − 03 E − 03 E − 03 E − 03 E − 01 E − 02 E − 01 E − 02
t/a m3/d m3/h m3/h m3/h m3/h L/s m3/h L/s m3/h L/s m3/h L/s
5.803 036 1.589 873 1.179 869 1.589 873 2.831 685 4.546 092 1.262 803 3.785 412 1.051 503 2.727 655 7.576 819 2.271 247 6.309 020
kmol/h kmol/s
4.535 924 E − 01 1.259 979 E − 04
Throughput (mole basis)
(lbm⋅mol)/h
Flow rate (mass basis)
U.K. ton/min U.S. ton/min U.K. ton/h U.S. ton/h U.K. ton/day U.S. ton/day million lbm/year U.K. ton/year U.S. ton/year lbm/s lbm/min lbm/h
kg/s kg/s kg/s kg/s kg/s kg/s kg/s kg/s kg/s kg/s kg/s kg/s
1.693 412 1.511 974 2.822 353 2.519 958 1.175 980 1.049 982 5.249 912 3.221 864 2.876 664 4.535 924 7.559 873 1.259 979
E + 01 E + 01 E − 01 E − 01 E − 02 E − 02 E + 00 E − 05 E − 05 E − 01 E − 03 E − 04
Flow rate (volume basis)
bbl/day
U.K. gal/h U.S. gal/h U.K. gal/min U.S. gal/min ft3/min ft3/s
m3/d L/s m3/d L/s m3/s L/s m3/s L/s dm3/s dm3/s dm3/s dm3/s dm3/s dm3/s
1.589 873 1.840 131 2.831 685 3.277 413 4.416 314 4.416 314 7.865 791 7.865 791 1.262 803 1.051 503 7.576 820 6.309 020 4.719 474 2.831 685
E − 01 E − 03 E − 02 E − 04 E − 05 E − 02 E − 06 E − 03 E − 03 E − 03 E − 02 E − 02 E − 01 E + 01
Flow rate (mole basis)
(lb⋅mol)/s (lb⋅mol)/h million scf/D
kmol/s kmol/s kmol/s
4.535 924 E − 01 1.259 979 E − 04 1.383 45 E − 02
Flow rate/length (mass basis)
lbm/(s⋅ft) lbm/(h⋅ft)
kg/(s⋅m) kg/(s⋅m)
1.488 164 E + 00 4.133 789 E − 04
Flow rate/length (volume basis)
U.K. gal/(min⋅ft) U.S. gal/(min⋅ft) U.K. gal/(h⋅in) U.S. gal/(h⋅in) U.K. gal/(h⋅ft) U.S. gal/(h⋅ft)
m2/s m2/s m2/s m2/s m2/s m2/s
Flow rate/area (mass basis)
lbm/(s⋅ft2) lbm/(h⋅ft2)
kg/(s⋅m2) kg/(s⋅m2)
Flow rate/area (volume basis)
ft3/(s⋅ft2) ft3/(min⋅ft2) U.K. gal/(h⋅in2) U.S. gal/(h⋅in2) U.K. gal/(min⋅ft2) U.S. gal/(min⋅ft2) U.K. gal/(h⋅ft2) U.S. gal/(h⋅ft2)
m/s m/s m/s m/s m/s m/s m/s m/s
Flow rate
3
ft /day bbl/h ft3/h
L/s L/s L/s L/s L/s L/s
m3/(s⋅m) m3/(s⋅m) m3/(s⋅m) m3/(s⋅m) m3/(s⋅m) m3/(s⋅m)
2.485 833 2.069 888 4.971 667 4.139 776 4.143 055 3.449 814
E − 04 E − 04 E − 05 E − 05 E − 06 E − 06
4.882 428 E + 00 1.356 230 E − 03 m3/(s⋅m2) m3/(s⋅m2) m3/(s⋅m2) m3/(s⋅m2) m3/(s⋅m2) m3/(s⋅m2) m3/(s⋅m2) m3/(s⋅m2)
3.048* 5.08* 1.957 349 1.629 833 8.155 621 6.790 972 1.359 270 1.131 829
E − 01 E − 03 E − 03 E − 03 E − 04 E − 04 E − 05 E − 05
1-7
TABLE 1-4
Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Continued ) Customary or commonly used unit
Quantity
SI unit
Alternate SI unit
Conversion factor; multiply customary unit by factor to obtain SI unit
Energy, work, power Energy, work
therm
E + 02 E + 05 E + 01 E + 01 E + 00 E + 03 E − 01 E + 00 E + 03 E − 01 E + 00 E + 03 E + 00 E − 04 E + 00 E − 04 E + 00 E − 03 E − 03 E − 03 E − 03 E − 05 E − 07
kcal cal ft⋅lbf lbf⋅ft J (lbf⋅ft2)/s2 erg
MJ kJ kWh MJ MJ kJ kWh MJ kJ kWh MJ kJ kJ kWh kJ kWh kJ kJ kJ kJ kJ kJ J
1.055 056 1.055 056 2.930 711 1.431 744 2.684 520 2.684 520 7.456 999 2.647 780 2.647 780 7.354 999 3.6* 3.6* 1.899 101 5.275 280 1.055 056 2.930 711 4.184* 4.184* 1.355 818 1.355 818 1.0* 4.214 011 1.0*
Impact energy
kgf⋅m lbf⋅ft
J J
9.806 650* E + 00 1.355 818 E + 00
Surface energy
erg/cm2
mJ/m2
1.0*
Specific-impact energy
(kgf⋅m)/cm2 (lbf⋅ft)/in2
J/cm2 J/cm2
9.806 650* E − 02 2.101 522 E − 03
Power
million Btu/h ton of refrigeration Btu/s kW hydraulic horsepower—hhp hp (electric) hp [(550 ft⋅lbf)/s] ch or CV Btu/min (ft⋅lbf)/s kcal/h Btu/h (ft⋅lbf)/min
MW kW kW kW kW kW kW kW kW kW W W W
2.930 711 3.516 853 1.055 056 1 7.460 43 7.46* 7.456 999 7.354 999 1.758 427 1.355 818 1.162 222 2.930 711 2.259 697
Power/area
Btu/(s⋅ft2) cal/(h⋅cm2) Btu/(h⋅ft2)
kW/m2 kW/m2 kW/m2
1.135 653 E + 01 1.162 222 E − 02 3.154 591 E − 03
Heat-release rate, mixing power
hp/ft3 cal/(h⋅cm3) Btu/(s⋅ft3) Btu/(h⋅ft3)
kW/m3 kW/m3 kW/m3 kW/m3
2.633 414 1.162 222 3.725 895 1.034 971
Cooling duty (machinery)
Btu/(bhp⋅h)
W/kW
Specific fuel consumption (mass basis)
lbm/(hp⋅h)
mg/J kg/kWh
kg/MJ
1.689 659 E − 01 6.082 774 E − 01
Specific fuel consumption (volume basis)
m3/kWh U.S. gal/(hp⋅h) U.K. pt/(hp⋅h)
dm3/MJ dm3/MJ dm3/MJ
mm3/J mm3/J mm3/J
2.777 778 E + 02 1.410 089 E + 00 2.116 806 E − 01
Fuel consumption
U.K. gal/mi U.S. gal/mi mi/U.S. gal mi/U.K. gal
dm3/100 km dm3/100 km km/dm3 km/dm3
L/100 km L/100 km km/L km/L
2.824 807 2.352 146 4.251 437 3.540 064
U.S. tonf⋅mi hp⋅h ch⋅h or CV⋅h kWh Chu Btu
1-8
E + 00
E − 01 E + 00 E + 00 E − 01 E − 01 E − 01 E − 01 E − 02 E − 03 E + 00 E − 01 E − 02
E + 01 E + 00 E + 01 E − 02
3.930 148 E − 01
E + 02 E + 02 E − 01 E − 01
TABLE 1-4
Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Continued ) Customary or commonly used unit
Quantity
SI unit
Alternate SI unit
Conversion factor; multiply customary unit by factor to obtain SI unit
in/s in/min
km/h km/h m/s cm/s m/s mm/s mm/s m/d mm/s mm/s
1.852* E + 00 1.609 344* E + 00 3.048* E − 01 3.048* E + 01 5.08* E − 03 8.466 667 E − 02 3.527 778 E − 03 3.048* E − 01 2.54* E + 01 4.233 333 E − 01
Corrosion rate
in/year (ipy) mil/year
mm/a mm/a
2.54* 2.54*
Rotational frequency
r/min
r/s rad/s
1.666 667 E − 02 1.047 198 E − 01
Acceleration (linear)
ft/s2
m/s2 cm/s2
3.048* 3.048*
Acceleration (rotational)
rpm/s
rad/s2
1.047 198 E − 01
Momentum
(lbm⋅ft)/s
(kg⋅m)/s
1.382 550 E − 01
Force
U.K. tonf U.S. tonf kgf (kp) lbf dyn
kN kN N N mN
9.964 016 E + 00 8.896 443 E + 00 9.806 650* E + 00 4.448 222 E + 00 1.0 E − 02
Bending moment, torque
U.S. tonf⋅ft kgf⋅m lbf⋅ft lbf⋅in
kN⋅m N⋅m N⋅m N⋅m
2.711 636 E + 00 9.806 650* E + 00 1.355 818 E + 00 1.129 848 E − 01
Bending moment/length
(lbf⋅ft)/in (lbf⋅in)/in
(N⋅m)/m (N⋅m)/m
5.337 866 E + 01 4.448 222 E + 00
Moment of inertia
lbm⋅ft2
Velocity (linear), speed
knot mi/h ft/s ft/min ft/h ft/day
MPa MPa MPa MPa kPa Pa
E − 01 E + 01
4.214 011 E − 02
kg⋅m2 2
E + 01 E − 02
N/mm2 N/mm2 N/mm2 N/mm2
1.378 951 E + 01 9.806 650* E + 00 9.576 052 E − 02 6.894 757 E − 03 4.788 026 E − 02 1.0* E − 01
Stress
U.S. tonf/in kgf/mm2 U.S. tonf/ft2 lbf/in2 (psi) lbf/ft2 (psf) dyn/cm2
Mass/length
lbm/ft
kg/m
1.488 164 E + 00
Mass/area structural loading, bearing capacity (mass basis)
U.S. ton/ft2 lbm/ft2
Mg/m2 kg/m2
9.764 855 E + 00 4.882 428 E + 00
Diffusivity
ft2/s m2/s ft2/h
m2/s mm2/s m2/s
9.290 304* E − 02 1.0* E + 06 2.580 64* E − 05
Thermal resistance
(°C⋅m2⋅h)/kcal (°F⋅ft2⋅h)/Btu
(K⋅m2)/kW (K⋅m2)/kW
8.604 208 E + 02 1.761 102 E + 02
Heat flux
Btu/(h⋅ft2)
kW/m2
3.154 591 E − 03
Thermal conductivity
(cal⋅cm)/(s⋅cm ⋅°C) (Btu⋅ft)/(h⋅ft2⋅°F)
W/(m⋅K) W/(m⋅K) (kJ⋅m)/(h⋅m2⋅K) W/(m⋅K) W/(m⋅K) W/(m⋅K)
4.184* 1.730 735 6.230 646 1.162 222 1.442 279 1.162 222
E + 02 E + 00 E + 00 E + 00 E − 01 E − 01
kW/(m2⋅K) kW/(m2⋅K) kW/(m2⋅K) kW/(m2⋅K) kJ/(h⋅m2⋅K) kW/(m2⋅K) kW/(m2⋅K)
4.184* 2.044 175 1.162 222 5.678 263 2.044 175 5.678 263 1.162 222
E + 01 E + 01 E − 02 E − 03 E + 01 E − 03 E − 03
Miscellaneous transport properties
2
(kcal⋅m)/(h⋅m2⋅°C) (Btu⋅in)/(h⋅ft2⋅°F) (cal⋅cm)/(h⋅cm2⋅°C) Heat-transfer coefficient
cal/(s⋅cm2⋅°C) Btu/(s⋅ft2⋅°F) cal/(h⋅cm2⋅°C) Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°R) kcal/(h⋅m2⋅°C)
1-9
TABLE 1-4
Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Continued ) Customary or commonly used unit
Quantity
SI unit
Alternate SI unit
kW/(m3⋅K) kW/(m3⋅K)
Conversion factor; multiply customary unit by factor to obtain SI unit
Volumetric heat-transfer coefficient
Btu/(s⋅ft3⋅°F) Btu/(h⋅ft3⋅°F)
6.706 611 E + 01 1.862 947 E − 02
Surface tension
dyn/cm
mN/m
Viscosity (dynamic)
(lbf⋅s)/in2 (lbf⋅s)/ft2 (kgf⋅s)/m2 lbm/(ft⋅s) (dyn⋅s)/cm2 cP lbm/(ft⋅h)
Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s Pa⋅s
Viscosity (kinematic)
ft2/s in2/s m2/h ft2/h cSt
m2/s mm2/s mm2/s m2/s mm2/s
9.290 304* E − 02 6.451 6* E + 02 2.777 778 E + 02 2.580 64* E − 05 1
Permeability
darcy millidarcy
µm2 µm2
9.869 233 E − 01 9.869 233 E − 04
Thermal flux
Btu/(h⋅ft2) Btu/(s⋅ft2) cal/(s⋅cm2)
W/m2 W/m2 W/m2
3.152 1.135 4.184
E + 00 E + 04 E + 04
Mass-transfer coefficient
(lb⋅mol)/[h⋅ft2(lb⋅mol/ft3)] (g⋅mol)/[s⋅m2(g⋅mol/L)]
m/s m/s
8.467 1.0
E − 05 E + 01
Admittance
S
S
1
Capacitance
µF
µF
1
Charge density
C/mm3
C/mm3
1
Conductance
S⍀
S S
1 1
1 (N⋅s)/m2 (N⋅s)/m2 (N⋅s)/m2 (N⋅s)/m2 (N⋅s)/m2 (N⋅s)/m2 (N⋅s)/m2
6.894 757 E + 03 4.788 026 E + 01 9.806 650* E + 00 1.488 164 E + 00 1.0* E − 01 1.0* E − 03 4.133 789 E − 04
Electricity, magnetism
(mho)
Conductivity
S/m ⍀ /m ⍀ m /m
S/m S/m mS/m
1 1 1
Current density
A/mm2
A/mm2
1
2
Displacement
C/cm
C/cm2
1
Electric charge
C
C
1
Electric current
A
A
1
Electric-dipole moment
C⋅m
C⋅m
1
Electric-field strength
V/m
V/m
1
Electric flux
C
C
1
Electric polarization
C/cm2
C/cm2
1
Electric potential
V mV
V mV
1 1
Electromagnetic moment
A⋅m2
A⋅m2
1
Electromotive force
V
V
1
Flux of displacement
C
C
1
Frequency
cycles/s
Hz
1
Impedance
Ω
Ω
1
Linear-current density
A/mm
A/mm
1
Magnetic-dipole moment
Wb⋅m
Wb⋅m
1
Magnetic-field strength
A/mm Oe gamma
A/mm A/m A/m
1 7.957 747 E + 01 7.957 747 E − 04
Magnetic flux
mWb
mWb
1
1-10
TABLE 1-4
Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Continued ) Customary or commonly used unit
Quantity Magnetic-flux density
mT G gamma
SI unit mT T nT
Alternate SI unit
Conversion factor; multiply customary unit by factor to obtain SI unit 1 1.0* 1
Magnetic induction
mT
mT
1
Magnetic moment
A⋅m2
A⋅m2
1
Magnetic polarization
mT
mT
1
Magnetic potential difference
A
A
1
Magnetic-vector potential
Wb/mm
Wb/mm
1
Magnetization
A/mm
A/mm
1
Modulus of admittance
S
S
1
Modulus of impedance
Ω
Ω
1
Mutual inductance
H
H
1
Permeability
µH/m
µH/m
1
E − 04
Permeance
H
H
1
Permittivity
µF/m
µF/m
1
Potential difference
V
V
1
Quantity of electricity
C
C
1
Reactance
Ω
Ω
1
Reluctance
H−1
H−1
1
Resistance
Ω
Ω
1
Resistivity
Ω⋅cm Ω⋅m
Ω⋅cm Ω⋅m
1 1
Self-inductance
mH
mH
1
Surface density of change
mC/m2
mC/m2
1
Susceptance
S
S
1
Volume density of charge
C/mm3
C/mm3
1
Absorbed dose
rad
Gy
1.0*
Acoustical energy
J
J
1
Acoustical intensity
W/cm2
W/m2
1.0*
Acoustical power
W
W
1
Sound pressure
N/m2
N/m2
1.0*
Illuminance
fc
lx
1.076 391 E + 01
Illumination
fc
lx
1.076 391 E + 01
Irradiance
W/m2
W/m2
1
Light exposure
fc⋅s
lx⋅s
1.076 391 E + 01
Luminance
cd/m2
cd/m2
1
Acoustics, light, radiation
Luminous efficacy
lm/W
lm/W
1
Luminous exitance
lm/m2
lm/m2
1
Luminous flux
lm
lm
1
Luminous intensity
cd
cd
1
Radiance
W/m2⋅sr
W/m2⋅sr
1
Radiant energy
J
J
1
Radiant flux
W
W
1
Radiant intensity
W/sr
W/sr
1
Radiant power
W
W
1
E − 02 E + 04
1-11
TABLE 1-4
Conversion Factors: U.S. Customary and Commonly Used Units to SI Units (Concluded ) Customary or commonly used unit
Quantity
SI unit
Alternate SI unit
Conversion factor; multiply customary unit by factor to obtain SI unit
Wavelength
Å
nm
1.0*
E − 01
Capture unit
10−3 cm−1
m−1
E + 01
m−1
m−1
1.0* 1 1
Ci
Bq
3.7*
E + 10
Radioactivity
10−3 cm−1
* An asterisk indicates that the conversion factor is exact. † Conversion factors for length, area, and volume are based on the international foot. The international foot is longer by 2 parts in 1 million than the U.S. Survey foot (land-measurement use). NOTE: The following unit symbols are used in the table: Unit symbol
Name
Unit symbol
Name
A a Bq C cd Ci d °C ° dyn F fc G g gr Gy H h ha Hz J K L, ᐉ, l
ampere annum (year) becquerel coulomb candela curie day degree Celsius degree dyne farad footcandle gauss gram grain gray henry hour hectare hertz joule kelvin liter
lm lx m min ′ N naut mi Oe Ω Pa rad r S s ″ sr St T t V W Wb
lumen lux meter minute minute newton U.S. nautical mile oersted ohm pascal radian revolution siemens second second steradian stokes tesla tonne volt watt weber
NOTE:
1-12
Copyright SPE-AIME, The SI Metric System of Units and SPE’s Tentative Metric Standard, Society of Petroleum Engineers, Dallas, 1977.
TABLE 1-5 Metric Conversion Factors as Exact Numerical Multiples of SI Units The first two digits of each numerical entry represent a power of 10. For example, the entry “−02 2.54” expresses the fact that 1 in = 2.54 × 10−2 m. To convert from abampere abcoulomb abfarad abhenry abmho abohm abvolt acre ampere (international of 1948) angstrom are astronomical unit atmosphere bar barn barrel (petroleum 42 gal) barye British thermal unit (ISO/ TC 12) British thermal unit (International Steam Table) British thermal unit (mean) British thermal unit (thermochemical) British thermal unit (39°F) British thermal unit (60°F) bushel (U.S.) cable caliber calorie (International Steam Table) calorie (mean) calorie (thermochemical) calorie (15°C) calorie (20°C) calorie (kilogram, International Steam Table) calorie (kilogram, mean) calorie (kilogram, thermochemical) carat (metric) Celsius (temperature) centimeter of mercury (0°C) centimeter of water (4°C) chain (engineer’s) chain (surveyor’s or Gunter’s) circular mil cord coulomb (international of 1948) cubit cup curie day (mean solar) day (sidereal) degree (angle) denier (international) dram (avoirdupois) dram (troy or apothecary) dram (U.S. fluid) dyne electron volt erg Fahrenheit (temperature)
Multiply by
To convert from
To
Multiply by
ampere coulomb farad henry mho ohm volt meter2 ampere
To
+01 1.00 +01 1.00 +09 1.00 −09 1.00 +09 1.00 −09 1.00 −08 1.00 +03 4.046 856 −01 9.998 35
meter meter2 meter newton/meter2 newton/meter2 meter2 meter3 newton/meter2 joule
−10 1.00 +02 1.00 +11 1.495 978 +05 1.013 25 +05 1.00 −28 1.00 −01 1.589 873 −01 1.00 +03 1.055 06
joule
+03 1.055 04
joule joule
+03 1.055 87 +03 1.054 350
joule joule meter3 meter meter joule
+03 1.059 67 +03 1.054 68 −02 3.523 907 +02 2.194 56 −04 2.54 +00 4.1868
joule joule joule joule joule
+00 4.190 02 +00 4.184 +00 4.185 80 +00 4.181 90 +03 4.186 8
joule joule
+03 4.190 02 +03 4.184
kilogram kelvin newton/meter2 newton/meter2 meter meter
−04 2.00 tK = tc + 273.15 +03 1.333 22 +01 9.806 38 +01 3.048 +01 2.011 68
meter3 meter meter newton/meter2 lumen/meter2 candela/meter2 meter meter/second2 meter3 meter3 meter3 tesla tesla ampere turn meter3 meter3 degree (angular) radian kilogram kilogram meter meter2 henry meter3 watt watt watt watt watt watt second (mean solar) second (mean solar) kilogram kilogram meter newton/meter2 newton/meter2 newton/meter2 newton/meter2 joule 1/meter joule
−05 2.957 352 −01 3.048 −01 3.048 006 +03 2.988 98 +01 1.076 391 +00 3.426 259 +02 2.011 68 −02 1.00 −03 4.546 087 −03 4.404 883 −03 3.785 411 −09 1.00 −04 1.00 −01 7.957 747 −04 1.420 652 −04 1.182 941 −01 9.00 −02 1.570 796 −05 6.479 891 −03 1.00 −01 1.016 +04 1.00 +00 1.000 495 −01 2.384 809 +02 7.456 998 +03 9.809 50 +02 7.46 +02 7.354 99 +02 7.457 +02 7.460 43 +03 3.60 +03 3.590 170 +01 5.080 234 +01 4.535 923 −02 2.54 +03 3.386 389 +03 3.376 85 +02 2.490 82 +02 2.4884 +00 1.000 165 +02 1.00 +03 4.186 74
meter2 meter3 coulomb
−10 5.067 074 +00 3.624 556 −01 9.998 35
meter meter3 disintegration/second second (mean solar) second (mean solar) radian kilogram/meter kilogram kilogram meter3 newton joule joule kelvin
fluid ounce (U.S.) foot foot (U.S. survey) foot of water (39.2°F) footcandle footlambert furlong gal (galileo) gallon (U.K. liquid) gallon (U.S. dry) gallon (U.S. liquid) gamma gauss gilbert gill (U.K.) gill (U.S.) grad grad grain gram hand hectare henry (international of 1948) hogshead (U.S.) horsepower (550 ft lbf/s) horsepower (boiler) horsepower (electric) horsepower (metric) horsepower (U.K.) horsepower (water) hour (mean solar) hour (sidereal) hundredweight (long) hundredweight (short) inch inch of mercury (32°F) inch of mercury (60°F) inch of water (39.2°F) inch of water (60°F) joule (international of 1948) kayser kilocalorie (International Steam Table) kilocalorie (mean) kilocalorie (thermochemical) kilogram mass kilogram-force (kgf) kilopond-force kip knot (international) lambert lambert langley lbf (pound-force, avoirdupois) lbm (pound-mass, avoirdupois) league (British nautical) league (international nautical) league (statute) light-year link (engineer’s) link (surveyor’s or Gunter’s) liter lux maxwell meter micrometer mil mile (U.S. statute) mile (U.K. nautical) mile (international nautical) mile (U.S. nautical) millibar millimeter of mercury (0°C)
joule joule kilogram newton newton newton meter/second candela/meter2 candela/meter2 joule/meter2 newton
+03 4.190 02 +03 4.184 +00 1.00 +00 9.806 65 +00 9.806 65 +03 4.448 221 −01 5.144 444 +04 1/π +03 3.183 098 +04 4.184 +00 4.448 221
kilogram
−01 4.535 923
meter meter
+03 5.559 552 +03 5.556
meter meter meter meter meter3 lumen/meter2 weber wavelengths Kr 86 meter meter meter meter meter meter newton/meter2 newton/meter2
+03 4.828 032 +15 9.460 55 −01 3.048 −01 2.011 68 −03 1.00 +00 1.00 −08 1.00 +06 1.650 763 −06 1.00 −05 2.54 +03 1.609 344 +03 1.853 184 +03 1.852 +03 1.852 +02 1.00 +02 1.333 224
Fahrenheit (temperature)
Celsius
farad (international of 1948) faraday (based on carbon 12) faraday (chemical) faraday (physical) fathom fermi (femtometer)
farad coulomb
−01 4.572 −04 2.365 882 +10 3.70 +04 8.64 +04 8.616 409 −02 1.745 329 −07 1.111 111 −03 1.771 845 −03 3.887 934 −06 3.696 691 −05 1.00 −19 1.602 10 −07 1.00 tK = (5/9)(tF + 459.67) tc = (5/9)(tF − 32) −01 9.995 05 +04 9.648 70
coulomb coulomb meter meter
+04 9.649 57 +04 9.652 19 +00 1.828 8 −15 1.00
1-13
TABLE 1-5 Metric Conversion Factors as Exact Numerical Multiples of SI Units (Concluded ) The first two digits of each numerical entry represent a power of 10. For example, the entry “−02 2.54” expresses the fact that 1 in = 2.54 × 10−2 To convert from minute (angle) minute (mean solar) minute (sidereal) month (mean calendar) nautical mile (international) nautical mile (U.S.) nautical mile (U.K.) oersted ohm (international of 1948) ounce-force (avoirdupois) ounce-mass (avoirdupois) ounce-mass (troy or apothecary) ounce (U.S. fluid) pace parsec pascal peck (U.S.) pennyweight perch phot pica (printer’s) pint (U.S. dry) pint (U.S. liquid) point (printer’s) poise pole pound-force (lbf avoirdupois) pound-mass (lbm avoirdupois) pound-mass (troy or apothecary) poundal quart (U.S. dry) quart (U.S. liquid) rad (radiation dose absorbed) Rankine (temperature) rayleigh (rate of photon emission) rhe rod roentgen rutherford second (angle)
1-14
To
Multiply by
radian second (mean solar) second (mean solar) second (mean solar) meter meter meter ampere/meter ohm newton kilogram kilogram meter3 meter meter newton/meter2 meter3 kilogram meter lumen/meter2 meter meter3 meter3 meter (newton-second)/meter2 meter newton
−04 2.908 882 +01 6.00 +01 5.983 617 +06 2.628 +03 1.852 +03 1.852 +03 1.853 184 +01 7.957 747 +00 1.000 495 −01 2.780 138 −02 2.834 952 −02 3.110 347 −05 2.957 352 −01 7.62 +16 3.083 74 +00 1.00 −03 8.809 767 −03 1.555 173 +00 5.0292 +04 1.00 −03 4.217 517 −04 5.506 104 −04 4.731 764 −04 3.514 598 −01 1.00 +00 5.0292 +00 4.448 221
kilogram
−01 4.535 923
kilogram
−01 3.732 417
newton meter3 meter3 joule/kilogram
−01 1.382 549 −03 1.101 220 −04 9.463 529 −02 1.00
kelvin 1/second-meter2
tK = (5/9)tR +10 1.00
meter2/(newtonsecond) meter coulomb/kilogram disintegration/second radian
+01 1.00 +00 5.0292 −04 2.579 76 +06 1.00 −06 4.848 136
To
Multiply by
second (ephemeris) second (mean solar)
To convert from
second second (ephemeris)
second (sidereal) section scruple (apothecary) shake skein slug span statampere statcoulomb statfarad stathenry statmho statohm statute mile (U.S.) statvolt stere stilb stoke tablespoon teaspoon ton (assay) ton (long) ton (metric) ton (nuclear equivalent of TNT) ton (register) ton (short, 2000 lb) tonne torr (0°C) township unit pole volt (international of 1948) watt (international of 1948) yard year (calendar) year (sidereal) year (tropical) year 1900, tropical, Jan., day 0, hour 12 year 1900, tropical, Jan., day 0, hour 12
second (mean solar) meter2 kilogram second meter kilogram meter ampere coulomb farad henry mho ohm meter volt meter3 candela/meter2 meter2/second meter3 meter3 kilogram kilogram kilogram joule meter3 kilogram kilogram newton/meter2 meter2 weber volt watt meter second (mean solar) second (mean solar) second (mean solar) second (ephemeris)
+00 1.000 000 Consult American Ephemeris and Nautical Almanac −01 9.972 695 +06 2.589 988 −03 1.295 978 −08 1.00 +02 1.097 28 +01 1.459 390 −01 2.286 −10 3.335 640 −10 3.335 640 −12 1.112 650 +11 8.987 554 −12 1.112 650 +11 8.987 554 +03 1.609 344 +02 2.997 925 +00 1.00 +04 1.00 −04 1.00 −05 1.478 676 −06 4.928 921 −02 2.916 666 +03 1.016 046 +03 1.00 +09 4.20 +00 2.831 684 +02 9.071 847 +03 1.00 +02 1.333 22 +07 9.323 957 −07 1.256 637 +00 1.000 330 +00 1.000 165 −01 9.144 +07 3.1536 +07 3.155 815 +07 3.155 692 +07 3.155 692
second
+07 3.155 692
TABLE 1-6
Alphabetical Listing of Common Conversions
To convert from Acres Acres Acres Acre-feet Ampere-hours (absolute) Angstrom units Angstrom units Angstrom units Atmospheres Atmospheres Atmospheres Atmospheres Atmospheres Atmospheres Atmospheres Atmospheres Bags (cement) Barrels (cement) Barrels (oil) Barrels (oil) Barrels (U.S. liquid) Barrels (U.S. liquid) Barrels per day Bars Bars Bars Board feet Boiler horsepower Boiler horsepower B.t.u. B.t.u. B.t.u. B.t.u. B.t.u. B.t.u. B.t.u. B.t.u. B.t.u. B.t.u. B.t.u. per cubic foot B.t.u. per hour B.t.u. per minute B.t.u. per pound B.t.u. per pound per degree Fahrenheit B.t.u. per pound per degree Fahrenheit B.t.u. per second B.t.u. per square foot per hour B.t.u. per square foot per minute B.t.u. per square foot per second for a temperature gradient of 1°F. per inch
To
Multiply by
To convert from
To
Multiply by
Square feet Square meters Square miles Cubic meters Coulombs (absolute) Inches Meters Microns Millimeters of mercury at 32°F Dynes per square centimeter Newtons per square meter Feet of water at 39.1°F Grams per square centimeter Inches of mercury at 32°F Pounds per square foot Pounds per square inch Pounds (cement) Pounds (cement) Cubic meters Gallons Cubic meters Gallons Gallons per minute Atmospheres Newtons per square meter Pounds per square inch Cubic feet B.t.u. per hour Kilowatts Calories (gram) Centigrade heat units (c.h.u. or p.c.u.) Foot-pounds Horsepower-hours Joules Liter-atmospheres Pounds carbon to CO2 Pounds water evaporated from and at 212°F Cubic foot-atmospheres Kilowatt-hours Joules per cubic meter Watts Horsepower Joules per kilogram Calories per gram per degree centigrade Joules per kilogram per degree Kelvin Watts Joules per square meter per second Kilowatts per square foot Calories, gram (15°C.), per square centimeter per second for a temperature gradient of 1°C. per centimeter
43,560 4074 0.001563 1233 3600 3.937 × 10−9 1 × 10−10 1 × 10−4 760 1.0133 × 106 101,325 33.90 1033.3 29.921 2116.3 14.696 94 376 0.15899 42 0.11924 31.5 0.02917 0.9869 1 × 105 14.504 1 ⁄12 33,480 9.803 252 0.55556 777.9 3.929 × 10−4 1055.1 10.41 6.88 × 10−5
B.t.u. (60°F.) per degree Fahrenheit Bushels (U.S. dry) Bushels (U.S. dry) Calories, gram Calories, gram Calories, gram Calories, gram Calories, gram Calories, gram, per gram per degree C. Calories, kilogram Calories, kilogram per second Candle power (spherical) Carats (metric) Centigrade heat units Centimeters Centimeters Centimeters Centimeters Centimeters Centimeters of mercury at 0°C. Centimeters of mercury at 0°C. Centimeters of mercury at 0°C Centimeters of mercury at 0°C. Centimeters of mercury at 0°C. Centimeters per second Centimeters of water at 4°C. Centistokes Circular mils Circular mils Circular mils Cords Cubic centimeters Cubic centimeters Cubic centimeters Cubic centimeters Cubic feet Cubic feet Cubic feet Cubic feet Cubic feet Cubic feet Cubic foot-atmospheres Cubic foot-atmospheres Cubic feet of water (60°F.) Cubic feet per minute Cubic feet per minute Cubic feet per second Cubic feet per second Cubic inches Cubic yards Curies Curies Degrees Drams (apothecaries’ or troy)
Calories per degree centigrade Cubic feet Cubic meters B.t.u. Foot-pounds Joules Liter-atmospheres Horsepower-hours Joules per kilogram per degree Kelvin Kilowatt-hours Kilowatts Lumens Grams B.t.u. Angstrom units Feet Inches Meters Microns Atmospheres Feet of water at 39.1°F. Newtons per square meter Pounds per square foot Pounds per square inch Feet per minute Newtons per square meter Square meters per second Square centimeters Square inches Square mils Cubic feet Cubic feet Gallons Ounces (U.S. fluid) Quarts (U.S. fluid) Bushels (U.S.) Cubic centimeters Cubic meters Cubic yards Gallons Liters Foot-pounds Liter-atmospheres Pounds Cubic centimeters per second Gallons per second Gallons per minute Million gallons per day Cubic meters Cubic meters Disintegrations per minute Coulombs per minute Radians Grams
453.6 1.2444 0.03524 3.968 × 10−3 3.087 4.1868 4.130 × 10−2 1.5591 × 10−6 4186.8 0.0011626 4.185 12.556 0.2 1.8 1 × 108 0.03281 0.3937 0.01 10,000 0.013158 0.4460 1333.2 27.845 0.19337 1.9685 98.064 1 × 10−6 5.067 × 10−6 7.854 × 10−7 0.7854 128 3.532 × 10−5 2.6417 × 10−4 0.03381 0.0010567 0.8036 28,317 0.028317 0.03704 7.481 28.316 2116.3 28.316 62.37 472.0 0.1247 448.8 0.64632 1.6387 × 10−5 0.76456 2.2 × 1012 1.1 × 1012 0.017453 3.888
0.001036 0.3676 2.930 × 10−4 37,260 0.29307 0.02357 2326 1 4186.8 1054.4 3.1546 0.1758 1.2405
1-15
TABLE 1-6
Alphabetical Listing of Common Conversions (Concluded )
To convert from Drams (avoirdupois) Dynes Ergs Faradays Fathoms Feet Feet per minute Feet per minute Feet per (second)2 Feet of water at 39.2°F. Foot-poundals Foot-poundals Foot-poundals Foot-pounds Foot-pounds Foot-pounds Foot-pounds Foot-pounds Foot-pounds Foot-pounds force Foot-pounds per second Foot-pounds per second Furlongs Gallons (U.S. liquid) Gallons Gallons Gallons Gallons Gallons Gallons per minute Gallons per minute Grains Grains Grains per cubic foot Grains per gallon Grams Grams Grams Grams Grams Grams Grams per cubic centimeter
1-16
To Grams Newtons Joules Coulombs (abs.) Feet Meters Centimeters per second Miles per hour Meters per (second)2 Newtons per square meter B.t.u. Joules Liter-atmospheres B.t.u. Calories, gram Foot-poundals Horsepower-hours Kilowatt-hours Liter-atmospheres Joules Horsepower Kilowatts Miles Barrels (U.S. liquid) Cubic meters Cubic feet Gallons (Imperial) Liters Ounces (U.S. fluid) Cubic feet per hour Cubic feet per second Grams Pounds Grams per cubic meter Parts per million Drams (avoirdupois) Drams (troy) Grains Kilograms Pounds (avoirdupois) Pounds (troy) Pounds per cubic foot
Multiply by 1.7719 1 × 10−5 1 × 10−7 96,500 6 0.3048 0.5080 0.011364 0.3048 2989 3.995 × 10−5 0.04214 4.159 × 10−4 0.0012856 0.3239 32.174 5.051 × 10−7 3.766 × 10−7 0.013381 1.3558 0.0018182 0.0013558 0.125 0.03175 0.003785 0.13368 0.8327 3.785 128 8.021 0.002228 0.06480 1⁄ 7000 2.2884 17.118 0.5644 0.2572 15.432 0.001 0.0022046 0.002679 62.43
To convert from Horsepower (British) Horsepower (metric) Horsepower (metric) Hours (mean solar) Inches Inches of mercury at 60°F Inches of water at 60°F Joules (absolute) Joules (absolute) Joules (absolute) Joules (absolute) Joules (absolute) Joules (absolute) Kilocalories Kilograms Kilograms force Kilograms per square centimeter Kilometers Kilowatt-hours Kilowatt-hours Kilowatts Knots (international) Knots (nautical miles per hour) Lamberts Liter-atmospheres Liter-atmospheres Liters Liters Liters Lumens Micromicrons Microns Microns Miles (nautical) Miles (nautical) Miles Miles Miles per hour Miles per hour Milliliters Millimeters
To Pounds water evaporated per hour at 212°F Foot-pounds per second Kilogram-meters per second Seconds Meters Newtons per square meter Newtons per square meter B.t.u. (mean) Calories, gram (mean) Cubic foot-atmospheres Foot-pounds Kilowatt-hours Liter-atmospheres Joules Pounds (avoirdupois) Newtons Pounds per square inch Miles B.t.u. Foot-pounds Horsepower Meters per second Miles per hour Candles per square inch Cubic foot-atmospheres Foot-pounds Cubic feet Cubic meters Gallons Watts Microns Angstrom units Meters Feet Miles (U.S. statute) Feet Meters Feet per second Meters per second Cubic centimeters Meters
Multiply by 2.64 542.47 75.0 3600 0.0254 3376.9 248.84 9.480 × 10−4 0.2389 0.3485 0.7376 2.7778 × 10−7 0.009869 4186.8 2.2046 9.807 14.223 0.6214 3414 2.6552 × 106 1.3410 0.5144 1.1516 2.054 0.03532 74.74 0.03532 0.001 0.26418 0.001496 1 × 10−6 1 × 104 1 × 10−6 6080 1.1516 5280 1609.3 1.4667 0.4470 1 0.001
TABLE 1-6
Alphabetical Listing of Common Conversions (Concluded )
To convert from Grams per cubic centimeter Grams per liter Grams per liter Grams per square centimeter Grams per square centimeter Hectares Hectares Horsepower (British) Horsepower (British) Horsepower (British) Horsepower (British) Horsepower (British) Horsepower (British) Horsepower (British) Pounds (avoirdupois) Pounds (avoirdupois) Pounds (avoirdupois) Pounds per cubic foot Pounds per cubic foot Pounds per square foot Pounds per square foot Pounds per square inch Pounds per square inch Pounds per square inch Pounds force Pounds force per square foot Pounds water evaporated from and at 212°F. Pound-centigrade units (p.c.u.) Quarts (U.S. liquid) Radians Revolutions per minute Seconds (angle) Slugs Slugs Slugs Square centimeters
To Pounds per gallon Grains per gallon Pounds per cubic foot Pounds per square foot Pounds per square inch Acres Square meters B.t.u. per minute B.t.u. per hour Foot-pounds per minute Foot-pounds per second Watts Horsepower (metric) Pounds carbon to CO2 per hour Grains Kilograms Pounds (troy) Grams per cubic centimeter Kilograms per cubic meter Atmospheres Kilograms per square meter Atmospheres Kilograms per square centimeter Newtons per square meter Newtons Newtons per square meter Horsepower-hours B.t.u. Cubic meters Degrees Radians per second Radians Gee pounds Kilograms Pounds Square feet
Multiply by 8.345 58.42 0.0624 2.0482 0.014223 2.471 10,000 42.42 2545 33,000 550 745.7 1.0139 0.175 7000 0.45359 1.2153 0.016018 16.018 4.725 × 10−4 4.882 0.06805 0.07031 6894.8 4.4482 47.88 0.379 1.8 9.464 × 10−4 57.30 0.10472 4.848 × 10−6 1 14.594 32.17 0.0010764
To convert from
To
Millimeters of mercury at 0°C. Millimicrons Mils Mils Minims (U.S.) Minutes (angle) Minutes (mean solar) Newtons Ounces (avoirdupois) Ounces (avoirdupois) Ounces (U.S. fluid) Ounces (troy) Pints (U.S. liquid) Poundals
Newtons per square meter Microns Inches Meters Cubic centimeters Radians Seconds Kilograms Kilograms Ounces (troy) Cubic meters Ounces (apothecaries’) Cubic meters Newtons
Square feet Square feet per hour Square inches Square inches Square yards Stokes Tons (long) Tons (long) Tons (metric) Tons (metric) Tons (metric) Tons (short) Tons (short) Tons (refrigeration) Tons (British shipping) Tons (U.S. shipping) Torr (mm. mercury, 0°C.) Watts Watts Watts Watt-hours Yards
Square meters Square meters per second Square centimeters Square meters Square meters Square meters per second Kilograms Pounds Kilograms Pounds Tons (short) Kilograms Pounds B.t.u. per hour Cubic feet Cubic feet Newtons per square meter B.t.u. per hour Joules per second Kilogram-meters per second Joules Meters
Multiply by 133.32 0.001 0.001 2.54 × 10−5 0.06161 2.909 × 10−4 60 0.10197 0.02835 0.9115 2.957 × 10−5 1.000 4.732 × 10−4 0.13826 0.0929 2.581 × 10−5 6.452 6.452 × 10−4 0.8361 1 × 10−4 1016 2240 1000 2204.6 1.1023 907.18 2000 12,000 42.00 40.00 133.32 3.413 1 0.10197 3600 0.9144
1-17
TABLE 1-7 Mass (M)
Length (L)
Common Units and Conversion Factors* 1 pound mass = 453.5924 grams = 0.45359 kilograms = 7000 grains 1 slug = 32.174 pounds mass 1 ton (short) = 2000 pounds mass 1 ton (long) = 2240 pounds mass 1 ton (metric) = 1000 kilograms = 2204.62 pounds mass 1 pound mole = 453.59 gram moles = 30.480 centimeters = 0.3048 meters 1 inch = 2.54 centimeters = 0.0254 meters 1 mile (U.S.) = 1.60935 kilometers 1 yard = 0.9144 meters 1 foot
Area (L2)
Volume (L3)
1 square foot = 929.0304 square centimeters = 0.09290304 square meters 1 square inch = 6.4516 square centimeters 1 square yard = 0.836127 square meters 1 cubic foot
1 gallon Time (θ) 1 hour
= 28,316.85 cubic centimeters = 0.02831685 cubic meters = 28.31685 liters = 7.481 gallons (U.S.) = 3.7853 liters = 231 cubic inches = 60 minutes = 3600 seconds
Temperature (T) 1 centigrade or Celsius degree = 1.8 Fahrenheit degree Temperature, Kelvin = T°C + 273.15 Temperature, Rankine = T°F + 459.7 Temperature, Fahrenheit = 9/5 T°C + 32 Temperature, centigrade or Celsius = 5/9 (T°F − 32) Temperature, Rankine = 1.8 T K Force (F) 1 pound force = 444,822.2 dynes = 4.448222 Newtons = 32.174 poundals Pressure (F/L2) Normal atmospheric pressure
1 atm = 760 millimeters of mercury at 0°C (density 13.5951 g/cm3) = 29.921 inches of mercury at 32°F = 14.696 pounds force/square inch = 33.899 feet of water at 39.1°F = 1.01325 × 106 dynes/square centimeter = 1.01325 × 105 Newtons/square meter Density (M/L3) 1 pound mass/cubic foot = 0.01601846 grams/cubic centimeter = 16.01846 kilogram/cubic meter Energy (H or FL) 1 British thermal unit = 251.98 calories = 1054.4 joules = 777.97 foot-pounds force = 10.409 liter-atmospheres = 0.2930 watt-hour 2 Diffusivity (L /θ) 1 square foot/hour = 0.258 cm2/s = 2.58 × 10−5 m2/s Viscosity (M/Lθ) 1 pound mass/foot hour = 0.00413 g/cm s 0.000413 kg/m s 1 centipoise = 0.01 poise = 0.01 g/cm s = 0.001 kg/m s = 0.000672 lbm/ft s = 0.0000209 lbfs/ft2 Thermal conductivity [H/θL2(T/L)] 2 1 Btu/hr ft (°F/ft) = 0.00413 cal/s cm2 (°C/cm) = 1.728 J/s m2 (°C/m) Heat transfer coefficient 1 Btu/hr ft2 °F = 5.678 J/s m2 °C Heat capacity (H/MT) 1 Btu/lbm °F = 1 cal/g °C = 4184 J/kg °C Gas constant 1.987 Btu/lbm mole °R = 1.987 cal/mol K = 82.057 atm cm3/mol K = 0.7302 atm ft3/lb mole °F = 10.73 (lbf /in.2) (ft3)/lb mole °R = 1545 (lbf /ft2) (ft3)/lb mole °R = 8.314 (N/m2) (m3)/mol K Gravitational acceleration g = 9.8066 m/s2 = 32.174 ft/s2
NOTE: U.S. customary units; or British units, on left and SI units on right. *Adapted from Faust et al., Principles of Unit Operations, John Wiley and Sons, 1980.
TABLE 1-8
Kinematic-Viscosity Conversion Formulas
Viscosity scale Saybolt Universal Saybolt Furol Redwood No. 1 Redwood Admiralty Engler
Range of t, sec 32 < t < 100 t > 100 25 < t < 40 t > 40 34 < t < 100 t > 100
Kinematic viscosity, stokes 0.00226t − 1.95/t 0.00220t − 1.35/t 0.0224t − 1.84/t 0.0216t − 0.60/t 0.00260t − 1.79/t 0.00247t − 0.50/t 0.027t − 20/t 0.00147t − 3.74/t
TABLE 1-9 Temp. scale
Values of the Gas-Law Constant Press. units
Vol. units
Kelvin atm. atm. mm. Hg bar kg/cm2 atm mm Hg
cm3 liters liters liters liters ft3 ft3
atm in Hg mm Hg lb/in2abs lb/ft2abs
ft3 ft3 ft3 ft3 ft3
Rankine
1-18
Wt. units
Energy units
R
g-moles g-moles g-moles g-moles g-moles g-moles g-moles g-moles lb-moles lb-moles lb-moles lb-moles lb-moles lb-moles lb-moles lb-moles lb-moles lb-moles lb-moles
calories joules (abs) joules (int) atm cm3 atm liters mm Hg-liters bar-liters kg/(cm2)(liters) atm-ft3 mm Hg-ft3 chu or pcu Btu hp-hr kw-hr atm-ft3 in Hg-ft3 mm Hg-ft3 (lb)(ft3)/in2 ft-lb
1.9872 8.3144 8.3130 82.057 0.08205 62.361 0.08314 0.08478 1.314 998.9 1.9872 1.9872 0.0007805 0.0005819 0.7302 21.85 555.0 10.73 1545.0
TABLE 1-10 United States Customary System of Weights and Measures Linear Measure 12 inches (in) or (″) = 1 foot (ft) or (′) 3 feet = 1 yard (yd) 16.5 feet = 1 rod (rd) 5.5 yards
冧 冧
5280 feet = 1 mile (mi) 320 rods 1 mil = 0.001 inch Nautical:
TABLE 1-11
Temperature Conversion Formulas
°F = (°C × 5/9) + 32 °C = (°F − 32) × 5/9 °R = °F + 459.69 °K = °C + 273.15 °K = °R × 5/9 Temperature difference, ⌬T °F = °C × 9/5 NOTE: An extensive table of temperature conversions may be found in the sixth edition of the Handbook (Table 1-12).
6080.2 feet = 1 nautical mile 6 feet = 1 fathom 120 fathoms = 1 cable length 1 knot = 1 nautical mile per hour 60 nautical miles = 1° of latitude
Square Measure 144 sq. inches (sq. in) or (in2) or (ⵧ″) = 1 sq. foot (ft2) or (ⵧ′) 9 sq. feet (ft2) (ⵧ′) = 1 sq. yard (yd2) 30.25 sq. yards = 1 sq. rod, pole, or perch 10 sq. chains 160 sq. rods = = 1 acre 43,560 sq. ft 640 acres = 1 sq. mile = 1 section 1 circular inch (area of circle of 1 inch diameter) = 0.7854 sq. inch 1 sq. inch = 1.2732 circular inch 1 circular mil = area of circle of 0.001 inch diameter 1,000,000 circular mils = 1 circular inch
冦
冧
Circular Measure 60 seconds (″) (sec) = 1 minute (min) or (′) 60 minutes (′) = 1 degree (°) 90 degrees (°) = 1 quadrant 360 degrees (°) = 1 circumference = 1 radian (rad.) 57.29578 degrees = 57° 17′ 44.81″
冦
Volume Measure Solid:
1728 cubic in (cu. in) (in3) = 1 cubic foot (cu. ft)(ft3) 27 cu. ft = 1 cubic yard (cu. yd) Dry Measure: 2 pints = 1 quart 8 quarts = 1 peck 4 pecks = 1 bushel 1 United States Winchester bushel = 2150.42 cubic inches Liquid: 4 gills = 1 pint (pt) 2 pints = 1 quart (qt) 4 quarts = 1 gallon (gal) 7.4805 gallons = 1 cubic foot Apothecaries’ Liquid: 60 minims (min. or ) = 1 fluid dram or drachm 8 drams ( ) = 1 fluid ounce 16 ounces (oz. ) = 1 pint Avoirdupois Weight 16 drams = 437.5 grains = 1 ounce (oz) 16 ounces = 7000 grains = 1 pound (lb) 100 pounds = 1 hundredweight (cwt) 2000 pounds = 1 short ton: 2240 pounds = 1 long ton Troy Weight 24 grains = 1 pennyweight (dwt) 20 pennyweights = 1 ounce (oz) 12 ounces = 1 pound (lb) Apothecaries’ Weight 20 grains (gr) = 1 scruple ( ) 3 scruples = 1 dram ( ) 8 drams = 1 ounce ( ) 12 ounces = 1 pound (lb)
1-19
TABLE 1-12
Specific Gravity, Degrees Baumé, Degrees API, Degrees Twaddell, Pounds per Gallon, Pounds per Cubic Foot* 145 140 sp gr 60°/60°F − 1 141.5 °Bé = 145 − ᎏ (heavier than H2O); °Bé = ᎏ − 130 (lighter than H2O); °Tw = ᎏᎏ °API = ᎏ − 131.5 sp gr sp gr 0.005 sp gr Lb per gal at 60°F wt in air
Lb per ft3 at 60°F wt in air
Sp gr 60°/ 60°
°Bé
104.33 102.38 100.47 98.58 96.73
4.9929 5.0346 5.0763 5.1180 5.1597
37.350 37.662 37.973 38.285 38.597
0.700 .705 .710 .715 .720
70.00 68.58 67.18 65.80 64.44
94.00 92.22 90.47 88.75 87.05
94.90 93.10 91.33 89.59 87.88
5.2014 5.2431 5.2848 5.3265 5.3682
39.910 39.222 39.534 39.845 40.157
.725 .730 .735 .740 .745
.650 .655 .660 .665 .670
85.38 83.74 82.12 80.53 78.96
86.19 84.53 82.89 81.28 79.69
5.4098 5.4515 5.4932 5.5349 5.5766
40.468 40.780 41.092 41.404 41.716
.675 .680 .685 .690 .695
77.41 75.88 74.38 72.90 71.44
78.13 76.59 75.07 73.57 72.10
5.6183 5.6600 5.7017 5.7434 5.7851
42.028 42.340 42.652 42.963 43.275
Lb per ft3 at 60°F wt in air
Sp gr 60°/ 60°
°Bé
Sp gr 60°/ 60°
°Bé
°API
0.600 .605 .610 .615 .620
103.33 101.40 99.51 97.64 95.81
.625 .630 .635 .640 .645
°API
Lb per gal at 60°F wt in air
Lb per ft3 at 60°F wt in air
Sp gr 60°/ 60°
°Bé
°API
70.64 69.21 67.80 66.40 65.03
5.8268 5.8685 5.9101 5.9518 5.9935
43.587 43.899 44.211 44.523 44.834
0.800 .805 .810 .815 .820
45.00 43.91 42.84 41.78 40.73
45.38 44.28 43.19 42.12 41.06
63.10 61.78 60.48 59.19 57.92
63.67 62.34 61.02 59.72 58.43
6.0352 6.0769 6.1186 6.1603 6.2020
45.146 45.458 45.770 46.082 46.394
.825 .830 .835 .840 .845
39.70 38.67 37.66 36.67 35.68
.750 .755 .760 .765 .770
56.67 55.43 54.21 53.01 51.82
57.17 55.92 54.68 53.47 52.27
6.2437 6.2854 6.3271 6.3688 6.4104
46.706 47.018 47.330 47.642 47.953
.850 .855 .860 .865 .870
.775 .780 .785 .790 .795
50.65 49.49 48.34 47.22 46.10
51.08 49.91 48.75 47.61 46.49
6.4521 6.4938 6.5355 6.5772 6.6189
47.265 48.577 48.889 49.201 49.513
.875 .880 .885 .890 .895
Lb per ft3 at 60°F wt in air
Sp gr 60°/ 60°
Lb per gal at 60°F wt in air
Lb per ft3 at 60°F wt in air
Sp gr 60°/ 60°
°Bé
°API
Lb per gal at 60°F wt in air
6.6606 6.7023 6.7440 6.7857 6.8274
49.825 50.137 50.448 50.760 51.072
0.900 .905 .910 .915 .920
25.56 24.70 23.85 23.01 22.17
25.72 24.85 23.99 23.14 22.30
7.4944 7.5361 7.5777 7.6194 7.6612
56.062 56.374 56.685 56.997 57.310
40.02 38.98 37.96 36.95 35.96
6.8691 6.9108 6.9525 6.9941 7.0358
51.384 51.696 52.008 52.320 52.632
.925 .930 .935 .940 .945
21.35 20.54 19.73 18.94 18.15
21.47 20.65 19.84 19.03 18.24
7.7029 7.7446 7.7863 7.8280 7.8697
57.622 57.934 58.246 58.557 58.869
34.71 33.74 32.79 31.85 30.92
34.97 34.00 33.03 32.08 31.14
7.0775 7.1192 7.1609 7.2026 7.2443
52.943 53.255 53.567 53.879 54.191
.950 .955 .960 .965 .970
17.37 16.60 15.83 15.08 14.33
17.45 16.67 15.90 15.13 14.38
7.9114 7.9531 7.9947 8.0364 8.0780
59.181 59.493 59.805 60.117 60.428
30.00 29.09 28.19 27.30 26.42
30.21 29.30 28.39 27.49 26.60
7.2860 7.3277 7.3694 7.4111 7.4528
54.503 54.815 55.127 55.438 55.750
.975 .980 .985 .990 .995 1.000
13.59 12.86 12.13 11.41 10.70 10.00
13.63 12.89 12.15 11.43 10.71 10.00
8.1197 8.1615 8.2032 8.2449 8.2866 8.3283
60.740 61.052 61.364 61.676 61.988 62.300 Lb per ft3 at 60°F. wt. in air
°Tw
Lb per gal at 60°F wt in air
°Tw
Lb per gal at 60°F wt in air
°Bé
°Tw
Lb per gal at 60°F wt in air
°Bé
°Tw
Lb per gal at 60°F wt in air
1.005 1.010 1.015 1.020 1.025
0.72 1.44 2.14 2.84 3.54
1 2 3 4 5
8.3700 8.4117 8.4534 8.4950 8.5367
62.612 62.924 63.236 63.547 63.859
1.255 1.260 1.265 1.270 1.275
29.46 29.92 30.38 30.83 31.27
51 52 53 54 55
10.4546 10.4963 10.5380 10.5797 10.6214
78.206 78.518 78.830 79.141 79.453
1.505 1.510 1.515 1.520 1.525
48.65 48.97 49.29 49.61 49.92
101 102 103 104 105
12.5392 12.5809 12.6226 12.6643 12.7060
93.800 94.112 94.424 94.735 95.047
1.755 1.760 1.765 1.770 1.775
62.38 62.61 62.85 63.08 63.31
151 152 153 154 155
14.6238 14.6655 14.7072 14.7489 14.7906
109.394 109.705 110.017 110.329 110.641
1.030 1.035 1.040 1.045 1.050
4.22 4.90 5.58 6.24 6.91
6 7 8 9 10
8.5784 8.6201 8.6618 8.7035 8.7452
64.171 64.483 64.795 65.107 65.419
1.280 1.285 1.290 1.295 1.300
31.72 32.16 32.60 33.03 33.46
56 57 58 59 60
10.6630 10.7047 10.7464 10.7881 10.8298
79.765 80.077 80.389 80.701 81.013
1.530 1.535 1.540 1.545 1.550
50.23 50.54 50.84 51.15 51.45
106 107 108 109 110
12.7477 12.7894 12.8310 12.8727 12.9144
95.359 95.671 95.983 96.295 96.606
1.780 1.785 1.790 1.795 1.800
63.54 63.77 63.99 64.22 64.44
156 157 158 159 160
14.8323 14.8740 14.9157 14.9574 14.9990
110.953 111.265 111.577 111.889 112.200
1.055 1.060 1.065 1.070 1.075
7.56 8.21 8.85 9.49 10.12
11 12 13 14 15
8.7869 8.8286 8.8703 8.9120 8.9537
65.731 66.042 66.354 66.666 66.978
1.305 1.310 1.315 1.320 1.325
33.89 34.31 34.73 35.15 35.57
61 62 63 64 65
10.8715 10.9132 10.9549 10.9966 11.0383
81.325 81.636 81.948 82.260 82.572
1.555 1.560 1.565 1.570 1.575
51.75 52.05 52.35 52.64 52.94
111 112 113 114 115
12.9561 12.9978 13.0395 13.0812 13.1229
96.918 97.230 97.542 97.854 98.166
1.805 1.810 1.815 1.820 1.825
64.67 64.89 65.11 65.33 65.55
161 162 163 164 165
15.0407 15.0824 15.1241 15.1658 15.2075
112.512 112.824 113.136 113.448 113.760
1.080 1.085 1.090 1.095 1.100
10.74 11.36 11.97 12.58 13.18
16 17 18 19 20
8.9954 9.0371 9.0787 9.1204 9.1621
67.290 67.602 67.914 68.226 68.537
1.330 1.335 1.340 1.345 1.350
35.98 36.39 36.79 37.19 37.59
66 67 68 69 70
11.0800 11.1217 11.1634 11.2051 11.2467
82.884 83.196 83.508 83.820 84.131
1.580 1.585 1.590 1.595 1.600
53.23 53.52 53.81 54.09 54.38
116 117 118 119 120
13.1646 13.2063 13.2480 13.2897 13.3313
98.478 98.790 99.102 99.414 99.725
1.830 1.835 1.840 1.845 1.850
65.77 65.98 66.20 66.41 66.62
166 167 168 169 170
15.2492 15.2909 15.3326 15.3743 15.4160
114.072 114.384 114.696 115.007 115.318
1.105 1.110 1.115 1.120 1.125
13.78 14.37 14.96 15.54 16.11
21 22 23 24 25
9.2038 9.2455 9.2872 9.3289 9.3706
68.849 69.161 69.473 69.785 70.097
1.355 1.360 1.365 1.370 1.375
37.99 38.38 38.77 39.16 39.55
71 72 73 74 75
11.2884 11.3301 11.3718 11.4135 11.4552
84.443 84.755 85.067 85.379 85.691
1.605 1.610 1.615 1.620 1.625
54.66 54.94 55.22 55.49 55.77
121 122 123 124 125
13.3730 13.4147 13.4564 13.4981 13.5398
100.037 100.349 100.661 100.973 101.285
1.855 1.860 1.865 1.870 1.875
66.83 67.04 67.25 67.46 67.67
171 172 173 174 175
15.4577 15.4993 15.5410 15.5827 15.6244
115.630 115.943 116.255 116.567 116.879
1.130 1.135 1.140 1.145 1.150
16.68 17.25 17.81 18.36 18.91
26 27 28 29 30
9.4123 9.4540 9.4957 9.5374 9.5790
70.409 70.721 71.032 71.344 71.656
1.380 1.385 1.390 1.395 1.400
39.93 40.31 40.68 41.06 41.43
76 77 78 79 80
11.4969 11.5386 11.5803 11.6220 11.6637
86.003 86.315 86.626 86.938 87.250
1.630 1.635 1.640 1.645 1.650
56.04 56.32 56.59 56.85 57.12
126 127 128 129 130
13.5815 13.6232 13.6649 13.7066 13.7483
101.597 101.909 102.220 102.532 102.844
1.880 1.885 1.890 1.895 1.900
67.87 68.08 68.28 68.48 68.68
176 177 178 179 180
15.6661 15.7078 15.7495 15.7912 15.8329
117.191 117.503 117.814 118.126 118.438
1.155 1.160 1.165 1.170 1.175
19.46 20.00 20.54 21.07 21.60
31 32 33 34 35
9.6207 9.6624 9.7041 9.7458 9.7875
71.968 72.280 72.592 72.904 73.216
1.405 1.410 1.415 1.420 1.425
41.80 42.16 42.53 42.89 43.25
81 82 83 84 85
11.7054 11.7471 11.7888 11.8304 11.8721
87.562 87.874 88.186 88.498 88.810
1.655 1.660 1.665 1.670 1.675
57.39 57.65 57.91 58.17 58.43
131 132 133 134 135
13.7900 13.8317 13.8734 13.9150 13.9567
103.156 103.468 103.780 104.092 104.404
1.905 1.910 1.915 1.920 1.925
68.88 69.08 69.28 69.48 69.68
181 182 183 184 185
15.8746 15.9163 15.9580 15.9996 16.0413
118.740 119.062 119.374 119.686 119.998
1.180 1.185 1.190 1.195 1.200
22.12 22.64 23.15 23.66 24.17
36 37 38 39 40
9.8292 9.8709 9.9126 9.9543 9.9960
73.528 73.840 74.151 74.463 74.775
1.430 1.435 1.440 1.445 1.450
43.60 43.95 44.31 44.65 45.00
86 87 88 89 90
11.9138 11.9555 11.9972 12.0389 12.0806
89.121 89.433 89.745 90.057 90.369
1.680 1.685 1.690 1.695 1.700
58.69 58.95 59.20 59.45 59.71
136 137 138 139 140
13.9984 14.0401 14.0818 14.1235 14.1652
104.715 105.027 105.339 105.651 105.963
1.930 1.935 1.940 1.945 1.950
69.87 70.06 70.26 70.45 70.64
186 187 188 189 190
16.0830 16.1247 16.1664 16.2081 16.2498
120.309 120.621 120.933 121.245 121.557
1.205 1.210 1.215 1.220 1.225
24.67 25.17 25.66 26.15 26.63
41 42 43 44 45
10.0377 10.0793 10.1210 10.1627 10.2044
75.087 75.399 75.711 76.022 76.334
1.455 1.460 1.465 1.470 1.475
45.34 45.68 46.02 46.36 46.69
91 92 93 94 95
12.1223 12.1640 12.2057 12.2473 12.2890
90.681 90.993 91.305 91.616 91.928
1.705 1.710 1.715 1.720 1.725
59.96 60.20 60.45 60.70 60.94
141 142 143 144 145
14.2069 14.2486 14.2903 14.3320 14.3737
106.275 106.587 106.899 107.210 107.522
1.955 1.960 1.965 1.970 1.975
70.83 71.02 71.21 71.40 71.58
191 192 193 194 195
16.2915 16.3332 16.3749 16.4166 16.4583
121.869 122.181 122.493 122.804 123.116
1.230 1.235 1.240 1.245 1.250
27.11 27.59 28.06 28.53 29.00
46 47 48 49 50
10.2461 10.2878 10.3295 10.3712 10.4129
76.646 76.958 77.270 77.582 77.894
1.480 1.485 1.490 1.495 1.500
47.03 47.36 47.68 48.01 48.33
96 97 98 99 100
12.3307 12.3724 12.4141 12.4558 12.4975
92.240 92.552 92.864 93.176 93.488
1.730 1.735 1.740 1.745 1.750
61.18 61.34 61.67 61.91 62.14
146 147 148 149 150
14.4153 14.4570 14.4987 14.5404 14.5821
107.834 108.146 108.458 108.770 109.082
1.980 1.985 1.990 1.995 2.000
71.77 71.95 72.14 72.32 72.50
196 197 198 199 200
16.5000 16.5417 16.5833 16.6250 16.6667
123.428 123.740 124.052 124.364 124.676
Sp gr 60°/ 60°
°Bé
Lb per ft3 at 60°F wt in air
Sp gr 60°/ 60°
Lb per ft3 at 60°F. wt. in air
*Prepared by Lewis V. Judson, Ph.D., Chief of Length Section of National Bureau of Standards with the advice and assistance of E. L. Peffer, B.S., A.M., late Chief of Capacity and Density Section, National Bureau of Standards.
1-20
TABLE 1-13
Wire and Sheet-Metal Gauges* Values in approximate decimals of an inch As a number of gauges are in use for various shapes and metals, it is advisable to state the thickness in thousandths when specifying gauge number.
Gauge number
American (AWG) or Brown & Sharpe (B & S) (for nonferrous wire and sheet)†
U.S. Steel Wire (Stl WG) or Washburn & Moen or Roebling or Am. Steel & Wire Co. [A. (steel) WG] (for steel wire)
Birmingham (BWG) (for steel wire) or Stubs Iron Wire (for iron or brass wire)‡
U.S. Standard (for sheet and plate metal, wrought iron)
0000000 000000 00000 0000 000 00 0
— — — 0.460 .410 .365 .325
0.4900 .4615 .4305 .3938 .3625 .3310 .3065
— — — 0.454 .425 .380 .340
1 2 3 4 5
.289 .258 .229 .204 .182
.2830 .2625 .2437 .2253 .2070
6 7 8 9 10
.162 .144 .128 .114 .102
11 12 13 14 15
U.S. Steel Wire (Stl WG) or Washburn & Moen or Roebling or Am. Steel & Wire Co. [A. (steel) WG] (for steel wire)
Birmingham (BWG) (for steel wire) or Stubs Iron Wire (for iron or brass wire)‡
U.S. Standard (for sheet and plate metal, wrought iron)
Standard Birmingham (BG) (for sheet and hoop metal)
Imperial Standard and Wire Gauge (SWG) (British legal standard)
Gauge number
Gauge number
American (AWG) or Brown & Sharpe (B & S) (for nonferrous wire and sheet)†
Standard Birmingham (BG) (for sheet and hoop metal)
Imperial Standard and Wire Gauge (SWG) (British legal standard)
Gauge number
— — — — — — —
0.6666 .6250 .5883 .5416 .5000 .4452 .3964
0.500 .464 .432 .400 .372 .348 .324
0000000 000000 00000 0000 000 00 0
26 27 28 29 30
0.0159 .0142 .0126 .0113 .0100
0.0181 .0173 .0162 .0150 .0140
0.018 .016 .014 .013 .012
0.0188 .0172 .0156 .0141 .0125
0.0196 .0175 .0156 .0139 .0123
0.018 .0164 .0148 .0136 .0124
26 27 28 29 30
.300 .284 .259 .238 .220
— — 0.239 .224 .209
.3532 .3147 .2804 .2500 .2225
.300 .276 .252 .232 .212
1 2 3 4 5
31 32 33 34 35
.0089 .0080 .0071 .0063 .0056
.0132 .0128 .0118 .0104 .0095
.010 .009 .008 .007 .005
.0109 .0102 .0094 .0086 .0078
.0110 .0098 .0087 .0077 .0069
.0116 .0108 .0100 .0092 .0084
31 32 33 34 35
.1920 .1770 .1620 .1483 .1350
.203 .180 .165 .148 .134
.194 .179 .164 .150 .135
.1981 .1764 .1570 .1398 .1250
.192 .176 .160 .144 .128
6 7 8 9 10
36 37 38 39 40
.0050 .0045 .0040 .0035 .0031
.0090 .0085 .0080 .0075 .0070
.004 — — — —
.0070 .0066 .0062 — —
.0061 .0054 .0048 .0043 .0039
.0076 .0068 .0060 .0052 .0048
36 37 38 39 40
.091 .081 .072 .064 .057
.1205 .1055 .0915 .0800 .0720
.120 .109 .095 .083 .072
.120 .105 .090 .075 .067
.1113 .0991 .0882 .0785 .0699
.116 .104 .092 .080 .072
11 12 13 14 15
41 42 43 44 45
— — — — —
.0066 .0062 .0060 .0058 .0055
— — — — —
— — — — —
.0034 .0031 .0027 .0024 .0022
.0044 .0040 .0036 .0032 .0028
41 42 43 44 45
16 17 18 19 20
.051 .045 .040 .036 .032
.0625 .0540 .0475 .0410 .0348
.065 .058 .049 .042 .035
.060 .054 .0478 .0418 .0359
.0625 .0556 .0495 .0440 .0392
.064 .056 .048 .040 .036
16 17 18 19 20
46 47 48 49 50
— — — — —
.0052 .0050 .0048 .0046 .0044
— — — — —
— — — — —
.0019 .0017 .0015 .0014 .0012
.0024 .0020 .0016 .0012 .0010
46 47 48 49 50
21 22 23 24 25
.0285 .0253 .0226 .0201 .0179
.0317 .0286 .0258 .0230 .0204
.032 .028 .025 .022 .020
.0329 .0299 .0269 .0239 .0209
.0349 .0313 .0278 .0248 .0220
.032 .028 .024 .022 .020
21 22 23 24 25
Metric wire gauge is 10 times the diameter in millimeters. *Courtesy of Dr. Lewis V. Judson with I. H. Fullmer, National Bureau of Standards. † Sometimes used for iron wire. ‡ Sometimes used for copper plate and for steel plate 12 gauge and heavier and for steel tubes.
1-21
TABLE 1-14
Fundamental Physical Constants
1 sec = 1.00273791 sidereal seconds g0 = 9.80665 m/sec2 1 liter = 0.001 cu. m 1 atm = 101,325 newtons/sq m 1 mm Hg (pressure) = (1⁄ 760) atm = 133.3224 newtons/sq m 1 int ohm = 1.000495 ⫾ 0.000015 abs ohm 1 int amp = 0.999835 ⫾ 0.000025 abs amp 1 int coul = 0.999835 ⫾ 0.000025 abs coul 1 int volt = 1.000330 ⫾ 0.000029 abs volt 1 int watt = 1.000165 ⫾ 0.000052 abs watt 1 int joule = 1.000165 ⫾ 0.000052 abs joule T0°C = 273.150 ⫾ 0.010°K (PV)0°CP=0 = (RT)0°C = 2271.16 ⫾ 0.04 abs joule/mole = 22,414.6 ⫾ 0.4 cu. cm atm/mole = 22.4146 ⫾ 0.0004 liter atm/mole R = 8.31439 ⫾ 0.00034 abs joule/deg mole = 1.98719 ⫾ 0.00013 cal/deg mole = 82.0567 ⫾ 0.0034 cu. cm atm/deg mole = 0.0820567 ⫾ 0.0000034 liter atm/deg mole ln 10 = 2.302585 R ln 10 = 19.14460 ⫾ 0.00078 abs joule/deg mole = 4.57567 ⫾ 0.00030 cal/deg mole N = (6.02283 ⫾ 0.0022) × 1023/mole h = (6.6242 ⫾ 0.0044) × 10−34 joule sec c = (2.99776 ⫾ 0.00008) × 108 m/sec (h2/8π2k) = (4.0258 ⫾ 0.0037) × 10−39 g sq cm deg (h/8π2c) = (2.7986 ⫾ 0.0018) × 10−39 g cm Z = Nhc = 11.9600 ⫾ 0.0036 abs joule cm/mole = 2.85851 ⫾ 0.0009 cal cm/mole (Z/R) = (hc/k) = c2 = 1.43847 ⫾ 0.00045 cm deg Ᏺ = 96,501.2 ⫾ 10.0 int coul/g-equiv or int joule/int volt g-equiv = 96,485.3 ⫾ 10.0 abs coul/g-equiv or abs joule/abs volt g-equiv = 23,068.1 ⫾ 2.4 cal/int volt g-equiv = 23,060.5 ⫾ 2.4 cal/abs volt g-equiv e = (1.60199 ⫾ 0.00060) × 10−19 abs coul = (1.60199 ⫾ 0.00060) × 10−20 abs emu = (4.80239 ⫾ 0.00180) × 10−10 abs esu 1 int electron-volt/molecule = 96,501.2 ⫾ 10 int joule/mole = 23,068.1 ⫾ 2.4 cal/mole 1 abs electron-volt/molecule = 96,485.3 ⫾ 10. abs joule/mole = 23,060.5 ⫾ 2.4 cal/mole 1 int electron-volt = (1.60252 ⫾ 0.00060) × 10−12 erg 1 abs electron-volt = (1.60199 ⫾ 0.00060) × 10−12 erg hc = (1.23916 ⫾ 0.00032) × 10−4 int electron-volt cm = (1.23957 ⫾ 0.00032) × 10−4 abs electron-volt cm k = (8.61442 ⫾ 0.00100) × 10−5 int electron-volt/deg = (8.61727 ⫾ 0.00100) × 10−5 abs electron-volt/deg = (R/N) = (1.38048 ⫾ 0.00050) × 10−23 joule/deg 1 IT cal = (1⁄ 860) = 0.00116279 int watt-hr = 4.18605 int joule = 4.18674 abs joule = 1.000654 cal 1 cal = 4.1840 abs joule = 4.1833 int joule = 41.2929 ⫾ 0.0020 cu. cm atm = 0.0412929 ⫾ 0.0000020 liter atm 1 IT cal/g = 1.8 Btu/lb 1 Btu = 251.996 IT cal = 0.293018 int watt-hr = 1054.866 int joule = 1055.040 abs joule = 252.161 cal 1 horsepower = 550 ft-lb (wt)/sec = 745.578 int watt = 745.70 abs watt 1 in = (1/0.3937) = 2.54 cm 1 ft = 0.304800610 m 1 lb = 453.5924277 g 1 gal = 231 cu. in = 0.133680555 cu. ft = 3.785412 × 10−3 cu. m = 3.785412 liter
1-22
sec = mean solar second Definition: g0 = standard gravity Definition: atm = standard atmosphere mm Hg (pressure) = standard millimeter mercury int = international; abs = absolute amp = ampere coul = coulomb
Absolute temperature of the ice point, 0°C PV product for ideal gas at 0°C R = gas constant per mole
ln = natural logarithm (base e) N = Avogadro number h = Planck constant c = velocity of light Constant in rotational partition function of gases Constant relating wave number and moment of inertia Z = constant relating wave number and energy per mole c2 = second radiation constant Ᏺ = Faraday constant
e = electronic charge
Constant relating wave number and energy per molecule k = Boltzmann constant Definition of IT cal: IT = International steam tables cal = thermochemical calorie Definition: cal = thermochemical calorie
Definition of Btu: Btu = IT British Thermal Unit
cal = thermochemical calorie Definition of horsepower (mechanical): lb (wt) = weight of 1 lb at standard gravity Definition of in: in = U.S. inch ft = U.S. foot (1 ft = 12 in) Definition; lb = avoirdupois pound Definition; gal = U.S. gallon
CONVERSION OF VALUES FROM U.S. CUSTOMARY UNITS TO SI UNITS American engineers are probably more familiar with the magnitude of physical entities in U.S. customary units than in SI units. Consequently, errors made in the conversion from one set of units to the other may go undetected. The following six examples will show how to convert the elements in six dimensionless groups. Proper conversions will result in the same numerical value for the dimensionless number. The dimensionless numbers used as examples are the Reynolds, Prandtl, Nusselt, Grashof, Schmidt, and Archimedes numbers. Table 1-7 provides a number of useful conversion factors. To make a conversion of an element in U.S. customary units to SI units, one multiplies the value of the U.S. customary unit, found on the left side in the table, by the equivalent value on the right side. For example, to convert 10 British thermal units to joules, one multiplies 10 by 1054.4 to obtain 10544 joules. In each example, the initial values of the factors are expressed in U.S. customary units, and the dimensionless value is calculated. Then the factors are converted to SI units, and the dimensionless value is recalculated. The two dimensionless values will be approximately the same. (Small variations occur due to the number of significant figures carried in the solution.)
Example 1. Calculation of a Reynolds Number DVρ NRe = ᎏ µ U.S. customary units D = 3 in. = 3⁄12 ft V = 6 ft/s ρ = 0.08 lbm/ft3 µ = 0.015 cp = (0.015)(0.000672) lbm/ft⋅s (3/12)(6)(0.08) NRe = ᎏᎏ = 11,904 (0.015)(0.000672) SI units D = (3)(0.0254) m V = (6)(0.3048) m/s ρ = (0.08)(16.018) kg/m3 µ = (0.015)(0.001) kg/m⋅s
(Difference due to rounding)
Example 4. Calculation of a Grashof Number NGr = L3ρ2gβ(∆T)/µ2 U.S. Customary units L = 3 ft ρ = 0.0725 lbm/ft3 g = 32.174 ft/s2 β = 0.00168/°R ∆T = 99 °R µ = 0.019 centipoise = 0.019 × 0.000672 lbm/ft⋅s = 1.277 × 10−5 lbm/ft⋅s (33) (0.0725)2(32.174) (0.00168) (99) = 4.66 × 109 NGr = ᎏᎏᎏᎏ (1.277 × 10−5)2 SI units L = (3)(0.3048) = 0.9144 m ρ = (0.0725)(16.018) = 1.1613 kg/m3 g = 9.807 m/s2 β = (0.00168)/(1.8) = 0.000933/°K ∆T = (99)(1.8) = 178.2 °K µ = (0.019)(0.001) = 1.9 × 10−5 kg/m⋅s (0.9144)3(1.1613)2(9.807)(0.000933)(178.2) = 4.66 × 109 NGr = ᎏᎏᎏᎏᎏ (1.9 × 10−5)2
Example 5. Calculation of a Schmidt Number µ NSc = ᎏ ρD
(3 × 0.0254) (6 × 0.3048) (0.08 × 16.018) NRe = ᎏᎏᎏᎏᎏ = 11,904 (0.015) (0.001)
Example 2. Calculation of a Prandtl Number Cp µ NPr = ᎏ k U.S. customary units γp = 0.47 Btu/lbm °F µ = 15 centipoise = (15) (0.000672) (3600) lbm/ft⋅hr k = 0.065 Btu/hr⋅ft2 (°F/ft) (0.47) (15 × 0.000672 × 3600) NPr = ᎏᎏᎏᎏ = 262.4 0.065 SI units γ = (0.47)(4184) J/kg °C µ = (15)(0.001) kg/m⋅s k = (0.065)(1.728) J/s⋅m2 (°C/m) (0.47) (4184) (15) (0.001) NPr = ᎏᎏᎏ = 262.6 (0.065) (1.728) (Difference due to rounding)
Example 3. Calculation of a Nusselt Number hD NNu = ᎏ k U.S. customary units h = 200 Btu/hr⋅ft2⋅°F D = 1.5 in. = 1.5/12 ft k = 0.07 Btu/hr⋅ft2 (°F/ft) (200)(1.5/12) NNu = ᎏᎏ = 357.1 0.07 SI units h = (200)(5.678) J/(s⋅m2⋅°C) D = (1.5)(0.0254) m k = (0.07)(1.728) J/s⋅m2 (°C/m)
(200) (5.678) (1.5) (0.0254) NNu = ᎏᎏᎏ = 357.7 (0.07) (1.728)
U.S. customary units µ = 0.02 centipoise = (0.02)(2.42) lbm/ft⋅hr ρ = 0.08 lbm/ft3 D = 1.0 ft2/hr (diffusivity) (0.02) (2.42) NSc = ᎏᎏ = 0.605 (0.08)(1.0) SI units µ = (0.02)(0.001) kg/m⋅s ρ = (0.08)(16.02) kg/m2 D = (1.0)(2.58 × 10−5) m2/s (0.02) (0.001) = 0.605 NSc = ᎏᎏᎏᎏ (0.08)(16.02)(1.0) (2.58 × 10−5)
Example 6. Calculation of an Archimedes Number d 3ρf(ρp − ρf)g NAr = ᎏᎏ µ2 U.S. customary units d = 2 mm = 2/[(1000)(0.3048)] = 0.00656 ft ρf = 0.0175 lbm/ft3 ρp = 168.5 lbm/ft3 g = 32.174 ft/s2 µ = 0.04 centipoise = 0.04 × 0.000672 = 2.688−5 lbm/ft⋅s (0.00656)3 (0.0175) (168.5 − 0.017) (32.174) = 37,064 NAr = ᎏᎏᎏᎏᎏ (2.688 × 10−5)2 SI units d = 2/1000 m ρp = 168.5 × 16.02 = 2699.37 kg/m3 ρf = 0.0175 × 16.02 = 0.2804 g/m3 g = 9.807 m/s2 µ = 0.04 × 0.001 = 4 × 10−5 kg/m⋅s (2/1000)3 (0.2804) (2699.37 − 0.28) (9.807) = 37,118 NAr = ᎏᎏᎏᎏᎏ (4 × 10−5)2 (Difference due to rounding)
1-23
MATHEMATICAL SYMBOLS TABLE 1-15
Mathematical Signs, Symbols, and Abbreviations
⫾ (⫿) : ⬋ < ⬏ > ⬐ ⬵ ∼ ⬑ ≠ ⬟ ∝ ∞ ∴ 兹苶苶 3 苶苶 兹 n 兹苶苶 ⬔ ⊥ 储 |x| log or log10 loge or ln e a° a′ a a″ a sin cos tan ctn or cot sec csc vers covers exsec sin−1 sinh cosh tanh sinh−1 f(x) or φ(x) ∆x 冱 dx dy/dx or y′ d2y/dx2 or y″ dny/dxn ∂y/∂x ∂ny/∂xn ∂ny ᎏ ∂x∂y
冮
冕
plus or minus (minus or plus) divided by, ratio sign proportional sign less than not less than greater than not greater than approximately equals, congruent similar to equivalent to not equal to approaches, is approximately equal to varies as infinity therefore square root cube root nth root angle perpendicular to parallel to numerical value of x common logarithm or Briggsian logarithm natural logarithm or hyperbolic logarithm or Naperian logarithm base (2.178) of natural system of logarithms an angle a degrees prime, an angle a minutes double prime, an angle a seconds, a second sine cosine tangent cotangent secant cosecant versed sine coversed sine exsecant anti sine or angle whose sine is hyperbolic sine hyperbolic cosine hyperbolic tangent anti hyperbolic sine or angle whose hyperbolic sine is function of x increment of x summation of differential of x derivative of y with respect to x second derivative of y with respect to x nth derivative of y with respect to x partial derivative of y with respect to x nth partial derivative of y with respect to x nth partial derivative with respect to x and y integral of
b
integral between the limits a and b
a
y˙ y¨ ∆ or ∇2
δ
冖
1-24
first derivative of y with respect to time second derivative of y with respect to time the “Laplacian” ∂2 ∂2 ∂2 ᎏ2 + ᎏ2 + ᎏ2 ∂x ∂y ∂z sign of a variation sign for integration around a closed path
冢
冣
TABLE 1-16 Alpha Beta Gamma Delta Epsilon Zeta Eta Theta Iota Kappa Lambda Mu
Greek Alphabet
= Α, α = A, a = Β, β = B, b = Γ, γ = G, g = ∆, δ = D, d = Ε, ε = E, e = Ζ, ζ = Z, z = Η, η = E, e = Θ, θ = Th, th = Ι, = I, i = Κ, κ = K, k = Λ, λ = L, l = Μ, µ = M, m
Nu Xi Omicron Pi Rho Sigma Tau Upsilon Phi Chi Psi Omega
= Ν, ν = N, n = Ξ, ξ = X, x = Ο, ο = O, o = Π, π = P, p = Ρ, ρ = R, r = Σ, σ = S, s = Τ, τ = T, t = ⌼, υ = U, u = Φ, φ = Ph, ph = Χ, χ = Ch, ch = Ψ, ψ = Ps, ps = Ω, ω = O, o
Section 2
Physical and Chemical Data*
Peter E. Liley, Ph.D., D.I.C., School of Mechanical Engineering, Purdue University. (physical and chemical data) George H. Thomson, AIChE Design Institute for Physical Property Data. (Tables 2-6, 2-30, 2-164, 2-193, 2-196, 2-198, 2-221) D.G. Friend, National Institutes of Standards and Technology, Boulder, CO. (Tables 2-333, 2-334, Figs. 2-25, 2-26) Thomas E. Daubert, Ph.D., Professor, Department of Chemical Engineering, The Pennsylvania State University. (Prediction and Correlation of Physical Properties) Evan Buck, M.S.Ch.E., Manager, Thermophysical Property Skill Center, Central Technology, Union Carbide Corporation. (Prediction and Correlation of Physical Properties)
GENERAL REFERENCES PHYSICAL PROPERTIES OF PURE SUBSTANCES Tables 2-1 2-2
Physical Properties of the Elements and Inorganic Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Properties of Organic Compounds. . . . . . . . . . . . . .
VAPOR PRESSURES OF PURE SUBSTANCES Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-3 Vapor Pressure of Water Ice from −15 to 0°C . . . . . . . . . . . . 2-4 Vapor Pressure of Liquid Water from −16 to 0°C . . . . . . . . . 2-5 Vapor Pressure of Liquid Water from 0 to 100°C . . . . . . . . . 2-6 Vapor Pressure of Inorganic and Organic Liquids. . . . . . . . . 2-6a Alphabetical Index to Substances in Tables 2-6, 2-30, 2-164, 2-193, 2-196, 2-198, and 2-221 . . . . . . . . . . . . . . . . . 2-7 Vapor Pressures of Inorganic Compounds, up to 1 atm . . . . 2-8 Vapor Pressures of Organic Compounds, up to 1 atm. . . . . . VAPOR PRESSURES OF SOLUTIONS Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables and Figures 2-9 Partial Pressures of Water over Aqueous Solutions of HCl . . 2-10 Partial Pressures of HCl over Aqueous Solutions of HCl . . .
2-11 2-7 2-28
2-12 2-13
2-48 2-48 2-48 2-48 2-49 2-50 2-55 2-57 2-61
2-76 2-76 2-76
2-14 2-15 2-16 2-17 2-18 2-19 2-20 2-21 2-22 2-23
Vapor Pressures of H3PO4 Aqueous: Partial Pressure of H2O Vapor (Fig. 2-1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapor Pressures of H3PO4 Aqueous: Weight of H2O in Saturated Air (Fig. 2-2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Pressures of H2O and SO2 over Aqueous Solutions of Sulfur Dioxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Water Partial Pressure, bar, over Aqueous Sulfuric Acid Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sulfur Trioxide Partial Pressure, bar, over Aqueous Sulfuric Acid Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sulfuric Acid Partial Pressure, bar, over Aqueous Sulfuric Acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Pressure, bar, of Aqueous Sulfuric Acid Solutions . . . . Partial Pressures of HNO3 and H2O over Aqueous Solutions of HNO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Pressures of H2O and HBr over Aqueous Solutions of HBr at 20 to 55°C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Pressures of HI over Aqueous Solutions of HI at 25°C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapor Pressures of the System: Water-Sulfuric Acid-Nitric Acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Vapor Pressures of Aqueous Solutions of CH3COOH . . Partial Pressures of H2O over Aqueous Solutions of HN3 . . Vapor Pressure of Aqueous Diethylene Glycol Solutions (Fig. 2-3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mole Percentages of H2O over Aqueous Solutions of NH3 . . Partial Pressures of NH3 over Aqueous Solutions of NH3 . . .
2-77 2-77 2-77 2-78 2-80 2-82 2-83 2-84 2-85 2-85 2-85 2-85 2-85 2-85 2-86 2-87
* The contributions of J.K. Fink, Argonne National Laboratory; U. Grigull, Tech. Universität, Munich, Germany; and H. Sato, Keio University, Japan, are acknowledged. 2-1
2-2 2-24 2-25 2-26 2-27
PHYSICAL AND CHEMICAL DATA Total Vapor Pressures of Aqueous Solutions of NH3 . . . . . . . Partial Pressures of H2O over Aqueous Solutions of Sodium Carbonate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Pressures of H2O and CH3OH over Aqueous Solutions of Methyl Alcohol . . . . . . . . . . . . . . . . . . . . . . . . . Partial Pressures of H2O over Aqueous Solutions of Sodium Hydroxide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
WATER-VAPOR CONTENT OF GASES Chart for Gases at High Pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Water Content of Air (Fig. 2-4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DENSITIES OF PURE SUBSTANCES Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-28 Density (kg/m3) of Water from 0 to 100°C. . . . . . . . . . . . . . . 2-29 Density (kg/m3) of Mercury from 0 to 350°C. . . . . . . . . . . . . 2-30 Densities of Inorganic and Organic Liquids . . . . . . . . . . . . . DENSITIES OF AQUEOUS INORGANIC SOLUTIONS Units and Units Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-31 Aluminum Sulfate [A12(SO4)3] . . . . . . . . . . . . . . . . . . . . . . . . . 2-32 Ammonia (NH3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-33 Ammonium Acetate (CH3COONH4) . . . . . . . . . . . . . . . . . . . . 2-34 Ammonium Bichromate [(NH4)2Cr2O7]. . . . . . . . . . . . . . . . . . 2-35 Ammonium Chloride (NH4Cl) . . . . . . . . . . . . . . . . . . . . . . . . . 2-36 Ammonium Chromate [(NH4)2CrO4]. . . . . . . . . . . . . . . . . . . . 2-37 Ammonium Nitrate (NH4NO3). . . . . . . . . . . . . . . . . . . . . . . . . 2-38 Ammonium Sulfate [(NH4)2SO4] . . . . . . . . . . . . . . . . . . . . . . . 2-39 Arsenic Acid (H3A3O4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-40 Barium Chloride (BaCl2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-41 Cadmium Nitrate [Cd(NO3)2]. . . . . . . . . . . . . . . . . . . . . . . . . . 2-42 Calcium Chloride (CaCl2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-43 Calcium Hydroxide [Ca(OH)2] . . . . . . . . . . . . . . . . . . . . . . . . . 2-44 Calcium Hypochlorite (CaOCl2). . . . . . . . . . . . . . . . . . . . . . . . 2-45 Calcium Nitrate [Ca(NO3)2] . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-46 Chromic Acid (CrO3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-47 Chromium Chloride (CrCl3) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-48 Copper Nitrate [Cu(NO3)2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-49 Copper Sulfate (CuSO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-50 Cuprous Chloride (Cu2Cl2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-51 Ferric Chloride (FeCl3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-52 Ferric Sulfate [Fe2(SO4)3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-53 Ferric Nitrate [Fe(NO3)3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-54 Ferrous Sulfate (FeSO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-55 Hydrogen Bromide (HBr). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-56 Hydrogen Cyanide (HCN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-57 Hydrogen Chloride (HCl). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-58 Hydrogen Fluoride (HF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-59 Hydrogen Peroxide (H2O2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-60 Hydrofluosilic Acid (H2SiF6). . . . . . . . . . . . . . . . . . . . . . . . . . . 2-61 Magnesium Chloride (MgCl2) . . . . . . . . . . . . . . . . . . . . . . . . . 2-62 Magnesium Sulfate (MgSO4) . . . . . . . . . . . . . . . . . . . . . . . . . . 2-63 Nickel Chloride (NiCl2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-64 Nickel Nitrate [Ni(NO3)2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-65 Nickel Sulfate (NiSO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-66 Nitric Acid (HNO3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-67 Perchloric Acid (HClO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-68 Phosphoric Acid (H3PO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-69 Potassium Bicarbonate (KHCO3) . . . . . . . . . . . . . . . . . . . . . . . 2-70 Potassium Bromide (KBr). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-71 Potassium Carbonate (K2CO3) . . . . . . . . . . . . . . . . . . . . . . . . . 2-72 Potassium Chromate (K2CrO4) . . . . . . . . . . . . . . . . . . . . . . . . . 2-73 Potassium Chlorate (KClO3) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-74 Potassium Chloride (KCl) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-75 Potassium Chrome Alum [K2Cr2(SO4)4]. . . . . . . . . . . . . . . . . . 2-76 Potassium Hydroxide (KOH) . . . . . . . . . . . . . . . . . . . . . . . . . . 2-77 Potassium Nitrate (KNO3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-78 Potassium Dichromate (K2Cr2O7). . . . . . . . . . . . . . . . . . . . . . . 2-79 Potassium Sulfate (K2SO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-80 Potassium Sulfite (K2SO3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-81 Sodium Acetate (NaC2H3O2) . . . . . . . . . . . . . . . . . . . . . . . . . . 2-82 Sodium Arsenate (Na3AsO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-83 Sodium Bichromate (Na2Cr2O7) . . . . . . . . . . . . . . . . . . . . . . . . 2-84 Sodium Bromide (NaBr). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-85 Sodium Formate (HCOONa) . . . . . . . . . . . . . . . . . . . . . . . . . . 2-86 Sodium Carbonate (Na2CO3) . . . . . . . . . . . . . . . . . . . . . . . . . . 2-87 Sodium Chlorate (NaClO3) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-88 2-89 2-89 2-89
2-90 2-90
2-91 2-91 2-93 2-94
2-99 2-99 2-99 2-99 2-99 2-99 2-99 2-99 2-99 2-99 2-99 2-100 2-100 2-100 2-100 2-100 2-100 2-100 2-100 2-100 2-100 2-100 2-100 2-101 2-101 2-101 2-101 2-101 2-101 2-101 2-101 2-101 2-101 2-101 2-101 2-101 2-101 2-102 2-103 2-103 2-103 2-103 2-104 2-104 2-104 2-104 2-104 2-104 2-104 2-104 2-104 2-104 2-104 2-104 2-104 2-104 2-104 2-105 2-105
2-88 2-89 2-90 2-91 2-92 2-93 2-94 2-95 2-96 2-97 2-98 2-99 2-100 2-101 2-102 2-103 2-104 2-105
Sodium Chloride (NaCl) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Chromate (Na2CrO4) . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Hydroxide (NaOH) . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Nitrate (NaNO3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Nitrite (NaNO2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Silicates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Sulfate (Na2SO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Sulfide (Na2S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Sulfite (Na2SO3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sodium Thiosulfate (Na2S2O3) . . . . . . . . . . . . . . . . . . . . . . . . Sodium Thiosulfate Pentahydrate (Na2S2O3⋅5H2O). . . . . . . . Stannic Chloride (SnCl4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stannous Chloride (SnCl2) . . . . . . . . . . . . . . . . . . . . . . . . . . . Sulfuric Acid (H2SO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zinc Bromide (ZnBr2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zinc Chloride (ZnCl2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zinc Nitrate [Zn(NO3)2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zinc Sulfate (ZnSO4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
DENSITIES OF AQUEOUS ORGANIC SOLUTIONS Units and Units Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-106 Formic Acid (HCOOH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-107 Acetic Acid (CH3COOH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-108 Oxalic Acid (H2C2O4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-109 Methyl Alcohol (CH3OH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-110 Ethyl Alcohol (C2H5OH). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-111 Densities of Mixtures of C2H5OH and H2O at 20°C . . . . . . . 2-112 Specific Gravity (60°/60°F [(15.56°/15.56°C)]) of Mixtures by Volume of C2H5OH and H2O . . . . . . . . . . . . . . 2-113 n-Propyl Alcohol (C3H7OH) . . . . . . . . . . . . . . . . . . . . . . . . . . 2-114 Isopropyl Alcohol (C3H7OH) . . . . . . . . . . . . . . . . . . . . . . . . . 2-115 Glycerol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-116 Hydrazine (N2H4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-117 Densities of Aqueous Solutions of Miscellaneous Organic Compounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DENSITIES OF MISCELLANEOUS MATERIALS Tables 2-118 Approximate Specific Gravities and Densities of Miscellaneous Solids and Liquids. . . . . . . . . . . . . . . . . . . . . 2-119 Density (kg/m3) of Selected Elements as a Function of Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SOLUBILITIES Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-120 Solubilities of Inorganic Compounds in Water at Various Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-121 Acetylene (C2H2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-122 Air. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-123 Ammonia (NH3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-124 Ammonia (NH3)—Low Pressures . . . . . . . . . . . . . . . . . . . . . 2-125 Carbon Dioxide (CO2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-126 Carbon Monoxide (CO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-127 Carbonyl Sulfide (COS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-128 Chlorine (Cl2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-129 Chlorine Dioxide (ClO2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-130 Ethane (C2H6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-131 Ethylene (C2H4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-132 Helium (He) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-133 Hydrogen (H2)—Temperature . . . . . . . . . . . . . . . . . . . . . . . . 2-134 Hydrogen (H2)—Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-135 Hydrogen Chloride (HCl). . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-136 Hydrogen Sulfide (H2S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-137 Methane (CH4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-138 Nitrogen (N2)—Temperature . . . . . . . . . . . . . . . . . . . . . . . . . 2-139 Nitrogen (N2)—Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-140 Oxygen (O2)—Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 2-141 Oxygen (O2)—Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-142 Ozone (O3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-143 Propylene (C3H6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-144 Sulfur Dioxide (SO2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . THERMAL EXPANSION Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Expansion of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-145 Linear Expansion of the Solid Elements . . . . . . . . . . . . . . . .
2-105 2-105 2-105 2-105 2-105 2-105 2-106 2-106 2-106 2-106 2-106 2-106 2-106 2-107 2-109 2-109 2-109 2-109
2-109 2-109 2-110 2-111 2-111 2-112 2-113 2-114 2-115 2-115 2-116 2-116 2-117
2-119 2-120 2-120 2-121 2-125 2-125 2-125 2-125 2-125 2-125 2-125 2-126 2-126 2-126 2-126 2-126 2-126 2-127 2-127 2-127 2-127 2-127 2-127 2-127 2-127 2-128 2-128 2-128
2-128 2-128 2-128 2-129
PHYSICAL AND CHEMICAL DATA 2-146 2-147 2-148
Linear Expansion of Miscellaneous Substances. . . . . . . . . . . Cubical Expansion of Liquids . . . . . . . . . . . . . . . . . . . . . . . . . Cubical Expansion of Solids . . . . . . . . . . . . . . . . . . . . . . . . . .
JOULE-THOMSON EFFECT Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-149 Additional References Available for the Joule-Thomson Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-150 Approximate Inversion-Curve Locus in Reduced Coordinates (Tr = T/Tc; Pr = P/Pc) . . . . . . . . . . . . . . . . . . . . . 2-151 Joule-Thomson Data for Air . . . . . . . . . . . . . . . . . . . . . . . . . . 2-152 Approximate Inversion-Curve Locus for Air . . . . . . . . . . . . . 2-153 Joule-Thomson Data for Argon . . . . . . . . . . . . . . . . . . . . . . . 2-154 Approximate Inversion-Curve Locus for Argon. . . . . . . . . . . 2-155 Joule-Thomson Data for Carbon Dioxide . . . . . . . . . . . . . . . 2-156 Approximate Inversion-Curve Locus for Carbon Dioxide . . 2-157 Approximate Inversion-Curve Locus for Deuterium . . . . . . 2-158 Approximate Inversion-Curve Locus for Ethane. . . . . . . . . . 2-159 Joule-Thomson Data for Helium . . . . . . . . . . . . . . . . . . . . . . 2-160 Approximate Inversion-Curve Locus for Normal Hydrogen. . 2-161 Approximate Inversion-Curve Locus for Methane . . . . . . . . 2-162 Joule-Thomson Data for Nitrogen . . . . . . . . . . . . . . . . . . . . . 2-163 Approximate Inversion-Curve Locus for Propane. . . . . . . . . CRITICAL CONSTANTS Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 2-164 Critical Constants and Acentric Factors of Inorganic and Organic Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . COMPRESSIBILITIES Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-165 Compressibility Factors for Air. . . . . . . . . . . . . . . . . . . . . . . . 2-166 Compressibility Factors for Argon . . . . . . . . . . . . . . . . . . . . . 2-167 Compressibility Factors for Carbon Dioxide . . . . . . . . . . . . . 2-168 Compressibility Factors for Carbon Monoxide . . . . . . . . . . . 2-169 Compressibility Factors for Ethanol. . . . . . . . . . . . . . . . . . . . 2-170 Compressibility Factors for Ethylene. . . . . . . . . . . . . . . . . . . 2-171 Compressibility Factors for Normal Hydrogen . . . . . . . . . . . 2-172 Compressibility Factors for KLEA 60 . . . . . . . . . . . . . . . . . . 2-173 Compressibility Factors for KLEA 61 . . . . . . . . . . . . . . . . . . 2-174 Compressibility Factors for KLEA 66 . . . . . . . . . . . . . . . . . . 2-175 Compressibility Factors for Krypton . . . . . . . . . . . . . . . . . . . 2-176 Compressibility Factors for Methane (R50) . . . . . . . . . . . . . 2-177 Compressibility Factors for Methanol . . . . . . . . . . . . . . . . . . 2-178 Compressibility Factors for Neon. . . . . . . . . . . . . . . . . . . . . . 2-179 Compressibility Factors for Nitrogen. . . . . . . . . . . . . . . . . . . 2-180 Compressibility Factors for Oxygen . . . . . . . . . . . . . . . . . . . . 2-181 Compressibility Factors for Refrigerant 32 . . . . . . . . . . . . . . 2-182 Compressibility Factors for Refrigerant 123 . . . . . . . . . . . . . 2-183 Compressibility Factors for Refrigerant 124 . . . . . . . . . . . . . 2-184 Compressibility Factors for Refrigerant 134a . . . . . . . . . . . . 2-185 Compressibility Factors for Water Substance (fps units) . . . 2-186 Compressibility Factors of Water Substance (SI units) . . . . . 2-187 Compressibility Factors for Xenon . . . . . . . . . . . . . . . . . . . . . 2-188 Compressibilities of Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . 2-189 Compressibilities of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . LATENT HEATS Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-190 Heats of Fusion and Vaporization of the Elements and Inorganic Compounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-191 Heats of Fusion of Miscellaneous Materials . . . . . . . . . . . . . 2-192 Heats of Fusion of Organic Compounds . . . . . . . . . . . . . . . . 2-193 Heats of Vaporization of Inorganic and Organic Compounds SPECIFIC HEATS OF PURE COMPOUNDS Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-194 Heat Capacities of the Elements and Inorganic Compounds 2-195 Specific Heat [kJ/(kg·K)] of Selected Elements. . . . . . . . . . . 2-196 Heat Capacities of Inorganic and Organic Liquids . . . . . . . . 2-197 Specific Heats of Organic Solids. . . . . . . . . . . . . . . . . . . . . . . 2-198 Heat Capacities of Inorganic and Organic Compounds in the Ideal Gas State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-130 2-131 2-131 2-132 2-132 2-133 2-133 2-133 2-133 2-134 2-134 2-134 2-134 2-134 2-135 2-135 2-135 2-135 2-135 2-136 2-136 2-140 2-140 2-140 2-140 2-141 2-141 2-141 2-142 2-142 2-142 2-143 2-143 2-143 2-144 2-144 2-144 2-145 2-145 2-145 2-146 2-146 2-146 2-147 2-148 2-149 2-149 2-150 2-150 2-151 2-153 2-154 2-156 2-161 2-161 2-161 2-169 2-170 2-175 2-178
2-199 Cp /Cv: Ratios of Specific Heats of Gases at 1-atm Pressure . . 2-200 Specific Heat Ratio, Cp /Cv, for Air . . . . . . . . . . . . . . . . . . . . .
2-3 2-183 2-183
SPECIFIC HEATS OF AQUEOUS SOLUTIONS Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-201 Acetic Acid (at 38°C). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-202 Ammonia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-203 Aniline (at 20°C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-204 Copper Sulfate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-205 Ethyl Alcohol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-206 Glycerol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-207 Hydrochloric Acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-208 Methyl Alcohol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-209 Nitric Acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-210 Phosphoric Acid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-211 Potassium Chloride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-212 Potassium Hydroxide (at 19°C) . . . . . . . . . . . . . . . . . . . . . . . 2-213 Normal Propyl Alcohol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-214 Sodium Carbonate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-215 Sodium Chloride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-216 Sodium Hydroxide (at 20°C) . . . . . . . . . . . . . . . . . . . . . . . . . 2-217 Sulfuric Acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-218 Zinc Sulfate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-184 2-184 2-184 2-184 2-184 2-184 2-184 2-184 2-184 2-184 2-185 2-185 2-185 2-185 2-185 2-185 2-185 2-185
SPECIFIC HEATS OF MISCELLANEOUS MATERIALS Tables 2-219 Specific Heats of Miscellaneous Liquids and Solids . . . . . . . 2-219a Oils (Animal, Vegetable, Mineral Oils). . . . . . . . . . . . . . . . . .
2-186 2-186
HEATS AND FREE ENERGIES OF FORMATION Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 2-220 Heats and Free Energies of Formation of Inorganic Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HEATS OF COMBUSTION Table 2-221 Enthalpies and Gibbs Energies of Formation, Entropies, and Net Enthalpies of Combustion of Inorganic and Organic Compounds at 298.15 K . . . . . . . . . . . . . . . . . . . . . 2-222 Ideal Gas Sensible Enthalpies, hT − h298 (kJ/kgmol), of Combustion Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-223 Ideal Gas Entropies, s°, kJ/kgmol⋅K, of Combustion Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-184 2-184
2-186 2-187
2-195 2-199 2-200
HEATS OF SOLUTION Tables 2-224 Heats of Solution of Inorganic Compounds in Water . . . . . . 2-201 2-225 Heats of Solution of Organic Compounds in Water (at Infinite Dilution and Approximately Room Temperature). . . . . . . . 2-204 THERMODYNAMIC PROPERTIES Explanation of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-226 Thermophysical Properties of Saturated Acetone . . . . . . . . . 2-227 Saturated Acetylene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-228 Saturated Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-229 Thermophysical Properties of Compressed Air . . . . . . . . . . . 2-230 Enthalpy and Psi Functions for Ideal-Gas Air . . . . . . . . . . . Temperature-Entropy Diagram for Air (Fig. 2-5) . . . . . . . . . 2-231 Air. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-232 Saturated Ammonia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy–Log-Pressure Diagram for Ammonia (Fig. 2-6) . . Enthalpy-Concentration Diagram for Aqueous Ammonia (Fig. 2-7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-233 Saturated Argon (R740) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-234 Thermodynamic Properties of Compressed Argon . . . . . . . . 2-235 Liquid-Vapor Equilibrium Data for the Argon-NitrogenOxygen System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-236 Thermodynamic Properties of the International Standard Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-237 Saturated Benzene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-238 Saturated Bromine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-239 Saturated Normal Butane (R600). . . . . . . . . . . . . . . . . . . . . . 2-240 Superheated Normal Butane . . . . . . . . . . . . . . . . . . . . . . . . .
2-205 2-205 2-205 2-206 2-206 2-207 2-208 2-212 2-213 2-214 2-214 2-215 2-216 2-217 2-217 2-218 2-221 2-221 2-222 2-222 2-223
2-4 2-241 2-242 2-243 2-244 2-245 2-246 2-247 2-248 2-249 2-250 2-251 2-252 2-253 2-254 2-255 2-256 2-257 2-258 2-259 2-260 2-261 2-262 2-263 2-264 2-265 2-266 2-267 2-268 2-269 2-270 2-271 2-272 2-273 2-274 2-275 2-276 2-277 2-278 2-279 2-280 2-281 2-282 2-283 2-284 2-285 2-286 2-287 2-288 2-289 2-290 2-291 2-292 2-293 2-294
2-295 2-296 2-297 2-298 2-299 2-300 2-301 2-302
PHYSICAL AND CHEMICAL DATA Saturated Carbon Dioxide . . . . . . . . . . . . . . . . . . . . . . . . . . . Superheated Carbon Dioxide . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Carbon Monoxide. . . . . . . . . . . . . . . . . . . . . . . . . . Temperature-Entropy Diagram for Carbon Monoxide (Fig. 2-8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermophysical Properties of Saturated Carbon Tetrachloride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Carbon Tetrafluoride (R14) . . . . . . . . . . . . . . . . . . Saturated Cesium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermophysical Properties of Saturated Chlorine . . . . . . . . Enthalpy–Log-Pressure Diagram for Chlorine (Fig. 2-9) . . . Saturated Chloroform (R20) . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Decane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Normal Deuterium. . . . . . . . . . . . . . . . . . . . . . . . . Saturated Deuterium Oxide . . . . . . . . . . . . . . . . . . . . . . . . . . Deuterium Oxide Gas at 1-kg/cm3 Pressure . . . . . . . . . . . . . Saturated Diphenyl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Ethane (R170) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superheated Ethane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Ethanol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy-Concentration Diagram for Aqueous Ethyl Alcohol (Fig. 2-10). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Ethylene (Ethene—R1150) . . . . . . . . . . . . . . . . . . Compressed Ethylene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Fluorine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluorine Gas at Atmospheric Pressure. . . . . . . . . . . . . . . . . . Flutec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Halon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Helium3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Helium4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superheated Helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Helium4 Gas at Atmospheric Pressure . . . . . . . . . . . . . . . . . . Saturated n-Heptane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hexane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Hydrazine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated n-Hydrogen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressed n-Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated para-Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Hydrogen Peroxide. . . . . . . . . . . . . . . . . . . . . . . . . Hydrogen Sulfide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy-Concentration Diagram for Aqueous Hydrogen Chloride at 1 atm (Fig. 2-11) . . . . . . . . . . . . . . . . . . . . . . . . Saturated Isobutane (R600a) . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Krypton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressed Krypton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Lithium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lithium Bromide—Water Solutions. . . . . . . . . . . . . . . . . . . . Saturated Mercury. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Methane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy–Log-Pressure Diagram for Mercury (Fig. 2-12) . . Superheated Methane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermophysical Properties of Saturated Methanol . . . . . . . . Thermodynamic Properties of Compressed Methanol . . . . . Saturated Methyl Chloride . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Neon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressed Neon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Nitrogen (R728) . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature-Entropy Diagram for Nitrogen (Fig. 2-13) . . . Thermophysical Properties of Nitrogen (R728) at Atmospheric Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Nitrogen Tetroxide . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Nitrous Oxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mollier Diagram for Nitrous Oxide (Fig. 2-14) . . . . . . . . . . . Nonane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Octane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Oxygen (R732) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature-Entropy Chart for Oxygen (Fig. 2-15) . . . . . . . Enthalpy-Concentration Diagram for Oxygen-Nitrogen Mixture at 1 atm (Fig. 2-16) . . . . . . . . . . . . . . . . . . . . . . . . . Pentane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Potassium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mollier Diagram for Potassium (Fig. 2-17) . . . . . . . . . . . . . . Saturated Propane (R290). . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Propylene (Propene, R1270) . . . . . . . . . . . . . . . . . Compressed Propylene (Propene, R1270). . . . . . . . . . . . . . . Saturated Refrigerant 11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy–Log-Pressure Diagram for Refrigerant 11 (Fig. 2-18) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy–Log-Pressure Diagram for Refrigerant 12 (Fig. 2-19) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-224 2-225 2-225
2-303 2-304 2-305
2-226 2-227 2-227 2-228 2-229 2-230 2-231 2-231 2-232 2-232 2-232 2-233 2-233 2-234 2-235 2-235 2-236 2-237 2-239 2-239 2-239 2-239 2-239 2-240 2-241 2-241 2-242 2-242 2-243 2-243 2-244 2-246 2-246 2-246 2-247 2-248 2-248 2-249 2-249 2-249 2-250 2-251 2-252 2-253 2-254 2-255 2-256 2-256 2-257 2-257 2-258 2-259 2-259 2-259 2-260 2-261 2-261 2-262 2-263 2-264 2-264 2-264 2-265 2-266 2-267 2-268 2-269 2-269 2-270 2-270
2-306 2-307 2-308 2-309 2-310 2-311 2-312 2-313 2-314 2-315 2-316 2-317 2-318 2-319 2-320 2-321 2-322 2-323 2-324 2-325 2-326 2-327 2-328 2-329 2-330 2-331 2-332 2-333 2-334 2-335 2-336 2-337 2-338 2-339 2-340 2-341 2-342 2-343 2-344 2-345
2-346 2-347
2-348 2-349 2-350 2-351 2-352 2-353 2-354 2-355 2-356
2-271
Saturated Refrigerant 13B1 . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 21. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy–Log-Pressure Diagram for Refrigerant 22 (Fig. 2-20) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermophysical Properties of Compressed R22 . . . . . . . . . . Saturated Refrigerant 23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermophysical Properties of Saturated Difluoromethane (R32) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specific Heat at Constant Pressure, Thermal Conductivity, Viscosity, and Prandtl of R32 Gas . . . . . . . . . . . . . . . . . . . . . Saturated SUVA MP 39. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUVA MP 39 at Atmospheric Pressure . . . . . . . . . . . . . . . . . Enthalpy–Log-Pressure Diagram for Refrigerant 32 (Fig. 2-21) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Properties of Saturated KLEA 60 . . . . . . . Thermodynamic Properties of Saturated KLEA 61 . . . . . . . Enthalpy–Log-Pressure Diagram for KLEA 60 (Fig. 2-22). . Enthalpy–Log-Pressure Diagram for KLEA 61 (Fig. 2-23). . Saturated SUVA HP 62 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUVA HP 62 at Atmospheric Pressure. . . . . . . . . . . . . . . . . . Thermodynamic Properties of Saturated KLEA 66 . . . . . . . Enthalpy–Log-Pressure Diagram for KLEA 66 (Fig. 2-24) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated SUVA MP 66. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUVA MP 66 at Atmospheric Pressure . . . . . . . . . . . . . . . . . Saturated SUVA HP 80 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUVA HP 80 at Atmospheric Pressure. . . . . . . . . . . . . . . . . . Saturated SUVA HP 81 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUVA HP 81 at Atmospheric Pressure. . . . . . . . . . . . . . . . . . Saturated Refrigerant 113. . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 114. . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 115. . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Properties of Refrigerant 123 . . . . . . . . . . . Saturated Refrigerant 124. . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy–Log-Pressure Diagram for Refrigerant 123 (Fig. 2-25) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermophysical Properties of Saturated Refrigerant 125 . . . Thermophysical Properties of Refrigerant 134a . . . . . . . . . . Enthalpy–Log-Pressure Diagram for Refrigerant 125 (Fig. 2-26) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermophysical Properties of Compressed Gaseous Refrigerant 134a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Refrigerant 141b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy–Log-Pressure Diagram for Refrigerant 134a (Fig. 2-27) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Refrigerant 142b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant R143a . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant R152a . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 216. . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 245. . . . . . . . . . . . . . . . . . . . . . . . . . . . Refrigerant C 318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 500. . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 502. . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 503. . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Refrigerant 504. . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Properties of Refrigerant 507 . . . . . . . . . . . Saturated Rubidium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermophysical Properties of Saturated Seawater . . . . . . . . Saturated Sodium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mollier Diagram for Sodium (Fig. 2-28) . . . . . . . . . . . . . . . . Enthalpy-Concentration Diagram for Aqueous Sodium Hydroxide at 1 atm (Fig. 2-29) . . . . . . . . . . . . . . . . . . . . . . . Saturated Sulfur Dioxide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic Properties of Saturated Sulfur Hexafluoride (SF6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy-Concentration Diagram for Aqueous Sulfuric Acid at 1 atm (Fig. 2-30) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy–Log-Pressure Diagram for Sulfur Hexafluoride (Fig. 2-31) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated SUVA AC 9000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Toluene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Solid/Vapor Water. . . . . . . . . . . . . . . . . . . . . . . . . . Saturated Water Substance–Temperature (fps units) . . . . . . Saturated Water Substance–Temperature (SI units) . . . . . . . Saturated Liquid Water–Miscellaneous Properties . . . . . . . . Thermodynamic Properties of Compressed Steam . . . . . . . . Density, Specific Heats at Constant Pressure and at Constant Volume and Velocity of Sound for Compressed Water, 1–1000 bar, 0–150°C . . . . . . . . . . . . . . . . . . . . . . . . . Specific Heat and Other Thermophysical Properties of Water Substance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-272 2-272 2-273 2-273 2-274 2-274 2-275 2-276 2-276 2-276 2-277 2-278 2-278 2-279 2-280 2-281 2-281 2-281 2-282 2-283 2-283 2-283 2-284 2-284 2-284 2-285 2-286 2-286 2-287 2-287 2-288 2-289 2-289 2-290 2-291 2-292 2-293 2-294 2-294 2-295 2-295 2-296 2-296 2-297 2-297 2-298 2-298 2-298 2-299 2-299 2-300 2-301 2-302 2-302 2-303 2-303 2-303 2-304 2-304 2-304 2-305 2-306 2-308 2-309 2-311 2-313
PHYSICAL AND CHEMICAL DATA 2-357 Thermodynamic Properties of Water Substance along the Melting Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-358 Saturated Xenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-359 Compressed Xenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-360 Surface Tension (N/m) of Saturated Liquid Refrigerants . . . 2-361 Velocity of Sound (m/s) in Gaseous Refrigerants at Atmospheric Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-362 Velocity of Sound (m/s) in Saturated Liquid Refrigerants. . . TRANSPORT PROPERTIES Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Units Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 2-363 Transport Properties of Selected Gases at Atmospheric Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-364 Viscosities of Gases: Coordinates for Use with Fig. 2-32. . . . Nomograph for Determining (a) Absolute Viscosity of a Gas as a Function of Temperature Near Ambient Pressure and (b) Relative Viscosity of a Gas Compared with Air (Fig. 2-32) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-365 Viscosities of Liquids: Coordinates for Use with Fig. 2-33 . . Nomograph for Viscosities of Liquids at 1 atm (Fig. 2-33) . . 2-366 Viscosity of Sucrose Solutions . . . . . . . . . . . . . . . . . . . . . . . . . Nomograph for Thermal Conductivity of Organic Liquids (Fig. 2-34) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-367 Thermal Conductivity Nomograph Coordinates . . . . . . . . . . 2-368 Prandtl Number of Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-369 Prandtl Number of Liquid Refrigerants. . . . . . . . . . . . . . . . . 2-370 Thermophysical Properties of Miscellaneous Saturated Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-371 Diffusivities of Pairs of Gases and Vapors (1 atm) . . . . . . . . . 2-372 Diffusivities in Liquids (25°C) . . . . . . . . . . . . . . . . . . . . . . . . 2-373 Thermal Conductivities of Some Building and Insulating Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-374 Thermal-Conductivity-Temperature Table for Metals . . . . . 2-375 Thermal Conductivity of Chromium Alloys . . . . . . . . . . . . . . 2-376 Thermal Conductivity of Some Alloys at High Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-377 Thermal Conductivities of Some Materials for Refrigeration and Building Insulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-378 Thermal Conductivities of Insulating Materials at High Temperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-379 Thermal Conductivities of Insulating Materials at Moderate Temperatures (Nusselt). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-380 Thermal Conductivities of Insulating Materials at Low Temperatures (Gröber) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-381 Thermal Diffusivity (m2/s) of Selected Elements . . . . . . . . . 2-382 Thermophysical Properties of Selected Nonmetallic Solid Substances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pure Component Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1 Estimate the Critical Temperature and Critical Pressure of 2-Butanol Using the Ambrose Method . . . . . . . . . . . . . . . . . . . . . Example 2 Estimate the Critical Temperature and Critical Pressure of 2-Butanol, Which Has an Experimental Normal Boiling Point of 372.7 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3 Estimate the Critical Volume of 2-Butanol . . . . . . . . . . . Critical Compressibility Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normal Freezing Temperature (Melting Point) . . . . . . . . . . . . . . . . . Normal Boiling Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 4 Estimate the Normal Boiling Point of 2-Butanol. . . . . . . Acentric Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miscellaneous Characterizing Constants . . . . . . . . . . . . . . . . . . . . . . . Vapor Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correlation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prediction Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 5 Estimate the Vapor Pressure of 1-Butene at 98°C. . . . . . Example 6 Estimate the Vapor Pressure of Tetralin at 150°C . . . . . . Example 7 Estimate the Vapor Pressure of Thiophene at 500 K. . . . Example 8 Estimate the Vapor Pressure of Acetaldehyde at 0°C . . . Ideal Gas Thermal Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-315 2-315 2-316 2-317 2-317 2-318
2-318 2-318 2-318 2-319 2-320
2-321 2-322 2-323 2-324 2-324 2-324 2-325 2-325 2-326 2-328 2-330 2-333 2-334 2-335 2-335 2-335 2-335 2-336 2-336 2-336 2-337
2-337 2-337 2-338 2-339 2-340 2-340 2-340 2-342 2-342 2-344 2-344 2-344 2-344 2-344 2-345 2-345 2-345 2-345 2-345 2-346 2-346 2-346 2-347 2-347 2-347
Heat Capacity, C po . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 9 Using Eq. 2-48 to Estimate the Ideal Gas Heat Capacity of Acetone (C3H6O) at 600 K . . . . . . . . . . . . . . . . . . . . . . . Enthalpy of Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 10 H°f 298 of 2-Butanol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gibbs Energy of Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 11 G°f 298 of Phenol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Entropy of Formation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy of Vaporization and Fusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy of Vaporization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 12 Estimate Hv of Propionaldehyde at 350 K . . . . . . . . . . Example 13 Estimate Hv of Ethyl Acetate . . . . . . . . . . . . . . . . . . . . Example 14 Estimate Hv of Ethyl Acetate at 450 K. . . . . . . . . . . . . Enthalpy of Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid and Liquid Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 15 Estimate Solid Heat Capacity of Dinenzothiophene. . . Liquid Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 16 Estimate Liquid Heat Capacity of 2-Butanol. . . . . . . . . Example 17 Estimate Liquid Heat Capacity of 1,4 Pentadiene . . . . Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correlation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapor Density Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 18 Estimate the Molar Volume of Isobutane at 155°C and 1.0 MPa Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 19 Estimate the Molar Volume of Isobutane at 155°C and 8.6 MPa Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Density Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 20 Estimate the Density of Saturated Liquid Propane at 0°C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 21 Estimate the Liquid Density of n-Nonane at 104.5°C and 6.893 MPa Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid Density Prediction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correlation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapor Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 22 Estimate the Vapor Viscosity of Propane at 101.3 kPa and 80°C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 23 Estimate the Vapor Viscosity of a Mixture of Propane and Methane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 24 Estimate the Vapor Viscosity of Isopropyl Alcohol at 251°C and Atmospheric Pressure . . . . . . . . . . . . . . . . . . . . . . . . . Example 25 Estimate the Vapor Viscosity of Carbon Dioxide at 350 K and a Total Pressure of 20 MPa. . . . . . . . . . . . . . . . . . . . . . . . Liquid Viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 26 Estimate the Liquid Viscosity of cis-1,4dimethylcyclohexane at 0°C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vapor and Liquid Thermal Conductivity. . . . . . . . . . . . . . . . . . . . . . . . . Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 27 Estimate Thermal Conductivity for n-Hexane. . . . . . . . Example 28 Estimate Thermal Conductivity of Carbon Dioxide at 370 K and Low Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 29 Estimate the Thermal Conductivity of Carbon Dioxide at 370 K and 10 MPa Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 30 Estimate Thermal Conductivity of a Mixture. . . . . . . . . Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 31 Estimate Thermal Conductivity of n-Octane at 373.15 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 32 Estimate Thermal Conductivity of n-Octane at 473.15 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 33 Estimate Thermal Conductivity of n-Butanol . . . . . . . . Example 34 Estimate the Thermal Conductivity of n-Propionaldehyde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 35 Estimate Thermal Conductivity of n-Butanol . . . . . . . . Example 36 Estimate Thermal Conductivity of a Mixture. . . . . . . . . Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 37 Estimate the Diffusivity of Benzene Vapor Diffusing into Air at 30°C and 96.5 kPa Total Pressure . . . . . . . . . . . . . . . . . . Example 38 Estimate the Diffusivity of Hydrogen (1) in Nitrogen (2) at 60°C and 17.23 MPa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 39 Estimate the Infinite Dilution Diffusivity of Propane (1) in Chlorobenzene (2) at 25°C. . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface Tension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 40 Estimate Surface Tension for Mercaptan. . . . . . . . . . . . Example 41 Estimate Surface Tension for Isobutytic Acid . . . . . . . . Example 42 Estimate Surface Tension of a Mixture . . . . . . . . . . . . . Example 43 Estimate Surface Tension of a Water-Methanol Mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flammability Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-5 2-347 2-348 2-348 2-348 2-348 2-348 2-348 2-349 2-349 2-350 2-350 2-350 2-350 2-351 2-351 2-351 2-351 2-351 2-354 2-355 2-355 2-355 2-355 2-357 2-358 2-359 2-360 2-361 2-361 2-362 2-362 2-363 2-363 2-363 2-364 2-365 2-366 2-367 2-367 2-367 2-367 2-368 2-368 2-368 2-368 2-369 2-369 2-369 2-369 2-370 2-370 2-370 2-370 2-371 2-371 2-371 2-372 2-372 2-372 2-372 2-373 2-374
GENERAL REFERENCES Considerations of reader interest, space availability, the system or systems of units employed, copyright considerations, etc., have all influenced the revision of material in previous editions for the present edition. Reference is made at numerous places to various specialized works and also, when appropriate, to more general works. A listing of general works may be useful to readers in need of further information. ASHRAE Handbook—Fundamentals, IP and SI editions, ASHRAE, Atlanta, various dates; Beaton, C.F. and G.F. Hewitt, Physical Property Data for the Design Engineer, Hemisphere, New York, 1989 (394 pp.); Benedek, P. and F. Olti, Computer-Aided Chemical Thermodynamics of Gases and Liquids, Wiley, New York, 1985 (731 pp.); Daubert, T.E. and R.P. Danner, Physical and Thermodynamic Properties of Pure Chemicals, 4 vols., Hemisphere, New York, 1989 (2030 pp.); suppl. 1, 1991 (456 pp.); suppl. 2, 1992 (736 pp.); Gmehling, J., Azeotropic Data, 2 vols., VCH Weinheim, Germany, 1994 (1900 pp.); Kaye, S.M., Encyclopedia of Explosives and Related Items, U.S. Army R&D command, Dover, NJ, 1980; King, M.B., Phase Equilibrium in Mixtures, Pergamon, Oxford, 1969; Lyman, W.J., W.F. Reehl et al., Handbook of Chemical Property Estimation Methods, McGraw-Hill, N.Y., 1982 (929 pp.); Ohse, R.W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Sci. Pubs., Oxford, England, 1985 (1020 pp.); Reid, R.C., J.M. Prausnitz et al., The Properties of Gases and Liquids, McGraw-Hill, New York, 1987 (742 pp.); Sterbacek, Z., B. Biskup et al., Calculation of Properties Using Corresponding States Methods, Elsevier, Amsterdam, 1979. Compilations of critical data include Ambrose, D., “Vapor-Liquid Critical Properties,” N.P.L. Teddington, Middx. rept. Chem 107, 1980 (62 pp.); Brule, M.R., L.L Lee et. al., Chem. Eng., 86, 25 (Nov. 19, 1979) 155–164; Kudchaker,
2-6
A.P., G.H. Alani et al., Chem. Revs., 68 (1968) 659–735; Matthews, J.F., Chem. Revs., 72 (1972) 71–100; Reid, R.C., J.M. Prausnitz et al., The Properties of Gases and Liquids, 4th ed., McGraw-Hill, New York, 1987 (741 pp.); Ohse, R.W. and H. von Tippelskirch, High Temp.—High Press., 9 (1977) 367–385; Young, D.A., “Phase Diagrams of the Elements,” UCRL Rept. 51902, 1975 (64 pp.); republished in expanded form by the University of California Press, 1991. Rothman, D. et al., Max Planck Inst. f. Stromungsforschung, Ber 6, 1978 (77 pp.). PUBLICATIONS ON THERMOCHEMISTRY Pedley, J.B., Thermochemical Data and Structures of Organic Compounds, 1, Thermodyn. Res. Ctr., Texas A&M Univ., 1994 (976 pp., 3000 cpds.); Frenkel, M., G.J. Kabo et al., Thermodynamics of Organic Compounds in the Gas State, 2 vols., Thermodyn. Res. Ctr., Texas A&M Univ., 1994 (1825 pp., 2000 cpds.); Barin, I., Thermochemical Data of Pure Substances, 2 vols., 2d ed., VCH Weinheim, Germany 1993 (1834 pp., 2400 substances); and Gurvich, L.V., I.V. Veyts et al., Thermodynamic Properties of Individual Substances, 3 vols., 4th ed., Hemisphere, New York, 1989, 1990, and 1993 (2520 pp.). See also Lide, D.R. and G.W.A. Milne, Handbook of Data on Organic Compounds, 7 vols., 3d ed., Chemical Rubber, Miami, 1993 (7000 pp.); Daubert, T.E., R.P. Danner et al., Physical and Thermodynamic Properties of Pure Chemicals: Data Compilation, extant 1995, Taylor & Francis, Bristol, PA, 1995; Database 11, N.I.S.T. Gaithersburg, MD. U.S. Bureau of Mines publications include Bulletins 584, 1960 (232 pp.); 592, 1961 (149 pp.); 595, 1961 (68 pp.); 654, 1970 (26 pp.) Chase, M.W., C.A. Davies et al., JANAF Thermochemical Tables, 3d ed., J. Phys. Chem. Ref. Data 14 suppl 1., 1986 (1896 pp.).
PHYSICAL PROPERTIES OF PURE SUBSTANCES TABLE 2-1
Physical Properties of the Elements and Inorganic Compounds* Abbreviations Used in the Table
a., acid A., specific gravity with reference to air = 1 abs., absolute ac., acetic acid act., acetone al., 95 percent ethyl alcohol alk, alkali (i.e., aq. NaOH or KOH) am., amyl (C5H11) amor., amorphous anh., anhydrous aq., aqueous or water aq. reg., aqua regia
atm., atmosphere or 760 mm. of mercury pressure bk., black brn., brown bz., benzene c., cold cb., cubic cc, cubic centimeter chl., chloroform col., colorless or white conc., concentrated cr., crystals or crystalline d., decomposes D., specific gravity with reference to hydrogen = 1
d. 50, decomposes at 50°C; 50 d., melts at 50°C with decomposition delq., deliquescent dil., dilute dk., dark eff., effloresces or efflorescent et., ethyl ether expl., explodes gel., gelatinous gly., glycerol (glycerin) gn., green h., hot hex., hexagonal
hyg., hygroscopic i., insoluble ign., ignites lq., liquid lt., light m. al., methyl alcohol mn., monoclinic nd., needles NH3, liquid ammonia NH4OH, ammonium hydroxide solution oct., octahedral or., orange pd., powder
Formula weights are based upon the International Atomic Weights of 1941 and are computed to the nearest hundredth. Refractive index, where given for a uniaxial crystal, is for the ordinary (ω) ray; where given for a biaxial crystal, the index given is for the median (β) value. Unless otherwise specified, the index is given for the sodium D-line (λ = 589.3 mµ). Specific gravity values are given at room temperatures (15° to 20°C) unless otherwise indicated by the small figures which follow the value: thus, “5.6 18° 4 ” indicates a specific gravity of 5.6 for the substance at 18°C referred to water at 4°C. In this table the values for the specific gravity of gases are given with reference to air (A) = 1, or hydrogen (D) = 1. Melting point is recorded in a certain case as “82 d.” and in some other case as “d. 82,” the distinction being made in this manner to indicate that the former is a melting point with decomposition at 82°C, while in the latter decomposition only occurs at 82°C. Where a value such as “−2H2O, 82” is given it indicates loss of 2 moles of water per formula weight of the compound at a temperature of 82°C. Boiling point is given at atmospheric pressure (760 mm. of mercury) unless otherwise indicated; thus, “8215 mm.” indicates the boiling point is 82°C when the pressure is 15 mm.
Name Aluminum acetate, normal acetate, basic bromide bromide carbide chloride chloride fluoride (fluellite) fluoride hydroxide nitrate nitride oxide oxide (corundum) phosphate
Formula
Formula weight
Color, crystalline form and refractive index
Al Al(C2H3O2)3 Al(OH)(C2H3O2)2 AlBr3 AlBr3·6H2O Al4C3 AlCl3
26.97 204.10 162.07 266.72 374.82 143.91 133.34
silv., cb. wh. pd. wh., amor. trig. col., delq. cr. yel., hex., 2.70 wh., delq., hex.
AlCl3·6H2O AlF3·H2O Al2F6·7H2O Al(OH)3 Al(NO3)3·9H2O Al2N2 Al2O3 Al2O3 AlPO4
241.44 101.99 294.05 77.99 375.14 81.96 101.94 101.94 121.95
col., delq., trig., 1.560 col., rhb., 1.490 wh., cr. pd. wh., mn. rhb., delq. yel., hex. col., hex., 1.67–8 wh., trig., 1.768 col., hex.
pl., plates pr., prisms or prismatic pyr., pyridine rhb., rhombic (orthorhombic) s., soluble satd., saturated sl., slightly soln., solution subl., sublimes sulf., sulfides tart. a., tartaric acid tet., tetragonal tr., transition tri., triclinic
trig., trigonal v., very vac., in vacuo vl., violet volt., volatile or volatilizes wh., white yel., yellow ∞, soluble in all proportions <, less than >, greater than 42, about or near 42 −3H2O, 100, loses 3 moles of water per formula weight at 100°C
Solubility is given in parts by weight (of the formula shown at the extreme left) per 100 parts by weight of the solvent; the small superscript indicates the temperature. In the case of gases the solubility is often expressed in some manner as “510° cc” which indicates that at 10°C, 5 cc. of the gas are soluble in 100 g. of the solvent. The symbols of the common mineral acids: H2SO4, HNO3, HCl, etc., represent dilute aqueous solutions of these acids. See also special tables on Solubility. REFERENCES: The information given in this table has been collected mainly from the following sources: Mellor, A Comprehensive Treatise on Inorganic and Theoretical Chemistry, Longmans, New York, 1922. Abegg, Handbuch der anorganischen Chemie, S. Hirzel, Leipzig, 1905. Gmelin-Kraut, Handbuch der anorganischen Chemie, 7th ed., Carl Winter, Heidelberg; 8th ed., Verlag Chemie, Berlin, 1924. Friend, Textbook of Inorganic Chemistry, Griffin, London, 1914. Winchell, Microscopic Character of Artificial Inorganic Solid Substances or Artificial Minerals, Wiley, New York, 1931. International Critical Tables, McGraw-Hill, New York, 1926. Tables annuelles internationales de constants et donnes numeriques, McGraw-Hill, New York. Annual Tables of Physical Constants and Numerical Data, National Research Council, Princeton, N.J., 1943. Comey and Hahn, A Dictionary of Chemical Solubilities, Macmillan, New York, 1921. Seidell, Solubilities of Inorganic and Metal Organic Compounds, Van Nostrand, New York, 1940.
Specific gravity 2.7020° 3.01 25° 4 2.95 2.44 25° 4 2.17 2.42 3.05 3.99 4.00 2.59
25° 4
Melting point, °C 660 d. 200 d. 97.5 d. 100 d. >2200 1945.2atm. d. −4H2O, 120 −2H2O, 300 73 21504atm. 1999 to 2032 1999 to 2032
Boiling point, °C 2056 268 752mm
182.7 ; subl. 178 −6H2O, 250 d. 134 d. >1400 2210
Solubility in 100 parts Cold water i. s. i. s. s. d. to CH4 69.8715° 400 sl. s. i. 0.00010418° v. s. d. slowly i. i. i.
Hot water i. d.
Other reagents s. HCl, H2SO4, alk.
s. d.
s.a.; i. NH4 salts s.al., act., CS2 s. al., CS2 s. a.; i. act. s. et., chl., CCl4; i. bz.
v. s.
50 al.; s. et.
s.
sl. s. i. v. s. d. i. i. i.
s. a., alk.; i. a. s. al., CS2 s. alk. d. v. sl. s. a., alk. v. sl. s. a., alk. s. a., alk.; i. ac.
*By N. A. Lange, Ph.D., Handbook Publishers, Inc., Sandusky, Ohio. Abridged from table of Physical Constants of Inorganic Compounds in Lange, “Handbook of Chemistry.” 2-7
TABLE 2-1
Physical Properties of the Elements and Inorganic Compounds (Continued )
Name Aluminum (Cont.) potassium silicate (muscovite) potassium silicate (orthoclase) Aluminum potassium tartrate sodium fluoride (cryolite) sodium silicate sulfate Alum, ammonium (tschermigite) ammonium chrome ammonium iron potassium (kalinite) potassium chrome sodium Ammonia† Ammonium acetate auricyanide bicarbonate bromide carbonate carbonate, carbamate
Formula 3Al2O3·K2O·6SiO2· 2H2O Al2O3·K2O·6SiO2 AlK(C4H4O6)2 AlF3·3NaF Al2O3·Na2O·6SiO2 Al2(SO4)3 Al2(SO4)3·(NH4)2SO4· 24H2O Cr2(SO4)3·(NH4)2SO4· 24H2O Fe2(SO4)3·(NH4)2SO4· 24H2O Al2(SO4)3·K2SO4· 24H2O Cr2(SO4)3·K2SO4· 24H2O Al2(SO4)3·Na2SO4· 24H2O NH3
Formula weight
Color, crystalline form and refractive index
Boiling point, °C
Solubility in 100 parts Cold water
mn., 1.590
2.9
d.
i.
556.49 362.21 209.96 524.29 342.12 906.64
col., mn., 1.524 col. wh., mn., 1.3389 col., tri., 1.529 wh. cr. col., oct., 1.4594
2.56
1450 (1150)
2.90 2.61 2.71 1.64 20° 4
1000 1100 d. 770 93.5
i. s. sl. s. i. 31.30° 3.90°
956.72
gn. or vl., oct., 1.4842
1.72
964.40
vl., oct., 1.485
1.71
40
col., mn., 1.4564
1.76 26° 4
92
998.84
red or gn., cb., 1.4814
1.83
89
916.56
col., oct., 1.4388
17.03
col. gas, 1.325 (lq.)
77.08 337.33 79.06 97.96 114.11 157.11
wh., hyg. cr. pl. mn. or rhb., 1.5358 col., cb., 1.7108 col. pl. wh. cr.
272.22
wh.
53.50 444.05 373.14 367.52 152.09 44.06 252.10 392.21 37.04 57.05 63.06
hydrosulfide hydroxide molybdate molybdate, heptanitrate (α), stable −16° to 32° nitrate (β), stable 32° to 84°
NH4HS NH4OH (NH4)2MoO4 (NH4)6Mo7O24·4H2O‡ NH4NO3 NH4NO3
51.11 35.05 196.03 1235.95 80.05 80.05
col., rhb. in soln. only mn. col., mn. col., tet., 1.611 col., rhb. or mn.
nitrite osmochloride oxalate oxalate, acid perchlorate persulfate phosphate, monobasic
NH4NO2 (NH4)2OsCl6 (NH4)2C2O4·H2O NH4HC2O4·H2O NH4ClO4 (NH4)2S2O8 NH4H2PO4
64.05 439.02 142.12 125.08 117.50 228.20 115.04
wh. nd. cb. col., rhb. col., trimetric col., rhb., 1.4833 wh., mn., 1.5016 col., tet., 1.5246
wh., cb., 1.639, 1.6426 yel., cb. tet. pink., cb. yel., mn. col., cb. or., mn. mn. wh., hex. wh., rhb., 1.390 col., mn., delq.
−20H2O, 120; −24H2O, 200
100 d.
948.76
chloride (salammoniac) chloroplatinate chloroplatinite chlorostannate chromate cyanide dichromate ferrocyanide fluoride fluoride, acid formate
2-8
Melting point, °C
796.40
NH4C2H3O2 NH4CN·Au(CN)3·H2O NH4HCO3 NH4Br (NH4)2CO3·H2O NH4HCO3· NH2CO2NH4‡ (NH4)2CO3· 2NH4HCO3·H2O NH4Cl (NH4)2PtCl6 (NH4)2PtCl4 (NH4)2SnCl6 (NH4)2CrO4 NH4CN (NH4)2Cr2O7 (NH4)4Fe(CN)6·6H2O NH4F NH4F·HF HCO2NH4
carbonate, sesqui-
Specific gravity
20° 4
1.675
−79°
0.817 0.5971 (A) 1.073 1.573 2.327 15° 4
1.53 3.065
2.4 1.91712° 0.79100° (A) 2.15
i. 89100° ∞ 100°
25°
124 −18H2O, 64.5
i. al.
50
106.4
0°
i. al. 45°
121.7
−77.7
−33.4
89.9
114 d. 200 d. 35–60 subl. 542 d. 58 subl.
d.
1484° s. 11.90° 6810° 10015° 2515°
v. s. 2730° 145.6100°
2015°
5049°
d. 350 d. d.
subl. 520
d. 180 36 d. 185 d.
12° 2.21 12 1.266
114–116
2.27
d.
1.66 25° 4 1.725 25° 4
169.6
1.69 2.93 20° 4 1.501 1.556 1.95 1.98 1.803 19° 4
expl.
d.
d. 180; subl. in vac. subl. 120
d. 210 d. 210
0°
0°
29.4 0.715° s. 33.315° 40.530° s. 47.230° s. v. s. v. s. 1020°
i. al.
7.4
96°
14.820° al.; s. et. s. al.; sl. s. act. i. al. i. al. s. al., et., act. i. al., CS2, NH3
6765°
77.3100° 1.25100° v. s.
s. NH3; sl. s. al., m. al. 0.005 al.
d. v. s. v. s. d.
sl. s. act., NH3; i. al. s. al. s. al.; i. act. i. al. s. al.; i. NH3
53180°
s. al.
d.
i. al., NH3 i. al.
v. s. s. d. 4425° 118.30° 365.835°
241.830° 58080°
s.
d.
3.820° al., 17.120° m. al.; v. s. NH3 s. al.
11.850°
sl. s. al.; i. NH3
46.9100° d. 173.2100°
220° al.; s. act.; i. et.
0°
d. d. d. 120
i. al.
∞93°
20
61
i. HCl d. a.
s. al.
25°
5.70°
Other reagents
s.
21.2
d. 17°
Hot water
2.5 s. 10.90° 58.20° 22.70°
s. al.
i. ac.
phosphate, dibasic phosphate, metaAmmonium phosphomolybdate silicofluoride sulfamate sulfate (mascagnite) sulfate, acid sulfide sulfide, pentasulfite sulfite, acid tartrate thiocyanate vanadate, metaAntimony chloride, tri- (butter of antimony)* oxide, tri- (valentinite) oxide, tri- (senarmontite) sulfide, tri- (stibnite)
(NH4)2HPO4 (NH4)4P4O12 (NH4)3PO4·12MoO3· 3H2O (?) (NH4)2SiF6 NH4·SO3NH2 (NH4)2SO4 NH4HSO4 (NH4)2S (NH4)2S5 (NH4)2SO3·H2O NH4HSO3 (NH4)2C4H4O6 NH4CNS NH4VO3 Sb
132.07 388.08 1930.55 178.14 114.12 132.14 115.11 68.14 196.38 134.16 99.11 184.15 76.12 116.99 121.76
col., mn., 1.53 col., mn. yel.
13115° s. 0.0315°
1.619 2.21 d.
cb., 1.3696 col. pl. col., rhb., 1.5230 col., rhb., 1.480 yel.-wh. or.-red pr. col., mn. rhb. col., mn. col., mn., 1.685 col. cr. tin wh., trig.
2.01
i.
17.5°
s. alk.; i. al., HNO3
1380
18.5 1340° 70.60° 100 v. s. s. 10012° s. 450° 1200° 0.4418° i.
87 17020° 3.0570° i.
73.4
220.2
601.60°
∞72°
1570
v. sl. s.
1.769 20° 4 1.78
132 235 d. 146.9 d.
1.41 2.03 12° 4 1.60 1.305 2.326 6.68425°
d. d. d. 149.6 d. 630.5
SbCl3
228.13
col., rhb., delq.
3.14 20° 4
Sb2O3 Sb2O3 Sb2S3
291.52 291.52 339.70
rhb., 2.35 cb., 2.087 bk., rhb., 4.046
5.67 5.2 4.64
656 652 550
subl. d. 160
i. act.
490
d. 170
0.00017
55.5 35750° 103.3100°
s. al.; i. act. i. al., act., CS2 v. sl. s. al.; i. act. 12025° NH3 i. al., act.
60°
sl. s. 18°
d.
sl. s. al. s. al., act., NH3, SO2 i. al., NH4Cl s. aq. reg., h. conc. H2SO4 s. al., HCl, HBr, H2C4H4O6 s. HCl, KOH, H2C4H4O6 s. HCl; alk., NH4HS, K2S; i. ac. s. HCl, alk., NH4HS
sulfide, pentatelluride, triAntimonyl potassium tartrate (tartar emetic) sulfate, normal sulfate, basic Argon
Sb2S5 Sb2Te3
403.82 626.35
golden gray
4.1200°
−2S, 135 629
i.
i.
(SbO)KC4H4O6·a H2O (SbO)2SO4 (SbO)2SO4·Sb2(OH)4 A
333.94 371.58 683.13 39.94
wh., rhb. wh. pd. wh. pd. col. gas
2.60 4.89
−a H2O, 100 −189.2
−185.7
35.7100° d. d. 2.2350° cc
s. gly.; i. al.
Arsenic (crystalline) (α) Arsenic (black) (β)
As4 As4
299.64 299.64
met., hex. bk., amor.
1.65−288°; 1.402−185.7°; 1.38 (A) 5.72714° 4.720°
5.268.7° d. i. 5.60° cc
81436atm.
subl. 615
i. i.
i. i.
s. HNO3 s. HNO3, aq. reg., aq. Cl2, h. alk.
Arsenic (yellow)(γ) acid, orthoacid, metaacid, pyropentoxide sulfide, di- (realgar)
As4 H3AsO4·a H2O HAsO3 H4As2O7 As2O5 As2S2
299.64 150.94 123.92 265.85 229.82 213.94
yel., cb. col., hyg. wh., hyg. col. wh., amor. red, mn., 2.68
−H2O, 160
50 H3AsO4 H3AsO4 76.7100° d.
s. alk.
d. 565
16.7 d. to form d. to form 59.50° i.
sulfide, pentaArsenious chloride (butter of arsenic) hydride (arsine) oxide (arsenolite) oxide (claudetite) oxide
As2S5
310.12
yel.
d. 500
0.0001360°
i.
s. HNO3, alk.
AsCl3 AsH3 As2O3 As2O3 As2O3
181.28 77.93 197.82 197.82 197.82
oily lq. col. gas col., cb., fibrous, 1.755 col., mn., 1.92 amor. or vitreous
130 −55; d. 230
d. 20 cc sl. s. sl. s. 1.210°
d. sl. s. sl. s. sl. s. 2.9340°
Auric chloride
AuCl3·2H2O
339.60
or. cr.
v. s.
v. s.
cyanide Aurous chloride cyanide Cf. also under Gold Barium acetate acetate bromide
Au(CN)3·6H2O AuCl AuCN
383.35 232.66 223.22
yel. cr. yel. cr.
7.4
v. s. d. i.
v. s. d. i.
s. HCl, HBr, PCl3 sl. s. alk. i. al., et. i. al., et. s. HCl, alk., Na2CO3; i. al., et. s. HCl, al., et.; sl. s. NH3 s. al. s. HCl, HBr; d. al. s. KCN; i. al., et.
Ba Ba(C2H3O2)2 Ba(C2H3O2)2·H2O BaBr2
137.36 255.45 273.46 297.19
silv. met. col. wh., tri. pr., 1.517 col.
3.5 2.468 2.19 4.781 24° 4
d. 58.80° 7530°(anh.) 980°
d. 75.0100° 7940°(anh.) 149100°
2.020° 2.0–2.5 4.086 (α)3.50619°; (β)3.25419° lq. 2.163 2.695 (A) 3.865 25° 4 3.85 3.738
d. 358 35.5 d. d. 206 (α)tr. 267; (β)307 −18 −113.5 subl. subl. 315 d. d. 50 AuCl3, 170 d.
d. 290
850
1140
−H2O, 41 847
d.
5.1515° gly. 2425° cc al.
s. alk., al. s. K2S, NaHCO3
s. a.; d. al. i. al. v. s. m. al.; v. sl. s. act.
*Usually the solution. †See special tables. ‡Usual commercial form. 2-9
TABLE 2-1
Physical Properties of the Elements and Inorganic Compounds (Continued )
Name Barium (Cont.) bromide carbonate (witherite) carbonate (α) carbonate (β) Barium chlorate chlorate chloride chloride chloride hydroxide hydroxide nitrate (nitrobarite) oxalate oxide peroxide peroxide phosphate, monobasic phosphate, dibasic phosphate, tribasic phosphate, pyrosilicofluoride sulfate (barite, barytes) sulfide, monosulfide, trisulfide, tetraBeryllium (glucinum) Bismuth carbonate, subchloride, dichloride, trinitrate nitrate, suboxide, trioxide, trioxide, trioxychloride
Formula
Formula weight
Color, crystalline form and refractive index
Specific gravity
Melting point, °C −2H2O, 100 tr. 811 to α tr. 982 to β 174090atm 414 d. 120 tr. 925 962 −2H2O, 100
BaBr2·2H2O BaCO3 BaCO3 BaCO3 Ba(ClO3)2 Ba(ClO3)2·H2O* BaCl2 BaCl2 BaCl2·2H2O† Ba(OH)2 Ba(OH)2·8H2O Ba(NO3)2 BaC2O4 BaO
333.22 197.37 197.37 197.37 304.27 322.29 208.27 208.27 244.31 171.38 315.50 261.38 225.38 153.36
col., mn., 1.7266 wh., rhb., 1.676 wh., hex. wh. col. col., mn., 1.577 col., mn., 1.7361 col., cb. col., mn., 1.646 col., mn. col., mn., 1.5017 col., cb., 1.572 wh. cr. col., cb., 1.98
BaO2* BaO2·8H2O BaH4(PO4)2 BaHPO4 Ba3(PO4)2 Ba2P2O7 BaSiF6 BaSO4
169.36 313.49 331.35 233.35 602.04 448.68 279.42 233.42
gray or wh. pd. pearly sc. tri. wh., rhb. nd., 1.635 wh., cb. wh., rhb. pr. col., rhb., 1.636
BaS BaS3 BaS4·2H2O Be(Gl) Bi
169.42 233.54 301.63 9.02 209.00
Bi2O3·CO2·H2O BiCl2(?) BiCl3* Bi(NO3)3·5H2O BiONO3·H2O Bi2O3 Bi2O3 Bi2O3 BiOCl
528.03 279.91 315.37 485.10 305.02 466.00 466.00 466.00 260.46
col., cb., 2.155 yel.-gn. red, rhb. gray, met., hex. silv. wh. or reddish, hex. wh. pd. bk. nd. wh. cr. col., tri. hex. pl. yel., rhb. yel., tet. yel., cb. wh., amor.
6.86 4.86 4.75 2.82 4.92815° 8.9 8.55 8.20 7.7215°
d. 163 230 d. 30 d. 260 820 860 tr. 704 185 d.
3.69 4.29
3.179 3.856 24° 4 3.097 24° 4 4.495 16° 2.188 3.24428° 2.658 5.72 4.958 4°
2.9 4.16515° 4.116° 3.920° 4.27915° 4.49915°
Boiling point, °C d. d. 1450
1560 1560
2.988 1.816 9.8020°
Hot water v. s. 0.0065100°
s. al. s. a.; i. al.
0.002218° 20.350° s. 310°
0.0065100° 84.880° s. 59100°
s. a.; i. al.
76.8100° 101.480°
sl. s. HCl, HNO3; i. al.
77.9 592
−8H2O, 550 d.
1923
2000
d. d. d.
tr. to mn. 1149
v. sl. s. 0.168 d. 0.015 i. 0.01 0.02617° 0.0001150°
2767 1450
d. s. 4115° i. i.
d. s. v. s. sl. s. d. i.
i. d. d. d. i. i. i. i. sl. s.
i.
i. i. i. i. sl. s.
2.660°
40.2100°
−O, 800 −8H2O, 100
1580 d. d. 400 d. 200 1284 271
d. 300 447 −5H2O, 80 1900
Other reagents
v. s. 0.002218°
39.30° 1.670° 5.615° 5.00° 0.00168° 1.50°
4.2515° 20°
Solubility in 100 parts Cold water
34.2100° 0.002424° 90.880°
0.09100° 0.00028530°
sl. s. al., act. sl. s. HCl, HNO3; i. al.
v. sl. s. al.; i. et. sl. s. a.; i. al. s. a., NH4Cl; i. al. s. HCl, HNO3, abs. al.; i. NH3, act. s. dil. a.; i. act. s. dil. a.; i. al., et., act. s. a. s. a., NH4 salts s. a. s. a., NH4 salts sl. s. HCl, NH4Cl; i. al. s. conc. H2SO4; 0.006, 3% HCl d. HCl; i. al. i. al., CS2 s. dil. a., alk. s. aq. reg., conc. H2SO4, HNO3 s. a. s. al. 4219° act.; s. a.; i. al. s. a. s. a. s. a. s. a. s. a.; i. act., NH3, H2C4H4O6 22.220° gly., 0.2425° et.; s. al. s. HNO3; i. al.
Boric acid
H3BO3
61.84
wh., tri.
1.43515°
Boron
B
10.82
2300
2550
i.
i.
B4C B2O3 B2O3·3H2O HBrO3 Br2
55.29 69.64 123.69 128.92 159.83
gray or bk., amor. or mn. bk. cr. col. glass, 1.459 tri., 1.456 col.; in soln. only rhb., or red lq.
2.32
carbide oxide oxide (sassolite) Bromic acid Bromine
2.54 1.85 1.49
2450 577 d. d. 100 −7.2
>3500 >1500
i. 1.10° sl. s. v. s. 4.220°
i. 15.7100° s. d. 3.1330°
i. a. s. a., al., gly.
hydrate Cadmium acetate acetate carbonate
Br2·10H2O Cd Cd(C2H3O2)2 Cd(C2H3O2)2·2H2O* CdCO3
339.99 112.41 230.50 266.53 172.42
red, oct. silv. met., hex. col. col., mn. wh., trig.
8.6520° 2.341 2.01 4.2584°
i.
wh., cb.
4.047 25° 4
s. a., NH4NO3 s. m. al. s. al. s. a., KCN, NH4 salts; i. NH3 1.5215° al.; i. et., act.
chloride
CdCl2
183.32
3.11920°; 5.87 (A)
d. 6.8 320.9 256 −H2O, 130 d. <500 568
58.78 767 d.
960
s. i. v. s. v. s. i. 0°
90
i. 147100°
s. al., et., alk., CS2
chloride cyanide hydroxide nitrate nitrate oxide oxide oxide, subCadmium sulfate sulfate sulfate sulfate sulfate sulfide (greenockite) Calcium acetate aluminate aluminum silicate (anorthite) arsenate bromide carbonate (aragonite) carbonate (calcite) chloride (hydrophilite) chloride chloride citrate cyanamide ferrocyanide fluoride (fluorite) formate hydride hydroxide hypochlorite hypophosphate lactate magnesium carbonate (dolomite) magnesium silicate (diopside) nitrate (nitrocalcite) nitrate nitride nitrite oxalate oxalate oxide
CdCl2.2a H2O Cd(CN)2 Cd(OH)2 Cd(NO3)2 Cd(NO3)2.4H2O* CdO CdO Cd2O CdSO4 CdSO4.H2O 3CdSO4.8H2O* CdSO4.H2O CdSO4.7H2O CdS Ca Ca(C2H2O2)2.H2O Ca(AlO2)2 CaO.Al2O3.2SiO2 Ca3(AsO4)2 CaBr2 CaCO3 CaCO3 CaCl2* CaCl2.H2O CaCl2.6H2O Ca3(C6H5O7)2.4H2O CaCN2 Ca2Fe(CN)6.12H2O CaF2 Ca(HCO2)2 CaH2 Ca(OH)2 Ca(ClO)2.4H2O Ca2P2O6.2H2O Ca(C3H5O3)2.5H2O
228.36 164.45 146.43 236.43 308.49 128.41 128.41 240.82 208.47 226.49 769.54 280.53 334.58 144.47 40.08 176.18 158.02 278.14 398.06 199.91 100.09 100.09 110.99 129.01 219.09 570.50 80.11 508.31 78.08 130.12 42.10 74.10 215.06 274.15 308.30
col., mn., 1.6513
3.327
wh., trig. col. col. nd. brn., cb. brn., amor, 2.49 gn., amor. rhb. mn. col., mn., 1.565 col. mn. yel.-or., hex., 2.506 silv. met., cb. wh. nd. col., rhb. or mn. tri., 1.5832 wh. pd. delq. nd. col., rhb., 1.6809 col., hex., 1.550 wh., delq., cb, 1.52 col., delq. col., trig., 1.417 col. nd. col., rhombohedral yel., tri., 1.5818 wh., cb., 1.4339 col., rhb. wh. cr. or pd. col., hex., 1.574 wh., feathery cr. granular col., eff.
4.79 15° 4
CaO.MgO.2CO2 CaO.MgO.2SiO2 Ca(NO3)2 Ca(NO3)2.4H2O* Ca3N2 Ca(NO2)2.H2O CaC2O4 CaC2O4.H2O CaO
184.42 216.52 164.10 236.16 148.26 150.11 128.10 146.12 56.08
trig., 1.68174 wh., mn. col., cb. col., mn., 1.498 brn. cr. delq., hex. col., cb. col. col., cb., 1.837
peroxide phosphate, monobasic phosphate, dibasic phosphate, tribasic phosphate, metaphosphate, pyrophosphate, pyro- (brushite) phosphide silicate (α) (pseudowollastonite)
CaO2.8H2O CaH4(PO4)2.H2O CaHPO4.2H2O Ca3(PO4)2 Ca(PO3)2 Ca2P2O7 Ca2P2O7.5H2O Ca3P2 CaSiO3
216.21 252.09 172.10 310.20 198.04 254.12 344.20 182.20 116.14
silicate (β) (wollastonite) sulfate (anhydrite)
CaSiO3 CaSO4
116.14 136.14
pearly, tet. wh., tri. wh., mn. pl. wh., amor. wh., tet., 1.588 col., biaxial, 1.60 wh., mn. red cr. col., pseudo hex., 1.6150 or mn.(?) col., mn., 1.610 col., rhb., 1.576, or mn, 1.50
2.455 17° 4 8.15 6.95 8.192 18° 4 4.691 24° 4 3.78620° 3.09 3.05 2.48 20° 4 4.58 20° 1.55
tr. 34 d. >200 d. 300 350 59.4
132
d. 900–1000 d. 1000 tr. 108 tr. 41.5
3.6720° 2.765
tr. 4 1750100atm 810 d. 1600 1551
3.353 25° 4 2.93 25° 2.711 4 2.152 15° 4
760 d. 825 1339103atm. 772
1810
1.6817°
29.92 −2H2O, 130
−6H2O, 200 −4H2O, 185
1.7 3.18020° 2.015 1.7 2.2
2.872 3.3 2.36 1.82 2.6317° 2.2334° 2.24° 2.2 3.32 2.220 16° 4 2.306 16° 4 3.14 2.82 3.09 2.25 2.5115° 2.905 2.915 2.96
subl. in N2, 980 1200 30
>1600
1330 d. d. 675 −H2O, 580 d. −2H2O, 200 −3H2O, 100
−8H2O, 100 −H2O, 100 d. 1670 975 1230
2850 expl. 275 d. 200
>1600 1540 tr. 1190 to α 1450(mn.)
180100°
76.50° s. 114.20° s. 350−5° 0.000001 d. 520° d.
60.8100° s. 127.660° s.
0.01325° 1250° 0.001220°† 0.001425° 59.50° s. v. s. 0.08518° s. d. s. 0.001618° 16.10° d. 0.1850° delq.; d. i. 10.5
32659.5° i. i.
Colloidal d. 45.580° i. 312105° 0.002100° 0.002100° 347260° s. v. s. 0.09626° d. 15090° 0.001726° 18.4100 0.077100° d. ∞
2.0515° m. al. s. a.; NH4OH, KCN s. a., NH4 salts; i. alk. v. s. a. s. al., NH3; i. HNO3 s. a., NH4 salts; i. alk. s. a., NH4 salts; i. alk. d. a., alk. i.act., NH3 i. al. i. al. i. al. s. a.; v. s. NH4OH s, a.; sl. s. al. sl. s. al. s. HCl s. dil. a. s. al., act.; sl. s. NH3 s. a., NH4Cl s. a., NH4Cl s. al. s. al. s. al. 0.006518° al. i. al. sl. s. a. i. al., et. d. a.; i. bz. s. NH4Cl d. a. s. HCl, H4P2O6 ∞h. al.; i. et.
18°
d. 730–760 1391 561 42.7 900 d. −H2O, 200 2570
16820° 0.024718° 0.0002625° 109.70° 2150° i. i.
tr. 1193 to rhb.
0.032 i. 1020° 2660° d. 770° 0.0006713° i. Forms Ca(OH)2 sl. s. 0.0224.5° 0.0025 i. i. sl. s. d. 0.009517° 0.29820°
i. 376151° v. s. d. 41790° 0.001495° i. d. d. 0.075100° d. i.
0.1619100°
1415° al.; s. amyl al., NH3 s. dil. a.; i. abs. al. s. 90% al. s. a.; i. ac. s. a.; i. ac s. a.; i. al. s. a. d.; i. al., et. s. a.; i. al., ac. i. a. s. a. s. a.; i. NH4Cl s. dil. a.; i. al., et. s. HCl s. a., Na2S2O3, NH4 salts
*Usual commercial form. †The solubility of CaCO3 in H2O is greatly increased by increasing the amount of CO2 in the H2O. 2-11
Physical Properties of the Elements and Inorganic Compounds (Continued )
TABLE 2-1
Name Calcium (Cont.) sulfate (gypsum)
Formula
Formula weight
Color, crystalline form and refractive index
CaSO4·2H2O
172.17
col., mn., 1.5226
Ca(SH)2·6H2O CaS CaSO3·2H2O CaC4H4O6·4H2O Ca(CNS)2·3H2O CaS2O3·6H2O CaWO4
214.31 72.14 156.17 260.22 210.28 260.30 288.00
col. pr. col., cb. wh., cr., 1.595 col., rhb. wh., delq. cr. col., tri., 1.56 wh., tet., 1.9200
C C C CO2
12.01 12.01 12.01 44.01
bk., amor. col., cb., 2.4195 bk., hex. col. gas
disulfide
CS2
76.13
col. lq.
monoxide
CO
28.01
col., poisonous, odorless gas
oxychloride (phosgene) oxysulfide
COCl2 COS
98.92 60.07
poisonous gas gas
sulfhydrate sulfide (oldhamite) sulfite tartrate thiocyanate thiosulfate tungstate (scheelite) Carbon, cf. table of organic compounds Carbon, amorphous Carbon, diamond Carbon, graphite dioxide
suboxide thionyl chloride Ceric hydroxide hydroxynitrate oxide sulfate Cerium
C3O2 CSCl2 2CeO2·3H2O Ce(OH)(NO3)3·3H2O CeO2 Ce(SO4)2·4H2O Ce
68.03 114.98 398.31 397.21 172.13 404.31 140.13
Cerous sulfate sulfate Cesium Chloric acid Chlorine
Ce2(SO4)3 Ce2(SO4)3·8H2O Ce HClO3·7H2O Cl2
568.44 712.57 132.91 210.58 70.91
hydrate Chloroplatinic acid Chlorostannic acid Chlorosulfonic acid Chromic acetate chloride chloride fluoride hydroxide
Cl2·8H2O H2PtCl6·6H2O H2SnCl6·6H2O HO·SO2·Cl Cr2(C2H3O2)6·2H2O CrCl3 CrCl3·6H2O* CrF3 Cr(OH)3
215.04 518.08 441.55 116.52 494.32 158.38 266.48 109.01 103.03
Cr(OH)3·2H2O Cr(NO3)3·9H2O* Cr(NO3)3·7a H2O Cr2O3 Cr2(SO4)3 Cr2(SO4)3·5H2O Cr2(SO4)3·15H2O Cr2(SO4)3·18H2O
139.07 400.18 373.15 152.02 392.20 482.28 662.44 716.49
hydroxide nitrate nitrate oxide sulfate sulfate sulfate sulfate
gas yel.-red lq. yel., gelatinous red, mn. wh. or pa. yel., cb. yel., rhb. steel gray, cb. or hex. wh., mn. or rhb. tri. silv. met., hex. lq. rhb., or gn.-yel. gas rhb. red-brn., delq. delq. col. lq. gn. pink, trig. vl. or gn., hex. pl. gn., rhb. gn. or blue, gelatinous gn. purple pr. purple, mn. dark gn., hex. rose pd. gn. vl. vl., cb., 1.564
Specific gravity 2.32 2.815°
Melting point, °C
Boiling point, °C
−1a H2O, 128
−2H2O, 163
d. 15 −2H2O, 100 d.
d. 650
Solubility in 100 parts Cold water
Hot water
0.2230°
0.25750°
v. s. d. 0.004318° 0.0370° s. 71.29° 0.2
v. s. d. 0.002790° 0.2285° v. s. d.
Other reagents s. a., gly., Na2S2O3, NH4 salts s. al. s. a. s. H2SO3 sl. s. al. v. s. al. i. al. s. NH4Cl; i. a.
1.87316° 6.06
d.
1.8–2.1 3.5120° 2.2620° lq. 1.101−87°; 1.53 (A); solid 1.56−79° 22° lq. 1.261 20 ; 2.63 (A) lq. 0.814−195° 4 ; 0.968 (A) 1.392 19° 4 lq. 1.24−87°; 2.10 (A) lq. 1.1140° 1.50915°
>3500 >3500 >3500 −56.65.2atm.
4200 4200 4200 subl. −78.5
i. i. i. 179.70° cc
i. i. i. 90.120° cc
i. a., alk. i. a., alk. i. a., alk. s. a., alk.
−108.6
46.3
0.20°
0.01450°
s. al.; et.
−207
−192
0.00440°; 3.50° cc
0.001850° 2.3220° cc
s. al., Cu2Cl2
−104 −138.2
8.2756mm −50.2760mm
v. s. sl. d. 1330° cc
d. 40.330° cc
s. ac., CCl4, bs.; d.a. v. s. alk., al.
−107
7761mm 73.5
d.
s. et. s. a.; sl. s. alk. carb.; i. alk
7.3 3.91 6.920° cb.; 6.7 hex. 3.91 2.88617° 1.9020° 1.28214.2° lq. 1.56−33.6°; 2.490° (A) 1.23 2.431 1.97128° 1.78725° 2.75715° 1.835 25° 4 3.8
5.21 3.012 1.86717° 1.722°
1950 645
1400
d. i. s. d. i.
i.
0°
−8H2O, 630 28.5 <−20 −101.6 d. 9.6 60 19.2 −80
670 d. 40 −34.6
151.5765mm 1200–1500 d.
subl. 83 >1000
d.
−2H2O, 100 36.5 100 1900
d. 100 d.
100
−10H2O, 100 −12H2O, 100
18.98 250° d. v. s. 1.460°; 31010° cc s. v. s. s. d. s. i.§ v. s. d. i. i. i. s. s. i. i.† s. s. 12020°
Slowly oxidized 0.4100° 7.640°
s. H2SO4, HCl s. dil. H2SO4 s. dil. a.; i. al.
s. a., al., NH3 0.57 ; 17730° cc 30°
v. s.
sl. s.
i. s. s. i. d. 67° d.
s. alk. s. alk. s. al., et. d. al.; i. CS2 4.7615° m. al. i. a., act., CS2 s. al.; i. et. sl. s. a.; i. al., NH3 s. a., alk.; sl. s. NH3 s. a., alk. s. a., alk., al., act. sl. s. a. i. a. s. al., H2SO4 sl. s. al. s. al.
sulfide Chromium
Cr2S3 Cr
200.02 52.01
brn.-bk. pd. gray, met., cb.
3.7719° 7.1
−S, 1350 1615
trioxide (chromic acid) Chromous chloride hydroxide oxide sulfate sulfide (daubrelite) Chromyl chloride Cobalt carbonyl sulfide, diCobaltic chloride chloride, dichro chloride, luteo chloride, praseo Cobaltic chloride, purpureo chloride, roseo hydroxide oxide sulfate sulfide Cobalto-cobaltic oxide Cobaltous acetate chloride chloride nitrate
CrO3 CrCl2 Cr(OH)2 CrO CrSO4·7H2O CrS CrO2Cl2 Co Co(CO)4 CoS2 CoCl3 Co(NH3)3Cl3·H2O Co(NH3)6Cl3 Co(NH3)4Cl3·H2O Co(NH3)5Cl3 Co(NH3)5Cl3·H2O Co(OH)3 Co2O3 Co2(SO4)3 Co2S3 Co3O4 Co(C2H3O2)2·4H2O CoCl2 CoCl2·6H2O* Co(NO3)2·6H2O
100.01 122.92 86.03 68.01 274.18 84.07 154.92 58.94 170.98 123.06 165.31 234.42 267.50 251.46 250.47 268.49 109.96 165.88 406.06 214.06 240.82 249.09 129.85 237.95 291.05
red, rhb. wh., delq. yel.-brn. bk. pd. blue bk. pd. dark red lq. silv. met., cb. or. cr. bk., cb. red cr.
2.70 2.75
197 d.
or., mn. gn., rhb. rhb. brick red bk. bk. blue cr. bk. cr. bk., cb. red-vl., mn., 1.542 blue cr. red, mn. red, mn., 1.4
1.7016 1.847 25° 1.819 25
CoO CoSO4 CoSO4·H2O
74.94 155.00 173.02
brn., cb. red pd. red pd., mn.(?), 1.639 red, mn., 1.483 brn. nd. yel.-red met., cb.
oxide sulfate sulfate sulfate (biebeorite) sulfide (syeporite) Copper Cupric acetate acetate aceto-arsenite (Paris green)
d. 3.97 1.92 8.920° 1.7318° 4.269 2.94
1550 −96.5 1480 51 subl.
5.18
d. 100 −1a H2O, 100 d. 900
4.8 6.07 1.705318.7° 3.356 25° 1.924 25 25° 1.883 25
−4H2O, 140 subl. 86 <100
5.68 3.71025° 3.13
d. 1800 d. 880 d.
25° 1.948 25 5.4518° 20° 8.92 1.930 20° 4 1.882
96.8 >1100 1083
−7H2O, 420
115
240 d.
1.98
d. i.
164.90° v. s. d. i. 12.350 i. d. i. i. i. s. s. 4.260° v. s. 0.2320° 16.120° i. i. d. i. i. s. 457° 116.50° 84.030°(anh.)
206.7100° v. s.
i. 25.60° s.
i.
i. d.
s. h. HNO3 s. HCl, dil. H2SO4; i. HNO3 s. H2SO4, al., et. sl. s. al.; i. et. s. conc. a. i. dil. HNO3 sl. s. al. v. s. a. s. et. s. a. s. al., et., CS2 s. HNO3, aq. reg.
s. 12.7446.5° 1.03146.5° 24.8716° i. i. i. s. 10596° 17780° 334.990° (anh.) i. 83100° s.
s. a.; al. i. al., NH4OH s. a.; i. al. i. al. sl. s. HCl s. a.; i. al. s. a. s. H2SO4 d. a. s. H2SO4; i. HCl, HNO3 s. a., al. 31 al.; 8.6 act. v. s. et., act. 10012.5° al.; s. act.; sl. s. NH3 s. a., NH4OH; i. al. 1.0418° m. al.; i. NH8
s.
20
7 al.; s. et.; gly. s. a., NH4OH
d. 110
33.80°
99.380°
s. a.
1.81 3.88
d. 150 d. 220
18.0521.5° i.
d. d.
i. 70.70°
d. 107.9100°
i. al. s. NH4OH, h. aq. NaHCO3 s. KCN; 0.03 aq. CO 5315° al.; 6815° m. al.
110.40° i. i. sl. s. i. i. 12.5 i. 16.7 38140° 243.70°
192.4100°
ammonium sulfate carbonate, basic (azurite)
CuSO4·4NH3·H2O 2CuCO3·Cu(OH)2
245.77 344.75
blue, tet., 1.670, 1.744 blue, rhb. blue, mn., 1.758
carbonate, basic (malachite) chloride (eriochalcite)
CuCO3·Cu(OH)2 CuCl2
221.17 134.48
dark gn., mn., 1.875 brn.-yel. pd.
3.9 3.054
d. 498
chloride chromate, basic cyanide dichromate ferricyanide ferrocyanide formate hydroxide lactate nitrate nitrate
CuCl2·2H2O CuCrO4·2CuO·2H2O Cu(CN)2 CuCr2O7·2H2O Cu3[Fe(CN)6]2 Cu2Fe(CN)6·7H2O Cu(HCO2)2 Cu(OH)2 Cu(C3H5O3)2·2H2O Cu(NO3)2·3H2O* Cu(NO3)2·6H2O
170.52 374.75 115.61 315.62 614.63 465.21 153.61 97.59 277.74 241.63 295.68
gn., rhb., 1.684 yel.-brn. yel.-gn. bk., tri. yel.-gn. red-brn. blue, mn. blue, gelatinous dark blue, mn. blue, delq. blue, rhb.
2.3922.4°
−2H2O, 110 −2H2O, 260 d. −2H2O, 100
277.51
1049 −6H2O, 110 d.
i. i.
3380° 0.0003818° i. s. 7.2 i.
ammonium chloride
dark gn., mn. gn.
117.6 2900 d. 52
20°
CoSO4·7H2O* CoS Cu Cu(C2H3O2)2 Cu(C2H3O2)2·H2O (CuOAs2O3)3· Cu(C2H3O2)2* CuCl2·2NH4Cl·2H2O
281.11 91.00 63.57 181.66 199.67 1013.83
2200
2.28618° 1.831 3.368
−H2O
2.0473.9° 2.074
114.5 −3H2O, 26.4
2300
Forms Cu2Cl2 993 d.
−HNO3, 170
i.
d. i. d. d. 45100° 66680° ∞
2.58° al. s. a., aq. reg. s. HNO3, h. H2SO4
s. al.; et., NH4Cl s. HNO3, NH4OH s. KCN, C5H5N s. a.; NH4OH s. NH4OH; i. HCl s. NH4OH; i. a., NH8 0.25 al. s. a., NH4OH, KCN, al. sl. s. al. 10012.5° al. s. al.
*Usual commercial form. †Also a soluble modification.
2-13
TABLE 2-1
Physical Properties of the Elements and Inorganic Compounds (Continued )
Name Cupric acetate (Cont.) oxide (paramelaconite) oxide (tenorite) oxychloride phosphide sulfate (hydrocyanite) sulfate (blue vitriol or chalcanthite) sulfide (covellite)
Formula
Specific gravity
249.71 95.63
tartate Cuprous ammonium iodide carbonate chloride (nantokite) cyanide
CuC4H4O6·3H2O CuI·NH4I·H2O Cu2CO3 Cu2Cl2 Cu2(CN)2
265.69 353.47 187.15 198.05 179.16
ferricyanide ferrocyanide fluoride hydroxide oxide (cuprite) Cuprous phosphide sulfide (chalcocite) sulfide Cyanogen
Cu3Fe(CN)6 Cu4Fe(CN)6 Cu2F2 CuOH Cu2O Cu6P2 Cu2S Cu2S C2N2
402.67 466.24 165.14 80.58 143.14 443.38 159.20 159.20 52.02
brn.-red brn.-red red cr. yel. red, cb., 2.705 gray-bk. bk., rhb. bk., cb. poisonous gas
Fe(OH)(C2H3O2)2
190.95
brn., amor.
FeCl3 FeCl3·6H2O* Fe4[Fe(CN)6]3
162.22 270.32 859.27
bk.-brn., hex. delq. red-yel., delq. dark blue
2.80411°
282 37 d.
Fe(OH)3 Fe(C3H5O3)3 Fe(NO3)3·6H2O Fe2O3
106.87 323.06 349.97 159.70
red-brn. brn., amor., delq. rhb., delq. red or bk., trig., 3.042 rhb., 1.814 yel., trig. yel., delq. gn. bk., cb., 2.42
3.4 to 3.9
−1a H2O, 500
sulfate sulfate (coquimbite) Ferroso-ferric chloride ferricyanide (Prussian green) oxide (magnetite; magnetic iron oxide) oxide, hydrated Ferrous ammonium sulfate
Fe2(SO4)3 Fe2(SO4)3·9H2O FeCl2·2FeCl3·18H2O Fe′′′ 4 Fe′′ 3 [Fe(CN)6]6 Fe3O4
399.88 562.02 775.49 1662.70 231.55 303.61 392.15
chloride (lawrencite)
Fe3O4·4H2O FeSO4·(NH4)2SO4· 6H2O FeCl2
chloroplatinate ferricyanide (Turnbull’s blue) ferrocyanide formate hydroxide nitrate oxide
FePtCl6·6H2O Fe3[Fe(CN)6]2 Fe2Fe(CN)6 Fe(HCO2)2·2H2O Fe(OH)2 Fe(NO3)2·6H2O FeO
571.92 591.47 323.66 181.92 89.87 287.96 71.85
2-14
126.76
bk. blue-gn., mn., 1.4915 gn.-yel., hex., 1.567 yel., hex. dark blue blue-wh., amor.
6.40 6.45
Melting point, °C
CuSO4·5H2O* CuS
hydroxide lactate nitrate oxide (hematite)
79.57 79.57 365.69 252.67 159.63
Color, crystalline form and refractive index bk., cb. bk., tri., 2.63 blue-gn. bk. gn.-wh., rhb., 1.733 blue, tri., 1.5368 blue, hex. or mn., 1.45 1 gn. pd. rhb. pl. yel. wh., cb., 1.973 wh., mn.
Cyanogen compounds, cf. table of organic compounds Ferric acetate, basic ammonium sulfate, cf. Alum chloride (molysite) chloride ferrocyanide (Prussian blue)
CuO CuO CuCl2·2CuO·4H2O Cu3P2 CuSO4
Formula weight
6.35 3.60615° 2.286 15.6° 4 4.6
d. 1026 d. 1026 −3H2O, 140 d. d. >600 −4H2O, 110 tr. 103
4.4 3.53 2.9
d. 422 474.5
3.4 6.0 6.4 to 6.8 5.6 5.80 lq. 0.866−17.2°; 1.806 (A)
Solubility in 100 parts Cold water i. i. i. i. 14.30°
Hot water i. i. 75.4100°
Other reagents s. a.; KCN, NH4Cl s. a., KCN, NH4Cl s. a. s. HNO3; i. HCl i. al.
Forms CuO, 650 −5H2O, 250 d. 220
24.3 0.00003318°
205100°
1.18° al. s. HNO3, KCN
0.1485°
1366 d.
0.0215° d. i. 1.5225° i.
s. a., KOH s. NH4I s. a., NH4OH s. HCl, NH4OH, al. s. KCN, HCl, NH4OH; sl. s. NH3 s. NH4OH; i. HCl s. NH4OH; i. NH4Cl s. HF, HCl, HNO3; i. al. s. a., NH4OH s. HCl, NH4Cl, NH4OH s. HNO3; i. HCl s. HNO3, NH4OH; i. act. s. HNO3, NH4OH; i. act. 230020° cc al.; 50018° cc et.
d.
908 −a H2O, 360 1235
−O, 1800
1100 1130 −34.4
−20.5
subl. 1100
0°
i. i. i. i. i. i. 0.000518° 0.000518° 45020° cc
i. i.
i. i.
i.
s. a.; al. v. s. al.; et. +HCl s. al., act., gly. s. HCl, conc. H2SO4; i. al., et. s. a.; i. al., et. i. et. s. al., act. s. HCl
74.40° 2460° i.
535.8100° ∞ d.
i. v. s. 1500° i.
i. v. s. ∞ d. d. s.
5.2
d. 50 d. 180 1538 d.
sl. s. 440 s. i. i.
i.
s. d. h. HCl i. al.
1.864
d. d.
i. 180°
i. 10075°
s. a. i. al.
64.410°
105.7100°
100 al.; s. act.; i. et.
v. s. i. i. sl. s. 0.00067 2000° i.
v. s.
20°
1.684 5.12
35 1560 d.
3.09718° 2.1
d. 480
2.7 2.714 d. 3.4 5.7
315 280
d.
delq.
d. lt. gn. cr. bk.
Boiling point, °C
60.5 1420
i. H2SO4, NH3 s. abs. al.
i. dil. a., al.
30025° i.
s. a., NH4Cl s. a.; i. alk.
phosphate (vivianite)
Fe3(PO4)2·8H2O
501.64
silicate sulfate (siderotilate) sulfate (copperas) sulfide cf. also under iron Fluoboric acid Fluorine
FeSiO3 FeSO4·5H2O FeSO4·7H2O* FeS
131.91 241.99 278.02 87.91
HBF4 F2
87.83 38.00
Fluosilicic acid Gadolinium Gallium bromide Glucinum cf. Beryllium Gold Gold, colloidal Gold salts cf. under Auric and Aurous Hafnium Helium Hydrazine formate hydrate hydrochloride hydrochloride, dinitrate nitrate, disulfate sulfate Hydrazoic acid (azoimide) Hydriodic acid Hydriodic acid Hydriodic acid Hydriodic acid Hydriodic acid Hydrobromic acid Hydrobromic acid
H2SiF6 Gd GaBr3
144.08 156.9 309.47
Au Au
197.20 197.20
yel. met., cb. blue to vl.
19.3
1063
2600
Hf He N2H4 N2H4·2HCO2H N2H4·H2O N2H4·HCl N2H4·2HCl N2H4·HNO3 N2H4·2HNO3 N2H4·a H2SO4 N2H4·H2SO4 HN3 HI HI·H2O HI·2H2O HI·3H2O HI·4H2O HBr HBr·H2O
178.6 4.00 32.05 124.10 50.06 68.51 104.98 95.06 158.08 81.09 130.12 43.03 127.93 145.94 163.96 181.98 199.99 80.92 98.94
hex. col. gas col. lq. cb. col. yel. lq. wh., cb. cr. nd. delq. pl. rhb. col. lq. col. gas col. lq. col. lq. col. lq. col. lq. col. gas; 1.325 (lq.) col. lq.
12.1 0.1368 (A) 1.011 15° 4
>1700 <−272.2 1.4 128 −40
>3200(?) −268.9 113.5
Hydrobromic acid Hydrobromic acid Hydrochloric acid Hydrochloric acid Hydrochloric acid Hydrochloric acid Hydrocyanic acid (prussic acid)
HBr (47.8% in H2O) HBr·2H2O HCl† HCl (45.2% in H2O) HCl·2H2O HCl·3H2O HCN
80.92 116.96 36.47 36.47 72.50 90.51 27.03
Hydrofluoric acid Hydrofluoric acid Hydrogen
HF HF (35.35% in H2O) H2
col. lq. wh. cr. col. gas; 1.256 (lq.) col. lq. col. lq. col. lq. poisonous gas or col. lq., 1.254 gas or col. lq. col. lq. col. gas or cb.
peroxide selenide sulfide Hydroxylamine hydrochloride nitrate sulfate
H2O2‡ H2Se H2S NH2OH NH2OH·HCl NH2OH·HNO3 NH2OH·a H2SO4
20.01 20.01 2.016 34.02 81.22 34.08 33.03 69.50 96.05 82.07
blue, mn., 1.592, 1.603 mn. gn., tri., 1.536 blue-gn., mn. bk., hex. col. lq. gn.-yel. gas
2.58 3.5 2.2 1.89914.8° 4.84 lq. 1.51−187°; 1.3115° (A)
1550
col. lq., 1.333 col. gas col. gas rhb., delq. col., mn. col. cr. col., mn.
1.0321° 1.42
1.378 4.40° (A) 1.715°
2.710° (A) 1.78 1.486 2.11−15° 1.2680° (A) 1.48 1.46 −18.3° 4 18°
0.697
0.98813.6° 1.15 lq. 0.0709−252.7° 0.06948 (A) 1.438 20° 4 2.12−42° 1.1895 (A) 1.3518° 1.6717°
i.
s. a.; i. ac.
64 1193
−5H2O, 300 −7H2O, 300 d.
s. 32.80° 0.00061618°
s. 14950°
i. al. i. al. s. a.; i. NH3
−223
130 d. −187
∞ d.
∞
s. al.
s.
s.
s.
s.
i. s.
i.
s. aq. reg., KCN; i. a. s. aq. reg., KCN; i. a.
0.970° cc ∞ s. ∞ v. s. s. 174.910° v. s. v. s. 3.05522° ∞ 4250010° cc ∞ ∞ ∞ ∞ 2210°
1.0850° cc ∞
Absorbed by Pt s. al.
∞ v. s. v. s. v. s.
∞ al.; i. et. sl. s. al. s. al.
delq. cr. 20°
i.
198 70.7 104 85 254 −80 −50.8 −43 −48 −36.5 −86
−11 −111 −15.35 0 −24.4 −14 −83 −35 −259.1 −0.89 −64 −82.9 34 151 48 170 d.
118.5739.5mm subl. 140 d. 37 −35.5 127774mm
−67
d. d. 26
∞ s. 82.30° ∞ ∞ ∞ ∞
19.4 120 −252.7
∞ 0° to 19.4° v. s. 2.10° cc
126 −85
760mm
151.4 −42 −59.6 56.522mm d. d. <100
∞ 3774° cc 4370° cc s. 83.317° v. s. 32.90°
27.6560° ∞ v. s.
130100°
s. 56.160°
i. al. v. sl. s. abs. al. ∞ al. s. al. ∞ al. s. al. s. al. s. al. s. al. Stable at −15.5° and 1 atm., and at −11.3° and 2.5 atm. s. al. s. al., et. s. al. s. al. s. al. ∞ al., et.
v. s. 0.8580° cc
sl. s. Fe, Pd, Pt
27022.5° cc 18640° cc d. v. s. d. 68.590°
s. a., et.; i. petr. et s. CS2, COCl2 9.5415° cc al.; s. CS2 s. a., al. s. al.; i. et. v. s. abs. al. v. sl. s. al.; i. et., abs. al.
*Usual commercial form. †Usual commercial form about 31 percent. ‡Usual commercial forms 3 or 30 percent. 2-15
TABLE 2-1
Physical Properties of the Elements and Inorganic Compounds (Continued )
Name
Formula
Hypobromous acid Illinium Indium Iodic acid
HBrO Il In HIO8
Iodine oxide, pentaIodoplatinic acid Iridium Iron, cast† pure steel white pig wrought carbide (cementite) carbonyl nitride
I2 I2O5 H2PtI6·9H2O Ir Fe Fe Fe Fe Fe Fe2C Fe(CO)5 Fe2N
silicide sulfide, di- (marcasite) sulfide, di- (pyrite) sulfide (pyrrhotite) Cf. also under ferric and ferrous Krypton Lanthanum Lead
FeSi FeS2 FeS2 Fe7S8
acetate acetate (sugar of lead) acetate acetate, basic acetate, basic acetate, basic arsenate, monobasic arsenate, dibasic (schultenite) arsenate, metaarsenate, pyroLead azide bromide
Formula weight 96.92 146(?) 114.76 175.93
Color, crystalline form and refractive index
Specific gravity
Melting point, °C
yel.
Boiling point, °C
Solubility in 100 parts Cold water
Hot water
4050mm
s.
d.
soft, tet. met. col., rhb.
7.320° 4.6290°
155 110 d.
1450
i. 2860°
i. 576101°
blue-bk., rhb. wh., trimetric brn., delq. mn. wh. met., cb. gray silv. met., cb. silv. gray gray gray pseudo hex. pa. yel. lq. gray
4.9320° 4.799 25° 4
113.5 d. 300
184.35
0.0956660°
22.420° 7.03 7.8620° 7.6 to 7.8 7.6 to 7.8 7.86 7.4 1.45721° 6.35
2350 1275 1535 1375 1075 1505 1837 −21 d. >560
>4800
0.01620° 187.412° s. d. i. i. i. i. i. i. i. i. d.
83.91 119.97 119.97 647.43
yel.-gray, oct. yel., rhb. yel., cb. hex.
6.1 20° 4 4.87 5.0 4.6 20° 4
d. d.
i. 0.00049 0.0005 i.
i.
tr. 450 1171 d. >700
Kr La Pb
83.70 138.92 207.21
col. gas lead gray silv. met., cb.
2.818 (A) 6.1520° 20° 11.337 20
−169 826 327.5
−151.8 1800 1620
11.050° cc d. i.
3.5760° cc
Pb(C2H3O2)2 Pb(C2H3O2)2·3H2O† Pb(C2H3O2)2IOH2O Pb2(C2H3O2)3OH Pb(C2H3O2)2· Pb(OH)2·H2O Pb(C2H3O2)2· 2Pb(OH)2 PbH4(AsO4)2 PbHAsO4 Pb(AsO3)2 Pb2As2O7 PbN6 PbBr2
325.30 379.35 505.46 608.56 584.54
wh. cr. wh., mn. wh., rhb. wh. wh. nd.
3.251 20° 4 2.55 1.689
280 −3H2O, 75 22
807.75
wh. nd.
489.06 347.13 453.03 676.24 291.26 367.05
tri., 1.82 wh., mn., 1.9097 hex. rhb., 2.03 col. nd. col., rhb.
4.46 5.94 6.4215° 15° 6.85 15 6.66
802 expl. 350 373
253.84 333.84 1120.91 193.10 55.85 55.85 55.85 55.85 55.85 179.56 195.90 125.71
15°
d. 140 d. >200
3000
102.5760mm
−H2O, 280
918
i. i. i. i. i. i. i.
i.
19.70° 45.6415° s. v. s. v. s.
22150° 200100° s.
5.55
18.2
d. i. d. i. i. 0.45540°
d. 0.05100° 4.75100°
sl. s.
PbCO3
267.22
wh., rhb., 2.0763
6.6
d. 315
0.0001120°
d.
2PbCO3·Pb(OH)2† PbCl2 PbCrO4 PbCrO4·PbO Pb(HCO2)2 3PbO·H2O Pb(NO3)2
775.67 278.12 323.22 546.43 297.25 687.65 331.23
6.14 5.80 6.12
d. 400 501 844
4.56 7.592 4.53
d. 190 −H2O, 130 d. 470
i. 0.6730° 0.00000720° i. 1.616° 0.014 38.80°
i. 3.34100° i. i. 18100° d.
oxide, suboxide, mono- (litharge)
Pb2O PbO
430.42 223.21
wh., hex. wh., rhb., 2.2172 yel., mn., 2.42 or.-yel. nd. wh., rhb. cb. col., cb. or mn., 1.7815 bk., amor. yel., tet.
8.34 9.53
d. red heat 888
i. 0.006818°
i.
oxide, mono (massicotite)
PbO
223.21
yel., rhb., 2.61
8.0
2-16
s. a. v. s. 87% al.; i. abs. al. et., chl. s. al., KI, et. i. abs. al., et., chl. sl. s. aq. reg., aq. Cl2 s. a.; i. alk. s. a.; i. alk. s. a.; i. alk. s. a.; i. alk. s. a.; i. alk. s. a. s. al., H2SO4, alk. s. HCl, H2SO4 i. aq. reg. i. dil. a. i. dil. a.
sl. s. al., bz. s. a. s. HNO3; i. c. HCl, H2SO4 s. gly.; v. sl. s. al. s. gly.; sl. s. al. sl. s. al. s. al.
carbonate (cerussite) carbonate, basic (hydrocerussite; white lead) chloride (cotunnite) chromate (crocoite) chromate, basic formate hydroxide nitrate
954760mm d.
Other reagents
138.8100°
s. al. s. HNO3 s. HNO3, NaOH s. HNO3 s. HCl, HNO3; i. sc. v. s. ac.; i. NH4OH s. a., KBr.; sl. s. NH3; i. al. s. a., alk.; i. NH3, al. s. ac.; sl. s. aq. CO2 sl. s. dil. HCl, NH3, i. al. s. a., alk.; i. NH3, ac. s. a., alk. i. al. s. a., alk. 8.822° al. s. a., alk. s. alk., PbAc, NH4Cl, CaCl2
oxide, mono-
PbO
223.21
amor.
9.2 to 9.5
oxide, red (minium) oxide, sesquioxide, di- (plattnerite) silicate sulfate (anglesite)
Pb3O4 Pb2O3 PbO2 PbSiO3 PbSO4
685.63 462.42 239.21 283.27 303.27
9.1
Pb(HSO4)2·H2O PbSO4·PbO PbS Pb(CNS)2 Li LiC7H5O2 LiBr
419.36 526.48 239.27 323.37 6.94 128.05 86.86
LiBr·2H2O Li2CO3 LiCl
122.89 73.89 42.40
citrate fluoride formate hydride hydroxide hydroxide nitrate nitrate oxide phosphate, monobasic phosphate, tribasic phosphate, tribasic salicylate sulfate sulfate sulfate, acid Lutecium Magnesium acetate acetate aluminate (spinel)
Li3C6H5O7·4H2O LiF LiHCO2·H2O LiH LiOH LiOH·H2O LiNO3 LiNO3·3H2O Li2O LiH2PO4 Li3PO4 Li3PO4·12H2O LiC7H5O3 Li2SO4 Li2SO4·H2O† LiHSO4 Lu Mg Mg(C2H3O2)2 Mg(C2H3O2)2·4H2O† MgO·Al2O3
281.98 25.94 69.97 7.95 23.95 41.96 68.95 123.00 29.88 103.94 115.80 331.99 144.05 109.94 127.96 104.01 174.99 24.32 142.41 214.47 142.26
red, amor. red-yel., amor. brn., tet., 2.229 col., mn., 1.961 wh., mn. or rhb., 1.8823 cr. col., mn. lead gray, cb., 3.912 col., mn. silv. met. cb. wh. leaflets wh., delq., cb., 1.784 wh. pr. col., mn., 1.567 wh., delq., cb., 1.662 wh. cr. wh., cb., 1.3915 col., rhb. wh., cb. wh. cr. col., mn. col., trig., 1.735 col. col., 1.644 col. wh., rhb. wh., trig. col. col., mn., 1.465 col., mn., 1.477 pr.
ammonium chloride ammonium phosphate (struvite) ammonium sulfate (boussingaultite) benzoate carbonate (magnesite) carbonate (nesquehonite) carbonate, basic (hydromagnesite) Magnesium chloride (chloromagnesite) chloride (bischofite) hydroxide (brucite) nitride oxide (magnesia; periclase) perchlorate
MgCl2·NH4Cl·6H2O MgNH4PO4·6H2O
256.83 245.44
sulfate, acid sulfate, basic (lanarkite) sulfide (galena) thiocyanate Lithium benzoate bromide bromide carbonate chloride
*See also a table of alloys. †Usual commercial form.
MgSO4·(NH4)2SO4· 6H2O Mg(C7H5O2)2·3H2O MgCO3 MgCO3·3H2O 3MgCO3·Mg(OH)2·3H2O MgCl2 MgCl2·6H2O† Mg(OH)2 Mg3N2 MgO Mg(ClO4)2†
i.
i.
d. 500 d. 360 d. 290 766
i. i. i. i.
i. i. i.
1336 5
3.464 25° 4
547
1265
0.00280° 0.000118° 0.004418° 0.0000918° 0.0520° d. 3325° 1430° (2H2O)
0.005640°
6.92 7.5 3.82 0.5320°
1170 d. 977 1120 d. 190 186
2.110° 2.068 25° 4
44 618 614
d. 1360
24620° 1.540° 670°
9.375 6.49 6.2
2.29521.5° 1.46 0.820 2.54 1.83 2.38 2.013 25° 4
d. 870 −H2O, 94 680 445 261 29.88
2.22 2.06 2.12313°
>100 837 100 d. 860 −H2O, 130 170.5
silv. met., hex. wh. wh., mn. pr., 1.491 col. cb., 1.718–23
1.7420° 1.42 1.454 3.6
651 323 80 2135
wh., rhb., delq. col., rhb., 1.496
1.456 1.715
−4H2O, 195 d. 100
2.461 2.53717.5° 1.645
1670 925 d. subl. <1000
1110
61.215° 0.2718° 49.20° d. 12.70° 22.310° 53.40° v. s. forms LiOH
s. alk., PbAc, NH4Cl, CaCl2 s. ac., h. HCl s. a., alk. s. ac., h. alk.; i. al. s. a.
i. s. d. 40100° 266100° (1H2O) 0.72100° 127.5100° 66.7100° 0.13535° 346.6104° 17.5100° 26.880° 19470° ∞
0.03418° v. sl. s. 12826° 35.340° 43.60° d.
v. sl. s. v. sl. s.
i. v. s. v. s. i.
sl. s. d. v. s. v. s.
16.7 0.02310°
s. 0.019580°
29.9100° 35100°
s. conc. a., NH4 salts; i. al. sl. s. H2SO4 sl. s. H2SO4 s. a.; i. alk. s. KCNS, HNO3 s. a., NH3 7.725°, 1078° al. s. al., act. s. al. s. dil. a.; i. al., act., NH3 2.4815° al.; s. et. sl. s. al., et. s. HF; i. act. sl. s. al., et. i. et. sl. s. al. sl. s. al. s. al., NH3
s. a., NH4Cl; i. act. v. s. al. i. act., 80% al. i. 80% al. s. a., NH4 salts 5.2515° m. al. v. s. al. v. sl. s. dil. HCl; i. dil. HNO3 s. a.; i. al.
360.62
col., mn.
1.72
>120
16.86
320.59 83.43 138.38 365.37
wh. pd. wh., trig. 1.700 col., rhb., 1.501 wh., rhb., 1.530
3.037 1.852 2.16
−3H2O, 110 d. 350 −H2O, 100 d.
4.525° (anh.) 0.0106 0.151819° 0.04
d. 0.011
s. act. s. a., aq. CO2; i. act., NH3 s. a., aq. CO2 s. a., NH4 salts; i. al.
95.23
col., hex., 1.675
2.32525°
712
52.80°
73100°
50 al.
203.33 58.34 100.98 40.32 223.23
wh., delq., mn., 1.507 wh., trig., 1.5617 gn.-yel., amor. col., cb., 1.7364 wh., delq.
1.56 2.4 3.65 2.6025°
118 d. d. d. 2800 d.
1412 d. 3600
0°
0°
281 0.000918° i. 0.00062 99.625°
100°
130 s.
100°
918 d. v. s.
50 al. s. NH4 salts, dil. a. s. a.; i. al. s. a., NH4 salts; i. al. 2425 al., 51.825° m. al.; 0.29 et.
Physical Properties of the Elements and Inorganic Compounds (Continued )
TABLE 2-1
Name
Formula
Formula weight
Color, crystalline form and refractive index
Magnesium chloride (Cont.) peroxide phosphate, pyrophosphate, pyropotassium chloride (carnallite) potassium sulfate (picromerite) silicofluoride sodium chloride sulfate sulfate (epsom salt; epsomite) Manganese acetate acetate carbonate (rhodocrosite)
MgO2 Mg2P2O7 Mg2P2O7·3H2O MgCl2·KCl·6H2O MgSO4·K2SO4·6H2O MgSiF6·6H2O MgCl2·NaCl·H2O MgSO4 MgSO4·7H2O* Mn Mn(C2H3O2)2 Mn(C2H3O2)2·4H2O* MnCO3
56.32 222.60 276.65 277.88 402.73 274.48 171.70 120.38 246.49 54.93 173.02 245.08 114.94
wh. pd. col., mn., 1.604 wh., amor. delq., rhb., 1.475 mn., 1.4629 col., trig., 1.3439 col. col. col., rhb., 1.4554 gray-pink met.
chloride (scacchite) chloride
MnCl2 MnCl2·4H2O*
125.84 197.91
chloride, perhydroxide (ous) (pyrochroite) hydroxide (ic) (manganite) nitrate oxide (ous) (manganosite) oxide (ic) oxide, di- (pyrolusite; polianite) sulfate (ous) sulfate (ous) (szmikite)
MnCl4 Mn(OH)2 Mn2O3·H2O Mn(NO3)2·6H2O MnO Mn2O3 MnO2*
196.76 88.95 175.88 287.04 70.93 157.86 86.93
rose, delq., cb. rose red, delq., mn. 1.575 gn. wh., trig. brn., rhb., 2.24 rose red, mn. gray-gn., cb., 2.16 brn.-bk., cb. bk., rhb.
MnSO4 MnSO4·H2O
150.99 169.01
red-wh. pa. pink, mn., 1.595
sulfate (ous)
MnSO4·2H2O
187.02
2.52615° 15°
pa. pink, mn. rose, trig., 1.817
Specific gravity
Melting point, °C
2.59822° 2.56 1.60 19.4° 4 2.15 1.788 17.5° 4
expl. 275 1383 −3H2O, 100 265 d. 72 d.
2.66 1.68 7.220° 1.74 20° 4 1.589 3.125
1185 70 d. 1260
2.977 25° 4 2.01
650 58.0
3.25818° 3.258 1.8221° 5.18 4.81 5.026
d. d. 25.8 1650 −0, 1080 −0, >230
3.235 2.87
sulfate (ous)
MnSO4·7H2O
277.10
pink, mn. or rhb.
2.092
sulfate (ic)
Mn2(SO4)3
398.04
gn., delq. cr.
3.24
Ma Hg(C2H3O2)2 HgBr2 HgCO3·2HgO HgCl2 Hg(CNO)2 Hg(OH)2 HgO HgCl2·3HgO HgSiF6·HgO·3H2O HgSO4 HgSO4·2HgO HgC2H3O2 HgBr Hg2CO3
98–99.5 318.70 360.44 693.84 271.52 284.65 234.63 216.61 921.35 613.33 296.67 729.89 259.65 280.53 461.23
11.5 3.270 6.053
2300 (?) d. 237
5.44 4.42
277 expl. −H2O, 175 d. 100 d. 260
MnSO4·3H2O
205.04
sulfate (ous)
MnSO4·4H2O*
223.05
sulfate (ous)
MnSO4·5H2O
241.07
sulfate (ous)
MnSO4·6H2O
259.09
Masurium Mercuric acetate bromide carbonate, basic chloride (corrosive sublimate) fulminate hydroxide oxide (montroydite) oxychloride (kleinite) silicofluoride, basic sulfate sulfate, basic (turpeth) Mercurous acetate bromide carbonate
2-18
2.356 pink, rhb. or mn., 1.518 pink, tri., 1.508
wh. pl. wh., rhb. brn.-red wh., rhb., 1.859 cb. yel. or red, rhb., 2.5 yel., hex. yel. nd. wh., rhb. yel., tet. wh. sc. wh., tet. yel. pd.
2.107 15°
2.103
11.14 7.93 6.47 6.44 7.307
d. d. subl. 345 d. 130
Solubility in 100 parts Cold water
Hot water
Other reagents
i. i. i. 64.519° d. 19.260° 64.817.5° s. 26.90° 72.40° d. s. s. 0.006525°
i. i. sl. s. d. 81.775° s. s. 68.3100° 17840°
63.40° 1518°
123.8100° ∞
s. al., m. al. s. aq. CO2, dil. a.; l. NH3, al. s. al.; i. et., NH3 s. al.; i. et.
129.5
s. 0.00220° i. 4260° i. i. i.
s. i. i. ∞ i. i. i.
s. al., et. s. a., NH4 salts; i. alk. s. h. H2SO4 v. s. al. s. a., NH4Cl s. a.; i. act. s. HCl; i. HNO3, act.
d. 850
530° 98.4748°
7350° 79.77100°
s. al.; i. et.
85.2735°
106.855°
1900
d.
700 Stable 57 to 117 Stable 40 to 57 Stable 30 to 40 Stable 18 to 30 Stable 8 to 18 Stable −5 to +8 Stable −10 to −5; 19 d. d. 160
sulfate (ous)
Boiling point, °C
1190 −H2O, 106; −4H2O, 200
5°
−4H2O, 450
74.22
99.3157°
13616°
16950°
5°
−7H2O, 280
322 304
s. 64.550°
200
2040°
2479°
176
d. HF s. al. s. al. s. dil. a.
i. al.
35°
142
0°
s. a. s. a.; i. alk. s. a.; i. al. d. al.
25114°
v. s.
d.
s. HCl, dil. H2SO4; l. conc. H2SO4, HNO3
2510° 0.520° i. 3.60° sl. s. i. 0.005225° i. d. d. 0.005 0.7513° 7 × 10−9 i.
100100° 25100°
s. al. sl. d. 25.20° al.; v. sl. s. et. s. aq. CO2, NH4Cl 3325° 99% al.; 33 et. s. NH4OH, al. s. a. s. a.; i. al. s. HCl s. a. s. a.; i. al., act., NH8 s. a.; i. al. s. H2SO4, HNO3; i. al. s. a.; i. al., act. s. NH4Cl
61.3100° i. 0.041100° d. 0.167100° d. i. d.
HgCl
236.07
wh., tet., 1.9733
7.150
302
383.7
0.00140°
0.000743°
iodide nitrate Mercurous oxide
HgI HgNO3·H2O Hg2O
327.53 280.63 417.22
yel., tet. wh. mn. bk.
7.70 4.7853.9° 9.8
290 d. 70 d. 100
subl. 140; 310d. expl.
2 × 10−8 v. s. i.
v. sl. s. d. 0.0007
sulfate Mercury† Molybdenum
Hg2SO4 Hg Mo
497.28 200.61 95.95
wh., mn. silv. lq. or hex.(?) gray, cb.
7.56 13.54620° 10.2
d. −38.87 2620 10
0.05516.5° i. i.
0.092100° i. i.
MoCl2
166.85
yel., amor.
3.714 25° 4
d.
i.
i.
3.578 25° 4
d.
chloride (calomel)
chloride, dichloride, tri-
MoCl3
202.32
dark red pd.
chloride, tetra-
MoCl4
237.78
brn., delq.
chloride, pentaoxide, tri- (molybdite) sulfide, di- (molybdenite) sulfide, trisulfide, tetraMolybdic acid Molybdic acid Neodymium Neon Neptunium Nickel
MoCl5 MoO3 MoS2 MoS3 MoS4 H2MoO4 H2MoO4·H2O Nd Ne 239
273.24 143.95 160.07 192.13 224.19 161.97 179.98 144.27 20.18
Np Ni
239 58.69 176.78 291.20 394.99 422.62 218.52 272.57 320.71 841.51 118.70 587.58
carbonyl chloride chloride
Ni(C2H3O2)2 NiCl2·NH4Cl·6H2O NiSO4·(NH4)2SO4· 6H2O Ni(BrO3)2·6H2O NiBr2 NiBr2·3H2O NiBr2·6NH3 NiPtBr6·6H2O NiCO3 2NiCO3·3Ni(OH)2· 4H2O Ni(CO)4 NiCl2 NiCl2·6H2O*
chloride, ammonia cyanide dimethylglyoxime
NiCl2·6NH3 Ni(CN)2·4H2O NiC8H14O4N4
231.80 182.79 288.91
formate hydroxide (ic) hydroxide (ous) nitrate nitrate, ammonia oxide, mono- (bunsenite) potassium cyanide sulfate
Ni(HCO2)2·2H2O Ni(OH)3 Ni(OH)2·d H2O Ni(NO3)2·6H2O Ni(NO3)2·4NH3·2H2O NiO Ni(CN)2·2KCN·H2O NiSO4
184.76 109.71 97.21 290.80 286.87 74.69 258.97 154.75
acetate ammonium chloride ammonium sulfate bromate bromide bromide bromide, ammonia bromoplatinate carbonate carbonate, basic
170.73 129.60 237.70
bk. cr. col., rhb. bk., hex., 4.7 red-brn. brn. pd. yel-wh., hex. yel., mn. yellowish col. gas
2.928 25° 4 19.5°
4.50 4.80114°
3.12415° 6.920° lq. 1.204−245.9° 0.674 (A)
s. lq. O2, al., act., bz.
i.
s. dil. HNO3; sl. s. H2SO4, HCl; i. NH3 i. al.
s.
d.
194
268
s.
795 1185 d. d. d. 115 −H2O, 70 840 −248.67
subl.
−2H2O, 200 −245.9
d. 18°
0.107 i. sl. s. i. v. sl. s. 0.13318° d. 2.60° cc 238
Produced by Neutron bombardment of U 1452 2900 i.
gn. pr. gn., delq., mn. blue-gn., mn., 1.5007 gn., cb. yel., delq. gn., delq. vl. pd. trig. lt. gn., rhb. lt. gn.
1.798 1.645 1.923
d.
2.575 4.64 28° 4
16.6 15025° 2.53.5°
v. s. 39.288°
v. sl. s. (NH4)2SO4
d. d. −3H2O, 200
28 112.80° 1999° v. s.
156100° 316100° d.
s. NH4OH s. al., et., NH4OH s. al., et., NH4OH i. c. NH4OH
d. d.
0.009325° i.
i. d.
s. a. s. a., NH4 salts
0.0189.8° 53.80° 180
i. 87.6100° v. s.
s. aq. reg., HNO3, al., et. s. NH4OH, al.; i. NH3 v. s. al.
s. i. i.
d. i. i.
s. NH4OH; i. al. s. KCN; i. dil. KCl s. abs. al., a.; i. ac., NH4OH
i. v. sl. s. ∞56.7°
s. a., NH4OH, NH4Cl s. a., NH4OH; i. alk. s. NH4OH; i. abs. al. i. al. s. a., NH4OH d. a. i. al., et., act.
1.837 3.715
gn. pl. scarlet red cr.
1.145° cc
d.
d.
8.9020
1.3117° 3.544
2.10679° i. s. i. sl. s. 2.1370°
i.
volt.
silv. met., cb.
lq. yel., delq. gn., delq., mn., 1.57
356.9 3700
s. aq. reg., Hg(NO3)2; sl. s. HNO3, HCl; i. al., etc. s. KI; i. al. s. HNO3; i. al., et. s. h. ac.; i. alk., dil. HCl, NH3 s. H2SO4, HNO3 s. HNO3; i. HCl s. h. conc. H2SO4; i. HCl, HF, NH3, dil. H2SO4, Hg s. HCl, H2SO4, NH4OH, al., et. s. HNO3, H2SO4; v. sl. s. al., et. s. HNO3, H2SO4; sl. s. al., et. s. HNO3, H2SO4; i. abs. al., et. s. a., NH4OH s. H2SO4, aq. reg. s. alk. sulfides s. alk. sulfides; i. NH3 s. NH4OH, H2SO4; i. NH s. a., NH4OH, NH4, salts
−25 subl.
43751mm 973
−4H2O, 200 subl. 250
d.
gn. cr. bk. lt. gn. gn., mn.
2.154 4.36 2.05
d. d. d. 56.7
gn.-bk., cb., 2.37 red yel., mn. yel., cb.
7.45 1.87511° 3.68
Forms Ni2O3 at 400 −H2O, 100 −SO3, 840
136.7
s. i. v. sl. s. 243.00° v. s. i. s. 27.20°
i. 76.7100°
*Usual commercial form. †See also Tables 2-28 and 2-280. 2-19
Physical Properties of the Elements and Inorganic Compounds (Continued )
TABLE 2-1
Name Nickel (Cont.) sulfate
Formula NiSO4·6H2O*
sulfate (morenosite) Nitric acid Nitric acid Nitric acid Nitro acid sulfite Nitrogen
NiSO4·7H2O HNO3 HNO3·H2O HNO3·3H2O NO2HSO3 N2
Nitrogen oxide, mono- (ous)
N2O
oxide, di- (ic)
NO or (NO)2
oxide, tri-
N2O3
oxide, tetra- (per- or di-)
NO2 or (NO2)2
Formula weight
Color, crystalline form and refractive index
262.85
gn. mn. or blue, tet., 1.5109 gn., rhb., 1.4893 col. lq. col. lq. col. lq. col., rhb. col. gas or cb. cr.
280.86 63.02 81.03 117.06 127.08 28.02 44.02
col. gas
30.01 (60.02) 76.02
col. gas red-brn. gas or blue lq. or solid yel. lq., col. solid, red-brn. gas wh., rhb.
oxide, penta-
N2O5
46.01 (92.02) 108.02
oxybromide oxychloride
NOBr NOCl
109.92 65.47
brn. lq. red-yel. lq. or gas
Nitroxyl chloride Osmium chloride, dichloride, trichloride, tetraOxygen
NO2Cl Os OsCl2 OsCl3 OsCl4 O2
81.47 190.2 261.11 296.57 332.03 32.00
yel.-brn. gas blue, hex. gn., delq. brn., cb. red-yel. nd. col. gas or hex. solid
Ozone
O3
Palladium bromide (ous) chloride chloride cyanide hydride Palladous dichlorodiammine Perchloric acid Perchloric acid Perchloric acid Periodic acid Periodic acid Permanganic acid Permolybdic acid Persulfuric acid Phosphamic acid Phosphatomolybdic acid Phosphine Phosphonium chloride 2-20
48.00
col. gas
Pd
106.70
silv. met., cb.
PdBr2 PdCl2 PdCl2·2H2O Pd(CN)2
266.53 177.61 213.65 158.74
brn. brn., cb. brn. pr. yel.
Pd2H Pd(NH3)2Cl2 HClO4 HClO4·H2O HClO4·2H2O* 73.6% anh. HIO4 HIO4·2H2O HMnO4 HMoO4·2H2O H2S2O8 PONH2·(OH)2 H7P(Mo2O7)6·28H2O PH3
214.41 211.68 100.46 118.48 136.50
met. red or yel., tet. unstable, col. lq fairly stable nd. stable lq., col.
PH4Cl
191.93 227.96 119.94 196.99 194.14 97.02 2365.88 34.00 70.47
wh. cr. delq., mn. exists only in solution wh. cr. hyg. cr. cb. yel. cb. col. gas wh., cb.
Specific gravity 2.07 1.948 1.502
Melting point, °C
1.448
18°
Solubility in 100 parts Cold water
tr. 53.3
−6H2O, 280
13150°
98–100 −42 −38 −18.5 73 d. −209.86
−6H2O, 103 86
0°
Hot water 280100° 30°
117.8 ∞ ∞ ∞
−195.8
63.5 ∞ ∞ 263−20° d. 2.350° cc
−90.7
130.520° cc
−151
7.340° cc
60.8224° cc 0.0100°
3.5
s.
−9.3
21.3
d.
1.026−252.5° 0.808−195.8° 12.50° (D) lq. 1.226−89° −102.3 1.530 (A) lq. 1.269−150.2° −161 1.0367 (A) 1.4472° −102 20°
Boiling point, °C
1.5520° cc
Other reagents v. s. NH4OH, al. s. al. expl. with al. d. al. d. al. s. H2SO4 sl. s. al. s. H2SO4, al. 26.6 cc al.; 3.5 cc H2SO4; s. aq. FeSO4 s. a., et. s. HNO3, H2SO4, chl., CS2
1.63
30
47
s.
>1.0 1.417−12° 2.31 (A) lq. 1.3214° 22.4820°
−55.5 −64.5
−2 −5.5
d. d.
<−30 2700
5 >5300
1.14−188° 1.426−252.5° 1.1053 (A) 1.71−183° 3.03−80° 1.658 (A) 12.020° 111550°
−218.4
−183
d i. s. d. sl. s. s. d. 4.890° cc
−251
−112
0.4940° cc
060° cc
s. oil turp., oil cinn.
1555
2200
i.
i.
i. s. s. i.
i. s. s. i.
s. aq. reg., h. H2SO4; i. NH3 s. HBr s. HCl, act., al. s. HCl, act., al. s. HCN, KCN, NH4OH; i. dil. a.
d. 560–600
500 d. d. 11.06 2.5 1.768 22° 4 1.88 1.71 25° 4
Forms HNO3 s. fuming H2SO4 i.
2.630° cc 1.7100° cc
sl. s. aq. reg., HNO3; i. NH3 s. NaCl, al., et. s. a., alk., al.; sl. s. et. s. HCl, al. sl. s. al., s. fused Ag
d.
−90°
lq. 0.746 1.146 (A)
−112 50 −17.8
1618mm d. 200
d. 138 d. 110
subl. 110
<60 d. 78 −132.5 46atm
28
s. s. s. v. s.
−25H2O, 140 −85
s. v. s. v. s. v. s. v. s. v. s. s. 2617° cc
subl.
d.
s. a., NH4OH s. al. v. s. d. v. s. d. v. s. d. 100°
i.
sl. s. al., et. d. al. i. al. s. HNO8 s. Cu2Cl2, al., et.
Phosphoric acid, hypoPhosphoric acid, meta-
H4P2O6 HPO3
161.99 79.99
cr. vitreous, delq.
2.2–2.5
55 subl.
d. 70
s. s.
Phosphoric acid, orthoPhosphoric acid, pyro-
H3PO4† H4P2O7
98.00 177.99
col., rhb. wh. nd.
1.834
42.35 61
−a H2O, 213
2340 80028°
Phosphorous acid, hypoPhosphorous acid, orthoPhosphorous acid, pyroPhosphorus, black Phosphorus, red Phosphorus, yellow
H3PO2 H3PO3 H4P2O6 P4 P4 P4
66.00 82.00 145.99 123.92 123.92 123.92
syrupy col. nd. rhombohedral red, cb. yel., hex., 2.1168
1.49318.8° 1.65121.2°
26.5 74 38
d. d. 200 d. 130 ign. in air, 400 ign. in air, 725 280
∞ 307.30° d. i. i. 0.0003
PCl3 PCl5
137.35 208.27
col., fuming lq. delq., tet.
−111.8 148 under pressure subl. 250 2
75.95760mm subl. 160
d. d.
21.4520° lq. 191755°
1755
4300
chloride, trichloride, pentaoxide, pentaoxychloride Phosphotungstic acid Platinum
P2O5 POCl2 P2O5·2WO3·42H2O Pt
141.96 153.35 3681.67 195.23
wh., delq., amor. col., fuming lq. yel.-gn. cr. silv. met., cb.
18.2°
2.69 2.2020° 1.8220°; lq. 1.74544.5° 1.574 20.8° 4 solid 1.6; 3.60295° (A) 2.387 1.675
59043atm 44.1; ign. 34
26°
Forms H3PO4 d. s. i.
107.2760mm
25°
45062° Forms H3PO4 v. s. Forms H3PO4 ∞ 73040°
i. CS2 s. alk.; i. CS2, NH3, et. 0.4 al.; 100010° CS2; 1.50°, 1081° bs.; s. NH3 s. et., chl., CS2 s. CS2, C6H5COCl
v. s.
s. H2SO4; i. NH3, act. d. al. s. al., et. s. aq. reg., fused alk.
i.
PtCl4
337.06
brn.
d. 370
140
v. s.
chloride (ous)
PtCl2
266.14
brn.
5.8711°
d. 581
i.
i.
PtCl4·8H2O Pt(CN)2 Pu Pu K
481.19 247.27 238 239 39.10
red, mn. yel.-brn.
2.43
−4H2O, 100
v. s. i.
v. s. i.
d.
KC2H3O2 KH(C2H3O2)2 K2(AlO2)2·3H2O KNH2 KH2AsO4 KAu(CN)4·1·5H2O KAu(CN)2 KHCO3 KHSO4 KBrO3 KBr K2CO3 K2CO3·2H2O 2K2CO3·3H3O KClO3 KCl K2PtCl6 K2CrO4 KCNO KCN K2Cr2O7 K3Fe(CN)6 K4Fe(CN)6·3H2O KHCO2 KH KHS KOH KIO3 KI
98.14 158.19 250.18 55.12 180.02 367.39 288.33 100.11 136.16 167.01 119.01 138.20 174.23 330.45 122.56 74.56 486.16 194.20 81.11 65.11 294.21 329.25 422.39 84.11 40.10 72.16 56.10 214.02 166.02
wh. pd. delq. nd. or pl. cr. yel.-grn. col., tet., 1.5674 pl. rhb. mn., 1.482 rhb., or mn., 1.480 trig. col., cb., 1.5594 wh., delq. pd., 1.531 rhb. mn. col., mn., 1.5167 col., cb., 1.4904 yel., cb., 1.825 yel., rhb., 1.7261 wh., tet. wh., cb., delq., 1.410 red, tri. red, mn. pr., 1.5689 yel., mn., 1.5772 col., rhb. cb., 1.453 wh., delq., rhb. wh., delq., rhb. col., mn. wh., cb., 1.6670
Forms KOH 39690°
acetate acetate, acid aluminate amide arsenate (monobasic) auricyanide aurocyanide bicarbonate bisulfate bromate bromide carbonate carbonate carbonate chlorate chloride (sylvite) chloroplatinate chromate (tarapacaite) cyanate cyanide dichromate ferricyanide ferrocyanide formate hydride hydrosulfide hydroxide iodate iodide
silv. met., cb.
238
Produced by deuteron bombardment on U Produced by neutron bombardment on U238 0.8620° 62.3 760 lq. 0.8342° 1.8 292 148 d. 200 2.867 2.17 2.35 3.2717.5° 2.7525° 2.29 2.043 2.13 2.32 1.988 3.499 2.73218° 2.048 1.5216° 2.69 1.84 1.85317° 1.91 0.80 2.0 2.044 3.89 3.13
338 288 d. 200 d. 100–200 210 370 d. 730 891 368 790 d. 250 975 634.5 398 d. −3HO2, 70 167.5 d. 455 380 560 723
subl. 400
d. 1380 d. d. 400 1500
d. d. 1320 1330
2170° d. s. d. 18.876° s. 14.3 22.40° 36.30° 3.110° 53.50° 105.50° 1830° 129.40° 3.30° 27.60° 0.740° 58.00° s. s. 4.90° 334.4° 27.812.2° 33118° d. s. 970° 4.730° 127.50°
s. al. v. s. al., et.
i. i. sl. s.
chloride (ic)
chloride (ic) cyanide (ous) Plutonium Plutonium Potassium
i. lq. CO2
d. v. s. v. s. 200100° 6060° 121.6100° 49.75100° 104100° 156100° 331100° 268100° 57100° 56.7100° 5.2100° 75.6100° d. 122.2108.8° 80100° 77.5100° 90.696.8° 65790° s. d. 178100° 32.2100° 208100°
s. al., act.; sl. s. NH2; i. et. s. HCl, NH4OH; sl. s. NH3; i. al., et. s. al., et. i. alk. s. a., al., Hg 33 al.; i. et. s. ac. s. alk.; i al. d. al.; 3.625° NH3 i. al. s. al. sl. s. al.; i. et. i. satd. K2CO3, al. d. al. sl. s. al.; i. act. sl. s. al., et. i. al. 0.83 al.; s. alk. s. al., alk. i. al., et. i. al. v. sl. s. al. s. gly.; 0.919.5° al.; 1.3 h. al. i. al. s. act.; sl. s. al.; i. NH3 s. act.; i. NH3, al., et. sl. s. al.; i. et. i. et., bz., CS2 s. al. v. s. al., et.; i. NH3 s. KI; i. al., NH3 420° al.; s. NH3; sl. s. et.
*One commercial form 70 to 72 per cent. †Common commercial form 85 per cent H3PO4 in aqueous solution. 2-21
TABLE 2-1
Physical Properties of the Elements and Inorganic Compounds (Continued )
Formula
Formula weight
Color, crystalline form and refractive index
Potassium (Cont.) iodide, triiodoplatinate manganate metabisulfite nitrate (saltpeter) nitrite oxalate oxalate, acid oxalate, acid oxide perchlorate permanganate persulfate phosphate, monobasic
KI3 K2PtI6 K2MnO4 K2S2O5 KNO3 KNO2 K2C2O4·H2O KHC2O4* KHC2O4·a H2O K2O KClO4 KMnO4 K2S2O3 KH2PO4
419.86 1034.94 197.12 222.31 101.10 85.10 184.23 128.12 137.13 94.19 138.55 158.03 270.31 136.09
phosphate, dibasic phosphate, tribasic phosphate, metaphosphate, metaphosphate, pyrophthalate, acid platinocyanide silicate silicate, tetrasulfate (arcanite) Potassium sulfate, pyrosulfide, monosulfite sulfite, acid tartrate tartrate, acid thiocyanate
K2HPO4 K3PO4 KPO3 K4P4O12·2H2O K4P2O7·3H2O KHC8H4O4 K2Pt(CN)4·3H2O K2SiO3 K2Si4O9·H2O K2SO4 K2S2O7 K2S·5H2O K2SO3·2H2O KHSO3 K2C4H4O6·a H2O KHC4H4O6* KCNS
174.18 212.27 118.08 508.34 384.39 204.22 431.54 154.25 352.45 174.25 254.31 200.33 194.28 120.16 235.27 188.18 97.17
thiosulfate thiosulfate Praseodymium Radium bromide Radon (Niton)
K2S2O3 3K2S2O3·H2O Pr Ra RaBr2 Rn
190.31 588.95 140.92 226.05 385.88 222.0
dark blue, delq., mn. cb. gn., rhb. mn., pl. col., rhb., 1.5038 pr. wh., mn. mn., 1.545 trimetric wh., cb. col., rhb., 1.4737 purple, rhb. wh., tri., 1.4669 col., delq., tet., 1.5095 wh., delq. wh., rhb. wh. pd. amor. delq. wh. cr. yel., rhb., 1.62 hyg. 1.521 rhb., 1.530 col., rhb., 1.4947 col. rhb., delq. wh., rhb. wh., mn. col., mn., 1.526 col., rhb. col., delq., mn., 1.660 col., cb. delq., mn. yel. wh., met. wh., mn. gas
Rhenium
Re
186.31
hex.
Rhodium chloride chloride Rubidium
Rh RhCl3 RhCl3·4H2O Rb
102.91 209.28 281.35 85.48
gray-wh., cb. red dark red silv. wh.
Ruthenium Ruthenium Samarium Scandium Selenic acid Selenic acid Selenium Selenium
Ru Ru Sm (also Sa) Sc H2SeO4 H2SeO4·H2O Se8 Se8
101.70 101.70 150.43 45.10 144.98 162.99 631.68 631.68
bk., porous gray, hex.
Name
2-22
hex. pr. nd. red pd., amor., 2.92 gray, trig., 3.00; red, hex.
Specific gravity 3.498 5.18 2.1110.6° 1.915 2.13 2.0 2.32 20° 4 2.524 11° 4 2.703 2.338 2.56417° 2.25814.5° 2.26414.5° 2.33 1.63 2.4516° 2.417 2.662 2.277
1.98 1.956 1.886 2.23 6.520° 5? 5.79 lq. 5.5; 111 (D)
Melting point, °C 45 d. 190 d. 150 tr. 129; 333 297 d. d. d.
Boiling point, °C d. 225
d. 400 d. 350
d. 400 d. <240 d. <100 256 d. 1340 tr. 450; 798 −2H2O, 100 −2H2O, 180 d. 976 d. 400 tr. 588 300 60 d. d. 190
1320 d. −3H2O, 300
−3H2O, 150 d.
172.3 d. 400 −H2O, 180 940 960 728 −71
d. 500
Solubility in 100 parts Cold water v. s. s. d. 250° 13.30° 2810° 28.70° 14.350° 2.20° Forms KOH 0.750° 2.830° 1.770° 14.80°
Hot water
s. KI, al. 12094° 246100° 413100° 83.2100° 48.1100° 51.5100° v. s. 21.8100° 32.3575° 1040° 83.590°
3325° 193.125° s. s. s. 10.225° sl. s. s. s. 7.350° s. s. 100 45.515° 12.517.5° 0.370° 1770°
v. s. v. s. s. 83 v. s. 36 v. s. s. s. 24.1100° d.
96.10°
311.290°
>100 91.575° 278100° 6.1100° 21720°
d. 1140 subl. 900 −62
1955 d. 450
>2500 subl. 800
lq. 1.47588.5; 1.5320° 8.6 12.220° 7.7 2.5? 2.950 15° 4 2.627 15° 4 4.2625° 4.80; 4.50
38.5 >1950 2450 >1300 1200 58 26 50 220
s. KOH sl. s. al.; i. et. 0.130° al.; i. et. v. s. NH3; sl. s. al.
s. al., et. 0.10520° m. al.; i. et. s. H2SO4; d. al. i. al. i. al. sl. s. al. i. al. s. a. i. al. s. al., et. i. al. i. al. i. al., act., CS2 s. al., gly.; i. et. sl. s. al.; i. NH3 i. abs. al. sl. s. al. s. a., alk.; i. al., ac. 20.822° act.; s. al. i. al.
d. d. +H2 7020° 510° cc
s. 8.560° cc
3440 12.5
Other reagents
d. a. s. al. i. HF, HCl; s. H2SO4; HNO3 sl. s. aq. reg., a. v. sl. s. alk.; i. aq. reg., a. s. HCl, al.; i. et. s. a., al.
i. i.
700
i. i. v. s. d.
>2700
i. i.
i. i.
sl. s. aq. reg., a.
2400 260 205 688 688
130030° v. s. i. i.
∞60°
s. H2SO4; d. al.; i. NH3
i. i.
s. CS2, H2SO4, CH2I2 s. CS2, H2SO4
Selenium Selenous acid Silicic acid, metaSilicic acid, orthoSilicon, crystalline
Se8 H2SeO3 H2SiO3 H4SiO4 Si
631.68 128.98 78.08 96.09 28.06
4.825° 3.004 15° 4 2.1–2.3 1.57617° 2.420°
cr. brn., amor. blue-bk., trig., 2.654 lf. or lq. col., fuming lq., 1.412 gas col. gas iridescent, amor. col., cb. or tet., 1.487
217 d.
688
2600
i. 900° i. sl. s. i.
i. 40090° i. sl. s. i.
1420
2.0–2.5
2600
i.
i.
2 3.17 1.580° 1.50
>2700 −1 −70
2600 subl. 2200 144760mm 57.6
i. i. d. d.
i. i.
3.57 (A) lq. 0.68−185° 2.2 2.32
−95.7 −185 1600–1750 1710
−651810mm −112760mm subl. 1750 2230
v. s. d. i. i. i.
Silicon, graphitic
Si
Silicon, amorphous carbide chloride, trichloride, tetra-
Si SiC Si2Cl6 SiCl4
28.06 40.07 268.86 169.89
SiF4 SiH4 SiO2·xH2O SiO2
104.06 32.09
SiO2 SiO2 SiO2 Ag AgBr
60.06 60.06 60.06 107.88 187.80
hex., 1.5442 trig., rhb., 1.469 silv. met., cb. pa. yel., cb., 2.252
2.20 2.65020° 2.26 10.520° 6.473 25° 4
tr. <1425 tr. 1670 960.5 434
2230 2230 2230 1950 d. 700
i. i. i. i. 0.0000220°
i. 0.00037100°
carbonate chloride (cerargyrite)
Ag2CO3 AgCl
275.77 143.34
yel. pd. wh., cb., 2.071
6.077 5.56
218 d. 455
1550
0.00320° 0.00008910°
0.05100° 0.00217100°
cyanide nitrate (lunar caustic) Sodium
AgCN AgNO3 Na
133.90 169.89 22.997
wh., 1.685 col., rhb., 1.744 silv. met, cb.
3.95 4.352 19° 4 0.9720°
−(CN)2, 320 212 97.5
444 d. 880
acetate acetate aluminate amide Sodium ammonium phosphate antimonate, metaarsenate arsenate, acid (monobasic) arsenate, acid (dibasic) arsenate, acid (dibasic) arsenite, acid benzoate bicarbonate bifluoride bisulfate bisulfite borate, tetraborate, tetra
NaC2H3O2 NaC2H3O2·3H2O NaAlO2 NaNH2 NaNH4HPO4·4H2O 2NaSbO3·7H2O Na3AsO4·12H2O NaH2AsO4·H2O Na2HAsO4·7H2O* Na2HAsO4·12H2O Na2HAs03 NaC7H5O2 NaHCO3 NaHF2 NaHSO4 NaHSO3 Na2B4O7 Na2B4O7·5H2O
82.04 136.09 81.97 39.02 209.09 511.63 424.09 181.94 312.02 402.10 169.91 144.11 84.01 62.00 120.06 104.06 201.27 291.35
wh., mn., 1.464 wh., mn. amor. olive gn. col., mn. cb. hex., 1.4589 rhb., 1.5535 col., mn., 1.4658 mn., 1.4496 col. col. cr. wh., mn., 1.500 col. cr. col., tri. col., mn., 1.526
1.528 1.45
324 58 1650 210 79 d.
col., rhb., 1.461
2.742 1.48 2.367 1.815
Na2B4O7·10H2O*
381.43
wh., mn., 1.4694
1.73
fluoride hydride (silane) oxide, di- (opal) oxide, di- (cristobalite) oxide, di- (lechatelierite) oxide, di- (quartz) oxide, di- (tridymite) Silver bromide (bromyrite)
borate, tetra- (borax)
28.06
steel gray hex. amor., 1.41 amor. gray, cb., 3.736
60.06
1.574
−3H2O, 120 400
1.759 2.535 1.871 1.72 1.87
86.3 d. 100 125 28
2.20
−CO2, 270 d. >315 d. 741
d., −H2O
75
−10H2O, 200
17.5°
bromate bromide bromide
NaBrO3 NaBr NaBr·2H2O
150.91 102.91 138.95
col., cb. col., cb., 1.6412 col., mn.
3.339 3.20517.5° 2.176
381 755 50.7
carbonate (soda ash) carbonate
Na2CO3 Na2CO3·H2O
106.00 124.02
2.533 1.55
851 −H2O, 100
carbonate carbonate (sal soda)
Na2CO3·7H2O Na2CO3·10H2O
232.12 286.16
wh. pd., 1.535 wh., rhb., 1.506– 1.509 rhb. or trig. wh., mn., 1.425
1.51 1.46
d. 35.1
−7H2O, 100 −12H2O, 100
0.00002220° 1220° d., forms NaOH 46.520° v. s. s. d. 16.7 0.03112.8° 26.717° s. 6115° 5.590.1° v. s. 62.525° 6.90° 3.720° 500° sl. s. 1.30° 2262° (anh.)
i. i. i. i.
952100° 170100° v. s. v. s. 100
s. HF; i. alk. s. HF; i. alk. s. HF; i. alk. s. HNO3, h. H2SO4; i. alk. 0.5118° NH4OH; s. KCN, Na2S2O3 s. NH4OH, Na2S2O3; i. al. s. NH4OH, KCN; sl. s. HCl s. NH4OH, KCN, HNO3 s. gly.; v. sl. s. al. i. bz.; d. al. 2.118° al. 7.825° abs. al. i. al. d. al. i. al. sl. s. al., NH4 salts; i. ac. 1.67 al., 5015° gly. sl. s. al. sl. s. al. 2.325°, 8.378° al. i. al.
1390
27.5 9020° 79.50° (anh.)
d.
7.10° s. s. 21.50°
s. 23830°
0°
s. HNO3, al., et. i. al., et.; d. KOH s. HF, h. alk., fused CaCl2 s. HF; i. alk.
v. s. 140.730° 76.9100° 16.460° s. 100100° s. 8.7940° 52.3100° (anh.) 20.380° (anh.) 90.9100° 121100° 118.380° (anh.) 48.5104° s.
1.30.5 (anh).
i. CS2; s. H2SO4 v. s. al.; i. NH3 s. alk.; i. NH4Cl s. alk.; i. NH4Cl s. HNO3 + HF, Ag; sl. s. Pb, Zn; i. HF s. HNO3 + HF, fused alk.; i. HF. s. HF, KOH s. fused alk.; i. a. d. alk. d. conc. H2SO4, al.
d. al.; i. NH3 i. al., act. i. al. s. gly.; i. abs. al. i. al. sl. s. al. sl. s. al. i. al., et. s. gly.; i. al., et. i. al.
*Usual commercial form. 2-23
Physical Properties of the Elements and Inorganic Compounds (Continued )
TABLE 2-1
Name Sodium ammonium phosphate (Cont.) carbonate, sesqui- (trona) chlorate
Formula
Na3H(CO3)2·2H2O NaClO3
Formula weight
Specific gravity
NaCl Na2CrO4 Na2CrO4·10H2O 2Na3C6H5O7·11H2O NaCN Na2Cr2O7·2H2O
58.45 162.00 342.16 714.36 49.02 298.05
ferricyanide ferrocyanide
Na3Fe(CN)6·H2O Na4Fe(CN)6·10H2O
298.97 484.11
red, delq. yel., mn.
1.458
fluoride (villiaumite) formate hydride
NaF NaHCO2 NaH
tet., 1.3258 wh., mn. silv. nd., 1.470
2.79 1.919 0.92
992 253 d. 800
hydrosulfide hydrosulfide hydrosulfite hydroxide hydroxide hypochlorite iodide iodide lactate nitrate (soda niter) nitrite
NaSH·2H2O NaSH·3H2O Na2S2O4·2H2O NaOH NaOH·3a H2O NaOCl NaI* NaI·2H2O NaC3H5O3 NaNO3 NaNO2
2.130
d. 22 d. 318.4 15.5 d. 651
1300
2.257 2.1680°
d. 308 271
d. 380 d. 320
oxide
Na2O
2.27
subl.
2-24
NaBO3·H2O NaClO4 NaClO4·H2O Na2O2* Na2O2·8H2O NaH2PO4·H2O* NaH2PO4·2H2O Na2HPO4·7H2O Na2HPO4·12H2O Na3PO4 Na3PO4·12H2O* Na4P4O12 Na4P2O7* Na4P2O7·10H2O Na2H2P2O7 Na2H2P2O7·6H2O NaKC4H4O6·4H2O Na2SiO3 Na2SiO3·9H2O Na4SiO4 Na2SiF6 Na2SnO3·3H2O Na2SO4 Na2SO4
92.10 110.11 210.15 40.00 103.06 74.45 149.92 185.95 112.07 85.01 69.01 61.99 99.83 122.45 140.47 77.99 222.12 138.01 156.03 268.09 358.17 163.97 380.16 407.91 265.95 446.11 221.97 330.07 282.23 122.05 284.20 184.05 188.05 266.74 142.05 142.05
col., delq., nd. rhb. col. cr. wh., delq. mn. pa. yel., in soln. only col., cb., 1.7745 col., mn. col., amor. col., trig., 1.5874 pa. yel., rhb. wh., delq. wh. pd. rhb., 1.4617 hex. yel.-wh. pd. wh., hex. col., rhb., 1.4852 col., rhb., 1.4629 col., mn., 1.4424 col., mn., 1.4361 wh. wh., trig., 1.4458 col. wh. mn., 1.4525 col., mn., 1.510 col., mn., 1.4645 rhb., 1.493 col., rhb., 1.520 rhb. col., hex., 1.530 wh., hex., 1.312 hex. tablets col., rhb., 1.477 col., mn.
2.163 2.723 1.483 1.857 23.5° 4 18°
2.52
3.6670° 2.448
2.02 2.805 2.040 1.91 1.679 1.52 2.53717.5° 1.62 2.476 2.45 1.82 1.862 1.848 1.790
2.679 2.698
d. 248
Boiling point, °C
chloride chromate chromate citrate cyanide dichromate
42.00 68.01 24.005
2.112 2.49015°
Melting point, °C
wh., mn., 1.5073 wh., cb., or trig., 1.5151 col., cb., 1.5443 yel., rhb. yel., delq., mn. wh., rhb. wh., cb., 1.452 red, mn., 1.6994
perborate perchlorate perchlorate peroxide peroxide phosphate, monobasic phosphate, monobasic phosphate, dibasic phosphate, dibasic phosphate, tribasic phosphate, tribasic phosphate, metaphosphate, pyrophosphate, pyrophosphate (pyrodisodium) phosphate (pyrodisodium) potassium tartrate silicate, metaSodium silicate, metasilicate, orthosilicofluoride stannate sulfate (thenardite) sulfate
226.05 106.45
Color, crystalline form and refractive index
800.4 392 19.9 −11H2O, 150 563.7 −2H2O, 84.6; 356 (anh.)
d. 40 482 d. d. 130 d. d. 30 −H2O, 100 60 d. 34.6 1340 73.4 616 d. 988 d. d. 220 70 to 80 1088 47 1018 d. d. 140 tr. 100 to mn. tr. 500 to hex.
d. 1413 d. 1496 d. 400
Solubility in 100 parts Cold water
130° 790°
42100° 230100° 0°
1390
d. 200 −12H2O, 180 −11H2O, 100
−4H2O, 215 −6H2O, 100
100°
35.7 320° v. s. 9125° 4810° 2380°
39.8 126100° ∞ 250100° 8235° 50880°
18.90° 17.920° (anh.)
67100° 6398.5° (anh.) 5100° 160100°
40° 440° d. d.
Hot water
s. s. 2220° 420° s. 260° 158.70° v. s. v. s. 730° 72.10°
s. s. d. 347100° v. s. 15856° 302100° v. s. v. s. 180100° 163.2100°
Forms NaOH sl. s. 1700° 20915° s. d. s. d. 710° 91.10° 18540° 4.30° 4.50° 28.315° s. 2.260° 5.40° 4.50° 6.90° 260° s. v. s. s. 0.440° 500° 50° 48.840°
d. 320100° 28450° d. d. 39083° 30840° 2000100° 76.730° 77100° ∞ s. 4596° 93100° 2140° 3640° 6626° s. d. v. s. s. 2.45100° 6750° 42100° 42.5100°
Other reagents
s. al. sl. s. al.; i. conc. HCl sl. s. al. i. al. s. NH3; sl. s. al. i. al. i. al. v. sl. s. al. sl. s. al.; i. et. i. bz., CS2, CCl4, NH3; s. molten metal s. al.; d. a. s. al.; d. a. i. al. v. s. al., et., gly.; i. act. v. s. al., act. v. s. NH3 s. al.; i. et. s. NH3; sl. s. gly., al. 0.320° et.; 0.3 abs. al.; 4.420° m. al.; v. s. NH3 d. al. s. gly., alk. s. al.; 51 m. al.; 52 act.; i. et. s. al. s. dil. a. i. al. i. al. i. CS2 s. a., alk. d. a. i. al., NH3 sl. s. al. i. Na or K salts, al. 2918°, a N NaOH i. al. i. al., act. i. al. d. HI; s. H2SO4
sulfate sulfate sulfate (Glauber’s salt) sulfide, monosulfide, tetrasulfide, pentasulfite sulfite tartrate thiocyanate thiosulfate thiosulfate (hypo) tungstate tungstate tungstate, parauranate vanadate vanadate, pyroStannic chloride
Na2SO4 Na2SO4.7H2O Na2SO4.10H2O Na2S Na2S4 Na2S5 Na2SO3 Na2SO3.7H2O Na2C4H4O6.2H2O NaCNS Na2S2O3 Na2S2O3.5H2O* Na2WO4 Na2WO4.2H2O* Na6W7O24.16H2O Na2UO4 Na3VO4.16H2O Na4V2O7 SnCl4
142.05 268.17 322.21 78.05 174.23 206.29 126.05 252.17 230.10 81.08 158.11 248.19 293.91 329.95 2097.68 348.06 472.20 305.89 260.53
col., hex. tet. col., mn., 1.396 pink or wh., amor. yel., cb. yel. hex. pr., 1.565 mn. rhb. delq., rhb., 1.625 mn. mn. pr., 1.5079 wh., rhb. wh., rhb. wh., tri. yel. col. nd. hex. col., fuming lq.
884 1.464 1.856 2.633 15° 4 1.561 1.818
2.226
866 (anh.) 654 −30.2
7.0
1127
SnO2
150.70
wh., tet., 1.9968
346.85
col., delq., hex.
SnBr2 SnCl2 SnCl2.2H2O* SnSO4 Sr
278.53 189.61 225.65 214.76 87.63
yel., rhb. wh., rhb. wh., tri. wh. cr. silv. met.
2.71
acetate carbonate (strontianite) chloride chloride hydroxide hydroxide
Sr(C2H3O2)2 SrCO3 SrCl2 SrCl2.6H2O* Sr(OH)2 Sr(OH)2.8H2O*
205.72 147.64 158.54 266.64 121.65 265.77
wh. cr. wh., rhb., 1.664 wh., cb., 1.6499 wh., rhb., 1.5364 wh., delq. col., tet., 1.499
2.099 3.70 3.052 1.93317° 3.625 1.90
nitrate nitrate oxide (strontia)
Sr(NO3)2* Sr(NO3)2.4H2O SrO
211.65 283.71 103.63
col., cb., 1.5878 wh., mn. col., cb., 1.870
2.986 2.2 4.7
SrO2 SrO2.8H2O SrSO4 Sr(HSO4)2 NH2SO3H S S8 S8 S2Br2 S2Cl2 SCl2 SCl4 SO2
119.63 263.76 183.69 281.77 97.09 32.06 256.48 256.48 223.95 135.03 102.97 173.89 64.06
wh. pd. wh. cr. col., rhb., 1.6237 col., granular wh., rhb. pa. yel. pd., 2.0–2.9 pa. yel., mn. pa. yel., rhb. red, fuming lq. red-yel. lq. dark red fuming lq. yel.-brn. lq. col. gas
oxide, tri-(β) Sulfuric acid Sulfuric acid
SO3 (SO3)2 H2SO4* H2SO4.H2O
80.06 160.12 98.08 116.09
col. pr. col., silky, nd. col., viscous lq. pr. or lq.
d.
d. 48.0 692 −2H2O, 100 −16H2O, 300
Sn(SO4)2.2H2O
oxide, tri-(α)
275 251.8 d. −7H2O, 150 287
sulfate
peroxide peroxide sulfate (celestite) sulfate, acid Sulfamic acid Sulfur, amorphous Sulfur, monoclinic Sulfur, rhombic Sulfur bromide, monochloride, monochloride, dichloride, tetraoxide, di-
−10H2O, 100
1.667 1.685 4.179 3.245 3.98714°
oxide (cassiterite)
Stannous bromide chloride chloride (tin salt) sulfate Strontium
32.4
5.1217° 15.5°
2.6
3.96 2.03 12° 4 2.046 1.96 2.07 2.635 1.687 1.621 15° 15 lq., 1.4340°; 2.264 (A) lq., 1.923; 2.75 (A) 1.9720° 1.834 18° 4 1.842 15° 4
215.5 246.8 37.7 −SO2, 360 800 149760atm 873 −4H2O, 61 375 −7H2O in dry air 570
114.1
620 623 d. 1150 d. −CO2, 1350 −6H2O, 100
19.420° 44.90° 3615° 15.410° s. s. 13.90° 34.72° 296° 11010° 500° 74.70° 57.580° 880° 8 i. v. s. s. s. i.
i. d.
s. 83.90° 118.70° 1919° d.
d. 269.815° ∞ 18100° Forms Sr(OH)2 36.497° 0.065100° 100.8100° 19840° 21.83100° 47.7100°
36.90° 0.001118° 43.50° 1040° 0.410° 0.900°
444.6 444.6 444.6 540.18mm 138 59 d. > −20 −10.0
16.83
44.6
d.
50 10.49 8.62
d. 340 290
Forms H2SO4 ∞ ∞
d. −8H2O, 100 1580 d. d. 205 d. 120 119.0 112.8 −46 −80 −78 −30 −75.5
d.
d.
v. s.
400° 62.20° Forms Sr(OH)2 0.00820° 0.01820° 0.01130° d. 200° i. i. i. d. d. d. d. 22.80°
2430
45.360° 202.626° 41234° 57.390° s. s. 28.384° 67.818° 6643° 225100° 23180° 301.860° 97100° 123.5100° d. i. d.
i. al. sl. s. al.; i. et. s. al. s. al. i. al., NH i. al. i. al. v. s. al. s. NH3; v. sl. s. al. sl. s. NH3; i. a., al. s. alk. carb., dil. a. i. al. i. al. s. abs. al., act., NH3; s. ∞ CS2 s. conc. H2SO4; i. alk.; NH4OH, NH3 s. dil. H2SO4, HCl; d. abs. al. s. C6H5N s. alk., abs. al., et. s. tart. a., alk., al. s. H2SO4 s. al., a. 0.2615° m. al. s. a., NH4 salts, aq. CO2 v. sl. s. act., abs. al.; i. NH3 s. NH4Cl s. NH4Cl; i. act.
10089° 12420°
s. NH3; 0.012 abs. al. i. HNO3 sl. s. al.; i. et.
d. d. 0.011432°
s. al., NH4Cl; i. act. s. al.; i. NH4OH sl. s. a.; i. dil. H2SO4, al. 1470° H2SO4 sl. s. al., act.; i. et. sl. s. CS2 s. CS2, al. 240°, 18155° CS2
4070° i. i. i.
s. CS2, et., bz. d. al. 4.550°
s. H2SO4; al., ac. s. H2SO4
∞ ∞
s. H2SO4 d. al. d. al.
*Usual commercial form. 2-25
Physical Properties of the Elements and Inorganic Compounds (Concluded )
TABLE 2-1
Name
Formula
Formula weight
Color, crystalline form and refractive index
Specific gravity
Melting point, °C
Boiling point, °C
Solubility in 100 parts Cold water
Hot water
Sulfuric acid Sulfuric acid, pyroSulfuric oxychloride Sulfurous oxybromide oxychloride Tantalum
H2SO4·2H2O H2S2O7 SO2Cl2 SOBr2 SOCl2 Ta
134.11 178.14 134.97 207.89 118.97 180.88
col. lq. cr. col. lq. or.-yel. lq. col. lq. bk.-gray, cb.
1.650 40° 1.920° 1.667 20° 4 2.6818° 1.638 16.6
−38.9 35 −54.1 −50 −104.5 2850
167 d. 69.1760mm 6840mm 78.8 >4100
∞ d. d. d. d. i.
∞
Tellurium
Te
127.61
met., hex.
(α) 6.24; (β) 6.00
452
1390
i.
i.
Terbium Thallium acetate chloride, monochloride, sesquichloride, trichloride, trisulfate (ic) sulfate (ous) sulfate, acid Thio, cf. sulfo or sulfur Thorium
Tb Tl TlC2H3O2 TlCl Tl2Cl3 TlCl3 TlCl3·4H2O Tl2(SO4)3·7H2O Tl2SO4 TlHSO4
159.20 204.39 263.43 239.85 515.15 310.76 382.83 823.07 504.84 301.46
blue-wh., tet. silky nd. wh., cb. yel., hex. hex. pl. nd. lf. col., rhb., 1.8671 trimorphous
11.85 3.68 7.00 5.9
303.5 110 430 400–500 25 37 −6H2O, 200 632 115 d.
1650 806 d. d. −4H2O, 100 d. d.
i. v. s. 0.210° 0.2615° v. s. 86.217° d. 2.700°
Th
232.12
cb.
11.2
1845
>3000
i.
i.
oxide, di- (thorianite) sulfate sulfate Thulium Tin
ThO2 Th(SO4)2 Th(SO4)2·9H2O Tm Sn
264.12 424.24 586.38 169.40 118.70
wh., cb.
9.69 4.22517° 2.77
>2800
4400
silv. met., tet.
7.31
231.85
2260
i. 0.740° sl. s. i. i.
5.2250° sl. s. i. i.
Tin
Sn
118.70
gray, cb.
5.750
Stable −163 to +18
2260
i.
i.
Tin salts, cf. stannic and stannous Titanic acid
H2TiO3
97.92
wh. pd.
i.
i.
i. d.
d.
s. s. i.
s. d. i.
i.
i.
Titanium chloride, di-
mn. pr.
Ti TiCl2
47.90 118.81
dark gray, cb. bk., delq.
chloride, trichloride, tetraoxide, di- (anatase)
TiCl3 TiCl4* TiO2
154.27 189.73 79.90
oxide, di- (brookite)
TiO2
vl., delq. col. lq. brn. or bk., tet., 2.534–2.564 brn. or bk., rhb., 2.586 col. if pure, tet., 2.615 gray-bk., cb.
oxide, di- (rutile) Tungsten carbide carbide
TiO2 W
79.90 79.90 183.92
6.77
17.5°
4.50
lq., 1.726 3.84
136.4
s. HCl, H2SO; sl. s. HNO3; i. HF, alk. s. h. H2SO4; i. alk.
s. HCl, H2SO4, dil. HNO3 h. aq KOH s. a., h. alk. solns. s. alk.; v. sl. s. dil. a.; i. al. s. a. i. CS2, et., chl. s. dil. HCl sl. s. alk.
1640 d.
<3000
i.
i.
s. H2SO4, alk.
3370
5900
i.
i.
6000 6000
i. i.
i. i.
i. i.
i. sl. s.
s. h. conc. KOH; sl. s. NH3, HNO3, aq. reg. s. F2; i. a. s. h. HNO3; sl. s. HCl, H2SO4 s. alk.; i. a. s. HF, alk., NH3
i.
i.
s. a., alk. carb.; i. alk.
i. d. i.
i.
s. a.; i. alk. d. a. s. HNO3, conc. H2SO4
18°
15.7 16.0618°
2777 2877
oxide, triTungstic acid (tungstite)
WO3 H2WO4
231.92 249.94
yel., rhb. yel., rhb. 2.24
7.16 5.5
Uranic acid
H2UO4
304.09
yel. pd.
5.92615°
wh. cr. cr. bk., rhb.
18.485 13° 4
>2130 −a H2O, 100; 1473 −H2O, 250 to 300 1133 2400 2176
2-26
s. al., et. s. al., et. s. dil. H2SO4
19.3
gray pd., cb. iron gray
238.07 512.14 270.07
d. d. 18.45100°
s. HNO3, H2SO4; i. NH3 v. s. al. sl. s. HCl; i. al., NH4OH
4.26
195.93 379.85
U U2C3 UO2
>3000
4.17
WC W2C
Uranium carbide oxide, di- (uraninite)
i. 1.8100° 1.9100°
v. sl. s. dil. H2SO4
−9H2O, 400
1800 Unstable in air d. 440 −30
i.
Other reagents d. al. d. al. s. ac.; d. al. s. bz., CS2, CCl4; d. act. s. bz., chl. s. fused alk., HF; i. HCl, HNO3, H2SO4 s. H2SO4, HNO3, KCN, KOH, aq. reg.; i. CS2
11.28 10.9
3500
i.
oxide (pitchblende) sulfate (ous) Uranyl acetate carbonate (rutherfordine) nitrate sulfate Vanadic acid, metaVanadic acid, pyroVanadium chloride, dichloride, trichloride, tetraoxide, dioxide, trioxide, tetraoxide, pentaoxychloride, monoVanadyl chloride chloride, dichloride, triWater†
U3O8 U(SO4)2·4H2O UO2(C2H3O2)2·2H2O UO2CO3 UO2(NO3)2·6H2O UO2SO4·3H2O HVO3 H4V2O7 V VCl2 VCl3 VCl4 V2O2 V2O3 V2O4 V2O5 VOCl (VO)2Cl VOCl2 VOCl3 H2O
842.21 502.25 424.19 330.08 502.18 420.18 99.96 217.93 50.95 121.86 157.23 192.78 133.90 149.90 165.90 181.90 102.41 169.36 137.86 173.32 18.016
Water, heavy Xenon
D2O Xe
20.029 131.30
olive gn. gn., rhb. yel., rhb. tet. yel., rhb., 1.4967 yel. cr. yel. scales pa. yel., amor. lt. gray, cb. gn., hex., delq. pink, tabular, delq. red lq. lt. gray cr. bk. cr. blue cr. red-yel., rhb. brn. pd. yel. cr. gn., delq. yel. lq. col. lq., 1.3330020°; hex. solid, 1.309 col. lq., 1.3284420° col. gas
Ytterbium Yttrium Zinc acetate acetate bromide carbonate
Yb Y Zn Zn(C2H3O2)2 Zn(C2H3O2)2·2H2O* ZnBr2 ZnCO3
173.04 88.92 65.38 183.47 219.50 225.21 125.39
dark gray, hex. silv. met., hex. mn. wh., mn., 1.494 rhb. wh., trig., 1.818
ZnCl2
136.29
wh., delq., 1.687, uniaxial col., rhb. col., rhb. cb.
chloride cyanide hydroxide iodide
Zn(CN)2 Zn(OH)2 ZnI2
117.42 99.40 319.22
nitrate oxide (zincite) oxide peroxide phosphide silicate
Zn(NO3)2·6H2O ZnO ZnO ZnO2 Zn3P2 ZnSiO3
297.49 81.38 81.38 97.38 258.10 141.44
sulfate (zincosite)
ZnSO4
sulfate sulfate sulfate (goslarite) sulfide (α) (wurzite) sulfide (β) (sphalerite)
ZnSO4·H2O ZnSO4·6H2O ZnSO4·7H2O* ZnS ZnS
161.44 179.46 269.54 287.55 97.44 97.44
sulfide (blende) sulfite Zirconium oxide, di- (baddeleyite) oxide, di- (free from Hf)
ZnS ZnSO3·2a H2O Zr ZrO2 ZrO2
97.44 190.48 91.22 123.22 123.22
col., tet. wh., hex., 2.004 wh., amor. yel. steel gray, cb. hex. or rhb.; glass, 1.650 wh., rhb., 1.669 col. mn. rhb., 1.4801 wh., hex., 2.356 wh., cb.; glass (?) 2.18–2.25 wh., granular mn. cb., pd. ign. easily yel. or brn., mn., 2.19 wh., mn.
7.31 15°
2.89 5.6 2.807 3.2816.5° 5.96 3.2318° 3.0018° 1.81630° 3.64 4.87 18° 4 4.399 3.357 18° 4 2.824 3.64 2.8813° 1.829 1.004° (lq.); 0.9150° (ice) 1.10720° lq., 3.06−109.1 2.7−140° 4.53 (A)
d. −4H2O, 300 −2H2O, 110 60.2 d. 100
118
1710
3000
d. −109 ign. 1970 1967 800
148.5755mm
d. 1750
d. in air
i. 2311° 9.217°
i. 963° d.
s. HNO3, H2SO4 s. dil. a. s. al., act.
170.30° 18.913.2° i. i. i. s. s. s. d. i. sl. s. i. 0.820° i. i. d. s. d.
∞60° 23025°
v. s. ac., al., et.; i. dil., alk. 4 al.; s. a. s. a., alk.; i. NH3 s. a., alk., NH4OH s. HNO3, H2SO4; i. aq., alk. s. al., et. s. abs. al., et. s. abs. al., et., chl., ac. s. a. s. HNO3, HF, alk. s. a., alk. s. a., alk.; i. abs. al. v. s. HNO3 s. HNO3 s. abs. al., dil. HNO3 s. al., et., ∞Br2 ∞ al.; sl. s. et.
i. d. d. i. s. i.
<−15 0
127.19 100
3.82 −140
101.42 −109.1
∞ 24.20° cc
∞ 7.350° cc
∞ al.; sl. s. et.
5.51 7.140 1.840 1.735 4.2194° 4.42
1490 419.4 242 237 394 −CO2, 300
2500 907 subl. in vac. −2H2O, 100 650
sl. d. i. 3025° 4025° 3900° 0.00115°
d. i. 44.6100° 66.6100° 670100°
2.91 25° 4
283
732
43225°
615100°
v. s. dil. a., h. KOH s. a., ac., alk. 2.825°, 16679° al. v. s. al. v. s. NH4OH, al., et. s. a., alk., NH4 salts; i. act., NH3 10012.5° al.; v. s. et.; i. NH3 s. KCN, NH3, alk.; i. al. s. a., alk., NH4OH s. a., al., NH3, aq. (NH4)2CO3 v. s. al. s. a., alk., NH4Cl; i. NH3
3.053 4.666 14.2° 4
d. 80 d. 125 446
2.065 14° 4 5.606 5.47 1.571 4.55 13° 4 3.52
36.4 >1800 >1800 expl. 212 >420 1437
3.74 15° 4 3.28 15° 4 2.072 15° 4 1.96616.5° 4.087 4.102 25° 4
d. 740 d. 238 −5H2O, 70 tr. 39 1850150atm tr. 1020
4.04 6.4 5.49 5.73
−2a H2O, 100 1700 2700
18°
624 −6H2O, 105
1100
−7H2O, 280 subl. 1185
d. 200 >2900 4300
0.0005 0.0005218° 4300°
sl. s.
324.5 0.0004218° 0.0004218° 0.0022 i. i.
∞36.4°
420° s. s. 115.20° 0.0006918° i.
61100° 89.5100° s. 653.6100° i. i.
sl. s. al.; s. gly.
i. 0.16 i. i. i.
i. d. i. i. i.
v. s. a.; i. ac. s. H2SO3, NH4OH; i. al. s. HF, aq. reg.; sl. s. a. s. H2SO4, HF s. H2SO4, HF
510100°
i. NH4OH; d. a. s. dil. a.
sl. s. al.; i. act.; NH3 sl. s. al.; i. act.; NH3 v. s. a.; i. ac. s. a.
*Usual commercial form. †Cf. special tables on water and steam, Tables 2-3, 2-4, 2-5, 2-185, 2-186 and 2-351 through 2-357. NOTE: °F = 9⁄ 5 °C + 32.
2-27
TABLE 2-2
Physical Properties of Organic Compounds* Abbreviations Used in the Table
(A), density referred to air al., ethyl alcohol amor., amorphous aq., aqua, water brn., brown bz., benzene c., cubic cc., cubic centimeter chl., chloroform col., colorless
cr., crystalline d., decomposes d-, dextrorotatory dl-, dextro-laevorotatory et., ethyl ether expl., explodes gn., green h., hot hex., hexagonal
i-, iso-, containing the group (CH3)2CHi., insoluble ign., ignites l-, laevorotatory lf., leaflets lq., liquid m-, meta mn., monoclinic n-, normal
nd., needles o-, ortho or., orange p-, para pd., powder pet., petroleum ether pl., plates pr., prisms rhb., rhombic s., soluble
This table of the physical properties includes the organic compounds of most general interest. For the properties of other organic compounds, reference must be made to larger tables in Lange’s Handbook of Chemistry (Handbook Publishers), Handbook of Chemistry and Physics (Chemical Rubber Publishing Co.), Van Nostrand’s Chemical Annual, International Critical Tables (McGraw-Hill), and similar works. The molecular weights are based on the 1941 atomic weight values. The densities are given for the temperature indicated and are usually referred to water at 4°C, e.g., 1.02895/4 a density of 1.028 at 95°C Name Abietic acid Acenaphthene Acetal Acet-aldehyde -aldehyde, par-aldehyde ammonia -amide -anilide -phenetidide (o-) (m-) -toluidide (o-) (p-) Acetic acid anhydride nitrile Acetone Acetonyl urea Acetophenone benzoyl hydride Acetyl-chloride -phenylenediamine (-p) Acetylene dichloride (cis) (trans) Aconitic acid Acridine Acrolein ethylene aldehyde Acrylic acid nitrile Adipic acid amide nitrile Adrenaline (1-) (3,4,1) Alanine (α) (dl-) Aldol acetaldol Alizarin Allyl alcohol bromide chloride thiocyanate (i) thiourea Aluminum ethoxide Amino-anthraquinone (α) (β) -azobenzene -benzoic acid (m-) (p-)
2-28
Synonym sylvic acid, abietinic acid naphthylene ethylene acetaldehyde diethylacetal ethanal paraldehyde ethanamide antifebrin o-ethoxyacetanilide acetyl-m-phenetidine N-tolylacetamide N-tolylacetamide ethanoic acid, vinegar acid acetyl oxide, acetic oxide methyl cyanide propanone, dimethyl ketone dimethyl hydantoin methyl-phenyl ketone ethanoyl chloride amino-acetanilide (p) ethyne; ethine 1,2-dichloroethene dioform equisetic acid; citridic acid acrylic aldehyde, propenal propenoic acid vinyl cyanide hexandioc acid, adipinic acid tetramethylene 1-suprarenine 2-hydroxybutyraldehyde Anthraquinoic acid propen-1-ol-3,propenyl alcohol 3-bromo-propene-1 3-chloro-propene-1 mustard oil thiosinamide
aminodracylic acid
s-, sec-, secondary silv., silvery sl., slightly subl., sublimes sym., symmetrical t-, tertiary tet., tetragonal tri., triclinic uns., unsymmetrical v., very
v. s., very soluble v. sl. s., very slightly soluble wh., white yel., yellow (+), right rotation >, greater than <, less than ∞, infinitely
referred to water at 4°C, the 4 being omitted when it is not clear whether the reference is to water at 4°C or at the temperature indicated by the upper figure. The melting and boiling points given have been selected from available data as probably the most accurate. The solubility is given in grams of the substance in 100 g. of the solvent. In the case of gases, the solubility is often expressed in some manner as “510 cc.” which indicates that, at 10°C, 5 cc. of the gas are soluble in 100 g. of the solvent.
Formula
Formula weight
Form and color
C20H30O2 C10H6(CH2)2 CH3CH(OC2H5)2 CH3CHO (C2H4O)3 CH3CHOHNH2 CH3CONH2 C6H5NHCOCH3 CH3CONHC6H4OC2H5 CH3CONHC6H4OC2H5 CH3C6H4NHCOCH3 CH3C6H4NHCOCH3 CH3CO2H (CH3CO)2O CH3CN CH3COCH3
(CH3)2 CH3COC6H5 CH3COCl C2H3ONHC6H4NH2 HC⯗CH CHCl:CHCl CHCl:CHCl C3H3(CO2H)3 C6H4 < (CH)(N) > C6H4 CH2:CHCHO CH2:CHCO2H CH2:CHCN (CH2CH2CO2H)2 (CH2CH2CONH2)2 (CH2CH2CN)2 C6H3(OH)2(CHOHCH2NHCH3) CH3CH(NH2)CO2H CH3CH(OH)CH2CO2H C6H4(CO)2C6H2(OH)2 CH2:CHCH2OH CH2:CHCH2Br CH2:CHCH2Cl CH2:CHCH2NCS CH2:CHCH2NHCSNH2 Al(OCH2CH3)3 C6H4(CO)2C6H3NH2 C6H4(CO)2C6H3NH2 C6H5N:NC6H4NH2 H2NC6H4CO2H H2NC6H4CO2H
302.44 154.20 118.17 44.05 132.16 61.08 59.07 135.16 179.21 179.21 149.19 149.19 60.05 102.09 41.05 58.08 128.13 120.14 78.50 150.18 26.04 96.95 96.95 174.11 179.21 56.06 72.06 53.06 146.14 144.17 108.14 183.20 89.09 88.10 240.20 58.08 120.99 76.53 99.15 116.18 164.15 223.22 223.22 197.23 137.13 137.13
lf. rhb./al. lq. col. lq. col. cr. col. cr. col. cr. rhb./al. lf./al. lf./al. rhb. rhb. or mn. col. lq. col. lq. col. lq. col. lq. tri./al. lf. col. lq. nd./aq. col. gas col. lq. col. lq. cr./aq. rhb./aq. al. col. lq. col. lq. col. lq. mn. pr. cr. pd. col. oil col. pd. nd./aq. col. lq. red rhb. col. lq. lq. col. lq. col. oil col. pr. pd. red nd. red nd. yel. mn. nd./aq. mn. pr.
Specific gravity 1.06995/95 0.82122/4 0.78318/4 0.99420/4 1.159 1.214 1.16815 1.21215 1.04920/4 1.08220/4 0.78320/4 0.79220/4 1.03315/15 1.10520/4 (A) 0.906 1.29115/4 1.26515/4 0.84120/4 1.06216/4 0.81120 1.36025/4 19/19
0.951
1.10320/4 0.85420/4 1.39820/4 0.93820/4 1.01320/4 1.21920/20 1.14220/0
1.5114°
Melting point, °C 182 95 −123.5 10.5–12 97 81(69.4) 113–4 79 96–7 110 153 16.7 −73 −41 −94.6 175 20.5 −112.0 162 −81.5891 −80.5 −50 192 d. 110–1 −87.7 12–13 −82 151–3 226–7 1 d. 207–11 295 d. 289–90 −129 −119.4 −136.4 −80 77–8 150–60 256 302 126–7 173–4 187–8
Boiling point, °C 278–9 102.2 20.2 124.4752 100–10 d. 222 305 >250 296 306–7 118.1 139.6 81.6–2.0 56.5 subl. 202.3749 51–2 −84760 60.3 48.4 346 52.5 141–2 78–9 26510 295 subl. >200 8320 430 96.6 70–1753 44.6 152 200–510 subl. subl. 225120
Solubility in 100 parts Water
Alcohol
Ether
i. i. 625 ∞ 1213 v. s. s. 0.56 i. sl. s. 0.8619 0.0922 ∞ 12 c. ∞ ∞ s. i. d. s. h. 100 cc.18 0.3520 0.6320 3315 sl. s. h. 40 ∞ s. 1.415 0.412 v. sl. s. 0.0320 2217 ∞ 0.03100 ∞ i. <0.1 0.2 30 d. i. i. sl. s. h. v. sl. s. 0.313
v. s. s. h. ∞ ∞ ∞ v. s. s. 2120 s. s. s. 1025 ∞ ∞ ∞ ∞ s. s. d. v. s. 600 cc.18 ∞ ∞ sl. s. s. s. ∞
v. s. s. chl. ∞ ∞ ∞ sl. s. v. sl. s. 7 25
v. s.
0.615
s. v. sl. s. v. sl. s. ∞ v. s. ∞ ∞ ∞ ∞ s. i. s. s. s. h. 210 1110
v. sl. s. i. i. s. v. s. ∞ ∞ ∞ ∞ v. sl. s. v. sl. s. s. i. s. 1.86 8.26
s. s. ∞ ∞ ∞ ∞ s. s. ∞ v. s. ∞ ∞ v. sl. s. s. s.
Amino-diphenylamine (p-) -G-acid (2-)(6-,8-), Na2 salt -mono-potassium salt -sodium salt -J-acid (2-)(5-,7-) -mono-potassium salt -naphthol sulfonic (1-,2-,4-)(α-) (1-,8-,4-) -phenol (o-) (m-) (p-) -toluene sulfonic acid (1-,2-,3-) (1-,4-,2-) (1-,4-,3-) (1-,2-,5-) Amyl acetate (n-) (i-) (s-) (s-) (t-) alcohol (n-) fusel oil, (s-,n-) methyl-propyl carbinol, (prim.-,i-) isobutyl carbinol, (s-,i-) (t-) (d-) -amine (n-) (s-,n-) (i-) (t-)
aniline (i-) benzoate (i-) bromide (n-) (i-) (t-) n-butyrate (n-) (i-) (t-) i-butyrate (i-) chloride (n-) (s-) (s-) (i-) (s-,i-) (t-) i-cyanide (i-) formate (n-) (i-) iodide (n-) (i-) (s-,n-) (t-) mercaptan (n-) (n-) (i-) phenol (t-)(p-) propionate (n-) (i-) (act.) salicylate (n-) Amyl i-valerate (i) (t-)
2-aminophenol 3-aminophenol p-hydroxyaniline
common amyl acetate α-Me-Bu-acetate di Et-carbinol acetate pentanol-1 pentanol-2 2-methyl-butanol-4 2-methyl-butanol-3 2-methyl-butanol-2 active amyl alcohol
1-NH2-2-Me-butane 3-amino pentane 3-NH2-2-Me-butane 1-bromopentane 4-Br-2-Me-butane 2-Br-2-Me-butane
1-chloropentane 2-chloropentane 3-chloropentane 4-Cl-2-Me-butane 3-Cl-2-Me-butane 2-Cl-2-Me-butane 1-Cl-2-Me-butane iso-caproic iso-nitrile 1-iodopentane 4-I-2-Me-butane 2-iodopentane 2-I-2-Me-butane pentanthiol-1 pentanthiol-3 2-Me-butanthiol-4 pentaphen
H2NC6H4NHC6H5 C10H5(NH2)(SO3Na)2 C10H5(NH2)S2O6HK C10H5(NH2)S2O6HNa C10H5(NH2)(SO3H)2 C10H5(NH2)S2O6HK C10H5OHNH2SO3Ha H2O NH2(OH)C10H5SO3H H2NC6H4OH H2NC6H4OH H2NC6H4OH C6H3(CH3)(NH2)SO3H C6H3(CH3)(NH2)SO3HH2O C6H3(CH3)(NH2)SO3Ha H2O C6H3(CH3)(NH2)SO3HH2O CH3CO2CH2(CH2)3CH3 CH3CO2CH2CH2CH(CH3)2 CH3CO2CH2CH(CH3)C2H5 CH3CO2CH(CH3)CH2C2H5 CH3CO2CH(C2H5)2 CH3CO2C(CH3)2C2H5 CH3(CH2)3CH2OH C2H5CH2CH(OH)CH3 (CH3)2CHCH2CH2OH (C2H5)2CHOH (CH3)2CHCH(OH)CH3 (CH3)2C(OH)C2H5 (CH3)3CCH2OH C2H5CH(CH3)CH2OH CH3(CH2)4NH2 (C3H7)(CH3)CHNH2 (CH3)2CH(CH2)2NH2 (C2H5)(CH3)2CNH2 C2H5CH(CH3)CH2NH2 (C2H5)2CHNH2 (CH3)2CHCH(CH3)NH2 C6H5NHC5H11 C6H5CO2C5H11 CH3(CH2)3CH2Br (CH3)2CH(CH2)2Br (CH3)2C(Br)C2H5 C2H5CH2CO2(CH2)4CH3 C2H5CH2CO2C5H11 C3H7CO2C(CH3)2C2H5 (CH3)2CHCO2C5H11 CH3(CH2)3CH2Cl C2H5CH2CHClCH3 (C2H5)2CHCl (CH3)2CH(CH2)2Cl (CH3)CHCHClCH3 (CH3)2CClC2H5 (CH3)(C2H5)CHCH2Cl (CH3)2CH(CH2)2NC HCO2CH2(CH2)3CH3 HCO2CH2CH2CH(CH3)2 CH3(CH2)3CH2I (CH3)2CHCH2CH2I C2H5CH2CHICH3 (CH3)2CIC2H5 C2H5CH(CH3)CH2I CH3(CH2)3CH2SH (C2H5)2CHSH (CH3)2CH(CH2)2SH C5H11C6H4OH C2H5CO2(CH3)4CH3 C2H5CO2(CH2)2CH(CH3)2 C2H5CO2C5H11 HOC6H4CO2C5H11 C4H9CO2C5H11 C4H9CO2C5H11
184.23 347.28 341.39 325.29 303.30 341.39 248.25 239.24 109.12 109.12 109.12 187.21 205.23 196.22 205.23 130.18 130.18 130.18 130.18 130.18 130.18 88.15 88.15 88.15 88.15 88.15 88.15 88.15 88.15 87.16 87.16 87.16 87.16 87.16 87.16 87.16 163.25 192.25 151.05 151.05 151.05 158.23 158.23 158.23 158.23 106.60 106.60 106.60 106.60 106.60 106.60 106.60 97.16 116.16 116.16 198.06 198.06 198.06 198.06 198.06 104.21 104.21 104.21 164.24 144.21 144.21 144.21 208.25 172.26 172.26
nd./aq. al.
67
354
col. nd. pr. lf. nd. mn. nd. tri./aq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. cr. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. col. lq. col. lq. col. lq. lq. col. lq. col. lq. col. lq. lq. col. lq. lq. col. lq. col. lq. lq. lq. lq. lq. lq. lq. lq. lq. lq. lq. lq. lq. col. lq. lq. cr. lq. col. lq. col. lq. lq. col. lq. col. lq.
173 122–3 184–6 d.
subl. subl.
d. 0.87920/20 0.87615/4 0.88013 0.9220 0.87120/4 0.87419 0.817 20/20 0.81020/20 0.81315/4 0.815 25/4 0.81919 0.809 20/4 0.81620/4 0.76619 0.749 20/4 0.75118/4 0.73125/4 0.75518 0.749 20/4 0.75718 0.92815/4 0.99214/14 1.21820/4 1.22017/15 1.21619/0 0.87115/4 0.86619/15 0.86515/0 0.8760/4 0.878 20/4 0.870 20/4 0.895 21 0.89320/4 0.8830 0.87120/4 0.88117.5 0.9020 0.882 20/4 1.51020/4 1.515 18/4 1.507 17/4 1.47119/15 1.524 20/4 0.857 20
−H2O, 120 −70.8
−78.5 −117.2 −11.9 52–3 −55 −105
−95 −73.2
−99
−72.9 −73.5 −93.5 −86
0.83520/4 0.87615/4 0.870 20/4 0.866 20/4 1.06515 0.85820/15 0.86114/0
93 −73.1
148.4737 142757 141–2 133.5 133 124.5 749 137.9 119.5 132.0 115.6 113–4 102 113–4 128 103–4 91–2 95 77–8 95–6 90–1 83–4 254.5 261746 129.7 120745 108765 186.4 178.6 164 168.8 108.4 96.7 97.3 99.7 758 91753 85.7 98–9 137–9 132 123.5 157.0 147 765 144–5 127 765 148 126767 105 120 265–7 168.7 160.2 5816 265 194 173–4
sl. s. v. sl. s. 12.820 2.718 10.0 20 3.418 v. s. v. sl. s. 1.7 0 2.6 0 1.10 0.97 11 0.5 20 0.47 311 v. sl. s. 0.315 v. sl. s. sl. s. sl. s. v. sl. s. 2.7 22 420 214 5.5 30 2.830 sl. s. sl. s. 3.630 s. ∞ ∞ ∞ ∞ ∞ ∞ i. i. i. 0.0216 i. 0.05 50 i. sl. s. i. i. i. i. i. i. i. i. i. v. sl. s. 0.322 i. i. i. i. i. i. i. i. sl. s. i. 0.125 v. sl. s. i. v. sl. s. sl. s.
s.
s.
4.30 s. 40
v. s. sl. s. i. bz.
i. i. ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. ∞ ∞ s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. s. s. ∞ ∞ ∞ s. s. s. ∞ s. s. s. s. s. ∞ ∞ s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. ∞ ∞ ∞ ∞ ∞ s.
∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. ∞ ∞ s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. s. ∞ ∞ ∞ s. s. s. ∞ ∞ s. s. s. s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. ∞ ∞ ∞ ∞ ∞ s.
*By N. A. Lange, Ph.D., Handbook Publishers, Inc., Sandusky, Ohio. Abridged from table of Physical Constants of Organic Compounds in Lange’s “Handbook of Chemistry.” 2-29
TABLE 2-2
Physical Properties of Organic Compounds (Continued )
Name Amylene (n-)(α-) (i-) (α-) (-n)(β-) (i-)(β-) Anethole (p-) Anhydroformald-aniline Aniline hydrochloride nitrate sulfate Anisal-acetone (p-) Anisic acid (p-) aldehyde (p-) Anisidine (o-) (m-) (p-) Anisole Anthracene Anthramine (α) (β) Anthranil Anthranilic acid (o-) Anthrapurpurin (1-,2-,7-) Anthraquinone disulfonate Na2 (1-,5-) (1-,8-) (2-,6-) (2-,7-) sulfonate Na (1-) (2-) Anthrarufin (1-,5-) Antipyrene Apiole Arabinose (α)(d- or l-) (dl-) Arachidic acid Arsanilic acid (p-) Asparagine (l-) Aspirin (o-) Atropic acid Auramine Aurine, coralline (4-,4′-) Azo-anisole (2-,2′-) benzene Azoxybenzene Barbituric acid Benzal acetone Benzaldehyde Benzamide Benzanilide Benzene sulfinic acid sulfonic acid sulfonic amide sulfonic chloride Benzidine (4-,4′-) disulfonic acid (2-,2′-) (3-,3′-) Benzil Benzoic acid anhydride nitrile
2-30
Synonym pentene-1 2-methyl-butene-3 2-methyl-butene-1 pentene-2 2-methyl-butene-2 p-propenyl anisole methylene aniline amino benzene, phenyl amine, cyanol aniline salt, aniline chloride MeO-benzalacetone 2-amino-anisole MeO-aniline(m) 4-amino anisole methyl phenyl ether paranaphthalene, anthracin green oil α-amino-anthracene β-amino-anthracene
diphenyleneketone, dihydrodiketoanthracene ρ-anthraquinone disulfonate x-anthraquinone disulfonate
1-ph-2,3-diMepyrazolone-5 1-allyl-2, 5-diMeO-3,4 methylenedioxybenzene eicosanoic acid
α-phenyl acrylic acid 4,4′-dimethylaminobenzophenomide diMeO-azobenzene diphenyldiimide malonyl urea Me-cinnamyl ketone artificial almond oil benzol, phenyl hydride, cyclohexatriene benzene sulfonamide benzene sulfonyl chloride
dibenzoyl phenyl cyanide
Formula weight
Form and color
Specific gravity
C2H5CH2CH:CH2 (CH3)2CHCH:CH2 (C2H5)(CH3)C:CH2 C2H5CH:CHCH3 (CH3)2C:CHCH3 CH3CH:CHC6H4OCH3 (CH2NC6H5)3 C6H5NH2
70.13 70.13 70.13 70.13 70.13 148.20 315.40 93.12
lq. col. lq. col. lq. col. lq. col. lq. lf./al. pr./al. col. oil
0.644 20 0.63215 0.667 0/0 0.650 20/4 0.66319/4 0.99120/20
C6H5NH2HCl C6H5NH2HNO3 (C6H5NH2)2H2SO4 CH3OC6H4CH:CHCOCH3 CH3OC6H4CO2H CH3OC6H4CHO CH3OC6H4NH2 CH3OC6H4NH2 CH3OC6H4NH2 CH3OC6H5 C6H4:(CH)2:C6H4
129.59 156.14 284.32 176.22 152.14 136.14 123.15 123.15 123.15 108.13 178.22
cr. rhb. lf./al. lf./et. mn./aq. col. oil col. lq. oil pl./aq. col. lq. col. mn.
1.222 4 1.356 4 1.377 4
C6H4:(CH)2:C6H3NH2 C6H4:(CH)2:C6H3NH2 C6H4:(NH)CO H2NC6H4CO2H C14H5O2(OH)3 C6H4:(CO)2:C6H4
193.24 193.24 119.12 137.13 256.20 208.20
yel./al. yel./al. col. oil col. rhb. or. nd./al. yel. rhb.
C14H6O2(SO3Na)25H2O C14H6O2(SO3Na)24H2O C14H6O2(SO3Na)27H2O C14H6O2(SO3Na)24H2O C14H7O2SO3Na C14H7O2SO3Na C14H6O3(OH)2 C11H12ON2 C12H14O2
502.38 484.37 538.41 484.37 310.25 310.25 240.20 188.22 222.23
yel. lf. yel. pr. col. cr. cr. yel. lf. silv. lf. yel. lf. mn./aq. col. nd.
1.088113/4 1.0220/4
CH2OH(CHOH)3CHO CH2OH(CHOH)3CHO CH3(CH2)18CO2H H2NC6H4.AsO3H2 HO2CC2H3(NH2)CONH2 CH3CO2C6H4OH C6H5C(:CH2)CO2H [(CH3)2NC6H4]2C:NH
150.13 150.13 312.52 217.04 132.12 180.15 148.15 267.36
rhb. pr.
1.585 20/4
col. lf. nd./aq. rhb. nd./aq. nd./aq. col./al.
(HOC6H4)2C:C6H4:O (CH3OC6H4N:)2 C6H5N:NC6H5 (C6H5)2N2O CO:(NHCO)2:CH22H2O C6H5CH:CHCOCH3 C6H5CHO C6H5CONH2 C6H5CONHC6H5 C6H6
290.30 242.27 182.22 198.22 164.12 146.18 106.12 121.13 197.23 78.11
red or. pr. or. mn. yel. rhb. col./aq. pl. col. lq. col. pr. lf./al. col. lq.
C6H5SO2H C6H5SO3H C6H5SO2NH2 C6H5SO2Cl NH2C6H4C6H4NH2 (C6H3(NH2)SO3H)23H2O (C6H3(NH2)SO3H)2 C6H5COCOC6H5 C6H5CO2H (C6H5CO)2O C6H5CN
142.17 158.17 157.18 176.62 184.23 398.40 344.35 210.22 122.12 226.22 103.12
pr./aq. col. nd. mn./aq. cr. cr./aq. pr./aq.
1.38415/15
83–4 65–6 156 14.5 128–9 d. >175
pr. mn. pr. rhb./et. col. lq.
1.2315 1.26615/4 1.19915/4 1.00125/6
95 121.7 42 −12.9
Formula
1.02220/4
1.385 4 1.123 20/4 1.09815/15 1.096 20/4 1.089 55/55 0.990 22/4 1.25 27/4
1.18715/4 1.438 20/4
1.543
15/4
1.20320/4 1.248 20/20 1.035 20/20 1.046 20/4 1.341 1.314 0.879 20/4
Melting point, °C −135 −139 −124 22.5 143 −6.2
Boiling point, °C
Solubility in 100 parts Water
Alcohol
Ether
30–1 20.5 771 31–2758 36.4 37–8 235.3 185 184.4
i. i. i. v. sl. s. i. v. sl. s. i. 3.618
∞ ∞ ∞ ∞ s. s. sl. s. ∞
∞ ∞ ∞ ∞ ∞ ∞ s. ∞
245
s. s. sl. s. v. s. v. s. ∞ ∞ s. s. s. 1.520
i. sl. s. i. v. s. v. s. ∞ ∞ s. s. s.
198 d. 190 d. 73–4 184.2 2.5 5.2 <−12 57.2 −37.3 217–8
275–80 247–8 225 251 243 154–5 340–2
1815 s. 514 i. 0.0319 v. sl. s. v. sl. s. v. sl. s. s. h. i. i.
130 238 <−18 144–5 369 286
subl. d. >215 subl. 462 379–81
i. i. sl. s. h. 0.35 14 sl. s. h. i.
s. sl. s. s. 1110 v. s. h. 0.0518
sl. s. s. 167 sl. s. v. sl. s.
i.
i.
subl. 319174 294
v. s. sl. s. 3.920 30.5 20 0.5320 0.8425 i. 100 25 i
v. sl. s. i. i. sl. s. 100 s.
i. i. i. s. sl. s. s.
460 16.910 i. v. s. h. 3.128 137 0.1 c. i.
0.59°
i.
s. h. v. s. h. i. c. s. s. 720
v. s. i.
i. i. i. i. s. h. i. 0.3 1.35 25 i. 0.07 22
s. s. 4.220 11.415 sl. s. s. ∞ 1725 430 s.
s. s. ∞ sl. s. sl. s. ∞
v. s. h. v. s. 0.4316 i. 1 h. 0.09 25 v. sl. s. i. 0.217 i. 1100
v. s. v. s. v. s. v. s. 1 h. i.
v. s. i. v. s. s. 2 i.
v. s. 4615 s. ∞
v. s. 6615 s. ∞
280 113(109) 30 159.5 164.5 77 232 227–35 135–6 106–7 136 310 d. 153 68 36 d. 245 41–2 −26 130 163 5.5
328 d. 235 267 d.
297 d. 260–2 179 290 117–910 80.1 d. > 100 d. 251.5 400740 348 d. 249.2 360 190.7
520 s. 2.320 s. s.
Benzoin (dl-) Benzophenone Benzotrichloride Benzoyl-benzoic acid (o-) -chloride -peroxide Benzyl acetate alcohol amine aniline benzoate butyrate chloride ether formate propionate Berberonic acid (2-,4-,5-) Biuret Borneol (dl-) (d- or l-) (iso-) Bornyl acetate (d-) Bromo-aniline (p-) -benzene -camphor (3-)(d-) -diphenyl (p-) -naphthalene (α-) (β-) -phenol (o-) (m-) (p-) -styrene (ω)(1) (2) -toluene (o-) (m-) (p-) Bromoform Butadiene (1-,2-) (1-,3-) Butadienyl acetylene Butane (i-) Butyl acetate (n-) (s-) (i-) (tert-) alcohol (n-) (s-) (i-) (tert-) amine (n-) (s-) (i-) (t-) p-aminophenol (N)(n) (N)(i-) aniline (n-) (i-) arsonic acid (n-) benzoate (n-) (i-) bromide (n-) (s-) (i-) (t-) butyrate (n-)(n-) (n-)(i-) (i-)(i-) caproate carbamate (i-) cellosolve (n-)
diphenyl ketone phenyl chloroform
phenyl carbinol ω-amino toluene phenyl-benzylamine ω-chlorotoluene dibenzyl ether
allophanamide
phenyl bromide α-bromocamphor α-naphthyl bromide β-naphthyl bromide
o-tolyl bromide tribromo-methane methyl-allene erythrene diethyl trimethyl-methane
butanol-1 butanol-2 2-methyl-propanol-1 2-methyl-propanol-2
1-bromo-butane 2-bromo-butane 1-Br-2-Me-propane 2-Br-2-Me-propane
2-BuO-ethanol-1
C6H5COCHOHC6H5 C6H5COC6H5 C6H5CCl3 C6H5COC6H4CO2HH2O C6H5COCl (C6H5CO)2O2 CH3CO2CH2C6H5 C6H5CH2OH C6H5CH2NH2 C6H5CH2NHC6H5 C6H5CO2CH2C6H5 C2H5CH2CO2CH2C6H5 C6H5CH2Cl (C6H5CH2)2O HCO2CH2C6H5 C2H5CO2CH2C6H5 C5H2N(CO2H)32H2O NH(CONH2)2 C10H17OH C10H17OH C10H17OH CH3CO2C10H17 BrC6H4NH2 C6H5Br BrC10H15O BrC6H4C6H5 C10H7Br C10H7Br BrC6H4OH BrC6H4OH BrC6H4OH C6H5CH:CHBr C6H5CH:CHBr CH3C6H4Br CH3C6H4Br CH3C6H4Br CHBr3 CH3CH:C:CH2 CH2:CHCH:CH2 CH2:(CH)2:CHC⯗CH CH3CH2CH2CH3 (CH3)2CHCH3 CH3CO2(CH2)2C2H5 CH3CO2CH(CH3)C2H5 CH3CO2CH2CH(CH3)2 CH3CO2C(CH3)3 C2H5CH2CH2OH C2H5CH(OH)CH3 (CH3)2CHCH2OH (CH3)3COH C2H5CH2CH2NH2 C2H5CH(NH2)CH3 (CH3)2CHCH2NH2 (CH3)3CNH2 C4H9NHC6H4OH C4H9NHC6H4OH C4H9NHC6H5 C4H9NHC6H5 C4H9AsO(OH)2 C6H5CO2C4H9 C6H5CO2C4H9 C2H5CH2CH2Br C2H5CH(Br)CH3 (CH3)2CHCH2Br (CH3)3CBr C2H5CH2CO2CH2CH2C2H5 C2H5CH2CO2CH2CH(CH3)2 (CH3)2CHCO2CH2CH(CH3)2 CH3(CH2)4CO2C4H9 NH2CO2CH2CH(CH3)2 C4H9OCH2CH2OH
212.24 182.21 195.48 244.24 140.57 242.22 150.17 108.13 107.15 183.24 212.24 178.22 126.58 198.25 136.14 164.20 247.16 103.08 154.24 154.24 154.24 196.28 172.03 157.02 231.14 233.11 207.07 207.07 173.02 173.02 173.02 183.05 183.05 171.04 171.04 171.04 252.77 54.09 54.09 78.11 58.12 58.12 116.16 116.16 116.16 116.16 74.12 74.12 74.12 74.12 73.14 73.14 73.14 73.14 165.23 165.23 149.23 149.23 182.04 178.22 178.22 137.03 137.03 137.03 137.03 144.21 144.21 144.21 172.26 117.15 118.17
mn. col. rhb. col. lq. tri./aq. col. lq. rhb./et. col. lq. col. lq. lq. mn. pr. nd. col. lq. col. lq. lq. col. lq. lq. tri. nd./al. col. cr. col. cr. col. cr. rhb./pet. rhb. col. lq. cr. cr./al. col. oil lf./al. col. lq. cr. tet. cr. lq. lq. col. lq. col. lq. cr./al. col. lq. lq. col. gas col. lq. col. gas col. gas col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. col. lq. col. lq. col. lq. col. lq. lq. oil col. lf. col. oil col. oil lq. lq. lq. lq. col. lq. col. lq. col. lq. col. lq. col. lf. col. lq.
1.08354 1.38014 1.21220/4 1.05717 1.04320/4 0.98220/4 1.065 25/25 1.1220/4 1.01616/18 1.100 20/20 1.03616 1.08123 1.03616/17 20/4
1.011 1.01120/4 0.99115 1.820 1.495 20/4 1.449 20/4 1.48220/4 1.605 0 1.55380 1.588 80 1.42220/4 1.427 20/4 1.42220/4 1.410 20/4 1.390 20/4 2.890 20/4 0.62120/4 0.773 20/4 0.600 0.600 0.882 20 0.865 25/4 0.87120/4 0.866 20/4 0.810 20/4 0.808 20/4 0.80517.5 0.779 26 0.739 25/4 0.724 20/4 0.73220/20 0.69818/4
133–7 48.5 −4.75 93(128) −0.5 108 d. −51.5 −15.3 37–8 21 238–40 −39 3.6 243 192–3 d. 210.5 208–9 212 29 63–4 −30.6 77–8 90–1 5–6 59 5.6 32–3 63.5 7 −7.5 −28 −39.8 28.5 8–9 −108.9 −135 −145 −76.3 −98.9 −79.9 −114.7 −108 25.5 −50 −104 −85 −67.5 71 79
0.940 20/4 1.005 25/25 0.997 25/25 1.277 20/4 1.25125/4 1.258 25/4 1.21120/4 0.87220/20 0.86318/4 0.875 0/4 0.8820/0 0.95676/4 0.90320/4
158–9 −22 −112.4 −112 −118.5 −16.2 −80.7 65
344768 305.4 220.7 197.2 expl. 213.5 204.7 184.5 306750 323–4 i. 179.4 295–8 202–3747 220–2 subl. 212–3 226–7 156.2 274 310 281.1 281–2 194–5 236–7 238 221 10826 181.8 183.7 184–5 150.5 18–9 −4.41 83–6 −0.6 −10 125 740 112744 118 95–6760 117 99.5 107–8 82.9 77.8 66772 68–9 45.2 235720 231–2 249–50 241.5 101.6 91.3 91.5 73.3 165.7736 156.9 148–9 204.3 206–7 171.2
v. sl. s. i. i. sl. s. d. i. i. 417 ∞ i. i. v. s. i. i. i. i. v. sl. s. 1.30 v. sl. s. v. sl. s. i. i. i. c. i. i. i. i. i. s. 1.415 i. i. i. i. i. 0.1 c. i. i. i. i. i. 0.7 i. 0.625 i. 915 12.520 1015 ∞ ∞ ∞ ∞ i. i. i. 0.0115 s. i. i. 0.0616 i. 0.0618 i. i. i. i. i. i. ∞
s. h. 6.515 s.
sl. s. 1513 s.
d. h. s. h. ∞ ∞ ∞
∞ s. ∞ ∞ ∞ s. ∞
∞ v. s. ∞ s. h. s.
∞ s. ∞
sl. s. h. s.
i.
v. s.
v. s.
s. v. s. s. 2026 s. s. 620 s. s. v. s. ∞ ∞ s. s. s. ∞ ∞ ∞
s. v. s. ∞ v. s. 34 25 ∞ v. s. ∞ s. v. s. ∞ ∞ ∞25 s. ∞25 ∞ ∞ ∞
s. s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞
s. s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞
v. s. v. s. s. s. ∞ ∞
v. s. v. s. i. s. ∞ ∞
∞ ∞ ∞ ∞ ∞
∞ ∞ ∞ ∞ ∞
s. ∞
s. ∞
2-31
TABLE 2-2
Physical Properties of Organic Compounds (Continued )
Name chloride (n-) (s-) (i-) (t-) dimethylbenzene (t-)(1-,3-,5-) formate (n-) (s-) (i-) furoate (n-) iodide (n-) (s-) (i-) (t-) lactate (n-) mercaptan (n-) (i-) (t-) methacrylate (n-) (i-) phenol (p-)(t-) propionate (n-) (s-) (i-) stearate (n-) (i-) iso-thiocyanate (n-) (i-) (s-)(d-) (t-) valerate (n-)(n-) (i-)(n-) (i-)(s-) (i-)(i-) Butylene (α-) (β-) Butyraldehyde (n-) (i-) Butyric acid (n-) (i-) amide (n-) (i-) anhydride (n-) (i-) anilide (n-) Caffeic acid (3-,4-) Caffeine Camphene (dl-) (d- or l-) Camphor (d-) Camphoric acid (d-) Cantharidine Capric acid Caproic acid (n-) (i-) Caprylic acid (n-) Carbazole Carbitol Carbon disulfide monoxide suboxide tetrabromide tetrachloride tetrafluoride Carbonyl sulfide Carminic acid Carvacrol (1-,2-,4-)
2-32
Synonym 1-chloro-butane 2-chloro-butane 1-Cl2-2-Me-propane 2-Cl2-2-Me-propane
1-iodo-butane 2-iodo-butane 1-iodo-2-Me-propane 2-iodo-2-Me-propane butanthiol-1 2-Me-propanthiol-1
butyl mustard oil iso-Bu mustard oil
butene-1 butene-2 2-Me-propanol butanoic acid 2-Me-propanoic acid n-butyramide iso-butyramide n-butyranilide
decanoic acid hexanoic acid 2-Me-pentanoic-5 acid octanoic acid diphenylenelimine, dibenzopyrrole diethylene glycol mono-Et ether
tetrabromomethane tetrachloromethane tetrafluoromethane
Formula C2H5CH2CH2Cl C2H5CHClCH3 (CH3)2CHCH2Cl (CH3)3CCl (CH3)3CC6H3:(CH3)2 HCO2CH2CH2C2H5 HCO2CH(CH3)C2H5 HCO2CH2CH(CH3)2 OC4H3CO2C4H9 C2H5CH2CH2I C2H5CHICH3 (CH3)2CHCH2I (CH3)3CI CH3CH(OH)CO2C4H3 C2H5CH2CH2SH (CH3)2CHCH2SH (CH3)3CSH CH2:C(CH3)CO2C4H9 CH2:C(CH3)CO2C4H9 (CH3)3CC6H4OH C2H5CO2C4H9 C2H5CO2C4H9 C2H5CO2C4H9 CH3(CH2)16CO2C4H9 CH3(CH2)16CO2C4H9 C2H5CH2CH2N:CS (CH3)2CHCH2N:CS C4H9N:CS (CH3)3CN:CS CH3(CH2)3CO2(CH2)3CH3 (CH3)2CHCH2CO2(CH2)3CH3 (CH3)2CHCH2CO2C4H9 C4H9CO2C4H9 C2H5CH:CH2 CH3CH:CHCH3 CH3CH2CH2CHO (CH3)2CHCHO C2H5CH2CO2H (CH3)2CHCO2H C2H5CH2CONH2 (CH3)2CHCONH2 (C2H5CH2CO)2O [(CH3)2CHCO]2O C3H7CONHC6H5 (HO)2C6H3C2H2CO2H C8H10O2N4H2O C10H16 C10H16 C10H16O C8H14(CO2H)2 C10H12O4 CH3(CH2)8CO2H CH3(CH2)4CO2H (CH3)2CH(CH2)2CO2H CH3(CH2)6CO2H (C6H4)2NH C2H5O(CH2)2O(CH2)2OH CS2 CO OC:C:CO CBr4 CCl4 CF4 COS C22H20O13 CH3C6H3(OH)CH(CH3)2
Formula weight
Form and color
Specific gravity
Melting point, °C
Boiling point, °C
92.57 92.57 92.57 92.57 162.26 102.13 102.13 102.13 168.19 184.03 184.03 184.03 184.03 146.18 90.18 90.18 90.18 142.19 142.19 150.21 130.18 130.18 130.18 340.57 340.57 115.19 115.19 115.19 115.19 158.23 158.23 158.23 158.23 56.10 56.10 72.10 72.10 88.10 88.10 87.12 87.12 158.19 158.19 163.21 180.15 212.21 136.23 136.23 152.23 200.23 196.20 172.26 116.16 116.16 144.21 167.20 134.17 76.13 28.01 68.03 331.67 153.84 88.01 60.07 492.40 150.21
col. lq. col. lq. col. lq. col. lq. col. lq. lq. lq. lq. col. lq. lq. lq. lq. lq. col. lq. col. lq. lq. lq. lq. lq. nd./aq. col. lq. col. lq. col. lq. col. lq. wax lq. lq. lq. lq. lq. lq. col. lq. col. lq. col. gas col. gas col. lq. col. lq. col. lq. col. lq. rhb. mn. pl. col. lq. col. lq. mn. pr. yel./aq. nd./al. cr. cr. trig. mn. cr. col. nd. oily lq. col. oil col. lf. lf. col. lq. col. lq. col. gas gas col. mn. col. lq. gas col. gas red pd. col. lq.
0.887 20 0.87120/4 0.88415 0.84715
−123.1 −131 −131.2 −26.5
77.9763 67.8767 68.9 51–2 200–2147 106.9 97 98.2 118–2025 129.9 118–9 120 99 75–66 97–8 88 65–7 155 155 236–8 146 132.5 136.8 220–525
0.9110 0.88220/4 0.885 20/4 1.056 20/4 1.617 20/4 1.595 20 1.606 20/4 1.370 19/15 0.968 0.837 25/4 0.836 20/4 0.889 15.6 0.889 15.6 0.908 112/4 0.88315 0.866 20/4 0.888 0/4 0.855 25/25 0.95611 0.96414/4 0.943 20/4 0.91910 0.87015/4 0.862 25/4 0.848 20/4 0.8740/4 0.69 0.817 20/4 0.79420/4 0.96420/4 0.949 20/4 1.032 1.013 0.968 20/20 0.950 25/4 1.134 1.2319 0.82278 0.845 50/4 0.999 9/9 1.186 0.889 87 0.922 20/4 0.925 20/4 0.910 20/4 0.990 20/20 1.263 20/4 0.81−195/4 1.1140 3.42 1.595 20/4 1.24−87 0.977 20/4
−95.3 −103.5 −104 −90.7 −34 −116 <−79
99 −89.55 −71.4 27.5 25
10.5 −93
−130 −127 −99 −65.9 −4.7 −47 115–6 129–30 −75 −53.5 92 195–213 237 50 42.7 178–9 187 212 31.5 −1.5 −35 16 244.8 −108.6 −207 −107 90.1(48) −22.6 −138.2 d. 136 0.5
165724 162 159–63 140770 186 168.8 163–4752 168.7 −5758 3746 75.7 64757 163.5757 154.5 216 216–20 199.5 181.5734 18915 d. subl. 160 159.6 209.1759 268–70 202761 207.7 237.5 354.8 201.9 46.3 −192 7761 189.5 76.8 −128 −50.2760 238
Solubility in 100 parts Water 0.0718 i. i. i. i. v. sl. s. sl. s. 1.122 i. i. i. i. i. sl. s. sl. s. v. sl. s. i. i. sl. s. i. i. i. 0.3 25 i. i. i. i. i. v. sl. s. i. i. i. i. 4 1120 ∞ 20 20 16.315 v. s. d. d. i. s. h. 2 i. i. 0.1 0.612 0.003 0.003 1.120 v. sl. s. 0.0715 i. ∞ 0.20 3.50 cc. d. 0.0230 0.0820 sl. s. 8014 cc. s. v. sl. s.
Alcohol
Ether
∞ ∞ ∞ ∞
∞ ∞ ∞ ∞
∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ v. s. s.
∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ v. s. s.
s. ∞ ∞ ∞ s.
s. ∞ ∞ ∞ s.
s. s. s. s. ∞ ∞ ∞ ∞ v. s.
s. s. s. s. ∞ ∞ ∞ ∞ v. s.
∞ ∞ ∞ ∞ s. s. d d s. s. 2 s. s. 12012 s.
∞ ∞ ∞ ∞ sl. s. sl. s. ∞ ∞ s. sl. s. 0.3 s. s. v. s.
s. s. s. s. 0.9214 v. s. ∞ s.
s. s. s. s. sl. s. s. ∞
s. ∞
s. s. ∞
s. s. ∞
s. v. sl. s. ∞
Carvacrylamine (2-,1-,4-) Carvone (d-) Cellosolve acetate Cellulose Cetyl acetate alcohol Chloral hydrate Chloranil Chloretone Chloro-acetanilide (p-) -acetic acid -acetone -acetophenone (ω-) -acetyl chloride -aniline (o-) (m-) (p-) -anthraquinone (1-) (2-) -benzaldehyde (o-) (m-) (p-) -benzene -benzoic acid (o-) (m-) (p-) -buta-1,3-diene (2-) (1-) -buta-1,2-diene (4-) -dimethylhydantoin -dinitrobenzene (α)(1-,2-)(4-) (α)(1-,3-)(4-) -diphenyl (o-) (m-) (p-) -hydroquinone -naphthalene (α-) (β-) -nitrobenzene (o-) (m-) (p-) -nitrotoluene (2-,4-) (2-,6-) -phenol (o-) (m-) (p-) -propionic acid (α)(dl-) -toluene (o-) (m-) (p-) Chloroform Chlorophyll (α-) Chloropicrin Cholesterol Chrysene Chrysoidine (2-,4-) Chrysophanic acid Cinchomeronic acid (3-,4-) Cineole, eucalyptole Cinnamic acid (cis-) (trans-) aldehyde Cinnamyl alcohol cinnamate Citraconic acid (cis-) Citral (α) Citric acid Citronellal (d-) Citronellol (d-) Coniine (d-)(2-)
H2NC6H3(CH3)C3H7 C10H14O C2H5O(CH2)2OH CH3CO2CH2CH2OC2H5 (C6H10O5)x CH3CO2(CH2)15CH3 CH3(CH2)14CH2OH CCl3CHO CCl3CH(OH)2 OC:(CClCCl)2:CO Cl3CC(OH)(CH3)2 CH3CO2NHC6H4Cl ClCH2CO2H CH3COCH2Cl C6H5COCH2Cl ClCH2COCl ClC6H4NH2 ClC6H4NH2 ClC6H4NH2 C6H4(CO)2C6H3Cl C6H4(CO)2C6H3Cl ClC6H4CHO ClC6H4CHO ClC6H4CHO C6H5Cl ClC6H4CO2H ClC6H4CO2H ClC6H4CO2H CH2:CClCH:CH2 CH2:CHCH:CHCl CH2:C:CHCH2Cl C(CH3)2N(Cl)CON(Cl)CO ClC6H3(NO2)2 ClC6H3(NO2)2 C6H5C6H4Cl C6H5C6H4Cl C6H5C6H4Cl ClC6H3(OH)2 C10H7Cl C10H7Cl ClC6H4NO2 ClC6H4NO2 ClC6H4NO2 CH3C6H3(NO2)(Cl) CH3C6H3(NO2)(Cl) ClC6H4OH ClC6H4OH ClC6H4OH CH3CHClCO2H CH3C6H4Cl CH3C6H4Cl CH3C6H4Cl CHCl3 C55H72O5N4Mg Cl3CNO2 C27H45OHH2O C18H12 C6H5N:NC6H3(NH2)2 C14H5(OH)2(CH3)O2 C5H3N(CO2H)2 C10H18O C6H5CH:CHCO2H C6H5CH:CHCO2H C6H5CH:CHCHO C6H5CH:CHCH2OH C8H7CO2C9H9 CH3C(CO2H):CHCO2H C9H15CHO C3H4(OH)(CO2H)3 C9H17CHO C10H20O C3H7C5H10N
149.23 150.21 90.12 132.16 162.14 284.47 242.43 147.40 165.42 245.89 177.47 169.61 94.50 92.53 154.59 112.95 127.57 127.57 127.57 242.65 242.65 140.57 140.57 140.57 112.56 156.57 156.57 156.57 88.54 88.54 88.54 197.03 202.56 202.56 188.65 188.65 188.65 144.56 162.61 162.61 157.56 157.56 157.56 171.56 171.56 128.56 128.56 128.56 108.53 126.58 126.58 126.58 119.39 893.48 164.39 404.65 228.28 212.25 254.23 167.12 154.24 148.15 148.15 132.15 134.17 264.31 130.10 152.23 192.12 154.24 156.26 127.22
oil col. lq. col. lq. col. lq. amor. nd. lf. col. lq. mn. pr. yel./bz. col. cr. rhb. col. cr. col. lq. rhb. col. lq. lq. lq. rhb. yel. nd. nd./al. nd. pr. pr. col. lq. mn./aq. pr. tri. col. lq. col. lq. col. lq. cr./et. rhb./et. cr. cr. lf. mn. col. lq. lf./al. mn. nd. yel./al. mn. pr. cr. cr. col. lq. nd. nd. col. lq. col. lq. col. lq. col. lq. col. lq. lq. rhb./al. col. rhb. yel. cr. yel./al. cr./HCl col. oil mn. pr. mn. pr. lq. nd. nd. or pr. nd. col. oil cr. col. oil col. oil col. lq.
0.99420 0.96120/4 0.93120/4 0.975 20/4 1.3–1.4 0.858 20 0.818 50/4 1.505 25/4 1.619 50/4 1.385 22 1.58 20/20 1.16216 1.32415 1.498 20/20 1.21320/4 1.216 20/4 1.42719 1.298 1.25015 1.196 61 1.107 20/4 1.544 25/4 1.496 25/4 1.54124 0.958 20/20 0.965 20/20 0.99120/20 1.5 20/20 1.697 22
1.194 20/4 1.26616 1.305 80/4 1.34350/4 1.298 91 1.256 80 1.24118/15 1.268 25 1.306 20/4 1.306 9 1.082 20/4 1.07220/4 1.070 20/4 1.489 20 1.65123/4 1.067
0.927 20 1.2844 1.245 1.110 20/20 1.040 35/35 1.08516.5 1.617 0.89017/4 1.54220/4 0.85517.5 0.84820/4 0.84717
−16 −70 22–3 49–50 −57 51.7 290 97 175–6 61.2 −44.5 58–9 0 −10.4 70–1 162 208–9 11 17–8 47.8 −45.2 141–2 158 242–3
130 39(36) 53(43) 34 89 77.5 106 −20 56–7 32.5 44.4(24) 83–4 38.2 37.5 7(0) 32–3 41–3 <−20 −34 −47.8 7.5 −63.5 d. −64 149–51 253–4 117.5 195 258–9 d. 1.5 68 133 −7.5 33 44 92–3 153 −2
241 230766 135.1 156.3 i. 20015 189.515 97.6768 d. 98 subl. 167 189.5 121 245–7 105 210.5 230767 230–1 subl. 208748 213–4 213748 132.1 subl. 59.4 69 88 315 d. 315 d. 267–8 284–5 282 263 sl. d. 259.3 264751 245.5753 235.6 242761 240718 238 175–6 214 217 186 159.5 161.6 162.2 61.2 i. 112.3766 subl. 448 subl. subl. d. 176–7 12519 300 252 sl. d. 257.5 229 d. 204–8 224–5 166–7
v. sl. s. i. ∞ 22 i. i. i. v. s. 47417 i. 0.8 c. sl. s. v. s. ∞ 0.11 d. i. i. s. h. i. i. v. sl. s. v. sl. s. s. h. 0.049 20 0.20825 0.04125 0.00825 v. sl. s. v. sl. s. d. 0.2125 i. i. i. i. i. v. s. i. i. i. i. i. i. i. 2.8520 2.6020 2.7120 ∞ i. i. i. 0.8220 s. 0.1718 0.2620 i. sl. s. h. i. c. v. sl. s. 1.915
s. ∞ ∞ ∞ i. v. sl. s. c. s. ∞ v. s. i. c. 111 s. s. ∞ v. s. d.
0.0418 v. sl. s. sl. s. i. 36025 i. 207.725 v. sl. s. v. sl. s. 1.1
s. sl. s. h.
s. ∞ ∞ ∞ s. ∞ s. i. c. s. v. s. s. ∞ v. s. s. s. s.
v. s. v. s. v. s. ∞ s. s. s. ∞ ∞
v. s. v. s. v. s. ∞ s. s. s. ∞ ∞
v. s. h. s. h.
v. s. s.
v. s. s. v. s. s. h. v. s. h. v. s. h.
v. s. ∞ v. s. s. v. s. v. s.
s. s. v. s. ∞ s. s. s. ∞
s. s. v. s. ∞ ∞ ∞ ∞ ∞
s. 1.117 0.116 s. s. h. sl. s. ∞
s. 18 v. sl. s. s. sl. s. i. ∞
2420 s. v. s. 4 c. s. ∞ 7615 ∞ ∞ v. s.
v. s. ∞ v. s. 33 s. ∞ 215 ∞ ∞ v. s.
2-33
TABLE 2-2
Physical Properties of Organic Compounds (Continued ) Name
Coumaric acid (o-) (p-) Coumarin Coumarone Creatine Creatinine Creosol (3-,1-,4-) Cresidine (1-,2-,4-) Cresol (o-) (m-) (p-) Cresyl benzoate (o-) (m-) (p-) Crotonic acid (α-) acid (β-)(cis-) aldehyde (α) Cumene Cumic acid (p-) Cumidine (p-) Cyanamide Cyanic acid Cyanoacetic acid Cyanogen bromide chloride Cyanuric acid Cyclo-butane -heptane -hexane -hexanol -hexanone -hexene -hexyl acetate amine bromide chloride -pentadiene (1-,3-) -pentane -pentanone -propane Cymene (o-) (m-) (p-) Cystine (l-) Dambose Decahydronaphthalene (cis-) (trans-) Decane (n-) Decyl alcohol Dextrin Diacetone alcohol Diamino-benzophenone (4-,4′-) -diphenylamine (4-,4′-) -diphenylmethane (4-,4′-) -diphenylurea (4-,4′-) Diamyl-amine (i-) ether (n-) (i-) Diamyl ketone (i-) phthalate (n-) (i-) tartrate (i-) Dianisidine (o-)(4-,3-)2 Diazo-aminobenzene -aminotoluene (2-,2′-) -methane
2-34
Formula HOC6H4CH:CHCO2H HOC6H4CH:CHCO2H C9H6O2 C8H6O C4H9N3O2H2O C4H7N3O CH3OC6H3(CH3)OH CH3(NH2)C6H3OCH3 CH3C6H4OH CH3C6H4OH CH3C6H4OH C6H5CO2C6H4CH3 C6H5CO2C6H4CH3 C6H5CO2C6H4CH3 CH3CH:CHCO2H CH3CH:CHCO2H CH3CH:CHCHO C6H5CH(CH3)2 (CH3)2CHC6H4CO2H (CH3)2CHC6H4NH2 H2NCN HOCN or HNCO CH2(CN)CO2H (CN)2 BrCN ClCN C3H3O3N32H2O CH2 < (CH2)2 > CH2 CH2 < (CH2CH2CH2)2 > CH2 < (CH2CH2)2 > CH2 CH2 < (CH2CH2)2 > CHOH CH2 < (CH2CH2)2 > CO (CH2CH2CH:)2 CH3CO2C6H11 CH2 < (CH2CH2)2 > CHNH2 CH2 < (CH2CH2)2 > CHBr CH2 < (CH2CH2)2 > CHCl CH2 < (CH:CH)2 > CH2 < (CH2CH2)2 > < (CH2CH2)2 > CO < CH2CH2CH2 > CH3C6H3CH(CH3)2 CH3C6H4CH(CH3)2 CH3C6H4CH(CH3)2 [SCH2CH(NH2)CO2H]2 C6H6(OH)6 C10H18 C10H18 CH3(CH2)8CH3 CH3(CH2)8CH2OH (C6H10O5)x (CH3)2C(OH)CH2COCH3 H2NC6H4COC6H4NH2 H2NC6H4NHC6H4NH2 H2NC6H4CH2C6H4NH2 (H2NC6H4NH)2CO [(CH3)2CHCH2CH2]2NH (C2H5CH2CH2CH2)2O [(CH3)2CH(CH2)2]2O [(CH3)2CHCH2CH2]2CO C6H4(CO2C5H11)2 C6H4(CO2C5H11)2 (HOCHCO2C5H11)2 [NH2(OCH3)C6H3]2 C6H5N:NNHC6H5 C7H7N:NNHC7H7 CH2:N2
Formula weight
Form and color
164.15 164.15 146.14 118.13 149.15 113.12 138.16 137.18 108.13 108.13 108.13 212.24 212.24 212.24 86.09 86.09 70.09 120.19 164.20 135.20 42.04 43.03 85.06 52.04 105.93 61.48 165.11 56.10 98.18 84.16 100.16 98.14 82.14 142.19 99.17 163.06 118.61 66.10 70.13 84.11 42.08 134.21 134.21 134.21 240.29 180.16 138.24 138.24 142.28 158.28 162.14 116.16 212.24 199.25 198.26 242.28 157.29 158.28 158.28 170.29 306.39 306.39 290.35 244.28 197.23 225.28 42.04
nd./aq. cr./aq. rhb./et. oil mn./aq. mn. pr. nd./pet. cr. lq. pr. lq. cr. cr. col. mn. nd. col. lq. col. lq. tri. lq. col. nd. gas col. lq. col. gas nd. gas mn./aq. col. gas oil col. lq. col. nd. col. oil lq. oil col. lq. col. lq. col. lq. col. lq. col. oil col. oil col. gas col. lq. col. lq. col. lq. pl. mn./aq. lq. lq. col. lq. col. oil amor. lq. yel. nd. lf./aq. nd./aq. cr. col. lq. col. lq. col. lq. yel. oil col. lq. col. lq. lq. col. lf. yel. lf. or. cr. gas
Specific gravity
0.93520/4 1.07815/15 1.09220/20 1.04820/4 1.03420/4 1.03520/4
0.96479.7 1.03115/4 0.85320/20 0.86220/4 1.1624 0.953 1.07348/4 1.1400 0.86617 2.01520/4 1.2220 1.7680/4 0.7030/4 0.81020/4 0.77920/4 0.96220/4 0.94719/4 0.81020/4 0.9850/4 0.86520/0 1.32420/20 0.97718/4 0.80519/4 0.74520/4 0.94820 0.720−79 0.87520/4 0.86220 0.85720/4 1.752 0.89518/4 0.87220/4 0.7302 0.83020/4 1.038 0.93125
0.76721/4 0.77420/4 0.77720/4 0.82125/4
Melting point, °C
Boiling point, °C
207–8 206–7 d. 70 <−18 295 260 d. 5.5 93–4 30.8 10.9 35–6
subl.
55 71.5 72 15.5 −69 −96.9 116–7 <−20 44–5 −80 65–6 −34.4 52 −6.5 >360 −50 −12 6.5 23.9 −45 −103.7
−43.9 −85 −93.3 −58.2 −126.6 <−25 −73.5 d. 258–61 253 −51 −32 −29.7 7 −47 237–9 158 93–4 subl. 310 −44 −69 14.6
1.03 1.06315/4 131.5 96–8 51 −145
290–1 173–4 221–2765 235 190.8 202.8 202 308 314 316 189 170–1 d. 102.2 152.5 subl. 225761 14019 −640 1080.2 −21 61.3750 12.5–13 d. 11–12726 118–20 80–1 160–1 155–6 83.3 174750 134 165714 142 41–2 49–50 129–30 −34749 177 175–6 176–7 31915 193.3 185.3 174.0 232.9 167.9 d. 249–5315 188–90 190 173.4 228 204–611 22540 19516 expl. −23
Solubility in 100 parts Water sl. s. c. s. h. 0.3 c. i. 1.418 8.716 v. sl. s. v. sl. s. 2.5 0.5 1.8 i. i. i. 8.315 ∞25 18 i. 0.0225 i. v. s. sl. s. s. 45020 cc. s. 250020 cc. 0.2717 i. i. i. 3.620 s. v. sl. s. i. i. i. i. i. i. v. sl. s. i. i. i. i. 0.0119 212 i. i. i. i. s. ∞ sl. s. h. sl. s. sl. s. c. v. sl. s. sl. s. i. i. i. i. i. i. i. 0.05 d.
Alcohol
Ether
s. v. s. h. v. s.
v. sl. s. v. s. s. s. i.
0.0117 116 ∞ s. ∞30 ∞ ∞36
∞ s. ∞30 ∞ ∞36
s. ∞ ∞ s.
∞ ∞ s.
v. s. s. 230020 cc. s. v. s. 0.122 v. s. ∞ s. s. v. s. ∞ s. s. ∞ s. s. s. s. i. i. s. s. ∞ s. i. ∞ s. s. s.
v. s. s. s. 50020 cc. s. 500020 cc.
∞ s. s. v. s. ∞ s. ∞ ∞ s. s. s. s. i. s. s. ∞ i. ∞ s. s. s.
s. ∞ ∞ s.
∞ ∞ ∞ s.
s.
s.
s. s. h.
s. v. s. s.
Dibenzothiazyl-disulfide (2-,2′-) Dibensoyl methane Dibensyl-amine -aniline ketone phthalate (o-) succinate Dibromo-benzene (o-) (m-) (p-) -diphenyl (4-,4′-) Dibutyl-adipate (n-) (i-) -amine (n-) (i-) -p-aminophenol (s-) -aniline (n-) carbonate (n-) (i-) (s-) ether (n-) (i-) (s-) ketone (n-) (i-) malate (l-)(n-) oxalate (n-) phthalate (n-) tartrate (d-)(n-) (d-)(i-) Dichloro-acetic acid -acetone (αα-) -aniline (2-,5-) -anthraquinone (1-,3-) (1-,4-) (1-,5-) (1-,6-) (1-,8-) (2-,3-) (2-,6-) (2-,7-) -benzene (o-) (m-) (p-) -butane (n-)(1-,4-) -diphenyl (4-,4′-) -ethane (1-,2-) -naphthalene (β-)(1-,4-) (γ-)(1-,5-) -nitrobenzene (2-,5-) -pentane (1-,5-) -phenol (2-,4-) Dichloramine T (p-) Dicyandiamide Diethanolamine Diethyl adipate -amine -aminophenol (m-) -aniline sulfonic acid (m-) carbonate diethyl malonate Diethyl dimethyl malonate glutarate ketone malonate -malonic acid -naphthylamine (α-) (β-) oxalate phthalate (o-) sulfate sulfide
(C6H4NSC)2S2 (C6H5CO)2CH2 (C6H5CH2)2NH C6H5N(CH2C6H5)2 (C6H5CH2)2CO C6H4(CO2CH2C6H5)2 (CH2CO2CH2C6H5)2 C6H4Br2 C6H4Br2 C6H4Br2 BrC6H4C6H4Br (CH2CH2CO2C4H9)2 (CH2CH2CO2C4H9)2 (C2H5CH2CH2)2NH [(CH3)2CHCH2]2NH (C4H9)2NC6H4OH C6H5N(C4H9)2 CO(OC4H9)2 CO(OC4H9)2 CO(OC4H9)2 (C2H5CH2CH2)2O [(CH3)2CHCH2]2O [C2H5(CH3)CH]2O (C2H5CH2CH2)2CO [(CH3)2CHCH2]2CO C2H4O(CO2C4H9)2 (CO2C4H9)2 C6H4(CO2C4H9)2 (CHOHCO2C4H9)2 (CHOHCO2C4H9)2 Cl2CHCO2H Cl2CHCOCH3 Cl2C6H3NH2 C6H4:(CO)2:C6H2Cl2 C6H4:(CO)2:C6H2Cl2 C6H3Cl:(CO)2:C6H3Cl C6H3Cl:(CO)2:C6H3Cl C6H3Cl:(CO)2:C6H3Cl C6H4:(CO)2:C6H2Cl2 C6H3Cl:(CO)2:C6H3Cl C6H3Cl:(CO)2:C6H3Cl C6H4Cl2 C6H4Cl2 C6H4Cl2 ClCH2(CH2)2CH2Cl ClC6H4C6H4Cl ClCH2CH2Cl C10H6Cl2 C10H6Cl2 Cl2C6H3NO2 ClCH2(CH2)3CH2Cl Cl2C6H3OH CH3C6H4SO2NCl2 H2NC(:NH)NHCN HN(CH2CH2OH)2 (CH2CH2CO2C2H5)2 (C2H5)2NH (C2H5)2NC6H4OH (C2H5)2NC6H5 (C2H5)2NC6H4SO3H OC(OC2H5)2 (C2H5)2C(CO2C2H5)2 (CH3)2C(CO2C2H5)2 CH2(CH2CO2C2H5)2 (C2H5)2CO CH2(CO2C2H5)2 (C2H5)2C(CO2H)2 C10H7N(C2H5)2 C10H7N(C2H5)2 (CO2C2H5)2 C6H4(CO2C2H5)2 O2S(OC2H5)2 (C2H5)2S
232.46 224.25 197.27 273.36 210.26 346.36 298.32 235.92 235.92 235.92 312.02 258.35 258.35 129.24 129.24 221.33 205.33 174.23 174.23 174.23 130.22 130.22 130.22 142.23 142.23 246.30 202.24 278.34 262.30 262.30 128.95 126.98 162.02 277.10 277.10 277.10 277.10 277.10 277.10 277.10 277.10 147.01 147.01 147.01 127.02 223.10 98.97 197.06 197.06 192.01 141.04 163.01 240.11 84.08 105.14 202.24 73.14 165.23 149.23 229.29 118.13 216.27 188.22 188.22 86.13 160.17 160.17 199.28 199.28 146.14 222.23 154.18 90.18
cr. rhb./al. col. oil pr./al. cr. pr./al. lf./al. col. lq. col. lq. pl./al. mn. pr. col. lq. col. lq. col. lq. col. lq. lq. lq. col. lq. col. lq. col. lq. lq. lq. lq. lq. oil lq. col. lq. col. lq. pr. cr. lq. lq. nd. yel. nd. yel. nd. yel. nd. yel. nd. yel. nd. yel. nd. yel. nd. yel. nd. col. lq. col. lq. col. mn. lq. pr. col. lq. nd./al. lf./al. tri./al. col. lq. nd. cr. mn. pl. pr. col. lq. col. lq. rhb. oil cr. col. lq. col. lq. col. lq. syrup col. lq. col. lq. pr./aq. col. oil col. oil col. lq. col. lq. col. lq. col. lq.
1.50 1.028 25/25
20/4
1.956 1.952 20/4 2.26118 1.897 0.965 20/4 0.950 25 0.768 20/20 0.74125/4
180 78 −26 70–1 34–5 42–3 45–6 1.8 −6.9 87–8 164–5 −38 −20 −70
0.924 20/4 0.91915 0.769 20/20 0.76215 0.756 21 0.82718/4 0.805 21/4 1.038 20/4 0.986 20/4 1.045 21 1.09815 1.03175/4 1.560 25/25 1.23415
1.305 20/4 1.288 20/4 1.458 21 1.442 0/4 1.256 20/20 1.300 76/4 1.669 22 1.094 25/4 1.383 60/25 1.4014 1.09720/4 1.00920/4 0.712 15/15 0.934 20/4 20/4
0.975 0.985 20/4 0.994 25/25 1.025 21 0.816 19/4 1.055 20/4 1.005 1.026 1.079 20/4 1.121 25/25 1.172 25/4 0.837 20/4
−98 −5.9 −29.6 22–2.5 73–4 9.7(−4) 50 208–9 187.5 251 203–4 202–3 268–70 282 210–11 −17.6 −24.8 53 −38.7 148 −35.3 67–8 107 54.6 45 83 207–8 28 −21 −38.9 78 −34.4 270 d. −43 −24 −42 −49.8 125 −40.6 −25 −99.5
d. 219–2118 268–71250 >300 330.6 27412 23814 221–2 219755 218.6758 355–60 18314 278–80 159761 139–40 17010 262.8 207740 190 178–80 142.4 122.5 121 187.7 168.1 170–118 245.5 340 200–318 323–5 194.4 120 251
179 172766 174764 161–3 315–9 83.7 286–7 740 subl. 266 180–1 209–10 d. 270 748 239–41761 55.5759 276–80 216 759
126 230 196.7 237 101.7 198.9 d. 170–80 285–90 318 186 298–9 210 92–3 754
i. i. i. i. i. v. sl. s. i. i. i. i. i. i. i. ∞ v. sl. s. i. i. i. i.
4.420 s. v. s. h.
s. s. s.
s. s. s. s. 1.6 v. sl. s. h. ∞
s. s. s. s. 7125 ∞
∞ s.
∞ s.
∞ s.
∞
<0.05 i. i. i. <0.06 v. sl. s. i. 0.04 25 i. v. sl. s. ∞ v. sl. s. v. sl. s. i. i. i. i. i. i. i. i. i. i. i.
∞ ∞ ∞ s. ∞
∞ ∞ ∞ v. s. ∞
s. ∞
s. ∞
∞ s. s. i. v. sl. s. sl. s.
∞ s. s.
∞ s. v. s.
∞ s. v. s.
i. 0.90 i. i. i. i. 0.45 20 sl. s. 2.318 ∞ 0.4380 v. s. s. 1.412 s. i. i. i. 0.88 20 4.7 20 2.08 20 65 16 i. i. v. sl. s. i. i. 0.3120
v. sl. s. ∞ v. sl. s. s. v. s. h. s. v. s.
425 ∞
1.318 ∞ s. ∞
0.0118 v. sl. s. s. ∞
s.
s.
∞ ∞ ∞ v. s. ∞ ∞ v. s. ∞ ∞ ∞ ∞ s. ∞
∞ ∞ ∞ s. ∞ ∞ v. s. ∞ ∞ ∞ ∞ ∞ ∞
v. sl. s.
sl. s. sl. s.
s. s. v. s.
2-35
TABLE 2-2
Physical Properties of Organic Compounds (Continued ) Name
tartrate (d-) -toluidine (o-) (m-) (p-) Diethyleneglycol dinitrate Difluorodichloromethane Diglycerol Dihydroxy-dinaphthyl (α-) (-2,-2′,-1,-1′) -diphenyl (4-,4′-) -ethyl formal (β-) -naphthalene (1-,5-) (1-,8-) Dimethoxy-benzene (p-) -diphenylamine (4-,4′-) -ethyl adipate Dimethyl adipate -amine -aminoasobenzene (p-) -aminoethanol -aminophenol (m-) -aniline sulfonic acid (m-) (p-) carbonate ether -formamide fumarate glutarate glyoxime -naphthalene (1-,4-) (2-,3-) -naphthylamine (α-) (β-) oxalate phthalate (o-) sulfate sulfide tartrate (d-) -vinyl-ethenyl carbinol Dinaphthyl (αα-) -methane (αα′-) (β,β′-) Dinitro-anisole (1-)(2-,4-) -benzene (o-) (m-) (p-) sulfonic acid (2-,4-)(1-) -benzoic acid (2-,4-) (3-,5-) -benzophenone (4-,4′-) -diphenyl (4-,4′-) (2-,4′-) -naphthalene (1-,5-) (1-,8-) Dinitro-phenol (2-,3-) (2-,4-) (2-,6-) -salicylic acid (3-,5-) -stilbene (4-,4′-) -toluene (2-,4-) (3-,4-) (3-,5-) Dioxane Dipentene
2-36
Formula (CHOHCO2C2H5)2 CH3C6H4N(C2H5)2 CH3C6H4N(C2H5)2 CH3C6H4N(C2H5)2 O(CH2CH2ONO2)2 F2CCl2 [(HO)2C3H5]2O (HOC10H6)2 (HOC10H6)2 (HOC6H4)2 CH2(OCH2CH2OH)2 C10H6(OH)2 C10H6(OH)2 (CH3O)2C6H4 HN(C6H4OCH3)2 (CH2)4(CO2C2H4OCH3)2 [(CH2)2CO2CH3]2 (CH3)2NH C6H5N:NC6H4N(CH3)2 (CH3)2NCH2CH2OH (CH3)2NC6H4OH (CH3)2NC6H5 (CH3)2NC6H4SO3H (CH3)2NC6H4SO3HH2O OC(OCH3)2 CH3OCH3 HCON(CH3)2 (:CHCO2CH3)2 (CH2)3(CO2CH3)2 (CH3C:NOH)2 C10H6(CH3)2 C10H6(CH3)2 C10H7N(CH3)2 C10H7N(CH3)2 (CO2CH3)2 C6H4(CO2CH3)2 (CH3O)2SO2 (CH3)2S (CHOHCO2CH3)2 (CH3)2COHC⯗CCH:CH2 C10H7C10H7 (C10H7)2CH2 (C10H7)2CH2 CH3OC6H3(NO2)2 C6H4(NO2)2 C6H4(NO2)2 C6H4(NO2)2 (NO2)2C6H3SO3H3H2O (NO2)2C6H3CO2H (NO2)2C6H3CO2H (NO2C6H4)2CO (NO2C6H4)2 (NO2C6H4)2 C10H6(NO2)2 C10H6(NO2)2 (NO2)2C6H3OH (NO2)C6H3OH (NO2)C6H3OH (NO2)2C6H2(OH)CO2HH2O (NO2C6H4CH:)2 (NO2)2C6H3CH3 (NO2)2C6H3CH3 (NO2)2C6H3CH3 O < (CH2CH2)2 > O C13H10
Formula weight
Form and color
Specific gravity
Melting point, °C
Boiling point, °C
206.19 163.25 163.25 163.25 196.12 120.92 166.17 286.31 286.31 186.20 136.15 160.16 160.16 138.16 229.26 262.30 174.19 45.08 225.28 89.14 137.18 121.18 201.24 219.25 90.08 46.07 73.09 144.12 160.17 116.12 156.22 156.22 171.23 171.23 118.09 194.18 126.13 62.13 178.14 110.15 254.31 268.34 268.34 198.13 168.11 168.11 168.11 302.22 212.12 212.12 272.21 244.20 244.20 218.16 218.16 184.11 184.11 184.11 246.13 270.24 182.13 182.13 182.13 88.10 136.23
lq. lq. lq. lq. lq. gas lq. pl./al. nd./al. rhb./al. lq. pr./aq. nd. lf. cr. lq. col. lq. col. lq. yel./al. col. lq. nd. yel. lq. cr. pr. col. lq. gas lq. col. tri. lq. col. cr. lq. lf./al. col. oil col. cr. col. mn. col. lq. col. oil oil cr. lq. lf./al. pr./al. nd./al. col. mn. col. mn. col. rhb. col. mn. pr. cr./aq. mn. pr. col. nd. nd./al. mn. nd. rhb. yel. mn. yel. rhb. yel. rhb. pl./aq. yel. lf. nd. nd. mn. pr. col. lq. col. lq.
1.204 20/4
17
0.924 15.5 1.377 25/4 1.486 −30
−11.3 −155
280 208–9755 231–2 228–9
1.25 1.154 25 1.053
55/55
1.075 15.6 1.063 20/4 0.680 0/4 0.887 20/4 0.956 20/4 1.070 20/4 0.94525 1.08915.6 1.016 20/4 1.042 20 1.03970/70 1.14854 1.18925/25 1.3520/4 0.84621/4 1.32820/4 0.887 20/4
1.341 20 1.5918 1.575 20/4 1.625 18
1.445 1.474 1.681 20 1.683 24
1.321 71 1.259 111 1.277 111 1.033 20/4 0.865 18
300 218 270–2 −5.3 258–60 140 56 103 10–1 −96 116–7 85 2.5 d. 266 257 0.5 −138.5 −58.3 102 −37 240–6 <−18 104 46 54 −26.8 −83.2 61.5 160 109 92 94–5 117–8 89.8 173–4 106–8 179–80 204–5 189 233 93.5 216 170–2 144–5 114–5 63–4 173 d. 210–6 70 60–1 92–3 9.5–10.5
−29.2 220–3010 subl. subl. 264 d. 212.6 145–502 11518 7.4 d. 135756 265–8 193 89–90 −23.7 152.8 192 13050 264–6 265767 274.5711 304–5 163.3 280734 188.3 37.3 280 150 240–412 >360 319774 300–2 299777 subl.
subl. d. subl.
300 subl. 101.1 178
Solubility in 100 parts Water sl. s. i. i. i. i. 5.7 cc.26 s. h. i. i. sl. s. ∞ sl. s. sl. s. h. v. sl. s. i. 5 i. v. s. i. ∞ sl. s. h. i. s. s. h. i. 3700 cc.18 ∞ i. 20
0.06 i. i. i. i. 6 0.43 v. sl. s. i. s. 620 i. i. i. sl. s. h. 0.01 c. 0.399 0.18100 s. 1.8525 s h. i. i. i. i. i. sl. s. 0.5 c. s. h. s. c. i. 0.0322 i. sl. s. ∞ i.
Alcohol
Ether
∞ s. s.
∞ s. s.
s.
s. i. v. s. v. s. v. s.
s. s. v. s. s. v. s.
v. s. v. s. v. s.
s. s.
s. s.
s. s.
s. s.
v. sl. s. ∞ s.
v. sl. s. ∞ s.
sl. s.
sl. s.
v. s.
v. s.
sl. s. s. s. s.
s. s. s.
∞ s. 20015
∞ s.
s. h. 0.8 c. s. 1.520 1.921 320 0.1821 s. s. v. s.
s. v. s.
v. sl. s. sl. s.
1.520 v. s. h. 0.2 c. v. s. h. 420 s. h. v. s. v. sl. s. 1.215
v. s. v. s. h. s. v. s. v. sl. s. 916
s. h. s.
s. s.
Diphenyl -amine carbonate -chloroarsine -ethane ether guanidine -methane phenylenediamine (p-) succinate sulfide sulfone urea (uns.) Diphenylene oxide Dipropyl adipate (n-) -amine (n-) (i-) aniline (n-) carbonate (n-) ether (n-) (i-) ketone (n-) (i-) oxalate (n-) (i-) Disalicylal ethylenediamine Ditolyl guanidine (o-) Divinyl acetylene Docosane (n-) Dodecane (n-) Dulcitol Durene (1-,2-,4-,5-) Elaidic acid Eosine Ephedrine (l-) Epichlorhydrin (α-) Epidichlorohydrin (α-) Erythritol (dl-) tetranitrate Ethane Ethanol-amine formamide Ether Ethyl abietate acetate acetoacetate alcohol -amine hydrochloride aniline sulfonic acid (m-) anisate (p-) anthranilate (o-) benzene benzoate -benzyl-aniline bromide butyrate (n-) (i-) caprate (n-) Ethyl caproate (n-) caprylate (n-) chloride chloroacetate chlorocarbonate cinnamate (trans-) cyanoacetate formate furoate (α) heptoate hypochlorite iodide lactate
C6H5C6H5 C6H5NHC6H5 O(COC6H5)2 (C6H5)2AsCl (C6H5CH2)2 C6H5OC6H5 (C6H5NH)2C:NH (C6H5)2CH2 (C6H5NH)2C6H4 (CH2CO2C6H5)2 (C6H5)2S (C6H5)2SO2 (C6H5)2NCONH2 < (C6H4)2O (CH2CH2CO2C3H7)2 (C2H5CH2)2NH [(CH3)2CH]2NH C6H5N(C3H7)2 O(COCH2C2H5)2 (C2H5CH2)2O [(CH3)2CH]2O (C2H5CH2)2CO [(CH3)2CH]2CO (CO2CH2C2H5)2 [CO2CH(CH3)2]2 [HOC6H4CH:NCH2]2 (C7H7NH)2C:NH (H2C:CHC⯗)2 CH3(CH2)20CH3 CH3(CH2)10CH3 CH2OH(CHOH)4CH2OH (CH3)4C6H2 C8H17CH:CH(CH2)7CO2H C20H8O5Br4 C6H5CHOHCH(CH3)NHCH3 C2H3OCH2Cl CH2:CClCH2Cl CH2OH(CHOH)2CH2OH C4H6(ONO2)4 CH3CH3 HOCH2CH2NH2 HCONHCH2CH2OH (CH3CH2)2O C19H29CO2C2H5 CH3CO2C2H5 CH3COCH2CO2C2H5 CH3CH2OH C2H5NH2 C2H5NH2HCl C6H5NHC2H5 C2H5NHC6H4SO3H CH3OC6H4CO2C2H5 NH2C6H4CO2C2H5 C6H5C2H5 C6H5CO2C2H5 C6H5N(C2H5)CH2C6H5 C2H5Br C2H5CH2CO2C2H5 (CH3)2CHCO2C2H5 CH3(CH2)8CO2C2H5 CH3(CH2)4CO2C2H5 CH3(CH2)6CO2C2H5 CH3CH2Cl ClCH2CO2C2H5 ClCO2CH2CH3 C6H5CH:CHCO2H CH2(CN)CO2C2H5 HCO2CH2CH3 OC4H3CO2C2H5 CH3(CH2)5CO2C2H5 ClOCH2CH3 CH3CH2I CH3CH(OH)CO2C2H5
154.20 169.22 214.21 264.57 182.25 170.20 211.26 168.23 260.32 270.27 186.26 218.26 212.24 168.18 230.30 101.19 101.19 177.28 146.18 102.17 102.17 114.18 114.18 174.19 174.19 268.30 239.31 78.11 310.59 170.33 182.17 134.21 282.45 647.93 165.23 92.53 110.98 122.12 302.12 30.07 61.08 89.09 74.12 330.49 88.10 130.14 46.07 45.08 81.55 121.18 201.24 180.20 165.19 106.16 150.17 211.29 108.98 116.16 116.16 200.31 144.21 172.26 64.52 122.55 108.53 176.21 113.11 74.08 140.13 158.23 80.52 155.98 118.13
col. mn. col. mn. nd./al. rhb. col. pr. col. rhb. mn./al. col. pr. cr. lf./al. col. lq. nd./aq. rhb. lf./al. col. lq. col. lq. col. lq. yel. oil col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. cr. cr. lq. cr. lq. mn. mn. lf./al. col. cr. cr./et. lq. col. lq. tet. pr. lf./al. col. gas col. oil lq. col. lq. lq. col. lq. col. lq. col. lq. col. lq. mn. lq. nd./aq. lq. cr. col. lq. col. lq. yel. oil col. lq. col. lq. col. lq. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lf. col. lq. yel. lq. col. lq. oil
0.992 73/4 1.160 20/20 1.272 14 1.583 40 0.978 50/50 1.073 20 1.001 26/4 1.119 15/15 1.248 25/4 1.276 0.979 20/4 0.739 20/4 0.722 22 0.910 20 0.968 22 0.744 21/0 0.725 21/0 0.822 20/4 0.806 20/4 1.038 0/0 1.34 1.1020/4 0.776 20/4 0.778 44/4 0.751 20/4 1.466 15 0.838 81/4 0.851 79/4 1.183 25/25 1.204 25 1.451 20/4 −88
0.546 1.022 20 1.169 25 0.708 25/4 1.020 20/20 0.901 20/4 1.025 20/4 0.789 20/4 0.689 15/15 1.216 0.963 20/4 1.103 25/25 1.117 20/4 0.867 20/4 1.052 15/15 1.034 18.5 1.431 20/4 0.879 20/4 0.871 20/4 0.859 28 0.873 20/20 0.878 17 0.917 6/6 1.159 20/4 1.138 20/4 1.049 20/4 1.062 20/4 0.923 20/4 1.117 21/4 0.872 20/20 1.013 −6/4 1.933 20/4 1.030 25/4
69–70 52.9 80 43–4 52–3 27 147–8 26–7 152 122–3 <−40 128–9 189 86–7 −20.3 −39.6 −61 −122 −60 −32.6 −51.7
254.9 302 302–6 d. 327 284 259 d. > 170 265 330 296–7 379 287–8 143–510 110–1 83.5743 245.4 168.2 91 69 144.2 123.7 213.5 190
125–6 178–9 44.5 −9.6 189 79–80 51–2 40 −25.6 126 61 −172 10.5 <−40 −116.3 −82.4 −45 −112 −80.6 108–9 −63.5 d. 294 7–8 13 −94.4 −34.6 −117.8 −93.3 −88.2 −20 −67.5 −45 −139 −26 −80.6 12 −22.5 −79 34 −66.1 expl. −105
85 224.515 214.5 290–53 193–5 288100 255 117756 94 329–31 expl. −88.6 171757 d. 34.6 2004 77.1 180755 78.4 16.6 204 269–70 266–8 136.2 211–2 28510 38.4 120–1 110–1 244.6758 165–6736 207–8753 13 144 94–5 271 208753 54760 195766 187–8 36752 72.4 155
i. 0.0325 i. 0.2 d. i. v. sl. s. v. sl. s. i. i. i. i. sl. s. h. v. sl. s. i. i. s. s. i. v. sl. s. sl. s. 0.2 0.43 v. sl. s. d. h. 0.0328 v. sl. s. i. i. i. 3.215 i. i. i. 5 <5 i. 60 i. c. 4.7 cc.20 ∞ ∞ 7.520 i. 8.515 1317 ∞ ∞ 24017 i. 2.1515 i. v. sl. s. 0.0115 0.0820 i. 1.060 0.6825 sl. s. 0.00220 i. i. 0.450 i. d. i. 225 1118 i. 0.02920 0.420 ∞
1020 5619.5 v. s. 20 s. s. 920 v. s.
6.620 s. s. s. v. s. ∞ sl. s. v. s. s. ∞
s. h. s. h. s. s. h. s. ∞ s. s.
s. v. s. s. ∞
∞ ∞ ∞ ∞
∞ ∞ ∞ ∞
s. h.
s.
4 h. v. s. v. sl. s. s. v. s. s. 500 ∞ ∞ sl. s. c. s. 150 cc. ∞
v. s. v. s. i. s. v. s.
s.
s. ∞ ∞ i. s. 1
∞ ∞ ∞ ∞ v. s. ∞ s. s. ∞ ∞ 18 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞
∞ ∞ ∞ ∞ i. ∞ s. s. ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞
2-37
TABLE 2-2
Physical Properties of Organic Compounds (Continued ) Name
Formula
laurate mercaptan methacrylate naphthylamine (α-) naphthyl ether (α-) nitrate nitrite oleate palmitate pelargonate propionate salicylate (o-) stearate toluate (o-) (m-) toluene sulfonate (p-) toluidine (o-) (p-) urea valerate (n-) (i-) Ethylal Ethylene bromide bromohydrin chlorobromide chlorohydrin diamine oxide Ethylidene diacetate Eugenol (1-,4-,3-) (i-)(1-,3-,4-) Fenchyl alcohol (dl-) (d-)(α-) (i-)(l-) Ferric dimethyl-dithiocarbamate Fluorene Fluorescein Fluoro-dichloromethane -trichloromethane Formaldehyde (m-) (p-) Formamide Formanilide Formic acid Fructose Fuchsin Fulminic acid Fumaric acid (trans-) Furfural Furfuran Furfuryl acetate alcohol butyrate propionate Furoic acid G-acid, K salt (2-)(6-,8-) Na salt (2-)(6-,8-) Galactose (d-)(α-) Gallic acid (3-,4-,5-) Gamma acid (2-,8-,6-) Geraniol Glucose (d-)(α-) (d-)(β-) Glucuronic acid Glutam(in)ic acid (dl-)
CH3(CH2)10CO2C2H5 CH3CH2SH CH2:C(CH3)CO2C2H5 C10H7NHC2H5 C10H7OC2H5 C2H5ONO2 C2H5ONO C17H33CO2C2H5 CH3(CH2)14CO2C2H5 CH3(CH2)7CO2C2H5 CH3CH2CO2C2H5 HOC6H4CO2C2H5 CH3(CH2)16CO2C2H5 CH3C6H4CO2C2H5 CH3C6H4CO2C2H5 CH3C6H4SO3C2H5 CH3C6H4NHC2H5 CH3C6H4NHC2H5 C2H5NHCONH2 CH3(CH2)3CO2C2H5 (CH3)2CH(CH2)2CO2C2H5 CH2(OC2H5)2 H2C:CH2 BrCH2CH2Br BrCH2CH2OH ClCH2CH2Br ClCH2CH2OH H2NCH2CH2NH2 < (CH2)2 > O CH3CH(O2CCH3)2 C3H5C6H3(OH)OCH3 C3H5C6H3(OCH3)OH C10H17OH C10H17OH C10H17OH Fe[SSCN(CH3)2]3 (C6H4)2 > CH2 C20H12O5 FCHCl2 Cl3CF HCHO (CH2O)3 (CH2O)xxH2O HCONH2 HCONHC6H5 HCO2H CH2OH(CHOH)3COCH2OH C20H19N3HCl C:NOH HO2CCH:CHCO2H C4H3OCHO C4H4O CH3CO2CH2C4H3O C4H3OCH2OH C3H7CO2CH2C4H3O C2H5CO2CH2C4H3O C4H3OCO2H HOC10H5(SO3K)2 HOC10H5(SO3Na)2 C5H11O5CHO (HO)3C6H2CO2HH2O C10H5(NH2)(OH)SO3H C9H15CH2OH C5H11O5CHO C6H12O6H2O CHO(CHOH)4CO2H [CHNH2(CH2)2](CO2H)2
2-38
Formula weight
Form and color
Specific gravity
Melting point, °C
Boiling point, °C
228.36 62.13 114.14 171.23 172.22 91.07 75.07 310.50 284.47 186.29 102.13 166.17 312.52 164.20 164.20 200.25 135.20 135.20 88.11 130.18 130.18 104.15 28.05 187.88 124.98 143.43 80.52 60.10 44.05 146.14 164.20 164.20 154.24 154.24 154.24 416.47 166.21 332.30 102.93 137.38 30.03 90.08 (30.03) 45.04 121.13 46.03 180.16 337.84 43.03 116.07 96.08 68.07 140.13 98.10 168.19 154.16 112.08 380.46 348.26 180.16 188.13 239.24 154.24 180.16 198.17 194.14 147.13
oil lq. col. lq. oil cr. col. lq. lq. oil col. nd. col. lq. col. lq. col. lq. col. cr. lq. lq. pr./al. lq. lq. nd. col. lq. col. lq. lq. col. gas col. lq. col. lq. lq. col. lq. col. lq. lq. col. lq. oil oil col. cr. col. pr. col. cr. cr. cr./al. yel. red gas col. lq. gas wh. amor. lq. mn. col. lq. nd./aq. red
0.868 13/4 0.839 20/4 0.913 15.6 1.060 20/4 1.061 20/20 1.100 25/4 0.900 15.5 0.867 25 0.858 25/4 0.866 17.5 0.891 20/4 1.136 15/4 0.848 36.3 1.032 25/25 1.030 20/20 1.166 48/4 0.948 25/4 0.942 25/4 1.213 18 0.877 20 0.867 20/4 0.824 25/4 0.57−102/4 2.180 20/4 1.772 20/4 1.689 19 1.213 20/4 0.900 20/20 0.887 7/4 1.061 12 1.070 15/15 1.091 15/15 0.935 40 0.964 20/4 0.961
−10.7 −121
269 36–7 118 303723 276.4 87–8 17 216–815 19110 227–8757 99.1 233–4 20110 227 231750 221.3 215–6 217
col. pr. lq. col. lq. col. oil oil col. lq. col. lq. mn. pr. cr. cr. pr. mn./aq. cr. col. lq. rhb. cr. cr. cr./aq.
1.635 20/4 1.159 20/4 0.937 20/4 1.118 20/4 1.129 25/4 1.053 20/4 1.109 20/4
1.203 0/4 1.4260 1.494 17.2 0.815 −20 1.1765 1.139 20/4 1.147 15/15 1.220 20/4 1.669 17.5 1.22
5.5 −102 <−15 24–5 −44.5 −72.6 1.3 33.4(31) <−10 33–4 <−15 92 −91.2 −99.3 −66.5 −169 10 −16.6 −69 8.5 −111.3 18.85 10.3 −10 35 45–7 61–2 d. 100–30 115–6 d. > 290 −127 −92 64 150–60 2 47 8.6 95–105 d. >200 286–7 −38.7
133–4
1.694 4/4 0.883 15 1.544 25 1.562 18/4 1.460
145.5 135 89 −103.9 131.5 150.3 106.7 128.8 117.2 13.5747 168740 253.5 267.5 201 201–2 201–2 ign. >150 293–5 14.5 24.9 −21 114.5759 subl. 193 216120 100.8
290 161.7760 31–2756 175–7 169.5752 212–3 195–6 230–2
165.5 d. 220 <−15 146 150 154 199 d.
230 d.
Solubility in 100 parts Water
Alcohol
Ether
i. 1.5 i. i. i. 1.355 v. sl. s. i. i. i. 2.420 i. i. i. i. i. i. i. v. s. 0.2425 0.1720 918 26 cc.0 0.4380 sl. s. 0.6980 ∞ ∞ ∞ sl. s. v. sl. s. v. sl. s. sl. s. sl. s. i. v. sl. s. i. v. sl. s. h. i. i. v. s. 2125 20–3018 ∞ sl. s. ∞ v. s. 0.3
s. s. s. s. s. ∞ ∞ ∞ s. ∞ ∞ ∞ s. ∞ ∞ s.
∞ s. s. s. s. ∞ ∞ ∞ s. ∞ ∞ ∞ s. ∞ ∞ s.
80 ∞ ∞ ∞ 360 cc. ∞ s.
i. ∞ ∞ ∞ s. ∞
∞ ∞ ∞ ∞ ∞ ∞
∞ 0.3 v. s.
s.
s.
s. h. s. h. s. ∞ v. s. s. i. ∞ v. s. ∞ 818 s.
s.
0.717 9.113 i. i. ∞ v. sl. s. v. sl. s. 3.615 825 3420 10.30 113
5.830 ∞ s. s. s. s. s. s.
0.725 ∞ s. s. s. ∞ ∞ s.
0.640 2815
2.515
i. 8217.5 15415 v. s. 1.5 20
∞ sl. s.
∞ i.
v. sl. s.
v. sl. s.
∞ ∞
s. ∞ v. s. s. i. v. sl. s. s. i.
Glutaric acid Glycerol acetate (mono-) (di-) nitrate (mono-) (α-) (β-) dinitrate (1-,3-) Glyceryl triacetate tribenzoate tributyrate tricaprate tricaproate tricaprylate trilaurate trimyristate trinitrate trinitrite trioleate tripalmitate tristearate Glycide Glycine, Glycocoll Glycol diacetate dibenzoate dibutyrate dicaprylate diformate dilaurate dinitrate dinitrite dipalmitate dipropionate ether formal formate (mono-) Glycolic acid Guaiacol (o-) Guanidine H-acid, Na salt (1-,8-,3-,6-) Heptacosane (n-) Heptane (n-) (i-)
Heptoic acid aldehyde Heptyl acetate (n-) alcohol (n-) mercaptan Hexachloro-benzene -ethane Hexacosane (n-) Hexadecane (n-) Hexaethylbenzene Hexamethylbenzene Hexamethylene-diamine -diisocyanate -glycol tetramine Hexane (n-) (i-) (neo-)
CH2(CH2CO2H)2 CH2OHCHOHCH2OH C5H10O4 (CH3CO2)2C3H5OH CH2OHCHOHCH2NO3 CH2OHCHNO3CH2OH CHOH(CH2ONO2)2 (CH3CO2)3C3H5 (C6H5CO2)3C3H5 (C2H5CH2CO2)3C3H5 [CH3(CH2)8CO2]3C3H5 [CH3(CH2)4CO2]3C3H5 [CH3(CH2)6CO2]3C3H5 [CH3(CH2)10CO2]3C3H5 [CH3(CH2)12CO2]3C3H5 CH2NO3CHNO3CH2NO3 CH2NO2CHNO2CH2NO2 (C17H33CO2)3C3H5 [CH3(CH2)14CO2]3C3H5 [CH3(CH2)16CO2]3C3H5 C2H3OCH2OH NH2CH2CO2H CH2OHCH2OH (CH3CO2CH2)2 (C6H5CO2CH2)2 (C3H7CO2CH2)2 (C7H15CO2CH2)2 (HCO2CH2)2 (C11H23CO2CH2)2 (O2NOCH2)2 (ONOCH2)2 (C15H31CO2CH2)2 (C2H5CO2CH2)2 (HOCH2CH2)2O < OCH2CH2OCH2 > HCO2CH2CH2OH HOCH2CO2H CH3OC6H4OH NH:C(NH2)2 C10H8O7NS2Na1a H2O CH3(CH2)25CH3 CH3(CH2)5CH3 (CH3)2CH(CH2)3CH3 C3H7CH(CH3)C2H5 (CH3)3CCH2C2H5 [(CH3)2CH]2CH2 (CH3)2C(C2H5)2 (C2H5)3CH (CH3)3CCH(CH3)2 CH3(CH2)5CO2H CH3(CH2)5CHO CH3CO2CH2(CH2)5CH3 CH3(CH2)5CH2OH [(CH3)2CH]2CHOH (C2H5CH2CH2)2CHOH CH3CH(SH)C5H11 C6Cl6 CCl3CCl3 CH3(CH2)24CH3 CH3(CH2)14CH3 C6(C2H5)6 C6(CH3)6 NH2(CH2)6NH2 OCN(CH2)6NCO HO(CH2)6OH (CH2)6N4 CH3(CH2)4CH3 (CH3)2CH(CH2)2CH3 (CH3)3CC2H5 (CH3)2CHCH(CH3)2 (C2H5)2CHCH3
132.11 92.09 134.13 176.17 137.09 137.09 182.09 218.20 404.40 302.36 554.83 386.51 470.67 638.98 723.14 227.09 179.09 885.40 807.29 891.45 74.08 75.07 62.07 146.14 270.27 202.24 314.45 118.09 426.66 152.07 120.07 538.87 174.19 106.12 74.08 90.08 76.05 124.13 59.07 368.31 380.72 100.20 100.20 100.20 100.20 100.20 100.20 100.20 100.20 130.18 114.18 158.24 116.20 116.20 116.20 132.26 284.80 236.76 366.69 226.43 246.42 162.26 116.20 168.19 118.17 140.19 86.17 86.17 86.17 86.17 86.17
col. cr. col. lq. col. oil col. lq. col. pr. lf. oil col. lq. nd. col. lq. col. cr. col. lq. col. lq. col. nd. lf. yel. oil yel. lq. col. oil col. nd. col. pr. col. lq. mn. col. lq. col. lq. rhb./et. col. lq. lq. lq. amor. yel. lq. lq. nd. lq. lq. lq. lq. nd./aq. pr. col. cr. cr. col. cr. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. lq. mn. rhb. cr. lf. pr./al. pl./al. lf. lq. nd./aq. col. rhb. col. lq. lq. lq. lq. lq.
1.429 15 1.260 50/4 1.20 20/4 1.178 15/15 1.4015 1.4015 1.4715 1.16117/4 1.228 12 1.032 20/4 0.921 40/4 0.987 20/4 0.954 20/4 0.894 60/4 0.885 60/6 1.601 15 1.291 10/16 0.915 15 0.866 80/4 0.862 80/4 1.114 16/16 1.161 1.113 19/4 1.109 14/4 1.024 0
97.5 17.9 40 58–9 54 <−30 −78 75–6 <−75 31(25) −25 8.3(−21) 45–6 56.5 13.3(2) −4 65.1 70.8(55)
20020 290 158165 175–640 155–60 155–60 146–815 258–9 d. 305–9
16015 150 sl. d. 24018 310–200.1 166 sl. d.
232–6 d. −15.6 −31 73–4
197.4 190.5 >360 240
22 1.482 21/2 1.216 0 1.045 25 1.118 20/20 1.060 20/4 1.199 15/4 1.140 15/15 0.780 60/4 0.684 20/4 0.679 20/4 0.687 20/4 0.674 20/4 0.675 20/4 0.693 20/4 0.698 20/4 0.690 20/4 0.918 20 0.850 20/ᐉ 0.874 16/16 0.824 20/4 0.829 20/4 0.820 20/4 0.835 20 2.044 24 2.091 20/4 0.779 57/4 0.774 20/4 0.831 130/4 1.0428 20/4
0.659 0.654 20/4 0.649 20/20 0.662 20/4 0.664 20/4
52–4 −20 <−15 71–2 −10.5 79(63) 28.3 50 59.5 −90.6 −118.2 −119.4 −125 −119.4 −135.0 −118.7 −25 −10 −42 34.6 −37 228–31 186–7 56.6 18.5 130 166 42 42 subl. −94 −153.7 −98.2 −129.8 −118
174 18820 expl. 114 96–8 2600.1 211–2 244.8 75–6 180 d. 205 27015 98.4760 90.0 91.8 79.1 80.8 86.0 93.5 80.8 221–2 155 191.5 759 175 756 140 156 174–5 765 309742 186777 26215 287.5 298.3 265 204–5 143–420 250 69 60.2 49.7 58.0760 63.2
63.920 ∞ v. s. s. 7015 7.1715 i. i. i. i. i. i. i. 0.1820 d. i. i. i. ∞ 23 c. ∞ 14.322 i. i. i. v. sl. s. i. 0.9225 i. i. sl. s. ∞ ∞ ∞ ∞ 1.715 v. s. 0.1720 i. 0.00515 i. i. i. i. i. i. i. 0.2515 0.0220 i. 0.18 25 v. sl. s. i. i. i. 0.00522 i. i. i. i. v. s. d. s. 8112 0.01415 i. i. i. i.
v. s. ∞ v. s. s. v. s. v. s. v. s. ∞ s. h. s. s. h. s. s. sl. s. c. s. 5020 d. sl. s. 0.00421 s. h. ∞ 0.1 c. ∞ ∞
v. s. i. sl. s. sl. s. v. sl. s. sl. s. v. s. ∞ s. s. v. s. s. s. v. s.
v. s.
∞ s. v. s. v. s. s. h. ∞ i. 1.0 ∞ s. v. s.
v. s. s. s. d. s. ∞ ∞
v. s. ∞ s. s. ∞ i.
9025 v. s. s.
v. s. v. s.
sl. s. s. s. s. s. s. s. s. s. ∞ s. ∞ ∞ s.
∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ s. ∞ s. ∞ ∞ s.
v. sl. s. h. v. s. v. sl. s. ∞ 0.7525 0.20 s. d. s. 3 5033
s. h. v. s. ∞ 825 v. s. sl. s. h. v. sl. s. ∞ s. s. s. s.
TABLE 2-2
Physical Properties of Organic Compounds (Continued ) Name
Hexyl acetate (n-) alcohol (n-) formate (n-) resorcinol (2-,4-) Hippuric acid Histidine (l-) Homophthalic acid (o-) Hydracrylic acid Hydro-cyanic acid -quinone (p-) Hydroxy-benzaldehyde (p-) -benzanilide (o-) -quinoline (2-)(α-) (8-)(o-) Indigo White Indole Indoxyl Iodo-benzene -phenol (p-) Iodoform Ionone (α-) (β-) Irone (β-) Isatin Isoprene Ketene Koch acid (1-)(3-,6-,8-) Lactic acid (dl-) anhydride Lactide (dl-) Lactose Lauric acid Laurone Lauryl alcohol Lead tetraethyl tetramethyl Lecithin (protagon) Lepidine (py-4) Leucine (l-) Levulinic acid Limonene (d- or l-) Linalool (d- or l-) Linalyl acetate Linoleic acid Maleic acid anhydride Malic acid (dl-) (d- or l-) Malonic acid Maltose Mandelic acid (dl-) Mannitol (d-) Mannose (d-) Margaric acid Mellitic acid Menthol (l-)(α-) Mercapto-benzothiazole (2-) -thiazoline (2-) Mercuric cyanide fulminate Mesityl oxide Mesitylene (1-,3-,5-) Metanilic acid (m-) Methane
2-40
Formula CH3CO2(CH2)5CH3 CH3(CH2)4CH2OH (CH3)2CHC(CH3)2OH (CH3)2COHCH2C2H5 HCO2CH2(CH2)4CH3 CH3(CH2)5C6H3(OH)2 C6H5CONHCH2CO2H C6H9O2N3 HO2CC6H4CH2CO2H HOCH2CH2CO2H HCN C6H4(OH)2 HOC6H4CHO HOC6H4CONHC6H5 C9H6NOH C9H6NOH [C6H4(CO)(NH)C:]2 C16H12O2N2 C8H7N C8H6NOH C6H5I IC6H4OH HCI3 C10H16:CHCOCH3 C10H16:CHCOCH3 C14H22O C6H4 < (CO)(N) > COH CH2:CHC(CH3):CH2 H2C:CO C10H4(NH2)S3O9HNa2 CH3CH(OH)CO2H C6H10O5 C6H8C4 C12H22O11H2O CH3(CH2)10CO2H [CH3(CH2)10]2CO CH3(CH2)10CH2OH Pb(CH2CH3)4 Pb(CH3)4 C42H48O9PN C9H6NCH3 (CH3)2CHCH2CH(NH2)CO2H CH3CO(CH2)2CO2H C10H16 C10H17OH CH3CO2C10H17 C17H31CO2H HO2CCH:CHCO2H < (CHCO)2 > O HO2CCH2CH(OH)CO2H HO2CCH2CH(OH)CO2H H2C(CO2H)2 C12H22O11H2O C6H5CH(OH)CO2H CH2OH(CHOH)4CH2OH CH2OH(CHOH)4CHO CH3(CH2)15CO2H C6(CO2H)6 C10H19OH < C6H4N:C(SH)S > < CH2N:C(SH)SCH2 > Hg(CN)2 Hg(ONC)2a H2O (CH3)2C:CHCOCH3 C6H3(CH3)3 H2NC6H4SO3H CH4
Formula weight
Form and color
Specific gravity
144.21 102.17 102.17 102.17 130.18 194.26 179.17 155.16 180.15 90.08 27.03 110.11 122.12 213.23 145.15 145.15 262.26 264.27 117.14 133.14 204.02 220.02 393.78 192.29 192.29 206.32 147.13 68.11 42.04 427.34 90.08 162.14 144.12 360.31 200.31 338.60 186.33 323.45 267.35 778.08 143.18 131.17 116.11 136.23 154.24 196.28 280.44 116.07 98.06 134.09 134.09 104.06 360.31 152.14 182.17 180.16 270.44 342.17 156.26 167.24 119.20 252.65 293.65 98.14 120.19 173.18 16.04
col. lq. col. lq. lq. lq. lq. col. nd. rhb. lf./aq. cr./aq. syrup lq. cr. nd./aq. pr./al. pr./al. pr. cr. gray lf./aq. yel. pr. col. lq. nd./aq. yel. hex. col. oil col. oil col. oil yel. red col. lq. col. gas cr. hyg. yel. oil tri./al. col. rhb. col. nd. pl. lf. col. lq. col. lq. wax lq. cr. lf. lq. col. oil col. lq. yel. oil mn. cr. col. cr. col. cr. col. tri. col. nd. rhb./aq. col. rhb. rhb. col. pl. nd./al. col. cr. nd. cr. cr. cr./aq. lq. col. lq. col. nd. gas
0.890 0/0 0.820 20/20 0.821 20/0 0.809 20/4 0.898 0 1.371 20/4
0.697 18 1.332 15 1.129 130
1.35
1.824 25/4 1.857 112 4.008 17 0.930 20 0.944 20 0.939 20
Melting point, °C −51.6 −14 −107 68–70 187–8 d. 287 175–80 −12 170.3 116–7 135 199–200 75–6 390–2 52 85 −28.5 93–4 119
0.681 20/4
200–1 −120 −151
1.249 15/4
16.8
10/4
0.862 1.525 20 0.869 50/4 0.809 69/4 0.831 24/4 1.659 18/4 1.995 20/4 1.086 20 1.29318 1.140 20/20 0.842 20/4 0.868 20 0.895 20 0.903 18/4 1.609 1.5 1.601 20/4 1.595 20/4 1.631 15 1.540 17 1.300 20/4 1.489 20/4 1.539 20/4 0.853 60 15/15
0.890 1.4220/4 1.50 4.003 22 4.4 0.858 20/4 0.865 20/4
0.415 −164
124.5 202 48(44) 69–70 24 −136 −27.5 150–200 d. 9–10 295 33.5 −96.9 −9.5 130.5 57–60 128–9 99–100 130–5 d. d. 118.1 166 132 60–1 286–8 42–3 179 106 d. 320 expl. −59 −45(−52) d. −182.6
Boiling point, °C 169.2 157.2 120–1 123762 153.6 1797 d. d. 25–6 285730 subl. d. subl. 266.6752 subl. 253–4 110 188.6 d. subl. 136.117 14018 14416 subl. 34 −56 12214 d. 250 255757 d. 225100 255–9 152291 110760 261–3 subl. 245–6 177 198–200 220762 d. 229–3016 135 d. 202 150 d. 140 d. d. 290–53 227100 d. 212 d.
130750 164.8 −161.4
Solubility in 100 parts Water i. 0.620 v. sl. s. v. sl. s. 0.05 0.420 s. s. h. ∞ 615 1.3831 v. sl. s. h. s. h. v. sl. s. c. i. i. s. h. s. 0.03420 sl. s. 0.0125 sl. s. sl. s. v. sl. s. s. h. i. d. 7.220 ∞ v. sl. s. v. sl. s. 1710 i. i. i. i. i. i. sl. s. 2.218 v. s. i. v. sl. s. v. sl. s. i. 7925 16.380 14426 v. s. 13816 10825 1620 1314 24817 i. v. s. 0.04 c. i. 1.660 12.515 0.0712 320 i. 215 0.420 cc.
Alcohol
Ether
v. s. ∞ ∞ ∞ ∞ v. s. s. h. v. sl. s. v. s.
v. s. ∞ ∞ ∞ ∞ s. 0.2518 i. sl. s.
∞ v. s.
∞ v. s.
s. v. s. s. i. s. s. h. s. s. v. s. 1.517 ∞ ∞ v. s. v. s. h. ∞ d.
s. v. s. sl. s. i. s. s. s. ∞ v. s. 13.625 ∞ ∞ v. s. sl. s. ∞ s.
∞ s. v. sl. s. c. i. s. i. c. s. sl. s. ∞ s. h. ∞
∞ s.
s. ∞ ∞ s. h. ∞
v. s. ∞ s. ∞ ∞ 7030
v. s. ∞ ∞ ∞ ∞ 825
v. s. v. s. 4225 v. sl. s. c. s. 0.0114 v. sl. s. 3228 v. s. v. s. s.
v. s. 8.415 815 i. s. i. i. v. s.
s. ∞ s. v. sl. s. 4720 cc.
i. s.
v. s. sl. s.
∞ ∞ v. sl. s. 10410 cc.
Methoxy-methoxyethanol Methyl acetate acrylic acid (α-) alcohol -amine -amine hydrochloride aniline anthracene (α-) (β-) anthranilate (o-) anthraquinone (2-) benzoate benzylaniline bromide butyrate (n-) (i-) caprate caproate (n-) caprylate cellosolve chloride chloroacetate chloroformate cinnamate cyclohexane ethyl carbonate ethyl ketone ethyl oxalate formate furoate glucamine glycolate heptoate hypochlorite iodide lactate laurate mercaptan methacrylate myristate naphthalene (α-) (β-) nitrate nitrite nonyl ketone (n-) oleate orange palmitate phosphine propionate propyl ketone (n-) salicylate (o-) stearate toluate (o-) (m-) (p-) Methyl toluidine (o-) (m-) (p-) valerate (n-) (i-) vinyl ketone Methylal Methylene-bis-(phenyl-4-isocyanate) bromide chloride dianiline iodide Michler’s hydrol (p-,p′-) ketone Morphine Mucic acid
CH3(OCH2)2CH2OH CH3CO2CH3 CH2:C(CH3)CO2H CH3OH CH3NH2 CH3NH2HCl C6H5NHCH3 C6H4:(CH)2:C6H3CH3 C6H4:(CH)2:C6H3CH3 NH2C6H4CO2CH3 C6H4:(CO)2:C6H3CH3 C6H5CO2CH3 C6H5N(CH3)CH2C6H5 CH3Br CH3(CH2)2CO2CH3 (CH3)2CHCO2CH3 CH3(CH2)8CO2CH3 CH3(CH2)4CO2CH3 CH3(CH2)6CO2CH3 CH3OCH2CH2OH CH3Cl ClCH2CO2CH3 ClCO2CH3 C6H5CH:CHCO2CH3 CH2 < (CH2CH2)2 > CHCH3 CH3OCOOC2H5 CH3.COC2H5 CH3OCOCO2C2H5 HCO2CH3 C4H3OCO2CH3 CH2OH(CHOH)4CH2NHCH3 HOCH2CO2CH3 CH3(CH2)5CO2CH3 ClOCH3 CH3I CH3CH(OH)CO2CH3 CH3(CH2)10CO2CH3 CH3SH CH2:C(CH3)CO2CH3 CH3(CH2)12CO2CH3 C10H7CH3 C10H7CH3 CH3ONO2 CH3ONO CH3(CH2)8COCH3 C17H33CO2CH3 (CH3)2NC6H4N2C6H4SO3Na CH3(CH2)14CO2CH3 CH3PH2 CH3CH2CO2CH3 CH3COCH2CH2CH3 HOC6H4CO2CH3 CH3(CH2)16CO2CH3 CH3C6H4CO2CH3 CH3C6H4CO2CH3 CH3C6H4CO2CH3 CH3C6H4NHCH3 CH3C6H4NHCH3 CH3C6H4NHCH3 CH3(CH2)3CO2CH3 (CH3)2CHCH2CO2CH3 CH3COCH:CH2 HCH(OCH3)2 (OCNC6H4)2CH2 CH2Br2 CH2Cl2 (C6H5NH)2CH2 CH2I2 [(CH3)2NC6H4]2CHOH [(CH3)2NC6H4]2CO C17H19O3NH2O (CHOHCHOHCO2H)2
106.12 74.08 86.09 32.04 31.06 67.52 107.15 192.25 192.25 151.16 222.23 136.14 197.27 94.95 102.13 102.13 186.29 130.18 158.23 76.09 50.49 108.53 94.50 162.18 98.18 104.10 72.10 132.11 60.05 126.11 195.21 90.08 144.21 66.49 141.95 104.10 214.34 48.10 100.11 242.39 142.19 142.19 77.04 61.04 170.29 296.48 327.33 270.44 48.03 88.10 86.13 152.14 298.49 150.17 150.17 150.17 121.18 121.18 121.18 116.16 116.16 70.09 76.09 250.25 173.86 84.94 198.26 267.87 270.36 268.35 303.35 210.14
lq. col. lq. pr. col. lq. col. gas pl./al. lq. lf./al. col. lf. col. lq. col. nd. col. lq. lq. gas col. lq. col. lq. lq. col. lq. col. lq. col. lq. gas col. lq. col. lq. cr. col. lq. lq. col. lq. lq. lq. col. lq. lq. lq. gas col. lq. lq. lq. gas lq. cr./al. oil mn. lq. gas col. oil oil red pd. col. cr. gas col. lq. col. lq. col. lq. col. cr. col. lq. col. lq. cr. lq. lq. lq. lq. col. lq. lq. col. lq. lq. col. lq. col. lq. cr. col. lq. gn. lf./al. pr./al. pd.
1.038 25 0.924 20/4 1.015 20/4 0.792 20/4 0.699 −11 1.23 0.989 20/4 1.047 99.4 1.181 0/4 1.168 19/4 1.087 25/25 1.732 0/0 0.898 20/4 0.891 20/4 0.904 0/0 0.887 18 0.965 20/4 0.952 0 1.236 20/4 1.236 15 1.042 36/0 0.769 20/4 1.002 27 0.805 20/4 1.156 0/0 0.974 20/4 1.179 21/4
<−70 −98.7 15–16 −97–8 −92.5 226–8 −57 86 207 24 176–7 −12.5 9.2 −93 <−95 −84.7 −18 −40 −97.7 −32.7 33.4 −126.3 −14.5 −85.9 −99.8
18
1.168 0.881 15/4 20/4
2.279 1.090 19
0.896 0 0.950 15.6 1.025 14/4 0.994 40/4 1.203 25 0.991 15 0.828 20/20 0.879 18
−64.4 5 −121 −48 18–9 −19 35–6 expl. 13.5 30–1
20/4
0.915 0.812 15/15 1.182 25/25 1.073 15 1.066 15 0.973 15 0.935 55/4 0.895 15/4 0.881 20/4 0.836 20/4 0.866 15/4 1.222 30 2.495 20/4 1.336 20/4 3.325 20/4 1.317
−87.5 −77.8 −8.3 38–9 <−50 33–4
−91 −104.8 −52.8 −96.7 65 5.7 96–7 174 254 d. 206–14
167.5 57.1 161–3 64.7 −6.7 758 23015 195.5 135.5 15 subl. 198–9 305–6 4.5 758 102.3 92.6 223–4 149.5 192–4 124–5 −24 130740 71–2 263 101 109.2 79.6 173.7 32 181.3 151.2 172–3 12726 42.4 144.8 14818 5.8752 100.3 295715 244.6 241–2 65 −12 228 190–110 19615 −14759 79.7 102 222.2 21515 213 215 217 206–7 206–7 211761 127.3 116.7764 81 42–3 210–213 98.5756 40–1 208–9 d. 180 d. >360 d.
∞ 3322 s. h. ∞ v. s. v. s. 0.0125 i. i. sl. s. i. 0.0230 i. v. sl. s. 1.7 v. sl. s. i. i. i. ∞ 28016 cc. v. sl. s. d. i. i. i. 3510 i. 3020 i.
∞ ∞ ∞ v. s. 23 h. s.
∞ ∞ ∞
v. sl. s. s. s. ∞ s. s. ∞ ∞ ∞ ∞ ∞ ∞ v. s. ∞ ∞ v. s. s. ∞ ∞ v. s. ∞ ∞
v. sl. s. s. s. ∞ s. s. ∞ ∞ ∞ ∞ ∞ ∞ v. s. ∞ ∞ v. s. s. ∞ ∞ v. s.
∞ s.
∞ s.
v. s.
v. s.
v. s. v. s. s. s. s. ∞
v. s. v. s. s. s. s. ∞
s. sl. s. ∞ ∞ ∞ s. ∞
∞ ∞ ∞ s. ∞
v. s. ∞ ∞ ∞ ∞ ∞
v. s. ∞ ∞ ∞ ∞ ∞
∞ d. ∞ ∞ s. ∞ s. h. sl. s. sl. s. i.
∞
i. ∞
∞
i. 1.815 ∞ i. s. i. i. i. i. sl. s. i. i. 0.2 c. i. i. 0.520 v. sl. s. 0.0730 i. i. i. i. i. i. i. v. sl. s. v. sl. s. >85 33 d. 1.170 220 i. 1.420 i. i. 0.0220 0.3314
s.
∞ ∞ s. ∞ s. v. sl. s. s. i.
TABLE 2-2
Physical Properties of Organic Compounds (Continued ) Name
Mustard gas Myricyl alcohol Myristic acid Myristyl alcohol Naphthalene disulfonic acid (1-,5-) (1-,6-) sulfonic acid (α-) (β-) Naphthasultam (1-,8-) disulfonate Na (1-,8-) (2-,4-) Naphthoic acid (α-) (β-) Naphthol (α-) (β-) sulfonic acid (α-)(1-,2-) (β-)(2-,6-) Naphthyl acetate (α-) (β-) amine (α-) (β-) amine hydrochloride (α-) (β-) amine sulfonic acid (1-,4-) (1-,5-) (1-,7-) (1-,8-) (2-,5-) (2-,6-) (2-,7-) isocyanate (α-) Nicotine Nicotinic acid (3-) (i-)(4-) Nitro-acetanilide (p-) -acetophenone (m-) -aminoanisole (4-,1-,2-) (5-,1-,2-) (3-,1-,4-) -aminophenol (4-,2-,1-) -aniline (o-) (m-) (p-) -anisole (o-) (p-) -anthraquinone (α-) -anthraquinone sulfonic acid (1-,5-) -benzal chloride (m-) -benzaldehyde (m-) Nitro-benzene -benzidine (2-) -benzoic acid (o-) (m-) (p-) -benzyl alcohol (m-) -benzyl bromide (p-) -chlorotoluene (1-,2-,6-) -cresol (1-,3-,4-) -cymene (1-,2-,4-) -dimethylaniline (o-) (m-) (p-) -diphenyl (o-) (p-) -diphenylamine (o-) -guanidine
2-42
Formula (ClCH2CH2)2S C31H63OH(?) CH3(CH2)12CO2H CH3(CH2)12CH2OH C10H8 C10H6(SO3H)2 C10H6(SO3H)2 C10H7SO3H2H2O C10H7SO2HH2O C10H7O2NS C10H5O8NS3Na22H2O C10H4O8NS3Na38a H2O C10H7CO2H C10H7CO2H C10H7OH C10H7OH HOC10H6SO3H HOC10H6SO3H CH3CO2C10H7 CH3CO2C10H7 C10H7NH2 C10H7NH2 C10H7NH2HCl C10H7NH2HCl NH2C10H6SO3H NH2C10H6SO3HH2O NH2C10H6SO3HH2O NH2C10H6SO3HH2O NH2C10H6SO3H NH2C10H6.SO3HH2O NH2C10H6SO3HH2O C10H7N:CO C10H14N2 C5H4NCO2H C5H4NCO2H CH3CONHC6H4NO2 CH3COC6H4NO2 NO2C6H3(OCH3)NH2 NO2C6H3(OCH3)NH2 NO2C6H3(OCH3)NH2 NO2C6H3(NH2)OH NO2C6H4NH2 NO2C6H4NH2 NO2C6H4NH2 CH3OC6H4NO2 CH3OC6H4NO2 C6H4:(CO)2:C6H3NO2 NO2C14H6O2SO3H NO2C6H4CHCl2 NO2C6H4CHO C6H5NO2 NH2C6H4C6H3(NH2)NO2 NO2C6H4CO2H NO2C6H4CO2H NO2C6H4CO2H NO2C6H4CH2OH NO2C6H4CH2Br CH3C6H3(NO2)Cl CH3C6H3(NO2)OH CH3C6H3(NO2)CH(CH3)2 NO2C6N4NHCH3 NO2C6H4NHCH3 NO2C6H4NHCH3 C6H5C6H4NO2 C6H5C6H4NO2 C6H5NHC6H4NO2 H2NC(NH)NHNO2
Formula weight
Form and color
Specific gravity
Melting point, °C
Boiling point, °C
159.08 452.82 228.36 214.38 128.16 288.28 288.28 244.26 226.24 205.22 445.35 584.45 172.17 172.17 144.16 144.16 224.22 224.22 186.20 186.20 143.18 143.18 179.65 179.65 223.24 241.26 241.26 241.26 223.24 241.26 241.26 169.17 162.23 123.11 123.11 180.16 165.14 168.15 168.15 168.15 154.12 138.12 138.12 138.12 153.13 153.13 253.20 333.26 206.03 151.12 123.11 229.23 167.12 167.12 167.12 153.13 216.04 171.58 153.13 179.21 166.18 166.18 166.18 199.20 199.20 214.22 104.07
oil cr. col. lf. cr. pl./al. lf. cr. cr. cr. nd. cr. lf. nd. mn. mn. mn. pl./aq. lf. nd./al. nd./al. rhb. lf./aq. nd. lf. nd. cr. cr. cr. cr. cr. cr. col. lq. oil nd./al. nd./aq. rhb. nd. red nd. yel. nd. red or. pr. yel. rhb. yel. rhb. yel. mn. col. cr. pr./al. nd. yel. cr. mn. nd./aq. yel. lq. red nd. tri./aq. mn. yel. mn. cr. nd./al. cr. yel. oil yel. oil red mn. yel. nd. rhb. nd./al. or. cr. nd./aq.
1.275 20/4 0.777 95 0.853 70/4 0.824 38/4 1.145 20/4
13–4 88 57–8 38 80.2 d. d. 125 90 125 177–8
217
1.077 100/4 1.224 4 1.217 4
1.123 25/25 1.061 98/4
160–1 184 96 122–3 >250 125 46–9 69–70 50 111–2
250.5100 16715 217.9
300 >300 278–80 285–6
300.8 306.1 subl.
d.
1.18 1.009 20/4
1.207 156 1.211 156 1.442 15 1.43 1.437 14 1.254 20/4 1.233 20
1.205
18/4
1.575 4/4 1.494 4/4 1.550 22/4
89/4
1.240 1.067 20/4 1.179 20/4 1.313 17 1.44
<−80 235.2 317 215–6 80–1 118 139–40 123 142–3 71.5 114 146–7 9.4 54 230 65 58 5.7 143 147.5 140–1 240–2 27 99–100 37.5 32 60–1 163–4 37 113–4 75–6 246–7
269–70 246730 subl. d. 202
284.1 306.4 331.7 272–3 274 2707 16423 210.9
subl. 175–803 238 12522 15215 151–380 280–5 320 340
Solubility in 100 parts Water 0.0725 i. i. <0.02 0.00325 10220 16420 v. s. 7730 s. h. v. s. v. s. v. sl. s. h. 0.00725 sl. s. h. 0.07425 v. s. h. v. s. sl. s. h. i. 0.17 c. v. s. h. 3.820 v. s. 0.2100 sl. s. 0.4625 0.42100 0.08 0.38100 0.28100 d. s. s. h. s. h. s. h. i. i.
Alcohol
Ether
s. v. sl. s. v. s. sl. s. 9.520 s. s. v. s.
s. v. s. v. s. s. v. s. i. i. sl. s.
sl. s.
s.
sl. s. s. h. s. v. s. v. s. v. s. s. s. v. s. s. s. v. s. i.
s. s. v. s. v. s. i. s. s. v. s. s. s. i.
s. ∞ s. h. sl. s. h. s. s. s.
s. ∞ v. sl. s. v. sl. s. s.
sl. s. sl. s. c. s. h. 0.1120 0.0819 0.1730 0.0630 i. s. i. 1.95112 0.1920 sl. s. h. 0.6520 0.24165 0.0215
s. v. s. v. s. 7.120 5.820 ∞ v. s. sl. s. i. v. s. h. v. s. h. v. s.
s. v. s. v. s. 7.920 6.120 ∞ v. s. v. sl. s. i. v. s. v. s. ∞
2811 3112 0.910
2211 2510 2.218
i. i. v. sl. s. i. v. sl. s. i. i. i. i.
219
v. s.
v. s.
v. s.
v. s. s. s. h. s. sl. s. c.
v. s. s.
sl. s.
v. sl. s.
9100
v. s. v. s.
-naphthalene (α-) (β-) -phenol (o-) (m-) (p-) -phenol sulfonic acid (1-,4-,2-) (1-,2-,4-) -phthalic acid (3-) (4-) -toluene (o-) (m-) (p-) -toluene sulfonic acid (1-,4-,2-) -toluidine (4-,1-,2-) (3-,1-,4-) Nitron Nitroso-dimethylaniline (p-) -naphthol (β-)(1-) Nonadecane (n-) Nonane (n-) Octadecane (n-) Octane (n-) (iso-) Octyl acetate (n-) (sec-) alcohol (n-) (sec-) Octylene (n-) Oleic acid Orcinol (1-,3-,5-) Oxalic acid Palmitic acid Pelargonic acid Penta-chloroethane -decane (n-) -erythritol Pentandiol Pentane (n-) (i-) (neo-) Phenacetin Phenanthrene Phenetidine (o-) (p-) Phenetole Phenol -phthalein -sulfonic acid (o-) Phenyl acetaldehyde acetic acid -acetylene aniline (o-) (p-) Phenyl-ethyl alcohol -glycine -hydrazine -hydrazine sulfonic acid (p-) isocyanate -methylpyrazolone (3-)(N-) -mustard oil naphthalene (α-) (β-) naphthylamine (α-) (β-) phenol (o-) (p-) propyl alcohol (γ-) quinoline (2-)(α-) (8-)(0-) salicylate, salol stearate urethane
C10H7NO2 C10H7NO2 NO2C6H4OH NO2C6H4OH NO2C6H4OH HOC6H3(NO2)SO3H3H2O HOC6H3(NO2)SO3H3H2O NO2C6H3(CO2H)2 NO2C6H3(CO2H)2 CH3C6H4NO2 CH3C6H4NO2 CH3C6H4NO2 CH3C6H3(NO2)SO3H2H2O NO2C6H3(CH3)NH2 NO2C6H3(CH3)NH2 C20H16N4 ONC6H4N(CH3)2 ONC10H6OH CH3(CH2)17CH3 CH3(CH2)7CH3 CH3(CH2)16CH3 CH3(CH2)6CH3 (CH3)3CCH2CH(CH3)2 CH3CO2CH2(CH2)6CH3 CH3CO2CH(CH3)C6H13 CH3(CH2)6CH2OH CH3(CH2)5CH(OH)CH3 CH3(CH2)5CH:CH2 C8H17CH:CH(CH2)7CO2H (HO)2C6H3CH3 HO2CCO2H2H2O CH3(CH2)14CO2H CH3(CH2)7CO2H CHCl2CCl3 CH3(CH2)13CH3 C(CH2OH)4 HOCH2(CH2)3CH2OH CH3(CH2)3CH3 (CH3)2CHCH2CH3 (CH3)2C(CH3)2 C2H5OC6H4NHCOCH3 < (C6H4CH)2 > C2H5OC6H4NH2 C2H5OC6H4NH2 C2H5OC6H5 C6H5OH C20H14O4 HOC6H4SO3He H2O C6H5CH2CHO C6H5CH2CO2H C6H5C:CH C6H5C6H4NH2 C6H5C6H4NH2 C6H5CH2CH2OH C6H5NHCH2CO2H C6H5NHNH2 H2NNHC6H4SO3H C6H5N:CO C4H5ON2C6H5 C6H5N:CS C10H7C6H5 C10H7C6H5 C10H7NHC6H5 C10H7NHC6H5 C6H5C6H4OH C6H5C6H4OH C6H5(CH2)3OH C6H5C9H6N C6H5C9H6N HOC6H4CO2C6H5 CH3(CH2)16CO2C6H5 C6H5NHCO2C2H5
173.16 173.16 139.11 139.11 139.11 273.22 273.22 211.13 211.13 137.13 137.13 137.13 253.23 152.15 152.15 312.36 150.18 173.16 268.51 128.25 254.48 114.22 114.22 172.26 172.26 130.22 130.22 112.21 282.45 124.13 126.07 256.42 158.23 202.31 212.41 136.15 104.15 72.15 72.15 72.15 179.21 178.22 137.18 137.18 122.16 94.11 318.31 187.68 120.14 136.14 102.13 169.22 169.22 122.16 151.16 108.14 188.20 119.12 174.20 135.18 204.26 204.26 219.27 219.27 170.20 170.20 136.19 205.25 205.25 214.21 360.56 165.19
yel./al. col./al. yel. mn. col. mn. yel. pr. nd. nd./aq. yel./aq. yel. cr. yel. lq. lq. rhb. pl./aq. yel. mn. red mn. yel. lf. gn. tri. brn. pr. cr. col. lq. cr. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. col. nd. pr./bz. col. mn. col. pl. col. oil col. lq. col. lq. cr. lq. col. lq. col. lq. col. lq. col. mn. pl./al. oil lq. col. lq. col. nd. col. rhb. cr. lq. lf. col. lq. cr. lf. col. oil cr. yel. oil cr./al. lq. pr./aq. col. lq. waxy lf./al. pr./al. rhb. nd. nd. oil nd. lq. rhb./al. cr. pl./al.
1.223 62 45
1.295 1.485 20 1.479 20
1.163 20/4 1.160 18/4 1.139 55/55 1.365 15 1.312 17
0.777 32/4 0.718 20/4 0.775 28/4 0.703 20/4 0.692 20/4 0.885 0/4 0.863 14/4 0.827 20/4 0.822 20/4 0.721 18/4 0.854 78/4 1.290 4 1.653 19/4 0.849 70/4 0.906 20/4 1.671 25/4 0.770 20/4 0.994 20/4 0.630 18/4 0.621 19 0.613 20/4 1.179 25 1.061 15 0.967 20/4 1.071 25/4 1.299 25/4 1.025 20 1.081 80/4 0.930 20/4 1.023 18/4 1.097 23/4 1.096 20/4 1.138 15/15 1.17 1.18 1.008 20/4 1.250 20/4 1.106
30/4
59–60 79 44–5 96–7 113–4 d. 110 51.5 222 164–5 −4.1 15–16 51.9 130 105–7 116–7 189–90 d. 86–7 109.5 32 −53.7 28 −56.5 −107.4 −38.5 −16 −38.6 14 107–8 101.5 63–4 12.5 −22 10 262 −129.7 −160.0 −20 134–5 99–100 <−21 3–4 −30.2 42–3 261–2 50 d. 76–7 −43 45–6 50–2 127 19.6 286 128 −21 45 102.5 62 107–8 56–7 164–5 <−18 86 42–3 52 52–3
304 16515 214.5 19470 subl.
222.3 230–1 237.7
330 150.5759 317 125.7 99.3760 210 195 194–5 179–80 126 285–6100 287–90 subl. 271.5100 253–4 162 270.5 27630 239.4 36.3 27.95 9.5 d. 340 228–9 254–5 172 181.4 193–4 265.5 142–3 299760 302 219–21750 243.5 166769 19117 219–20 336–7 345–6 335258 399.5 275 305–8 235–7 363 283187 172–312 26715 237–8
i. i. 1.08100 1.3520 1.625 v. s. v. s. 2.0525 v. s. 0.0780 0.0580 0.0480 47.728 v. sl. s. sl. s. h. i. i. 0.120 i. i. i. 0.00216 i. i. i. 0.05425 0.09625 i. i. v. s. s. i. v. sl. s. 0.0520 i. 5.615 ∞ 0.03616 i. i. 0.720 i. i. i. i. 8.215 0.220 v. s. v. sl. s. 1.6620 i. v. sl. s. s. h. 1.620 s. sl. s. h. 0.612 d. 120 i. i. i. 0.0860 0.460 i. i. sl. s. sl. s. sl. s. 0.01525 i. i. c.
s. v. s. v. s. v. s. v. s. v. s. v. s. v. s. h. v. s. ∞ ∞ 8.615 v. s. s. s. s. h. s. 2.418 sl. s. sl. s. sl. s. sl. s. sl. s. s. s. ∞ ∞ ∞ ∞ v. s. s. 920 s. ∞ v. s. v. sl. s. ∞ ∞ s. 40 h. 10 h. s. s. ∞ ∞ 1025 v. s. ∞ v. s. ∞ s. s. s. s. ∞ sl. s. d. v. s. h. s. v. s. sl. s. s. v. s. h. s. s. ∞ s. h. s. v. s. s.
s. v. s. v. s. s. v. s. sl. s. sl. s. s. ∞ ∞ 80.815 v. s. s. v. sl. s. s. s. s. s. s. s. s. s. ∞ ∞ ∞ ∞ v. s. 1.3 s. s. ∞ v. s. i. ∞ ∞ s. 1.625 v. s. s. s. ∞ ∞ 5.9 c. ∞ v. s. ∞ s. s. ∞ sl. s. ∞ v. s. v. sl. s. s. v. s. sl. s. s. v. s. h. s. s. ∞ s. s. s. s.
2-43
TABLE 2-2
Physical Properties of Organic Compounds (Continued ) Name
Phenylene-diamine (o-) (m-) (p-) Phloroglucinol (1-,3-,5-) Phorone Phosgene Phthalic acid (o-) (m-)(iso-) anhydride (o-) nitrile (o-) Phthalide Phthalimide (o-) Picoline (α-) (β-) (γ-) Picramic acid (1-,2-,4-,6-) Picric acid (2-,4-,6-) Picryl chloride (2-,4-,6-) Pinacol Pinacoline Pinene (α-)(dl-) hydrochloride Pinol (dl-) Piperidine carboxylic acid (α-)(dl-) Piperidinium pentamethylene dithiocarbamate Propane Propionic acid aldehyde anhydride Propyl acetate (n-) (i-) alcohol (n-) (i-) amine (n-) (i-) aniline (n-) benzoate (n-) (i-) bromide (n-) (i-) n-butyrate (n-) i-butyrate (n-) n-butyrate (i-) i-butyrate (i-) chloride (n-) (i-) Propyl formate (n-) (i-) furoate (n-) lactate (n-) (i-) mercaptan (n-) (i-) propionate (n-) (i-) thiocyanate (i-) n-valerate (n-) i-valerate (n-) i-valerate (i-) Propylene bromide chlorohydrin chloride glycol oxide Protocatechuic acid (3-,4-)
2-44
Formula C6H4(NH2)2 C6H4(NH2)2 C6H4(NH2)2 C6H3(OH)32H2O [(CH3)2C:CH]2CO OCCl2 C6H4(CO2H)2 C6H4(CO2H)2 C6H4 < (CO)2 > O C6H4(CN)2 C6H4(CH2)(CO) > O C6H4 < (CO)2 > NH C5H4NCH3 C5H4NCH3 C5H4NCH3 HOC6H2(NH2)(NO2)2 HOC6H2(NO2)3 ClC6H2(NO2)3 [(CH3)2COH]2 CH3COC(CH3)3 C10H16 C10H17Cl C10H16O CH2 < (CH2CH2)2 > NH HO2CCH < (CH2CH2)2 > NH (CH2)5CS2HHN(CH2)5 CH3CH2CH3 CH3CH2CO2H CH3CH2CHO (CH3CH2CO)2O CH2CO2CH2CH2CH3 CH3CO2CH(CH3)2 CH3CH2CH2OH (CH3)2CHOH CH3CH2CH2NH2 (CH3)2CHNH2 C6H5NHCH2CH2CH3 C6H5CO2CH2CH2CH3 C6H5CO2CH(CH3)2 CH3CH2CH2Br (CH3)2CHBr C2H5CH2CO2CH2C2H5 (CH3)2CHCO2CH2C2H5 C2H5CH2CO2CH(CH3)2 (CH3)2CHCO2CH(CH3)2 CH3CH2CH2Cl (CH3)2CHCl HCO2CH2CH2CH3 HCO2CH(CH3)2 C4H3OCO2C3H7 CH3CH(OH)CO2CH2C2H5 CH3CH(OH)CO2CH(CH3)2 CH3CH2CH2SH (CH3)2CHSH C2H5CO2CH2C2H5 C2H5CO2CH(CH3)2 (CH3)2CHCNS CH3(CH2)3CO2CH2C2H5 (CH3)2CHCH2CO2C3H7 (CH3)2CHCH2CO2C3H7 CH3CH:CH2 CH3CHBrCH2Br CH3CHClCH2OH CH3CHClCH2Cl CH3CH(OH)CH2OH CH3(CHCH2)O (HO)2C6H3CO2HH2O
Formula weight
Form and color
108.14 108.14 108.14 162.14 138.20 98.92 166.13 166.13 148.11 128.13 134.13 147.13 93.12 93.12 93.12 199.12 229.11 247.56 118.17 100.16 136.23 172.69 152.23 85.15 129.16 232.41 44.09 74.08 58.08 130.14 102.13 102.13 60.09 60.09 59.11 59.11 135.20 164.20 164.20 123.00 123.00 130.18 130.18 130.18 130.18 78.54 78.54 88.10 88.10 154.16 132.16 132.16 76.15 76.15 116.16 116.16 101.16 144.21 144.21 144.21 42.08 201.91 94.54 112.99 6.09 58.08 172.13
lf./aq. rhb. mn. rhb. yel. pr. gas mn./aq. nd./aq. rhb. cr. nd./aq. cr./et. col. lq. col. lq. lq. red nd. yel. rhb. yel. mn. col. nd. col. lq. col. lq. lf. lq. lq. cr. cr. gas col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. col. lq. lq. lq. col. lq. col. lq. lq. lq. col. lq. col. lq. gas col. lq. col. lq. col. lq. col. oil col. lq. nd./aq.
Specific gravity 1.139 15/15 0.885 20/4 1.392 19/4 1.593 20/4 1.527
4
1.164 99/4 0.950 15/4 0.961 15/4 0.957 15/4 1.763 20/4 1.797 20 0.967 15 0.800 16 0.878 20/4 0.953 20/20 0.860 20/4 1.13 0.585 −45/4 0.992 20/4 0.807 20/4 1.012 20/4 0.886 20/4 0.874 20/20 0.804 20/4 0.78920/4 0.718 20/20 0.694 15/4 0.949 18 1.021 25/25 1.010 25/25 1.353 20/4 1.310 20/4 0.879 15 0.884 0/4 0.865 18 0.869 0/4 0.890 20/4 0.859 20 0.901 20/4 0.873 20/4 1.075 26/4 0.836 25/4 0.809 25/4 0.883 20/4 0.893 0 0.963 20 0.874 15 0.863 20/4 0.854 17 0.609 −47/4 1.933 20/4 1.103 20 1.159 20/20 1.040 19.4 0.831 20/20 1.542 4/4
Melting point, °C
Boiling point, °C
103–4 62.8 140 117 28 −104 208 330 130.8 141 73(65) 238 −70
256–8 284–7 267 subl. 197.2743 8.2756 d. subl. 284.5
169 121.8 83 43(38) −52.5 −55 131–2 −9 264 175 −187.1 −22 −81 −45 −92.5 −73.4 −127 −85.8 −83 −101 −51.6 −109.9 −89 −95.2
−122.8 −117 −92.9
−112 −130.7 −76 −70.7 −185 −55.5 <−70 199 d.
290 subl. 128.8 143.5 143.1 expl. d. 171–2789 106.2 154–6 207–8 183–4 106 −42.2 141.1 49.5740 168.8780 101.6 88.4 97.8 82.5 49–50761 33–4 222 231 218.5 70.8 60 142.7 134–5 128 120.8 46.4 36.5 81.3 68-71751 211 122-3150 167.5 67-8 58-60 122-3 109-11750 152-3754 67.5 155.9 142756 −48749 141.6 133-4 96.8 188-9 35
Solubility in 100 parts Water 73381 35.125 669107 1.1325 0.150 v. sl. s. 0.7025 0.2100 v. sl. s. sl. s. c. v. sl. s. 0.0425 v. s. ∞ ∞ 0.1422 1.2320 0.01815 sl. s. c. 2.515 v. sl. s. i.
Alcohol
Ether
v. s. v. s. s. v. s. s.
v. s. s. s. v. s. s.
1218 s. s.
0.6815
s. 5 ∞ ∞ ∞ s. 620 4.817 v. s. s. s. 33 s. ∞
sl. s. s. h. ∞ ∞ ∞ sl. s. 113 717 v. s. s. ∞ s. s.
∞ s. 628 6.518 cc. ∞ 2020 d. 1.616 320 ∞ ∞ ∞ ∞ i. i. i. 0.2520 0.3220 0.1717 v. sl. s. v. sl. s. v. sl. s. 0.2720 0.3120 12.222 2.122 v. sl. s. s. s. v. sl. s. v. sl. s. 0.5625 0.625 i. i. i.
s. ∞ ∞ d. ∞ ∞ ∞ ∞ ∞ ∞ v. s. s. s. ∞ ∞ ∞
v. s. ∞ ∞
∞ ∞ ∞ ∞ s. s. s. s. ∞ ∞ ∞ ∞ ∞ ∞
∞ ∞ ∞ ∞ ∞ s. s. s. ∞ ∞ ∞ ∞ ∞ ∞
44.6 cc. 0.2520 s. 0.2720 ∞ 3320 1.8214
1200 cc. s. s. v. s. ∞ ∞ v. s.
v. s. s. v. s. 8 ∞ s.
∞ ∞ ∞ ∞ ∞ ∞ v. s. s. s. ∞ ∞ ∞
Pulegol (iso-)(d-) Pulegone Pyrazole Pyrazoline Pyrazolone Pyrene Pyridazine Pyridine Pyrocatechol (o-) Pyrogallol (1-,2-,3-) Pyrone Pyrrole Pyrrolidine Pyrroline Pyruvic acid Quercitrin Quinaldine (py-2) Quinoline (iso-) -diol (1-,3-) Quinone (p-) R-acid Ca salt (2-)(3-,6-) K salt Na salt Raffinose Resorcinol (m-) Retene Rhamnose (β-) Ricinoleic acid Rosaniline Rosolic acid Saccharin Safrole (1-,3-,4-) (iso-)(1-,3-,4-) Salicylic acid (o-) aldehyde (o-) Saligenin Schaeffer’s salt, Ca K Na Semicarbazide hydrochloride Skatole (3-) Sodium methylate Sorbitol Sorbose (d- or l-) Starch Stearic acid amide Styrene Suberic acid Succinic acid Sucrose Sulfanilic acid (p-) Sylvestrene (d-) Tartaric acid (meso-) (racemic) (d- or l-) Tartronic acid Terephthalic acid (p-) Terpin hydrate (cis-) Terpineol (α-)(d- or l-) (dl-) Terpinyl acetate (α-)(dl-) Tetrabromo-ethane (sym) (uns) Tetrachloro-ethane (sym) (uns) -ethylene Tetracosane (n-) Tetradecane (n-) Tetraethyl-thiuram disulfide
C10H17OH C10H16O NHN:CHCH:CH NHN:CHCH2CH2 NHCOCH2CH:N C16H10 N2 < (CHCH)2 > CH < (CHCH)2 > N C6H4(OH)2 C6H3(OH)3 CO < (CHCH)2 > O < (CH:CH)2 > NH < (CH2CH2)2 > NH < (CHCH2)2 > NH CH3COCO2H C21H20O112H2O CH3C9H6N C9H7N C9H7N C6H4CH:C(OH)N:C(OH) CO < (CHCH)2 > CO HOC10H5(SO3)2Ca HOC10H5(SO3K)2 HOC10H5(SO3Na)2 C18H32O165H2O C6H4(OH)2 C18H18 CH3(CHOH)4CHOH2O C17H32(OH)CO2H C20H21ON3 C20H16O3 C6H4(CO)(SO2) > NH CH2:CHCH2C6H3:O2CH2 CH3CH:CHC6H3:O2CH2 HOC6H4CO2H HOC6H4CHO HOC6H4CH2OH (HOC10H6SO3)2Ca5H2O HOC10H6SO3K HOC10H6SO3Na NH2CONHNH2 NH2CONHNH3Cl CH3C8H6N CH3ONa [CH2OH(CHOH)2]2 C6H12O6 (C6H10O5)x CH3(CH2)16CO2H CH3(CH2)16CONH2 C6H5CH:CH2 HO2C(CH2)6CO2H HO2C(CH2)2CO2H C12H22O11 H2NC6H4SO3H C10H16 (CHOHCO2H)2 (CHOHCO2H)2H2O (CHOHCO2H)2 CH(OH)(CO2H)2a H2O C6H4(CO2H)2 C10H20O2H2O C10H18O C10H18O CH3CO2C10H17 Br2CHCHBr2 Br3CCH2Br Cl2CHCHCl2 Cl3CCH2Cl Cl2C:CCl2 CH3(CH2)22CH3 CH3(CH2)12CH3 [(C2H5)2NCS]2S2
154.24 152.23 68.08 70.09 84.08 202.24 80.09 79.10 110.11 126.11 96.08 67.09 71.12 69.10 88.06 484.40 143.18 129.15 129.15 161.15 108.09 342.35 380.46 348.26 594.52 110.11 234.32 182.17 298.45 319.39 304.33 183.18 162.18 162.18 138.12 122.12 124.13 576.59 262.31 246.21 75.07 111.54 131.17 54.03 182.17 180.16 162.14 284.47 283.48 104.14 174.19 118.09 342.30 173.18 136.23 150.09 168.10 150.09 129.07 166.13 190.28 154.24 154.24 196.28 345.70 345.70 167.86 167.86 165.85 338.64 198.38 296.52
col. lq. col. lq. nd./et. lq. nd. yel. pr. lq. col. lq. nd./aq. nd. cr. lq. lq. lq. col. lq. yel. nd. lq. lq. pl. cr. yel. mn. cr. cr. cr. cr./aq. col. rhb. lf./al. col. mn. lq. col. nd. red lf. mn. col. mn. col. lq. mn. col. oil rhb./aq. cr. cr. cr. pr./al. pr. lf. pd. cr. rhb. amor. mn. col. cr. col. lq. nd./aq. col. mn. col. mn. col. cr. lq. cr. tri. mn. pr./aq. cr. rhb. col. cr. col. cr. lq. col. lq. col. lq. col. lq. lq. col. lq. cr. col. lq. cr.
0.911 20/4 0.932 20/20 70 1.277 0/4 1.107 20/4 0.982 20/4 1.344 4 1.453 4 1.190 40.3 0.948 20/4 0.852 22.5 0.910 20/4 1.267 20/4 1.059 20/4 1.095 20 1.099 21/4 1.318 20/4
1.465 0 1.272 15 1.1316 1.47120/4 0.954 16
1.100 20/4 1.122 20/4 1.443 20/4 1.153 25/4 1.161 25
1.654 15 1.5021 0.847 69.3 0.903 20/4 1.266 25/4 1.572 25/4 1.588 15 0.863 20/4 1.737 1.697 20/4 1.760 20/4 1.510 0.935 15 0.935 20/20 0.966 20/4 2.964 20/4 2.875 20/4 1.600 20/4 1.588 20/4 1.624 15/4 0.779 51/4 0.765 20/4 1.17
165 149-50 −8 −42 104-5 133-4 32.5
13.6 182-5 −1 −15 24.6 237 115.7
119 110.7 98-9 126 4-5 186 d. 308-10 d. 225-8 11.2 6-7 159 -7 86-7
96 173 d. 95 d. 300 110-2 165 d. 70-1 108-9 −31 140-4 189-90 170-86 d. d. > 280
86-9 10 224754 186-8 144 subl. d. >360 208 115-6 240-5 309 215-7 131 87-8 90-1 165 244-5750 237.1747 240.5763 subl.
d. 130 276.5 390-4 226-810 subl. 233-4 252-3 21120 196.5 subl.
755
265-6
291110 25112 145-6 279100 235 d. 176-7
159-60 205-6 168-70 d. 155-8 subl. 117 38-40 35 < −50 −1.0 0 −36 −19 51.1 5.5 70
d. subl. d. 219-21 218-9752 220 d. 15154 10413 146.3 129-30 120.8 324 252.5
v. sl. s. i. s. ∞ s. i. ∞ ∞ 45.120 40 13 v. sl. s. i. ∞ v. s. ∞ 0.04 20 v. sl. s. 6 sl. s. v. sl. s. sl. s. h. 30.6 25 29.5 25 25.2 25 14.3 20 14712 i. 60.8 21 i. v. sl. s. 0.1225 0.4 25 i. i. 0.223 1.7 86 6.615 4.7620 3.4625 6.2925 v. s. v. s. 0.05 c. d. v. s. 5517 i. 0.0325 i. v. sl. s. 0.1416 6.820 1790 0.810 12015 20.620 13920 v. s. 0.001 c. 0.415 i. i. i. i. 20
0.29 i. 0.0220 i. i.
∞ s. ∞ v. s. 3 h. s. ∞ v. s. s. s. s. ∞ ∞ ∞ s. ∞ s.
0.120 v. s. 69 h.
∞ s. sl. s. v. sl. s. v. s. s. s. v. s. s. v. s. s. ∞ ∞ ∞ sl. s. s. ∞ s. s.
∞ sl. s. v. s. h. 3.1 c. s. ∞ 4915 ∞ v. s.
v. s. v. s. h. i. ∞ i. sl. s. 1.05 c. ∞ ∞ 5115 ∞ v. s.
v. s. sl. s. s.
i. i. s.
v. s. h. sl. s. i. 220 s. h. ∞ s. 9.915 0.9 v. sl. s.
i. 6g s. h: ∞ 0.815 1.215 i. v. sl. s.
20 2515 v. s. sl. s. h. 1015 v. s. v. s. 20 ∞ s. ∞ ∞ ∞ v. s.
0.09 0.415 i. i. 115 v. s. v. s. ∞ ∞ ∞ ∞ s. v. s.
2-45
TABLE 2-2
Physical Properties of Organic Compounds (Concluded ) Name
Tetrafluoro-ethylene Tetrahydro-furan -furfuryl alcohol -pyran Tetralin Tetramethyl-thiuram disulfide Tetryl (2-,4-,6-) Theobromine Thio-acetic acid -aniline (4-, 4′-) -carbanilide -naphthol (β-) -phenol -salicylic acid (o-) -urea Thiophene Thymol (5-,2-,1-) Tolidine (0-)(3-,3′-,4-,4′-) Toluene sulfonic acid (o-) (p-) sulfonic amide (p-) sulfonic chloride (p-) Toluic acid (o-) (m-) (p-) Toluidine (o-) (m-) (p-) hydrochloride (o-) sulfonic acid (1-,2-,3-) Toluylenediamine (1-,2-,4-) Tolylene diisocyanate (1-,2-,4-) Trehalose Triamylamine (n-) (i-) Tributyl-amine (n-) phosphite Trichloro-acetic acid -benzene (s-)(1-,3-,5-) -ethane (1-,1-,1-) -ethylene -phenol Tricosane (n-) Tricresyl phosphate (o-) Tridecane (n-) Triethanol amine Triethyl-amine -benzene (1-,3-,5-) (1-,2-,4-) borate citrate Triethylene glycol Trifluoro-chloromethane chloroethylene -trichloroethane Trimethoxybutane (1-,3-,3-) Trimethylamine Trimethylene bromide chloride glycol Trinitro-benzene (1-,3-,5-) -benzoic acid (2-,4-,6-) -tert-butylxylene -naphthalene (α-)(1-,3-,5-) (β-)(1-,3-,8-) (γ-)(1-,4-,5-)
2-46
Formula F2C:CF2 CH2(CH2)2CH2O C4H7OCH2OH CH2(CH2)3CH2O C6H4CH2(CH2)2CH2 [(CH3)2NCS]2S2 (NO2)3C6H2N(CH3)NO2 C7H8O2N4 CH3COSH (NH2C6H4)2S (C6H5NH)2CS C10H7SH C6H5SH HSC6H4CO2H NH2CSNH2 < (CH:CH)2 > S (CH3)(C3H7)C6H3OH [CH3(NH2)C6H3]2 C6H5CH3 CH3C6H4SO3H2H2O CH3C6H4SO3HH2O CH3C6H4SO2NH2 CH3C6H4SO2Cl CH3C6H4CO2H CH3C6H4CO2H CH3C6H4CO2H CH3C6H4NH2 CH3C6H4NH2 CH3C6H4NH2 CH3C6H4NH3Cl CH3(NH2)C6H3SO3H CH3C6H3(NH2)2 CH3C6H3(NCO)2 C12H22O112H2O [CH3(CH2)3CH2]3N [(CH3)2CH(CH2)2]3N [CH3(CH2)2CH2]3N [CH3(CH2)3O]3P Cl3CCO2H C6H3Cl3 Cl3CCH3 Cl2C:CHCl Cl3C6H2OH CH3(CH2)21CH3 OP(OC6H4CH3)3 CH3(CH2)11CH3 (HOCH2CH2)3N (CH3CH2)3N (C2H5)3C6H3 (C2H5)3C6H3 B(OCH2CH3)3 HOC3H4(CO2C2H5)3 (CH2OCH2CH2OH)2 CF3Cl F2C:CFCl Cl2CFCClF2 CH2(OCH3)CH2C(OCH3)2CH3 (CH3)3N BrCH2CH2CH2Br ClCH2CH2CH2Cl HOCH2CH2CH2OH C6H3(NO2)3 (NO2)3C6H2CO2H (NO2)3C6(CH3)2C4H9 C10H5(NO2)3 C10H5(NO2)3 C10H5(NO2)3
Formula weight
Form and color
Specific gravity
Melting point, °C
Boiling point, °C
100.02 72.10 102.13 86.13 132.20 240.41 287.15 180.17 76.11 216.29 228.30 160.22 110.17 154.18 76.12 84.13 150.21 212.28 92.13 208.23 190.21 171.21 190.64 136.14 136.14 136.14 107.15 107.15 107.15 143.62 187.21 122.17 174.15 378.33 227.42 227.42 185.34 250.32 163.40 181.46 133.42 131.40 197.46 324.61 368.36 184.35 149.19 101.19 162.26 162.26 146.00 276.28 150.17 104.47 116.48 187.39 148.20 59.11 201.91 112.99 76.09 213.11 257.12 297.26 263.16 263.16 263.16
gas col. lq. col. lq. lq. col. lq. cr. yel. mn. rhb. yel. lq. nd./aq. rhb./al. cr./al. col. lq. yel. nd. rhb./al. col. lq. cr. lf. col. lq. cr. mn. mn. tri. cr./aq. pr./aq. cr./aq. col. lq. col. lq. cr. mn. pr. cr. rhb. lq. rhb./al. lq. col. lq. col. lq. lq. cr. nd. lq. col. lq. nd. lf. lq. col. lq. col. lq. col. oil lq. lq. lq. oil col. lq. gas gas lq. lq. gas lq. lq. oil col. rhb. rhb./aq. nd./al. rhb. cr./al. yel. cr.
1.58−78 0.88821/4 1.05020/4 0.88120/4 0.97318/4 1.29 1.5719
−142.5 −65
−76.3 65-6 177-8743 88 206764
1.07410 1.324 1.07423/4 1.40520/4 1.07015/4 0.97225/25 20/4
0.866
1.062115/4 1.054112/4 0.99920/4 0.98920/4 1.04620/4
1.2328
−31 155-6 130.5 330 < −17 108 154 81 164 180-2 −30 51.5 128-9 −95 d. 104-5 137 69 104-5 110-1 179-80 −16.3 −31.5 44-5 218-20 99
expl. 93 d. 286-8 168-9 subl. d. 84 232752 110.8 128.80 146-70 134.510 259751 263 274-5 199.7 203.3 200.3 242 283-5 134.520
97 0.78620/4 0.77820/20 0.92520/4 1.61746/15 1.32526/4 1.46620/20 1.49075/4 0.77948/4 0.75720/4 1.12620/20 0.72920/20 0.86120/4 0.88217/4 0.86420/20 1.13720/4 1.12520/20 1.726−130 1.57620/4 0.932 0.662 −5 1.987 15/4 1.201 15 1.060 20/4 1.688 20/4
58 63.5 −73 68-9 47.7 −6.2 20-1 −114.8
−5 −182 −157.5 −35 −124 −34.4 121 210-20 d. 110 122-3 218-9 148-9
240-5 235 216.5761 122-312 195.5754 208.5764 74.1 87.2 246 23415 234 277-9150 89.4 215 217-8755 120 294 290 −80 −27.9 47.6 63-525 3.5 167.5 123-5 214 d.
Solubility in 100 parts Water 0.0130 s. ∞ s. i. i. i. 0.0615 s. sl. s. h. i. v. sl. s. v. sl. s. sl. s. h. 9.213 i. 0.0919 v. sl. s. 0.0516 v. s. v. s. 0.29 i. 2.17100 1.6100 1.3100 1.525 sl. s. 0.7421 s. 0.9711 s. h. d. s. h. i. i. i. i. 12025 i. i. 0.125 0.0925 i. i. i. ∞ ∞ > 190 i. i. d. i. ∞ d. i. d. 4119 0.1730 0.2725 ∞ 0.0315 2.0524 i. i. 0.02100 i.
Alcohol
Ether
s. ∞
s. ∞
s.
s.
s. h. 0.06 c. ∞ s. v. s. v. s. v. s. s. s. s. v. s. s. s. s. s. 7.45 s. v. s. v. s. v. s. ∞ ∞ v. s. sl. s.
s. 0.03 h. ∞ s. v. s. v. s. ∞ sl. s. v. s. s. ∞
s. v. s. v. s. ∞ ∞ v. s.
s. d. sl. s. h.
i.
s.
s.
∞
s. sl. s. ∞ ∞ v. s.
∞ ∞ v. s.
v. s. ∞ ∞ s. s.
v. s. sl. s. ∞ s. s.
∞ ∞
∞ v. sl. s.
∞
∞
s. s. s. ∞ 1.918
s. s. s. 1.518
sl. s. s. 0.0523 0.1119
0.1315 0.419
s.
s.
-phenol (2-,3-,6-) -toluene (β-)(2-,3-,4-) (γ-)(2-,4-,5-) (α-)(2-,4-,6-) Trional Triphenyl-arsine carbinol guanidine (α-) methane methyl phosphate Tripropylamine (n-) Undecane (n-) Urea nitrate Uric acid Valeric acid (n-) (i-) aldehyde (n-) (i-) amide (n-) (i-) Vanillic acid (3-,4-,1-) alcohol (3-,4-,1-) hyl-thiuram disulfide Vanillin (3-,4-,1-) Veratrole (o-) Vinyl acetate (poly-) acetic acid acetylene alcohol (poly-) chloride propionate Xylene (o-) (m-) (p-) sulfonic acid (1-,4-,2-) Xylidine (1:2)(3-) (1:2)(4-) (1:3)(2-) (1:3)(4-) (1:3)(5-) (1:4)(2-) Xylose (l-)(+) Xylylene dichloride (p-) Zinc diethyl dimethyl dimethyl-dithiocarbamate NOTE:
(NO2)3C6H2OH CH3C6H2(NO2)3 CH3C6H2(NO2)3 CH3C6H3(NO2)3 C2H5(CH3)C(SO2C2H5)3 (C6H5)3As (C6H5)3COH C6H5N:C(NHC6H5)2 (C6H5)3CH (C6H5)3C . . . OP(OC6H5)3 (CH3CH2CH2)3N CH3(CH2)3CH3 H2NCONH2 CO(NH2)2HNO3 C5H4O3N4 C2H5CH2CH2CO2H (CH3)2CHCH2CO2H C2H5CH2CH2CHO (CH3)2CHCH2CHO C2H5CH2CH2CONH2 (CH3)2CHCH2CONH2 CH3O(OH)C6H3CO2H CH3O(OH)C6H3CH2OH [(C2H5)2NCS]2S2 CH3O(OH)C6H3CHO C6H4(OCH3)2 CH3CO2CH:CH2 (CH3CO2CH:CH2)x CH2:CHCH2CO2H CH2:CHC:CH CH2:CHOH (CH2:CHOH)x CH2:CHCl C2H5CO2CH:CH2 C6H4(CH3)2 C6H4(CH3)2 C6H4(CH3)2 (CH3)2C6H3SO3H2H2O (CH3)2C6H3NH2 (CH3)2C6H3NH2 (CH3)2C6H3NH2 (CH3)2C6H3NH2 (CH3)2C6H3NH2 (CH3)2C6H3NH2 CH2OH(CHOH)3CHO C6H4(CH2Cl)2 Zn(CH2CH3)2 Zn(CH3)2 Zn[S2CN(CH3)2]2
229.11 227.13 227.13 227.13 242.34 306.21 260.32 287.35 244.32 243.31 326.28 143.27 156.30 60.06 123.07 168.11 102.13 102.13 86.13 86.13 101.15 101.15 168.14 154.16 296.52 152.14 138.16 86.09 (86.09) 86.09 52.07 44.06 (44.06) 62.50 100.11 106.16 106.16 106.16 222.25 121.18 121.18 121.18 121.18 121.18 121.18 150.13 175.06 123.50 95.45 305.79
nd. cr. yel. pl. cr./al. pl./al. pl. cr. rhb./al. cr. col. cr. pr./al. col. lq. col. lq. col. pr. col. mn. cr. col. lq. col. lq. lq. col. lq. mn. pl. mn. nd./aq. mn./aq. cr. mn. cr. col. lq. col. lq. gas gas lq. col. lq. col. lq. col. lq. col. lf. lq. pr. lq. lq. oil oil nd. mn. col. lq. col. lq.
1.620 20/4 1.620 20/4 1.654 1.199 85/4 1.306 1.188 20/4 1.13 1.014 99/4 1.206 58/4 0.757 20/4 0.741 20/4 1.335 20/4 1.893 20 0.939 20/4 0.931 20/20 0.819 11 0.803 17 1.023 0.965 20/4 1.17 1.056 1.091 15/15 0.932 20/4 1.1920 1.013 15/15 0.705 1.5
117-8 112 104 80.8 76 59-60 162.5 144-5 93.4 145-7 49-50 −93.5 −25.6 132.7 152 d. d. −34.5 −37.6 −92 −51 106 135-7 207 115 70 81-2 22.5 < −60 100-25 −39
1.320 0.908 25/25
d. >200 −160
0.881 20/4 0.867 17/4 0.861 20/4
−25 −47.4 13.2 86 < −15 49-50
0.991 15 1.076 17.5 0.980 15 0.978 20/4 0.972 20/4 0.979 21/4 1.535 0 1.417 0 1.182 18 1.386 11 2.0040/4
15.5 153-4 100.5 −28 −40 248-50
expl. expl. expl. d. >360 >360 d. 359754 d. 24511 156.5 194.5 d. 187 176 103.4 92.5 232 subl. d. 285 207.1 72-3 163 5.5 −12 93-5 144 139.3 138.5 1490.1 223 224-6 216-7 213-4 221-2 215789 240-5 d. 118 46
s. h. i. i. 0.0120 0.315 i. i. i. i. i. i. v. sl. s. i. 10017 v. s. h. 0.06 h. 3.316 4.220 v. sl. s. sl. s. v. s. s. 0.1214 v. s. h. i. 114 v. sl. s. 220 i. s. 0.670.6 s. sl. s. v. sl. s. i. i. i. s. v. sl. s. v. sl. s. v. sl. s. v. sl. s. v. sl. s. v. sl. s. 11720 i. d. d. i.
v. s. sl. s. c. s. h. 1.522 50 s. v. s. 40 v. s. h. sl. s. h. 15525 ∞ ∞ 2020 s. i. ∞ ∞ s. s. v. s. s. v. s. v. s.
v. s. s. v. s. 533 6.615 v. s. v. s.
v. s. s. ∞
v. s. s. ∞
∞
∞
v. s. v. s. ∞ ∞ sl. s. i. ∞ ∞ s. s. v. s. s. v. s. v. s.
s.
v. s.
s. s. s.
∞ ∞ v. s.
s.
s.
v. sl. s. s. d. d.
i. v. sl. s.
°F = 9⁄5 °C + 32.
2-47
2-48
PHYSICAL AND CHEMICAL DATA
VAPOR PRESSURES OF PURE SUBSTANCES UNITS CONVERSIONS
ADDITIONAL REFERENCES
For this subsection, the following units conversions are applicable:
Additional compilations of vapor-pressure data include Boublik, Fried, and Hala, The Vapor Pressures of Pure Substances, Elsevier, Amsterdam, 1984. See also Hirata, Ohe, and Nagahama, Computer Aided Data Book of Vapor-Liquid Equilibria, Kodansha/Elsevier, Tokyo, 1975; Weishaupt, Landolt-Börnstein New Series Group IV, vol. 3; Thermodynamic Equilibria of Boiling Mixtures, Springer-Verlag, Berlin, 1975; Wichterle, Linek, and Hala, Vapor-Liquid Equilibrium Data Bibliography, Elsevier, Amsterdam, 1973; suppl. 1, 1976; suppl. 2, 1982.
°F = 9⁄5 °C + 32. To convert millimeters of mercury to pounds-force per square inch, multiply by 0.01934.
TABLE 2-3
Vapor Pressure of Water Ice from -15 to 0°C* mmHg
TABLE 2-4
Vapor Pressure of Liquid Water from -16 to 0°C* mmHg
t, °C
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
t, °C
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
−14 −13 −12 −11 −10
1.361 1.490 1.632 1.785 1.950
1.348 1.477 1.617 1.769 1.934
1.336 1.464 1.602 1.753 1.916
1.324 1.450 1.588 1.737 1.899
1.312 1.437 1.574 1.722 1.883
1.300 1.424 1.559 1.707 1.866
1.288 1.411 1.546 1.691 1.849
1.276 1.399 1.532 1.676 1.833
1.264 1.386 1.518 1.661 1.817
1.253 1.373 1.504 1.646 1.800
−15 −14 −13 −12 −11
1.436 1.560 1.691 1.834 1.987
1.425 1.547 1.678 1.819 1.971
1.414 1.534 1.665 1.804 1.955
1.402 1.522 1.651 1.790 1.939
1.390 1.511 1.637 1.776 1.924
1.379 1.497 1.624 1.761 1.909
1.368 1.485 1.611 1.748 1.893
1.356 1.472 1.599 1.734 1.878
1.345 1.460 1.585 1.720 1.863
1.334 1.449 1.572 1.705 1.848
−9 −8 −7 −6 −5
2.131 2.326 2.537 2.765 3.013
2.112 2.306 2.515 2.742 2.987
2.093 2.285 2.493 2.718 2.962
2.075 2.266 2.472 2.695 2.937
2.057 2.246 2.450 2.672 2.912
2.039 2.226 2.429 2.649 2.887
2.021 2.207 2.408 2.626 2.862
2.003 2.187 2.387 2.603 2.838
1.985 2.168 2.367 2.581 2.813
1.968 2.149 2.346 2.559 2.790
−10 −9 −8 −7 −6
2.149 2.326 2.514 2.715 2.931
2.134 2.307 2.495 2.695 2.909
2.116 2.289 2.475 2.674 2.887
2.099 2.271 2.456 2.654 2.866
2.084 2.254 2.437 2.633 2.843
2.067 2.236 2.418 2.613 2.822
2.050 2.219 2.399 2.593 2.800
2.034 2.201 2.380 2.572 2.778
2.018 2.184 2.362 2.553 2.757
2.001 2.167 2.343 2.533 2.736
−4 −3 −2 −1 −0
3.280 3.568 3.880 4.217 4.579
3.252 3.539 3.848 4.182 4.542
3.225 3.509 3.816 4.147 4.504
3.198 3.480 3.785 4.113 4.467
3.171 3.451 3.753 4.079 4.431
3.144 3.422 3.722 4.045 4.395
3.117 3.393 3.691 4.012 4.359
3.091 3.364 3.660 3.979 4.323
3.065 3.336 3.630 3.946 4.287
3.039 3.308 3.599 3.913 4.252
−5 −4 −3 −2 −1
3.163 3.410 3.673 3.956 4.258
3.139 3.384 3.647 3.927 4.227
3.115 3.359 3.620 3.898 4.196
3.092 3.334 3.593 3.871 4.165
3.069 3.309 3.567 3.841 4.135
3.046 3.284 3.540 3.813 4.105
3.022 3.259 3.514 3.785 4.075
3.000 3.235 3.487 3.757 4.045
2.976 3.211 3.461 3.730 4.016
2.955 3.187 3.436 3.702 3.986
*For data at 0(0.2)−30(2)−98°C see p. 2324, Handbook of Chemistry and Physics, 40th ed., Chemical Rubber Publishing Co.
−0
4.579 4.546 4.513 4.480 4.448 4.416 4.385 4.353 4.320 4.289
*Computed from the above table with the aid of the thermodynamic equation pw −1.1489t log10 = − 1.330 × 10−5t2 + 9.084 × 10−8t3 273.1 + t pi
TABLE 2-5
Vapor Pressure of Liquid Water from 0 to 100°C* mmHg
t, °C
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 1 2 3 4
4.579 4.926 5.294 5.685 6.101
4.613 4.962 5.332 5.725 6.144
4.647 4.998 5.370 5.766 6.187
4.681 5.034 5.408 5.807 6.230
4.715 5.070 5.447 5.848 6.274
4.750 5.107 5.486 5.889 6.318
4.785 5.144 5.525 5.931 6.363
4.820 5.181 5.565 5.973 6.408
4.855 5.219 5.605 6.015 6.453
4.890 5.256 5.645 6.058 6.498
5 6 7 8 9
6.543 7.013 7.513 8.045 8.609
6.589 7.062 7.565 8.100 8.668
6.635 7.111 7.617 8.155 8.727
6.681 7.160 7.669 8.211 8.786
6.728 7.209 7.722 8.267 8.845
6.775 7.259 7.775 8.323 8.905
6.822 7.309 7.828 8.380 8.965
6.869 7.360 7.882 8.437 9.025
6.917 7.411 7.936 8.494 9.086
6.965 7.462 7.990 8.551 9.147
10 11 12 13 14
9.209 9.844 10.518 11.231 11.987
9.271 9.910 10.588 11.305 12.065
9.333 9.976 10.658 11.379 12.144
9.395 10.042 10.728 11.453 12.223
9.458 10.109 10.799 11.528 12.302
9.521 10.176 10.870 11.604 12.382
9.585 10.244 10.941 11.680 12.462
9.649 10.312 11.013 11.756 12.543
9.714 10.380 11.085 11.833 12.624
9.779 10.449 11.158 11.910 12.706
15 16 17 18 19
12.788 13.634 14.530 15.477 16.477
12.870 13.721 14.622 15.575 16.581
12.953 13.809 14.715 15.673 16.685
13.037 13.898 14.809 15.772 16.789
13.121 13.987 14.903 15.871 16.894
13.205 14.076 14.997 15.971 16.999
13.290 14.166 15.092 16.071 17.105
13.375 14.256 15.188 16.171 17.212
13.461 14.347 15.284 16.272 17.319
13.547 14.438 15.380 16.374 17.427
20 21 22 23 24
17.535 18.650 19.827 21.068 22.377
17.644 18.765 19.948 21.196 22.512
17.753 18.880 20.070 21.324 22.648
17.863 18.996 20.193 21.453 22.785
17.974 19.113 20.316 21.583 22.922
18.085 19.231 20.440 21.714 23.060
18.197 19.349 20.565 21.845 23.198
18.309 19.468 20.690 21.977 23.337
18.422 19.587 20.815 22.110 23.476
18.536 19.707 20.941 22.243 23.616
25 26 27 28 29
23.756 25.209 26.739 28.349 30.043
23.897 25.359 26.897 28.514 30.217
24.039 25.509 27.055 28.680 30.392
24.182 25.660 27.214 28.847 30.568
24.326 25.812 27.374 29.015 30.745
24.471 25.964 27.535 29.184 30.923
24.617 26.117 27.696 29.354 31.102
24.764 26.271 27.858 29.525 31.281
24.912 26.426 28.021 29.697 31.461
25.060 26.582 28.185 29.870 31.642
30 31 32 33 34
31.824 33.695 35.663 37.729 39.898
32.007 33.888 35.865 37.942 40.121
32.191 34.082 36.068 38.155 40.344
32.376 34.276 36.272 38.369 40.569
32.561 34.471 36.477 33.584 40.796
32.747 34.667 36.683 38.801 41.023
32.934 34.864 36.891 39.018 41.251
33.122 35.062 37.099 39.237 41.480
33.312 35.261 37.308 39.457 41.710
33.503 35.462 37.518 39.677 41.942
35 36 37 38 39
42.175 44.563 47.067 49.692 52.442
42.409 44.808 47.324 49.961 52.725
42.644 45.054 47.582 50.231 53.009
42.880 45.301 47.841 50.502 53.294
43.117 45.549 48.102 50.774 53.580
43.355 45.799 48.364 51.048 53.867
43.595 46.050 48.627 51.323 54.156
43.836 46.302 48.891 51.600 54.446
44.078 46.556 49.157 51.879 54.737
44.320 46.811 49.424 52.160 55.030
40 41 42 43 44
55.324 58.34 61.50 64.80 68.26
55.61 58.65 61.82 65.14 68.61
55.91 58.96 62.14 65.48 68.97
56.21 59.27 62.47 65.82 69.33
56.51 59.58 62.80 66.16 69.69
56.81 59.90 63.13 66.51 70.05
57.11 60.22 63.46 66.86 70.41
57.41 60.54 63.79 67.21 70.77
57.72 60.86 64.12 67.56 71.14
58.03 61.18 64.46 67.91 71.51
45 46 47 48 49
71.88 75.65 79.60 83.71 88.02
72.25 76.04 80.00 84.13 88.46
72.62 76.43 80.41 84.56 88.90
72.99 76.82 80.82 84.99 89.34
73.36 77.21 81.23 85.42 89.79
73.74 77.60 81.64 85.85 90.24
74.12 78.00 82.05 86.28 90.69
74.50 78.40 82.46 86.71 91.14
74.88 78.80 82.87 87.14 91.59
75.26 79.20 83.29 87.58 92.05
t, °C
0
1
2
3
4
5
6
7
8
9
50 60 70 80
92.51 149.38 233.7 355.1
97.20 156.43 243.9 369.7
102.09 163.77 254.6 384.9
107.20 171.38 265.7 400.6
112.51 179.31 277.2 416.8
118.04 187.54 289.1 433.6
123.80 196.09 301.4 450.9
129.82 204.96 314.1 468.7
136.08 214.17 327.3 487.1
142.60 223.73 341.0 506.1
90 91 92 93 94
525.76 546.05 566.99 588.60 610.90
527.76 548.11 569.12 590.80 613.17
529.77 550.18 571.26 593.00 615.44
531.78 552.26 573.40 595.21 617.72
533.80 554.35 575.55 597.43 620.01
535.82 556.44 577.71 599.66 622.31
537.86 558.53 579.87 601.89 624.61
539.90 560.64 582.04 604.13 626.92
541.95 562.75 584.22 606.38 629.24
544.00 564.87 586.41 608.64 631.57
95 96 97 98 99 100 101
633.90 657.62 682.07 707.27 733.24 760.00 787.57
636.24 660.03 684.55 709.83 735.88 762.72 790.37
638.59 662.45 687.04 712.40 738.53 765.45 793.18
640.94 664.88 689.54 714.98 741.18 768.19 796.00
643.30 667.31 692.05 717.56 743.85 770.93 796.82
645.67 669.75 694.57 720.15 746.52 773.68 801.66
648.05 672.20 697.10 722.75 749.20 776.44 804.50
650.43 674.66 699.63 725.36 751.89 779.22 807.35
652.82 677.12 702.17 727.98 754.58 782.00 810.21
655.22 679.69 704.71 730.61 757.29 784.78 813.06
*From the Physikalisch-technische Reichsanstalt, Holborn, Scheel, and Henning, Wärmetabellen, Friedrich Vieweg & Sohn, Brunswick, 1909. By permission. For data at 50(0.2)101.8°C, see Handbook of Chemistry and Physics, 40th ed., p. 2326, Chemical Rubber Publishing Co. For a tabulation of temperature for pressures 700(1)779 mm Hg, see Atack, Handbook of Chemical Data, p. 117, Reinhold, New York, 1957. For a tabulation of pressure for 105(5)200(10)370°C, see Atack, p. 134, and for 100(1)374°C, see Handbook of Chemistry and Physics, 40th ed., pp. 2328–2330, Chemical Rubber Publishing Co.
2-49
TABLE 2-6 Cmpd. no.
Vapor Pressure of Inorganic and Organic Liquids CAS no.
C1
C2
C3
C4
C5
Tmin, K
Ps at Tmin
Tmax, K
Ps at Tmax
1 2 3 4 5 6 7 8 9 10
Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane
CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22
74828 74840 74986 106978 109660 110543 142825 111659 111842 124185
39.205 51.857 59.078 66.343 78.741 104.65 87.829 96.084 109.35 112.73
−1324.4 −2598.7 −3492.6 −4363.2 −5420.3 −6995.5 −6996.4 −7900.2 −9030.4 −9749.6
−3.4366 −5.1283 −6.0669 −7.046 −8.8253 −12.702 −9.8802 −11.003 −12.882 −13.245
3.1019E−05 1.4913E−05 1.0919E−05 9.4509E−06 9.6171E−06 1.2381E−05 7.2099E−06 7.1802E−06 7.8544E−06 7.1266E−06
2 2 2 2 2 2 2 2 2 2
90.69 90.35 85.47 134.86 143.42 177.83 182.57 216.38 219.66 243.51
1.1687E+04 1.1273E+00 1.6788E−04 6.7441E−01 6.8642E−02 9.0169E−01 1.8269E−01 2.1083E+00 4.3058E−01 1.3930E+00
190.56 305.32 369.83 425.12 469.7 507.6 540.2 568.7 594.6 617.7
4.5897E+06 4.8522E+06 4.2135E+06 3.7699E+06 3.3642E+06 3.0449E+06 2.7192E+06 2.4673E+06 2.3054E+06 2.0908E+06
11 12 13 14 15 16 17 18 19 20
n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane n-Eicosane
C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40 C20H42
1120214 112403 629505 629594 629629 544763 629787 593453 629925 112958
131 137.47 137.45 140.47 135.57 156.06 156.95 157.68 182.54 203.66
−15.855 −16.698 −16.543 −16.859 −16.022 −18.941 −18.966 −18.954 −22.498 −25.525
8.1871E−06 8.0906E−06 7.1275E−06 6.5877E−06 5.6136E−06 6.8172E−06 6.4559E−06 5.9272E−06 7.4008E−06 8.8382E−06
2 2 2 2 2 2 2 2 2 2
247.57 263.57 267.76 279.01 283.07 291.31 295.13 301.31 305.04 309.58
4.0836E−01 6.1534E−01 2.5096E−01 2.5268E−01 1.2884E−01 9.2265E−02 4.6534E−02 3.3909E−02 1.5909E−02 9.2574E−03
639 658 675 693 708 723 736 747 758 768
1.9493E+06 1.8223E+06 1.6786E+06 1.5693E+06 1.4743E+06 1.4106E+06 1.3438E+06 1.2555E+06 1.2078E+06 1.1746E+06
21 22 23 24 25 26 27
2-Methylpropane 2-Methylbutane 2,3-Dimethylbutane 2-Methylpentane 2,3-Dimethylpentane 2,3,3-Trimethylpentane 2,2,4-Trimethylpentane
C4H10 C5H12 C6H14 C6H14 C7H16 C8H18 C8H18
75285 78784 79298 107835 565593 560214 540841
100.18 72.35 77.235 77.36 78.282 83.105 87.868
−4841.9 −5010.9 −5695.9 −5791.7 −6347 −6903.7 −6831.7
−13.541 −7.883 −8.5109 −8.4912 −8.502 −9.1858 −9.9783
2.0063E−02 8.9795E−06 8.0163E−06 7.7939E−06 6.4169E−06 6.4703E−06 7.7729E−06
1 2 2 2 2 2 2
113.54 113.25 145.19 119.55 160 172.22 165.78
1.4051E−02 1.1569E−04 1.5081E−02 9.2204E−06 1.2631E−02 1.6820E−02 1.6187E−02
408.14 460.43 499.98 497.5 537.35 573.5 543.96
3.6199E+06 3.3709E+06 3.1255E+06 3.0192E+06 2.8823E+06 2.8116E+06 2.5630E+06
28 29 30 31 32 33 34 35
Ethylene Propylene 1-Butene cis-2-Butene trans-2-Butene 1-Pentene 1-Hexene 1-Heptene
C2H4 C3H6 C4H8 C4H8 C4H8 C5H10 C6H12 C7H14
74851 115071 106989 590181 624646 109671 592416 592767
74.242 57.263 68.49 102.62 70.589 120.15 85.3 92.68
−2707.2 −3382.4 −4350.2 −5260.3 −4530.4 −6192.4 −6171.7 −7055.2
−9.8462 −5.7707 −7.4124 −13.764 −7.7229 −16.597 −9.702 −10.679
2.2457E−02 1.0431E−05 1.0503E−05 1.9183E−02 1.0928E−05 2.1922E−02 8.9604E−06 8.4459E−06
1 2 2 1 2 1 2 2
104 87.89 87.8 134.26 167.62 107.93 133.39 154.27
1.2361E+02 9.3867E−04 7.1809E−07 2.4051E−01 7.4729E+01 3.5210E−06 2.5272E−04 1.2810E−03
282.34 365.57 419.95 435.58 428.63 464.78 504.03 537.29
5.0296E+06 4.6346E+06 4.0391E+06 4.2388E+06 4.0811E+06 3.5557E+06 3.1397E+06 2.8225E+06
36 37 38 39 40 41 42 43 44
1-Octene 1-Nonene 1-Decene 2-Methylpropene 2-Methyl-1-butene 2-Methyl-2-butene 1,2-Butadiene 1,3-Butadiene 2-Methyl-1,3-butadiene
C8H16 C9H18 C10H20 C4H8 C5H10 C5H10 C4H6 C4H6 C5H8
111660 124118 872059 115117 563462 513359 590192 106990 78795
97.57 144.45 78.808 102.5 97.33 82.605 39.714 73.522 79.656
−7836 −9676.2 −8367.9 −5021.8 −5631.8 −5606.6 −3769.9 −4564.3 −5239.6
−11.272 −19.446 −7.9553 −13.88 −12.589 −9.4236 −2.6407 −8.1958 −9.4314
7.7267E−06 1.8031E−02 8.7442E−18 2.0296E−02 1.5395E−02 1.0512E−05 6.9379E−18 1.1580E−05 9.5850E−03
2 1 6 1 1 2 6 2 1
171.45 191.78 206.89 132.81 135.58 139.39 136.95 164.25 127.27
2.7570E−03 8.5514E−03 1.7308E−02 6.2213E−01 1.9687E−02 1.9447E−02 4.4720E−01 6.9110E+01 2.4768E−03
566.65 593.25 616.4 417.9 465 471 452 425.17 484
2.5735E+06 2.3308E+06 2.2092E+06 3.9760E+06 3.4544E+06 3.3769E+06 4.3613E+06 4.3041E+06 3.8509E+06
45 46 47 48 49 50
Acetylene Methylacetylene Dimethylacetylene 3-Methyl-1-butyne 1-Pentyne 2-Pentyne
C2H2 C3H4 C4H6 C5H8 C5H8 C5H8
74862 74997 503173 598232 627190 627214
172.06 119.42 66.592 69.459 82.805 137.29
−5318.5 −5364.5 −4999.8 −5250 −5683.8 −7447.1
−27.223 −16.81 −6.8387 −7.1125 −9.4301 −19.01
5.4619E−02 2.5523E−02 6.6793E−06 7.9289E−17 1.0767E−05 2.1415E−02
1 1 2 6 2 1
192.4 170.45 240.91 183.45 167.45 163.83
1.2603E+05 3.7264E+02 6.1212E+03 4.3551E+01 2.3990E+00 2.0462E−01
308.32 402.39 473.2 463.2 481.2 519
6.1467E+06 5.6206E+06 4.8699E+06 4.1986E+06 4.1701E+06 4.0198E+06
51 52 53 54 55 56
1-Hexyne 2-Hexyne 3−Hexyne 1-Heptyne 1-Octyne Vinylacetylene1
C6H10 C6H10 C6H10 C7H12 C8H14 C4H4
693027 764352 928494 628717 629050 689974
133.2 123.71 47.091 66.447 82.353 55.682
−7492.9 −7639 −5104 −6395.6 −7240.6 −4439.3
−18.405 −16.451 −3.6371 −6.3848 −9.1843 −5.0136
2.2062Ε−02 1.6495Ε−02 5.1621Ε−04 1.1250E−17 5.8038E−03 1.9650E−17
1 1 1 6 1 6
141.25 183.65 170.05 192.22 193.55 173.15
3.9157Ε−04 5.4026Ε−01 2.1950Ε−01 6.7026E−01 1.0092E−01 6.6899E+01
516.2 549 544 559 585 454
3.6352Ε+06 3.5301Ε+06 3.5397Ε+06 3.1343E+06 2.8202E+06 4.8874E+06
2-50
Name
Formula
−11143 −11976 −12549 −13231 −13478 −15015 −15557 −16093 −17897 −19441
57 58 59 60 61 62 63 64 65 66
Cyclopentane Methylcyclopentane Ethylcyclopentane Cyclohexane Methylcyclohexane 1,1-Dimethylcyclohexane Ethylcyclohexane Cyclopentene 1-Methylcyclopentene Cyclohexene
C5H10 C6H12 C7H14 C6H12 C7H14 C8H16 C8H16 C5H8 C6H10 C6H10
287923 96377 1640897 110827 108872 590669 1678917 142290 693890 110838
51.434 79.673 88.622 116.51 92.611 81.184 80.208 49.88 52.732 88.184
−4770.6 −6086.6 −7011 −7103.3 −7077.8 −6927 −7203.2 −4649.7 −5286.9 −6624.9
−4.3515 −8.7933 −10.038 −15.49 −10.684 −8.8498 −8.6023 −4.1191 −4.4509 −10.059
1.9605E−17 7.4046E−06 7.4481E−06 1.6959E−02 8.1239E−06 5.4580E−06 4.5901E−06 1.9564E−17 1.0883E−17 8.2566E−06
6 2 2 1 2 2 2 6 6 2
179.28 130.73 134.71 279.69 146.58 239.66 161.84 138.13 146.62 169.67
9.4420E+00 6.7059E−05 3.7061E−06 5.3802E+03 1.5256E−04 6.0584E+01 3.5747E−04 1.6884E−02 3.9787E−03 1.0377E−01
511.76 532.79 569.52 553.58 572.19 591.15 609.15 507 542 560.4
4.5028E+06 3.7808E+06 3.3970E+06 4.0958E+06 3.4828E+06 2.9387E+06 3.0411E+06 4.8062E+06 4.1303E+06 4.3922E+06
67 68 69 70 71 72 73
Benzene Toluene o-Xylene m-Xylene p-Xylene Ethylbenzene Propylbenzene
C6H6 C7H8 C8H10 C8H10 C8H10 C8H10 C9H12
71432 108883 95476 108383 106423 100414 103651
83.918 80.877 90.356 84.782 85.475 88.09 136.83
−6517.7 −6902.4 −7948.7 −7598.3 −7595.8 −7688.3 −9544.8
−9.3453 −8.7761 −10.081 −9.2612 −9.378 −9.7708 −18.190
7.1182E−06 5.8034E−06 5.9756E−06 5.5445E−06 5.6875E−06 5.8844E−06 1.6590E−02
2 2 2 2 2 2 1
278.68 178.18 247.98 225.3 286.41 178.15 324.18
4.7620E+03 4.2348E−02 2.1968E+01 3.2099E+00 5.8144E+02 4.0140E−03 2.0014E+03
562.16 591.8 630.33 617.05 616.23 617.2 638.32
4.8819E+06 4.1012E+06 3.7424E+06 3.5286E+06 3.4984E+06 3.5968E+06 3.2001E+06
74 75 76 77 78 79 80 81
1,2,4-Trimethylbenzene Isopropylbenzene 1,3,5-Trimethylbenzene p-Isopropyltoluene Naphthalene Biphenyl Styrene m-Terphenyl
C9H12 C9H12 C9H12 C10H14 C10H8 C12H10 C8H8 C18H14
95636 98828 108678 99876 91203 92524 100425 92068
60.658 143.62 48.603 107.71 62.447 76.811 105.93 88.044
−7260.4 −9687.7 −6545.2 −9402.7 −8109 −9878.5 −8685.9 −13367
−5.3772 −19.305 −3.6412 −12.545 −5.5571 −7.4384 −12.42 −8.6482
4.5816E−18 1.7703E−02 1.9307E−18 6.6661E−06 2.0800E−18 2.0436E−18 7.5583E−06 8.7874E−19
6 1 6 2 6 6 2 6
229.33 177.14 228.42 205.25 353.43 342.2 242.54 360
7.9735E−01 3.8034E−04 1.1889E+00 9.9261E−03 9.9229E+02 9.3752E+01 1.0613E+01 1.0112E+00
649.13 631.1 637.36 653.15 748.35 789.26 636 924.85
3.2533E+06 3.1837E+06 3.1119E+06 2.7957E+06 3.9941E+06 3.8615E+06 3.8234E+06 3.5297E+06
82 83 84 85 86 87 88
Methanol Ethanol 1-Propanol 1-Butanol 2-Butanol 2-Propanol 2-Methyl-2-propanol
CH4O C2H6O C3H8O C4H10O C4H10O C3H8O C4H10O
67561 64175 71238 71363 78922 67630 75650
81.768 74.475 88.134 93.173 152.54 76.964 172.31
−6876 −7164.3 −8498.6 −9185.9 −11111 −7623.8 −11590
−8.7078 −7.327 −9.0766 −9.7464 −19.025 −7.4924 −22.118
7.1926E−06 3.1340E−06 8.3303E−18 4.7796E−18 1.0426E−05 5.9436E−18 1.3709E−05
2 2 6 6 2 6 2
175.47 159.05 146.95 184.51 158.45 185.28 298.97
1.1147E−01 4.8459E−04 3.0828E−07 5.7220E−04 1.1323E−06 3.6606E−02 5.9356E+03
512.64 513.92 536.78 563.05 536.05 508.3 506.21
8.1402E+06 6.1171E+06 5.1214E+06 4.3392E+06 4.2014E+06 4.7908E+06 3.9910E+06
89 90 91 92 93 94 95 96
1-Pentanol 2-Methyl-1-butanol 3-Methyl-1-butanol 1-Hexanol 1-Heptanol Cyclohexanol Ethylene glycol 1,2-Propylene glycol
C5H12O C5H12O C5H12O C6H14O C7H16O C6H12O C2H6O2 C3H8O2
71410 137326 123513 111273 111706 108930 107211 57556
168.96 410.44 107.02 117.31 160.08 135.01 79.276 212.8
−12659 −20262 −10237 −11239 −14095 −12238 −10105 −15420
−21.366 −62.366 −11.695 −13.149 −19.211 −15.702 −7.521 −28.109
1.1591E−05 6.3353E−02 6.8003E−18 9.3676E−18 1.7043E−17 1.0349E−17 7.3408E−19 2.1564E−05
2 1 6 6 6 6 6 2
195.56 203 155.95 228.55 239.15 296.6 260.15 213.15
3.1816E−04 3.7992E−04 2.1036E−08 3.7401E−02 1.6990E−02 7.9382E+01 2.4834E−01 9.2894E−05
586.15 565 577.2 611.35 631.9 650 719.7 626
3.8657E+06 3.8749E+06 3.9013E+06 3.4557E+06 3.1810E+06 4.2456E+06 7.7100E+06 6.0413E+06
97 98 99 100
Phenol o-Cresol m-Cresol p-Cresol
C6H6O C7H8O C7H8O C7H8O
108952 95487 108394 106445
95.444 210.88 95.403 118.53
−10113 −13928 −10581 −11957
−10.09 −29.483 −10.004 −13.293
6.7603E−18 2.5182E−02 4.3032E−18 8.6988E−18
6 1 6 6
314.06 304.19 285.39 307.93
1.8798E+02 6.5326E+01 5.8624E+00 3.4466E+01
694.25 697.55 705.85 704.65
6.0585E+06 5.0583E+06 4.5221E+06 5.1507E+06
101 102 103 104 105 106 107 108 109 110 111 112
Dimethyl ether Methyl ethyl ether Methyl n-propyl ether Methyl isopropyl ether Methyl-n-butyl ether Methyl isobutyl ether Methyl tert-butyl ether Diethyl ether Ethyl propyl ether Ethyl isopropyl ether Methyl phenyl ether Diphenyl ether
C2H6O C3H8O C4H10O C4H10O C5H12O C5H12O C5H12O C4H10O C5H12O C5H12O C7H8O C12H10O
115106 540670 557175 598538 628284 625445 1634044 60297 628320 625547 100663 101848
44.704 205.79 50.83 55.096 102.04 58.165 55.875 136.9 143.11 57.723 128.06 59.969
−3.4444 −28.739 −4.1773 −4.8689 −12.278 −5.2568 −4.9604 −19.254 −18.751 −5.2136 −16.693 −5.1538
5.4574E−17 3.5317E−05 9.4076E−18 2.9518E−17 1.2131E−05 2.0194E−17 1.9123E−17 2.4508E−02 2.0620E−05 2.2998E−17 1.4919E−02 1.9983E−18
6 2 6 6 2 6 6 1 2 6 1 6
131.65 160 133.97 127.93 157.48 150 164.55 156.85 145.65 140 235.65 300.03
3.0496E+00 5.3423E−01 4.8875E−03 2.4971E−03 1.9430E−02 1.9801E−02 5.3566E−01 3.9545E−01 7.3931E−04 4.3092E−03 2.4466E+00 7.0874E+00
400.1 437.8 476.3 464.5 510 497 497.1 466.7 500.23 489 645.6 766.8
5.2735E+06 4.4658E+06 3.7721E+06 3.8892E+06 3.3089E+06 3.4130E+06 3.4106E+06 3.6412E+06 3.3729E+06 3.4145E+06 4.2731E+06 3.0971E+06
−3525.6 −9834.5 −4781.7 −4793.2 −6954.9 −5362.1 −5131.6 −6954.3 −8353.7 −5236.9 −9307.7 −8585.5
2-51
TABLE 2-6 Cmpd. no.
Vapor Pressure of Inorganic and Organic Liquids (Continued ) CAS no.
C1
C2
C3
C4
C5
Tmin, K
Ps at Tmin
Tmax, K
Ps at Tmax
113 114 115 116 117 118 119 120 121 122
Formaldehyde Acetaldehyde 1-Propanal 1-Butanal 1-Pentanal 1-Hexanal 1-Heptanal 1-Octanal 1-Nonanal 1-Decanal
CH2O C2H4O C3H6O C4H8O C5H10O C6H12O C7H14O C8H16O C9H18O C10H20O
50000 75070 123386 123728 110623 66251 111717 124130 124196 112312
101.51 193.69 80.581 99.33 149.58 81.507 107.17 250.25 337.71 201.64
−4917.2 −8036.7 −5896.1 −7083.6 −8890 −7776.8 −9070.3 −16162 −18506 −15133
−13.765 −29.502 −8.9301 −11.733 −20.697 −8.4516 −12.503 −33.927 −50.224 −26.264
2.2031E−02 4.3678E−02 8.2236E−06 1.0027E−05 2.2101E−02 1.5143E−17 7.4446E−06 2.2349E−05 4.7345E−02 1.4625E−05
1 1 2 2 1 6 2 2 1 2
181.15 150.15 170 176.75 182 217.15 229.8 246 255.15 267.15
8.8700E+02 3.2320E−01 1.3133E+00 3.1699E−01 5.2282E−02 1.2473E+00 1.1177E+00 4.1640E−01 3.4172E−01 4.8648E−01
408 466 504.4 537.2 566.1 591 617 638.1 658 674.2
6.5935E+06 5.5652E+06 4.9189E+06 4.3232E+06 3.9685E+06 3.4607E+06 3.1829E+06 2.9704E+06 2.7430E+06 2.5989E+06
123 124 125 126 127 128 129 130 131 132 133 134
Acetone Methyl ethyl ketone 2-Pentanone Methyl isopropyl ketone 2-Hexanone Methyl isobutyl ketone 3-Methyl-2-pentanone 3-Pentanone Ethyl isopropyl ketone Diisopropyl ketone Cyclohexanone Methyl phenyl ketone
C3H6O C4H8O C5H10O C5H10O C6H12O C6H12O C6H12O C5H10O C6H12O C7H14O C6H10O C8H8O
67641 78933 107879 563804 591786 108101 565617 96220 565695 565800 108941 98862
69.006 72.698 84.635 308.74 65.841 153.23 64.641 44.286 206.77 96.919 95.118 62.688
−5599.6 −6143.6 −7078.4 −13693 −7042 −10055 −6457.4 −5415.1 −12537 −8014.2 −8300.4 −8088.8
−7.0985 −7.5779 −9.3 −47.557 −6.1376 −19.848 −6.218 −3.0913 −27.894 −11.093 −10.796 −5.5434
6.2237E−06 5.6476E−06 6.2702E−06 5.7002E−02 7.2196E−18 1.6426E−05 3.4543E−06 1.8580E−18 2.2462E−05 7.3452E−06 6.5037E−06 2.0774E−18
2 2 2 1 6 2 2 6 2 2 2 6
178.45 186.48 196.29 181.15 217.35 189.15 167.15 234.18 200 204.81 242 292.81
2.7851E+00 1.3904E+00 7.5235E−01 2.2648E−02 1.5111E+00 3.3536E−02 3.2662E−03 7.3422E+01 6.0339E−02 3.9036E−01 6.9667E+00 3.5899E+01
508.2 535.5 561.08 553 587.05 571.4 573 560.95 567 576 653 709.5
4.7091E+06 4.1201E+06 3.7062E+06 3.8413E+06 3.3120E+06 3.2659E+06 3.3213E+06 3.6993E+06 3.3424E+06 3.0606E+06 4.0126E+06 3.8451E+06
135 136 137 138 139 140 141
Formic acid Acetic acid Propionic acid n-Butyric acid Isobutyric acid Benzoic acid Acetic anhydride
CH2O2 C2H4O2 C3H6O2 C4H8O2 C4H8O2 C7H6O2 C4H6O3
64186 64197 79094 107926 79312 65850 108247
50.323 53.27 54.552 93.815 110.38 88.513 100.95
−5378.2 −6304.5 −7149.4 −9942.2 −10540 −11829 −8873.2
−4.203 −4.2985 −4.2769 −9.8019 −12.262 −8.6826 −11.451
3.4697E−06 8.8865E−18 1.1843E−18 9.3124E−18 1.4310E−17 2.3248E−19 6.1316E−06
2 6 6 6 6 6 2
281.45 289.81 252.45 267.95 227.15 395.45 200.15
2.4024E+03 1.2769E+03 1.3142E+01 6.7754E+00 7.8244E−02 7.9550E+02 2.1999E−02
588 591.95 600.81 615.7 605 751 606
5.8074E+06 5.7390E+06 4.6080E+06 4.0705E+06 3.6834E+06 4.4691E+06 3.9702E+06
142 143 144 145 146 147 148
Methyl formate Methyl acetate Methyl propionate Methyl n-butyrate Ethyl formate Ethyl acetate Ethyl propionate
C2H4O2 C3H6O2 C4H8O2 C5H10O2 C3H6O2 C4H8O2 C5H10O2
107313 79209 554121 623427 109944 141786 105373
77.184 61.267 70.717 71.87 73.833 66.824 105.64
−5606.1 −5618.6 −6439.7 −6885.7 −5817 −6227.6 −8007
−8.392 −5.6473 −6.9845 −7.0944 −7.809 −6.41 −12.477
7.8468E−06 2.1080E−17 2.0129E−17 1.4903E−17 6.3200E−06 1.7914E−17 9.0000E−06
2 6 6 6 2 6 2
174.15 175.15 185.65 187.35 193.55 189.6 199.25
6.8808E+00 1.0170E+00 6.3409E−01 1.3435E−01 1.8119E+01 1.4318E+00 7.7988E−01
487.2 506.55 530.6 554.5 508.4 523.3 546
5.9829E+06 4.6948E+06 4.0278E+06 3.4797E+06 4.7080E+06 3.8502E+06 3.3365E+06
149 150 151 152 153 154 155
Ethyl n-butyrate n-Propyl formate n-Propyl acetate n-Butyl acetate Methyl benzoate Ethyl benzoate Vinyl acetate
C6H12O2 C4H8O2 C5H10O2 C6H12O2 C8H8O2 C9H10O2 C4H6O2
105544 110747 109604 123864 93583 93890 108054
57.661 104.08 115.16 71.34 82.976 53.024 57.406
−6346.5 −7535.9 −8433.9 −7285.8 −9226.1 −7676.8 −5702.8
−5.032 −12.348 −13.934 −6.9459 −8.4427 −4.1593 −5.0307
8.2534E−18 9.6020E−06 1.0346E−05 9.9895E−18 5.9115E−18 1.6850E−18 1.1042E−17
6 2 2 6 6 6 6
175.15 180.25 178.15 199.65 260.75 238.45 180.35
1.0390E−02 2.1101E−01 1.7113E−02 1.4347E−01 1.8653E+00 1.4385E−01 7.0586E−01
571 538 549.73 579.15 693 698 519.13
2.9352E+06 4.0310E+06 3.3657E+06 3.1097E+06 3.5896E+06 3.2190E+06 3.9298E+06
156 157 158 159 160 161
Methylamine Dimethylamine Trimethylamine Ethylamine Diethylamine Triethylamine
CH5N C2H7N C3H9N C2H7N C4H11N C6H15N
74895 124403 75503 75047 109897 121448
75.206 71.738 134.68 81.56 49.314 56.55
−5082.8 −5302 −6055.8 −5596.9 −4949 −5681.9
−8.0919 −7.3324 −19.415 −9.0779 −3.9256 −4.9815
8.1130E−06 6.4200E−17 2.8619E−02 8.7920E−06 9.1978E−18 1.2363E−17
2 6 1 2 6 6
179.69 180.96 156.08 192.15 223.35 158.45
1.7671E+02 7.5575E+01 9.9206E+00 1.5183E+02 3.7411E+02 1.0646E−02
430.05 437.2 433.25 456.15 496.6 535.15
7.4139E+06 5.2583E+06 4.1020E+06 5.5937E+06 3.6744E+06 3.0373E+06
162 163 164 165 166
n-Propylamine di-n-Propylamine Isopropylamine Diisopropylamine Aniline
C3H9N C6H15N C3H9N C6H15N C6H7N
107108 142847 75310 108189 62533
58.398 54 136.66 462.84 66.287
−5312.7 −6018.5 −7201.5 −18227 −8207.1
−5.2876 −4.4981 −18.934 −73.734 −6.0132
1.9913E−06 9.9684E−18 2.2255E−02 9.2794E−02 2.8414E−18
2 6 1 1 6
188.36 210.15 177.95 176.85 267.13
1.3004E+01 3.6942E+00 7.7251E+00 4.4724E−03 7.1322E+00
496.95 550 471.85 523.1 699
4.7381E+06 3.1113E+06 4.5404E+06 3.1987E+06 5.3514E+06
2-52
Name
Formula
167 168
N-Methylaniline N,N-Dimethylaniline
C7H9N C8H11N
100618 121697
70.843 51.352
−8517.5 −7160
−6.7007 −4.0127
5.6411E−18 8.1481E−07
6 2
216.15 275.6
1.0207E−02 1.7940E+01
701.55 687.15
5.1935E+06 3.6262E+06
169 170 171 172
Ethylene oxide Furan Thiophene Pyridine
C2H4O C4H4O C4H4S C5H5N
75218 110009 110021 110861
91.944 74.738 89.171 82.154
−5293.4 −5417 −6860.3 −7211.3
−11.682 −8.0636 −10.104 −8.8646
1.4902E−02 7.4700E−06 7.4769E−06 5.2528E−06
1 2 2 2
160.65 187.55 234.94 231.51
7.7879E+00 5.0026E+01 1.8538E+02 2.0535E+01
469.15 490.15 579.35 619.95
7.2553E+06 5.5497E+06 5.7145E+06 5.6356E+06
173 174 175 176 177 178 179 180
Formamide N,N-Dimethylformamide Acetamide N-Methylacetamide Acetonitrile Propionitrile n-Butyronitrile Benzonitrile
CH3NO C3H7NO C2H5NO C3H7NO C2H3N C3H5N C4H7N C7H5N
75127 68122 60355 79163 75058 107120 109740 100470
100.3 82.762 125.81 79.128 58.302 82.699 66.32 55.463
−10763 −7955.5 −12376 −9523.9 −5385.6 −6703.5 −6714.9 −7430.8
−10.946 −8.8038 −14.589 −7.7355 −5.4954 −9.1506 −6.3087 −4.548
3.8503E−06 4.2431E−06 5.0824E−06 3.1616E−18 5.3634E−06 7.5424E−06 1.3516E−17 1.7501E−18
2 2 2 6 2 2 6 6
275.6 212.72 353.33 301.15 229.32 180.26 161.25 260.4
1.0350E+00 1.9532E−01 3.3637E+02 2.8618E+01 1.8694E+02 1.6936E−01 6.1777E−04 5.1063E+00
771 649.6 761 718 545.5 564.4 582.25 699.35
7.7514E+06 4.3653E+06 6.5688E+06 4.9973E+06 4.8517E+06 4.1906E+06 3.7870E+06 4.2075E+06
181 182 183 184 185 186 187 188 189
Methyl mercaptan Ethyl mercaptan n-Propyl mercaptan n-Butyl mercaptan Isobutyl mercaptan sec-Butyl mercaptan Dimethyl sulfide Methyl ethyl sulfide Diethyl sulfide
CH4S C2H6S C3H8S C4H10S C4H10S C4H10S C2H6S C3H8S C4H10S
74931 75081 107039 109795 513440 513531 75183 624895 352932
54.15 65.551 62.165 65.382 61.736 60.649 83.485 79.07 60.867
−4337.7 −5027.4 −5624 −6262.4 −5909.2 −5785.9 −5711.7 −6114.1 −5969.6
−4.8127 −6.6853 −5.8595 −6.2585 −5.7554 −5.6113 −9.4999 −8.631 −5.5979
4.5000E−17 6.3208E−06 2.0597E−17 1.4943E−17 1.5119E−17 1.5877E−17 9.8449E−06 6.5333E−06 1.4530E−17
6 2 6 6 6 6 2 2 6
150.18 125.26 159.95 157.46 128.31 133.02 174.88 167.23 169.2
3.1479E+00 1.1384E−03 6.5102E−02 2.3532E−03 4.7502E−06 3.3990E−05 7.9009E+00 2.2456E−01 4.3401E−02
469.95 499.15 536.6 570.1 559 554 503.04 533 557.15
7.2309E+06 5.4918E+06 4.6272E+06 3.9730E+06 4.0603E+06 4.0598E+06 5.5324E+06 4.2610E+06 3.9629E+06
190 191 192 193 194 195 196 197
Fluoromethane Chloromethane Trichloromethane Tetrachloromethane Bromomethane Fluoroethane Chloroethane Bromoethane
CH3F CH3Cl CHCl3 CCl4 CH3Br C2H5F C2H5Cl C2H5Br
593533 74873 67663 56235 74839 353366 75003 74964
59.123 64.697 146.43 78.441 72.586 56.639 70.159 62.217
−3043.7 −4048.1 −7792.3 −6128.1 −4698.6 −3576.5 −4786.7 −5113.3
−6.1845 −6.8066 −20.614 −8.5766 −7.9966 −5.5801 −7.5387 −5.9761
1.6637E−05 1.0371E−05 2.4578E−02 6.8465E−06 1.1553E−05 9.8969E−06 9.3370E−06 4.7174E−17
2 2 1 2 2 2 2 6
131.35 175.43 207.15 250.33 179.47 129.95 134.8 154.55
4.3287E+02 8.7091E+02 5.2512E+01 1.1225E+03 1.9544E+02 8.3714E+00 1.1658E−01 3.7155E−01
317.42 416.25 536.4 556.35 467 375.31 460.35 503.8
5.8754E+06 6.6905E+06 5.5543E+06 4.5436E+06 7.9972E+06 5.0060E+06 5.4578E+06 6.2903E+06
198 199 200 201 202 203 204 205
1-Chloropropane 2-Chloropropane 1,1-Dichloropropane 1,2-Dichloropropane Vinyl chloride Fluorobenzene Chlorobenzene Bromobenzene
C3H7Cl C3H7Cl C3H6Cl2 C3H6Cl2 C2H3Cl C6H5F C6H5Cl C6H5Br
540545 75296 78999 78875 75014 462066 108907 108861
79.24 46.854 83.495 65.955 91.432 51.915 54.144 63.749
−5718.8 −4445.5 −6661.4 −6015.6 −5141.7 −5439 −6244.4 −7130.2
−8.789 −3.6533 −9.2386 −6.5509 −10.981 −4.2896 −4.5343 −5.879
8.4486E−06 1.3260E−17 6.7652E−06 4.3172E−06 1.4318E−05 8.7527E−18 4.7030E−18 5.2136E−18
2 6 2 2 2 6 6 6
150.35 155.97 200 172.71 119.36 230.94 227.95 242.43
6.9630E−02 9.0844E−01 4.5248E+00 8.2532E−02 1.9178E−02 1.5142E+02 8.4456E+00 7.8364E+00
503.15 489 560 572 432 560.09 632.35 670.15
4.5812E+06 4.5097E+06 4.2394E+06 4.2319E+06 5.7495E+06 4.5437E+06 4.5293E+06 4.5196E+06
206 207 208 209 210 211 212 213 214 215 216 217 218
Air3 Hydrogen Helium-44 Neon Argon Fluorine Chlorine Bromine Oxygen Nitrogen Ammonia Hydrazine Nitrous oxide
H2 He Ne Ar F2 Cl2 Br2 O2 N2 NH3 N2H4 N2O
132259100 1333740 7440597 7440019 7440371 7782414 7782505 7726956 7782447 7727379 7664417 302012 10024972
21.662 12.69 11.533 29.755 42.127 42.393 71.334 108.26 51.245 58.282 90.483 76.858 96.512
−692.39 −94.896 −8.99 −271.06 −1093.1 −1103.3 −3855 −6592 −1200.2 −1084.1 −4669.7 −7245.2 −4045
−0.39208 1.1125 0.6724 −2.6081 −4.1425 −4.1203 −8.5171 −14.16 −6.4361 −8.3144 −11.607 −8.22 −12.277
4.7574E−03 3.2915E−04 2.7430E−01 5.2700E−04 5.7254E−05 5.7815E−05 1.2378E−02 1.6043E−02 2.8405E−02 4.4127E−02 1.7194E−02 6.1557E−03 2.8860E−05
1 2 1 2 2 2 1 1 1 1 1 1 2
59.15 13.95 1.76 24.56 83.78 53.48 172.12 265.85 54.36 63.15 195.41 274.69 182.3
5.6421E+03 7.2116E+03 1.4625E+03 4.3800E+04 6.8721E+04 2.5272E+02 1.3660E+03 5.8534E+03 1.4754E+02 1.2508E+04 6.1111E+03 4.0847E+02 8.6908E+04
132.45 33.19 5.2 44.4 150.86 144.12 417.15 584.15 154.58 126.2 405.65 653.15 309.57
3.7934E+06 1.3154E+06 2.2845E+05 2.6652E+06 4.8963E+06 5.1674E+06 7.7930E+06 1.0276E+07 5.0206E+06 3.3906E+06 1.1301E+07 1.4731E+07 7.2782E+06
2-53
TABLE 2-6 Cmpd. no. 219 220 221 222 223 224 225 226 227 228 229 230 231
Vapor Pressure of Inorganic and Organic Liquids (Concluded ) Name Nitric oxide Cyanogen Carbon monoxide Carbon dioxide Carbon disulfide Hydrogen fluoride Hydrogen chloride Hydrogen bromide2 Hydrogen cyanide Hydrogen sulfide Sulfur dioxide Sulfur trioxide Water
Formula NO C2N2 CO CO2 CS2 HF HCl HBr HCN H2S SO2 SO3 H2O
CAS no.
C1
C2
C3
C4
C5
Tmin, K
Ps at Tmin
Tmax, K
Ps at Tmax
10102439 460195 630080 124389 75150 7664393 7647010 10035106 74908 7783064 7446095 7446119 7732185
72.974 88.589 45.698 140.54 67.114 59.544 104.27 29.315 36.75 85.584 47.365 180.99 73.649
−2650 −5059.9 −1076.6 −4735 −4820.4 −4143.8 −3731.2 −2424.5 −3927.1 −3839.9 −4084.5 −12060 −7258.2
−8.261 −10.483 −4.8814 −21.268 −7.5303 −6.1764 −15.047 −1.1354 −2.1245 −11.199 −3.6469 −22.839 −7.3037
9.7000E−15 1.5403E−05 7.5673E−05 4.0909E−02 9.1695E−03 1.4161E−05 3.1340E−02 2.3806E−18 3.8948E−17 1.8848E−02 1.7990E−17 7.2350E−17 4.1653E−06
6 2 2 1 1 2 1 6 6 1 6 6 2
109.5 245.25 68.15 216.58 161.11 189.79 158.97 185.15 259.83 187.68 197.67 289.95 273.16
2.1956E+04 7.3385E+04 1.5430E+04 5.1867E+05 1.4944E+00 3.3683E+02 1.3522E+04 2.9501E+04 1.8687E+04 2.2873E+04 1.6743E+03 2.0934E+04 6.1056E+02
180.15 400.15 132.92 304.21 552 461.15 324.65 363.15 456.65 373.53 430.75 490.85 647.13
6.5156E+06 5.9438E+06 3.4940E+06 7.3896E+06 8.0408E+06 6.4872E+06 8.3564E+06 8.4627E+06 5.3527E+06 8.9988E+06 7.8596E+06 8.1919E+06 2.1940E+07
All substances are listed in alphabetical order in Table 2-6a. Compiled from Daubert, T. E., R. P. Danner, H. M. Sibul, and C. C. Stebbins, DIPPR Data Compilation of Pure Compound Properties, Project 801 Sponsor Release, July, 1993, Design Institute for Physical Property Data, AIChE, New York, NY; and from Thermodynamics Research Center, “Selected Values of Properties of Hydrocarbons and Related Compounds,” Thermodynamics Research Center Hydrocarbon Project, Texas A&M University, College Station, Texas (extant 1994). Temperatures are in K; vapor pressures are in Pa. Pa × 9.869233E−06 = atm; Pa × 1.450377E−04 = psia; vapor pressure = exp [C1 + (C2/T) + C3 × ln (T) + C4 × T C5 ]. 1 Decomposes violently on heating. Forms explosive peroxides with air or oxygen. Polymerizes under pressure and heat. 2 Coefficients are hypothetical above the decomposition temperature. 3 At the bubble point. 4 Exhibits superfluid properties below 2.2 K.
2-54
VAPOR PRESSURES OF PURE SUBSTANCES
2-55
TABLE 2-6a Alphabetical Index to Substances in Tables 2-6, 2-30, 2-164, 2-193, 2-196, 2-198, and 2-221 Name Acetaldehyde Acetamide Acetic acid Acetic anhydride Acetone Acetonitrile Acetophenone Acetylene Air Ammonia Aniline Anisole Argon Benzene Benzoic acid Benzonitrile Biphenyl Bromine Bromobenzene Bromoethane Bromomethane 1,2-Butadiene 1,3-Butadiene n-Butane 1-Butanol 2-Butanol 1-Butene cis-2-Butene trans-2-Butene n-Butyl acetate n-Butyl mercaptan sec-Butyl mercaptan Butyraldehyde n-Butyric acid n-Butyronitrile Carbon dioxide Carbon disulfide Carbon monoxide Carbon tetrachloride Chlorine Chlorobenzene Chloroethane Chloroform Chloromethane 1-Chloropropane 2-Chloropropane o-Cresol m-Cresol p-Cresol Cumene Cyanogen Cyclohexane Cyclohexanol Cyclohexanone Cyclohexene Cyclopentane Cyclopentene p-Cymene 1-Decanal n-Decane 1-Decene 1,1-Dichloropropane 1,2-Dichloropropane Diethylamine Diethyl ether Diethyl sulfide Diisopropylamine Diisopropyl ketone Dimethylacetylene Dimethylamine N,N-Dimethylaniline 2,3-Dimethylbutane 1,1-Dimethylcyclohexane Dimethyl ether
Synonym Ethanal Ethanoic acid 2-Propanone Methyl cyanide Methyl phenyl ketone
Methyl phenyl ether
Phenyl cyanide 1,1′-Biphenyl Ethyl bromide Methyl bromide
sec-Butyl alcohol Z-2-Butene E-2-Butene 1-Butanethiol 2-Butanethiol Butanal Propyl cyanide
Tetrachloromethane Ethyl chloride Trichloromethane Methyl chloride Propyl chloride Isopropyl chloride 2-Methylphenol 3-Methylphenol 4-Methylphenol Isopropylbenzene Cyclohexyl alcohol Cyclohexyl ketone
p-Isopropyltoluene
Ethyl ether Ethyl sulfide 2,4-Dimethyl-3-pentanone 2-Butyne N,N-Dimethylbenzamine Diisopropyl Methyl ether
Cmpd. no.
Formula
114 175 136 141 123 177 134 45 206 216 166 111 210
C2H4O C2H5NO C2H4O2 C4H6O3 C3H6O C2H3N C8H8O C2H2
67 140 180 79 213 205 197 194 42 43 4 85 86 30 31 32 152 184 186 116 138 179
C6H6 C7H6O2 C7H5N C12H10 Br2 C6H5Br C2H5Br CH3Br C4H6 C4H6 C4H10 C4H10O C4H10O C4H8 C4H8 C4H8 C6H12O2 C4H10S C4H10S C4H8O C4H8O2 C4H7N
222 223 221 193 212 204 196 192 191 198 199 98 99 100 75 220 60 94 133 66 57 64 77
CO2 CS2 CO CCl4 Cl2 C6H5Cl C2H5Cl CHCl3 CH3Cl C3H7Cl C3H7Cl C7H8O C7H8O C7H8O C9H12 C2N2 C6H12 C6H12O C6H10O C6H10 C5H10 C5H8 C10H14
122 10 38 200 201 160 108 189 165 132 47 157 168 23 62 101
C10H20O C10H22 C10H20 C3H6Cl2 C3H6Cl2 C4H11N C4H10O C4H10S C6H15N C7H14O C4H6 C2H7N C8H11N C6H14 C8H16 C2H6O
NH3 C6H7N C7H8O Ar
Name
Synonym
1-Hexene 1-Hexyne 2-Hexyne 3-Hexyne Hydrazine Hydrogen Hydrogen bromide Hydrogen chloride Hydrogen cyanide Hydrogen fluoride Hydrogen sulfide
Cmpd. no.
Formula
34 51 52 53 217 207 226 225 227 224 228
C6H12 C6H10 C6H10 C6H10 N2H4 H2 HBr HCl HCN HF H2S
Isobutyl mercaptan Isobutyric acid Isooctane Isoprene Isopropylamine
2-Methyl-1-propanethiol 2-Methylpropanoic acid 2,2,4-Trimethylpentane 2-Methyl-1,3-butadiene
185 139 27 44 164
C4H10S C4H8O2 C8H18 C5H8 C3H9N
Mesitylene Methane Methanol N-Methylacetamide Methyl acetate Methylacetylene Methylamine N-Methylaniline Methyl benzoate 2-Methylbutane 2-Methyl-1-butanol 3-Methyl-1-butanol 2-Methyl-1-butene 2-Methyl-2-butene Methyl butyl ether 3-Methyl-1-butyne Methyl butyrate Methylcyclohexane Methylcyclopentane 1-Methylcyclopentene Methyl ethyl ether Methyl ethyl ketone Methyl ethyl sulfide Methyl formate Methyl isobutyl ether Methyl isobutyl ketone Methyl isopropyl ether Methyl isopropyl ketone Methyl mercaptan 2-Methylpentane 3-Methyl-2-pentanone 2-Methylpropane 2-Methyl-2-propanol 2-Methylpropene Methyl propionate Methyl propyl ether Methyl tert-butyl ether
1,3,5-Trimethylbenzene
76 1 82 176 143 46 156 167 153 22 90 91 40 41 105 48 145 61 58 65 102 124 188 142 106 128 104 126 181 24 129 21 88 39 144 103 107
C9H12 CH4 CH4O C3H7NO C3H6O2 C3H4 CH5N C7H9N C8H8O2 C5H12 C5H12O C5H12O C5H10 C5H10 C5H12O C5H8 C5H10O2 C7H14 C6H12 C6H10 C3H8O C4H8O C3H8S C2H4O2 C5H12O C6H12O C4H10O C5H10O CH4S C6H14 C6H12O C4H10 C4H10O C4H8 C4H8O2 C4H10O C5H12O
78 209 219 215 218 19 121 9 37
C10H8 Ne NO N2 N2O C19H40 C9H18O C9H20 C9H18
18 120 8 36 55 214
C18H38 C8H16O C8H18 C8H16 C8H14 O2
15 117 5 89
C15H32 C5H10O C5H12 C5H12O
Naphthalene Neon Nitric oxide Nitrogen Nitrous oxide n-Nonadecane 1-Nonanal n-Nonane 1-Nonene n-Octadecane 1-Octanal n-Octane 1-Octene 1-Octyne Oxygen n-Pentadecane 1-Pentanal n-Pentane 1-Pentanol
Methyl alcohol
Isopentane Isoamyl alcohol Amylene
2-Butanone 2-Thiabutane
Methanethiol Isohexane Methyl sec-butyl ketone Isobutane tert-Butyl alcohol Isobutene
n-Nonaldehyde
n-Octaldehyde
Valeraldehyde n-Amyl alcohol
2-56
PHYSICAL AND CHEMICAL DATA
TABLE 2-6a Alphabetical Index to Substances in Tables 2-6, 2-30, 2-164, 2-193, 2-196, 2-198, and 2-221 (Concluded ) Name N,N-Dimethylformamide 2,3-Dimethylpentane Dimethyl sulfide Diphenyl ether n-Dodecane n-Eicosane Ethane Ethanol Ethyl acetate Ethylamine Ethylbenzene Ethyl benzoate Ethyl butyrate Ethylcyclohexane Ethylcyclopentane Ethylene Ethylene glycol Ethylene oxide Ethyl formate Ethyl isopropyl ether Ethyl isopropyl ketone Ethyl mercaptan Ethyl propionate Ethyl propyl ether Fluorine Fluorobenzene Fluoroethane Fluoromethane Formaldehyde Formamide Formic acid Furan Helium-4 n-Heptadecane 1-Heptanal n-Heptane 1-Heptanol 1-Heptene 1-Heptyne n-Hexadecane 1-Hexanal n-Hexane 1-Hexanol 2-Hexanone
Synonym
Methyl sulfide
Ethyl alcohol Phenylethane
1,2-Ethanediol 1,2-Epoxyethane
Ethanethiol
Ethyl fluoride Methyl fluoride Methanal Methanoic acid
n-Heptaldehyde n-Heptyl alcohol
Caproaldehyde n-Hexyl alcohol Methyl n-butyl ketone
Cmpd. no.
Formula
174 25 187 112 12
C3H7NO C7H16 C2H6S C12H10O C12H26
20 2 83 147 159 72 154 149 63 59 28 95 169 146 110 131 182 148 109
C20H42 C2H6 C2H6O C4H8O2 C2H7N C8H10 C9H10O2 C6H12O2 C8H16 C7H14 C2H4 C2H6O2 C2H4O C3H6O2 C5H12O C6H12O C2H6S C5H10O2 C5H12O
211 203 195 190 113 173 135 170
F2 C6H5F C2H5F CH3F CH2O CH3NO CH2O2 C4H4O
208 17 119 7 93 35 54 16 118 6 92 127
He C17H36 C7H14O C7H16 C7H16O C7H14 C7H12 C16H34 C6H12O C6H14 C6H14O C6H12O
Name
Synonym
Cmpd. no.
Formula
Methyl n-propyl ketone Diethyl ketone
125 130 33 49 50 97 115 3 84 87 137 178 151 162 163 73 29 96 150 183 172
C5H10O C5H10O C5H10 C5H8 C5H8 C6H6O C3H6O C3H8 C3H8O C3H8O C3H6O2 C3H5N C5H10O2 C3H9N C6H15N C9H12 C3H6 C3H8O2 C4H8O2 C3H8S C5H5N
Styrene Sulfur dioxide Sulfur trioxide
80 229 230
C8H8 SO2 SO3
m-Terphenyl n-Tetradecane Thiophene Toluene n-Tridecane Triethylamine Trimethylamine 1,2,4-Trimethylbenzene 2,3,3-Trimethylpentane
81 14 171 68 13 161 158 74 26
C18H14 C14H30 C4H4S C7H8 C13H28 C6H15N C3H9N C9H12 C8H18
11
C11H24
Vinyl acetate Vinylacetylene Vinyl chloride
155 56 202
C4H6O2 C4H4 C2H3Cl
Water
231
H2O
69 70 71
C8H10 C8H10 C8H10
2-Pentanone 3-Pentanone 1-Pentene 1-Pentyne 2-Pentyne Phenol 1-Propanal n-Propane 1-Propanol 2-Propanol n-Propionic acid n-Propionitrile n-Propyl acetate n-Propylamine di-n-Propylamine n-Propylbenzene Propylene 1,2-Propylene glycol n-Propyl formate n-Propyl mercaptan Pyridine
Propionaldehyde n-Propyl alcohol Isopropyl alcohol Ethyl cyanide
1,2-Propanediol Propanethiol
n-Undecane
o-Xylene m-Xylene p-Xylene
1,2-Dimethylbenzene 1,3-Dimethylbenzene 1,4-Dimethylbenzene
VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-7
2-57
Vapor Pressures of Inorganic Compounds, up to 1 atm* Compound
Pressure, mm Hg 1
Name Aluminum borohydride bromide chloride fluoride iodide oxide Ammonia heavy Ammonium bromide carbamate chloride cyanide hydrogen sulfide iodide Antimony tribromide trichloride pentachloride triiodide trioxide Argon Arsenic Arsenic tribromide trichloride trifluoride pentafluoride trioxide Arsine Barium Beryllium borohydride bromide chloride iodide Bismuth tribromide trichloride Diborane hydrobromide Borine carbonyl triamine Boron hydrides dihydrodecaborane dihydrodiborane dihydropentaborane tetrahydropentaborane tetrahydrotetraborane Boron tribromide trichloride trifluoride Bromine pentafluoride Cadmium chloride fluoride iodide oxide Calcium Carbon (graphite) dioxide disulfide monoxide oxyselenide oxysulfide selenosulfide subsulfide tetrabromide tetrachloride tetrafluoride Cesium bromide chloride fluoride iodide
5
10
20
Al Al(BH4)3 AlBr3 Al2Cl6 AlF3 AlI3 Al2O3 NH3 ND3 NH4Br N2H6CO2 NH4Cl NH4CN NH4HS NH4I Sb SbBr3 SbCl3 SbCl5 SbI3 Sb4O6 A As AsBr3 AsCl3 AsF3 AsF5 As2O3 AsH3 Ba Be(BH4)2 BeBr2 BeCl2 BeI2 Bi BiBr3 BiCl3 B2H5Br BH3CO B3N3H6 B10H14 B2H6 B5H9 B5H11 B4H10 BBr3 BCl3 BF3 Br2 BrF5 Cd CdCl2 CdF2 CdI2 CdO Ca C CO2 CS2 CO COSe COS CSeS C3S2 CBr4 CCl4 CF4 Cs CsBr CsCl CsF CsI
40
60
100
200
400
760
Melting point, °C
1749 −3.9 176.1 152.0 1422 294.5 2665 −68.4 −67.4 320.0 26.7 271.5 −0.5 0.0 331.8 1223 203.5 143.3 114.1 303.5 957 −200.5 518 145.2 70.9 13.2 −84.3 332.5 −98.0 1301 58.6 405 411 411 1271 360 343 −29.0 −95.3 +4.0
1844 +11.2 199.8 161.8 1457 322.0 2766 −57.0 −57.0 345.3 37.2 293.2 +9.6 +10.5 355.8 1288 225.7 165.9
1947 28.1 227.0 171.6 1496 354.0 2874 −45.4 −45.4 370.8 48.0 316.5 20.5 21.8 381.0 1364 250.2 192.2
2056 45.9 256.3 180.2 1537 385.5 2977 −33.6 −33.4 396.0 58.3 337.8 31.7 33.3 404.9 1440 275.0 219.0
660 −64. 97. 192.4 1040
333.8 1085 −195.6 548 167.7 89.2 26.7 −75.5 370.0 −87.2 1403 69.0 427 435 435 1319 392 372 −15.4 −85.5 18.5
368.5 1242 −190.6 579 193.6 109.7 41.4 −64.0 412.2 −75.2 1518 79.7 451 461 461 1370 425 405 0.0 −74.8 34.3
401.0 1425 −185.6 610 220.0 130.4 56.3 −52.8 457.2 −62.1 1638 90.0 474 487 487 1420 461 441 +16.3 −64.0 50.6
142.3 −120.9 +9.6 20.1 −28.1 33.5 −32.4 −123.0 +9.3 −4.5 611 797 1486 640 1341 1207 4373 −100.2 −5.1 −205.7 −61.7 −85.9 28.3 109.9 119.7 23.0 −150.7 509 1072 1069 1025 1055
163.8 −111.2 24.6 34.8 −14.0 50.3 −18.9 −115.9 24.3 +9.9 658 847 1561 688 1409 1288 4516 −93.0 +10.4 −201.3 −49.8 −75.0 45.7 130.8 139.7 38.3 −143.6 561 1140 1139 1092 1124
−99.6 40.8 51.2 +0.8 70.0 −3.6 −108.3 41.0 25.7 711 908 1651 742 1484 1388 4660 −85.7 28.0 −196.3 −35.6 −62.7 65.2
−86.5 58.1 67.0 16.1 91.7 +12.7 −100.7 58.2 40.4 765 967 1751 796 1559 1487 4827 −78.2 46.5 −191.3 −21.9 −49.9 85.6
163.5 57.8 −135.5 624 1221 1217 1170 1200
189.5 76.7 −127.7 690 1300 1300 1251 1280
Temperature, °C
Formula 1284 81.3 100.0 1238 178.0 2148 −109.1
1421 −52.2 103.8 116.4 1298 207.7 2306 −97.5
1487 −42.9 118.0 123.8 1324 225.8 2385 −91.9
1555 −32.5 134.0 131.8 1350 244.2 2465 −85.8
1635 −20.9 150.6 139.9 1378 265.0 2549 −79.2
198.3 −26.1 160.4 −50.6 −51.1 210.9 886 93.9 49.2 22.7 163.6 574 −218.2 372 41.8 −11.4
234.5 −10.4 193.8 −35.7 −36.0 247.0 984 126.0 71.4 48.6 203.8 626 −213.9 416 70.6 +11.7
252.0 −2.9 209.8 −28.6 −28.7 263.5 1033 142.7 85.2 61.8 223.5 666 −210.9 437 85.2 +23.5
270.6 +5.3 226.1 −20.9 −20.8 282.8 1084 158.3 100.6 75.8 244.8 729 −207.9 459 101.3 36.0
−117.9 212.5 −142.6
−108.0 242.6 −130.8 984 19.8 325 328 322 1099 261 242 −75.3 −127.3 −45.0
−103.1 259.7 −124.7 1049 28.1 342 346 341 1136 282 264 −66.3 −121.1 −35.3
−98.0 279.2 −117.7 1120 36.8 361 365 361 1177 305 287 −56.4 −114.1 −25.0
290.0 14.0 245.0 −12.6 −12.3 302.8 1141 177.4 117.8 91.0 267.8 812 −204.9 483 118.7 50.0 −2.5 −92.4 299.2 −110.2 1195 46.2 379 384 382 1217 327 311 −45.4 −106.6 −13.2
1684 −13.4 161.7 145.4 1398 277.8 2599 −74.3 −74.0 303.8 19.6 256.2 −7.4 −7.0 316.0 1176 188.1 128.3 101.0 282.5 873 −202.9 498 130.0 58.7 +4.2 −88.5 310.3 −104.8 1240 51.7 390 395 394 1240 340 324 −38.2 −101.9 −5.8
3586 −134.3 −73.8 −222.0 −117.1 −132.4 −47.3 14.0
80.8 −149.5 −40.4 −29.9 −73.1 −20.4 −75.2 −145.4 −32.8 −51.0 455 618 1231 481 1100 926 3828 −124.4 −54.3 −217.2 −102.3 −119.8 −26.5 41.2
90.2 −144.3 −30.7 −19.9 −64.3 −10.1 −66.9 −141.3 −25.0 −41.9 484 656 1286 512 1149 983 3946 −119.5 −44.7 −215.0 −95.0 −113.3 −16.0 54.9
100.0 −138.5 −20.0 −9.2 −54.8 +1.5 −57.9 −136.4 −16.8 −32.0 516 695 1344 546 1200 1046 4069 −114.4 −34.3 −212.8 −86.3 −106.0 −4.4 69.3
−50.0 −184.6 279 748 744 712 738
−30.0 −174.1 341 838 837 798 828
−19.6 −169.3 375 887 884 844 873
−8.2 −164.3 409 938 934 893 923
117.4 −131.6 −8.0 +2.7 −44.3 14.0 −47.8 −131.0 −8.0 −21.0 553 736 1400 584 1257 1111 4196 −108.6 −22.5 −210.0 −76.4 −98.3 +8.6 85.6 96.3 +4.3 −158.8 449 993 989 947 976
127.8 −127.2 −0.4 10.2 −37.4 22.1 −41.2 −127.6 −0.6 −14.0 578 762 1436 608 1295 1152 4273 −104.8 −15.3 −208.1 −70.2 −93.0 17.0 96.0 106.3 12.3 −155.4 474 1026 1023 980 1009
+1.0 289 291 283 1021 −93.3 −139.2 −63.0 60.0 −159.7 −50.2 −90.9 −41.4 −91.5 −154.6 −48.7 −69.3 394 1112 416 1000
*Compiled from the extended tables published by D. R. Stull in Ind. Eng. Chem., 39, 517 (1947).
2050 −77.7 −74.0 520 36 630.5 96.6 73.4 2.8 167 656 −189.2 814 −18 −5.9 −79.8 312.8 −116.3 850 123 490 405 488 271 218 230 −104.2 −137.0 −58.2 99.6 −169 −47.0 −119.9 −45 −107 −126.8 −7.3 −61.4 320.9 568 520 385 851 −57.5 −110.8 −205.0 −138.8 −75.2 +0.4 90.1 −22.6 −183.7 28.5 636 646 683 621
2-58
PHYSICAL AND CHEMICAL DATA
TABLE 2-7
Vapor Pressures of Inorganic Compounds, up to 1 atm (Continued ) Compound
Name Chlorine fluoride trifluoride monoxide dioxide heptoxide Chlorosulfonic acid Chromium carbonyl oxychloride Cobalt chloride nitrosyl tricarbonyl Columbium fluoride Copper Cuprous bromide chloride iodide Cyanogen bromide chloride fluoride Deuterium cyanide Fluorine oxide Germanium bromide chloride hydride Trichlorogermane Tetramethylgermane Digermane Trigermane Gold Helium para-Hydrogen Hydrogen bromide chloride cyanide fluoride iodide oxide (water) sulfide disulfide selenide telluride Iodine heptafluoride Iron pentacarbonyl Ferric chloride Ferrous chloride Krypton Lead bromide chloride fluoride iodide oxide sulfide Lithium bromide chloride fluoride iodide Magnesium chloride Manganese chloride Mercury Mercuric bromide chloride iodide Molybdenum hexafluoride oxide
Pressure, mm Hg 1
5
10
20
−118.0 −98.5
−106.7 −143.4 −80.4 −81.6
−45.3 32.0 1616 36.0 −18.4
−23.8 53.5 1768 58.0 +3.2
−101.6 −139.0 −71.8 −73.1 −59.0 −13.2 64.0 1845 68.3 13.8
−93.3 −134.3 −62.3 −64.3 −51.2 −2.1 75.3 1928 79.5 25.7
86.3 1879 718 702 656 −76.8 −10.0 −53.8 −118.5 −46.7 −214.1 −182.3 56.8 −15.0 −145.3 −13.0 −45.2 −60.1 −0.9 2154 −271.3 −261.3 −121.8 −135.6 −47.7 −65.8 −102.3 11.2 −116.3 −15.2 −97.9 −75.4 73.2 −63.0 2039 +4.6 235.5 700 −187.2 1162 610 648 904 571 1085 975 881 888 932 1211 841 743 930 1505 778 184.0 179.8 180.2 204.5 3535 −40.8 814
−1.3 103.0 1970 777 766 716 −70.1 −1.0 −46.1 −112.8 −38.8 −211.0 −177.8 71.8 −4.1 −139.2 −3.0 −35.0 −49.9 +11.8 2256 −271.1 −260.4 −115.4 −130.0 −39.7 −56.0 −94.5 22.1 −109.7 −5.1 −91.8 −67.8 84.7 −54.5 2128 16.7 246.0 737 −182.9 1234 646 684 950 605 1134 1005 940 939 987 1270 883 789 988 1583 825 204.6 194.3 195.8 220.0 3690 −32.0 851
60
100
200
400
760
Melting point, °C
−71.7 −120.8 −34.7 −39.4 −29.4 29.1 105.3 2139 108.0 58.0 843 29.0 148.5 2207 951 960 907 −51.8 22.6 −24.9 −97.0 −17.5 −202.7 −165.8 113.2 27.5 −120.3 26.5 −6.3 −20.3 47.9 2521 −270.3 −257.9 −97.7 −114.0 −17.8 −28.2 −72.1 51.6 −91.6 22.0 −74.2 −45.7 116.5 −31.9 2360 50.3 272.5 842 −171.8 1421 745 784 1080 701 1265 1108 1097 1076 1129 1425 993 909 1142 1792 960 261.7 237.8 237.0 261.8 4109 −8.0 955
−60.2 −114.4 −20.7 −26.5 −17.8 44.6 120.0 2243 121.8 75.2 904 44.4 172.2 2325 1052 1077 1018 −42.6 33.8 −14.1 −89.2 −5.4 −198.3 −159.0 135.4 44.4 −111.2 41.6 +8.8 −4.7 67.0 2657 −269.8 −256.3 −88.1 −105.2 −5.3 −13.2 −60.3 66.5 −82.3 35.3 −65.2 −32.4 137.3 −20.7 2475 68.0 285.0 897 −165.9 1519 796 833 1144 750 1330 1160 1178 1147 1203 1503 1049 967 1223 1900 1028 290.7 262.7 256.5 291.0 4322 +4.1 1014
−47.3 −107.0 −4.9 −12.5 −4.0 62.2 136.1 2361 137.2 95.2 974 62.0 198.0 2465 1189 1249 1158 −33.0 46.0 −2.3 −80.5 +10.0 −193.2 −151.9 161.6 63.8 −100.2 58.3 26.0 +13.3 88.6 2807 −269.3 −254.5 −78.0 −95.3 +10.2 +2.5 −48.3 83.0 −71.8 49.6 −53.6 −17.2 159.8 −8.3 2605 86.1 298.0 961 −159.0 1630 856 893 1219 807 1402 1221 1273 1226 1290 1591 1110 1034 1316 2029 1108 323.0 290.0 275.5 324.2 4553 17.2 1082
−33.8 −100.5 +11.5 +2.2 +11.1 78.8 151.0 2482 151.0 117.1 1050 80.0 225.0 2595 1355 1490 1336 −21.0 61.5 +13.1 −72.6 26.2 −187.9 −144.6 189.0 84.0 −88.9 75.0 44.0 31.5 110.8 2966 −268.6 −252.5 −66.5 −84.8 25.9 19.7 −35.1 100.0 −60.4 64.0 −41.1 −2.0 183.0 +4.0 2735 105.0 319.0 1026 −152.0 1744 914 954 1293 872 1472 1281 1372 1310 1382 1681 1171 1107 1418 2151 1190 357.0 319.0 304.0 354.0 4804 36.0 1151
−100.7 −145 −83 −116 −59 −91 −80 1615
Temperature, °C
Formula Cl2 ClF ClF3 Cl2O ClO2 Cl2O7 HSO3Cl Cr Cr(CO)6 CrO2Cl2 CoCl2 Co(CO)3NO CbF5 Cu Cu2Br2 Cu2Cl2 Cu2I2 C2N2 CNBr CNCl CNF DCN F2 F2O GeBr4 GeCl4 GeH4 GeHCl3 Ge(CH3)4 Ge2H6 Ge3H8 Au He H2 HBr HCl HCN H2F2 HI H2O H2S HSSH H2Se H2Te I2 IF7 Fe Fe(CO)5 Fe2Cl6 FeCl2 Kr Pb PbBr2 PbCl2 PbF2 PbI2 PbO PbS Li LiBr LiCl LiF LiI Mg MgCl2 Mn MnCl2 Hg HgBr2 HgCl2 HgI2 Mo MoF6 MoO3
40
1628 572 546 −95.8 −35.7 −76.7 −134.4 −68.9 −223.0 −196.1 −45.0 −163.0 −41.3 −73.2 −88.7 −36.9 1869 −271.7 −263.3 −138.8 −150.8 −71.0 −123.3 −17.3 −134.3 −43.2 −115.3 −96.4 38.7 −87.0 1787 194.0 −199.3 973 513 547 479 943 852 723 748 783 1047 723 621 778 1292 126.2 136.5 136.2 157.5 3102 −65.5 734
1795 666 645 610 −83.2 −18.3 −61.4 −123.8 −54.0 −216.9 −186.6 43.3 −24.9 −151.0 −22.3 −54.6 −69.8 −12.8 2059 −271.5 −261.9 −127.4 −140.7 −55.3 −74.7 −109.6 +1.2 −122.4 −24.4 −103.4 −82.4 62.2 −70.7 1957 −6.5 221.8 −191.3 1099 578 615 861 540 1039 928 838 840 880 1156 802 702 877 1434 736 164.8 165.3 166.0 189.2 3393 −49.0 785
−84.5 −128.8 −51.3 −54.3 −42.8 +10.3 87.6 2013 91.2 38.5 770 +11.0 121.5 2067 844 838 786 −62.7 +8.6 −37.5 −106.4 −30.1 −207.7 −173.0 88.1 +8.0 −131.6 +8.8 −23.4 −38.2 26.3 2363 −270.7 −259.6 −108.3 −123.8 −30.9 −45.0 −85.6 34.0 −102.3 +6.0 −84.7 −59.1 97.5 −45.3 2224 30.3 256.8 779 −178.4 1309 686 725 1003 644 1189 1048 1003 994 1045 1333 927 838 1050 1666 879 228.8 211.5 212.5 238.2 3859 −22.1 892
−79.0 −125.3 −44.1 −48.0 −37.2 +18.2 95.2 2067 98.3 46.7 801 18.5 133.2 2127 887 886 836 −57.9 14.7 −32.1 −102.3 −24.7 −205.6 −170.0 98.8 16.2 −126.7 16.2 −16.2 −30.7 35.5 2431 −270.6 −258.9 −103.8 −119.6 −25.1 −37.9 −79.8 41.5 −97.9 12.8 −80.2 −53.7 105.4 −39.4 2283 39.1 263.7 805 −175.7 1358 711 750 1036 668 1222 1074 1042 1028 1081 1372 955 868 1088 1720 913 242.0 221.0 222.2 249.0 3964 −16.2 917
735 −11 75.5 1083 504 422 605 −34.4 58 −6.5 −12 −223 −223.9 26.1 −49.5 −165 −71.1 −88 −109 −105.6 1063 −259.1 −87.0 −114.3 −13.2 −83.7 −50.9 0.0 −85.5 −89.7 −64 −49.0 112.9 5.5 1535 −21 304 −156.7 327.5 373 501 855 402 890 1114 186 547 614 870 446 651 712 1260 650 −38.9 237 277 259 2622 17 795
VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-7
2-59
Vapor Pressures of Inorganic Compounds, up to 1 atm (Continued ) Compound
Name Neon Nickel carbonyl chloride Nitrogen Nitric oxide Nitrogen dioxide Nitrogen pentoxide Nitrous oxide Nitrosyl chloride fluoride Osmium tetroxide (yellow) (white) Oxygen Ozone Phosgene Phosphorus (yellow) (violet) tribromide trichloride pentachloride Phosphine Phosphonium bromide chloride iodide Phosphorus trioxide pentoxide oxychloride thiobromide thiochloride Platinum Potassium bromide chloride fluoride hydroxide iodide Radon Rhenium heptoxide Rubidium bromide chloride fluoride iodide Selenium dioxide hexafluoride oxychloride tetrachloride Silicon dioxide tetrachloride tetrafluoride Trichlorofluorosilane Iodosilane Diiodosilane Disiloxan Trisilane Trisilazane Tetrasilane Octachlorotrisilane Hexachlorodisiloxane Hexachlorodisilane Tribromosilane Trichlorosilane Trifluorosilane Dibromosilane Difluorosilane Monobromosilane Monochlorosilane Monofluorosilane Tribromofluorosilane Dichlorodifluorosilane Trifluorobromosilane
Pressure, mm Hg 1
5
10
20
40
−257.3 1810
−255.5 1979
−254.6 2057
−253.7 2143
671 −226.1 −184.5 −55.6 −36.8 −143.4
731 −221.3 −180.6 −42.7 −23.0 −133.4
759 −219.1 −178.2 −36.7 −16.7 −128.7
789 −216.8 −175.3 −30.4 −10.0 −124.0
−132.0 3.2 −5.6 −219.1 −180.4 −92.9 76.6 237 7.8 −51.6 55.5
−120.3 22.0 +15.6 −213.4 −168.6 −77.0 111.2 271 34.4 −31.5 74.0
−114.3 31.3 26.0 −210.6 −163.2 −69.3 128.0 287 47.8 −21.3 83.2
−107.8 41.0 37.4 −207.5 −157.2 −60.3 146.2 306 62.4 −10.2 92.5
−43.7 −91.0 −25.2 384
−28.5 −79.6 −9.0 39.7 424
50.0 −18.3 2730 341 795 821 885 719 745 −144.2 212.5 297 781 792 921 748 356 157.0 −118.6 34.8 74.0 1724
72.4 +4.6 3007 408 892 919 988 814 840 −132.4 237.5 358 876 887 982 839 413 187.7 −105.2 59.8 96.3 1835
−63.4 −144.0 −92.6
−44.1 −134.8 −76.4 −53.0 3.8 −95.8 −49.7 −49.9 −6.2 74.7 17.8 27.4 −8.0 −62.6 −142.7 −40.0 −136.0 −85.7 −104.3 −145.5 −25.4 −110.5
−21.2 −74.0 −1.1 53.0 442 2.0 83.6 16.1 3146 443 940 968 1039 863 887 −126.3 248.0 389 923 937 1016 884 442 202.5 −98.9 71.9 107.4 1888 1732 −34.4 −130.4 −68.3 −47.7 18.0 −88.2 −40.0 −40.4 +4.3 89.3 29.4 38.8 +3.4 −53.4 −138.2 −29.4 −130.4 −77.3 −97.7 −141.2 −15.1 −102.9
−13.3 −68.0 +7.3 67.8 462 13.6 95.5 29.0 3302 483 994 1020 1096 918 938 −119.2 261.0 422 975 990 1052 935 473 217.5 −92.3 84.2 118.1 1942 1798 −24.0 −125.9 −59.0 −33.4 34.1 −79.8 −29.0 −30.0 15.8 104.2 41.5 51.5 16.0 −43.8 −132.9 −18.0 −124.3 −68.3 −90.1 −136.3 −3.7 −94.5
−252.6 2234 −23.0 821 −214.0 −171.7 −23.9 −2.9 −118.3 −60.2 −100.3 51.7 50.5 −204.1 −150.7 −50.3 166.7 323 79.0 +2.3 102.5 −129.4 −5.0 −61.5 16.1 84.0 481 27.3 108.0 42.7 3469 524 1050 1078 1156 976 995 −111.3 272.0 459 1031 1047 1096 991 506 234.1 −84.7 98.0 130.1 2000 1867 −12.1 −120.8 −48.8 −21.8 52.6 −70.4 −16.9 −18.5 28.4 121.5 55.2 65.3 30.0 −32.9 −127.3 −5.2 −117.6 −57.8 −81.8 −130.8 +9.2 −85.0
100
200
400
760
Melting point, °C
−251.0 2364 −6.0 866 −209.7 −166.0 −14.7 7.4 −110.3 −46.3 −88.8 71.5 71.5 −198.8 −141.0 −35.6 197.3 349 103.6 21.0 117.0 −118.8 7.4 −52.0 29.3 108.3 510 47.4 126.3 63.8 3714 586 1137 1164 1245 1064 1080 −99.0 289.0 514 1114 1133 1168 1072 554 258.0 −73.9 118.0 147.5 2083 1969 +5.4 −113.3 −33.2 −4.4 79.4 −55.9 +1.6 −1.1 47.4 146.0 75.4 85.4 51.6 −16.4 −118.7 14.1 −107.3 −42.3 −68.5 −122.4 28.6 −70.3
−249.7 2473 +8.8 904 −205.6 −162.3 −5.0 15.6 −103.6 −34.0 −79.2 89.5 89.5 −194.0 −132.6 −22.3 222.7 370 125.2 37.6 131.3 −109.4 17.6 −44.0 39.9 129.0 532 65.0 141.8 82.0 3923 643 1212 1239 1323 1142 1152 −87.7 307.0 563 1186 1207 1239 1141 594 277.0 −64.8 134.6 161.0 2151 2053 21.0 −170.2 −19.3 +10.7 101.8 −43.5 17.8 +14.0 63.6 166.2 92.5 102.2 70.2 −1.8 −111.3 31.6 −98.3 −28.6 −57.0 −115.2 45.7 −58.0 −69.8
−248.1 2603 25.8 945 −200.9 −156.8 +8.0 24.4 −96.2 −20.3 −68.2 109.3 109.3 −188.8 −122.5 −7.6 251.0 391 149.7 56.9 147.2 −98.3 28.0 −35.4 51.6 150.3 556 84.3 157.8 102.3 4169 708 1297 1322 1411 1233 1238 −75.0 336.0 620 1267 1294 1322 1223 637 297.7 −55.2 151.7 176.4 2220 2141 38.4 −100.7 −4.0 27.9 125.5 −29.3 35.5 31.0 81.7 189.5 113.6 120.6 90.2 +14.5 −102.8 50.7 −87.6 −13.3 −44.5 −106.8 64.6 −45.0 −55.9
−246.0 2732 42.5 987 −195.8 −151.7 21.0 32.4 −85.5 −6.4 −56.0 130.0 130.0 −183.1 −111.1 +8.3 280.0 417 175.3 74.2 162.0 −87.5 38.3 −27.0 62.3 173.1 591 105.1 175.0 124.0 4407 774 1383 1407 1502 1327 1324 −61.8 362.4 679 1352 1381 1408 1304 680 317.0 −45.8 168.0 191.5 2287 2227 56.8 −94.8 +12.2 45.4 149.5 −15.4 53.1 48.7 100.0 211.4 135.6 139.0 111.8 31.8 −95.0 70.5 −77.8 +2.4 −30.4 −98.0 83.8 −31.8 −41.7
−248.7 1452 −25 1001 −210.0 −161 −9.3 30 −90.9 −64.5 −134 56 42 −218.7 −251 −104 44.1 590 −40 −111.8
Temperature, °C
Formula Ne Ni Ni(CO)4 NiCl2 N2 NO NO2 N2O5 N 2O NOCl NOF OsO4 OsO4 O2 O3 COCl2 P P PBr3 PCl3 PCl5 PH3 PH4Br PH4Cl PH4I P4O6 P4O10 POCl3 PSBr3 PSCl3 Pt K KBr KCl KF KOH KI Rn Re2O7 Rb RbBr RbCl RbF RbI Se SeO2 SeF6 SeOCl2 SeCl4 Si SiO2 SiCl4 SiF4 SiFCl3 SiH3I SiH2I2 (SiH3)2O Si3H8 (SiH3)3N Si4H10 Si3Cl3 (SiCl3)2O Si2Cl6 SiHBr3 SiHCl3 SiHF3 SiH2Br2 SiH2F2 SiH3Br SiH3Cl SiH3F SiFBr3 SiF2Cl2 SiF3Br
60
−112.5 −68.9 −68.7 −27.7 46.3 −5.0 +4.0 −30.5 −80.7 −152.0 −60.9 −146.7 −117.8 −153.0 −46.1 −124.7
−251.9 2289 −15.9 840 −212.3 −168.9 −19.9 +1.8 −114.9 −54.2 −95.7 59.4 59.4 −201.9 −146.7 −44.0 179.8 334 89.8 10.2 108.3 −125.0 +0.3 −57.3 21.9 94.2 493 35.8 116.0 51.8 3574 550 1087 1115 1193 1013 1030 −106.2 280.0 482 1066 1084 1123 1026 527 244.6 −80.0 106.5 137.8 2036 1911 −4.8 −117.5 −42.2 −14.3 64.0 −64.2 −9.0 −11.0 36.6 132.0 63.8 73.9 39.2 −25.8 −123.7 +3.2 −113.3 −51.1 −76.0 −127.2 17.4 −78.6
−132.5 −28.5 22.5 569 2 38 −36.2 1755 62.3 730 790 880 380 723 −71 296 38.5 682 715 760 642 217 340 −34.7 8.5 1420 1710 −68.8 −90 −120.8 −57.0 −1.0 −144.2 −117.2 −105.7 −93.6 −33.2 −1.2 −73.5 −126.6 −131.4 −70.2 −93.9 −82.5 −139.7 −70.5
2-60
PHYSICAL AND CHEMICAL DATA
TABLE 2-7
Vapor Pressures of Inorganic Compounds, up to 1 atm (Concluded ) Compound
Pressure, mm Hg
Name
Formula
Trifluorochlorosilane Hexafluorodisilane Dichlorofluorobromosilane Dibromochlorofluorosilane Silane Disilane Silver chloride iodide Sodium bromide chloride cyanide fluoride hydroxide iodide Strontium Strontium oxide Sulfur monochloride hexafluoride Sulfuryl chloride Sulfur dioxide trioxide (α) trioxide (β) trioxide (γ) Tellurium chloride fluoride Thallium Thallous bromide chloride iodide Thionyl bromide Thionyl chloride Tin Stannic bromide Stannous chloride Stannic chloride iodide hydride Tin tetramethyl trimethyl-ethyl trimethyl-propyl Titanium chloride Tungsten Tungsten hexafluoride Uranium hexafluoride Vanadyl trichloride Xenon Zinc chloride fluoride diethyl Zirconium bromide chloride iodide
SiF3Cl Si2F6 SiFCl2Br SiFClBr2 SiH4 Si2H6 Ag AgCl AgI Na NaBr NaCl NaCN NaF NaOH NaI Sr SrO S S2Cl2 SF6 SO2Cl2 SO2 SO3 SO3 SO3 Te TeCl4 TeF6 Tl TlBr TlCl TlI SOBr2 SOCl2 Sn SnBr4 SnCl2 SnCl4 SnI4 SnH4 Sn(CH3)4 Sn(CH3)3·C2H5 Sn(CH3)3·C3H7 TiCl4 W WF6 UF6 VOCl3 Xe Zn ZnCl2 ZnF2 Zn(C2H5)2 ZrBr4 ZrCl4 ZrI4
1
5
10
20
40
−144.0 −81.0 −86.5 −65.2 −179.3 −114.8 1357 912 820 439 806 865 817 1077 739 767
−133.0 −68.8 −68.4 −45.5 −168.6 −99.3 1500 1019 927 511 903 967 928 1186 843 857 847 2198 223.0 +15.7 −120.6 −35.1 −83.0 −23.7 −19.2 −2.0 605
−127.0 −63.1 −59.0 −35.6 −163.0 −91.4 1575 1074 983 549 952 1017 983 1240 897 903 898 2262 243.8 27.5 −114.7 −24.8 −76.8 −16.5 −12.3 +4.3 650 233 −92.4 983 522 517 531 31.0 −21.9 1703 72.7 391 +10.0 175.8 −118.5 −20.6 +3.8 21.8 21.3 4507 −49.2 −13.8 12.2 −152.8 593 508 1086 +11.7 250 230 311
−120.5 −57.0 −48.8 −24.5 −156.9 −82.7 1658 1134 1045 589 1005 1072 1046 1300 953 952 953 2333 264.7 40.0 −108.4 −13.4 −69.7 −9.1 −4.9 11.1 697 253 −86.0 1040 559 550 567 44.1 −10.5 1777 88.1 420 22.0 196.2 −111.2 −9.3 16.1 34.0 34.2 4690 −41.5 −5.2 26.6 −147.1 632 536 1129 24.2 266 243 329
−112.8 −50.6 −37.0 −12.0 −150.3 −72.8 1743 1200 1111 633 1063 1131 1115 1363 1017 1005 1018 2410 288.3 54.1 −101.5 −1.0 −60.5 −1.0 +3.2 17.9 753 273 −78.4 1103 598 589 607 58.8 +2.2 1855 105.5 450 35.2 218.8 −102.3 +3.5 30.0 48.5 48.4 4886 −33.0 +4.4 40.0 −141.2 673 566 1175 38.0 281 259 344
100
200
400
760
Melting point, °C
−108.2 −46.7 −29.0 −4.7 −146.3 −66.4 1795 1242 1152 662 1099 1169 1156 1403 1057 1039 1057
−101.7 −41.7 −19.5 +6.3 −140.5 −57.5 1865 1297 1210 701 1148 1220 1214 1455 1111 1083 1111
−91.7 −34.2 −3.2 23.0 −131.6 −44.6 1971 1379 1297 758 1220 1296 1302 1531 1192 1150 1192
−81.0 −26.4 +15.4 43.0 −122.0 −29.0 2090 1467 1400 823 1304 1379 1401 1617 1286 1225 1285
−70.0 −18.9 35.4 59.5 −111.5 −14.3 2212 1564 1506 892 1392 1465 1497 1704 1378 1304 1384
305.5 63.2 −96.8 +7.2 −54.6 +4.0 8.0 21.4 789 287 −73.8 1143 621 612 631 68.3 10.4 1903 116.2 467 43.5 234.2 −96.6 11.7 38.4 57.5 58.0 5007 −27.5 10.4 49.8 −137.7 700 584 1207 47.2 289 268 355
327.2 75.3 −90.9 17.8 −46.9 10.5 14.3 28.0 838 304 −67.9 1196 653 645 663 80.6 21.4 1968 131.0 493 54.7 254.2 −89.2 22.8 50.0 69.8 71.0 5168 −20.3 18.2 62.5 −132.8 736 610 1254 59.1 301 279 369
359.7 93.5 −82.3 33.7 −35.4 20.5 23.7 35.8 910 330 −57.3 1274 703 694 712 99.0 37.9 2063 152.8 533 72.0 283.5 −78.0 39.8 67.3 88.0 90.5 5403 −10.0 30.0 82.0 −125.4 788 648 1329 77.0 318 295 389
399.6 115.4 −72.6 51.3 −23.0 32.6 32.6 44.0 997 360 −48.2 1364 759 748 763 119.2 56.5 2169 177.7 577 92.1 315.5 −65.2 58.5 87.6 109.6 112.7 5666 +1.2 42.7 103.5 −117.1 844 689 1417 97.3 337 312 409
444.6 138.0 −63.5 69.2 −10.0 44.8 44.8 51.6 1087 392 −38.6 1457 819 807 823 139.5 75.4 2270 204.7 623 113.0 348.0 −52.3 78.0 108.8 131.7 136.0 5927 17.3 55.7 127.2 −108.0 907 732 1497 118.0 357 331 431
−142 −18.6 −112.3 −99.3 −185 −132.6 960.5 455 552 97.5 755 800 564 992 318 651 800 2430 112.8 −80 −50.2 −54.1 −73.2 16.8 32.3 62.1 452 224 −37.8 3035 460 430 440 −52.2 −104.5 231.9 31.0 246.8 −30.2 144.5 −149.9
60
Temperature, °C
2068 183.8 −7.4 −132.7 −95.5 −39.0 −34.0 −15.3 520 −111.3 825 440 −6.7 −52.9 1492 316 −22.7 −140.0 −51.3 −30.0 −12.0 −13.9 3990 −71.4 −38.8 −23.2 −168.5 487 428 970 −22.4 207 190 264
−98.8 931 490 487 502 +18.4 −32.4 1634 58.3 366 −1.0 156.0 −125.8 −31.0 −7.6 +10.7 +9.4 4337 −56.5 −22.0 +0.2 −158.2 558 481 1055 0.0 237 217 297
−30 3370 −0.5 69.2 −111.6 419.4 365 872 −28 450 437 499
VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8
2-61
Vapor Pressures of Organic Compounds, up to 1 atm* Pressure, mm Hg Compound
1
Name
Formula
Acenaphthalene Acetal Acetaldehyde Acetamide Acetanilide Acetic acid anhydride Acetone Acetonitrile Acetophenone Acetyl chloride Acetylene Acridine Acrolein (2-propenal) Acrylic acid Adipic acid Allene (propadiene) Allyl alcohol (propen-1-ol-3) chloride (3-chloropropene) isopropyl ether isothiocyanate n-propyl ether 4-Allylveratrole iso-Amyl acetate n-Amyl alcohol iso-Amyl alcohol sec-Amyl alcohol (2-pentanol) tert-Amyl alcohol sec-Amylbenzene iso-Amyl benzoate bromide (1-bromo-3-methylbutane) n-butyrate formate iodide (1-iodo-3-methylbutane) isobutyrate Amyl isopropionate iso-Amyl isovalerate n-Amyl levulinate iso-Amyl levulinate nitrate 4-tert-Amylphenol Anethole Angelonitrile Aniline 2-Anilinoethanol Anisaldehyde o-Anisidine (2-methoxyaniline) Anthracene Anthraquinone Azelaic acid Azelaldehyde Azobenzene Benzal chloride (α,α-Dichlorotoluene) Benzaldehyde Benzanthrone Benzene Benzenesulfonylchloride Benzil Benzoic acid anhydride Benzoin Benzonitrile Benzophenone Benzotrichloride (α,α,α-Trichlorotoluene) Benzotrifluoride (α,α,α-Trifluorotoluene) Benzoyl bromide chloride nitrile Benzyl acetate alcohol
C12H10 C6H14O2 C2H4O C2H5NO C8H9NO C2H4O2 C4H6O3 C3H6O C2H3N C8H8O C2H3OCl C2H2 C13H9N C3H4O C3H4O2 C6H10O4 C3H4 C3H6O C3H5Cl C6H12O C4H5NS C6H12O C11H14O2 C7H14O2 C5H12O C5H12O C5H12O C5H12O C11H16 C12H16O2 C5H11Br C9H18O2 C6H12O2 C5H11I C9H18O2 C8H16O2 C10H20O2 C10H18O3 C10H18O3 C5H11NO3 C11H16O C10H12O C5H7N C6H7N C8H11NO C8H8O2 C7H9NO C14H10 C14H8O2 C9H16O4 C9H18O C12H10N2 C7H6Cl2 C7H6O C17H10O C6H6 C6H5ClO2S C14H10O2 C7H6O2 C14H10O3 C14H12O2 C7H5N C13H10O C7H5Cl3 C7H5F3 C7H5BrO C7H5ClO C8H5NO C9H10O2 C7H8O
5
10
20
114.8 −2.3 −65.1 92.0 146.6 +6.3 24.8 −40.5 −26.6 64.0 −35.0 −133.0 165.8 −46.0 27.3 191.0 −108.0 +0.2 −52.0 −23.1 +25.3 −18.2 113.9 +23.7 34.7 30.9 22.1 +7.2 55.8 104.5 +2.1 47.1 +5.4 +21.9 40.1 33.7 54.4 110.0 104.0 28.8 109.8 91.6 +15.0 57.9 134.3 102.6 88.0 173.5 219.4 210.4 58.4 135.7 64.0 50.1 274.5 −19.6 96.5 165.2 119.5 180.0 170.2 55.3 141.7 73.7 −10.3 75.4 59.1 71.7 73.4 80.8
131.2 +8.0 −56.8 105.0 162.0 17.5 36.0 −31.1 −16.3 78.0 −27.6 −128.2 184.0 −36.7 39.0 205.5 −101.0 10.5 −42.9 −12.9 38.3 −7.9 127.0 35.2 44.9 40.8 32.2 17.2 69.2 121.6 13.6 59.9 17.1 34.1 52.8 46.3 68.6 124.0 118.8 40.3 125.5 106.0 28.0 69.4 149.6 117.8 101.7 187.2 234.2 225.5 71.6 151.5 78.7 62.0 297.2 −11.5 112.0 183.0 132.1 198.0 188.1 69.2 157.6 87.6 −0.4 89.8 73.0 85.5 87.6 92.6
148.7 19.6 −47.8 120.0 180.0 29.9 48.3 −20.8 −5.0 92.4 −19.6 −122.8 203.5 −26.3 52.0 222.0 −93.4 21.7 −32.8 −1.8 52.1 +3.7 142.8 47.8 55.8 51.7 42.6 27.9 83.8 139.7 26.1 74.0 30.0 47.6 66.6 60.0 83.8 139.7 134.4 53.5 142.3 121.8 41.0 82.0 165.7 133.5 116.1 201.9 248.3 242.4 85.0 168.3 94.3 75.0 322.5 −2.6 129.0 202.8 146.7 218.0 207.0 83.4 175.8 102.7 12.2 105.4 87.6 100.2 102.3 105.8
40
60
100
200
400
760
197.5 50.1 −22.6 158.0 227.2 63.0 82.2 +7.7 27.0 133.6 +3.2 −107.9 256.0 +2.5 86.1 265.0 −72.5 50.0 −4.5 29.0 89.5 35.8 183.7 83.2 85.8 80.7 70.7 55.3 124.1 186.8 60.4 113.1 65.4 84.4 104.4 97.6 125.1 180.5 177.0 88.6 189.0 164.2 77.5 119.9 209.5 176.7 155.2 250.0 285.0 286.5 123.0 216.0 138.3 112.5 390.0 26.1 174.5 255.8 186.2 270.4 258.0 123.5 224.4 144.3 45.3 147.7 128.0 141.0 144.0 141.7
222.1 66.3 −10.0 178.3 250.5 80.0 100.0 22.7 43.7 154.2 16.1 −100.3 284.0 17.5 103.3 287.8 −61.3 64.5 10.4 44.3 108.0 52.6 204.0 101.3 102.0 95.8 85.7 69.7 145.2 210.2 78.7 133.2 83.2 103.8 124.2 117.3 146.1 203.1 198.1 106.7 213.0 186.1 96.3 140.1 230.6 199.0 175.3 279.0 314.6 309.6 142.1 240.0 160.7 131.7 426.5 42.2 198.0 283.5 205.8 299.1 284.4 144.1 249.8 165.6 62.5 169.2 149.5 161.3 165.5 160.0
250.0 84.0 +4.9 200.0 277.0 99.0 119.8 39.5 62.5 178.0 32.0 −92.0 314.3 34.5 122.0 312.5 −48.5 80.2 27.5 61.7 129.8 71.4 226.2 121.5 119.8 113.7 102.3 85.7 168.0 235.8 99.4 155.3 102.7 125.8 146.0 138.4 169.5 227.4 222.7 126.5 239.5 210.5 117.7 161.9 254.5 223.0 197.3 310.2 346.2 332.8 163.4 266.1 187.0 154.1
277.5 102.2 20.2 222.0 303.8 118.1 139.6 56.5 81.8 202.4 50.8 −84.0 346.0 52.5 141.0 337.5 −35.0 96.6 44.6 79.5 150.7 90.5 248.0 142.0 137.8 130.6 119.7 101.7 193.0 262.0 120.4 178.6 123.3 148.2 168.8 160.2 194.0 253.2 247.9 147.5 266.0 235.3 140.0 184.4 279.6 248.0 218.5 342.0 379.9 356.5 185.0 293.0 214.0 179.0
60.6 224.0 314.3 227.0 328.8 313.5 166.7 276.8 189.2 82.0 193.7 172.8 185.0 189.0 183.0
80.1 251.5 347.0 249.2 360.0 343.0 190.6 305.4 213.5 102.2 218.5 197.2 208.0 213.5 204.7
Temperature, °C −23.0 −81.5 65.0 114.0 −17.2 1.7 −59.4 −47.0 37.1 −50.0 −142.9 129.4 −64.5 +3.5 159.5 −120.6 −20.0 −70.0 −43.7 −2.0 −39.0 85.0 0.0 +13.6 +10.0 +1.5 −12.9 29.0 72.0 −20.4 21.2 −17.5 −2.5 14.8 +8.5 27.0 81.3 75.6 +5.2 62.6 −8.0 34.8 104.0 73.2 61.0 145.0 190.0 178.3 33.3 103.5 35.4 26.2 225.0 −36.7 65.9 128.4 96.0 143.8 135.6 28.2 108.2 45.8 −32.0 47.0 32.1 44.5 45.0 58.0
168.2 31.9 −37.8 135.8 199.6 43.0 62.1 −9.4 +7.7 109.4 −10.4 −116.7 224.2 −15.0 66.2 240.5 −85.2 33.4 −21.2 +10.9 67.4 16.4 158.3 62.1 68.0 63.4 54.1 38.8 100.0 158.3 39.8 90.0 44.0 62.3 81.8 75.5 100.6 155.8 151.7 67.6 160.3 139.3 55.8 96.7 183.7 150.5 132.0 217.5 264.3 260.0 100.2 187.9 112.1 90.1 350.0 +7.6 147.7 224.5 162.6 239.8 227.6 99.6 195.7 119.8 25.7 122.6 103.8 116.6 119.6 119.8
181.2 39.8 −31.4 145.8 211.8 51.7 70.8 −2.0 15.9 119.8 −4.5 −112.8 238.7 −7.5 75.0 251.0 −78.8 40.3 −14.1 18.7 76.2 25.0 169.6 71.0 75.5 71.0 61.5 46.0 110.4 171.4 48.7 99.8 53.3 71.9 91.7 85.2 110.3 165.2 162.6 76.3 172.6 149.8 65.2 106.0 194.0 161.7 142.1 231.8 273.3 271.8 110.0 199.8 123.4 99.6 368.8 15.4 158.2 238.2 172.8 252.7 241.7 109.8 208.2 130.0 34.0 133.4 114.7 127.0 129.8 129.3
Melting point, °C 95 −123.5 81 113.5 16.7 −73 −94.6 −41 20.5 −112.0 −81.5 110.5 −87.7 14 152 −136 −129 −136.4 −80
−117.2 −11.9
93 22.5 −6.2 2.5 5.2 217.5 286 106.5 68 −16.1 −26 174 +5.5 14.5 95 121.7 42 132 −12.9 48.5 −21.2 −29.3 0 −0.5 33.5 −51.5 −15.3
*Compiled from the extended tables published by D. R. Stull in Ind. Eng. Chem., 39, 517 (1947). For information on fuels see Hibbard, N.A.C.A. Research Mem. E56I21, 1956. For methane see Johnson (ed.), WADD-TR-60-56, 1960.
2-62
PHYSICAL AND CHEMICAL DATA
TABLE 2-8
Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name
Benzylamine Benzyl bromide (α-bromotoluene) chloride (α-chlorotoluene) cinnamate Benzyldichlorosilane Benzyl ethyl ether phenyl ether isothiocyanate Biphenyl 1-Biphenyloxy-2,3-epoxypropane d-Bornyl acetate Bornyl n-butyrate formate isobutyrate propionate Brassidic acid Bromoacetic acid 4-Bromoanisole Bromobenzene 4-Bromobiphenyl 1-Bromo-2-butanol 1-Bromo-2-butanone cis-1-Bromo-1-butene trans-1-Bromo-1-butene 2-Bromo-1-butene cis-2-Bromo-2-butene trans-2-Bromo-2-butene 1,4-Bromochlorobenzene 1-Bromo-1-chloroethane 1-Bromo-2-chloroethane 2-Bromo-4,6-dichlorophenol 1-Bromo-4-ethyl benzene (2-Bromoethyl)-benzene 2-Bromoethyl 2-chloroethyl ether (2-Bromoethyl)-cyclohexane 1-Bromoethylene Bromoform (tribromomethane) 1-Bromonaphthalene 2-Bromo-4-phenylphenol 3-Bromopyridine 2-Bromotoluene 3-Bromotoluene 4-Bromotoluene 3-Bromo-2,4,6-trichlorophenol 2-Bromo-1,4-xylene 1,2-Butadiene (methyl allene) 1,3-Butadiene n-Butane iso-Butane (2-methylpropane) 1,3-Butanediol 1,2,3-Butanetriol 1-Butene cis-2-Butene trans-2-Butene 3-Butenenitrile iso-Butyl acetate n-Butyl acrylate alcohol iso-Butyl alcohol sec-Butyl alcohol tert-Butyl alcohol iso-Butyl amine n-Butylbenzene iso-Butylbenzene sec-Butylbenzene tert-Butylbenzene iso-Butyl benzoate n-Butyl bromide (1-bromobutane) iso-Butyl n-butyrate carbamate Butyl carbitol (diethylene glycol butyl ether) n-Butyl chloride (1-chlorobutane) iso-Butyl chloride
1
5
10
20
29.0 32.2 22.0 173.8 45.3 26.0 95.4 79.5 70.6 135.3 46.9 74.0 47.0 70.0 64.6 209.6 54.7 48.8 +2.9 98.0 23.7 +6.2 −44.0 −38.4 −47.3 −39.0 −45.0 32.0 −36.0 −28.8 84.0 30.4 48.0 36.5 38.7 −95.4
70.0
54.8 59.6 47.8 206.3 70.2 52.0 127.7 107.8 101.8 169.9 75.7 103.4 74.8 99.8 93.7 241.7 81.6 77.8 27.8 133.7 45.4 30.0 −23.2 −17.0 −27.0 −17.9 −24.1 59.5 −18.0 −7.0 115.6 42.5 76.2 63.2 66.6 −77.8 22.0 117.5 135.4 42.0 49.7 50.8 47.5 146.2 65.0 −72.7 −87.6 −85.7 −94.1 67.5 132.0 −89.4 −81.1 −84.0 +2.9 +1.4 +23.5 +20.0 +11.6 +7.2 −3.0 −31.0 48.8 40.5 44.2 39.0 93.6 −11.2 30.0 83.7 95.7
67.7 73.4 60.8 221.5 83.2 65.0 144.0 121.8 117.0 187.2 90.2 118.0 89.3 114.0 108.0 256.0 94.1 91.9 40.0 150.6 55.8 41.8 −12.8 −6.4 −16.8 −7.2 −13.8 72.7 −9.4 +4.1 130.8 74.0 90.5 76.3 80.5 −68.8 34.0 133.6 152.3 55.2 62.3 64.0 61.1 163.2 78.8 −64.2 −79.7 −77.8 −86.4 85.3 146.0 −81.6 −73.4 −76.3 14.1 12.8 35.5 30.2 21.7 16.9 +5.5 −21.0 62.0 53.7 57.0 51.7 108.6 −0.3 42.2 96.4 107.8
81.8 88.3 75.0 239.3 96.7 79.6 160.7 137.0 134.2 205.8 106.0 133.8 104.0 130.0 123.7 272.9 108.2 107.8 53.8 169.8 67.2 54.2 −1.4 +5.4 −5.3 +4.6 −2.4 87.8 0.0 16.0 147.7 90.2 105.8 90.8 95.8 −58.8 48.0 150.2 171.8 69.1 76.0 78.1 75.2 181.8 94.0 −54.9 −71.0 −68.9 −77.9 100.0 161.0 −73.0 −64.6 −67.5 26.6 25.5 48.6 41.5 32.4 27.3 14.3 −10.3 76.3 67.8 70.6 65.6 124.2 +11.6 56.1 110.1 120.5
97.3 104.8 90.7 255.8 111.8 95.4 180.1 153.0 152.5 226.3 123.7 150.7 121.2 147.2 140.4 290.0 124.0 125.0 68.6 190.8 79.5 68.2 +11.5 18.4 +7.2 17.7 +10.5 103.8 +10.4 29.7 165.8 108.5 123.2 106.6 113.0 −48.1 63.6 170.2 193.8 84.1 91.0 93.9 91.8 200.5 110.6 −44.3 −61.3 −59.1 −68.4 117.4 178.0 −63.4 −54.7 −57.6 40.0 39.2 63.4 53.4 44.1 38.1 24.5 +1.3 92.4 83.3 86.2 80.8 141.8 24.8 71.7 125.3 135.5
C4H9Cl C4H9Cl
−49.0 −53.8
−28.9 −34.3
−18.6 −24.5
−7.4 −13.8
+5.0 −1.9
60
100
200
400
760
107.3 115.6 100.5 267.0 121.3 105.5 192.6 163.8 165.2 239.7 135.7 161.8 131.7 157.6 151.2 301.5 133.8 136.0 78.1 204.5 87.0 77.3 19.8 27.2 15.4 26.2 18.7 114.8 17.0 38.0 177.6 121.0 133.8 116.4 123.7 −41.2 73.4 183.5 207.0 94.1 100.0 104.1 102.3 213.0 121.6 −37.5 −55.1 −52.8 −62.4 127.5 188.0 −57.2 −48.4 −51.3 48.8 48.0 72.6 60.3 51.7 45.2 31.0 8.8 102.6 93.3 96.0 90.6 152.0 33.4 81.3 134.6 146.0
120.0 129.8 114.2 281.5 133.5 118.9 209.2 177.7 180.7 255.0 149.8 176.4 145.8 172.2 165.7 316.2 146.3 150.1 90.8 221.8 97.6 89.2 30.8 38.1 26.3 37.5 29.9 128.0 28.0 49.5 193.2 135.5 148.2 129.8 138.0 −31.9 85.9 198.8 224.5 107.8 112.0 117.8 116.4 229.3 135.7 −28.3 −46.8 −44.2 −54.1 141.2 202.5 −48.9 −39.8 −42.7 60.2 59.7 85.1 70.1 61.5 54.1 39.8 18.8 116.2 107.0 109.5 103.8 166.4 44.7 94.0 147.2 159.8
140.0 150.8 134.0 303.8 152.0 139.6 233.2 198.0 204.2 280.4 172.0 198.0 166.4 194.2 187.5 336.8 165.8 172.7 110.1 248.2 112.1 107.0 47.8 55.7 42.8 54.5 46.5 149.5 44.7 66.8 216.5 156.5 169.8 150.0 160.0 −17.2 106.1 224.2 251.0 127.7 133.6 138.0 137.4 253.0 156.4 −14.2 −33.9 −31.2 −41.5 161.0 222.0 −36.2 −26.8 −29.7 78.0 77.6 104.0 84.3 75.9 67.9 52.7 32.0 136.9 127.2 128.8 123.7 188.2 62.0 113.9 165.7 181.2
161.3 175.2 155.8 326.7 173.0 161.5 259.8 220.4 229.4 309.8 197.5 222.2 190.2 218.2 211.2 359.6 186.7 197.5 132.3 277.7 128.3 126.3 66.8 75.0 61.9 74.0 66.0 172.6 63.4 86.0 242.0 182.0 194.0 172.3 186.2 −1.1 127.9 252.0 280.2 150.0 157.3 160.0 160.2 278.0 181.0 +1.8 −19.3 −16.3 −27.1 183.8 243.5 −21.7 −12.0 −14.8 98.0 97.5 125.2 100.8 91.4 83.9 68.0 50.7 159.2 149.6 150.3 145.8 212.8 81.7 135.7 186.0 205.0
184.5 198.5 179.4 350.0 194.3 185.0 287.0 243.0 254.9 340.0 223.0 247.0 214.0 243.0 235.0 382.5 208.0 223.0 156.2 310.0 145.0 147.0 86.2 94.7 81.0 93.9 85.5 196.9 82.7 106.7 268.0 206.0 219.0 195.8 213.0 +15.8 150.5 281.1 311.0 173.4 181.8 183.7 184.5 305.8 206.7 18.5 −4.5 −0.5 −11.7 206.5 264.0 −6.3 +3.7 +0.9 119.0 118.0 147.4 117.5 108.0 99.5 82.9 68.6 183.1 172.8 173.5 168.5 237.0 101.6 156.9 206.5 231.2
13.0 +5.9
24.0 16.0
40.0 32.0
58.8 50.0
77.8 68.9
Temperature, °C
Formula C7H9N C7H7Br C7H7Cl C16H14O2 C7H8Cl2Si C9H12O C13H12O C8H7NS C12H10 C15H14O2 C12H20O2 C14H24O2 C11H18O2 C14H24O2 C13H22O2 C22H42O2 C2H3BrO2 C7H7BrO C6H5Br C12H9Br C4H9BrO C4H7BrO C4H7Br C4H7Br C4H7Br C4H7Br C4H7Br C6H4BrCl C2H4BrCl C2H4BrCl C6H3BrCl2O C8H9Br C8H9Br C4H8BrClO C8H15Br C2H3Br CHBr3 C10H7Br C12H9BrO C5H4BrN C7H7Br C7H7Br C7H7Br C6H2BrCl3O C8H9Br C4H6 C4H6 C4H10 C4H10 C4H10O2 C4H10O3 C4H8 C4H8 C4H8 C4H5N C6H12O2 C7H12O2 C4H10O C4H10O C4H10O C4H10O C4H11N C10H14 C10H14 C10H14 C10H14 C11H14O2 C4H9Br C8H16O2 C5H11NO2 C8H18O3
40
84.2 100.0 16.8 24.4 14.8 10.3 112.4 37.5 −89.0 −102.8 −101.5 −109.2 22.2 102.0 −104.8 −96.4 −99.4 −19.6 −21.2 −0.5 −1.2 −9.0 −12.2 −20.4 −50.0 22.7 14.1 18.6 13.0 64.0 −33.0 +4.6
Melting point, °C −4 −39 39
69.5 29
61.5 49.5 12.5 −30.7 90.5
−100.3 −133.4 −111.2 −114.6 16.6 −16.6 68 −45.0
−138 8.5 5.5 95 −28 39.8 28.5 +9.5 −108.9 −135 −145 77 −130 −138.9 −105.4 −98.9 −64.6 −79.9 −108 −114.7 25.3 −85.0 −88.0 −51.5 −75.5 −58 −112.4 65 −123.1 −131.2
VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8
2-63
Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name
Formula
sec-Butyl chloride (2-Chlorobutane) tert-Butyl chloride sec-Butyl chloroacetate 2-tert-Butyl-4-cresol 4-tert-Butyl-2-cresol iso-Butyl dichloroacetate 2,3-Butylene glycol (2,3-butanediol) 2-Butyl-2-ethylbutane-1,3-diol 2-tert-Butyl-4-ethylphenol n-Butyl formate iso-Butyl formate sec-Butyl formate sec-Butyl glycolate iso-Butyl iodide (1-iodo-2-methylpropane) isobutyrate isovalerate levulinate naphthylketone (1-isovaleronaphthone) 2-sec-Butylphenol 2-tert-Butylphenol 4-iso-Butylphenol 4-sec-Butylphenol 4-tert-Butylphenol 2-(4-tert-Butylphenoxy)ethyl acetate 4-tert-Butylphenyl dichlorophosphate
C4H9Cl C4H9Cl C6H11ClO2 C11H16O C11H16O C6H10Cl2O2 C4H10O2 C10H22O2 C12H15O C5H10O2 C5H10O2 C5H10O2 C6H12O3 C4H9I C8H16O2 C9H18O2 C9H16O3 C15H16O C10H14O C10H14O C10H14O C10H14O C10H14O C14H20O3 C10H13Cl2 O2P C11H14O C7H14O2 C12H18O C12H18O C12H18O C12H18O C4H8O2 C4H8O2 C4H7N C11H14O C10H16 C10H16O2 C10H16O C10H19N C10H20O C10H20O2 C6H12O2 C6H12O2 C6H10O2 C6H11N C8H18O C8H16O C8H16O2 C8H15N C12H9N CO2 CS2 CO COSe COS CBr4 CCl4 CF4 C10H14O C10H14O C10H12O2 C2HCl3O C2H3Cl3O2
tert-Butyl phenyl ketone (pivalophenone) iso-Butyl propionate 4-tert-Butyl-2,5-xylenol 4-tert-Butyl-2,6-xylenol 6-tert-Butyl-2,4-xylenol 6-tert-Butyl-3,4-xylenol Butyric acid iso-Butyric acid Butyronitrile iso-Valerophenone Camphene Campholenic acid d-Camphor Camphylamine Capraldehyde Capric acid n-Caproic acid iso-Caproic acid iso-Caprolactone Capronitrile Capryl alcohol (2-octanol) Caprylaldehyde Caprylic acid (octanoic acid) Caprylonitrile Carbazole Carbon dioxide disulfide monoxide oxyselenide (carbonyl selenide) oxysulfide (carbonyl sulfide) tetrabromide tetrachloride tetrafluoride Carvacrol Carvone Chavibetol Chloral (trichloroacetaldehyde) hydrate (trichloroacetaldehyde hydrate) Chloranil Chloroacetic acid anhydride 2-Chloroaniline 3-Chloroaniline 4-Chloroaniline Chlorobenzene 2-Chlorobenzotrichloride (2-α,α,α-tetrachlorotoluene)
C6Cl4O2 C2H3ClO2 C4H4Cl2O3 C6H6ClN C6H6ClN C6H6ClN C6H5Cl C7H4Cl4
1
5
10
20
40
−60.2
−39.8
−29.2
−17.7
17.0 70.0 74.3 28.6 44.0 94.1 76.3 −26.4 −32.7 −34.4 28.3 −17.0 +4.1 16.0 65.0 136.0 57.4 56.6 72.1 71.4 70.0 118.0 96.0
41.8 98.0 103.7 54.3 68.4 122.6 106.2 −4.7 −11.4 −13.3 53.6 +5.8 28.0 41.2 92.1 167.9 86.0 84.2 100.9 100.5 99.2 150.0 129.6
54.6 112.0 118.0 67.5 80.3 136.8 121.0 +6.1 −0.8 −3.1 66.0 17.0 39.9 53.8 105.9 184.0 100.8 98.1 115.5 114.8 114.0 165.8 146.0
68.2 127.2 134.0 81.4 93.4 151.2 137.0 18.0 +11.0 +8.4 79.8 29.8 52.4 67.7 120.2 201.6 116.1 113.0 130.3 130.3 129.5 183.3 164.0
−5.0 −19.0 83.6 143.9 150.8 96.7 107.8 167.8 154.0 31.6 24.1 21.3 94.2 42.8 67.2 82.7 136.2 219.7 133.4 129.2 147.2 147.8 146.0 201.5 184.3
57.8 −2.3 88.2 74.0 70.3 83.9 25.5 14.7 −20.0 58.3
85.7 +20.9 119.8 103.9 100.2 113.6 49.8 39.3 +2.1 87.0
97.6 41.5 45.3 51.9 125.0 71.4 66.2 38.3 9.2 32.8 73.4 92.3 43.0
125.7 68.6 74.0 78.8 142.0 89.5 83.0 66.4 34.6 57.6 92.0 114.1 67.6
99.0 32.3 135.0 119.0 115.0 127.0 61.5 51.2 13.4 101.4 47.2 139.8 82.3 83.7 92.0 152.2 99.5 94.0 80.3 47.5 70.0 101.2 124.0 80.4
114.3 44.8 151.0 135.0 131.0 143.0 74.0 64.0 25.7 116.8 60.4 153.9 97.5 97.6 106.3 165.0 111.8 107.0 95.7 61.7 83.3 110.2 136.4 94.6
130.4 58.5 169.8 152.2 148.5 159.7 88.0 77.8 38.4 133.8 75.7 170.0 114.0 112.5 122.2 179.9 125.0 120.4 112.3 76.9 98.0 120.0 150.6 110.6
60
100
200
400
760
+3.4 −11.4 93.0 153.7 161.7 106.6 116.3 178.0 165.4 39.8 32.4 29.6 104.0 51.8 75.9 92.4 147.0 231.5 143.9 140.0 157.0 157.9 156.0 212.8 197.2
14.2 −1.0 105.5 167.0 176.2 119.8 127.8 191.9 179.0 51.0 43.4 40.2 116.4 63.5 88.0 105.2 160.2 246.7 157.3 153.5 171.2 172.4 170.2 228.0 214.3
31.5 +14.6 124.1 187.8 197.8 139.2 145.6 212.0 200.3 67.9 60.0 56.8 135.5 81.0 106.3 124.8 181.8 269.7 179.7 173.8 192.1 194.3 191.5 250.3 240.0
50.0 32.6 146.0 210.0 221.8 160.0 164.0 233.5 223.8 86.2 79.0 75.2 155.6 100.3 126.3 146.4 205.5 294.0 203.8 196.3 214.7 217.6 214.0 277.6 268.2
68.0 51.0 167.8 232.6 247.0 183.0 182.0 255.0 247.8 106.0 98.2 93.6 177.5 120.4 147.5 168.7 229.9 320.0 228.0 219.5 237.0 242.1 238.0 304.4 299.0
140.8 67.6 180.3 163.6 158.2 170.0 96.5 86.3 47.3 144.6 85.0 180.0 124.0 122.0 132.0 189.8 133.3 129.6 123.2 86.8 107.4 126.0 160.0 121.2 248.2 −104.8 −15.3 −208.1 −70.2 −93.0 106.3 12.3 −155.4 155.3 143.8 170.7 29.1 46.2
154.0 79.5 195.0 176.0 172.0 184.0 108.0 98.0 59.0 158.0 97.9 193.7 138.0 134.6 145.3 200.0 144.0 141.4 137.2 99.8 119.8 133.9 172.2 134.8 265.0 −100.2 −5.1 −205.7 −61.7 −85.9 119.7 23.0 −150.7 169.7 157.3 185.5 40.2 55.0
Temperature, °C
175.0 197.7 220.0 97.0 116.4 136.8 217.5 241.3 265.3 196.0 217.8 239.8 192.3 214.2 236.5 204.5 226.7 249.5 125.5 144.5 163.5 115.8 134.5 154.5 76.7 96.8 117.5 180.1 204.2 228.0 117.5 138.7 160.5 212.7 234.0 256.0 157.9 182.0 209.2 153.0 173.8 195.0 164.8 186.3 208.5 217.1 240.3 268.4 160.8 181.0 202.0 158.3 181.0 207.7 157.8 182.1 207.0 119.7 141.0 163.7 138.0 157.5 178.5 145.4 156.5 168.5 190.3 213.9 237.5 155.2 179.5 204.5 292.5 323.0 354.8 −93.0 −85.7 −78.2 +10.4 28.0 46.5 −201.3 −196.3 −191.3 −49.8 −35.6 −21.9 −75.0 −62.7 −49.9 139.7 163.5 189.5 38.3 57.8 76.7 −143.6 −135.5 −127.7 191.2 213.8 237.0 179.6 203.5 227.5 206.8 229.8 254.0 57.8 77.5 97.7 68.0 82.1 96.2
Melting point, °C −131.3 −26.5
22.5
−95.3 −90.7 −80.7
99
−71
−74 −47 50 178.5 31.5 −1.5 −35 −38.6 16 244.8 −57.5 −110.8 −205.0
−134.3 −73.8 −222.0 −117.1 −132.4
−124.4 −54.3 −217.2 −102.3 −119.8
−119.5 −44.7 −215.0 −95.0 −113.3
−114.4 −34.3 −212.8 −86.3 −106.0
−50.0 −184.6 70.0 57.4 83.6 −37.8 −9.8
−30.0 −174.1 98.4 86.1 113.3 −16.0 +10.0
−19.6 −169.3 113.2 100.4 127.0 −5.0 19.5
−8.2 −164.3 127.9 116.1 143.2 +7.2 29.2
−108.6 −22.5 −210.0 −76.4 −98.3 96.3 +4.3 −158.8 145.2 133.0 159.8 20.2 39.7
70.7 43.0 67.2 46.3 63.5 59.3 −13.0
89.3 68.3 94.1 72.3 89.8 87.9 +10.6
97.8 81.0 108.0 84.8 102.0 102.1 22.2
106.4 94.2 122.4 99.2 116.7 117.8 35.3
116.1 109.2 138.2 115.6 133.6 135.0 49.7
122.0 118.3 148.0 125.7 144.1 145.8 58.3
129.5 130.7 159.8 139.5 158.0 159.9 70.7
140.3 149.0 177.8 160.0 179.5 182.3 89.4
151.3 169.0 197.0 183.7 203.5 206.6 110.0
162.6 189.5 217.0 208.8 228.5 230.5 132.2
290 61.2 46 0 −10.4 70.5 −45.2
69.0
101.8
117.9
135.8
155.0
167.8
185.0
208.0
233.0
262.1
28.7
−138.8 90.1 −22.6 −183.7 +0.5 −57 51.7
2-64
PHYSICAL AND CHEMICAL DATA
TABLE 2-8
Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name
2-Chlorobenzotrifluoride (2-chloro-α,α,α-trifluorotoluene) 2-Chlorobiphenyl 4-Chlorobiphenyl α-Chlorocrotonic acid Chlorodifluoromethane Chlorodimethylphenylsilane 1-Chloro-2-ethoxybenzene 2-(2-Chloroethoxy) ethanol bis-2-Chloroethyl acetacetal 1-Chloro-2-ethylbenzene 1-Chloro-3-ethylbenzene 1-Chloro-4-ethylbenzene 2-Chloroethyl chloroacetate 2-Chloroethyl 2-chloroisopropyl ether 2-Chloroethyl 2-chloropropyl ether 2-Chloroethyl α-methylbenzyl ether Chloroform (trichloromethane) 1-Chloronaphthalene 4-Chlorophenethyl alcohol 2-Chlorophenol 3-Chlorophenol 4-Chlorophenol 2-Chloro-3-phenylphenol 2-Chloro-6-phenylphenol Chloropicrin (trichloronitromethane) 1-Chloropropene 2-Chloropyridine 3-Chlorostyrene 4-Chlorostyrene 1-Chlorotetradecane 2-Chlorotoluene 3-Chlorotoluene 4-Chlorotoluene Chlorotriethylsilane 1-Chloro-1,2,2-trifluoroethylene Chlorotrifluoromethane Chlorotrimethylsilane trans-Cinnamic acid Cinnamyl alcohol Cinnamylaldehyde Citraconic anhydride cis-α-Citral d-Citronellal Citronellic acid Citronellol Citronellyl acetate Coumarin o-Cresol (2-cresol; 2-methylphenol) m-Cresol (3-cresol; 3-methylphenol) p-Cresol (4-cresol; 4-methylphenol) cis-Crotonic acid trans-Crotonic acid cis-Crotononitrile trans-Crotononitrile Cumene 4-Cumidene Cuminal Cuminyl alcohol 2-Cyano-2-n-butyl acetate Cyanogen bromide chloride iodide Cyclobutane Cyclobutene Cyclohexane Cyclohexaneethanol Cyclohexanol Cyclohexanone 2-Cyclohexyl-4,6-dinitrophenol Cyclopentane Cyclopropane Cymene
1
5
10
20
60
100
200
400
760
88.3 197.0 212.5 155.9 −76.4 124.7 141.8 139.5 150.7 110.0 113.6 116.0 140.0 115.8 125.6 164.8 10.4 180.4 188.1 106.0 143.0 150.0 237.0 237.1 53.8 −15.1 104.6 121.2 122.0 215.5 94.7 96.3 96.6 82.3 −66.7 −111.7 +6.0 232.4 177.8 177.7 145.4 160.0 140.1 195.4 159.8 161.0 216.5 127.4 138.0 140.0 116.3 128.0 50.1 62.8 88.1 158.0 160.0 176.2 133.8 −51.8 22.6 −24.9 97.6 −32.8 −41.2 25.5 142.7 103.7 90.4 229.0 −1.3 −70.0 110.8
108.3 219.6 237.8 173.8 −65.8 145.5 162.0 157.2 169.8 130.2 133.8 137.0 159.8 135.7 146.3 186.3 25.9 204.2 210.0 126.4 164.8 172.0 261.3 261.6 71.8 +1.3 125.0 142.2 143.5 240.3 115.0 116.6 117.1 101.6 −55.0 −102.5 21.9 253.3 199.8 199.3 165.8 181.8 160.0 214.5 179.8 178.8 240.0 146.7 157.3 157.3 133.9 146.0 68.0 81.1 107.3 180.0 182.8 197.9 152.2 −42.6 33.8 −14.1 111.5 −18.9 −27.8 42.0 161.7 121.7 110.3 248.7 +13.8 −59.1 131.4
130.0 243.8 264.5 193.2 −53.6 168.6 185.5 176.5 190.5 152.2 156.7 159.8 182.2 156.5 169.8 210.8 42.7 230.8 234.5 149.8 188.7 196.0 289.4 289.5 91.8 18.0 147.7 165.7 166.0 267.5 137.1 139.7 139.8 123.6 −41.7 −92.7 39.4 276.7 224.6 222.4 189.8 205.0 183.8 236.6 201.0 197.8 264.7 168.4 179.0 179.4 152.2 165.5 88.0 101.5 129.2 203.2 206.7 221.7 173.4 −33.0 46.0 −2.3 126.1 −3.4 −12.2 60.8 183.5 141.4 132.5 269.8 31.0 −46.9 153.5
152.2 267.5 292.9 212.0 −40.8 193.5 208.0 196.0 212.6 177.6 181.1 184.3 205.0 180.0 194.1 235.0 61.3 259.3 259.3 174.5 214.0 220.0 317.5 317.0 111.9 37.0 170.2 190.0 191.0 296.0 159.3 162.3 162.3 146.3 −27.9 −81.2 57.9 300.0 250.0 246.0 213.5 228.0 206.5 257.0 221.5 217.0 291.0 190.8 202.8 201.8 171.9 185.0 108.0 122.8 152.4 227.0 232.0 246.6 195.2 −21.0 61.5 +13.1 141.1 +12.9 +2.4 80.7 205.4 161.0 155.6 291.5 49.3 −33.5 177.2
Temperature, °C
Formula C7H4ClF3 C12H9Cl C12H9Cl C4H5ClO2 CHClF2 C8H11ClSi C8H9ClO C4H9ClO2 C6H12Cl2O2 C8H9Cl C8H9Cl C8H9Cl C4H6Cl2O2 C5H10Cl2O C5H10Cl2O C10H13ClO CHCl3 C10H7Cl C8H9ClO C6H5ClO C6H5ClO C6H5ClO C12H9ClO C12H9ClO CCl3NO2 C3H5Cl C5H4ClN C8H7Cl C8H7Cl C14H29Cl C7H7Cl C7H7Cl C7H7Cl C6H15ClSi C2ClF3 CClF3 C3H9ClSi C9H8O2 C9H10O C9H8O C5H4O3 C10H16O C10H18O C10H18O2 C10H20O C12H22O2 C9H6O2 C7H8O C7H8O C7H8O C4H6O2 C4H6O2 C4H5N C4H5N C9H12 C9H13N C10H12O C10H14O C7H11NO2 C2N2 CBrN CClN CIN C4H8 C4H6 C6H12 C8H16O C6H12O C6H10O C12H14N2O5 C5H10 C3H6 C10H14
40
0.0 89.3 96.4 70.0 −122.8 29.8 45.8 53.0 56.2 17.2 18.6 19.2 46.0 24.7 29.8 62.3 −58.0 80.6 84.0 12.1 44.2 49.8 118.0 119.8 −25.5 −81.3 13.3 25.3 28.0 98.5 +5.4 +4.8 +5.5 −4.9 −116.0 −149.5 −62.8 127.5 72.6 76.1 47.1 61.7 44.0 99.5 66.4 74.7 106.0 38.2 52.0 53.0 33.5
24.7 109.8 129.8 95.6 −110.2 56.7 72.8 78.3 83.7 43.0 45.2 46.4 72.1 50.1 56.5 91.4 −39.1 104.8 114.3 38.2 72.0 78.2 152.2 153.7 −3.3 −63.4 38.8 51.3 54.5 131.8 30.6 30.3 31.0 +19.8 −102.5 −139.2 −43.6 157.8 102.5 105.8 74.8 90.0 71.4 127.3 93.6 100.2 137.8 64.0 76.0 76.5 57.4
−29.0 −19.5 +2.9 60.0 58.0 74.2 42.0 −95.8 −35.7 −76.7 25.2 −92.0 −99.1 −45.3 50.4 21.0 +1.4 132.8 −68.0 −116.8 17.3
−7.1 +3.5 26.8 88.2 87.3 103.7 68.7 −83.2 −18.3 −61.4 47.2 −76.0 −83.4 −25.4 77.2 44.0 26.4 161.8 −49.6 −104.2 43.9
37.1 134.7 146.0 108.0 −103.7 70.0 86.5 90.7 97.6 56.1 58.1 60.0 86.0 63.0 70.0 106.0 −29.7 118.6 129.0 51.2 86.1 92.2 169.7 170.7 +7.8 −54.1 51.7 65.2 67.5 148.2 43.2 43.2 43.8 32.0 −95.9 −134.1 −34.0 173.0 117.8 120.0 88.9 103.9 84.8 141.4 107.0 113.0 153.4 76.7 87.8 88.6 69.0 80.0 +4.0 15.0 38.3 102.2 102.0 118.0 82.0 −76.8 −10.0 −53.8 57.7 −67.9 −75.4 −15.9 90.0 56.0 38.7 175.9 −40.4 −97.5 57.0
50.6 151.2 164.0 121.2 −96.5 84.7 101.5 104.1 112.2 70.3 73.0 75.5 100.0 77.2 84.8 121.8 −19.0 134.4 145.0 65.9 101.7 108.1 186.7 189.8 20.0 −44.0 65.8 80.0 82.0 166.2 56.9 57.4 57.8 45.5 −88.2 −128.5 −23.2 189.5 133.7 135.7 103.8 119.4 99.8 155.6 121.5 126.0 170.0 90.5 101.4 102.3 82.0 93.0 16.4 27.8 51.5 117.8 117.9 133.8 96.2 −70.1 −1.0 −46.1 68.6 −58.7 −66.6 −5.0 104.0 68.8 52.5 191.2 −30.1 −90.3 71.1
65.9 169.9 183.8 135.6 −88.6 101.2 117.8 118.4 127.8 86.2 89.2 91.8 116.0 92.4 101.5 139.6 −7.1 153.2 162.0 82.0 118.0 125.0 207.4 208.2 33.8 −32.7 81.7 96.5 98.0 187.0 72.0 73.0 73.5 60.2 −79.7 −121.9 −11.4 207.1 151.0 152.2 120.3 135.9 116.1 171.9 137.2 140.5 189.0 105.8 116.0 117.7 96.0 107.8 30.0 41.8 66.1 134.2 135.2 150.3 111.8 −62.7 +8.6 −37.5 80.3 −48.4 −56.4 +6.7 119.8 83.0 67.8 206.7 −18.6 −82.3 87.0
75.4 182.1 196.0 144.4 −83.4 111.5 127.8 127.5 138.0 96.4 99.6 102.0 126.2 102.2 111.8 150.0 +0.5 165.6 173.5 92.0 129.4 136.1 219.6 220.0 42.3 −25.1 91.6 107.2 108.5 199.8 81.8 83.2 83.3 69.5 −74.1 −117.3 −4.0 217.8 162.0 163.7 131.3 146.3 126.2 182.1 147.2 149.7 200.5 115.5 125.8 127.0 104.5 116.7 38.5 50.9 75.4 145.0 146.0 161.7 121.5 −57.9 14.7 −32.1 88.0 −41.8 −50.0 14.7 129.8 91.8 77.5 216.0 −11.3 −77.0 97.2
Melting point, °C −6.0 34 75.5 −160
−80.2 −53.3 −62.6
−63.5 −20 7 32.5 42 +6 −64 −99.0 −15.0 +0.9 +7.3 −157.5 133 33 −7.5
70 30.8 10.9 35.5 15.5 72 −96.0
−34.4 58 −6.5 −50 +6.6 23.9 −45.0 −93.7 −126.6 −68.2
VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8
2-65
Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name
cis-Decalin trans-Decalin Decane Decan-2-one 1-Decene Decyl alcohol Decyltrimethylsilane Dehydroacetic acid Desoxybenzoin Diacetamide Diacetylene (1,3-butadiyne) Diallyldichlorosilane Diallyl sulfide Diisoamyl ether oxalate sulfide Dibenzylamine Dibenzyl ketone (1,3-diphenyl2-propanone) 1,4-Dibromobenzene 1,2-Dibromobutane dl-2,3-Dibromobutane meso-2,3-Dibromobutane 1,2-Dibromodecane Di(2-bromoethyl) ether α,β-Dibromomaleic anhydride 1,2-Dibromo-2-methylpropane 1,3-Dibromo-2-methylpropane 1,2-Dibromopentane 1,2-Dibromopropane 1,3-Dibromopropane 2,3-Dibromopropene 2,3-Dibromo-1-propanol Diisobutylamine 2,6-Ditert-butyl-4-cresol 4,6-Ditert-butyl-2-cresol 4,6-Ditert-butyl-3-cresol 2,6-Ditert-butyl-4-ethylphenol 4,6-Ditert-butyl-3-ethylphenol Diisobutyl oxalate 2,4-Ditert-butylphenol Dibutyl phthalate sulfide Diisobutyl d-tartrate Dicarvacryl-mono-(6-chloro-2-xenyl) phosphate Dicarvacryl-2-tolyl phosphate Dichloroacetic acid 1,2-Dichlorobenzene 1,3-Dichlorobenzene 1,4-Dichlorobenzene 1,2-Dichlorobutane 2,3-Dichlorobutane 1,2-Dichloro-1,2-difluoroethylene Dichlorodifluoromethane Dichlorodiphenyl silane Dichlorodiisopropyl ether Di(2-chloroethoxy) methane Dichloroethoxymethylsilane 1,2-Dichloro-3-ethylbenzene 1,2-Dichloro-4-ethylbenzene 1,4-Dichloro-2-ethylbenzene cis-1,2-Dichloroethylene trans-1,2-Dichloro ethylene Di(2-chloroethyl) ether Dichlorofluoromethane 1,5-Dichlorohexamethyltrisiloxane Dichloromethylphenylsilane 1,1-Dichloro-2-methylpropane 1,2-Dichloro-2-methylpropane 1,3-Dichloro-2-methylpropane 2,4-Dichlorophenol 2,6-Dichlorophenol
1
5
10
20
22.5 −0.8 16.5 44.2 14.7 69.5 67.4 91.7 123.3 70.0 −82.5 +9.5 −9.5 18.6 85.4 43.0 118.3 125.5
50.1 +30.6 42.3 71.9 40.3 97.3 96.4 122.0 156.2 95.0 −68.0 34.8 +14.4 44.3 116.0 73.0 149.8 159.8
64.2 47.2 55.7 85.8 53.7 111.3 111.0 137.3 173.5 108.0 −61.2 47.4 26.6 57.0 131.4 87.6 165.6 177.6
79.8 65.3 69.8 100.7 67.8 125.8 126.5 153.0 192.0 122.6 −53.8 61.3 39.7 70.7 147.7 102.7 182.2 195.7
97.2 85.7 85.5 117.1 83.3 142.1 144.0 171.0 212.0 138.2 −45.9 76.4 54.2 86.3 165.7 120.0 200.2 216.6
C6H4Br2 61.0 C4H8Br2 7.5 C4H8Br2 +5.0 C4H8Br2 +1.5 C10H20Br2 95.7 C4H8Br2O 47.7 C4H2Br2O3 50.0 C4H8Br2 −28.8 C4H8Br2 14.0 C5H10Br2 19.8 C3H6Br2 −7.0 C3H6Br2 +9.7 C3H4Br2 −6.0 C3H6Br2O 57.0 C8H19N −5.1 C15H24O 85.8 C15H24O 86.2 C15H24O 103.7 C16H26O 89.1 C16H26O 111.5 C10H18O4 63.2 C14H22O 84.5 C16H22O4 148.2 C8H18S +21.7 C12H22O6 117.8 C32H34ClO4P 204.2
79.3 33.2 30.0 26.6 123.6 75.3 78.0 −3.0 40.0 45.4 +17.3 35.4 +17.9 84.5 +18.4 116.2 117.3 135.2 121.4 142.6 91.2 115.4 182.1 51.8 151.8 234.5
87.7 46.1 41.6 39.3 137.3 88.5 92.0 +10.5 53.0 58.0 29.4 48.0 30.0 98.2 30.6 131.0 132.4 150.0 137.0 157.4 105.3 130.0 198.2 66.4 169.0 249.3
103.6 60.0 56.4 53.2 151.0 103.6 106.7 25.7 67.5 72.0 42.3 62.1 43.2 113.5 43.7 147.0 149.0 167.0 154.0 174.0 120.3 146.0 216.2 80.5 188.0 264.5
C27H33O4P 180.2 C2H2Cl2O2 44.0 C6H4Cl2 20.0 C6H4Cl2 12.1 C6H4Cl2 C4H8Cl2 −23.6 C4H8Cl2 −25.2 C2Cl2F2 −82.0 CCl2F2 −118.5 C12H10Cl2Si 109.6 C6H12Cl2O 29.6 C5H10Cl2O2 53.0 C8H8Cl2OSi −33.8 C8H8Cl2 46.0 C8H8Cl2 47.0 C8H8Cl2 38.5 C2H2Cl2 −58.4 C2H2Cl2 −65.4 C4H8Cl2O 23.5 CHCl2F −91.3 C6H18Cl2 26.0 O2Si3 C7H8Cl2Si 35.7 C4H8Cl2 −31.0 C4H8Cl2 −25.8 C4H8Cl2 −3.0 C6H4Cl2O 53.0 C6H4Cl2O 59.5
209.3 69.8 46.0 39.0 −0.3 −3.0 −65.6 −104.6 142.4 55.2 80.4 −12.1 75.0 77.2 68.0 −39.2 −47.2 49.3 −75.5 52.0
221.8 82.6 59.1 52.0 54.8 +11.5 +8.5 −57.3 −97.8 158.0 68.2 94.0 −1.3 90.0 92.3 83.2 −29.9 −38.0 62.0 −67.5 65.1
63.5 −8.4 −4.2 +20.6 80.0 87.6
77.4 +2.6 +6.7 32.0 92.8 101.0
60
Melting point, °C
100
200
400
760
108.0 98.4 95.5 127.8 93.5 152.0 154.3 181.5 224.5 148.0 −41.0 86.3 63.7 96.0 177.0 130.6 212.2 229.4
123.2 114.6 108.6 142.0 106.5 165.8 169.5 197.5 241.3 160.6 −34.0 99.7 75.8 109.6 192.2 145.3 227.3 246.6
145.4 136.2 128.4 163.2 126.7 186.2 191.0 219.5 265.2 180.8 −20.9 119.4 94.8 129.0 215.0 166.4 249.8 272.3
169.9 160.1 150.6 186.7 149.2 208.8 215.5 244.5 293.0 202.0 −6.1 142.0 116.1 150.3 240.0 191.0 274.3 301.7
194.6 186.7 174.1 211.0 172.0 231.0 240.0 269.0 321.0 223.0 +9.7 165.3 138.6 173.4 265.0 216.0 300.0 330.5
−43.3 −30.7 −29.7 +3.5
120.8 76.0 72.0 68.0 167.4 119.8 123.5 42.3 83.5 87.4 57.2 77.8 57.8 129.8 57.8 164.1 167.4 185.3 172.1 192.3 137.5 164.3 235.8 96.0 208.5 280.5
131.6 86.0 82.0 78.0 177.5 130.0 133.8 53.7 93.7 97.4 66.4 87.8 67.0 140.0 67.0 175.2 179.0 196.1 183.9 204.4 147.8 175.8 247.8 105.8 221.6 290.7
146.5 99.8 95.3 91.7 190.2 144.0 147.7 68.8 107.4 110.1 78.7 101.3 79.5 153.0 79.2 190.0 194.0 211.0 198.0 218.0 161.8 190.0 263.7 118.6 239.5 304.9
168.5 120.2 115.7 111.8 209.6 165.0 168.0 92.1 117.8 130.2 97.8 121.7 98.0 173.8 97.6 212.8 217.5 233.0 220.0 241.7 183.5 212.5 287.0 138.0 264.7 323.8
192.5 143.5 138.0 134.2 229.8 188.0 192.0 119.8 150.6 151.8 118.5 144.1 119.5 196.0 118.0 237.6 243.4 257.1 244.0 264.6 205.8 237.0 313.5 159.0 294.0 342.0
218.6 166.3 160.5 157.3 250.4 212.5 215.0 149.0 174.6 175.0 141.6 167.5 141.2 219.0 139.5 262.5 269.3 282.0 268.6 290.0 229.5 260.8 340.0 182.0 324.0 361.0
87.5 −64.5
237.0 96.3 73.4 66.2 69.2 24.5 21.2 −48.3 −90.1 176.0 82.2 109.5 +11.3 105.9 109.6 99.8 −19.4 −28.0 76.0 −58.6 79.0
251.5 111.8 89.4 82.0 84.8 37.7 35.0 −38.2 −81.6 195.5 97.3 125.5 24.4 123.8 127.5 118.0 −7.9 −17.0 91.5 −48.8 94.8
260.3 121.5 99.5 92.2 95.2 47.8 43.9 −31.8 −76.1 207.5 106.9 135.8 32.6 135.0 139.0 129.0 −0.5 −10.0 101.5 −42.6 105.0
272.5 134.0 112.9 105.0 108.4 60.2 56.0 −23.0 −68.6 223.8 119.7 149.6 44.1 149.8 153.3 144.0 +9.5 −0.2 114.5 −33.9 118.2
290.0 152.3 133.4 125.9 128.3 79.7 74.0 −10.0 −57.0 248.0 139.0 170.0 61.0 172.0 176.0 166.2 24.6 +14.3 134.0 −20.9 138.3
309.8 173.7 155.8 149.0 150.2 100.8 94.2 +5.0 −43.9 275.5 159.8 192.0 80.3 197.0 201.7 191.5 41.0 30.8 155.4 −6.2 160.2
330.0 194.4 179.0 173.0 173.9 123.5 116.0 20.9 −29.8 304.0 182.7 215.0 100.6 222.1 226.6 216.3 59.0 47.8 178.5 +8.9 184.0
92.4 14.6 18.7 44.8 107.7 115.5
109.5 28.2 32.0 58.6 123.4 131.6
120.0 37.0 40.2 67.5 133.5 141.8
134.2 48.2 51.7 78.8 146.0 154.6
155.5 65.8 68.9 96.1 165.2 175.5
180.2 85.4 87.8 115.4 187.5 197.7
205.5 106.0 108.0 135.0 210.0 220.0
Temperature, °C
Formula C10H18 C10H18 C10H22 C10H20O C10H20 C10H22O C13H30Si C8H8O4 C14H12O C4H7NO2 C4H2 C6H10Cl2Si C6H10S C10H22O C12H22O4 C10H22S C14H15N C15H14O
40
+7 60 78.5 −34.9 −83
−26 34.5
−34.5
−70.3 −55.5 −34.4 −70
−79.7 73.5
9.7 −17.6 −24.2 53.0 −80.4 −112
−40.8 −76.4 −61.2 −80.5 −50.0 −135 −53.0
45.0
2-66
PHYSICAL AND CHEMICAL DATA
TABLE 2-8
Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name
α,α-Dichlorophenylacetonitrile Dichlorophenylarsine 1,2-Dichloropropane 2,3-Dichlorostyrene 2,4-Dichlorostyrene 2,5-Dichlorostyrene 2,6-Dichlorostyrene 3,4-Dichlorostyrene 3,5-Dichlorostyrene 1,2-Dichlorotetraethylbenzene 1,4-Dichlorotetraethylbenzene 1,2-Dichloro-1,1,2,2-tetrafluoroethane Dichloro-4-tolylsilane 3,4-Dichloro-α,α,α-trifluorotoluene Dicyclopentadiene Diethoxydimethylsilane Diethoxydiphenylsilane Diethyl adipate Diethylamine N-Diethylaniline Diethyl arsanilate 1,2-Diethylbenzene 1,3-Diethylbenzene 1,4-Diethylbenzene Diethyl carbonate cis-Diethyl citraconate Diethyl dioxosuccinate Diethylene glycol Diethyleneglycol-bis-chloroacetate Diethylene glycol dimethyl ether Di(2-methoxyethyl) ether glycol ethyl ether Diethyl ether ethylmalonate fumarate glutarate Diethylhexadecylamine Diethyl itaconate ketone (3-pentanone) malate maleate malonate mesaconate oxalate phthalate sebacate 2,5-Diethylstyrene Diethyl succinate isosuccinate sulfate sulfide sulfite d-Diethyl tartrate dl-Diethyl tartrate 3,5-Diethyltoluene Diethylzinc 1-Dihydrocarvone Dihydrocitronellol 1,4-Dihydroxyanthraquinone Dimethylacetylene (2-butyne) Dimethylamine N,N-Dimethylaniline Dimethyl arsanilate Di(α-methylbenzyl) ether 2,2-Dimethylbutane 2,3-Dimethylbutane Dimethyl citraconate 1,1-Dimethylcyclohexane cis-1,2-Dimethylcyclohexane trans-1,2-Dimethylcyclohexane trans-1,3-Dimethylcyclohexane cis-1,3-Dimethylcyclohexane cis-1,4-Dimethylcyclohexane trans-1,4-Dimethylcyclohexane
1
5
10
20
56.0 61.8 −38.5 61.0 53.5 55.5 47.8 57.2 53.5 105.6 91.7 −95.4 46.2 11.0 −19.1 111.5 74.0
84.0 100.0 −17.0 90.1 82.2 83.9 75.7 86.0 82.2 138.7 126.1 −80.0 71.7 38.3 34.1 +2.4 142.8 106.6
49.7
78.0
98.1 116.0 −6.1 104.6 97.4 98.2 90.0 100.4 97.4 155.0 143.8 −72.3 84.2 52.2 47.6 13.3 157.6 123.0 −33.0 91.9
113.8 133.1 +6.0 120.5 111.8 114.0 105.5 116.2 111.8 172.5 162.0 −63.5 97.8 67.3 62.0 25.3 174.3 138.3 −22.6 107.2
130.0 151.0 19.4 137.8 129.2 131.0 122.4 133.7 129.2 192.2 183.2 −53.7 113.2 84.0 77.9 38.0 193.2 154.6 −11.3 123.6
38.0 22.3 20.7 20.7 −10.1 59.8 70.0 91.8 148.3
62.6 48.7 46.8 47.1 +12.3 88.3 98.0 120.0 180.0
74.8 62.0 59.9 60.3 23.8 103.0 112.0 133.8 195.8
88.0 76.4 74.5 74.7 36.0 118.2 126.8 148.0 212.0
C6H14O3 C6H14O3 C4H10O C9H16O4 C8H12O4 C9H16O4 C20H43N C9H14O4 C5H10O C8H14O5 C8H12O4 C7H12O4 C9H14O4 C6H10O4 C12H14O4 C14H26O4 C12H16 C8H14O4 C8H14O4 C4H10O4S C4H10S C4H10O3S C8H14O6 C8H14O6 C11H16 C4H10Zn C10H16O C10H22O C14H8O4 C4H6 C2H7N C8H11N C8H12AsNO3 C16H18O C6H14 C6H14 C7H10O4 C8H16 C8H16 C8H16 C8H16 C8H16 C8H16 C8H16
13.0 45.3 −74.3 50.8 53.2 65.6 139.8 51.3 −12.7 80.7 57.3 40.0 62.8 47.4 108.8 125.3 49.7 54.6 39.8 47.0 −39.6 10.0 102.0 100.0 34.0 −22.4 46.6 68.0 196.7 −73.0 −87.7 29.5 15.0 96.7 −69.3 −63.6 50.8 −24.4 −15.9 −21.1 −19.4 −22.7 −20.0 −24.3
37.6 72.0 −56.9 77.8 81.2 94.7 175.8 80.2 +7.5 110.4 85.6 67.5 91.0 71.8 140.7 156.2 78.4 83.0 66.7 74.0 −18.6 34.2 133.0 131.7 61.5 0.0 75.5 91.7 239.8 −57.9 −72.2 56.3 39.6 128.3 −50.7 −44.5 78.2 −1.4 +7.3 +1.7 +3.4 0.0 +3.2 −1.7
50.0 85.8 −48.1 91.6 95.3 109.7 194.0 95.2 17.2 125.3 100.0 81.3 105.3 83.8 156.0 172.1 92.6 96.6 80.0 87.7 −8.0 46.4 148.0 147.2 75.3 +11.7 90.0 103.0 259.8 −50.5 −64.6 70.0 51.8 144.0 −41.5 −34.9 91.8 +10.3 18.4 13.0 14.9 +11.2 14.5 +10.1
63.0 100.3 −38.5 106.0 110.2 125.4 213.5 111.0 27.9 141.2 115.3 95.9 120.3 96.8 173.6 189.8 108.5 111.7 94.7 102.1 +3.5 59.7 164.2 163.8 90.2 24.2 106.0 115.0 282.0 −42.5 −56.0 84.8 65.0 160.3 −31.1 −24.1 106.5 23.0 31.1 25.6 27.4 23.6 27.1 22.6
60
100
200
400
760
141.0 163.2 28.0 149.0 140.0 142.0 133.3 144.6 140.0 204.8 195.8 −47.5 122.6 95.0 88.0 46.3 205.0 165.8 −4.0 133.8
154.5 178.9 39.4 163.5 153.8 155.8 147.6 158.2 153.8 220.7 212.0 −39.1 135.5 109.2 101.7 57.6 220.0 179.0 +6.0 147.3
176.2 202.8 57.0 185.7 176.0 178.0 169.0 181.5 176.0 245.6 238.5 −26.3 153.5 129.0 121.8 74.2 243.8 198.2 21.0 168.2
199.5 228.8 76.0 210.0 200.0 202.5 193.5 205.7 200.0 272.8 265.8 −12.0 175.2 150.5 144.2 93.2 259.7 219.1 38.0 192.4
223.5 256.5 96.8 235.0 225.0 227.0 217.0 230.0 225.0 302.0 296.5 +3.5 196.3 172.8 166.6 113.5 296.0 240.0 55.5 215.5
102.6 92.5 90.4 91.1 49.5 135.7 143.8 164.3 229.0
111.8 102.6 100.7 101.3 57.9 146.2 153.7 174.0 239.5
123.8 116.2 114.4 115.3 69.7 160.0 167.7 187.5 252.0
141.9 136.7 134.8 136.1 86.5 182.3 188.0 207.0 271.5
161.0 159.0 156.9 159.0 105.8 206.5 210.8 226.5 291.8
181.0 183.5 181.1 183.8 125.8 230.3 233.5 244.8 313.0
77.5 116.7 27.7 122.4 126.7 142.8 235.0 128.2 39.4 157.8 131.8 113.3 137.3 110.6 192.1 207.5 125.8 127.8 111.0 118.0 16.1 74.2 182.3 181.7 107.0 38.0 123.7 127.6 307.4 −33.9 −46.7 101.6 79.7 179.6 −19.5 −12.4 122.6 37.3 45.3 39.7 41.4 37.5 41.1 36.5
86.8 126.8 −21.8 132.4 137.7 153.2 248.5 139.9 46.7 169.0 142.4 123.0 147.9 119.7 204.1 218.4 136.8 138.2 121.4 128.6 24.2 83.8 194.0 193.2 117.7 47.2 134.7 136.7 323.3 −27.8 −40.7 111.9 88.6 191.5 −12.1 −4.9 132.7 45.7 54.4 48.7 50.4 46.4 50.1 45.4
99.5 140.3 −11.5 146.0 151.1 167.8 265.5 154.3 56.2 183.9 156.0 136.2 161.6 130.8 219.5 234.4 151.0 151.1 134.8 142.5 35.0 96.3 208.5 208.0 131.7 59.1 149.7 145.9 344.5 −18.8 −32.6 125.8 101.0 206.8 −2.0 +5.4 145.8 57.9 66.8 61.0 62.5 58.5 62.3 57.6
118.0 159.0 +2.2 166.0 172.2 189.5 292.8 177.5 70.6 205.3 177.8 155.5 183.2 147.9 243.0 255.8 173.2 171.7 155.1 162.5 51.3 115.8 230.4 230.0 152.4 77.0 171.8 160.2 377.8 −5.0 −20.4 146.5 119.8 229.7 +13.4 21.1 165.8 76.2 85.6 79.6 81.0 76.9 80.8 76.0
138.5 180.3 17.9 188.7 195.8 212.8 324.6 203.1 86.3 229.5 201.7 176.8 205.8 166.2 267.5 280.3 198.0 193.8 177.7 185.5 69.7 137.0 254.8 254.3 176.5 97.3 197.0 176.8 413.0 +10.6 −7.1 169.2 140.3 254.8 31.0 39.0 188.0 97.2 107.0 100.9 102.1 97.8 101.9 97.0
159.8 201.9 34.6 211.5 218.5 237.0 355.0 227.9 102.7 253.4 225.0 198.9 229.0 185.7 294.0 305.5 223.0 216.5 201.3 209.5 88.0 159.0 280.0 280.0 200.7 118.0 223.0 193.5 450.0 27.2 +7.4 193.1 160.5 281.0 49.7 58.0 210.5 119.5 129.7 123.4 124.4 120.1 124.3 119.3
Temperature, °C
Formula C8H5Cl2N C6H5AsCl2 C3H6Cl2 C8H6Cl2 C8H6Cl2 C8H6Cl2 C8H6Cl2 C8H6Cl2 C8H6Cl2 C14H20Cl2 C14H20Cl2 C2Cl2F4 C7H8Cl2Si C7H3Cl2F3 C10H8 C6H16O2Si C16H20O2Si C10H18O4 C4H11N C10H15N C10H16As NO3 C10H14 C10H14 C10H14 C5H10O3 C9H14O4 C8H10O6 C4H10O3 C8H12Cl2O5
40
Melting point, °C
−94 −12.1 32.9 −21 −38.9 −34.4 −31.4 −83.9 −43.2 −43
−116.3 +0.6
−42 −49.8 −40.6 1.3 −20.8 −25.0 −99.5 17 −28 194 −32.5 −96 +2.5 −99.8 −128.2 −34 −50.0 −88.0 −92.0 −76.2 −87.4 −36.9
VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8
2-67
Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name
Dimethyl ether 2,2-Dimethylhexane 2,3-Dimethylhexane 2,4-Dimethylhexane 2,5-Dimethylhexane 3,3-Dimethylhexane 3,4-Dimethylhexane Dimethyl itaconate 1-Dimethyl malate Dimethyl maleate malonate trans-Dimethyl mesaconate 2,7-Dimethyloctane Dimethyl oxalate 2,2-Dimethylpentane 2,3-Dimethylpentane 2,4-Dimethylpentane 3,3-Dimethylpentane 2,3-Dimethylphenol (2,3-xylenol) 2,4-Dimethylphenol (2,4-xylenol) 2,5-Dimethylphenol (2,5-xylenol) 3,4-Dimethylphenol (3,4-xylenol) 3,5-Dimethylphenol (3,5-xylenol) Dimethylphenylsilane Dimethyl phthalate 3,5-Dimethyl-1,2-pyrone 4,6-Dimethylresorcinol Dimethyl sebacate 2,4-Dimethylstyrene 2,5-Dimethylstyrene α,α-Dimethylsuccinic anhydride Dimethyl sulfide d-Dimethyl tartrate dl-Dimethyl tartrate N,N-Dimethyl-2-toluidine N,N-Dimethyl-4-toluidine Di(nitrosomethyl) amine Diosphenol 1,4-Dioxane Dipentene Diphenylamine Diphenyl carbinol (benzhydrol) chlorophosphate disulfide 1,2-Diphenylethane (dibenzyl) Diphenyl ether 1,1-Diphenylethylene trans-Diphenylethylene 1,1-Diphenylhydrazine Diphenylmethane Diphenyl sulfide Diphenyl-2-tolyl thiophosphate 1,2-Dipropoxyethane 1,2-Diisopropylbenzene 1,3-Diisopropylbenzene Dipropylene glycol Dipropyleneglycol monobutyl ether isopropyl ether Di-n-propyl ether Diisopropyl ether Di-n-propyl ketone (4-heptanone) Di-n-propyl oxalate Diisopropyl oxalate Di-n-propyl succinate Di-n-propyl d-tartrate Diisopropyl d-tartrate Divinyl acetylene (1,5-hexadiene-3-yne) 1,3-Divinylbenzene Docosane n-Dodecane 1-Dodecene n-Dodecyl alcohol Dodecylamine Dodecyltrimethylsilane Elaidic acid
1
5
10
20
−115.7 −29.7 −23.0 −26.9 −26.7 −25.8 −22.1 69.3 75.4 45.7 35.0 46.8 +6.3 20.0 −49.0 −42.0 −48.0 −45.9 56.0 51.8 51.8 66.2 62.0 +5.3 100.3 78.6 49.0 104.0 34.2 29.0 61.4 −75.6 102.1 100.4 28.8 50.1 +3.2 66.7 −35.8 14.0 108.3 110.0 121.5 131.6 86.8 66.1 87.4 113.2 126.0 76.0 96.1 159.7 −38.8 40.0 34.7 73.8 64.7 46.0 −43.3 −57.0 23.0 53.4 43.2 77.5 115.6 103.7 −45.1 32.7 157.8 47.8 47.2 91.0 82.8 91.2 171.3
−101.1 −7.9 −1.1 −5.3 −5.5 −4.4 +0.2 94.0 104.0 73.0 59.8 74.0 30.5 44.0 −28.7 −20.8 −27.4 −25.0 83.8 78.0 78.0 93.8 89.2 30.3 131.8 107.6 76.8 139.8 61.9 55.9 88.1 −58.0 133.2 131.8 54.1 74.3 27.8 95.4 −12.8 40.4 141.7 145.0 160.5 164.0 119.8 97.8 119.6 145.8 159.3 107.4 129.0 179.8 −10.3 67.8 62.3 102.1 92.0 72.8 −22.3 −37.4 44.4 80.2 69.0 107.6 147.7 133.7 −24.4 60.0 195.4 75.8 74.0 120.2 111.8 122.1 206.7
−93.3 +3.1 +9.9 +5.2 +5.3 +6.1 11.3 106.6 118.3 86.4 72.0 87.8 42.3 56.0 −18.7 −10.3 −17.1 −14.4 97.6 91.3 91.3 107.7 102.4 42.6 147.6 122.0 90.7 156.2 75.8 69.0 102.0 −49.2 148.2 147.5 66.2 86.7 40.0 109.0 −1.2 53.8 157.0 162.0 182.0 180.0 136.0 114.0 135.0 161.0 176.1 122.8 145.0 201.6 +5.0 81.8 76.0 116.2 106.0 86.2 −11.8 −27.4 55.0 93.9 81.9 122.2 163.5 148.2 −14.0 73.8 213.0 90.0 87.8 134.7 127.8 137.7 223.5
−85.2 15.0 22.1 17.2 17.2 18.2 23.5 119.7 133.8 101.3 85.0 102.1 55.8 69.4 −7.5 +1.1 −5.9 −2.9 112.0 105.0 105.0 122.0 117.0 56.2 164.0 136.4 105.8 175.8 90.8 84.0 116.3 −39.4 164.3 164.0 80.2 100.0 53.7 124.0 +12.0 68.2 175.2 180.9 203.8 197.0 153.7 130.8 151.8 179.8 194.0 139.8 162.0 215.5 22.3 96.8 91.2 131.3 120.4 100.8 0.0 −16.7 66.2 108.6 95.6 138.0 180.4 164.0 −2.8 88.7 233.5 104.6 102.4 150.0 141.6 153.8 242.3
60
100
200
400
760
−62.7 48.2 56.0 50.6 50.5 52.5 57.7 153.7 175.1 140.4 121.9 141.5 93.9 104.8 23.9 33.3 25.4 29.3 152.2 143.0 143.0 161.0 156.0 94.2 210.0 177.5 147.3 222.6 132.3 124.7 155.3 −12.0 208.8 209.5 118.1 140.3 90.3 165.6 45.1 108.3 222.8 227.5 265.0 241.3 202.8 178.8 198.6 227.4 242.5 186.3 211.8 252.5 74.2 138.7 132.3 169.9 159.8 140.3 33.0 13.7 96.0 148.1 132.6 180.3 227.0 207.3 29.5 130.0 286.0 146.2 142.3 192.0 182.1 199.5 288.0
−50.9 65.7 73.8 68.1 68.0 70.0 75.6 171.0 196.3 160.0 140.0 161.0 114.0 123.3 40.3 50.1 41.8 46.2 173.0 161.5 161.5 181.5 176.2 114.2 232.7 198.0 167.8 245.0 153.2 145.6 175.8 +2.6 230.5 232.3 138.3 161.6 110.0 186.2 62.3 128.2 247.5 250.0 299.5 262.6 227.8 203.3 222.8 251.7 267.2 210.7 236.8 270.3 103.8 159.8 153.7 189.9 180.0 160.0 50.3 30.0 111.2 168.0 151.2 202.5 250.1 228.2 46.0 151.4 314.2 167.2 162.2 213.0 203.0 222.0 312.4
−37.8 85.6 94.1 88.2 87.9 90.4 96.0 189.8 219.5 182.2 159.8 183.5 136.0 143.3 59.2 69.4 60.6 65.5 196.0 184.2 184.2 203.6 197.8 136.4 257.8 221.0 192.0 269.6 177.5 168.7 197.5 18.7 255.0 257.4 161.5 185.4 131.3 209.5 81.8 150.5 274.1 275.6 337.2 285.8 255.0 230.7 249.8 278.3 294.0 237.5 263.9 290.0 140.0 184.3 177.6 210.5 203.8 183.1 69.5 48.2 127.3 190.3 171.8 226.5 275.6 251.8 64.4 175.2 343.5 191.0 185.5 235.7 225.0 248.0 337.0
−23.7 106.8 115.6 109.4 109.1 112.0 117.7 208.0 242.6 205.0 180.7 206.0 159.7 163.3 79.2 89.8 80.5 86.1 218.0 211.5 211.5 225.2 219.5 159.3 283.7 245.0 215.0 293.5 202.0 193.0 219.5 36.0 280.0 282.0 184.8 209.5 153.0 232.0 101.1 174.6 302.0 301.0 378.0 310.0 284.0 258.5 277.0 306.5 322.2 264.5 292.5 310.0 180.0 209.0 202.0 231.8 227.0 205.6 89.5 67.5 143.7 213.5 193.5 250.8 303.0 275.0 84.0 199.5 376.0 216.2 208.0 259.0 248.0 273.0 362.0
Temperature, °C
Formula C2H6O C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C7H10O4 C6H10O5 C6H8O4 C5H8O4 C7H10O4 C10H22 C4H6O4 C7H16 C7H16 C7H16 C7H16 C8H10O C8H10O C8H10O C8H10O C8H10O C8H12Si C10H10O4 C7H8O2 C8H10O2 C12H22O4 C10H12 C10H12 C6H8O3 C2H6S C6H10O6 C6H10O6 C9H13N C9H13N C2H5N3O2 C10H16O2 C4H8O2 C10H16 C12H11N C13H12O C12H10ClPO3 C12H10S2 C14H14 C12H10O C14H12 C14H12 C12H12N2 C13H12 C12H10S C18H17O3PS C8H18O2 C12H18 C12H18 C6H14O3 C10H22O3 C9H20O3 C6H14O C6H14O C7H14O C8H14O4 C8H14O4 C10H18O4 C10H18O6 C10H18O6 C6H6 C10H10 C22H46 C12H26 C12H24 C12H26O C12H27N C15H34Si C18H34O2
40 −76.2 28.2 35.6 30.5 30.4 31.7 37.1 133.7 150.1 117.2 100.0 118.0 71.2 83.6 +5.0 13.9 +6.5 +9.9 129.2 121.5 121.5 138.0 133.3 71.4 182.8 152.7 122.5 196.0 107.7 100.2 132.3 −28.4 182.4 182.4 95.0 116.3 68.2 141.2 25.2 84.3 194.3 200.0 227.9 214.8 173.7 150.0 170.8 199.0 213.5 157.8 182.8 230.6 42.3 114.0 107.9 147.4 136.3 117.0 +13.2 −4.5 78.1 124.6 110.5 154.8 199.7 181.8 +10.0 105.5 254.5 121.7 118.6 167.2 157.4 172.1 260.8
−70.4 36.7 44.2 39.0 38.9 40.4 45.8 142.6 160.4 127.1 109.7 127.8 80.8 92.8 13.0 22.1 14.5 18.1 139.5 131.0 131.0 148.0 143.5 81.3 194.0 163.8 133.2 208.0 118.0 110.7 142.4 −21.4 193.8 193.8 105.2 126.4 77.7 151.3 33.8 94.6 206.9 212.0 244.2 226.2 186.0 162.0 183.4 211.5 225.9 170.2 194.8 240.4 55.8 124.3 118.2 156.5 146.3 126.8 21.6 +3.4 85.8 134.8 120.0 166.0 211.7 192.6 18.1 116.0 268.3 132.1 128.5 177.8 168.0 184.2 273.0
Melting point, °C −138.5
−90.7 38 −62 −52.8 −123.7 −135 −119.5 −135.0 75 25.5 74.5 62.5 68 51.5 38
−83.2 61.5 89 −61
10 52.9 68.5 61 51.5 27 124 44 26.5
−105
−122 −60 −32.6
−66.9 44.5 −9.6 −31.5 24 51.5
2-68
PHYSICAL AND CHEMICAL DATA
TABLE 2-8
Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound
1
5
10
20
−16.5 −69.0 206.7 52.6 −159.5 36.3 −50.9 167.0 −43.4 28.5 −92.5 −29.5 47.0 −29.0 −31.3 −82.3 52.0 38.5 29.7 33.7 33.5 −9.8 44.0 107.6 −74.3 10.6 −18.4 −24.3 118.2 11.0 107.8 133.2 −89.8 +1.0 −5.1 +6.6 87.6 28.3 31.5 67.8 −14.5 −32.2 9.6 76.0 98.3 −168.3 112.0 −4.0 −11.0 −27.0 −44.5 53.0 −33.5
+5.6 −50.0 239.7 80.0 −148.5 63.1 −31.0 198.2 −23.5 54.0 −76.7 −8.7 70.7 −6.4 −12.0 −66.4 80.0 66.4 55.9 60.3 60.2 +13.9 72.0 136.4 −56.4 35.8 +4.0 −2.4 149.8 35.8 65.8 131.8 168.2 −73.9 25.4 +18.0 30.2 108.5 55.5 58.4 93.5 +9.2 −10.8 34.0 106.3 130.2 −158.3 142.4 +19.0 +10.5 +4.7 −24.0 79.7 −10.2
16.6 −40.3 254.5 93.7 −142.9 76.2 −20.7 213.5 −13.5 67.3 −68.7 +2.0 82.0 +5.0 −2.3 −58.3 93.8 80.6 69.0 73.9 73.9 25.9 86.0 150.3 −47.5 48.0 15.3 +8.4 165.0 48.0 77.8 143.7 186.0 −65.8 37.5 29.9 41.9 134.0 68.8 72.0 106.0 20.6 −0.1 46.3 121.7 146.0 −153.2 158.0 30.3 21.5 18.6 −13.6 92.1 +1.6
29.0 −29.5 270.6 108.4 −136.7 91.0 −9.8 230.0 −3.0 81.1 −59.9 13.0 94.4 17.7 +8.0 −48.6 109.0 96.0 83.1 88.5 88.5 38.6 101.4 166.8 −37.8 61.8 27.8 20.6 181.8 61.7 91.0 155.5 205.5 −56.8 50.4 42.0 54.3 150.3 83.6 86.7 119.8 33.4 +11.7 59.5 137.7 162.8 −147.6 173.5 42.5 33.0 32.7 −2.4 105.8 14.7
42.0 −17.3 289.1 124.6 −129.8 107.2 +3.7 247.0 +9.1 96.2 −50.0 26.0 108.1 31.8 19.0 −39.8 125.7 113.2 98.8 104.8 104.7 52.8 118.2 181.8 −26.7 77.0 41.5 33.8 199.8 76.3 105.6 168.8 226.5 −47.0 65.2 56.0 68.2 169.2 99.9 103.3 133.8 47.6 25.0 74.0 154.4 182.0 −141.3 191.0 56.0 45.8 48.0 +10.0 120.0 29.7
50.6 −9.7 300.2 135.2 −125.4 127.5 11.5 258.3 16.6 106.0 −43.4 33.5 116.7 40.6 26.0 −33.4 136.0 123.6 109.0 115.5 115.4 61.8 129.0 191.9 −19.5 86.7 50.1 42.3 211.5 85.8 114.8 177.3 239.8 −40.6 74.0 65.2 77.3 181.2 110.2 113.8 142.1 56.7 33.4 83.6 166.0 193.7 −137.3 201.8 64.1 53.8 57.9 18.1 129.5 39.0
62.0 +1.2 314.4 148.5 −119.3 131.4 22.1 273.5 27.0 118.5 −34.9 44.5 127.5 53.0 34.9 −25.1 149.8 137.3 122.3 129.2 128.4 74.1 143.2 205.0 −10.0 99.8 62.0 53.5 226.6 98.4 126.2 187.9 256.8 −32.0 86.0 76.6 89.3 196.0 124.3 127.2 152.8 69.0 45.0 96.1 180.3 209.8 −131.8 215.0 75.0 62.5 70.4 29.4 141.8 51.8
C4H10O2
−48.0
−26.2
−15.3
−3.0
+10.7
19.7
31.8
50.0
70.8
93.0
C3H8O2
−13.5
+10.2
22.0
34.3
47.8
56.4
68.0
85.3
104.3
124.4
−89.7 40.5 −117.0 −60.5 37.6 14.3 −20.0 50.0 −60.7 −112.5 −54.4 27.8 47.3 −76.7 26.5 −91.0
−73.8 67.3 −103.8 −42.2 63.8 38.8 +2.1 77.7 −41.9 −98.4 −34.3 57.3 74.0 −59.1 51.0 −75.6
−65.7 80.2 −97.7 −33.0 77.1 50.5 12.8 91.8 −32.3 −91.7 −24.3 72.1 87.3 −50.2 63.2 −67.8
−56.6 94.6 −90.0 −22.7 91.5 63.9 25.0 106.3 −21.9 −84.1 −13.1 88.0 101.8 −40.7 76.1 −59.1
−46.9 110.3 −81.8 −11.5 107.5 78.1 38.5 123.7 −10.2 −75.8 −0.9 106.0 117.7 −29.8 91.0 −49.4
−40.7 120.6 −76.4 −4.3 117.5 87.6 47.1 134.0 −2.9 −70.4 +7.2 117.8 127.6 −22.4 100.0 −43.3
−32.1 133.8 −69.3 −5.4 130.4 99.8 58.9 147.9 +7.2 −63.2 18.0 131.8 141.3 −13.0 112.0 −34.8
−19.5 153.2 −58.0 20.0 150.1 117.8 76.7 168.2 22.4 −52.0 34.1 149.8 160.2 +1.5 130.0 −22.0
−4.9 175.6 −45.5 37.1 172.5 138.0 97.0 192.2 39.8 −39.5 52.3 167.3 183.0 17.7 149.8 −7.8
+10.7 198.0 −32.0 54.3 195.0 158.2 118.5 216.0 57.4 −26.5 72.4 184.0 206.2 35.0 170.0 +7.5
Name
Formula
Epichlorohydrin 1,2-Epoxy-2-methylpropane Erucic acid Estragole (p-methoxy allyl benzene) Ethane Ethoxydimethylphenylsilane Ethoxytrimethylsilane Ethoxytriphenylsilane Ethyl acetate acetoacetate Ethylacetylene (1-butyne) Ethyl acrylate α-Ethylacrylic acid α-Ethylacrylonitrile Ethyl alcohol (ethanol) Ethylamine 4-Ethylaniline N-Ethylaniline 2-Ethylanisole 3-Ethylanisole 4-Ethylanisole Ethylbenzene Ethyl benzoate benzoylacetate bromide α-bromoisobutyrate n-butyrate isobutyrate Ethylcamphoronic anhydride Ethyl isocaproate carbamate carbanilate Ethylcetylamine Ethyl chloride chloroacetate chloroglyoxylate α-chloropropionate trans-cinnamate 3-Ethylcumene 4-Ethylcumene Ethyl cyanoacetate Ethylcyclohexane Ethylcyclopentane Ethyl dichloroacetate N,N-diethyloxamate N-Ethyldiphenylamine Ethylene Ethylene-bis-(chloroacetate) Ethylene chlorohydrin (2-chloroethanol) diamine (1,2-ethanediamine) dibromide (1,2-dibromethane) dichloride (1,2-dichloroethane) glycol (1,2-ethanediol) glycol diethyl ether (1,2-diethoxyethane) glycol dimethyl ether (1,2-dimethoxyethane) glycol monomethyl ether (2-methoxyethanol) oxide Ethyl α-ethylacetoacetate fluoride formate 2-furoate glycolate 3-Ethylhexane 2-Ethylhexyl acrylate Ethylidene chloride (1,1-dichloroethane) fluoride (1,1-difluoroethane) Ethyl iodide Ethyl l-leucinate Ethyl levulinate Ethyl mercaptan (ethanethiol) Ethyl methylcarbamate Ethyl methyl ether
C3H5ClO C4H8O C22H42O2 C10H12O C2H6 C10H16OSi C5H14OSi C20H20OSi C4H8O2 C6H10O3 C4H6 C5H8O2 C5H8O2 C5H7N C2H6O C2H7N C8H11N C8H11N C9H12O C9H12O C9H12O C8H10 C9H10O2 C11H12O3 C2H5Br C6H11BrO2 C6H12O2 C6H12O2 C11H16O5 C8H16O2 C3H7NO2 C9H11NO2 C18H39N C2H5Cl C4H7ClO2 C4H5ClO3 C5H9ClO2 C11H12O2 C11H16 C11H16 C5H7NO2 C8H16 C7H14 C4H6Cl2O2 C8H15NO3 C14H15N C2H4 C6H8Cl2O4 C2H5ClO C2H8N2 C2H4Br2 C2H4Cl2 C2H6O2 C6H14O2
C2H4O C8H14O3 C2H5F C3H6O2 C7H8O3 C4H8O3 C8H18 C11H20O2 C2H4Cl2 C2H4F2 C2H5I C8H17NO2 C7H12O3 C2H6S C4H9NO2 C3H8O
40
60
100
200
400
760
Temperature, °C 79.3 98.0 117.9 17.5 36.0 55.5 336.5 358.8 381.5 168.7 192.0 215.0 −110.2 −99.7 −88.6 151.5 175.0 199.5 38.1 56.3 75.7 295.0 319.5 344.0 42.0 59.3 77.1 138.0 158.2 180.8 −21.6 −6.9 +8.7 61.5 80.0 99.5 144.0 160.7 179.2 71.6 92.2 114.0 48.4 63.5 78.4 −12.3 +2.0 16.6 170.6 194.2 217.4 156.9 180.8 204.0 142.1 164.2 187.1 149.7 172.8 196.5 149.2 172.3 196.5 92.7 113.8 136.2 164.8 188.4 213.4 223.8 244.7 265.0 +4.5 21.0 38.4 119.7 141.2 163.6 79.8 100.0 121.0 71.0 90.0 110.0 248.5 272.8 298.0 117.8 139.2 160.4 144.2 164.0 184.0 203.8 220.0 237.0 283.3 313.0 342.0 −18.6 −3.9 +12.3 103.8 123.8 144.2 94.5 114.7 135.0 107.2 126.2 146.5 219.3 245.0 271.0 145.4 168.2 193.0 148.3 171.8 195.8 169.8 187.8 206.0 87.8 109.1 131.8 62.4 82.3 103.4 115.2 135.9 156.5 202.8 226.5 252.0 233.0 258.8 286.0 −123.4 −113.9 −103.7 237.3 259.5 283.5 91.8 110.0 128.8 81.0 99.0 117.2 89.8 110.1 131.5 45.7 64.0 82.4 158.5 178.5 197.3 71.8 94.1 119.5
Melting point, °C −25.6 33.5 −183.2
−82.4 −45 −130 −71.2 −112 −80.6 −4 −63.5
−94.9 −34.6 −117.8 −93.3 −88.2 49 52.5 −139 −26 12
−111.3 −138.6
−169 −69 8.5 10 −35.3 −15.6
−111.3 −79 34
−96.7 −117 −105 −121
VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8
2-69
Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name
1-Ethylnaphthalene Ethyl α-naphthyl ketone (1-propionaphthone) Ethyl 3-nitrobenzoate 3-Ethylpentane 4-Ethylphenetole 2-Ethylphenol 3-Ethylphenol 4-Ethylphenol Ethyl phenyl ether (phenetole) Ethyl propionate Ethyl propyl ether Ethyl salicylate 3-Ethylstyrene 4-Ethylstyrene Ethylisothiocyanate 2-Ethyltoluene 3-Ethyltoluene 4-Ethyltoluene Ethyl trichloroacetate Ethyltrimethylsilane Ethyltrimethyltin Ethyl isovalerate 2-Ethyl-1,4-xylene 4-Ethyl-1,3-xylene 5-Ethyl-1,3-xylene Eugenol iso-Eugenol Eugenyl acetate Fencholic acid d-Fenchone dl-Fenchyl alcohol Fluorene Fluorobenzene 2-Fluorotoluene 3-Fluorotoluene 4-Fluorotoluene Formaldehyde Formamide Formic acid trans-Fumaryl chloride Furfural (2-furaldehyde) Furfuryl alcohol Geraniol Geranyl acetate Geranyl n-butyrate Geranyl isobutyrate Geranyl formate Glutaric acid Glutaric anhydride Glutaronitrile Glutaryl chloride Glycerol Glycerol dichlorohydrin (1,3-dichloro-2-propanol) Glycol diacetate Glycolide (1,4-dioxane-2,6-dione) Guaicol (2-methoxyphenol) Heneicosane Heptacosane Heptadecane Heptaldehyde (enanthaldehyde) n-Heptane Heptanoic acid (enanthic acid) 1-Heptanol Heptanoyl chloride (enanthyl chloride) 2-Heptene Heptylbenzene Heptyl cyanide (enanthonitrile) Hexachlorobenzene Hexachloroethane Hexacosane Hexadecane 1-Hexadecene n-Hexadecyl alcohol (cetyl alcohol)
1
5
10
20
70.0
101.4
116.8
133.8
152.0
C13H12O C9H9NO4 C7H16 C10H14O C8H10O C8H10O C8H10O C8H10O C5H10O2 C5H12O C9H10O3 C10H12 C10H12 C3H5NS C9H12 C9H12 C9H12 C4H5Cl3O2 C5H14Si C5H14Sn C7H14O2 C10H14 C10H14 C10H14 C10H12O2 C10H12O2 C12H14O3 C10H16O2 C10H16O C10H18O C13H10 C6H5F C7H7F C7H7F C7H7F CH2O CH3NO CH2O2 C4H2Cl2O2 C5H4O2 C5H6O2 C10H18O C12H20O2 C14H24O2 C14H24O2 C11H18O2 C5H8O4 C5H6O3 C5H6N2 C5H6Cl2O2 C3H8O3 C3H6Cl2O
124.0 108.1 −37.8 48.5 46.2 60.0 59.3 18.1 −28.0 −64.3 61.2 28.3 26.0 13.2 9.4 7.2 7.6 20.7 −60.6 −30.0 −6.1 25.7 26.3 22.1 78.4 86.3 101.6 101.7 28.0 45.8 −43.4 −24.2 −22.4 −21.8
155.5 140.2 −17.0 75.7 73.4 86.8 86.5 43.7 −7.2 −45.0 90.0 55.0 52.7 +10.6 34.8 32.3 32.7 45.5 −41.4 −7.6 +17.0 52.0 53.0 48.8 108.1 117.0 132.3 128.7 54.7 70.3 129.3 −22.8 −2.2 −0.3 +0.3
70.5 −20.0 +15.0 18.5 31.8 69.2 73.5 96.8 90.9 61.8 155.5 100.8 91.3 56.1 125.5 28.0
96.3 −5.0 38.5 42.6 56.0 96.8 102.7 125.2 119.6 90.3 183.8 133.3 123.7 84.0 153.8 52.2
171.0 155.0 −6.8 89.5 87.0 100.2 100.2 56.4 +3.4 −35.0 104.2 68.3 66.3 22.8 47.6 44.7 44.9 57.7 −31.8 +3.8 28.7 65.6 66.4 62.1 123.0 132.4 148.0 142.3 68.3 82.1 146.0 −12.4 +8.9 +11.0 11.8 −88.0 109.5 +2.1 51.8 54.8 68.0 110.0 117.9 139.0 133.0 104.3 196.0 149.5 140.0 97.8 167.2 64.7
188.1 173.6 +4.7 103.8 101.5 114.5 115.0 70.3 14.3 −24.0 119.3 82.8 80.8 36.1 61.2 58.2 58.5 70.6 −21.0 16.1 41.3 79.8 80.6 76.5 138.7 149.0 164.2 155.8 83.0 95.6 164.2 −1.2 21.4 23.4 24.0 −79.6 122.5 10.3 65.0 67.8 81.0 125.6 133.0 153.8 147.9 119.8 210.5 166.0 156.5 112.3 182.2 78.0
206.9 192.6 17.5 119.8 117.9 130.0 131.3 86.6 27.2 −12.0 136.7 99.2 97.3 50.8 76.4 73.3 73.6 85.5 −9.0 30.0 55.2 96.0 97.2 92.6 155.8 167.0 183.0 171.8 99.5 110.8 185.2 +11.5 34.7 37.0 37.8 −70.6 137.5 24.0 79.5 82.1 95.7 141.8 150.0 170.1 164.0 136.2 226.3 185.5 176.4 128.3 198.0 93.0
C6H10O4 C4H4O4 C7H8O2 C21H44 C27H56 C17H36 C7H14O C7H16 C7H14O2 C7H16O C7H13ClO C7H14 C13H20 C7H13N C6Cl6 C2Cl6 C26H54 C16H34 C16H32 C16H34O
38.3
64.1 103.0 79.1 188.0 248.6 145.2 32.7 −12.7 101.3 64.3 54.6 −14.1 94.6 47.8 149.3 49.8 240.0 135.2 131.7 158.3
77.1 116.6 92.0 205.4 266.8 160.0 43.0 −2.1 113.2 74.7 64.6 −3.5 110.0 61.6 166.4 73.5 257.4 149.8 146.2 177.8
90.8 132.0 106.0 223.2 284.6 177.7 54.0 +9.5 125.6 85.8 75.0 +8.3 126.0 76.3 185.7 87.6 275.8 164.7 162.0 197.8
106.1 148.6 121.6 243.4 305.7 195.8 66.3 22.3 139.5 99.8 86.4 21.5 144.0 92.6 206.0 102.3 295.2 181.3 178.8 219.8
100
200
400
760
Melting point, °C
164.1
180.0
204.6
230.8
258.1
−27
218.2 205.0 25.7 129.8 127.9 139.8 141.7 95.4 35.1 −4.0 147.6 109.6 107.6 59.8 86.0 82.9 83.2 94.4 −1.2 38.4 64.0 106.2 107.4 103.0 167.3 178.2 194.0 181.5 109.8 120.2 197.8 19.6 43.7 45.8 46.5 −65.0 147.0 32.4 89.0 91.5 104.0 151.5 160.3 180.2 174.0 147.2 235.5 196.2 189.5 139.1 208.0 102.0
233.5 220.3 36.9 143.5 141.8 152.0 154.2 108.4 45.2 +6.8 161.5 123.2 121.5 71.9 99.0 95.9 96.3 107.4 +9.2 50.0 75.9 120.0 121.2 116.5 182.2 194.0 209.7 194.0 123.6 132.3 214.7 30.4 55.3 57.5 58.1 −57.3 157.5 43.8 101.0 103.4 115.9 165.3 175.2 193.8 187.7 160.7 247.0 212.5 205.5 151.8 220.1 114.8
255.5 244.6 53.8 163.2 161.6 171.8 175.0 127.9 61.7 23.3 183.7 144.0 142.0 90.0 119.0 115.5 116.1 125.8 25.0 67.3 93.8 140.2 141.8 137.4 204.7 217.2 232.5 215.0 144.0 150.0 240.3 47.2 73.0 75.4 76.0 −46.0 175.5 61.4 120.0 121.8 133.1 185.6 196.3 214.0 207.6 182.6 265.0 236.5 230.0 172.4 240.0 133.3
280.2 270.6 73.0 185.7 184.5 193.3 197.4 149.8 79.8 41.6 207.0 167.2 165.0 110.1 141.4 137.8 136.4 146.0 42.8 87.6 114.0 163.1 164.4 159.6 228.3 242.3 257.4 237.8 166.8 173.2 268.6 65.7 92.8 95.4 96.1 −33.0 193.5 80.3 140.0 141.8 151.8 207.8 219.8 235.0 228.5 205.8 283.5 261.0 257.3 195.3 263.0 153.5
306.0 298.0 93.5 208.0 207.5 214.0 219.0 172.0 99.1 61.7 231.5 191.5 189.0 131.0 165.1 161.3 162.0 167.0 62.0 108.8 134.3 186.9 188.4 183.7 253.5 267.5 282.0 264.1 191.0 201.0 295.0 84.7 114.0 116.0 117.0 −19.5 210.5 100.6 160.0 161.8 170.0 230.0 243.3 257.4 251.0 230.0 303.0 287.0 286.2 217.0 290.0 174.3
115.8 158.2 131.0 255.3 318.3 207.3 74.0 30.6 148.5 108.0 93.5 30.0 154.8 103.0 219.0 112.0 307.8 193.2 190.8 234.3
128.0 173.2 144.0 272.0 333.5 223.0 84.0 41.8 160.0 119.5 102.7 41.3 170.2 116.8 235.5 124.2 323.2 208.5 205.3 251.7
147.8 194.0 162.7 296.5 359.4 247.8 102.0 58.7 179.5 136.6 116.3 58.6 193.3 137.7 258.5 143.1 348.4 231.7 226.8 280.2
168.3 217.0 184.1 323.8 385.0 274.5 125.5 78.0 199.6 155.6 130.7 78.1 217.8 160.0 283.5 163.8 374.6 258.3 250.0 312.7
190.5 240.0 205.0 350.5 410.6 303.0 155.0 98.4 221.5 175.8 145.0 98.5 244.0 184.6 309.4 185.6 399.8 287.5 274.0 344.0
60
Temperature, °C
Formula C12H12
40
52.4 152.6 211.7 115.0 12.0 −34.0 78.0 42.4 34.2 −35.8 64.0 21.0 114.4 32.7 204.0 105.3 101.6 122.7
47 −118.6 −45 −4 46.5 −30.2 −72.6 1.3 −5.9 −95.5
−99.3
−10 295 19 5 35 113 −42.1 −80 −110.8 −92 8.2
97.5
17.9 −31 97 28.3 40.4 59.5 22.5 −42 −90.6 −10 34.6
230 186.6 56.6 18.5 4 49.3
2-70
PHYSICAL AND CHEMICAL DATA
TABLE 2-8
Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name
Formula
n-Hexadecylamine (cetylamine) Hexaethylbenzene n-Hexane 1-Hexanol 2-Hexanol 3-Hexanol 1-Hexene n-Hexyl levulinate n-Hexyl phenyl ketone (enanthophenone) Hydrocinnamic acid Hydrogen cyanide (hydrocyanic acid) Hydroquinone 4-Hydroxybenzaldehyde α-Hydroxyisobutyric acid α-Hydroxybutyronitrile 4-Hydroxy-3-methyl-2-butanone 4-Hydroxy-4-methyl-2-pentanone 3-Hydroxypropionitrile Indene Iodobenzene Iodononane 2-Iodotoluene α-Ionone Isoprene Lauraldehyde Lauric acid Levulinaldehyde Levulinic acid d-Limonene Linalyl acetate Maleic anhydride Menthane 1-Menthol Menthyl acetate benzoate formate Mesityl oxide Methacrylic acid Methacrylonitrile Methane Methanethiol Methoxyacetic acid N-Methylacetanilide Methyl acetate acetylene (propyne) acrylate alcohol (methanol) Methylamine N-Methylaniline Methyl anthranilate benzoate 2-Methylbenzothiazole α-Methylbenzyl alcohol Methyl bromide 2-Methyl-1-butene 2-Methyl-2-butene Methyl isobutyl carbinol (2-methyl4-pentanol) n-butyl ketone (2-hexanone) isobutyl ketone (4-methyl-2-pentanone) n-butyrate isobutyrate caprate caproate caprylate chloride chloroacetate cinnamate α-Methylcinnamic acid Methylcyclohexane Methylcyclopentane Methylcyclopropane Methyl n-decyl ketone (n-dodecan-2-one) dichloroacetate N-Methyldiphenylamine
C16H35N C18H30 C6H14 C6H14O C6H14O C6H14O C6H12 C11H20O3 C13H18O C9H10O2 CHN C6H6O2 C7H6O2 C4H8O3 C5H9NO C5H10O2 C6H12O2 C3H5NO C9H8 C6H5I C9H19I C7H7I C13H20O C5H8 C12H24O C12H24O2 C5H8O2 C5H8O3 C10H16 C12H20O2 C4H2O3 C10H20 C10H20O C12H22O2 C17H24O2 C11H20O2 C6H10O C4H6O2 C4H5N CH4 CH4S C3H6O3 C9H11NO C3H6O2 C3H4 C4H6O2 CH4O CH5N C7H9N C8H9NO2 C8H8O2 C8H7NS C8H10O CH3Br C5H10 C5H10 C6H14O C6H12O C6H12O C5H10O2 C5H10O2 C11H22O2 C7H14O2 C9H18O2 CH3Cl C3H5ClO2 C10H10O2 C10H10O2 C7H14 C6H12 C4H8 C12H24O C3H4Cl2O2 C13H13N
1
5
10
20
40
60
100
123.6
157.8 134.3 −34.5 47.2 34.8 25.7 −38.0 120.0 130.3 133.5 −55.3 153.3 153.2 98.5 65.8 69.3 46.7 87.8 44.3 50.6 96.2 65.9 108.8 −62.3 108.4 150.6 54.9 128.1 40.4 82.5 63.4 35.7 83.2 85.8 154.2 75.8 +14.1 48.5 −23.3 −199.0 −75.3 79.3 103.8 −38.6 −97.5 −23.6 −25.3 −81.3 62.8 109.0 64.4 97.5 75.2 −80.6 −72.8 −57.0
176.0 150.3 −25.0 58.2 45.0 36.7 −28.1 134.7 145.5 148.7 −47.7 163.5 169.7 110.5 77.8 81.0 58.8 102.0 58.5 64.0 109.0 79.8 123.0 −53.3 123.7 166.0 68.0 141.8 53.8 96.0 78.7 48.3 96.0 100.0 170.0 90.0 26.0 60.0 −12.5 −195.5 −67.5 92.0 118.6 −29.3 −90.5 −13.5 −16.2 −73.8 76.2 124.2 77.3 111.2 88.0 −72.8 −64.3 −47.9
195.7 168.0 −14.1 70.3 55.9 49.0 −17.2 150.2 161.0 165.0 −39.7 174.6 186.8 123.8 90.7 94.0 72.0 117.9 73.9 78.3 123.0 95.6 139.0 −43.5 140.2 183.6 82.7 154.1 68.2 111.4 95.0 62.7 110.3 115.4 186.3 105.8 37.9 72.7 −0.6 −191.8 −58.8 106.5 135.1 −19.1 −82.9 −2.7 −6.0 −65.9 90.5 141.5 91.8 125.5 102.1 −64.0 −54.8 −37.9
215.7 187.7 −2.3 83.7 67.9 62.2 −5.0 167.8 178.9 183.3 −30.9 192.0 206.0 138.0 104.8 108.2 86.7 134.1 90.7 94.4 138.1 112.4 155.6 −32.6 157.8 201.4 98.3 169.5 84.3 127.7 111.8 78.3 126.1 132.1 204.3 123.0 51.7 86.4 +12.8 −187.7 −49.2 122.0 152.2 −7.9 −74.3 +9.2 +5.0 −56.9 106.0 159.7 107.8 141.2 117.8 −54.2 −44.1 −26.7
228.8 199.7 +5.4 92.0 76.0 70.7 +2.8 179.0 189.8 194.0 −25.1 203.0 217.5 146.4 113.9 117.4 96.0 144.7 100.8 105.0 147.7 123.8 166.3 −25.4 168.7 212.7 108.4 178.0 94.6 138.1 122.0 88.6 136.1 143.2 215.8 133.8 60.4 95.3 21.5 −185.1 −43.1 131.8 164.2 −0.5 −68.8 17.3 12.1 −51.3 115.8 172.0 117.4 150.4 127.4 −48.0 −37.3 −19.4
245.8 216.0 15.8 102.8 87.3 81.8 13.0 193.6 204.2 209.0 −17.8 216.5 233.5 157.7 125.0 129.0 108.2 157.7 114.7 118.3 159.8 138.1 181.2 −16.0 184.5 227.5 121.8 190.2 108.3 151.8 135.8 102.1 149.4 156.7 230.4 148.0 72.1 106.6 32.8 −181.4 −34.8 144.5 179.8 +9.4 −61.3 28.0 21.2 −43.7 129.8 187.8 130.8 163.9 140.3 −39.4 −28.0 −9.9
+22.1 28.8 +19.7 −5.5 −13.0 93.5 30.0 61.7 −99.5 19.0 108.1 155.0 −14.0 −33.8 −80.6 106.0 26.7 134.0
33.3 38.8 30.0 +5.0 −2.9 108.0 42.0 74.9 −92.4 30.0 123.0 169.8 −3.2 −23.7 −72.8 120.4 38.1 149.7
45.4 50.0 40.8 16.7 +8.4 123.0 55.4 89.0 −84.8 41.5 140.0 185.2 +8.7 −12.8 −64.0 136.0 50.7 165.8
58.2 62.0 52.8 29.6 21.0 139.0 70.0 105.3 −76.0 54.5 157.9 201.8 22.0 −0.6 −54.2 152.4 64.7 184.0
67.0 69.8 60.4 37.4 28.9 148.6 79.7 115.3 −70.4 63.0 170.0 212.0 30.5 +7.2 −48.0 163.8 73.6 195.4
78.0 79.8 70.4 48.0 39.6 161.5 91.4 128.0 −63.0 73.5 185.8 224.8 42.1 17.9 −39.3 177.5 85.4 210.1
200
400
760
Temperature, °C
−53.9 24.4 14.6 +2.5 −57.5 90.0 100.0 102.2 −71.0 132.4 121.2 73.5 41.0 44.6 22.0 58.7 16.4 24.1 70.0 37.2 79.5 −79.8 77.7 121.0 28.1 102.0 14.0 55.4 44.0 +9.7 56.0 57.4 123.2 47.3 −8.7 25.5 −44.5 −205.9 −90.7 52.5 −57.2 −111.0 −43.7 −44.0 −95.8 36.0 77.6 39.0 70.0 49.0 −96.3 −89.1 −75.4 −0.3 +7.7 −1.4 −26.8 −34.1 63.7 +5.0 34.2 −2.9 77.4 125.7 −35.9 −53.7 −96.0 77.1 3.2 103.5
272.2 300.4 330.0 241.7 268.5 298.3 31.6 49.6 68.7 119.6 138.0 157.0 103.7 121.8 139.9 98.3 117.0 135.5 29.0 46.8 66.0 215.7 241.0 266.8 225.0 248.3 271.3 230.8 255.0 279.8 −5.3 +10.2 25.9 238.0 262.5 286.2 256.8 282.6 310.0 175.2 193.8 212.0 142.0 159.8 178.8 146.5 165.5 185.0 126.8 147.5 167.9 178.0 200.0 221.0 135.6 157.8 181.6 139.8 163.9 188.6 179.0 199.3 219.5 160.0 185.7 211.0 202.5 225.2 250.0 −1.2 +15.4 32.6 207.8 231.8 257.0 249.8 273.8 299.2 142.0 164.0 187.0 208.3 227.4 245.8 128.5 151.4 175.0 173.3 196.2 220.0 155.9 179.5 202.0 122.7 146.0 169.5 168.3 190.2 212.0 178.8 202.8 227.0 253.2 277.1 301.0 169.8 194.2 219.0 90.0 109.8 130.0 123.9 142.5 161.0 50.0 70.3 90.3 −175.5 −168.8 −161.5 −22.1 −7.9 +6.8 163.5 184.2 204.0 202.3 227.4 253.0 24.0 40.0 57.8 −49.8 −37.2 −23.3 43.9 61.8 80.2 34.8 49.9 64.7 −32.4 −19.7 −6.3 149.3 172.0 195.5 212.4 238.5 266.5 151.4 174.7 199.5 183.2 204.5 225.5 159.0 180.7 204.0 −26.5 −11.9 +3.6 −13.8 +2.5 20.2 +4.9 21.6 38.5 94.9 94.3 85.6 64.3 55.7 181.6 109.8 148.1 −51.2 90.5 209.6 245.0 59.6 34.0 −26.0 199.0 103.2 232.8
113.5 111.0 102.0 83.1 73.6 202.9 129.8 170.0 −38.0 109.5 235.0 266.8 79.6 52.3 −11.3 222.5 122.6 257.0
131.7 127.5 119.0 102.3 92.6 224.0 150 193.0 −24.0 130.3 263.0 288.0 100.9 71.8 +4.5 246.5 143.0 282.0
Melting point, °C 130 −95.3 −51.6 −98.5 48.5 −13.2 170.3 115.5 79 −47 −2 −28.5
−146.7 44.5 48 33.5 −96.9 58 42.5 54.5 −59 15 −182.5 −121 102 −98.7 −102.7 −97.8 −93.5 −57 24 −12.5 15.4 −93 −135 −133 −56.9 −84.7 −84.7 −18 −40 −97.7 −31.9 33.4 −126.4 −142.4
−7.6
VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8
2-71
Vapor Pressures of Organic Compounds, up to 1 atm* (Continued ) Pressure, mm Hg Compound Name
Formula
Methyl n-dodecyl ketone (2-tetradecanone) Methylene bromide (dibromomethane) chloride (dichloromethane) Methyl ethyl ketone (2-butanone) 2-Methyl-3-ethylpentane 3-Methyl-3-ethylpentane Methyl fluoride formate α-Methylglutaric anhydride Methyl glycolate 2-Methylheptadecane 2-Methylheptane 3-Methylheptane 4-Methylheptane 2-Methyl-2-heptene 6-Methyl-3-hepten-2-ol 6-Methyl-5-hepten-2-ol 2-Methylhexane 3-Methylhexane Methyl iodide laurate levulinate methacrylate myristate α-naphthyl ketone (1-acetonaphthone) β-naphthyl ketone (2-acetonaphthone) n-nonyl ketone (undecan-2-one) palmitate n-pentadecyl ketone (2-heptdecanone) 2-Methylpentane 3-Methylpentane 2-Methyl-1-pentanol 2-Methyl-2-pentanol Methyl n-pentyl ketone (2-heptanone) phenyl ether (anisole) 2-Methylpropene Methyl propionate 4-Methylpropiophenone 2-Methylpropionyl bromide Methyl propyl ether n-propyl ketone (2-pentanone) isopropyl ketone (3-Methyl-2-butanone) 2-Methylquinoline Methyl salicylate α-Methyl styrene 4-Methyl styrene Methyl n-tetradecyl ketone (2-hexadecanone) thiocyanate isothiocyanate undecyl ketone (2-tridecanone) isovalerate Monovinylacetylene (butenyne) Myrcene Myristaldehyde Myristic acid (tetradecanoic acid) Naphthalene 1-Naphthoic acid 2-Naphthoic acid 1-Naphthol 2-Naphthol 1-Naphthylamine 2-Naphthylamine Nicotine 2-Nitroaniline 3-Nitroaniline 4-Nitroaniline 2-Nitrobenzaldehyde 3-Nitrobenzaldehyde Nitrobenzene Nitroethane Nitroglycerin Nitromethane 2-Nitrophenol 2-Nitrophenyl acetate
C14H28O CH2Br2 CH2Cl2 C4H8O C8H18 C8H18 CH3F C2H4O2 C6H8O3 C3H6O3 C18H38 C8H18 C8H18 C8H18 C8H16 C8H16O C8H16O C7H16 C7H16 CH3I C13H26O2 C6H10O3 C5H8O2 C15H30O2 C12H10O C12H10O C11H22O C17H34O2 C17H34O C6H14 C6H14 C6H14O C6H14O C7H14O C7H8O C4H8 C4H8O2 C10H12O C4H7BrO C4H10O C5H10O C5H10O C10H9N C8H8O3 C9H10 C9H10 C16H32O C2H3NS C2H3NS C13H26O C6H12O2 C4H4 C10H16 C14H28O C14H28O2 C10H8 C11H8O2 C11H8O2 C10H8O C10H8O C10H9N C10H9N C10H14N2 C6H6N2O2 C6H6N2O2 C6H6N2O2 C7H5NO3 C7H5NO3 C6H5NO2 C2H5NO2 C3H5N3O9 CH3NO2 C6H5NO3 C8H7NO4
1
5
10
20
40
99.3 −35.1 −70.0 −48.3 −24.0 −23.9 −147.3 −74.2 93.8 +9.6 119.8 −21.0 −19.8 −20.4 −16.1 41.6 41.9 −40.4 −39.0
130.0 −13.2 −52.1 −28.0 −1.8 −1.4 −137.0 −57.0 125.4 33.7 152.0 +1.3 +2.6 +1.5 +6.7 65.0 66.0 −19.5 −18.1 −55.0 117.9 66.4 −10.0 145.7 146.3 152.3 95.5 166.8 161.6 −41.7 −39.8 38.0 +16.8 43.6 30.0 −96.5 −21.5 89.3 38.4 −54.3 +8.0 −1.0 104.0 81.6 34.0 42.0
145.5 −2.4 −43.3 −17.7 +9.5 +9.9 −131.6 −48.6 141.8 45.3 168.7 12.3 13.3 12.4 17.8 76.7 77.8 −9.1 −7.8 −45.8 133.2 79.7 +1.0 160.8 161.5 168.5 108.9 184.3 178.0 −32.1 −30.1 49.6 27.6 55.5 42.2 −81.9 −11.8 103.8 50.6 −45.4 17.9 +8.3 119.0 95.3 47.1 55.1
161.3 +9.7 −33.4 −6.5 21.7 22.3 −125.9 −39.2 157.7 58.1 186.0 24.4 25.4 24.5 30.4 89.3 90.4 +2.3 +3.6 −35.6 149.0 93.7 11.0 177.8 178.4 185.7 123.1 202.0 196.4 −21.4 −19.4 61.6 38.8 67.7 55.8 −73.4 −1.0 120.2 64.1 −35.4 28.5 18.3 134.0 110.0 61.8 69.2
179.8 23.3 −22.3 +6.0 35.2 36.2 −119.1 −28.7 177.5 72.3 204.8 37.9 38.9 38.0 44.0 102.7 104.0 14.9 16.4 −24.2 166.0 109.5 25.5 195.8 196.8 203.8 139.0
151.5 +9.8 −8.3 117.0 +2.9 −77.7 40.0 132.0 174.1 74.2 184.0 189.7 125.5 128.6 137.7 141.6 91.8 135.7 151.5 177.6 117.7 127.4 71.6 +1.5 167 −7.9 76.8 128.0
167.3 21.6 +5.4 131.8 14.0 −70.0 53.2 148.3 190.8 85.8 196.8 202.8 142.0 145.5 153.8 157.6 107.2 150.4 167.8 194.4 133.4 142.8 84.9 12.5 188 +2.8 90.4 142.0
184.6 34.5 20.4 147.8 26.4 −61.3 67.0 166.2 207.6 101.7 211.2 216.9 158.0 161.8 171.6 175.8 123.7 167.7 185.5 213.2 150.0 159.0 99.3 24.8 210 14.1 105.8 155.8
60
100
200
400
760
191.4 31.6 −15.7 14.0 43.9 45.0 −115.0 −21.9 189.9 81.8 216.3 46.6 47.6 46.6 52.8 111.5 112.8 23.0 24.5 −16.9 176.8 119.3 34.5 207.5 208.6 214.7 148.6
206.0 42.3 −6.3 25.0 55.7 57.1 −109.0 −12.9 205.0 93.7 231.5 58.3 59.4 58.3 64.6 122.6 123.8 34.1 35.6 −7.0 190.8 133.0 47.0 222.6 223.8 229.8 161.0
228.2 58.5 +8.0 41.6 73.6 75.3 −99.9 +0.8 229.1 111.8 254.5 76.0 77.1 76.1 82.3 139.5 140.0 50.8 52.4 +8.0
253.3 79.0 24.1 60.0 94.0 96.2 −89.5 16.0 255.5 131.7 279.8 96.2 97.4 96.3 102.2 156.6 156.6 69.8 71.6 25.3
278.0 98.6 40.7 79.6 115.6 118.3 −78.2 32.0 282.5 151.5 306.5 117.6 118.9 117.7 122.5 175.5 174.3 90.0 91.9 42.4
153.4 63.0 245.3 246.7 251.6 181.2
175.8 82.0 269.8 270.5 275.8 202.3
197.7 101.0 295.8 295.5 301.0 224.0
214.3 −9.7 −7.3 74.7 51.3 81.2 70.7 −63.8 +11.0 138.0 79.4 −24.3 39.8 29.6 150.8 126.2 77.8 85.0
226.7 −1.9 +0.1 83.4 58.8 89.8 80.1 −57.7 18.7 149.3 88.8 −17.4 47.3 36.2 161.7 136.7 88.3 95.0
242.0 +8.1 10.5 94.2 69.2 100.0 93.0 −49.3 29.0 164.2 101.6 −8.1 56.8 45.5 176.2 150.0 102.2 108.6
265.8 24.1 26.5 111.3 85.0 116.1 112.3 −36.7 44.2 187.4 120.5 +6.0 71.0 59.0 197.8 172.6 121.8 128.7
291.7 41.6 44.2 129.8 102.6 133.2 133.8 −22.2 61.8 212.7 141.7 22.5 86.8 73.8 211.7 197.5 143.0 151.2
319.5 60.3 63.3 147.9 121.2 150.2 155.5 −6.9 79.8 238.5 163.0 39.1 103.3 88.9 246.5 223.2 165.4 175.0
203.7 49.0 38.2 165.7 39.8 −51.7 82.6 186.0 223.5 119.3 225.0 231.5 177.8 181.7 191.5 195.7 142.1 186.0 204.2 234.2 168.8 177.7 115.4 38.0 235 27.5 122.1 172.8
215.0 58.1 47.5 176.6 48.2 −45.3 92.6 198.3 237.2 130.2 234.5 241.3 190.0 193.7 203.8 208.1 154.7 197.8 216.5 245.9 180.7 189.5 125.8 46.5 251 35.5 132.6 181.7
230.5 70.4 59.3 191.5 59.8 −37.1 106.0 214.5 250.5 145.5 245.8 252.7 206.0 209.8 220.0 224.3 169.5 213.0 232.1 261.8 196.2 204.3 139.9 57.8
254.4 89.8 77.5 214.0 77.3 −24.1 126.0 240.4 272.3 167.7 263.5 270.3 229.6 234.0 244.9 249.7 193.8 236.3 255.3 284.5 220.0 227.4 161.2 74.8
279.8 110.8 97.8 238.3 96.7 −10.1 148.3 267.9 294.6 193.2 281.4 289.5 255.8 260.6 272.2 277.4 219.8 260.0 280.2 310.2 246.8 252.1 185.8 94.0
307.0 132.9 119.0 262.5 116.7 +5.3 171.5 297.8 318.0 217.9 300.0 308.5 282.5 288.0 300.8 306.1 247.3 284.5 305.7 336.0 273.5 278.3 210.6 114.0
46.6 146.4 194.1
63.5 167.6 213.0
82.0 191.0 233.5
101.2 214.5 253.0
Temperature, °C
87.8 39.8 −30.5 115.0 115.6 120.2 68.2 134.3 129.6 −60.9 −59.0 15.4 −4.5 19.3 +5.4 −105.1 −42.0 59.6 13.5 −72.2 −12.0 −19.9 75.3 54.0 7.4 16.0 109.8 −14.0 −34.7 86.8 −19.2 −93.2 14.5 99.0 142.0 52.6 156.0 160.8 94.0 104.3 108.0 61.8 104.0 119.3 142.4 85.8 96.2 44.4 −21.0 127 −29.0 49.3 100.0
Melting point, °C −52.8 −96.7 −85.9 −114.5 −90 −99.8
−109.5 −120.8 −121.1
−118.2 −64.4 5 18.5 55.5 15 30 −154 −118 −103 −37.3 −140.3 −87.5
−77.8 −92 −1 −8.3 −23.2
−51 35.5 28.5
23.5 57.5 80.2 160.5 184 96 122.5 50 111.5 71.5 114 146.5 40.9 58 +5.7 −90 11 −29 45
2-72
PHYSICAL AND CHEMICAL DATA
TABLE 2-8
Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name
1-Nitropropane 2-Nitropropane 2-Nitrotoluene 3-Nitrotoluene 4-Nitrotoluene 4-Nitro-1,3-xylene (4-nitro-m-xylene) Nonacosane Nonadecane n-Nonane 1-Nonanol 2-Nonanone Octacosane Octadecane n-Octane n-Octanol (1-octanol) 2-Octanone n-Octyl acrylate iodide (1-Iodooctane) Oleic acid Palmitaldehyde Palmitic acid Palmitonitrile Pelargonic acid Pentachlorobenzene Pentachloroethane Pentachloroethylbenzene Pentachlorophenol Pentacosane Pentadecane 1,3-Pentadiene 1,4-Pentadiene Pentaethylbenzene Pentaethylchlorobenzene n-Pentane iso-Pentane (2-methylbutane) neo-Pentane (2,2-dimethylpropane) 2,3,4-Pentanetriol 1-Pentene α-Phellandrene Phenanthrene Phenethyl alcohol (phenyl cellosolve) 2-Phenetidine Phenol 2-Phenoxyethanol 2-Phenoxyethyl acetate Phenyl acetate Phenylacetic acid Phenylacetonitrile Phenylacetyl chloride Phenyl benzoate 4-Phenyl-3-buten-2-one Phenyl isocyanate isocyanide Phenylcyclohexane Phenyl dichlorophosphate m-Phenylene diamine (1,3-phenylenediamine) Phenylglyoxal Phenylhydrazine N-Phenyliminodiethanol 1-Phenyl-1,3-pentanedione 2-Phenylphenol 4-Phenylphenol 3-Phenyl-1-propanol Phenyl isothiocyanate Phorone iso-Phorone Phosgene (carbonyl chloride) Phthalic anhydride Phthalide Phthaloyl chloride 2-Picoline Pimelic acid α-Pinene β-Pinene
1
5
10
20
40
C3H7NO2 −9.6 C3H7NO2 −18.8 C7H7NO2 50.0 C7H7NO2 50.2 C7H7NO2 53.7 C8H9NO2 65.6 C29H60 234.2 C19H40 133.2 C9H20 +1.4 C9H20O 59.5 C9H18O 32.1 C28H58 226.5 C18H38 119.6 C8H18 −14.0 C8H18O 54.0 C8H16O 23.6 C11H20O2 58.5 C8H17I 45.8 C18H34O2 176.5 C16H32O 121.6 C16H32O2 153.6 C16H31N 134.3 C9H18O2 108.2 C6HCl5 98.6 C2HCl5 +1.0 C8H5Cl5 96.2 C6HCl5O C25H52 194.2 C15H32 91.6 C5H8 −71.8 C5H8 −83.5 C16H26 86.0 C16H25Cl 90.0 C5H12 −76.6 C5H12 −82.9 C5H12 −102.0 C5H12O3 155.0 C5H10 −80.4 C10H16 20.0 C14H10 118.2 C8H10O2 58.2 C8H11NO 67.0 C6H6O 40.1 C8H10O2 78.0 C10H12O3 82.6 C8H8O2 38.2 C8H8O2 97.0 C8H7N 60.0 C8H7ClO 48.0 C13H10O2 106.8 C10H10O 81.7 C7H5NO 10.6 C7H5N 12.0 C12H16 67.5 C6H5Cl2O2P 66.7
+13.5 +4.1 79.1 81.0 85.0 95.0 269.8 166.3 25.8 86.1 59.0 260.3 152.1 +8.3 76.5 48.4 87.7 74.8 208.5 154.6 188.1 168.3 126.0 129.7 27.2 130.0
25.3 15.8 93.8 96.0 100.5 109.8 286.4 183.5 38.0 99.7 72.3 277.4 169.6 19.2 88.3 60.9 102.0 90.0 223.0 171.8 205.8 185.8 137.4 144.3 39.8 148.0
230.0 121.0 −53.8 −66.2 120.0 123.8 −62.5 −65.8 −85.4 189.3 −63.3 45.7 154.3 85.9 94.7 62.5 106.6 113.5 64.8 127.0 89.0 75.3 141.5 112.2 36.0 37.0 96.5 95.9
248.2 135.4 −45.0 −57.1 135.8 140.7 −50.1 −57.0 −76.7 204.5 −54.5 58.0 173.0 100.0 108.6 73.8 121.2 128.0 78.0 141.3 103.5 89.0 157.8 127.4 48.5 49.7 111.3 110.0
37.9 28.2 109.6 112.8 117.7 125.8 303.6 200.8 51.2 113.8 87.2 295.4 187.5 31.5 101.0 74.3 117.8 105.9 240.0 190.0 223.8 204.2 149.8 160.0 53.9 166.0 192.2 266.1 150.2 −34.8 −47.7 152.4 158.1 −40.2 −47.3 −67.2 220.5 −46.0 72.1 193.7 114.8 123.7 86.0 136.0 144.5 92.3 156.0 119.4 103.6 177.0 143.8 62.5 63.4 126.4 125.9
51.8 41.8 126.3 130.7 136.0 143.3 323.2 220.0 66.0 129.0 103.4 314.2 207.4 45.1 115.2 89.8 135.6 123.8 257.2 210.0 244.4 223.8 163.7 178.5 69.9 186.2 211.2 285.6 167.7 −23.4 −37.0 171.9 178.2 −29.2 −36.5 −56.1 239.6 −34.1 87.8 215.8 130.5 139.9 100.1 152.2 162.3 108.1 173.6 136.3 119.8 197.6 161.3 77.7 78.3 144.0 143.4
C6H8N2 C8H6O2 C6H8N2 C10H15NO2 C11H12O2 C12H10O C12H10O C9H12O C7H5NS C9H14O C9H14O CCl2O C8H4O3 C8H6O2 C8H4Cl2O2 C6H7N C7H12O4 C10H16 C10H16
71.8 145.0 98.0 100.0
131.2 75.0 101.6 179.2 128.5 131.6
74.7 47.2 42.0 38.0 −92.9 96.5 95.5 86.3 −11.1 163.4 −1.0 +4.2
102.4 75.6 68.3 66.7 −77.0 121.3 127.7 118.3 +12.6 196.2 +24.6 30.0
147.0 87.8 115.8 195.8 144.0 146.2 176.2 116.0 89.8 81.5 81.2 −69.3 134.0 144.0 134.2 24.4 212.0 37.3 42.3
163.8 100.7 131.5 213.4 159.9 163.3 193.8 131.2 115.5 95.6 96.8 −60.3 151.7 161.3 151.0 37.4 229.3 51.4 58.1
182.5 115.5 148.2 233.0 178.0 180.3 213.0 147.4 122.5 111.3 114.5 −50.3 172.0 181.0 170.0 51.2 247.0 66.8 71.5
60
200
400
760
60.5 50.3 137.6 142.5 147.9 153.8 334.8 232.8 75.5 139.0 113.8 326.8 219.7 53.8 123.8 99.0 145.6 135.4 269.8 222.6 256.0 236.6 172.3 190.1 80.0 199.0 223.4 298.4 178.4 −16.5 −30.0 184.2 191.0 −22.2 −29.6 −49.0 249.8 −27.1 97.6 229.9 141.2 149.8 108.4 163.2 174.0 118.1 184.5 147.7 129.8 210.8 172.6 87.7 88.0 154.2 153.6
72.3 62.0 151.5 156.9 163.0 168.5 350.0 248.0 88.1 151.3 127.4 341.8 236.0 65.7 135.2 111.7 159.1 150.0 286.0 239.5 271.5 251.5 184.4 205.5 93.5 216.0 239.6 314.0 194.0 −6.7 −20.6 200.0 208.0 −12.6 −20.2 −39.1 263.5 −17.7 110.6 249.0 154.0 163.5 121.4 176.5 189.2 131.6 198.2 161.8 143.5 227.8 187.8 100.6 101.0 169.3 168.0
90.2 80.0 173.7 180.3 186.7 191.7 373.2 271.8 107.5 170.5 148.2 364.8 260.6 83.6 152.0 130.4 180.2 173.3 309.8 264.1 298.7 277.1 203.1 227.0 114.0 241.8 261.8 339.0 216.1 +8.0 −6.7 224.1 230.3 +1.9 −5.9 −23.7 284.5 −3.4 130.6 277.1 175.0 184.0 139.0 197.6 211.3 151.2 219.5 184.2 163.8 254.0 211.0 120.8 120.8 191.3 189.8
110.6 99.8 197.7 206.8 212.5 217.5 397.2 299.8 128.2 192.1 171.2 388.9 288.0 104.0 173.8 151.0 204.0 199.3 334.7 292.3 326.0 304.5 227.5 251.6 137.2 269.3 285.0 365.4 242.8 24.7 +8.3 250.2 257.2 18.5 +10.5 −7.1 307.0 +12.8 152.0 308.0 197.5 207.0 160.0 221.0 235.0 173.5 243.0 208.5 186.0 283.5 235.4 142.7 142.3 214.6 213.0
131.6 120.3 222.3 231.9 238.3 244.0 421.8 330.0 150.8 213.5 195.0 412.5 317.0 125.6 195.2 172.9 227.0 225.5 360.0 321.0 353.8 332.0 253.5 276.0 160.5 299.0 309.3 390.3 270.5 42.1 26.1 277.0 285.0 36.1 27.8 +9.5 327.2 30.1 175.0 340.2 219.5 228.0 181.9 245.3 259.7 195.9 265.5 233.5 210.0 314.0 261.0 165.6 165.0 240.0 239.5
−108 −93 −4.1 15.5 51.9 +2 63.8 32 −53.7 −5 −19 61.6 28 −56.8 −15.4 −16
194.0 124.2 158.7 245.3 189.8 192.2 225.3 156.8 133.3 121.4 125.6 −44.0 185.3 193.5 182.2 59.9 258.2 76.8 81.2
209.9 136.2 173.5 260.6 204.5 205.9 240.9 170.3 147.7 134.0 140.6 −35.6 202.3 210.0 197.8 71.4 272.0 90.1 94.0
233.0 153.8 195.4 284.5 226.7 227.9 263.2 191.2 169.6 153.5 163.3 −22.3 228.0 234.5 222.0 89.0 294.5 110.2 114.1
259.0 173.5 218.2 311.3 251.2 251.8 285.5 212.8 194.0 175.3 188.7 −7.6 256.8 261.8 248.3 108.4 318.5 132.3 136.1
285.5 193.5 243.5 337.8 276.5 275.0 308.0 235.0 218.5 197.2 215.2 +8.3 284.5 290.0 275.8 128.8 342.1 155.0 158.3
62.8 73 19.5
Temperature, °C
Formula
99.8
Melting point, °C
100
−45.9 14 34 64.0 31 12.5 85.5 −22 188.5 53.3 10
−129.7 −159.7 −16.6
99.5 40.6 11.6 −6.7 76.5 −23.8 70.5 41.5 +7.5
56.5 164.5 −21.0 28 −104 130.8 73 88.5 −70 103 −55
VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8
2-73
Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name
Piperidine Piperonal Propane Propenylbenzene Propionamide Propionic acid anhydride Propionitrile Propiophenone n-Propyl acetate iso-Propyl acetate n-Propyl alcohol (1-propanol) iso-Propyl alcohol (2-propanol) n-Propylamine Propylbenzene Propyl benzoate n-Propyl bromide (1-bromopropane) iso-Propyl bromide (2-bromopropane) n-Propyl n-butyrate isobutyrate iso-Propyl isobutyrate Propyl carbamate n-Propyl chloride (1-chloropropane) iso-Propyl chloride (2-chloropropane) iso-Propyl chloroacetate Propyl chloroglyoxylate Propylene Propylene glycol (1,2-Propanediol) Propylene oxide n-Propyl formate iso-Propyl formate 4,4′-iso-Propylidenebisphenol n-Propyl iodide (1-iodopropane) iso-Propyl iodide (2-iodopropane) n-Propyl levulinate iso-Propyl levulinate Propyl mercaptan (1-propanethiol) 2-iso-Propylnaphthalene iso-Propyl β-naphthyl ketone (2-isobutyronaphthone) 2-iso-Propylphenol 3-iso-Propylphenol 4-iso-Propylphenol Propyl propionate 4-iso-Propylstyrene Propyl isovalerate Pulegone Pyridine Pyrocatechol Pyrocaltechol diacetate (1,2-phenylene diacetate) Pyrogallol Pyrotartaric anhydride Pyruvic acid Quinoline iso-Quinoline Resorcinol Safrole Salicylaldehyde Salicylic acid Sebacic acid Selenophene Skatole Stearaldehyde Stearic acid Stearyl alcohol (1-octadecanol) Styrene Styrene dibromide [(1,2-dibromoethyl) benzene] Suberic acid Succinic anhydride Succinimide Succinyl chloride α-Terpineol Terpenoline
1
5
10
20
87.0 −128.9 17.5 65.0 4.6 20.6 −35.0 50.0 −26.7 −38.3 −15.0 −26.1 −64.4 6.3 54.6 −53.0 −61.8 −1.6 −6.2 −16.3 52.4 −68.3 −78.8 +3.8 9.7 −131.9 45.5 −75.0 −43.0 −52.0 193.0 −36.0 −43.3 59.7 48.0 −56.0 76.0
−7.0 117.4 −115.4 43.8 91.0 28.0 45.3 −13.6 77.9 −5.4 −17.4 +5.0 −7.0 −46.3 31.3 83.8 −33.4 −42.5 +22.1 +16.8 +5.8 77.6 −50.0 −61.1 28.1 32.3 −120.7 70.8 −57.8 −22.7 −32.7 224.2 −13.5 −22.1 86.3 74.5 −36.3 107.9
+3.9 132.0 −108.5 57.0 105.0 39.7 57.7 −3.0 92.2 +5.0 −7.2 14.7 +2.4 −37.2 43.4 98.0 −23.3 −32.8 34.0 28.3 17.0 90.0 −41.0 −52.0 40.2 43.5 −112.1 83.2 −49.0 −12.6 −22.7 240.8 −2.4 −11.7 99.9 88.0 −26.3 123.4
15.8 148.0 −100.9 71.5 119.0 52.0 70.4 +8.8 107.6 16.0 +4.2 25.3 12.7 −27.1 56.8 114.3 −12.4 −22.0 47.0 40.6 29.0 103.2 −31.0 −42.0 53.9 55.6 −104.7 96.4 −39.3 −1.7 −12.1 255.5 +10.0 0.0 114.0 102.4 −15.4 140.3
29.2 165.7 −92.4 87.7 134.8 65.8 85.6 22.0 124.3 28.8 17.0 36.4 23.8 −16.0 71.6 131.8 −0.3 −10.1 61.5 54.3 42.4 117.7 −19.5 −31.0 68.7 68.8 −96.5 111.2 −28.4 +10.8 −0.2 273.0 23.6 +13.2 130.1 118.1 −3.2 159.0
60
100
200
400
760
37.7 177.0 −87.0 97.8 144.3 74.1 94.5 30.1 135.0 37.0 25.1 43.5 30.5 −9.0 81.1 143.3 +7.5 −2.5 70.3 63.0 51.4 126.5 −12.1 −23.5 78.0 77.2 −91.3 119.9 −21.3 18.8 +7.5 282.9 32.1 21.6 140.6 127.8 +4.6 171.4
49.0 191.7 −79.6 111.7 156.0 85.8 107.2 41.4 149.3 47.8 35.7 52.8 39.5 +0.5 94.0 157.4 18.0 +8.0 82.6 73.9 62.3 138.3 −2.5 −13.7 90.3 88.0 −84.1 132.0 −12.0 29.5 17.8 297.0 43.8 32.8 154.0 141.8 15.3 187.6
66.2 214.3 −68.4 132.0 174.2 102.5 127.8 58.2 170.2 64.0 51.7 66.8 53.0 15.0 113.5 180.1 34.0 23.8 101.0 91.8 80.2 155.8 +12.2 +1.3 108.8 104.7 −73.3 149.7 +2.1 45.3 33.6 317.5 61.8 50.0 175.6 161.6 31.5 211.8
85.7 238.5 −55.6 154.7 194.0 122.0 146.0 77.7 194.2 82.0 69.8 82.0 67.8 31.5 135.7 205.2 52.0 41.5 121.7 112.0 100.0 175.8 29.4 18.1 128.0 123.0 −60.9 168.1 17.8 62.6 50.5 339.0 81.8 69.5 198.0 185.2 49.2 238.5
106.0 263.0 −42.1 179.0 213.0 141.1 167.0 97.1 218.0 101.8 89.0 97.8 82.5 48.5 159.2 231.0 71.0 60.0 142.7 133.9 120.5 195.0 46.4 36.5 148.6 150.0 −47.7 188.2 34.5 81.3 68.3 360.5 102.5 89.5 221.2 208.2 67.4 266.0
Temperature, °C
Formula C5H11N C8H6O3 C3H8 C9H10 C3H7NO C3H6O2 C6H10O3 C3H5N C9H10O C5H10O2 C5H10O2 C3H8O C3H8O C3H9N C9H12 C10H12O2 C3H7Br C3H7Br C7H14O2 C7H14O2 C7H14O2 C4H9NO2 C3H7Cl C3H7Cl C5H9ClO2 C5H7ClO3 C3H6 C3H8O2 C3H6O C4H8O2 C4H8O2 C15H16O2 C3H7I C3H7I C8H14O3 C8H14O3 C3H8S C13H14
40
C14H14O C9H12O C9H12O C9H12O C6H12O2 C11H14 C8H16O2 C10H16O C5H5N C6H6O2
133.2 56.6 62.0 67.0 −14.2 34.7 +8.0 58.3 −18.9
165.4 83.8 90.3 94.7 +8.0 62.3 32.8 82.5 +2.5 104.0
181.0 97.0 104.1 108.0 19.4 76.0 45.1 94.0 13.2 118.3
197.7 111.7 119.8 123.4 31.6 91.2 58.0 106.8 24.8 134.0
215.6 127.5 136.2 139.8 45.0 108.0 72.8 121.7 38.0 150.6
227.0 137.7 146.6 149.7 53.8 118.4 82.3 130.2 46.8 161.7
242.3 150.3 160.2 163.3 65.2 132.8 95.0 143.1 57.8 176.0
264.0 170.1 182.0 184.0 82.7 153.9 113.9 162.5 75.0 197.7
288.2 192.6 205.0 206.1 102.0 178.0 135.0 189.8 95.6 221.5
313.0 214.5 228.0 228.2 122.4 202.5 155.9 221.0 115.4 245.5
C10H10O4 C6H6O3 C5H6O3 C3H4O3 C9H7N C9H7N C6H6O2 C10H10O2 C7H6O2 C7H6O3 C10H18O4 C4H4Se C9H9N C18H36O C18H36O2 C18H36O C8H8
98.0 69.7 21.4 59.7 63.5 108.4 63.8 33.0 113.7 183.0 −39.0 95.0 140.0 173.7 150.3 −7.0
129.8 151.7 99.7 45.8 89.6 92.7 138.0 93.0 60.1 136.0 215.7 −16.0 124.2 174.6 209.0 185.6 +18.0
145.7 167.7 114.2 57.9 103.8 107.8 152.1 107.6 73.8 146.2 232.0 −4.0 139.6 192.1 225.0 202.0 30.8
161.8 185.3 130.0 70.8 119.8 123.7 168.0 123.0 88.7 156.8 250.0 +9.1 154.3 210.6 243.4 220.0 44.6
179.8 204.2 147.8 85.3 136.7 141.6 185.3 140.1 105.2 172.2 268.2 24.1 171.9 230.8 263.3 240.4 59.8
191.6 216.3 158.6 94.1 148.1 152.0 195.8 150.3 115.7 182.0 279.8 33.8 183.6 244.2 275.5 252.7 69.5
206.5 232.0 173.8 106.5 163.2 167.6 209.8 165.1 129.4 193.4 294.5 47.0 197.4 260.0 291.0 269.4 82.0
228.7 255.3 196.1 124.7 186.2 190.0 230.8 186.2 150.0 210.0 313.2 66.7 218.8 285.0 316.5 293.5 101.3
253.3 281.5 221.0 144.7 212.3 214.5 253.4 210.0 173.7 230.5 332.8 89.8 242.5 313.8 343.0 320.3 122.5
278.0 309.0 247.4 165.0 237.7 240.5 276.5 233.0 196.5 256.0 352.3 114.3 266.2 342.5 370.0 349.5 145.2
C8H8Br2 C8H14O4 C4H4O3 C4H5NO2 C4H4Cl2O2 C10H18O C10H16
86.0 172.8 92.0 115.0 39.0 52.8 32.3
115.6 205.5 115.0 143.2 65.0 80.4 58.0
129.8 219.5 128.2 157.0 78.0 94.3 70.6
145.2 238.2 145.3 174.0 91.8 109.8 84.8
161.8 254.6 163.0 192.0 107.5 126.0 100.0
172.2 265.4 174.0 203.0 117.2 136.3 109.8
186.3 279.8 189.0 217.4 130.0 150.1 122.7
207.8 300.5 212.0 240.0 149.3 171.2 142.0
230.0 322.8 237.0 263.5 170.0 194.3 163.5
254.0 345.5 261.0 287.5 192.5 217.5 185.0
Melting point, °C −9 37 −187.1 −30.1 79 −22 −45 −91.9 21 −92.5 −127 −85.8 −83 −99.5 −51.6 −109.9 −89.0 −95.2
−122.8 −117 −185 −112.1 −92.9 −98.8 −90 −112
15.5 26 61 −76
−42 105 133 13.6 −15 24.6 110.7 11.2 −7 159 134.5 95 63.5 69.3 58.5 −30.6 142 119.6 125.5 17 35
2-74
PHYSICAL AND CHEMICAL DATA
TABLE 2-8
Vapor Pressures of Organic Compounds, up to 1 atm (Continued ) Pressure, mm Hg Compound Name
Formula
1,1,1,2-Tetrabromoethane 1,1,2,2-Tetrabromoethane Tetraisobutylene Tetracosane 1,2,3,4-Tetrachlorobenzene 1,2,3,5-Tetrachlorobenzene 1,2,4,5-Tetrachlorobenzene 1,1,2,2-Tetrachloro-1,2-difluoroethane 1,1,1,2-Tetrachloroethane 1,1,2,2-Tetrachloroethane 1,2,3,5-Tetrachloro-4-ethylbenzene Tetrachloroethylene 2,3,4,6-Tetrachlorophenol 3,4,5,6-Tetrachloro-1,2-xylene Tetradecane Tetradecylamine Tetradecyltrimethylsilane Tetraethoxysilane 1,2,3,4-Tetraethylbenzene Tetraethylene glycol Tetraethylene glycol chlorohydrin Tetraethyllead Tetraethylsilane Tetralin 1,2,3,4-Tetramethylbenzene 1,2,3,5-Tetramethylbenzene 1,2,4,5-Tetramethylbenzene 2,2,3,3-Tetramethylbutane Tetramethylene dibromide (1,4-dibromobutane) Tetramethyllead Tetramethyltin Tetrapropylene glycol monoisopropyl ether Thioacetic acid (mercaptoacetic acid) Thiodiglycol (2,2′-thiodiethanol) Thiophene Thiophenol (benzenethiol) α-Thujone Thymol Tiglaldehyde Tiglic acid Tiglonitrile Toluene Toluene-2,4-diamine 2-Toluic nitrile (2-tolunitrile) 4-Toluic nitrile (4-tolunitrile) 2-Toluidine 3-Toluidine 4-Toluidine 2-Tolyl isocyanide 4-Tolylhydrazine Tribromoacetaldehyde 1,1,2-Tribromobutane 1,2,2-Tribromobutane 2,2,3-Tribromobutane 1,1,2-Tribromoethane 1,2,3-Tribromopropane Triisobutylamine Triisobutylene 2,4,6-Tritertbutylphenol Trichloroacetic acid Trichloroacetic anhydride Trichloroacetyl bromide 2,4,6-Trichloroaniline 1,2,3-Trichlorobenzene 1,2,4-Trichlorobenzene 1,3,5-Trichlorobenzene 1,2,3-Trichlorobutane 1,1,1-Trichloroethane 1,1,2-Trichloroethane Trichloroethylene Trichlorofluoromethane 2,4,5-Trichlorophenol 2,4,6-Trichlorophenol
C2H2Br4 C2H2Br4 C16H32 C24H50 C6H2Cl4 C6H2Cl4 C6H2Cl4 C2Cl4F2 C2H2Cl4 C2H2Cl4 C8H6Cl4 C2Cl4 C6H2Cl4O C8H6Cl4 C14H30 C14H31N C17H38Si C8H20O4Si C14H22 C8H18O5 C8H17ClO4 C8H20Pb C8H20Si C10H12 C10H14 C10H14 C10H14 C8H18 C4H8Br2 C4H12Pb C4H12Sn C15H32O5 C2H4O2S C4H10O2S C4H4S C6H6S C10H16O C10H14O C5H8O C5H8O2 C5H7N C7H8 C7H10N2 C8H7N C8H7N C7H9N C7H9N C7H9N C8H7N C7H10N2 C2HBr3O C4H7Br3 C4H7Br3 C4H7Br3 C2H3Br3 C3H5Br3 C12H27N C12H24 C18H30O C2HCl3O2 C4Cl6O3 C2BrCl3O C6H4Cl3N C6H3Cl3 C6H3Cl3 C6H3Cl3 C4H7Cl3 C2H3Cl3 C2H3Cl3 C2HCl3 CCl3F C6H3Cl3O C6H3Cl3O
1
5
10
20
58.0 65.0 63.8 183.8 68.5 58.2
83.3 95.5 93.7 219.6 99.6 89.0
95.7 110.0 108.5 237.6 114.7 104.1
108.5 126.0 124.5 255.3 131.2 121.6
−37.5 −16.3 −3.8 77.0 −20.6 100.0 94.4 76.4 102.6 120.0 16.0 65.7 153.9 110.1 38.4 −1.0 38.0 42.6 40.6 45.0 −17.4
−16.0 +7.4 +20.7 110.0 +2.4 130.3 125.0 106.0 135.8 150.7 40.3 96.2 183.7 141.8 63.6 +23.9 65.3 68.7 65.8 65.0 +3.2
−5.0 19.3 33.0 126.0 13.8 145.3 140.3 120.7 152.0 166.2 52.6 111.6 197.1 156.1 74.8 36.3 79.0 81.8 77.8 74.6 13.5
32.0 −29.0 −51.3 116.6 60.0 42.0 −40.7 18.6 38.3 64.3 −25.0 52.0 −25.5 −26.7 106.5 36.7 42.5 44.0 41.0 42.0 25.2 82.4 18.5 45.0 41.0 38.2 32.6 47.5 32.3 18.0 95.2 51.0 56.2 −7.4 134.0 40.0 38.4
58.8 −6.8 −31.0 147.8 87.7 96.0 −20.8 43.7 65.7 92.8 −1.6 77.8 −2.4 −4.4 137.2 64.0 71.3 69.3 68.0 68.2 51.0 110.0 45.0 73.5 69.0 66.0 58.0 75.8 57.4 44.0 126.1 76.0 85.3 +16.7 157.8 70.0 67.3 63.8 27.2 −32.0 −2.0 −22.8 −67.6 102.1 105.9
72.4 +4.4 −20.6 163.0 101.5 128.0 −10.9 56.0 79.3 107.4 +10.0 90.2 +9.2 +6.4 151.7 77.9 85.8 81.4 82.0 81.8 64.0 123.8 58.0 87.8 83.2 79.8 70.6 90.0 69.8 56.5 142.0 88.2 99.6 29.3 170.0 85.6 81.7 78.0 40.0 −21.9 +8.3 −12.4 −59.0 117.3 120.2
40
60
100
200
400
760
Temperature, °C
+0.5 −52.0 −24.0 −43.8 −84.3 72.0 76.5
Melting point, °C
+6.7 32.1 46.2 143.7 26.3 161.0 156.0 135.6 170.0 183.5 65.8 127.7 212.3 172.6 88.0 50.0 93.8 95.8 91.0 88.0 24.6
123.2 144.0 142.2 276.3 149.2 140.0 146.0 19.8 46.7 60.8 162.1 40.1 179.1 174.2 152.7 189.0 201.5 81.1 145.8 228.0 190.0 102.4 65.3 110.4 111.5 105.8 104.2 36.8
132.0 155.1 152.6 288.4 160.0 152.0 157.7 28.1 56.0 70.0 175.0 49.2 190.0 185.8 164.0 200.2 213.3 90.7 156.7 237.8 200.5 111.7 74.8 121.3 121.8 115.4 114.8 44.5
144.0 170.0 167.5 305.2 175.7 168.0 173.5 38.6 68.0 83.2 191.6 61.3 205.2 200.5 178.5 215.7 227.8 103.6 172.4 250.0 214.7 123.8 88.0 135.3 135.7 128.3 128.1 54.8
161.5 192.5 190.0 330.5 198.0 193.7 196.0 55.0 87.2 102.2 215.3 79.8 227.2 223.0 201.8 239.8 250.0 123.5 196.0 268.4 236.5 142.0 108.0 157.2 155.7 149.9 149.5 70.2
181.0 217.5 214.6 358.0 225.5 220.0 220.5 73.1 108.2 124.0 243.0 100.0 250.4 248.3 226.8 264.6 275.0 146.2 221.4 288.0 258.2 161.8 130.2 181.8 180.0 173.7 172.1 87.4
200.0 243.5 240.0 386.4 254.0 246.0 245.0 92.0 130.5 145.9 270.0 120.8 275.0 273.5 252.5 291.2 300.0 168.5 248.0 307.8 281.5 183.0 153.0 207.2 204.4 197.9 195.9 106.3
87.6 16.6 −9.3 179.8 115.8 165.0 0.0 69.7 93.7 122.6 23.2 103.8 22.1 18.4 167.9 93.0 101.7 95.1 96.7 95.8 78.2 138.6 72.1 103.2 98.6 94.6 84.2 105.8 83.0 70.0 158.0 101.8 114.3 42.1 182.6 101.8 97.2 93.7 55.0 −10.8 21.6 −1.0 −49.7 134.0 135.8
104.0 30.3 +3.5 197.7 131.8 210.0 +12.5 84.2 110.0 139.8 37.0 119.0 36.7 31.8 185.7 110.0 109.5 110.0 113.5 111.5 94.0 154.1 87.8 120.2 116.0 111.8 100.0 122.8 97.8 86.7 177.4 116.3 131.2 57.2 195.8 119.8 114.8 110.8 71.5 +1.6 35.2 +11.9 −39.0 151.5 152.2
115.1 39.2 11.7 209.0 142.0 240.5 20.1 93.9 120.2 149.8 45.8 127.8 46.0 40.3 196.2 120.8 130.0 119.8 123.8 121.5 104.0 165.0 97.5 131.6 127.0 122.2 110.0 134.0 107.3 96.7 188.0 125.9 141.8 66.7 204.5 131.5 125.7 121.8 82.0 9.5 44.0 20.0 −32.3 162.5 163.5
128.7 50.8 22.8 223.3 154.0 285 30.5 106.6 134.0 164.1 57.7 140.5 58.2 51.9 211.5 135.0 145.2 133.0 136.7 133.7 117.7 178.0 110.2 146.0 141.8 136.3 123.5 148.0 119.7 110.0 203.0 137.8 155.2 79.5 214.6 146.0 140.0 136.0 96.2 20.0 55.7 31.4 −23.0 178.0 177.8
149.8 68.8 39.8 245.0
173.8 89.0 58.5 268.3
197.5 110.0 78.0 292.7
−20 −27.5
46.5 125.8 154.2 185.5 75.4 158.0 77.8 69.5 232.8 156.0 167.3 153.0 157.6 154.0 137.8 198.0 130.0 167.8 163.5 157.8 143.5 170.0 138.0 130.2 226.2 155.4 176.2 98.4 229.8 168.2 162.0 157.7 118.0 36.2 73.3 48.0 −9.1 201.5 199.0
64.7 146.7 177.8 209.2 95.5 179.2 99.7 89.5 256.0 180.0 193.0 176.2 180.6 176.9 159.9 219.5 151.6 192.0 188.0 182.2 165.4 195.0 157.8 153.0 250.6 175.2 199.8 120.2 246.4 193.5 187.7 183.0 143.0 54.6 93.0 67.0 +6.8 226.5 222.5
84.4 168.0 201.0 231.8 116.4 198.5 122.0 110.6 280.0 205.2 217.6 199.7 203.3 200.4 183.5 242.0 174.0 216.2 213.8 206.5 188.4 220.0 179.0 179.0 276.3 195.6 223.0 143.0 262.0 218.5 213.0 208.4 169.0 74.1 113.9 86.7 23.7 251.8 246.0
−38.3
51.1 46.5 54.5 139 26.5 −68.7 −36 −19.0 69.5 5.5
11.6 −136 −31.0 −6.2 −24.0 79.5 −102.2
−16.5
51.5 64.5 −95.0 99 −13 29.5 −16.3 −31.5 44.5 65.5
−26 16.5 −22 57 78 52.5 17 63.5 −30.6 −36.7 −73 62 68.5
VAPOR PRESSURES OF PURE SUBSTANCES TABLE 2-8
2-75
Vapor Pressures of Organic Compounds, up to 1 atm (Concluded ) Pressure, mm Hg Compound Name
Tri-2-chlorophenylthiophosphate 1,1,1-Trichloropropane 1,2,3-Trichloropropane 1,1,2-Trichloro-1,2,2-trifluoroethane Tricosane Tridecane Tridecanoic acid Triethoxymethylsilane Triethoxyphenylsilane 1,2,4-Triethylbenzene 1,3,4-Triethylbenzene Triethylborine Triethyl camphoronate citrate Triethyleneglycol Triethylheptylsilane Triethyloctylsilane Triethyl orthoformate phosphate Triethylthallium Trifluorophenylsilane Trimethallyl phosphate 2,3,5-Trimethylacetophenone Trimethylamine 2,4,5-Trimethylaniline 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene 1,3,5-Trimethylbenzene 2,2,3-Trimethylbutane Trimethyl citrate Trimethyleneglycol (1,3-propanediol) 1,2,4-Trimethyl-5-ethylbenzene 1,3,5-Trimethyl-2-ethylbenzene 2,2,3-Trimethylpentane 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane 2,3,4-Trimethylpentane 2,2,4-Trimethyl-3-pentanone Trimethyl phosphate 2,4,5-Trimethylstyrene 2,4,6-Trimethylstyrene Trimethylsuccinic anhydride Triphenylmethane Triphenylphosphate Tripropyleneglycol Tripropyleneglycol monobutyl ether Tripropyleneglycol monoisopropyl ether Tritolyl phosphate Undecane Undecanoic acid 10-Undecenoic acid Undecan-2-ol n-Valeric acid iso-Valeric acid γ-Valerolactone Valeronitrile Vanillin Vinyl acetate 2-Vinylanisole 3-Vinylanisole 4-Vinylanisole Vinyl chloride (1-chloroethylene) cyanide (acrylonitrile) fluoride (1-fluoroethylene) Vinylidene chloride (1,1-dichloroethene) 4-Vinylphenetole 2-Xenyl dichlorophosphate 2,4-Xyaldehyde 2-Xylene (2-xylene) 3-Xylene (3-xylene) 4-Xylene (4-xylene) 2,4-Xylidine 2,6-Xylidine
1
5
10
20
217.2
231.2
246.7
261.7
−7.0 33.7 −49.4 206.3 98.3 166.3 +22.8 98.8 74.2 76.0
+4.2 46.0 −40.3 223.0 104.0 181.0 34.6 112.6 88.5 90.2 −148.0 166.0 144.0 158.1 114.6 120.6 40.5 82.1 51.7 +0.8 149.8 122.3 −73.8 109.0 55.9 50.7 47.4 −18.8 160.4 100.6 84.6 80.5 +3.9 −4.3 +6.9 +7.1 46.4 67.8 91.6 79.7 97.4 197.0 249.8 140.5 147.0 127.3 198.0 73.9 149.0 156.3 112.8 79.8 71.3 79.8 30.0 154.0 −18.0 81.0 83.0 85.7 −83.7 −20.3 −132.2 −51.2 105.6 187.0 99.0 32.1 28.3 27.3 93.0 87.0
16.2 59.3 −30.0 242.0 120.2 195.8 47.2 127.2 104.0 105.8 −140.6 183.6 171.1 174.0 130.3 137.7 53.4 97.8 67.7 12.3 169.8 137.5 −65.0 123.7 69.9 64.5 61.0 −7.5 177.2 115.5 99.7 96.0 16.0 +7.5 19.2 19.3 57.6 83.0 107.1 94.8 113.8 206.8 269.7 155.8 161.8 143.7 213.2 85.6 166.0 172.0 127.5 93.1 84.0 95.2 43.3 170.5 −7.0 94.7 97.2 100.0 −75.7 −9.0 −125.4 −41.7 120.3 205.0 114.0 45.1 41.1 40.1 107.6 102.7
29.9 74.0 −18.5 261.3 137.7 212.4 61.7 143.5 121.7 122.6 −131.4 201.8 190.4 191.3 148.0 155.7 67.5 115.7 85.4 25.4 192.0 154.2 −55.2 139.8 85.4 79.8 76.1 +5.2 194.2 131.0 106.0 113.2 29.5 20.7 33.0 32.9 69.8 100.0 124.2 111.8 131.0 215.5 290.3 173.7 179.8 161.4 229.7 104.4 185.6 188.7 143.7 107.8 98.0 101.9 57.8 188.7 +5.3 110.0 112.5 116.0 −66.8 +3.8 −118.0 −31.1 136.3 223.8 129.7 59.5 55.3 54.4 123.8 120.2
60
100
200
400
760
271.5
283.8
302.8
322.0
341.3
38.3 83.6 −11.2 273.8 148.2 222.0 70.4 153.2 132.2 133.4 −125.2 213.5 202.5 201.5 158.2 168.0 76.0 126.3 95.7 33.2 207.0 165.7 −48.8 149.5 95.3 89.5 85.8 13.3 205.5 141.1 126.3 123.8 38.1 29.1 41.8 41.6 77.3 110.0 135.5 122.3 142.2 221.2 305.2 184.6 190.2 173.2 239.8 115.2 197.2 199.5 153.7 116.6 107.3 122.4 66.9 199.8 13.0 119.8 122.3 126.1 −61.1 11.8 −113.0 −24.0 146.4 236.0 139.8 68.8 64.4 63.5 133.7 131.5
50.0 96.1 −1.7 289.8 162.5 236.0 82.7 167.5 146.8 147.7 −116.0 228.6 217.8 214.6 174.0 184.3 88.0 141.6 112.1 44.2 225.7 179.7 −40.3 162.0 108.8 102.8 98.9 24.4 219.6 153.4 140.3 137.9 49.9 40.7 53.8 53.4 87.6 124.0 149.8 136.8 156.5 228.4 322.5 199.0 204.4 187.8 252.2 128.1 212.5 213.5 167.2 128.3 118.9 136.5 78.6 214.5 23.3 132.3 135.3 139.7 −53.2 22.8 −106.2 −15.0 159.8 251.5 152.2 81.3 76.8 75.9 146.8 146.0
67.7 115.6 +13.5 313.5 185.0 255.2 101.0 188.0 168.3 168.3 −101.0 250.8 242.2 235.2 196.0 208.0 106.0 163.7 136.0 60.1 255.0 201.3 −27.0 182.3 129.0 122.7 118.6 41.2 241.3 172.8 160.3 158.4 67.8 58.1 72.0 71.3 102.2 145.0 171.8 157.8 179.8 239.7 349.8 220.2 224.4 209.7 271.8 149.3 237.8 232.8 187.7 146.0 136.2 157.7 97.7 237.3 38.4 151.0 154.0 159.0 −41.3 38.7 −95.4 −1.0 180.0 275.3 172.3 100.2 95.5 94.6 166.4 168.0
87.5 137.0 30.2 339.8 209.4 276.5 121.8 210.5 193.7 193.2 −81.0 276.0 267.5 256.6 221.0 235.0 125.7 187.0 163.5 78.7 288.5 224.3 −12.5 203.7 152.0 145.4 141.0 60.4 264.2 193.8 184.5 183.5 88.2 78.0 92.7 91.8 118.4 167.8 196.1 182.3 205.5 249.8 379.2 244.3 247.0 232.8 292.7 171.9 262.8 254.0 209.8 165.0 155.2 182.3 118.7 260.0 55.5 172.1 175.8 182.0 −28.0 58.3 −84.0 +14.8 202.8 301.5 194.1 121.7 116.7 115.9 188.3 193.7
108.2 158.0 47.6 366.5 234.0 299.0 143.5 233.5 218.0 217.5 −56.2 301.0 294.0 278.3 247.0 262.0 146.0 211.0 192.1 98.3 324.0 247.5 +2.9 234.5 176.1 169.2 164.7 80.9 287.0 214.2 208.1 208.0 109.8 99.2 114.8 113.5 135.0 192.7 221.2 207.0 231.0 259.2 413.5 267.2 269.5 256.6 313.0 195.8 290.0 275.0 232.0 184.4 175.1 207.5 140.8 285.0 72.5 194.0 197.5 204.5 −13.8 78.5 −72.2 31.7 225.0 328.5 215.5 144.4 139.1 138.3 211.5 217.9
Temperature, °C
Formula C18H12Cl3O3 188.2 PS C3H5Cl3 −28.8 C3H5Cl3 +9.0 C2Cl3F3 −68.0 C23H48 170.0 C13H28 59.4 C13H26O2 137.8 C7H18O3Si −1.5 C12H20O3Si 71.0 C12H18 46.0 C12H18 47.9 C6H15B C15H26O6 C12H20O7 107.0 C6H14O4 114.0 C13H30Si 70.0 C14H32Si 73.7 C7H16O3 +5.5 C6H15O4P 39.6 C6H15Tl +9.3 C6H5F3Si −31.0 C12H21PO4 93.7 C11H14O 79.0 C3H9N −97.1 C9H13N 68.4 C9H12 16.8 C9H12 13.6 C9H12 9.6 C7H16 C9H14O7 106.2 C3H8O2 59.4 C11H16 43.7 C11H16 38.8 C8H18 −29.0 C8H18 −36.5 C8H18 −25.8 C8H18 −26.3 C8H16O 14.7 C3H9O4P 26.0 C11H14 48.1 C11H14 37.5 C7H10O3 53.5 C19H16 169.7 C18H15O4P 193.5 C9H20O4 96.0 C13H28O4 101.5 C12H26O4 82.4 C21H21O4P 154.6 C11H24 32.7 C11H22O2 101.4 C11H20O2 114.0 C11H24O 71.1 C5H10O2 42.2 C5H10O2 34.5 C5H8O2 37.5 C5H9N −6.0 C8H8O3 107.0 C4H6O2 −48.0 C9H10O 41.9 C9H10O 43.4 C9H10O 45.2 C2H3Cl −105.6 C3H3N −51.0 C2H3F −149.3 C2H2Cl2 −77.2 C10H12O 64.0 C12H9Cl2PO 138.2 C9H10O 59.0 C8H10 −3.8 C8H10 −6.9 C8H10 −8.1 C8H11N 52.6 C8H11N 44.0
40
150.2 138.7 144.0 99.8 104.8 29.2 67.8 37.6 −9.7 131.0 108.0 −81.7 95.9 42.9 38.3 34.7 146.2 87.2 71.2 67.0 −7.1 −15.0 −3.9 −4.1 36.0 53.7 77.0 65.7 82.6 188.4 230.4 125.7 131.6 112.4 184.2 59.7 133.1 142.8 99.0 67.7 59.6 65.8 +18.1 138.4 −28.0 68.0 69.9 72.0 −90.8 −30.7 −138.0 −60.0 91.7 171.1 85.9 +20.2 +16.8 +15.5 79.8 72.6
Melting point, °C
−77.7 −14.7 −35 47.7 −6.2 41
135
−63.0
−117.1 67 −25.5 −44.1 −44.8 −25.0 78.5
−112.3 −107.3 −101.5 −109.2
93.4 49.4
−25.6 29.5 24.5 −34.5 −37.6 81.5
−153.7 −82 −160.5 −122.5 75 −25.2 −47.9 +13.3
2-76
PHYSICAL AND CHEMICAL DATA
VAPOR PRESSURES OF SOLUTIONS UNITS CONVERSIONS
To convert cubic feet to cubic meters, multiply by 0.02832. To convert bars to pounds-force per square inch, multiply by 14.504. To convert bars to kilopascals, multiply by 1 × 102.
For this subsection, the following units conversions are applicable: °F = 9⁄5°C + 32. To convert millimeters of mercury to pounds-force per square inch, multiply by 0.01934.
TABLE 2-9
Partial Pressures of Water over Aqueous Solutions of HCl* log10 pmm = A − B/T, (T in K), which, however, agrees only approximately with the table. The table is more nearly correct. Partial pressure of H2O, mmHg, °C
% HCl
A
B
0°
5°
10°
15°
20°
25°
30°
35°
40°
45°
50°
60°
6 10 14 18 20
8.99156 8.99864 8.97075 8.98014 8.97877
2282 2295 2300 2323 2334
4.18 3.84 3.39 2.87 2.62
6.04 5.52 4.91 4.21 3.83
8.45 7.70 6.95 5.92 5.40
11.7 10.7 9.65 8.26 7.50
15.9 14.6 13.1 11.3 10.3
21.8 20.0 18.0 15.4 14.1
29.1 26.8 24.1 20.6 19.0
39.4 35.5 31.9 27.5 25.1
50.6 47.0 42.1 36.4 33.3
66.2 61.5 55.3 47.9 43.6
86.0 80.0 72.0 62.5 57.0
139 130 116 102 93.5
22 24 26 28 30
9.02708 8.96022 9.01511 8.97611 9.00117
2363 2356 2390 2395 2422
2.33 2.05 1.76 1.50 1.26
3.40 3.04 2.60 2.24 1.90
4.82 4.31 3.71 3.21 2.73
6.75 6.03 5.21 4.54 3.88
9.30 8.30 7.21 6.32 5.41
12.6 11.4 9.95 8.75 7.52
17.1 15.4 13.5 11.8 10.2
22.8 20.4 18.0 15.8 13.7
30.2 27.1 24.0 21.1 18.4
39.8 35.7 31.7 27.9 24.3
52.0 46.7 41.5 36.5 32.0
32 34 36 38 40 42
9.03317 9.07143 9.11815 9.20783 9.33923 9.44953
2453 2487 2526 2579 2647 2709
1.04 0.85 0.68 0.53 0.41 0.31
1.57 1.29 1.03 0.81 0.63 0.48
2.27 1.87 1.50 1.20 0.94 0.72
3.25 2.70 2.19 1.75 1.37 1.06
4.55 3.81 3.10 2.51 2.00 1.56
6.37 5.35 4.41 3.60 2.88 2.30
11.7 9.95 8.33 6.92 5.68 4.60
15.7 13.5 11.4 9.52 7.85 6.45
21.0 18.1 15.4 13.0 10.7 8.90
27.7 24.0 20.4 17.4 14.5 12.1
8.70 7.32 6.08 5.03 4.09 3.28
70°
80°
90°
100° 110°
220 204 185 162 150
333 310 273 248 230
492 463 425 374 345
715 677 625 550 510
960 892 783 729
85.6 77.0 69.0 60.7 53.5
138 124 112 99.0 87.5
211 194 173 154 136
317 290 261 234 207
467 426 387 349 310
670 611 555 499 444
46.5 40.5 34.8 29.6 25.0 21.2
76.5 66.5 57.0 49.1 42.1 35.8
120 104 90.0 77.5 67.3 57.2
184 161 140 120 105 89.2
275 243 212 182 158 135
396 355 311 266 230 195
*Accuracy, ca. 2 percent for solutions of 15 to 30 percent HCl between 0 and 100°; for solutions of > 30 percent HCl the accuracy is ca. 5 percent at the lower temperatures and ca. 15 percent at the higher temperatures. Below 15 percent HCl, the accuracy is ca. 5 percent at the lower temperatures and higher strengths to ca. 15 to 20 percent at the lower strengths and perhaps 15 to 20 percent at the higher temperatures and lower strengths.
TABLE 2-10
Partial Pressures of HCl over Aqueous Solutions of HCl*
log10 pmm = A − B/T, (T in K), which, however, agrees only approximately with the table. The table is more nearly correct. mmHg, °C
% HCl
A
2 4 6 8 10
11.8037 11.6400 11.2144 11.0406 10.9311
4736 0.0000117 0.000023 0.000044 0.000084 0.000151 0.000275 0.00047 0.00083 0.00140 0.00380 0.0100 0.0245 0.058 0.132 0.280 4471 0.000018 0.000036 .000069 .000131 .00024 .00044 .00077 .00134 .0023 .00385 .0064 .0165 .0405 .095 .21 .46 .93 4202 .000066 .000125 .000234 .000425 .00076 .00131 .00225 .0038 .0062 .0102 .0163 .040 .094 .206 .44 .92 1.78 4042 .000118 .000323 .000583 .00104 .00178 .0031 .00515 .0085 .0136 .022 .0344 .081 .183 .39 .82 1.64 3.10 3908 .00042 .00075 .00134 .00232 .00395 .0067 .0111 .0178 .0282 .045 .069 .157 .35 .73 1.48 2.9 5.4
12 14 16 18 20
10.7900 10.6954 10.6261 10.4957 10.3833
3765 3636 3516 3376 3245
22 24 26 28 30
10.3172 10.2185 10.1303 10.0115 9.8763
3125 .0734 2995 .175 2870 .41 2732 1.0 2593 2.4
.119 .277 .64 1.52 3.57
.187 .43 .98 2.27 5.23
.294 .66 1.47 3.36 7.60
.45 1.00 2.17 4.90 10.6
.68 1.49 3.20 7.05 15.1
1.02 2.17 4.56 9.90 21.0
32 34 36 38 40
9.7523 9.6061 9.5262 9.4670 9.2156
2457 2316 2229 2094 1939
5.7 13.1 29.0 63.0 130
8.3 18.8 41.0 87.0 176
11.8 26.4 56.4 117 233
16.8 36.8 78 158 307
23.5 50.5 105.5 210 399
32.5 68.5 142 277 515
44.5 92 188 360 627
42 44 46
8.9925 1800 253 8.8621 1681 510 940
332 655
430 840
560
709
900
B
0°
.00099 .0024 .0056 .0135 .0316
5°
.00175 .00415 .0095 .0225 .052
10°
.00305 .0071 .016 .037 .084
15°
.0052 .0118 .0265 .060 .132
20°
.0088 .0196 .0428 .095 .205
25°
.0145 .0316 .0685 .148 .32
30°
.0234 .050 .106 .228 .48
35°
.037 .078 .163 .345 .72
40°
45°
50°
60°
70°
80°
90°
100°
110°
.058 .121 .247 .515 1.06
.091 .185 .375 .77 1.55
.136 .275 .55 1.11 2.21
.305 .60 1.17 2.3 4.4
.66 1.25 2.40 4.55 8.5
1.34 2.50 4.66 8.6 15.6
2.65 4.8 8.8 15.7 28.1
5.1 9.0 16.1 28 49
9.3 16.0 28 48 83
1.50 3.14 6.50 13.8 28.6
2.18 4.5 9.2 19.1 39.4
3.14 6.4 12.7 26.4 53
4.42 8.9 17.5 35.7 71
8.6 16.9 32.5 64 124
16.3 31.0 58.5 112 208
29.3 54.5 100 188 340
52 94 169 309 542
90 157 276 493 845
146 253 436 760
60.0 122 246 465 830
81 161 322 598
107 211 416 758
141 273 535 955
238 450 860
390 720
623
970
*Accuracy, ca. 2 percent for solutions of 15 to 30 percent HCl between 0 and 100°; for solutions of > 30 percent HCl the accuracy is ca. 5 percent at the lower temperatures and ca. 15 percent at the higher temperatures. Below 15 percent HCl, the accuracy is ca. 5 percent at the lower temperatures and higher strengths to ca. 15 to 20 percent at the lower strengths and perhaps 15 to 20 percent at the higher temperatures and lower strengths.
VAPOR PRESSURES OF SOLUTIONS
FIG. 2-1 Vapor pressures of H3PO4 aqueous: partial pressure of H2O vapor. (Courtesy of Victor Chemical Works, Stauffer Chemical Company; measurements by W. H. Woodstock.)
TABLE 2-11 g SO2 / 100 g H2O
Temperature, ∞C 10
20
0.01 0.05 0.10 0.15 0.20
0.02 0.38 1.15 2.10 3.17
0.04 0.66 1.91 3.44 5.13
0.07 1.07 3.03 5.37 7.93
0.25 0.30 0.40 0.50 1.00
4.34 5.57 8.17 10.9 25.8
6.93 8.84 12.8 17.0 39.5
8.00 10.00 15.00 20.00
Vapor pressures of H3PO4 aqueous: weight of H2O in saturated air. (Courtesy of Victor Chemical Works, Stauffer Chemical Company; measurements by W. H. Woodstock.)
FIG. 2-2
Partial Pressures of H2O and SO2 over Aqueous Solutions of Sulfur Dioxide* Partial pressures of H2O and SO2, mmHg, °C 0
2.00 3.00 4.00 5.00 6.00
2-77
10.6 13.5 19.4 25.6 58.4
30
40
50
60
90
120
0.12 1.68 4.62 8.07 11.8
0.19 2.53 6.80 11.7 17.0
0.29 3.69 9.71 16.5 23.8
0.43 5.24 13.5 22.7 32.6
1.21 12.9 31.7 52.2 73.7
2.82 27.0 63.9 104 145
15.7 19.8 28.3 37.1 83.7
58.6 93.2 129 165 202
88.5 139 192 245 299
129 202 277 353 430
183 285 389 496 602
275 351 542 735
407 517 796
585 741
818
22.5 28.2 40.1 52.3 117
31.4 39.2 55.3 72.0 159
42.8 53.3 74.7 96.8 212
95.8 118 164 211 454
253 393 535 679 824
342 530 720
453 700
955
*Extracted with permission from J. Chem Eng. Data 8, 1963: 333–336. Copyright 1963 American Chemical Society.
186 229 316 404 856
2-78
PHYSICAL AND CHEMICAL DATA
TABLE 2-12 °C 0 10 20 30 40 50 60 70 80 90
Water Partial Pressure, bar, over Aqueous Sulfuric Acid Solutions* Weight percent, H2SO4
10.0 .582E−02 .117E−01 .223E−01 .404E−01 .703E−01 .117 .189 .296 .449 .664
20.0 .534E−02 .107E−01 .205E−01 .373E−01 .649E−01 .109 .175 .275 .417 .617
30.0 .448E−02 .909E−02 .174E−01 .319E−01 .558E−01 .939E−01 .152 .239 .365 .542
40.0 .326E−02 .670E−02 .130E−01 .241E−01 .427E−01 .725E−01 .119 .188 .290 .434
50.0
60.0
.193E−02 .405E−02 .802E−02 .151E−01 .272E−01 .470E−01 .782E−01 .126 .196 .298
.836E−03 .180E−02 .367E−02 .710E−02 .131E−01 .232E−01 .395E−01 .651E−01 .104 .161
75.0
80.0
85.0
.207E−03 .467E−03 .995E−03 .201E−02 .387E−02 .715E−02 .127E−01 .217E−01 .360E−01 .578E−01
70.0
.747E−04 .175E−03 .388E−03 .811E−03 .162E−02 .309E−02 .565E−02 .997E−02 .170E−01 .281E−01
.197E−04 .490E−04 .115E−04 .253E−03 .531E−03 .106E−02 .204E−02 .376E−02 .668E−02 .115E−01
.343E−05 .952E−05 .245E−04 .589E−04 .133E−03 .286E−03 .584E−03 .114E−02 .213E−02 .383E−02
.905E−01 .138 .206 .301 .481 .605 .837 1.138 1.525 2.017
.452E−01 .708E−01 .108 .162 .236 .339 .478 .662 .902 1.212
.192E−01 .312E−01 .493E−01 .760E−01 .115 .170 .246 .350 .489 .673
.666E−02 .112E−01 .183E−01 .291E−01 .451E−01 .682E−01 .101 .147 .208 .291
100 110 120 130 140 150 160 170 180 190
.957 1.349 1.863 2.524 3.361 4.404 5.685 7.236 9.093 11.289
.891 1.258 1.740 2.361 3.149 4.132 5.342 6.810 8.571 10.658
.786 1.113 1.544 2.101 2.810 3.697 4.793 6.127 7.731 9.640
.634 .904 1.264 1.732 2.333 3.090 4.031 5.185 6.584 8.259
.441 .638 .903 1.253 1.708 2.289 3.021 3.930 5.045 6.397
.244 .360 .519 .734 1.020 1.392 1.870 2.475 3.233 4.169
200 210 220 230 240 250 260 270 280 290
13.861 16.841 20.264 24.160 28.561 33.494 38.984 45.055 51.726 59.015
13.107 15.951 19.225 22.960 27.188 31.939 37.240 43.116 49.590 56.681
11.887 14.505 17.529 20.992 24.927 29.364 34.334 39.865 45.984 52.715
10.245 12.576 15.287 18.414 21.992 26.056 30.642 35.784 41.514 47.865
8.020 9.948 12.217 14.864 17.929 21.452 25.472 30.030 35.168 40.926
5.312 6.696 8.354 10.322 12.641 15.351 18.496 22.121 26.274 31.003
2.632 3.395 4.331 5.466 6.831 8.458 10.382 12.640 15.269 18.311
1.606 2.101 2.714 3.467 4.381 5.480 6.788 8.333 10.142 12.242
.913 1.220 1.609 2.096 2.699 3.435 4.326 5.395 6.663 8.155
.401 .542 .724 .952 1.237 1.587 2.012 2.525 3.136 3.857
300 66.934 310 75.495 320 84.705 330 94.567 340 105.083 350 116.251
64.407 72.781 81.816 91.518 101.894 112.946
60.081 68.100 76.792 86.172 96.252 107.043
54.868 62.553 70.947 80.077 89.969 100.646
47.346 54.470 62.337 70.988 80.463 90.802
36.360 42.395 49.164 56.721 65.123 74.426
21.808 25.804 30.343 35.473 41.240 47.692
14.665 17.438 20.591 24.153 28.154 32.622
9.897 11.912 14.227 16.867 19.855 23.217
4.701 5.680 6.806 8.093 9.551 11.193
*Vermeulen, Dong, Robinson, Nguyen, and Gmitro, AIChE meeting, Anaheim, Calif., 1982; and private communication from Prof. Theodore Vermeulen, Chemical Engineering Dept., University of California, Berkeley.
VAPOR PRESSURES OF SOLUTIONS TABLE 2-12
2-79
Water Partial Pressure, bar, over Aqueous Sulfuric Acid Solutions (Concluded ) Weight percent, H2SO4
°C
90.0
92.0
94.0
96.0
97.0
98.0
98.5
99.0
99.5
100.0
0 10 20 30 40 50 60 70 80 90
.518E−06 .159E−05 .448E−05 .117E−04 .285E−04 .652E−04 .141E−03 .290E−03 .569E−03 .107E−02
.242E−06 .762E−06 .220E−05 .587E−05 .146E−04 .341E−04 .754E−04 .158E−03 .316E−03 .606E−03
.107E−06 .344E−06 .101E−05 .275E−05 .696E−05 .166E−04 .372E−04 .795E−04 .162E−03 .315E−03
.401E−07 .130E−06 .390E−06 .108E−05 .278E−05 .672E−05 .154E−04 .334E−04 .691E−04 .137E−03
.218E−07 .713E−07 .215E−06 .598E−06 .155E−05 .379E−05 .875E−05 .192E−04 .400E−04 .801E−04
.980E−08 .323E−07 .978E−07 .275E−06 .720E−06 .177E−05 .413E−05 .912E−05 .192E−04 .388E−04
.569E−08 .188E−07 .572E−07 .161E−06 .424E−06 .105E−05 .245E−05 .544E−05 .115E−04 .234E−04
.268E−08 .888E−08 .271E−07 .766E−07 .202E−06 .503E−06 .118E−05 .263E−05 .559E−05 .114E−04
.775E−09 .258E−08 .789E−08 .224E−07 .595E−07 .149E−06 .350E−06 .784E−06 .168E−05 .343E−05
.196E−09 .655E−09 .201E−08 .575E−08 .153E−07 .384E−07 .910E−07 .205E−06 .439E−06 .903E−06
100 110 120 130 140 150 160 170 180 190
.194E−02 .338E−02 .571E−02 .938E−02 .150E−01 .233E−01 .354E−01 .526E−01 .766E−01 .110
.112E−02 .198E−02 .341E−02 .569E−02 .923E−02 .146E−01 .225E−01 .340E−01 .502E−01 .729E−01
.590E−03 .107E−02 .186E−02 .315E−02 .519E−02 .832E−02 .130E−01 .199E−01 .298E−01 .438E−01
.261E−03 .479E−03 .851E−03 .146E−02 .245E−02 .399E−02 .633E−02 .983E−02 .149E−01 .222E−01
.154E−03 .285E−03 .511E−03 .886E−03 .149E−02 .245E−02 .393E−02 .614E−02 .941E−02 .141E−01
.752E−04 .141E−03 .254E−03 .445E−03 .757E−03 .125E−02 .202E−02 .319E−02 .492E−02 .744E−02
.455E−04 .855E−04 .155E−03 .278E−03 .467E−03 .776E−03 .126E−02 .199E−02 .309E−02 .469E−02
.223E−04 .420E−04 .766E−04 .135E−03 .232E−03 .387E−03 .629E−03 .999E−03 .155E−02 .236E−02
.674E−05 .128E−04 .233E−04 .414E−04 .711E−04 .119E−03 .194E−03 .309E−03 .482E−03 .735E−03
.178E−05 .339E−05 .623E−05 .111E−04 .191E−04 .321E−04 .526E−04 .840E−04 .131E−03 .201E−03
.631E−01 .894E−01 .125 .171 .232 .310 .409 .534 .689 .880
.325E−01 .467E−01 .660E−01 .918E−01 .126 .170 .227 .300 .391 .505
.208E−01 .300E−01 .427E−01 .598E−01 .825E−01 .112 .151 .200 .263 .341
.110E−01 .161E−01 .230E−01 .325E−01 .451E−01 .618E−01 .835E−01 .111 .147 .192
.698E−02 .102E−01 .147E−01 .208E−01 .290E−01 .398E−01 .540E−01 .723E−01 .957E−01 .125
.352E−02 .516E−02 .743E−02 .105E−01 .147E−01 .202E−01 .274E−01 .366E−01 .485E−01 .634E−01
.110E−02 .161E−02 .232E−02 .329E−02 .460E−02 .633E−02 .858E−02 .115E−01 .152E−01 .199E−01
.300E−03 .442E−03 .638E−03 .906E−03 .127E−02 .174E−02 .237E−02 .317E−02 .420E−02 .548E−02
.248 .316 .400 .502 .624 .770
.162 .208 .264 .331 .413 .511
.820E−01 .105 .133 .167 .208 .256
.257E−01 .328E−01 .415E−01 .520E−01 .646E−01 .795E−01
.708E−02 .905E−02 .114E−01 .143E−01 .178E−01 .218E−01
200 210 220 230 240 250 260 270 280 290
.154 .213 .290 .389 .514 .673 .870 1.112 1.407 1.763
.104 .146 .201 .273 .366 .485 .635 .822 1.052 1.335
300 310 320 330 340 350
2.190 2.696 3.292 3.990 4.801 5.738
1.676 2.088 2.578 3.159 3.843 4.641
1.112 1.394 1.732 2.133 2.608 3.164
.646 .817 1.025 1.274 1.571 1.922
.437 .556 .701 .875 1.083 1.331
2-80
PHYSICAL AND CHEMICAL DATA
TABLE 2-13
Sulfur Trioxide Partial Pressure, bar, over Aqueous Sulfuric Acid Solutions* Weight percent, H2SO4
°C
10.0
20.0
30.0
40.0
50.0
60.0
70.0
75.0
80.0
85.0
0 10 20 30 40 50 60 70 80 90
.644E−29 .149E−27 .278E−26 .426E−25 .549E−24 .602E−23 .573E−22 .477E−21 .352E−20 .233E−19
.103E−27 .223E−26 .394E−25 .577E−24 .714E−23 .757E−22 .699E−21 .567E−20 .410E−19 .266E−18
.205E−26 .395E−25 .626E−24 .832E−23 .941E−22 .921E−21 .789E−20 .599E−19 .408E−18 .250E−17
.688E−25 .113E−23 .156E−22 .181E−21 .181E−20 .158E−19 .122E−18 .843E−18 .524E−17 .296E−16
.368E−23 .522E−22 .621E−21 .630E−20 .555E−19 .429E−18 .294E−17 .181E−16 .101E−15 .516E−15
.341E−21 .415E−20 .426E−19 .376E−18 .288E−17 .195E−16 .118E−15 .643E−15 .319E−14 .145E−13
.784E−19 .796E−18 .685E−17 .509E−16 .331E−15 .191E−14 .985E−14 .461E−13 .197E−12 .775E−12
.174E−17 .158E−16 .121E−15 .808E−15 .473E−14 .246E−13 .116E−12 .492E−12 .192E−11 .693E−11
.531E−16 .417E−15 .280E−14 .164E−13 .851E−13 .395E−12 .165E−11 .634E−11 .223E−10 .731E−10
.229E−14 .141E−13 .767E−13 .371E−12 .162E−11 .643E−11 .234E−10 .791E−10 .249E−09 .734E−09
100 110 120 130 140 150 160 170 180 190
.139E−18 .756E−18 .377E−17 .174E−16 .743E−16 .297E−15 .111E−14 .393E−14 .131E−13 .415E−13
.157E−17 .844E−17 .418E−16 .191E−15 .815E−15 .325E−14 .122E−13 .430E−13 .144E−12 .458E−12
.140E−16 .719E−16 .340E−15 .150E−14 .615E−14 .237E−13 .862E−13 .296E−12 .967E−12 .301E−11
.153E−15 .730E−15 .323E−14 .133E−13 .517E−13 .188E−12 .649E−12 .212E−11 .622E−11 .197E−10
.242E−14 .105E−13 .424E−13 .160E−12 .569E−12 .191E−11 .608E−11 .184E−10 .532E−10 .147E−09
.606E−13 .236E−12 .858E−12 .293E−11 .943E−11 .287E−10 .833E−10 .231E−09 .610E−09 .155E−08
.283E−11 .961E−11 .307E−10 .922E−10 .262E−09 .710E−09 .183E−08 .453E−08 .107E−07 .246E−07
.232E−10 .729E−10 .215E−09 .601E−09 .159E−08 .403E−08 .974E−08 .226E−07 .505E−07 .109E−06
.223E−09 .641E−09 .174E−08 .446E−08 .109E−07 .256E−07 .575E−07 .125E−06 .260E−06 .527E−06
.204E−08 .538E−08 .135E−07 .324E−07 .745E−07 .165E−06 .351E−06 .725E−06 .145E−05 .282E−05
200 210 220 230 240 250 260 270 280 290
.125E−12 .362E−12 .100E−11 .265E−11 .678E−11 .167E−10 .399E−10 .920E−10 .206E−09 .449E−09
.139E−11 .404E−11 .112E−10 .301E−10 .777E−10 .193E−09 .466E−09 .109E−08 .247E−08 .545E−08
.893E−11 .254E−10 .695E−10 .183E−09 .465E−09 .114E−08 .272E−08 .628E−08 .141E−07 .308E−07
.561E−10 .154E−09 .405E−09 .103E−08 .253E−08 .602E−08 .139E−07 .312E−07 .683E−07 .145E−06
.391E−09 .100E−08 .246E−08 .587E−08 .135E−07 .303E−07 .660E−07 .140E−06 .288E−06 .580E−06
.379E−08 .894E−08 .204E−07 .450E−07 .965E−07 .201E−06 .408E−06 .807E−06 .156E−05 .295E−05
.542E−07 .116E−06 .240E−06 .482E−06 .944E−06 .180E−05 .336E−05 .612E−05 .109E−04 .191E−04
.228E−06 .462E−06 .911E−06 .175E−05 .328E−05 .600E−05 .108E−04 .189E−04 .326E−04 .553E−04
.103E−05 .198E−05 .368E−05 .668E−05 .119E−04 .206E−04 .352E−04 .590E−04 .973E−04 .158E−03
.534E−05 .986E−05 .178E−04 .314E−04 .543E−04 .923E−04 .154E−03 .253E−03 .408E−03 .649E−03
300 310 320 330 340 350
.953E−09 .197E−08 .397E−08 .782E−08 .151E−07 .285E−07
.117E−07 .245E−07 .502E−07 .100E−06 .196E−06 .376E−06
.657E−07 .136E−06 .277E−06 .551E−06 .107E−05 .204E−05
.302E−06 .614E−06 .122E−05 .237E−05 .452E−05 .846E−05
.114E−05 .220E−05 .414E−05 .766E−05 .139E−04 .246E−04
.546E−05 .990E−05 .176E−04 .308E−04 .529E−04 .893E−04
.329E−04 .556E−04 .923E−04 .151E−03 .243E−03 .387E−03
.921E−04 .151E−03 .245E−03 .391E−03 .617E−03 .963E−03
.253E−03 .398E−03 .621E−03 .956E−03 .145E−02 .219E−02
.102E−02 .158E−02 .242E−02 .367E−02 .550E−02 .815E−02
*Vermeulen, Dong, Robinson, Nguyen, and Gmitro, AIChE meeting, Anaheim, Calif., 1982; and private communication from Prof. Theodore Vermeulen, Chemical Engineering Dept., University of California, Berkeley.
VAPOR PRESSURES OF SOLUTIONS TABLE 2-13
2-81
Sulfur Trioxide Partial Pressure, bar, over Aqueous Sulfuric Acid Solutions (Concluded ) Weight percent, H2SO4
°C
90.0
92.0
94.0
96.0
97.0
98.0
98.5
99.0
99.5
0 10 20 30 40 50 60 70 80 90
.671E−13 .345E−12 .159E−11 .664E−11 .254E−10 .897E−10 .294E−09 .904E−09 .261E−08 .712E−08
.216E−12 .107E−11 .475E−11 .192E−10 .709E−10 .242E−09 .771E−09 .230E−08 .643E−08 .171E−07
.677E−12 .326E−11 .141E−10 .557E−10 .201E−09 .669E−09 .207E−08 .602E−08 .165E−07 .426E−07
.240E−11 .114E−10 .482E−10 .186E−09 .655E−09 .214E−08 .647E−08 .184E−07 .492E−07 .124E−06
.500E−11 .234E−10 .986E−10 .376E−09 .131E−08 .424E−08 .127E−07 .357E−07 .946E−07 .237E−06
.124E−10 .578E−10 .241E−09 .911E−09 .315E−08 .101E−07 .299E−07 .833E−07 .218E−06 .541E−06
.224E−10 .104E−09 .433E−09 .163E−08 .562E−08 .179E−07 .528E−07 .146E−06 .381E−06 .940E−06
.502E−10 .232E−09 .961E−09 .360E−08 .123E−07 .391E−07 .115E−06 .316E−06 .820E−06 .201E−05
.182E−09 .839E−09 .346E−08 .129E−07 .440E−07 .139E−06 .405E−06 .111E−05 .286E−05 .698E−05
.755E−09 .347E−08 .142E−07 .528E−07 .179E−06 .560E−06 .163E−05 .444E−05 .114E−04 .276E−04
100 110 120 130 140 150 160 170 180 190
.184E−07 .456E−07 .108E−06 .244E−06 .533E−06 .112E−05 .229E−05 .453E−05 .870E−05 .163E−04
.430E−07 .103E−06 .238E−06 .526E−06 .112E−05 .230E−05 .459E−05 .886E−05 .166E−04 .304E−04
.105E−06 .247E−06 .555E−06 .120E−05 .250E−05 .504E−05 .983E−05 .186E−04 .343E−04 .615E−04
.300E−06 .689E−06 .152E−05 .321E−05 .656E−05 .129E−04 .247E−04 .459E−04 .829E−04 .146E−03
.565E−06 .128E−05 .280E−05 .586E−05 .118E−04 .231E−04 .438E−04 .806E−04 .144E−03 .252E−03
.127E−05 .287E−05 .619E−05 .128E−04 .257E−04 .497E−04 .932E−04 .170E−03 .301E−03 .520E−03
.220E−05 .494E−05 .106E−04 .219E−04 .435E−04 .837E−04 .156E−03 .283E−03 .499E−03 .859E−03
.470E−05 .105E−04 .224E−04 .459E−04 .910E−04 .174E−03 .324E−03 .586E−03 .103E−02 .177E−02
.162E−04 .359E−04 .764E−04 .156E−03 .308E−03 .588E−03 .109E−02 .196E−02 .343E−02 .587E−02
.638E−04 .141E−03 .298E−03 .606E−03 .119E−02 .226E−02 .416E−02 .746E−02 .130E−01 .222E−01
200 210 220 230 240 250 260 270 280 290
.297E−04 .528E−04 .919E−04 .157E−03 .261E−03 .428E−03 .690E−03 .109E−02 .170E−02 .261E−02
.543E−04 .946E−04 .161E−03 .269E−03 .441E−03 .708E−03 .112E−02 .174E−02 .266E−02 .401E−02
.108E−03 .185E−03 .309E−03 .508E−03 .819E−03 .130E−02 .202E−02 .309E−02 .466E−02 .694E−02
.251E−03 .422E−03 .694E−03 .112E−02 .178E−02 .276E−02 .423E−02 .638E−02 .948E−02 .139E−01
.429E−03 .714E−03 .117E−02 .187E−02 .293E−02 .453E−02 .688E−02 .103E−01 .152E−01 .221E−01
.878E−03 .145E−02 .235E−02 .373E−02 .582E−02 .891E−02 .134E−01 .200E−01 .293E−01 .423E−01
.144E−02 .237E−02 .383E−02 .605E−02 .939E−02 .143E−01 .215E−01 .319E−01 .465E−01 .670E−01
.296E−02 .486E−02 .781E−02 .123E−01 .191E−01 .291E−01 .437E−01 .646E−01 .943E−01 .136
.981E−02 .161E−01 .258E−01 .405E−01 .627E−01 .955E−01 .143 .212 .309 .444
.370E−01 .603E−01 .965E−01 .152 .234 .356 .532 .786 1.144 1.646
300 310 320 330 340 350
.395E−02 .589E−02 .868E−02 .126E−01 .181E−01 .258E−01
.595E−02 .873E−02 .126E−01 .181E−01 .255E−01 .357E−01
.102E−01 .148E−01 .211E−01 .299E−01 .418E−01 .578E−01
.201E−01 .287E−01 .405E−01 .565E−01 .780E−01 .107
.318E−01 .451E−01 .632E−01 .877E−01 .120 .164
.604E−01 .852E−01 .119 .164 .224 .303
.953E−01 .134 .186 .256 .348 .470
.193 .272 .378 .520 .708 .956
.632 .889 1.236 1.703 2.323 3.142
100.0
2.339 3.289 4.575 6.303 8.603 11.640
TABLE 2-14
Sulfuric Acid Partial Pressure, bar, over Aqueous Sulfuric Acid* Weight Percent, H2SO4
°C
10.0
20.0
30.0
40.0
50.0
60.0
70.0
75.0
80.0
85.0
0 10 20 30 40 50 60 70 80 90
.576E−21 .634E−20 .588E−19 .468E−18 .324E−17 .197E−16 .107E−15 .526E−15 .235E−14 .960E−14
.843E−20 .874E−19 .769E−18 .584E−17 .389E−16 .229E−15 .121E−14 .581E−14 .254E−13 .102E−12
.141E−18 .131E−17 .104E−16 .721E−16 .441E−15 .241E−14 .119E−13 .535E−13 .221E−12 .844E−12
.344E−17 .276E−16 .193E−15 .119E−14 .649E−14 .320E−13 .144E−12 .592E−12 .225E−11 .798E−11
.109E−15 .769E−15 .474E−14 .259E−13 .127E−12 .562E−12 .228E−11 .851E−11 .295E−10 .956E−10
.438E−14 .273E−13 .149E−12 .725E−12 .317E−11 .126E−10 .462E−10 .156E−09 .492E−09 .145E−08
.249E−12 .135E−11 .649E−11 .278E−10 .108E−09 .380E−09 .124E−08 .373E−08 .105E−07 .279E−07
.200E−11 .101E−10 .447E−10 .178E−09 .643E−09 .212E−08 .646E−08 .183E−07 .485E−07 .121E−06
.161E−10 .743E−10 .305E−09 .113E−08 .379E−08 .117E−07 .334E−07 .888E−07 .222E−06 .522E−06
.121E−09 .490E−09 .179E−08 .594E−08 .181E−07 .513E−07 .135E−06 .336E−06 .786E−06 .175E−05
100 110 120 130 140 150 160 170 180 190
.353E−13 .127E−12 .418E−12 .129E−11 .375E−11 .103E−10 .272E−10 .682E−10 .164E−09 .378E−09
.381E−12 .132E−11 .432E−11 .132E−10 .385E−10 .106E−09 .279E−09 .702E−09 .170E−08 .394E−08
.300E−11 .997E−11 .312E−10 .924E−10 .259E−09 .694E−09 .178E−08 .436E−08 .103E−07 .234E−07
.264E−10 .824E−10 .243E−09 .678E−09 .181E−08 .460E−08 .112E−07 .264E−07 .599E−07 .131E−06
.291E−09 .835E−09 .227E−08 .589E−08 .146E−07 .346E−07 .789E−07 .174E−06 .369E−06 .760E−06
.402E−08 .106E−07 .264E−07 .631E−07 .144E−06 .316E−06 .670E−06 .137E−05 .271E−05 .521E−05
.698E−07 .166E−06 .375E−06 .814E−06 .169E−05 .340E−05 .659E−05 .124E−04 .225E−04 .400E−04
.287E−06 .644E−06 .138E−05 .285E−05 .565E−05 .108E−04 .200E−04 .359E−04 .627E−04 .107E−03
.117E−05 .249E−05 .508E−05 .995E−05 .188E−04 .343E−04 .608E−04 .104E−03 .175E−03 .286E−03
.371E−05 .752E−05 .147E−04 .277E−04 .503E−04 .889E−04 .152E−03 .255E−03 .416E−03 .663E−03
200 210 220 230 240 250 260 270 280 290
.842E−09 .181E−08 .376E−08 .758E−08 .148E−07 .283E−07 .526E−07 .954E−07 .169E−06 .294E−06
.883E−08 .191E−07 .401E−07 .817E−07 .162E−06 .312E−06 .588E−06 .108E−05 .194E−05 .342E−05
.514E−07 .109E−06 .226E−06 .455E−06 .889E−06 .170E−05 .316E−05 .577E−05 .103E−04 .180E−04
.278E−06 .573E−06 .115E−05 .224E−05 .427E−05 .793E−05 .144E−04 .257E−04 .450E−04 .771E−04
.152E−05 .295E−05 .559E−05 .103E−04 .186E−04 .329E−04 .569E−04 .965E−04 .161E−03 .263E−03
.975E−05 .178E−04 .316E−04 .549E−04 .935E−04 .156E−03 .255E−03 .411E−03 .650E−03 .101E−02
.691E−04 .117E−03 .193E−03 .311E−03 .494E−03 .770E−03 .118E−02 .178E−02 .265E−02 .389E−02
.177E−03 .288E−03 .459E−03 .717E−03 .110E−02 .166E−02 .247E−02 .362E−02 .524E−02 .750E−02
.457E−03 .715E−03 .110E−02 .166E−02 .245E−02 .358E−02 .516E−02 .733E−02 .103E−01 .143E−01
.104E−02 .159E−02 .239E−02 .354E−02 .515E−02 .740E−02 .105E−01 .147E−01 .203E−01 .278E−01
300 310 320 330 340 350
.500E−06 .834E−06 .137E−05 .220E−05 .349E−05 .544E−05
.591E−05 .100E−04 .167E−04 .273E−04 .440E−04 .698E−04
.309E−04 .522E−04 .865E−04 .141E−03 .227E−03 .360E−03
.130E−03 .215E−03 .352E−03 .565E−03 .895E−03 .140E−02
.424E−03 .672E−03 .105E−02 .162E−02 .246E−02 .369E−02
.156E−02 .236E−02 .352E−02 .519E−02 .757E−02 .109E−01
.563E−02 .805E−02 .114E−01 .159E−01 .221E−01 .303E−01
.106E−01 .148E−01 .205E−01 .281E−01 .382E−01 .516E−01
.196E−01 .266E−01 .359E−01 .480E−01 .636E−01 .836E−01
.376E−01 .504E−01 .670E−01 .883E−01 .116 .150
°C
90.0
92.0
94.0
96.0
97.0
98.0
98.5
99.0
99.5
100.0
0 10 20 30 40 50 60 70 80 90
.534E−09 .200E−08 .677E−08 .211E−07 .607E−07 .163E−06 .411E−06 .976E−06 .220E−05 .473E−05
.803E−09 .296E−08 .993E−08 .306E−07 .870E−07 .231E−06 .575E−06 .135E−05 .302E−05 .642E−05
.112E−08 .409E−08 .136E−07 .415E−07 .117E−06 .309E−06 .765E−06 .179E−05 .396E−05 .835E−05
.148E−08 .540E−08 .179E−07 .543E−07 .153E−06 .400E−06 .985E−06 .229E−05 .504E−05 .106E−04
.167E−08 .609E−08 .201E−07 .611E−07 .171E−06 .449E−06 .110E−05 .256E−05 .562E−05 .118E−04
.187E−08 .679E−08 .224E−07 .680E−07 .191E−06 .498E−06 .122E−05 .283E−05 .622E−05 .130E−04
.196E−08 .714E−08 .236E−07 .714E−07 .200E−06 .523E−06 .128E−05 .297E−05 .652E−05 .136E−04
.206E−08 .750E−08 .247E−07 .749E−07 .210E−06 .548E−06 .134E−05 .310E−05 .681E−05 .143E−04
.217E−08 .788E−08 .260E−07 .786E−07 .220E−06 .574E−06 .140E−05 .325E−05 .712E−05 .149E−04
.228E−08 .827E−08 .273E−07 .824E−07 .230E−06 .600E−06 .147E−05 .339E−05 .743E−05 .155E−04
100 110 120 130 140 150 160 170 180 190
.973E−05 .192E−04 .366E−04 .672E−04 .120E−03 .207E−03 .348E−03 .572E−03 .917E−03 .144E−02
.131E−04 .256E−04 .482E−04 .879E−04 .155E−03 .266E−03 .444E−03 .723E−03 .115E−02 .179E−02
.169E−04 .328E−04 .614E−04 .111E−03 .195E−03 .332E−03 .550E−03 .889E−03 .140E−02 .217E−02
.213E−04 .412E−04 .767E−04 .138E−03 .241E−03 .408E−03 .673E−03 .108E−02 .170E−02 .262E−02
.237E−04 .457E−04 .849E−04 .153E−03 .266E−03 .449E−03 .740E−03 .119E−02 .186E−02 .286E−02
.261E−04 .503E−04 .935E−04 .168E−03 .292E−03 .493E−03 .810E−03 .130E−02 .204E−02 .312E−02
.274E−04 .527E−04 .977E−04 .175E−03 .304E−03 .514E−03 .844E−03 .135E−02 .212E−02 .325E−02
.285E−04 .549E−04 .102E−03 .182E−03 .316E−03 .534E−03 .876E−03 .140E−02 .220E−02 .336E−02
.298E−04 .572E−04 .106E−03 .190E−03 .329E−03 .554E−03 .909E−03 .145E−02 .227E−02 .348E−02
.310E−04 .595E−04 .110E−03 .197E−03 .341E−03 .574E−03 .941E−03 .150E−02 .235E−02 .359E−02
200 210 220 230 240 250 260 270 280 290
.221E−02 .333E−02 .494E−02 .719E−02 .103E−01 .146E−01 .203E−01 .279E−01 .380E−01 .510E−01
.273E−02 .408E−02 .601E−02 .869E−02 .124E−01 .174E−01 .240E−01 .329E−01 .444E−01 .592E−01
.329E−02 .490E−02 .715E−02 .103E−01 .146E−01 .203E−01 .279E−01 .380E−01 .510E−01 .676E−01
.395E−02 .585E−02 .850E−02 .122E−01 .171E−01 .238E−01 .326E−01 .441E−01 .589E−01 .778E−01
.431E−02 .637E−02 .924E−02 .132E−01 .186E−01 .257E−01 .352E−01 .475E−01 .633E−01 .835E−01
.470E−02 .693E−02 .100E−01 .143E−01 .201E−01 .278E−01 .380E−01 .513E−01 .683E−01 .900E−01
.488E−02 .720E−02 .104E−01 .149E−01 .209E−01 .289E−01 .394E−01 .531E−01 .706E−01 .930E−01
.505E−02 .744E−02 .108E−01 .153E−01 .215E−01 .297E−01 .405E−01 .545E−01 .725E−01 .954E−01
.522E−02 .768E−02 .111E−01 .158E−01 .221E−01 .305E−01 .416E−01 .560E−01 .744E−01 .978E−01
.538E−02 .791E−02 .114E−01 .162E−01 .227E−01 .314E−01 .427E−01 .574E−01 .762E−01 .100
300 310 320 330 340 350
.678E−01 .892E−01 .116 .150 .192 .243
.782E−01 .102 .132 .170 .216 .272
.888E−01 .115 .149 .190 .240 .301
.102 .132 .169 .214 .270 .337
.109 .141 .180 .228 .287 .358
.117 .151 .193 .245 .307 .383
.121 .156 .199 .252 .317 .394
.124 .160 .204 .258 .328 .402
.127 .164 .209 .263 .330 .410
.130 .167 .213 .269 .386 .417
Weight percent, H2SO4
*Vermeulen, Dong, Robinson, Nguyen, and Gmitro, AIChE meeting, Anaheim, CA, 1982; and private communication from Prof. Theodore Vermeulen, Chemical Engineering Dept., University of California, Berkeley. 2-82
TABLE 2-15
Total Pressure, bar, of Aqueous Sulfuric Acid Solutions* Weight percent, H2SO4
°C 0 10 20 30 40 50 60 70 80 90
10.0 .582E−02 .117E−01 .223E−01 .404E−01 .703E−01 .117 .189 .296 .449 .664
20.0 .534E−02 .107E−01 .205E−01 .373E−01 .649E−01 .109 .175 .275 .417 .617
30.0 .448E−02 .909E−02 .174E−01 .319E−01 .558E−01 .939E−01 .152 .239 .365 .542
40.0 .326E−02 .670E−02 .130E−01 .241E−01 .427E−01 .725E−01 .119 .188 .290 .434
50.0 .193E−02 .405E−02 .802E−02 .151E−01 .272E−01 .470E−01 .782E−01 .126 .196 .298
60.0 .836E−03 .180E−02 .367E−02 .710E−02 .131E−01 .232E−01 .395E−01 .651E−01 .104 .161
70.0
75.0
80.0
85.0
.207E−03 .467E−03 .995E−03 .201E−02 .387E−02 .715E−02 .127E−01 .217E−01 .360E−01 .578E−01
.747E−04 .175E−03 .388E−03 .811E−03 .162E−02 .309E−02 .565E−02 .997E−01 .170E−01 .281E−01
.197E−04 .490E−04 .115E−03 .253E−03 .531E−03 .106E−02 .204E−02 .376E−02 .668E−02 .115E−01
.343E−05 .952E−05 .245E−04 .589E−04 .134E−03 .286E−03 .584E−03 .114E−02 .213E−02 .383E−02
.905E−01 .138 .206 .301 .431 .605 .837 1.138 1.525 2.017
.452E−01 .708E−01 .108 .162 .236 .339 .478 .662 .902 1.212
.192E−01 .312E−01 .493E−01 .760E−01 .115 .170 .246 .350 .489 .673
.666E−02 .112E−01 .183E−01 .291E−01 .451E−01 .683E−01 .101 .147 .209 .292
100 110 120 130 140 150 160 170 180 190
.957 1.349 1.863 2.524 3.361 4.404 5.685 7.236 9.093 11.289
.891 1.258 1.740 2.361 3.149 4.132 5.342 6.810 8.571 10.658
.786 1.113 1.544 2.101 2.810 3.697 4.793 6.127 7.731 9.640
.634 .904 1.264 1.732 2.333 3.090 4.031 5.185 6.584 8.259
.441 .638 .903 1.253 1.708 2.289 3.021 3.930 5.045 6.397
.244 .360 .519 .734 1.020 1.392 1.870 2.475 3.233 4.169
200 210 220 230 240 250 260 270 280 290
13.861 16.841 20.264 24.160 28.561 33.494 38.984 45.055 51.726 59.015
13.107 15.951 19.225 22.960 27.188 31.939 37.240 43.116 49.590 56.681
11.887 14.505 17.529 20.992 24.927 29.364 34.334 39.865 45.984 52.715
10.245 12.576 15.287 18.414 21.992 26.056 30.642 35.784 41.514 47.866
8.020 9.948 12.217 14.864 17.929 21.452 25.472 30.030 35.168 40.926
5.312 6.696 8.354 10.322 12.641 15.351 18.496 22.122 26.275 31.004
2.633 3.396 4.331 5.466 6.832 8.459 10.384 12.642 15.272 18.315
1.606 2.101 2.715 3.468 4.382 5.481 6.791 8.337 10.147 12.250
.913 1.221 1.610 2.098 2.701 3.439 4.332 5.402 6.673 8.170
.402 .544 .726 .956 1.242 1.594 2.023 2.540 3.157 3.886
300 310 320 330 340 350
66.934 75.495 84.705 94.567 105.083 116.251
64.407 72.781 81.816 91.518 101.894 112.947
60.081 68.101 76.792 86.172 96.252 107.043
54.869 62.553 70.947 80.078 89.970 100.647
47.347 54.470 62.338 70.990 80.466 90.806
36.361 42.398 49.168 56.727 65.130 74.437
21.814 25.812 30.355 35.489 41.262 47.723
14.675 17.453 20.611 24.182 28.193 32.674
9.916 11.939 14.264 16.916 19.920 23.303
4.740 5.732 6.876 8.185 9.672 11.351
Weight percent, H2SO4 °C
90.0
92.0
94.0
96.0
97.0
98.0
98.5
99.0
99.5
100.0
0 10 20 30 40 50 60 70 80 90
.518E−06 .159E−05 .449E−05 .117E−04 .385E−04 .653E−04 .141E−03 .291E−03 .571E−03 .107E−02
.243E−06 .765E−06 .221E−05 .590E−05 .147E−04 .344E−04 .759E−04 .159E−03 .319E−03 .612E−03
.109E−06 .348E−06 .102E−05 .279E−05 .708E−05 .169E−04 .380E−04 .813E−04 .166E−03 .324E−03
.416E−07 .136E−06 .407E−06 .113E−05 .293E−05 .712E−05 .164E−04 .357E−04 .742E−04 .148E−03
.235E−07 .774E−07 .235E−06 .659E−06 .173E−05 .425E−05 .987E−05 .218E−04 .458E−04 .921E−04
.117E−07 .391E−07 .121E−06 .344E−06 .914E−06 .228E−05 .538E−05 .120E−04 .257E−04 .524E−04
.768E−08 .261E−07 .812E−07 .234E−06 .630E−06 .159E−05 .379E−05 .856E−05 .184E−04 .390E−04
.479E−08 .166E−07 .528E−07 .155E−06 .425E−06 .109E−05 .264E−05 .605E−05 .132E−04 .277E−04
.313E−08 .113E−07 .373E−07 .114E−06 .323E−06 .861E−06 .216E−05 .514E−05 .117E−04 .253E−04
.323E−08 .124E−07 .435E−07 .141E−06 .425E−06 .120E−05 .319E−05 .804E−05 .193E−04 .441E−04
100 110 120 130 140 150 160 170 180 190
.195E−02 .340E−02 .575E−02 .944E−02 .151E−01 .235E−01 .357E−01 .532E−01 .775E−01 .111
.113E−02 .201E−02 .346E−02 .578E−02 .939E−02 .149E−01 .230E−01 .347E−01 .514E−01 .747E−01
.607E−03 .110E−02 .192E−02 .327E−02 .539E−02 .866E−02 .136E−01 .208E−01 .312E−01 .460E−01
.283E−03 .521E−03 .929E−03 .161E−02 .270E−02 .441E−02 .703E−02 .110E−01 .167E−01 .250E−01
.178E−03 .332E−03 .598E−03 .104E−02 .177E−02 .293E−02 .471E−02 .741E−02 .114E−01 .172E−01
.103E−03 .194E−03 .354E−03 .626E−03 .107E−02 .180E−02 .293E−02 .466E−02 .726E−02 .111E−01
.751E−04 .143E−03 .263E−03 .470E−03 .815E−03 .137E−02 .226E−02 .363E−02 .571E−02 .880E−02
.555E−04 .107E−03 .201E−03 .363E−03 .639E−03 .109E−02 .183E−02 .299E−02 .478E−02 .749E−02
.527E−04 .106E−03 .206E−03 .387E−03 .708E−03 .126E−02 .219E−02 .372E−02 .619E−02 .101E−01
.966E−04 .204E−03 .414E−03 .314E−03 .155E−02 .287E−02 .516E−02 .905E−02 .155E−01 .260E−01
.665E−01 .944E−01 .132 .182 .247 .331 .439 .575 .744 .954
.367E−01 .530E−01 .752E−01 .105 .145 .197 .264 .351 .460 .597
.255E−01 .371E−01 .531E−01 .749E−01 .104 .143 .193 .258 .341 .446
.166E−01 .245E−01 .354E−01 .505E−01 .710E−01 .985E−01 .135 .153 .245 .324
.133E−01 .198E−01 .289E−01 .417E−01 .592E−01 .830E−01 .115 .157 .213 .285
.115E−01 .175E−01 .260E−01 .382E−01 .553E−01 .790E−01 .112 .156 .215 .295
.161E−01 .253E−01 .392E−01 .596E−01 .895E−01 .132 .193 .279 .398 .562
.427E−01 .687E−01 .109 .169 .258 .389 .577 .846 1.225 1.751
200 210 220 230 240 250 260 270 280 290
.156 .216 .295 .396 .525 .688 .881 1.141 1.447 1.817
.107 .150 .207 .282 .379 .503 .660 .856 1.099 1.398
300 310 320 330 340 350
2.261 2.791 3.417 4.153 5.011 6.006
1.761 2.199 2.723 3.347 4.084 4.949
1.211 1.524 1.901 2.353 2.889 3.523
.767 .977 1.234 1.545 1.919 2.366
.578 .742 .944 1.191 1.491 1.852
.425 .553 .713 .911 1.156 1.456
.379 .498 .649 .840 1.078 1.374
.399 .536 .714 .944 1.239 1.614
.785 1.085 1.486 2.018 2.718 3.631
2.476 3.465 4.800 6.586 8.957 12.079
*Vermeulen, Dong, Robinson, Nguyen, and Gmitro, AIChE meeting, Anaheim, Calif., 1982; and private communication from Prof. Theodore Vermeulen, Chemical Engineering Dept., University of California, Berkeley. 2-83
2-84
PHYSICAL AND CHEMICAL DATA
TABLE 2-16
Partial Pressures of HNO3 and H2O over Aqueous Solutions of HNO3 mmHg Percentages are weight % HNO3 in solution. 20%
°C
25%
HNO3
H2O
HNO3
30% H2O
HNO3
35% H2O
0 5 10 15 20
4.1 5.7 8.0 10.9 15.2
3.8 5.4 7.6 10.3 14.2
3.6 5.0 7.1 9.7 13.2
25 30 35 40 45
20.6 27.6 36.5 47.5 62
19.2 25.7 33.8 44 57.5
17.8 23.8 31.1 41 53
0.09
0.11 .17
40%
HNO3
H2O
45%
HNO3
H2O
3.3 4.6 6.5 8.9 12.0 16.2 21.7 28.3 37.7 48
0.09 .13 .20 .28
0.12 .17 .25 .36 .52
50%
HNO3
H2O
HNO3
3.0 4.2 5.8 8.0 10.8
H2O
0.10 .15
2.6 3.6 5.0 6.9 9.4
0.12 .18 .27
2.1 3.0 4.2 5.8 7.9
14.6 19.5 25.5 33.5 43
.23 .33 .48 .68 .96
12.7 16.9 22.3 29.3 38.0
.39 .56 .80 1.13 1.57
10.7 14.4 19.0 25.0 32.5
49.5 62.5 80 100 126
2.18 2.95 4.05 5.46 7.25
42.5 54 70 88 110
50 55 60 65 70
0.09 .13 .19 .27
80 100 128 162 200
.13 .18 .28 .40 .54
75 94 121 151 187
.25 .35 .51 .71 1.00
69 87 113 140 174
.42 .59 .85 1.18 1.63
63 79 102 127 159
.75 1.04 1.48 2.05 2.80
56 71 90 114 143
1.35 1.83 2.54 3.47 4.65
75 80 85 90 95
.38 .53 .74 1.01 1.37
250 307 378 458 555
.77 1.05 1.44 1.95 2.62
234 287 352 426 517
1.38 1.87 2.53 3.38 4.53
217 267 325 393 478
2.26 3.07 4.15 5.50 7.32
198 243 297 359 436
3.80 5.10 6.83 9.0 11.7
178 218 268 325 394
6.20 8.15 10.7 13.7 17.8
158 195 240 292 355
9.6 12.5 16.3 20.9 26.8
138 170 211 258 315
100 105 110 115 120
1.87 2.50
675 800
3.50 4.65
628 745
6.05 7.90
580 690
530 631 755
15.5 20.0 25.7 32.5
480 573 688 810
23.0 29.2 37.0 46
430 520 625 740
34.2 43.0 54.5 67 84
383 463 560 665 785
°C
HNO3
H2O
HNO3
H2O
HNO3
H2O
HNO3
H2O
0 5 10 15 20
0.14 .21 .31 .45
1.8 2.5 3.5 4.9 6.7
0.19 .28 .41 .59 .84
1.5 2.1 3.0 4.1 5.6
0.41 .60 .86 1.21 1.68
1.3 1.8 2.6 3.5 4.9
0.79 1.12 1.58 2.18 3.00
1.1 1.6 2.2 3.0 4.1
25 30 35 40 45
.66 .93 1.30 1.82 2.50
9.1 12.2 16.1 21.3 28.0
1.21 1.66 2.28 3.10 4.20
7.7 10.3 13.6 18.1 23.7
2.32 3.17 4.26 5.70 7.55
6.6 8.8 11.6 15.5 20.0
4.10 5.50 7.30 9.65 12.6
5.5 7.4 9.8 12.8 16.7
50 55 60 65 70
3.41 4.54 6.15 8.18 10.7
36.3 46 60 76 95
5.68 7.45 9.9 13.0 16.8
31 39 51 64 81
10.0 12.8 16.8 21.7 27.5
26.0 33.0 43.0 54.5 68
16.5 21.0 27.1 34.5 43.3
21.8 27.3 35.3 44.5 56
75 80 85 90 95
13.9 18.0 23.0 29.4 37.3
120 148 182 223 272
21.8 27.5 34.8 43.7 55.0
102 126 156 192 233
35.0 43.5 54.5 67.5 83.5
100 105 110 115 120 125
47 58.5 73 90 110
331 400 485 575 685
69.5 84.5 103 126 156 187
285 345 417 495 590 700
55%
60%
9.7 12.7 16.5
65%
103 124 152 181 218 260
70%
80% HNO3
90%
100%
H2O
HNO3
H2O
HNO3
2 3 4 6 8
1.2 1.7 2.4
5.5 8 11 15 20
10.5 14 18.5 24.5 32
3.2 4 5.5 7 9.5
27 36 47 62 80
1 1.3 1.8 2.4 3
57 77 102 133 170
11 15 22 30 42
41 52 67 85 106
12 15 20 25 31
103 127 157 192 232
4 5 6.5 8 10
215 262 320 385 460 540 625 720 820
86 106 131 160 195
54.5 67.5 83 103 125
70 86 107 130 158
130 158 192 230 278
38 48 60 73 89
282 338 405 480 570
13 16 20 24 29
238 288 345 410 490 580
152 183 221 262 312 372
192 231 278 330 393 469
330 392 465 545 640
108 129 155 185 219
675 790
35 42
VAPOR PRESSURES OF SOLUTIONS TABLE 2-17 Partial Pressures of H2O and HBr over Aqueous Solutions of HBr at 20 to 55°C mmHg 20°C
% HBr
HBr
32 34 36 38 40 42 44 46 48 50 52 54 56 58 60
25°C H2O
0.09 .23 .71 2.2 6.8 21
6.2 4.5 3.3 2.4 1.7 1.3
HBr
50°C H2O
0.0016 .0022 .0033 .0061 .011 .023 .048 .10 .13 .37 1.1 3.2 9.3 27
HBr
8.2 6.1 4.5 3.3 2.4 1.9
1.3 3.2 7.2 17 40 91
TABLE 2-20 Total Vapor Pressures of Aqueous Solutions of CH3COOH Percentages of weight % acetic acid in the solution mmHg
55°C
H2O
HBr
30.2 24.3 19.3 16.0 13.3 10.4
H2O
2.0 4.6 10.2 23.0 51 115 260
2-85
°C
25%
50%
75%
20 25 30 35 40
16.3 22.1 29.6 39.4 51.7
15.7 21.4 28.8 38.3 50.2
15.3 20.8 27.8 36.6 48.1
45 50 55 60 65
67.0 87.2 110 141 178
65.0 85.0 107 138 172
62.0 80.1 102 130 162
70 75 80 85 90
223 277 342 419 510
216 269 331 407 497
203 251 310 376 458
95 100
618 743
602 725
550 666
38 31 25 21 18 14 11.4
TABLE 2-18 Partial Pressures of HI over Aqueous Solutions of HI at 25°C mmHg %HI pHI
4 0.00064
46 0.0010
48 0.0022
50 0.0050
52 0.013
54 0.035
56 0.10
TABLE 2-19 Vapor Pressures of the System: Water-Sulfuric Acid-Nitric Acid For these data reference must be made to the graphs of International Critical Tables, vol. 3, pp. 306–308.
TABLE 2-21
Vapor pressure of aqueous diethylene glycol solutions. (Courtesy of Carbide and Carbon Chemicals Corp.)
FIG. 2-3
Partial Pressures of H2O over Aqueous Solutions of NH3* Pressures are in pounds per square inch absolute Molal concentration of ammonia in the solutions in percentages (Weight concentration of ammonia in the solution in percentages)
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
t, °F
(0)
32 40 50 60 70
0.09 0.084 .12 .115 .18 .17 .26 .24 .36 .34
0.079 .108 .16 .23 .32
0.074 .101 .15 .21 .30
0.070 .095 .14 .20 .28
0.065 .089 .13 .19 .26
0.060 .083 .12 .17 .25
0.056 .076 .11 .16 .23
0.051 .070 .10 .15 .21
0.047 .064 .094 .13 .19
0.042 .058 .085 .12 .17
0.038 .052 .076 .11 .15
0.034 .046 .068 .097 .14
0.030 .040 .059 .085 .12
0.025 .035 .051 .073 .10
0.021 .029 .042 .061 .086
0.017 .023 .034 .049 .069
0.013 .015 .025 .037 .052
0.008 .012 .017 .024 .034
0.004 .006 .008 .012 .017
80 90 100 110 120
.51 .48 .70 .66 .95 .90 1.27 1.20 1.69 1.60
.45 .63 .85 1.14 1.51
.42 .58 .79 1.07 1.42
.40 .55 .74 1.00 1.33
.37 .51 .69 .93 1.24
.34 .47 .64 .86 1.15
.32 .44 .59 .80 1.06
.29 .40 .55 .73 .97
.27 .37 .50 .67 .89
.24 .33 .45 .60 .80
.22 .30 .41 .54 .72
.19 .26 .36 .48 .64
.17 .23 .31 .42 .56
.14 .20 .27 .36 .48
.12 .16 .22 .30 .40
.096 .13 .18 .24 .32
.072 .10 .13 .18 .24
.048 .066 .090 .120 .160
.024 .033 .045 .061 .081
130 140 150 160 170
2.22 2.89 3.72 4.74 5.99
2.10 2.73 3.51 4.48 5.66
1.98 2.57 3.31 4.22 5.34
1.86 2.42 3.11 3.97 5.02
1.74 2.26 2.91 3.71 4.70
1.62 2.11 2.72 3.46 4.38
1.51 1.96 2.52 3.22 4.07
1.39 1.81 2.33 2.97 3.75
1.28 1.66 2.14 2.73 3.45
1.17 1.52 1.95 2.49 3.15
1.05 1.37 1.76 2.25 2.84
.95 1.23 1.59 2.02 2.56
.84 1.10 1.41 1.80 2.28
.74 .96 1.24 1.58 1.99
.63 .82 1.06 1.35 1.71
.53 .69 .88 1.12 1.42
.42 .55 .71 .90 1.13
.32 .41 .53 .67 1.85
.210 .270 .350 .450 .570
.100 .140 .180 .220 .300
180 7.51 7.10 190 9.34 8.83 200 11.53 10.90 210 14.12 13.35 220 17.19 16.25
6.69 8.32 10.27 12.58 15.32
6.30 7.82 9.65 11.82 14.39
5.89 7.32 9.04 11.07 13.48
5.49 6.83 8.43 10.32 12.57
5.10 6.34 7.83 9.59 11.67
4.71 5.86 7.23 8.86 10.78
4.33 5.38 6.64 8.13 9.90
3.94 4.91 6.06 7.42 9.03
3.57 4.44 5.48 6.71 8.17
3.21 3.99 4.93 6.04 7.31
2.85 3.55 4.38 5.34
2.50 3.10 3.81
2.14 2.65
1.77
1.42
1.06
230 20.78 19.64 240 24.97 23.60 250 29.83 28.20
18.51 22.25 26.58
17.40 20.91 25.00
16.29 19.58 23.39
15.19 18.26 21.82
14.11 16.95 20.25
13.03 15.66 18.71
11.97 14.38 17.18
10.91 13.12 15.67
9.87 11.86
(4.74) (9.50) (14.29) (19.10) (23.94) (28.81) (33.71) (38.64) (43.59) (48.57) (53.58) (58.62) (63.69) (68.79) (73.91) (79.07) (84.26) (89.47) (94.72)
*Wilson, Univ. Ill., Eng. Expt. Sta. Bull. 146.
TABLE 2-22
Mole Percentages of H2O over Aqueous Solutions of NH3* Molal concentration of ammonia in the solutions in percentages (Weight concentration of ammonia in the solutions in percentages)
t, °F
0 (0)
5 (4.74)
10 (9.50)
15 (14.29)
20 (19.10)
25 (23.94)
30 (28.81)
35 (33.71)
40 (38.64)
45 (43.59)
50 (48.57)
55 (53.58)
60 (58.62)
65 (63.69)
70 (68.79)
75 (73.91)
80 (79.07)
85 (84.26)
90 (89.47)
95 (94.72)
32 40 50 60 70
100 100 100 100 100
24.3 25.3 26.6 27.9 29.1
13.2 14.1 15.2 16.2 17.4
7.63 8.15 9.09 9.50 10.30
4.43 4.73 5.24 5.69 6.14
2.50 2.74 3.03 3.42 3.65
1.43 1.59 1.78 1.97 2.27
0.856 .943 1.060 1.210 1.390
0.514 .581 .652 .777 .873
0.335 .372 .434 .481 .569
0.216 .248 .290 .331 .383
0.151 .172 .202 .238 .266
0.109 .124 .148 .172 .205
0.0816 .0914 .1095 .1290 .1510
0.0585 .0706 .0838 .0986 .112
0.0457 .0533 .0630 .0754 .0882
0.0345 .0395 .0477 .0566 .0656
0.0249 .0243 .0332 .0406 .0474
0.0146 .0185 .0215 .0251 .0296
0.00689 .00879 .00959 .01125 .0135
80 90 100 110 120
100 100 100 100 100
31.6 32.7 34.4 35.9 37.5
18.5 20.0 21.0 22.2 23.4
11.20 12.00 12.90 13.80 14.70
6.89 7.40 7.92 8.59 9.22
4.08 4.47 4.85 5.29 5.75
2.45 2.73 3.00 3.30 3.63
1.550 1.730 1.890 2.110 2.320
.978 1.100 1.250 1.370 1.520
.659 .742 .834 .932 1.044
.444 .505 .574 .644 .714
.323 .366 .420 .466 .529
.230 .267 .307 .347 .395
.1750 .2020 .2290 .2640 .3020
.130 .157 .179 .208 .233
.103 .115 .135 .157 .180
.0772 .0884 .104 .118 .135
.0528 .0647 .0714 .0846 .0970
.0351 .0408 .0473 .0540 .0619
.0167 .0194 .0226 .0262 .0300
130 140 150 160 170
100 100 100 100 100
39.0 40.7 42.3 44.1 45.6
24.5 25.8 27.1 28.3 29.6
15.60 16.50 17.50 18.40 19.40
9.85 10.50 11.20 11.90 12.70
6.18 6.69 7.19 7.69 8.22
3.95 4.28 4.63 5.01 5.38
2.550 2.790 3.080 3.300 3.580
1.690 1.860 2.040 2.230 2.430
1.160 1.286 1.410 1.550 1.700
.811 .906 1.004 1.110 1.220
.596 .663 .741 .818 .904
.444 .501 .558 .617 .689
.3430 .3840 .4320 .4800 .5300
.263 .297 .334 .372 .414
.205 .232 .257 .287 .320
.154 .175 .197 .218 .242
.1117 .124 .140 .154 .174
.0703 .0786 .0892 .1005 .112
.0339 .0385 .0439 .0499 .0567
180 190 200 210 220
100 100 100 100 100
47.3 48.7 50.4 52.1 53.7
30.9 32.2 33.4 34.7 36.1
20.40 21.40 22.30 23.40 24.40
13.40 14.10 14.90 15.70 16.40
8.76 9.31 9.88 10.45 11.05
5.78 6.18 6.59 7.03 7.48
3.870 4.160 4.470 4.780 5.100
2.640 2.860 3.080 3.310 3.560
1.850 2.020 2.190 2.360 2.540
1.340 1.460 1.580 1.720 1.860
.994 1.087 1.187 1.272 1.390
.756 .830 .907 .983
.5860 .6420 .7010
.456 .501
.352
.268
.192
230 240 250
100 100 100
55.2 56.8 58.4
37.3 38.6 39.8
25.40 26.50 27.50
17.30 18.00 18.80
11.63 12.24 12.88
7.91 8.36 8.82
5.440 5.780 6.120
3.810 4.060 4.340
2.730 2.920 3.120
2.000 2.150
*Wilson, Univ. Ill., Eng. Expt. Sta. Bull. 146.
2-86
100 (100.00) 0.00
VAPOR PRESSURES OF SOLUTIONS TABLE 2-23
2-87
Partial Pressures of NH3 over Aqueous Solutions of NH3* Pressures are in pounds per square inch absolute Molal concentration of ammonia in the solutions in percentages (Weight concentration of ammonia in the solutions in percentages)
t, °F
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 (4.74) (9.50) (14.29) (19.10) (23.94) (28.81) (33.71) (38.64) (43.59) (48.57) (53.58) (58.62) (63.69) (68.79) (73.91) (79.07) (84.26) (89.47) (94.72)
32 40 50 60 70
0.26 .33 .47 .62 .83
0.52 .66 .89 1.19 1.52
0.90 1.14 1.50 2.00 2.60
1.51 1.92 2.53 3.21 4.28
2.67 3.16 4.16 5.36 6.87
4.27 5.13 6.63 8.48 10.76
6.54 7.98 10.24 13.06 16.33
8.93 11.98 15.24 19.15 23.84
14.13 17.14 21.56 26.92 33.20
19.36 23.33 29.17 36.14 44.25
25.12 30.15 37.46 46.12 56.29
31.13 37.15 45.86 56.22 68.32
36.74 43.69 53.79 65.81 79.42
42.69 49.56 60.82 73.99 89.26
45.92 54.40 66.63 80.90 97.42
49.26 58.31 71.26 86.44 104.01
52.13 61.62 75.22 91.04 109.55
54.89 64.77 79.05 95.67 114.83
58.01 68.31 83.40 100.65 120.61
80 90 100 110 120
1.04 1.36 1.72 2.14 2.67
1.98 2.52 3.20 4.00 4.95
3.34 4.25 5.34 6.65 8.21
5.45 6.88 8.60 10.64 13.09
8.69 10.89 13.53 16.65 20.30
13.52 16.76 20.68 25.21 30.54
20.29 25.04 30.57 37.01 44.56
29.40 35.94 43.57 52.43 62.62
40.69 49.45 59.49 71.20 84.44
53.84 64.99 77.85 92.59 109.40
67.97 81.61 97.27 115.16 135.48
82.36 98.35 116.81 137.62 161.44
95.52 113.79 134.70 158.42 185.14
107.06 127.22 150.23 176.18 205.81
116.42 138.18 162.94 190.85 222.28
124.20 147.02 173.22 203.02 236.05
130.57 154.46 181.97 212.71 247.14
136.35 161.74 190.13 222.22 258.24
143.70 169.73 199.17 232.79 270.02
130 140 150 160 170
3.28 3.97 4.78 5.68 6.75
6.09 7.41 8.92 10.70 12.67
10.05 12.21 14.70 17.57 20.85
15.93 19.23 23.09 27.45 32.41
24.58 29.43 35.09 41.56 48.89
36.74 43.77 51.91 61.03 71.48
53.16 62.97 74.28 86.91 101.09
74.27 87.53 102.51 119.37 138.30
99.69 116.72 136.15 157.71 181.95
128.45 149.93 173.64 200.45 230.36
158.45 184.17 212.91 244.98 280.54
188.16 218.18 251.24 288.38 329.42
215.14 248.70 286.00 327.82 373.61
238.70 275.33 316.24 361.75 411.59
257.87 297.12 340.82 389.08 442.28
272.88 314.45 360.39 411.30 466.67
286.08 328.99 376.57 429.73 487.85
298.46 342.93 392.45 447.35 507.63
311.80 358.46 409.62 466.38 528.50
180 7.90 190 9.23 200 10.70 210 12.26 220 14.02
14.96 17.55 20.45 23.68 27.15
24.56 28.78 33.49 38.76 44.61
38.13 44.49 51.58 59.65 68.43
57.19 66.49 76.90 88.48 101.24
83.07 96.22 110.85 126.83 144.74
116.97 134.89 154.58 176.24 200.46
159.37 182.72 208.56 236.97 268.30
208.66 238.39 270.94 307.08 346.07
263.43 299.86 340.02 383.99 431.43
319.89 363.11 410.17 462.36 518.19
374.25 424.15 478.62 537.56
424.10 479.40 539.79
466.26 526.15
500.63
528.08
551.24
230 15.95 240 17.92 250 20.12
31.09 35.40 40.09
51.06 58.00 65.74
78.14 89.02 100.69
115.45 130.94 147.66
164.17 185.79 209.37
226.67 255.26 286.89
302.53 339.72 380.42
389.29 435.78 486.73
483.53 540.44
*Wilson, Univ. Ill., Eng. Expt. Sta. Bull. 146.
TABLE 2-24
Total Vapor Pressures of Aqueous Solutions of NH3* Pressures are in pounds per square inch absolute Molal concentration of ammonia in the solutions in percentages (Weight concentration of ammonia in the solutions in percentages)
t, °F
0 (0)
5 (4.74)
10 (9.50)
15 (14.29)
20 (19.10)
25 (23.94)
30 (28.81)
35 (33.71)
40 (38.64)
45 (43.59)
50 (48.57)
55 (53.58)
60 (58.62)
65 (63.69)
70 (68.79)
75 (73.91)
80 (79.07)
85 (84.26)
90 (89.47)
95 (94.72)
100 (100.00)
32 40 50 60 70
0.09 .12 .18 .26 .36
0.34 .45 .64 .86 1.17
0.60 .77 1.05 1.42 1.84
0.97 1.24 1.65 2.21 2.90
1.58 2.01 2.67 3.51 4.56
2.60 3.25 4.29 5.55 7.13
4.20 5.21 6.75 8.65 11.01
6.54 8.06 10.35 13.22 16.56
9.93 12.05 15.34 19.30 24.05
14.18 17.20 21.65 27.05 33.39
19.40 23.39 29.26 36.26 44.42
25.16 30.20 37.54 46.23 56.44
31.16 37.20 45.93 56.32 68.46
36.77 43.73 53.85 65.90 79.54
42.72 49.60 60.87 74.06 89.36
45.94 54.43 66.67 80.96 97.51
49.28 58.33 71.29 86.49 104.08
52.14 61.64 75.25 91.08 109.60
54.90 64.78 79.07 95.69 114.86
58.01 68.32 83.41 100.66 120.63
62.29 73.32 89.19 107.6 128.8
80 90 100 110 120
.51 .70 .95 1.27 1.69
1.52 2.02 2.62 3.34 4.27
2.43 3.15 4.05 5.14 6.46
3.76 4.83 6.13 7.72 9.63
5.85 7.43 9.34 11.64 14.42
9.06 11.40 14.22 17.58 21.54
13.86 17.23 21.32 26.07 31.69
20.61 25.48 31.16 37.81 45.62
29.69 36.34 44.12 53.16 63.59
40.96 49.82 59.99 71.87 85.33
54.08 65.32 78.30 93.19 110.2
68.19 81.91 97.68 115.7 136.2
82.55 98.61 117.17 138.10 162.08
95.69 114.02 135.01 158.84 185.70
107.20 127.42 150.50 176.54 206.29
116.54 138.34 163.16 191.15 222.68
124.30 147.15 173.40 203.26 236.37
130.64 154.56 182.10 212.89 247.38
136.40 161.81 190.22 222.34 258.40
143.72 169.76 199.22 232.85 270.1
153.0 180.6 211.9 247.0 286.4
130 140 150 160 170
2.22 2.89 3.72 4.74 5.99
5.38 6.70 8.29 10.16 12.41
8.07 9.98 12.23 14.92 18.01
11.91 14.63 17.81 21.54 25.87
17.67 21.49 26.00 31.16 37.11
26.20 31.54 37.81 45.02 53.27
38.25 45.73 54.43 64.25 75.55
54.55 64.78 76.61 89.88 104.84
75.55 89.19 104.65 122.10 141.75
100.86 118.24 138.1 160.2 185.1
129.5 151.3 175.4 202.7 233.2
159.0 185.4 214.5 247.0 283.1
189.00 219.28 252.65 290.18 331.7
215.88 249.66 287.24 329.4 375.6
239.33 276.15 317.3 363.1 413.3
258.40 297.81 341.7 390.2 443.7
273.3 315.0 361.1 412.2 467.8
286.4 329.4 377.1 430.4 488.7
298.67 343.2 392.8 447.8 508.2
311.9 358.6 409.8 466.6 528.8
330.3 379.1 432.2 492.8 558.4
180 190 200 210 220
7.51 9.34 11.53 14.12 17.19
15.00 18.06 21.60 25.61 30.27
21.65 25.87 30.72 36.26 42.47
30.86 36.60 43.14 50.58 59.00
44.02 51.81 60.62 70.72 81.91
62.68 73.32 85.33 98.80 113.81
88.17 102.56 118.68 136.42 156.41
121.68 140.75 161.81 185.10 211.24
163.7 188.1 215.2 245.1 278.2
212.6 243.3 277.0 314.5 355.1
267.0 304.3 345.5 390.7 439.6
323.1 367.1 415.1 468.4 525.5
377.1 427.7 483.0 542.9
426.6 482.5 543.6
468.4 528.8
502.4
529.5
552.3
230 240 250
20.78 24.97 29.83
35.59 41.52 48.32
49.60 57.65 66.67
68.46 78.91 90.74
94.43 108.60 124.08
130.64 149.20 169.48
178.28 202.74 229.62
239.70 270.92 305.60
314.5 354.1 397.6
400.2 448.9 502.4
493.4 552.3
*Wilson, Univ. Ill., Eng. Expt. Sta. Bull. 146.
2-88
VAPOR PRESSURES OF SOLUTIONS TABLE 2-25 Partial Pressures of H2O over Aqueous Solutions of Sodium Carbonate mmHg %Na2CO3 t, °C
0
5
10
15
0 10 20 30 40 50 60 70 80 90 100
4.5 9.2 17.5 31.8 55.3 92.5 149.5 239.8 355.5 526.0 760.0
4.5 9.0 17.2 31.2 54.2 90.7 146.5 235 348 516 746
8.8 16.8 30.4 53.0 88.7 143.5 230.5 342 506 731
16.3 29.6 57.6 86.5 139.9 225 334 494 715
20
28.8 50.2 84.1 136.1 219 325 482 697
25
27.8 48.4 81.2 131.6 211.5 315 467 676
30
26.4 46.1 77.5 125.7 202.5 301 447 648
2-89
TABLE 2-26 Partial Pressures of H2O and CH3OH over Aqueous Solutions of Methyl Alcohol* 39.9°C
Mole fraction CH3OH
PH2O, mmHg
PCH3OH , mmHg
0 14.99 17.85 21.07 27.31 31.06 40.1 47.0 55.8 68.9 86.0 100.0
54.7 39.2 38.5 37.2 35.8 34.9 32.8 31.5 27.3 20.7 10.1 0
0 66.1 75.5 85.2 100.6 108.8 127.7 141.6 158.4 186.6 225.2 260.7
59.4°C
Mole fraction CH3OH
PH2O, mmHg
PCH3OH , mmHg
0 22.17 27.40 33.24 39.80 47.08 55.5 69.2 78.5 85.9 100.0
145.4 106.9 102.2 96.6 91.7 84.8 76.9 57.8 43.8 30.1 0
0 210.1 240.2 272.1 301.9 335.6 373.7 439.4 486.6 526.9 609.3
*International Critical Tables, vol. 3, McGraw-Hill, p. 290.
TABLE 2-27 Conc. g NaOH/ 100 g H2O 0 5 10 20 30 40 50 60 70 80 90 100 120 140 160 180 200 250 300 350 400 500 700 1000 2000 4000 8000
Partial Pressures of H2O over Aqueous Solutions of Sodium Hydroxide mmHg Temperature, °C 0
20
40
60
80
100
120
160
200
250
300
350
4.6 4.4 4.2 3.6 2.9 2.2
17.5 16.9 16.0 13.9 11.3 8.7 6.3 4.4 3.0 2.0 1.3 0.9
55.3 53.2 50.6 44.2 36.6 28.7 20.7 15.5 10.9 7.6 5.2 3.6 1.7
149.5 143.5 137.0 120.5 101.0 81.0 62.5 47.0 34.5 24.5 17.5 12.5 6.3 3.0 1.5
355.5 341.5 325.5 288.5 246.0 202.0 160.5 124.0 94.0 70.5 53.0 38.5 20.5 11.0 6.0 3.5 2.0 0.5 0.1
760.0 730.0 697.0 621.0 537.0 450.0 368.0 294.0 231.0 179.0 138.0 105.0 61.0 35.5 20.5 12.0 7.0 2.0 0.5
1,489 1,430 1,365 1,225 1,070 920 770 635 515 415 330 262 164 102 63 40 25 8 2.7 0.9
4,633 4,450 4,260 3,860 3,460 3,090 2,690 2,340 2,030 1,740 1,490 1,300 915 765 470 340 245 110 50 23 11
11,647 11,200 10,750 9,800 8,950 8,150 7,400 6,750 6,100 5,500 5,000 4,500 3,650 2,980 2,430 1,980 1,620 985 610 380 240 100
29,771 28,600 27,500 25,300 23,300 21,500 19,900 18,400 17,100 15,800 14,700 13,650 11,800 10,300 8,960 7,830 6,870 5,000 3,690 2,750 2,080 1,210 440
64,200 61,800 59,300 54,700 50,800 47,200 44,100 41,200 38,700 36,300 34,200 32,200 28,800 25,900 23,300 21,200 19,200 15,400 12,500 10,300 8,600 6,100 3,300 1,470 150
123,600 118,900 114,100 105,400 98,000 91,600 85,800 80,700 76,000 71,900 68,100 64,600 58,600 53,400 49,000 45,100 41,800 35,000 29,800 25,700 22,400 17,500 11,500 6,800 1,760 120 7
2-90
PHYSICAL AND CHEMICAL DATA
WATER-VAPOR CONTENT OF GASES CHART FOR GASES AT HIGH PRESSURES The accompanying figure is useful in determining the water-vapor content of air at high pressure in contact with liquid water.
Water content of air, °C = (°F − 32) × 5⁄ 9. (Landsbaum, Dadds, and Stutzman. Reprinted from vol. 47, January 1955 issue of Ind. Eng. Chem. [p. 192]. Copyright 1955 by the American Chemical Society and reproduced by permission of the copyright owner.)
FIG. 2-4
DENSITIES OF PURE SUBSTANCES
2-91
DENSITIES OF PURE SUBSTANCES UNITS CONVERSIONS For this subsection, the following units conversions are applicable:
To convert kilograms per cubic meter to pounds per cubic foot, multiply by 0.06243.
°F = 9⁄ 5 °C + 32.
TABLE 2-28
Density (kg/m3) of Water from 0 to 100°C* ρ, kg/m3
t, °C
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 1 2 3 4
999.839 999.898 999.940 999.964 999.972
999.846 999.903 999.943 999.966 999.972
999.852 999.908 999.946 999.967 999.972
999.859 999.913 999.949 999.968 999.971
999.865 999.917 999.952 999.969 999.971
999.871 999.921 999.954 999.970 999.970
999.877 999.925 999.956 999.971 999.969
999.882 999.929 999.959 999.971 999.968
999.888 999.933 999.961 999.972 999.967
999.893 999.936 999.962 999.972 999.965
5 6 7 8 9
999.964 999.940 999.901 999.848 999.781
999.962 999.937 999.897 999.842 999.773
999.960 999.934 999.892 999.836 999.765
999.958 999.930 999.887 999.829 999.758
999.956 999.926 999.882 999.823 999.750
999.954 999.923 999.877 999.816 999.742
999.951 999.919 999.871 999.809 999.734
999.949 999.915 999.866 999.802 999.725
999.946 999.910 999.860 999.795 999.717
999.943 999.906 999.854 999.788 999.708
10 11 12 13 14
999.699 999.605 999.497 999.377 999.244
999.691 999.595 999.486 999.364 999.230
999.682 999.584 999.474 999.351 999.216
999.672 999.574 999.462 999.338 999.202
999.663 999.563 999.451 999.325 999.188
999.654 999.553 999.439 999.312 999.173
999.644 999.542 999.426 999.299 999.159
999.635 999.531 999.414 999.285 999.144
999.625 999.520 999.402 999.272 999.129
999.615 999.509 999.389 999.258 999.114
15 16 17 18 19
999.099 998.943 998.775 998.595 998.405
999.084 998.926 998.757 998.577 998.385
999.069 998.910 998.740 998.558 998.366
999.054 999.894 998.722 998.539 998.346
999.038 998.877 998.704 998.520 998.326
999.022 998.860 998.686 998.502 998.306
999.007 998.843 998.668 998.482 998.286
998.991 998.826 998.650 998.463 998.265
998.975 998.809 998.632 998.444 998.245
998.958 998.792 998.614 998.425 998.224
20 21 22 23 24
998.204 997.992 997.770 997.538 997.296
998.183 997.971 997.747 997.515 997.272
998.162 997.949 997.725 997.491 997.247
998.141 997.927 997.702 997.467 997.222
998.120 997.905 997.679 997.443 997.197
998.099 997.883 997.656 997.419 997.172
998.078 997.860 997.632 997.394 997.146
998.057 997.838 997.609 997.370 997.121
998.035 997.816 997.585 997.345 997.096
998.014 997.793 997.562 997.321 997.070
25 26 27 28 29
997.045 996.783 996.513 996.233 995.945
997.019 996.757 996.485 996.205 995.915
996.993 996.730 996.458 996.176 995.886
996.967 996.703 996.430 996.148 995.856
996.941 996.676 996.402 996.119 995.827
996.915 996.649 996.374 996.090 995.797
996.889 996.622 996.346 996.061 995.767
996.863 996.595 996.318 996.032 995.737
996.836 996.568 996.290 996.003 995.707
996.810 996.540 996.262 995.974 995.677
30 31 32 33 34
995.647 995.341 995.026 994.703 994.371
995.617 995.310 994.997 994.670 994.338
995.586 995.278 994.962 994.637 994.304
995.556 995.247 994.930 994.604 994.270
995.526 995.216 994.898 994.571 994.236
995.495 995.184 994.865 994.538 994.202
995.464 995.153 994.833 994.505 994.168
995.433 995.121 994.801 994.472 994.134
995.403 995.090 994.768 994.438 994.100
995.372 995.058 994.735 994.405 994.066
35 36 37 38 39
994.032 993.684 993.328 992.965 992.594
993.997 993.648 993.292 992.928 992.557
993.963 993.613 993.256 992.891 992.519
993.928 993.578 993.220 992.855 992.481
993.893 993.543 993.184 992.818 992.444
993.859 993.507 993.148 992.780 992.406
993.824 993.471 993.111 992.743 992.368
993.789 993.436 993.075 992.706 992.330
993.754 993.400 993.038 992.669 992.292
993.719 993.364 993.002 992.631 992.254
40 41 42 43 44
992.215 991.830 991.436 991.036 990.628
992.177 991.791 991.396 990.995 990.587
992.139 991.751 991.357 990.955 990.546
992.100 991.712 991.317 990.914 990.504
992.062 992.673 991.277 990.873 990.463
992.023 991.634 991.237 990.833 990.421
991.985 991.594 991.197 990.792 990.380
991.946 991.555 991.157 990.751 990.338
991.907 991.515 991.116 990.710 990.297
992.868 991.476 991.076 990.669 990.255
45 46 47 48 49
990.213 989.792 989.363 988.928 988.485
990.171 989.749 989.320 988.884 988.441
990.129 989.706 989.276 988.840 988.396
990.087 989.664 989.233 988.796 988.352
990.045 989.621 989.190 988.752 988.307
990.003 989.578 989.146 988.707 988.262
989.961 989.535 989.103 988.663 988.217
989.919 989.492 989.059 988.619 988.172
989.876 989.449 989.015 988.574 988.127
989.834 989.406 988.971 988.530 988.082
*From “Water: Density at Atmospheric Pressure and Temperatures from 0 to 100°C,” Tables of Standard Handbook Data, Standartov, Moscow, 1978. To conserve space, only a few tables of density values are given. The reader is reminded that density values may be found as the reciprocal of the specific volume values tabulated in the “Thermodynamic Properties: Tables” subsection.
2-92
PHYSICAL AND CHEMICAL DATA
TABLE 2-28
Density (kg/m3) of Water from 0 to 100°C (Concluded ) ρ, kg/m3
t, °C
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
50 51 52 53 54
988.037 987.581 987.120 986.652 986.177
987.992 987.536 987.073 986.604 986.129
987.946 987.490 987.027 986.557 986.081
987.901 987.444 986.980 986.510 986.033
987.844 987.398 986.933 986.463 985.985
987.810 987.351 986.886 986.415 985.937
987.764 987.305 986.840 986.368 985.889
987.719 987.259 986.793 986.320 985.841
987.673 987.213 986.746 986.272 985.793
987.627 987.166 986.699 986.225 985.745
55 56 57 58 59
985.696 985.219 984.716 984.217 983.712
985.648 985.160 984.666 984.167 983.661
985.599 985.111 984.617 984.116 983.610
985.551 985.062 984.567 984.066 983.559
985.502 985.013 984.517 984.016 983.508
985.454 984.963 984.467 983.965 983.457
985.405 984.914 984.417 983.914 983.406
985.356 984.865 984.367 983.864 983.354
985.307 984.815 984.317 983.813 983.303
985.258 984.766 984.267 983.762 983.252
60 61 62 63 64
983.200 982.683 982.160 981.631 981.097
983.149 982.631 982.108 981.578 981.043
983.097 982.579 982.055 981.525 980.989
983.046 982.527 982.002 981.472 980.935
982.994 982.475 981.949 981.418 980.881
982.943 982.422 981.897 981.365 980.827
982.891 982.370 981.844 981.311 980.773
982.839 982.318 981.791 981.258 980.719
982.787 982.265 981.738 981.204 980.665
982.735 982.213 981.685 981.151 980.611
65 66 67 68 69
980.557 980.011 979.459 978.902 978.339
980.502 979.956 979.403 978.846 978.283
980.443 979.901 979.348 978.790 978.226
980.393 979.846 979.293 978.734 978.170
980.339 979.791 979.237 978.678 978.113
980.284 979.736 979.181 978.621 978.056
980.230 979.680 979.126 978.565 977.999
980.175 979.625 979.070 978.509 977.942
980.120 979.570 979.014 978.452 977.885
980.065 979.515 978.958 978.396 977.828
70 71 72 73 74
977.771 977.198 976.619 976.035 975.445
977.714 977.140 976.561 975.976 975.386
977.657 977.082 976.503 975.917 975.327
977.600 977.025 976.444 975.858 975.267
977.543 976.967 976.386 975.800 975.208
977.485 976.909 976.327 975.741 975.148
977.428 976.851 976.269 975.682 975.089
977.370 976.793 976.211 975.623 975.029
977.313 976.735 976.152 975.564 974.970
977.255 976.677 976.093 975.504 974.910
75 76 77 78 79
974.850 974.250 973.645 973.025 972.419
974.791 974.190 973.584 972.974 972.358
974.731 974.130 973.524 972.912 972.296
974.671 974.069 973.463 972.851 972.234
974.611 974.009 973.402 972.789 972.172
974.551 973.948 973.341 972.728 972.110
974.491 973.888 973.280 972.666 972.048
974.431 973.827 973.218 972.605 971.986
974.371 973.767 973.157 972.543 971.923
974.311 973.706 973.096 972.481 971.861
80 81 82 83 84
971.799 971.173 970.543 969.907 969.267
971.737 971.110 970.479 969.843 969.202
971.674 971.048 970.416 969.772 969.138
971.612 970.985 970.353 969.715 969.073
971.549 970.922 970.289 969.652 969.009
971.487 970.859 970.226 969.587 968.944
971.424 970.796 970.162 969.523 968.880
971.361 970.732 970.098 969.459 968.815
971.299 970.669 970.035 969.395 968.751
971.236 970.606 969.971 969.331 968.686
85 86 87 88 89
968.621 967.971 967.316 966.656 965.991
968.556 967.906 967.250 966.589 965.924
968.491 967.840 967.184 966.523 965.857
968.427 967.775 967.118 966.457 965.790
968.362 967.709 967.052 966.390 965.723
968.297 967.641 966.986 966.324 965.656
969.232 967.578 966.920 966.257 965.589
968.166 967.513 966.854 966.191 965.522
968.101 967.447 966.788 966.124 965.455
968.036 967.381 966.722 966.057 965.388
90 91 92 93 94
965.321 964.647 963.967 963.284 962.595
965.254 954.579 963.899 963.215 962.526
965.187 964.511 963.831 963.146 962.457
965.119 964.443 963.763 963.077 962.387
965.052 964.376 963.694 963.009 962.318
964.984 964.308 963.626 962.940 962.249
964.917 964.240 963.558 962.871 962.180
964.849 964.172 963.489 962.802 962.110
964.782 964.104 963.421 962.733 962.041
964.714 964.036 963.352 962.664 961.971
95 96 97 98 99
961.902 961.204 960.501 959.794 959.082
961.832 961.134 960.431 959.723 959.010
961.762 961.064 960.360 959.652 958.939
961.693 960.993 960.289 959.581 958.867
961.693 960.923 960.219 959.510 958.796
961.553 960.853 960.148 959.438 958.724
961.483 960.783 960.077 959.367 958.653
961.414 960.712 960.006 959.296 958.581
961.344 960.642 959.936 959.225 958.509
961.274 960.572 959.865 959.153 958.431
100
958.365
DENSITIES OF PURE SUBSTANCES TABLE 2-29
2-93
Density (kg/m3) of Mercury from 0 to 350°C* Density, kg /m3
t, °C
2
3
4
5
6
7
8
9
0 10 20 30 40
13595.08 13570.44 13545.87 13521.36 13496.92
0
13592.61 13567.98 13543.41 13518.91 13494.48
1
13590.14 13565.52 13540.96 13516.47 13492.04
13587.68 13563.06 13538.51 13514.02 13489.60
13585.21 13560.60 13536.06 13511.58 13487.16
13582.75 13558.14 13533.61 13509.13 13484.72
13580.29 13555.69 13531.16 13506.69 13482.29
13577.82 13553.23 13528.71 13504.25 13479.85
13575.36 13550.78 13526.26 13501.80 13477.41
13572.90 13548.32 13523.81 13499.36 13474.98
50 60 70 80 90
13472.54 13448.22 13423.96 13399.75 13375.59
13470.11 13445.80 13421.54 13397.34 13373.18
13467.67 13443.37 13419.12 13394.92 13370.77
13465.24 13440.94 13416.69 13392.50 13368.36
13462.81 13438.51 13414.27 13390.08 13365.94
13460.38 13436.09 13411.85 13387.67 13363.53
13457.94 13433.66 13409.43 13385.25 13361.12
13455.51 13431.23 13407.01 13382.84 13358.71
13453.08 13428.81 13404.59 13380.42 13356.30
13450.65 13426.39 13402.17 13378.01 13353.89
100 110 120 130 140
13351.5 13327.4 13303.4 13279.4 13255.4
13349.1 13325.0 13301.0 13277.0 13253.0
13346.7 13322.6 13298.6 13274.6 13250.6
13344.3 13320.2 13296.2 13272.2 13248.2
13341.9 13317.8 13293.8 13269.8 13245.8
13339.4 13315.4 13291.4 13267.4 13243.4
13337.0 13313.0 13288.9 13265.0 13241.0
13334.6 13310.6 13286.6 13262.6 13238.7
13332.2 13308.2 13284.2 13260.2 13236.3
13329.8 13305.8 13281.8 13257.8 13233.9
150 160 170 180 190
13231.5 13207.6 13183.7 13159.8 13136.0
13229.1 13205.2 13181.3 13157.4 13133.6
13226.7 13202.8 13178.9 13155.0 13131.2
13224.3 13200.4 13176.5 13152.6 13128.3
13221.9 13198.0 13174.1 13150.3 13126.4
13219.5 13195.6 13171.7 13147.9 13124.0
13217.1 13193.2 13169.4 13145.5 13121.7
13214.7 13190.8 13167.0 13143.1 13119.3
13212.4 13188.5 13164.6 13140.7 13116.9
13210.0 13186.1 13162.2 13138.3 13114.5
200 210 220 230 240
13112.1 13088.3 13064.5 13040.6 13016.8
13109.7 13085.9 13062.1 13038.3 13014.5
13107.4 13083.5 13059.7 13035.9 13012.1
13105.0 13081.1 13057.3 13033.5 13009.7
13102.6 13078.8 13054.9 13031.1 13007.3
13100.2 13076.4 13052.6 13028.7 13004.9
13097.8 13074.0 13050.2 13026.4 13002.5
13095.4 13071.6 13047.8 13024.0 13000.2
13093.1 13069.2 13045.4 13021.6 12997.8
13090.7 13066.8 13043.0 13019.2 12995.4
250 260 270 280 290
12993.0 12969.2 12945.4 12921.5 12897.7
12990.6 12966.8 12943.0 12919.1 12895.3
12988.3 12964.4 12940.6 12916.7 12892.9
12985.9 12962.0 12938.2 12914.4 12890.5
12983.5 12959.7 12935.8 12912.0 12888.1
12981.1 12957.3 12933.4 12909.6 12885.7
12978.7 12954.9 12931.1 12907.2 12883.3
12976.3 12952.5 12928.7 12904.8 12880.9
12974.0 12950.1 12926.3 12902.4 12878.5
12971.6 12947.7 12923.9 12900.0 12876.2
300 310 320 330 340
12873.8 12849.9 12825.9 12801.9 12777.8
12871.4 12847.5 12823.5 12799.5 12775.4
12869.0 12845.1 12821.1 12797.1 12773.0
12866.6 12842.7 12818.7 12794.7 12770.6
12864.2 12840.3 12816.3 12792.3 12768.2
12861.8 12837.9 12813.9 12789.9 12765.8
12859.4 12835.5 12811.5 12787.5 12763.4
12857.0 12833.1 12809.1 12785.1 12761.0
12854.6 12830.7 12806.7 12782.7 12758.6
12852.2 12828.3 12804.3 12780.2 12756.1
350
12753.7
*From “Mercury—Density and Thermal Expansion at Atmospheric Pressure and Temperatures from 0 to 350°C,” Tables of Standard Handbook Data, Standartov, Moscow, 1978. The density values obtainable from those cited for the specific volume of the saturated liquid in the “Thermodynamic Properties” subsection show minor differences. No attempt was made to adjust either set.
TABLE 2-30 Cmpd. no.
Densities of Inorganic and Organic Liquids Formula
CAS no.
Mol. wt.
C1
C2
C3
C4
Tmin, K
Density at Tmin
Tmax, K
Density at Tmax
1 2 3 4 5 6 7 8 9 10
Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane
CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22
74828 74840 74986 106978 109660 110543 142825 111659 111842 124185
16.043 30.070 44.097 58.123 72.150 86.177 100.204 114.231 128.258 142.285
2.9214 1.9122 1.3757 1.0677 0.84947 0.70824 0.61259 0.53731 0.48387 0.42831
0.28976 0.27937 0.27453 0.27188 0.26726 0.26411 0.26211 0.26115 0.26147 0.25745
190.56 305.32 369.83 425.12 469.7 507.6 540.2 568.7 594.6 617.7
0.28881 0.29187 0.29359 0.28688 0.27789 0.27537 0.28141 0.28034 0.28281 0.28912
90.69 90.35 85.47 134.86 143.42 177.83 182.57 216.38 219.66 243.51
28.18 21.64 16.583 12.62 10.474 8.747 7.6998 6.6558 6.007 5.3811
190.56 305.32 369.83 425.12 469.7 507.6 540.2 568.7 594.6 617.7
10.082 6.845 5.011 3.927 3.178 2.682 2.337 2.058 1.851 1.664
11 12 13 14 15 16 17 18 19 20
n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane n-Eicosane
C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40 C20H42
1120214 112403 629505 629594 629629 544763 629787 593453 629925 112958
156.312 170.338 184.365 198.392 212.419 226.446 240.473 254.500 268.527 282.553
0.39 0.35541 0.3216 0.30545 0.28445 0.26807 0.2545 0.23864 0.22451 0.21624
0.25678 0.25511 0.2504 0.2535 0.25269 0.25287 0.254 0.25272 0.25133 0.25287
639 658 675 693 708 723 736 747 758 768
0.2913 0.29368 0.3071 0.30538 0.30786 0.31143 0.31072 0.31104 0.3133 0.31613
247.57 263.57 267.76 279.01 283.07 291.31 295.13 301.31 305.04 309.58
4.9362 4.5132 4.2035 3.8924 3.6471 3.4187 3.2241 3.0466 2.8933 2.7496
21 22 23 24 25 26 27
2-Methylpropane 2-Methylbutane 2,3-Dimethylbutane 2-Methylpentane 2,3-Dimethylpentane 2,3,3-Trimethylpentane 2,2,4-Trimethylpentane
C4H10 C5H12 C6H14 C6H14 C7H16 C8H18 C8H18
75285 78784 79298 107835 565593 560214 540841
58.123 72.150 86.177 86.177 100.204 114.231 114.231
1.0463 0.9079 0.76929 0.73335 0.7229 0.6028 0.5886
0.27294 0.2761 0.27524 0.2687 0.28614 0.27446 0.27373
408.14 460.43 499.98 497.5 537.35 573.5 543.96
0.27301 0.28673 0.27691 0.28361 0.2713 0.2741 0.2846
113.54 113.25 145.19 119.55 160.00 172.22 165.78
28 29 30 31 32 33 34 35
Ethylene Propylene 1-Butene cis-2-Butene trans-2-Butene 1-Pentene 1-Hexene 1-Heptene
C2H4 C3H6 C4H8 C4H8 C4H8 C5H10 C6H12 C7H14
74851 115071 106989 590181 624646 109671 592416 592767
28.054 42.081 56.108 56.108 56.108 70.134 84.161 98.188
2.0961 1.4094 1.0972 1.1609 1.1426 0.9038 0.7389 0.63734
0.27657 0.26465 0.2649 0.27104 0.27095 0.26648 0.26147 0.26319
282.34 365.57 419.95 435.58 428.63 464.78 504.03 537.29
0.29147 0.295 0.29043 0.2816 0.2854 0.2905 0.2902 0.27375
36 37 38 39 40 41 42 43 44
1-Octene 1-Nonene 1-Decene 2-Methylpropene 2-Methyl-1-butene 2-Methyl-2-butene 1,2-Butadiene 1,3-Butadiene 2-Methyl-1,3-butadiene1
C8H16 C9H18 C10H20 C4H8 C5H10 C5H10 C4H6 C4H6 C5H8
111660 124118 872059 115117 563462 513359 590192 106990 78795
112.215 126.242 140.269 56.108 70.134 70.134 54.092 54.092 68.119
0.5871 0.4945 0.44244 1.1454 0.91619 0.93322 1.187 1.2384 0.95673
0.27005 0.26108 0.25838 0.2725 0.26752 0.27251 0.26114 0.2725 0.26488
566.65 593.25 616.4 417.9 465 471 452 425.17 484
45 46 47 48 49 50
Acetylene Methylacetylene Dimethylacetylene 3-Methyl-1-butyne 1-Pentyne 2-Pentyne
C2H2 C3H4 C4H6 C5H8 C5H8 C5H8
74862 74997 503173 598232 627190 627214
26.038 40.065 54.092 68.119 68.119 68.119
2.4091 1.6086 1.1717 0.94575 0.8491 0.92099
0.27223 0.26448 0.25895 0.26008 0.2352 0.25419
51 52 53
1-Hexyne 2-Hexyne 3-Hexyne1
C6H10 C6H10 C6H10
693027 764352 928494
82.145 82.145 82.145
0.84427 0.76277 0.78045
0.27185 0.25248 0.26065
2-94
Name
639 658 675 693 708 723 736 747 758 768
1.519 1.393 1.284 1.205 1.126 1.060 1.002 0.944 0.893 0.855
12.575 10.776 9.0343 9.2041 7.8746 7.0934 6.9163
408.14 460.43 499.98 497.5 537.35 573.5 543.96
3.833 3.288 2.795 2.729 2.526 2.196 2.150
104.00 87.89 87.80 134.26 167.62 107.93 133.39 154.27
23.326 18.143 14.326 13.895 13.1 11.543 9.6388 8.1759
282.34 365.57 419.95 435.58 428.63 464.78 504.03 537.29
7.579 5.326 4.142 4.283 4.217 3.392 2.826 2.422
0.27187 0.27319 0.28411 0.28186 0.28164 0.26031 0.3065 0.28813 0.28571
171.45 191.78 206.89 132.81 135.58 139.39 136.95 164.25 127.27
7.1247 6.333 5.7131 13.506 11.332 11.218 15.123 14.061 12.205
566.65 593.25 616.4 417.9 465 471 452 425.17 484
2.174 1.894 1.712 4.203 3.425 3.425 4.546 4.545 3.612
308.32 402.39 473.2 463.2 481.2 519
0.28477 0.279 0.27289 0.30807 0.353 0.31077
192.40 170.45 240.91 183.45 167.45 163.83
23.692 19.027 13.767 11.519 12.532 12.24
308.32 402.39 473.2 463.2 481.2 519
8.850 6.082 4.525 3.636 3.610 3.623
516.2 549 544
0.2771 0.31611 0.28571
141.25 183.65 170.05
10.23 10.133 10.021
516.2 549 544
3.106 3.021 2.994
54 55 56
1-Heptyne 1-Octyne Vinylacetylene2
C7H12 C8H14 C4H4
628717 629050 689974
96.172 110.199 52.076
0.67366 0.59229 1.2703
0.26003 0.26118 0.26041
559 585 454
0.29804 0.29357 0.297
192.22 193.55 173.15
8.4987 7.478 15.664
559 585 454
2.591 2.268 4.878
57 58 59 60 61 62
C5H10 C6H12 C7H14 C6H12 C7H14 C8H16
287923 96377 1640897 110827 108872 590669
70.134 84.161 98.188 84.161 98.188 112.215
1.124 0.84798 0.7193 0.8908 0.735 0.55873
0.28859 0.27042 0.26936 0.27396 0.27041 0.25143
511.76 532.79 569.52 553.58 572.19 591.15
0.2506 0.28276 0.2777 0.2851 0.2927 0.27758
179.28 130.73 134.71 279.69 146.58 239.66
11.883 10.492 9.018 9.3797 9.018 7.3417
511.76 532.79 569.52 553.58 572.19 591.15
3.895 3.136 2.670 3.252 2.718 2.222
63 64 65 66
Cyclopentane Methylcyclopentane Ethylcyclopentane Cyclohexane Methylcyclohexane 1,1-Dimethylcyclohexane Ethylcyclohexane Cyclopentene 1-Methylcyclopentene Cyclohexene
C8H16 C5H8 C6H10 C6H10
1678917 142290 693890 110838
112.215 68.119 82.145 82.145
0.61587 1.1035 0.88824 0.92997
0.26477 0.27035 0.26914 0.27056
609.15 507 542 560.4
0.28054 0.28699 0.27874 0.28943
161.84 138.13 146.62 169.67
7.8679 13.47 10.98 11.16
609.15 507 542 560.4
2.326 4.082 3.300 3.437
67 68 69 70 71 72 73
Benzene Toluene o-Xylene m-Xylene p-Xylene Ethylbenzene Propylbenzene
C6H6 C7H8 C8H10 C8H10 C8H10 C8H10 C9H12
71432 108883 95476 108383 106423 100414 103651
78.114 92.141 106.167 106.167 106.167 106.167 120.194
1.0162 0.8488 0.69883 0.69555 0.6816 0.6952 0.57695
0.2655 0.26655 0.26113 0.26204 0.25963 0.26037 0.25395
562.16 591.8 630.33 617.05 616.23 617.2 638.32
0.28212 0.2878 0.27429 0.27602 0.2768 0.2844 0.283
278.68 178.18 247.98 225.30 286.41 178.15 183.15
11.421 10.495 8.6285 8.6505 8.1616 9.0568 7.8942
562.16 591.8 630.33 617.05 616.23 617.2 638.32
3.828 3.184 2.676 2.654 2.625 2.670 2.272
74 75 76 77 78 79 80 81
1,2,4-Trimethylbenzene Isopropylbenzene 1,3,5-Trimethylbenzene p-Isopropyltoluene Naphthalene6 Biphenyl Styrene m-Terphenyl
C9H12 C9H12 C9H12 C10H14 C10H8 C12H10 C8H8 C18H14
95636 98828 108678 99876 91203 92524 100425 92068
120.194 120.194 120.194 134.221 128.174 154.211 104.152 230.309
0.60394 0.604 0.59879 0.51036 0.61674 0.5039 0.7397 0.30826
0.25955 0.25912 0.25916 0.25383 0.25473 0.25273 0.2603 0.23669
649.13 631.1 637.36 653.15 748.35 789.26 636 924.85
0.27716 0.2914 0.27968 0.28816 0.27355 0.281 0.3009 0.29678
229.33 177.14 228.42 205.25 333.15 342.20 242.54 360.00
7.6895 7.9496 7.6154 6.8779 7.7543 6.4395 9.1088 4.5223
649.13 631.1 637.36 653.15 748.35 789.26 636 924.85
2.327 2.331 2.311 2.011 2.421 1.994 2.842 1.302
82 83 84 85 86 87 88
Methanol Ethanol 1-Propanol 1-Butanol 2-Butanol 2-Propanol 2-Methyl-2-propanol
CH4O C2H6O C3H8O C4H10O C4H10O C3H8O C4H10O
67561 64175 71238 71363 78922 67630 75650
32.042 46.069 60.096 74.123 74.123 60.096 74.123
2.288 1.648 1.235 0.965 0.966 1.24 0.9212
0.2685 0.27627 0.27136 0.2666 0.26064 0.27342 0.2544
512.64 513.92 536.78 563.05 536.05 508.3 506.21
0.2453 0.2331 0.24 0.24419 0.2746 0.2353 0.276
175.47 159.05 146.95 184.51 158.45 185.28 298.97
27.912 19.413 15.231 12.016 12.57 14.547 10.555
512.64 513.92 536.78 563.05 536.05 508.3 506.21
8.521 5.965 4.551 3.620 3.706 4.535 3.621
89 90 91 92 93 94 95 96
1-Pentanol 2-Methyl-1-butanol 3-Methyl-1-butanol 1-Hexanol 1-Heptanol Cyclohexanol Ethylene glycol 1,2-Propylene glycol
C5H12O C5H12O C5H12O C6H14O C7H16O C6H12O C2H6O2 C3H8O2
71410 137326 123513 111273 111706 108930 107211 57556
88.150 88.150 88.150 102.177 116.203 100.161 62.068 76.095
0.8164 0.82046 0.837 0.70617 0.60481 0.8243 1.3151 1.0923
0.2673 0.26829 0.27375 0.26901 0.2632 0.26546 0.25125 0.26106
586.15 565 577.2 611.35 631.9 650 719.7 626
0.2506 0.2322 0.22951 0.2479 0.273 0.2848 0.2187 0.20459
195.56 203.00 155.95 228.55 239.15 296.60 260.15 213.15
10.057 10.017 10.204 8.4506 7.421 9.4693 18.31 14.363
586.15 565 577.2 611.35 631.9 650 719.7 626
3.054 3.058 3.058 2.625 2.298 3.105 5.234 4.184
97 98 99 100
Phenol o-Cresol m-Cresol p-Cresol
C6H6O C7H8O C7H8O C7H8O
108952 95487 108394 106445
94.113 108.140 108.140 108.140
1.3798 1.0861 0.9061 1.1503
0.31598 0.30624 0.28268 0.31861
694.25 697.55 705.85 704.65
0.32768 0.30587 0.2707 0.30104
314.06 304.19 285.39 307.93
11.244 9.5751 9.6115 9.4494
694.25 697.55 705.85 704.65
4.367 3.547 3.205 3.610
101 102 103 104 105 106 107 108
Dimethyl ether Methyl ethyl ether Methyl-n-propyl ether Methyl isopropyl ether Methyl-n-butyl ether Methyl isobutyl ether1 Methyl tert-butyl ether Diethyl ether
C2H6O C3H8O C4H10O C4H10O C5H12O C5H12O C5H12O C4H10O
115106 540670 557175 598538 628284 625445 1634044 60297
46.069 60.096 74.123 74.123 88.150 88.150 88.150 74.123
1.5693 1.2635 1.0124 1.0318 0.8281 0.8252 0.82157 0.9554
0.2679 0.27878 0.27942 0.28478 0.27245 0.27282 0.27032 0.26847
400.1 437.8 476.3 464.5 510 497 497.1 466.7
0.2882 0.2744 0.2555 0.2444 0.2827 0.2857 0.2829 0.2814
131.65 160.00 133.97 127.93 157.48 150.00 164.55 156.85
18.95 13.995 11.696 11.568 9.8068 9.7673 9.7682 11.487
400.1 437.8 476.3 464.5 510 497 497.1 466.7
5.858 4.532 3.623 3.623 3.040 3.025 3.039 3.559
2-95
TABLE 2-30 Cmpd. no.
Densities of Inorganic and Organic Liquids (Continued ) CAS no.
Mol. wt.
C1
C2
C3
C4
Tmin, K
Density at Tmin
Tmax, K
Density at Tmax
109 110 111 112
Ethyl propyl ether Ethyl isopropyl ether Methyl phenyl ether Diphenyl ether
C5H12O C5H12O C7H8O C12H10O
628320 625547 100663 101848
88.150 88.150 108.140 170.211
0.7908 0.82049 0.77488 0.52133
0.266 0.26994 0.26114 0.26218
500.23 489 645.6 766.8
0.292 0.30381 0.28234 0.31033
145.65 140.00 235.65 300.03
9.8474 9.9117 9.6675 6.2648
500.23 489 645.6 766.8
2.973 3.040 2.967 1.988
113 114 115 116 117 118 119 120 121 122
Formaldehyde3 Acetaldehyde 1-Propanal 1-Butanal 1-Pentanal 1-Hexanal 1-Heptanal 1-Octanal 1-Nonanal 1-Decanal
CH2O C2H4O C3H6O C4H8O C5H10O C6H12O C7H14O C8H16O C9H18O C10H20O
50000 75070 123386 123728 110623 66251 111717 124130 124196 112312
30.026 44.053 58.080 72.107 86.134 100.161 114.188 128.214 142.241 156.268
1.9415 1.6994 1.296 1.0361 0.83871 0.71899 0.62649 0.56833 0.49587 0.46802
0.22309 0.26167 0.26439 0.26731 0.26252 0.26531 0.26376 0.26939 0.26135 0.27146
408 466 504.4 537.2 566.1 591 617 638.1 658 674.2
0.28571 0.2913 0.29471 0.28397 0.29444 0.27628 0.29221 0.26975 0.30736 0.26869
181.15 150.15 170.00 176.75 182.00 217.15 229.80 246.00 255.15 267.15
30.945 21.499 15.929 12.589 10.534 8.7243 7.6002 6.6637 6.0165 5.3834
408 466 504.4 537.2 566.1 591 617 638.1 658 674.2
8.703 6.494 4.902 3.876 3.195 2.710 2.375 2.110 1.897 1.724
123 124 125 126
C3H6O C4H8O C5H10O C5H10O
67641 78933 107879 563804
58.080 72.107 86.134 86.134
1.2332 0.93767 0.90411 0.8374
0.25886 0.25035 0.27207 0.26204
508.2 535.5 561.08 553
0.2913 0.29964 0.30669 0.2857
178.45 186.48 196.29 181.15
15.683 12.663 10.398 10.565
508.2 535.5 561.08 553
4.764 3.745 3.323 3.196
127 128 129 130 131 132 133 134
Acetone Methyl ethyl ketone 2-Pentanone Methyl isopropyl ketone1 2-Hexanone Methyl isobutyl ketone 3-Methyl-2-pentanone1 3-Pentanone Ethyl isopropyl ketone Diisopropyl ketone Cyclohexanone Methyl phenyl ketone
C6H12O C6H12O C6H12O C5H10O C6H12O C7H14O C6H10O C8H8O
591786 108101 565617 96220 565695 565800 108941 98862
100.161 100.161 100.161 86.134 100.161 114.188 98.145 120.151
0.70659 0.71791 0.6969 0.71811 0.66469 0.56213 0.8663 0.64417
0.26073 0.26491 0.2587 0.24129 0.24527 0.23385 0.26941 0.24863
587.05 571.4 573 560.95 567 576 653 709.5
0.2963 0.28544 0.2857 0.27996 0.34305 0.2618 0.2977 0.28661
217.35 189.15 167.15 234.18 200.00 204.81 242.00 292.81
8.7505 8.8579 9.1722 10.102 9.0933 8.7779 10.081 8.5581
587.05 571.4 573 560.95 567 576 653 709.5
2.710 2.710 2.694 2.976 2.710 2.404 3.216 2.591
135 136 137 138 139 140 141
Formic acid Acetic acid Propionic acid n-Butyric acid Isobutyric acid Benzoic acid1 Acetic anhydride
CH2O2 C2H4O2 C3H6O2 C4H8O2 C4H8O2 C7H6O2 C4H6O3
64186 64197 79094 107926 79312 65850 108247
46.026 60.053 74.079 88.106 88.106 122.123 102.090
1.938 1.4486 1.1041 0.89213 0.88575 0.71587 0.86852
0.24225 0.25892 0.25659 0.25938 0.25736 0.24812 0.25187
588 591.95 600.81 615.7 605 751 606
0.24435 0.2529 0.26874 0.24909 0.26265 0.2857 0.31172
281.45 289.81 252.45 267.95 227.15 395.45 200.15
26.806 17.492 13.933 11.087 11.42 8.8935 11.643
588 591.95 600.81 615.7 605 751 606
8.000 5.595 4.303 3.440 3.442 2.885 3.448
142 143 144 145 146 147 148
Methyl formate Methyl acetate Methyl propionate Methyl n-butyrate Ethyl formate Ethyl acetate Ethyl propionate
C2H4O2 C3H6O2 C4H8O2 C5H10O2 C3H6O2 C4H8O2 C5H10O2
107313 79209 554121 623427 109944 141786 105373
60.053 74.079 88.106 102.133 74.079 88.106 102.133
1.525 1.13 0.9147 0.76983 1.1343 0.8996 0.7405
0.2634 0.2593 0.2594 0.26173 0.26168 0.25856 0.25563
487.2 506.55 530.6 554.5 508.4 523.3 546
0.2806 0.2764 0.2774 0.26879 0.2791 0.278 0.2795
174.15 175.15 185.65 187.35 193.55 189.60 199.25
18.811 14.475 11.678 9.7638 14.006 11.478 9.6317
487.2 506.55 530.6 554.5 508.4 523.3 546
5.790 4.358 3.526 2.941 4.335 3.479 2.897
149 150 151 152 153 154 155
Ethyl n-butyrate n-Propyl formate n-Propyl acetate n-Butyl acetate Methyl benzoate Ethyl benzoate Vinyl acetate
C6H12O2 C4H8O2 C5H10O2 C6H12O2 C8H8O2 C9H10O2 C4H6O2
105544 110747 109604 123864 93583 93890 108054
116.160 88.106 102.133 116.160 136.150 150.177 86.090
0.63566 0.915 0.73041 0.669 0.53944 0.4883 0.9591
0.25613 0.26134 0.25456 0.26028 0.23519 0.23878 0.2593
571 538 549.73 579.15 693 698 519.13
0.27829 0.28 0.27666 0.309 0.2676 0.28487 0.27448
175.15 180.25 178.15 199.65 260.75 238.45 180.35
8.4912 11.59 9.7941 8.3747 8.2133 7.2924 12.287
571 538 549.73 579.15 693 698 519.13
2.482 3.501 2.869 2.570 2.294 2.045 3.699
156 157 158 159 160
Methylamine Dimethylamine Trimethylamine Ethylamine Diethylamine
CH5N C2H7N C3H9N C2H7N C4H11N
74895 124403 75503 75047 109897
31.057 45.084 59.111 45.084 73.138
1.39 1.5436 1.0116 1.1477 0.85379
0.21405 0.27784 0.25683 0.23182 0.25675
430.05 437.2 433.25 456.15 496.6
0.2275 0.2572 0.2696 0.26053 0.27027
179.69 180.96 156.08 192.15 223.35
25.378 16.964 13.144 17.588 10.575
430.05 437.2 433.25 456.15 496.6
6.494 5.556 3.939 4.951 3.325
2-96
Name
Formula
161 162 163 164 165 166 167 168
Triethylamine n-Propylamine di-n-Propylamine Isopropylamine Diisopropylamine Aniline N-Methylaniline N,N-Dimethylaniline
C6H15N C3H9N C6H15N C3H9N C6H15N C6H7N C7H9N C8H11N
121448 107108 142847 75310 108189 62533 100618 121697
101.192 59.111 101.192 59.111 101.192 93.128 107.155 121.182
0.7035 0.9195 0.659 1.2801 0.6181 1.0405 0.6527 0.4923
0.27386 0.23878 0.26428 0.2828 0.25786 0.2807 0.24324 0.22868
535.15 496.95 550 471.85 523.1 699 701.55 687.15
0.2872 0.2461 0.2766 0.2972 0.271 0.29236 0.25374 0.2335
158.45 188.36 210.15 177.95 176.85 267.13 216.15 275.60
8.2843 13.764 7.9929 13.561 8.0541 11.176 9.7244 7.9705
535.15 496.95 550 471.85 523.1 699 701.55 687.15
2.569 3.851 2.494 4.527 2.397 3.707 2.683 2.153
169 170 171 172
Ethylene oxide Furan Thiophene Pyridine
C2H4O C4H4O C4H4S C5H5N
75218 110009 110021 110861
44.053 68.075 84.142 79.101
1.836 1.1339 1.2875 0.9815
0.26024 0.24741 0.28195 0.24957
469.15 490.15 579.35 619.95
0.2696 0.2612 0.3077 0.29295
160.65 187.55 234.94 231.51
23.477 15.702 13.431 13.193
469.15 490.15 579.35 619.95
7.055 4.583 4.566 3.933
173 174
CH3NO C3H7NO
75127 68122
45.041 73.095
1.2486 0.89615
0.20352 0.23478
771 649.6
0.25178 0.28091
275.60 212.72
25.488 13.954
771 649.6
6.135 3.817
175 176 177 178 179 180
Formamide5 N,N-Dimethylformamide Acetamide N-Methylacetamide Acetonitrile Propionitrile n-Butyronitrile Benzonitrile
C2H5NO C3H7NO C2H3N C3H5N C4H7N C7H5N
60355 79163 75058 107120 109740 100470
59.068 73.095 41.053 55.079 69.106 103.123
1.016 0.88268 1.3064 1.0224 0.87533 0.73136
0.21845 0.23568 0.22597 0.23452 0.24331 0.24793
761 718 545.5 564.4 582.25 699.35
0.26116 0.27379 0.28678 0.2804 0.28586 0.2841
353.33 301.15 229.32 180.26 161.25 260.40
16.936 13.012 20.628 16.027 13.047 10.009
761 718 545.5 564.4 582.25 699.35
4.651 3.745 5.781 4.360 3.598 2.950
181 182 183 184 185 186 187 188 189
Methyl mercaptan Ethyl mercaptan n-Propyl mercaptan n-Butyl mercaptan Isobutyl mercaptan sec-Butyl mercaptan Dimethyl sulfide Methyl ethyl sulfide Diethyl sulfide
CH4S C2H6S C3H8S C4H10S C4H10S C4H10S C2H6S C3H8S C4H10S
74931 75081 107039 109795 513440 513531 75183 624895 352932
48.109 62.136 76.163 90.189 90.189 90.189 62.136 76.163 90.189
1.9323 1.3047 1.0714 0.89458 0.88801 0.89137 1.4029 1.067 0.82413
0.28018 0.2694 0.27214 0.27463 0.27262 0.27365 0.27991 0.27101 0.26333
469.95 499.15 536.6 570.1 559 554 503.04 533 557.15
0.28523 0.27866 0.29481 0.28512 0.29522 0.2953 0.2741 0.29363 0.27445
150.18 125.26 159.95 157.46 128.31 133.02 174.88 167.23 169.20
21.564 16.242 12.716 10.585 10.851 10.761 15.556 12.672 10.476
469.95 499.15 536.6 570.1 559 554 503.04 533 557.15
6.897 4.843 3.937 3.257 3.257 3.257 5.012 3.937 3.130
190 191 192 193 194 195 196 197
Fluoromethane Chloromethane Trichloromethane Tetrachloromethane Bromomethane Fluoroethane Chloroethane Bromoethane
CH3F CH3Cl CHCl3 CCl4 CH3Br C2H5F C2H5Cl C2H5Br
593533 74873 67663 56235 74839 353366 75003 74964
34.033 50.488 119.377 153.822 94.939 48.060 64.514 108.966
2.1854 1.817 1.0841 0.99835 1.6762 1.6525 2.176 1.1908
0.24725 0.25877 0.2581 0.274 0.26141 0.27099 0.3377 0.25595
317.42 416.25 536.4 556.35 467 375.31 460.35 503.8
0.27558 0.2833 0.2741 0.287 0.28402 0.2442 0.3361 0.29152
131.35 175.43 209.63 250.33 179.47 129.95 134.80 154.55
29.526 22.347 13.702 10.843 20.64 19.785 16.934 15.833
317.42 416.25 536.4 556.35 467 375.31 460.35 503.8
8.839 7.022 4.200 3.644 6.412 6.098 6.444 4.653
198 199 200 201 202 203 204 205
1-Chloropropane 2-Chloropropane 1,1-Dichloropropane1 1,2-Dichloropropane Vinyl chloride Fluorobenzene Chlorobenzene Bromobenzene
C3H7Cl C3H7Cl C3H6Cl2 C3H6Cl2 C2H3Cl C6H5F C6H5Cl C6H5Br
540545 75296 78999 78875 75014 462066 108907 108861
78.541 78.541 112.986 112.986 62.499 96.104 112.558 157.010
1.087 1.1202 0.91064 0.89833 1.5115 1.0146 0.8711 0.8226
0.26832 0.27669 0.26561 0.26142 0.2707 0.27277 0.26805 0.26632
503.15 489 560 572 432 560.09 632.35 670.15
0.28055 0.27646 0.28571 0.2868 0.2716 0.28291 0.2799 0.2821
150.35 155.97 200.00 172.71 119.36 230.94 227.95 242.43
13.328 12.855 11.03 11.526 18.481 11.374 10.385 9.9087
503.15 489 560 572 432 560.09 632.35 670.15
4.051 4.049 3.429 3.436 5.584 3.720 3.250 3.089
206 207 208 209 210 211 212 213 214 215 216 217
Air Hydrogen Helium-44 Neon Argon Fluorine Chlorine Bromine Oxygen Nitrogen Ammonia Hydrazine
H2 He Ne Ar F2 Cl2 Br2 O2 N2 NH3 N2H4
132259100 1333740 7440597 7440019 7440371 7782414 7782505 7726956 7782447 7727379 7664417 302012
28.951 2.016 4.003 20.180 39.948 37.997 70.905 159.808 31.999 28.014 17.031 32.045
2.8963 5.414 7.2475 7.3718 3.8469 4.2895 2.23 2.1872 3.9143 3.2091 3.5383 1.0516
0.26733 0.34893 0.41865 0.3067 0.2881 0.28587 0.27645 0.29527 0.28772 0.2861 0.25443 0.16613
132.45 33.19 5.2 44.4 150.86 144.12 417.15 584.15 154.58 126.2 405.65 653.15
0.27341 0.2706 0.24096 0.2786 0.29783 0.28776 0.2926 0.3295 0.2924 0.2966 0.2888 0.1898
59.15 13.95 2.20 24.56 83.78 53.48 172.12 265.85 54.35 63.15 195.41 274.69
33.279 38.487 37.115 61.796 35.491 44.888 24.242 20.109 40.77 31.063 43.141 31.934
132.45 33.19 5.2 44.4 150.86 144.12 417.15 584.15 154.58 126.2 405.65 653.15
10.834 15.516 17.312 24.036 13.353 15.005 8.067 7.408 13.605 11.217 13.907 6.330 2-97
TABLE 2-30 Cmpd. no. 218 219 220 221 222 223 224 225 226 227 228 229 230 231
Densities of Inorganic and Organic Liquids (Concluded ) Name
Nitrous oxide Nitric oxide Cyanogen Carbon monoxide Carbon dioxide Carbon disulfide Hydrogen fluoride Hydrogen chloride Hydrogen bromide1 Hydrogen cyanide Hydrogen sulfide Sulfur dioxide Sulfur trioxide Water7
Formula N2O NO C2N2 CO CO2 CS2 HF HCl HBr HCN H2S SO2 SO3 H2O
CAS no.
Mol. wt.
C1
C2
C3
C4
Tmin, K
10024972 10102439 460195 630080 124389 75150 7664393 7647010 10035106 74908 7783064 7446095 7446119 7732185
44.013 30.006 52.036 28.010 44.010 76.143 20.006 36.461 80.912 27.026 34.082 64.065 80.064 18.015
2.781 5.246 1.0761 2.897 2.768 1.7968 2.5635 3.342 2.832 1.3413 2.7672 2.106 1.4969 5.459
0.27244 0.3044 0.20984 0.27532 0.26212 0.28749 0.1766 0.2729 0.2832 0.18589 0.27369 0.25842 0.19013 0.30542
309.57 180.15 400.15 132.92 304.21 552 461.15 324.65 363.15 456.65 373.53 430.75 490.85 647.13
0.2882 0.242 0.20635 0.2813 0.2908 0.3226 0.3733 0.3217 0.28571 0.28206 0.29015 0.2895 0.4359 0.081
182.30 109.50 245.25 68.15 216.58 161.11 189.79 158.97 185.15 259.83 187.68 197.67 289.95 273.16
Density at Tmin 27.928 44.487 18.513 30.18 26.828 19.064 60.203 34.854 27.985 27.202 29.13 25.298 24.241 55.583
Tmax, K
Density at Tmax
309.57 180.15 400.15 132.92 304.21 552 461.15 324.65 363.15 456.65 373.53 430.75 490.85 333.15
10.208 17.234 5.128 10.522 10.560 6.250 14.516 12.246 10.000 7.216 10.111 8.150 7.873 54.703
All substances are listed in alphabetical order in Table 2-6a. Compiled from Daubert, T. E., R. P. Danner, H. M. Sibul, and C. C. Stebbins, DIPPR Data Compilation of Pure Compound Properties, Project 801 Sponsor Release, July, 1993, Design Institute for Physical Property Data, AIChE, New York, NY; and from Thermodynamics Research Center, “Selected Values of Properties of Hydrocarbons and Related Compounds,” Thermodynamics Research Center Hydrocarbon Project, Texas A&M University, College Station, Texas (extant 1994). Temperatures are in kelvins. Liquid densities are in kmol/m3. Density formulas: kmol/m3 × (mol. wt./1E+03) = g/cm3; kmol/m3 × (mol. wt./1.601846E+01) = lb/ft3. C The liquid density equation is C1/C2[1 + (1 − T/C3) 4] unless otherwise noted. 1 The modified Rackett equation, density = (Pc /RTc)/ZRA1 + [1 − (T/Tc)]2/7, was used. See Spencer, C. F., and R. P. Danner, “Improved Equation for Prediction of Saturated Liquid Density,” J. Chem. Eng. Data 17, 236 (1972). 2 Decomposes violently on heating. Forms explosive peroxides with air or oxygen. Polymerizes under pressure and heat. 3 For the hypothetical pure liquid. 4 Exhibits superfluid properties below 2.2 K. 5 Coefficients are hypothetical above the decomposition temperature. 6 Lower limit is for the undercooled liquid. 7 For the temperature range 333.15 to 403.15 K, use the coefficients: C1 = 4.9669E+00, C2 = 2.7788E−01, C3 = 6.4713E+02, C4 = 1.8740E−01. For the temperature range 403.15 to 647.13 K, use C1 = 4.3910E+00, C2 = 2.4870E−01, C3 = 6.4713E+02, C4 = 2.5340E−01.
2-98
DENSITIES OF AQUEOUS INORGANIC SOLUTIONS
2-99
DENSITIES OF AQUEOUS INORGANIC SOLUTIONS UNITS AND UNITS CONVERSIONS Densities are given in grams per cubic centimeter. To convert to pounds per cubic foot, multiply by 62.43. °F = 9⁄ 5 °C + 32. ADDITIONAL REFERENCES For more detailed data on densities see International Critical Tables: tabular index, vol. 3, p. 1; abrasives, vol. 2, p. 87; air, moist, vol. 1, p. 71; building stones, vol. 2, p. 52; clays, vol. 2, p. 56; coals, vol. 2, p. 135; compounds, vol. 1. pp. 106, 176, 313, 341; elements, vol. 1, pp. 102, 340; fibers, vol. 2, p. 237; gases and vapors, vol. 3, pp. 3, 345; glass, vol. 2, p. 93; liquids and vitreous solids, vol. 3, p. 22; vol. 1, pp. 102, 340; vol. 2, pp. 456, 463; vol. 3, pp. 20, 35; liquid coolants and saturated TABLE 2-31
Aluminum Sulfate [Al2(SO4)3] d 15 4
%
d 15 4
1 2 4 8 12
1.0093 1.0195 1.0404 1.0837 1.1293
16 20 24 26
1.1770 1.2272 1.2803 1.3079
0°C
5°C
10°C
20°C
25°C
d 15 4
%
0.9943 0.9954 0.9959 0.9958 0.9955 0.9939 0.993 32 0.889 .9906 .9915 .9919 .9917 .9913 .9895 .988 36 .877 .9834 .9840 .9842 .9837 .9832 .9811 .980 40 .865 0.970 .9701 .9701 .9695 .9686 .9677 .9651 .964 45 .849 .958 .9576 .9571 .9561 .9548 .9534 .9501 .948 50 .832 .947 .9461 .9450 .9435 .9420 .9402 .9362 .934 60 .796 .9353 .9335 .9316 .9296 .9275 .9229 70 .755 .9249 .9226 .9202 .9179 .9155 .9101 80 .711 .9150 .9122 .9094 .9067 .9040 .8980 90 .665 .9101 .9070 .9040 .9012 .8983 .8920 100 .618
TABLE 2-33 Ammonium Acetate* (CH3COONH4)
TABLE 2-34 Ammonium Bichromate [(NH4)2Cr2O7]
%
d425
%
d412
1 2 4 8 12 16 20 24 28 30 35 40 45
0.9992 1.0013 1.0055 1.0136 1.0216 1.0294 1.0368 1.0439 1.0507 1.0540 1.0618 1.0691 1.0760
1 2 4 8 12 16 20
1.0051 1.0108 1.0223 1.0463 1.0715 1.0981 1.1263
*For data at 16°C for 3(1)52 percent see Atack Handbook of Chemical Data, p. 33, Reinhold, New York, 1957. TABLE 2-35
%
°C
d 4t
3.80 10.52 19.75 28.04
20 13 13.7 19.6
1.0219 1.0627 1.1189 1.1707
Ammonia (NH3)
% −15°C −10°C −5°C 1 2 4 8 12 16 20 24 28 30
TABLE 2-36 Ammonium Chromate [(NH4)2CrO4]
%
TABLE 2-32
vapors are available from WADC-TR-59-598, 1959; plastics are collected in the Handbook of Chemistry and Physics, Chemical Rubber Publishing Co.: solid helium, neon, argon, fluorine, and methane data are given by Johnson (ed.), WADD-TR-60-56, 1960; temperatures of maximum solubility, vol. 3, p. 107; metals, vol. 2, p. 463; oils, fats, and waxes, vol. 2, p. 201; orthobaric, vol. 3, pp. 202, 228, 237, 244; petroleums, vol. 2, pp. 137, 144; plastics, vol. 2, p. 296; porcelains, vol. 2, pp. 68, 75; refrigerating brines, vol. 2, p. 327; rubber, vol. 2, pp. 255, 259; soaps, vol. 5, p. 447; metallic solid solutions, vol. 2, p. 358; solids, vol. 3, pp. 43, 45; vol. 2, p. 456; vol. 3, p. 21; solutions and mixtures, vol. 3, pp. 17, 51, 95, 104, 107, 111, 125, 130; woods, vol. 2, p. 1. Also see the Handbook of Chemistry and Physics, Chemical Rubber Publishing Co., 40th ed., etc.
TABLE 2-37
Ammonium Nitrate (NH4NO3)
%
0°C
10°C
25°C
40°C
60°C
80°C
1.0 2.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 30.0 40.0 50.0
1.0043 1.0088 1.0178 1.0358 1.0539 1.0721 1.0905 1.1090 1.1277 1.1371 1.1862 1.2380
1.0039 1.0082 1.0168 1.0340 1.0515 1.0691 1.0870 1.1051 1.1234 1.1327 1.1810 1.2320
1.0011 1.0051 1.0132 1.0297 1.0464 1.0633 1.0806 1.0982 1.1161 1.1252 1.1727 1.2229
0.9961 1.0000 1.0079 1.0238 1.0400 1.0565 1.0734 1.0907 1.1082 1.1171 1.1640 1.2136
0.9870 .9908 .9985 1.0142 1.0301 1.0462 1.0627 1.0796 1.0968 1.1055 1.1515 1.2006
0.9755 .9793 .9869 1.0024 1.0181 1.0342 1.0506 1.0673 1.0844 1.0931 1.1385 1.1868
TABLE 2-38
Ammonium Sulfate [(NH4)2SO4]
%
0°C
20°C
40°C
80°C
100°C
1 2 4 8 12 16 20 24 28 35 40 50
1.0061 1.0124 1.0248 1.0495 1.0740 1.0980 1.1215 1.1448 1.1677 1.2072 1.2350 1.2899
1.0041 1.0101 1.0220 1.0456 1.0691 1.0924 1.1154 1.1383 1.1609 1.2800 1.2277 1.2825
0.9980 1.0039 1.0155 1.0387 1.0619 1.0849 1.1077 1.1304 1.1529 1.1919 1.2196 1.2745
0.9777 .9836 .9953 1.0187 1.0421 1.0653 1.0883 1.1111 1.1338 1.1731 1.2011 1.2568
0.9644 .9705 .9826 1.0066 1.0303 1.0539 1.0772 1.1003 1.1232 1.1629 1.1910 1.2466
Ammonium Chloride (NH4Cl)
%
0°C
10°C
20°C
30°C
50°C
80°C
100°C
1 2 4 8 12 16 20 24
1.0033 1.0067 1.0135 1.0266 1.0391 1.0510 1.0625 1.0736
1.0029 1.0062 1.0126 1.0251 1.0370 1.0485 1.0596 1.0705
1.0013 1.0045 1.0107 1.0227 1.0344 1.0457 1.0567 1.0674
0.9987 1.0018 1.0077 1.0195 1.0310 1.0422 1.0532 1.0641
0.9910 .9940 .9999 1.0116 1.0231 1.0343 1.0454 1.0564
0.9749 .9780 .9842 .9963 1.0081 1.0198 1.0312 1.0426
0.9617 .9651 .9718 .9849 .9975 1.0096 1.0213 1.0327
TABLE 2-39
Arsenic Acid (H3A3O4)
%
d 15 4
%
d 15 4
1 2 6 10 16
1.0057 1.0124 1.0398 1.0681 1.1128
20 30 40 50 60 70
1.1447 1.2331 1.3370 1.4602 1.6070 1.7811
2-100
PHYSICAL AND CHEMICAL DATA
TABLE 2-40
Barium Chloride (BaCl2)
TABLE 2-46
Chromic Acid (CrO3)
%
0°C
20°C
40°C
60°C
80°C
100°C
%
d 15 4
%
d 15 4
2 4 8 12 16 20 24 26
1.0181 1.0368 1.0760 1.1178 1.1627 1.2105
1.0159 1.0341 1.0721 1.1128 1.1564 1.2031 1.2531 1.2793
1.0096 1.0275 1.0648 1.1047 1.1478 1.1938 1.2430 1.2688
1.0004 1.0181 1.0551 1.0948 1.1373 1.1828 1.2316 1.2571
0.9890 1.0066 1.0434 1.0827 1.1249 1.1702 1.2186 1.2440
0.9755 .9931 1.0299 1.0692 1.1113 1.1563 1.2045 1.2298
1 2 6 10 16
1.006 1.014 1.045 1.076 1.127
20 26 30 40 50 60
1.163 1.220 1.260 1.371 1.505 1.663
TABLE 2-41
TABLE 2-47
d 18 4
%
d 18 4
2 4 8 12 16
1.0154 1.0326 1.0683 1.1061 1.1468
20 25 30 40 50
1.1904 1.2488 1.3124 1.4590 1.6356
TABLE 2-42 % −5°C 2 4 8 12 16 20 25 30 35 40
Cadmium Nitrate [Cd(NO3)2]
%
1.0708 1.1083 1.1471 1.1874
Chromium Chloride (CrCl3) d 18 4
%
Violet
Green
Equilibrium mixture of violet and green
1 2 4 8 12 14
1.0076 1.0166 1.0349 1.0724 1.1114 1.1316
1.0071 1.0157 1.0332 1.0691 1.1065
1.0075 1.0165 1.0347 1.0722 1.1111
Calcium Chloride (CaCl2)
0°C
20°C
30°C
40°C
60°C
1.0171 1.0346 1.0703 1.1072 1.1454 1.1853 1.2376 1.2922
1.0148 1.0316 1.0659 1.1015 1.1386 1.1775 1.2284 1.2816 1.3373 1.3957
1.0120 1.0286 1.0626 1.0978 1.1345 1.1730 1.2236 1.2764 1.3316 1.3895
1.0084 1.0249 1.0586 1.0937 1.1301 1.1684 1.2186 1.2709 1.3255 1.3826
0.9994 1.0158 1.0492 1.0840 1.1202 1.1581 1.2079 1.2597 1.3137 1.3700
80°C 100°C 120°C* 140°C 0.9881 1.0046 1.0382 1.0730 1.1092 1.1471 1.1965 1.2478 1.3013 1.3571
0.9748 .9915 1.0257 1.0610 1.0973 1.1352 1.1846 1.2359 1.2893 1.3450
0.9596 .9765 1.0111 1.0466 1.0835 1.1219
0.9428 .9601 .9954 1.0317 1.0691 1.1080
TABLE 2-48
Copper Nitrate [Cu(NO3)2]
%
d 20 4
%
d 20 4
1 2 4 8
1.007 1.015 1.032 1.069
12 16 20 25
1.107 1.147 1.189 1.248
*Corrected to atmospheric pressure.
TABLE 2-43 Calcium Hydroxide [Ca(OH)2]
TABLE 2-44 Calcium Hypochlorite* (CaOCl2)
TABLE 2-49 (CuSO4)
Copper Sulfate
TABLE 2-50 Cuprous Chloride (Cu2Cl2)
%
d 15 4
d 425
% total salt
d 15 4
%
0°C
20°C
40°C
%
0°C
20°C
40°C
0.05 .10 .15
0.99979 1.00044 1.00110
0.99773 .99838 .99904
2 4 6 8 10 12
1.0169 1.0345 1.0520 1.0697 1.0876 1.1060
1 4 8 12 16 18
1.0104 1.0429 1.0887 1.1379
1.0086 1.0401 1.084 1.1308 1.180 1.206
1.0024 1.0332 1.0764 1.1222
1 4 8 12 16 20
1.0095 1.0387 1.0788 1.1208 1.1653 1.2121
1.0072 1.036 1.0754 1.1165 1.1595 1.2052
1.002 1.0305 1.0682 1.107 1.151 1.1953
*CaOCl2 = 89.15% CaCl2 = 7.31% Ca(ClO3)2 = 0.26% Ca(OH)2 = 2.92%.
TABLE 2-45
Calcium Nitrate [Ca(NO3)2]
TABLE 2-51
Ferric Chloride (FeCl3)
%
6°C
18°C
25°C
30°C
%
0°C
10°C
20°C
30°C
2* 4 8 12 16 20 25 30 35 40 45 68*
1.0157 1.0316 1.0641 1.0979 1.1330 1.1694 1.2168
1.0137 1.0291 1.0608 1.0937 1.1279 1.1636 1.2106 1.260 1.311 1.365 1.422 1.747
1.0120 1.0272 1.0585 1.0911 1.1250 1.1602 1.2065
1.0105 1.0256 1.0565 1.0887 1.1224 1.1575 1.2032
1.741
1.736
1 2 4 8 12 16 20 25 30 35 40 45 50
1.0086 1.0174 1.0347 1.0703 1.1088 1.1475 1.1870 1.2400 1.2970 1.3605 1.4280
1.0084 1.0168 1.0341 1.0692 1.1071 1.1449 1.1847 1.2380 1.2950 1.3580 1.4235 1.4920 1.5610
1.0068 1.0152 1.0324 1.0669 1.1040 1.1418 1.1820 1.2340 1.2910 1.3530 1.4175 1.4850 1.5510
1.0040 1.0122 1.0292 1.0636 1.1006 1.1386 1.1786 1.2290 1.2850 1.3475 1.4115
*Supercooled tetrahydrate (m.p. 41.4°C).
DENSITIES OF AQUEOUS INORGANIC SOLUTIONS TABLE 2-52 [Fe2(SO4)3]
Ferric Sulfate
TABLE 2-53 [Fe(NO3)3]
17.5
%
d4
1 2 4 8 12 16 20 30 40 50 60
1.0072 1.0157 1.0327 1.0670 1.1028 1.1409 1.1811 1.3073 1.4487 1.6127 1.7983
TABLE 2-54 Ferrous Sulfate (FeSO4) % 0.2 0.4 0.8 1.0 4.0 8.0 12.0 16.0 20.0
Ferric Nitrate
d4
d4
%
d4
%
d4
1 2 4 8 12 16 20 25
1.0065 1.0144 1.0304 1.0636 1.0989 1.1359 1.1748 1.2281
5 10 20 30 40 50 60 70 80 90 95 100
1.020 1.040 1.080 1.119 1.159 1.198 1.235 1.258 1.259 1.178 1.089 1.0005
1.017 1.035 1.070 1.101 1.130 1.155
1 2 4 6 8 10 12 14 16 18 20 22 24
1.0022 1.0058 1.0131 1.0204 1.0277 1.0351 1.0425 1.0499 1.0574 1.0649 1.0725 1.0802 1.0880
26 28 30 35 40 45 50 55 60 70 80 90 100
1.0959 1.1040 1.1122 1.1327 1.1536 1.1749 1.1966 1.2188 1.2416 1.2897 1.3406 1.3931 1.4465
4
10
25
TABLE 2-60
20°C
%
d4
d4
d4
%
1.0090 1.0380 1.0790 1.1235 1.1690 1.2150
1.00068 1.00275 1.00645 1.0085 1.0375 1.0785 1.1220 1.1675 1.2135
1.0002 1.0022 1.0062 1.0082
1.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0 40.0 50.0 60.0 65.0
1.0073 1.0146 1.0295 1.0448 1.0604 1.0764 1.0928 1.1097 1.1272 1.1453 1.1640 1.1832 1.2030 1.2235 1.2446 1.2663 1.3877 1.5305 1.6950 1.7854
1.0068 1.0139 1.0285 1.0435 1.0589 1.0747 1.0910 1.1078 1.1251 1.1430 1.1615 1.1806 1.2003 1.2206 1.2415 1.2630 1.3838 1.5257 1.6892 1.7792
1.0041 1.0111 1.0255 1.0402 1.0552 1.0707 1.0867 1.1032 1.1202 1.1377 1.1557 1.1743 1.1935 1.2134 1.2340 1.2552 1.3736 1.5127 1.6731 1.7613
1 2 4 8 12
15
1 2 4 8 12 16 82 90 100
0.998 .996 .993 .984 .971 .956 .752 .724 .691
TABLE 2-57
18
%
18
18°C
d4
18
d4
15°C
%
0
%
TABLE 2-55 Hydrogen Bromide (HBr)
TABLE 2-56 Hydrogen Cyanide (HCN)
TABLE 2-59 Hydrogen Peroxide (H2O2)
TABLE 2-58 Hydrogen Fluoride (HF) 20
2-101
Hydrofluosilic Acid (H2SiF6) 17.5
d4
1.0080 1.0161 1.0324 1.0661 1.1011
TABLE 2-61
Hydrogen Chloride (HCl)
−5°C
0°C
10°C
20°C
40°C
60°C
80°C
100°C
1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
1.0048 1.0104 1.0213 1.0321 1.0428 1.0536 1.0645 1.0754 1.0864 1.0975 1.1087 1.1200 1.1314 1.1426 1.1537 1.1648
1.0052 1.0106 1.0213 1.0319 1.0423 1.0528 1.0634 1.0741 1.0849 1.0958 1.1067 1.1177 1.1287 1.1396 1.1505 1.1613
1.0048 1.0100 1.0202 1.0303 1.0403 1.0504 1.0607 1.0711 1.0815 1.0920 1.1025 1.1131 1.1238 1.1344 1.1449 1.1553
1.0032 1.0082 1.0181 1.0279 1.0376 1.0474 1.0574 1.0675 1.0776 1.0878 1.0980 1.1083 1.1187 1.1290 1.1392 1.1493 1.1593 1.1691 1.1789 1.1885 1.1980
0.9970 1.0019 1.0116 1.0211 1.0305 1.0400 1.0497 1.0594 1.0692 1.0790 1.0888 1.0986 1.1085 1.1183 1.1280 1.1376
0.9881 .9930 1.0026 1.0121 1.0215 1.0310 1.0406 1.0502 1.0598 1.0694 1.0790 1.0886 1.0982 1.1076 1.1169 1.1260
0.9768 0.9819 0.9919 1.0016 1.0111 1.0206 1.0302 1.0398 1.0494 1.0590 1.0685 1.0780 1.0874 1.0967 1.1058 1.1149
0.9636 .9688 .9791 .9892 .9992 1.0090 1.0188 1.0286 1.0383 1.0479 1.0574 1.0668 1.0761 1.0853 1.0942 1.1030
d4
16 20 25 30 34
1.1373 1.1748 1.2235 1.2742 1.3162
Magnesium Chloride (MgCl2)
%
0°C
20°C
40°C
60°C
80°C
100°C
2 4 8 12 16 20 25 30
1.0168 1.0338 1.0683 1.1035 1.1395 1.1764 1.2246 1.2754
1.0146 1.0311 1.0646 1.0989 1.1342 1.1706 1.2184 1.2688
1.0084 1.0248 1.0580 1.0921 1.1272 1.1635 1.2111 1.2614
0.9995 1.0159 1.0493 1.0836 1.1188 1.1552 1.2031 1.2535
0.9883 1.0050 1.0388 1.0735 1.1092 1.1460 1.1942 1.2451
0.9753 .9923 1.0269 1.0622 1.0984 1.1359 1.1847 1.2360
TABLE 2-62
%
17.5
%
Magnesium Sulfate (MgSO4)
%
0°C
20°C
30°C
40°C
50°C
60°C
80°C
2 4 8 12 16 20 26
1.0210 1.0423 1.0858 1.1309 1.1777 1.2264 1.3032
1.0186 1.0392 1.0816 1.1256 1.1717 1.2198 1.2961
1.0158 1.0362 1.0782 1.1220 1.1679 1.2159 1.2922
1.0123 1.0326 1.0743 1.1179 1.1637 1.2117 1.2879
1.0081 1.0283 1.0700 1.1135 1.1592 1.2072 1.2836
1.0032 1.0234 1.0650 1.1083
0.9916 1.0118 1.0534 1.0968
TABLE 2-63 Nickel Chloride (NiCl2) 18
TABLE 2-64 Nickel Nitrate [Ni(NO3)2] 20
TABLE 2-65 Nickel Sulfate (NiSO4 ) 18
%
d4
%
d4
%
d4
1 2 4 8 12 16 20 30
1.0082 1.0179 1.0375 1.0785 1.1217 1.1674 1.2163 1.353
1 2 4 8 12 16 20 30 35
1.0065 1.0150 1.0325 1.0688 1.1070 1.1480 1.191 1.311 1.377
1 2 4 8 12 16 18
1.0091 1.0198 1.0415 1.0852 1.1325 1.1825 1.2090
2-102
PHYSICAL AND CHEMICAL DATA
TABLE 2-66 %
Nitric Acid (HNO3)
0°C
5°C
10°C
15°C
20°C
25°C
30°C
40°C
50°C
60°C
80°C
100°C
1 2 3 4
1.0058 1.0117 1.0176 1.0236
1.00572 1.01149 1.01730 1.02315
1.00534 1.01099 1.01668 1.02240
1.00464 1.01018 1.01576 1.02137
1.00364 1.00909 1.01457 1.02008
1.00241 1.00778 1.01318 1.01861
1.0009 1.0061 1.0114 1.0168
0.9973 1.0025 1.0077 1.0129
0.9931 .9982 1.0033 1.0084
0.9882 .9932 .9982 1.0033
0.9767 .9816 .9865 .9915
0.9632 .9681 .9730 .9779
5 6 7 8 9
1.0296 1.0357 1.0418 1.0480 1.0543
1.02904 1.03497 1.0410 1.0471 1.0532
1.02816 1.03397 1.0399 1.0458 1.0518
1.02702 1.03272 1.0385 1.0443 1.0502
1.02563 1.03122 1.0369 1.0427 1.0485
1.02408 1.02958 1.0352 1.0409 1.0466
1.0222 1.0277 1.0333 1.0389 1.0446
1.0182 1.0235 1.0289 1.0344 1.0399
1.0136 1.0188 1.0241 1.0295 1.0349
1.0084 1.0136 1.0188 1.0241 1.0294
.9965 1.0015 1.0066 1.0117 1.0169
.9829 .9879 .9929 .9980 1.0032
10 11 12 13 14
1.0606 1.0669 1.0733 1.0797 1.0862
1.0594 1.0656 1.0718 1.0781 1.0845
1.0578 1.0639 1.0700 1.0762 1.0824
1.0561 1.0621 1.0681 1.0742 1.0803
1.0543 1.0602 1.0661 1.0721 1.0781
1.0523 1.0581 1.0640 1.0699 1.0758
1.0503 1.0560 1.0618 1.0676 1.0735
1.0455 1.0511 1.0567 1.0624 1.0681
1.0403 1.0458 1.0513 1.0568 1.0624
1.0347 1.0401 1.0455 1.0509 1.0564
1.0221 1.0273 1.0326 1.0379 1.0432
1.0083 1.0134 1.0186 1.0238 1.0289
15 16 17 18 19
1.0927 1.0992 1.1057 1.1123 1.1189
1.0909 1.0973 1.1038 1.1103 1.1168
1.0887 1.0950 1.1014 1.1078 1.1142
1.0865 1.0927 1.0989 1.1052 1.1115
1.0842 1.0903 1.0964 1.1026 1.1088
1.0818 1.0879 1.0940 1.1001 1.1062
1.0794 1.0854 1.0914 1.0974 1.1034
1.0739 1.0797 1.0855 1.0913 1.0972
1.0680 1.0737 1.0794 1.0851 1.0908
1.0619 1.0675 1.0731 1.0787 1.0843
1.0485 1.0538 1.0592 1.0646 1.0700
1.0341 1.0393 1.0444 1.0496 1.0547
20 21 22 23 24
1.1255 1.1322 1.1389 1.1457 1.1525
1.1234 1.1300 1.1366 1.1433 1.1501
1.1206 1.1271 1.1336 1.1402 1.1469
1.1178 1.1242 1.1306 1.1371 1.1437
1.1150 1.1213 1.1276 1.1340 1.1404
1.1123 1.1185 1.1247 1.1310 1.1374
1.1094 1.1155 1.1217 1.1280 1.1343
1.1031 1.1090 1.1150 1.1210 1.1271
1.0966 1.1024 1.1083 1.1142 1.1201
1.0899 1.0956 1.1013 1.1070 1.1127
1.0754 1.0808 1.0862 1.0917 1.0972
1.0598 1.0650 1.0701 1.0753 1.0805
25 26 27 28 29
1.1594 1.1663 1.1733 1.1803 1.1874
1.1569 1.1638 1.1707 1.1777 1.1847
1.1536 1.1603 1.1670 1.1738 1.1807
1.1503 1.1569 1.1635 1.1702 1.1770
1.1469 1.1534 1.1600 1.1666 1.1733
1.1438 1.1502 1.1566 1.1631 1.1697
1.1406 1.1469 1.1533 1.1597 1.1662
1.1332 1.1394 1.1456 1.1519 1.1582
1.1260 1.1320 1.1381 1.1442 1.1503
1.1185 1.1244 1.1303 1.1362 1.1422
1.1027 1.1083 1.1139 1.1195 1.1251
1.0857 1.0910 1.0963 1.1016 1.1069
30 31 32 33 34
1.1945 1.2016 1.2088 1.2160 1.2233
1.1917 1.1988 1.2059 1.2131 1.2203
1.1876 1.1945 1.2014 1.2084 1.2155
1.1838 1.1906 1.1974 1.2043 1.2113
1.1800 1.1867 1.1934 1.2002 1.2071
1.1763 1.1829 1.1896 1.1963 1.2030
1.1727 1.1792 1.1857 1.1922 1.1988
1.1645 1.1708 1.1772 1.1836 1.1901
1.1564 1.1625 1.1687 1.1749 1.1812
1.1482 1.1542 1.1602 1.1662 1.1723
1.1307 1.1363 1.1419 1.1476 1.1533
1.1122 1.1175 1.1228 1.1281 1.1335
35 36 37 38 39
1.2306 1.2375 1.2444 1.2513 1.2581
1.2275 1.2344 1.2412 1.2479 1.2546
1.2227 1.2294 1.2361 1.2428 1.2494
1.2183 1.2249 1.2315 1.2381 1.2446
1.2140 1.2205 1.2270 1.2335 1.2399
1.2098 1.2163 1.2227 1.2291 1.2354
1.2055 1.2119 1.2182 1.2245 1.2308
1.1966 1.2028 1.2089 1.2150 1.2210
1.1876 1.1936 1.1995 1.2054 1.2112
1.1784 1.1842 1.1899 1.1956 1.2013
1.1591 1.1645 1.1699 1.1752 1.1805
1.1390 1.1440 1.1490 1.1540 1.1589
40 41 42 43 44
1.2649 1.2717 1.2786 1.2854 1.2922
1.2613 1.2680 1.2747 1.2814 1.2880
1.2560 1.2626 1.2692 1.2758 1.2824
1.2511 1.2576 1.2641 1.2706 1.2771
1.2463 1.2527 1.2591 1.2655 1.2719
1.2417 1.2480 1.2543 1.2606 1.2669
1.2370 1.2432 1.2494 1.2556 1.2618
1.2270 1.2330 1.2390 1.2450 1.2510
1.2170 1.2229 1.2287 1.2345 1.2403
1.2069 1.2126 1.2182 1.2238 1.2294
1.1858 1.1911 1.1963 1.2015 1.2067
1.1638 1.1687 1.1735 1.1783 1.1831
45 46 47 48 49
1.2990 1.3058 1.3126 1.3194 1.3263
1.2947 1.3014 1.3080 1.3147 1.3214
1.2890 1.2955 1.3021 1.3087 1.3153
1.2836 1.2901 1.2966 1.3031 1.3096
1.2783 1.2847 1.2911 1.2975 1.3040
1.2732 1.2795 1.2858 1.2921 1.2984
1.2680 1.2742 1.2804 1.2867 1.2929
1.2570 1.2630 1.2690 1.2750 1.2811
1.2461 1.2519 1.2577 1.2635 1.2693
1.2350 1.2406 1.2462 1.2518 1.2575
1.2119 1.2171 1.2223 1.2275 1.2328
1.1879 1.1927 1.1976 1.2024 1.2073
50 51 52 53 54
1.3327 1.3391 1.3454 1.3517 1.3579
1.3277 1.3339 1.3401 1.3462 1.3523
1.3215 1.3277 1.3338 1.3399 1.3459
1.3157 1.3218 1.3278 1.3338 1.3397
1.3100 1.3160 1.3219 1.3278 1.3336
1.3043 1.3102 1.3160 1.3218 1.3275
1.2987 1.3045 1.3102 1.3159 1.3215
1.2867 1.2923 1.2978 1.3033 1.3087
1.2748 1.2802 1.2856 1.2909 1.2961
1.2628 1.2680 1.2731 1.2782 1.2833
1.2377 1.2425 1.2473 1.2521 1.2568
1.2118 1.2163 1.2208 1.2252 1.2296
55 56 57 58 59
1.3640 1.3700 1.3759 1.3818 1.3875
1.3583 1.3642 1.3700 1.3757 1.3813
1.3518 1.3576 1.3634 1.3691 1.3747
1.3455 1.3512 1.3569 1.3625 1.3680
1.3393 1.3449 1.3505 1.3560 1.3614
1.3331 1.3386 1.3441 1.3495 1.3548
1.3270 1.3324 1.3377 1.3430 1.3482
1.3141 1.3194 1.3246 1.3298 1.3348
1.3013 1.3064 1.3114 1.3164 1.3213
1.2883 1.2932 1.2981 1.3029 1.3077
1.2615 1.2661 1.2706 1.2751 1.2795
1.2339 1.2382 1.2424 1.2466 1.2507
60 61 62 63 64
1.3931 1.3986 1.4039 1.4091
1.3868 1.3922 1.3975 1.4027 1.4078
1.3801 1.3855 1.3907 1.3958 1.4007
1.3734 1.3787 1.3838 1.3888 1.3936
1.3667 1.3719 1.3769 1.3818 1.3866
1.3600 1.3651 1.3700 1.3748 1.3795
1.3533 1.3583 1.3632 1.3679 1.3725
1.3398 1.3447 1.3494 1.3540
1.3261 1.3308 1.3354 1.3398
1.3124 1.3169 1.3213 1.3255
1.2839 1.2881 1.2922 1.2962
1.2547 1.2587 1.2625 1.2661
DENSITIES OF AQUEOUS INORGANIC SOLUTIONS Nitric Acid (HNO3) (Concluded )
TABLE 2-66 %
0°C
5°C
10°C
15°C
20°C
25°C
30°C
65 66 67 68 69
1.4128 1.4177 1.4224 1.4271 1.4317
1.4055 1.4103 1.4150 1.4196 1.4241
1.3984 1.4031 1.4077 1.4122 1.4166
1.3913 1.3959 1.4004 1.4048 1.4091
1.3841 1.3887 1.3932 1.3976 1.4019
1.3770 1.3814 1.3857 1.3900 1.3942
70 71 72 73 74
1.4362 1.4406 1.4449 1.4491 1.4532
1.4285 1.4328 1.4371 1.4413 1.4454
1.4210 1.4252 1.4294 1.4335 1.4376
1.4134 1.4176 1.4218 1.4258 1.4298
1.4061 1.4102 1.4142 1.4182 1.4221
1.3983 1.4023 1.4063 1.4103 1.4142
75 76 77 78 79
1.4573 1.4613 1.4652 1.4690 1.4727
1.4494 1.4533 1.4572 1.4610 1.4647
1.4415 1.4454 1.4492 1.4529 1.4565
1.4337 1.4375 1.4413 1.4450 1.4486
1.4259 1.4296 1.4333 1.4369 1.4404
1.4180 1.4217 1.4253 1.4288 1.4323
80 81 82 83 84
1.4764 1.4800 1.4835 1.4869 1.4903
1.4683 1.4718 1.4753 1.4787 1.4820
1.4601 1.4636 1.4670 1.4704 1.4737
1.4521 1.4555 1.4589 1.4622 1.4655
1.4439 1.4473 1.4507 1.4540 1.4572
1.4357 1.4391 1.4424 1.4456 1.4487
85 86 87 88 89
1.4936 1.4968 1.4999 1.5029 1.5058
1.4852 1.4883 1.4913 1.4942 1.4970
1.4769 1.4799 1.4829 1.4858 1.4885
1.4686 1.4716 1.4745 1.4773 1.4800
1.4603 1.4633 1.4662 1.4690 1.4716
1.4518 1.4548 1.4577 1.4605 1.4631
90 91 92 93 94
1.5085 1.5111 1.5136 1.5156 1.5177
1.4997 1.5023 1.5048 1.5068 1.5088
1.4911 1.4936 1.4960 1.4979 1.4999
1.4826 1.4850 1.4873 1.4892 1.4912
1.4741 1.4766 1.4789 1.4807 1.4826
1.4656 1.4681 1.4704 1.4722 1.4741
95 96 97 98 99 100
1.5198 1.5220 1.5244 1.5278 1.5327 1.5402
1.5109 1.5130 1.5152 1.5187 1.5235 1.5310
1.5019 1.5040 1.5062 1.5096 1.5144 1.5217
1.4932 1.4952 1.4974 1.5008 1.5056 1.5129
1.4846 1.4867 1.4889 1.4922 1.4969 1.5040
1.4761 1.4781 1.4802 1.4835 1.4881 1.4952
TABLE 2-67 15
%
d4
1 2 4 6 8 10 12 14 16 18 20 22 24 26
1.0050 1.0109 1.0228 1.0348 1.0471 1.0597 1.0726 1.0589 1.0995 1.1135 1.1279 1.1428 1.1581 1.1738
2-103
Perchloric Acid (HClO4) 20
d4
TABLE 2-69
40°C
50°C
d4
%
d4
d4
d4
°C
1%
2%
4%
1.0020 1.0070 1.0169 1.0270 1.0372 1.0475
0.9933 0.9986 0.9906 1.0205 1.0320 1.0440 1.0560 1.0680 1.0810 1.0940 1.1070 1.1205 1.1345 1.1490
28 30 32 34 36 38 40 45 50 55 60 65 70
1.1900 1.2067 1.2239 1.2418 1.2603 1.2794 1.2991 1.3521 1.4103 1.4733 1.5389 1.6059 1.6736
1.1851 1.2013 1.2183 1.2359 1.2542 1.2732 1.2927 1.3450 1.4018 1.4636 1.5298 1.5986 1.6680
1.1645 1.1800 1.1960 1.2130 1.2310 1.2490 1.2680 1.3180 1.3730 1.4320 1.4950 1.5620 1.6290
0 10 15 20 30 40 50 60 80 100
1.0066 1.0064 1.0058 1.0049 1.0024 0.9990 .9949 .9901 .9786 .9653
1.0134 1.0132 1.0125 1.0117 1.0092 1.0058 1.0017 0.9969 .9855 .9722
1.0270 1.0268 1.0260 1.0252 1.0228 1.0195 1.0154 1.0106 0.9993 .9860
1.1697
50
15
20
50
80°C
100°C
6%
8%
10%
1.0396
1.0534
1.0674
Potassium Bicarbonate (KHCO3)
d4
25
60°C
TABLE 2-70 Potassium Bromide (KBr) TABLE 2-68
Phosphoric Acid (H3PO4)
°C
2%
6%
14%
0 10 20 30 40
1.0113 1.0109 1.0092 1.0065 1.0029
1.0339 1.0330 1.0309 1.0279 1.0241
1.0811 1.0792 1.0764 1.0728 1.0685
20%
26%
35%
50%
75%
1.1192 1.1167 1.1567 1.221 1.341 1.1134 1.1529 1.216 1.335 1.579 1.1094 1.1484 1.211 1.329 1.572 1.1048
100%
1.870 1.862
%
d 20 4
1 2 6 12 20 30 40
1.0054 1.0127 1.0426 1.0903 1.1601 1.2593 1.3746
2-104
PHYSICAL AND CHEMICAL DATA
TABLE 2-71
Potassium Carbonate (K2CO3)
TABLE 2-77
Potassium Nitrate (KNO3)
%
0°C
10°C
20°C
40°C
60°C
80°C
100°C
%
0°C
10°C
20°C
40°C
60°C
80°C
100°C
1 2 4 8 12 16 20 24 28 30 35 40 45 50
1.0094 1.0189 1.0381 1.0768 1.1160 1.1562 1.1977 1.2405 1.2846 1.3071 1.3646 1.4244 1.4867 1.5517
1.0089 1.0182 1.0369 1.0746 1.1131 1.1530 1.1941 1.2366 1.2804 1.3028 1.3600 1.4195 1.4815 1.5462
1.0072 1.0163 1.0345 1.0715 1.1096 1.1490 1.1898 1.2320 1.2756 1.2979 1.3548 1.4141 1.4759 1.5404
1.0010 1.0098 1.0276 1.0640 1.1013 1.1399 1.1801 1.2219 1.2652 1.2873 1.3440 1.4029 1.4644 1.5285
0.9919 1.0005 1.0180 1.0538 1.0906 1.1290 1.1690 1.2106 1.2538 1.2759 1.3324 1.3913 1.4528 1.5169
0.9803 .9889 1.0063 1.0418 1.0786 1.1170 1.1570 1.1986 1.2418 1.2640 1.3206 1.3795 1.4408 1.5048
0.9670 .9756 .9951 1.0291 1.0663 1.1049 1.1451 1.1869 1.2301 1.2522 1.3089 1.3678 1.4290 1.4928
1 2 4 8 12 16 20 24
1.00654 1.01326 1.02677 1.05419 1.08221
1.00615 1.01262 1.02566 1.05226 1.07963
1.00447 1.01075 1.02344 1.04940 1.07620 1.10392 1.13261 1.16233
0.99825 1.00430 1.01652 1.04152 1.06740 1.09432 1.12240 1.15175
0.9890 .9949 1.0068 1.0313 1.0567 1.0831 1.1106 1.1391
0.9776 .9834 .9951 1.0192 1.0442 1.0703 1.0974 1.1256
0.9641 .9699 .9816 1.0056 1.0304 1.0562 1.0831 1.1110
TABLE 2-72 Potassium Chromate (K2Cr O4)
TABLE 2-73 (KClO3)
Potassium Chlorate
%
d4
d4
°C
1%
2%
3%
4%
1 2 4 8 12 16 20 24 28 30
1.0073 1.0155 1.0321 1.0659 1.1009
1.0066 1.0147 1.0311 1.0647 1.0999 1.1366 1.1748 1.2147 1.2566 1.2784
0 10 20 30 40 60 80 100
1.0061 1.0059 1.0045 1.0020 0.9986 .9895 .9781 .9646
1.0124 1.0122 1.0109 1.0085 1.0051 0.9959 .9845 .9709
1.0189 1.0187 1.0174 1.0151 1.0116 1.0024 0.9910 .9774
1.0256 1.0254 1.0241 1.0218 1.0183 1.0091 0.9977 .9840
15
TABLE 2-74
18
Potassium Chloride (KCl)
%
0°C
20°C
25°C
40°C
60°C
80°C
100°C
1.0 2.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0
1.00661 1.01335 1.02690 1.05431 1.08222 1.11068 1.13973
1.00462 1.01103 1.02391 1.05003 1.07679 1.10434 1.13280 1.16226
1.00342 1.00977 1.02255 1.04847 1.07506 1.10245 1.13072 1.15995
0.99847 1.00471 1.01727 1.04278 1.06897 1.09600 1.12399 1.15299 1.18304
0.9894 .9956 1.0080 1.0333 1.0592 1.0861 1.1138 1.1425 1.1723
0.9780 .9842 .9966 1.0219 1.0478 1.0746 1.1024 1.1311 1.1609
0.9646 .9708 .9634 1.0888 1.0350 1.0619 1.0897 1.1185 1.1483
%
110°C
120°C
130°C
140°C
3.79 7.45 13.62
0.9733 .9978 1.0388
0.9663 .9899 1.0313
0.9583 .9827 1.0238
0.9502 .9745 1.0159
TABLE 2-75 Potassium Chrome Alum [K2Cr2(SO4)4]
TABLE 2-78 Potassium Dichromate (K2Cr2O7) 20
d 15 4
%
d 15 4
1 2 6 10 14 20 30 40 50
1.007 1.016 1.052 1.089 1.129 1.193 1.315 1.456 1.615
1.0 2.0 4.0 6.0 8.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 51.7
1.0083 1.0175 1.0359 1.0544 1.0730 1.0918 1.1396 1.1884 1.2387 1.2905 1.3440 1.3991 1.4558 1.5143 1.5355 (sat’d. soln.)
20
%
d4
%
d4
1 2 4 6 8 10
1.0052 1.0122 1.0264 1.0408 1.0554 1.0703
1 2 4 6 8 10
1.0063 1.0145 1.0310 1.0477 1.0646 1.0817
TABLE 2-80 Potassium Sulfite (K2SO3) 15
TABLE 2-81 Sodium Acetate (NaC2H3O2) 20
%
d4
%
d4
1 2 4 8 12 16 20 24 26
1.0073 1.0155 1.0322 1.0667 1.1026 1.1402 1.1793 1.2197 1.2404
1 2 4 8 12 18 20 26 28
1.0033 1.0084 1.0186 1.0392 1.0598 1.0807 1.1021 1.1351 1.1462
TABLE 2-82 Sodium Arsenate (Na3AsO4) 17
TABLE 2-83 Sodium Bichromate (Na2Cr2O7) 15
%
d4
%
d4
1 2 4 8 10 12
1.0097 1.0207 1.0431 1.0892 1.1130 1.1373
1 2 4 8 12 16 20 24 28 30 35 40 45 50
1.006 1.013 1.027 1.056 1.084 1.112 1.140 1.166 1.193 1.207 1.244 1.279 1.312 1.342
TABLE 2-76 Potassium Hydroxide (KOH)
%
TABLE 2-79 Potassium Sulfate (K2SO4)
TABLE 2-84 Sodium Bromide (NaBr) 17
TABLE 2-85 Sodium Formate (HCOONa) 25
%
d4
%
d4
1 2 4 8 10 12 20 30 40
1.0060 1.0139 1.0298 1.0631 1.0803 1.0981 1.1745 1.2841 1.4138
1 2 4 8 12 16 20 24 28 30 35 40
1.003 1.009 1.022 1.048 1.074 1.100 1.127 1.155 1.184 1.199 1.236 1.274
DENSITIES OF AQUEOUS INORGANIC SOLUTIONS TABLE 2-86
Sodium Carbonate (Na2CO3)
TABLE 2-90
2-105
Sodium Hydroxide (NaOH)
%
0°C
10°C
20°C
30°C
40°C
60°C
80°C
100°C
%
0°C
15°C
20°C
40°C
60°C
80°C
100°C
1 2 4 8 12 14 16 18 20 24 28 30
1.0109 1.0219 1.0439 1.0878 1.1319 1.1543
1.0103 1.0210 1.0423 1.0850 1.1284 1.1506
1.0086 1.0190 1.0398 1.0816 1.1244 1.1463
1.0058 1.0159 1.0363 1.0775 1.1200 1.1417 1.1636 1.1859 1.2086 1.2552 1.3031 1.3274
1.0022 1.0122 1.0323 1.0732 1.1150 1.1365
0.9929 1.0027 1.0223 1.0625 1.1039 1.1251
0.9814 .9910 1.0105 1.0503 1.0914 1.1125
0.9683 .9782 .9980 1.0380 1.0787 1.0996
1 2 4 8 12 16 20 24 28 32 36 40 44 48 50
1.0124 1.0244 1.0482 1.0943 1.1399 1.1849 1.2296 1.2741 1.3182 1.3614 1.4030 1.4435 1.4825 1.5210 1.5400
1.01065 1.02198 1.04441 1.08887 1.13327 1.17761 1.22183 1.26582 1.3094 1.3520 1.3933 1.4334 1.4720 1.5102 1.5290
1.0095 1.0207 1.0428 1.0869 1.1309 1.1751 1.2191 1.2629 1.3064 1.3490 1.3900 1.4300 1.4685 1.5065 1.5253
1.0033 1.0139 1.0352 1.0780 1.1210 1.1645 1.2079 1.2512 1.2942 1.3362 1.3768 1.4164 1.4545 1.4922 1.5109
0.9941 1.0045 1.0254 1.0676 1.1101 1.1531 1.1960 1.2388 1.2814 1.3232 1.3634 1.4027 1.4405 1.4781 1.4967
0.9824 .9929 1.0139 1.0560 1.0983 1.1408 1.1833 1.2259 1.2682 1.3097 1.3498 1.3889 1.4266 1.4641 1.4827
0.9693 .9797 1.0009 1.0432 1.0855 1.1277 1.1700 1.2124 1.2546 1.2960 1.3360 1.3750 1.4127 1.4503 1.4690
TABLE 2-87
Sodium Chlorate (NaClO3) 18
18
%
d4
%
d4
1 2 4 6 8 10 12 14 16
1.0053 1.0121 1.0258 1.0397 1.0538 1.0681 1.0827 1.0977 1.1131
18 20 22 24 26 28 30 32 34
1.1288 1.1449 1.1614 1.1782 1.1953 1.2128 1.2307 1.2491 1.2680
TABLE 2-88
TABLE 2-91
Sodium Chloride (NaCl)
%
0°C
10°C
25°C
40°C
60°C
80°C
100°C
1 2 4 8 12 16 20 24 26
1.00747 1.01509 1.03038 1.06121 1.09244 1.12419 1.15663 1.18999 1.20709
1.00707 1.01442 1.02920 1.05907 1.08946 1.12056 1.15254 1.18557 1.20254
1.00409 1.01112 1.02530 1.05412 1.08365 1.11401 1.14533 1.17776 1.19443
0.99908 1.00593 1.01977 1.04798 1.07699 1.10688 1.13774 1.16971 1.18614
0.9900 .9967 1.0103 1.0381 1.0667 1.0962 1.1268 1.1584 1.1747
0.9785 .9852 .9988 1.0264 1.0549 1.0842 1.1146 1.1463 1.1626
0.9651 .9719 .9855 1.0134 1.0420 1.0713 1.1017 1.1331 1.1492
Sodium Nitrate (NaNO3)
%
0°C
20°C
40°C
60°C
80°C
100°C
1 2 4 8 12 16 20 24 28 30 35 40 45
1.0071 1.0144 1.0290 1.0587 1.0891 1.1203 1.1526 1.1860 1.2204 1.2380 1.2834 1.3316
1.0049 1.0117 1.0254 1.0532 1.0819 1.1118 1.1429 1.1752 1.2085 1.2256 1.2701 1.3175 1.3683
0.9986 1.0050 1.0180 1.0447 1.0724 1.1013 1.1314 1.1629 1.1955 1.2122 1.2560 1.3027 1.3528
0.9894 .9956 1.0082 1.0340 1.0609 1.0892 1.1187 1.1496 1.1816 1.1980 1.2413 1.2875 1.3371
0.9779 .9840 .9964 1.0218 1.0481 1.0757 1.1048 1.1351 1.1667 1.1830 1.2258 1.2715 1.3206
0.9644 .9704 .9826 1.0078 1.0340 1.0614 1.0901 1.1200 1.1513 1.1674 1.2100 1.2555 1.3044
TABLE 2-89 Sodium Chromate (Na2CrO4)
TABLE 2-93
TABLE 2-92 Sodium Nitrite (NaNO2)
18
15
%
d4
%
d4
1 2 4 8 12 16 20 24 26
1.0074 1.0164 1.0344 1.0718 1.1110 1.1518 1.1942 1.2383 1.2611
1 2 4 8 12 16 20
1.0058 1.0125 1.0260 1.0535 1.0816 1.1103 1.1394
Sodium Silicates Concentration, % 1
2
4
8
10
14
Na2O/3.9SiO2 Na2O/3.36SiO2 Na2O/2.40SiO2 Na2O/2.44SiO2 Na2O/2.06SiO2 Na2O/1.69SiO2
20
24
30
36
40
45
50
1.445 1.450
1.520
1.594
20
Formula
d4 1.006 1.006 1.007
1.014 1.014 1.016
1.030 1.030 1.034
1.063 1.065 1.071
1.080 1.083 1.090
1.116 1.120 1.130
1.172 1.179
1.211 1.222
1.275 1.290
1.365
1.007 1.007
1.016 1.017
1.035 1.036
1.073 1.077
1.093 1.098
1.134 1.141
1.200 1.210
1.247 1.259
1.309 1.321 1.337
1.387 1.397 1.424
2-106
PHYSICAL AND CHEMICAL DATA TABLE 2-94 %
0°C
1 2 4 8 12 16 20 24
1.0094 1.0189 1.0381 1.0773 1.1174 1.1585 1.2008 1.2443
TABLE 2-95 Sodium Sulfide (Na2S)
Sodium Sulfate (Na2SO4) 20°C
30°C
40°C
60°C
80°C
100°C
1.0073 1.0164 1.0348 1.0724 1.1109 1.1586 1.1915 1.2336
1.0046 1.0135 1.0315 1.0682 1.1062 1.1456 1.1865 1.2292
1.0010 1.0098 1.0276 1.0639 1.1015 1.1406 1.1813 1.2237
0.9919 1.0007 1.0184 1.0544 1.0915 1.1299 1.1696
0.9805 .9892 1.0068 1.0426 1.0795 1.1176 1.1569
0.9671 .9758 .9934 1.0292 1.0661 1.1042
TABLE 2-96 Sodium Sulfite (Na2SO3)
TABLE 2-97 Sodium Thiosulfate (Na2S2O3)
%
d 18 4
%
d 19 4
%
d 20 4
1 2 4 8 12 16 18
1.0098 1.0211 1.0440 1.0907 1.1388 1.1885 1.2140
1 2 4 8 12 16 18
1.0078 1.0172 1.0363 1.0751 1.1146 1.1549 1.1755
1 2 4 8 12 16 20 24 28 30 35 40
1.0065 1.0148 1.0315 1.0654 1.1003 1.1365 1.1740 1.2128 1.2532 1.2739 1.3273 1.3827
TABLE 2-98 Sodium Thiosulfate Pentahydrate (Na2S2O3⋅5H2O) %
d 19 4
1 2 4 8 12 16 20 24 28 30 40 50
1.0052 1.0105 1.0211 1.0423 1.0639 1.0863 1.1087 1.1322 1.1558 1.1676 1.2297 1.2954
TABLE 2-99 Stannic Chloride (SnCl4) %
d 15 4
1 2 4 8 12 16 20 24 28 30 35 40 45 50 55 60 65 70
1.007 1.015 1.031 1.064 1.099 1.135 1.173 1.212 1.255 1.278 1.337 1.403 1.475 1.555 1.644 1.742 1.851 1.971
TABLE 2-100 Stannous Chloride (SnCl2) %
d 15 4
1 2 4 8 12 16 20 24 28 30 35 40 45 50 55 60 65
1.0068 1.0146 1.0306 1.0638 1.0986 1.1353 1.1743 1.2159 1.2603 1.2837 1.3461 1.4145 1.4897 1.5729 1.6656 1.7695 1.8865
DENSITIES OF AQUEOUS INORGANIC SOLUTIONS TABLE 2-101 %
2-107
Sulfuric Acid (H2SO4)
0°C
10°C
15°C
20°C
25°C
30°C
40°C
50°C
60°C
80°C
100°C
1 2 3 4
1.0074 1.0147 1.0219 1.0291
1.0068 1.0138 1.0206 1.0275
1.0060 1.0129 1.0197 1.0264
1.0051 1.0118 1.0184 1.0250
1.0038 1.0104 1.0169 1.0234
1.0022 1.0087 1.0152 1.0216
0.9986 1.0050 1.0113 1.0176
0.9944 1.0006 1.0067 1.0129
0.9895 .9956 1.0017 1.0078
0.9779 .9839 .9900 .9961
0.9645 .9705 .9766 .9827
5 6 7 8 9
1.0364 1.0437 1.0511 1.0585 1.0660
1.0344 1.0414 1.0485 1.0556 1.0628
1.0332 1.0400 1.0469 1.0539 1.0610
1.0317 1.0385 1.0453 1.0522 1.0591
1.0300 1.0367 1.0434 1.0502 1.0571
1.0281 1.0347 1.0414 1.0481 1.0549
1.0240 1.0305 1.0371 1.0437 1.0503
1.0192 1.0256 1.0321 1.0386 1.0451
1.0140 1.0203 1.0266 1.0330 1.0395
1.0022 1.0084 1.0146 1.0209 1.0273
.9888 .9950 1.0013 1.0076 1.0140
10 11 12 13 14
1.0735 1.0810 1.0886 1.0962 1.1039
1.0700 1.0773 1.0846 1.0920 1.0994
1.0681 1.0753 1.0825 1.0898 1.0971
1.0661 1.0731 1.0802 1.0874 1.0947
1.0640 1.0710 1.0780 1.0851 1.0922
1.0617 1.0686 1.0756 1.0826 1.0897
1.0570 1.0637 1.0705 1.0774 1.0844
1.0517 1.0584 1.0651 1.0719 1.0788
1.0460 1.0526 1.0593 1.0661 1.0729
1.0338 1.0403 1.0469 1.0536 1.0603
1.0204 1.0269 1.0335 1.0402 1.0469
15 16 17 18 19
1.1116 1.1194 1.1272 1.1351 1.1430
1.1069 1.1145 1.1221 1.1298 1.1375
1.1045 1.1120 1.1195 1.1271 1.1347
1.1020 1.1094 1.1168 1.1243 1.1318
1.0994 1.1067 1.1141 1.1215 1.1290
1.0968 1.1040 1.1113 1.1187 1.1261
1.0914 1.0985 1.1057 1.1129 1.1202
1.0857 1.0927 1.0998 1.1070 1.1142
1.0798 1.0868 1.0938 1.1009 1.1081
1.0671 1.0740 1.0809 1.0879 1.0950
1.0537 1.0605 1.0674 1.0744 1.0814
20 21 22 23 24
1.1510 1.1590 1.1670 1.1751 1.1832
1.1453 1.1531 1.1609 1.1688 1.1768
1.1424 1.1501 1.1579 1.1657 1.1736
1.1394 1.1471 1.1548 1.1626 1.1704
1.1365 1.1441 1.1517 1.1594 1.1672
1.1335 1.1410 1.1486 1.1563 1.1640
1.1275 1.1349 1.1424 1.1500 1.1576
1.1215 1.1288 1.1362 1.1437 1.1512
1.1153 1.1226 1.1299 1.1373 1.1448
1.1021 1.1093 1.1166 1.1239 1.1313
1.0885 1.0957 1.1029 1.1102 1.1176
25 26 27 28 29
1.1914 1.1996 1.2078 1.2160 1.2243
1.1848 1.1929 1.2010 1.2091 1.2173
1.1816 1.1896 1.1976 1.2057 1.2138
1.1783 1.1862 1.1942 1.2023 1.2104
1.1750 1.1829 1.1909 1.1989 1.2069
1.1718 1.1796 1.1875 1.1955 1.2035
1.1653 1.1730 1.1808 1.1887 1.1966
1.1588 1.1665 1.1742 1.1820 1.1898
1.1523 1.1599 1.1676 1.1753 1.1831
1.1388 1.1463 1.1539 1.1616 1.1693
1.1250 1.1325 1.1400 1.1476 1.1553
30 31 32 33 34
1.2326 1.2409 1.2493 1.2577 1.2661
1.2255 1.2338 1.2421 1.2504 1.2588
1.2220 1.2302 1.2385 1.2468 1.2552
1.2185 1.2267 1.2349 1.2432 1.2515
1.2150 1.2232 1.2314 1.2396 1.2479
1.2115 1.2196 1.2278 1.2360 1.2443
1.2046 1.2126 1.2207 1.2289 1.2371
1.1977 1.2057 1.2137 1.2218 1.2300
1.1909 1.1988 1.2068 1.2148 1.2229
1.1771 1.1849 1.1928 1.2008 1.2088
1.1630 1.1708 1.1787 1.1866 1.1946
35 36 37 38 39
1.2746 1.2831 1.2917 1.3004 1.3091
1.2672 1.2757 1.2843 1.2929 1.3016
1.2636 1.2720 1.2805 1.2891 1.2978
1.2599 1.2684 1.2769 1.2855 1.2941
1.2563 1.2647 1.2732 1.2818 1.2904
1.2526 1.2610 1.2695 1.2780 1.2866
1.2454 1.2538 1.2622 1.2707 1.2793
1.2383 1.2466 1.2550 1.2635 1.2720
1.2311 1.2394 1.2477 1.2561 1.2646
1.2169 1.2251 1.2334 1.2418 1.2503
1.2027 1.2109 1.2192 1.2276 1.2361
40 41 42 43 44
1.3179 1.3268 1.3357 1.3447 1.3538
1.3103 1.3191 1.3280 1.3370 1.3461
1.3065 1.3153 1.3242 1.3332 1.3423
1.3028 1.3116 1.3205 1.3294 1.3384
1.2991 1.3079 1.3167 1.3256 1.3346
1.2953 1.3041 1.3129 1.3218 1.3308
1.2880 1.2967 1.3055 1.3144 1.3234
1.2806 1.2893 1.2981 1.3070 1.3160
1.2732 1.2819 1.2907 1.2996 1.3086
1.2589 1.2675 1.2762 1.2850 1.2939
1.2446 1.2532 1.2619 1.2707 1.2796
45 46 47 48 49
1.3630 1.3724 1.3819 1.3915 1.4012
1.3553 1.3646 1.3740 1.3835 1.3931
1.3515 1.3608 1.3702 1.3797 1.3893
1.3476 1.3569 1.3663 1.3758 1.3854
1.3437 1.3530 1.3624 1.3719 1.3814
1.3399 1.3492 1.3586 1.3680 1.3775
1.3325 1.3417 1.3510 1.3604 1.3699
1.3251 1.3343 1.3435 1.3528 1.3623
1.3177 1.3269 1.3362 1.3455 1.3549
1.3029 1.3120 1.3212 1.3305 1.3399
1.2886 1.2976 1.3067 1.3159 1.3253
50 51 52 53 54
1.4110 1.4209 1.4310 1.4412 1.4515
1.4029 1.4128 1.4228 1.4329 1.4431
1.3990 1.4088 1.4188 1.4289 1.4391
1.3951 1.4049 1.4148 1.4248 1.4350
1.3911 1.4009 1.4109 1.4209 1.4310
1.3872 1.3970 1.4069 1.4169 1.4270
1.3795 1.3893 1.3991 1.4091 1.4191
1.3719 1.3816 1.3914 1.4013 1.4113
1.3644 1.3740 1.3837 1.3936 1.4036
1.3494 1.3590 1.3687 1.3785 1.3884
1.3348 1.3444 1.3540 1.3637 1.3735
55 56 57 58 59
1.4619 1.4724 1.4830 1.4937 1.5045
1.4535 1.4640 1.4746 1.4852 1.4959
1.4494 1.4598 1.4703 1.4809 1.4916
1.4453 1.4557 1.4662 1.4768 1.4875
1.4412 1.4516 1.4621 1.4726 1.4832
1.4372 1.4475 1.4580 1.4685 1.4791
1.4293 1.4396 1.4500 1.4604 1.4709
1.4214 1.4317 1.4420 1.4524 1.4629
1.4137 1.4239 1.4342 1.4446 1.4551
1.3984 1.4085 1.4187 1.4290 1.4393
1.3834 1.3934 1.4035 1.4137 1.4240
60 61 62 63 64
1.5154 1.5264 1.5375 1.5487 1.5600
1.5067 1.5177 1.5287 1.5398 1.5510
1.5024 1.5133 1.5243 1.5354 1.5465
1.4983 1.5091 1.5200 1.5310 1.5421
1.4940 1.5048 1.5157 1.5267 1.5378
1.4898 1.5006 1.5115 1.5225 1.5335
1.4816 1.4923 1.5031 1.5140 1.5250
1.4735 1.4842 1.4950 1.5058 1.5167
1.4656 1.4762 1.4869 1.4977 1.5086
1.4497 1.4602 1.4708 1.4815 1.4923
1.4344 1.4449 1.4554 1.4660 1.4766
2-108
PHYSICAL AND CHEMICAL DATA
TABLE 2-101
Sulfuric Acid (H2SO4) (Concluded )
%
0°C
10°C
15°C
20°C
25°C
30°C
40°C
50°C
60°C
80°C
100°C
65 66 67 68 69
1.5714 1.5828 1.5943 1.6059 1.6176
1.5623 1.5736 1.5850 1.5965 1.6081
1.5578 1.5691 1.5805 1.5920 1.6035
1.5533 1.5646 1.5760 1.5874 1.5989
1.5490 1.5602 1.5715 1.5829 1.5944
1.5446 1.5558 1.5671 1.5785 1.5899
1.5361 1.5472 1.5584 1.5697 1.5811
1.5277 1.5388 1.5499 1.5611 1.5724
1.5195 1.5305 1.5416 1.5528 1.5640
1.5031 1.5140 1.5249 1.5359 1.5470
1.4873 1.4981 1.5089 1.5198 1.5307
70 71 72 73 74
1.6293 1.6411 1.6529 1.6648 1.6768
1.6198 1.6315 1.6433 1.6551 1.6670
1.6151 1.6268 1.6385 1.6503 1.6622
1.6105 1.6221 1.6338 1.6456 1.6574
1.6059 1.6175 1.6292 1.6409 1.6526
1.6014 1.6130 1.6246 1.6363 1.6480
1.5925 1.6040 1.6155 1.6271 1.6387
1.5838 1.5952 1.6067 1.6182 1.6297
1.5753 1.5867 1.5981 1.6095 1.6209
1.5582 1.5694 1.5806 1.5919 1.6031
1.5417 1.5527 1.5637 1.5747 1.5857
75 76 77 78 79
1.6888 1.7008 1.7128 1.7247 1.7365
1.6789 1.6908 1.7026 1.7144 1.7261
1.6740 1.6858 1.6976 1.7093 1.7209
1.6692 1.6810 1.6927 1.7043 1.7158
1.6644 1.6761 1.6878 1.6994 1.7108
1.6597 1.6713 1.6829 1.6944 1.7058
1.6503 1.6619 1.6734 1.6847 1.6959
1.6412 1.6526 1.6640 1.6751 1.6862
1.6322 1.6435 1.6547 1.6657 1.6766
1.6142 1.6252 1.6361 1.6469 1.6575
1.5966 1.6074 1.6181 1.6286 1.6390
80 81 82 83 84
1.7482 1.7597 1.7709 1.7815 1.7916
1.7376 1.7489 1.7599 1.7704 1.7804
1.7323 1.7435 1.7544 1.7649 1.7748
1.7272 1.7383 1.7491 1.7594 1.7693
1.7221 1.7331 1.7437 1.7540 1.7639
1.7170 1.7279 1.7385 1.7487 1.7585
1.7069 1.7177 1.7281 1.7382 1.7479
1.6971 1.7077 1.7180 1.7279 1.7375
1.6873 1.6978 1.7080 1.7179 1.7274
1.6680 1.6782 1.6882 1.6979 1.7072
1.6493 1.6594 1.6692 1.6787 1.6878
85 86 87 88 89
1.8009 1.8095 1.8173 1.8243 1.8306
1.7897 1.7983 1.8061 1.8132 1.8195
1.7841 1.7927 1.8006 1.8077 1.8141
1.7786 1.7872 1.7951 1.8022 1.8087
1.7732 1.7818 1.7897 1.7968 1.8033
1.7678 1.7763 1.7842 1.7914 1.7979
1.7571 1.7657 1.7736 1.7809 1.7874
1.7466 1.7552 1.7632 1.7705 1.7770
1.7364 1.7449 1.7529 1.7602 1.7669
1.7161 1.7245 1.7324 1.7397 1.7464
1.6966 1.7050 1.7129 1.7202 1.7269
90 91 92 93 94
1.8361 1.8410 1.8453 1.8490 1.8520
1.8252 1.8302 1.8346 1.8384 1.8415
1.8198 1.8248 1.8293 1.8331 1.8363
1.8144 1.8195 1.8240 1.8279 1.8312
1.8091 1.8142 1.8188 1.8227 1.8260
1.8038 1.8090 1.8136 1.8176 1.8210
1.7933 1.7986 1.8033 1.8074 1.8109
1.7829 1.7883 1.7932 1.7974 1.8011
1.7729 1.7783 1.7832 1.7876 1.7914
1.7525 1.7581 1.7633 1.7681
1.7331 1.7388 1.7439 1.7485
95 96 97 98 99 100
1.8544 1.8560 1.8569 1.8567 1.8551 1.8517
1.8439 1.8457 1.8466 1.8463 1.8445 1.8409
1.8388 1.8406 1.8414 1.8411 1.8393 1.8357
1.8337 1.8355 1.8364 1.8361 1.8342 1.8305
1.8286 1.8305 1.8314 1.8310 1.8292 1.8255
1.8236 1.8255 1.8264 1.8261 1.8242 1.8205
1.8137 1.8157 1.8166 1.8163 1.8145 1.8107
1.8040 1.8060 1.8071 1.8068 1.8050 1.8013
1.7944 1.7965 1.7977 1.7976 1.7958 1.7922
%
d 5.96 4
%
d 13.00 4
d 18.00 4
0.005 .01 .02 .03 .04
1.000 0140 1.000 0576 1.000 1434 1.000 2276 1.000 3104
0.05 .1 .2 .3 .4
0.999 810 1.000 185 1.000 912 1.001 623 1.002 326
0.999 028 .999 400 1.000 119 1.000 820 1.001 512
.05 .06 .07 .08 .09
1.000 3920 1.000 4726 1.000 5523 1.000 6313 1.000 7098
.5 .6 .8 1.0 1.2
1.003 023 1.003 716 1.005 090 1.006 452 1.007 807
1.002 197 1.002 877 1.004 227 1.005 570 1.006 909
.10 .15 .20 .25 .30
1.000 7880 1.001 1732 1.001 5514 1.001 9254 1.002 2961
1.4 1.6 1.8 2.0 2.2
1.009 159 1.010 510 1.011 860 1.013 209 1.014 557
1.008 247 1.009 583 1.010 918 1.012 252 1.013 586
.35 .40 .45 .50
1.002 6639 1.003 0292 1.003 3923 1.003 7534
2.4
1.015 904
1.014 919
DENSITIES OF AQUEOUS ORGANIC SOLUTIONS TABLE 2-102
Zinc Bromide (ZnBr2)
%
0°C
20°C
40°C
60°C
80°C
100°C
2 4 8 12 16
1.0188 1.0381 1.0777 1.1186 1.1609
1.0167 1.0354 1.0738 1.1135 1.1544
1.0102 1.0285 1.0660 1.1046 1.1445
1.0008 1.0187 1.0554 1.0932 1.1320
0.9890 1.0065 1.0422 1.0789 1.1169
0.9751 0.9921 1.0270 1.0629 1.1000
20 30 40 50 60 65
1.2043 1.3288 1.477 1.661 1.891 2.026
1.1965 1.3170 1.462 1.643 1.869 2.002
1.1855 1.3030 1.445 1.623 1.845 1.976
1.1720 1.2868 1.427 1.602 1.822 1.951
1.1560 1.2688 1.406 1.579 1.797 1.924
1.1382 1.2489 1.385 1.555 1.771 1.898
TABLE 2-103
TABLE 2-104 [Zn(NO3)2]
Zinc Nitrate
TABLE 2-105 (ZnSO4)
2-109
Zinc Sulfate
%
18°C
%
18°C
%
20°C
2 4 6 8 10 12 14 16
1.0154 1.0322 1.0496 1.0675 1.0859 1.1048 1.1244 1.1445
18 20 25 30 35 40 45 50
1.1652 1.1865 1.2427 1.3029 1.3678 1.4378 1.5134 1.5944
2 4 6 8 10 12 14 16
1.019 1.0403 1.0620 1.0842 1.1071 1.1308 1.1553 1.1806
Zinc Chloride (ZnCl2)
%
0°C
20°C
40°C
60°C
80°C
100°C
2 4 8 12 16
1.0192 1.0384 1.0769 1.1159 1.1558
1.0167 1.0350 1.0715 1.1085 1.1468
1.0099 1.0274 1.0624 1.0980 1.1350
1.0003 1.0172 1.0508 1.0853 1.1212
0.9882 1.0044 1.0369 1.0704 1.1055
0.9739 .9894 1.0211 1.0541 1.0888
20 30 40 50 60 70
1.1970 1.3062 1.4329 1.5860
1.1866 1.2928 1.4173 1.5681 1.749 1.962
1.1736 1.2778 1.4003 1.5495
1.1590 1.2614 1.3824 1.5300
1.1428 1.2438 1.3637 1.5097
1.1255 1.2252 1.3441 1.4892
DENSITIES OF AQUEOUS ORGANIC SOLUTIONS* From International Critical Tables, vol. 3, pp. 115–129. All compositions are in weight percent in vacuo. All density values are d4t = g/mL in vacuo.
UNITS AND UNITS CONVERSIONS Unless otherwise noted, densities are given in grams per cubic centimeter. To convert to pounds per cubic foot, multiply by 62.43. °F = 9⁄ 5 °C + 32 TABLE 2-106
*For gasoline and aircraft fuels see Hibbard, NACA Res. Mem. E56I21 (declassified 1958).
Formic Acid (HCOOH)
%
0°C
15°C
20°C
30°C
%
0°C
15°C
20°C
30°C
%
0°C
15°C
20°C
30°C
%
0°C
15°C
20°C
30°C
0 1 2 3 4
0.9999 1.0028 1.0059 1.0090 1.0120
0.9991 1.0019 1.0045 1.0072 1.0100
0.9982 1.0019 1.0044 1.0070 1.0093
0.9957 0.9980 1.0004 1.0028 1.0053
25 26 27 28 29
1.0706 1.0733 1.0760 1.0787 1.0813
1.0627 1.0652 1.0678 1.0702 1.0726
1.0609 1.0633 1.0656 1.0681 1.0705
1.0540 1.0564 1.0587 1.0609 1.0632
50 51 52 53 54
1.1349 1.1374 1.1399 1.1424 1.1448
1.1225 1.1248 1.1271 1.1294 1.1318
1.1207 1.1223 1.1244 1.1269 1.1295
1.1098 1.1120 1.1142 1.1164 1.1186
75 76 77 78 79
1.1953 1.1976 1.1999 1.2021 1.2043
1.1794 1.1816 1.1837 1.1859 1.1881
1.1769 1.1785 1.1801 1.1818 1.1837
1.1636 1.1656 1.1676 1.1697 1.1717
5 6 7 8 9
1.0150 1.0179 1.0207 1.0237 1.0266
1.0124 1.0151 1.0177 1.0204 1.0230
1.0115 1.0141 1.0170 1.0196 1.0221
1.0075 1.0101 1.0125 1.0149 1.0173
30 31 32 33 34
1.0839 1.0866 1.0891 1.0916 1.0941
1.0750 1.0774 1.0798 1.0821 1.0844
1.0729 1.0753 1.0777 1.0800 1.0823
1.0654 1.0676 1.0699 1.0721 1.0743
55 56 57 58 59
1.1472 1.1497 1.1523 1.1548 1.1573
1.1341 1.1365 1.1388 1.1411 1.1434
1.1320 1.1342 1.1361 1.1381 1.1401
1.1208 1.1230 1.1253 1.1274 1.1295
80 81 82 83 84
1.2065 1.2088 1.2110 1.2132 1.2154
1.1902 1.1924 1.1944 1.1965 1.1985
1.1806 1.1876 1.1896 1.1914 1.1929
1.1737 1.1758 1.1778 1.1798 1.1817
10 11 12 13 14
1.0295 1.0324 1.0351 1.0379 1.0407
1.0256 1.0281 1.0306 1.0330 1.0355
1.0246 1.0271 1.0296 1.0321 1.0345
1.0197 1.0221 1.0244 1.0267 1.0290
35 36 37 38 39
1.0966 1.0993 1.1018 1.1043 1.1069
1.0867 1.0892 1.0916 1.0940 1.0964
1.0847 1.0871 1.0895 1.0919 1.0940
1.0766 1.0788 1.0810 1.0832 1.0854
60 61 62 63 64
1.1597 1.1621 1.1645 1.1669 1.1694
1.1458 1.1481 1.1504 1.1526 1.1549
1.1424 1.1448 1.1473 1.1493 1.1517
1.1317 1.1338 1.1360 1.1382 1.1403
85 86 87 88 89
1.2176 1.2196 1.2217 1.2237 1.2258
1.2005 1.2025 1.2045 1.2064 1.2084
1.1953 1.1976 1.1994 1.2012 1.2028
1.1837 1.1856 1.1875 1.1893 1.1910
15 16 17 18 19
1.0435 1.0463 1.0491 1.0518 1.0545
1.0380 1.0405 1.0430 1.0455 1.0480
1.0370 1.0393 1.0417 1.0441 1.0464
1.0313 1.0336 1.0358 1.0381 1.0404
40 41 42 43 44
1.1095 1.1122 1.1148 1.1174 1.1199
1.0988 1.1012 1.1036 1.1060 1.1084
1.0963 1.0990 1.1015 1.1038 1.1062
1.0876 1.0898 1.0920 1.0943 1.0965
65 66 67 68 69
1.1718 1.1742 1.1766 1.1790 1.1813
1.1572 1.1595 1.1618 1.1640 1.1663
1.1543 1.1565 1.1584 1.1604 1.1628
1.1425 1.1446 1.1467 1.1489 1.1510
90 91 92 93 94
1.2278 1.2297 1.2316 1.2335 1.2354
1.2102 1.2121 1.2139 1.2157 1.2174
1.2044 1.2059 1.2078 1.2099 1.2117
1.1927 1.1945 1.1961 1.1978 1.1994
20 21 22 23 24
1.0571 1.0598 1.0625 1.0652 1.0679
1.0505 1.0532 1.0556 1.0580 1.0604
1.0488 1.0512 1.0537 1.0561 1.0585
1.0427 1.0451 1.0473 1.0496 1.0518
45 46 47 48 49
1.1224 1.1249 1.1274 1.1299 1.1324
1.1109 1.1133 1.1156 1.1179 1.1202
1.1085 1.1108 1.1130 1.1157 1.1185
1.0987 1.1009 1.1031 1.1053 1.1076
70 71 72 73 74
1.1835 1.1858 1.1882 1.1906 1.1929
1.1685 1.1707 1.1729 1.1751 1.1773
1.1655 1.1677 1.1702 1.1728 1.1752
1.1531 1.1552 1.1573 1.1595 1.1615
95 96 97 98 99
1.2372 1.2390 1.2408 1.2425 1.2441
1.2191 1.2208 1.2224 1.2240 1.2257
1.2140 1.2158 1.2170 1.2183 1.2202
1.2008 1.2022 1.2036 1.2048 1.2061
100
1.2456
1.2273
1.2212
1.2073
2-110
PHYSICAL AND CHEMICAL DATA
TABLE 2-107 %
Acetic Acid (CH3COOH)
0°C
10°C
15°C
20°C
25°C
30°C
40°C
%
0°C
10°C
15°C
20°C
25°C
30°C
40°C
0 1 2 3 4
0.9999 1.0016 1.0033 1.0051 1.0070
0.9997 1.0013 1.0029 1.0044 1.0060
0.9991 1.0006 1.0021 1.0036 1.0051
0.9982 .9996 1.0012 1.0025 1.0040
0.9971 .9987 1.0000 1.0013 1.0027
0.9957 .9971 .9984 .9997 1.0011
0.9922 .9934 .9946 .9958 .9970
50 51 52 53 54
1.0729 1.0738 1.0748 1.0757 1.0765
1.0654 1.0663 1.0671 1.0679 1.0687
1.0613 1.0622 1.0629 1.0637 1.0644
1.0575 1.0582 1.0590 1.0597 1.0604
1.0534 1.0542 1.0549 1.0555 1.0562
1.0492 1.0499 1.0506 1.0512 1.0518
1.0408 1.0414 1.0421 1.0427 1.0432
5 6 7 8 9
1.0088 1.0106 1.0124 1.0142 1.0159
1.0076 1.0092 1.0108 1.0124 1.0140
1.0066 1.0081 1.0096 1.0111 1.0126
1.0055 1.0069 1.0083 1.0097 1.0111
1.0041 1.0055 1.0068 1.0081 1.0094
1.0024 1.0037 1.0050 1.0063 1.0076
.9982 .9994 1.0006 1.0018 1.0030
55 56 57 58 59
1.0774 1.0782 1.0790 1.0798 1.0805
1.0694 1.0701 1.0708 1.0715 1.0722
1.0651 1.0658 1.0665 1.0672 1.0678
1.0611 1.0618 1.0624 1.0631 1.0637
1.0568 1.0574 1.0580 1.0586 1.0592
1.0525 1.0531 1.0536 1.0542 1.0547
1.0438 1.0443 1.0448 1.0453 1.0458
10 11 12 13 14
1.0177 1.0194 1.0211 1.0228 1.0245
1.0156 1.0171 1.0187 1.0202 1.0217
1.0141 1.0155 1.0170 1.0184 1.0199
1.0125 1.0139 1.0154 1.0168 1.0182
1.0107 1.0120 1.0133 1.0146 1.0159
1.0089 1.0102 1.0115 1.0127 1.0139
1.0042 1.0054 1.0065 1.0077 1.0088
60 61 62 63 64
1.0813 1.0820 1.0826 1.0833 1.0838
1.0728 1.0734 1.0740 1.0746 1.0752
1.0684 1.0690 1.0696 1.0701 1.0706
1.0642 1.0648 1.0653 1.0658 1.0662
1.0597 1.0602 1.0607 1.0612 1.0616
1.0552 1.0557 1.0562 1.0566 1.0571
1.0462 1.0466 1.0470 1.0473 1.0477
15 16 17 18 19
1.0262 1.0278 1.0295 1.0311 1.0327
1.0232 1.0247 1.0262 1.0276 1.0291
1.0213 1.0227 1.0241 1.0255 1.0269
1.0195 1.0209 1.0223 1.0236 1.0250
1.0172 1.0185 1.0198 1.0210 1.0223
1.0151 1.0163 1.0175 1.0187 1.0198
1.0099 1.0110 1.0121 1.0132 1.0142
65 66 67 68 69
1.0844 1.0850 1.0856 1.0860 1.0865
1.0757 1.0762 1.0767 1.0771 1.0775
1.0711 1.0716 1.0720 1.0725 1.0729
1.0666 1.0671 1.0675 1.0678 1.0682
1.0621 1.0624 1.0628 1.0631 1.0634
1.0575 1.0578 1.0582 1.0585 1.0588
1.0480 1.0483 1.0486 1.0489 1.0491
20 21 22 23 24
1.0343 1.0358 1.0374 1.0389 1.0404
1.0305 1.0319 1.0333 1.0347 1.0361
1.0283 1.0297 1.0310 1.0323 1.0336
1.0263 1.0276 1.0288 1.0301 1.0313
1.0235 1.0248 1.0260 1.0272 1.0283
1.0210 1.0222 1.0233 1.0244 1.0256
1.0153 1.0164 1.0174 1.0185 1.0195
70 71 72 73 74
1.0869 1.0874 1.0877 1.0881 1.0884
1.0779 1.0783 1.0786 1.0789 1.0792
1.0732 1.0736 1.0738 1.0741 1.0743
1.0685 1.0687 1.0690 1.0693 1.0694
1.0637 1.0640 1.0642 1.0644 1.0645
1.0590 1.0592 1.0594 1.0595 1.0596
1.0493 1.0495 1.0496 1.0497 1.0498
25 26 27 28 29
1.0419 1.0434 1.0449 1.0463 1.0477
1.0375 1.0388 1.0401 1.0414 1.0427
1.0349 1.0362 1.0374 1.0386 1.0399
1.0326 1.0338 1.0349 1.0361 1.0372
1.0295 1.0307 1.0318 1.0329 1.0340
1.0267 1.0278 1.0289 1.0299 1.0310
1.0205 1.0215 1.0225 1.0234 1.0244
75 76 77 78 79
1.0887 1.0889 1.0891 1.0893 1.0894
1.0794 1.0796 1.0797 1.0798 1.0798
1.0745 1.0746 1.0747 1.0747 1.0747
1.0696 1.0698 1.0699 1.0700 1.0700
1.0647 1.0648 1.0648 1.0648 1.0648
1.0597 1.0598 1.0598 1.0598 1.0597
1.0499 1.0499 1.0499 1.0498 1.0497
30 31 32 33 34
1.0491 1.0505 1.0519 1.0532 1.0545
1.0440 1.0453 1.0465 1.0477 1.0489
1.0411 1.0423 1.0435 1.0446 1.0458
1.0384 1.0395 1.0406 1.0417 1.0428
1.0350 1.0361 1.0372 1.0382 1.0392
1.0320 1.0330 1.0341 1.0351 1.0361
1.0253 1.0262 1.0272 1.0281 1.0289
80 81 82 83 84
1.0895 1.0895 1.0895 1.0895 1.0893
1.0798 1.0797 1.0796 1.0795 1.0793
1.0747 1.0745 1.0743 1.0741 1.0738
1.0700 1.0699 1.0698 1.0696 1.0693
1.0647 1.0646 1.0644 1.0642 1.0638
1.0596 1.0594 1.0592 1.0589 1.0585
1.0495 1.0493 1.0490 1.0487 1.0483
35 36 37 38 39
1.0558 1.0571 1.0584 1.0596 1.0608
1.0501 1.0513 1.0524 1.0535 1.0546
1.0469 1.0480 1.0491 1.0501 1.0512
1.0438 1.0449 1.0459 1.0469 1.0479
1.0402 1.0412 1.0422 1.0432 1.0441
1.0371 1.0380 1.0390 1.0399 1.0408
1.0298 1.0306 1.0314 1.0322 1.0330
85 86 87 88 89
1.0891 1.0887 1.0883 1.0877 1.0872
1.0790 1.0787 1.0783 1.0778 1.0773
1.0735 1.0731 1.0726 1.0721 1.0715
1.0689 1.0685 1.0680 1.0675 1.0668
1.0635 1.0630 1.0626 1.0620 1.0613
1.0582 1.0576 1.0571 1.0564 1.0557
1.0479 1.0473 1.0467 1.0460 1.0453
40 41 42 43 44
1.0621 1.0633 1.0644 1.0656 1.0667
1.0557 1.0568 1.0578 1.0588 1.0598
1.0522 1.0532 1.0542 1.0551 1.0561
1.0488 1.0498 1.0507 1.0516 1.0525
1.0450 1.0460 1.0469 1.0477 1.0486
1.0416 1.0425 1.0433 1.0441 1.0449
1.0338 1.0346 1.0353 1.0361 1.0368
90 91 92 93 94
1.0865 1.0857 1.0848 1.0838 1.0826
1.0766 1.0758 1.0749 1.0739 1.0727
1.0708 1.0700 1.0690 1.0680 1.0667
1.0661 1.0652 1.0643 1.0632 1.0619
1.0605 1.0597 1.0587 1.0577 1.0564
1.0549 1.0541 1.0530 1.0518 1.0506
1.0445 1.0436 1.0426 1.0414 1.0401
45 46 47 48 49
1.0679 1.0689 1.0699 1.0709 1.0720
1.0608 1.0618 1.0627 1.0636 1.0645
1.0570 1.0579 1.0588 1.0597 1.0605
1.0534 1.0542 1.0551 1.0559 1.0567
1.0495 1.0503 1.0511 1.0518 1.0526
1.0456 1.0464 1.0471 1.0479 1.0486
1.0375 1.0382 1.0389 1.0395 1.0402
95 96 97 98 99
1.0813 1.0798 1.0780 1.0759 1.0730
1.0714
1.0652 1.0632 1.0611 1.0590 1.0567
1.0605 1.0588 1.0570 1.0549 1.0524
1.0551 1.0535 1.0516 1.0495 1.0468
1.0491 1.0473 1.0454 1.0431 1.0407
1.0386 1.0368 1.0348 1.0325 1.0299
100
1.0697
1.0545
1.0498
1.0440
1.0380
1.0271
DENSITIES OF AQUEOUS ORGANIC SOLUTIONS TABLE 2-108
TABLE 2-109
2-111
Oxalic Acid (H2C2O4)
%
d 17.5 4
%
d 17.5 4
1 2 4
1.0035 1.0070 1.0140
8 10 12
1.0280 1.0350 1.0420
Methyl Alcohol (CH3OH)*
%
0°C
10°C
20°C
15°C
%
0°C
10°C
20°C
15°C
%
0°C
10°C
20°C
15°C
0 1 2 3 4
0.9999 .9981 .9963 .9946 .9930
0.9997 .9980 .9962 .9945 .9929
15.56°C 0.9990 .9973 .9955 .9938 .9921
0.9982 .9965 .9948 .9931 .9914
0.99913 .99727 .99543 .99370 .99198
35 36 37 38 39
0.9534 .9520 .9505 .9490 .9475
0.9484 .9469 .9453 .9437 .9420
15.56°C 0.9456 .9440 .9422 .9405 .9387
0.9433 .9416 .9398 .9381 .9363
0.94570 .94404 .94237 .94067 .93894
70 71 72 73 74
0.8869 .8847 .8824 .8801 .8778
0.8794 .8770 .8747 .8724 .8699
15.56°C 0.8748 .8726 .8702 .8678 .8653
0.8715 .8690 .8665 .8641 .8616
0.87507 .87271 .87033 .86792 .86546
5 6 7 8 9
.9914 .9899 .9884 .9870 .9856
.9912 .9896 .9881 .9865 .9849
.9904 .9889 .9872 .9857 .9841
.9896 .9880 .9863 .9847 .9831
.99029 .98864 .98701 .98547 .98394
40 41 42 43 44
.9459 .9443 .9427 .9411 .9395
.9403 .9387 .9370 .9352 .9334
.9369 .9351 .9333 .9315 .9297
.9345 .9327 .9309 .9290 .9272
.93720 .93543 .93365 .93185 .93001
75 76 77 78 79
.8754 .8729 .8705 .8680 .8657
.8676 .8651 .8626 .8602 .8577
.8629 .8604 .8579 .8554 .8529
.8592 .8567 .8542 .8518 .8494
.86300 .86051 .85801 .85551 .85300
10 11 12 13 14
.9842 .9829 .9816 .9804 .9792
.9834 .9820 .9805 .9791 .9778
.9826 .9811 .9796 .9781 .9766
.9815 .9799 .9784 .9768 .9754
.98241 .98093 .97945 .97802 .97660
45 46 47 48 49
.9377 .9360 .9342 .9324 .9306
.9316 .9298 .9279 .9260 .9240
.9279 .9261 .9242 .9223 .9204
.9252 .9234 .9214 .9196 .9176
.92815 .92627 .92436 .92242 .92048
80 81 82 83 84
.8634 .8610 .8585 .8560 .8535
.8551 .8527 .8501 .8475 .8449
.8503 .8478 .8452 .8426 .8400
.8469 .8446 .8420 .8394 .8366
.85048 .84794 .84536 .84274 .84009
15 16 17 18 19
.9780 .9769 .9758 .9747 .9736
.9764 .9751 .9739 .9726 .9713
.9752 .9738 .9723 .9709 .9695
.9740 .9725 .9710 .9696 .9681
.97518 .97377 .97237 .97096 .96955
50 51 52 53 54
.9287 .9269 .9250 .9230 .9211
.9221 .9202 .9182 .9162 .9142
.9185 .9166 .9146 .9126 .9106
.9156 .9135 .9114 .9094 .9073
.91852 .91653 .91451 .91248 .91044
85 86 87 88 89
.8510 .8483 .8456 .8428 .8400
.8422 .8394 .8367 .8340 .8314
.8374 .8347 .8320 .8294 .8267
.8340 .8314 .8286 .8258 .8230
.83742 .83475 .83207 .82937 .82667
20 21 22 23 24
.9725 .9714 .9702 .9690 .9678
.9700 .9687 .9673 .9660 .9646
.9680 .9666 .9652 .9638 .9624
.9666 .9651 .9636 .9622 .9607
.96814 .96673 .96533 .96392 .96251
55 56 57 58 59
.9191 .9172 .9151 .9131 .9111
.9122 .9101 .9080 .9060 .9039
.9086 .9065 .9045 .9024 .9002
.9052 .9032 .9010 .8988 .8968
.90839 .90631 .90421 .90210 .89996
90 91 92 93 94
.8374 .8347 .8320 .8293 .8266
.8287 .8261 .8234 .8208 .8180
.8239 .8212 .8185 .8157 .8129
.8202 .8174 .8146 .8118 .8090
.82396 .82124 .81849 .81568 .81285
25 26 27 28 29
.9666 .9654 .9642 .9629 .9616
.9632 .9618 .9604 .9590 .9575
.9609 .9595 .9580 .9565 .9550
.9592 .9576 .9562 .9546 .9531
.96108 .95963 .95817 .95668 .95518
60 61 62 63 64
.9090 .9068 .9046 .9024 .9002
.9018 .8998 .8977 .8955 .8933
.8980 .8958 .8936 .8913 .8890
.8946 .8924 .8902 .8879 .8856
.89781 .89563 .89341 .89117 .88890
95 96 97 98 99
.8240 .8212 .8186 .8158 .8130
.8152 .8124 .8096 .8068 .8040
.8101 .8073 .8045 .8016 .7987
.8062 .8034 .8005 .7976 .7948
.80999 .80713 .80428 .80143 .79859
30 31 32 33 34
.9604 .9590 .9576 .9563 .9549
.9560 .9546 .9531 .9516 .9500
.9535 .9521 .9505 .9489 .9473
.9515 .9499 .9483 .9466 .9450
.95366 .95213 .95056 .94896 .94734
65 66 67 68 69
.8980 .8958 .8935 .8913 .8891
.8911 .8888 .8865 .8842 .8818
.8867 .8844 .8820 .8797 .8771
.8834 .8811 .8787 .8763 .8738
.88662 .88433 .88203 .87971 .87739
100
.8102
.8009
.7959
.7917
.79577
*It should be noted that the values for 100 percent do not agree with some data available elsewhere, e.g., American Institute of Physics Handbook, McGraw-Hill, New York, 1957. Also, see Atack, Handbook of Chemical Data, Reinhold, New York, 1957.
2-112
PHYSICAL AND CHEMICAL DATA
TABLE 2-110
Ethyl Alcohol (C2H5OH)*
%
10°C
15°C
20°C
25°C
30°C
35°C
40°C
%
10°C
15°C
20°C
25°C
30°C
35°C
40°C
0 1 2 3 4
0.99973 785 602 426 258
0.99913 725 542 365 195
0.99823 636 453 275 103
0.99708 520 336 157 .98984
0.99568 379 194 014 .98839
0.99406 217 031 .98849 672
0.99225 034 .98846 663 485
50 51 52 53 54
0.92126 .91943 723 502 279
0.91776 555 333 110 .90885
0.91384 160 .90936 711 485
0.90985 760 534 307 079
0.90580 353 125 .89896 667
0.90168 .89940 710 479 248
0.89750 519 288 056 .88823
5 6 7 8 9
098 .98946 801 660 524
032 .98877 729 584 442
.98938 780 627 478 331
817 656 500 346 193
670 507 347 189 031
501 335 172 009 .97846
311 142 .97975 808 641
55 56 57 58 59
055 .90831 607 381 154
659 433 207 .89980 752
258 031 .89803 574 344
.89850 621 392 162 .88931
437 206 .88975 744 512
016 .88784 552 319 085
589 356 122 .87888 653
10 11 12 13 14
393 267 145 026 .97911
304 171 041 .97914 790
187 047 .97910 775 643
043 .97897 753 611 472
.97875 723 573 424 278
685 527 371 216 063
475 312 150 .96989 829
60 61 62 63 64
.89927 698 468 237 006
523 293 062 .88830 597
113 .88882 650 417 183
699 446 233 .87998 763
278 044 .87809 574 337
.87851 615 379 142 .86905
417 180 .86943 705 466
15 16 17 18 19
800 692 583 473 363
669 552 433 313 191
514 387 259 129 .96997
334 199 062 .96923 782
133 .96990 844 697 547
.96911 760 607 452 294
670 512 352 189 023
65 66 67 68 69
.88774 541 308 074 .87839
364 130 .87895 660 424
.87948 713 477 241 004
527 291 054 .86817 579
100 .86863 625 387 148
667 429 190 .85950 710
227 .85987 747 407 266
20 21 22 23 24
252 139 024 .96907 787
068 .96944 818 689 558
864 729 592 453 312
639 495 348 199 048
395 242 087 .95929 769
134 .95973 809 643 476
.95856 687 516 343 168
70 71 72 73 74
602 365 127 .86888 648
187 .86949 710 470 229
.86766 527 287 047 .85806
340 100 .85859 618 376
.85908 667 426 184 .84941
470 228 .84986 743 500
025 .84783 540 297 053
25 26 27 28 29
665 539 406 268 125
424 287 144 .95996 844
168 020 .95867 710 548
.95895 738 576 410 241
607 442 272 098 .94922
306 133 .94955 774 590
.94991 810 625 438 248
75 76 77 78 79
408 168 .85927 685 442
.85988 747 505 262 018
564 322 079 .84835 590
134 .84891 647 403 158
698 455 211 .83966 720
257 013 .83768 523 277
.83809 564 319 074 .82827
30 31 32 33 34
.95977 823 665 502 334
686 524 357 186 011
382 212 038 .94860 679
067 .94890 709 525 337
741 557 370 180 .93986
403 214 021 .93825 626
055 .93860 662 461 257
80 81 82 83 84
197 .84950 702 453 203
.84772 525 277 028 .83777
344 096 .83848 599 348
.83911 664 415 164 .82913
473 224 .82974 724 473
029 .82780 530 279 027
578 329 079 .81828 576
35 36 37 38 39
162 .94986 805 620 431
.94832 650 464 273 079
494 306 114 .93919 720
146 .93952 756 556 353
790 591 390 186 .92979
425 221 016 .92808 597
051 .92843 634 422 208
85 86 87 88 89
.83951 697 441 181 .82919
525 271 014 .82754 492
095 .82840 583 323 062
660 405 148 .81888 626
220 .81965 708 448 186
.81774 519 262 003 .80742
322 067 .80811 552 291
40 41 42 43 44
238 042 .93842 639 433
.93882 682 478 271 062
518 314 107 .92897 685
148 .92940 729 516 301
770 558 344 128 .91910
385 170 .91952 733 513
.91992 774 554 332 108
90 91 92 93 94
654 386 114 .81839 561
227 .81959 688 413 134
.81797 529 257 .80983 705
362 094 .80823 549 272
.80922 655 384 111 .79835
478 211 .79941 669 393
028 .79761 491 220 .78947
45 46 47 48 49
226 017 .92806 593 379
.92852 640 426 211 .91995
472 257 041 .91823 604
085 .91868 649 429 208
692 472 250 028 .90805
291 069 .90845 621 396
.90884 660 434 207 .89979
95 96 97 98 99
278 .80991 698 399 094
.80852 566 274 .79975 670
424 138 .79846 547 243
.79991 706 415 117 .78814
555 271 .78981 684 382
114 .78831 542 247 .77946
670 388 100 .77806 507
100
.79784
360
.78934
506
075
641
203
*For data from −78° to 78°C, see p. 2-142, Table 2N-5, American Institute of Physics Handbook, McGraw-Hill, New York, 1957.
DENSITIES OF AQUEOUS ORGANIC SOLUTIONS TABLE 2-111 % alcohol by weight
2-113
Densities of Mixtures of C2H5OH and H2O at 20°C g/mL Tenths of %
0
1
2
3
4
5
6
7
0 1 2 3 4
0.99823 636 453 275 103
804 618 435 257 087
785 599 417 240 070
766 581 399 222 053
748 562 381 205 037
729 544 363 188 020
710 525 345 171 003
692 507 327 154 *987
5 6 7 8 9
.98938 780 627 478 331
922 765 612 463 316
906 749 597 449 301
890 734 582 434 287
874 718 567 419 273
859 703 553 404 258
843 688 538 389 244
827 673 523 374 229
10 11 12 13 14
187 047 .97910 775 643
172 033 896 761 630
158 019 883 748 617
144 006 869 735 604
130 *992 855 722 591
117 *978 842 709 578
103 *964 828 696 565
089 *951 815 683 552
15 16 17 18 19
514 387 259 129 .96997
501 374 246 116 984
488 361 233 103 971
475 349 220 089 957
462 336 207 076 944
450 323 194 063 931
438 310 181 050 917
425 297 168 037 904
412 284 155 024 891
20 21 22 23 24
864 729 592 453 312
850 716 578 439 297
837 702 564 425 283
823 688 551 411 269
810 675 537 396 254
796 661 523 382 240
783 647 509 368 225
769 634 495 354 211
756 620 481 340 196
25 26 27 28 29
168 020 .95867 710 548
153 005 851 694 532
139 *990 836 678 516
124 *975 820 662 499
109 *959 805 646 483
094 *944 789 630 466
080 *929 773 613 450
30 31 32 33 34
382 212 038 .94860 679
365 195 020 842 660
349 178 003 824 642
332 161 *985 806 624
315 143 *967 788 605
298 126 *950 770 587
35 36 37 38 39
494 306 114 .93919 720
475 287 095 899 700
456 268 075 879 680
438 249 056 859 660
419 230 036 840 640
40 41 42 43 44
518 314 107 .92897 685
498 294 086 876 664
478 273 065 855 642
458 253 044 834 621
45 46 47 48 49
472 257 041 .91823 604
450 236 019 801 582
429 214 *997 780 560
408 193 *976 758 538
Tenths of % 0
1
2
3
4
5
6
7
50 51 52 53 54
0.91384 160 .90936 711 485
361 138 914 689 463
339 116 891 666 440
317 093 869 644 417
295 071 846 621 395
272 049 824 598 372
250 026 801 576 349
228 206 183 004 *981 *959 779 756 734 553 531 508 327 304 281
796 642 493 345 201
55 56 57 58 59
258 031 .89803 574 344
236 008 780 551 321
213 *985 757 528 298
190 *962 734 505 275
167 *939 711 482 252
145 *917 688 459 229
122 *894 665 436 206
099 076 054 *871 *848 *825 643 620 597 413 390 367 183 160 137
075 061 *937 *923 801 788 670 657 539 526
60 61 62 63 64
113 .88882 650 417 183
090 859 626 393 160
067 836 603 370 136
044 812 580 347 113
021 789 557 323 089
*998 766 533 300 066
*975 743 510 277 042
*951 *928 *905 720 696 673 487 463 440 253 230 206 019 *995 *972
400 272 142 010 877
65 66 67 68 69
.87948 713 477 241 004
925 689 454 218 *981
901 666 430 194 *957
878 642 406 170 *933
854 619 383 147 *909
831 595 359 123 *885
807 572 336 099 *862
784 760 737 548 524 501 312 288 265 075 052 028 *838 *814 *790
742 606 467 326 182
70 71 72 73 74
.86766 527 287 047 .85806
742 503 263 022 781
718 479 239 *998 757
694 455 215 *974 733
671 431 191 *950 709
647 407 167 *926 685
623 383 143 *902 661
599 575 551 339 335 311 119 095 071 *878 *854 *830 636 612 588
065 *914 757 597 433
050 035 *898 *883 742 726 581 565 416 400
75 76 77 78 79
564 322 079 .84835 590
540 297 055 811 566
515 273 031 787 541
491 467 443 419 394 370 346 249 225 200 176 152 128 103 006 *982 *958 *933 *909 *884 *860 762 738 713 689 664 640 615 517 492 467 443 418 393 369
281 108 *932 752 568
264 091 *914 734 550
247 230 074 056 *896 *878 715 697 531 512
80 81 82 83 84
344 096 .83848 599 348
319 072 823 574 323
294 047 798 549 297
270 022 773 523 272
245 *997 748 498 247
220 *972 723 473 222
196 *947 698 448 196
171 146 121 *923 *898 *873 674 649 624 423 398 373 171 146 120
400 211 017 820 620
382 192 *997 800 599
363 172 *978 780 579
344 325 153 134 *958 *939 760 740 559 539
85 86 87 88 89
095 .82840 583 323 062
070 815 557 297 035
044 789 531 271 009
019 763 505 245 *983
*994 738 479 219 *956
*968 712 453 193 *930
*943 686 427 167 *903
*917 *892 *866 660 635 609 401 375 349 140 114 088 *877 *850 *824
437 232 023 812 600
417 212 002 791 579
396 191 *981 770 557
376 170 *960 749 536
356 335 149 129 *939 *918 728 707 515 493
90 91 92 93 94
.81797 529 257 .80983 705
770 502 230 955 677
744 475 203 928 649
717 448 175 900 621
690 421 148 872 593
664 394 120 844 565
637 366 093 817 537
386 171 *954 736 516
365 150 *932 714 494
343 128 *910 692 472
322 106 *889 670 450
300 279 085 063 *867 *845 648 626 428 406
95 96 97 98 99
424 138 .79846 547 243
395 109 816 517 213
367 080 787 487 182
338 051 757 456 151
310 281 253 224 195 166 022 *993 *963 *934 *905 *875 727 698 668 638 608 578 426 396 365 335 305 274 120 089 059 028 *997 *966
100
.78934
*Indicates change in the first two decimal places.
8
% alcohol by weight
9
673 655 489 471 310 292 137 120 *971 *954 811 658 508 360 215
610 339 066 789 509
8
583 312 038 761 480
9
556 285 010 733 452
2-114
PHYSICAL AND CHEMICAL DATA
TABLE 2-112 % alcohol by volume at 60°F 0 1 2 3 4
Specific Gravity (60°/60°F [(15.56°/15.56°C)]) of Mixtures by Volume of C2H5OH and H2O
Tenths of % 0
1
1.00000 *985 0.99850 835 703 688 559 545 419 405
2
3
4
5
6
7
8
9
*970 820 674 531 391
*955 806 659 516 378
*940 791 645 502 364
*925 776 630 488 350
*910 761 616 474 336
*895 747 602 460 323
*880 732 587 446 309
865 717 573 432 296
176 163 047 035 *923 *911 803 791 684 672
% alcohol by volume at 60°F
Tenths of % 0
1
2
3
4
5
6
50 51 52 53 54
0.93426 230 031 .92830 626
407 210 011 810 605
387 190 *991 789 585
368 171 *971 769 564
348 151 *951 749 544
328 131 *931 728 523
309 111 *911 708 502
55 56 57 58 59
419 210 .91999 784 565
398 189 978 762 543
377 168 956 741 521
357 147 935 719 499
336 126 914 697 477
315 105 892 675 455
294 084 871 653 433
7
8
9
289 270 250 091 071 051 *890 *870 *850 688 667 647 482 461 440
5 6 7 8 9
282 150 022 .98899 779
269 137 009 887 767
255 124 *997 875 755
242 111 *984 863 743
228 098 *972 851 731
215 085 *960 838 720
202 073 *947 826 708
189 060 *935 814 696
10 11 12 13 14
661 544 430 319 210
649 532 419 308 200
637 521 408 297 190
625 509 396 286 179
614 498 385 275 168
602 487 374 264 157
590 475 363 254 147
579 464 352 243 136
567 452 341 232 125
556 441 330 221 115
60 61 62 63 64
344 120 .90893 664 434
322 097 870 641 411
299 075 847 618 388
277 052 825 595 365
255 030 802 572 341
232 007 779 549 318
210 *984 756 526 295
15 16 17 18 19
104 .97998 895 794 694
093 988 885 784 684
083 977 875 774 674
072 967 864 764 664
062 956 854 754 654
051 946 844 744 645
040 936 834 734 635
030 925 824 724 625
019 915 814 714 615
009 905 804 704 605
65 66 67 68 69
202 .89967 729 489 245
179 943 705 465 220
155 920 681 441 196
132 896 657 416 171
108 872 633 392 147
085 848 609 368 122
061 825 585 343 098
038 801 561 319 073
014 *991 777 753 537 513 295 270 048 024
20 21 22 23 24
596 496 395 293 189
586 486 385 283 179
576 476 375 272 168
566 466 365 262 158
556 456 354 252 147
546 446 344 241 137
536 436 334 231 126
526 425 324 221 116
516 415 313 210 105
506 405 303 200 095
70 71 72 73 74
.88999 751 499 244 .87987
974 725 474 218 961
950 700 448 193 935
925 675 423 167 910
900 650 397 141 884
875 625 372 116 858
850 600 346 090 832
825 574 321 064 806
801 549 296 039 780
25 26 27 28 29
084 .96978 870 760 648
073 967 859 749 637
063 957 848 738 625
052 946 837 727 614
042 935 826 715 603
031 924 815 704 591
020 914 804 693 580
010 903 793 682 568
*999 *988 892 881 782 771 671 659 557 546
75 76 77 78 79
728 465 199 .86929 656
702 439 172 902 629
676 412 145 875 601
650 386 118 847 574
623 359 092 820 546
597 332 065 793 518
571 306 038 766 491
545 518 492 279 252 226 011 *984 *957 738 711 684 463 435 408
30 31 32 33 34
534 418 296 170 041
522 406 284 157 028
511 394 271 144 015
499 382 259 132 002
488 370 246 119 *988
476 358 234 106 *975
464 346 221 093 *962
453 334 209 080 *948
441 429 321 309 196 183 067 054 *935 *921
80 81 82 83 84
380 100 .85817 531 240
352 072 789 502 211
324 044 760 473 181
296 015 732 444 152
269 *987 703 415 122
241 *959 674 386 093
213 *931 646 357 063
185 157 129 *902 *874 *846 617 588 560 328 299 270 033 004 *974
35 36 37 38 39
.95908 770 628 482 332
894 756 614 467 317
881 742 599 452 302
867 728 585 437 286
854 714 570 423 271
840 700 556 408 256
826 685 541 393 240
812 671 526 378 225
784 643 497 347 194
85 86 87 88 89
.84944 642 336 025 .83707
914 612 305 *994 675
884 581 274 *962 643
854 551 243 *930 610
824 520 212 *899 578
794 490 181 *867 545
764 459 150 *835 513
734 703 673 428 398 367 119 088 056 *803 *771 *739 480 447 415
40 41 42 43 44
178 020 .94858 693 524
162 004 842 676 507
147 *988 825 660 490
131 *972 809 643 473
115 *956 792 626 455
100 *940 776 609 438
084 *923 759 592 421
068 *907 743 575 403
052 036 *891 *875 726 710 558 541 386 369
90 91 92 93 94
382 049 .82705 351 .81984
349 015 670 315 947
315 *981 635 279 909
282 *947 600 243 871
249 *913 565 206 834
216 *879 529 170 796
183 *845 494 133 757
150 116 083 *810 *776 *741 458 423 387 096 059 022 719 681 642
45 46 47 48 49
351 174 .93993 808 619
334 156 975 789 600
316 138 956 771 581
298 120 938 752 562
281 102 920 733 543
263 084 901 714 523
245 066 883 695 504
228 048 864 676 485
95 96 97 98 99
603 206 .80792 356 .79889
564 165 750 311 841
525 125 707 265 792
486 084 664 219 743
446 042 620 173 693
407 001 577 127 643
367 *960 533 080 593
327 287 247 *918 *876 *834 489 445 401 033 *985 *937 543 492 441
100
389
*Indicates change in first two decimal places.
798 657 512 362 209
210 030 845 657 465
192 011 827 638 446
273 062 849 631 410
252 041 827 610 388
231 020 806 588 366
188 165 143 *962 *939 *916 733 710 687 503 480 457 272 249 225
776 524 270 013 754
DENSITIES OF AQUEOUS ORGANIC SOLUTIONS
2-115
TABLE 2-113 n-Propyl Alcohol (C3H7OH) %
0°C
15°C
30°C
%
0°C
15°C
30°C
%
0°C
15°C
30°C
%
0°C
15°C
30°C
%
0°C
15°C
30°C
0 1 2 3 4
0.9999 .9982 .9967 .9952 .9939
0.9991 .9974 .9960 .9944 .9929
0.9957 .9940 .9924 .9908 .9893
20 21 22 23 24
0.9789 .9776 .9763 .9748 .9733
0.9723 .9705 .9688 .9670 .9651
0.9643 .9622 .9602 .9583 .9563
40 41 42 43 44
0.9430 .9411 .9391 .9371 .9352
0.9331 .9310 .9290 .9269 .9248
0.9226 .9205 .9184 .9164 .9143
60 61 62 63 64
0.9033 .9013 .8994 .8974 .8954
0.8922 .8902 .8882 .8861 .8841
0.8807 .8786 .8766 .8745 .8724
80 81 82 83 84
0.8634 .8614 .8594 .8574 .8554
0.8516 .8496 .8475 .8454 .8434
0.8394 .8373 .8352 .8332 .8311
5 6 7 8 9
.9926 .9914 .9904 .9894 .9883
.9915 .9902 .9890 .9877 .9864
.9877 .9862 .9848 .9834 .9819
25 26 27 28 29
.9717 .9700 .9682 .9664 .9646
.9633 .9614 .9594 .9576 .9556
.9543 .9522 .9501 .9481 .9460
45 46 47 48 49
.9332 .9311 .9291 .9272 .9252
.9228 .9207 .9186 .9165 .9145
.9122 .9100 .9079 .9057 .9036
65 66 67 68 69
.8934 .8913 .8894 .8874 .8854
.8820 .8800 .8779 .8759 .8739
.8703 .8682 .8662 .8641 .8620
85 86 87 88 89
.8534 .8513 .8492 .8471 .8450
.8413 .8393 .8372 .8351 .8330
.8290 .8269 .8248 .8227 .8206
10 11 12 13 14
.9874 .9865 .9857 .9849 .9841
.9852 .9840 .9828 .9817 .9806
.9804 .9790 .9775 .9760 .9746
30 31 32 33 34
.9627 .9608 .9589 .9570 .9550
.9535 .9516 .9495 .9474 .9454
.9439 .9418 .9396 .9375 .9354
50 51 52 53 54
.9232 .9213 .9192 .9173 .9153
.9124 .9104 .9084 .9064 .9044
.9015 .8994 .8973 .8952 .8931
70 71 72 73 74
.8835 .8815 .8795 .8776 .8756
.8719 .8700 .8680 .8659 .8639
.8600 .8580 .8559 .8539 .8518
90 91 92 93 94
.8429 .8408 .8387 .8364 .8342
.8308 .8287 .8266 .8244 .8221
.8185 .8164 .8142 .8120 .8098
15 16 17 18 19
.9833 .9825 .9817 .9808 .9800
.9793 .9780 .9768 .9752 .9739
.9730 .9714 .9698 .9680 .9661
35 36 37 38 39
.9530 .9511 .9491 .9471 .9450
.9434 .9413 .9392 .9372 .9351
.9333 .9312 .9289 .9269 .9247
55 56 57 58 59
.9132 .9112 .9093 .9073 .9053
.9023 .9003 .8983 .8963 .8942
.8911 .8890 .8869 .8849 .8828
75 76 77 78 79
.8736 .8716 .8695 .8675 .8655
.8618 .8598 .8577 .8556 .8536
.8497 .8477 .8456 .8435 .8414
95 96 97 98 99
.8320 .8296 .8272 .8248 .8222
.8199 .8176 .8153 .8128 .8104
.8077 .8054 .8031 .8008 .7984
100
.8194
.8077
.7958
TABLE 2-114
Isopropyl Alcohol (C3H7OH)
%
0°C
15°C*
15°C*
20°C
30°C
%
0°C
0 1 2 3 4
0.9999 .9980 .9962 .9946 .9930
0.9991 .9973 .9956 .9938 .9922
0.99913 .9972 .9954 .9936 .9920
0.9982 .9962 .9944 .9926 .9909
0.9957 .9939 .9921 .9904 .9887
35 36 37 38 39
0.9557 .9536 .9514 .9493 .9472
5 6 7 8 9
.9916 .9902 .9890 .9878 .9866
.9906 .9892 .9878 .9864 .9851
.9904 .9890 .9875 .9862 .9849
.9893 .9877 .9862 .9847 .9833
.9871 .9855 .9839 .9824 .9809
40 41 42 43 44
.9450 .9428 .9406 .9384 .9361
10 11 12 13 14
.9856 .9846 .9838 .9829 .9821
.9838 .9826 .9813 .9802 .9790
.98362 .9824 .9812 .9800 .9788
.9820 .9808 .9797 .9876 .9776
.9794 .9778 .9764 .9750 .9735
45 46 47 48 49
15 16 17 18 19
.9814 .9806 .9799 .9792 .9784
.9779 .9768 .9756 .9745 .9730
.9777 .9765 .9753 .9741 .9728
.9765 .9754 .9743 .9731 .9717
.9720 .9705 .9690 .9675 .9658
20 21 22 23 24
.9777 .9768 .9759 .9749 .9739
.9719 .9704 .9690 .9675 .9660
.97158 .9703 .9689 .9674 .9659
.9703 .9688 .9669 .9651 .9634
25 26 27 28 29
.9727 .9714 .9699 .9684 .9669
.9643 .9626 .9608 .9590 .9570
.9642 .9624 .9605 .9586 .9568
30 31 32 33 34
.9652 .9634 .9615 .9596 .9577
.9551
.95493 .9530 .9510 .9489 .9468
15°C*
15°C*
20°C
30°C
%
0°C
15°C*
15°C*
20°C
30°C
0.9446 .9424 .9401 .9379 .9356
0.9419 .9399 .9377 .9355 .9333
0.9338 .9315 .9292 .9269 .9246
70 71 72 73 74
0.8761 .8738 .8714 .8691 .8668
0.8639 .8615 .8592 .8568 .8545
0.86346 .8611 .8588 .8564 .8541
0.8584 .8560 .8537 .8513 .8489
0.8511 .8487 .8464 .8440 .8416
.93333 .9311 .9288 .9266 .9243
.9310 .9287 .9264 .9239 .9215
.9224 .9201 .9177 .9154 .9130
75 76 77 78 79
.8644 .8621 .8598 .8575 .8551
.8521 .8497 .8474 .8450 .8426
.8517 .8493 .8470 .8446 .8422
.8464 .8439 .8415 .8391 .8366
.8392 .8368 .8344 .8321 .8297
.9338 .9315 .9292 .9270 .9247
.9220 .9197 .9174 .9150 .9127
.9191 .9165 .9141 .9117 .9093
.9106 .9082 .9059 .9036 .9013
80 81 82 83 84
.8528 .8503 .8479 .8456 .8432
.8403 .8379 .8355 .8331 .8307
.83979 .8374 .8350 .8326 .8302
.8342 .8317 .8292 .8268 .8243
.8273 .8248 .8224 .8200 .8175
50 51 52 53 54
.9224 .9201 .9178 .9155 .9132
.91043 .9081 .9058 .9035 .9011
.9069 .9044 .9020 .8996 .8971
.8990 .8966 .8943 .8919 .8895
85 86 87 88 89
.8408 .8384 .8360 .8336 .8311
.8282 .8259 .8234 .8209 .8184
.8278 .8254 .8229 .8205 .8180
.8219 .8194 .8169 .8145 .8120
.8151 .8127 .8201 .8078 .8053
.9642 .9624 .9606 .9587 .9569
55 56 57 58 59
.9109 .9086 .9063 .9040 .9017
.8988 .8964 .8940 .8917 .8893
.8946 .8921 .8896 .8874 .8850
.8871 .8847 .8823 .8800 .8777
90 91 92 93 94
.8287 .8262 .8237 .8212 .8186
.8161 .8136 .8110 .8085 .8060
.81553 .8130 .8104 .8079 .8052
.8096 .8072 .8047 .8023 .7998
.8029 .8004 .7979 .7954 .7929
.9615 .9597 .9577 .9558 .9540
.9549 .9529 .9509 .9488 .9467
60 61 62 63 64
.8994 .8970 .8947 .8924 .8901
0.8829 .8805 .8781
.88690 .8845 .8821 .8798 .8775
.8825 .8800 .8776 .8751 .8727
.8752 .8728 .8704 .8680 .8656
95 96 97 98 99
.8160 .8133 .8106 .8078 .8048
.8034 .8008 .7981 .7954 .7926
.8026 .7999 .7972 .7945 .7918
.7973 .7949 .7925 .7901 .7877
.7904 .7878 .7852 .7826 .7799
.9520 .9500 .9481 .9460 .9440
.9446 .9426 .9405 .9383 .9361
65 66 67 68 69
.8878 .8854 .8831 .8807 .8784
.8757 .8733 .8710 .8686 .8662
.8752 .8728 .8705 .8682 .8658
.8702 .8679 .8656 .8632 .8609
.8631 .8607 .8583 .8559 .8535
100
.8016
.7896
.78913
.7854
.7770
*Two different observers; see International Critical Tables, vol. 3, p. 120.
2-116
PHYSICAL AND CHEMICAL DATA
TABLE 2-115
Glycerol* Density
Density
Glycerol, %
15°C
15.5°C
20°C
25°C
30°C
Glycerol, %
15°C
15.5°C
20°C
25°C
30°C
Glycerol, % 15°C
100 99 98 97 96
1.26415 1.26160 1.25900 1.25645 1.25385
1.26381 1.26125 1.25865 1.25610 1.25350
1.26108 1.25850 1.25590 1.25335 1.25080
1.15802 1.25545 1.25290 1.25030 1.24770
1.25495 1.25235 1.24975 1.24710 1.24450
65 64 63 62 61
1.17030 1.16755 1.16480 1.16200 1.15925
1.17000 1.16725 1.16445 1.16170 1.15895
1.16750 1.16475 1.16205 1.15930 1.15655
1.16475 1.16200 1.15925 1.15655 1.15380
1.16195 1.15925 1.15650 1.15375 1.15100
30 29 28 27 26
95 94 93 92 91
1.25130 1.24865 1.24600 1.24340 1.24075
1.25095 1.24830 1.24565 1.24305 1.24040
1.24825 1.24560 1.24300 1.24035 1.23770
1.24515 1.24250 1.23985 1.23725 1.23460
1.24190 1.23930 1.23670 1.23410 1.23150
60 59 58 57 56
1.15650 1.15370 1.15095 1.14815 1.14535
1.15615 1.15340 1.15065 1.14785 1.14510
1.15380 1.15105 1.14830 1.14555 1.14280
1.15105 1.14835 1.14560 1.14285 1.14015
1.14830 1.14555 1.14285 1.14010 1.13740
90 89 88 87 86
1.23810 1.23545 1.23280 1.23015 1.22750
1.23775 1.23510 1.23245 1.22980 1.22710
1.23510 1.23245 1.22975 1.22710 1.22445
1.23200 1.22935 1.22665 1.22400 1.22135
1.22890 1.22625 1.22360 1.22095 1.21830
55 54 53 52 51
1.14260 1.13980 1.13705 1.13425 1.13150
1.14230 1.13955 1.13680 1.13400 1.13125
1.14005 1.13730 1.13455 1.13180 1.12905
1.13740 1.13465 1.13195 1.12920 1.12650
85 84 83 82 81
1.22485 1.22220 1.21955 1.21690 1.21425
1.22445 1.22180 1.21915 1.21650 1.21385
1.22180 1.21915 1.21650 1.21380 1.21115
1.21870 1.21605 1.21340 1.21075 1.20810
1.21565 1.21300 1.21035 1.20770 1.20505
50 49 48 47 46
1.12870 1.12600 1.12325 1.12055 1.11780
1.12845 1.12575 1.12305 1.12030 1.11760
1.12630 1.12360 1.12090 1.11820 1.11550
80 79 78 77 76
1.21160 1.20885 1.20610 1.20335 1.20060
1.21120 1.20845 1.20570 1.20300 1.20025
1.20850 1.20575 1.20305 1.20030 1.19760
1.20545 1.20275 1.20005 1.19735 1.19465
1.20240 1.19970 1.19705 1.19435 1.19170
45 44 43 42 41
1.11510 1.11235 1.10960 1.10690 1.10415
1.11490 1.11215 1.10945 1.10670 1.10400
75 74 73 72 71
1.19785 1.19510 1.19235 1.18965 1.18690
1.19750 1.19480 1.19205 1.18930 1.18655
1.19485 1.19215 1.18940 1.18670 1.18395
1.19195 1.18925 1.18650 1.18380 1.18110
1.18900 1.18635 1.18365 1.18100 1.17830
40 39 38 37 36
1.10145 1.09875 1.09605 1.09340 1.09070
70 69 68 67 66
1.18415 1.18135 1.17860 1.17585 1.17305
1.18385 1.18105 1.17830 1.17555 1.17275
1.18125 1.17850 1.17575 1.17300 1.17025
1.17840 1.17565 1.17295 1.17020 1.16745
1.17565 1.17290 1.17020 1.16745 1.16470
35 34 33 32 31
1.08800 1.08530 1.08265 1.07995 1.07725
20°C
25°C
30°C
1.07455 1.07195 1.06935 1.06670 1.06410
1.07435 1.07175 1.06915 1.06655 1.06390
1.07270 1.07010 1.06755 1.06495 1.06240
1.07070 1.06815 1.06560 1.06305 1.06055
1.06855 1.06605 1.06355 1.06105 1.05855
25 24 23 22 21
1.06150 1.05885 1.05625 1.05365 1.05100
1.06130 1.05870 1.05610 1.05350 1.05090
1.05980 1.05720 1.05465 1.05205 1.04950
1.05800 1.05545 1.05290 1.05035 1.04780
1.05605 1.05350 1.05100 1.04850 1.04600
1.13470 1.13195 1.12925 1.12650 1.12380
20 19 18 17 16
1.04840 1.04590 1.04335 1.04085 1.03835
1.04825 1.04575 1.04325 1.04075 1.03825
1.04690 1.04440 1.04195 1.03945 1.03695
1.04525 1.04280 1.04035 1.03790 1.03545
1.04350 1.04105 1.03860 1.03615 1.03370
1.12375 1.12110 1.11840 1.11575 1.11310
1.12110 1.11845 1.11580 1.11320 1.11055
15 14 13 12 11
1.03580 1.03330 1.03080 1.02830 1.02575
1.03570 1.03320 1.03070 1.02820 1.02565
1.03450 1.03200 1.02955 1.02705 1.02455
1.03300 1.03055 1.02805 1.02560 1.02315
1.03130 1.02885 1.02640 1.02395 1.02150
1.11280 1.11010 1.10740 1.10470 1.10200
1.11040 1.10775 1.10510 1.10240 1.09975
1.10795 1.10530 1.10265 1.10005 1.09740
10 9 8 7 6
1.02325 1.02085 1.01840 1.01600 1.01360
1.02315 1.02075 1.01835 1.01590 1.01350
1.02210 1.01970 1.01730 1.01495 1.01255
1.02070 1.01835 1.01600 1.01360 1.01125
1.01905 1.01670 1.01440 1.01205 1.00970
1.10130 1.09860 1.09590 1.09320 1.09050
1.09930 1.09665 1.09400 1.09135 1.08865
1.09710 1.09445 1.09180 1.08915 1.08655
1.09475 1.09215 1.08955 1.08690 1.08430
5 4 3 2 1
1.01120 1.00875 1.00635 1.00395 1.00155
1.01110 1.00870 1.00630 1.00385 1.00145
1.01015 1.00780 1.00540 1.00300 1.00060
1.00890 1.00655 1.00415 1.00180 0.99945
1.00735 1.00505 1.00270 1.00035 0.99800
1.08780 1.08515 1.08245 1.07975 1.07705
1.08600 1.08335 1.08070 1.07800 1.07535
1.08390 1.08125 1.07860 1.07600 1.07335
1.08165 1.07905 1.07645 1.07380 1.07120
0
0.99913 0.99905 0.99823 0.99708 0.99568
*Bosart and Snoddy, Ind. Eng. Chem., 20, (1928): 1378.
TABLE 2-116
Density 15.5°C
Hydrazine (N2H4)
%
d 15 4
%
d 15 4
1 2 4 8 12 16 20 24 28
1.0002 1.0013 1.0034 1.0077 1.0121 1.0164 1.0207 1.0248 1.0286
30 40 50 60 70 80 90 100
1.0305 1.038 1.044 1.047 1.046 1.040 1.030 1.011
DENSITIES OF AQUEOUS ORGANIC SOLUTIONS TABLE 2-117
2-117
Densities of Aqueous Solutions of Miscellaneous Organic Compounds*
d (resp., dw, ds) = density of the solution (resp., water; resp., the pure liquid solute) in g/mL; ps (resp., pw) = wt % of solute (resp., water) in the solution; range = range of applicability of the equation. Section A Name
Formula
t, °C
Acetaldehyde Acetamide
C2H4O C2H5NO
Acetone
C3H6O
Acetonitrile Allyl alcohol Benzenepentacarboxylic acid Butyl alcohol (n-)
C2H3N C3H6O C11H6O10 C4H10O
Butyric acid (n-)
C4H8O2
Chloral hydrate
C2H3Cl3O2
Chloroacetic acid
C2H3ClO2
Citric acid (hydrate)
C6H3O7 + H2O
Dichloroacetic acid
C2H2Cl2O2
Diethylamine hydrochloride Ethylamine hydrochloride
C4H12ClN C2H8ClN
Ethylene glycol
C2H6O2
Ethyl ether tartrate Formaldehyde Formamide
C4H10O C8H14O6 CH2O CH3NO
Furfural Isoamyl alcohol
C5H4O2 C5H12O
Isobutyl alcohol
C4H10O
Isobutyric acid
C4H8O2
Isovaleric acid Lactic acid Maleic acid
C5H10O2 C3H6O C4H4O4
Malic acid
C4H6O5
Malonic acid Methyl acetate
C3H4O4 C3H6O2
glucoside (α-)
C7H14O6
Nicotine Nitrophenol (p-)
C10H14N2 C6H5NO3
Oxalic acid
C2H2O4
Phenol
C6H6O
Phenylglycolic acid Picoline (α-) (β-)
C8H8O3 C6H7N C6H7N
Propionic acid
C3H6O2
Pyridine Resorcinol Succinic acid
C5H5N C6H6O2 C4H6O4
Tartaric acid (d, l, or dl)
C4H6O6
*From “International Critical Tables,” vol. 3, pp. 111–114.
18 15 0 4 15 20 25 15 0 25 20 18 25 0 15 30 20 25 18 20 25 21 21 0 15 20 25 15 15 25 20 25 20 15 20 15 18 25 25 25 25 20 25 20 20 0 30 20 15 0 15 17.5 20 25 15 80 25 25 25 18 25 25 18 25 15 17.5 20 30 40 50 60
d = d w + Aps + Bp s2 + Cp3s Range, ps
A
B
0– 30 0– 6 0–100 0–100 0–100 0–100 0–100 0– 16 0– 89 0– 0.6 0– 7.9 0– 10 0– 62 0– 70 0– 78 0– 90 0– 32 0– 86 0– 50 0– 30 0– 97 0– 36 0– 65 0–100 0– 6 0– 5 0– 4.5 0– 95 0– 40 22– 96 0– 8 0– 8 0– 2.5 0– 8 0– 8 0– 9 0– 9 0– 12 0– 5 0– 9 0– 40 0– 40 0– 40 0– 40 0– 20 26– 51 26– 51 0– 60 0– 1.5 0– 4 0– 4 0– 9 0– 4 0– 4 0– 5 0– 65 0– 11 0– 70 0– 60 0– 10 0– 40 0– 60 0– 52 0– 5.5 0– 15 0– 50 0– 50 0– 50 0– 50 0– 50 0– 50
+0.03255 +0.03639 −0.03856 −0.027648 −0.021009 −0.021233 −0.021171 −0.021175 −0.033729 +0.025615 −0.021651 +0.03414 +0.035135 +0.024489 +0.024455 +0.024401 +0.023648 +0.023602 +0.023824 +0.024427 +0.024427 +0.0334 +0.021193 +0.021483 +0.02133 −0.02221 −0.02221 +0.022367 +0.022518 +0.021217 +0.021827 +0.021664 +0.02155 −0.02146 −0.02169 +0.0352 +0.0345 +0.0337 +0.03253 +0.02231 +0.0234 +0.023933 +0.023736 +0.02389 +0.0340 +0.023336 +0.023151 +0.03642 +0.023216 +0.025898 +0.02494 +0.02494 +0.025264 +0.025108 +0.02111 +0.03462 +0.02207 −0.04386 −0.04683 +0.0395 +0.039245 +0.03229 +0.02201 +0.02304 +0.024482 +0.024455 +0.024432 +0.024335 +0.024265 +0.024205 +0.024155
−0.0516 +0.04171 −0.05449 −0.041193 −0.059682 −0.053529 −0.05904 −0.042024 −0.041232 −0.02117 +0.04285 +0.04131 −0.04166 +0.042802 +0.042198 +0.041887 +0.05302 +0.05552 +0.041141 +0.05537 +0.05537 +0.0676 −0.05307 +0.052992 −0.05108 +0.0448 +0.0435 +0.05358 −0.05658 +0.053199 +0.05366 +0.0421 +0.043 +0.056 +0.0438
−0.04282 +0.05186 +0.0575 +0.05957 +0.04175 +0.041066 −0.0574 +0.05996 +0.05975 +0.05454 −0.0455 −0.033185 −0.058 −0.058 −0.031996 −0.031607 −0.04283 −0.0686 +0.0423 −0.051405 −0.0513 −0.04172 −0.0599 −0.05204 +0.05519 +0.04185 +0.04185 +0.041837 +0.04185 +0.04185 +0.04185 +0.04185
C
−0.07588 +0.08272 −0.08624 −0.075327 −0.0856 +0.072984
+0.0611 −0.071291 +0.074366 +0.076549 +0.0722 +0.0717 +0.077534 +0.077534 −0.0747 −0.075248
−0.076005 +0.06542 −0.072529
+0.01544 +0.08978 −0.07687 +0.0441 +0.04254 +0.04208
−0.074167 +0.07361 −0.0828 −0.0819
2-118
PHYSICAL AND CHEMICAL DATA
TABLE 2-117
Densities of Aqueous Solutions of Miscellaneous Organic Compounds (Concluded ) d = d w + Aps + Bp s2 + Cp3s (Cont.)
Section A Name
t, °C
Formula
Tetraethyl ammonium chloride Thiourea
C8H20ClN CH4N2S
Trichloroacetic acid
C2HCl3O2
Triethylamine hydrochloride
C6H16ClN
Trimethyl carbinol
C4H10O
Urea
CH4N2O
Urethane Valeric acid (n-)
C3H7NO2 C5H10O2
Name
Formula C4H10O C4H8O2 C4H10O
Isobutyl alcohol
C4H10O
Isobutyric acid Nicotine Picoline (α-) (β-) Pyridine Trimethyl carbinol
C4H8O2 C10H14N2 C6H7N C6H7N C5H5N C4H10O
Name
Chloral hydrate
C2H3Cl3O2
Ethyl tartrate
C7H14O6
Furfural
C5H4O2
Pyridine
C5H5N
B
C
+0.056 +0.05374 +0.04153 +0.041387 +0.056119 +0.05558 −0.041908 −0.04176 −0.044802 +0.051552 +0.053712 −0.041817 −0.05245 −0.0427
+0.07122
t, °C
Range, pw
A
B
20 25 25 0 15 26 20 25 25 25 20
0–20 0–38 0– 1.1 0–14 0–16 0–80 0–40 0–30 0–40 0–40 0–20
+0.022103 +0.021854 +0.0234 +0.022437 +0.02224 +0.021808 +0.02199 +0.022715 +0.021925 +0.021157 +0.022287
−0.04113 −0.042314 +0.0336 −0.04285 −0.04129 −0.042358 −0.04331 −0.04393 −0.04352 −0.05536 +0.05275
+0.061038 −0.0869 +0.07957 +0.07887 +0.051216 +0.072573 −0.072285 +0.051379 −0.073437
d = ds + Apw + Bp 2w + Cp 3w
ds
C
+0.061253 +0.07315 +0.0625 −0.062
dt = do + At + Bt2
ps
do
Range, °C
A
B
76.60 80.95 2.00 10.00 5.00 10.00 25.00 4.62 5.69 6.56 9.34 21.20 29.50 40.40
0.9122 0.8614 1.0094 1.0476 1.0150 1.0270 1.0665 1.0125 1.0140 1.0155 1.0055 1.0115 1.0145 1.0182
0–45 0–43 7–80 7–80 15–80 15–80 15–80 22–74 22–74 22–74 11–73 14–73 12–72 9–74
−0.038 −0.037292 −0.042597 −0.047955 −0.032103 −0.032116 −0.03401 −0.03232 −0.03221 −0.03211 −0.03171 −0.03378 −0.03463 −0.03605
−0.0527 −0.0675 −0.054313 −0.054253 −0.052544 −0.062929 −0.0523 −0.05254 −0.05268 −0.05290 −0.053615 −0.05248 −0.05235 −0.05167
Formula C3H6O C4H10O
A +0.031884 +0.022995 +0.02499 +0.025053 +0.025051 +0.046 −0.02117 −0.021286 +0.023213 +0.022718 +0.022702 +0.022728 +0.021278 +0.0334
0.8097 0.9534 0.7077 0.8170 0.8055 0.9425 1.0093 0.9404 0.9515 0.9776 0.7856 Section C
Allyl alcohol Butyl alcohol (n-)
0– 63 0– 7 0– 61 10– 30 0– 94 0– 54 0–100 0–100 0– 12 0– 51 0– 35 0– 10 0– 56 0– 3
Section B Butyl alcohol (n-) Butyric acid (n-) Ethyl ether
Range, ps
21 15 12.5 20 25 21 20 25 14.8 18 20 25 20 25
DENSITIES OF MISCELLANEOUS MATERIALS
2-119
DENSITIES OF MISCELLANEOUS MATERIALS TABLE 2-118
Approximate Specific Gravities and Densities of Miscellaneous Solids and Liquids*
Water at 4°C and normal atmospheric pressure taken as unity. For more detailed data on any material, see the section dealing with the properties of that material.
Substance
Sp. gr.
Aver. weight lb/ft 3
Sp. gr.
Aver. weight lb/ft 3
Metals, Alloys, Ores Aluminum, cast-hammered bronze Brass, cast-rolled Bronze, 7.9 to 14% Sn phosphor
2.55–2.80 7.7 8.4–8.7 7.4–8.9 8.88
165 481 534 509 554
Timber, Air-dry Apple Ash, black white Birch, sweet, yellow Cedar, white, red
0.66–0.74 0.55 0.64–0.71 0.71–0.72 0.35
44 34 42 44 22
Copper, cast-rolled ore, pyrites German silver Gold, cast-hammered coin (U.S.)
8.8–8.95 4.1–4.3 8.58 19.25–19.35 17.18–17.2
556 262 536 1205 1073
Cherry, wild red Chestnut Cypress Elm, white Fir, Douglas
0.43 0.48 0.45–0.48 0.56 0.48–0.55
27 30 29 35 32
Iridium Iron, gray cast cast, pig wrought spiegeleisen
21.78–22.42 7.03–7.13 7.2 7.6–7.9 7.5
1383 442 450 485 468
balsam Hemlock Hickory Locust Mahogany
0.40 0.45–0.50 0.74–0.80 0.67–0.77 0.56–0.85
25 29 48 45 44
ferro-silicon ore, hematite ore, limonite ore, magnetite slag
6.7–7.3 5.2 3.6–4.0 4.9–5.2 2.5–3.0
437 325 237 315 172
Maple, sugar white Oak, chestnut live red, black
0.68 0.53 0.74 0.87 0.64–0.71
43 33 46 54 42
Lead ore, galena Manganese ore, pyrolusite Mercury
11.34 7.3–7.6 7.42 3.7–4.6 13.6
710 465 475 259 849
white Pine, Norway Oregon red Southern white
0.77 0.55 0.51 0.48 0.61–0.67 0.43
48 34 32 30 38–42 27
8.97 8.9 21.5 10.4–10.6 7.83 7.80 7.70–7.73 7.2–7.5 6.4–7.0 19.22
555 537 1330 656 489 487 481 459 418 1200
Poplar Redwood, California Spruce, white, red Teak, African Indian Walnut, black Willow
0.43 0.42 0.45 0.99 0.66–0.88 0.59 0.42–0.50
27 26 28 62 48 37 28
6.9–7.2 3.9–4.2
440 253
Various Solids Cereals, oats, bulk barley, bulk corn, rye, bulk wheat, bulk Cork
49 50 75 94 112
26 39 45 48 15
0.789 0.796 1.20 1.50 1.80
0.51 0.62 0.73 0.77 0.22–0.26
Various Liquids Alcohol, ethyl (100%) methyl (100%) Acid, muriatic, 40% nitric, 91% sulfuric, 87% Chloroform Ether Lye, soda, 66% Oils, vegetable mineral, lubricants
1.500 0.736 1.70 0.91–0.94 0.88–0.94
95 46 106 58 57
Cotton, flax, hemp Fats Flour, loose pressed Glass, common
1.47–1.50 0.90–0.97 0.40–0.50 0.70–0.80 2.40–2.80
93 58 28 47 162
2.45–2.72 2.90–3.00 3.2–4.7 0.32 0.86–1.02
161 184 247 20 59
54 62.428 59.830 56 8
plate or crown crystal dint Hay and straw, bales Leather
0.861–0.867 1.0 0.9584 0.88–0.92 0.125 1.02–1.03
64
Paper Potatoes, piled Rubber, caoutchouc goods Salt, granulated, piled
0.70–1.15 0.67 0.92–0.96 1.0–2.0 0.77
58 44 59 94 48
Ashlar Masonry Bluestone Granite, syenite, gneiss Limestone Marble Sandstone
2.3–2.6 2.4–2.7 2.1–2.8 2.4–2.8 2.0–2.6
153 159 153 162 143
Saltpeter Starch Sulfur Wool
1.07 1.53 1.93–2.07 1.32
67 96 125 82
Rubble Masonry Bluestone Granite, syenite, gneiss Limestone Marble Sandstone
2.2–2.5 2.3–2.6 2.0–2.7 2.3–2.7 1.9–2.5
147 153 147 156 137
Monel metal, rolled Nickel Platinum, cast-hammered Silver, cast-hammered Steel, cold-drawn machine tool Tin, cast-hammered cassiterite Tungsten Zinc, cast-rolled blende
Substance
Turpentine Water, 4°C max. density 100°C ice snow, fresh fallen sea water
*From Marks, Mechanical Engineers’ Handbook, McGraw-Hill.
Sp. gr.
Aver. weight lb/ft 3
Dry Rubble Masonry Granite, syenite, gneiss Limestone, marble Sandstone, bluestone
1.9–2.3 1.9–2.1 1.8–1.9
130 125 110
Brick Masonry Hard brick Medium brick Soft brick Sand-lime brick
1.8–2.3 1.6–2.0 1.4–1.9 1.4–2.2
128 112 103 112
Concrete Masonry Cement, stone, sand slag, etc. cinder, etc.
2.2–2.4 1.9–2.3 1.5–1.7
144 130 100
0.64–0.72 1.5 0.85–1.00 1.4–1.9 2.08–2.25
40–45 94 53–64 103 94–135
Portland cement Slags, bank slag bank screenings machine slag slag sand
3.1–3.2 1.1–1.2 1.5–1.9 1.5 0.8–0.9
196 67–72 98–117 96 49–55
Earth, etc., Excavated Clay, dry damp plastic and gravel, dry Earth, dry, loose dry, packed moist, loose moist, packed mud, flowing mud, packed Riprap, limestone
1.0 1.76 1.6 1.2 1.5 1.3 1.6 1.7 1.8 1.3–1.4
63 110 100 76 95 78 96 108 115 80–85
1.4 1.7 1.4–1.7 1.6–1.9 1.89–2.16
90 105 90–105 100–120 126
1.28 1.44 0.96 1.00 1.12 1.00
80 90 60 65 70 65
2.1–2.8 4.50 2.7–3.2 2.55 2.5–2.6
153 281 184 159 159
Borax Chalk Clay, marl Dolomite Feldspar, orthoclase
1.7–1.8 1.8–2.8 1.8–2.6 2.9 2.5–2.7
109 143 137 181 162
Gneiss Granite Greenstone, trap Gypsum, alabaster Hornblende Limestone Marble Magnesite Phosphate rock, apatite Porphyry
2.7–2.9 2.6–2.7 2.8–3.2 2.3–2.8 3.0 2.1–2.86 2.6–2.86 3.0 3.2 2.6–2.9
175 165 187 159 187 155 170 187 200 172
Substance
Various Building Materials Ashes, cinders Cement, Portland, loose Lime, gypsum, loose Mortar, lime, set Portland cement
Riprap, sandstone Riprap, shale Sand, gravel, dry, loose gravel, dry, packed gravel, wet Excavations in Water Clay River mud Sand or gravel and clay Soil Stone riprap Asbestos Barytes Basait Bauxite Bluestone
Minerals
2-120
PHYSICAL AND CHEMICAL DATA
TABLE 2-118
Approximate Specific Gravities and Densities of Miscellaneous Solids and Liquids (Concluded )
Water at 4°C and normal atmospheric pressure taken as unity. For more detailed data on any material, see the section dealing with the properties of that material.
Substance
Sp. gr.
Minerals (Cont.) Pumice, natural Quartz, flint Sandstone Serpentine Shale, slate Soapstone, talc Syenite Stone, Quarried, Piled Basalt, granite, gneiss Greenstone, hornblende Limestone, marble, quartz Sandstone Shale NOTE:
Aver. weight lb/ft3
0.37–0.90 2.5–2.8 2.0–2.6 2.7–2.8 2.6–2.9
40 165 143 171 172
2.6–2.8 2.6–2.7
169 165
1.5 1.7 1.5 1.3 1.5
96 107 95 82 92
Substance
Aver. weight lb/ft 3
Sp. gr.
Bituminous Substances Asphaltum Coal, anthracite bituminous lignite peat, turf, dry
1.1–1.5 1.4–1.8 1.2–1.5 1.1–1.4 0.65–0.85
81 97 84 78 47
charcoal, pine charcoal, oak coke Graphite Paraffin
0.28–0.44 0.47–0.57 1.0–1.4 1.64–2.7 0.87–0.91
23 33 75 135 56
Substance
Sp. gr.
Aver. weight lb/ft3
Bituminous Substances (Cont.) Petroleum refined (kerosene) benzine gasoline Pitch Tar, bituminous
0.87 0.78–0.82 0.73–0.75 0.70–0.75 1.07–1.15 1.20
54 50 46 45 69 75
Coal and Coke, Piled Coal, anthracite bituminous, lignite peat, turf charcoal coke
0.75–0.93 0.64–0.87 0.32–0.42 0.16–0.23 0.37–0.51
47–58 40–54 20–26 10–14 23–32
To convert pounds per cubic foot to kilograms per cubic meter, multiply by 16.02. °F = 9⁄ 5 °C + 32.
TABLE 2-119
Density (kg/m3) of Selected Elements as a Function of Temperature Element symbol
Temperature, K*
Al
Be†
Cr
Cu
Au
Ir
Fe
Pb
Mo
Ni
Pt
Ag
Zn†
50 100 150 200 250
2736 2732 2726 2719 2710
3650 3640 3630 3620 3610
7160 7155 7150 7145 7140
9019 9009 8992 8973 8951
19,490 19,460 19,420 19,380 19,340
22,600 22,580 22,560 22,540 22,520
7910 7900 7890 7880 7870
11,570 11,520 11,470 11,430 11,380
10,260 10,260 10,250 10,250 10,250
8960 8950 8940 8930 8910
21,570 21,550 21,530 21,500 21,470
10,620 10,600 10,575 10,550 10,520
7280 7260 7230 7200 7170
300 400 500 600 800
2701 2681 2661 2639 2591
3600 3580 3555 3530
7135 7120 7110 7080 7040
8930 8885 8837 8787 8686
19,300 19,210 19,130 19,040 18,860
22,500 22,450 22,410 22,360 22,250
7860 7830 7800 7760 7690
11,330 11,230 11,130 11,010 10,430
10,240 10,220 10,210 10,190 10,160
8900 8860 8820 8780 8690
21,450 21,380 21,330 21,270 21,140
10,490 10,430 10,360 10,300 10,160
7135 7070 7000 6935 6430
1000 1200 1400 1600 1800
2365 2305 2255
7000 6945 6890 6760 6700
8568 8458 7920 7750 7600
18,660 18,440 17,230 16,950
22,140 22,030 21,920 21,790 21,660
7650 7620 7520 7420 7320
10,190 9,940
10,120 10,080 10,040 10,000 9,950
8610 8510 8410 8320 7690
21,010 20,870 20,720 20,570 20,400
10,010 9,850 9,170 8,980
6260
21,510
7030
9,900
7450
20,220
2000
7460
NOTE:
Above the horizontal line the condensed phase is solid; below the line, it is liquid. *°R = 9⁄ 5 K. †Polycrystalline form tabulated. Similar tables for an additional 45 elements appear in the Handbook of Heat Transfer, 2d ed., McGraw-Hill, New York, 1984.
SOLUBILITIES UNITS CONVERSIONS For this subsection, the following units conversions are applicable: °F = 9⁄5 °C + 32. To convert cubic centimeters to cubic feet, multiply by 3.532 × 10−5.
To convert millimeters of mercury to pounds-force per square inch, multiply by 0.01934. To convert grams per liter to pounds per cubic foot, multiply by 6.243 × 10−2.
TABLE 2-120
Solubilities of Inorganic Compounds in Water at Various Temperatures*
This table shows the amount of anhydrous substance that is soluble in 100 g of water at the temperature in degrees Celsius as indicated; when the name is followed by †, the value is expressed in grams of substance in 100 cm3 of saturated solution. Solid phase gives the hydrated form in equilibrium with the saturated solution. Solid phase
Substance
Formula
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Aluminum chloride sulfate Ammonium aluminum sulfate bicarbonate bromide chloride chloroplatinate chromate chromium sulfate dichromate dihydrogen phosphite hydrogen phosphate iodide magnesium phosphate manganese phosphate nitrate oxalate perchlorate† persulfate sulfate thiocyanate vanadate (meta) Antimonious fluoride sulfide Arsenic oxide Arsenious sulfide
AlCl3 Al2(SO4)3 (NH4)2Al2(SO4)4 NH4HCO3 NH4Br NH4Cl (NH4)2PtCl6 (NH4)2CrO4 (NH4)2Cr2(SO4)4 (NH4)2Cr2O7 NH4H2PO3 (NH4)2HPO4 NH4I NH4MgPO4 NH4MnPO4 NH4NO3 (NH4)2C2O4 NH4ClO4† (NH4)2S2O8 (NH4)2SO4 NH4CNS NH4VO3 SbF3 Sb2S3 As2O5 As2S3
27 28 29
Barium acetate acetate carbonate
Ba(C2H3O2)2 Ba(C2H3O2)2 BaCO3
3H2O 1H2O
30 31 32 33 34 35 36 37 38
chlorate chloride chromate hydroxide iodide iodide nitrate nitrite oxalate
Ba(ClO3)2 BaCl2 BaCrO4 Ba(OH)2 BaI2 BaI2 Ba(NO3)2 Ba(NO2)2 BaC2O4
1H2O 2H2O
3H2O
39 40 41 42 43 44 45 46 47 48 49 50 51
perchlorate sulfate Beryllium sulfate sulfate sulfate Boric acid Boron oxide Bromine Cadmium chloride chloride chloride cyanide hydroxide
Ba(ClO4)2 BaSO4 BeSO4 BeSO4 BeSO4 H3BO3 B2O3 Br2 CdCl2 CdCl2 CdCl2 Cd(CN)2 Cd(OH)2
52 53 54
sulfate Calcium acetate acetate
CdSO4 Ca(C2H3O2)2 Ca(C2H3O2)2
6H2O 18H2O 24H2O
0°C 31.2 2.1 11.9 60.6 29.4
10°C 33.5 4.99 15.8 68 33.3 0.7
10.7825°
24II2O 171 6H2O 7H2O 1H2O
154.2 0.023 118.3 2.2 11.56 58.2 70.6 119.8
163.2
3.1 73.0 144
384.7
8H2O 6H2O 2H2O
20°C 69.8615° 36.4 7.74 21 75.5 37.2
59.5 5.17×10−5 at 18° 59
62.1 63 0.00168°
20.34 31.6 0.0002 1.67 170.2 5.0
1H2O
26.95 33.3 0.00028 2.48 185.7 7.0 0.00168°
205.8 1.15 × 10−4
2.0 × 10−4
19014.5° 13115 172.3 0.052 0 192 4.4 20.85 75.4 170 0.48 444.7 0.00017518° 65.8
4H2O 2aH2O 1H2O
2H2O 1H2O
3.57 1.5 3.4 125.1
50°C
60°C
70°C
80°C
90°C
100°C
46.1 14.88
52.2 20.10
59.2 26.70
66.1
73.0
80.8
89.0 109.796°
91.1 45.8
99.2 50.4
107.8 55.2
116.8 60.2
126 65.6
135.6 71.3
145.6 77.3 1.25
190.5 0.036 0 297.0 8.0 30.58
199.6 0.030
208.9 0.040 0 421.0
218.7 0.016 0.005 499.0
228.8 0.019 0.007 580.0
40.4 47.17 26031° 181.4 241.8 5.9 78.0 207.7 0.84 563.6
81.0
69.5
71.2
1.32
344.0 10.3
39.05
740.0
48.19
88.0 1.78
250.3 871.0 57.01
95.3
103.3
75.1
76.7
3.05 73.0
0.002218° 33.80 35.7 0.00037 3.89 203.1 9.2 67.5 0.002218° 289.1 2.4 × 10−4
5.04 2.2 3.20
75 0.0024 at 24.2° 41.70 38.2 0.00046 5.59 219.6 11.6
79
77
74
74
49.61 40.7
43.6
66.81 46.4
49.4
8.22
13.12
20.94
231.9 14.2
17.1
358.7
426.3
75 84.84 52.4
104.9 58.8
101.4
247.3 20.3
261.0 27.0 205.8
271.7 34.2 300
0.0024 at 24.2° 2.85 × 10−4 52 43.78 6.60
495.2
562.3
60.67 46.74 8.72 4.0
62 11.54
14.81 6.2
16.73
84.76 23.75 9.5
83 98 30.38
100 110 40.25 15.7
3.13 132.1
135.1
76.48 37.4
40.4 10.94 27 83.2 41.4
40°C
71
6H2O 4H2O 2H2O 2.66 1.1 4.22 97.59 90.01
30°C
76.00 36.0
134.5 1.715° 76.60 34.7
135.3
136.5
140.4
147.0
2.6 × 10−4 at 25° 33.8
78.54 33.2
83.68 32.7
63.13
60.77
31.1
29.7
33.5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
*By N. A. Lange. Abridged from “Table of Solubilities of Inorganic Compounds in Water at Various Temperatures” in Lange, Handbook of Chemistry, 10th ed., McGraw-Hill, New York, 1961. For tables of the solubility of gases in water at various temperatures, Atack (Handbook of Chemical Data, Reinhold, New York, 1957) gives values at closer temperature intervals, usually 1 or 5°C, than are tabulated here. For materials marked by ‡, additional data are given in tables subsequent to this one. For the solubility of various hydrocarbons in water at high pressures see J. Chem. Eng. Data, 4, 212 (1959).
2-121
TABLE 2-120
Solubilities of Inorganic Compounds in Water at Various Temperatures (Continued )
Substance
Formula
1 2 3 4 5 6 7 8 9 10 11
Calcium bicarbonate chloride chloride fluoride hydroxide nitrate nitrate nitrate nitrite nitrite oxalate
Ca(HCO3)2 CaCl2 CaCl2 CaF2 Ca(OH)2 Ca(NO3)2 Ca(NO3)2 Ca(NO3)2 Ca(NO2)2 Ca(NO2)2 CaC2O4
12 13 14 15 16 17 18 19 20 21 22 23 24
sulfate Carbon dioxide, 760 mm ‡ monoxide, 760 mm ‡ Cesium chloride nitrate sulfate Chlorine, 760 mm ‡ Chromic anhydride Cuprio chloride nitrate nitrate sulfate sulfide
CaSO4 CO2 CO CsCl CsNO3 Cs2SO4 Cl2 CrO3 CuCl2 Cu(NO3)2 Cu(NO3)2 CuSO4 CuS
25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62
Cuprous chloride Ferric chloride Ferrous chloride chloride nitrate sulfate sulfate Hydrobromic acid, 760 mm Hydrochloric acid, 760 mm Iodine Lead acetate bromide carbonate chloride chromate fluoride nitrate sulfate Magnesium bromide chloride hydroxide nitrate sulfate sulfate sulfate Manganous sulfate sulfate sulfate sulfate Mercurous chloride Molybdic oxide Nickel chloride nitrate nitrate sulfate sulfate Nitric oxide, 760 mm Nitrous oxide
CuCl FeCl3 FeCl2 FeCl2 Fe(NO3)2 FeSO4 FeSO4 HBr HCl I2 Pb(C2H3O2)2 PbBr2 PbCO3 PbCl2 PbCrO4 PbF2 Pb(NO3)2 PbSO4 MgBr2 MgCl2 Mg(OH)2 Mg(NO3)2 MgSO4 MgSO4 MgSO4 MnSO4 MnSO4 MnSO4 MnSO4 HgCl MoO3 NiCl2 Ni(NO3)2 Ni(NO3)2 NiSO4 NiSO4 NO N2O
2-122
Solid phase 6H2O 2H2O 4H2O 3H2O
0°C 16.15 59.5 0.185 102.0
10°C 65.0 0.176 115.3
20°C 16.60 74.5 0.001618° 0.165 129.3
30°C
40°C
50°C
17.05
60°C
70°C
17.50
80°C
90°C
17.95
18.40
102 0.001726° 0.153 152.6
136.8 0.141 195.9 237.5
0.128
0.116
141.7 0.106
147.0 0.094
152.7 0.085
62.07
2H2O
0.1759 0.3346 0.0044 161.4 9.33 167.1 1.46 164.9 70.7 81.8
2H2O 6H2O 3H2O 5H2O
6.7 × 10−4 at 13° 0.1928 0.2318 0.0035 174.7 14.9 173.1 0.980 73.76 95.28 17.4
74.4
81.9 64.5
6.8 × 10−4 at 25° 0.1688 0.0028 186.5 23.0 178.7 0.716 77.0 125.1 20.7 3.3 × 10−5 at 18° 1.5225° 91.8
9.5 × 10−4 at 50° 0.2090 0.1257 0.0024 197.3 33.9 184.1 0.562
132.6
80.34
14 × 10−4 at 95° 0.2097 0.0973 0.0021 208.0 47.2 189.9 0.451 174.0 83.8
25
159.8 28.5
33.3
77.3
315.1 82.5
73.0
0.0761 0.0018 218.5 64.4 194.9 0.386 182.1 87.44
0.2047 0.0576 0.0015 229.7 83.8 199.9 0.324
151.9
363.6 244.8
0.1966 0.0013 239.5 107.0 205.0 0.274
0.0010 250.0 134.0 210.3 0.219
91.2
99.2
178.8 40
207.8 55
88.7
525.8 100
0.0006 260.1 163.0 214.9 0.125 217.5
71.02 15.65
20.51
83.8 26.5
535.7
210.3
0.029 0.4554 0.6728
6H2O 6H2O 6H2O 7H2O 6H2O 1H2O 7H2O 5H2O 4H2O 1H2O
38.8 0.0028 91.0 52.8
0.060 48.3 0.0035 94.5 53.5
40.2
48.6
0.85 0.00011 0.99 7 × 10−6 0.064 56.5 0.0041 96.5 54.5 0.000918°
40.8 53.23
30.9 42.2 60.01 59.5
53.9 79.58
59.5
27.22
32
1.53
1.94
2.36
3.34
4.75
1.20
1.45
1.70
1.98
2.62
3.34
0.068 66 0.0049 99.2
40.8 45.3
62.9 64.5
67.76 66.44 0.264 68.9
75 0.0056 101.6 57.5
130 56.1
85
95
104.1
107.5 61.0
0.00757 0.1705
115
68.8 0.0007 0.476 73.3 122.2
38.8
50.4
53.5
59.5
64.2 62.9
69.0
74.0 68.3
72.6 58.17
55.0
52.0
48.0
42.5
34.0
0.687 78.3
1.206 82.2
2.055 85.2
113.7 66.0
84.74 45.6
120.2 73.0 137.0
163.1 0.00984
37.3
63.3 0.056
35.5 44.5
0.0002 0.138 64.2 96.31
43.6
67.3 0.04 55.0425° 1.15
171.5 59.6 0.078
66.55
0.00014 2H2O 6H2O 6H2O 3H2O 7H2O 6H2O
32.9
198
3H2O
105.8
165.6 50.9
221.2 82.3
0.1619 0 0 270.5 197.0 220.3 0 206.8 107.9 75.4
105.3 6H2O 7H2O 1H2O
0.077
76.68
14.3
4H2O
159
281.5 358.7
4H2O 2H2O
100°C
2.106 87.6
169.1
235.1
42.46 0.00618 0.1211
0.00517
0.00440
50.15 0.00376
54.80 0.00324
59.44 0.00267
63.17 0.00199
0.00114
76.7 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
Potassium acetate acetate alum bicarbonate bisulfate bitartrate carbonate chlorate chloride chromate dichromate ferricyanide hydroxide hydroxide nitrate nitrite perchlorate permanganate persulfate† sulfate thiocyanate Silver cyanide nitrate sulfate Sodium acetate acetate bicarbonate carbonate carbonate chlorate chloride chromate chromate chromate dichromate dichromate dihydrogen phosphate dihydrogen phosphate dihydrogen phosphate hydrogen arsenate hydrogen phosphate hydrogen phosphate hydrogen phosphate hydrogen phosphate hydroxide hydroxide hydroxide hydroxide nitrate nitrite oxalate phosphate, tripyrophosphate sulfate sulfate sulfate sulfide sulfide sulfide sulfite sulfite tetraborate tetraborate vanadate (meta)
KC2H3O2 KC2H3O2 K2SO4⋅Al2(SO4)3 KHCO3 KHSO4 KHC4H4O6 K2CO3 KClO3 KCl K2CrO4 K2Cr2O7 K3Fe(CN)6 KOH KOH KNO3 KNO2 KClO4 KMnO4 K2S2O8† K2SO4 KCNS AgCN AgNO3 Ag2SO4 NaC2H3O2 NaC2H3O2 NaHCO3 Na2CO3 Na2CO3 NaClO8 NaCl Na2CrO4 Na2CrO4 Na2CrO4 Na2Cr2O7 Na2Cr2O7 NaH2PO4 NaH2PO4 NaH2PO4 Na2HAsO4 Na2HPO4 Na2HPO4 Na2HPO4 Na2HPO4 NaOH NaOH NaOH NaOH NaNO3 NaNO2 Na2C2O4 Na3PO4 Na4P2O7 Na2SO4 Na2SO4 Na2SO4 Na2S Na2S Na2S Na2SO3 Na2SO3 Na2B4O7 Na2B4O7 NaVO8
1aH2O aH2O 24H2O
2H2O
2H2O 1H2O
†
3H2O 10H2O 1H2O 10H2O 4H2O
216.7 3.0 22.4 36.3 0.32 105.5 3.3 27.6 58.2 5 31 97
233.9 4.0 27.7 0.40 108 5 31.0 60.0 7 36 103
255.6 5.9 33.2 51.4 0.53 110.5 7.4 34.0 61.7 12 43 112
13.3 278.8 0.75 2.83 1.62 7.35 177.0
20.9
122 0.573 36.3 119 6.9 7
170 0.695 40.8 121 8.15 12.5
31.6 298.4 1.80 6.4 4.49 11.11 217.5 2.2 × 10−5 222 0.796 46.5 123.5 9.6 21.5
79 35.7 31.70
89 35.8 50.17
101 36.0 88.7
1.05 4.4 2.60 9.22
283.8 8.39 39.1 0.90 113.7 10.5 37.0 63.4 20 50 126 45.8 2.6 9.0 7.19 12.97 300 0.888 54.5 126 11.1 38.8 50.5 113 36.3 88.7
2H2O
163.0
2H2O 1H2O
57.9
323.3 11.70 45.4 67.3 1.32 116.9 14 40.0 65.2 26 60 63.9 334.9 4.4 12.56 9.89 14.76
337.3 17.00
350 24.75 60.0
1.83 121.2 19.3 42.6 66.8 34
2.46 126.8 24.5 45.5 68.6 43 66
140 85.5
110.0
364.8 40.0
133.1 48.3 70.4 52
138
380.1 71.0 4.6 139.8 38.5 51.1 72.1 61
169
396.3 109.0
147.5 54.0 73.9 70
202
6.5 16.89
9 22.2
11.8
14.8
18
16.50
18.17
19.75
21.4
22.8
376 0.979 65.5 129.5 12.7
455 1.08 83 134 14.45
525 1.15 139 139.5 16.4
1.22
669 1.30
48.5 126 36.6
140 37.0
46.4 155 37.3
104
114.6
95.96
177.8
244.8
146
153
172 37.8
45.8 189 38.4
123.0 316.7
124.8 376.2
1.36 161
39.0
121.6 6.95 155.7 57 56.7 75.6 80 82.6104 178 246 412.8 21.8 24.1 952 1.41 170 45.5 230 39.8 125.9 426.3
12H2O 12H2O 7H2O 2H2O
69.9
85.2
106.5
138.2
15.5 3.6
26.5 7.7
37 20.8
47
158.6 7.3 1.67
179.3 65
190.3
207.3 85
225.3
82.9
88.1
92.4
102.9
246.6
51.8 80.2
102.2 4H2O 3aH2O 1H2O
42 51.5 109
119
129
145
174
104 98.4
114 104.1
124
313
12H2O 10H2O 10H2O 7H2O 9H2O 5aH2O 6H2O 7H2O
73 72.1
80 78.0
1.5 3.16 5.0 19.5
13.9
96 91.6
4.1 3.95 9.0 30
88 84.5 3.7 11 6.23 19.4 44
15.42
18.8
22.5
20
26.9
36
20 9.95 40.8
31 13.50
43 17.45
55 21.83
81 30.04
347 180 163.2 6.33 108 40.26
48.8 28.5
46.7
45.3
43.7
42.5
39.82 36.4
42.69 39.1
28.2 10.5
28.8 20.3
28 10H2O 5H2O 2H2O
1.3
1.6
2.7 15.325°
3.9
148 132.6
45.73 43.31
68.4
59.23 57.28
28.3 24.4
30.2
51.40 49.14
31.5
41
52.5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
2-123
TABLE 2-120
Solubilities of Inorganic Compounds in Water at Various Temperatures (Concluded )
Substance 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Sodium vanadate (meta) Stannous chloride sulfate Strontium acetate acetate chloride chloride nitrate nitrate nitrate sulfate Sulfur dioxide, 760 mm † Thallium sulfate Thorium sulfate sulfate sulfate sulfate Zinc chlorate chlorate nitrate nitrate sulfate sulfate sulfate
2-124
Formula NaVO3 SnCl2 SnSO4 Sr(C2H3O2)2 Sr(C2H3O2)2 SrCl2 SrCl2 Sr(NO3)2 Sr(NO3)2 Sr(NO3)2 SrSO4 SO2 Tl2SO4 Th(SO4)2 Th(SO4)2 Th(SO4)2 Th(SO4)2 ZnClO3 ZnClO3 Zn(NO3)2 Zn(NO3)2 ZnSO4 ZnSO4 ZnSO4
Solid phase
0°C
10°C
9H2O 8H2O 6H2O 4H2O 6H2O 4H2O 6H2O 3H2O 7H2O 6H2O 1H2O
36.9 43.5
43.61 42.95 47.7
41.6 52.9
52.7 40.1
64.0 70.5
0.0113 22.83 2.70 0.74 1.0 1.50
0.0114 11.29 4.87 1.38 1.62 1.90
145.0
30°C
21.1025° 269.815° 19
83.9 4H2O aH2O 6H2O 2H2O 1H2O 4H2O
20°C
16.21 3.70 0.98 1.25
40°C
50°C
26.23
60°C
70°C
80°C
32.97
36.9
38.875°
36.24
36.10
85.9
90.5
90°C
100°C
18 39.5 58.7
88.6 0.0114 7.81 6.16 1.995
65.3
37.35 72.4
81.8
83.8
97.2
90.1
2.998
5.41
4.5 9.21 5.22
4.04
2.54
2.45
36.4 130.4
100.8 139
93.8
96
98
100
10.92
12.74
14.61
16.53
18.45
6.64 1.63
1.09
86.6
83.7
80.8
152.5 200.3 118.3
94.78
209.2
223.2
273.1
206.9 41.9
47
54.4 70.1
76.8
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
SOLUBILITIES The H in solubility tables (2-121 to 2-144) is the proportionality constant for the expression of Henry’s law, p = Hx, where x = mole fraction of the solute in the liquid phase; p = partial pressure of the solute in the gas phase, expressed in atmospheres; and H = a proportionality constant expressed in units of atmospheres of solute pressure in the gas phase per unit concentration of the solute in the liquid phase. (The unit of concentration of the solute in the liquid phase is moles solute per mole solution.)
TABLE 2-124
Ammonia (NH3)—Low Pressures
Weight NH3 per 100 weights H2O
0.105
0.244
0.32
0.38
0.576
0.751
1.02
Partial pressure NH3, mm. Hg, at 25°C
0.791
1.83
2.41
2.89
4.41
5.80
7.96
Weight NH3 per 100 weights H2O
1.31
1.53
1.71
1.98
2.11
2.58
2.75
10.31
11.91
13.46
15.75
16.94
20.86
22.38
Partial pressure NH3, mm. Hg, at 25°C TABLE 2-121
Acetylene (C2H2)
t, °C
0
5
10
15
20
25
30
10−3 × H*
0.72
0.84
0.96
1.08
1.21
1.33
1.46
2-125
“Landolt-Börnstein Physikalische-chemische Tabellen,” Eg. I, p. 303, 1927. Phase-equilibrium data for the binary system NH3-H2O are given by Clifford and Hunter, J. Phys. Chem., 37, 101 (1933).
International Critical Tables, vol. 3, p. 260, McGraw-Hill, 1928. *H. See footnote for Table 2-122. TABLE 2-125
TABLE 2-122
Carbon Dioxide (CO2)
Weight of CO2 per 100 weights of H2O* Total pressure, atm 12°C 18°C 25°C 31.04°C 35°C 40°C 50°C 75°C 100°C
Air
t, °C
0
5
10
15
20
25
30
35
10−4 × H*
4.32
4.88
5.49
6.07
6.64
7.20
7.71
8.23
t, °C
40
45
50
60
70
80
90
100
10−4 × H*
8.70
9.11
9.46
10.1
10.5
10.7
10.8
10.7
International Critical Tables, vol. 3, p. 257. *H is calculated from the absorption coefficients of O2 and N2, taking into consideration the correction for constant argon content.
25 50 75 100 150 200 300 400 500 700
7.03 7.18 7.27 7.59
3.86 6.33 6.69 6.72 7.07
7.86 8.12
7.35 7.77
5.38 6.17 6.28
7.54
2.80 4.77 5.80 5.97 6.25 6.48
2.56 4.39 5.51 5.76 6.03 6.29
2.30 4.02 5.10 5.50 5.81 6.28
7.27 7.65
7.06 7.51
6.89 7.26
1.92 3.41 4.45 5.07 5.47 5.76 6.20 6.58
1.35 2.49 3.37 4.07 4.86 5.27 5.83 6.30
1.06 2.01 2.82 3.49 4.49 5.08 5.84 6.40
7.58
7.43
7.61
*In the original, concentration is expressed in cubic centimeters of CO2 (reduced to 0°C and 1 atm) dissolved in 1 g of water. TABLE 2-123 Weight NH3 per 100 weights H2O 100 90 80 70 60 50 40 30 25 20 15 10 7.5 5 4 3 2.5 2 1.6 1.2 1.0 0.5
Ammonia (NH3) Partial pressure of NH3, mm. Hg 0°C
947 785 636 500 380 275 190 119 89.5 64 42.7 25.1 17.7 11.2
10°C
20°C
987 1450 780 1170 600 945 439 686 301 470 190 298 144 227 103.5 166 70.1 114 41.8 69.6 29.9 50.0 19.1 31.7 16.1 24.9 11.3 18.2 15.0 12.0
*Extrapolated values.
25°C 30°C
23.5 19.4 15.3 12.0 9.1 7.4 3.4
719 454 352 260 179 110 79.7 51.0 40.1 29.6 24.4 19.3 15.3 11.5
40°C
3300 2760 2130 1520 1065 692 534 395 273 167 120 76.5 60.8 45 (37.6)* (30.0) (24.1) (18.3) (15.4)
50°C 60°C
825 596 405 247 179 115 91.1 67.1 (55.7) (44.5) (35.5) (26.7) (22.2)
834 583 361 261 165 129.2 94.3 77.0 61.0 48.7 36.3 30.2
TABLE 2-126
Carbon Monoxide (CO) 10−4 × H
Partial pressure of CO, mm Hg
17.7°C
19.0°C
900 2000 3000 4000 5000 6000 7000 8000
4.77 4.77 4.77 4.78 4.80 4.82 4.86 4.88
4.88 4.91 4.93 4.95 4.97 4.98 5.02 5.08
International Critical Tables, vol. 3, p. 260.
TABLE 2-127
Carbonyl Sulfide (COS)
t °C
0
5
10
15
20
25
30
10−3 × H
0.92
1.17
1.48
1.82
2.19
2.59
3.04
International Critical Tables, vol. 3, p. 261.
2-126
PHYSICAL AND CHEMICAL DATA
TABLE 2-128 Partial pressure of Cl2, mm Hg
Chlorine (Cl2)
TABLE 2-129
Solubility, g of Cl2 per liter 10°C
20°C
30°C
40°C
50°C
5 10 30 50 100
0.488 .679 1.221 1.717 2.79
0.451 .603 1.024 1.354 2.08
0.438 .575 .937 1.210 1.773
0.424 .553 .873 1.106 1.573
0.412 .532 .821 1.025 1.424
0.398 .512 .781 .962 1.313
150 200 250 300 350
3.81 4.78 5.71
2.73 3.35 3.95 4.54 5.13
2.27 2.74 3.19 3.63 4.06
1.966 2.34 2.69 3.03 3.35
1.754 2.05 2.34 2.61 2.86
1.599 1.856 2.09 2.31 2.53
400 450 500 550 600
5.71 6.26 6.85 7.39 7.97
4.48 4.88 5.29 5.71 6.12
3.69 3.98 4.30 4.60 4.91
3.11 3.36 3.61 3.84 4.08
2.74 2.94 3.14 3.33 3.52
650 700 750 800 900
8.52 9.09 9.65 10.21
6.52 6.90 7.29 7.69 8.46
5.21 5.50 5.80 6.08 6.68
4.32 4.54 4.77 4.99 5.44
3.71 3.89 4.07 4.27 4.62
9.27 10.84 13.23 17.07 21.0
7.27 8.42 10.14 13.02 15.84
5.89 6.81 8.05 10.22 12.32
4.97 5.67 6.70 8.38 10.03
18.73 21.7 24.7 27.7 30.8
14.47 16.62 18.84 20.7 23.3
11.70 13.38 15.04 16.75 18.46
Cl2.8H2O2 separates
3000 3500 4000 4500 5000 Partial pressure of Cl2, mm Hg
70°C
80°C
90°C
1 3 5 7 10 11 12 13 14 15 16
100°C
110°C
5 10 30 50 100
0.383 .492 .743 .912 1.228
0.369 .470 .704 .863 1.149
0.351 .447 .671 .815 1.085
0.339 .431 .642 .781 1.034
0.326 .415 .627 .747 .987
0.316 .402 .598 .722 .950
150 200 250 300 350
1.482 1.706 1.914 2.10 2.28
1.382 1.580 1.764 1.932 2.10
1.294 1.479 1.642 1.793 1.940
1.227 1.396 1.553 1.700 1.831
1.174 1.333 1.480 1.610 1.736
1.137 1.276 1.413 1.542 1.661
400 450 500 550 600
2.47 2.64 2.80 2.97 3.13
2.25 2.41 2.55 2.69 2.83
2.08 2.22 2.35 2.47 2.59
1.965 2.09 2.21 2.32 2.43
1.854 1.972 2.08 2.19 2.29
1.773 1.880 1.986 2.09 2.19
650 700 750 800 900
3.29 3.44 3.59 3.75 4.04
2.97 3.10 3.23 3.37 3.63
2.72 2.84 2.96 3.08 3.30
2.55 2.66 2.76 2.87 3.08
2.41 2.50 2.60 2.69 2.89
2.28 2.37 2.47 2.56 2.74
1000 1200 1500 2000 2500
4.36 4.92 5.76 7.14 8.48
3.88 4.37 5.09 6.26 7.40
3.53 3.95 4.58 5.63 6.61
3.28 3.67 4.23 5.17 6.05
3.07 3.43 3.95 4.78 5.59
2.91 3.25 3.74 4.49 5.25
3000 3500 4000 4500 5000
9.83 11.22 12.54 13.88 15.26
8.52 9.65 10.76 11.91 13.01
7.54 8.53 9.52 10.46 11.42
6.92 7.79 8.65 9.49 10.35
6.38 7.16 7.94 8.72 9.48
5.97 6.72 7.42 8.13 8.84
0°C
5°C
10°C
15°C
20°C
30°C
40°C
2.00 6.00 10.0 14.0 20.0
1.50 4.7 7.8 10.9 15.5 17.0 18.6 20.3
1.25 3.85 6.30 8.95 12.8 14.0 15.3 16.6 18.0 19.2 20.3
1.00 3.20 5.25 7.35 10.5 11.7 12.8 13.8 14.9 16.0 17.0
0.90 2.70 4.30 6.15 8.80 9.70 10.55 11.5 12.3 13.2 14.2
0.60 1.95 3.20 4.40 6.30 7.00 7.50 8.20 8.80 9.50 10.1
0.46 1.30 2.25 3.20 4.50 5.00 5.45 5.85 6.35 6.80 7.20
Ishi, Chem. Eng. (Japan), 22, 153 (1958).
TABLE 2-130
Ethane (C2H6)
t, °C
0
5
10
15
20
25
30
35
10−4 × H
1.26
1.55
1.89
2.26
2.63
3.02
3.42
3.83
t, °C
40
45
50
60
70
80
90
100
10−4 × H
4.23
4.63
5.00
5.65
6.23
6.61
6.87
6.92
International Critical Tables, vol. 3, p. 261.
TABLE 2-131
Solubility, g of Cl2 per liter 60°C
Weight of ClO2, grams per liter of solution
Vol % of ClO2 in gas phase
0°C
1000 1200 1500 2000 2500
Chlorine Dioxide (ClO2)
Ethylene (C2H4)
t, °C
0
5
10
15
20
25
30
10−3 × H
5.52
6.53
7.68
8.95
10.2
11.4
12.7
International Critical Tables, vol. 3, p. 260.
TABLE 2-132
Helium (He)
t, °C
0
10
20
30
40
50
10−4 × H
12.9
12.6
12.5
12.4
12.1
11.5
See also Pray, Schweickert, and Minnich, Ind. Eng. Chem., 44, 1146 (1952).
TABLE 2-133
Hydrogen (H2)—Temperature
t, °C
0
5
10
15
20
25
30
35
10−4 × H
5.79
6.08
6.36
6.61
6.83
7.07
7.29
7.42
t, °C
40
45
50
60
70
80
90
100
10−4 × H
7.51
7.60
7.65
7.65
7.61
7.55
7.51
7.45
“International Critical Tables,” vol. 3, p. 256. See also Pray, Schweickert, and Minnich, Ind. Eng. Chem., 44, 1146 (1952).
SOLUBILITIES TABLE 2-134
Hydrogen (H2)—Pressure
TABLE 2-137
10−4 × H
Partial pressure H2, mm Hg
19.5°C
900 1100 2000 3000 4000 5000 6000 7000 8200 8250
23°C
7.42 7.75 7.76 7.77 7.81 7.89 8.00 8.16 8.41
7.42 7.43 7.47 7.56 7.70 7.87
2-127
Methane (CH4)
t, °C
0
5
10
15
20
25
30
35
10−4 × H
2.24
2.59
2.97
3.37
3.76
4.13
4.49
4.86
t, °C
40
45
50
60
70
80
90
100
10−4 × H
5.20
5.51
5.77
6.26
6.66
6.82
6.92
7.01
International Critical Tables, vol. 3, p. 260.
TABLE 2-138
8.17
International Critical Tables, vol. 3, p. 256.
Nitrogen (N2)—Temperature*
t, °C
0
5
10
15
20
25
30
35
10−4 × H
5.29
5.97
6.68
7.38
8.04
8.65
9.24
9.85
t, °C
40
45
50
60
70
80
90
100
10−4 × H
10.4
10.9
11.3
12.0
12.5
12.6
12.6
12.6
“International Critical Tables,” vol. 3, p. 256. See also Pray, Schweickert, and Minnich, Ind. Eng. Chem., 44, 1146 (1952). *Atmospheric nitrogen = 98.815 vol. % N2 + 1.185 vol. % A. TABLE 2-135
Hydrogen Chloride (HCl)
Weights of HCl per 100 weights of H2O
TABLE 2-139
Partial pressure of HCl, mm Hg 0°C
78.6 66.7 56.3 47.0 38.9 31.6 25.0 19.05 13.64 8.70 4.17 2.04
10°C
510 130 29.0 5.7 1.0 0.175 .0316 .0056 .00099 .000118 .000018
Weights of HCl per 100 weights of H2O
20°C
840 233 56.4 11.8 2.27 0.43 .084 .016 .00305 .000583 .000069 .0000117
30°C
399 105.5 23.5 4.90 1.00 0.205 .0428 .0088 .00178 .00024 .000044
Nitrogen (N2)—Pressure
627 188 44.5 9.90 2.17 0.48 .106 .0234 .00515 .00077 .000151
900 2000 3000 4000 5000 6000 7000 8100 8200
80°C
535 141 35.7 8.9 2.21 0.55 .136 .0344 .0064 .00140
110°C
623 188 54.5 15.6 4.66 1.34 0.39 .095 .0245
760 253 83 28 9.3 3.10 0.93 .280
Oxygen (O2)—Temperature
t, °C
0
5
10
15
20
25
30
35
10−4 × H
2.55
2.91
3.27
3.64
4.01
4.38
4.75
5.07
t, °C
40
45
50
60
70
80
90
100
10−4 × H
5.35
5.63
5.88
6.29
6.63
6.87
6.99
7.01
International Critical Tables, vol. 3, p. 257. Pray, Schweickert, and Minnich [Ind. Eng. Chem., 44, 1146 (1952)] give H = 4.46 × 10−4 at 25°C and other values up to 343°C.
Hydrogen Sulfide (H2S)
t, °C
0
5
10
15
20
25
30
35
10−2 × H
2.68
3.15
3.67
4.23
4.83
5.45
6.09
6.76
t, °C
40
45
50
60
70
80
90
100
7.45
9.08 9.15 9.25 9.38 9.49 9.62 9.75 9.91
TABLE 2-141
Oxygen (O2)—Pressure
Partial pressure of O2, mm Hg
10 × H
24.9°C
8.24 8.32 8.41 8.49 8.59 8.74 8.86 9.04
See also Goodman and Krase [Ind. Eng. Chem., 23, 401 (1931)] for values up to 169°C and 300 atm.
TABLE 2-140
Enthalpy and phase-equilibrium data for the binary system HCl-H2O are given by Van Nuys, Trans. Am. Inst. Chem. Engrs., 39, 663 (1943).
−2
19.4°C
Partial pressure of HCl, mm Hg 50°C
78.6 66.7 56.3 47.0 38.9 31.6 25.0 19.05 13.64 8.70 4.17 2.04
TABLE 2-136
10−4 × H
Partial pressure of N2, mm Hg
8.14
8.84
10.3
International Critical Tables, vol. 3, p. 259.
11.9
13.5
14.4
14.8
800 900 2000 3000 4000 5000 6000 7000 8150 8200
10−4 × H 23°C
25.9°C 4.79
4.58 4.59 4.60 4.68 4.73 4.80 4.88 4.98
4.80 4.83 4.88 4.92 4.98 5.05 5.16
International Critical Tables, vol. 3, p. 257. See also Trans. Am. Soc. Mech. Engrs., 76, 69 (1954) for solubility of O2 for 100°F < T < 650°F, 300 < P < 2000 lb/in2.
2-128
PHYSICAL AND CHEMICAL DATA
TABLE 2-142 t, °C
0
10−3 × H 1.94
Ozone (O3)
TABLE 2-143
5
10
15
20
25
30
35
2.18
2.48
2.88
3.76
4.57
5.98
8.18
40
50
12.0 27.4
International Critical Tables, vol. 3, p. 257.
TABLE 2-144
Propylene (C3H6)
t, °C
2
6
10
14
18
10−3 × H
3.04
3.84
4.46
5.06
5.69
International Critical Tables, vol. 3, p. 260.
Partial Vapor Pressure of Sulfur Dioxide over Water, mm Hg Temperature, °C
g SO2 / 100 g H2O
0
10
0.01 0.05 0.10 0.15 0.20
0.02 0.38 1.15 2.10 3.17
0.04 0.66 1.91 3.44 5.13
0.25 0.30 0.40 0.50 1.00
4.34 5.57 8.17 10.9 25.8
6.93 8.84 12.8 17.0 39.5
2.00 3.00 4.00 5.00 6.00 8.00 10.00 15.00 20.00
20 0.07 1.07 3.03 5.37 7.93 10.6 13.5 19.4 25.6 58.4
30
40
50
60
90
120
0.12 1.68 4.62 8.07 11.8
0.19 2.53 6.80 11.7 17.0
0.29 3.69 9.71 16.5 23.8
0.43 5.24 13.5 22.7 32.6
1.21 12.9 31.7 52.2 73.7
2.82 27.0 63.9 104 145
15.7 19.8 28.3 37.1 83.7
58.6 93.2 129 165 202
88.5 139 192 245 299
129 202 277 353 430
183 285 389 496 602
275 351 542 735
407 517 796
585 741
818
22.5 28.2 40.1 52.3 117
31.4 39.2 55.3 72.0 159
42.8 53.3 74.7 96.8 212
95.8 118 164 211 454
253 393 535 679 824
342 530 720
453 700
955
186 229 316 404 856
Condensed from Rabe, A. E. and Harris, J. F., J. Chem. Eng. Data, 8 (3), 333–336, 1963. Copyright © American Chemical Society and reproduced by permission of the copyright owner.
THERMAL EXPANSION UNITS CONVERSIONS For this subsection, the following units conversion is applicable:
2, p. 93; metals, vol. 2, p. 459; petroleums, vol. 2, p. 145; porcelains, vol. 2, pp. 70, 78; refractory materials, vol. 2, p. 83; solid insulators, vol. 2, p. 310.
°F = 9⁄ 5 °C + 32. THERMAL EXPANSION OF GASES ADDITIONAL REFERENCES The tables given under this subject are reprinted by permission from the Smithsonian Tables. For more detailed data on thermal expansion, see International Critical Tables: tabular index, vol. 3, p. 1; abrasives, vol. 2, p. 87; alloys, vol. 2, p. 463; building stones, vol. 2, p. 54; carbons, vol. 2, p. 303; elements, vol. 1, p. 102; enamels, vol. 2, p. 115; glass, vol.
No tables of the coefficients of thermal expansion of gases are given in this edition. The coefficient at constant pressure, 1/υ(∂υ/∂T)p, for an ideal gas is merely the reciprocal of the absolute temperature. For a real gas or liquid, both it and the coefficient at constant volume, 1/p (∂p/∂T)v, should be calculated either from the equation of state or from tabulated PVT data.
THERMAL EXPANSION TABLE 2-145
2-129
Linear Expansion of the Solid Elements*
C is the true expansion coefficient at the given temperature; M is the mean coefficient between given temperatures; where one temperature is given, the true coefficient at that temperature is indicated; α and β are coefficients in formula lt = l0(1 + αt + βt2); l0 is length at 0°C (unless otherwise indicated, when, if x is the reference temperature, lt = lx[1 + α(t − tx) + β(t − tx)2]; lt is length at t°C). Element
Temp. °C
C × 104
Aluminum Aluminum Antimony Arsenic Bismuth Cadmium Cadmium Carbon, diamond graphite Chromium Cobalt Copper Copper Gold Gold Indium Iodine Iridium Iridium Iron, soft cast wrought steel Lead (99.9)
20 300 20 20 20 0 0 50 50
0.224 0.284 0.136 0.05 0.014 0.54 0.20⊥ 0.012 0.06
20 20 200 20
0.123 0.162 0.170 0.140
40
0.417
20
0.065
40 20 20 20
0.1210 0.118 0.119 0.114
Magnesium
100 280 20
0.291 0.343 0.254
Manganese
20
0.233
Molybdenum†
20
0.053
Nickel
20
0.126
Osmium Palladium
40 20
0.066 0.1173
Platinum
20 20
0.0887 0.0893
40 40 0 40 20 20
0.0850 0.0963 0.439 0.0763 0.1846 0.195
Potassium Rhodium Ruthenium Selenium Silicon Silver Sodium Steel, 36.4Ni Tantalum†
20
Tellurium Thallium Tin
20 40 20 20 27 20‡ 20‡ 20
Tungsten† Zinc
0.065 0.016 0.302 0.214 0.305 0.0444 0.643 0.125⊥ 0.358
Temp. range, °C
M × 104
100 500 20
0.235 0.311 0.080⊥
20 −180, −140 −180, −140
0.103⊥ 0.59 0.117⊥
20, 100
0.068
100 300 17, 100 −191, 17
0.166 0.175 0.143 0.132
−190,
0.837
17
Temp. range, °C
α × 104
β × 106
0,
500
0.22
0.009
20, 20,
100 100
0.526 0.214⊥
20, 6, 0,
500 121 625
0.086 0.121 0.161
0.0064 0.0040
0,
520
0.142
0.0022
0.0636 0.0679
0.0032 0.0011
0, 80 1070, 1720 0, 100
20, 100 20, 200 −100, 20, 0, −190, 0, 25, 25, 0,
0, 6,
+ 20 100 100 0 100 100 500 100
0.11
0.291 0.300 0.240 0.260 0.228 0.159 0.052 0.049 0.055 0.130
50 21
0.83 0.0876
0, 100 −3, +18 0, 100
0.660 0.0249 0.197
−190, 20, 20, −78, 0,
−17 260 340 0 100 20
0.622 0.031 0.055 0.059 0.0655 0.272⊥
20 100 −100 100 100
0.154⊥ 0.045 0.656 0.639 0.141⊥
0, 0, 0, 100,
750 750 750 240
0.1158 0.1170 0.1118 0.269
0.0053 0.0053 0.0053 0.011
+ 20,
500
0.2480
0.0096
20, 300 −142, 19 19, +305
0.216 0.0515 0.0501
0.0121 0.0057 0.0014
−190, + 20 + 20, +300 500, 1000
0.1308 0.1236 0.1346
0.0166 0.0066 0.0033
−190, 0, −190, 0, 0,
+100 1000 −100 + 80 1000
0.1152 0.1167 0.0875 0.0890 0.0887
0.00517 0.0022 0.00314 0.00121 0.00132
−75, −112
0.0746
−75, −67 0, 875 20, 500 0, 50 260, 500 340, 500 20, 400
0.0182 0.1827 0.1939 0.72 0.144 0.136 0.0646
8, 0, −140, +20, +20,
0.00479 0.00295
0.0009
95
0.2033
0.0263
−105, +502 + 0, 400
0.0428 0.354
0.00058 0.010
*Smithsonian Tables. For more complete tabulations see Table 142, Smithsonian Physical Tables, 9th ed., 1954; Handbook of Chemistry and Physics, 40th ed., pp. 2239–2245. Chemical Rubber Publishing Co.; Goldsmith, and Waterman, WADC-TR-58-476, 1959; Johnson (ed.), WADD-TR-60-56, 1960, etc. †Molybdenum, 300° to 2500°C; lt = l300[1 + 5.00 × 10−6(t − 300) + 10.5 × 10−10(t − 300)2] Tantalum, 300° to 2800°C; lt = l300[1 + 6.60 × 10−6(t − 300) + 5.2 × 10−10(t − 300)2] Tungsten, 300° to 2700°C; lt = l300[1 + 4.44 × 10−6(t − 300) + 4.5 × 10−10(t − 300)2] Beryllium, 20° to 100°C; 12.3 × 10−6 per °C. Columbium, 0° to 100°C; 7.2 × 10−6 per °C. Tantalum, 20° to 100°C; 6.6 × 10−6 per °C. ‡Two errors in the data of zinc have been corrected. These values were taken from Grüneisen and Goens, Z. Physik., 29, 141 (1924).
2-130
PHYSICAL AND CHEMICAL DATA
TABLE 2-146
Linear Expansion of Miscellaneous Substances*
The coefficient of cubical expansion may be taken as three times the linear coefficient. In the following table, t is the temperature or range of temperature, and C, the coefficient of expansion. Substance
t°C
Amber
0−30 0−09 20−60
Bakelite, bleached Brass: Cast 0−100 Wire 0−100 Wire 0−100 71.5 Cu + 27.7 Zn + 0.3 Sn + 0.5 Pb 40 71 Cu + 29 Zn 0−100 Bronze: 3 Cu + 1 Sn 16.6−100 3 Cu + 1 Sn 16.6−350 3 Cu + 1 Sn 16.6−957 86.3 Cu + 9.7 Sn + 4 Zn 40 97.6 Cu + hard 0−80 2.2 Sn + soft 0−80 0.2 P Caoutchouc Caoutchouc 16.7−25.3 Celluloid 20−70 Constantan 4−29 Duralumin, 94Al 20−100 20−300 Ebonite 25.3−35.4 Fluorspar, CaF2 0−100 German silver 0−100 Gold-platinum, 2 Au + 1 Pt 0−100 Gold-copper, 2 Au + 1 Cu 0−100 Glass: Tube 0−100 Tube 0−100 Plate 0−100 Crown (mean) 0−100 Crown (mean) 50−60 Flint 50−60 Jena ther- 16III 0−100 mometer normal
{
}
C × 104 0.50 0.61 0.22 0.1875 0.1930 0.1783 to 0.193 0.1859 0.1906 0.1844 0.2116 0.1737 0.1782 0.1713 0.1708 0.657 to 0.686 0.770 1.00 0.1523 0.23 0.25 0.842 0.1950 0.1836 0.1523 0.1552 0.0833 0.0828 0.0891 0.0897 0.0954 0.0788 0.081
Substance
t°C
C × 104
Jena thermometer 59III Jena thermometer 59III Gutta percha Ice Iceland spar: Parallel to axis Perpendicular to axis Lead tin (solder) 2 Pb + 1 Sn Limestone Magnalium Manganin Marble Monel metal
0−100 −191 to +16 20 −20 to −1
0.058 0.424 1.983 0.51
0−80 0−80
0.2631 0.0544
0−100 25−100 12−39 15−100 25−100 25−600 0−16 16−38 38−49
0.2508 0.09 0.238 0.181 0.117 0.14 0.16 1.0662 1.3030 4.7707
40
0.0884
0−100 20−790 1000−1400
0.1523 0.0413 0.0553
0−80 −190 to + 16 0−80 −190 to + 16 16 to 500 16 to 1000 40 0 −160 0−100 25−100 25−600
0.0797 0.0521 0.1337 −0.0026 0.0057 0.0058 0.4040 0.691 0.300 0.1933 0.037 0.136
Paraffin Paraffin Paraffin Platinum-iridium, 10 Pt + 1 Ir Platinum-silver, 1 Pt + 2 Ag Porcelain Porcelain Bayeux Quartz: Parallel to axis Parallel to axis Perpend. to axis Quartz glass Quartz glass Quartz glass Rock salt Rubber, hard Rubber, hard Speculum metal Steel, 0.14 C, 34.5 Ni
Substance
t°C
Topas: Parallel to lesser horizontal axis 0−100 Parallel to greater horizontal axis 0−100 Parallel to vertical axis 0−100 Tourmaline: Parallel to longitudinal axis 0−100 Parallel to horizontal axis 0−100 Type metal 16.6−254 Vulcanite 0−18 Wedgwood ware 0−100 Wood: Parallel to fiber: Ash 0−100 Beech 2.34 Chestnut 2.34 Elm 2.34 Mahogany 2.34 Maple 2.34 Oak 2.34 Pine 2.34 Walnut 2.34 Across the fiber: Beech 2.34 Chestnut 2.34 Elm 2.34 Mahogany 2.34 Maple 2.34 Oak 2.34 Pine 2.34 Walnut 2.34 Wax white 10−26 Wax white 26−31 Wax white 31−43 Wax white 43−57
C × 104
0.0832 0.0836 0.0472 0.0937 0.0773 0.1952 0.6360 0.0890 0.0951 0.0257 0.0649 0.0565 0.0361 0.0638 0.0492 0.0541 0.0658 0.614 0.325 0.443 0.404 0.484 0.544 0.341 0.484 2.300 3.120 4.860 15.227
*Smithsonian Tables. For a more complete tabulation see Tables 143, 144. Smithsonian Physical Tables. 9th ed., 1954, also reprinted in American Institute of Physics Handbook, McGraw-Hill, New York, 1957; Handbook of Chemistry and Physics, 40th ed., pp. 2239–2245, Chemical Rubber Publishing Co. For data on many solids prior to 1926, see Gruneisen, Handbuch der Physik, vol. 10, pp. 1–52, 1926, translation available as N.A.S.A. RE 2-18-59W, 1959. For eight plastic solids below 300 K, see Scott, Cryogenic Engineering, p. 331, Van Nostrand, Princeton, NJ, 1959. For 11 other materials to 300 K, see Scott, loc. cit., p. 333. For quartz and silica, see Cook, Brit. J. Appl. Phys., 7, 285 (1956).
THERMAL EXPANSION TABLE 2-147
Cubical Expansion of Liquids*
TABLE 2-148
If V0 is the volume at 0°, then at t° the expansion formula is Vt = V0(1 + αt + βt2 + γ t3). The table gives values of α, β, and γ, and of C, the true coefficient of cubical expansion at 20° for some liquids and solutions. The temperature range of the observation is ∆t. Values for the coefficient of cubical expansion of liquids can be derived from the tables of specific volumes of the saturated liquid given as a function of temperature later in this section. Liquid
Range
α × 103
β × 106
Acetic acid Acetone Alcohol: Amyl Ethyl, 30% by volume Ethyl, 50% by volume Ethyl, 99.3% by volume Ethyl, 500 atm. pressure Ethyl, 3000 atm. pressure Methyl Benzene Bromine Calcium chloride: 5.8% solution 40.9% solution Carbon disulfide 500 atm. pressure 3000 atm. pressure Carbon tetrachloride Chloroform Ether Glycerin Hydrochloric acid, 33.2% solution Mercury Olive oil Pentane Potassium chloride, 24.3% solution Phenol Petroleum, 0.8467 density Sodium chloride, 20.6% solution Sodium sulfate, 24% solution Sulfuric acid: 10.9% solution 100.0% Turpentine Water
16−107 0−54
1.0630 1.3240
0.12636 3.8090
− 15−80 18−39 0−39
0.9001 0.2928 0.7450
0.6573 10.790 1.85
27−46
1.012
2.20
0−40
0.866
0−40 0−61 11−81 0−59
0.524 1.1342 1.17626 1.06218
18−25 17−24 −34−60 0−50 0−50 0−76 0−63 −15−38 0−33 0−100
γ × 108
C × 108 at 20°
1.0876 1.071 − 0.87983 1.487 1.18458 0.902 −11.87 0.730 1.12
1.3635 1.27776 1.87714
0.8741 1.199 0.80648 1.237 −0.30854 1.132
0.07878 0.42383 1.13980 0.940 0.581 1.18384 1.10715 1.51324 0.4853
4.2742 0.8571 1.37065
0.250 0.458 1.91225 1.218
0.89881 4.66473 2.35918 0.4895
1.35135 1.236 − 1.74328 1.273 4.00512 1.656 0.505
0−33
0.4460 0.18182 0.6821 1.4646
0.215 0.0078 1.1405 3.09319
− 0.539 1.6084
16−25 36−157
0.2695 0.8340
2.080 0.10732
0.4446
24−120
0.8994
1.396
0.955
0−29
0.3640
1.237
0.414
11−40
0.3599
1.258
0.410
0−30 0.2835 2.580 0−30 0.5758 −0.432 − 9−106 0.9003 1.9595 0−33 −0.06427 8.5053
0.455 0.18186 0.721 1.608 0.353 1.090
0.387 0.558 − 0.44998 0.973 − 6.7900 0.207
*Smithsonian Tables, Table 269. For a detailed discussion of mercury data, see Cook, Brit. J. Appl. Phys., 7, 285 (1956). For data on nitrogen and argon, see Johnson (ed.), WADD-TR-60-56, 1960. Bromoform1 7.7 − 50°C. Vt = 0.34204[1 + 0.00090411(t − 7.7) + 0.0000006766(t − 7.7)2] 0.34204 in the specific volume of bromoform at 7.7°C. Glycerin2 −62 to 0°C. Vt = V0(1 + 4.83 × 10−4t − 0.49 × 10−6t2) 0 − 80°C. Vt = V0(1 + 4.83 × 10−4t + 0.49 × 10−6t2) 3 Mercury 0 − 300°C. Vt − V0[1 + 10−8(18153.8t + 0.7548t2 + 0.001533t2 + 0.00000536t4)] 1 Sherman and Sherman, J. Am. Chem. Soc., 50, 1119 (1928). (An obvious error in their equation has been corrected.) 2 Samsoen, Ann. phys., (10) 9, 91 (1928). 3 Harlow, Phil. Mag., (7) 7, 674 (1929).
2-131
Cubical Expansion of Solids*
If v2 and v1 are the volumes at t2 and t1, respectively, then v2 = v1(1 + C∆t), C being the coefficient of cubical expansion and ∆t the temperature interval. Where only a single temperature is stated, C represents the true coefficient of cubical expansion at that temperature. Substance
t or ∆t
C × 104
Antimony Beryl Bismuth Copper† Diamond Emerald Galena Glass, common tube hard Jena, borosilicate 59 III pure silica Gold Ice Iron Lead† Paraffin Platinum Porcelain, Berlin chloride nitrate sulfate Quartz Rock salt Rubber Silver Sodium Stearic acid Sulfur, native Tin Zinc†
0−100 0−100 0−100 0−100 40 40 0−100 0−100 0−100 20−100 0−80 0−100 −20 to −1 0−100 0−100 20 0−100 20 0−100 0−100 20 0−100 50−60 20 0−100 20 33.8−45.4 13.2−50.3 0−100 0−100
0.3167 0.0105 0.3948 0.4998 0.0354 0.0168 0.558 0.276 0.214 0.156 0.0129 0.4411 1.1250 0.3550 0.8399 5.88 0.265 0.0814 1.094 1.967 1.0754 0.3840 1.2120 4.87 0.5831 2.13 8.1 2.23 0.6889 0.8928
*Smithsonian Tables, Table 268. †See additional data below. Aluminum1 100 − 530°C. V = V0(1 + 2.16 × 10−5t + 0.95 × 10−8t2) 1 Cadmium 130 − 270°C. V = V0(1 + 8.04 × 10−5t + 5.9 × 10−8t2) 1 Copper 110 − 300°C. V = V0(1 + 1.62 × 10−5t + 0.20 × 10−8t2) Colophony2 0 − 34°C. V = V0(1 + 2.21 × 10−4t + 0.31 × 10−6t2) 34 − 150°C. V = V34[1 + 7.40 × 10−4(t − 34) + 5.91 × 10−6(t − 34)2] 1 Lead 100 − 280°C. V = V0(1 + 1.60 × 10−5t + 3.2 × 10−8t2) 2 Shellac 0 − 46°C. V = V0(1 + 2.73 × 10−4t + 0.39 × 10−6t2) 46 − 100°C. V = V46[1 + 13.10 × 10−4(t − 46) + 0.62 × 10−6(t − 46)2] Silica (vitreous)3 0 − 300°C. Vt = V0[1 + 10−8(93.6t + 0.7776t2 − 0.003315t2 + 0.000005244t4) Sugar (cane, amorphous)2 0 − 67°C. Vt = V0(1 + 2.34 × 10−4t + 0.14 × 10−6t2) 67 − 160°C. Vt = V67[1 + 5.02 × 10−4(t − 67) + 0.43 × 10−6(t − 67)2] Zinc1 120 − 360°C. Vt = V0(1 + 8.50 × 10−5t + 3.9 × 10−8t2) 1 2 3
Uffelmann, Phil. Mag., (7) 10, 633 (1930). Samsoen, Ann. phys., (10) 9, 83 (1928). Harlow, Phil. Mag., (7) 7, 674 (1929).
2-132
PHYSICAL AND CHEMICAL DATA
JOULE-THOMSON EFFECT UNITS CONVERSIONS For this subsection, the following units conversions are applicable: To convert the Joule-Thomson coefficient, µ, in degrees Celsius per atmosphere to degrees Fahrenheit per atmosphere, multiply by 1.8.
TABLE 2-149
°F = 9⁄ 5 °C + 32; °R = 9⁄ 5 K To convert bars to pounds-force per square inch, multiply by 14.504; to convert bars to kilopascals, multiply by 1 × 102.
Additional References Available for the Joule-Thomson Coefficient Temp. range, °C
Pressure range, atm Gas Air Ammonia Argon Benzene Butane Carbon dioxide Carbon monoxide Deuterium Dowtherm A Ethane Ethylene Helium Hydrogen
>200
<0
0–10
10–50
50–200
12, 15, 16 19, 35 28 39 31 26 7, 8, 28 37 17
12, 15, 19 35
15, 19, 35
19, 35
39 31 26 7, 8, 37
39 31
39
7, 8, 37
7, 8, 37
1,* 22, 24 25
17 1,* 22, 24, 25
46 45 1, 38 24, 30
17 22, 24, 25 1* 46 45 1, 38 22, 24, 25 30 6
6 33 13, 40
Methane Mixtures Natural gas Nitrogen
13, 28, 40
13, 40
Nitrous oxide Pentane Propane Steam
26, 34, 44 41 28, 29, 42
34 43 29, 42, 47
38 24, 30
34 42, 47
1, 38 22, 24, 25 30 33 13
33 13, 40
0–300 12, 15, 16 19, 35 28 39 31 26 7, 8, 9, 10 37 17 46 45 9, 10 38 24 6 9, 11 33 9, 10, 13 28, 40 9, 10 26, 34, 44 43 28, 29, 42 45
>300
Unclassified 3, 4, 18 2, 3
31
46 48
13
19
29, 42, 47
29, 47
*See also 14 (generalized chart); 18 (review, to 1919); 20–22; 23 (review, to 1948); 27 (review, to 1905); 32, 36, 41, 50. REFERENCES: 1. Baehr. Z. Elektrochem., 60, 515 (1956). 2. Beattie, J. Math. Phys., 9, 11 (1930). 3. Beattie, Phys. Rev., 35, 643 (1930). 4. Bradley and Hale, Phys. Rev., 29, 258 (1909). 5. Brown and Dean, Bur. Stand. J. Res., 60, 161 (1958). 6. Budenholzer, Sage, et al., Ind. Eng. Chem., 29, 658 (1937). 7. Burnett, Phys. Rev., 22, 590 (1923). 8. Burnett, Univ. Wisconsin Bull. 9(6), 1926. 9. Charnley, Ph.D. thesis. University of Manchester, 1952. 10. Charnley, Isles, et al., Proc. R. Soc. (London), A217, 133 (1953). 11. Charnley, Rowlinson, et al., Proc. R. Soc. (London), A230, 354 (1955). 12. Dalton, Commun. Phys. Lab. Univ. Leiden, no. 109c, 1909. 13. Deming and Deming, Phys. Rev., 48, 448 (1935). 14. Edmister, Pet. Refiner, 28, 128 (1949). 15. Eucken, Clusius, et al., Z. Tech. Phys., 13, 267 (1932). 16. Eumorfopoulos and Rai, Phil. Mag., 7, 961 (1926). 17. Huang, Lin, et al., Z. Phys., 100, 594 (1936). 18. Hoxton, Phys. Rev., 13, 438 (1919). 19. Ishkin and Kaganev, J. Tech. Phys. U.S.S.R., 26, 2323 (1956). 20. Isles, Ph.D. thesis, Leeds University. 21. Jenkin and Pye, Phil. Trans. R. Soc. (London), A213, 67 (1914); A215, 353 (1915). 22. Johnston, J. Am. Chem. Soc., 68, 2362 (1946). 23. Johnston, Trans. Am. Soc. Mech. Eng., 70, 651 (1948). 24. Johnston, Bezman, et al., J. Am. Chem. Soc., 68, 2367 (1946). 25. Johnston, Swanson, et al., J. Am. Chem. Soc., 68, 2373 (1946). 26. Kennedy, Sage, et al., Ind. Eng. Chem., 28, 718 (1936). 27. Kester, Phys. Rev., 21, 260 (1905). 28. Keyes and Collins, Proc. Nat. Acad. Sci., 18, 328 (1932). 29. Kleinschmidt, Mech. Eng., 45, 165 (1923); 48, 155 (1926). 30. Koeppe, Kältetechnik, 8, 275 (1956). 31. Lindsay and Brown, Ind. Eng. Chem., 27, 817 (1935). 32. Noell, dissertation, Munich, 1914, Forschungsdienst, 184, p. 1, 1916. 33. Palienko, Tr. Inst. Ispol’ z. Gaza, Akad. Nauk Ukr. SSR, no. 4, p. 87, 1956. 34. Pattee and Brown, Ind. Eng. Chem., 26, 511, (1934). 35. Roebuck, Proc. Am. Acad. Arts Sci., 60, 537 (1925); 64, 287 (1930). 36. Roebuck, see 49 below, 37. Roebuck and Murrell, Phys. Rev., 55, 240 (1939). 38. Roebuck and Osterberg, Phys. Rev., 37, 110 (1931); 43, 60 (1933). 39. Roebuck and Osterberg, Phys. Rev., 46, 785 (1934). 40. Roebuck and Osterberg, Phys. Rev., 48, 450 (1935). 41. Roebuck, Murrell, et al., J. Am. Chem. Soc., 64, 400 (1942). 42. Sage, unpublished data, California Institute of Technology, 1959. 43. Sage and Lacy, Ind. Eng. Chem., 27, 1484 (1934). 44. Sage, Kennedy, et al., Ind. Eng. Chem., 28, 601 (1936). 45. Sage, Webster, et al., Ind. Eng. Chem., 29, 658 (1937). 46. Ullock, Gaffert, et al., Trans. Am. Inst. Chem. Eng., 32, 73 (1936). 47. Yang, Ind. Eng. Chem., 45, 786 (1953). 48. Zelmanov, J. Phys. U.S.S.R., 3, 43 (1940). 49. Roebuck, recalculated data. 50. Michels et al., van der Waals laboratory publications. Gunn, Cheuh, and Prausnitz, Cryogenics, 6, 324 (1966), review equations relating the inversion temperatures and pressures. The ability of various equations of state to relate these was also discussed by Miller, Ind. Eng. Chem. Fundam., 9, 585 (1970); and Juris and Wenzel, Am. Inst. Chem. Eng. J., 18, 684 (1972). Perhaps the most detailed review is that of Hendricks, Peller, and Baron. NASA Tech. Note D 6807, 1972.
JOULE-THOMSON EFFECT
2-133
TABLE 2-150 Approximate Inversion-Curve Locus in Reduced Coordinates (Tr = T/Tc; Pr = P/Pc)* Pr
0
0.5
1
1.5
2
2.5
3
4
TrL TrU
0.782 4.984
0.800 4.916
0.818 4.847
0.838 4.777
0.859 4.706
0.880 4.633
0.903 4.550
0.953 4.401
Pr
5
6
7
8
9
10
11
11.79
TrL TrU
1.01 4.23
1.08 4.06
1.16 3.88
1.25 3.68
1.35 3.45
1.50 3.18
1.73 2.86
2.24 2.24
*Calculated from the best three-constant equation recommended by Miller, Ind. Eng. Chem. Fundam., 9, 585 (1970). TrL refers to the lower curve, and TrU, to the upper curve.
TABLE 2-151
Joule-Thomson Data for Air* t, °C
P, atm 1 20 60 100 140 180 200
−150
−100
−75
−50
−25
0
25
50
75
100
150
200
250
0.0450 .0185 − .0070 − .0255 − .0330
0.5895 .5700 .4820 .2775 .1360 .0655 .0440
0.4795 .4555 .3835 .2880 .1855 .1136 .0855
0.3910 .3690 .3195 .2505 .1825 .1270 .1065
0.3225 .3010 .2610 .2130 .1650 .1240 .1090
0.2745 .2580 .2200 .1820 .1450 .1100 .0950
0.2320 .2173 .1852 .1550 .1249 .0959
0.1956 .1830 .1571 .1310 .1070 .0829
0.1614 .1508 .1293 .1087 .0889 .0707
0.1355 .1258 .1062 .0884 .0726 .0580
0.0961 .0883 .0732 .0600 .0482 .0376
0.0645 .0580 .0453 .0343 .0250 .0174
0.0409 .0356 .0254 .0165 .0092 .0027
*Free of water and CO2. Extracted from Table 261, Smithsonian Physical Tables, 9th rev. ed., Washington, DC, 1954. These data are corrected from earlier publications. µ in °C/atm.
TABLE 2-152 P, bar TL, K TU, K
Approximate Inversion-Curve Locus for Air
0 (112)* 653
25
50
75
100
125
150
175
200
225
114 641
117 629
120 617
124 606
128 594
132 582
137 568
143 555
149 541
P, bar
250
275
300
325
350
375
400
425
432
TL, K TU, K
156 526
164 509
173 491
184 470
197 445
212 417
230 386
265 345
300 300
*Hypothetical low-pressure limit.
TABLE 2-153
Joule-Thomson Data for Argon* Pressure, atm
t, °C
1
−150 −125 −100 −75 −50
20
60
100
140
180
200
1.812 1.112 0.8605 .7100 .5960
1.102 0.8485 .6895 .5720
−0.0025 .1250 .6900 .5910 .4963
−0.0277 .0415 .2820 .4225 .3970
−0.0403 .0090 .1137 .2480 .2840
−0.0595 −.0100 .0560 .1537 .2037
−0.0640 − .0165 .0395 .1215 .1860
−25 0 25 50 75
.5045 .4307 .3720 .3220 .2695
.4805 .4080 .3490 .3015 .2557
.4210 .3600 .3077 .2650 .2285
.3460 .3010 .2628 .2297 .1993
.2763 .2505 .2213 .1947 .1710
.2140 .2050 .1890 .1700 .1505
.1950 .1883 .1745 .1580 .1415
100 125 150 200 250
.2413 .2105 .1845 .1377 .0980
.2277 .1980 .1720 .1280 .0910
.1975 .1707 .1485 .1102 .0785
.1715 .1480 .1285 .0950 .0665
.1490 .1300 .1123 .0823 .0555
.1320 .1153 .0998 .0715 .0485
.1255 .1100 .0945 .0675 .0468
300
.0643
.0607
.0530
.0445
.0370
.0370
.0276
*Extracted from Table 263, Smithsonian Physical Tables, 9th rev. ed., Washington, DC, 1954. These data are corrected from an earlier publication. µ in °C/atm.
2-134
PHYSICAL AND CHEMICAL DATA TABLE 2-154
Approximate Inversion-Curve Locus for Argon
P, bar
0
25
50
75
100
125
150
175
200
225
TL, K TU, K
94 765
97 755
101 744
105 736
109 726
113 716
118 705
123 694
128 683
134 671
P, bar
250
275
300
325
350
375
400
425
450
475
TL, K TU, K
141 657
148 643
158 627
170 610
183 591
201 569
222 544
248 515
288 478
375 375
TABLE 2-155
Joule-Thomson Data for Carbon Dioxide* Pressure, atm
t, °C
1
20
60
73
100
140
180
200
−75 −50 0 50 100
2.4130 1.2900 0.8950 .6490
−0.0200 − .0140 1.4020 .8950 .6375
−0.0200 − .0150 .0370 .8800 .6080
−0.0232 − .0165 .0310 .8225 .5920
−0.0228 − .0160 .0215 .5570 .5405
−0.0240 − .0183 .0115 .1720 .4320
−0.0250 − .0228 .0085 .1025 .3000
−0.0290 − .0248 .0045 .0930 .2555
125 150 200 250 300
.5600 .4890 .3770 .3075 .2650
.5450 .4695 .3575 .2885 .2425
.5160 .4430 .3400 .2625 .2080
.5068 .4380 .3325 .2565 .2002
.4750 .4155 .3150 .2420 .1872
.4130 .3760 .2890 .2235 .1700
.3230 .3102 .2600 .2045 .1540
.2915 .2910 .2455 .1975 .1505
*Extracted from Table 266, Smithsonian Physical Tables, 9th rev. ed., Washington, DC, 1954. These data are corrected from an earlier publication. µ in °C/atm.
TABLE 2-156
Approximate Inversion-Curve Locus for Carbon Dioxide*
P, bar
50
100
150
200
250
300
350
400
450
TL, K TU, K
243 1290
251 1261
258 1233
266 1205
272 1175
283 1146
293 1111
302 1076
312 1045
P, bar
500
550
600
650
700
750
800
850
884
TL, K TU, K
325 1015
338 983
351 950
365 914
383 878
403 840
441 796
496 739
608 608
*Interpolated from Vukalovich and Altunin’s interpolation of data of Price, Ind. Eng. Chem., 47, 1691 (1955). TL = lower inversion temperature, and TU = upper inversion temperature.
TABLE 2-157 P, bar TL, K TU, K
Approximate Inversion-Curve Locus for Deuterium
0 (31)* 216
25
50
75
100
125
150
175
194
34 202
38 189
43 178
49 168
56 157
65 146
77 131
108 108
*Hypothetical low-pressure limit.
TABLE 2-158 P, bar
0
TL, K
Approximate Inversion-Curve Locus for Ethane 25
50
75
100
125
150
175
200
225
249
255
262
269
275
282
290
297
306
P, bar
250
275
300
325
350
375
400
425
450
475
TL, K
315
325
335
345
357
370
383
398
415
432
P, bar
500
525
550
575
600
TL, K
453
477
505
545
626
JOULE-THOMSON EFFECT TABLE 2-159
2-135
Joule-Thomson Data for Helium*
T, K
160
180
200
220
240
260
280
300
µ
−0.0574
−0.0587
−0.0594
−0.0601
−0.0608
−0.0614
−0.0619
−0.0625
T, K
320
340
360
380
400
420
440
460
µ
−0.0629
−0.0634
−0.0637
−0.0640
−0.0643
−0.0645
−0.0645
−0.0643
T, K
480
500
520
540
560
580
600
µ
−0.0640
−0.0636
−0.0630
−0.0622
−0.0611
−0.0587
−0.0540
*Interpolated and converted from data in Table 262, Smithsonian Physical Tables, 9th rev. ed., Washington, DC, 1954. These data are corrected from those in an earlier publication. µ is in °C/atm. Below about 200 atm, little change in the coefficient with pressure occurs.
TABLE 2-160
Approximate Inversion-Curve Locus for Normal Hydrogen
P, bar
0
25
50
75
100
125
150
164
TL, K TU, K
(28)* 202
32 193
38 183
44 171
52 157
61 141
73 119
92 92
*Hypothetical low-pressure limit.
TABLE 2-161 P, bar
Approximate Inversion-Curve Locus for Methane
25
TL, K
50
75
100
125
150
175
200
225
250
275
300
161
166
172
176
182
189
195
202
209
217
225
P, bar
325
350
375
400
425
450
475
500
525
534
TL, K TU, K
234
243
254
265
277
292
309 505
331 474
365 437
400 400
TABLE 2-162
Joule-Thomson Data for Nitrogen* Pressure, atm
t, °C
1
20
33.5
60
100
140
180
200
−150 −125 −100 −75 −50
1.2659 0.8557 .6490 .5033 .3968
1.1246 0.7948 .5958 .4671 .3734
0.1704 .7025 .5494 .4318 .3467
0.0601 .4940 .4506 .3712 .3059
0.0202 .1314 .2754 .2682 .2332
−0.0056 .0498 .1373 .1735 .1676
−0.0211 .0167 .0765 .1026 .1120
−0.0284 .0032 .0587 .0800 .0906
−25 0 25 50 75
.3224 .2656 .2217 .1855 .1555
.3013 .2494 .2060 .1709 .1421
.2854 .2377 .1961 .1621 .1336
.2528 .2088 .1729 .1449 .1191
.2001 .1679 .1400 .1164 .0941
.1506 .1316 .1105 .0915 .0740
.1101 .1015 .0874 .0732 .0583
.0932 .0891 .0779 .0666 .0543
100 125 150 200 250
.1292 .1070 .0868 .0558 .0331
.1173 .0973 .0776 .0472 .0256
.1100 .0904 .0734 .0430 .0230
.0975 .0786 .0628 .0372 .0160
.0768 .0621 .0482 .0262 .0071
.0582 .0459 .0348 .0168 .0009
.0462 .0347 .0248 .0094 −.0037
.0419 .0326 .0228 .0070 −.0058
300
.0140
.0096
.0050
−.0013
−.0075
−.0129
−.0160
−.0171
*Extracted from Table 264, Smithsonian Physical Tables, 9th rev. ed., Washington, DC, 1954. These data are corrected from an earlier publication. µ in °C/atm.
TABLE 2-163
Approximate Inversion-Curve Locus for Propane
P, bar
0
25
50
75
100
125
150
175
200
225
250
275
TL, K
(296)*
303
311
318
327
336
345
355
365
374
389
403
P, bar
300
325
350
375
400
425
450
475
500
525
541
TL, K
418
435
452
473
495
521
551
586
628
686
780
*Hypothetical low-pressure limit.
2-136
PHYSICAL AND CHEMICAL DATA
CRITICAL CONSTANTS ADDITIONAL REFERENCES Other data and estimation techniques for the elements are contained in Gates and Thodos, Am. Inst. Chem. Eng. J., 6 (1960):50–54; and Ohse and von Tippelskirch, High Temperatures—High Pressures, 9 TABLE 2-164 Cmpd. no.
(1977):367–385. For inorganic substances see Mathews, Chem. Rev., 72 (1972):71–100; for organics see Kudchaker, Alani, and Zwolinski, Chem. Rev., 68 (1968):659–735; and for fluorocarbons see Advances in Fluorine Chemistry, App. B, Butterworth. Washington, 1963, pp. 173–175.
Critical Constants and Acentric Factors of Inorganic and Organic Compounds Name
Formula
CAS no.
Mol. wt.
Tc, K
Pc × 1E-06 Pa
Vc, m3/Kmol
Zc
Acentric factor
1 2 3 4 5 6 7 8 9 10
Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane
CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22
74828 74840 74986 106978 109660 110543 142825 111659 111842 124185
16.043 30.070 44.097 58.123 72.150 86.177 100.204 114.231 128.258 142.285
190.564 305.32 369.83 425.12 469.7 507.6 540.2 568.7 594.6 617.7
4.59 4.85 4.21 3.77 3.36 3.04 2.72 2.47 2.31 2.09
0.099 0.146 0.200 0.255 0.315 0.373 0.428 0.486 0.540 0.601
0.286 0.279 0.273 0.272 0.271 0.269 0.259 0.254 0.252 0.245
0.011 0.098 0.149 0.197 0.251 0.304 0.346 0.396 0.446 0.488
11 12 13 14 15 16 17 18 19 20
n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane n-Eicosane
C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40 C20H42
1120214 112403 629505 629594 629629 544763 629787 593453 629925 112958
156.312 170.338 184.365 198.392 212.419 226.446 240.473 254.500 268.527 282.553
639 658 675 693 708 723 736 747 758 768
1.95 1.82 1.68 1.57 1.47 1.41 1.34 1.26 1.21 1.17
0.658 0.718 0.779 0.830 0.888 0.943 0.998 1.059 1.119 1.169
0.242 0.239 0.233 0.226 0.222 0.221 0.219 0.214 0.215 0.215
0.530 0.577 0.617 0.643 0.685 0.721 0.771 0.806 0.851 0.912
21 22 23 24 25 26 27
2-Methylpropane 2-Methylbutane 2,3-Dimethylbutane 2-Methylpentane 2,3-Dimethylpentane 2,3,3-Trimethylpentane 2,2,4-Trimethylpentane
C4H10 C5H12 C6H14 C6H14 C7H16 C8H18 C8H18
75285 78784 79298 107835 565593 560214 540841
58.123 72.150 86.177 86.177 100.204 114.231 114.231
408.14 460.43 499.98 497.5 537.35 573.5 543.96
3.62 3.37 3.13 3.02 2.88 2.81 2.56
0.261 0.304 0.358 0.366 0.396 0.455 0.465
0.278 0.268 0.269 0.267 0.255 0.268 0.264
0.177 0.226 0.246 0.279 0.292 0.289 0.301
28 29 30 31 32 33 34 35
Ethylene Propylene 1-Butene cis-2-Butene trans-2-Butene 1-Pentene 1-Hexene 1-Heptene
C2H4 C3H6 C4H8 C4H8 C4H8 C5H10 C6H12 C7H14
74851 115071 106989 590181 624646 109671 592416 592767
28.054 42.081 56.108 56.108 56.108 70.134 84.161 98.188
282.34 365.57 419.95 435.58 428.63 464.78 504.03 537.29
5.03 4.63 4.04 4.24 4.08 3.56 3.14 2.82
0.132 0.188 0.241 0.233 0.237 0.295 0.354 0.413
0.283 0.286 0.279 0.273 0.272 0.271 0.265 0.261
0.086 0.137 0.190 0.204 0.216 0.236 0.280 0.330
36 37 38 39 40 41 42 43 44
1-Octene 1-Nonene 1-Decene 2-Methylpropene 2-Methyl-1-butene 2-Methyl-2-butene 1,2-Butadiene 1,3-Butadiene 2-Methyl-1,3-butadiene
C8H16 C9H18 C10H20 C4H8 C5H10 C5H10 C4H6 C4H6 C5H8
111660 124118 872059 115117 563462 513359 590192 106990 78795
112.215 126.242 140.269 56.108 70.134 70.134 54.092 54.092 68.119
566.65 593.25 616.4 417.9 465 471 452 425.17 484
2.57 2.33 2.21 3.98 3.45 3.38 4.36 4.30 3.85
0.460 0.528 0.584 0.238 0.292 0.292 0.220 0.220 0.277
0.251 0.249 0.252 0.272 0.261 0.252 0.255 0.268 0.265
0.377 0.417 0.478 0.192 0.237 0.272 0.166 0.192 0.158
45 46 47 48 49 50
Acetylene Methylacetylene Dimethylacetylene 3-Methyl-1-butyne 1-Pentyne 2-Pentyne
C2H2 C3H4 C4H6 C5H8 C5H8 C5H8
74862 74997 503173 598232 627190 627214
26.038 40.065 54.092 68.119 68.119 68.119
308.32 402.39 473.2 463.2 481.2 519
6.15 5.62 4.87 4.20 4.17 4.02
0.113 0.164 0.221 0.275 0.277 0.276
0.271 0.276 0.274 0.300 0.289 0.257
0.188 0.216 0.239 0.308 0.290 0.174
51 52 53 54 55 56
1-Hexyne 2-Hexyne 3-Hexyne 1-Heptyne 1-Octyne Vinylacetylene
C6H10 C6H10 C6H10 C7H12 C8H14 C4H4
693027 764352 928494 628717 629050 689974
82.145 82.145 82.145 96.172 110.199 52.076
516.2 549 544 559 585 454
3.64 3.53 3.54 3.13 2.82 4.89
0.322 0.331 0.334 0.386 0.441 0.205
0.273 0.256 0.261 0.260 0.256 0.265
0.335 0.221 0.219 0.272 0.323 0.109
CRITICAL CONSTANTS TABLE 2-164
2-137
Critical Constants and Acentric Factors of Inorganic and Organic Compounds (Continued ) Mol. wt.
Tc, K
Pc × 1E-06 Pa
287923 96377 1640897 110827 108872 590669 1678917 142290 693890 110838
70.134 84.161 98.188 84.161 98.188 112.215 112.215 68.119 82.145 82.145
511.76 532.79 569.52 553.58 572.19 591.15 609.15 507 542 560.4
4.50 3.78 3.40 4.10 3.48 2.94 3.04 4.81 4.13 4.39
C6H6 C7H8 C8H10 C8H10 C8H10 C8H10 C9H12
71432 108883 95476 108383 106423 100414 103651
78.114 92.141 106.167 106.167 106.167 106.167 120.194
562.16 591.8 630.33 617.05 616.23 617.2 638.32
1,2,4-Trimethylbenzene Isopropylbenzene 1,3,5-Trimethylbenzene p-Isopropyltoluene Naphthalene Biphenyl Styrene m-Terphenyl
C9H12 C9H12 C9H12 C10H14 C10H8 C12H10 C8H8 C18H14
95636 98828 108678 99876 91203 92524 100425 92068
120.194 120.194 120.194 134.221 128.174 154.211 104.152 230.309
82 83 84 85 86 87 88
Methanol Ethanol 1-Propanol 1-Butanol 2-Butanol 2-Propanol 2-Methyl-2-propanol
CH4O C2H6O C3H8O C4H10O C4H10O C3H8O C4H10O
67561 64175 71238 71363 78922 67630 75650
89 90 91 92 93 94 95 96
1-Pentanol 2-Methyl-1-butanol 3-Methyl-1-butanol 1-Hexanol 1-Heptanol Cyclohexanol Ethylene glycol 1,2-Propylene glycol
C5H12O C5H12O C5H12O C6H14O C7H16O C6H12O C2H6O2 C3H8O2
97 98 99 100
Phenol o-Cresol m-Cresol p-Cresol
C6H6O C7H8O C7H8O C7H8O
101 102 103 104 105 106 107 108 109 110 111 112
Dimethyl ether Methyl ethyl ether Methyl n-propyl ether Methyl isopropyl ether Methyl n-butyl ether Methyl isobutyl ether Methyl tert-butyl ether Diethyl ether Ethyl propyl ether Ethyl isopropyl ether Methyl phenyl ether Diphenyl ether
113 114 115 116 117 118 119 120 121 122
Formaldehyde Acetaldehyde 1-Propanal 1-Butanal 1-Pentanal 1-Hexanal 1-Heptanal 1-Octanal 1-Nonanal 1-Decanal
Cmpd. no.
Name
Zc
Acentric factor
57 58 59 60 61 62 63 64 65 66
Cyclopentane Methylcyclopentane Ethylcyclopentane Cyclohexane Methylcyclohexane 1,1-Dimethylcyclohexane Ethylcyclohexane Cyclopentene 1-Methylcyclopentene Cyclohexene
C5H10 C6H12 C7H14 C6H12 C7H14 C8H16 C8H16 C5H8 C6H10 C6H10
0.257 0.319 0.374 0.308 0.368 0.450 0.430 0.245 0.303 0.291
0.272 0.272 0.269 0.274 0.269 0.269 0.258 0.279 0.278 0.274
0.196 0.230 0.271 0.212 0.236 0.233 0.246 0.196 0.232 0.216
67 68 69 70 71 72 73
Benzene Toluene o-Xylene m-Xylene p-Xylene Ethylbenzene Propylbenzene
4.88 4.10 3.74 3.53 3.50 3.60 3.20
0.261 0.314 0.374 0.377 0.381 0.375 0.440
0.273 0.262 0.267 0.259 0.260 0.263 0.265
0.209 0.262 0.311 0.325 0.320 0.301 0.344
74 75 76 77 78 79 80 81
649.13 631.1 637.36 653.15 748.35 789.26 636 924.85
3.25 3.18 3.11 2.80 3.99 3.86 3.82 3.53
0.430 0.429 0.433 0.497 0.413 0.502 0.352 0.768
0.259 0.260 0.254 0.256 0.265 0.295 0.254 0.352
0.380 0.322 0.397 0.366 0.296 0.367 0.295 0.561
32.042 46.069 60.096 74.123 74.123 60.096 74.123
512.64 513.92 536.78 563.05 536.05 508.3 506.21
8.14 6.12 5.12 4.34 4.20 4.79 3.99
0.117 0.168 0.220 0.276 0.270 0.221 0.276
0.224 0.240 0.252 0.256 0.254 0.250 0.262
0.566 0.643 0.617 0.585 0.574 0.670 0.613
71410 137326 123513 111273 111706 108930 107211 57556
88.150 88.150 88.150 102.177 116.203 100.161 62.068 76.095
586.15 565 577.2 611.35 631.9 650 719.7 626
3.87 3.87 3.90 3.46 3.18 4.25 7.71 6.04
0.327 0.327 0.327 0.381 0.435 0.322 0.191 0.239
0.260 0.270 0.266 0.259 0.263 0.253 0.246 0.277
0.592 0.678 0.586 0.572 0.592 0.371 0.487 1.102
108952 95487 108394 106445
94.113 108.140 108.140 108.140
694.25 697.55 705.85 704.65
6.06 5.06 4.52 5.15
0.229 0.282 0.312 0.277
0.240 0.246 0.240 0.244
0.438 0.438 0.444 0.507
C2H6O C3H8O C4H10O C4H10O C5H12O C5H12O C5H12O C4H10O C5H12O C5H12O C7H8O C12H10O
115106 540670 557175 598538 628284 625445 1634044 60297 628320 625547 100663 101848
46.069 60.096 74.123 74.123 88.150 88.150 88.150 74.123 88.150 88.150 108.140 170.211
400.1 437.8 476.3 464.5 510 497 497.1 466.7 500.23 489 645.6 766.8
5.27 4.47 3.77 3.89 3.31 3.41 3.41 3.64 3.37 3.41 4.27 3.10
0.171 0.221 0.276 0.276 0.329 0.331 0.329 0.281 0.336 0.329 0.337 0.503
0.271 0.271 0.263 0.278 0.257 0.273 0.272 0.264 0.273 0.276 0.268 0.244
0.192 0.229 0.264 0.280 0.335 0.310 0.264 0.281 0.347 0.306 0.353 0.441
CH2O C2H4O C3H6O C4H8O C5H10O C6H12O C7H14O C8H16O C9H18O C10H20O
50000 75070 123386 123728 110623 66251 111717 124130 124196 112312
30.026 44.053 58.080 72.107 86.134 100.161 114.188 128.214 142.241 156.268
408 466 504.4 537.2 566.1 591 617 638.1 658 674.2
6.59 5.57 4.92 4.32 3.97 3.46 3.18 2.97 2.74 2.60
0.115 0.154 0.204 0.258 0.313 0.369 0.421 0.474 0.527 0.580
0.223 0.221 0.239 0.250 0.264 0.260 0.261 0.265 0.264 0.269
0.282 0.292 0.256 0.278 0.347 0.387 0.427 0.474 0.514 0.582
Formula
CAS no.
Vc, m3/Kmol
2-138
PHYSICAL AND CHEMICAL DATA
TABLE 2-164
Critical Constants and Acentric Factors of Inorganic and Organic Compounds (Continued ) Mol. wt.
Tc, K
Pc × 1E-06 Pa
67641 78933 107879 563804 591786 108101 565617 96220 565695 565800 108941 98862
58.080 72.107 86.134 86.134 100.161 100.161 100.161 86.134 100.161 114.188 98.145 120.151
508.2 535.5 561.08 553 587.05 571.4 573 560.95 567 576 653 709.5
4.71 4.12 3.71 3.84 3.31 3.27 3.32 3.70 3.34 3.06 4.01 3.85
CH2O2 C2H4O2 C3H6O2 C4H8O2 C4H8O2 C7H6O2 C4H6O3
64186 64197 79094 107926 79312 65850 108247
46.026 60.053 74.079 88.106 88.106 122.123 102.090
588 591.95 600.81 615.7 605 751 606
Methyl formate Methyl acetate Methyl propionate Methyl n-butyrate Ethyl formate Ethyl acetate Ethyl propionate
C2H4O2 C3H6O2 C4H8O2 C5H10O2 C3H6O2 C4H8O2 C5H10O2
107313 79209 554121 623427 109944 141786 105373
60.053 74.079 88.106 102.133 74.079 88.106 102.133
149 150 151 152 153 154 155
Ethyl n-butyrate n-Propyl formate n-Propyl acetate n-Butyl acetate Methyl benzoate Ethyl benzoate Vinyl acetate
C6H12O2 C4H8O2 C5H10O2 C6H12O2 C8H8O2 C9H10O2 C4H6O2
105544 110747 109604 123864 93583 93890 108054
156 157 158 159 160 161
Methylamine Dimethylamine Trimethylamine Ethylamine Diethylamine Triethylamine
CH5N C2H7N C3H9N C2H7N C4H11N C6H15N
162 163 164 165 166 167 168
n-Propylamine di-n-Propylamine Isopropylamine Diisopropylamine Aniline N-Methylaniline N,N-Dimethylaniline
169 170 171 172
Cmpd. no.
Name
Formula
Zc
Acentric factor
123 124 125 126 127 128 129 130 131 132 133 134
Acetone Methyl ethyl ketone 2-Pentanone Methyl isopropyl ketone 2-Hexanone Methyl isobutyl ketone 3-Methyl-2-pentanone 3-Pentanone Ethyl isopropyl ketone Diisopropyl ketone Cyclohexanone Methyl phenyl ketone
C3H6O C4H8O C5H10O C5H10O C6H12O C6H12O C6H12O C5H10O C6H12O C7H14O C6H10O C8H8O
0.210 0.267 0.301 0.313 0.369 0.369 0.371 0.336 0.369 0.416 0.311 0.386
0.234 0.247 0.239 0.261 0.250 0.254 0.259 0.267 0.262 0.266 0.230 0.252
0.307 0.320 0.345 0.349 0.395 0.389 0.386 0.340 0.394 0.411 0.308 0.365
135 136 137 138 139 140 141
Formic acid Acetic acid Propionic acid n-Butyric acid Isobutyric acid Benzoic acid Acetic anhydride
5.81 5.74 4.61 4.07 3.68 4.47 3.97
0.125 0.179 0.232 0.291 0.291 0.347 0.290
0.148 0.208 0.214 0.231 0.213 0.248 0.229
0.317 0.463 0.574 0.682 0.612 0.603 0.450
142 143 144 145 146 147 148
487.2 506.55 530.6 554.5 508.4 523.3 546
5.98 4.69 4.03 3.48 4.71 3.85 3.34
0.173 0.229 0.284 0.340 0.231 0.287 0.345
0.255 0.256 0.259 0.257 0.257 0.254 0.254
0.254 0.326 0.349 0.378 0.282 0.363 0.391
116.160 88.106 102.133 116.160 136.150 150.177 86.090
571 538 549.73 579.15 693 698 519.13
2.94 4.03 3.37 3.11 3.59 3.22 3.93
0.403 0.286 0.349 0.389 0.436 0.489 0.270
0.249 0.257 0.257 0.251 0.272 0.271 0.246
0.399 0.310 0.390 0.410 0.421 0.477 0.348
74895 124403 75503 75047 109897 121448
31.057 45.084 59.111 45.084 73.138 101.192
430.05 437.2 433.25 456.15 496.6 535.15
7.41 5.26 4.10 5.59 3.67 3.04
0.154 0.180 0.254 0.202 0.301 0.389
0.319 0.260 0.289 0.298 0.268 0.266
0.279 0.293 0.210 0.283 0.300 0.316
C3H9N C6H15N C3H9N C6H15N C6H7N C7H9N C8H11N
107108 142847 75310 108189 62533 100618 121697
59.111 101.192 59.111 101.192 93.128 107.155 121.182
496.95 550 471.85 523.1 699 701.55 687.15
4.74 3.11 4.54 3.20 5.35 5.19 3.63
0.260 0.401 0.221 0.417 0.270 0.373 0.465
0.298 0.273 0.256 0.307 0.248 0.332 0.295
0.280 0.446 0.276 0.388 0.381 0.480 0.403
Ethylene oxide Furan Thiophene Pyridine
C2H4O C4H4O C4H4S C5H5N
75218 110009 110021 110861
44.053 68.075 84.142 79.101
469.15 490.15 579.35 619.95
7.26 5.55 5.71 5.64
0.142 0.218 0.219 0.254
0.264 0.297 0.260 0.278
0.201 0.205 0.195 0.239
173 174 175 176 177 178 179 180
Formamide N,N-Dimethylformamide Acetamide N-Methylacetamide Acetonitrile Propionitrile n-Butyronitrile Benzonitrile
CH3NO C3H7NO C2H5NO C3H7NO C2H3N C3H5N C4H7N C7H5N
75127 68122 60355 79163 75058 107120 109740 100470
45.041 73.095 59.068 73.095 41.053 55.079 69.106 103.123
771 649.6 761 718 545.5 564.4 582.25 699.35
7.75 4.37 6.57 5.00 4.85 4.19 3.79 4.21
0.163 0.262 0.215 0.267 0.173 0.229 0.278 0.339
0.197 0.212 0.223 0.224 0.185 0.205 0.217 0.245
0.410 0.312 0.419 0.437 0.340 0.325 0.371 0.352
181 182 183 184 185 186 187 188 189
Methyl mercaptan Ethyl mercaptan n-Propyl mercaptan n-Butyl mercaptan Isobutyl mercaptan sec-Butyl mercaptan Dimethyl sulfide Methyl ethyl sulfide Diethyl sulfide
CH4S C2H6S C3H8S C4H10S C4H10S C4H10S C2H6S C3H8S C4H10S
74931 75081 107039 109795 513440 513531 75183 624895 352932
48.109 62.136 76.163 90.189 90.189 90.189 62.136 76.163 90.189
469.95 499.15 536.6 570.1 559 554 503.04 533 557.15
7.23 5.49 4.63 3.97 4.06 4.06 5.53 4.26 3.96
0.145 0.206 0.254 0.307 0.307 0.307 0.200 0.254 0.320
0.268 0.273 0.263 0.257 0.268 0.271 0.264 0.244 0.273
0.158 0.188 0.232 0.272 0.253 0.251 0.194 0.209 0.294
CAS no.
Vc, m3/Kmol
CRITICAL CONSTANTS TABLE 2-164 Cmpd. no.
2-139
Critical Constants and Acentric Factors of Inorganic and Organic Compounds (Concluded ) Name
Formula
CAS no.
Mol. wt.
Tc, K
Pc × 1E-06 Pa
Vc, m3/Kmol
Zc
Acentric factor
190 191 192 193 194 195 196 197
Fluoromethane Chloromethane Trichloromethane Tetrachloromethane Bromomethane Fluoroethane Chloroethane Bromoethane
CH3F CH3Cl CHCl3 CCl4 CH3Br C2H5F C2H5Cl C2H5Br
593533 74873 67663 56235 74839 353366 75003 74964
34.033 50.488 119.377 153.822 94.939 48.060 64.514 108.966
317.42 416.25 536.4 556.35 467 375.31 460.35 503.8
5.88 6.69 5.55 4.54 8.00 5.01 5.46 6.29
0.113 0.142 0.238 0.274 0.156 0.164 0.155 0.215
0.252 0.275 0.296 0.270 0.321 0.263 0.221 0.323
0.198 0.154 0.228 0.191 0.192 0.218 0.206 0.259
198 199 200 201 202 203 204 205
1-Chloropropane 2-Chloropropane 1,1-Dichloropropane 1,2-Dichloropropane Vinyl chloride Fluorobenzene Chlorobenzene Bromobenzene
C3H7Cl C3H7Cl C3H6Cl2 C3H6Cl2 C2H3Cl C6H5F C6H5Cl C6H5Br
540545 75296 78999 78875 75014 462066 108907 108861
78.541 78.541 112.986 112.986 62.499 96.104 112.558 157.010
503.15 489 560 572 432 560.09 632.35 670.15
4.58 4.51 4.24 4.23 5.75 4.54 4.53 4.52
0.247 0.247 0.292 0.291 0.179 0.269 0.308 0.324
0.270 0.274 0.266 0.259 0.287 0.262 0.265 0.263
0.228 0.196 0.253 0.256 0.106 0.247 0.251 0.251
206 207 208 209 210 211 212 213 214 215 216 217 218
Air Hydrogen Helium-4 Neon Argon Fluorine Chlorine Bromine Oxygen Nitrogen Ammonia Hydrazine Nitrous oxide
H2 He Ne Ar F2 Cl2 Br2 O2 N2 NH3 N2H4 N2O
132259100 1333740 7440597 7440019 7440371 7782414 7782505 7726956 7782447 7727379 7664417 302012 10024972
28.951 2.016 4.003 20.180 39.948 37.997 70.905 159.808 31.999 28.014 17.031 32.045 44.013
132.45 33.19 5.2 44.4 150.86 144.12 417.15 584.15 154.58 126.2 405.65 653.15 309.57
3.79 1.32 0.23 2.67 4.90 5.17 7.79 10.28 5.02 3.39 11.30 14.73 7.28
0.092 0.064 0.058 0.042 0.075 0.067 0.124 0.135 0.074 0.089 0.072 0.158 0.098
0.318 0.307 0.305 0.300 0.292 0.287 0.279 0.286 0.287 0.288 0.241 0.429 0.277
0.000 −0.215 −0.388 −0.038 0.000 0.053 0.073 0.128 0.020 0.037 0.253 0.315 0.143
219 220 221 222 223 224 225 226 227 228 229 230 231
Nitric oxide Cyanogen Carbon monoxide Carbon dioxide Carbon disulfide Hydrogen fluoride Hydrogen chloride Hydrogen bromide Hydrogen cyanide Hydrogen sulfide Sulfur dioxide Sulfur trioxide Water
NO C2N2 CO CO2 CS2 HF HCl HBr HCN H2S SO2 SO3 H2O
10102439 460195 630080 124389 75150 7664393 7647010 10035106 74908 7783064 7446095 7446119 7732185
30.006 52.036 28.010 44.010 76.143 20.006 36.461 80.912 27.026 34.082 64.065 80.064 18.015
180.15 400.15 132.92 304.21 552 461.15 324.65 363.15 456.65 373.53 430.75 490.85 647.13
6.52 5.94 3.49 7.39 8.04 6.49 8.36 8.46 5.35 9.00 7.86 8.19 21.94
0.058 0.195 0.095 0.095 0.160 0.069 0.082 0.100 0.139 0.099 0.123 0.127 0.056
0.252 0.348 0.300 0.277 0.280 0.117 0.253 0.280 0.195 0.287 0.269 0.255 0.228
0.585 0.276 0.048 0.224 0.118 0.383 0.134 0.069 0.407 0.096 0.244 0.423 0.343
All substances are listed in alphabetical order in Table 2-6a. Compiled from Daubert, T. E., R. P. Danner, H. M. Sibul, and C. C. Stebbins, DIPPR Data Compilation of Pure Compound Properties, Project 801 Sponsor Release, July, 1993, Design Institute for Physical Property Data, AIChE, New York, NY; and from Ambrose, D. “Vapour-Liquid Critical Properties”, Report Chem 107, National Physical Laboratory, Teddington, UK, October, 1979. In order to ensure thermodynamic consistency, in almost all cases these properties are calculated from Tc and the vapor pressure and liquid density correlation coefficients listed in those tables. This means that there will be slight differences between the values listed here and those in the DIPPR tables. Most of the differences are less than 1%, and almost all the rest are less than the estimated accuracy of the quantity in question. The atomic weights used, taken from J. Phys. Chem. Ref. Data 22(6), 1993, are C = 12.011, H = 1.00794, O = 15.9994, N = 14.00674, S = 32.066, F = 18.9984, Cl = 35.4527, Br = 79.904, and I = 126.90447. The value of the gas constant, R, used here is 8314.51 J/(kmol·K), as given by E. R. Cohen and B. N. Taylor in J. Phys. Chem. Ref. Data 17, 1988. K − 273.15 = °C; 1.8 × K − 459.67 = °F; Pa × 9.869233E-06 = atm; Pa × 1.450377E-04 = psia j; m3/kmol × (1E + 03/mol. wt.) = cm3/g; m3/kmol × (1.601846E + 01/mol wt) = ft3/lb.
2-140
PHYSICAL AND CHEMICAL DATA
COMPRESSIBILITIES reminded that compressibilities can be calculated from the pressure—volume (or density)—temperature tables of the subsection “Thermodynamic Properties.”
INTRODUCTION The increasing ranges of pressure and temperature of interest to technology for an ever-increasing number of substances would necessitate additional tables in this subsection as well as in the subsection “Thermodynamic Properties.” Space restrictions preclude this. Hence, in the present revision, an attempt was made to update the fluidcompressibility tables for selected fluids and to omit tables for other fluids. The reader is thus referred to the fourth edition for tables on miscellaneous gases at 0°C, acetylene, ammonia, ethane, ethylene, hydrogen-nitrogen mixtures, and methyl chloride. The reader is also
TABLE 2-165
UNITS CONVERSIONS For this subsection, the following units conversions are applicable: °R = 9⁄5 K. To convert bars to pounds-force per cubic inch, multiply by 14.504. To convert bars to kilopascals, multiply by 1 × 102.
Compressibility Factors for Air* Pressure, bar
Temp., K
1
5
10
20
40
60
80
100
150
200
250
300
400
500
75 80 90 100 120
0.0052 0.9764 0.9797 0.9880
0.0260 0.0250 0.0236 0.8872 0.9373
0.0519 0.0499 0.0471 0.0453 0.8660
0.1036 0.0995 0.0940 0.0900 0.6730
0.2063 0.1981 0.1866 0.1782 0.1778
0.3082 0.2958 0.2781 0.2635 0.2557
0.4094 0.3927 0.3686 0.3498 0.3371
0.5099 0.4887 0.4581 0.4337 0.4132
0.7581 0.7258 0.6779 0.6386 0.5964
1.0025 0.9588 0.8929 0.8377 0.7720
1.1931 1.1098 1.0395 0.9530
1.4139 1.3110 1.2227 1.1076
1.7161 1.5937 1.5091
2.1105 1.9536 1.7366
140 160 180 200 250
0.9927 0.9951 0.9967 0.9978 0.9992
0.9614 0.9748 0.9832 0.9886 0.9957
0.9205 0.9489 0.9660 0.9767 0.9911
0.8297 0.8954 0.9314 0.9539 0.9822
0.5856 0.7803 0.8625 0.9100 0.9671
0.3313 0.6603 0.7977 0.8701 0.9549
0.3737 0.5696 0.7432 0.8374 0.9463
0.4340 0.5489 0.7084 0.8142 0.9411
0.5909 0.6340 0.7180 0.8061 0.9450
0.7699 0.7564 0.7986 0.8549 0.9713
0.9114 0.8840 0.9000 0.9311 1.0152
1.0393 1.0105 1.0068 1.0185 1.0702
1.3202 1.2585 1.2232 1.2054 1.1990
1.5903 1.4970 1.4361 1.3944 1.3392
300 350 400 450 500
0.9999 1.0000 1.0002 1.0003 1.0003
0.9987 1.0002 1.0012 1.0016 1.0020
0.9974 1.0004 1.0025 1.0034 1.0034
0.9950 1.0014 1.0046 1.0063 1.0074
0.9917 1.0038 1.0100 1.0133 1.0151
0.9901 1.0075 1.0159 1.0210 1.0234
0.9903 1.0121 1.0229 1.0287 1.0323
0.9930 1.0183 1.0312 1.0374 1.0410
1.0074 1.0377 1.0533 1.0614 1.0650
1.0326 1.0635 1.0795 1.0913 1.0913
1.0669 1.0947 1.1087 1.1183 1.1183
1.1089 1.1303 1.1411 1.1463 1.1463
1.2073 1.2116 1.2117 1.2090 1.2051
1.3163 1.3015 1.2890 1.2778 1.2667
600 800 1000
1.0004 1.0004 1.0004
1.0022 1.0020 1.0018
1.0039 1.0038 1.0037
1.0081 1.0077 1.0068
1.0164 1.0157 1.0142
1.0253 1.0240 1.0215
1.0340 1.0321 1.0290
1.0434 1.0408 1.0365
1.0678 1.0621 1.0556
1.0920 1.0844 1.0744
1.1172 1.1061 1.0948
1.1427 1.1283 1.1131
1.1947 1.1720 1.1515
1.2475 1.2150 1.1889
*Calculated from values of pressure, volume (or density), and temperature in Vasserman, Kazavchinskii, and Rabinovich, Thermophysical Properties of Air and Air Components, Moscow, Nauka, 1966, and NBS-NSF Trans. TT 70-50095, 1971; and Vasserman and Rabinovich, Thermophysical Properties of Liquid Air and Its Components, Moscow, 1968, and NBS-NSF Trans. 69-55092, 1970.
TABLE 2-166
Compressibility Factors for Argon*
Temp., K
1
5
10
20
40
60
80
100
200
300
400
500
100 150 200 250 300
0.9773 0.9932 0.9972 0.9988 0.9995
0.0183 0.9647 0.9857 0.9935 0.9969
0.0366 0.9273 0.9713 0.9869 0.9941
0.0729 0.8447 0.9419 0.9741 0.9884
0.1449 0.6101 0.8810 0.9494 0.9777
0.2162 0.2249 0.8208 0.9263 0.9686
0.2867 0.2781 0.7624 0.9056 0.9611
0.3567 0.3324 0.7121 0.8877 0.9552
0.6975 0.5934 0.6870 0.8590 0.9533
1.0267 0.8387 0.8360 0.9207 0.9950
1.3470 1.0732 1.0051 1.0262 1.0673
1.6932 1.2995 1.1982 1.1479 1.1786
400 500 600 800 1000
1.0001 1.0002 1.0003 1.0003 1.0002
0.9997 1.0007 1.0012 1.0012 1.0013
0.9998 1.0012 1.0025 1.0023 1.0022
0.9999 1.0034 1.0046 1.0050 1.0050
1.0004 1.0071 1.0094 1.0102 1.0096
1.0018 1.0113 1.0143 1.0151 1.0142
1.0031 1.0154 1.0198 1.0205 1.0193
1.0056 1.0205 1.0250 1.0258 1.0239
1.0280 1.0501 1.0553 1.0532 1.0484
1.0656 1.0874 1.0904 1.0830 1.0736
1.1157 1.1301 1.1291 1.1147 1.0999
1.1976 1.1997 1.1933 1.1707 1.1497
Pressure, bar
*Calculated from PVT values tabulated in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standard Press, Moscow, 1976. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).
COMPRESSIBILITIES TABLE 2-167
2-141
Compressibility Factors for Carbon Dioxide* Pressure, bar
Temp., °C
1
5
10
20
40
60
80
100
200
300
400
500
0 50 100 150 200
0.9933 0.9964 0.9977 0.9985 0.9991
0.9658 0.9805 0.9883 0.9927 0.9953
0.9294 0.9607 0.9764 0.9853 0.9908
0.8496 0.9195 0.9524 0.9705 0.9818
0.8300 0.9034 0.9416 0.9640
0.7264 0.8533 0.9131 0.9473
0.5981 0.8022 0.8854 0.9313
0.4239 0.7514 0.8590 0.9170
0.5891 0.7651 0.8649
0.6420 0.7623 0.8619
0.8235 0.8995
0.9098 0.9621
250 300 350 400 450
0.9994 0.9996 0.9998 0.9999 1.0000
0.9971 0.9982 0.9991 0.9997 1.0000
0.9943 0.9967 0.9983 0.9994 1.0003
0.9886 0.9936 0.9964 0.9989 1.0005
0.9783 0.9875 0.9938 0.9982 1.0013
0.9684 0.9822 0.9914 0.9979 1.0023
0.9593 0.9773 0.9896 0.9979 1.0038
0.9511 0.9733 0.9882 0.9984 1.0056
0.9253 0.9640 0.9895 1.0073 1.0070
0.9294 0.9746 1.0053 1.0266 1.0412
0.9508 1.0030 1.0340 1.0559 1.0709
1.0096 1.0464 1.0734 1.0928 1.1067
500 600 700 800 900
1.0000 1.0000 1.0003 1.0002 1.0002
1.0004 1.0007 1.0010 1.0009 1.0009
1.0008 1.0013 1.0017 1.0019 1.0020
1.0015 1.0030 1.0036 1.0040 1.0041
1.0035 1.0062 1.0073 1.0082 1.0083
1.0056 1.0093 1.0161 1.0122 1.0128
1.0079 1.0129 1.0155 1.0168 1.0171
1.0107 1.0168 1.0198 1.0212 1.0221
1.0282 1.0386 1.0436 1.0458 1.0463
1.0522 1.0648 1.0707 1.0731 1.0726
1.0820 1.0948 1.1000 1.1016 1.1012
1.1165 1.1277 1.1318 1.1324 1.1303
1000
1.0002
1.0009
1.0021
1.0042
1.0084
1.0128
1.0172
1.0218
1.0460
1.0725
1.0725
1.1274
*Calculated from density-pressure-temperature data in Vukalovitch and Altunin, Thermophysical Properties of Carbon Dioxide, Atomizdat, Moscow, 1965, and Collet’s, London, 1968, translation.
TABLE 2-168
Compressibility Factors for Carbon Monoxide*
Temp., K
1
4
7
10
40
70
100
200 250 300 350 400
0.9973 0.9989 0.9997 1.0000 1.0002
0.9893 0.9957 0.9987 1.0002 1.0010
0.9813 0.9926 0.9977 1.0003 1.0017
0.9734 0.9896 0.9968 1.0005 1.0025
0.9632 0.9907 1.0042 1.0042
0.9896 1.0112 1.0112
0.9935 1.0216 1.0216
450 500 600 700 800
1.0003 1.0004 1.0005 1.0005 1.0004
1.0014 1.0016 1.0018 1.0018 1.0017
1.0025 1.0029 1.0032 1.0032 1.0030
1.0035 1.0041 1.0045 1.0045 1.0044
1.0152 1.0172 1.0186 1.0183 1.0175
1.0285 1.0314 1.0332 1.0325 1.0309
1.0433 1.0469 1.0485 1.0470 1.0445
900 1000 1500 2000 2500
1.0004 1.0004 1.0003 1.0002 1.0002
1.0017 1.0016 1.0012 1.0009 1.0007
1.0029 1.0027 1.0021 1.0016 1.0013
1.0041 1.0039 1.0029 1.0022 1.0018
1.0166 1.0156 1.0115 1.0088 1.0071
1.0291 1.0273 1.0200 1.0155 1.0124
1.0418 1.0391 1.0286 1.0221 1.0178
3000
1.0002
1.0006
1.0010
1.0015
1.0059
1.0104
1.0148
Pressure, atm
*From Hilsenrath et al., N.B.S. Circ. 564, 1955. Some of the above values have been rounded to four decimal places. Values at 10-K increments below 1000 K and at 50 K increments for higher temperatures appear in the original, also for pressures below atmospheric.
TABLE 2-169
Compressibility Factors for Ethanol
Temp., K
0.1
0.5
1.013
10
20
50
100
250
500
300 350 400 450 500
0.0022
0.0023
0.0024
0.999 1.000 1.000
0.993 0.997 0.997
0.986 0.991 0.994
0.0229 0.0215 0.0204 0.908 0.941
0.0458 0.0411 0.0408 0.874
0.114 0.107 0.101 0.101 0.122
0.228 0.208 0.201 0.198 0.214
0.565 0.509 0.490 0.472 0.473
1.11 1.03 0.95 0.898 0.868
600 700 800 900 1000
1.000 1.000 1.000 1.000 1.000
0.998 0.999 1.000 1.000 1.000
0.997 0.999 0.999 1.000 1.000
0.972 0.985 0.992 0.996 0.998
0.943 0.971 0.984 0.992 0.997
0.948 0.973 0.988 0.993
0.672 0.902 0.953 0.981 0.990
0.470 0.760 0.890 0.962 1.002
0.868 0.921 0.988 1.04 1.08
Pressure, bar
Rounded and interpolated from Thermodynamics Research Center tables, Texas A&M University.
2-142
PHYSICAL AND CHEMICAL DATA TABLE 2-170
Compressibility Factors for Ethylene Temperature, K
Pressure, bar
110
150
200
250
300
350
400
450
500
1 5 10 15 20
0.0047 0.0237 0.0472 0.0710 0.0946
0.0038 0.0189 0.0378 0.0566 0.0754
0.9808 0.0162 0.0323 0.0484 0.0644
0.9902 0.9495 0.8946 0.8320 0.7578
0.9944 0.9717 0.9425 0.9121 0.8804
0.9966 0.9828 0.9659 0.9479 0.9299
0.9979 0.9894 0.9785 0.9679 0.9574
0.9986 0.9935 0.9867 0.9749 0.9734
0.9991 0.9959 0.9919 0.9876 0.9833
30 40 60 80 100
0.1418 0.1889 0.2831 0.3767 0.4702
0.1129 0.1504 0.2251 0.2994 0.3734
0.0963 0.1280 0.1910 0.2533 0.3150
0.0950 0.1251 0.1838 0.2410 0.2968
0.8122 0.7342 0.5235 0.3302 0.3480
0.8936 0.8560 0.7791 0.7023 0.6359
0.9357 0.9144 0.8730 0.9056 0.9220
0.9603 0.9477 0.9231 0.9009 0.8825
0.9754 0.9677 0.9541 0.9428 0.9321
150 200 250 300 400
0.7030 0.9337 1.1636 1.3917 1.8441
0.5567 0.7382 0.9179 1.0960 1.4475
0.4671 0.6161 0.7630 0.9075 1.1910
0.4324 0.5630 0.6904 0.8148 1.0565
0.4528 0.5641 0.6740 0.7816 0.9909
0.5842 0.6347 0.7110 0.7969 0.9726
0.7483 0.7499 0.7895 0.8479 0.9849
0.8523 0.8494 0.8710 0.9095 1.0142
0.9167 0.9184 0.9343 0.9631 1.0450
500
solid
1.7934
1.4679
1.2908
1.1932
1.1468
1.1304
1.1341
1.1436
Calculated from Jacobsen, R.T., M. Jahangiri, et al., Ethylene, Blackwell Sci. Publs., Oxford, 1988 (299 pp.).
TABLE 2-171
Compressibility Factors for Normal Hydrogen* Pressure, bar
Temp., K
1
10
20
40
60
80
100
200
400
600
800
1000
20 40 60 80 100
0.0169 0.9848 0.9955 0.9986 0.9998
0.1680 0.8340 0.9562 0.9776 0.9979
0.3302 0.6311 0.9169 0.9763 0.9976
0.6430 0.5240 0.8608 0.9655 1.0022
0.9434 0.6627 0.8498 0.9676 1.0133
1.2346 0.8118 0.8832 0.9842 1.0280
1.5166 0.9590 0.9432 1.0138 1.0528
2.844 1.650 1.347 1.257 1.225
2.878 2.158 1.834 1.659
3.993 2.902 2.389 2.095
5.034 3.598 2.907 2.512
6.019 4.263 3.404 2.902
200 300 400 500 600
1.0007 1.0005 1.0004 1.0004 1.0003
1.0066 1.0059 1.0048 1.0040 1.0034
1.0134 1.0117 1.0096 1.0080 1.0068
1.0275 1.0236 1.0192 1.0160 1.0136
1.0422 1.0357 1.0289 1.0240 1.0204
1.0575 1.0479 1.0386 1.0320 1.0272
1.0734 1.0603 1.0484 1.0400 1.0340
1.163 1.124 1.098 1.080 1.068
1.355 1.253 1.196 1.159 1.133
1.555 1.383 1.293 1.236 1.197
1.753 1.510 1.388 1.311 1.259
1.936 1.636 1.481 1.385 1.320
800 1000 2000
1.0002 1.0002 1.0009
1.0026 1.0021 1.0013
1.0052 1.0042 1.0023
1.0104 1.0084 1.0044
1.0156 1.0126 1.0065
1.0208 1.0168 1.0086
1.0259 1.0209 1.0107
1.051 1.041 1.021
1.100 1.080 1.040
1.147 1.117 1.057
1.193 1.153 1.073
1.237 1.187 1.088
*Calculated from PVT tables of McCarty, Hord, and Roder, NBS Monogr. 168, 1981.
TABLE 2-172
Compressibility Factors for KLEA 60
Temp., K
1
Pressure, bar 5
10
15
20
25
30
250 260 270 280 290
0.9687 0.9780 0.9803 0.9824 0.9848
0.9099 0.9199
300 310 320 330 340
0.9867 0.9872 0.9884 0.9894 0.9905
0.9284 0.9359 0.9425 0.9484 0.9537
0.8459 0.8637 0.8790 0.8908 0.9026
0.7800 0.8066 0.8299 0.8488
0.7577 0.7888
0.6700 0.7184
0.6305
350
0.9920
0.9582
0.9139
0.8663
0.8145
0.7570
0.6908
Zsat Tsat
0.9712 234.0
0.9022 273.1
0.8361 295.0
0.7777 309.5
0.7224 320.7
0.6677 329.8
0.6118 337.6
Zsat
Psat
0.9494 0.9315 0.9098 0.8839 0.8538
2.08 3.11 4.49 6.30 8.62
0.8175 0.7756 0.7261 0.6666
11.55 15.19 19.66 25.10
Converted and interpolated from “Thermodynamic Properties of KLEA 60,” British units, © ICI Chemicals and Polymers, 1993 (20 pp.). Reproduced by permission. KLEA 60 is R32/125/134a (20/40/40 wt %).
COMPRESSIBILITIES TABLE 2-173
Compressibility Factors for KLEA 61
Temp., K
1
2-143
Pressure, bar 5
10
15
20
25
30
250 260 270 280 290
0.9746 0.9773 0.9798 0.9787 0.9838
0.9067 0.9185
300 310 320 330 340
0.9854 0.9868 0.9881 0.9892 0.9903
0.9270 0.9348 0.9416 0.9481 0.9529
0.8431 0.8615 0.8772 0.8909 0.9027
0.7755 0.8042 0.8280 0.8484
0.7148 0.7518 0.7934
0.6659 0.7174
0.6312
350
0.9917
0.9577
0.9131
0.8653
0.8134
0.7565
0.6916
Zsat Tsat
0.9686 230.0
0.8944 269.0
0.8237 290.9
0.7602 305.5
0.7003 316.7
0.6399 325.9
0.5780 333.7
Zsat
Psat
0.9381 0.9172 0.8920 0.8622 0.8272
2.46 3.63 5.18 7.19 9.75
0.7868 0.7377 0.6801 0.6087
12.21 16.88 21.68 27.50
Converted and interpolated from “Thermodynamic Properties of KLEA 61,” British units, © ICI Chemicals and Polymers, 1993 (23 pp.). Reproduced by permission. KLEA 61 is R32/125/134a (10/70/20 wt %).
TABLE 2-174
Compressibility Factors for KLEA 66
Temp., K
1
Pressure, bar 5
10
15
20
25
30
250 260 270 280 290
0.974 0.9772 0.9796 0.9838 0.9858
0.9089 0.9209
300 310 320 330 340
0.9872 0.9883 0.9896 0.9907 0.9917
0.9287 0.9359 0.9431 0.9490 0.9540
0.8461 0.8663 0.8786 0.8910 0.9035
0.8056 0.8292 0.8492
0.7551 0.7878
0.7147
350
0.9926
0.9588
0.9137
0.8659
0.8127
0.7542
0.6843
Zsat Tsat
0.9719 236.1
0.9044 275.5
0.8397 297.6
0.7827 312.1
0.7289 323.5
0.6759 332.6
0.6220 340.4
Zsat
Psat
0.9541 0.9374 0.9172 0.8931 0.8645
1.89 2.84 4.12 5.81 7.98
0.8328 0.7920 0.7462 0.6918 0.6255
10.73 14.15 18.37 23.50 29.73
Converted and interpolated from “Thermodynamic properties of KLEA 66,” British units, © ICI Chemicals and Polymers, 1993 (20 pp.). Reproduced by permission. KLEA 66 is R32/125/134a (23/25/52 wt %).
TABLE 2-175
Compressibility Factors for Krypton*
Temp., K
1
5
10
20
40
60
80
100
200
300
400
500
150 200 250 300 350
0.9837 0.9933 0.9966 0.9982 0.9989
0.9155 0.9648 0.9841 0.9899 0.9949
0.0310 0.9278 0.9635 0.9800 0.9897
0.0618 0.8459 0.9265 0.9595 0.9793
0.1227 0.6039 0.8468 0.9197 0.9522
0.1829 0.1870 0.7605 0.8807 0.9415
0.2423 0.2393 0.6680 0.8437 0.9250
0.3012 0.2903 0.5810 0.8097 0.9110
0.5875 0.5313 0.5785 0.7337 0.8774
0.8636 0.7568 0.7461 0.7954 0.8992
1.1315 0.9730 0.9197 0.9302 0.9799
1.3932 1.1820 1.0891 1.0627 1.0664
400 450 500 600 800
0.9993 0.9998 0.9998 1.0000 1.0002
0.9967 0.9985 0.9992 1.0003 1.0010
0.9933 0.9969 0.9984 1.0005 1.0020
0.9867 0.9939 0.9970 1.0012 1.0041
0.9746 0.9886 0.9942 1.0025 1.0079
0.9635 0.9838 0.9921 1.0043 1.0122
0.9539 0.9800 0.9910 1.0064 1.0170
0.9459 0.9774 0.9906 1.0091 1.0214
0.9323 0.9663 1.0019 1.0301 1.0475
0.9570 1.0011 1.0311 1.0618 1.0779
1.0150 1.0543 1.0732 1.1000 1.1112
1.0910 1.1142 1.1258 1.1431 1.1147
1000
1.0002
1.0013
1.0023
1.0045
1.0091
1.0135
1.0184
1.0230
1.0486
1.0767
1.1063
1.1369
Pressure, bar
*Calculated from PVT values tabulated in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).
2-144
PHYSICAL AND CHEMICAL DATA
TABLE 2-176
Compressibility Factors for Methane (R50)*
Temp., K
1
5
10
20
40
60
80
100
200
300
400
500
150 200 250 300 350
0.9854 0.9936 0.9965 0.9983 0.9991
0.9225 0.9676 0.9838 0.9915 0.9954
0.8275 0.9339 0.9680 0.9830 0.9911
0.0714 0.8599 0.9352 0.9667 0.9825
0.1411 0.6784 0.8682 0.9343 0.9662
0.2093 0.3559 0.8020 0.9047 0.9520
0.2763 0.3172 0.7386 0.8783 0.9401
0.3423 0.3618 0.6854 0.8556 0.9306
0.6599 0.6141 0.6899 0.8280 0.9227
0.9623 0.8568 0.8554 0.9154 0.9800
1.2537 1.0887 1.0359 1.0432 1.0723
1.5363 1.3122 1.2155 1.1829 1.1804
400 450 500 600 800
0.9995 0.9997 0.9999 1.0000 1.0003
0.9977 0.9989 0.9997 1.0009 1.0017
0.9953 0.9979 0.9995 1.0020 1.0034
0.9912 0.9963 0.9995 1.0039 1.0068
0.9835 0.9935 0.9996 1.0081 1.0130
0.9772 0.9917 1.0005 1.0125 1.0197
0.9726 0.9911 1.0022 1.0171 1.0263
0.9696 0.9916 1.0048 1.0217 1.0330
0.9779 1.0098 1.0285 1.0540 1.0678
1.0245 1.0528 1.0699 1.0969 1.1068
1.0986 1.1152 1.1248 1.1470 1.1496
1.1859 1.1899 1.1899 1.2019 1.1951
1000
1.0004
1.0014
1.0035
1.0071
1.0141
1.0207
1.0274
1.0342
1.0678
1.1033
1.1400
1.1790
Pressure, bar
*Calculated from PVT values tabulated in Goodwin, NBS Tech. Note 653, 1974, for temperatures up to 500 K, and from PVT values tabulated in Zhuravlev. Thermophysical Properties of Gaseous and Liquid Methane, Standartov, Moscow, 1969, and NBS-NSF transl. TT 70-50097, 1970.
TABLE 2-177
Compressibility Factors for Methanol Pressure, bar
Temp., K
0.1
0.5
1.0133
10
20
50
100
150
200
250
300
400
500
200 250 300 350 400
0.0002 0.0002 0.9792 0.9844 0.9872
0.0011 0.0009 0.0008 0.9713 0.9795
0.0022 0.0019 0.0017 0.9551 0.9722
0.0219 0.0185 0.0164 0.0150 0.0142
0.0438 0.0370 0.0327 0.0298 0.0283
0.1091 0.0923 0.0813 0.0742 0.0702
0.2174 0.1837 0.1617 0.1473 0.1386
0.3250 0.2743 0.2413 0.2193 0.2056
0.4319 0.3643 0.3201 0.2904 0.2714
0.5381 0.4535 0.3981 0.3606 0.3362
0.6437 0.5422 0.4755 0.4301 0.4000
0.8531 0.7176 0.6284 0.5671 0.5253
1.6030 0.8909 0.7791 0.7016 0.6478
450 500 600 700 800
0.9890 0.9903 0.9922 0.9934 0.9964
0.9835 0.9859 0.9889 0.9907 0.9920
0.9792 0.9828 0.9867 0.9889 0.9904
0.9145 0.9525 0.9756 0.9816 0.9838
0.7989 0.9081 0.9643 0.9778 0.9818
0.0701 0.6799 0.9042 0.9541 0.9711
0.1366 0.1505 0.7629 0.8932 0.9411
0.2007 0.2110 0.6275 0.8392 0.9156
0.2629 0.2699 0.5255 0.8027 0.9025
0.3238 0.3271 0.4921 0.7797 0.8994
0.3834 0.3829 0.5010 0.7675 0.9026
0.4997 0.4912 0.5606 0.7713 0.9205
0.6128 0.5959 0.6358 0.7993 0.9485
Goodwin, R.D., J. Phys. Chem. Ref. Data, 16 (4), 799, 1987.
TABLE 2-178
Compressibility Factors for Neon*
Temp., K
1
5
10
20
40
60
80
100
200
300
400
500
50 100 150 200 250
0.9913 0.9993 1.0002 1.0003 1.0001
0.9472 0.9970 1.0017 1.0023 1.0022
0.9083 0.9949 1.0036 1.0049 1.0045
0.8013 0.9913 1.0078 1.0100 1.0097
0.3810 0.9854 1.0162 1.0204 1.0198
0.4398 0.9245 1.0262 1.0318 1.0295
0.4984 0.9864 1.0375 1.0427 1.0403
0.5850 0.9930 1.0497 1.0551 1.0502
0.9864 1.0796 1.1236 1.1191 1.1057
1.3659 1.2197 1.2131 1.1909 1.1633
1.7289 1.3796 1.3113 1.2655 1.2223
2.0794 1.5473 1.4150 1.3422 1.2822
300 400 500 600 800
1.0000 1.0000 1.0000 1.0000 1.0000
1.0020 1.0017 1.0014 1.0012 1.0009
1.0041 1.0036 1.0029 1.0024 1.0018
1.0091 1.0074 1.0058 1.0049 1.0043
1.0181 1.0151 1.0124 1.0107 1.0081
1.0277 1.0216 1.0188 1.0160 1.0123
1.0369 1.0301 1.0252 1.0214 1.0163
1.0469 1.0376 1.0316 1.0267 1.0206
1.0961 1.0771 1.0641 1.0542 1.0413
1.1476 1.1172 1.0963 1.0814 1.0622
1.1997 1.1575 1.1291 1.1091 1.0829
1.2520 1.1981 1.1621 1.1369 1.1039
1000
1.0000
1.0007
1.0014
1.0034
1.0068
1.0098
1.0132
1.0165
1.0330
1.0500
1.0670
1.0836
Pressure, bar
*Calculated from PVT values tabulated in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).
COMPRESSIBILITIES
2-145
TABLE 2-179
Compressibility Factors for Nitrogen*
Temp., K
1
5
10
20
40
60
80
100
200
300
400
500
70 80 90 100 120
0.0057 0.9593 0.9722 0.9798 0.9883
0.0287 0.0264 0.0251 0.8910 0.9397
0.0573 0.0528 0.0500 0.0487 0.8732
0.1143 0.1053 0.0996 0.0966 0.7059
0.2277 0.2093 0.1975 0.1905 0.1975
0.3400 0.3122 0.2938 0.2823 0.2822
0.4516 0.4140 0.3888 0.3720 0.3641
0.5623 0.5148 0.4826 0.4605 0.4438
1.1044 1.0061 0.9362 0.8840 0.8188
1.6308 1.4797 1.3700 1.2852 1.1684
Solid 1.9396 1.7890 1.6707 1.5015
Solid 2.3879 2.1962 2.0441 1.8223
140 160 180 200 250
0.9927 0.9952 0.9967 0.9978 0.9992
0.9635 0.9766 0.9846 0.9897 0.9960
0.9253 0.9529 0.9690 0.9791 0.9924
0.8433 0.9042 0.9381 0.9592 0.9857
0.6376 0.8031 0.8782 0.9212 0.9741
0.4251 0.7017 0.8125 0.8882 0.9655
0.4278 0.6304 0.7784 0.8621 0.9604
0.4799 0.6134 0.7530 0.8455 0.9589
0.7942 0.8107 0.8550 0.9067 1.0048
1.0996 1.0708 1.0669 1.0760 1.1143
1.3920 1.3275 1.2893 1.2683 1.2501
1.6726 1.5762 1.5105 1.4631 1.3962
300 350 400 450 500
0.9998 1.0001 1.0002 1.0003 1.0004
0.9990 1.0007 1.0011 1.0018 1.0020
0.9983 1.0011 1.0024 1.0033 1.0040
0.9971 1.0029 1.0057 1.0073 1.0081
0.9964 1.0069 1.0125 1.0153 1.0167
0.9973 1.0125 1.0199 1.0238 1.0257
1.0000 1.0189 1.0283 1.0332 1.0350
1.0052 1.0271 1.0377 1.0430 1.0451
1.0559 1.0810 1.0926 1.0973 1.0984
1.1422 1.1560 1.1609 1.1606 1.1575
1.2480 1.2445 1.2382 1.2303 1.2213
1.3629 1.3405 1.3216 1.3043 1.2881
600 800 1000
1.0004 1.0004 1.0003
1.0021 1.0017 1.0015
1.0040 1.0036 1.0034
1.0084 1.0074 1.0067
1.0173 1.0157 1.0136
1.0263 1.0237 1.0205
1.0355 1.0320 1.0275
1.0450 1.0402 1.0347
1.0951 1.0832 1.0714
1.1540 1.1264 1.1078
1.2028 1.1701 1.1449
1.2657 1.2140 1.1814
Pressure, bar
*Computed from pressure-volume-temperature tables in the Vasserman monographs referenced under Table 2-165. TABLE 2-180
Compressibility Factors for Oxygen* Pressure, bar
Temp., K
1
5
10
20
40
60
80
100
200
300
400
500
75 80 90 100 120
0.0043 0.0041 0.0038 0.9757 0.9855
0.0213 0.0203 0.0188 0.0177 0.9246
0.0425 0.0406 0.0376 0.0354 0.8367
0.0849 0.0811 0.0750 0.0705 0.0660
0.1693 0.1616 0.1494 0.1404 0.1302
0.2533 0.2418 0.2233 0.2096 0.1935
0.3368 0.3214 0.2966 0.2783 0.2558
0.4200 0.4007 0.3696 0.3464 0.3173
0.8301 0.7912 0.7281 0.6798 0.6148
1.2322 1.1738 1.0780 1.0040 0.8999
1.6278 1.5495 1.4211 1.3206 1.1762
2.0175 1.9196 1.7580 1.6309 1.4456
140 160 180 200 250
0.9911 0.9939 0.9960 0.9970 0.9987
0.9535 0.9697 0.9793 0.9853 0.9938
0.9034 0.9379 0.9579 0.9705 0.9870
0.7852 0.8689 0.9134 0.9399 0.9736
0.1334 0.6991 0.8167 0.8768 0.9477
0.1940 0.3725 0.7696 0.8140 0.9237
0.2527 0.2969 0.5954 0.7534 0.9030
0.3099 0.3378 0.5106 0.6997 0.8858
0.5815 0.5766 0.6043 0.6720 0.8563
0.8374 0.8058 0.8025 0.8204 0.9172
1.0832 1.0249 0.9990 0.9907 1.0222
1.3214 1.2364 1.1888 1.1623 1.1431
300 350 400 450 500
0.9994 0.9998 1.0000 1.0002 1.0002
0.9968 0.9990 1.0000 1.0007 1.0011
0.9941 0.9979 1.0000 1.0015 1.0022
0.9884 0.9961 1.0000 1.0024 1.0038
0.9771 0.9919 1.0003 1.0048 1.0075
0.9676 0.9890 1.0011 1.0074 1.0115
0.9597 0.9870 1.0022 1.0106 1.0161
0.9542 0.9870 1.0045 1.0152 1.0207
0.9560 1.0049 1.0305 1.0445 1.0523
0.9972 1.0451 1.0718 1.0859 1.0927
1.0689 1.1023 1.1227 1.1334 1.1380
1.1572 1.1722 1.1816 1.1859 1.1866
600 800 1000
1.0003 1.0003 1.0003
1.0014 1.0014 1.0013
1.0024 1.0026 1.0026
1.0052 1.0055 1.0053
1.0102 1.0109 1.0101
1.0153 1.0164 1.0149
1.0207 1.0219 1.0198
1.0266 1.0271 1.0253
1.0582 1.0565 1.0507
1.0961 1.0888 1.0783
1.1374 1.1231 1.1072
1.1803 1.1582 1.1369
*Calculated from pressure-volume-temperature tables in the Vasserman monographs listed under Table 2-165. TABLE 2-181
Compressibility Factors for Refrigerant 32* Pressure, bar
Temp., K
1
5
10
15
20
25
30
40
50
230 240 250 260 270
0.9656 0.9711 0.9755 0.9791 0.9819
0.8865 0.9036
280 290 300 310 320
0.9844 0.9864 0.9880 0.9894 0.9904
0.9180 0.9285 0.9376 0.9453 0.9518
0.8210 0.8476 0.8686 0.8358 0.8998
0.7899 0.8197 0.8436
0.7439 0.7812
0.7089
330 340 350
0.9914 0.9923 0.9932
0.9573 0.9619 0.9655
0.9118 0.9203 0.9296
0.8628 0.8790 0.8932
0.8102 0.8338 0.8534
0.7518 0.7846 0.8115
0.6851 0.7316 0.7671
0.6021 0.6675
0.5312
0.9595 221.2
0.8843 258.8
0.8202 279.8
0.7670 293.8
0.7191 304.6
0.6722 313.5
0.6303 321.1
0.5427 333.9
0.4467 344.3
Zsat Tsat
Zsat
Psat
0.9453 0.9278 0.9062 0.8811 0.8522
1.54 2.40 3.60 5.22 7.34
0.8194 0.7822 0.7401 0.6922 0.6370
10.07 13.51 17.76 22.95 29.21
0.5719 0.4905 0.3702
36.72 45.66 56.35
*Converted and interpolated from British units shown in Thermodynamic properties of KLEA 32, ICI Chemicals and Polymers, 1993. Reproduced by permission.
2-146
PHYSICAL AND CHEMICAL DATA
TABLE 2-182
Compressibility Factors for Refrigerant 123
Temp., °C
1
40 50 60 70 80
0.9639 0.9682 0.9717 0.9745 0.9766
0.9248 0.9327 0.9401
100 120 140 160 180
0.9804 0.9839 0.9861 0.9886 0.9908
0.9501 0.9591 0.9650 0.9714 0.9762
0.9197 0.9146 0.9282 0.9406 0.9518
0.8355 0.8667 0.8915 0.9077 0.9254
0.8140 0.8503 0.8747 0.8970
0.8023 0.8398 0.8709
0.7479 0.8026 0.8402
0.6916 0.7600 0.8072
0.7134 0.7712
0.6553 0.7346
200 225 250
0.9924 0.9938 0.9954
0.9806 0.9846 0.9885
0.9602 0.9692 0.9
0.9388 0.9526 0.9651
0.9174 0.9378 0.9528
0.8931 0.9170 0.9382
0.8688 0.8972 0.9229
0.8422 0.8800 0.9101
0.8163 0.8566 0.8930
Zsat T
0.9575 27.5
0.9210 55.4
0.9730 80.8
0.8292 97.9
0.7947 111.1
0.7654 122.0
0.7229 131.4
0.7110 139.7
Pressure, bar 2.5
5
7.5
10
12.5
15
17.5
20
22.5
25
Zsat
Psat
0.9427 0.9294 0.9134 0.8950 0.8727
1.54 2.13 2.96 3.78 4.90
0.6841
0.8262 0.7640 0.6890 0.5820 0.3926
7.87 12.01 17.59 24.92 34.54
0.7882 0.8401 0.8771
0.7539 0.8157 0.8581
— — —
— — —
0.6564 147.2
0.6206 154.0
0.5821 160.2
— —
— —
20
22.5
25
Dashes indicate inaccessible states; blanks indicate no available data. TABLE 2-183
Compressibility Factors for Refrigerant 124
Temp., °C
1
Pressure, bar 2.5
−20 −10 0 10 20
0.9573 0.9641 0.9693
30 40 50 60 80
0.9736 0.9675 0.9798 0.9820 0.9854
0.9313 0.9396 0.9473 0.9534 0.9633
100 120 140 160 180
0.9880 0.9899 0.9917
0.9700 0.9749 0.9794 0.9825
5
7.5
0.9468 −12.4
0.9071 11.9
12.5
15
17.5
0.8728 0.8889 0.9017 0.9226
0.8229 0.8462 0.8820
0.8366
0.9370 0.9478 0.9575 0.9645 0.9690
0.9040 0.9206 0.9357 0.9464 0.9536
0.8710 0.8935 0.9138 0.9247 0.9382
0.8314 0.8634 0.8884 0.9061 0.9213
0.7918 0.8329 0.8641 0.8868 0.9056
0.7463 0.8022 0.8391 0.8644 0.8857
0.6950 0.7682 0.8105 0.8489 0.8634
0.6380 0.7285 0.7896 0.8285 0.8574
0.6878 0.7647 0.7868 0.8379
0.9601
0.9471 0.9589 0.9650
0.9338 0.9488 0.9573
0.9211 0.9391 0.9443
0.9042 0.9223 0.9412
0.8951 0.9160 0.9333
0.8783 0.9040 0.9252
0.8647 0.8947 0.9174
0.8185 48.7
0.7830 60.2
0.7488 69.6
0.7157 77.8
0.6825 85.0
0.6484 91.4
0.6279 96.3
0.5788 102.6
200 225 250 Zsat T
10
0.8605 34.0
0.7251
Zsat
Psat
0.9562 0.9431 0.9284 0.9243 0.8920
0.72 1.10 1.63 2.34 3.27
0.8828 0.8427 0.8151 0.7803 0.7024
4.45 5.93 7.75 9.96 15.74
0.5955 0.3912
23.75 34.70
Dashes indicate inaccessible states; blanks indicate no available data. TABLE 2-184
Compressibility Factors for Refrigerant 134a Pressure, bar
Temp., °C
1
−10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 satn. sat. T
0.9622 0.9710 0.9752 0.9778 0.9817 0.9839 0.9857 0.9872 0.9886 0.9897 0.9908 0.9916 0.9920 0.9924 0.9927 0.9929 0.9931 0.9567 −26.37
5
0.8819 0.8973 0.9098 0.9206 0.9296 0.9376 0.9442 0.9495 0.9543 0.9592 0.9638 0.9673 0.9691 0.9727 0.8741 15.74
10
0.8005 0.8280 0.8449 0.8678 0.8828 0.8954 0.9062 0.9151 0.9235 0.9308 0.9370 0.9428 0.7989 39.39
15
0.7361 0.7917 0.8137 0.8390 0.8555 0.8630 0.8802 0.8949 0.9040 0.8877 0.7017 55.23
Dashes indicate inaccessible states; blanks indicate no available data.
20
0.6853 0.7327 0.7682 0.7965 0.8144 0.8386 0.8553 0.8694 0.8817 0.6704 67.49
25
0.6290 0.6860 0.7335 0.7630 0.7915 0.8165 0.8350 0.8495 0.6094 77.57
30
0.5832 0.6557 0.7046 0.7418 0.7716 0.7964 0.8173 0.5415 86.20
40
0.5732 0.6249 0.6771 0.7169 0.7489 0.4442 100.35
50
Zsat
Psat
0.4530 0.4885 0.5645 0.6303 0.6783 — —
0.9316 0.9119 0.8888 0.8621 0.8314 0.7963 0.7560 0.7098 0.6562 0.5911 0.5054 0.3462 — — — — — — —
2.005 2.926 4.144 5.716 7.701 10.17 13.18 16.82 21.17 26.38 32.45 39.72 — — — — — — —
TABLE 2-185 Compressibility Factors for Water Substance (fps units)* Pressure, lb/in2 abs.
Temp., °F 400
600
800
1000
1200
1400
1600
1800
2000
2200
2400
2600
2800
3000
3200
3400
3600
3800
4000
10 15 20 40 60
0.9965 0.9943 0.9930 0.9861 0.9788
0.9989 0.9972 0.9970 0.9940 0.9910
0.9992 0.9986 0.9981 0.9967 0.9951
0.9995 0.9993 0.9991 0.9981 0.9973
0.9999 0.9997 0.9995 0.9990 0.9984
0.9999 0.9998 0.9996 0.9994 0.9991
0.9999 0.9999 0.9998 0.9996 0.9994
1.0000 0.9999 0.9999 0.9998 0.9997
1.0000 1.0000 1.0000 0.9999 0.9999
1.0000 1.0000 1.0000 0.9999 0.9999
1.0001 1.0001 1.0001 1.0001 1.0001
1.0006 1.0004 1.0003 1.0003 1.0003
1.0012 1.0012 1.0011 1.0010 1.0009
1.0024 1.0022 1.0020 1.0018 1.0018
1.0053 1.0042 1.0036 1.0028 1.0024
1.0084 1.0072 1.0065 1.0054 1.0048
1.0145 1.0124 1.0112 1.0090 1.0080
1.0211 1.0188 1.0173 1.0139 1.0120
1.0332 1.0295 1.0269 1.0214 1.0186
80 100 150 200 400
0.9714 0.9469 0.9435 0.9216
0.9878 0.9848 0.9770 0.9690 0.9356
0.9935 0.9919 0.9879 0.9839 0.9675
0.9963 0.9954 0.9931 0.9908 0.9817
0.9979 0.9974 0.9960 0.9947 0.9893
0.9987 0.9985 9.9976 0.9968 0.9935
0.9992 0.9990 0.9985 0.9980 0.9960
0.9996 0.9995 0.9993 0.9991 0.9982
0.9998 0.9998 0.9997 0.9996 0.9992
0.9999 0.9999 0.9998 0.9998 0.9998
1.0001 1.0001 1.0001 1.0001 1.0002
1.0003 1.0004 1.0004 1.0005 1.0007
1.0008 1.0007 1.0006 1.0007 1.0011
1.0016 1.0015 1.0014 1.0015 1.0017
1.0023 1.0022 1.0021 1.0021 1.0023
1.0044 1.0042 1.0039 1.0037 1.0033
1.0073 1.0067 1.0059 1.0055 1.0049
1.0108 1.0099 1.0087 1.0080 1.0070
1.0170 1.0157 1.0137 1.0126 1.0105
0.8989 0.8586 0.8138 0.6702
0.9509 0.9336 0.9162 0.8695 0.8188
0.9725 0.9633 0.9540 0.9305 0.9067
0.9839 0.9790 0.9733 0.9600 0.9468
0.9904 0.9872 0.9841 0.9764 0.9687
0.9942 0.9925 0.9905 0.9859 0.9813
0.9973 0.9964 0.9955 0.9932 0.9900
0.9988 0.9985 0.9981 0.9971 0.9958
0.9997 0.9996 0.9994 0.9992 0.9990
1.0002 1.0003 1.0004 1.0007 1.0010
1.0008 1.0010 1.0012 1.0017 1.0023
1.0014 1.0016 1.0019 1.0026 1.0034
1.0019 1.0022 1.0025 1.0033 1.0042
1.0026 1.0029 1.0032 1.0040 1.0049
1.0034 1.0036 1.0039 1.0048 1.0058
1.0048 1.0049 1.0052 1.0059 1.0068
1.0066 1.0065 1.0066 1.0072 1.0082
1.0097 1.0094 1.0092 1.0096 1.0104
0.5608
0.8060 0.7042 0.6185 0.5699
0.8942 0.8442 0.8003 0.7657
0.9392 0.9121 0.8883 0.8693
0.9647 0.9497 0.9371 0.9274
0.9836 0.9771 0.9714 0.9668
0.9930 0.9907 0.9895 0.9890
0.9989 0.9991 1.0004 1.0025
1.0024 1.0048 1.0075 1.0105
1.0050 1.0081 1.0118 1.0158
1.0069 1.0110 1.0152 1.0196
1.0082 1.0128 1.0172 1.0220
1.0093 1.0139 1.0188 1.0240
1.0106 1.0152 1.0204 1.0258
1.0118 1.0165 1.0216 1.0271
1.0132 1.0179 1.0229 1.0284
1.0149 1.0195 1.0242 1.0298
600 800 1,000 1,500 2,000 4,000 6,000 8,000 10,000
*Calculated by P. E. Liley from various steam tables for the lower temperatures and from Paper B-11 by P. H. Kesselman and Yu. I. Blank, 7th. Int. Conf. Properties of Steam, Tokyo, 1968, for the higher temperatures.
2-147
TABLE 2-186 Temperature, K
Compressibility Factors of Water Substance (SI units)* Pressure, bar 1
5
10
15
20
25
30
40
50
60
80
100
150
200
250
300
400
500
600
800
400 450 500 550 600 650
0.990 0.993 0.996 0.997 0.998 0.999
0.003 0.003 0.980 0.985 0.990 0.992
0.006 0.006 0.958 0.969 0.979 0.984
0.009 0.009 0.930 0.956 0.970 0.977
0.012 0.012 0.901 0.939 0.961 0.968
0.014 0.014 0.878 0.922 0.948 0.959
0.017 0.016 0.016 0.904 0.935 0.958
0.023 0.022 0.021 0.865 0.910 0.937
0.029 0.027 0.026 0.822 0.885 0.919
0.035 0.033 0.031 0.773 0.858 0.902
0.046 0.043 0.042 0.042 0.798 0.864
0.058 0.054 0.052 0.052 0.726 0.824
0.086 0.080 0.077 0.077 0.082 0.702
0.114 0.107 0.102 0.102 0.107 0.514
0.143 0.134 0.127 0.126 0.131 0.177
0.171 0.159 0.152 0.150 0.155 0.183
0.227 0.206 0.201 0.181 0.201 0.221
0.282 0.255 0.249 0.198 0.246 0.260
0.336 0.304 0.297 0.289 0.290 0.303
0.445 0.402 0.390 0.378 0.375 0.383
1000 0.552 0.498 0.482 0.464 0.457 0.460
700 750 800 850 900 950
1.000 1.000 1.000 1.000 1.000 1.000
0.994 0.996 0.997 0.997 0.998 0.998
0.988 0.991 0.993 0.995 0.997 0.997
0.984 0.988 0.991 0.992 0.993 0.994
0.976 0.981 0.985 0.989 0.992 0.994
0.967 0.975 0.982 0.984 0.989 0.993
0.966 0.971 0.976 0.981 0.986 0.991
0.952 0.961 0.970 0.977 0.982 0.985
0.941 0.955 0.966 0.973 0.979 0.983
0.929 0.945 0.957 0.967 0.974 0.980
0.900 0.927 0.945 0.957 0.965 0.973
0.876 0.907 0.929 0.946 0.958 0.967
0.800 0.856 0.892 0.917 0.936 0.950
0.716 0.801 0.853 0.889 0.915 0.933
0.618 0.743 0.813 0.860 0.893 0.916
0.503 0.682 0.773 0.831 0.872 0.901
0.326 0.557 0.693 0.775 0.830 0.867
0.316 0.465 0.620 0.715 0.792 0.839
0.340 0.435 0.568 0.679 0.760 0.816
0.406 0.456 0.538 0.631 0.714 0.780
0.476 0.509 0.561 0.629 0.700 0.761
1000 1200 1400 1600 1800 2000
1.000 1.000 1.000 1.000 1.001 1.003
0.999 1.000 1.000 1.000 1.001 1.002
0.998 0.999 1.000 1.000 1.001 1.002
0.995 0.998 1.000 1.000 1.000 1.002
0.995 0.998 1.000 1.000 1.000 1.002
0.994 0.997 1.000 1.000 1.000 1.002
0.993 0.997 1.000 1.000 1.000 1.002
0.990 0.995 0.999 1.000 1.000 1.002
0.987 0.994 0.998 1.000 1.000 1.002
0.985 0.994 0.998 1.000 1.001 1.003
0.978 0.992 0.998 1.000 1.002 1.003
0.973 0.990 0.997 1.000 1.003 1.004
0.960 0.986 0.996 1.001 1.003 1.004
0.948 0.982 0.995 1.002 1.004 1.006
0.935 0.975 0.995 1.002 1.005 1.008
0.923 0.968 0.994 1.004 1.008 1.011
0.900 0.961 0.993 1.006 1.011 1.014
0.878 0.957 0.992 1.009 1.014 1.018
0.859 0.949 0.994 1.012 1.017 1.021
0.831 0.942 0.996 1.015 1.021 1.032
0.816 0.937 0.998 1.020 1.031 1.043
*Calculated by P. E. Liley from various steam tables for the lower temperatures and from Pap. B-11 by P. H. Kesselman and Yu. I. Blank, 7th Internal Conference on the Properties of Steam, Tokyo, 1968, for the higher temperatures.
2-148
COMPRESSIBILITIES TABLE 2-187 Temperature, K
2-149
Compressibility Factors for Xenon* Pressure, bar 1
5
10
20
40
60
80
100
200
300
400
500
200 250 300 350 400
0.9831 0.9911 0.9949 0.9967 0.9977
0.9088 0.9545 0.9736 0.9834 0.9892
0.0293 0.9052 0.9465 0.9669 0.9183
0.0584 0.7887 0.8885 0.9322 0.9562
0.1162 0.1114 0.7517 0.8473 0.9128
0.1733 0.1642 0.5492 0.7840 0.8696
0.2300 0.2158 0.2794 0.7039 0.8278
0.2861 0.2663 0.3016 0.6249 0.7888
0.5601 0.5074 0.5021 0.5645 0.6916
0.8253 0.7355 0.6997 0.7124 0.7642
1.0833 0.9546 0.8886 0.8706 0.8850
1.3356 1.1670 1.0707 1.0269 1.0148
450 500 600 800 1000
0.9989 0.9982 0.9996 1.0000 1.0000
0.9928 0.9951 0.9979 0.9998 1.0004
0.9856 0.9902 0.9957 1.0002 1.0015
0.9714 0.9810 0.9917 1.0004 1.0031
0.9429 0.9623 0.9841 1.0012 1.0144
0.9163 0.9452 0.9772 1.0020 1.0101
0.8911 0.9293 0.9715 1.0034 1.0133
0.8679 0.9156 0.9667 1.0054 1.0172
0.7335 0.8774 0.9596 1.0213 1.0394
0.8331 0.8953 0.9791 1.0476 1.0669
0.9187 0.9572 1.0211 1.0818 1.0979
1.0224 1.0412 1.0799 1.1222 1.1331
*Calculated from PVT values tabulated in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).
TABLE 2-188
Compressibilities of Liquids*
At the constant temperature T, the compressibility β = (1/V 0)(dV/dP). In general as P increases, β decreases rapidly at first and then slowly; the change of β with T is large at low pressures but very small at pressures above 1000 to 2000 megabars. 1 megabar = 0.987 atm. = 106 dynes/cm2 based upon the older usage, 1 bar = 1 dyne/cm2. The use of the bar as a pressure unit is not encouraged.
Substance
Temp., °C
Pressure, megabars
Acetone Acetone Acetone Acetone Amyl alcohol alcohol, iso. alcohol, iso. alcohol, n alcohol, n alcohol, n alcohol, n Benzene Benzene Benzene Bromine Bromine Butyl alcohol, iso alcohol, iso alcohol, iso alcohol, iso alcohol, iso alcohol, iso Carbon bisulfide bisulfide bisulfide bisulfide tetrachloride tetrachloride Chloroform Chloroform Dichloroethylsulfide Dichloroethylsulfide Ethyl acetate acetate
14 20 20 40 14 20 20 20 20 20 40 17 20 20 20 20 18 20 20 20 20 20 16 20 20 20 20 20 20 20 32 32 13 20
23 500 1,000 12,000 23 200 400 500 1,000 12,000 12,000 5 200 400 200 400 8 200 400 500 1,000 12,000 21 500 1,000 12,000 200 400 200 400 1,000 2,000 23 200
Compressibility per megabar β × 106
Substance
111 61 52 9 88 84 70 61 46 8 8 89 77 67 56 51 97 81 64 56 46 8 86 57 48 6 86 73 83 70 34 24 103 90
Ethyl acetate alcohol alcohol alcohol alcohol bromide bromide bromide bromide bromide chloride chloride chloride chloride ether ether ether ether iodide iodide iodide iodide iodide Gallium Glycerol Hexane Hexane Kerosene Kerosene Kerosene Mercury Mercury Mercury Mercury
Temp., °C
Pressure, megabars
Compressibility per megabar β × 106
Substance
20 14 20 20 20 20 20 20 20 20 15 20 20 20 25 20 20 20 20 20 20 20 20 30 15 20 20 20 20 20 20 22 22 22
400 23 500 1,000 12,000 200 400 500 1,000 12,000 23 500 1,000 12,000 23 500 1,000 12,000 200 400 500 1,000 12,000 300 5 200 400 500 1,000 12,000 300 500 1,000 12,000
75 100 63 54 8 100 82 70 54 8 151 102 66 8 188 84 61 10 81 69 64 50 8 3.97 22 117 91 55 45 8 3.95 3.97 3.91 2.37
Methyl alcohol alcohol alcohol alcohol alcohol alcohol Nitric acid Oils: Almond Castor Linseed Olive Rapeseed Phosphorus trichloride trichloride trichloride trichloride Propyl alcohol (n) alcohol (n) alcohol (n?) alcohol (n?) alcohol (n?) Toluene Toluene Turpentine Water Water Water Water Water Water Water Xylene, meta meta
Temp., °C
Pressure, megabars
Compressibility per megabar β × 106
15 20 20 20 20 20 0
23 200 400 500 1,000 12,000 17
103 95 80 65 54 8 32
15 15 15 15 20 10 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 40 40 40 20 20
5 5 5 5
53 46 51 55 59 71 63 47 8 77 67 65 47 7 74 64 74 49 43 41 39 38 33 9 69 60
250 500 1,000 12,000 200 400 500 1,000 12,000 200 400 13 200 400 500 500 1,000 12,000 200 400
* Smithsonian Tables, Table 106. Scott (Cryogenic Engineering, Van Nostrand, Princeton, NJ, 1959) gives data for liquid nitrogen (p. 283), oxygen (p. 276), and hydrogen (p. 303). For a convenient index to the high-pressure work of Bridgman, see American Institute of Physics Handbook, p. 2-163, McGraw-Hill, New York, 1957.
2-150
PHYSICAL AND CHEMICAL DATA TABLE 2-189
Compressibilities of Solids
Many data on the compressibility of solids obtained prior to 1926 are contained in Gruneisen, Handbuch der Physik, vol. 10, Springer, Berlin, 1926, pp. 1–52; also available as translation, NASA RE 2-18-59W, 1959. See also Tables 271, 273, 276, 278, and other material in Smithsonian Physical Tables, 9th ed., 1954. For a review of high-pressure work to 1946, see Bridgman, Rev. Mod. Phys., 18, 1 (1946).
LATENT HEATS UNITS CONVERSIONS For this subsection, the following units conversions are applicable: °F = 9⁄ 5 °C + 32. To convert calories per gram-mole to British thermal units per
pound-mole, multiply by 1.799; to convert calories per gram to British thermal units per pound, multiply by 1.799. To convert millimeters of mercury to pounds-force per square inch, multiply by 1.934 × 10−2.
LATENT HEATS TABLE 2-190
2-151
Heats of Fusion and Vaporization of the Elements and Inorganic Compounds*
Unless stated otherwise, the values have been taken from the compilations by K. K. Kelley on “Heats of Fusion of Inorganic Compounds,” U.S. Bur. Mines Bull. 393 (1936), and “The Free Energies of Vaporization and Vapor Pressures of Inorganic Substances,” U.S. Bur. Mines Bull. 383 (1935).
Substance Aluminum Al Al2Br6 Al2Cl6 AlF3·3NaF Al2I6 Al2O3 Antimony Sb SbBr3 SbCl3 SbCl5 Sb4O6 Sb4S6 Argon A Arsenic As AsBr3 AsCl3 AsF5 As4O6 Barium Ba BaBr2 BaCl2 BaF2 Ba(NO3)2 Ba3(PO4)2 BaSO4 Beryllium Be Bismuth Bi BiBr3 BiCl3 Bi2O3 Bi2S5 Boron BBr3 BCl3 BF3 B2H6 B3H10 B5H9 B5H11 B10H14 B2H5Br B3N3H6 Bromine Br2 BrF5 Cadmium Cd CdBr2 CdCl2 CdF2 CdI2 CdO CdSO4 Calcium Ca CaBr2 CaCO3 CaCl2 CaF2 Ca(NO3)2 CaO CaO·Al2O3·2SiO2 CaO·MgO·2SiO2 CaO·SiO2 CaSO4 Carbon C (graphite) CBr4 CCl4 CF4 CH4 C2N2 CNBr CNCl
mp, °C
Heat of fusion,a,b cal/mole
660.0 97.5 192.5 1000 191.0 2045
2,550 5,420 16,960 16,380 7,960 (26,000)
2057 256.4 180.2c
61,020 10,920 26,750c
385.5 3000
15,360
630.5 97 73.4 4 655 546
4,770 3,510 3,030 2,400 (27,000) 11,200
1440
46,670
219 172d 1425
10,360 11,570 17,820
−185.8
1,590
610c
31,000c
122 −52.8 457.2
7,570 4,980 14,300
−189.3 814 31 −16 −80.7 313
290 (6,620) 2,810 2,420 2,800 8,000
704 847 960 1287 595 1730 1350
(1,400)e 6,000 5,370 3,000 (5,980) 18,600 9,700
1280
2,500e
271.3
2,505
224 817 747
2,600 6,800 8,900
−128 −165.5 −119.8 −46.9
480
99.7 −104 −58
7,800
−7.2 −61.3
2,580 1,355
320.9 568 568 1110 387
1,460 (5,000) 5,300 (5,400) 3,660
1000
4,790
851 730 1282 782 1392 561 2707 1550 1392 1512 1297
2,230 4,180 (12,700) 6,100 4,100 5,120 (12,240) 29,400 (18,200) 13,400 6,700
bp at 1 atm, °C
1638
35,670
1420 461 441
18,020 17,350
91.3 12.5 −100.9 −92.4 16 58 67 f 16 50.4
7,300 5,680 4,620 3,685 6,470 7,700 8,500 11,600 6,230 7,670
58.0 40.4
7,420 7,470
765
23,870
967
29,860
796 1559c
25,400 53,820c
1487
36,580
3600 90 −24.0
11,000e 1,050 644
−182.5 −27.8 52 −5
224 1,938u
77 −127.9 −161.4 −21.1
2,240
13
*See also subsection “Thermodynamic Properties.”
Heat of vaporization,a,b cal/mole
7,280 3,110 2,040 5,576u 11,010c 6,300
Substance Carbon (Cont.) CNF CNI CO CO2 COS COCl2 CS2 Cerium Ce Cesium Cs CsBr CsCl CsF CsI CsNO3 Chlorine Cl2 ClF ClF3 Cl2O ClO2 Cl2O7 Chromium Cr CrO2Cl2 Cobalt Co CoCl2 Copper Cu Cu2Br2 Cu2Cl2 CuI Cu2(CN)2 Cu2O CuO Cu2S Fluorine F2 F2O Gallium Ga Germanium Ge GeH4 Ge2H6 Ge3H8 GeHCl3 GeBr4 GeCl4 Ge(CH3)4 Gold Au Helium He Hydrogen H2 HBr HCl HCN HF (HF)6 HI H2O H22O (= D2O) H2O2 HNO3 H3PO2 H3PO3 H3PO4 H4P2O6 H2S H2S2 H2SO4 H2Se H2SeO4 H2Te Indium In
mp, °C
Heat of fusion,a,b cal/mole
−205.0 −57.5 −138.8
200 1,900 1,129 k
−112.0
1,049 l
775
2,120
28.4
500
642 715
3,600 (2,450)
407
3,250
−101.0
1,531m
1550
3,930
1490 727
3,660 7,390
1083.0
3,110
430
4,890
473 1230 1447 1127
(5,400) (13,400) 2,820 5,500
−223 29.8
bp at 1 atm, °C −72.8 141 −191.5 −78.4c −50.2 8.0
5,780c 13,980c 1,444 6,030 c, r 4,423 k 5,990
690 1300 1300 1251 1280
16,320 35,990 35,690 34,330 35,930
−34.1 −101 11.3 2.0 10.9 79
5,890 6,280 7,100 8,480
2475 117
8,250
1050
27,170
2595 1355 1490 1336
72,810 16,310 11,920 15,940
−188.2 −144.8 1,336
959 −165 −109 −105.6 −71 26.1 −49.5 −88
(8,300)
1063.0
3,030
−271.4 −259.2 −86.9 −114.2 −13.2 −83.0
28 575 476 2,009i 1,094
−50.8 0.0 3.8 −2 −47 17.4 74 42.4 55 −85.5 −87.6 10.5
686 1,436 1,501s 2,520c 600 2,310 3,070 2,520 8,300 568t 1,805 2,360
58 −48.9
3,450 1,670
156.4
781
Heat of vaporization,a,b cal/mole
4,878 m
1,640 2,650
2071 −89.1 31.4 110.6 75g 189 84 44 2966
3,580 5,900 7,550 8,000 8,560 7,030 6,460 81,800
−268.4
22
−252.7 −66.7 −85.0 25.7 33.3 51.2
216 4,210 3,860 6,027i 7,460 5,020
100.0 101.4 158
9,729 h,q 9,945 r,q 10,270
−60.3
4,463 t
−41.3
4,880
−2.2
5,650
2-152
PHYSICAL AND CHEMICAL DATA
TABLE 2-190
Substance Iodine I2 ICl(α) ICl(β) IF7 Iron Fe FeCl2 Fe2Cl6 Fe(CO)5 FeO FeS Krypton Kr Lead Pb PbBr2 PbCl2 PbF2 PbI2 PbMoO4 PbO PbS PbSO4 PbWO4 Lithium Li LiBO2 LiBr LiCl LiF LiI LiOH Li2MoO4 LiNO3 Li2SiO3 Li4SiO4 Li2SO4 Li2WO4 Magnesium Mg MgBr2 MgCl2 MgF2 MgO Mg3(PO4)2 MgSiO3 MgSO4 MgZn2 Manganese Mn MnCl2 MnSiO3 MnTiO3 Mercury Hg HgBr2 HgCl2 HgI2 HgSO4 Molybdenum Mo MoF6 MoO3 Neon Ne Nickel Ni NiCl2 Ni(CO)4 Ni2S Ni3S2 Nitrogen N2 NF3 NH3 NH4CNS NH4NO3 N2O NO N2O4 N2O5 NOCl Osmium OsF8 OsO4 (yellow) OsO4 (white) Oxygen O2 O3
Heats of Fusion and Vaporization of the Elements and Inorganic Compounds (Continued ) mp, °C 113.0 17.2 13.9
Heat of fusion,a,b cal/mole
bp at 1 atm, °C
Heat of vaporization,a,b cal/mole
3,650 2,660 2,270
183 4c
7,460c
1530 677 304 −21 1380 1195
3,560 7,800 20,590 3,250 (7,700) 5,000
2735 1026 319 105
84,600 30,210 12,040 9,000
−157
360e
152.9
10,390
2,310e
327.4 488 498 824 412 1065 890 1114 1087 1123
1,224 4,290 5,650 1,860 5,970 (25,800) 2,820 4,150 9,600 (15,200)
1744 914 954 1293 872
42,060 27,700 29,600 38,300 24,850
1472 1281
51,310 (50,000)
179 845 552 614 847 440 462 705
1,100 (5,570) 2,900 3,200 (2,360) (1,420) 2,480 4,200
1372
32,250
1310 1382 1681 1171
35,420 35,960 50,970 40,770
1177 1249 857 742
7,210 7,430 3,040 (6,700)
650 711 712 1221 2642 1184 1524 1127 589
2,160 8,300 8,100 5,900 18,500 (11,300) 14,700 3,500 (8,270)
1220 650 1274 1404
3,450 7,340 (8,200) (7,960)
2152 1190
557 3,960 4,150 4,500 (1,440)
361 319 304 354
13,980 14,080 14,080 14,260
(6,660) 2,500 (2,500)
(4800) 36 1151
(128,000) 6,000
−38.9 241 277 250 850 2622 17 745 −248.5
1107 1418
32,520 32,690
55,150 29,630
77
−246.0
440e
1455
4,200
2730 987c 42.5
87,300 48,360c 7,000
645 790
(2,980) 5,800 −195.8 −129.0 −33.4
1,336 3,000 5,581n
−88.5 −151.7 30 32.4 −6.4
3,950 3,307 7,040 13,800c 6,140
−210.0 −77.7 146 169.6 −90.8 −163.6 −13
56 42 −218.9
172 1,352n (4,700) 1,460 1,563 550 5,540
4,060 2,340 106
47.4 130
6,840 9,450
−183.0 −111
1,629 2,880
Substance Palladium Pd Phosphorus P4 (yellow) P4 (violet) P4 (black) PCl3 PH3 P4O6 P4O10(α) P4O10(β) POCl3 P2S3 Platinum Pt Potassium K KBO2 KBr KCl KCN KCNS K2CO3 K2CrO4 K2Cr2O7 KF KI K2MoO4 KNO3 KOH KPO3 K3PO4 K4P2O7 K2SO4 K2TiO3 K2WO4 Praseodymium Pr Radon Rn Rhenium Re Re2O7 Re2O8 Rubidium Rb RbBr RbCl RbF RbI RbNO3 Selenium Se2 Se6 SeF6 SeO2 SeOCl2 Silicon Si SiCl4 Si2Cl6 Si3Cl8 (SiCl3)2O SiF4 Si2F6 SiF3Cl SiF2Cl2 SiH4 Si2H6 Si3H8 Si4H10 SiH3Br SiH2Br2 SiHCl3 (SiH3)3N (SiH3)2O SiO2 (quartz) SiO2 (cristobalite) Silver Ag AgBr AgCl AgCN AgI AgNO3 Ag2S Ag2SO4 Sodium Na NaBO2
mp, °C
Heat of fusion,a,b cal/mole
1554
4,120
bp at 1 atm, °C
44.2
615
−133.8 23.8 569
270o 3,360 17,080
1.1
3,110
1773.5
4,700
(4400)
(107,000)
63.5 947 742 770 623 179 897 984 398 857 682 922 338 360 817 1340 1092 1074 810 927
574 (5,700) 5,000 6,410 (3,500) 2,250 7,800 6,920 8,770 6,500 4,100 (4,000) 2,840 (2,000) 2,110 8,900 14,000 8,100 (10,600) (4,400)
776
18,920
1383 1407
37,060 38,840
1324
34,690
1327
30,850
932
2,700
−71 (3000) 296 147
15,340 3,800
39.1 677 717 833 638 305
525 3,700 4,400 4,130 2,990 1,340
217
1,220
10
1,010
1427 −67.6 −1
9,470 1,845
−33
280 417c 453c 74.2 −87.7 174 591 358c 105.1 508
Heat of vaporization,a,b cal/mole
12,520 25,600c 33,100 7,280 3,489 o 10,380 20,670 8,380
−61.8
4,010
362.4
18,060
679 1352 1381 1408 1304 753 736 −45.8c 317c 168 2290 56.8 139 211.4 135.6 −94.8c −18.9c −70.1 −31.5 −111.6 −14.3 53.1 100 2.4 70.5 31.8 48.7 −15.4 2230
18,110 37,120 36,920 39,510 35,960 25,490 20,600 6,350c 20,900
6,860 12,340 8,820 6,130c 10,400c 4,460 5,080 2,960 5,110 6,780 8,890 5,650 6,840 6,360 6,850 5,350
−18.5 −138 −144 −185 −132.5 −117 −93.5 −93.8 −70.0 −126.5 −105.6 −144 1470 1700
3,900
960.5 430 455 350 557 209 842 657
2,700 2,180 3,155 2,750 2,250 2,755 3,360 (4,300)
2212
60,720
1564
42,520
1506
34,450
97.7 966
630 8,660
914
23,120
3,400 2,100
LATENT HEATS TABLE 2-190
2-153
Heats of Fusion and Vaporization of the Elements and Inorganic Compounds (Concluded )
Substance Sodium (Cont.) NaBr NaCl NaClO3 NaCN NaCNS Na2CO3 NaF NaI Na2MoO4 NaNO3 NaOH aNa2O·aAl2O3·3SiO2 NaPO3 Na4P2O7 Na2S Na2SiO3 Na2Si2O5 Na2SO4 Na2WO4 Strontium Sr SrBr2 SrCl2 SrF2 Sr3(PO4)2 Sulfur S (rhombic) S (monoclinic) S2Cl2 SF6 SO2 SO3(α) SO3(β) SO3(γ) SOBr2 SOCl2 SO2Cl2 Tellurium Te TeCl4 TeF6
mp, °C
Heat of fusion,a,b cal/mole
bp at 1 atm, °C
Heat of vaporization,a,b cal/mole
747 800 255 562 323 854 992 662 687 310 322 1107 988 970 920 1087 884 884 702
6,140 7,220 5,290 (4,400) 4,450 7,000 7,000 5,240 3,600 3,760 2,000 13,150 (5,000) (13,700) (1,200) 10,300 8,460 5,830 5,800
1392 1465
37,950 40,810
1500
37,280
757 643 872 1400 1770
2,190 4,780 4,100 4,260 18,500
1384
112.8 119.2 −75.5 17 32.4 62.2
453
1,769p 2,060 2,890 6,310
3,230
1704
53,260
1378
33,610
444.6
2,200
138 −63.5c −5.0 44.8
8,720 5,600c 5,960p 10,190
139.5 75.4 69.2
9,920 7,600 7,760
1090 392 −38.6c
16,830 6,700c
Substance Thallium Tl TlBr TlCl Tl2CO3 TlI TlNO3 Tl2S Tl2SO4 Tin Sn4 SnBr2 SnBr4 SnCl2 SnCl4 Sn(CH3)4 SnH4 SnI4 Titanium TiBr4 TiCl4 TiO2 Tungsten W WF6 Uranium UF6 Xenon Xe Zinc Zn ZnCl2 Zn(C2H5)2 ZnO ZnS Zirconium ZrBr4 ZrCl4 ZrI4 ZrO2
a
k
b
l
c
m
Values in parentheses are uncertain. For the freezing point or the normal boiling point unless otherwise stated. Sublimation. d Decomposes at about 75°C; value obtained by extrapolation. e Bichowsky and Rossini, “Thermochemistry of the Chemical Substances,” Reinhold, New York (1936). f Decomposes before the normal boiling point is reached. g Decomposes at about 40°C; value obtained by extrapolation. h See also pp. 2-304 through 2-307 on steam table. i Giauque and Ruehrwein, J. Am. Chem. Soc., 61 (1939): 2626. j Giauque and Egan, J. Chem. Phys., 5 (1937): 45.
TABLE 2-191
mp, °C
Alloys 30.5 Pb + 69.5 Sn 36.9 Pb + 63.1 Sn 63.7 Pb + 36.3 Sn 77.8 Pb + 22.2 Sn 1 Pb + 9 Sn 24 Pb + 27.3 Sn + 48.7 Bi 25.8 Pb + 14.7 Sn + 52.4 Bi + 7 Cd Silicates Anorthite (CaAl2Si2O8) Orthoclase (KAlSi2O8) Microcline (KAlSi3O8) Wollastonite (CaSiO8) Malacolite (Ca8MgSi4O12) Diopside (CaMgSi2O4) Olivine (Mg2SiO4) Fayalite (Fe2SiO4) Spermaceti Wax (bees’)
bp at 1 atm, °C
Heat of vaporization,a,b cal/mole
302.5 460 427 273 440 207 449 632
1,030 5,990 4,260 4,400 3,125 2,290 3,000 5,500
1457 819 807
38,810 23,800 24,420
823
25,030
231.8 232 30 247 −33.2
1,720 (1,700) 3,000 3,050 2,190
2270
68,000
−149.8 143.5
(4,300)
38.2 −23 1825
(2,060) 2,240 (11,400)
136
3390 −0.4
(8,400) 1,800
(5900) 17.3
(176,000) 6,350
55.1c
9,990c
−111.5 419.5 283 1975 1645
2715
740 1,595 (5,500)
623 113 78.3 −52.3
20,740 8,330 7,320 4,420
8,350
−108.0
3,110
907 732 118
27,430 28,710 8,960
357c 311c 431c
25,800c 25,290c 29,030c
4,470 (9,000)
20,800
Kemp and Giauque, J. Am. Chem. Soc., 59 (1937): 79. Brown and Manov, J. Am. Chem. Soc., 59 (1937): 500. Giauque and Powell, J. Am. Chem. Soc. 61 (1939): 1970. n Overstreet and Giauque, J. Am. Chem. Soc 59 (1937): 254. o Stephenson and Giauque, J. Chem. Phys., 5 (1937): 149. p Giauque and Stephenson, J. Am. Chem. Soc., 60 (1938): 1389. q Osborne, Stimson, and Ginnings, Bur. Standards J. Research, 23, 197 (1939): 261. r Miles and Menzies, J. Am. Chem. Soc., 58 (1936): 1067. s Long and Kemp, J. Am. Chem. Soc., 58 (1936): 1829. t Giauque and Blue, J. Am. Chem. Soc., 58 (1936): 831. u Ruehrwein and Giauque, J. Am. Chem. Soc., 61 (1939): 2940.
Heats of Fusion of Miscellaneous Materials
Material
Heat of fusion,a,b cal/mole
mp, °C
Heat of fusion, cal/g
183 179 177.5 176.5 236 98.8 75.5
17 15.5 11.6 9.54 28 6.85 8.4
43.9 61.8
100 100 83 100 94 100 130 85 37.0 42.3
2-154
PHYSICAL AND CHEMICAL DATA
TABLE 2-192
Heats of Fusion of Organic Compounds
The values for the hydrocarbons are from the tables of the American Petroleum Institute Research Project 44 at the National Bureau of Standards, with some from Parks and Huffman, Ind. Eng. Chem., 23, 1138 (1931). The values for the nonhydrocarbon compounds were recalculated from data in International Critical Tables, vol. 5. Hydrocarbon compounds Paraffins Methane Ethane Propane n-Butane 2-Methylpropane n-Pentane 2-Methylbutane 2,2-Dimethylpropane n-Hexane 2-Methylpentane 2,2-Dimethylbutane 2,3-Dimethylbutane n-Heptane 2-Methylhexane 3-Ethylpentane 2,2-Dimethylpentane 2,4-Dimethylpentane 3,3-Dimethylpentane 2,2,3-Trimethylbutane n-Octane 2-Methylheptane 3-Methylpentane 4-Methylheptane 2,2-Dimethylhexane 2,5-Dimethylhexane 3,3-Dimethylhexane 2-Methyl-3-ethylpentane 3-Methyl-3-ethylpentane 2,2,3-Trimethylpentane 2,2,4-Trimethylpentane 2,3,3-Trimethylpentane 2,3,4-Trimethylpentane 2,2,3,3-Tetramethylbutane n-Nonane n-Decane n-Undecane n-Dodecane Eicosane Pentacosane Tritriacontane Aromatics Benzene Methylbenzene (Toluene) Ethylbenzene o-Xylene m-Xylene p-Xylene n-Propylbenzene Isopropylbenzene 1-Methyl-2-ethylbenzene
Formula
mp, °C
Heat of fusion, cal/g
CH4 C2H6 C3H8 C4H10 C4H10 C5H12 C5H12 C5H12 C6H14 C6H14 C6H14 C6H14 C7H16 C7H16 C7H16 C7H16 C7H16 C7H16 C7H16 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C8H18 C9H20 C10H22 C11H24 C12H26 C20H42 C25H52 C33H68
−182.48 −183.23 −187.65 −138.33 −159.60 −129.723 −159.890 −16.6 −95.320 −153.680 −99.73 −128.41 −90.595 −118.270 −118.593 −123.790 −119.230 −134.46 −24.96 −56.798 −109.04 −120.50 −120.955 −121.18 −91.200 −126.10 −114.960 −90.870 −112.27 −107.365 −100.70 −109.210 +100.69 −53.9 −30.0 −25.9 −9.6 +36.4 +53.3 +71.1
14.03 22.712 19.100 19.167 18.668 27.874 17.076 10.786 36.138 17.407 1.607 2.251 33.513 21.158 22.555 13.982 15.968 16.856 5.250 43.169 21.458 23.795 22.692 24.226 26.903 14.9 23.690 22.657 18.061 19.278 3.204 19.392 14.900 41.2 48.3 34.1 51.3 52.0 53.6 54.0
C6H6 C7H8 C8H10 C8H10 C8H10 C8H10 C9H12 C9H12 C9H12
+5.533 −94.991 −94.950 −25.187 −47.872 +13.263 −99.500 −96.028 −80.833
30.100 17.171 20.629 30.614 26.045 38.526 16.97 19.22 21.13 Heat of fusion, cal/g
Formula
mp, °C
Acetic acid Acetone Acrylic acid Allo-cinnamic acid Aminobenzoic acid (o-) (m-) (p-) Amyl alcohol Anethole Aniline Anthraquinone Apiol Azobenzene Azoxybenzene
C2H4O2 C3H6O C3H4O2 C9H8O2 C7H7NO2 C7H7NO2 C7H7NO2 C5H12O C10H12O C6H5NH2 C14H8O2 C12H14O4 C12H10N2 C12H10N2O
16.7 −95.5 12.3 68 145 179.5 188.5 −78.9 22.5 −6.3 284.8 29.5 67.1 36
46.68 23.42 37.03 27.35 35.48 38.03 36.46 26.65 25.80 27.09 37.48 25.80 28.91 21.62
Benzil Benzoic acid Benzophenone Benzylaniline Bromocamphor Bromochlorbenzene (o-) (m-) (p-) Bromoiodobenzene (o-) (m-) (p-) Bromol hydrate Bromophenol (p-) Bromotoluene (p-)
C14H10O2 C7H8O2 C13H10O C13H13N C10H15BrO C6H4BrCl C6H4BrCl C6H4BrCl C6H4BrI C6H4BrI C6H4BrI C2H3Br3O2 C6H5BrO C7H7Br
95.2 122.45 47.85 32.37 78 −12.6 −21.2 64.6 21 9.3 90.1 46 63.5 28
22.15 33.90 23.53 21.86 41.57 15.41 15.29 23.41 12.18 10.27 16.60 16.90 20.50 20.86
Nonhydrocarbon compounds
Hydrocarbon compounds Aromatics—(Cont.) 1-Methyl-3-ethylbenzene 1-Methyl-4-ethylbenzene 1,2,3-Trimethylbenzene 1,2,4-Trimethylbenzene 1,3,5-Trimethylbenzene Naphthalene Camphene Durene Isodurene Prehnitene p-Cymene n-Butyl benzene tert-Butyl benzene β-Methyl naphthalene Diphenyl Hexamethyl benzene Diphenyl methane Anthracene Phenanthrene Tolane Stilbene Dibenzil Triphenyl methane Alkyl cyclohexanes Cyclohexane Methylcyclohexane Alkyl cyclopentanes Cyclopentane Methylcyclopentane Ethylcyclopentane 1,1-Dimethylcyclopentane cis-1,2-Dimethylcyclopentane trans-1,2-Dimethylcyclopentane trans-1,3-Dimethylcyclopentane Monoolefins Ethene (Ethylene) Propene (Propylene) 1-Butene cis-2-Butene trans-2-Butene 2-Methylpropene (isobutene) 1-Pentene cis-2-pentene trans-2-pentene 2-Methyl-1-butene 3-Methyl-1-butene 2-Methyl-2-butene Acetylenes Acetylene 2-Butyne (dimethylacetylene) Nonhydrocarbon compounds
mp, °C
Heat of fusion, cal/g
C9H12 C9H12 C9H12 C9H12 C9H12 C10H8 C10H12 C10H14 C10H14 C10H14 C10H14 C10H14 C10H14 C11H10 C12H10 C12H18 C13H12 C14H10 C14H10 C14H10 C14H12 C14H14 C19H16
−95.55 −62.350 −25.375 −43.80 −44.720 +80.0 +51 +79.3 −24.0 −7.7 −68.9 −88.5 −58.1 +34.1 +68.6 +165.5 +25.2 +216.5 +96.3 +60 +124 +51.4 +92.1
15.14 25.29 16.64 24.54 18.97 36.0 57 37.4 23.0 20.0 17.1 19.5 14.9 20.1 28.8 30.4 26.4 38.7 25.0 28.7 40.0 30.7 21.1
C6H12 C7H14
+6.67 −126.58
7.569 16.429
C5H10 C6H12 C7H14 C7H14 C7H14 C7H14 C7H14
−93.80 −142.445 −138.435 −69.73 −53.85 −117.57 −133.680
2.068 19.68 11.10 3.36 3.87 15.68 17.93
C2H4 C3H6 C4H8 C4H8 C4H8 C4H8 C5H10 C5H10 C5H10 C5H10 C5H10 C5H10
−169.15 −185.25 −185.35 −138.91 −105.55 −140.35 −165.27 −151.363 −140.235 −137.560 −168.500 −133.780
28.547 17.054 16.393 31.135 41.564 25.265 16.82 24.239 26.536 26.879 18.009 25.738
C2H2 C4H6
−81.5 −132.23
23.04 40.808
Formula
Formula
mp, °C
Heat of fusion, cal/g
Butyl alcohol (n-) (t-) Butyric acid (n-)
C4H10O C4H10O C4H8O2
−89.2 25.4 −5.7
29.93 21.88 30.04
Capric acid (n-) Caprylic acid (n-) Carbazole Carbon tetrachloride Carvoxime (d-) (l-) (dl-) Cetyl alcohol Chloracetic acid (α-) (β-) Chloral alcoholate hydrate Chloroaniline (p-) Chlorobenzoic acid (o-) (m-) (p-) Chloronitrobenzene (m-) (p-) Cinnamic acid anhydride Cresol (p-) Crotonic acid (α-) (cis-) Cyanamide Cyclohexanol
C10H20O2 C8H16O2 C12H9N CCl4 C10H15NO C10H15NO C10H15NO C16H34O C2H3ClO2 C2H3ClO2 C4H7Cl3O2 C2H3Cl3O2 C6H6ClN C7H5ClO2 C7H5ClO2 C7H5ClO2 C6H4ClNO2 C6H4ClNO2 C9H8O2 C18H14O3 C7H8O C4H6O2 C4H6O2 CH2N2 C6H12O
31.99 16.3 243 −22.8 71.5 71 91 49.27 61.2 56 9 47.4 71 140.2 154.25 239.7 44.4 83.5 133 48 34.6 72 71.2 44 25.46
38.87 35.40 42.05 41.57 23.29 23.41 24.61 33.80 31.06 35.12 24.03 33.18 37.15 39.30 36.41 49.21 29.38 31.51 36.50 28.14 26.28 25.32 34.90 49.81 4.19
LATENT HEATS TABLE 2-192
2-155
Heats of Fusion of Organic Compounds (Concluded ) Heat of fusion, cal/g
Formula
mp, °C
Dibromobenzene (o-) (m-) (p-) Dibromophenol (2, 4-) Dichloroacetic acid Dichlorobenzene (o-) (m-) ( p-) Dihydroxybenzene (o-) (m-) (p-) Di-iodobenzene (o-) (m-) (p-) Dimethyl tartrate (dl-) (d-) pyrone Dinitrobenzene (o-) (m-) ( p-) Dinitrotoluene (2, 4-) Dioxane Diphenyl amine
C6H4Br2 C6H4Br2 C6H4Br2 C6H4Br2O C2H2Cl2O2 C6H4Cl2 C6H4Cl2 C6H4Cl2 C6H6O2 C6H6O2 C6H6O2 C6H4I2 C6H4I2 C6H4I2 C6H10O6 C6H10O6 C7H8O2 C6H4N2O4 C6H4N2O4 C6H4N2O4 C7H6N2O4 C4H8O2 C12H11N
1.8 −6.9 86 12 −4(?) −16.7 −24.8 53.13 104.3 109.65 172.3 23.4 34.2 129 87 49 132 116.93 89.7 173.5 70.14 11.0 52.98
12.78 13.38 20.55 13.97 14.21 21.02 20.55 29.67 49.40 46.20 58.77 10.15 11.54 16.20 35.12 21.50 56.14 32.25 24.70 39.99 26.40 34.85 25.23
Elaidic acid Ethyl acetate alcohol Ethylene dibromide Ethyl ether
C18H34O2 C4H8O2 C2H6O C2H4Br2 C4H10O
44.4 83.8 −114.4 10.012 −116.3
52.08 28.43 25.76 13.52 23.54
Formic acid
CH2O2
Glutaric acid Glycerol Glycol, ethylene
Nonhydrocarbon compounds
8.40
58.89
C6H8O4 C3H8O3 C2H6O2
97.5 18.07 −11.5
37.39 47.49 43.26
Hydrazo benzene Hydrocinnamic acid Hydroxyacetanilide
C12H12N2 C9H10O2 C8H9NO2
134 48 91.3
22.89 28.14 33.59
Iodotoluene (p-) Isopropyl alcohol ether
C7H7I C3H8O C6H14O
34 −88.5 −86.8
18.75 21.08 25.79
Lauric acid (n-) Levulinic acid
C12H24O2 C5H8O3
43.22 33
43.72 18.97
Menthol (l-) (α) Methyl alcohol Myristic acid Methyl cinnamate fumarate oxalate phenylpropiolate succinate
C10H20O CH4O C14H28O2 C10H10O2 C6H8O4 C4H6O4 C10H8O2 C6H10O4
43.5 −97.8 53.86 36 102 54.35 18 19.5
18.63 23.7 47.49 26.53 57.93 42.64 22.86 35.72
Formula
mp, °C
Heat of fusion, cal/g
Naphthol (α-) (β-) Naphthylamine (α-) Nitroaniline (o-) (m-) (p-) Nitrobenzene Nitrobenzoic acid (o-) (m-) (p-) Nitronaphthalene Nitrophenol (o-)
C10H8O C10H8O C10H9N C6H6N2O2 C6H6N2O2 C6H6N2O2 C6H5NO2 C7H5NO4 C7H5NO4 C7H5NO4 C10H7NO2 C6H5NO3
95.0 120.6 50 71.2 114.0 147.3 5.85 145.8 141.1 239.2 56.7 45.13
38.94 31.30 22.34 27.88 40.97 36.46 22.52 40.06 27.59 52.80 25.44 26.76
Palmitic acid Paraldehyde Pelargic acid (n-) (β-) Pelargonic acid (n-) (α-) Phenol Phenylacetic acid Phenylhydrazine Propyl ether (n)
C16H32O2 C6H12O3 C9H18O2 C9H18O2 C6H6O C8H8O2 C6H8N2 C6H14O
61.82 10.5 12.35 40.92 76.7 19.6 −126.1
39.18 25.02 39.04 30.63 29.03 25.44 36.31 20.66
Quinone
C6H4O2
115.7
40.85
Stearic acid Succinic anhydride Succinonitrile
C18H30O2 C4H4O3 C4H4N2
68.82 119 54.5
47.54 48.74 11.71
Tetrachloroxylene (o-) (p-) Thiophene Thiosinamine Thymol Toluic acid (o-) (m-) (p-) Toluidine (p-) Tribromophenol (2, 4, 6-) Trichloroacetic acid Trinitroglycerol Trinitrotoluene (2, 4, 6-) Tristearin
C8H6Cl4 C8H6Cl4 C4H4S C4H8N2S C10H14O C8H8O2 C8H8O2 C8H8O2 C7H9N C6H3Br3O C2HCl3O2 C3H5N3O9 C7H5N3O6 C57H110O6
86 95 −39.4 77 51.5 103.7 108.75 179.6 43.3 93 57.5 12.3 80.83 70.8, 54.5
21.02 22.10 14.11 33.45 27.47 35.40 27.59 39.90 39.90 13.38 8.60 23.02 22.34 45.63
Undecylic acid (α-) (n-) (β-) (n-) Urethane
C11H22O2 C11H22O2 C3H7NO2
28.25 48.7
32.20 42.91 40.85
Veratrol
C8H10O2
22.5
27.45
Xylene dibromide (o-) (m-) dichloride (o-) (m-) (p-)
C8H8Br2 C8H8Br2 C8H8Cl2 C8H8Cl2 C8H8Cl2
Nonhydrocarbon compounds
95 77 55 34 100
24.25 21.45 29.03 26.64 32.73
TABLE 2-193 Cmpd. no.
Heats of Vaporization of Inorganic and Organic Compounds Name
Formula
CAS no.
Mol wt
C1 × 1E−07
C2
C3
C4
Tmin, K
∆Hv at Tmin × 1E−07
Tmax, K
∆Hv at Tmax
1 2 3 4 5 6 7 8 9 10
Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane
CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22
74828 74840 74986 106978 109660 110543 142825 111659 111842 124185
16.043 30.070 44.097 58.123 72.150 86.177 100.204 114.231 128.258 142.285
1.0194 2.1091 2.9209 3.6238 3.9109 4.4544 5.0014 5.5180 6.0370 6.6126
0.26087 0.60646 0.78237 0.8337 0.38681 0.39002 0.38795 0.38467 0.38522 0.39797
−0.14694 −0.55492 −0.77319 −0.82274 0 0 0 0 0 0
0.22154 0.32799 0.39246 0.39613 0 0 0 0 0 0
90.69 90.35 85.47 134.86 143.42 177.83 182.57 216.38 219.66 243.51
0.8724 1.7879 2.4787 2.8684 3.3968 3.7647 4.2619 4.5898 5.0545 5.4168
190.56 305.32 369.83 425.12 469.7 507.6 540.2 568.7 594.6 617.7
0 0 0 0 0 0 0 0 0 0
11 12 13 14 15 16 17 18 19 20
n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane n-Eicosane
C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40 C20H42
1120214 112403 629505 629594 629629 544763 629787 593453 629925 112958
156.312 170.338 184.365 198.392 212.419 226.446 240.473 254.500 268.527 282.553
7.2284 7.7337 8.4339 9.0539 9.6741 10.1560 10.4730 10.9690 11.6740 12.8600
0.40607 0.40681 0.4257 0.44467 0.45399 0.45726 0.4374 0.44327 0.45865 0.50351
0 0 0 0 0 0 0 0 0 0.32986
0 0 0 0 0 0 0 0 0 −0.42184
247.57 263.57 267.76 279.01 283.07 291.31 295.13 301.31 305.04 309.58
5.9240 6.2802 6.8015 7.2002 7.6728 8.0225 8.3699 8.7246 9.2185 9.5933
639 658 675 693 708 723 736 747 758 768
0 0 0 0 0 0 0 0 0 0
21 22 23 24 25 26 27
2-Methylpropane 2-Methylbutane 2,3-Dimethylbutane 2-Methylpentane 2,3-Dimethylpentane 2,3,3-Trimethylpentane 2,2,4-Trimethylpentane
C4H10 C5H12 C6H14 C6H14 C7H16 C8H18 C8H18
75285 78784 79298 107835 565593 560214 540841
58.123 72.150 86.177 86.177 100.204 114.231 114.231
3.1667 3.7700 4.1404 4.2780 4.6536 4.9910 4.7721
0.3855 0.3952 0.38124 0.384 0.37579 0.383 0.37992
0 0 0 0 0 0 0
0 0 0 0 0 0 0
113.54 113.25 145.19 119.55 160 172.22 165.78
2.7927 3.3720 3.6328 3.8495 4.0747 4.3530 4.1565
408.14 460.43 499.98 497.5 537.35 573.5 543.96
0 0 0 0 0 0 0
28 29 30 31 32 33 34 35
Ethylene Propylene 1-Butene cis-2-Butene trans-2-Butene 1-Pentene 1-Hexene 1-Heptene
C2H4 C3H6 C4H8 C4H8 C4H8 C5H10 C6H12 C7H14
74851 115071 106989 590181 624646 109671 592416 592767
28.054 42.081 56.108 56.108 56.108 70.134 84.161 98.188
2.8694 3.2300 3.4190 3.3320 3.7740 4.3236 4.8120
0.3746 0.8375 0.3747 0.3754 0.3736 0.37647 0.3788 0.3685
0 −0.9216 0 0 0 0 0 0
0 0.5012 0 0 0 0 0 0
104 87.89 87.8 134.26 167.62 107.93 133.39 154.27
1.6025 2.4031 2.9582 2.9773 2.7684 3.4166 3.8483 4.2478
282.34 365.57 419.95 435.58 428.63 464.78 504.03 537.29
0 0 0 0 0 0 0 0
36 37 38 39 40 41 42 43 44
1-Octene 1-Nonene 1-Decene 2-Methylpropene 2-Methyl-1-butene 2-Methyl-2-butene 1,2-Butadiene 1,3-Butadiene 2-Methyl-1,3-butadiene
C8H16 C9H18 C10H20 C4H8 C5H10 C5H10 C4H6 C4H6 C5H8
111660 124118 872059 115117 563462 513359 590192 106990 78795
112.215 126.242 140.269 56.108 70.134 70.134 54.092 54.092 68.119
5.3980 5.9940 6.4898 3.2720 3.9091 3.9121 3.5220 3.2580 3.9310
0.3835 0.3953 0.39187 0.383 0.39866 0.3634 0.395 0.373 0.425
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
171.45 191.78 206.89 132.81 135.58 139.39 136.95 164.25 127.27
4.7013 5.1366 5.5289 2.8262 3.4072 3.4437 3.0540 2.7155 3.4529
566.65 593.25 616.4 417.9 465 471 452 425.17 484
0 0 0 0 0 0 0 0 0
45 46 47 48 49 50
Acetylene Methylacetylene Dimethylacetylene 3-Methyl-1-butyne 1-Pentyne 2-Pentyne
C2H2 C3H4 C4H6 C5H8 C5H8 C5H8
74862 74997 503173 598232 627190 627214
26.038 40.065 54.092 68.119 68.119 68.119
2.3795 3.2775 3.8560 3.7920 3.9540 4.4158
0.375 0.3997 0.3737 0.3565 0.3512 0.44347
0 0 0 0 0 0
0 0 0 0 0 0
192.4 170.45 240.91 183.45 167.45 163.83
1.6488 2.6297 2.9557 3.1681 3.4025 3.7321
308.32 402.39 473.2 463.2 481.2 519
0 0 0 0 0 0
51 52
1-Hexyne 2-Hexyne
C6H10 C6H10
693027 764352
82.145 82.145
4.5740 4.9110
0.3698 0.4392
0 0
0 0
141.25 183.65
4.0640 4.1067
516.2 549
0 0
2-156
53 54 55 56
3-Hexyne 1-Heptyne 1-Octyne Vinylacetylene1
C6H10 C7H12 C8H14 C4H4
928494 628717 629050 689974
82.145 96.172 110.199 52.076
4.8080 5.0514 5.6306 3.6490
0.436 0.41163 0.4148 0.4
0 0 0 0.043
0 0 0 0
170.05 192.22 193.55 173.15
4.0831 4.2470 4.7663 2.9876
544 559 585 454
0 0 0 0
57 58 59 60 61 62 63 64 65 66
Cyclopentane Methylcyclopentane Ethylcyclopentane Cyclohexane Methylcyclohexane 1,1-Dimethylcyclohexane Ethylcyclohexane Cyclopentene 1-Methylcyclopentene Cyclohexene
C5H10 C6H12 C7H14 C6H12 C7H14 C8H16 C8H16 C5H8 C6H10 C6H10
287923 96377 1640897 110827 108872 590669 1678917 142290 693890 110838
70.134 84.161 98.188 84.161 98.188 112.215 112.215 68.119 82.145 82.145
3.8900 4.3600 4.8288 4.4940 4.7534 5.0402 5.3832 3.8107 4.3541 4.4405
0.361 0.38531 0.37809 0.3974 0.39461 0.4036 0.41763 0.3543 0.36805 0.37479
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
179.28 130.73 134.71 279.69 146.58 239.66 161.84 138.13 146.62 169.67
3.3292 3.9118 4.3604 3.3977 4.2295 4.0862 4.7318 3.4046 3.8769 3.8791
511.76 532.79 569.52 553.58 572.19 591.15 609.15 507 542 560.4
0 0 0 0 0 0 0 0 0 0
67 68 69 70 71 72 73
Benzene Toluene o-Xylene m-Xylene p-Xylene Ethylbenzene Propylbenzene
C6H6 C7H8 C8H10 C8H10 C8H10 C8H10 C9H12
71432 108883 95476 108383 106423 100414 103651
78.114 92.141 106.167 106.167 106.167 106.167 120.194
4.7500 5.0144 5.5330 5.4600 5.3740 5.4640 5.7663
0.45238 0.3859 0.377 0.3726 0.3656 0.392 0.3956
0.0534 0 0 0 0 0 −8.9129E−03
−0.1181 0 0 0 0 0 0
278.68 178.18 247.98 225.3 286.41 178.15 215.03
3.4909 4.3670 4.5826 4.6097 4.2761 4.7811 5.0574
562.16 591.8 630.33 617.05 616.23 617.2 574.54
0 0 0 0 0 0 2.4695E+07
74 75 76 77 78 79 80 81
1,2,4-Trimethylbenzene Isopropylbenzene 1,3,5-Trimethylbenzene p-Isopropyltoluene Naphthalene Biphenyl Styrene m-Terphenyl
C9H12 C9H12 C9H12 C10H14 C10H8 C12H10 C8H8 C18H14
95636 98828 108678 99876 91203 92524 100425 92068
120.194 120.194 120.194 134.221 128.174 154.211 104.152 230.309
5.9126 5.7950 6.0380 6.3314 7.0510 7.5736 5.7260 10.1230
0.35632 0.3956 0.37999 0.40289 0.4612 0.3975 0.4055 0.3767
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
229.33 177.14 228.42 205.25 353.43 342.2 242.54 360
5.0621 5.0869 5.1010 5.4387 5.2508 6.0420 4.7128 8.4070
649.13 631.1 637.36 653.15 748.35 789.26 636 924.85
0 0 0 0 0 0 0 0
82 83 84 85 86 87 88
Methanol Ethanol 1-Propanol 1-Butanol 2-Butanol 2-Propanol 2-Methyl-2-propanol
CH4O C2H6O C3H8O C4H10O C4H10O C3H8O C4H10O
67561 64175 71238 71363 78922 67630 75650
32.042 46.069 60.096 74.123 74.123 60.096 74.123
5.2390 5.6900 6.3300 6.7390 7.2560 6.3080 7.7320
0.3682 0.3359 0.3575 0.173 0.4774 0.3921 0.5645
0 0 0 0.2915 0 0 0
0 0 0 0 0 0 0
175.47 159.05 146.95 184.51 158.45 185.28 298.97
4.4900 5.0245 5.6460 6.0575 6.1383 5.2807 4.6703
512.64 513.92 536.78 563.05 536.05 508.3 506.21
0 0 0 0 0 0 0
89 90 91 92 93 94 95 96
1-Pentanol 2-Methyl-1-butanol 3-Methyl-1-butanol 1-Hexanol 1-Heptanol Cyclohexanol Ethylene glycol 1,2-Propylene glycol
C5H12O C5H12O C5H12O C6H14O C7H16O C6H12O C2H6O2 C3H8O2
71410 137326 123513 111273 111706 108930 107211 57556
88.150 88.150 88.150 102.177 116.203 100.161 62.068 76.095
8.3100 7.7839 8.0815 8.5980 9.6900 9.2440 8.2900 8.0700
0.511 0.45313 0.50185 0.513 0.572 0.64825 0.4266 0.295
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
195.56 203 155.95 228.55 239.15 296.6 260.15 213.15
6.7533 6.3619 6.8999 6.7623 7.3822 6.2273 6.8461 7.1374
586.15 565 577.2 611.35 631.9 650 719.7 626
0 0 0 0 0 0 0 0
97 98 99 100
Phenol o-Cresol m-Cresol p-Cresol
C6H6O C7H8O C7H8O C7H8O
108952 95487 108394 106445
94.113 108.140 108.140 108.140
7.3060 7.1979 8.0082 8.4942
0.4246 0.40317 0.45314 0.50234
0 0 0 0
0 0 0 0
314.06 304.19 285.39 307.93
5.6577 5.7135 6.3326 6.3649
694.25 697.55 705.85 704.65
0 0 0 0
101 102 103 104 105 106 107
Dimethyl ether Methyl ethyl ether Methyl n-propyl ether Methyl isopropyl ether Methyl-n-butyl ether Methyl isobutyl ether Methyl tert-butyl ether
C2H6O C3H8O C4H10O C4H10O C5H12O C5H12O C5H12O
115106 540670 557175 598538 628284 625445 1634044
46.069 60.096 74.123 74.123 88.150 88.150 88.150
2.9940 3.5300 3.9795 3.9305 4.5328 4.2678 4.2024
0.3505 0.376 0.3729 0.3711 0.3824 0.37995 0.37826
0 0 0 0 0 0 0
0 0 0 0 0 0 0
131.65 160 133.97 127.93 157.48 150 164.55
2.6032 2.9751 3.5184 3.4876 3.9358 3.7232 3.6096
400.1 437.8 476.3 464.5 510 497 497.1
0 0 0 0 0 0 0 2-157
TABLE 2-193 Cmpd. no.
Heats of Vaporization of Inorganic and Organic Compounds (Continued ) Name
Formula
CAS no.
Mol wt
C1 × 1E−07
C2
C3
C4
Tmin, K
∆Hv at Tmin × 1E−07
Tmax, K
∆Hv at Tmax
108 109 110 111 112
Diethyl ether Ethyl propyl ether Ethyl isopropyl ether Methyl phenyl ether Diphenyl ether
C4H10O C5H12O C5H12O C7H8O C12H10O
60297 628320 625547 100663 101848
74.123 88.150 88.150 108.140 170.211
4.0600 5.4380 4.2580 5.8662 6.8243
0.3868 0.60624 0.37221 0.37127 0.30877
0 0 0 0 0
0 0 0 0 0
156.85 145.65 140 235.65 300.03
3.4651 4.4140 3.7556 4.9560 5.8546
466.7 500.23 489 645.6 766.8
0 0 0 0 0
113 114 115 116 117 118 119 120 121 122
Formaldehyde Acetaldehyde 1-Propanal 1-Butanal 1-Pentanal 1-Hexanal 1-Heptanal 1-Octanal 1-Nonanal 1-Decanal
CH2O C2H4O C3H6O C4H8O C5H10O C6H12O C7H14O C8H16O C9H18O C10H20O
50000 75070 123386 123728 110623 66251 111717 124130 124196 112312
30.026 44.053 58.080 72.107 86.134 100.161 114.188 128.214 142.241 156.268
3.0760 4.6070 4.1492 4.6403 5.1478 5.6661 6.1299 6.8347 7.3363 7.9073
0.2954 0.62 0.36751 0.3849 0.37541 0.38533 0.37999 0.41039 0.41735 0.4129
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
181.15 150.15 170 176.75 182 217.15 229.8 246 255.15 267.15
2.5863 3.6199 3.5675 3.9797 4.4502 4.7495 5.1353 5.5966 5.9779 6.4201
408 466 504.4 537.2 566.1 591 617 638.1 658 674.2
0 0 0 0 0 0 0 0 0 0
123 124 125 126 127 128 129 130 131 132 133 134
Acetone Methyl ethyl ketone 2-Pentanone Methyl isopropyl ketone 2-Hexanone Methyl isobutyl ketone 3-Methyl-2-pentanone 3-Pentanone Ethyl isopropyl ketone Diisopropyl ketone Cyclohexanone Methyl phenyl ketone
C3H6O C4H8O C5H10O C5H10O C6H12O C6H12O C6H12O C5H10O C6H12O C7H14O C6H10O C8H8O
67641 78933 107879 563804 591786 108101 565617 96220 565695 565800 108941 98862
58.080 72.107 86.134 86.134 100.161 100.161 100.161 86.134 100.161 114.188 98.145 120.151
4.2150 4.6220 5.1740 5.1400 5.6770 5.4000 5.1130 5.2359 5.3880 5.5980 5.5500 6.6104
0.3397 0.355 0.39422 0.3858 0.3817 0.383 0.3395 0.40465 0.40616 0.3774 0.3538 0.37425
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
178.45 186.48 196.29 250 217.35 189.15 167.15 234.18 200 204.81 242 292.81
3.6390 3.9704 4.3663 4.0753 4.7584 4.6294 4.5480 4.2075 4.5154 4.7426 4.7114 5.4166
508.2 535.5 561.08 553 587.05 571.4 573 560.95 567 576 653 709.5
0 0 0 0 0 0 0 0 0 0 0 0
135 136 137 138 139 140 141
Formic acid Acetic acid Propionic acid n-Butyric acid Isobutyric acid Benzoic acid2 Acetic anhydride
CH2O2 C2H4O2 C3H6O2 C4H8O2 C4H8O2 C7H6O2 C4H6O3
64186 64197 79094 107926 79312 65850 108247
46.026 60.053 74.079 88.106 88.106 122.123 102.090
2.3700 2.0265 2.7290 7.4996 4.4967 10.1900 6.3520
1.999 0.11911 0.06954 2.333 1.1615 0.478 0.3986
−5.1503 −1.3487 −1.0423 −3.8644 −2.4573 0 0
3.331 1.4227 1.1152 2.016 1.5823 0 0
281.45 289.81 252.45 267.95 227.15 395.45 200.15
1.9532 2.3185 2.9964 4.1566 3.6179 7.1277 5.4139
588 591.95 600.81 615.7 605 751 606
0 0 0 0 0 0 0
142 143 144 145 146 147 148
Methyl formate Methyl acetate Methyl propionate Methyl n-butyrate Ethyl formate Ethyl acetate Ethyl propionate
C2H4O2 C3H6O2 C4H8O2 C5H10O2 C3H6O2 C4H8O2 C5H10O2
107313 79209 554121 623427 109944 141786 105373
60.053 74.079 88.106 102.133 74.079 88.106 102.133
4.1030 4.4920 5.0080 5.3781 4.5909 4.9330 5.3325
0.3825 0.3685 0.3959 0.39523 0.4123 0.3847 0.401
0 0 0 0 0 0 0
0 0 0 0 0 0 0
174.15 175.15 185.65 187.35 193.55 189.6 199.25
3.4644 3.8418 4.2231 4.5694 3.7679 4.1490 4.4449
487.2 506.55 530.6 554.5 508.4 523.3 546
0 0 0 0 0 0 0
149 150 151 152 153 154 155
Ethyl n-butyrate n-Propyl formate n-Propyl acetate n-Butyl acetate Methyl benzoate Ethyl benzoate Vinyl acetate
C6H12O2 C4H8O2 C5H10O2 C6H12O2 C8H8O2 C9H10O2 C4H6O2
105544 110747 109604 123864 93583 93890 108054
116.160 88.106 102.133 116.160 136.150 150.177 86.090
5.6419 4.9687 5.4327 5.7800 6.9650 6.3400 4.7700
0.37985 0.4025 0.407 0.3935 0.4061 0.2911 0.3765
0 0 0 0 0 0 0
0 0 0 0 0 0 0
175.15 180.25 178.15 199.65 260.75 238.45 180.35
4.9090 4.2162 4.6322 4.8943 5.7500 5.6137 4.0619
571 538 549.73 579.15 693 698 519.13
0 0 0 0 0 0 0
156 157 158
Methylamine Dimethylamine Trimethylamine
CH5N C2H7N C3H9N
74895 124403 75503
31.057 45.084 59.111
3.8580 4.0900 3.3050
0.404 0.42005 0.354
0 0 0
0 0 0
179.69 180.96 156.08
3.1006 3.2678 2.8216
430.05 437.2 433.25
0 0 0
2-158
159 160 161
Ethylamine Diethylamine Triethylamine
C2H7N C4H11N C6H15N
75047 109897 121448
45.084 73.138 101.192
4.2750 4.6133 4.6640
0.5857 0.42628 0.3663
−0.332 0 0
0.169 0 0
192.15 223.35 158.45
3.2955 3.5761 4.1011
456.15 496.6 535.15
0 0 0
162 163 164 165 166 167 168
n-Propylamine di-n-Propylamine Isopropylamine Diisopropylamine Aniline N-Methylaniline N,N-Dimethylaniline
C3H9N C6H15N C3H9N C6H15N C6H7N C7H9N C8H11N
107108 142847 75310 108189 62533 100618 121697
59.111 101.192 59.111 101.192 93.128 107.155 121.182
4.4488 5.4280 4.4041 5.0070 7.1950 6.3860 6.7900
0.39494 0.3665 0.43325 0.4362 0.458 0.3104 0.4053
0 0 0 0 0 0 0
0 0 0 0 0 0 0
188.36 210.15 177.95 176.85 267.13 216.15 275.6
3.6857 4.5500 3.5874 4.1823 5.7710 5.6961 5.5162
496.95 550 471.85 523.1 699 701.55 687.15
0 0 0 0 0 0 0
169 170 171 172
Ethylene oxide Furan Thiophene Pyridine
C2H4O C4H4O C4H4S C5H5N
75218 110009 110021 110861
44.053 68.075 84.142 79.101
3.6652 4.0050 4.5793 5.1740
0.37878 0.3995 0.38557 0.38865
0 0 0 0
0 0 0 0
160.65 196.29 234.94 231.51
3.1271 3.2647 3.7472 4.3144
469.15 490.15 579.35 619.95
0 0 0 0
173 174 175 176 177 178 179 180
Formamide3 N,N-Dimethylformamide Acetamide N-Methylacetamide Acetonitrile Propionitrile n-Butyronitrile Benzonitrile
CH3NO C3H7NO C2H5NO C3H7NO C2H3N C3H5N C4H7N C7H5N
75127 68122 60355 79163 75058 107120 109740 100470
45.041 73.095 59.068 73.095 41.053 55.079 69.106 103.123
7.3580 5.9217 8.1070 7.3402 4.3511 4.9348 5.2200 6.2615
0.3564 0.37996 0.42 0.38974 0.34765 0.41873 0.165 0.35427
0 0 0 0 0 0 0.6692 0
0 0 0 0 0 0 −0.539 0
275.7 212.72 353.15 301.15 229.32 180.26 161.25 260.4
6.2844 5.0931 6.2386 5.9384 3.5996 4.2005 4.7223 5.3091
771 649.6 761 718 545.5 564.4 582.25 699.35
0 0 0 0 0 0 0 0
181 182 183 184 185 186 187 188 189
Methyl mercaptan Ethyl mercaptan n-Propyl mercaptan n-Butyl mercaptan Isobutyl mercaptan sec-Butyl mercaptan Dimethyl sulfide Methyl ethyl sulfide Diethyl sulfide
CH4S C2H6S C3H8S C4H10S C4H10S C4H10S C2H6S C3H8S C4H10S
74931 75081 107039 109795 513440 513531 75183 624895 352932
48.109 62.136 76.163 90.189 90.189 90.189 62.136 76.163 90.189
3.4448 3.8440 4.4782 4.9702 4.7420 4.6432 3.8690 4.4740 4.7182
0.37427 0.37534 0.41073 0.41199 0.40535 0.399 0.3694 0.4097 0.3643
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
150.18 125.26 159.95 157.46 128.31 133.02 174.88 167.23 169.2
2.9825 3.4489 3.8723 4.3505 4.2664 4.1614 3.3042 3.8344 4.1353
469.95 499.15 536.6 570.1 559 554 503.04 533 557.15
0 0 0 0 0 0 0 0 0
190 191 192 193 194 195 196 197
Fluoromethane Chloromethane Trichloromethane Tetrachloromethane Bromomethane Fluoroethane Chloroethane Bromoethane
CH3F CH3Cl CHCl3 CCl4 CH3Br C2H5F C2H5Cl C2H5Br
593533 74873 67663 56235 74839 353366 75003 74964
34.033 50.488 119.377 153.822 94.939 48.060 64.514 108.966
2.4708 2.9745 4.1860 4.3252 3.1690 2.7617 3.5240 3.9004
0.37014 0.353 0.3584 0.37688 0.3015 0.32162 0.3652 0.38012
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
131.35 175.43 209.63 250.33 179.47 129.95 134.8 154.55
2.0276 2.4520 3.5047 3.4528 2.7379 2.4089 3.1052 3.3933
317.42 416.25 536.4 556.35 467 375.31 460.35 503.8
0 0 0 0 0 0 0 0
198 199 200 201 202 203 204 205
1-Chloropropane 2-Chloropropane 1,1-Dichloropropane 1,2-Dichloropropane Vinyl chloride Fluorobenzene Chlorobenzene Bromobenzene
C3H7Cl C3H7Cl C3H6Cl2 C3H6Cl2 C2H3Cl C6H5F C6H5Cl C6H5Br
540545 75296 78999 78875 75014 462066 108907 108861
78.541 78.541 112.986 112.986 62.499 96.104 112.558 157.010
3.9890 3.8871 4.7740 4.6750 3.4125 4.5820 5.1480 5.5520
0.37956 0.38043 0.39204 0.36529 0.4513 0.3717 0.36614 0.37694
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
150.35 155.97 200 172.71 119.36 230.94 227.95 242.43
3.4862 3.3586 4.0147 4.0997 2.9491 3.7605 4.3707 4.6875
503.15 489 560 572 432 560.09 632.35 670.15
0 0 0 0 0 0 0 0
206 207 208 209 210 211 212 213
Air Hydrogen Helium-4 Neon Argon Fluorine Chlorine Bromine
H2 He Ne Ar F2 Cl2 Br2
132259100 1333740 7440597 7440019 7440371 7782414 7782505 7726956
28.951 2.016 4.003 20.180 39.948 37.997 70.905 159.808
0.8474 0.1013 0.0125 0.2389 0.8731 0.8876 3.0680 4.0000
0.3822 0.698 1.3038 0.3494 0.3526 0.34072 0.8458 0.351
0 −1.817 −2.6954 0 0 0 −0.9001 0
0 1.447 1.7098 0 0 0 0.453 0
59.15 13.95 2.2 24.56 83.78 53.48 172.12 265.85
0.6759 0.0913 0.0097 0.1803 0.6561 0.7578 2.2878 3.2323
132.45 33.19 5.2 44.4 150.86 144.12 417.15 584.15
0 0 0 0 0 0 0 0 2-159
TABLE 2-193 Cmpd. no.
Heats of Vaporization of Inorganic and Organic Compounds (Concluded ) Name
Formula
CAS no.
Mol wt
C1 × 1E−07
C2
C3
C4
Tmin, K
∆Hv at Tmin × 1E−07
Tmax, K
∆Hv at Tmax
214 215 216 217 218
Oxygen Nitrogen Ammonia Hydrazine Nitrous oxide
O2 N2 NH3 N2H4 N2O
7782447 7727379 7664417 302012 10024972
31.999 28.014 17.031 32.045 44.013
0.9008 0.7491 3.1523 5.9794 2.3215
0.4542 0.40406 0.3914 0.9424 0.384
−0.4096 −0.317 −0.2289 −1.398 0
0.3183 0.27343 0.2309 0.8862 0
54.36 63.15 195.41 274.69 182.3
0.7742 0.6024 2.5298 4.5238 1.6502
154.58 126.2 405.65 653.15 309.57
0 0 0 0 0
219 220 221 222 223 224 225 226 227 228 229 230 231
Nitric oxide Cyanogen Carbon monoxide Carbon dioxide Carbon disulfide Hydrogen fluoride Hydrogen chloride Hydrogen bromide Hydrogen cyanide2 Hydrogen sulfide Sulfur dioxide Sulfur trioxide Water
NO C2N2 CO CO2 CS2 HF HCl HBr HCN H2S SO2 SO3 H2O
10102439 460195 630080 124389 75150 7664393 7647010 10035106 74908 7783064 7446095 7446119 7732185
30.006 52.036 28.010 44.010 76.143 20.006 36.461 80.912 27.026 34.082 64.065 80.064 18.015
2.1310 3.3840 0.8585 2.1730 3.4960 13.4510 2.2093 2.4850 3.3490 2.5676 3.6760 7.3370 5.2053
0.4056 0.3707 0.4921 0.382 0.2986 13.36 0.3466 0.39 0.2053 0.37358 0.4 0.5647 0.3199
0 0 −0.326 −0.4339 0 −23.383 0 0 0 0 0 0 −0.212
0 0 0.2231 0.42213 0 10.785 0 0 0 0 0 0 0.25795
109.5 245.25 68.13 216.58 161.11 277.56 158.97 185.15 259.83 187.68 197.67 289.95 273.16
1.4578 2.3803 0.6517 1.5202 3.1537 0.7104 1.7498 1.8817 2.8176 1.9782 2.8753 4.4303 4.4733
180.15 400.15 132.5 304.21 552 461.15 324.65 363.15 456.65 373.53 430.75 490.85 647.13
0 0 915280 0 0 0 0 0 0 0 0 0 0
All substances are listed in alphabetical order in Table 2-6a. Compiled from Daubert, T. E., R. P. Danner, H. M. Sibul, and C. C. Stebbins, DIPPR Data Compilation of Pure Compound Properties, Project 801 Sponsor Release, July, 1993, Design Institute for Physical Property Data, AIChE, New York, NY; and from Thermodynamics Research Center, “Selected Values of Properties of Hydrocarbons and Related Compounds,” Thermodynamics Research Center Hydrocarbon Project, Texas A&M University, College Station, Texas (extant 1994). Temperatures are expressed in kelvins; heats of vaporization, in J/kmol. J/ kmol × 2.390E−04 = cal/gmol; J/ kmol × 4.302106E−04 = Btu/lbmol. The heat of vaporization equation used is ∆Hv = C1 × (1 − Tr)C2 + C3 × Tr + C4 × Tr × Tr. Tr is the reduced temperature, T/Tc. 1 Coefficients are hypothetical; compound decomposes violently on heating. 2 For the monomer. 3 Equation coefficients are hypothetical above the decomposition temperature. 2-160
SPECIFIC HEATS OF PURE COMPOUNDS
2-161
SPECIFIC HEATS OF PURE COMPOUNDS To convert kilojoules per kilogram-kelvin to British thermal units per pound-degree Rankine, multiply by 0.2388.
UNITS CONVERSIONS For this subsection, the following units conversions are applicable: °F = 9⁄ 5 °C + 32 °F = 1.8 K
ADDITIONAL REFERENCES
To convert calories per gram-kelvin to British thermal units per pound-degree Rankine, multiply by 1.0; to convert calories per grammole-kelvin to British thermal units per pound-mole-degree Rankine, multiply by 1.0. TABLE 2-194
Heat Capacities of the Elements and Inorganic Compounds* Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol
Range of temperature, K
Uncertainty, %
c l c l c l c c c c l c l c c, sillimanite c, disthene c, andalusite c, mullite c c c
4.80 + 0.00322T 7.00 18.74 + 0.01866T 29.5 13.25 + 0.02800T 31.2 76 19.3 50.5 38.63 + 0.04760T − 449200/T 2 142 16.88 + 0.02266T 28.8 22.08 + 0.008971T − 522500/T 2 40.79 + 0.004763T − 992800/T 2 41.81 + 0.005283T − 1211000/T 2 43.96 + 0.001923T − 1086000/T 2 59.65 + 0.0670T 113.2 + 0.0652T 63.5 235
273–931 931–1273 273–370 370–407 273–465 465–504 288–327 288–326 288–326 273–1273 1273–1373 273–464 464–480 273–1973 273–1573 273–1673 273–1573 273–576 273–575 273–373 288–325
1 5 3 5 3 3 ? ? ? 2 ? 3 5 3 3 2 3 5 3 ? ?
c l c c c c c
5.51 + 0.00178T 7.15 17.2 + 0.0293T 10.3 + 0.0511T 19.1 + 0.0171T 22.6 + 0.0162T 24.2 + 0.0132T
273–903 903–1273 273–370 273–346 273–929 273–1198 273–821
2 5 ? ? ? ? ?
Substance Aluminum1 Al AlBr3 AlCl3 AlCl3·6H2O AlF3 AlF3·3aH2O AlF3·3NaF AlI3 Al2O3 Al2O3·SiO2 3Al2O3·2SiO2 4Al2O3·3SiO2 Al2(SO4)3 Al2(SO4)3·18H2O Antimony Sb SbBr3 SbCl3 Sb2O3 Sb2O4 Sb2S3 Argon2 A Arsenic As AsCl3 As2O3 As2S3 Barium BaCl2 BaCl2·H2O BaCl2·2H2O Ba(ClO3)2·H2O BaCO3 BaMoO4 Ba(NO3)2 BaSO4 Beryllium3,4 Be BeO BeO·Al2O3 BeSO4
Additional data are contained in the subsection “Thermodynamic Properties.” Data on water are also contained in that subsection. Additional tables for water are found in Eng. Sci. Data Item 68008, 251 Regent Street, London, England, which contains about 5000 values from 1 to 1000 bar, 0 to 1500°C.
State†
g
4.97
c l c c
5.17 + 0.00234T 31.9 8.37 + 0.0486T 25.8
273–1168 286–371 273–548 293–373
All
5 ? ? ?
0
c c c c c, α c, β c c c
17.0 + 0.00334T 28.2 37.3 51 17.26 + 0.0131T 30.0 34 39.8 21.35 + 0.0141T
273–1198 273–307 273–307 289–320 273–1083 1083–1255 273–297 285–371 273–1323
? ? ? ? 5 15 ? ? 5
c c c c
4.698 + 0.001555T − 121000/T 2 8.69 + 0.00365T − 313000/T 2 25.4 20.8
273–1173 273–1175 273–373 273–373
1 5 ? ?
*From Kelley, U.S. Bur. Mines Bull. 371, 1934. For a revision see Kelley, U.S. Bur. Mines Bull. 477, 1948. Data for many elements and compounds are given by Johnson (ed.), WADD-TR-60-56, 1960, for cryogenic temperatures. Tabulated data for gases can be obtained from many of the references cited in the “Thermodynamic Properties” subsection and other tables in this section. Thinh, Duran, et al., Hydrocarbon Process., 50, 98 (January 1971), review previous equation fits and give newer fits for 408 hydrocarbons and related compounds. Later publications include Duran, Thinh, et al., Hydrocarbon Process., 55, 153 (August 1976); Thompson, J. Chem. Eng. Data, 22(4), 431 (1977); and Passut and Danner, Ind. Eng. Chem. Process Des. Dev., 11, 543 (1972); 13, 193 (1974). † The symbols in this column have the following meaning; c, crystal; l, liquid; g, gas; gls, glass.
TABLE 2-194
Heat Capacities of the Elements and Inorganic Compounds (Continued )
Substance Bismuth4 Bi Bi2O3 Bi2S3 Boron B B2O3 BN Bromine Br2 Cadmium Cd CdO CdS CdSO4·8/3H2O Calcium Ca CaCl2 CaCO3 CaF2 CaMg(CO3)2 CaMoO4 CaO Ca(OH)2 CaO·Al2O3·2SiO2 CaO·MgO·2SiO2 CaO·SiO2 CaP2O6 CaSO4 CaSO4·2H2O CaWO4 Carbon5 C CH4 CO6 CO2 CS2 Cerium Ce CeO2 Ce2(MoO4)3 Ce2(SO4)3 Ce2(SO4)3·5H2O Cesium Cs CsBr CsCl CsF CsI Chlorine Cl2 Chromium4 Cr CrCl3 Cr2O3 CrSb CrSb2 Cr2(SO4)3 Cobalt4 Co CoAs2·CoS2 CoSb Co2Sn CoS CoSO4·7H2O 2-162
State†
Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol
Range of temperature, K
Uncertainty, %
c l c c
5.38 + 0.00260T 7.60 23.27 + 0.01105T 30.4
273–544 544–1273 273–777 284–372
3 3 2 ?
c gls gls c
1.54 + 0.00440T 5.14 + 0.0320T 30.4 1.61 + 0.00400T
273–1174 273–513 513–623 273–1173
5 3 3 5
g
9.00
300–2000
5
c l c c c
5.46 + 0.002466T 7.13 9.65 + 0.00208T 12.9 + 0.00090T 51.3
273–594 594–973 273–2086 273–1273 293
1 5 ? ? ?
c c c c c c c c c c, anorthite gls c, diopside gls c, wollastonite c, pseudowollastonite gls c c c c
5.31 + 0.00333T 6.29 + 0.00140T 16.9 + 0.00386T 19.68 + 0.01189T − 307600/T 2 14.7 + 0.00380T 40.1 33 10.00 + 0.00484T − 108000/T 2 21.4 63.13 + 0.01500T − 1537000/T 2 67.41 + 0.01048T − 1874000/T 2 54.46 + 0.005746T − 1500000/T 2 51.68 + 0.009724T − 1308000/T 2 27.95 + 0.002056T − 745600/T 2 25.48 + 0.004132T − 488100/T 2 23.16 + 0.009672T − 487100/T 2 39.5 18.52 + 0.02197T − 156800/T 2 46.8 27.9
273–673 673–873 273–1055 273–1033 273–1651 299–372 273–297 273–1173 276–373 273–1673 273–973 273–1573 273–973 273–1573 273–1673 273–973 287–371 273–1373 282–373 292–322
2 2 ? 3 ? ? ? 2 ? 1 1 1 1 1 1 1 ? 5 ? ?
c, graphite c, diamond g g g l
2.673 + 0.002617T − 116900/T 2 2.162 + 0.003059T − 130300/T 2 5.34 + 0.0115T 6.60 + 0.00120T 10.34 + 0.00274T − 195500/T 2 18.4
273–1373 273–1313 273–1200 273–2500 273–1200 293
2 3 2 1a 1a ?
c c c c c
5.88 + 0.00123T 15.1 96 66.4 131.6
273–908 273–373 273–297 273–373 273–319
? ? ? ? ?
c l g c c c c
1.96 + 0.0182T 8.00 4.97 12.6 + 0.00259T 11.7 + 0.00309T 11.3 + 0.00285T 11.6 + 0.00268T
273–301 302 All 273–909 273–752 273–957 273–894
3 3 0 ? ? ? ?
g
8.28 + 0.00056T
273–2000
1a
c l c c c c c
4.84 + 0.00295T 9.70 23 26.0 + 0.00400T 12.3 + 0.00120T 19.2 + 0.00184T 67.4
273–1823 1823–1923 286–319 273–2263 273–1383 273–949 273–373
5 10 ? ? ? ? ?
c l c c c c c
5.12 + 0.00333T 8.40 32.9 11.7 + 0.00156T 15.83 + 0.00950T 10.6 + 0.00251T 96
273–1763 1763–1873 283–373 273–1464 273–903 273–1373 286–303
5 5 ? ? 2 ? ?
TABLE 2-194
Heat Capacities of the Elements and Inorganic Compounds (Continued )
Substance Copper7 Cu CuAl CuAl2 Cu3Al CuI CuI2 CuO CuO·SiO2·H2O CuS Cu2S CuS·FeS Cu2Sb Cu2Sb Cu2Se Cu3Si CuSO4 CuSO4·H2O CuSO4·3H2O CuSO4·5H2O Fluorine8 F2 Gallium Ga2O3 Ga2(SO4)3 Germanium4 Ge Gold Au AuSb2 Helium9 He Hydrogen10 H H2 HBr HCl HI H2O H2S H2S2O7 Indium In Iodine I2 Iridium Ir Iron4 Fe
FeAs2 Fe3C FeCO3 FeO Fe2O3 Fe3O4 Fe2O3·3H2O FeS FeS2 FeSi Fe2SiO4 FeSO4 Fe2(SO4)3 FeSO4·4H2O FeSO4·7H2O Krypton Kr
State†
Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol
Range of temperature, K
Uncertainty, %
273–1357 1357–1573 273–733 273–773 273–775 273–675 274–328 273–810 293–323 273–1273 273–376 376–1173 292–321 273–573 273–693 273–383 383–488 273–1135 282 282 282 282
1 3 2 2 2 ? ? 2 ? ? 3 2 ? 2 2 5 5 ? ? ? ? ?
c l c c c c c c c c c, α c, β c c c c, α c, β c c c c c
5.44 + 0.001462T 7.50 9.88 + 0.00500T 16.78 + 0.00366T 19.61 + 0.01054T 12.1 + 0.00286T 20.1 10.87 + 0.003576T − 150600/T 2 29 10.6 + 0.00264T 9.38 + 0.0312T 20.9 24 13.73 + 0.01350T 21.79 + 0.00900T 20.85 20.35 20.3 + 0.00587T 24.1 31.3 49.0 67.2
g
6.50 + 0.00100T
300–3000
5
c c
18.2 + 0.0252T 62.4
273–923 273–373
? ?
273–1336 1336–1573 273–628 628–713
2 5 1 ?
c c l c, α c, βγ
5.61 + 0.00144T 7.00 17.12 + 0.00465T 11.47 + 0.01756T
g
4.97
g g g g g l g g c l
4.97 6.62 + 0.00081T 6.80 + 0.00084T 6.70 + 0.00084T 6.93 + 0.00083T See Tables 2-355 through 2-357 8.22 + 0.00015T + 0.00000134T 2 7.20 + 0.00360T 27 58
All 273–2500 273–2000 273–2000 273–2000
0 2 2 1a 2
300–2500 300–600 281 308
? 8 ? ?
g
9.00
300–2000
5
c
5.50 + 0.00148T
273–1873
1
c, α c, β c, γ c, δ l c c c c c c c c, α c, β c c c c c c c
4.13 + 0.00638T 6.12 + 0.00336T 8.40 10.0 8.15 17.8 25.17 + 0.00223T 22.7 12.62 + 0.001492T − 76200/T 2 24.72 + 0.01604T − 423400/T 2 41.17 + 0.01882T − 979500/T 2 47.8 2.03 + 0.0390T 12.05 + 0.00273T 10.7 + 0.01336T 10.54 + 0.00458T 33.57 + 0.01907T − 879700/T 2 22 66.2 63.6 96
273–1041 1041–1179 1179–1674 1674–1803 1803–1873 283–373 273–1173 293–368 273–1173 273–1097 273–1065 286–373 273–411 411–1468 273–773 273–903 273–1161 293–373 273–373 282 291–319
3 3 5 5 5 ? 10 ? 2 2 2 ? 5 3 ? 2 2 ? ? ? ?
g
4.97
All
0
c
All
0 2-163
2-164
PHYSICAL AND CHEMICAL DATA TABLE 2-194
Heat Capacities of the Elements and Inorganic Compounds (Continued )
Substance Lanthanum La La2O3 La2(MoO4)3 La2(SO4)3 La2(SO4)3·9H2O Lead4 Pb Pb3(AsO4)2 PbB2O4 PbB4O7 PbBr2 PbCl2 2PbCl2·NH4Cl PbCO3 PbCrO4 PbF2 PbI2 PbMoO4 Pb(NO3)2 PbO PbO2 Pb2P2O7 PbS PbSO4 PbS2O3 PbWO4 Lithium Li LiBr LiBr·H2O LiCl LiCl·H2O LiF LiI LiI·H2O LiI·2H2O LiI·3H2O LiNO3 Magnesium4 Mg MgAg Mg4Al3 MgAu Mg2Au Mg3Au MgCl2 MgCl2·6H2O MgCO3 MgCu2 Mg2Cu MgNi2 MgO MgO·Al2O3 MgO·SiO2 6MgO·MgCl2·8B2O3 Mg(OH)2 Mg3Sb2 Mg2Si MgSO4 MgSO4·H2O MgSO4·6H2O MgSO4·7H2O
State†
Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol
Range of temperature, K
Uncertainty, %
c c c c c
5.91 + 0.00100T 22.6 + 0.00544T 86 66.9 152
273–1009 273–2273 273–307 273–373 273–319
? ? ? ? ?
c l c c c c l c l c c c c c l c c c c c c c c c
5.77 + 0.00202T 6.8 65.5 26.5 41.4 18.13 + 0.00310T 27.4 15.88 + 0.00835T 27.2 53.1 21.1 29.1 16.5 + 0.00412T 18.66 + 0.00293T 32.3 30.4 36.4 10.33 + 0.00318T 12.7 + 0.00780T 48.3 10.63 + 0.00401T 26.4 29 35
273–600 600–1273 286–370 288–371 289–371 273–761 761–860 273–771 771–851 293 286–320 292–323 273–1091 273–648 648–776 292–322 286–320 273–544 273–? 284–371 273–873 293–372 293–373 273–297
2 5 ? ? ? 2 10 2 10 ? ? ? ? 2 20 ? ? 2 ? ? 3 ? ? ?
c g c c c c c c c c c c l
0.68 + 0.0180T 4.97 11.5 + 0.00302T 22.6 11.0 + 0.00339T 23.6 8.20 + 0.00520T 12.5 + 0.00208T 23.6 32.9 43.2 9.17 + 0.0360T 26.8
273–459 All 273–825 278–318 273–887 279–360 273–1117 273–723 277–359 277–345 277–347 273–523 523–575
10 0 ? ? ? ? ? ? ? ? ? 5 5
c l c c c c c c c c c c c c c c, amphibole c, pyroxene gls c, α c, β c c c c c c c
6.20 + 0.00133T − 67800/T 2 7.4 10.58 + 0.00412T 34.4 + 0.0198T 11.3 + 0.00189T 16.2 + 0.00451T 21.2 + 0.00614T 17.3 + 0.00377T 77.1 16.9 14.96 + 0.00776T 15.5 + 0.00652T 15.87 + 0.00692T 10.86 + 0.001197T − 208700/T 2 28 25.60 + 0.004380T − 674200/T 2 23.35 + 0.008062T − 558800/T 2 23.30 + 0.007734T − 542000/T 2 58.7 + 0.408T 107.2 + 0.2876T 18.2 28.2 + 0.00560T 15.4 + 0.00415T 26.7 33 80 89
273–923 923–1048 273–905 273–736 273–1433 273–1073 273–1103 273–991 292–342 290 273–903 273–843 273–903 273–2073 288–319 273–1373 273–773 273–973 273–538 538–623 292–323 273–1234 273–1343 296–372 282 282 291–319
1 10 2 ? ? ? ? ? ? ? 3 ? 2 2 ? 1 1 1 5 5 ? ? ? ? ? ? ?
SPECIFIC HEATS OF PURE COMPOUNDS TABLE 2-194
Heat Capacities of the Elements and Inorganic Compounds (Continued ) Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol
Range of temperature, K
Uncertainty, %
c, α c, β c, γ l c c c c c c c c c c
3.76 + 0.00747T 5.06 + 0.00395T 4.80 + 0.00422T 11.0 16.2 + 0.00520T 7.79 + 0.0421T + 0.0000090T 2 7.43 + 0.01038T − 0.00000362T 2 10.33 + 0.0530T − 0.0000257T 2 19.25 + 0.0538T − 0.0000209T 2 1.92 + 0.0471T − 0.0000297T 2 31 10.21 + 0.00656T − 0.00000242T 2 27.5 78
273–1108 1108–1317 1317–1493 1493–1673 273–923 273–773 273–1923 273–1173 273–1773 273–773 291–322 273–1883 293–373 290–319
5 5 5 10 ? ? ? ? ? ? ? ? ? ?
l g g c c c c c, α c, β c c c
6.61 4.97 9.00 11.05 + 0.00370T 15.3 + 0.0103T 25 11.4 + 0.00461T 17.4 + 0.004001T 20.2 11.5 10.9 + 0.00365T 31.0
273–630 All 300–2000 273–798 273–553 285–319 273–563 273–403 403–523 278–371 273–853 273–307
1 0 5 ? ? ? ? 3 3 ? ? ?
c c c
5.69 + 0.00188T − 50300/T 2 15.1 + 0.0121T 19.7 + 0.00315T
273–1773 273–1068 273–729
5 ? ?
g
4.97
c, α c, β l c c c c c c c c
4.26 + 0.00640T 6.99 + 0.000905T 8.55 11.3 + 0.00215T 9.25 + 0.00640T 15.8 + 0.00329T 10.0 + 0.00312T 20.78 + 0.0102T 33.4 82 11.00 + 0.00433T
g g c c, α c, β c c c g
Substance Manganese Mn
MnCl2 MnCO3 MnO Mn2O3 Mn3O4 MnO2 Mn2O3·H2O MnS MnSO4 MnSO4·5H2O Mercury11 Hg Hg2 HgCl HgCl2 Hg(CN)2 HgI HgI2 HgO HgS Hg2SO4 Molybdenum Mo MoO3 MoS2 Neon12 Ne Nickel4 Ni NiO NiS Ni2Si NiSi Ni3Sn NiSO4 NiSO4·6H2O NiTe Nitrogen13 N2 NH3 NH4Br NH4Cl NH4I NH4NO3 (NH4)2SO4 NO Osmium Os Oxygen14 O2 Palladium Pd Phosphorus P PCl3 P4O10 Platinum4 Pt Potassium K
State†
All
0
273–626 626–1725 1725–1903 273–1273 273–597 273–1582 273–1273 273–904 293–373 291–325 273–700
2 5 10 ? 3 ? ? 2 ? ? 2
6.50 + 0.00100T 6.70 + 0.00630T 22.8 9.80 + 0.0368T 5.0 + 0.0340T 17.8 31.8 51.6 8.05 + 0.000233T − 156300/T 2
300–3000 300–800 274–328 273–457 457–523 273–328 273–293 275–328 300–5000
3 1a ? 5 5 ? ? ? 2
c
5.686 + 0.000875T
273–1877
1
g
8.27 + 0.000258T − 187700/T
300–5000
1
2
c
5.41 + 0.00184T
273–1822
2
c, yellow c, red l l c g
5.50 0.21 + 0.0180T 6.6 28.7 15.72 + 0.1092T 73.6
273–317 273–472 317–373 284–371 273–631 631–1371
5 10 10 ? 2 3
c
5.92 + 0.00116T
273–1873
1
c l
5.24 + 0.00555T 7.7
273–336 336–373
5 5
2-165
2-166
PHYSICAL AND CHEMICAL DATA TABLE 2-194
Heat Capacities of the Elements and Inorganic Compounds (Continued )
Substance Potassium—(Cont.) K K2 KAsO3 KBO2 K2B4O7 KBr KCl KClO3 KClO4 2KCl·CuCl2.2H2O 2KCl·PtCl4 2KCl·SnCl4 2KCl·ZnCl2 2KCN·Zn(CN)2 K2CO3 K2CrO4 K2Cr2O7 KF K4Fe(CN)6 K4Fe(CN)6·3H2O KH2AsO4 KH2PO4 KHSO4 KMnO4 KNO3 K2O·Al2O3·3SiO2
K4P2O7 K2SO4 K2S2O3 K2SO4·Al2(SO4)3·24H2O K2SO4·Cr2(SO4)3·24H2O K2SO4·MgSO4·6H2O K2SO4·NiSO4·6H2O K2SO4·ZnSO4·6H2O Prometheum Pr Radon Rn Rhenium Re Rhodium Rh Rubidium Rb RbBr RbCl Rb2CO3 RbF RbI Scandium Sc2O3 Sc2(SO4)3 Selenium Se Silicon Si SiC SiCl4 SiO2
Silver4 Ag
State† g g c c c c c c c c c c c c c c c l c c c c c c c c c l c, orthoclase gls, orthoclase c, microcline gls, microcline c c c c c c c c
Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol 4.97 9.00 25.3 12.6 + 0.0126T 51.3 11.49 + 0.00360T 10.93 + 0.00376T 25.7 26.3 63 55 54.5 43.4 57.4 29.9 35.9 42.80 + 0.0410T 96.9 10.8 + 0.00284T 80.1 114.5 32 28.3 30 28 6.42 + 0.0530T 28.8 29.5 69.26 + 0.00821T − 2331000/T 2 69.81 + 0.01053 − 2403000/T 2 65.65 + 0.01102T − 1748000/T 2 64.83 + 0.01438T − 1641000/T 2 63.1 33.1 37 352 324 106 107 120
Range of temperature, K
Uncertainty, %
All 300–2000 290–372 273–1220 290–372 273–543 273–1043 289–371 287–318 292–323 286–319 292–323 279–319 277–319 296–372 289–371 273–671 671–757 273–1129 273–319 273–310 289–319 290–320 292–324 287–318 273–401 401–611 611–683 273–1373 273–1373 273–1373 273–1373 290–371 287–371 293–373 292–322 292–324 292–323 289–319 293–317
0 5 ? ? ? 2 2 ? ? ? ? ? ? ? ? ? 5 5 ? ? ? ? ? ? ? 10 5 10 1a 1a 1a 1a ? ? ? ? ? ? ? ?
c g
4.97
c
6.30 + 0.00053T
273–2273
All
0 ?
c
5.40 + 0.00219T
273–1877
2
c l c c c c c
3.27 + 0.0131T 7.85 11.6 + 0.00255T 11.5 + 0.00249T 28.4 11.3 + 0.00256T 11.6 + 0.00263T
273–312 312–373 273–954 273–987 291–320 273–1048 273–913
2 5 ? ? ? ? ?
c c
21.1 62.0
273–373 273–373
? ?
c l
4.53 + 0.00550T 8.35
273–490 490–570
2 3
c c l c, quartz, α c, quartz, β c, cristobalite, α c, cristobalite, β gls
5.74 + 0.000617T − 101000/T 2 8.89 + 0.00291T − 284000/T 2 32.4 10.87 + 0.008712T − 241200/T 2 10.95 + 0.00550T 3.65 + 0.0240T 17.09 + 0.000454T − 897200/T 2 12.80 + 0.00447T − 302000/T 2
273–1174 273–1629 293–373 273–848 848–1873 273–523 523–1973 273–1973
2 2 ? 1 3a 2a 2 3a
c l
5.60 + 0.00150T 8.2
273–1234 1234–1573
1 3
SPECIFIC HEATS OF PURE COMPOUNDS TABLE 2-194
Heat Capacities of the Elements and Inorganic Compounds (Continued )
Substance Silver—(Cont.) Ag3Al Ag2Al AgAl12 AgBr AgCl AgCNO AgI AgNO3 Ag3PO4 Ag2S Ag3Sb Ag2Se Sodium15 Na NaBO2 Na2B4O7 Na2B4O7·10H2O NaBr NaCl NaClO3 NaCNO Na2CO3 NaF Na2HPO4·7H2O Na2HPO4·12H2O NaI NaNO3 Na2O·Al2O3·3SiO2 NaPO3 Na4P2O7 Na2SO4 Na2S2O3 Na2S2O3·5H2O Sodium-potassium alloys15 Strontium SrBr2 SrBr2·H2O SrBr2·6H2O SrCl2 SrCl2·H2O SrCl2·2H2O SrCO3 SrI2 SrI2·H2O SrI2·2H2O SrI2·6H2O SrMoO4 Sr(NO3)2 SrSO4 Sulfur16 S S2 S2Cl2 SO2 Tantalum Ta Tellurium Te Thallium Tl
State†
Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol
Range of temperature, K
Uncertainty, %
273–902 273–903 273–768 273–703 703–836 273–728 728–806 273–353 273–423 273–433 433–482 482–541 293–325 273–448 448–597 273–694 273–406 406–460
2 2 5 6 5 2 5 ? 6 2 5 5 ? 5 5 5 5 5
273–371 371–451 All 273–1239 289–371 292–323 273–543 273–1074 1073–1205 273–528 528–572 273–353 288–371 273–1261 275–307 275–307 273–936 273–583 583–703 273–1373 273–1173 290–319 290–371 289–371 273–307 273–307
1a 2 0 ? ? ? 2 2 3 3 5 ? ? ? ? ? ? 5 10 1 1 ? ? ? ? ?
c c c c l c l c c, α c, α c, β l c c, α c, β c c, α c, β
22.56 + 0.00570T 16.85 + 0.00450T 58.62 + 0.0575T 8.58 + 0.0141T 14.9 9.60 + 0.00929T 14.05 18.7 8.58 + 0.0141T 18.83 + 0.0160T 25.7 30.2 37.5 18.8 21.8 19.53 + 0.0160T 20.2 20.4
c l g c c c c c l c l c c c c c c c l c, albite gls c c c c c l
5.01 + 0.00536T 7.50 4.97 10.4 + 0.0199T 47.9 147 11.74 + 0.00233T 10.79 + 0.00420T 15.9 9.48 + 0.0468T 31.8 13.1 28.9 10.4 + 0.00289T 86.6 133.4 12.5 + 0.00162T 4.56 + 0.0580T 37.2 63.78 + 0.01171T − 1678000/T 2 61.25 + 0.01768T − 1545000/T 2 22.1 60.7 32.8 34.9 86.2
c c c c c c c c c c c c c c
18.1 + 0.00311T 28.9 82.1 18.2 + 0.00244T 28.7 38.3 21.8 18.6 + 0.00304T 28.5 39.1 84.9 37 38.3 26.2
273–923 277–370 276–327 273–1143 276–365 277–366 281–371 273–783 276–363 275–336 275–333 273–297 290–320 293–369
? ? ? ? ? ? ? ? ? ? ? ? ? ?
c, rhombic c, monoclinic g l g
3.63 + 0.00640T 4.38 + 0.00440T 8.58 + 0.00030T 27.5 7.70 + 0.00530T − 0.00000083T 2
273–368 368–392 300–2500 273–332 300–2500
3 3 5 ? 2a
c
5.91 + 0.00099T
273–1173
2
c
5.19 + 0.00250T
273–600
3
c, α c, β
5.32 + 0.00385T 8.12
273–500 500–576
1 1
2-167
2-168
PHYSICAL AND CHEMICAL DATA TABLE 2-194
Heat Capacities of the Elements and Inorganic Compounds (Concluded )
Substance Thallium—(Cont.) Tl TlBr TlCl Thorium Th ThO2 Th(SO4)2 Tin4 Sn SnAu SnCl2 SnCl4 SnO SnO2 SnPt SnS SnS2 Titanium Ti TiCl4 TiO2 Tungsten W WO3 Uranium U U3O8 Vanadium V Xenon Xe Zinc4 Zn ZnCl2 ZnO ZnS ZnSb ZnSO4 ZnSO4·H2O ZnSO4·6H2O ZnSO4·7H2O Zirconium ZrO2 ZrO2·SiO2 1
State†
Heat capacity at constant pressure (T = K; 0°C = 273.1 K), cal/deg mol
Range of temperature, K
Uncertainty, %
l c l c l
7.12 12.53 + 0.00100T 16.0 12.56 + 0.00088T 14.2
576–773 273–733 733–800 273–700 700–803
3 10 10 5 10
c c c
6.40 14.6 + 0.00507T 41.2
273–373 273–1273 273–373
? ? ?
c l c c l c c c c c
5.05 + 0.00480T 6.6 11.79 + 0.00233T 16.2 + 0.00926T 38.4 9.40 + 0.00362T 13.94 + 0.00565T − 252000/T 2 11.49 + 0.00190T 12.1 + 0.00165T 20.5 + 0.00400T
273–504 504–1273 273–581 273–520 286–371 273–1273 273–1373 273–1318 273–1153 273–873
2 10 1 ? ? ? ? 1 ? ?
c l c
8.91 + 0.00114T − 433000/T 2 35.7 11.81 + 0.00754T − 41900/T 2
273–713 285–372 273–713
3 ? 3
c c
5.65 + 0.00866 16.0 + 0.00774T
273–2073 273–1550
1 ?
c c
6.64 59.8
273–372 276–314
? ?
c
5.57 + 0.00097T
273–1993
?
g
4.97
c l c c c c c c c c
5.25 + 0.00270T 7.59 + 0.00055T 15.9 + 0.00800T 11.40 + 0.00145T − 182400/T 2 12.81 + 0.00095T − 194600/T 2 11.5 + 0.00313T 28 34.7 80.8 100.2
273–692 692–1122 273–638 273–1573 273–1173 273–810 293–373 282 282 273–307
1 3 ? 1 5 ? ? ? ? ?
c c
11.62 + 0.01046T − 177700/T 2 26.7
273–1673 297–372
5 ?
All
0
See also Table 2-195. Data to 298 K are also given by Scott, Cryogenic Engineering, Van Nostrand, Princeton, N.J., 1959. For liquid and gas data, see Johnson (ed.), WADD-TR-60-56, 1960. 3 Stalder, NACA Tech. Note 4141, 1957 (Fig. 5), gives data from 400 to 2600°R. 4 See also Table 2-195. 5 For data from 400 to 5500°R see Stalder, NACA Tech. Note 4141, 1975 (Fig. 4). 6 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-60-56, 1960. 7 For data from 400 to 2350°R see Stalder, NACA Tech. Note 4141, 1957. 8 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-60-56, 1960. 9 For liquid and gas data, see Johnson (ed.), WADD-TR-60-56, 1960. 10 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-60-56, 1960. 11 See also Table 2-195; Douglas, Ball, et al., Bur. Stand. J. Res., 46 (1951): 334; Busey and Giaque, J. Am. Chem. Soc., 75 (1953): 806; Sheldon, ASME Pap. 49-A-30, 1949. 12 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-56-60, 1960. 13 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-56-60, 1960. 14 For solid, liquid, and gas data, see Johnson (ed.), WADD-TR-56-60, 1960. Ozone: For liquid see Brabets and Waterman, J. Chem. Phys., 28 (1958): 1212. 15 For data on liquid Na-K alloys to 1500°F and for liquid Na to 1460°F, see Lubarsky and Kaufman, NACA Rep. 1270, 1956. 16 See also Evans and Wagman, Bur. Stand. J. Res. 49 (1952): 141; Gratch, OTS PB 124957, 1950; Guthrie, Scott, et al., J. Am. Chem. Soc., 76 (1954): 1488. 2
SPECIFIC HEATS OF PURE COMPOUNDS TABLE 2-195
2-169
Specific Heat [kJ/(kg·K)] of Selected Elements Temperature, K
Symbol
4
6
8
10
20
40
60
80
100
200
250
300
400
600
800
Al Be Bi Cr Co
0.00026 0.00008 0.00054 0.00016 0.00036
0.00050
0.00088
0.214
0.357
0.00541 0.00050 0.00085
0.0089 0.0014 0.0340 0.0021 0.0048
0.0775
0.00220 0.00029 0.00059
0.00140 0.00028 0.01040 0.00081 0.00121
0.0729 0.0107 0.0404
0.092 0.059 0.110
0.102 0.127 0.184
0.481 0.195 0.109 0.190 0.234
0.797 1.109 0.120 0.382 0.376
0.859 1.537 0.121 0.424 0.406
0.902 1.840 0.122 0.450 0.426
0.949 2.191 0.123 0.501 0.451
1.042 2.605 0.142 0.565 0.509
1.134 2.823 0.136 0.611 0.543
Cu Ge Au Ir Fe
0.00011
0.00024
0.00018
0.00047
0.00048 0.00037 0.00126
0.137 0.108 0.084
0.203 0.153 0.100
0.00061
0.00090
0.0076 0.0129 0.0163 0.0021 0.0039
0.059 0.0619 0.0569
0.00038
0.00086 0.00081 0.00255 0.00032 0.00127
0.0276
0.086
0.154
0.254 0.192 0.109 0.090 0.216
0.357 0.286 0.124 0.122 0.384
0.377 0.305 0.127 0.128 0.422
0.386 0.323 0.129 0.131 0.450
0.396 0.343 0.131 0.133 0.491
0.431 0.364 0.136 0.140 0.555
0.448 0.377 0.141 0.146 0.692
Pb Mg Hg Mo Ni
0.00075 0.00034 0.00417 0.00011 0.00054
0.00242 0.00080 0.01420 0.00019 0.00086
0.00747 0.00155 0.01820 0.00032 0.00121
0.01350 0.00172 0.02250 0.00050 0.00178
0.0531 0.0148 0.0515 0.0029 0.0058
0.0944 0.138 0.0895 0.0236 0.0380
0.108 0.336 0.107 0.061 0.103
0.114 0.513 0.116 0.105 0.173
0.118 0.648 0.121 0.140 0.232
0.125 0.929 0.136 0.223 0.383
0.127 0.985 0.141 0.241 0.416
1.129 1.005 0.139 0.248 0.444
0.132 1.082 0.136 0.261 0.490
0.142 1.177 0.135 0.280 0.590
1.263 0.104 0.292 0.530
Pt Ag Sn Zn
0.00019 0.00016 0.00024 0.00011
0.00028 0.00035 0.00127 0.00029
0.00067 0.00093 0.00423 0.00096
0.00112 0.00186 0.00776 0.00250
0.0077 0.0159 0.0400 0.0269
0.0382 0.0778 0.108 0.123
0.069 0.133 0.149 0.205
0.088 0.166 0.173 0.258
0.101 0.187 0.189 0.295
0.127 0.225 0.214 0.366
0.132 0.232 0.220 0.380
0.134 0.236 0.222 0.389
0.136 0.240 0.245 0.404
0.140 0.251 0.257 0.435
0.146 0.264 0.257 0.479
TABLE 2-196 Cmpd. no.
Heat Capacities of Inorganic and Organic Liquids Name
Formula
CAS no.
Mol wt
C1
C2
C3
C4
C5
Tmin, K
Cp at Tmin × 1E−05
Tmax, K
Cp at Tmax × 1E−05
1 2 3 4 5 6 7 8 9 10
Methane (eqn. 2) Ethane (eqn. 2) Propane (eqn. 2) n-Butane (eqn. 2) n-Pentane n-Hexane n-Heptane (eqn. 2) n-Octane n-Nonane n-Decane
CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22
74828 74840 74986 106978 109660 110543 142825 111659 111842 124185
16.043 30.070 44.097 58.123 72.150 86.177 100.204 114.231 128.258 142.285
6.5708E+01 4.4009E+01 6.2983E+01 6.4730E+01 1.5908E+05 1.7212E+05 6.1260E+01 2.2483E+05 3.8308E+05 2.7862E+05
3.8883E+04 8.9718E+04 1.1363E+05 1.6184E+05 −2.7050E+02 −1.8378E+02 3.1441E+05 −1.8663E+02 −1.1398E+03 −1.9791E+02
−2.5795E+02 9.1877E+02 6.3321E+02 9.8341E+02 9.9537E−01 8.8734E−01 1.8246E+03 9.5891E−01 2.7101E+00 1.0737E+00
6.1407E+02 −1.8860E+03 −8.7346E+02 −1.4315E+03 0 0 −2.5479E+03 0 0 0
0 0 0 0 0 0 0 0 0 0
90.69 92 85.47 134.86 143.42 177.83 182.57 216.38 219.66 243.51
0.5361 0.6855 0.8488 1.1380 1.4076 1.6750 1.9989 2.2934 2.6348 2.9409
190 290 360 420 390 460 520 460 325 460
11 12 13 14 15 16 17 18 19 20
n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane n-Eicosane
C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40 C20H42
1120214 112403 629505 629594 629629 544763 629787 593453 629925 112958
156.312 170.338 184.365 198.392 212.419 226.446 240.473 254.500 268.527 282.553
2.9398E+05 5.0821E+05 3.5018E+05 3.5314E+05 3.4691E+05 3.7035E+05 3.7697E+05 3.9943E+05 3.4257E+05 3.5272E+05
−1.1498E+02 −1.3687E+03 −1.0470E+02 2.9130E+01 2.1954E+02 2.3147E+02 3.4782E+02 3.7464E+02 7.6208E+02 8.0732E+02
9.6936E−01 3.1015E+00 1.0022E+00 8.6116E−01 6.5632E−01 6.8632E−01 5.7895E−01 5.8156E−01 2.0481E−01 2.1220E−01
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
247.57 263.57 267.76 279.01 283.07 291.31 295.13 301.31 305.04 309.58
3.2493 3.6292 3.9400 4.2831 4.6165 4.9602 5.3005 5.6511 5.9409 6.2299
433.42 330 508.62 526.73 543.84 560.01 575.3 589.86 603.05 616.93
4.2624 3.9429 5.5619 6.0741 6.6042 7.1521 7.6869 8.2276 8.7663 9.3154
21 22 23 24 25 26 27
2-Methylpropane 2-Methylbutane 2,3-Dimethylbutane 2-Methylpentane 2,3-Dimethylpentane1 2,3,3-Trimethylpentane 2,2,4-Trimethylpentane
C4H10 C5H12 C6H14 C6H14 C7H16 C8H18 C8H18
75285 78784 79298 107835 565593 560214 540841
58.123 72.150 86.177 86.177 100.204 114.231 114.231
1.7237E+05 1.0830E+05 1.2945E+05 1.4222E+05 1.4642E+05 3.8862E+05 9.5275E+04
−1.7839E+03 1.4600E+02 1.8500E+01 −4.7830E+01 5.9200E+01 −1.4395E+03 6.9670E+02
1.4759E+01 −2.9200E−01 6.0800E−01 7.3900E−01 6.0400E−01 3.2187E+00 −1.3765E+00
−4.7909E−02 1.5100E−03 0 0 0 0 2.1734E−03
5.8050E−05 0 0 0 0 0 0
113.54 113.25 145.19 119.55 90 280 165.78
0.9961 1.2328 1.4495 1.4706 1.5664 2.3791 1.8285
380 310 331.13 333.41 380 320 520
2.0725 1.7048 2.0224 2.0842 2.5613 2.5757 3.9095
28 29 30 31 32 33 34 35
Ethylene Propylene 1-Butene cis-2-Butene trans-2-Butene 1-Pentene 1-Hexene 1-Heptene
C2H4 C3H6 C4H8 C4H8 C4H8 C5H10 C6H12 C7H14
74851 115071 106989 590181 624646 109671 592416 592767
28.054 42.081 56.108 56.108 56.108 70.134 84.161 98.188
2.4739E+05 1.1720E+05 1.3589E+05 1.2668E+05 1.1276E+05 1.5467E+05 1.9263E+05 1.8997E+05
−4.4280E+03 −3.8632E+02 −4.7739E+02 −6.5470E+01 −1.0470E+02 −4.2600E+02 −5.7116E+02 −1.5670E+02
4.0936E+01 1.2348E+00 2.1835E+00 −6.4000E−01 5.2140E−01 1.9640E+00 2.4004E+00 3.4300E−01
−1.6970E−01 0 −2.2230E−03 2.9120E−03 0 −1.8038E−03 −1.9758E−03 1.5222E−03
2.6816E−04 0 0 0 0 0 0 0
103.97 87.89 87.8 134.26 167.62 107.93 133.39 154.27
0.7013 0.9279 1.0930 1.1340 1.0986 1.2930 1.5446 1.7955
252.7 298.15 300 350 274.03 310 336.63 330
0.9758 1.1178 1.2917 1.5022 1.2322 1.5761 1.9700 2.3032
36 37 38 39 40 41 42 43 44
1-Octene 1-Nonene 1-Decene 2-Methylpropene 2-Methyl-1-butene2 2-Methyl-2-butene2 1,2-Butadiene 1,3-Butadiene 2-Methyl-1,3-butadiene
C8H16 C9H18 C10H20 C4H8 C5H10 C5H10 C4H6 C4H6 C5H8
111660 124118 872059 115117 563462 513359 590192 106990 78795
112.215 126.242 140.269 56.108 70.134 70.134 54.092 54.092 68.119
3.7930E+05 2.5875E+05 3.1950E+05 8.7680E+04 1.4951E+05 1.5160E+05 1.3515E+05 1.2886E+05 1.4148E+05
−2.1175E+03 −3.5450E+02 −5.7621E+02 2.1710E+02 −2.4763E+02 −2.6672E+02 −3.1114E+02 −3.2310E+02 −2.8870E+02
8.2362E+00 1.3126E+00 1.7087E+00 −9.1530E−01 9.1849E−01 9.0847E−01 9.7007E−01 1.0150E+00 1.0910E+00
−9.0093E−03 0 0 2.2660E−03 0 0 −1.5230E−04 3.2000E−05 0
0 0 0 0 0 0 0 0 0
171.45 191.78 206.89 132.81 135.58 139.39 136.95 165 130.32
2.1295 2.3904 2.7343 1.0568 1.3282 1.3207 1.1034 1.0333 1.2239
315 420.02 443.75 343.15 304.31 311.71 290 350 307.2
2.4793 3.4142 4.0027 1.4596 1.5921 1.5673 1.2279 1.4148 1.5575
45 46 47 48 49 50
Acetylene Methylacetylene Dimethylacetylene 3-Methyl-1-butyne 1-Pentyne 2-Pentyne
C2H2 C3H4 C4H6 C5H8 C5H8 C5H8
74862 74997 503173 598232 627190 627214
26.038 40.065 54.092 68.119 68.119 68.119
2.0011E+05 7.9791E+04 8.8153E+04 1.0520E+05 8.6200E+04 6.8671E+04
−1.1988E+03 8.9490E+01 1.2416E+02 1.9110E+02 2.5660E+02 2.4666E+02
3.0027E+00 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
192.4 200 240.91 200 200 200
0.8061 0.9769 1.1806 1.4342 1.3752 1.1800
250 249.94 300.13 299.49 313.33 329.27
0.8808 1.0216 1.2542 1.6243 1.6660 1.4989
2-170
14.9780 1.2444 2.6079 5.0822 2.0498 2.7534 4.0657 3.4189 2.9890 4.1478
51 52 53 54 55 56
1-Hexyne 2-Hexyne 3-Hexyne 1-Heptyne 1-Octyne Vinylacetylene3
C6H10 C6H10 C6H10 C7H12 C8H14 C4H4
693027 764352 928494 628717 629050 689974
82.145 82.145 82.145 96.172 110.199 52.076
9.3000E+04 9.4860E+04 8.2795E+04 8.5122E+04 9.1748E+04 6.8720E+04
3.2600E+02 2.5415E+02 2.8340E+02 4.0247E+02 4.7140E+02 1.3500E+02
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
200 300 300 192.22 193.55 200
1.5820 1.7110 1.6781 1.6248 1.8299 0.9572
344.48 357.67 354.35 372.93 399.35 278.25
2.0530 1.8576 1.8322 2.3522 2.8000 1.0628
57 58 59 60 61 62 63 64 65 66
Cyclopentane Methylcyclopentane Ethylcyclopentane Cyclohexane Methylcyclohexane 1,1-Dimethylcyclohexane Ethylcyclohexane Cyclopentene 1-Methylcyclopentene Cyclohexene
C5H10 C6H12 C7H14 C6H12 C7H14 C8H16 C8H16 C5H8 C6H10 C6H10
287923 96377 1640897 110827 108872 590669 1678917 142290 693890 110838
70.134 84.161 98.188 84.161 98.188 112.215 112.215 68.119 82.145 82.145
1.2253E+05 1.5592E+05 1.7852E+05 −2.2060E+05 1.3134E+05 1.3450E+05 1.3236E+05 1.2538E+05 5.3271E+04 1.0585E+05
−4.0380E+02 −4.9000E+02 −5.1835E+02 3.1183E+03 −6.3100E+01 8.7650E+00 7.2740E+01 −3.4970E+02 3.2792E+02 −6.0000E+01
1.7344E+00 2.1383E+00 2.3255E+00 −9.4216E+00 8.1250E−01 8.1151E−01 6.4738E−01 1.1430E+00 0 6.8000E−01
−1.0975E−03 −1.5585E−03 −1.6818E−03 1.0687E−02 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
179.28 130.73 134.71 279.69 146.58 239.66 161.84 138.13 200 169.67
0.9956 1.2492 1.4678 1.4836 1.3955 1.8321 1.6109 0.9888 1.1885 1.1525
322.4 366.48 301.82 400 320 392.7 404.95 317.38 348.64 356.12
1.3584 1.8682 1.8767 2.0323 1.9435 2.6309 2.6798 1.2953 1.6760 1.7072
67 68 69 70 71 72 73
Benzene Toluene o-Xylene m-Xylene p-Xylene Ethylbenzene Propylbenzene
C6H6 C7H8 C8H10 C8H10 C8H10 C8H10 C9H12
71432 108883 95476 108383 106423 100414 103651
78.114 92.141 106.167 106.167 106.167 106.167 120.194
1.2944E+05 1.4014E+05 3.6500E+04 1.7555E+05 −3.5500E+04 1.3316E+05 2.3477E+05
−1.6950E+02 −1.5230E+02 1.0175E+03 −2.9950E+02 1.2872E+03 4.4507E+01 −8.0022E+02
6.4781E−01 6.9500E−01 −2.6300E+00 1.0880E+00 −2.5990E+00 3.9645E−01 3.4037
0 0 3.0200E−03 0 2.4260E−03 0 −3.1739E−03
0 0 0 0 0 0 0
278.68 178.18 248 225.3 286.41 178.15 173.59
1.3251 1.3507 1.7315 1.6330 1.7697 1.5367 1.8182
353.24 500 415 360 600 409.35 370
1.5040 2.3774 2.2166 2.0873 3.2520 2.1781 2.4389
74 75 76 77 78 79 80 81
1,2,4-Trimethylbenzene Isopropylbenzene 1,3,5-Trimethylbenzene p-Isopropyltoluene Naphthalene Biphenyl Styrene m-Terphenyl
C9H12 C9H12 C9H12 C10H14 C10H8 C12H10 C8H8 C18H14
95636 98828 108678 99876 91203 92524 100425 92068
120.194 120.194 120.194 134.221 128.174 154.211 104.152 230.309
1.7880E+05 1.8290E+05 1.4805E+05 1.4560E+05 2.9800E+04 1.2177E+05 1.1334E+05 1.9567E+05
−1.2847E+02 −1.7400E+02 1.9700E+01 2.4870E+02 5.2750E+02 4.2930E+02 2.9020E+02 5.9407E+02
8.3741E−01 9.1200E−01 6.2260E−01 1.8700E−01 0 0 −6.0510E−01 0
0 0 0 0 0 0 1.3567E−03 0
0 0 0 0 0 0 0 0
229.33 177.14 228.42 205.25 353.43 342.2 242.54 360
1.9338 1.8069 1.8503 2.0452 2.1623 2.6868 1.6749 4.0954
350 500 350 450.28 491.14 533.37 418.31 650
2.3642 3.2390 2.3121 2.9550 2.8888 3.5075 2.2816 5.8182
82 83 84 85 86 87 88
Methanol Ethanol 1-Propanol 1-Butanol 2-Butanol 2-Propanol 2-Methyl-2-propanol
CH4O C2H6O C3H8O C4H10O C4H10O C3H8O C4H10O
67561 64175 71238 71363 78922 67630 75650
32.042 46.069 60.096 74.123 74.123 60.096 74.123
1.0580E+05 1.0264E+05 1.5876E+05 1.9120E+05 2.0670E+05 7.2355E+05 −9.2546E+05
−3.6223E+02 −1.3963E+02 −6,3500E+02 −7.3040E+02 −1.0204E+03 −8.0950E+03 7.8949E+03
9.3790E−01 −3.0341E−02 1.9690E+00 2.2998E+00 3.2900E+00 3.6662E+01 −1.7661E+01
0 2.0386E−03 0 0 0 −6.6395E−02 1.3617E−02
0 0 0 0 0 4.4064E−05 0
175.47 159.05 146.95 184.51 158.45 185.28 298.96
0.7112 0.8787 1.0797 1.3473 1.2762 1.1189 2.2016
400 390 400 390.81 372.7 480 460
1.1097 1.6450 2.1980 2.5701 2.8340 2.8122 2.9455
89 90 91 92 93 94 95 96
1-Pentanol 2-Methyl-1-butanol 3-Methyl-1-butanol 1-Hexanol 1-Heptanol Cyclohexanol Ethylene glycol 1,2-Propylene glycol
C5H12O C5H12O C5H12O C6H14O C7H16O C6H12O C2H6O2 C3H8O2
71410 137326 123513 111273 111706 108930 107211 57556
88.150 88.150 88.150 102.177 116.203 100.161 62.068 76.095
2.0120E+05 8.2937E+04 −5.3777E+04 4.8466E+05 4.3790E+05 −4.0000E+04 3.5540E+04 5.8080E+04
−6.5130E+02 4.5998E+02 8.8342E+02 −2.7613E+03 −2.0947E+03 8.5300E+02 4.3678E+02 4.4520E+02
2.2750E+00 0.0000E+00 0 6.5555E+00 5.2090E+00 0 −1.8486E−01 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
200.14 250 295.52 228.55 239.15 296.6 260.15 213.15
1.6198 1.9793 2.0729 1.9599 2.3487 2.1300 1.3666 1.5297
389.15 401.85 350 320 370 434 493.15 460.75
2.9227 2.6778 2.5542 2.7233 3.7597 3.3020 2.0598 2.6321
Phenol o-Cresol m-Cresol p-Cresol
C6H6O C7H8O C7H8O C7H8O
108952 95487 108394 106445
94.113 108.140 108.140 108.140
1.0172E+05 −1.8515E+05 −2.4670E+05 2.5998E+05
3.1761E+02 3.1480E+03 3.2568E+03 −1.1123E+03
0 −8.0367E+00 −7.4202E+00 4.9427E+00
0 7.2540E−03 6.0467E−03 −5.4367E−03
0 0 0 0
314.06 304.2 285.39 307.93
2.0147 2.3297 2.1895 2.2740
425 400 400 400
2.3670 2.5243 2.5578 2.5794
97 98 99 100
2-171
TABLE 2-196
Heat Capacities of Inorganic and Organic Liquids (Continued )
Cmpd. no.
Name
Formula
CAS no.
Mol wt
C1
C2
C3
101 102 103 104 105 106 107 108 109 110 111 112
Dimethyl ether Methyl ethyl ether Methyl-n-propyl ether Methyl isopropyl ether Methyl-n-butyl ether Methyl isobutyl ether Methyl tert-butyl ether Diethyl ether Ethyl propyl ether Ethyl isopropyl ether Methyl phenyl ether Diphenyl ether
C2H6O C3H8O C4H10O C4H10O C5H12O C5H12O C5H12O C4H10O C5H12O C5H12O C7H8O C12H10O
115106 540670 557175 598538 628284 625445 1634044 60297 628320 625547 100663 101848
46.069 60.096 74.123 74.123 88.150 88.150 88.150 74.123 88.150 88.150 108.140 170.211
1.1010E+05 1.2977E+05 1.4411E+05 1.4344E+05 1.7785E+05 5.1380E+04 1.4012E+05 4.4400E+04 1.0368E+05 1.0625E+05 1.5094E+05 1.3416E+05
−1.5747E+02 −3.3196E+02 −1.0209E+02 −1.5407E+02 −1.7157E+02 4.5040E+02 −9.0000E+00 1.3010E+03 7.2630E+02 2.9215E+02 9.3455E+01 4.4767E+02
5.1853E−01 1.3869E+00 5.8113E−01 7.2550E−01 7.4379E−01 0 5.6300E−01 −5.5000E+00 −2.6047E+00 0 2.3602E−01 0
113 114 115 116 117 118 119 120 121 122
Formaldehyde4 Acetaldehyde 1-Propanal 1-Butanal 1-Pentanal 1-Hexanal 1-Heptanal 1-Octanal 1-Nonanal 1-Decanal
CH2O C2H4O C3H6O C4H8O C5H10O C6H12O C7H14O C8H16O C9H18O C10H20O
50000 75070 123386 123728 110623 66251 111717 124130 124196 112312
30.026 44.053 58.080 72.107 86.134 100.161 114.188 128.214 142.241 156.268
6.1900E+04 1.1510E+05 9.9306E+04 6.5682E+04 1.1205E+05 1.1770E+05 2.2236E+05 1.3065E+05 1.3682E+05 1.5046E+05
2.8300E+01 −4.3300E+02 1.1573E+02 1.3291E+03 2.5778E+02 3.2952E+02 −1.0517E+02 4.6361E+02 5.3129E+02 5.8663E+02
123 124 125 126 127 128 129 130 131 132 133 134
Acetone Methyl ethyl ketone 2-Pentanone Methyl isopropyl ketone 2-Hexanone Methyl isobutyl ketone 3-Methyl-2-pentanone 3-Pentanone Ethyl isopropyl ketone Diisopropyl ketone Cyclohexanone Methyl phenyl ketone
C3H6O C4H8O C5H10O C5H10O C6H12O C6H12O C6H12O C5H10O C6H12O C7H14O C6H10O C8H8O
67641 78933 107879 563804 591786 108101 565617 96220 565695 565800 108941 98862
58.080 72.107 86.134 86.134 100.161 100.161 100.161 86.134 100.161 114.188 98.145 120.151
1.3560E+05 1.3230E+05 1.9459E+05 1.8361E+05 2.7249E+05 1.2492E+05 9.9815E+04 1.9302E+05 8.3630E+04 1.7927E+05 1.0980E+05 7.2692E+04
135 136 137 138 139 140 141
Formic acid Acetic acid Propionic acid n-Butyric acid Isobutyric acid Benzoic acid Acetic anhydride
CH2O2 C2H4O2 C3H6O2 C4H8O2 C4H8O2 C7H6O2 C4H6O3
64186 64197 79094 107926 79312 65850 108247
46.026 60.053 74.079 88.106 88.106 122.123 102.090
142 143 144 145 146 147 148
Methyl formate Methyl acetate Methyl propionate Methyl-n-butyrate Ethyl formate Ethyl acetate Ethyl propionate
C2H4O2 C3H6O2 C4H8O2 C5H10O2 C3H6O2 C4H8O2 C5H10O2
107313 79209 554121 623427 109944 141786 105373
60.053 74.079 88.106 102.133 74.079 88.106 102.133
2-172
C5
Tmin, K
Cp at Tmin × 1E−05
Tmax, K
Cp at Tmax × 1E−05
0 0 0 0 0 0 0 8.7630E−03 4.0957E−03 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
131.65 218.9 133.97 127.93 157.48 300 164.55 156.92 145.65 298.15 298.15 300.03
0.9836 1.2356 1.4086 1.3560 1.6928 1.8650 1.5388 1.4698 1.6686 1.9335 1.9978 2.6847
250 328.35 312.2 310 343.35 370 328.35 460 320 326.15 484.2 570
1.0314 1.7030 1.6888 1.6540 2.0663 2.1803 1.9786 3.3202 2.0358 2.0153 2.5153 3.8933
0 1.4250E+00 0 −7.1579E+00 0 0 6.5074E−01 0 0 0
0 0 0 1.2755E−02 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
204 150.15 200 176.75 200 217.15 229.8 246 255.15 267.15
0.6767 0.8221 1.2245 1.4741 1.6361 1.8926 2.3256 2.4470 2.7238 3.0718
234 294 328.75 300 376.15 401.45 381.25 447.15 468.15 488.15
0.6852 1.1097 1.3735 1.6459 2.0901 2.4999 2.7685 3.3795 3.8554 4.3682
−1.7700E+02 2.0087E+02 −2.6386E+02 −2.6885E+02 −7.9070E+02 3.0410E+02 3.4672E+02 −1.7643E+02 3.9900E+02 2.8370E+01 2.6150E+02 3.3783E+02
2.8370E−01 −9.5970E−01 7.6808E−01 8.6080E−01 2.5834E+00 0 0 5.6690E−01 0 5.3750E−01 0 3.5572E−01
6.8900E−04 1.9533E−03 0 0 −2.0040E−03 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
178.45 186.48 196.29 181.15 220.87 298.15 298.15 234.18 298.15 204.81 290 298.2
1.1696 1.4905 1.7239 1.6316 2.0228 2.1559 2.0319 1.8279 2.0259 2.0763 1.8563 2.0506
329.44 373.15 375.46 367.55 382.62 390 390.55 375.14 425 410 486.5 532.12
1.3271 1.7511 2.0380 2.0108 2.3590 2.4352 2.3523 2.0661 2.5320 2.8126 2.3702 3.5318
7.8060E+04 1.3964E+05 2.1366E+05 2.3770E+05 1.2754E+05 −5.4800E+03 3.6600E+04
7.1540E+01 −3.2080E+02 −7.0270E+02 −7.4640E+02 −6.5350E+01 6.4712E+02 5.1100E+02
0 8.9850E−01 1.6605E+00 1.8290E+00 8.2867E−01 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
281.45 289.81 252.45 267.95 270 395.45 250
0.9820 1.2213 1.4209 1.6902 1.7031 2.5042 1.6435
380 391.05 414.32 436.42 427.65 450 350
1.0525 1.5159 2.0756 2.6031 2.5114 2.8572 2.1545
1.3020E+05 6.1260E+04 7.1140E+04 1.0293E+05 8.0000E+04 2.2623E+05 7.6330E+04
−3.9600E+02 2.7090E+02 3.3550E+02 1.2910E+02 2.2360E+02 −6.2480E+02 4.0010E+02
1.2100E+00 0 0 6.2516E−01 0 1.4720E+00 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
174.15 253.4 300 277.25 254.2 189.6 298.15
1.2991 1.7179 1.8678 1.3684 1.6068 1.9562
304.9 373.4 390 415.87 374.2 350.21 410
1.2195 1.6241 2.0198 2.6474 1.6367 1.8796 2.4037
C4
149 150 151 152 153 154 155
Ethyl-n-butyrate n-Propyl formate n-Propyl acetate n-Butyl acetate Methyl benzoate Ethyl benzoate Vinyl acetate
C6H12O2 C4H8O2 C5H10O2 C6H12O2 C8H8O2 C9H10O2 C4H6O2
105544 110747 109604 123864 93583 93890 108054
116.160 88.106 102.133 116.160 136.150 150.177 86.090
8.2434E+04 7.5700E+04 8.3400E+04 1.1730E+05 1.1950E+05 1.2450E+05 1.3630E+05
4.2245E+02 3.2610E+02 3.8410E+02 3.5220E+02 2.9400E+02 3.7060E+02 −1.0617E+02
2.0992E−01 0 0 0 0 0 7.5175E−01
156 157 158 159 160 161
Methylamine Dimethylamine Trimethylamine Ethylamine Diethylamine Triethylamine
CH5N C2H7N C3H9N C2H7N C4H11N C6H15N
74895 124403 75503 75047 109897 121448
31.057 45.084 59.111 45.084 73.138 101.192
9.2520E+04 −2.1487E+05 1.3605E+05 1.2170E+05 1.0133E+05 1.1148E+05
3.7450E+01 3.7872E+03 −2.8800E+02 3.8993E+01 2.4318E+02 3.6813E+02
0 −1.3781E+01 9.9130E−01 0 0 0
162 163 164 165 166 167 168
n-Propylamine di-n-Propylamine Isopropylamine Diisopropylamine Aniline N-Methylaniline N,N-Dimethylaniline
C3H9N C6H15N C3H9N C6H15N C6H7N C7H9N C8H11N
107108 142847 75310 108189 62533 100618 121697
59.111 101.192 59.111 101.192 93.128 107.155 121.182
1.3953E+05 4.9120E+04 −3.2469E+04 9.8434E+04 1.4150E+05 1.2850E+05 4.1860E+04
7.8000E+01 5.6224E+02 1.9771E+03 4.2904E+02 1.7120E+02 1.0020E+02 5.2750E+02
169 170 171 172
Ethylene oxide Furan Thiophene Pyridine
C2H4O C4H4O C4H4S C5H5N
75218 110009 110021 110861
44.053 68.075 84.142 79.101
1.4471E+05 1.1437E+05 8.1350E+04 1.0785E+05
173 174 175 176 177 178 179 180
Formamide5 N,N-Dimethylformamide Acetamide N-Methylacetamide Acetonitrile Propionitrile n-Butyronitrile Benzonitrile
CH3NO C3H7NO C2H5NO C3H7NO C2H3N C3H5N C4H7N C7H5N
75127 68122 60355 79163 75058 107120 109740 100470
45.041 73.095 59.068 73.095 41.053 55.079 69.106 103.123
181 182 183 184 185 186 187 188 189
Methyl mercaptan Ethyl mercaptan n-Propyl mercaptan n-Butyl mercaptan Isobutyl mercaptan sec-Butyl mercaptan2 Dimethyl sulfide Methyl ethyl sulfide Diethyl sulfide
CH4S C2H6S C3H8S C4H10S C4H10S C4H10S C2H6S C3H8S C4H10S
74931 75081 107039 109795 513440 513531 75183 624895 352932
190 191 192 193 194 195 196 197
Fluoromethane2 Chloromethane Trichloromethane Tetrachloromethane Bromomethane Fluoroethane Chloroethane Bromoethane
CH3F CH3Cl CHCl3 CCl4 CH3Br C2H5F C2H5Cl C2H5Br
593533 74873 67663 56235 74839 353366 75003 74964
0 0 0 0 0 0 0
0 0 0 0 0 0 0
285.5 298.15 274.7 289.58 260.75 238.45 259.56
2.2015 1.7293 1.8891 2.1929 1.9616 2.1287 1.5939
428.25 398.15 404.7 429.58 472.65 486.55 389.35
3.0185 2.0554 2.3885 2.6860 2.5846 3.0482 2.0892
0 1.6924E−02 0 0 0 0
0 0 0 0 0 0
179.69 180.96 156.08 192.15 223.35 200
0.9925 1.1947 1.1525 1.2919 1.5564 1.8511
266.82 298.15 276.02 289.73 328.6 361.92
1.0251 1.3779 1.3208 1.3300 1.8124 2.4471
0 0 −7.0145E+00 0 0 3.7400E−01 0
0 0 8.6913E−03 0 0 0 0
0 0 0 0 0 0 0
188.36 277.9 177.95 275 267.13 216.15 343.58
1.5422 2.0537 1.4621 2.1642 1.8723 1.6763 2.2310
340 407.9 320 357.05 457.15 469.02 513.58
1.6605 2.7846 1.6671 2.5162 2.1976 2.5777 3.1277
−7.5887E+02 −2.1569E+02 1.2980E+02 −3.4787E+01
2.8261E+00 7.2691E−01 −3.9000E−03 3.9565E−01
−3.0640E−03 0 0 0
0 0 0 0
160.65 187.55 234.94 231.51
0.8303 0.9949 1.1163 1.2100
283.85 304.5 357.31 388.41
0.8693 1.1609 1.2723 1.5403
6.3400E+04 1.4790E+05 1.0230E+05 6.2600E+04 9.7582E+04 1.1819E+05 1.0400E+05 7.6900E+04
1.5060E+02 −1.0600E+02 1.2870E+02 2.4340E+02 −1.2220E+02 −1.2098E+02 1.7400E+02 3.1420E+02
0 3.8400E−01 0 0 3.4085E−01 4.2075E−01 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
292 273.82 354.15 359 229.32 180.26 161.25 260.4
1.0738 1.4767 1.4788 1.4998 0.8748 1.1005 1.3206 1.5872
493 466.44 571 538.5 354.75 370.5 390.75 464.15
1.3765 1.8200 1.7579 1.9367 0.9713 1.3112 1.7199 2.2274
48.109 62.136 76.163 90.189 90.189 90.189 62.136 76.163 90.189
1.1530E+05 1.3467E+05 1.6733E+05 2.3219E+05 1.7336E+05 1.9789E+05 1.4695E+05 1.6124E+05 2.3852E+05
−2.6323E+02 −2.3439E+02 −3.1910E+02 −8.0435E+02 −2.1732E+02 −4.9154E+02 −3.8006E+02 −2.8861E+02 −1.0384E+03
6.0412E−01 5.9656E−01 8.1270E−01 2.7063E+00 7.0933E−01 1.7219E+00 1.2035E+00 7.8179E−01 4.0587E+00
0 0 0 −2.3017E−03 0 −1.2499E−03 −8.4787E−04 0 −4.4691E−03
0 0 0 0 0 0 0 0 0
150.18 125.26 159.95 157.46 128.31 133.02 174.88 167.23 181.95
0.8939 1.1467 1.3708 1.6365 1.5715 1.6003 1.1276 1.3484 1.5703
298.15 315.25 340.87 390 361.64 370 310.48 339.8 322.08
0.9052 1.2007 1.5299 1.9359 1.8754 1.8844 1.1959 1.5344 1.7579
34.033 50.488 119.377 153.822 94.939 48.060 64.514 108.966
7.4746E+04 9.6910E+04 1.2485E+05 −7.5270E+05 1.2973E+05 8.3303E+04 1.2790E+05 9.4364E+04
−1.3232E+02 −2.0790E+02 −1.6634E+02 8.9661E+03 −5.9654E+02 6.5454E+01 −3.4515E+02 −1.0912E+02
5.3772E−01 3.7456E−01 4.3209E−01 −3.0394E+01 2.1600E+00 0 9.1500E−01 4.4032E−01
0 4.8800E−04 0 3.4455E−02 −2.4234E−03 0 0 0
0 0 0 0 0 0 0 0
140 175.43 233.15 250.33 184.45 200 134.8 160
0.6676 0.7460 1.0956 1.2763 0.7798 0.9639 0.9800 0.8818
220 373.15 366.48 388.71 276.71 281.48 340 320
0.7166 0.9684 1.2192 1.6374 0.7870 1.0173 1.1632 1.0453
2-173
TABLE 2-196 Cmpd. no.
Heat Capacities of Inorganic and Organic Liquids (Concluded ) Name
Formula
CAS no.
Mol wt
C1
C2
C3
540545 75296 78999 78875 75014 462066 108907 108861
78.541 78.541 112.986 112.986 62.499 96.104 112.558 157.010
9.6344E+04 6.9362E+04 7.0010E+04 1.1094E+05 −1.0320E+04 −9.9120E+05 −1.3075E+06 1.2160E+05
1.1752E+02 2.1501E+02 2.6660E+02 8.3496E+00 3.2280E+02 1.1734E+04 1.5338E+04 −9.4500E+00
0 0 0 4.7218E−01 0 −4.0669E+01 −5.3974E+01 3.5800E−01
C5
Tmin, K
Cp at Tmin × 1E−05
Tmax, K
Cp at Tmax × 1E−05
0 0 0 0 0 4.7333E−02 6.3483E−02 0
0 0 0 0 0 0 0 0
230 200 280 286 200 239.99 227.95 293.15
1.2337 1.1236 1.4466 1.5195 0.5424 1.3675 1.3617 1.4960
319.67 308.85 420 429 400 319.99 360 495.08
1.3391 1.3577 1.8198 2.0142 1.1880 1.5018 1.8101 2.0467
C4
198 199 200 201 202 203 204 205
1-Chloropropane 2-Chloropropane 1,1-Dichloropropane 1,2-Dichloropropane Vinyl chloride Fluorobenzene Chlorobenzene Bromobenzene
C3H7Cl C3H7Cl C3H6Cl2 C3H6Cl2 C2H3Cl C6H5F C6H5Cl C6H5Br
206 207 208 209 210 211 212 213 214 215 216 217 218
Air Hydrogen (eqn. 2) Helium-46 Neon Argon Fluorine Chlorine Bromine Oxygen Nitrogen Ammonia (eqn. 2) Hydrazine Nitrous oxide
H2 He Ne Ar F2 Cl2 Br2 O2 N2 NH3 N2H4 N 2O
132259100 1333740 7440597 7440019 7440371 7782414 7782505 7726956 7782447 7727379 7664417 302012 10024972
28.951 2.016 4.003 20.180 39.948 37.997 70.905 159.808 31.999 28.014 17.031 32.045 44.013
−2.1446E+05 6.6653E+01 3.8722E+05 1.0341E+06 1.3439E+05 −9.4585E+04 6.3936E+04 3.7570E+04 1.7543E+05 2.8197E+05 6.1289E+01 7.9815E+04 6.7556E+04
9.1851E+03 6.7659E+03 −4.6557E+05 −1.3877E+05 −1.9894E+03 7.5299E+03 4.6350E+01 3.2850E+02 −6.1523E+03 −1.2281E+04 8.0925E+04 5.0929E+01 5.4373E+01
−1.0612E+02 −1.2363E+02 2.1180E+05 7.1540E+03 1.1043E+01 −1.3960E+02 −1.6230E−01 −6.7000E−01 1.1392E+02 2.4800E+02 7.9940E+02 4.3379E−02 0
4.1616E−01 4.7827E+02 −4.2494E+04 −1.6255E+02 0 1.1301E+00 0 0 −9.2382E−01 −2.2182E+00 −2.6510E+03 0 0
0 0 3.2129E+03 1.3841E+00 0 −3.3241E−03 0 0 2.7963E−03 7.4902E−03 0 0 0
75 13.95 2.2 24.56 83.78 58 172.12 265.9 54.36 63.15 203.15 274.69 182.3
0.5307 0.1262 0.1087 0.3666 0.4523 0.5541 0.6711 0.7755 0.5365 0.5593 0.7575 0.9708 0.7747
115 32 4.6 40 135 98 239.12 305.37 142 112 401.15 653.15 200
0.7132 1.3122 0.2965 0.6980 0.6708 0.5966 0.6574 0.7541 0.9066 0.7960 4.1847 1.3158 0.7843
219 220 221 222 223 224 225 226 227 228 229 230 231
Nitric oxide Cyanogen Carbon monoxide (eqn. 2) Carbon dioxide Carbon disulfide Hydrogen fluoride Hydrogen chloride Hydrogen bromide Hydrogen cyanide Hydrogen sulfide (eqn. 2) Sulfur dioxide Sulfur trioxide Water
NO C2N2 CO CO2 CS2 HF HCl HBr HCN H 2S SO2 SO3 H 2O
10102439 460195 630080 124389 75150 7664393 7647010 10035106 74908 7783064 7446095 7446119 7732185
30.006 52.036 28.010 44.010 76.143 20.006 36.461 80.912 27.026 34.082 64.065 80.064 18.015
−2.9796E+06 3.1322E+06 6.5429E+01 −8.3043E+06 8.5600E+04 6.2520E+04 4.7300E+04 5.7720E+04 9.5398E+04 6.4666E+01 8.5743E+04 2.5809E+05 2.7637E+05
7.6602E+04 −2.4320E+04 2.8723E+04 1.0437E+05 −1.2200E+02 −2.2302E+02 9.0000E+01 9.9000E+00 −1.9752E+02 4.9354E+04 5.7443E+00 0.0000E+00 −2.0901E+03
−6.5259E+02 4.8844E+01 −8.4739E+02 −4.3333E+02 5.6050E−01 6.2970E−01 0 0 3.8830E−01 2.2493E+01 0 0 8.1250E+00
1.8879E+00 0 1.9596E+03 6.0052E−01 −1.4520E−03 0 0 0 0 −1.6230E+03 0 0 −1.4116E−02
0 0 0 0 2.0080E−06 0 0 0 0 0 0 0 9.3701E−06
109.5 245.25 68.15 220 161.11 189.79 165 185.15 259.83 187.68 197.67 303.15 273.16
0.6229 1.0557 0.5912 0.7827 0.7577 0.4288 0.6215 0.5955 0.7029 0.6733 0.8688 2.5809 0.7615
150 300 132 290 552 292.67 185 206.45 298.85 370 350 303.15 533.15
1.9909 2.3216 6.4799 1.6603 1.3125 0.5119 0.6395 0.5976 0.7105 4.9183 0.8775 2.5809 0.8939
All substances are listed in alphabetical order in Table 2-6a. Compiled from Daubert, T. E., R. P. Danner, H. M. Sibul, and C. C. Stebbins, DIPPR Data Compilation of Pure Compound Properties, Project 801 Sponsor Release, July, 1993, Design Institute for Physical Property Data, AIChE, New York, NY; and from Thermodynamics Research Center, “Selected Values of Properties of Hydrocarbons and Related Compounds,” Thermodynamics Research Center Hydrocarbon Project, Texas A&M University, College Station, Texas (extant 1994). Temperatures are expressed in kelvins; liquid heat capacities are in J/kmol-K. J/(kmol·K) × 2.390E−04 = cal/(gmol·°C); J/(kmol·K) × 2.390059E−04 = Btu/(lbmol·°F). Equation 1, heat capacity = C1 + C2 × T + C3 × T 2 + C4 × T 3 + C5 × T 4, should be used except as otherwise specified. Equation 2 is heat capacity = C12/t + C2 − (2 × C1 × C3)t − (C1 × C4)t2 − (C32/3)t3 − (C3 × C4/2)t4 − (C42/5)t5. t = (1 − Tr) and Tr is the reduced temperature, T/Tc. 1 Coefficients are for the monomer and are hypothetical above 473 K. 2 For the saturated heat capacity. 3 Coefficients are hypothetical; compound decomposes violently on heating. 4 Coefficients are hypothetical and are based on predicted data. 5 Coefficients are hypothetical. 6 Exhibits superfluid properties below 2.2 K. 2-174
SPECIFIC HEATS OF PURE COMPOUNDS TABLE 2-197
Specific Heats of Organic Solids Recalculated from International Critical Tables, vol. 5, pp. 101–105
Compound
Formula
Acetic acid Acetone Aminobenzoic acid (o-) (m-) (p-) Aniline Anthracene
C2H4O2 C3H6O C7H7NO2 C7H7NO2 C7H7NO2 C6H7N C14H10
Anthraquinone Apiol Azobenzene
C14H8O2 C12H14O4 C12H10N2
Benzene
C6H6
Benzoic acid Benzophenone
C7H6O2 C13H10O
Betol
C17H12O3
Bromoiodobenzene (o-) (m-) (p-) Bromonaphthalene (β-) Bromophenol
C6H4BrI C6H4BrI C6H4BrI C10H7Br C6H5BrO
Camphene Capric acid Caprylic acid Carbon tetrachloride
C10H16 C10H20O2 C8H16O2 CCl4
Cerotic acid Chloral alcoholate hydrate Chloroacetic acid Chlorobenzoic acid (o-) (m-) (p-) Chlorobromobenzene (o-) (m-) (p-) Crotonic acid Cyamelide Cyanamide Cyanuric acid
C27H54O2 C4H7Cl3O2 C2H3Cl3O2 C2H3ClO2 C7H5ClO2 C7H5ClO2 C7H5ClO2 C6H4BrCl C6H4BrCl C6H4BrCl C4H6O2 C3H3N3O3 CH2N2 C3H3N3O3
Dextrin Dextrose
(C6H10O5)x C6H12O6
Dibenzyl Dibromobenzene (o-) (m-) (p-) Dichloroacetic acid Dichlorobenzene (o-) (m-) (p-) Dicyandiamide
C14H14 C6H4Br2 C6H4Br2 C6H4Br2 C2H2Cl2O2 C6H4Cl2 C6H4Cl2 C6H4Cl2 C2H4N4
Temperature, °C −200 to +25 −210 to −80 85 to mp 120 to mp 128 to mp
sp ht, cal/g °C
50 100 150 0 to 270 10 28
0.330 + 0.00080t 0.540 + 0.0156t 0.254 + 0.00136t 0.253 + 0.00122t 0.287 + 0.00088t 0.741 0.308 0.350 0.382 0.258 + 0.00069t 0.299 0.330
−250 −225 −200 −150 −100 −50 0 20 to mp −150 −100 −50 0 +20 −150 −100 0 +50 −50 to 0 −75 to −15 −40 to 50 41 32
0.0399 0.0908 0.124 0.170 0.227 0.299 0.375 0.287 + 0.00050t 0.115 0.172 0.220 0.275 0.303 0.129 0.167 0.248 0.308 0.143 + 0.00025t 0.143 0.116 + 0.00032t 0.260 0.263
35 8 −2 −240 −200 −160 −120 −80 −40 15 78 32 60 80 to mp 94 to mp 180 to mp −34 −52 −40 38 to 70 40 20 40
0.380 0.695 0.628 0.013 0.081 0.131 0.162 0.182 0.201 0.387 0.509 0.213 0.363 0.228 + 0.00084t 0.232 + 0.00073t 0.242 + 0.00055t 0.192 0.150 0.150 0.520 + 0.00020t 0.263 0.547 0.318
0 to 90 −250 −200 −100 0 20 28 −36 −25 −50 to +50
0.291 + 0.00096t 0.016 0.077 0.160 0.277 0.300 0.363 0.248 0.134 0.139 + 0.00038t 0.406 0.185 0.186 0.219 + 0.0021t 0.456
−48.5 −52 −50 to +53 0 to 204
2-175
2-176
PHYSICAL AND CHEMICAL DATA TABLE 2-197
Specific Heats of Organic Solids (Continued ) Recalculated from International Critical Tables, vol. 5, pp. 101–105
Compound
Formula
Dihydroxybenzene (o-) (m-) (p-)
C6H6O2 C6H6O2 C6H6O2
Di-iodobenzene (o-) (m-) (p-) Dimethyl oxalate Dimethylpyrene Dinitrobenzene (o-) (m-) (p-) Diphenyl Diphenylamine Dulcitol
C6H4I2 C6H4I2 C6H4I2 C4H6O4 C7H8O2 C6H4N2O4 C6H4N2O4 C6H4N2O4 C12H10 C12H11N C6H14O6
Erythritol Ethyl alcohol
C4H10O4 C2H6O (crystalline)
(vitreous)
Temperature, °C
sp ht, cal/g °C
−163 to mp −160 to mp −250 −240 −220 −200 −150 to mp −50 to +15 −52 to −42 −50 to +80 10 to 50 50 −160 to mp −160 to mp 119 to mp 40 26 20
0.278 + 0.00098t 0.269 + 0.00118t 0.025 0.038 0.061 0.081 0.268 + 0.00093t 0.109 + 0.00026t 0.100 + 0.00026t 0.101 + 0.00026t 0.212 + 0.0044t 0.368 0.252 + 0.00083t 0.248 + 0.00077t 0.259 + 0.00057t 0.385 0.337 0.282
60 −190 −180 −160 −140 −130 −190 −180 −175 −170 −190 to −40
0.351 0.232 0.248 0.282 0.318 0.376 0.260 0.296 0.380 0.399 0.366 + 0.00110t
Ethylene glycol
C2H6O2
Formic acid
CH2O2
−22 0
0.387 0.430
Glutaric acid Glycerol
C5H8O4 C3H8O3
20 −265 −260 −250 −220 −200 −100 0
0.299 0.009 0.022 0.047 0.085 0.115 0.217 0.330
Hexachloroethane Hexadecane Hydroxyacetanilide
C2Cl6 C16H34 C8H9NO2
Iodobenzene Isopropyl alcohol
C6H5I C3H8O
Lactose Lauric acid Levoglucosane Levulose
C12H22O11 C12H22O11·H2O C12H24O2 C6H10O5 C6H12O6
Malonic acid Maltose Mannitol Melamine Myristic acid Naphthalene Naphthol (α-) (β-) Naphthylamine (α-) Nitroaniline (o-) (m-) (p-) Nitrobenzoic acid (o-) (m-) (p-) Nitronaphthalene
25 41 to mp
0.174 0.495 0.249 + 0.00154t
40 −200 to −160
0.191 0.051 + 0.00165t
20 20 −30 to +40 40 20
0.287 0.299 0.430 + 0.000027t 0.607 0.275
C3H4O4 C12H22O11 C6H14O6 C3H6N6 C14H28O2
20 20 0 to 100 40 0 to 35
0.275 0.320 0.313 + 0.00025t 0.351 0.381 + 0.00545t
C10H8 C10H8O C10H8O C10H9N C6H6N2O2 C6H6N2O2 C6H6N2O2 C7H5NO4 C7H5NO4 C7H5NO4 C10H7NO2
−130 to mp 50 to mp 61 to mp 0 to 50 −160 to mp −160 to mp −160 to mp −163 to mp 66 to mp −160 to mp 0 to 55
0.281 + 0.00111t 0.240 + 0.00147t 0.252 + 0.00128t 0.270 + 0.0031t 0.269 + 0.000920t 0.275 + 0.000946t 0.276 + 0.001000t 0.256 + 0.00085t 0.258 + 0.00091t 0.247 + 0.00077t 0.236 + 0.00215t
SPECIFIC HEATS OF PURE COMPOUNDS TABLE 2-197
Specific Heats of Organic Solids (Concluded ) Recalculated from International Critical Tables, vol. 5, pp. 101–105
Compound
Formula
Temperature, °C
sp ht, cal/g °C
Oxalic acid
C2H2O4 C2H2O4.2H2O
−200 to +50 −200 −100 0 +50 100
0.259 + 0.00076t 0.117 0.239 0.338 0.385 0.416
Palmitic acid
C16H32O2
Phenol Phthalic acid Picric acid
C6H6O C8H6O4 C6H3N3O7
Propionic acid Propyl alcohol (n-)
C3H6O2 C3H8O
Pyrotartaric acid
C6H8O4
−180 −140 −100 −50 0 +20 14 to 26 20 −100 0 +50 100 120 −33 −200 −175 −150 −130 20
0.167 0.208 0.251 0.306 0.382 0.430 0.561 0.232 0.165 0.240 0.263 0.297 0.332 0.726 0.170 0.363 0.471 0.497 0.301
Quinhydrone
C12H10O4
Quinone
C6H4O2
−250 −225 −200 −100 0 −250 −225 −200 −150 to mp
0.017 0.061 0.098 0.191 0.256 0.031 0.082 0.113 0.282 + 0.00083t
Salol Stearic acid Succinic acid Sucrose Sugar (cane)
C13H10O3 C18H36O2 C4H6O4 C12H22O11 C12H22O11
32 15 0 to 160 20 22 to 51
0.289 0.399 0.248 + 0.00153t 0.299 0.301
Tartaric acid Tartaric acid
C4H6O6 C4H6O6·H2O
Tetrachloroethylene Tetryl
C2Cl4 C7H5N5O8
1 Tetryl + 1 picric acid 1 Tetryl + 2 TNT
C13H8N8O15 C21H15N11O20
Thymol Toluic acid (o-) (m-) (p-) Toluidine (p-)
C10H14O C8H8O2 C8H8O2 C8H8O2 C7H9N
Trichloroacetic acid Trimethyl carbinol Trinitrotoluene
C2HCl3O2 C4H10O C7H5N3O6
36 −150 −100 −50 0 +50 −40 to 0 −100 −50 0 +100 −100 to +100 −100 0 +50 0 to 49 54 to mp 54 to mp 130 to mp 0 20 40 solid −4 −100 −50 0 +100 −185 to +23 20 to 50 0 to 91
0.287 0.112 0.170 0.231 0.308 0.366 0.198 + 0.00018t 0.182 0.199 0.212 0.236 0.253 + 0.00072t 0.172 0.280 0.325 0.315 + 0.0031t 0.277 + 0.00120t 0.239 + 0.00195t 0.271 + 0.00106t 0.337 0.387 0.440 0.459 0.559 0.170 0.253 0.311 0.385 0.241 0.423 0.189 + 0.0027t
Trinitroxylene
C8H7N3O6
Triphenylmethane
C19H16
Urea
CH4N2O
20
0.320
2-177
TABLE 2-198 Cmpd. no.
Heat Capacities of Inorganic and Organic Compounds in the Ideal Gas State Name
Formula
CAS no.
Mol wt.
C1 × 1E−05
C2 × 1E−05
C3 × 1E−03
C4 × 1E−05
C5
Tmin, K
Cp at Tmin × 1E−05
Tmax, K
Cp at Tmax × 1E−05
1 2 3 4 5 6 7 8 9 10
Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane
CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22
74828 74840 74986 106978 109660 110543 142825 111659 111842 124185
16.043 30.070 44.097 58.123 72.150 86.177 100.204 114.231 128.258 142.285
0.3330 0.4033 0.5192 0.7134 0.8805 1.0440 1.2015 1.3554 1.5175 1.6720
0.7993 1.3422 1.9245 2.4300 3.0110 3.5230 4.0010 4.4310 4.9150 5.3530
2.0869 1.6555 1.6265 1.6300 1.6502 1.6946 1.6766 1.6356 1.6448 1.6141
0.4160 0.7322 1.1680 1.5033 1.8920 2.3690 2.7400 3.0540 3.4700 3.7820
991.96 752.87 723.6 730.42 747.6 761.6 756.4 746.4 749.6 742
50 200 200 200 200 200 200 200 200 200
0.3330 0.4256 0.5632 0.7673 0.9404 1.1117 1.2828 1.4529 1.6257 1.7967
1500 1500 1500 1500 1500 1500 1500 1500 1500 1500
0.8890 1.4562 2.0556 2.6602 3.2927 3.8620 4.4283 4.9764 5.5407 6.0932
11 12 13 14 15 16 17 18 19 20
n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane n-Eicosane
C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40 C20H42
1120214 112403 629505 629594 629629 544763 629787 593453 629925 112958
156.312 170.338 184.365 198.392 212.419 226.446 240.473 254.500 268.527 282.553
1.9529 2.1295 2.1496 2.3082 2.4679 2.6283 2.7878 2.9502 3.1062 3.2481
6.0998 6.6330 7.3045 7.8678 8.4212 8.9733 9.5247 10.0340 10.5750 11.0900
1.7087 1.7155 1.6695 1.6823 1.6865 1.6912 1.6935 0.7711 0.7679 1.6360
4.1302 4.5161 4.9998 5.4486 5.8537 6.2640 6.6651 −4.3012 −4.5661 7.4500
775.4 777.5 741.02 743.1 743.6 744.41 744.57 916.73 −912.03 −726.27
200 200 200 200 200 200 200 200 200 200
2.0594 2.2442 2.3156 2.4864 2.6586 2.8312 3.0034 3.1800 3.3533 3.5235
1500 1500 1500 1500 1500 1500 1500 1500 1500 1500
6.8342 7.4325 8.0251 8.6225 9.2209 9.8182 10.4160 11.0160 11.6130 12.2110
21 22 23 24 25 26 27
2-Methylpropane 2-Methylbutane 2,3-Dimethylbutane 2-Methylpentane 2,3-Dimethylpentane 2,3,3-Trimethylpentane 2,2,4-Trimethylpentane
C4H10 C5H12 C6H14 C6H14 C7H16 C8H18 C8H18
75285 78784 79298 107835 565593 560214 540841
58.123 72.150 86.177 86.177 100.204 114.231 114.231
0.6549 0.7460 0.7772 0.9030 0.8544 0.9820 1.1390
2.4776 3.2650 4.0320 3.8010 4.5772 5.4020 5.2860
1.5870 1.5450 1.5440 1.6020 1.5181 1.5310 1.5940
1.5750 1.9230 2.5080 2.4530 2.9740 3.4930 3.3510
−706.99 666.7 −649.95 −691.6 641.01 639.9 677.94
200 200 200 200 200 200 200
0.7218 0.8546 0.9363 1.0192 1.0550 1.2194 1.3139
1500 1500 1500 1500 1500 1500 1500
2.6656 3.3792 4.0353 3.9617 4.5983 5.3754 5.3769
28 29 30 31 32 33 34 35
Ethylene Propylene 1-Butene cis-2-Butene trans-2-Butene 1-Pentene 1-Hexene 1-Heptene
C2H4 C3H6 C4H8 C4H8 C4H8 C5H10 C6H12 C7H14
74851 115071 106989 590181 624646 109671 592416 592767
28.054 42.081 56.108 56.108 56.108 70.134 84.161 98.188
0.3338 0.4339 0.5998 0.5765 0.6592 0.7595 0.9180 1.0775
0.9479 1.5200 2.0846 2.1150 2.0700 2.5525 3.0220 3.4900
1.5960 1.4250 1.5884 1.6299 1.6733 1.5820 1.5742 1.5705
0.5510 0.7860 1.2940 1.2872 1.2510 1.6660 2.0320 2.4030
740.8 623.9 707.3 739.1 742.2 713 715 717.4
60 130 200 200 200 200 200 200
0.3338 0.4388 0.6547 0.6199 0.7004 0.8273 0.9995 1.1723
1500 1500 1500 1500 1500 1500 1500 1500
1.0987 1.6836 2.2853 2.2715 2.2904 2.8467 3.4088 3.9706
36 37 38 39 40 41 42 43 44
1-Octene 1-Nonene 1-Decene 2-Methylpropene 2-Methyl-1-butene 2-Methyl-2-butene 1,2-Butadiene 1,3-Butadiene 2-Methyl-1,3-butadiene
C8H16 C9H18 C10H20 C4H8 C5H10 C5H10 C4H6 C4H6 C5H8
111660 124118 872059 115117 563462 513359 590192 106990 78795
112.215 126.242 140.269 56.108 70.134 70.134 54.092 54.092 68.119
1.2355 1.3950 1.7573 0.6125 0.8703 0.8192 0.5750 0.5095 0.6527
3.9570 4.4255 5.1710 2.0660 2.5556 2.6038 1.6476 1.7050 2.2993
1.5640 1.5624 1.7664 1.5450 1.7757 1.7593 1.5270 1.5324 1.4943
2.7669 3.1370 3.6210 1.2057 1.7636 1.7195 0.9900 1.3370 1.5164
718.17 719.6 803.02 676 807.82 800.93 677.3 685.6 −647.15
200 200 200 200 200 200 200 200 200
1.3440 1.5168 1.8333 0.6763 0.9060 0.8559 0.6269 0.5756 0.7508
1500 1500 1500 1500 1500 1500 1500 1500 1500
4.5322 5.0938 5.8682 2.2814 2.8923 2.8709 1.9202 1.9555 2.5571
45 46 47 48 49 50
Acetylene Methylacetylene Dimethylacetylene 3-Methyl-1-butyne 1-Pentyne 2-Pentyne
C2H2 C3H4 C4H6 C5H8 C5H8 C5H8
74862 74997 503173 598232 627190 627214
26.038 40.065 54.092 68.119 68.119 68.119
0.3199 0.4478 0.6534 0.8274 0.7530 0.7074
0.5424 1.0917 1.6179 2.1377 2.0905 2.2229
1.5940 1.5508 1.7837 1.7550 1.5307 1.5570
0.4325 0.6750 1.0242 1.5149 1.3780 1.3125
607.1 658.2 821.4 782 672.8 690.78
200 200 200 200 200 200
0.3566 0.4882 0.6721 0.8646 0.8276 0.7700
1500 1500 1500 1500 1500 1500
0.7575 1.3293 1.9148 2.5255 2.4754 2.5052
51 52 53 54
1-Hexyne 2-Hexyne 3-Hexyne 1-Heptyne
C6H10 C6H10 C6H10 C7H12
693027 764352 928494 628717
82.145 82.145 82.145 96.172
0.9129 1.0360 0.9376 1.0712
2.5577 3.0090 3.0150 3.0258
1.5290 2.1160 1.9057 1.5273
1.7370 2.1060 1.9860 2.0975
683 902.4 817 689.62
200 300 300 200
1.0004 1.2215 1.1909 1.1721
1500 1500 1500 1500
3.0371 3.1894 3.1889 3.5985
2-178
55 56
1-Octyne Vinylacetylene
C8H14 C4H4
629050 689974
110.199 52.076
1.2307 0.5598
3.4942 1.2141
1.5280 1.6102
2.4617 0.8908
694.81 −710.4
200 200
1.3448 0.5967
1500 1500
4.1604 1.5590
57 58 59 60 61 62 63 64 65 66
Cyclopentane Methylcyclopentane Ethylcyclopentane Cyclohexane Methylcyclohexane 1,1-Dimethylcyclohexane Ethylcyclohexane Cyclopentene 1-Methylcyclopentene Cyclohexene
C5H10 C6H12 C7H14 C6H12 C7H14 C8H16 C8H16 C5H8 C6H10 C6H10
287923 96377 1640897 110827 108872 590669 1678917 142290 693890 110838
70.134 84.161 98.188 84.161 98.188 112.215 112.215 68.119 82.145 82.145
0.4160 0.6646 0.8205 0.4320 0.9227 1.0776 1.1059 0.4807 0.6941 0.5817
3.0140 3.5070 4.0342 3.7350 4.1150 4.6718 4.6306 2.5159 3.0209 3.1717
1.4617 1.5892 1.5670 1.1920 1.6504 1.6540 1.6628 1.5803 1.6903 1.5435
1.8095 2.3526 2.6697 1.6350 2.9006 3.3397 3.2990 1.7454 2.1209 2.1273
−668.8 727.13 715.52 −530.1 779.48 792.5 781.1 718.37 781.56 701.62
100 200 200 100 200 200 200 150 200 150
0.4165 0.7510 0.9272 0.4366 0.9953 1.1535 1.1875 0.4918 0.7464 0.5978
1500 1500 1500 1500 1500 1500 1500 1500 1500 1500
2.9298 3.5495 4.1472 3.6516 4.3180 4.9543 4.9184 2.5619 3.1496 3.2132
67 68 69 70 71 72 73
Benzene Toluene o-Xylene m-Xylene p-Xylene Ethylbenzene Propylbenzene (eqn. 3)
C6H6 C7H8 C8H10 C8H10 C8H10 C8H10 C9H12
71432 108883 95476 108383 106423 100414 103651
78.114 92.141 106.167 106.167 106.167 106.167 120.194
0.4442 0.5814 0.8521 0.7568 0.7512 0.7844 −21.4827
2.3205 2.8630 3.2954 3.3924 3.3970 3.3990 3.8070
1.4946 1.4406 1.4944 1.4960 1.4928 1.5590 54701
1.7213 1.8980 2.1150 2.2470 2.2470 2.4260 −0.001713
−678.15 −650.43 -675.8 −675.9 −675.1 −702 0
200 200 200 200 200 200 200
0.5340 0.7016 0.9643 0.8759 0.8710 0.8912 1.0802
1500 1500 1500 1500 1500 1500 1500
2.4169 3.0029 3.5965 3.5920 3.5923 3.6147 4.1537
74 75 76 77 78 79 80 81
1,2,4-Trimethylbenzene Isopropylbenzene 1,3,5-Trimethylbenzene p-Isopropyltoluene Naphthalene Biphenyl Styrene m-Terphenyl
C9H12 C9H12 C9H12 C10H14 C10H8 C12H10 C8H8 C18H14
95636 98828 108678 99876 91203 92524 100425 92068
120.194 120.194 120.194 134.221 128.174 154.211 104.152 230.309
1.0106 1.0810 0.9154 1.3186 0.6805 0.9060 0.8930 1.6397
3.8314 3.7932 3.9270 4.3036 3.5494 4.2634 2.1503 6.0125
1.5010 1.7505 1.4980 1.7734 1.4262 1.4553 0.7720 1.6902
2.3950 3.0027 2.5090 3.2570 2.5984 3.1550 0.9990 5.1314
678.3 794.8 676.9 811.9 650.1 661.2 2442 757.5
200 200 200 200 200 200 100 298.15
1.1354 1.1480 1.0474 1.3825 0.8454 1.0913 0.8931 2.4618
1500 1500 1500 1500 1500 1500 1500 1500
4.1854 4.1808 4.1807 4.7952 3.7359 4.5581 3.2416 6.6678
82 83 84 85 86 87 88
Methanol Ethanol 1-Propanol 1-Butanol 2-Butanol 2-Propanol 2-Methyl-2-propanol
CH4O C2H6O C3H8O C4H10O C4H10O C3H8O C4H10O
67561 64175 71238 71363 78922 67630 75650
32.042 46.069 60.096 74.123 74.123 60.096 74.123
0.3925 0.4920 0.6190 0.7454 0.8202 0.5723 0.7704
0.8790 1.4577 2.0213 2.5907 2.5220 1.9100 2.5390
1.9165 1.6628 1.6293 1.6073 1.6010 1.4210 1.5502
0.5365 0.9390 1.2956 1.7320 1.5864 1.2155 1.6690
896.7 744.7 727.4 712.4 −704.15 626 −679.3
200 200 200 200 200 150 200
0.3980 0.5224 0.6665 0.8162 0.8890 0.5924 0.8567
1500 1500 1500 1500 1500 1500 1500
1.0533 1.6576 2.2458 2.8509 2.8513 2.1792 2.8508
89 90 91 92 93 94 95 96
1-Pentanol 2-Methyl-1-butanol 3-Methyl-1-butanol 1-Hexanol 1-Heptanol Cyclohexanol Ethylene glycol 1,2-Propylene glycol
C5H12O C5H12O C5H12O C6H14O C7H16O C6H12O C2H6O2 C3H8O2
71410 137326 123513 111273 111706 108930 107211 57556
88.150 88.150 88.150 102.177 116.203 100.161 62.068 76.095
0.9060 1.0890 1.1060 1.0625 1.2215 0.9043 0.8200 2.0114
3.0620 2.1850 2.2100 3.5210 3.9910 2.5771 1.2780 0.8082
1.6054 0.8530 0.8760 1.5835 1.5800 0.7882 1.6980 1.8656
2.1150 1.4000 1.2200 2.4620 2.8350 1.3068 0.9290 −2.4404
−717.97 2906 2940 715.75 717.7 1952.2 −754 279.98
200 298.15 298.15 200 200 200 200 298.15
0.9890 1.3247 1.3213 1.1607 1.3330 0.9648 0.8481 1.0218
1500 1500.1 1200.15 1500 1500 1500 1500 1000.15
3.4133 3.4718 3.1770 3.9726 4.5346 3.8251 1.8521 2.1175
97 98 99 100
Phenol o-Cresol m-Cresol p-Cresol
C6H6O C7H8O C7H8O C7H8O
108952 95487 108394 106445
94.113 108.140 108.140 108.140
0.4340 0.7988 0.7515 0.7384
2.4450 2.8530 2.0900 2.9080
1.1520 1.4765 0.6666 1.4559
1.5120 2.0420 1.2120 2.0910
−507 −664.7 2214 −650.42
100 200 200 200
0.4401 0.9158 0.8701 0.8707
1500 1500 1500 1500
2.6045 3.2163 3.2075 3.2102
101 102 103 104 105 106
Dimethyl ether Methyl ethyl ether Methyl-n-propyl ether Methyl isopropyl ether Methyl-n-butyl ether Methyl isobutyl ether
C2H6O C3H8O C4H10O C4H10O C5H12O C5H12O
115106 540670 557175 598538 628284 625445
46.069 60.096 74.123 74.123 88.150 88.150
0.5148 0.6868 0.9215 0.8923 0.8205 0.7284
1.4420 1.9959 2.3943 2.4765 3.0869 3.1713
1.6034 1.5534 1.6936 1.6960 1.3864 1.3520
0.7747 1.1168 1.4896 1.5598 1.7886 1.8948
725.4 692.04 797.79 791.4 613.87 585.14
200 200 298 200 300 300
0.5436 0.7396 1.1251 0.9280 1.3300 1.3200
1500 1500 1200 1500 1200 1200
1.6581 2.2931 2.6391 2.8696 3.1994 3.1987 2-179
TABLE 2-198 Cmpd. no.
Heat Capacities of Inorganic and Organic Compounds in the Ideal Gas State (Continued ) Mol wt.
C1 × 1E−05
C2 × 1E−05
C3 × 1E−03
C4 × 1E−05
C5
1634044 60297 628320 625547 100663 101848
88.150 74.123 88.150 88.150 108.140 170.211
0.9933 0.8621 1.1320 1.0953 0.7637 1.0985
3.0667 2.5510 2.9400 3.0032 2.9377 4.3412
1.7426 1.5413 1.8270 1.7988 1.6051 1.6222
2.0764 1.4370 2.0550 2.1311 2.1700 3.6455
795.59 −688.9 −852 817.35 751.2 743.62
CH2O C2H4O C3H6O C4H8O C5H10O C6H12O C7H14O C8H16O C9H18O C10H20O
50000 75070 123386 123728 110623 66251 111717 124130 124196 112312
30.026 44.053 58.080 72.107 86.134 100.161 114.188 128.214 142.241 156.268
0.3327 0.4451 0.7174 0.8966 1.0743 1.2320 1.4040 1.6088 1.7347 1.9641
0.4954 1.0687 1.9140 2.3731 2.8363 2.2146 2.5907 4.2180 4.5115 5.1412
1.8666 1.6141 2.0144 1.9754 1.9549 0.8400 0.8315 1.9126 1.7120 1.8989
0.2808 0.6135 1.1708 1.5866 2.0146 1.2190 1.3120 3.2780 3.3256 4.1278
Acetone Methyl ethyl ketone 2-Pentanone Methyl isopropyl ketone 2-Hexanone Methyl isobutyl ketone 3-Methyl-2-pentanone 3-Pentanone Ethyl isopropyl ketone Diisopropyl ketone Cyclohexanone Methyl phenyl ketone
C3H6O C4H8O C5H10O C5H10O C6H12O C6H12O C6H12O C5H10O C6H12O C7H14O C6H10O C8H8O
67641 78933 107879 563804 591786 108101 565617 96220 565695 565800 108941 98862
58.080 72.107 86.134 86.134 100.161 100.161 100.161 86.134 100.161 114.188 98.145 120.151
0.5704 0.7840 0.9005 1.5914 1.0940 1.2270 1.0028 0.9690 1.2400 1.0869 0.5776 0.8540
1.6320 2.1032 2.7085 1.7640 1.8070 2.1950 3.3169 2.4907 3.2000 4.0540 3.3535 2.3340
1.6070 1.5488 1.6592 1.2076 0.6890 0.8420 1.6900 1.4177 1.9670 1.7802 1.2202 0.8310
135 136 137 138 139 140 141
Formic acid1 Acetic acid2 Propionic acid2 n-Butyric acid2 Isobutyric acid2 Benzoic acid Acetic anhydride
CH2O2 C2H4O2 C3H6O2 C4H8O2 C4H8O2 C7H6O2 C4H6O3
64186 64197 79094 107926 79312 65850 108247
46.026 60.053 74.079 88.106 88.106 122.123 102.090
0.3381 0.4020 0.6959 1.4880 0.7469 0.7759 0.7130
0.7593 1.3675 1.7778 1.3522 2.4356 2.6455 2.2220
142 143 144 145 146 147 148
Methyl formate Methyl acetate Methyl propionate Methyl n-butyrate Ethyl formate Ethyl acetate Ethyl propionate
C2H4O2 C3H6O2 C4H8O2 C5H10O2 C3H6O2 C4H8O2 C5H10O2
107313 79209 554121 623427 109944 141786 105373
60.053 74.079 88.106 102.133 74.079 88.106 102.133
0.5060 0.5550 0.7765 0.8940 0.5370 0.9981 0.9370
149 150 151 152 153 154 155
Ethyl n-butyrate n-Propyl formate n-Propyl acetate n-Butyl acetate Methyl benzoate Ethyl benzoate Vinyl acetate
C6H12O2 C4H8O2 C5H10O2 C6H12O2 C8H8O2 C9H10O2 C4H6O2
105544 110747 109604 123864 93583 93890 108054
116.160 88.106 102.133 116.160 136.150 150.177 86.090
156 157 158 159
Methylamine Dimethylamine Trimethylamine Ethylamine
CH5N C2H7N C3H9N C2H7N
74895 124403 75503 75047
31.057 45.084 59.111 45.084
Name
Formula
107 108 109 110 111 112
Methyl tert-butyl ether Diethyl ether Ethyl propyl ether Ethyl isopropyl ether Methyl phenyl ether Diphenyl ether
C5H12O C4H10O C5H12O C5H12O C7H8O C12H10O
113 114 115 116 117 118 119 120 121 122
Formaldehyde Acetaldehyde 1-Propanal 1-Butanal 1-Pentanal 1-Hexanal 1-Heptanal 1-Octanal 1-Nonanal 1-Decanal
123 124 125 126 127 128 129 130 131 132 133 134
2-180
CAS no.
Cp at Tmin × 1E−05
Tmax, K
Cp at Tmax × 1E−05
200 200 298.15 298.15 300 300
1.0394 0.9316 1.3538 1.3620 1.1302 1.7298
1500 1500 1500 1200 1200 1200
3.4321 2.9244 3.4535 3.2289 3.0226 4.5143
934.9 737.8 930.6 904.13 890.44 2205 2201 869 810.96 862.51
50 200 200 200 200 200 200 200 200 200
0.3327 0.4660 0.7266 0.9119 1.0960 1.2672 1.4479 1.6504 1.8005 2.0192
1500 1500 1500 1500 1500 1500 1500 1500 1500 1500
0.7113 1.2994 2.1149 2.6775 3.2404 3.7314 4.2863 4.9286 5.4439 6.0539
0.9680 1.1855 1.8012 −407.4000 1.4740 1.1910 2.3000 1.3010 2.3460 2.9786 1.5700 0.7730
731.5 693 743.96 10.503 1772 2460 770.7 646.7 896 791.6 586.92 2227
200 200 200 300 200 298.15 300 200 298.15 300 200 298.15
0.6049 0.8397 0.9591 1.1291 1.1815 1.4755 1.3604 1.0536 1.4479 1.5102 0.7321 1.1313
1500 1500 1500 1500 1200 1500.15 1200 1500 1200 1500 1500 1500
1.8820 2.4816 3.0797 2.9991 3.3207 3.6532 3.4275 3.0358 3.4234 4.3093 3.4870 3.2797
1.1925 1.2620 1.7098 1.1460 1.7150 1.7925 1.6203
0.3180 0.7003 1.2654 −678.0000 1.8484 2.2382 1.6760
550 569.7 −763.78 6.98 757.75 835.9 746.5
50 50 298.15 298.15 298.15 200 200
0.3381 0.4020 0.8938 1.1533 1.0427 0.8126 0.7665
1500 1500 1500 1200.1 1200 1500 1500
0.9933 1.5756 2.1248 2.4716 2.5383 2.9712 2.5675
1.2190 1.7820 2.4420 2.9100 1.8860 2.0931 2.8290
1.6370 1.2600 1.7140 1.5700 1.2070 2.0226 1.6480
0.8940 0.8530 1.8180 2.0730 0.8640 1.8030 2.1550
743 562 716 678.3 496 928.05 724.7
250 298 300 298 100 200 300
0.5888 0.8489 1.1242 1.3461 0.5412 1.0126 1.3377
1500 1500 1200 1200 1500 1500 1200
1.5109 2.0754 2.5276 3.0766 2.1485 2.6594 3.0569
1.1150 0.8710 1.7994 1.1684 0.9396 1.0944 0.5360
3.3910 2.4470 1.7530 3.7690 2.5590 4.1794 2.1190
1.6705 1.9254 1.1960 1.9560 0.8250 0.8838 1.1980
2.5180 1.8880 −4.1200 2.8180 1.3600 −1.6090 1.1470
298 298.15 298.15 300 300 300 100
1.5583 1.1022 1.3594 1.5358 1.2586 1.4598 0.5404
1200 1500 1500 1200 1200 1500 1500
3.6213 2.7484 3.2024 3.6724 3.3569 4.2540 2.3750
0.4100 0.5565 0.7107 0.5940
1.0578 1.6384 1.5051 1.6180
1.7080 1.7341 0.7966 1.8120
0.6836 1.0899 0.8454 1.0780
150 200 200 200
0.4136 0.5812 0.7439 0.6139
1500 1500 1500 1500
1.2388 1.8585 2.4322 1.8528
733.6 −821.3 108.2 811.2 3000 −1183.1 510 735 793.04 2187.6 820
Tmin, K
160 161
Diethylamine Triethylamine
C4H11N C6H15N
109897 121448
73.138 101.192
0.9102 1.2766
2.6740 2.5559
1.7190 0.8094
1.7926 1.4829
794.94 2231.7
200 200
0.9502 1.3278
1500 1500
3.0519 4.2046
162 163 164 165 166 167 168
n-Propylamine di-n-Propylamine Isopropylamine Diisopropylamine Aniline N-Methylaniline N,N-Dimethylaniline
C3H9N C6H15N C3H9N C6H15N C6H7N C7H9N C8H11N
107108 142847 75310 108189 62533 100618 121697
59.111 101.192 59.111 101.192 93.128 107.155 121.182
0.7608 1.2114 0.6855 1.1384 0.6533 0.7796 0.8742
2.1049 2.6127 2.1876 2.5747 2.5192 3.0280 2.7204
1.7256 0.7896 1.5831 0.7384 1.4608 1.5203 0.7242
1.3936 1.6903 1.3855 1.6200 1.8870 2.3280 1.1300
789.03 2394.4 691.76 2143 −653.1 699.8 1949
200 300 200 300 200 300 300
1500 1500 1500 1500 1500 1500 1500
2.4353 4.2484 2.4540 4.1941 2.8047 3.3641 3.8844
169 170 171 172
Ethylene oxide Furan Thiophene Pyridine
C2H4O C4H4O C4H4S C5H5N
75218 110009 110021 110861
44.053 68.075 84.142 79.101
0.3346 0.3727 0.4040 0.4413
1.2116 1.6606 1.6270 2.0830
1.6084 1.5112 1.4564 1.4783
0.8241 1.3145 1.3212 1.5330
737.3 686 649 676.8
50 200 200 200
0.7933 1.5900 0.7510 1.5995 0.7705 1.2602 1.3903 0.0000 0.3346 0.4376 0.4884 0.5220
1500 1500 1500 1500
1.3297 1.7940 1.8097 2.2194
173 174 175 176 177 178 179 180
Formamide N,N-Dimethylformamide Acetamide N-Methylacetamide Acetonitrile Propionitrile n-Butyronitrile Benzonitrile
CH3NO C3H7NO C2H5NO C3H7NO C2H3N C3H5N C4H7N C7H5N
75127 68122 60355 79163 75058 107120 109740 100470
45.041 73.095 59.068 73.095 41.053 55.079 69.106 103.123
0.3822 0.7220 0.3420 0.6116 0.4191 0.5357 0.6906 0.7186
0.9300 1.7830 1.2940 2.0290 0.8876 1.4617 1.9996 2.2700
1.8450 1.5320 1.0750 1.7683 1.5818 1.5530 1.5494 1.4669
0.6900 1.3100 0.6400 1.3302 0.5032 0.9120 1.3146 1.6930
850 762 502 835.5 699.8 678.2 675 −680.77
150 200 100 300 100 200 200 200
0.3833 0.7594 0.3448 0.7698 0.4192 0.5832 0.7607 0.8053
1500 1500 1500 1500 1500 1500 1500 1500
1.1203 2.2596 1.4997 2.2209 1.1285 1.7235 2.3273 2.6706
181 182 183 184 185 186 187 188 189
Methyl mercaptan Ethyl mercaptan n-Propyl mercaptan n-Butyl mercaptan Isobutyl mercaptan sec-Butyl mercaptan Dimethyl sulfide Methyl ethyl sulfide Diethyl sulfide
CH4S C2H6S C3H8S C4H10S C4H10S C4H10S C2H6S C3H8S C4H10S
74931 75081 107039 109795 513440 513531 75183 624895 352932
48.109 62.136 76.163 90.189 90.189 90.189 62.136 76.163 90.189
0.4146 0.5576 0.7474 0.9248 0.9142 0.9237 0.6037 0.7508 0.9429
0.8307 1.3617 1.9523 2.7795 2.4513 2.5166 1.3747 1.9577 2.6863
1.5890 1.5221 1.6310 1.6837 1.6265 1.6109 1.6410 1.6424 1.7624
0.4612 0.8073 1.2112 1.5974 1.6157 1.5641 0.7988 1.1949 1.6752
716.7 687.5 750.92 758.68 745.8 739.2 −743.5 749.19 −798.3
200 200 200 200 200 200 200 273.16 200
0.4329 0.5970 0.7848 0.9714 0.9660 0.9763 0.6298 0.9004 0.9794
1500 1500 1500 1500 1500 1500 1500 1500 1500
1.0781 1.6729 2.3216 3.1008 2.9095 2.9615 1.6949 2.3178 3.0338
190 191 192 193 194 195 196 197
Fluoromethane Chloromethane Trichloromethane Tetrachloromethane Bromomethane Fluoroethane Chloroethane Bromoethane
CH3F CH3Cl CHCl3 CCl4 CH3Br C2H5F C2H5Cl C2H5Br
593533 74873 67663 56235 74839 353366 75003 74964
34.033 50.488 119.377 153.822 94.939 48.060 64.514 108.966
0.3329 0.3409 0.3942 0.3758 0.3377 0.4437 0.4568 0.4719
0.7399 0.7246 0.6573 0.7054 0.7150 1.3119 1.2967 1.2787
1.8639 1.7230 0.9280 0.5121 1.5780 1.6422 1.5992 1.5957
0.4608 0.4480 0.4930 0.4850 0.4175 0.8544 0.8590 0.8517
891.16 780.5 399.6 236.1 691.4 738.77 708.8 703.87
50 150 100 100 100 200 100 200
0.3329 0.3424 0.4048 0.4730 0.3378 0.4726 0.4569 0.5089
1500 1500 1500 1500 1500 1500 1500 1500
0.9024 0.9097 1.0063 1.0662 0.9107 1.5008 1.5112 1.5121
198 199 200 201 202 203 204 205
1-Chloropropane 2-Chloropropane 1,1-Dichloropropane 1,2-Dichloropropane Vinyl chloride Fluorobenzene Chlorobenzene Bromobenzene
C3H7Cl C3H7Cl C3H6Cl2 C3H6Cl2 C2H3Cl C6H5F C6H5Cl C6H5Br
540545 75296 78999 78875 75014 462066 108907 108861
78.541 78.541 112.986 112.986 62.499 96.104 112.558 157.010
0.6210 0.6181 0.7145 0.7866 0.4236 0.6265 0.8011 0.7210
1.8430 1.8023 1.7344 1.7429 0.8735 2.1646 2.3100 2.0640
1.6290 1.5438 1.5240 1.7157 1.6492 1.5640 2.1570 1.6504
1.2337 1.1893 1.2230 1.2627 0.6556 1.7278 2.0460 1.6870
724 685.93 674.2 765.1 739.07 −724.29 −897.6 765.3
200 200 150 200 200 200 200 200
0.6674 0.6768 0.7268 0.8217 0.4457 0.6914 0.8219 0.7679
1500 1500 1500 1500 1500 1500 1500 1500
2.1126 2.1023 2.1609 2.1894 1.1423 2.4736 2.5327 2.4628
206 207 208 209 210 211
Air Hydrogen3 Helium-4 (eqn 2) Neon Argon Fluorine
H2 He Ne Ar F2
132259100 1333740 7440597 7440019 7440371 7782414
28.951 2.016 4.003 20.180 39.948 37.997
0.2896 0.2762 0.2079 0.2079 0.2079 0.2912
0.0939 0.0956 0 0 0 0.1013
3.0120 2.4660 0 0 0 1.4530
0.0758 0.0376 0 0 0 0.0941
1484 567.6 0 0 0 662.91
50 250 100 100 100 50
0.2896 0.2843 0.2079 0.2079 0.2079 0.2912
1500 1500 1500 1500 1500 1500
0.3496 0.3225 0.2079 0.2079 0.2079 0.3812 2-181
TABLE 2-198 Cmpd. no.
Heat Capacities of Inorganic and Organic Compounds in the Ideal Gas State (Concluded ) Name
Formula
CAS no.
Mol wt.
C1 × 1E−05
C2 × 1E−05
C3 × 1E−03
C4 × 1E−05
C5
Tmin, K
Cp at Tmin × 1E−05
Tmax, K
Cp at Tmax × 1E−05
212 213 214 215 216 217 218
Chlorine Bromine Oxygen Nitrogen Ammonia Hydrazine Nitrous oxide
Cl2 Br2 O2 N2 NH3 N2H4 N2O
7782505 7726956 7782447 7727379 7664417 302012 10024972
70.905 159.808 31.999 28.014 17.031 32.045 44.013
0.2914 0.3011 0.2910 0.2911 0.3343 0.3871 0.2934
0.0918 0.0801 0.1004 0.0861 0.4898 0.8576 0.3236
0.9490 0.7514 2.5265 1.7016 2.0360 1.7228 1.1238
0.1003 0.1078 0.0936 0.0010 0.2256 0.5664 0.2177
425 314.6 1153.8 909.79 882 733.53 479.4
50 100 50 50 100 200 100
0.2914 0.3090 0.2910 0.2911 0.3343 0.4070 0.2948
1500 1500 1500 1500 1500 1500 1500
0.3793 0.3794 0.3653 0.3484 0.6647 1.0571 0.5828
219 220 221 222 223 224 225 226 227 228 229 230 231
Nitric oxide (eqn 2) Cyanogen Carbon monoxide Carbon dioxide Carbon disulfide Hydrogen fluoride Hydrogen chloride Hydrogen bromide Hydrogen cyanide Hydrogen sulfide Sulfur dioxide Sulfur trioxide Water
NO C2N2 CO CO2 CS2 HF HCl HBr HCN H2S SO2 SO3 H2O
10102439 460195 630080 124389 75150 7664393 7647010 10035106 74908 7783064 7446095 7446119 7732185
30.006 52.036 28.010 44.010 76.143 20.006 36.461 80.912 27.026 34.082 64.065 80.064 18.015
0.3498 0.3545 0.2911 0.2937 0.3010 0.2913 0.2916 0.2912 0.3013 0.3329 0.3338 0.3341 0.3336
−3.5320E−04 0.5015 0.0877 0.3454 0.3338 0.0933 0.0905 0.0953 0.3171 0.2609 0.2586 0.4968 0.2679
7.7290E−05 1.0570 3.0851 1.4280 0.8960 2.9050 2.0938 2.1420 1.6102 0.9134 0.9328 0.8732 2.6105
−5.7357E−10 0.4520 0.0846 0.2640 0.2893 0.0020 −0.0011 0.0157 0.2179 −0.1798 0.1088 0.2856 0.0890
1.4526E−08 −396 1538.2 588 374.7 1326 120 1400 626 949.4 423.7 393.74 1169
100 100 60 50 100 50 50 50 100 100 100 100 100
0.3217 0.3648 0.2911 0.2937 0.3100 0.2913 0.2914 0.2912 0.3014 0.3329 0.3354 0.3408 0.3336
1500 1500 1500 5000 1500 1500 1500 1500 1500 1500 1500 1500 2273.15
0.3586 0.8100 0.3521 0.6335 0.6148 0.3224 0.3406 0.3479 0.5522 0.5143 0.5695 0.7967 0.5276
All substances are listed in alphabetical order in Table 2-6a. Compiled from Daubert, T. E., R. P. Danner, H. M. Sibul, and C. C. Stebbins, DIPPR Data Compilation of Pure Compound Properties, Project 801 Sponsor Release, July, 1993, Design Institute for Physical Property Data, AIChE, New York, NY; and from Thermodynamics Research Center, “Selected Values of Properties of Hydrocarbons and Related Compounds,” Thermodynamics Research Center Hydrocarbon Project, Texas A&M University, College Station, Texas (extant 1994). Temperatures are expressed in kelvins; heat capacities, in J/kmol-K. J/(kmol·K) × 2.390E−04 = cal/(gmol·°C); J/(kmol·K) × 2.390059E−04 = Btu/(lbmol·°F). cosh unless otherwise specified. + C4 T T
C3 C3 Use heat capacity = C1 + C2 sinh T T
2
C5
C5
2
Equation 2 is heat capacity = C1 + C2 × T + C3 × T 2 + C4 × T 3 + C5 × T 4. Equation 3 is heat capacity = C1 + C2 × ln T + C3/T + C4 × T. 1 For the monomer. Monomer and dimer are in equilibrium below 600 K. 2 For the monomer. 3 For equilibrium mixture of ortho and para hydrogen. 2-182
SPECIFIC HEATS OF PURE COMPOUNDS
2-183
TABLE 2-199 Cp /Cv : Ratios of Specific Heats of Gases at 1-atm Pressure*
Compound
Formula
Acetaldehyde Acetic acid Acetylene
C2H4O C2H4O2 C2H2
Air
Ammonia Argon
NH3 A
Temperature, °C
Ratio of specific heats, (γ) = Cp /Cv
30 136 15 −71 925 17 −78 −118 15 15 −180 0–100
1.14 1.15 1.26 1.31 1.36 1.403 1.408 1.415 1.310 1.668 1.76 (?) 1.67
Benzene Bromine
C6H6 Br2
90 20–350
1.10 1.32
Carbon dioxide
CO2
disulfide monoxide
CS2 CO
1.304 1.37 1.21 1.404 1.41 1.355 1.15 1.256 1.08
Chlorine Chloroform Cyanogen Cyclohexane
Cl2 CHCl3 (CN)2 C6H12
15 −75 100 15 −180 15 100 15 80
Dichlorodifluormethane
CCl2F2
25
1.139
Ethane
C2H6
Ethyl alcohol ether
C2H6O C4H10O
Ethylene
C2H4
100 15 −82 90 35 80 100 15 −91
1.19 1.22 1.28 1.13 1.08 1.086 1.18 1.255 1.35
Helium Hexane (n-) Hydrogen
He C6H14 H2
bromide chloride
HBr HCl
cyanide
HCN
−180 80 15 −76 −181 20 15 100 65 140 210
1.660 1.08 1.410 1.453 1.597 1.42 1.41 1.40 1.31 1.28 1.24
Compound
Formula
Hydrogen (Cont.) iodide sulfide
HI H2S
Iodine Isobutane
I2 C4H10
Krypton
Kr
Mercury Methane
Hg CH4
Methyl acetate alcohol ether Methylal
C3H6O2 CH4O C2H6O C3H8O2
Neon Nitric oxide
Ne NO
Nitrogen
N2
Nitrous oxide
N2O
Oxygen
O2
Pentane (n-) Phosphorus Potassium
C5H12 P K
Sodium Sulfur dioxide
Na SO2
Xenon
Xe
Temperature, °C
Ratio of specific heats, (γ) = Cp /Cv
20–100 15 −45 −57
1.40 1.32 1.30 1.29
185 15
1.30 1.11
19
1.68
360 600 300 15 −80 −115 15 77 6–30 13 40
1.67 1.113 1.16 1.31 1.34 1.41 1.14 1.203 1.11 1.06 1.09
19 15 −45 −80 15 −181 100 15 −30 −70
1.64 1.400 1.39 1.38 1.404 1.47 1.28 1.303 1.31 1.34
15 −76 −181
1.401 1.415 1.45
86 300 850
1.086 1.17 1.77
750–920 15
1.68 1.29
19
1.66
*From International Critical Tables, vol. 5, pp. 80–82.
TABLE 2-200
Specific Heat Ratio, Cp /Cv , for Air Pressure, bar
Temperature, K
1
10
20
40
60
80
100
150
200
250
300
400
500
600
800
1000
150 200 250 300 350
1.410 1.406 1.403 1.402 1.399
1.510 1.452 1.429 1.418 1.411
1.668 1.505 1.457 1.436 1.422
2.333 1.630 1.517 1.470 1.446
4.120 1.781 1.577 1.505 1.467
3.973 1.943 1.640 1.537 1.488
3.202 2.093 1.699 1.570 1.509
2.507 2.274 1.816 1.640 1.553
2.243 2.236 1.877 1.687 1.589
2.091 2.140 1.896 1.716 1.612
1.988 2.050 1.885 1.730 1.627
1.851 1.920 1.836 1.727 1.640
1.768 1.832 1.782 1.707 1.638
1.712 1.771 1.743 1.683 1.629
1.654 1.682 1.681 1.645 1.605
1.639 1.619 1.636 1.619 1.585
400 450 500 600 800
1.395 1.392 1.387 1.377 1.353
1.404 1.397 1.391 1.378 1.355
1.412 1.404 1.395 1.382 1.357
1.429 1.416 1.406 1.386 1.359
1.444 1.428 1.414 1.392 1.361
1.460 1.438 1.421 1.398 1.365
1.472 1.449 1.430 1.403 1.366
1.505 1.471 1.448 1.413 1.372
1.529 1.490 1.463 1.423 1.375
1.548 1.505 1.474 1.432 1.381
1.563 1.518 1.484 1.439 1.384
1.579 1.533 1.499 1.448 1.392
1.584 1.541 1.507 1.457 1.397
1.580 1.542 1.510 1.461 1.401
1.567 1.537 1.510 1.465 1.406
1.555 1.528 1.504 1.466 1.409
1000
1.336
1.337
1.338
1.339
1.342
1.343
1.343
1.345
1.348
1.350
1.354
1.358
1.361
1.365
1.368
1.372
Calculated from Cp, Cv values of Sychev, V. V., A. A. Vasserman, et al., “Thermodynamic Properties of Air,” Standartov, Moscow, 1978 and Hemisphere, New York, 1988 (276 pp.).
2-184
PHYSICAL AND CHEMICAL DATA
SPECIFIC HEATS OF AQUEOUS SOLUTIONS UNITS CONVERSIONS
ADDITIONAL REFERENCES
For this subsection, the following units conversions are applicable: °F = 9⁄5 °C + 32. To convert calories per gram-degree Celsius to British thermal units per pound-degree Fahrenheit, multiply by 1.0.
For additional data, see International Critical Tables, vol. 5, pp. 115–116, 122–125.
TABLE 2-201
TABLE 2-208
Acetic Acid (at 38°C)
Mole % acetic acid Cal/g °C
TABLE 2-202
0 1.0
6.98 0.911
30.9 0.73
54.5 0.631
100 0.535
Ammonia Specific heat, cal/g °C
Mole % NH3
2.4°C
20.6°C
41°C
61°C
0 10.5 20.9 31.2 41.4
1.01 0.98 .96 .956 .985
1.0 0.995 .99 1.0
0.995 1.06 1.03
1.0 1.02
TABLE 2-203
TABLE 2-204
100 0.497
5°C
20°C
40°C
5.88 12.3 27.3 45.8 69.6 100
1.02 0.975 .877 .776 .681 .576
1.0 0.982 .917 .811 .708 .60
0.995 .98 .92 .83 .726 .617
TABLE 2-209
95 0.52
90.5 0.53
82.3 0.56
75.2 0.581
Copper Sulfate
Composition CuSO4 + 50H2O CuSO4 + 200H2O CuSO4 + 400H2O TABLE 2-205
Specific heat, cal/g °C Mole % CH3OH
Aniline (at 20°C)
Mol % aniline Cal/g °C
Methyl Alcohol
Temperature
Specific heat, cal/g °C
12° to 15°C 12° to 14°C 13° to 17°C
0.848 .951 .975
Nitric Acid
% HNO3 by Weight
Specific Heat at 20°C, cal/g °C
0 10 20 30 40 50 60 70 80 90
1.000 0.900 .810 .730 .675 .650 .640 .615 .575 .515
Ethyl Alcohol Specific heat, cal/g °C
Mole % C2H5OH 4.16 11.5 37.0 61.0 100.0 TABLE 2-206
3°C
23°C
41°C
1.05 1.02 0.805 .67 .54
1.02 1.03 0.86 .727 .577
1.02 1.03 0.875 .748 .621
Glycerol Specific heat, cal/g °C
Mole % C3H5(OH)3
15°C
32°C
2.12 4.66 11.5 22.7 43.9 100.0
0.961 .929 .851 .765 .67 .555
0.960 .924 .841 .758 .672 .576
TABLE 2-207
Hydrochloric Acid Specific heat, cal/g °C
Mole % HCl 0.0 9.09 16.7 20.0 25.9
0°C
10°C
20°C
40°C
60°C
1.00 0.72 .61 .58 .55
0.72 .605 .575
0.74 .631 .591
0.75 .645 .615
0.78 .67 .638 .61
TABLE 2-210
Phosphoric Acid*
%H2PO4
Cp at 21.3°C cal/g °C
%H3PO4
Cp at 21.3°C cal/g °C
2.50 3.80 5.33 8.81 10.27 14.39 16.23 19.99 22.10 24.56 25.98 28.15 29.96 32.09 33.95 36.26 38.10 40.10 42.08 44.11 46.22 48.16 49.79
0.9903 .9970 .9669 .9389 .9293 .8958 .8796 .8489 .8300 .8125 .8004 .7856 .7735 .7590 .7432 .7270 .7160 .7024 .6877 .6748 .6607 .6475 .6370
50.00 52.19 53.72 56.04 58.06 60.23 62.10 64.14 66.13 68.14 69.97 69.50 71.88 73.71 75.79 77.69 79.54 80.00 82.00 84.00 85.98 88.01 89.72
0.6350 .6220 .6113 .5972 .5831 .5704 .5603 .5460 .5349 .5242 .5157 .5160 .5046 .4940 .4847 .4786 .4680 .4686 .4593 .4500 .4419 .4359 .4206
*Z. Physik. Chem., A167, 42 (1933).
SPECIFIC HEATS OF AQUEOUS SOLUTIONS TABLE 2-211
Potassium Chloride
TABLE 2-215
Sodium Chloride Specific heat, cal/g °C
Specific heat, cal/g °C Mole % KCl
6°C
20°C
33°C
40°C
Mole % NaCl
0.99 3.85 5.66 7.41
0.945 .828 .77
0.947 .831 .775 .727
0.947 .835 .778
0.947 .837 .775
0.249 .99 2.44 9.09
TABLE 2-212
Potassium Hydroxide (at 19°C)
Mole % KOH Cal/g °C
TABLE 2-213
0 1.0
0.497 0.975
1.64 0.93
TABLE 2-216 4.76 0.814
9.09 0.75
Normal Propyl Alcohol
Mole % NaOH Cal/g °C
TABLE 2-217
Mole % C3H7OH
5°C
20°C
40°C
1.55 5.03 11.4 23.1 41.2 73.0 100.0
1.03 1.07 1.035 0.877 .75 .612 .534
1.02 1.06 1.032 0.90 .78 .645 .57
1.01 1.03 0.99 .91 .815 .708 .621
% Na2CO3 by weight 0.000 1.498 2.000 2.901 4.000 5.000 6.000 8.000 10.000 13.790 13.840 20.000 25.000
Sodium Carbonate* Temperature, °C 17.6
30.0
76.6
98.0
0.9992 .9807
0.9986
1.0098
1.0084
.9786 .9597
20°C
33°C
57°C
0.96 .91 .805
0.99 .97 .915 .81
0.97 .915 .81
0.923 .82
0 1.0
0.5 0.985
1.0 0.97
9.09 0.835
16.7 0.80
28.6 0.784
37.5 0.782
Sulfuric Acid*
%H2SO4
Cp at 20°C, cal/g °C
%H2SO4
Cp at 20°C, cal/g °C
0.34 0.68 1.34 2.65 3.50 5.16 9.82 15.36 21.40 22.27 23.22 24.25 25.39 26.63 28.00 29.52 30.34 31.20 33.11
0.9968 .9937 .9877 .9762 .9688 .9549 .9177 .8767 .8339 .8275 .8205 .8127 .8041 .7945 .7837 .7717 .7647 .7579 .7422
35.25 37.69 40.49 43.75 47.57 52.13 57.65 64.47 73.13 77.91 81.33 82.49 84.48 85.48 89.36 91.81 94.82 97.44 100.00
0.7238 .7023 .6770 .6476 .6153 .5801 .5420 .5012 .4628 .4518 .4481 .4467 .4408 .4346 .4016 .3787 .3554 .3404 .3352
*Vinal and Craig, Bur. Standards J. Research, 24, 475 (1940).
.9594 .9428
6°C
Sodium Hydroxide (at 20°C)
Specific heat, cal/g °C
TABLE 2-214
2-185
0.9761 .9392
.9183 .9086 .8924
*J. Chem. Soc. 3062–3079 (1931).
.9452 .8881 .8631
TABLE 2-218 .8936 .8615
0.8911
Zinc Sulfate
Composition
Temperature
Specific heat, cal/g °C
ZnSO4 + 50H2O ZnSO4 + 200H2O
20° to 52°C 20° to 52°C
0.842 .952
2-186
PHYSICAL AND CHEMICAL DATA
SPECIFIC HEATS OF MISCELLANEOUS MATERIALS TABLE 2-219 and Solids
Specific Heats of Miscellaneous Liquids
Material Alumina Alundum Asbestos Asphalt Bakelite Brickwork Carbon (gas retort) (see under Graphite) Cellulose Cement, Portland Clinker Charcoal (wood) Chrome brick Clay Coal tar oils Coal tars Coke Concrete Cryolite Diamond Fireclay brick Fluorspar Gasoline Glass (crown) (flint) (pyrex) (silicate) wool Granite Graphite Gypsum Kerosene Limestone Litharge Magnesia Magnesite brick Marble Porcelain, fired Berlin Porcelain, green Berlin Porcelain, fired earthenware Porcelain, green earthenware
Specific heat, cal/g °C 0.2 (100°C); 0.274 (1500°C) 0.186 (100°C) 0.25 0.22 0.3 to 0.4 About 0.2 0.168 (26° to 76°C) 0.314 (40° to 892°C) 0.387 (56° to 1450°C) 0.204 0.32 0.186 0.242 0.17 0.224 0.26 to 0.37 0.34 (15° to 90°C) 0.35 (40°C); 0.45 (200°C) 0.265 (21° to 400°C) 0.359 (21° to 800°C) 0.403 (21° to 1300°C) 0.156 (70° to 312°F); 0.219 (72° to 1472°F) 0.253 (16° to 55°C) 0.147 0.198 (100°C); 0.298 (1500°C) 0.21 (30°C) 0.53 0.16 to 0.20 0.117 0.20 0.188 to 0.204 (0 to 100°C) 0.24 to 0.26 (0 to 700°C) 0.157 0.20 (20° to 100°C) 0.165 (26° to 76°C); 0.390 (56° to 1450°C) 0.259 (16° to 46°C) 0.47 0.217 0.055 0.234 (100°C); 0.188 (1500°C) 0.222 (100°C); 0.195 (1500°C) 0.21 (18°C) 0.189 (60°C) 0.185 (60°C) 0.186 (60°C) 0.181 (60°C)
TABLE 2-219 Specific Heats of Miscellaneous Liquids and Solids (Concluded ) Specific heat, cal/g °C
Material Pyrex glass Pyrites (copper) Pyrites (iron) Pyroxylin plastics Quartz Rubber (vulcanized) Sand Silica Silica brick Silicon carbide brick Silk Steel Stone Stoneware (common) Turpentine Wood (Oak) Woods, miscellaneous Wool Zirconium oxide
0.20 0.131 (30°C) 0.136 (30°C) 0.34 to 0.38 0.17 (0°C); 0.28 (350°C) 0.415 0.191 0.316 0.202 (100°C); 0.195 (1500°C) 0.202 (100°C) 0.33 0.12 about 0.2 0.188 (60°C) 0.42 (18°C) 0.570 0.45 to 0.65 0.325 0.11 (100°C); 0.179 (1500°C)
TABLE 2-219a Oils (Animal, Vegetable, Mineral Oils) 15 Cp[cal/(g ⋅ °C) = A/ d 4 + B(t − 15)
where d = density, g/cm3. °F = 9⁄5 °C + 32; to convert calories per gram-degree Celsius to British thermal units per pound-degree Fahrenheit, multiply by 1.0; to convert grams per cubic centimeter to pounds per cubic foot, multiply by 62.43. Oils
A
Castor Citron Fatty drying non-drying semidrying oils (except castor) Naphthene base Olive Paraffin base Petroleum oils
0.500
HEATS AND FREE ENERGIES OF FORMATION UNITS CONVERSIONS °F = 9⁄5 °C + 32; to convert kilocalories per gram-mole to British thermal units per pound-mole, multiply by 1.799 × 10−3.
B
0.0007 (0.438 at 54°C) 0.440 0.0007 0.450 0.0007 0.445 0.0007 0.450 0.0007 0.405 0.0009 (0.47 at 7°C) 0.425 0.0009 0.415 0.0009
HEATS AND FREE ENERGIES OF FORMATION TABLE 2-220
2-187
Heats and Free Energies of Formation of Inorganic Compounds
The values given in the following table for the heats and free energies of formation of inorganic compounds are derived from (a) Bichowsky and Rossini, “Thermochemistry of the Chemical Substances,” Reinhold, New York, 1936; (b) Latimer, “Oxidation States of the Elements and Their Potentials in Aqueous Solution,” PrenticeHall, New York, 1938; (c) the tables of the American Petroleum Institute Research Project 44 at the National Bureau of Standards; and (d) the tables of Selected Values of Chemical Thermodynamic Properties of the National Bureau of Standards. The reader is referred to the preceding books and tables for additional details as to methods of calculation, standard states, and so on.
Compound Aluminum Al AlBr3 Al4C3 AlCl3 AlF3 AlI3 AlN Al(NH4)(SO4)2 Al(NH4)(SO4)2·12H2O Al(NO3)3·6H2O Al(NO3)3·9H2O Al2O3 Al(OH)3 Al2O3·SiO2 Al2O3·SiO2 Al2O3·SiO2 3Al2O3·2SiO2 Al2S3 Al2(SO4)3 Al2(SO4)3·6H2O Al2(SO4)3·18H2O Antimony Sb SbBr3 SbCl3 SbCl5 SbF3 SbI3 Sb2O3 Sb2O4 Sb2O5 Sb2S3 Arsenic As AsBr3 AsCl3 AsF3 AsH3 AsI3 As2O3 As2O5 As2S3 Barium Ba BaBr2 BaCl2 Ba(ClO3)2 Ba(ClO4)2 Ba(CN)2 Ba(CNO)2 BaCN2 BaCO3 BaCrO4
State† c c aq c c aq, 600 c aq c aq c c c c c c, corundum c c, sillimanite c, disthene c, andalusite c, mullite c c aq c c c c c l c c c, I, orthorhombic c, II, octahedral c c c, black
Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole 0.00 −123.4 −209.5 −30.8 −163.8 −243.9 −329 −360.8 −72.8 −163.4 −57.7 −561.19 −1419.36 −680.89 −897.59 −399.09 −304.8 −648.7 −642.4 −642.0 −1874 −121.6 −820.99 −893.9 −1268.15 −2120 0.00 −59.9 −91.3 −104.8 −216.6 −22.8 −165.4 −166.6 −213.0 −230.0 −38.2
c c l l g c c c c amorphous
0.00 −45.9 −80.2 −223.76 43.6 −13.6 −154.1 −217.9 −20 −34.76
c c aq, 400 c aq, 300 c aq, 1600 c aq, 800 c c aq c c, witherite c
0.00 −180.38 −185.67 −205.25 −207.92 −176.6 −170.0 −210.2
*For footnotes see end of table.
−48 −212.1 −63.6 −284.2 −342.2
Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole 0.00 −189.2 −29.0 −209.5 −312.6 −152.5 −50.4 −486.17 −1179.26 −526.32 −376.87 −272.9
−739.53 −759.3 −1103.39
Compound Barium (Cont.) BaF2 BaH2 Ba(HCO3)2 BaI2 Ba(IO3)2 BaMoO4 Ba3N2 Ba(NO2)2 Ba(NO3)2 BaO Ba(OH)2 BaO·SiO2 Ba3(PO4)2 BaPtCl6 BaS BaSO3 BaSO4 BaWO4 Beryllium Be BeBr2
0.00 −77.8
BeCl2 BeI2
−146.0 −186.6 −196.1 −36.9 0.00 −70.5 −212.27 37.7 −134.8 −183.9 −20 0.00 −183.0 −196.5 −134.4 −155.3 −180.7 −271.4
Be3N2 BeO Be(OH)2 BeS BeSO4 Bismuth Bi BiCl3 BiI3 BiO Bi2O3 Bi(OH)3 Bi2S3 Bi2(SO4)3 Boron B BBr3 BCl3 BF3 B2H6 BN B2O3 B(OH)3 B2S3 Bromine Br2 BrCl
State†
Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole
c aq, 1600 c aq c aq, 400 c aq c c c aq c aq, 600 c c aq, 400 c c c c c c c
−287.9 −284.6 −40.8 −459 −144.6 −155.17 −264.5 −237.50 −370 −90.7 −184.5 −179.05 −236.99 −227.74 −133.0 −225.9 −237.76 −363 −992 −284.9 −111.2 −282.5 −340.2 −402
c c aq c aq c aq c c c c c aq
0.00 −79.4 −142 −112.6 −163.9 −39.4 −112 −134.5 −145.3 −215.6 −56.1 −281
c c aq c aq c c c c c
0.00 −90.5 −101.6 −24 −27 −49.5 −137.1 −171.1 −43.9 −607.1
c l g g g g c c gls c c
0.00 −52.7 −44.6 −94.5 −265.2 7.5 −32.1 −302.0 −297.6 −260.0 −56.6
l g g
0.00 7.47 3.06
Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole −265.3 −31.5 −414.4 −158.52 −198.35
−150.75 −189.94
−209.02
−313.4 0.00 −127.9 −141.4 −103.4 −122.4 −138.3
−254.8 0.00 −76.4
−43.2 −117.9 −39.1 0.00 −50.9 −90.8 −261.0 19.9 −27.2 −282.9 −280.3 −229.4 0.00 0.931 −0.63
2-188
PHYSICAL AND CHEMICAL DATA
TABLE 2-220
Heats and Free Energies of Formation of Inorganic Compounds (Continued )
Compound Cadmium Cd CdBr2 CdCl2 Cd(CN)2 CdCO3 CdI2 Cd3N2 Cd(NO3)2 CdO Cd(OH)2 CdS CdSO4 Calcium Ca CaBr2 CaC2 CaCl2 CaCN2 Ca(CN)2 CaCO3 CaCO3·MgCO3 CaC2O4 Ca(C2H3O2)2 CaF2 CaH2 CaI2 Ca3N2 Ca(NO3)2 Ca(NO3)2·2H2O Ca(NO3)2·3H2O Ca(NO3)2·4H2O CaO Ca(OH)2 CaO·SiO2 CaS CaSO4 CaSO4·aH2O CaSO4·2H2O CaWO4 Carbon C CO CO2 Cerium Ce CeN Cesium Cs CsBr CsCl
State†
Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole
c c aq, 400 c aq, 400 c c c aq, 400 c aq, 400 c c c c aq, 400
0.00 −75.8 −76.6 −92.149 −96.44 36.2 −178.2 −48.40 −47.46 39.8 −115.67 −62.35 −135.0 −34.5 −222.23 −232.635
c c aq, 400 c c aq c c aq c, calcite c, aragonite c c c aq c aq c c aq, 400 c c aq, 400 c c c c c aq, 800 c, II, wollastonite c, I, pseudowollastonite c c, insoluble form c, soluble form α c, soluble form β c c c
0.00 −162.20 −187.19 −14.8 −190.6 −209.15 −85 −43.3 −289.5 −289.54 −558.8 −332.2 −356.3 −364.1 −290.2 −286.5 −46 −128.49 −156.63 −103.2 −224.05 −228.29 −367.95 −439.05 −509.43 −151.7 −235.58 −239.2 −377.9 −376.6 −114.3 −338.73 −336.58 −335.52 −376.13 −479.33 −387
Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole 0.00 −70.7 −67.6 −81.889 −81.2 −163.2 −43.22 −71.05 −55.28 −113.7 −33.6 −194.65 0.00 −181.86 −16.0 −179.8 −195.36 −54.0 −270.8 −270.57
−311.3 −264.1 −35.7 −157.37 −88.2 −177.38 −293.57 −351.58 −409.32 −144.3 −213.9 −207.9 −357.5 −356.6 −113.1 −311.9 −309.8 −308.8 −425.47
c, graphite c, diamond g g
0.00 0.453 −26.416 −94.052
0.00 0.685 −32.808 −94.260
c c
0.00 −78.2
0.00 −70.8
0.00 −97.64 −91.39 −106.31 −102.01
0.00
c c aq, 500 c aq, 400
−94.86 −101.61
Compound Cesium (Cont.) Cs2CO3 CsF CsH CsHCO3 CsI CsNH2 CsNO3 Cs2O CsOH Cs2S Cs2SO4 Chlorine Cl2 ClF ClO ClO2 ClO3 Cl2O Cl2O7 Chromium Cr CrBr3 Cr3C2 Cr4C CrCl2 CrF2 CrF3 CrI2 CrO3 Cr2O3 Cr2(SO4)3 Cobalt Co CoBr2 Co3C CoCl2 CoCO3 CoF2 CoI2 Co(NO3)2 CoO Co3O4 Co(OH)2 Co(OH)3 CoS Co2S3 CoSO4 Columbium Cb Cb2O5 Copper Cu CuBr CuBr2 CuCl CuCl2
State† c c aq, 400 c c aq, 2000 c aq, 400 c c aq, 400 c c aq, 200 c c aq g g g g g g g c aq c c c aq c c c aq c c aq
Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole −271.88 −131.67 −140.48 −12 −230.6 −226.6 −83.91 −75.74 −28.2 −121.14 −111.54 −82.1 −100.2 −117.0 −87 −344.86 −340.12 0.00 −25.7 33 24.7 37 18.20 63 0.00 −21.008 −16.378 −103.1 −152 −231 −63.7 −139.3 −268.8
Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole
−135.98 −7.30 −210.56 −82.61 −96.53 −107.87 −316.66 0.00 29.5 22.40 0.00 −122.7 −21.20 −16.74 −93.8 −102.1
−64.1 −249.3 −626.3
c c aq c c aq, 400 c aq c aq c aq c c c c c c c aq, 400
0.00 −55.0 −73.61 9.49 −76.9 −95.58 −172.39 −172.98 −24.2 −43.15 −102.8 −114.9 −57.5 −196.5 −131.5 −177.0 −22.3 −40.0 −216.6
c c
0.00 −462.96
0.00
0.00 −26.7 −34.0 −42.4 −31.4 −48.83 −64.7
0.00 −23.8
c c c aq c c aq, 400
0.00 −61.96 7.08 −66.6 −75.46 −155.36 −144.2 −37.4 −65.3 −108.9 −142.0 −19.8 −188.9
−33.25 −24.13
HEATS AND FREE ENERGIES OF FORMATION TABLE 2-220
Heats and Free Energies of Formation of Inorganic Compounds (Continued )
Compound Copper (Cont.) CuClO4 Cu(ClO3)2 Cu(ClO4)2 CuI CuI2 Cu3N Cu(NO3)2 CuO Cu2O Cu(OH)2 CuS Cu2S CuSO4 Cu2SO4 Erbium Er Er(OH)3 Fluorine F2 F2O Gallium Ga GaBr3 GaCl3 GaN Ga2O Ga2O3 Germanium Ge Ge3N4 GeO2 Gold Au AuBr AuBr3 AuCl AuCl3 AuI Au2O3 Au(OH)3 Hafnium Hf HfO2 Hydrogen H3AsO3 H3AsO4 HBr HBrO HBrO3 HCl HCN HClO HClO3 HClO4 HC2H3O2 H2C2O4 HCOOH
2-189
State† aq aq, 400 aq c c aq c c aq, 200 c c c c c c aq, 800 c aq
Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole −28.3 −17.8 −4.8 −11.9 17.78 −73.1 −83.6 −38.5 −43.00 −108.9 −11.6 −18.97 −184.7 −200.78 −179.6
Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole 1.34 15.4 −5.5 −16.66 −8.76 −36.6 −31.9 −38.13 −85.5 −11.69 −20.56 −158.3 −160.19 −152.0
Compound Hydrogen (Cont.) H2CO3 HF HI HIO HIO3 HN3 HNO3 HNO3·H2O HNO3·3H2O H2O H2O2 H3PO2
c c
0.00 −326.8
0.00
g g
0.00 5.5
0.00 9.7
H3PO3 H3PO4 H2S
c c c c c c
0.00 −92.4 −125.4 −26.2 −84.3 −259.9
0.00
c c c
0.00 −15.7 −128.6
0.00
c c c aq c c aq c c c
0.00 −3.4 −14.5 −11.0 −8.3 −28.3 −32.96 0.2 11.0 −100.6
0.00
c c
0.00 −271.1
0.00 −258.2
aq c aq g aq, 400 aq aq g aq, 400 g aq, 100 aq, 400 aq aq, 660 l aq, 400 c aq, 300 l aq, 200
−175.6 −214.9 −214.8 −8.66 −28.80 −25.4 −11.51 −22.063 −39.85 31.1 24.2 −28.18 −23.4 −31.4 −116.2 −116.74 −196.7 −194.6 −97.8 −98.0
−153.04
H2S2 H2SO3 H2SO4 H2Se H2SeO3 H2SeO4
24.47
H2SiO3 H4SiO4 H2Te H2TeO3
4.21 −0.76 18.71
H2TeO4 Indium In InBr3 InCl3
−183.93 −12.72 −24.58 −19.90 5.00 −22.778 −31.330 27.94 26.55 −19.11 −0.25 −10.70 −93.56 −96.8 −165.64 −82.7 −85.1
InI3 InN In2O3 Iodine I2 IBr ICl ICl3 I2O5 Iridium Ir IrCl IrCl2 IrCl3 IrF6 IrO2 Iron Fe FeBr2
State†
Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole
Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole
aq g aq, 200 g aq, 400 aq c aq g g l aq, 400 l l g l l aq, 200 c aq c aq c aq, 400 g aq, 2000 l aq, 200 l aq, 400 g aq c aq c aq, 400 c c g c aq aq
−167.19 −64.2 −75.75 6.27 −13.47 −38 −56.77 −54.8 70.3 −31.99 −41.35 −49.210 −112.91 −252.15 −57.7979 −68.3174 −45.16 −45.80 −145.5 −145.6 −232.2 −232.2 −306.2 −309.32 −4.77 −9.38 −3.6 −146.88 −193.69 −212.03 20.5 18.1 −126.5 −122.4 −130.23 −143.4 −267.8 −340.6 36.9 −145.0 −145.0 −165.6
c c aq c aq c aq c c
0.00 −97.2 −112.9 −128.5 −145.6 −56.5 −67.2 −4.8 −222.47
c g g g c c
0.00 14.88 10.05 4.20 −21.8 −42.5
0.00 4.63 1.24 −1.32 −6.05
c c c c l c
0.00 −20.5 −40.6 −60.5 −130 −40.14
0.00 −16.9 −32.0 −46.5
c, α c aq, 540
0.00 −57.15 −78.7
−149.0 −64.7 0.365 −12.35 −23.33 −32.25 78.50 −17.57 −19.05 −78.36 −193.70 −54.6351 −56.6899 −28.23 −31.47 −120.0 −204.0 −270.0 −7.85 −128.54 17.0 18.4 −101.36 −247.9 33.1 −115.7
0.00 −97.2 −117.5 −60.5
0.00 −69.47
2-190
PHYSICAL AND CHEMICAL DATA
TABLE 2-220
Heats and Free Energies of Formation of Inorganic Compounds (Continued )
Compound Iron (Cont.) FeBr3 Fe3C Fe(CO)5 FeCO3 FeCl2 FeCl3 FeF2 FeI2 FeI3 Fe4N Fe(NO3)2 Fe(NO3)3 FeO Fe2O3 Fe3O4 Fe(OH)2 Fe(OH)3 FeO·SiO2 Fe2P FeSi FeS FeS2 FeSO4 Fe2(SO4)3 FeTiO3 Lanthanum La LaCl3 La3H8 LaN La2O3 LaS2 La2S3 La2(SO4)3 Lead Pb PbBr2 PbCO3 Pb(C2H3O2)2 PbC2O4 PbCl2 PbF2 PbI2 Pb(NO3)2 PbO PbO2 Pb3O4 Pb(OH)2 PbS PbSO4 Lithium Li LiBr LiBrO3 Li2C2 LiCN LiCNO
State† aq c l c, siderite c aq c aq, 2000 aq, 1200 c aq aq c aq aq, 800 c c c c c c c c c c, pyrites c, marcasite c aq, 400 aq, 400 c, ilmenite
Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole −95.5 5.69 −187.6 −172.4 −81.9 −100.0 −96.4 −128.5 −177.2 −24.2 −47.7 −49.7 −2.55 −118.9 −156.5 −64.62 −198.5 −266.9 −135.9 −197.3 −273.5 −13 −19.0 −22.64 −38.62 −33.0 −221.3 −236.2 −653.3 −295.51
Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole −76.26 4.24 −154.8 −72.6 −83.0 −96.5 −151.7 −45 −39.5 0.862 −72.8 −81.3 −59.38 −179.1 −242.3 −115.7 −166.3
Compound Lithium (Cont.) LiC2H3O2 Li2CO3 LiCl LiClO3 LiClO4 LiF LiH LiHCO3 LiI LiIO3 Li3N LiNO3 Li2O Li2O2 LiOH
−23.23 −35.93 −195.5 −196.4 −533.4 −277.06
c c aq c c c c c aq
0.00 −253.1 −284.7 −160 −72.0 −539 −148.3 −351.4 −972
0.00
c c aq c, cerussite c aq, 400 c c aq c c c aq, 400 c, red c, yellow c c c c c
0.00 −66.24 −56.4 −167.6 −232.6 −234.2 −205.3 −85.68 −82.5 −159.5 −41.77 −106.88 −99.46 −51.72 −50.86 −65.0 −172.4 −123.0 −22.38 −218.5
0.00 −62.06 −54.97 −150.0
c c aq, 400 aq c aq aq
0.00 −83.75 −95.40 −77.9 −13.0 −31.4 −101.2
0.00
LiOH·H2O Li2O·SiO2 Li2Se Li2SO4 Li2SO4·H2O Magnesium Mg Mg(AsO4)2 MgBr2
−64.6
−184.40 −75.04 −68.47 −148.1 −41.47 −58.3 −45.53 −43.88 −52.0 −142.2 −102.2 −21.98 −192.9
−95.28 −65.70 −31.35 −94.12
Mg(CN)2 MgCN2 Mg(C2H3O2)2 MgCO3 MgCl2 MgCl2·H2O MgCl2·2H2O MgCl2·4H2O MgCl2·6H2O MgF2 MgI2 MgMoO4 Mg3N2 Mg(NO3)2 Mg(NO3)2·2H2O Mg(NO3)2·6H2O MgO MgO·SiO2 Mg(OH)2 MgS MgSO4 MgTe MgWO4 Manganese Mn MnBr2 Mn3C
State†
Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole
aq c aq, 1900 c aq, 278 aq aq c aq, 400 c aq, 2000 c aq, 400 aq c c aq, 400 c c aq c aq, 400 c gls c aq c aq, 400 c
−183.9 −289.7 −293.1 −97.63 −106.45 −87.5 −106.3 −145.57 −144.85 −22.9 −231.1 −65.07 −80.09 −121.3 −47.45 −115.350 −115.88 −142.3 −151.9 −159 −116.58 −121.47 −188.92 −374 −84.9 −95.5 −340.23 −347.02 −411.57
c c aq c aq, 400 aq c aq c c aq, 400 c c c c c c aq, 400 c c c aq, 400 c c c c c, ppt. c, brucite c aq c aq, 400 c c
0.00 −731.3 −749 −123.9 −167.33 −39.7 −61 −344.6 −261.7 −153.220 −189.76 −230.970 −305.810 −453.820 −597.240 −263.8 −86.8 −136.79 −329.9 −115.2 −188.770 −209.927 −336.625 −624.48 −143.84 −347.5 −221.90 −223.9 −84.2 −108 −304.94 −325.4 −25 −345.2
c, α c aq c
0.00 −91 −106 1.1
Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole −160.00 −269.8 −267.58 −102.03 −70.95 −81.4 −136.40 −210.98 −83.03 −102.95 −37.33 −96.95 −138.0 −106.44 −108.29
−105.64 −314.66 −375.07 0.00 −630.14 −156.94 −29.08 −286.38 −241.7 −143.77 −205.93 −267.20 −387.98 −505.45 −132.45 −100.8 −140.66 −160.28 −496.03 −136.17 −326.7 −200.17 −193.3 −277.7 −283.88
0.00 −97.8 1.26
HEATS AND FREE ENERGIES OF FORMATION TABLE 2-220
Heats and Free Energies of Formation of Inorganic Compounds (Continued )
Compound Manganese (Cont.) Mn(C2H3O2)2 MnCO3 MnC2O4 MnCl2 MnF2 MnI2 Mn5N2 Mn(NO3)2 Mn(NO3)2.6H2O MnO MnO2 Mn2O3 Mn3O4 MnO.SiO2 Mn(OH)2 Mn(OH)3 Mn3(PO4)2 MnSe MnS MnSO4 Mn2(SO4)3 Mercury Hg HgBr HgBr2 Hg(C2H3O2)2 HgCl2 HgCl Hg2Cl2 Hg(CN)2 HgC2O4 HgH HgI2 HgI Hg2I2 Hg(NO3)2 Hg2(NO3)2 HgO Hg2O HgS HgSO4 Hg2SO4 Molybdenum Mo Mo2C Mo2N MoO2 MoO3 MoS2 MoS3 Nickel Ni NiBr2 Ni3C Ni(C2H3O2)2 Ni(CN)2 NiCl2
2-191
State†
Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole
c aq c c c aq, 400 aq, 1200 c aq c c aq, 400 c c c c c c c c c c c, green c aq, 400 c aq
−270.3 −282.7 −211 −240.9 −112.0 −128.9 −206.1 −49.8 −76.2 −57.77 −134.9 −148.0 −557.07 −92.04 −124.58 −229.5 −331.65 −301.3 −163.4 −221 −736 −26.3 −47.0 −254.18 −265.2 −635 −657
l g c aq c aq c aq g c c aq, 1110 c g c, red g c aq aq c, red c, yellow ppt. c c, black c c
0.00 23 −40.68 −38.4 −196.3 −192.5 −53.4 −50.3 19 −63.13 62.8 66.25 −159.3 57.1 −25.3 33 −28.88 −56.8 −58.5 −21.6 −20.8 −21.6 −10.7 −166.6 −177.34
c c c c c c c
0.00 4.36 −8.3 −130 −180.39 −56.27 −61.48
c c aq c aq aq c
0.00 −53.4 −72.6 9.2 −249.6 230.9 −75.0
Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole
Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole
Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole
aq, 400 c aq c aq c aq, 200 c c c c c aq, 200
−94.34 −157.5 −171.6 −22.4 −42.0 −101.5 −113.5 −58.4 −129.8 −163.2 −20.4 −216 −231.3
−74.19
g g g aq, 200 c aq c aq, 400 c aq c aq aq c aq c aq, 400 c aq c aq c aq c aq c aq, 500 aq aq, 400 c aq, 400 l l c g g g g c l g
0.00 −27 −10.96 −19.27 −64.57 −60.27 −148.1 −148.58 −0.7 3.6 −17.8 −12.3 −223.4 −266.3 −260.6 −75.23 −71.20 −69.4 −63.2 −276.9 −271.3 −111.6 −110.2 −48.43 −44.97 −87.40 −80.89 −87.59 −55.21 −281.74 −279.33 12.06 −57.96 −232.2 19.55 21.600 7.96 2.23 −10.0 11.6 12.8
Compound
State†
Nickel (Cont.) −227.2 −192.5 −102.2 −180.0 −73.3 −46.49 −101.1 −441.2 −86.77 −111.49 −209.9 −306.22 −282.1 −143.1 −190 −27.5 −48.0 −228.41
NiF2 NiI2 Ni(NO3)2 NiO Ni(OH)2 Ni(OH)3 NiS NiSO4 Nitrogen N2 NF3 NH3 NH4Br NH4C2H3O2 NH4CN NH4CNS
0.00 18 −38.8 −9.74 −139.2 −42.2 −23.25 14
(NH4)2CO3 (NH4)2C2O4 NH4Cl NH4ClO4 (NH4)2CrO4 NH4F NH4I NH4NO3
52.25 −24.0 23 −26.53 −13.09 −15.65 −13.94 −12.80 −8.80 −149.12 0.00 2.91 −118.0 −162.01 −54.19 −57.38 0.00 −60.7 8.88 −190.1 66.3
NH4OH (NH4)2S (NH4)2SO4 N2H4 N2H4·H2O N2H4·H2SO4 N2O NO NO2 N2O4 N2O5 NOBr NOCl Osmium Os OsO4 Oxygen O2 O3 Palladium Pd PdO Phosphorus P P
−142.9 −36.2 −64.0 −51.7 −105.6
−187.6 0.00 −3.903 −43.54 −108.26 20.4 4.4 −164.1 −196.2 −48.59 −21.1 −209.3 −84.7 −31.3
−14.50 −215.06 −214.02
24.82 20.719 12.26 23.41 19.26 16.1
c c g
0.00 −93.6 −80.1
0.00 −70.9 −68.1
g g
0.00 33.88
0.00 38.86
c c
0.00 −20.40
0.00
c, white (“yellow”) c, red (“violet”) g
0.00 −4.22 150.35
0.00 −1.80 141.88
2-192
PHYSICAL AND CHEMICAL DATA
TABLE 2-220
Heats and Free Energies of Formation of Inorganic Compounds (Continued )
Compound Phosphorus (Cont.) P2 P4 PBr3 PBr5 PCl3 PCl5 PH3 PI3 P2O5 POCl3 Platinum Pt PtBr4 PtCl2 PtCl4 PtI4 Pt(OH)2 PtS PtS2 Potassium K K3AsO3 K3AsO4 KH2AsO4 KBr KBrO3 KC2H3O2 KCl KClO3 KClO4 KCN KCNO KCNS K2CO3 K2C2O4 K2CrO4 K2Cr2O7 KF K3Fe(CN)6 K4Fe(CN)6 KH KHCO3 KI KIO3 KIO4 KMnO4 K2MoO4
State† g g l c g l g g c c g c c aq c c aq c c c c c aq aq c c aq, 400 c aq, 1667 c aq, 400 c aq, 400 c aq, 400 c aq, 400 c aq, 400 c aq c aq, 400 c aq, 400 c aq, 400 c aq, 400 c aq, 400 c aq, 180 c aq c aq c c aq, 2000 c aq, 500 c aq, 400 aq c aq, 400 aq, 880
Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole 33.82 13.2 −45 −60.6 −70.0 −76.8 −91.0 2.21 −10.9 −360.0 −138.4 0.00 −40.6 −50.7 −34 −62.6 −82.3 −18 −87.5 −20.18 −26.64 0.00 −323.0 −390.3 −271.2 −94.06 −89.19 −81.58 −71.68 −173.80 −177.38 −104.348 −100.164 −93.5 −81.34 −103.8 −101.14 −28.1 −25.3 −99.6 −94.5 −47.0 −41.07 −274.01 −280.90 −319.9 −315.5 −333.4 −328.2 −488.5 −472.1 −134.50 −138.36 −48.4 −34.5 −131.8 −119.9 −10 −229.8 −224.85 −78.88 −73.95 −121.69 −115.18 −98.1 −192.9 −182.5 −364.2
Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole 24.60 5.89 −65.2 −63.3 −73.2 −1.45 −127.2 0.00
Compound Potassium (Cont.) KNH2 KNO2 KNO3 K2O K2O·Al2O3·SiO2 K2O·Al2O3·SiO2 KOH K3PO3 K3PO4 KH2PO4 K2PtCl4 K2PtCl6
−67.9 −18.55 −24.28 0.00 −355.7 −236.7 −90.8 −92.0 −60.30 −156.73 −97.76 −98.76 −69.30 −72.86 −28.08 −90.85 −44.08
K2Se K2SeO4 K2S K2SO3 K2SO4 K2SO4·Al2(SO4)3 K2SO4·Al2(SO4)3· 24H2O K2S2O6 Rhenium Re ReF6 Rhodium Rh RhO Rh2O Rh2O3 Rubidium Rb RbBr
−264.04
RbCN Rb2CO3
−293.1
RbCl
−306.3 −440.9 −133.13
−5.3 −207.71 −77.37 −79.76 −101.87 −99.68 −169.1 −168.0 −342.9
RbF RbHCO3 RbI RbNH2 RbNO3 Rb2O Rb2O2 RbOH Ruthenium Ru RuS2 Selenium Se
State†
Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole
Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole
c aq c aq, 400 c c, leucite gls c, adularia c, microcline gls c aq, 400 aq aq c c aq c aq, 9400 c aq aq c aq, 400 c aq c aq, 400 c
−28.25 −86.0 −118.08 −109.79 −86.2 −1379.6 −1368.2 −1784.5 −1784.5 −1747 −102.02 −114.96 −397.5 −478.7 −362.7 −254.7 −242.6 −299.5 −286.1 −74.4 −83.4 −267.1 −121.5 −110.75 −267.7 −269.7 −342.65 −336.48 −1178.38
c c
−2895.44 −418.62
−2455.68
c g
0.00 −274
0.00
c c c c
0.00 −21.7 −22.7 −68.3
0.00
0.00 −95.82 −45.0 −90.54 −25.9 −273.22 −282.61 −105.06 −53.6 −101.06 −133.23 −139.31 −230.01 −225.59 −81.04 −31.2 −74.57 −27.74 −119.22 −110.52 −82.9 −107 −101.3 −115.8
0.00
c c g aq, 500 aq c aq, 220 c g aq, ∞ c aq, 400 c aq, 2000 c g aq, 400 c c aq, 400 c c c aq, 200 c c c, I, hexagonal
−75.9 −94.29 −93.68
−105.0 −443.3 −326.1 −226.5 −263.6 −99.10 −240.0 −111.44 −251.3 −314.62 −310.96 −1068.48
−52.50 −93.38 −263.78 −98.48 −57.9 −100.13 −134.5 −209.07 −40.5 −81.13 −95.05
−106.39
0.00 −46.99
0.00 −44.11
0.00
0.00
HEATS AND FREE ENERGIES OF FORMATION TABLE 2-220
Compound
Heats and Free Energies of Formation of Inorganic Compounds (Continued )
State†
Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole
Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole
SiF4 SiH4 SiI4 Si3N4 SiO2
Silver Ag AgBr Ag2C2 AgC2H3O2 AgCN Ag2CO3 Ag2C2O4 AgCl AgF AgI AgIO3 AgNO2 AgNO3 Ag2O Ag2S Ag2SO4 Sodium Na Na3AsO3 Na3AsO4 NaBr NaBrO NaBrO3 NaC2H3O2 NaCN NaCNO NaCNS Na2CO3 NaCO2NH2 Na2C2O4 NaCl NaClO3 NaClO4
Compound
State†
Sodium (Cont.)
Selenium (Cont.) Se2Cl2 SeF6 SeO2 Silicon Si SiBr4 SiC SiCl4
2-193
c, II, red, monoclinic l g c
0.2 −22.06 −246 −56.33
−13.73 −222
c l c l g g g c c c, cristobalite, 1600° form c, cristobalite, 1100° form c, quartz c, tridymite
0.00 −93.0 −28 −150.0 −142.5 −370 −14.8 −29.8 −179.25 −202.62
0.00
c c c c aq c c c c c aq, 400 c c c aq c aq, 6500 c c c aq
0.00 −23.90 84.5 −95.9 −91.7 33.8 −119.5 −158.7 −30.11 −48.7 −53.1 −15.14 −42.02 −11.6 −2.9 −29.4 −24.02 −6.95 −5.5 −170.1 −165.8
c aq, 500 c aq, 500 c aq, 400 aq aq, 400 c aq, 400 c aq, 200 c aq c aq, 400 c aq, 1000 c c aq, 600 c aq, 400 c aq, 400 c
0.00 −314.61 −366 −381.97 −86.72 −86.33 −78.9 −68.89 −170.45 −175.450 −22.47 −22.29 −96.3 −91.7 −39.94 −38.23 −269.46 −275.13 −142.17 −313.8 −309.92 −98.321 −97.324 −83.59 −78.42 −101.12
−27.4 −133.9 −133.0 −360 −9.4 −154.74
Na2Cr2O7 NaF NaH NaHCO3 NaI NaIO3 Na2MoO4 NaNO2 NaNO3
−202.46 −203.35 −203.23
Na2CrO4
−190.4 0.00 −23.02 −70.86 38.70 −103.0 −25.98 −47.26 −16.17 −24.08 3.76 9.99 −7.66 −7.81 −2.23 −7.6 −146.8 −139.22 0.00 −341.17 −87.17 −57.59 −152.31 −23.24 −86.00 −39.24 −249.55 −251.36 −283.42 −91.894 −93.92 −62.84
Na2O Na2O2 Na2O·SiO2 Na2O·Al2O3·3SiO2 Na2O·Al2O3·4SiO2 NaOH Na3PO3 Na3PO4 Na2PtCl4 Na2PtCl6 Na2Se Na2SeO4 Na2S Na2SO3 Na2SO4 Na2SO4·10H2O Na2WO4 Strontium Sr SrBr2 Sr(C2H3O2)2 Sr(CN)2 SrCO3 SrCl2 SrF2 Sr(HCO3)2 SrI2 Sr3N2 Sr(NO3)2 SrO SrO·SiO2 SrO2 Sr2O Sr(OH)2 Sr3(PO4)2 SrS
aq, 476 c aq, 800 aq, 1200 c aq, 400 c c aq c aq, ∞ aq, 400 c aq c aq c aq, 400 c c c c, natrolite c c aq, 400 aq, 1000 c aq, 400 aq c aq c aq, 440 c aq, 800 c aq, 400 c aq, 800 c aq, 1100 c c aq c c aq, 400 c aq aq c c aq, 400 c aq c aq, 400 c c aq, 400 c gls c c c aq, 800 c aq c
Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole
Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole
−97.66 −319.8 −323.0 −465.9 −135.94 −135.711 −14 −226.0 −222.1 −69.28 −71.10 −112.300 −364 −358.7 −86.6 −83.1 −111.71 −106.880 −99.45 −119.2 −383.91 −1180 −1366 −101.96 −112.193 −389.1 −457 −471.9 −237.2 −272.1 −280.9 −59.1 −78.1 −254 −261.5 −89.8 −105.17 −261.2 −264.1 −330.50 −330.82 −1033.85 −391 −381.5
−73.29
0.00 −171.0 −187.24 −358.0 −364.4 −59.5 −290.9 −197.84 −209.20 −289.0 −459.1 −136.1 −156.70 −91.4 −233.2 −228.73 −140.8 −364 −153.3 −153.6 −228.7 −239.4 −980 −985 −113.1
−296.58 −431.18 −129.0 −128.29 −9.30 −202.66 −202.87 −74.92 −94.84 −333.18 −71.04 −87.62 −88.84 −90.06 −105.0 −361.49 −90.60 −100.18 −428.74 −216.78
−89.42 −230.30 −101.76 −240.14 −241.58 −302.38 −301.28 −870.52 −345.18 0.00 −182.36 −311.80 −54.50 −271.9 −195.86 −413.76 −157.87 −76.5 −185.70 −133.7 −139.0 −208.27 −881.54
2-194
PHYSICAL AND CHEMICAL DATA
TABLE 2-220
Heats and Free Energies of Formation of Inorganic Compounds (Continued )
Compound
State†
Strontium (Cont.) SrSO4 SrWO4 Sulfur S
S2 S6 S8 S2Br2 SCl4 S2Cl2 S2Cl4 SF6 SO SO2 SO3
SO2Cl2 Tantalum Ta TaN Ta2O5 Tellurium Te TeBr4 TeCl4 TeF6 TeO2 Thallium Tl TlBr TlCl TlCl3 TlF TlI TlNO3 Tl2O Tl2O3 TlOH Tl2S Tl2SO4 Thorium Th ThBr4 ThC2 ThCl4 ThI4 Th3N4 ThO2 Th(OH)4 Th(SO4)2
aq c aq, 400 c
Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole −120.4 −345.3 −345.0 −393
Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole −109.78 −309.30
0.00 −0.071 0.257
0.00 0.023 0.072 0.071 43.57 19.36 13.97 12.770
53.25 31.02 27.78 27.090 −4 −13.7 −14.2 −24.1 −262 19.02 −70.94 −94.39 −103.03 −105.09 −105.92 −109.34 −82.04 −89.80
−237 12.75 −71.68 −88.59 −88.28 −88.22 −88.34 −88.98 −74.06 −75.06
c c c
0.00 −51.2 −486.0
0.00 −45.11 −453.7
c c c g c
0.00 −49.3 −77.4 −315 −77.56
0.00
c c aq c aq c aq aq c aq c aq c c c aq c c aq, 800
0.00 −41.5 −28.0 −49.37 −38.4 −82.4 −91.0 −77.6 −31.1 −12.7 −58.2 −48.4 −43.18 −120 −57.44 −53.9 −22 −222.8 −214.1
c c aq c c aq aq c c c, “soluble” c aq
0.00 −281.5 −352.0 −45.1 −335 −392 −292.0 −309.0 −291.6 −336.1 −632 −668.1
−5.90
−57.4 −292 −64.66 0.00 −39.43 −32.34 −44.46 −39.09 −44.25 −73.46 −31.3 −20.09 −36.32 −34.01 −45.54 −45.35 −197.79 −191.62 0.00
SnBr2
SnCl2 SnCl4 SnI2 SnO SnO2 Sn(OH)2 Sn(OH)4 SnS Titanium Ti TiC TiCl4 TiN TiO2 Tungsten W WO2 WO3 WS2 Uranium U UC2 UCl3 UCl4 U3N4 UO2 UO2(NO3)2·6H2O UO3 U3O8 Vanadium V VCl2 VCl3 VCl4 VN V2O2 V2O3 V2O4 V2O5 Zinc Zn ZnSb ZnBr2 Zn(C2H3O2)2 Zn(CN)2 ZnCO3 ZnCl2
−295.31
ZnF2 ZnI2
−322.32 −246.33 −282.3 −280.1
Zn(NO3)2 ZnO ZnO·SiO2 Zn(OH)2 ZnS ZnSO4
−549.2
c, II, tetragonal c, III, “gray,” cubic c aq c aq c aq l aq c aq c c c c c
0.00 0.6 −61.4 −60.0 −94.8 −110.6 −83.6 −81.7 −127.3 −157.6 −38.9 −33.3 −67.7 −138.1 −136.2 −268.9 −18.61
−68.94 −110.4 −124.67
c c l c c, III, rutil amorphous
0.00 −110 −181.4 −80.0 −225.0 −214.1
0.00 −109.2 −165.5 −73.17 −211.9 −201.4
c c c c
0.00 −130.5 −195.7 −84
0.00 −118.3 −177.3
c c c c c c c c c
0.00 −29 −213 −251 −274 −256.6 −756.8 −291.6 −845.1
0.00
c c l l c c c c c
0.00 −147 −187 −165 −41.43 −195 −296 −342 −373
c c c aq, 400 c aq, 400 c c c aq, 400 aq c aq aq, 400 c, hexagonal c c, rhombic c, wurtzite c aq, 400
0.00 −3.6 −77.0 −93.6 −259.4 −269.4 17.06 −192.9 −99.9 −115.44 −192.9 −50.50 −61.6 −134.9 −83.36 −282.6 −153.66 −45.3 −233.4 −252.12
Compound Tin Sn
SnBr4 c, rhombic c, monoclinic l, λ l, λµ equilibrium g g g g l l l l g g g g l c, α c, β c, γ g l
State†
Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole
Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole 0.00 1.1 −55.43 −97.66
−30.95 −60.75 −123.6 −115.95 −226.00
−249.6 −242.2 −617.8
0.00
−35.08 −277 −316 −342 0.00 −3.88 −72.9 −214.4 −173.5 −88.8 −166.6 −49.93 −87.7 −76.19 −44.2 −211.28
HEATS OF COMBUSTION TABLE 2-220
2-195
Heats and Free Energies of Formation of Inorganic Compounds (Concluded )
Compound
State†
Zirconium Zr ZrC ZrCl4 ZrN
c c c c
Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole 0.00 −29.8 −268.9 −82.5
Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole 0.00 −34.6 −75.9
Compound Zirconium (Cont.) ZrO2 Zr(OH)4 ZrO(OH)2
Heat of formation‡§ ∆H (formation) at 25°C, kcal/mole
State† c, monoclinic c c
−258.5 −411.0 −337
Free energy of formation¶ ∆F (formation) at 25°C, kcal/mole −244.6 −307.6
† The physical state is indicated as follows: c, crystal (solid); l, liquid; g, gas; gls, glass or solid supercooled liquid; aq, in aqueous solution. A number following the symbol aq applies only to the values of the heats of formation (not to those of free energies of formation); and indicates the number of moles of water per mole of solute; when no number is given, the solution is understood to be dilute. For the free energy of formation of a substance in aqueous solution, the concentration is always that of the hypothetical solution of unit molality. ‡ The increment in heat content, ∆H, in the reaction of forming the given substance from its elements in their standard states. When ∆H is negative, heat is evolved in the process, and, when positive, heat is absorbed. § The heat of solution in water of a given solid, liquid, or gaseous compound is given by the difference in the value for the heat of formation of the given compound in the solid, liquid, or gaseous state and its heat of formation in aqueous solution. The following two examples serve as an illustration of the procedure: (1) For NaCl(c) and NaCl(aq, 400H2O), the values of ∆H(formation) are, respectively, −98.321 and −97.324 kg-cal per mole. Subtraction of the first value from the second gives ∆H = 0.998 kg-cal per mole for the reaction of dissolving crystalline sodium chloride in 400 moles of water. When this process occurs at a constant pressure of 1 atm, 0.998 kg-cal of energy are absorbed. (2) For HCl(g) and HCl(aq, 400H2O), the values for ∆H(formation) are, respectively, −22.06 and −39.85 kg-cal per mole. Subtraction of the first from the second gives ∆H = −17.79 kg-cal per mole for the reaction of dissolving gaseous hydrogen chloride in 400 moles of water. At a constant pressure of 1 atm, 17.79 kg-cal of energy are evolved in this process. The increment in the free energy, ∆F, in the reaction of forming the given substance in its standard state from its elements in their standard states. The standard states are: for a gas, fugacity (approximately equal to the pressure) of 1 atm; for a pure liquid or solid, the substance at a pressure of 1 atm; for a substance in aqueous solution, the hypothetical solution of unit molality, which has all the properties of the infinitely dilute solution except the property of concentration. ¶ The free energy of solution of a given substance from its normal standard state as a solid, liquid, or gas to the hypothetical one molal state in aqueous solution may be calculated in a manner similar to that described in footnote § for calculating the heat of solution.
HEATS OF COMBUSTION TABLE 2-221 Enthalpies and Gibbs Energies of Formation, Entropies, and Net Enthalpies of Combustion of Inorganic and Organic Compounds at 298.15 K
Mol wt
Ideal gas enthalpy of formation, J/kmol × 1E-07
Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane
CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20
74828 74840 74986 106978 109660 110543 142825 111659 111842
16.043 30.070 44.097 58.123 72.150 86.177 100.204 114.231 128.258
−7.4520 −8.3820 −10.4680 −12.5790 −14.6760 −16.6940 −18.7650 −20.8750 −22.8740
−5.0490 −3.1920 −2.4390 −1.6700 −0.8813 −0.0066 0.8165 1.6000 2.4980
1.8627 2.2912 2.7020 3.0991 3.4945 3.8874 4.2798 4.6723 5.0640
−0.8026 −1.4286 −2.0431 −2.6573 −3.2449 −3.8551 −4.4647 −5.0742 −5.6846
10 11 12 13 14 15 16 17 18 19
n-Decane n-Undecane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane
C10H22 C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40
124185 1120214 112403 629505 629594 629629 544763 629787 593453 629925
142.285 156.312 170.338 184.365 198.392 212.419 226.446 240.473 254.500 268.527
−24.9460 −27.0430 −29.0720 −31.1770 −33.2440 −35.3110 −37.4170 −39.4450 −41.5120 −43.5790
3.3180 4.1160 4.9810 5.7710 6.5990 7.4260 8.2160 9.0830 9.9100 10.7400
5.4570 5.8493 6.2415 6.6337 7.0259 7.4181 7.8102 8.2023 8.5945 8.9866
−6.2942 −6.9036 −7.5137 −8.1229 −8.7328 −9.3424 −9.9515 −10.5618 −11.1715 −11.7812
20 21 22 23 24 25 26 27 28 29
n-Eicosane 2-Methylpropane 2-Methylbutane 2,3-Dimethylbutane 2-Methylpentane 2,3-Dimethylpentane 2,3,3-Trimethylpentane 2,2,4-Trimethylpentane Ethylene Propylene
C20H42 C4H10 C5H12 C6H14 C6H14 C7H16 C8H18 C8H18 C2H4 C3H6
112958 75285 78784 79298 107835 565593 560214 540841 74851 115071
282.553 58.123 72.150 86.177 86.177 100.204 114.231 114.231 28.054 42.081
−45.6460 −13.4180 −15.3700 −17.6800 −17.4550 −19.4100 −21.8450 −22.4010 5.2510 1.9710
11.5700 −2.0760 −1.4050 −0.3125 −0.5338 0.5717 1.8280 1.3940 6.8440 6.2150
9.3787 2.9539 3.4374 3.6592 3.8089 4.1455 4.2702 4.2296 2.1920 2.6660
−12.3908 −2.6490 −3.2395 −3.8476 −3.8492 −4.4608 −5.0688 −5.0653 −1.3230 −1.9257
Cmpd. no. 1 2 3 4 5 6 7 8 9
Name
Formula
CAS no.
Ideal gas Gibbs energy Ideal gas Standard net enthalpy of formation, entropy, of combustion, J/kmol × 1E-07 J/(kmol·K) × 1E-05 J/kmol × 1E-09
2-196
PHYSICAL AND CHEMICAL DATA
TABLE 2-221 Enthalpies and Gibbs Energies of Formation, Entropies, and Net Enthalpies of Combustion of Inorganic and Organic Compounds (Continued ) Cmpd. no.
Name
Formula
CAS no.
Mol wt
Ideal gas enthalpy of formation, J/kmol × 1E-07
Ideal gas Gibbs energy Ideal gas Standard net enthalpy of formation, entropy, of combustion, J/kmol × 1E-07 J/(kmol·K) × 1E-05 J/kmol × 1E-09
30 31 32 33 34 35 36 37 38 39
1-Butene cis-2-Butene trans-2-Butene 1-Pentene 1-Hexene 1-Heptene 1-Octene 1-Nonene 1-Decene 2-Methylpropene
C4H8 C4H8 C4H8 C5H10 C6H12 C7H14 C8H16 C9H18 C10H20 C4H8
106989 590181 624646 109671 592416 592767 111660 124118 872059 115117
56.108 56.108 56.108 70.134 84.161 98.188 112.215 126.242 140.269 56.108
−0.0540 −0.7400 −1.1000 −2.1300 −4.2000 −6.2800 −8.3600 −10.4000 −12.4700 −1.7100
7.0270 6.5360 6.3160 7.8450 8.7390 9.4830 10.3000 11.1500 11.9800 5.8080
3.0775 3.0120 2.9650 3.4699 3.8389 4.2549 4.6469 5.0399 5.4319 2.9309
−2.5408 −2.5339 −2.5303 −3.1296 −3.7394 −4.3489 −4.9606 −5.5684 −6.1781 −2.5242
40 41 42 43 44 45 46 47 48 49
2-Methyl-1-butene 2-Methyl-2-butene 1,2-Butadiene 1,3-Butadiene 2-Methyl-1,3-butadiene Acetylene Methylacetylene Dimethylacetylene 3-Methyl-1-butyne 1-Pentyne
C5H10 C5H10 C4H6 C4H6 C5H8 C2H2 C3H4 C4H6 C5H8 C5H8
563462 513359 590192 106990 78795 74862 74997 503173 598232 627190
70.134 70.134 54.092 54.092 68.119 26.038 40.065 54.092 68.119 68.119
−3.5300 −4.1800 16.2300 10.9240 7.5730 22.8200 18.4900 14.5700 13.8000 14.4400
6.6680 6.0450 19.8600 14.9720 14.5896 21.0680 19.3840 18.4900 20.7200 21.0300
3.3950 3.3860 2.9300 2.7889 3.1564 2.0081 2.4836 2.8330 3.1890 3.2980
−3.1159 −3.1088 −2.4617 −2.4090 −2.9842 −1.2570 −1.8487 −2.4189 −3.0460 −3.0510
50 51 52 53 54 55 56 57 58 59
2-Pentyne 1-Hexyne 2-Hexyne 3-Hexyne 1-Heptyne 1-Octyne Vinylacetylene Cyclopentane Methylcyclopentane Ethylcyclopentane
C5H8 C6H10 C6H10 C6H10 C7H12 C8H14 C4H4 C5H10 C6H12 C7H14
627214 693027 764352 928494 628717 629050 689974 287923 96377 1640897
68.119 82.145 82.145 82.145 96.172 110.199 52.076 70.134 84.161 98.188
12.5100 12.3700 10.5000 10.6000 10.3000 8.2300 30.4600 −7.7030 −10.6200 −12.6900
19.0700 21.8500 19.9000 19.9000 22.7000 23.5000 30.6000 3.8850 3.6300 4.4800
3.3084 3.6940 3.7200 3.7600 4.0850 4.4780 2.7940 2.9290 3.3990 3.7830
−3.0291 −3.6610 −3.6400 −3.6400 −4.2717 −4.8815 −2.3620 −3.0709 −3.6741 −4.2839
60 61 62 63 64 65 66 67 68 69
Cyclohexane Methylcyclohexane 1,1-Dimethylcyclohexane Ethylcyclohexane Cyclopentene 1-Methylcyclopentene Cyclohexene Benzene Toluene o-Xylene
C6H12 C7H14 C8H16 C8H16 C5H8 C6H10 C6H10 C6H6 C7H8 C8H10
110827 108872 590669 1678917 142290 693890 110838 71432 108883 95476
84.161 98.188 112.215 112.215 68.119 82.145 82.145 78.114 92.141 106.167
−12.3300 −15.4800 −18.1000 −17.1500 3.3100 −0.3800 −0.4600 8.2880 5.0170 1.9080
3.1910 2.7330 3.5229 3.9550 11.0500 10.3800 10.7700 12.9600 12.2200 12.2000
2.9728 3.4330 3.6501 3.8260 2.9127 3.2640 3.1052 2.6930 3.2099 3.5383
−3.6560 −4.2571 −4.8639 −4.8705 −2.9393 −3.5340 −3.5320 −3.1360 −3.7340 −4.3330
70 71 72 73 74 75 76 77 78 79
m-Xylene p-Xylene Ethylbenzene Propylbenzene 1,2,4-Trimethylbenzene Isopropylbenzene 1,3,5-Trimethylbenzene p-Isopropyltoluene Naphthalene Biphenyl
C8H10 C8H10 C8H10 C9H12 C9H12 C9H12 C9H12 C10H14 C10H8 C12H10
108383 106423 100414 103651 95636 98828 108678 99876 91203 92524
106.167 106.167 106.167 120.194 120.194 120.194 120.194 134.221 128.174 154.211
1.7320 1.8030 2.9920 0.7910 −1.3800 0.4000 −1.5900 −2.9000 15.0580 18.2420
11.8760 12.1400 13.0730 13.8090 11.7100 13.7900 11.8100 13.3520 22.4080 28.0230
3.5854 3.5223 3.6063 3.9843 3.9610 3.8600 3.8560 4.2630 3.3315 3.9367
−4.3318 −4.3330 −4.3450 −4.9542 −4.9307 −4.9510 −4.9291 −5.5498 −4.9809 −6.0317
80 81 82 83 84 85 86 87 88 89
Styrene m-Terphenyl Methanol Ethanol 1-Propanol 1-Butanol 2-Butanol 2-Propanol 2-Methyl-2-propanol 1-Pentanol
C8H8 C18H14 CH4O C2H6O C3H8O C4H10O C4H10O C3H8O C4H10O C5H12O
100425 92068 67561 64175 71238 71363 78922 67630 75650 71410
104.152 230.309 32.042 46.069 60.096 74.123 74.123 60.096 74.123 88.150
14.7400 27.6600 −20.0940 −23.4950 −25.5200 −27.4600 −29.2900 −27.2700 −31.2400 −29.8737
21.3900 42.3000 −16.2320 −16.7850 −15.9900 −15.0300 −16.9600 −17.3470 −17.7600 −14.6022
3.4510 5.2630 2.3988 2.8064 3.2247 3.6148 3.6469 3.0920 3.2630 4.0250
−4.2190 −9.0530 −0.6382 −1.2350 −1.8438 −2.4560 −2.4408 −1.8300 −2.4239 −3.0605
90 91 92 93 94 95 96
2-Methyl-1-butanol 3-Methyl-1-butanol 1-Hexanol 1-Heptanol Cyclohexanol Ethylene glycol 1,2-Propylene glycol
C5H12O C5H12O C6H14O C7H16O C6H12O C2H6O2 C3H8O2
137326 123513 111273 111706 108930 107211 57556
88.150 88.150 102.177 116.203 100.161 62.068 76.095
−30.2085 −30.2100 −31.6500 −33.6400 −28.6200 −38.7500 −42.1500
−14.6709 −14.5000 −13.4400 −12.5300 −10.9500 −30.2600 −30.4000
3.9351 3.8770 4.4010 4.7919 3.2770 3.2350 3.5200
−3.0620 −3.0623 −3.6766 −4.2860 −3.4639 −1.0590 −1.6476
HEATS OF COMBUSTION
2-197
TABLE 2-221 Enthalpies and Gibbs Energies of Formation, Entropies, and Net Enthalpies of Combustion of Inorganic and Organic Compounds (Continued ) Cmpd. no.
Name
Formula
CAS no.
Mol wt
Ideal gas enthalpy of formation, J/kmol × 1E-07
Ideal gas Gibbs energy Ideal gas Standard net enthalpy of formation, entropy, of combustion, J/kmol × 1E-07 J/(kmol·K) × 1E-05 J/kmol × 1E-09
97 98 99
Phenol o-Cresol m-Cresol
C6H6O C7H8O C7H8O
108952 95487 108394
94.113 108.140 108.140
−9.6399 −12.8570 −13.2300
−3.2637 −3.5430 −4.0190
3.1481 3.5259 3.5604
−2.9210 −3.5280 −3.5278
100 101 102 103 104 105 106 107 108 109
p-Cresol Dimethyl ether Methyl ethyl ether Methyl n-propyl ether Methyl isopropyl ether Methyl n-butyl ether Methyl isobutyl ether Methyl tert-butyl ether Diethyl ether Ethyl propyl ether
C7H8O C2H6O C3H8O C4H10O C4H10O C5H12O C5H12O C5H12O C4H10O C5H12O
106445 115106 540670 557175 598538 628284 625445 1634044 60297 628320
108.140 46.069 60.096 74.123 74.123 88.150 88.150 88.150 74.123 88.150
−12.5350 −18.4100 −21.6400 −23.8200 −25.2000 −25.8100 −26.6000 −28.3500 −25.2100 −27.2200
−3.1660 −11.2800 −11.7100 −11.1000 −12.1800 −10.1700 −10.7000 −11.7500 −12.2100 −11.5200
3.5075 2.6670 3.0881 3.5200 3.4160 3.9010 3.8100 3.5780 3.4230 3.8810
−3.5226 −1.3284 −1.9314 −2.5174 −2.5311 −3.1282 −3.1220 −3.1049 −2.5035 −3.1200
110 111 112 113 114 115 116 117 118 119
Ethyl isopropyl ether Methyl phenyl ether Diphenyl ether Formaldehyde Acetaldehyde 1-Propanal 1-Butanal 1-Pentanal 1-Hexanal 1-Heptanal
C5H12O C7H8O C12H10O CH2O C2H4O C3H6O C4H8O C5H10O C6H12O C7H14O
625547 100663 101848 50000 75070 123386 123728 110623 66251 111717
88.150 108.140 170.211 30.026 44.053 58.080 72.107 86.134 100.161 114.188
−28.5800 −6.7900 5.2000 −10.8600 −16.6200 −18.6300 −20.7000 −22.7800 −24.8600 −26.9400
−12.6400 2.2700 17.5000 −10.2600 −13.3100 −12.4600 −11.6300 −10.7100 −10.0050 −9.1910
3.8000 3.6100 4.1300 2.1866 2.6420 3.0440 3.4365 3.8289 4.2214 4.6138
−3.1030 −3.6072 −5.8939 −0.5268 −1.1045 −1.6857 −2.3035 −2.9100 −3.5200 −4.1360
120 121 122 123 124 125 126 127 128 129
1-Octanal 1-Nonanal 1-Decanal Acetone Methyl ethyl ketone 2-Pentanone Methyl isopropyl ketone 2-Hexanone Methyl isobutyl ketone 3-Methyl-2-pentanone
C8H16O C9H18O C10H20O C3H6O C4H8O C5H10O C5H10O C6H12O C6H12O C6H12O
124130 124196 112312 67641 78933 107879 563804 591786 108101 565617
128.214 142.241 156.268 58.080 72.107 86.134 86.134 100.161 100.161 100.161
−29.0200 −31.0900 −33.1700 −21.5700 −23.9000 −25.9200 −26.2400 −27.9826 −28.8000 −28.1000
−8.3770 −7.5530 −6.7390 −15.1300 −14.7000 −13.8300 −13.9000 −13.0081 −13.5000 −12.9000
5.0063 5.3988 5.7912 2.9540 3.3940 3.7860 3.6990 4.1786 4.0700 4.1200
−4.7400 −5.3500 −5.9590 −1.6590 −2.2680 −2.8796 −2.8770 −3.4900 −3.4900 −3.4900
130 131 132 133 134 135 136 137 138 139
3-Pentanone Ethyl isopropyl ketone Diisopropyl ketone Cyclohexanone Methyl phenyl ketone Formic acid Acetic acid Propionic acid n-Butyric acid Isobutyric acid
C5H10O C6H12O C7H14O C6H10O C8H8O CH2O2 C2H4O2 C3H6O2 C4H8O2 C4H8O2
96220 565695 565800 108941 98862 64186 64197 79094 107926 79312
86.134 100.161 114.188 98.145 120.151 46.026 60.053 74.079 88.106 88.106
−25.7900 −28.6100 −31.1400 −22.6100 −8.6700 −37.8600 −43.2800 −45.3500 −47.5800 −48.4100
−13.4400 −13.3000 −13.2000 −8.6620 −0.1364 −35.1000 −37.4600 −36.6700 −36.0000 −36.2100
3.7000 4.0690 4.5700 3.2200 3.8450 2.4870 2.8250 3.2300 3.6200 3.4120
−2.8804 −3.4860 −4.0950 −3.2990 −3.9730 −0.2115 −0.8146 −1.3950 −2.0077 −2.0004
140 141 142 143 144 145 146 147 148 149
Benzoic acid Acetic anhydride Methyl formate Methyl acetate Methyl propionate Methyl n-butyrate Ethyl formate Ethyl acetate Ethyl propionate Ethyl n-butyrate
C7H6O2 C4H6O3 C2H4O2 C3H6O2 C4H8O2 C5H10O2 C3H6O2 C4H8O2 C5H10O2 C6H12O2
65850 108247 107313 79209 554121 623427 109944 141786 105373 105544
122.123 102.090 60.053 74.079 88.106 102.133 74.079 88.106 102.133 116.160
−29.4100 −57.2500 −35.2400 −41.1900 −42.7500 −45.0700 −38.8300 −44.4500 −46.3600 −48.5500
−21.4200 −47.3400 −29.5000 −32.4200 −31.1000 −30.5300 −30.3100 −32.8000 −31.9300 −31.2200
3.6900 3.8990 2.8520 3.1980 3.5960 3.9880 3.2820 3.5970 4.0250 4.4170
−3.0951 −1.6750 −0.8924 −1.4610 −2.0780 −2.6860 −1.5070 −2.0610 −2.6740 −3.2840
150 151 152 153 154 155 156 157 158 159
n-Propyl formate n-Propyl acetate n-Butyl acetate Methyl benzoate Ethyl benzoate Vinyl acetate Methylamine Dimethylamine Trimethylamine Ethylamine
C4H8O2 C5H10O2 C6H12O2 C8H8O2 C9H10O2 C4H6O2 CH5N C2H7N C3H9N C2H7N
110747 109604 123864 93583 93890 108054 74895 124403 75503 75047
88.106 102.133 116.160 136.150 150.177 86.090 31.057 45.084 59.111 45.084
−40.7600 −46.4800 −48.5600 −28.7900 −32.6000 −31.4900 −2.2970 −1.8450 −2.4310 −4.7150
−29.3600 −32.0400 −31.2600 −18.1000 −19.0500 −22.7900 3.2070 6.8390 9.8990 3.6160
3.6780 4.0230 4.4250 4.1400 4.5500 3.2800 2.4330 2.7296 2.8700 2.8480
−2.0410 −2.6720 −3.2800 −3.7720 −4.4100 −1.9500 −0.9751 −1.6146 −2.2449 −1.5874
160 161 162
Diethylamine Triethylamine n-Propylamine
C4H11N C6H15N C3H9N
109897 121448 107108
73.138 101.192 59.111
−7.1420 −9.5800 −7.0500
7.3080 11.4100 4.1700
3.5220 4.0540 3.2420
−2.8003 −4.0405 −2.1650
2-198
PHYSICAL AND CHEMICAL DATA
TABLE 2-221 Enthalpies and Gibbs Energies of Formation, Entropies, and Net Enthalpies of Combustion of Inorganic and Organic Compounds (Continued ) Cmpd. no.
Name
Formula
CAS no.
Mol wt
Ideal gas enthalpy of formation, J/kmol × 1E-07
Ideal gas Gibbs energy Ideal gas Standard net enthalpy of formation, entropy, of combustion, J/kmol × 1E-07 J/(kmol·K) × 1E-05 J/kmol × 1E-09
163 164 165 166 167 168 169
di-n-Propylamine Isopropylamine Diisopropylamine Aniline N-Methylaniline N,N-Dimethylaniline Ethylene oxide
C6H15N C3H9N C6H15N C6H7N C7H9N C8H11N C2H4O
142847 75310 108189 62533 100618 121697 75218
101.192 59.111 101.192 93.128 107.155 121.182 44.053
−11.6000 −8.3800 −15.0000 8.7100 8.8000 10.0500 −5.2630
8.6800 3.1920 5.7900 16.6800 20.2000 24.7728 −1.3230
4.2900 3.1240 4.1200 3.1980 3.4100 3.6600 2.4299
−4.0189 −2.1566 −3.9900 −3.2390 −3.9000 −4.5250 −1.2180
170 171 172 173 174 175 176 177 178 179
Furan Thiophene Pyridine Formamide N,N-Dimethylformamide Acetamide N-Methylacetamide Acetonitrile Propionitrile n-Butyronitrile
C4H4O C4H4S C5H5N CH3NO C3H7NO C2H5NO C3H7NO C2H3N C3H5N C4H7N
110009 110021 110861 75127 68122 60355 79163 75058 107120 109740
68.075 84.142 79.101 45.041 73.095 59.068 73.095 41.053 55.079 69.106
−3.4800 11.5440 14.0370 −19.2200 −19.1700 −23.8300 −24.0000 7.4040 5.1800 3.4058
0.0823 12.6620 19.0490 −14.7100 −8.8400 −15.9600 −13.5000 9.1868 9.7495 10.8658
2.6714 2.7865 2.8278 2.4857 3.2600 2.7220 3.2000 2.4329 2.8614 3.2543
−1.9959 −2.4352 −2.6721 −0.5021 −1.7887 −1.0741 −1.7100 −1.1904 −1.8007 −2.4148
180 181 182 183 184 185 186 187 188 189
Benzonitrile Methyl mercaptan Ethyl mercaptan n-Propyl mercaptan n-Butyl mercaptan Isobutyl mercaptan sec-Butyl mercaptan Dimethyl sulfide Methyl ethyl sulfide Diethyl sulfide
C7H5N CH4S C2H6S C3H8S C4H10S C4H10S C4H10S C2H6S C3H8S C4H10S
100470 74931 75081 107039 109795 513440 513531 75183 624895 352932
103.123 48.109 62.136 76.163 90.189 90.189 90.189 62.136 76.163 90.189
21.8823 −2.2900 −4.6300 −6.7500 −8.7800 −9.6900 −9.6600 −3.7240 −5.9600 −8.3470
26.0872 −0.9800 −0.4814 0.2583 1.1390 0.5982 0.5120 0.7302 1.1470 1.7780
3.2104 2.5500 2.9610 3.3650 3.7520 3.6280 3.6670 2.8585 3.3320 3.6800
−3.5224 −1.1517 −1.7366 −2.3458 −2.9554 −2.9490 −2.9490 −1.7449 −2.3531 −2.9607
190 191 192 193 194 195 196 197 198 199
Fluoromethane Chloromethane Trichloromethane Tetrachloromethane Bromomethane Fluoroethane Chloroethane Bromoethane 1-Chloropropane 2-Chloropropane
CH3F CH3Cl CHCl3 CCl4 CH3Br C2H5F C2H5Cl C2H5Br C3H7Cl C3H7Cl
593533 74873 67663 56235 74839 353366 75003 74964 540545 75296
34.033 50.488 119.377 153.822 94.939 48.060 64.514 108.966 78.541 78.541
−23.4300 −8.1960 −10.2900 −9.5810 −3.7700 −26.4400 −11.2260 −6.3600 −13.3180 −14.4770
−21.0400 −5.8440 −7.0100 −5.3540 −2.8190 −21.2300 −6.0499 −2.5820 −5.2610 −6.1360
2.2273 2.3418 2.9560 3.0991 2.4580 2.6440 2.7578 2.8730 3.1547 3.0594
−0.5219 −0.6754 −0.3800 −0.2653 −0.7054 −1.1270 −1.2849 −1.2850 −1.8670 −1.8630
200 201 202 203 204 205 206 207 208 209
1,1-Dichloropropane 1,2-Dichloropropane Vinyl chloride Fluorobenzene Chlorobenzene Bromobenzene Air Hydrogen Helium-4 Neon
C3H6Cl2 C3H6Cl2 C2H3Cl C6H5F C6H5Cl C6H5Br H2 He Ne
78999 78875 75014 462066 108907 108861 132259100 1333740 7440597 7440019
112.986 112.986 62.499 96.104 112.558 157.010 28.951 2.016 4.003 20.180
−15.0800 −16.2800 2.8450 −11.6566 5.1090 10.5018 0 0 0 0
−6.5200 −8.0180 4.1950 −6.9036 9.8290 13.8532 0 0 0 0
3.4480 3.5480 2.7354 3.0263 3.1403 3.2439 1.9900 1.3057 1.2604 1.4622
−1.7200 −1.7070 −1.1780 −2.8145 −2.9760 −3.0192 0 −0.2418 0 0
210 211 212 213 214 215 216 217 218 219
Argon Fluorine Chlorine Bromine Oxygen Nitrogen Ammonia Hydrazine Nitrous oxide Nitric oxide
Ar F2 Cl2 Br2 O2 N2 NH3 N2H4 N2O NO
7440371 7782414 7782505 7726956 7782447 7727379 7664417 302012 10024972 10102439
39.948 37.997 70.905 159.808 31.999 28.014 17.031 32.045 44.013 30.006
0 0 0 3.0910 0 0 −4.5898 9.5353 8.2050 9.0250
0 0 0 0.3140 0 0 −1.6400 15.9170 10.4160 8.6570
1.5474 2.0268 2.2297 2.4535 2.0504 1.9150 1.9266 2.3861 2.1985 2.1060
0 0 0 0 0 0 −0.3168 −5.3420 −0.0820 −0.0902
220 221 222 223 224 225 226 227
Cyanogen Carbon monoxide Carbon dioxide Carbon disulfide Hydrogen fluoride Hydrogen chloride Hydrogen bromide Hydrogen cyanide
C2N2 CO CO2 CS2 HF HCl HBr HCN
460195 630080 124389 75150 7664393 7647010 10035106 74908
52.036 28.010 44.010 76.143 20.006 36.461 80.912 27.026
30.9072 −11.0530 −39.3510 11.6900 −27.3300 −9.2310 −3.6290 13.5143
29.7553 −13.7150 −39.4370 6.6800 −27.5400 −9.5300 −5.3340 12.4725
2.4146 1.9756 2.1368 2.3790 1.7367 1.8679 1.9859 2.0172
−1.0961 −0.2830 0 −1.0769 0.1524 −0.0286 −0.0690 −0.6233
HEATS OF COMBUSTION
2-199
TABLE 2-221 Enthalpies and Gibbs Energies of Formation, Entropies, and Net Enthalpies of Combustion of Inorganic and Organic Compounds (Concluded ) Cmpd. no. 228 229 230 231
Name
Formula
Hydrogen sulfide Sulfur dioxide Sulfur trioxide Water
H2S SO2 SO3 H2O
CAS no. 7783064 7446095 7446119 7732185
Mol wt
Ideal gas enthalpy of formation, J/kmol × 1E-07
34.082 64.065 80.064 18.015
−2.0630 −29.6840 −39.5720 −24.1814
Ideal gas Gibbs energy Ideal gas Standard net enthalpy of formation, entropy, of combustion, J/kmol × 1E-07 J/(kmol·K) × 1E-05 J/kmol × 1E-09 −3.3440 −30.0120 −37.0950 −22.8590
−0.5180 0 0.0989 0
2.0560 2.4810 2.5651 1.8872
All substances are listed in alphabetical order in Table 2-6a. Compiled from Daubert, T. E., R. P. Danner, H. M. Sibul, and C. C. Stebbins, DIPPR Data Compilation of Pure Compound Properties, Project 801 Sponsor Release, July, 1993, Design Institute for Physical Property Data, AIChE, New York, NY; and from Thermodynamics Research Center, “Selected Values of Properties of Hydrocarbons and Related Compounds,” Thermodynamics Research Center Hydrocarbon Project, Texas A&M University, College Station, Texas (extant 1994). The compounds are considered to be formed from the elements in their standard states at 298.15 K and 101,325 Pa. These include C (graphite) and S (rhombic). Enthalpy of combustion is the net value for the compound in its standard state at 298.15K and 101,325 Pa. Products of combustion are taken to be CO2 (gas), H2O (gas), F2 (gas), Cl2 (gas), Br2 (gas), I2 (gas), SO2 (gas), N2 (gas), H3PO4 (solid), and SiO2 (crystobalite). J/kmol × 2.390E-04 = cal/gmol; J/kmol × 4.302106E-04 = Btu/lbmol. J/(kmol·K) × 2.390E-04 = cal/(gmol·°C); J/(kmol·K) × 2.390059E-04 = Btu/(lbmol·°F). TABLE 2-222
Ideal Gas Sensible Enthalpies, hT - h298 (kJ/kgmol), of Combustion Products
Temperature, K
CO
CO2
H
OH
H2
N
NO
NO2
N2
N2O
O
O2
SO2
H2O
200 240 260 280 298.15
−2858 −1692 −1110 −529 0
−3414 −2079 −1383 −665 0
−2040 −1209 −793 −377 0
−2976 −1756 −1150 −546 0
−2774 −1656 −1091 −522 0
−2040 −1209 −793 −378 0
−2951 −1743 −1142 −543 0
−3495 −2104 −1392 −672 0
−2857 −1692 −1110 −528 0
−3553 −2164 −1438 −692 0
−2186 −1285 −840 −398 0
−2868 −1703 −1118 −533 0
−3736 −2258 −1496 −718 0
−3282 −1948 −1279 −609 0
300 320 340 360 380
54 638 1221 1805 2389
69 823 1594 2382 3184
38 454 870 1285 1701
55 654 1251 1847 2442
53 630 1209 1791 2373
38 454 870 1286 1701
55 652 1248 1845 2442
68 816 1571 2347 3130
54 636 1219 1802 2386
72 854 1654 2470 3302
41 478 913 1346 1777
54 643 1234 1828 2425
74 881 1702 2538 3387
62 735 1410 2088 2769
400 420 440 460 480
2975 3563 4153 4643 5335
4003 4835 5683 6544 7416
2117 2532 2948 3364 3779
3035 3627 4219 4810 5401
2959 3544 4131 4715 5298
2117 2533 2949 3364 3780
3040 3638 4240 4844 5450
3927 4735 5557 6392 7239
2971 3557 4143 4731 5320
4149 5010 5884 6771 7670
2207 2635 3063 3490 3918
3025 3629 4236 4847 5463
4250 5126 6015 6917 7831
3452 4139 4829 5523 6222
500 550 600 650 700
5931 7428 8942 10477 12023
8305 10572 12907 15303 17754
4196 5235 6274 7314 8353
5992 7385 8943 10423 11902
5882 6760 8811 10278 11749
4196 5235 6274 7314 8353
6059 7592 9144 10716 12307
8099 10340 12555 14882 17250
5911 7395 8894 10407 11937
8580 10897 13295 15744 18243
4343 5402 6462 7515 8570
6084 7653 9244 10859 12499
8758 11123 13544 16022 18548
6925 8699 10501 12321 14192
750 800 850 900 950
13592 15177 16781 18401 20031
20260 22806 25398 28030 30689
9392 10431 11471 12510 13550
13391 14880 16384 17888 19412
13223 14702 16186 17676 19175
9329 10431 11471 12510 13550
13919 15548 17195 18858 20537
19671 22136 24641 27179 29749
13481 15046 16624 18223 19834
20791 23383 26014 28681 31381
9620 10671 11718 12767 13812
14158 15835 17531 19241 20965
21117 23721 26369 29023 31714
16082 18002 19954 21938 23954
1000 1100 1200 1300 1400
21690 25035 28430 31868 35343
33397 38884 44473 50148 55896
14589 16667 18746 20824 22903
20935 24024 27160 30342 33569
20680 23719 26797 29918 33082
14589 16667 18746 20824 22903
22229 25653 29120 32626 36164
32344 37605 42946 48351 53808
21463 24760 28109 31503 34936
34110 39647 45274 50976 56740
14860 16950 19039 21126 23212
22703 26212 29761 33344 36957
34428 39914 45464 51069 56718
26000 30191 34506 38942 43493
1500 1600 1700 1800 1900
38850 42385 45945 49526 53126
61705 67569 73480 79431 85419
24982 27060 29139 31217 33296
36839 40151 43502 46889 50310
36290 39541 42835 46169 49541
24982 27060 29139 31218 33296
39729 43319 46929 50557 54201
59309 64846 70414 76007 81624
38405 41904 45429 48978 52548
62557 68420 74320 80254 86216
25296 27381 29464 31547 33630
40599 44266 47958 51673 55413
62404 68123 73870 79642 85436
48151 52908 57758 62693 67706
2000 2100 2200 2300 2400
56744 60376 64021 67683 71324
91439 97488 103562 109660 115779
35375 37453 39532 41610 43689
53762 57243 60752 64285 67841
52951 56397 59876 63387 66928
35375 37454 39534 41614 43695
57859 61530 65212 68904 72606
87259 92911 98577 104257 109947
56137 59742 63361 66995 70640
92203 98212 104240 110284 116344
35713 37796 39878 41962 44045
59175 62961 66769 70600 74453
91250 97081 102929 108792 114669
72790 77941 83153 88421 93741
2500 2600 2700 2800 2900
74985 78673 82369 86074 89786
121917 128073 134246 140433 146636
45768 47846 49925 52004 54082
71419 75017 78633 82267 85918
70498 74096 77720 81369 85043
45777 47860 49945 52033 54124
76316 80034 83759 87491 91229
115648 121357 127075 132799 138530
74296 77963 81639 85323 89015
122417 128501 134596 140701 146814
46130 48216 50303 52391 54481
78328 82224 86141 90079 94036
120559 126462 132376 138302 144238
99108 104520 109973 115464 120990
3000 3500 4000 4500 5000
93504 112185 130989 149895 168890
152852 184109 215622 247354 279283
56161 66554 75947 87340 97733
89584 108119 126939 145991 165246
88740 107555 126874 146660 166876
56218 66769 77532 88614 100111
94973 113768 132671 151662 170730
144267 173020 201859 230756 259692
92715 111306 130027 148850 167763
152935 183636 214453 245348 276299
56574 67079 77675 88386 99222
98013 118165 188705 159572 180749
150184 180057 210145 240427 270893
126549 154768 183552 212764 242313
Converted and usually rounded off from JANAF Thermochemical Tables, NSRDS-NBS-37, 1971 (1141 pp.)
2-200
PHYSICAL AND CHEMICAL DATA
TABLE 2-223 Temperature, K
Ideal Gas Entropies, s°, kJ/kgmol·K, of Combustion Products CO
CO2
H
OH
H2
N
NO
NO2
N2
N2O
O
O2
SO2
H2O
200 240 260 280 298.15
186.0 191.3 193.7 195.3 197.7
200.0 206.0 208.8 211.5 213.8
106.4 110.1 111.8 113.3 114.7
171.6 177.1 179.5 181.8 183.7
119.4 124.5 126.8 129.2 130.7
145.0 148.7 150.4 151.9 153.3
198.7 204.1 206.6 208.8 210.8
225.9 232.2 235.0 237.7 240.0
180.0 185.2 187.6 189.8 191.6
205.6 211.9 214.8 217.5 220.0
152.2 156.2 158.0 159.7 161.1
193.5 198.7 201.1 203.3 205.1
233.0 239.9 242.8 245.8 248.2
175.5 181.4 184.1 186.6 188.8
300 320 340 360 380
197.8 199.7 201.5 203.2 204.7
214.0 216.5 218.8 221.0 223.2
114.8 116.2 117.4 118.6 119.7
183.9 185.9 187.7 189.4 191.0
130.9 132.8 134.5 136.2 137.7
153.4 154.8 156.0 157.2 158.3
210.9 212.9 214.7 216.4 218.0
240.3 242.7 245.0 247.2 249.3
191.8 193.7 195.5 197.2 198.7
220.2 222.7 225.2 227.5 229.7
161.2 162.6 163.9 165.2 166.3
205.3 207.2 209.0 210.7 212.5
248.5 251.1 253.6 256.0 258.2
189.0 191.2 193.3 195.2 197.1
400 420 440 460 480
206.2 207.7 209.0 210.4 211.6
225.3 227.3 229.3 231.2 233.1
120.8 121.8 122.8 123.7 124.6
192.5 194.0 195.3 196.6 197.9
139.2 140.6 141.9 143.2 144.5
159.4 160.4 161.4 162.3 163.1
219.5 221.0 222.3 223.7 225.0
251.3 253.2 255.1 257.0 258.8
200.2 201.5 202.9 204.2 205.5
231.9 234.0 236.0 238.0 239.9
167.4 168.4 169.4 170.4 171.3
213.8 215.3 216.7 218.0 219.4
260.4 262.5 264.6 266.6 268.5
198.8 200.5 202.0 203.6 205.1
500 550 600 650 700
212.8 215.7 218.3 220.8 223.1
234.9 239.2 243.3 247.1 250.8
125.5 127.5 129.3 131.0 132.5
199.1 201.8 204.4 206.8 209.0
145.7 148.6 151.1 153.4 155.6
164.0 166.0 167.8 169.4 171.0
226.3 229.1 231.9 234.4 236.8
260.6 264.7 268.8 272.6 276.0
206.7 209.4 212.2 214.6 216.9
241.8 246.2 250.4 254.3 258.0
172.2 174.2 176.1 177.7 179.3
220.7 223.7 226.5 229.1 231.5
270.5 274.9 279.2 283.1 286.9
206.5 210.5 213.1 215.9 218.7
750 800 850 900 950
225.2 227.3 229.2 231.1 232.8
255.4 257.5 260.6 263.6 266.5
133.9 135.2 136.4 137.7 138.8
211.1 213.0 214.8 216.5 218.1
157.6 159.5 161.4 163.1 164.7
172.5 173.8 175.1 176.3 177.4
239.0 241.1 243.0 245.0 246.8
279.3 282.5 285.5 288.4 291.3
219.0 221.0 223.0 224.8 226.5
261.5 264.8 268.0 271.1 274.0
180.7 182.1 183.4 184.6 185.7
233.7 235.9 237.9 239.9 241.8
290.4 293.8 297.0 300.1 303.0
221.3 223.8 226.2 228.5 230.6
1000 1100 1200 1300 1400
234.5 237.7 240.7 243.4 246.0
269.3 274.5 279.4 283.9 288.2
139.9 141.9 143.7 145.3 146.9
219.7 222.7 225.4 228.0 230.3
166.2 169.1 171.8 174.3 176.6
178.5 180.4 182.2 183.9 185.4
248.4 251.8 254.8 257.6 260.2
293.9 298.9 303.6 307.9 311.9
228.2 231.3 234.2 236.9 239.5
276.8 282.1 287.0 291.5 295.8
186.8 188.8 190.6 192.3 193.8
243.6 246.9 250.0 252.9 255.6
305.8 311.0 315.8 320.3 324.5
232.7 236.7 240.5 244.0 247.4
1500 1600 1700 1800 1900
248.4 250.7 252.9 254.9 256.8
292.2 296.0 299.6 303.0 306.2
148.3 149.6 150.9 152.1 153.2
232.6 234.7 236.8 238.7 240.6
178.8 180.9 182.9 184.8 186.7
186.9 188.2 189.5 190.7 191.8
262.7 265.0 267.2 269.3 271.3
315.7 319.3 322.7 325.9 328.9
241.9 244.1 246.3 248.3 250.2
299.8 303.6 307.2 310.6 313.8
195.3 196.6 197.9 199.1 200.2
258.1 260.4 262.7 264.8 266.8
328.4 332.1 335.6 338.9 342.0
250.6 253.7 256.6 259.5 262.2
2000 2100 2200 2300 2400
258.7 260.5 262.2 263.8 265.4
309.3 312.2 315.1 317.8 320.4
154.3 155.3 156.3 157.2 158.1
242.3 244.0 245.7 247.2 248.7
188.4 190.1 191.7 193.3 194.8
192.9 193.9 194.8 195.8 196.7
273.1 274.9 276.6 278.3 279.8
331.8 334.5 337.2 339.7 342.1
252.1 253.8 255.5 257.1 258.7
316.9 319.8 322.6 325.3 327.9
201.3 202.3 203.2 204.2 205.0
268.7 270.6 272.4 274.1 275.7
345.0 347.9 350.6 353.2 355.7
264.8 267.3 269.7 272.0 274.3
2500 2600 2700 2800 2900
266.9 268.3 269.7 271.0 272.3
322.9 325.3 327.6 329.9 332.1
158.9 159.7 160.5 161.3 162.0
250.2 251.6 253.0 254.3 255.6
196.2 197.7 199.0 200.3 201.6
197.5 198.3 199.1 199.9 200.6
281.4 282.8 284.2 285.6 286.9
344.5 346.7 348.9 350.9 352.9
260.2 261.6 263.0 264.3 265.6
330.4 332.7 335.0 337.3 339.4
205.9 206.7 207.5 208.3 209.0
277.3 278.8 280.3 281.7 283.1
358.1 360.4 362.6 364.8 366.9
276.5 278.6 380.7 282.7 284.6
3000 3500 4000 4500 5000
273.6 279.4 284.4 288.8 292.8
334.2 343.8 352.2 359.7 366.4
162.7 165.9 168.7 171.1 173.3
256.8 262.5 267.6 272.1 276.1
202.9 208.7 213.8 218.5 222.8
201.3 204.6 207.4 210.1 212.5
288.2 294.0 299.0 303.5 307.5
354.9 363.8 371.5 378.3 384.4
266.9 272.6 277.6 282.1 286.0
341.5 350.9 359.2 366.5 373.0
209.7 212.9 215.8 218.3 220.6
284.4 290.7 296.2 301.1 305.5
368.9 378.1 386.1 393.3 399.7
286.5 295.2 302.9 309.8 316.0
Usually rounded off from JANAF Thermochemical Tables, NSRDS-NBS-37, 1971 (1141 pp.). Equilibrium constants can be calculated by combining ∆h°f values from Table 2-221, hT − h298 from Table 2-222, and s° values from the above, using the formula ln kp = −∆G/(RT), where ∆G = ∆h°f + (hT − h298) − T °. s
HEATS OF SOLUTION
2-201
HEATS OF SOLUTION TABLE 2-224
Heats of Solution of Inorganic Compounds in Water
Heat evolved, in kilogram-calories per gram formula weight, on solution in water at 18°C. Computed from data in Bichowsky and Rossini, Thermochemistry of Chemical Substances, Reinhold, New York, 1936.
Substance
Dilution*
Formula
Heat, kg-cal/ g-mole
Aluminum bromide chloride
aq 600 600 aq aq aq aq aq aq aq aq ∞ aq 600 aq ∞ aq ∞ 800 aq aq aq aq aq
AlBr3 AlCl3 AlCl3·6H2O AlF3 AlF3·aH2O AlF3·3aH2O AlI3 Al2(SO4)3 Al2(SO4)3·6H2O Al2(SO4)3·18H2O NH4Br NH4Cl (NH4)2CrO4 (NH4)2Cr2O7 NH4I NH4NO3 NH4BO3·H2O (NH4)2SO4 NH4HSO4 (NH4)2SO3 (NH4)2SO3·H2O SbF3 SbI3 H3AsO4
+85.3 +77.9 +13.2 +31 +19.0 −1.7 +89.0 +126 +56.2 +6.7 −4.45 −3.82 −5.82 −12.9 −3.56 −6.47 −9.0 −2.75 +0.56 −1.2 −4.13 −1.7 −0.8 −0.4
∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ aq aq aq ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ aq aq aq aq aq aq aq aq aq
Ba(BrO3)2·H2O BaBr2 BaBr2·H2O BaBr2·2H2O Ba(ClO3)2 Ba(ClO3)2·H2O BaCl2 BaCl2·H2O BaCl2.2H2O Ba(CN)2 Ba(CN)2·H2O Ba(CN)2·2H2O Ba(IO3)2 Ba(IO3)2·H2O BaI2 BaI2·H2O BaI2·2H2O BaI2·2aH2O BaI2·7H2O Ba(NO3)2 Ba(ClO4)2 Ba(ClO4)2·3H2O BaS BeBr2 BeCl2 BeI2 BeSO4 BeSO4·H2O BeSO4·2H2O BeSO4·4H2O BiI3 H3BO3
−15.9 +5.3 −0.8 −3.87 −6.7 −10.6 +2.4 −2.17 −4.5 +1.5 −2.4 −4.9 −9.1 −11.3 +10.5 +2.7 +0.14 −0.58 −6.61 −10.2 −2.8 −10.5 +7.2 +62.6 +51.1 +72.6 +18.1 +13.5 +7.9 +1.1 +3 −5.4
400 400 400 400 400 400 400 400 400 400 ∞ ∞
CdBr2 CdBr2·4H2O CdCl2 CdCl2·H2O CdCl2·2aH2O Cd(NO3)2·H2O Cd(NO3)2·4H2O CdSO4 CdSO4·H2O CdSO4·2wH2O Ca(C2H3O2)2 Ca(C2H3O2)2·H2O
+0.4 −7.3 +3.1 +0.6 −3.00 +4.17 −5.08 +10.69 +6.05 +2.51 +7.6 +6.5
fluoride iodide sulfate Ammonium bromide chloride chromate dichromate iodide nitrate perborate sulfate sulfate, acid sulfite Antimony fluoride iodide Arsenic acid Barium bromate bromide chlorate chloride cyanide iodate iodide
nitrate perchlorate sulfide Beryllium bromide chloride iodide sulfate
Bismuth iodide Boric acid Cadmium bromide chloride nitrate sulfate Calcium acetate
Substance Calcium—(Cont.) bromide
Dilution*
Formula
Heat, kg-cal/ g-mole
∞ ∞ ∞ ∞ ∞ ∞ ∞ 400 ∞ ∞ ∞ ∞ ∞ ∞ ∞ aq aq ∞ ∞ ∞ aq
+24.86 −0.9 +4.9 +12.3 +12.5 +2.4 −4.11 +0.7 +28.0 +1.8 +4.1 +0.7 −3.2 −4.2 −7.99 −0.6 −1 +5.1 +3.6 −0.18 +18.6 +5.3 +2.0 +5.7 +18.4 −1.25 +18.5 +9.8 −2.9 +18.8 +15.0 −1.4 −3.6 +2.4 +0.5 +10.3 −2.6 −10.7 +15.9 +9.3 +3.65 −2.85 +11.6
Cuprous sulfate
aq
CaBr2 CaBr2·6H2O CaCl2 CaCl2·H2O CaCl2·2H2O CaCl2·4H2O CaCl2·6H2O Ca(CHO2)2 CaI2 CaI2·8H2O Ca(NO3)2 Ca(NO3)2·H2O Ca(NO3)2·2H2O Ca(NO3)2·3H2O Ca(NO3)2·4H2O Ca(H2PO4)2·H2O CaHPO4·2H2O CaSO4 CaSO4·aH2O CaSO4·2H2O CrCl2 CrCl2·3H2O CrCl2·4H2O CrI2 CoBr2 CoBr2·6H2O CoCl2 CoCl2·2H2O CoCl2·6H2O CoI2 CoSO4 CoSO4·6H2O CoSO4·7H2O Cu(C2H3O2)2 Cu(CHO2)2 Cu(NO3)2 Cu(NO3)2·3H2O Cu(NO3)2·6H2O CuSO4 CuSO4·H2O CuSO4·3H2O CuSO4·5H2O Cu2SO4
Ferric chloride
1000 1000 1000 800 aq 400 400 400 aq 400 400 400 400
FeCl3 FeCl3·2aH2O FeCl3·6H2O Fe(NO3)3·9H2O FeBr2 FeCl2 FeCl2·2H2O FeCl2·4H2O FeI2 FeSO4 FeSO4·H2O FeSO4·4H2O FeSO4·7H2O
+31.7 +21.0 +5.6 −9.1 +18.0 +17.9 +8.7 +2.7 +23.3 +14.7 +7.35 +1.4 −4.4
400 400 aq aq aq 400 ∞ ∞ ∞ ∞ ∞
Pb(C2H3O2)2 Pb(C2H3O2)2·3H2O PbBr2 PbCl2 Pb(CHO2)2 Pb(NO3)2 LiBr LiBr·H2O LiBr·2H2O LiBr·3H2O LiCl
+1.4 −5.9 −10.1 −3.4 −6.9 −7.61 +11.54 +5.30 +2.05 −1.59 +8.66
chloride
formate iodide nitrate
phosphate, monodibasic sulfate Chromous chloride iodide Cobaltous bromide chloride iodide sulfate Cupric acetate formate nitrate sulfate
nitrate Ferrous bromide chloride iodide sulfate
Lead acetate bromide chloride formate nitrate Lithium bromide
chloride
aq aq aq 400 400 400 aq 400 400 400 aq aq 200 200 200 800
*The numbers represent moles of water used to dissolve 1 g formula weight of substance; ∞ means “infinite dilution”; and aq means “aqueous solution of unspecified dilution.”
2-202
PHYSICAL AND CHEMICAL DATA
TABLE 2-224
Heats of Solution of Inorganic Compounds in Water (Continued )
Substance Lithium—(Cont.)
fluoride hydroxide iodide
nitrate sulfate Magnesium bromide chloride
iodide nitrate phosphate sulfate
sulfide Manganic nitrate sulfate Manganous acetate bromide chloride formate iodide
sulfate Mercuric acetate bromide chloride nitrate Mercurous nitrate Nickel bromide Nickel chloride
iodide nitrate sulfate
Dilution*
Formula
Heat, kg-cal/ g-mole
∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞
LiCl·H2O LiCl·2H2O LiCl·3H2O LiF LiOH LiOH·fH2O LiOH·H2O LiI LiI·aH2O LiI·H2O LiI·2H2O LiI·3H2O LiNO3 LiNO3·3H2O Li2SO4 Li2SO4·H2O
+4.45 +1.07 −1.98 −0.74 +4.74 +4.39 +9.6 +14.92 +10.08 +6.93 +3.43 −0.17 +0.466 −7.87 +6.71 +3.77
∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ aq ∞ ∞ ∞ ∞ ∞ ∞ aq 400 400 400 aq aq aq aq aq aq 400 400 400 aq aq aq aq aq aq aq 400 400 400 aq aq aq aq aq
MgBr2 MgBr2·H2O MgBr2·6H2O MgCl2 MgCl2·2H2O MgCl2·4H2O MgCl2·6H2O MgI2 Mg(NO3)2·6H2O Mg3(PO4)2 MgSO4 MgSO4·H2O MgSO4·2H2O MgSO4·4H2O MgSO4·6H2O MgSO4·7H2O MgS Mn(NO3)2 Mn(NO3)2·3H2O Mn(NO3)2·6H2O Mn2(SO4)3 Mn(C2H3O2)2 Mn(C2H3O2)2·4H2O MnBr2 MnBr2·H2O MnBr2·4H2O MnCl2 MnCl2·2H2O MnCl2·4H2O Mn(CHO2)2 Mn(CHO2)2·2H2O MnI2 MnI2·H2O MnI2·2H2O MnI2·4H2O MnI2·6H2O MnSO4 MnSO4·H2O MnSO4·7H2O Hg(C2H3O2)2 HgBr2 HgCl2 Hg(NO3)2·aH2O Hg2(NO3)2·2H2O
+43.7 +35.9 +19.8 +36.3 +20.8 +10.5 +3.4 +50.2 −3.7 +10.2 +21.1 +14.0 +11.7 +4.9 +0.55 −3.18 +25.8 +12.9 −3.9 −6.2 +22 +12.2 +1.6 +15 +14.4 +16.1 +16.0 +8.2 +1.5 +4.3 −2.9 +26.2 +24.1 +22.7 +19.9 +21.2 +13.8 +11.9 −1.7 −4.0 −2.4 −3.3 −0.7 −11.5
aq aq 800 800 800 800 aq 200 200 200 200
NiBr2 NiBr2·3H2O NiCl2 NiCl2·2H2O NiCl2·4H2O NiCl2·6H2O NiI2 Ni(NO3)2 Ni(NO3)2·6H2O NiSO4 NiSO4·7H2O
+19.0 +0.2 +19.23 +10.4 +4.2 −1.15 +19.4 +11.8 −7.5 +15.1 −4.2
Substance
Dilution*
Phosphoric acid, orthopyroPotassium acetate aluminum sulfate
∞ 400 aq aq aq ∞ 800 ∞ aq aq ∞ aq ∞
+2.79 −0.1 +25.9 +4.65 +3.55 +48.5 +26.6 −10.1 −5.1 −10.13 −5.13 +6.58 +4.25 −0.43 −10.31 −4.404 −4.9 +55 +42 +33 +7 −9.5 −3.0 −17.8 +3.96 −1.85 −6.05 +0.86 +1.21 +12.91 +4.27 +3.48 +0.86 −6.93 −5.23 −8.633 −4.6 −7.5 −12.94 −10.4 +4.7 −11.0 −10.22 −6.32 −3.10 −11.0 +1.8 +1.37 −6.08 −13.0 −4.5
aq 200 ∞ ∞ 500 500 1800 900 900 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 800 800 800 200 200
AgC2H3O2 AgNO3 NaC2H3O2 NaC2H3O2·3H2O Na3AsO4 Na3AsO4·12H2O NaHCO3 Na2B4O7 Na2B4O7·10H2O NaBr NaBr·2H2O Na2CO3 Na2CO3·H2O Na2CO3·7H2O Na2CO3·10H2O NaClO3 NaCl Na2CrO4 Na2CrO4·4H2O Na2CrO4·10H2O NaCN NaCN·aH2O
−5.4 −4.4 +4.085 −4.665 +15.6 −12.61 −4.1 +10.0 −16.8 −0.58 −4.57 +5.57 +2.19 −10.81 −16.22 −5.37 −1.164 +2.50 −7.52 −16.0 −0.37 −0.92
2000 ∞ ∞ ∞
chlorate chloride chromate chrome sulfate
∞ ∞ 2185 600
cyanide dichromate fluoride
200 1600 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 400
hydroxide
iodate iodide nitrate oxalate perchlorate permanganate phosphate, dihydrogen pyrosulfite sulfate sulfate, acid sulfide sulfite thiocyanate thionate, dithiosulfate Silver acetate nitrate Sodium acetate arsenate bicarbonate borate, tetrabromide carbonate
chlorate chloride chromate cyanide
Heat, kg-cal/ g-mole
H3PO4 H3PO4·aH2O H4P2O7 H4P2O7·1aH2O KC2H3O2 KAl(SO4)2 KAl(SO4)2·3H2O KAl(SO4)2·12H2O KHCO3 KBrO3 KBr K2CO3 K2CO3·aH2O K2CO3·1aH2O KClO3 KCl K2CrO4 KCr(SO4)2 KCr(SO4)2·H2O KCr(SO4)2·2H2O KCr(SO4)2·6H2O KCr(SO4)2·12H2O KCN K2Cr2O7 KF KF·2H2O KF·4H2O KHS KHS·dH2O KOH KOH·eH2O KOH·H2O KOH·7H2O KIO3 KI KNO3 K2C2O4 K2C2O4·H2O KClO4 KMnO4 KH2PO4 K2S2O5 K2S2O5·aH2O K2SO4 KHSO4 K2S K2SO3 K2SO3·H2O KCNS K2S2O6 K2S2O3
400 400 aq aq ∞ 600 600
bicarbonate bromate bromide carbonate
hydrosulfide
Formula
HEATS OF SOLUTION TABLE 2-224
Heats of Solution of Inorganic Compounds in Water (Concluded )
Substance
Dilution*
Formula
200 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 600 ∞ aq ∞ 1600 1600 1600 1600 1600 1600 600 600 800 800 1600 1600 1200 1200 ∞ ∞ 800 800 ∞ ∞ ∞ ∞ ∞ ∞ ∞
NaCN·2H2O NaF NaHS NaHS·2H2O NaOH NaOH·aH2O NaOH·wH2O NaOH·eH2O NaOH·H2O NaI NaI·2H2O NaPO3 NaNO3 NaNO2 NaClO4 Na2HPO4 Na3PO4 Na3PO4·12H2O Na2HPO4·2H2O Na2HPO4·7H2O Na2HPO4·12H2O NaH2PO3 NaH2PO3·2aH2O Na2HPO3 Na2HPO3·5H2O Na4P2O7 Na4P2O7·10H2O Na2H2P2O7 Na2H2P2O7·6H2O Na2SO4 Na2SO4·10H2O NaHSO4 NaHSO4·H2O Na2S Na2S·4aH2O Na2S·5H2O Na2S·9H2O Na2SO3 Na2SO3·7H2O NaCNS
Sodium—(Cont.) fluoride hydrosulfide Sodium hydroxide
iodide metaphosphate nitrate nitrite perchlorate phosphate di triphosphate di diphosphite, monodipyrophosphate disulfate sulfate, acid sulfide
sulfite thiocyanate NOTE:
2-203
Heat, kg-cal/ g-mole −4.41 −0.27 +4.62 −1.49 +10.18 +8.17 +7.08 +6.48 +5.17 +1.57 −3.89 +3.97 −5.05 −3.6 −4.15 +5.21 +13 −15.3 −0.82 −12.04 −23.18 +0.90 −5.29 +9.30 −4.54 +11.9 −11.7 −2.2 −14.0 +0.28 −18.74 +1.74 +0.15 +15.2 +0.09 −6.54 −16.65 +2.8 −11.1 −1.83
Substance Sodium—(Cont.) thionate, diSodium thiosulfate Stannic bromide Stannous bromide iodide Strontium acetate bromide
chloride
iodide
nitrate sulfate Sulfuric acid, pyroZinc acetate bromide chloride iodide nitrate sulfate
To convert kilocalories per gram-mole to British thermal units per pound-mole, multiply by 1.799 × 10−3.
Dilution*
Formula
Heat, kg-cal/ g-mole
aq aq aq aq aq aq aq ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞
Na2S2O6 Na2S2O6·2H2O Na2S2O3 Na2S2O3·5H2O SnBr4 SnBr2 SnI2 Sr(C2H3O2)2 Sr(C2H3O2)2·aH2O SrBr2 SrBr2·H2O SrBr2·2H2O SrBr2·4H2O SrBr2·6H2O SrCl2 SrCl2·H2O SrCl2·2H2O SrCl2·6H2O SrI2 SrI2·H2O SrI2·2H2O SrI2·6H2O Sr(NO3)2 Sr(NO3)2·4H2O SrSO4 H2S2O7
−5.80 −11.86 +2.0 −11.30 +15.5 −1.6 −5.8 +6.2 +5.9 +16.4 +9.25 +6.5 +0.4 −6.1 +11.54 +6.4 +2.95 −7.1 +20.7 +12.65 +10.4 −4.5 −4.8 −12.4 +0.5 −18.08
400 400 400 400 400 aq 400 400 400 400 400 400
Zn(C2H3O2)2 Zn(C2H3O2)2·H2O Zn(C2H3O2)2·2H2O ZnBr2 ZnCl2 ZnI2 Zn(NO3)2·3H2O Zn(NO3)2·6H2O ZnSO4 ZnSO4·H2O ZnSO4·6H2O ZnSO4·7H2O
+9.8 +7.0 +3.9 +15.0 +15.72 +11.6 −5 −6.0 +18.5 +10.0 −0.8 −4.3
2-204
PHYSICAL AND CHEMICAL DATA TABLE 2-225 Heats of Solution of Organic Compounds in Water (at Infinite Dilution and Approximately Room Temperature) Recalculated and rearranged from International Critical Tables, vol. 5, pp. 148–150. (g⋅cal)/(g⋅mol) = Btu/(lb⋅mol) × 1.799.
Solute Acetic acid (solid), C2H4O2 Acetylacetone, C5H8O2 Acetylurea, C3H6N2O2 Aconitic acid, C6H6O6 Ammonium benzoate, C7H9NO2 picrate succinate (n-) Aniline, hydrochloride, C6H8ClN Barium picrate Benzoic acid, C7H6O2 Camphoric acid, C10H16O4 Citric acid, C6H8O7 Dextrin, C12H20O10 Fumaric acid, C4H4O4 Hexamethylenetetramine, C6H12N4 Hydroxybenzamide (m-), C7H7NO2 (m-), (HCl) (o-), C7H7NO2 (p-) Hydroxybenzoic acid (o-), C7H6O3 (p-), C7H6O3 Hydroxybenzyl alcohol (o-), C7H8O2 Inulin, C36H62O31 Isosuccinic acid, C4H6O4 Itaconic acid, C5H6O4 Lactose, C12H22O11·H2O Lead picrate (2H2O) Magnesium picrate (8H2O) Maleic acid, C4H4O4 Malic acid, C4H6O5 Malonic acid, C3H4O4 Mandelic acid, C8H2O3 Mannitol, C6H14O6 Menthol, C10H20O Nicotine dihydrochloride, C10H16Cl2N2 Nitrobenzoic acid (m-), C7H5NO4 (o-), C7H5NO4 (p-), C7H5NO4 Nitrophenol (m-), C6H5NO3 (o-), C6H5NO3 (p-), C6H5NO3
Heat of Solution, G-cal/g-mole Solute* −2,251 −641 −6,812 −4,206 −2,700 −8,700 −3,489 −2,732 −4,708 −6,501 −502 −5,401 268 −5,903 4,780 −4,161 −7,003 −4,340 −5,392 −6,350 −5,781 −3,203 −96 −3,420 −5,922 −3,705 −7,098 −13,193 14,699 −15,894 −4,441 −3,150 −4,493 −3,090 −5,260 0 6,561 −5,593 −5,306 −8,891 −5,210 −6,310 −4,493
Solute Oxalic acid, C2H2O4 (2H2O) Phenol (solid), C6H6O Phthalic acid, C8H6O4 Picric acid, C6H3N3O7 Piperic acid, C12H10O4 Piperonylic acid, C8H6O4 Potassium benzoate citrate tartrate (n-) (0.5 H2O) Pyrogallol, C6H6O3 Pyrotartaric acid Quinone Raffinose, C18H32O16 (5H2O) Resorcinol, C6H6O2 Silver malonate (n-) Sodium citrate (tri-) picrate potassium tartrate (4H2O) succinate (n-) (6H2O) tartrate (n-) (2H2O) Strontium picrate (6H2O) Succinic acid, C4H6O4 Succinimide, C4H5NO2 Sucrose, C12H22O11 Tartaric acid (d-) Thiourea, CH4N2S Urea, CH4N2O acetate formate nitrate oxalate Vanillic acid Vanillin Zinc picrate (8H2O)
Heat of Solution, G-cal/g-mole Solute* −2,290 −8,485 −2,605 −4,871 −7,098 −10,492 −9,106 −1,506 2,820 −5,562 −3,705 −5,019 −3,991 −9,703 −3,960 −9,799 5,270 −6,441 −1,817 −12,342 2,390 −10,994 −1,121 −5,882 7,887 −14,412 −6,405 −4,302 −1,319 −3,451 −5,330 −3,609 −8,795 −7,194 −10,803 −17,806 −5,160 −5,210 −11,496 −15,894
* + denotes heat evolved, and − denotes heat absorbed. All values are positive unless otherwise noted. The data in the International Critical Tables were calculated by E. Anderson.
THERMODYNAMIC PROPERTIES
2-205
THERMODYNAMIC PROPERTIES EXPLANATION OF TABLES The following subsection presents information on the thermodynamic properties of a number of fluids. In some cases transport properties are also included. Notation cp = specific heat e = specific internal energy h = enthalpy k = thermal conductivity p = pressure s = specific entropy t = temperature T = absolute temperature u = specific internal energy µ = viscosity v = specific volume f = subscript denoting saturated liquid g = subscript denoting saturated vapor UNITS CONVERSIONS For this subsection, the following units conversions are applicable: cp, specific heat: To convert kilojoules per kilogram-kelvin to British thermal units per pound–degree Fahrenheit, multiply by 0.23885. e, internal energy: To convert kilojoules per kilogram to British thermal units per pound, multiply by 0.42992. g, gravity acceleration: To convert meters per second squared to feet per second squared, multiply by 3.2808. h, enthalpy: To convert kilojoules per kilogram to British thermal units per pound, multiply by 0.42992. k, thermal conductivity: To convert watts per meter-kelvin to British thermal unit–feet per hour–square foot–degree Fahrenheit, multiply by 0.57779. p, pressure: To convert bars to kilopascals, multiply by 1 × 102; to convert bars to pounds-force per square inch, multiply by 14.504; and to convert millimeters of mercury to pounds-force per square inch, multiply by 0.01934. s, entropy: to convert kilojoules per kilogram-kelvin to British thermal units per pound–degree Rankine, multiply by 0.23885.
t, temperature: °F = 9⁄5 °C + 32. T, absolute temperature: °R = 9⁄ 5 K. u, internal energy: to convert kilojoules per kilogram to British thermal units per pound, multiply by 0.42992. µ, viscosity: to convert pascal-seconds to pound-force–seconds per square foot, multiply by 0.020885; to convert pascal-seconds to cp, multiply by 1000. v, specific volume: to convert cubic meters per kilogram to cubic feet per pound, multiply by 16.018. ρ, density: to convert kilograms per cubic meter to pounds per cubic foot, multiply by 0.062428. ADDITIONAL REFERENCES Bretsznajder, Prediction of Transport and Other Physical Properties of Fluids, Pergamon, New York, 1971. D’Ans and Lax, Handbook for Chemists and Physicists (in German), 3 vols., Springer-Verlag, Berlin. Engineering Data Book, Natural Gas Processors Suppliers Association, Tulsa, Okla. Ganic, Hartnett, and Rohsenow, Handbook of Heat Transfer, 2d ed., McGraw-Hill, New York, 1984. Gray, American Institute of Physics Handbook, 3d ed., McGraw-Hill, New York, 1972. Kay and Laby, Tables of Physical and Chemical Constants, Longman, London, various editions and dates. Landolt-Börnstein Tables, many volumes and dates, Springer-Verlag, Berlin. Lange, Handbook of Chemistry, McGraw-Hill, New York, various editions and dates. Partington, Advanced Treatise on Physical Chemistry, 5 vols., Longman, London, 1950. Raznjevic, Handbook of Thermodynamic Tables and Charts, McGraw-Hill, New York, 1976 and other editions. Reynolds, Thermodynamic Properties in SI, Department of Mechanical Engineering, Stanford University, 1979. Stephan and Lucas, Viscosity of Dense Fluids, Plenum, New York and London, 1979. Selected Values of Properties of Chemical Compounds and Selected Values of the Properties of Hydrocarbons and Related Compounds, Thermodynamics Research Center, Texas A&M University, College Station, looseleaf, intermittent publication. Vargaftik, Tables of the Thermophysical Properties of Gases and Liquids, Wiley, New York, 1975. Vargaftik, Filippov, Tarzimanov, and Totskiy, Thermal Conductivity of Liquids and Gases (in Russian), Standartov, Moscow, 1978. Weast, Handbook of Chemistry and Physics, Chemical Rubber Co., Boca Raton, FL, annually.
2-206
PHYSICAL AND CHEMICAL DATA
TABLE 2-226
Thermophysical Properties of Saturated Acetone
Temperature, K
Pressure, bar
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/kg⋅K
sg, kJ/kg⋅K
c pf, kJ/kg⋅K
µ f, 10−6 Pa⋅s
k f, W/m⋅K
Pr
300 310 320 329.3 b 330
0.318 0.482 0.710 1.013 1.040
0.001 0.001 0.001 0.001 0.001
261 285 309 333 335
1.415 0.942 0.645 0.456 0.448
−67 −46 −22 0 2
466 476 490 506 506
−0.213 −0.144 −0.068 0 0.003
1.561 1.540 1.531 1.537 1.521
2.29 2.29
232 231
0.141 0.141
3.77 3.75
340 350 360 370 380
1.52 2.04 2.74 3.60 4.52
0.001 0.001 0.001 0.001 0.001
359 383 408 435 464
0.311 0.237 0.179 0.138 0.110
25 51 78 103 127
509 529 543 554 566
0.075 0.150
1.514 1.516
2.33 2.38 2.43 2.48 2.53
212 200 187 176 165
0.137 0.132 0.128 0.124 0.119
3.61 3.61 3.55 3.52 3.51
390 400 410 420 430
5.87 7.31 8.94 10.82 13.64
0.001 0.001 0.001 0.001 0.001
495 528 564 604 647
0.0854 0.0684 0.0556 0.0454 0.0356
151 184 207 231 256
577 588 598 608 618
2.59 2.65 2.73 2.82 2.92
153 141 130 119 109
0.115 0.111 0.107 0.103 0.099
3.45 3.37 3.32 3.26 3.21
440 450 460 470 480
16.37 19.42 22.79 27.52 32.52
0.001 0.001 0.001 0.001 0.001
695 748 81 88 98
0.0292 0.0240 0.0199 0.0159 0.0130
281 308 337 365 396
625 632 637 641 638
3.03 3.15 3.29 3.45 3.76
99 90 80 71 64
0.095 0.092 0.088 0.083 0.077
3.16 3.08 2.99 2.95 3.13
490 500 508.2c
37.73 43.08 47.61
0.002 15 0.002 46 0.003 67
vf, m3/kg
0.0091 0.0063 0.0037
b = normal boiling point; c = critical point P, v, h, and s interpolated and converted from Heat Exchanger Design Handbook, vol. 5, Hemisphere, Washington, DC, 1983 and reproduced in Beaton, C. F. and G. F. Hewitt, Physical Property Data for the Design Engineer, Hemisphere, New York, 1989 (394 pp.). Other values compiled by P. E. Liley An enthalpy-pressure diagram to 1000 psia, 250–500 °F appears in J. Chem. Eng. Data 7, 1 (1962): 75–78.
TABLE 2-227
Saturated Acetylene*
Temperature, K
Pressure, bar
162.0 169.3 173.9 180.0 184.3
vcond, m3/kg
vg, m3/kg
hcond, kJ/kg
hg, kJ/kg
scond, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
0.101 0.203 0.304 0.507 0.709
5.081 2.644 1.805 1.116 0.810
158 173 182 194 203
983 994 999 1007 1011
2.967 3.039 3.095 3.161 3.216
8.062 7.889 7.797 7.672 7.596
189.1 192.4t
1.013 1.283
0.5780 0.4617
214 221
1015 1018
3.272 3.312
7.511 7.455
192.4t 200.9 209.4
1.283 2.027 3.040
0.00164 0.00165 0.00169
0.4617 0.3011 0.2074
378 411 445
1018 1027 1035
4.127 4.296 4.461
7.455 7.362 7.280
221.5 230.4 240.7 253.2 263.0
5.066 7.093 10.13 15.20 20.27
0.00174 0.00179 0.00186 0.00195 0.00204
0.1264 0.0907 0.0635 0.0420 0.0309
493 528 565 602 628
1046 1052 1058 1061 1061
4.684 4.837 4.990 5.133 5.231
7.180 7.111 7.037 6.947 6.878
271.6 278.9 284.9 290.4 300.0
25.33 30.40 35.46 40.53 50.66
0.00213 0.00223 0.00232 0.00242 0.00270
0.0240 0.0193 0.0159 0.0133 0.0093
654 680 704 727 778
1060 1057 1051 1041 1017
5.326 5.414 5.494 5.576 5.737
6.822 6.767 6.716 6.658 6.534
307.8 308.7c
60.80 62.47
0.00335 0.00434
0.0061 0.0043
850 908
968 908
5.965 6.158
6.351 6.158
*Values recalculated into SI units from those of Din. Thermodynamic Functions of Gases, vol. 2, Butterworth, London, 1956. Above the solid line the condensed phase is solid; below the line it is liquid. t = triple point; c = critical point.
THERMODYNAMIC PROPERTIES TABLE 2-228 T, K
2-207
Saturated Air*
Pf, bar
Pg, bar
vf, m3/kg
vg, m3/kg
h f, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
5.55 3.73 2.57 1.82 1.313
−159.2 −155.2 −151.4 −147.8 −144.2
59.7 61.7 63.6 65.5 67.4
2.528 2.585 2.641 2.696 2.747
6.255 6.164 6.080 6.002 5.929
cpf, kJ/(kg⋅K)
µ f, 10−4 Pa⋅s
k f, W/(m⋅K)
3.25 2.98 2.75 2.54 2.36
0.180 0.176 0.173 0.169 0.166
60 62 64 66 68
0.123 0.174 0.239
0.071 0.104 0.147
1.040.–3 1.050.–3 1.060.–3 1.070.–3 1.080.–3
70 72 74 76 78
0.323 0.429 0.560 0.721 0.915
0.205 0.280 0.376 0.495 0.644
1.089.–3 1.101.–3 1.113.–3 1.125.–3 1.136.–3
0.968 0.728 0.556 0.431 0.339
−140.6 −137.1 −133.5 −129.9 −126.3
69.2 71.0 72.8 74.5 76.2
2.797 2.847 2.895 2.941 2.988
5.862 5.799 5.740 5.685 5.634
1.817 1.827 1.838 1.849 1.861
2.21 2.07 1.95 1.84 1.74
0.163 0.160 0.156 0.152 0.148
80 82 84 86 88
1.146 1.420 1.741 2.114 2.544
0.825 1.043 1.305 1.614 1.976
1.146.–3 1.160.–3 1.173.–3 1.187.–3 1.201.–3
0.270 0.217 0.177 0.145 0.120
−122.6 −118.8 −115.0 −111.2 −107.4
77.8 79.4 80.9 82.3 83.6
3.034 3.079 3.123 3.167 3.209
5.585 5.540 5.496 5.454 5.414
1.873 1.885 1.898 1.912 1.927
1.65 1.58 1.51 1.44 1.38
0.145 0.142 0.139 0.135 0.132
90 92 94 96 98
3.036 3.596 4.229 4.940 5.736
2.397 2.884 3.441 4.075 4.792
1.216.–3 1.231.–3 1.247.–3 1.265.–3 1.283.–3
0.1002 0.0843 0.0713 0.0607 0.0520
−103.5 −99.5 −95.5 −91.5 −87.5
84.8 85.9 87.0 87.9 88.7
3.251 3.293 3.335 3.376 3.416
5.376 5.340 5.304 5.270 5.236
1.944 1.962 1.982 2.003 2.027
1.32 1.27 1.23 1.18 1.14
0.128 0.125 0.121 0.117 0.114
100 105 110 115 120
6.621 9.265 12.59 16.68 21.61
5.599 8.056 11.22 15.21 20.14
1.302.–3 1.355.–3 1.418.–3 1.495.–3 1.596.–3
0.0447 0.0312 0.0222 0.0159 0.0115
−83.3 −72.8 −61.9 −50.3 −37.5
89.3 90.2 90.1 88.4 84.8
3.456 3.553 3.649 3.747 3.850
5.204 5.124 5.045 4.964 4.877
2.053 2.137 2.264 2.477 2.916
1.10 1.02 0.95 0.87 0.75
0.110 0.102 0.093 0.084 0.076
125 130 132.55c
27.43 34.16
26.14 33.32 37.69
1.757.–3 2.075.–3 3.196.–3
0.0081 0.0054 0.0032
−22.0 0.4 37.4
78.2 66.1 37.4
3.969 4.136 4.410
4.776 4.644 4.410
4.585
0.42
∞
0.067 ∞
*Liquid properties extracted or converted from Vasserman and Rabinovich, Thermophysical Properties of Liquid Air and Its Components, Moscow, 1968, and NBSNSF transl. TT 69-55092, 1970. Copyrighted material. Reproduced by permission. Vapor properties extracted or converted from Vasserman, Kazavchinskii, and Rabinovich, Thermophysical Properties of Air and Its Components, Nauka, Moscow, 1966, and NBS-NSF transl. TT 70-50095, 1971. Copyrighted material. Reproduced by permission. Note that on pages 150–151 of the TT 69-55092 publication certain values of TT 70-50095 were adjusted. As a complete retabulation was not given, the tables here are based upon the two separate publications, as indicated. See also Table 2-235 for the argon-oxygen-nitrogen equilibrium data. c = critical point. The notation 1.040.–3 signifies 1.040 × 10−3.
2-208
PHYSICAL AND CHEMICAL DATA
TABLE 2-229 Pressure, bar 1v h s Cp µ k
Thermophysical Properties of Compressed Air* Temperature, K 80
Mix
90
100
120
140
160
180
200
220
240
260
280
300
0.251 87.9 5.650 1.044 0.064 0.0084
0.281 98.3 5.759 1.032 0.071 0.0093
0.340 118.8 5.946 1.020 0.085 0.0112
0.399 139.1 6.103 1.014 0.097 0.0129
0.457 159.3 6.238 1.010 0.109 0.0147
0.515 179.5 6.357 1.008 0.121 0.0164
0.537 199.7 6.463 1.007 0.133 0.0181
0.631 219.8 6.559 1.006 0.144 0.0198
0.688 239.9 6.647 1.006 0.154 0.0214
0.746 260.0 6.727 1.006 0.165 0.0231
0.803 280.2 6.802 1.006 0.175 0.0247
0.861 300.3 6.871 1.007 0.185 0.0263
5v h s Cp µ k
0.00115 −122.3 3.031 1.868 1.794 0.146
0.00122 −103.3 3.250 1.941 1.163 0.128
0.0509 90.6 5.246 1.212 0.077 0.0103
0.0646 113.6 5.455 1.107 0.087 0.0119
0.0773 135.3 5.623 1.065 0.098 0.0135
0.0895 156.4 5.763 1.045 0.110 0.0151
0.102 177.1 5.885 1.033 0.122 0.0168
0.114 197.7 5.994 1.025 0.134 0.0185
0.125 218.1 6.092 1.020 0.145 0.0201
0.137 238.5 6.180 1.017 0.155 0.0217
0.149 258.8 6.262 1.015 0.165 0.0234
0.160 279.1 6.337 1.013 0.175 0.0250
0.172 299.4 6.406 1.013 0.185 0.0265
10 v h s Cp µ k
0.00115 −122.0 3.028 1.863 1.816 0.146
0.00121 −103.1 3.246 1.932 1.177 0.128
0.00130 −83.2 3.452 2.041 0.838 0.111
0.0298 106.2 5.214 1.270 0.089 0.0126
0.0370 130.2 5.398 1.146 0.101 0.0141
0.0436 152.5 5.548 1.093 0.112 0.0157
0.0499 174.1 5.675 1.065 0.124 0.0173
0.0561 195.2 5.786 1.049 0.135 0.0189
0.0621 216.1 5.885 1.038 0.146 0.0205
0.0681 236.7 5.975 1.031 0.156 0.0221
0.0741 257.3 6.058 1.026 0.166 0.0237
0.0800 277.8 6.134 1.023 0.176 0.0253
0.0859 298.3 6.204 1.201 0.186 0.0268
20 v h s Cp µ k
0.00114 −121.3 3.022 1.853 1.859 0.147
0.00121 −102.5 3.239 1.916 1.205 0.130
0.00129 −82.9 3.442 2.010 0.857 0.112
0.0116 85.2 4.882 2.237 0.098 0.0152
0.0167 118.5 5.140 1.390 0.106 0.0157
0.0206 144.3 5.312 1.215 0.116 0.0169
0.0241 167.7 5.450 1.141 0.127 0.0182
0.0274 190.1 5.568 1.101 0.137 0.0197
0.0306 211.9 5.672 1.076 0.148 0.0212
0.0337 233.2 5.765 1.061 0.158 0.0228
0.0368 254.3 5.849 1.050 0.168 0.0243
0.0398 275.2 5.927 1.042 0.178 0.0258
0.0428 296.0 5.998 1.037 0.187 0.0273
40 v h s Cp µ k
0.00114 −120.0 3.011 1.834 1.943 0.149
0.00120 −101.4 3.225 1.886 1.261 0.132
0.00128 −82.2 3.424 1.958 0.896 0.115
0.00153 −39.8 3.807 2.432 0.516 0.0814
0.0058 83.6 4.745 3.193 0.132 0.0460
0.0090 125.3 5.025 1.610 0.129 0.0201
0.0114 154.3 5.196 1.335 0.135 0.0206
0.0131 179.7 5.330 1.221 0.144 0.0217
0.0148 203.5 5.444 1.159 0.154 0.0229
0.0165 226.3 5.543 1.122 0.163 0.0242
0.0182 248.5 5.632 1.097 0.172 0.0256
0.0198 270.2 5.712 1.081 0.182 0.0270
0.0214 291.7 5.786 1.068 0.191 0.0284
60 v h s Cp µ k
0.00113 −118.6 3.000 1.818 2.028 0.150
0.00119 −100.3 3.211 1.860 1.318 0.134
0.00126 −81.4 3.407 1.915 0.936 0.117
0.00147 −40.8 3.773 2.205 0.559 0.0861
0.00222 22.8 4.260 4.808 0.277 0.0480
0.00505 90.0 4.798 2.338 0.153 0.0360
0.00687 132.6 5.020 1.594 0.149 0.0238
0.00833 163.9 5.174 1.361 0.154 0.0240
0.00963 191.1 5.298 1.249 0.161 0.0248
0.0108 216.1 5.404 1.186 0.169 0.0258
0.0120 240.0 5.497 1.146 0.178 0.0270
0.0131 263.1 5.581 1.119 0.186 0.0283
0.0142 285.6 5.657 1.100 0.195 0.0296
80 v h s Cp µ k
0.00113 −117.2 2.989 1.802 2.12 0.152
0.00119 −99.1 3.198 1.838 1.38 0.134
0.00126 −80.4 3.391 1.881 0.977 0.120
0.00145 −41.3 3.745 2.078 0.597 0.0901
0.00188 9.0 4.138 2.992 0.356 0.0599
0.00327 78.4 4.597 3.029 0.194 0.0420
0.00480 125.3 4.875 1.887 0.167 0.0278
0.00601 158.7 5.051 1.510 0.166 0.0268
0.00706 187.1 5.186 1.342 0.170 0.0269
0.00803 212.9 5.299 1.250 0.177 0.0276
0.00894 237.3 5.396 1.194 0.184 0.0286
0.00981 260.8 5.484 1.156 0.191 0.0296
0.0107 283.7 5.562 1.130 0.200 0.0308
100 v h s Cp µ k
0.00112 −115.8 2.978 1.789 2.21 0.154
0.00118 −97.8 3.186 1.818 1.44 0.137
0.00125 −79.4 3.376 1.852 1.02 0.122
0.00142 −41.3 3.721 1.992 0.631 0.0936
0.00174 3.9 4.076 2.506 0.405 0.0669
0.00252 61.7 4.457 2.874 0.249 0.0500
0.00366 111.8 4.753 2.114 0.193 0.0327
0.00467 148.8 4.949 1.650 0.181 0.0299
0.00556 179.4 5.095 1.431 0.181 0.0293
0.00637 206.7 5.214 1.311 0.185 0.0295
0.00713 232.2 5.315 1.239 0.191 0.0302
0.00785 256.4 5.406 1.191 0.198 0.0311
0.00855 279.9 5.486 1.158 0.205 0.0320
150 v h s Cp µ k
0.00111 −112.2 2.954 1.789 2.44 0.157
0.00116 −94.5 3.157 1.818 1.60 1.142
0.00122 −76.6 3.342 1.852 1.13 0.127
0.00137 −40.1 3.673 1.992 0.709 0.101
0.00158 0.5 3.988 2.506 0.490 0.0785
0.00194 45.2 4.287 2.874 0.349 0.0588
0.00247 89.5 4.548 2.114 0.266 0.0455
0.00309 129.2 4.757 1.650 0.229 0.0389
0.00369 163.2 4.919 1.431 0.215 0.0360
0.00425 193.4 5.051 1.311 0.211 0.0348
0.00478 221.0 5.161 1.239 0.212 0.0346
0.00529 247.0 5.257 1.267 0.215 0.0349
0.00578 271.8 5.343 1.220 0.220 0.0354
200 v h s Cp µ k
0.00110 −108.5 2.930 1.733 2.70 0.161
0.00115 −91.2 3.130 1.747 1.78 0.146
0.00120 −73.6 3.312 1.761 1.25 0.132
0.00133 −38.0 3.634 1.809 0.782 0.107
0.00150 0.2 3.931 1.905 0.561 0.0868
0.00174 40.2 4.198 1.988 0.420 0.0691
0.00206 79.8 4.432 1.953 0.331 0.0559
0.00245 117.6 4.631 1.814 0.279 0.0476
0.00287 152.2 4.796 1.643 0.253 0.0429
0.00328 183.6 4.932 1.501 0.241 0.0405
0.00368 212.5 5.048 1.396 0.236 0.0393
0.00407 239.6 5.149 1.321 0.235 0.0389
0.00446 265.5 5.238 1.266 0.237 0.0389
250 v h s Cp µ k
0.00109 −104.8 2.909 1.712 2.96 0.165
0.00114 −87.6 3.106 1.722 1.97 0.150
0.00119 −70.3 3.285 1.733 1.39 0.137
0.00130 −35.4 3.601 1.767 0.855 0.113
0.00144 1.3 3.886 1.824 0.625 0.0935
0.00162 38.9 4.138 1.854 0.476 0.0769
0.00186 75.8 4.355 1.831 0.385 0.0641
0.00214 111.7 4.544 1.748 0.327 0.0552
0.00244 145.6 4.706 1.635 0.292 0.0495
0.00276 177.1 4.843 1.522 0.272 0.0460
0.00307 206.6 4.961 1.427 0.262 0.0441
0.00338 234.3 5.064 1.353 0.257 0.0430
0.00368 260.8 5.155 1.297 0.256 0.0426
*For sources, units, and remarks, see Table 2-228. v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K); cp = specific heat at constant pressure, kJ/(kg⋅K); µ = viscosity, 10−4 Pa⋅s; and k = thermal conductivity, W/(m⋅K). For specific heat ratio, see Table 2-200; for Prandtl number, see Table 2-369.
THERMODYNAMIC PROPERTIES
2-209
Temperature, K 350
400
450
500
600
800
1000
1200
1400
1600
1800
2000
2500
1.005 350.7 7.026 1.009 0.208 0.0301
1.148 401.2 7.161 1.014 0.230 0.0336
1.292 452.1 7.282 1.021 0.251 0.0371
1.436 503.4 7.389 1.030 0.270 0.0404
1.723 607.5 7.579 1.051 0.306 0.0466
2.297 822.5 7.888 1.099 0.370 0.0577
2.872 1046.8 8.138 1.141 0.424 0.0681
3.446 1278 8.349 1.175 0.473 0.0783
4.020 1515 8.531 1.207 0.527 0.0927
4.594 1764 8.695 1.248 0.584 0.106
5.168 2017 8.844 1.286 0.637 0.120
5.743 2279 8.983 1.337 0.689 0.137
7.200 3011 9.308 1.665 0.818 0.222
0.201 350.0 6.563 1.014 0.208 0.0303
0.230 400.8 6.698 1.017 0.230 0.0338
0.259 451.8 6.818 1.024 0.251 0.0372
0.288 503.2 6.927 1.032 0.270 0.0405
0.345 607.4 7.116 1.053 0.306 0.0467
0.460 822.6 7.426 1.100 0.370 0.0578
0.575 1046.9 7.676 1.142 0.425 0.0681
0.690 1279 7.887 1.175 0.473 0.0783
0.805 1516 8.069 1.208 0.527 0.0927
0.920 1764 8.233 1.248 0.584 0.106
1.034 2017 8.382 1.285 0.637 0.120
1.149 2278 8.520 1.326 0.689 0.136
1.438 2981 8.832 1.516 0.818 0.195
0.101 349.2 6.361 1.019 0.209 0.0305
0.115 400.2 6.497 1.021 0.231 0.0340
0.130 451.4 6.618 1.027 0.252 0.0374
0.144 502.9 6.727 1.034 0.271 0.0407
0.173 607.3 6.917 1.055 0.306 0.0469
0.231 822.7 7.226 1.100 0.370 0.0579
0.288 1047.2 7.477 1.142 0.425 0.0682
0.345 1279 7.688 1.175 0.473 0.0784
0.403 1516 7.870 1.208 0.527 0.0927
0.460 1765 8.034 1.248 0.584 0.106
0.518 2018 8.183 1.284 0.637 0.120
0.575 2279 8.321 1.324 0.689 0.135
0.720 2974 8.630 1.481 0.817 0.187
0.0503 347.7 6.158 1.030 0.210 0.0309
0.0577 399.1 6.295 1.029 0.232 0.0344
0.0650 450.7 6.417 1.033 0.253 0.0377
0.0723 502.4 6.526 1.039 0.272 0.0410
0.0868 607.2 6.716 1.057 0.307 0.0471
0.116 823.0 7.027 1.102 0.371 0.0581
0.145 1047.7 0.277 1.143 0.425 0.0685
0.173 1280 7.489 1.176 0.474 0.0787
0.202 1517 7.671 1.209 0.527 0.0928
0.231 1766 7.835 1.249 0.584 0.106
0.260 2019 7.984 1.284 0.637 0.120
0.288 2279 8.121 1.322 0.689 0.135
0.360 2970 8.428 1.456 0.817 0.181
0.0252 344.6 5.950 1.051 0.213 0.0318
0.0290 397.0 6.090 1.044 0.235 0.0351
0.0327 449.2 6.212 1.044 0.255 0.0384
0.0364 501.5 6.323 1.049 0.274 0.0416
0.0438 606.9 6.515 1.063 0.309 0.0476
0.0583 823.7 6.826 0.105 0.372 0.0584
0.0728 1048.8 7.077 1.145 0.426 0.0687
0.0872 1281 7.289 1.177 0.474 0.0789
0.102 1519 7.473 1.210 0.527 0.0928
0.116 1768 7.636 1.249 0.584 0.106
0.130 2021 7.785 1.284 0.637 0.120
0.145 2281 7.922 1.322 0.689 0.135
0.181 2969 8.229 1.438 0.817 0.177
0.0169 340.4 5.824 1.072 0.217 0.0328
0.0194 394.0 5.967 1.059 0.237 0.0359
0.0220 447.1 6.091 1.055 0.257 0.0391
0.0245 500.6 6.202 1.057 0.275 0.0422
0.0294 606.8 6.396 1.069 0.310 0.0481
0.0392 824.3 6.708 1.108 0.373 0.0588
0.0489 1050.0 6.960 1.147 0.427 0.0690
0.0585 1283 7.172 1.178 0.475 0.0790
0.0681 1521 7.355 1.210 0.527 0.0929
0.0776 1770 7.520 1.249 0.584 0.106
0.0872 2023 7.669 1.286 0.637 0.120
0.0968 2284 7.806 1.322 0.689 0.134
0.1207 2969 8.112 1.430 0.817 0.176
0.0127 339.0 5.733 1.091 0.220 0.0337
0.0147 393.1 5.878 1.073 0.240 0.0368
0.0166 446.5 6.004 1.066 0.259 0.0398
0.0185 499.8 6.116 1.065 0.278 0.0428
0.0223 606.7 6.311 1.075 0.312 0.0486
0.0296 825.1 6.624 1.111 0.374 0.0592
0.0369 1051.1 6.877 1.149 0.428 0.0693
0.0442 1284 7.089 1.180 0.475 0.0793
0.0513 1522 7.273 1.210 0.527 0.0929
0.0585 1772 7.437 1.249 0.584 0.106
0.0657 2025 7.586 1.286 0.637 0.120
0.0729 2285 7.723 1.322 0.689 0.134
0.0908 2971 8.029 1.426 0.817 0.175
0.0102 336.5 5.661 1.110 0.224 0.0347
0.0118 391.3 5.807 1.087 0.243 0.0376
0.0134 445.3 5.935 1.076 0.262 0.0405
0.0149 499.0 6.048 1.073 0.280 0.0434
0.0180 606.6 6.244 1.080 0.314 0.0491
0.0239 825.8 6.559 1.114 0.375 0.0595
0.0298 1052.4 6.812 1.151 0.429 0.0696
0.0356 1286 7.024 1.181 0.477 0.0795
0.0413 1524 7.208 1.211 0.527 0.0930
0.0470 1774 7.373 1.250 0.584 0.106
0.0528 2027 7.522 1.288 0.637 0.120
0.0584 2288 7.659 1.323 0.689 0.134
0.0729 2972 7.964 1.423 0.817 0.175
0.00695 330.9 5.525 1.151 0.235 0.0374
0.00806 387.5 5.677 1.117 0.252 0.0398
0.00914 442.9 5.807 1.099 0.270 0.0424
0.0102 497.5 5.922 1.092 0.286 0.0451
0.0123 606.6 6.121 1.093 0.318 0.0504
0.0163 827.8 6.439 1.121 0.379 0.0605
0.0202 1055.5 6.693 1.155 0.431 0.0703
0.0241 1290 6.906 1.184 0.478 0.0801
0.0279 1529 7.092 1.213 0.527 0.0932
0.0317 1779 7.256 1.252 0.584 0.106
0.0356 2033 7.405 1.290 0.637 0.120
0.0394 2294 7.543 1.325 0.689 0.133
0.0490 2977 7.848 1.418
0.00534 326.5 5.426 1.184 0.248 0.0400
0.00620 384.5 5.581 1.141 0.262 0.0420
0.00702 440.9 5.715 1.119 0.278 0.0423
0.00783 496.6 5.831 1.108 0.293 0.0467
0.00940 607.0 6.033 1.104 0.324 0.0517
0.0125 829.9 6.353 1.128 0.382 0.0614
0.0154 1058.7 6.608 1.160 0.434 0.0711
0.0184 1294 6.822 1.187 0.481 0.0808
0.0212 1533 7.009 1.214 0.528 0.0934
0.0241 1783 7.173 1.254 0.585 0.106
0.0269 2038 7.323 1.292 0.638 0.120
0.0298 2299 7.460 1.326
0.0370 2982 7.765 1.415
0.00440 323.2 5.348 1.208 0.262 0.0429
0.00509 382.3 5.506 1.161 0.273 0.0443
0.00576 439.6 5.641 1.135 0.286 0.0462
0.00642 496.0 5.760 1.121 0.301 0.0484
0.00770 607.6 5.963 1.115 0.329 0.0531
0.0102 832.2 6.286 1.135 0.386 0.0624
0.0126 1062.0 6.542 1.164 0.437 0.0718
0.0149 1298 6.757 1.190 0.483 0.0814
0.0172 1538 6.944 1.216 0.528 0.0937
0.0195 1789 7.108 1.256 0.585 0.106
0.0218 2043 7.258 1.294
0.0241 2304 7.396 1.328
0.0298 2988 7.701 1.414
2-210
PHYSICAL AND CHEMICAL DATA
TABLE 2-229 Pressure, bar
Thermophysical Properties of Compressed Air (Concluded ) Temperature, K 90
100
120
140
160
180
200
220
240
260
280
300
0.00112 −84.0 3.083 1.703 2.18 0.154
0.00117 −67.0 3.260 1.713 1.53 0.141
0.00127 −32.4 3.572 1.740 0.932 0.118
0.00139 3.1 3.849 1.769 0.687 0.0996
0.00155 39.2 4.090 1.777 0.529 0.0836
0.00173 74.5 4.298 1.751 0.433 0.0710
0.00195 109.0 4.480 1.689 0.370 0.0619
0.00219 142.0 4.637 1.607 0.329 0.0555
0.00243 173.2 4.773 1.518 0.303 0.0514
0.00269 202.7 4.891 1.438 0.288 0.0487
0.00294 230.8 4.995 1.370 0.280 0.0471
0.00318 257.7 5.088 1.316 0.276 0.0462
400 v h s Cp µ k
0.00110 −76.6 3.042 1.674 2.63 0.161
0.00114 −59.8 3.216 1.686 1.86 0.149
0.00123 −25.9 3.523 1.704 1.10 0.127
0.00133 8.3 3.788 1.702 0.802 0.110
0.00145 42.4 4.016 1.685 0.631 0.0946
0.00158 75.8 4.214 1.654 0.500 0.0823
0.00173 108.5 4.386 1.607 0.446 0.0729
0.00189 140.1 4.537 1.550 0.397 0.0660
0.00206 170.5 4.669 1.490 0.364 0.0610
0.00224 199.7 4.786 1.431 0.341 0.0574
0.00242 227.8 4.890 1.378 0.325 0.0550
0.00260 254.8 4.983 1.331 0.316 0.0533
500 v h s Cp µ k
0.00109 −69.0 3.005 1.655 3.13 0.167
0.00112 −52.3 3.177 1.670 2.24 0.156
0.00120 −18.7 3.482 1.686 1.31 0.135
0.00128 14.4 3.743 1.667 0.924 0.119
0.00138 47.4 3.966 1.644 0.710 0.104
0.00148 79.8 4.151 1.598 0.0560 0.0916
0.00160 111.4 4.317 1.557 0.512 0.0822
0.00173 142.0 4.463 1.509 0.459 0.0749
0.00186 171.7 4.593 1.461 0.420 0.0694
0.00199 200.5 4.708 1.415 0.391 0.0653
0.00213 228.4 4.811 1.371 0.370 0.0622
0.00227 255.4 4.905 1.331 0.356 0.0599
0.00151 116.0 2.263 1.525
0.00161 146.1 4.406 1.480 0.516 0.0828
0.00172 175.3 4.533 1.438 0.472 0.0769
0.00183 203.6 4.646 1.398 0.439 0.0724
0.00194 231.2 4.749 1.361 0.414 0.0689
0.00205 258.1 4.842 1.327 0.396 0.0662
0.00147 157.4 4.318 1.445
0.00155 185.9 4.442 1.406
0.0964
0.0901
0.00163 213.7 4.553 1.372 0.529 0.0850
0.00171 240.3 4.653 1.342 0.497 0.0809
0.00179 267.3 4.745 1.314 0.473 0.0776
0.00151 226.4 4.482 1.355
0.00157 253.2 4.582 1.327
0.0961
0.0916
0.00163 279.5 4.672 1.303 0.546 0.0878
300 v h s Cp µ k
600 v h s Cp µ k 800 v h s Cp µ k 1000 v h s Cp µ k
80 0.00108 −101.0 2.888 1.694 3.24 0.168
0.0903
THERMODYNAMIC PROPERTIES
2-211
Temperature, K 350
400
450
500
600
800
1000
1200
1400
1600
1800
2000
2500
0.00379 320.9 5.283 1.226 0.276 0.0457
0.00437 380.9 5.443 1.176 0.284 0.0466
0.00493 438.9 5.580 1.148 0.296 0.0481
0.00548 495.9 5.700 1.133 0.308 0.0501
0.00656 608.5 5.906 1.124 0.335 0.0544
0.00864 834.5 6.230 1.140 0.390 0.0634
0.0107 1065.3 6.488 1.168 0.440 0.0726
0.0126 1302 6.703 1.193 0.485 0.0820
0.0145 1542 6.891 1.217 0.529 0.0940
0.0164 1794 7.056 1.257
0.0183 2049 7.206 1.298
0.0202 2310 7.344 1.330
0.0250 2993 7.648 1.413
0.00304 319.1 5.181 1.246 0.307 0.0513
0.00348 380.0 5.344 1.195 0.308 0.0512
0.00390 439.0 5.483 1.166 0.315 0.0521
0.00432 496.8 5.605 1.149 0.325 0.0535
0.00514 611.0 5.813 1.138 0.348 0.0571
0.00673 839.4 6.142 1.151 0.398 0.0653
0.00826 1072.0 6.401 1.176 0.446 0.0740
0.00977 1310 6.618 1.199 0.490 0.0832
0.0111 1552 6.808 1.222
0.0126 1804 6.972 1.258
0.0140 2059 7.123 1.301
0.0155 2321 7.261 1.333
0.0190 3004 7.566 1.412
0.00262 319.9 5.103 1.255 0.338 0.0568
0.00296 381.3 5.267 1.206 0.333 0.0557
0.00330 440.8 5.408 1.176 0.336 0.0560
0.00364 499.1 5.531 1.159 0.343 0.0569
0.00430 614.3 5.741 1.148 0.361 0.0598
0.00558 844.6 6.072 1.159 0.407 0.0672
0.00683 1078.8 6.333 1.183 0.452 0.0755
0.00804 1318 6.550 1.205 0.495 0.0844
0.00911 1561 6.743 1.226
0.0103 1814 6.907 1.265
0.0114 2070 7.058 1.306
0.0126 2332 7.196 1.337
0.0154 3015 7.501 1.412
0.00234 322.6 5.041 1.258 0.370 0.0620
0.00262 384.2 5.205 1.211 0.359 0.0602
0.00290 444.0 5.346 1.182 0.358 0.0598
0.00318 502.6 5.470 1.166 0.361 0.0603
0.00374 618.5 5.681 1.154 0.375 0.0625
0.00481 850.1 6.014 1.166 0.416 0.0691
0.00586 1085.5 6.277 1.189 0.459 0.0770
0.00689 1326 6.495 1.210 0.501 0.0857
0.00776 1570 6.690 1.231
0.00873 1824 6.854 1.267
0.00970 2080 7.005 1.310
0.0107 2343 7.144 1.341
0.0130 3026 7.449 1.412
0.00200 331.6 4.943 1.257 0.432 0.0718
0.00221 393.8 5.108 1.216 0.411 0.0688
0.00242 453.4 5.250 1.188 0.402 0.0673
0.00263 512.3 5.374 1.172 0.399 0.0669
0.00304 625.8 5.586 1.161 0.405 0.0679
0.00385 862.0 5.922 1.175 0.436 0.0730
0.00465 1099.3 6.136 1.198 0.474 0.0800
0.00544 1341 6.407 1.219 0.512 0.0881
0.00608 1588 6.605 1.240
0.00681 1844 6.769 1.275
0.00754 2101 6.921 1.318
0.00826 2365 7.060 1.347
0.0101 3049 7.366 1.412
0.00180 343.4 4.869 1.254 0.494 0.0810
0.00196 405.1 5.034 1.217 0.463 0.0768
0.00213 465.3 5.176 1.192 0.446 0.0744
0.00230 524.4 5.300 1.175 0.438 0.0733
0.00262 641.2 5.513 1.164 0.435 0.0732
0.00328 875.1 5.850 1.179 0.456 0.0768
0.00392 1113.3 6.115 1.204 0.489 0.0830
0.00455 1356 6.337 1.225 0.524 0.0906
0.00507 1606 6.539 1.248
0.00565 1863 6.703 1.283
0.00624 2121 6.856 1.325
0.00681 2386 6.995 1.354
0.00825 3071 7.302 1.413
2-212
PHYSICAL AND CHEMICAL DATA TABLE 2-230
Enthalpy and Psi Functions for Ideal-Gas Air*
T, K
h, kJ/kg
Ψ
T, K
h, kJ/kg
Ψ
T, K
h, kJ/kg
Ψ
200 210 220 230 240
200.0 210.0 220.0 230.1 240.1
−0.473 −0.400 −0.329 −0.262 −0.197
650 660 670 680 690
659.8 670.5 681.1 691.8 702.5
1.339 1.364 1.388 1.412 1.436
1200 1220 1240 1260 1280
1278 1301 1325 1349 1372
2.376 2.406 2.435 2.463 2.491
250 260 270 280 290
250.1 260.1 270.1 280.1 290.2
−0.135 −0.076 −0.018 0.037 0.090
700 710 720 730 740
713.3 724.0 734.8 745.6 756.4
1.459 1.482 1.505 1.528 1.550
1300 1320 1340 1360 1380
1396 1420 1444 1467 1491
2.519 2.547 2.574 2.601 2.627
300 310 320 330 340
300.2 310.3 320.3 330.4 340.4
0.142 0.191 0.240 0.286 0.332
750 760 770 780 790
767.3 778.2 789.1 800.0 811.0
1.572 1.594 1.615 1.637 1.658
1400 1420 1440 1460 1480
1515 1539 1563 1587 1612
2.653 2.679 2.705 2.730 2.755
350 360 370 380 390
350.5 360.6 370.7 380.8 390.9
0.376 0.419 0.461 0.502 0.541
800 810 820 830 840
821.9 832.9 844.0 855.0 866.1
1.679 1.699 1.720 1.740 1.760
1500 1520 1540 1560 1580
1636 1660 1684 1709 1738
2.779 2.803 2.827 2.851 2.875
400 410 420 430 440
401.0 411.2 421.3 431.5 441.7
0.580 0.618 0.655 0.691 0.727
850 860 870 880 890
877.2 888.3 899.4 910.6 921.8
1.780 1.800 1.819 1.838 1.857
1600 1620 1640 1660 1680
1758 1782 1806 1831 1855
2.898 2.921 2.944 2.966 2.988
450 460 470 480 490
451.8 462.1 472.3 482.5 492.8
0.761 0.795 0.829 0.861 0.893
900 910 920 930 940
933.0 944.2 955.4 966.7 978.0
1.876 1.895 1.914 1.932 1.950
1700 1720 1740 1760 1780
1880 1905 1929 1954 1979
3.010 3.032 3.054 3.075 3.096
500 510 520 530 540
503.1 513.4 523.7 534.0 544.4
0.925 0.956 0.986 1.016 1.045
950 960 970 980 990
989.3 1000.6 1011.9 1023.3 1034.7
1.969 1.987 2.004 2.022 2.039
1800 1820 1840 1860 1880
2003 2028 2053 2078 2102
3.117 3.138 3.158 3.178 3.198
550 560 570 580 590
554.8 565.2 575.6 586.1 596.5
1.074 1.102 1.130 1.158 1.185
1000 1020 1040 1060 1080
1046.1 1068.9 1091.9 1114.9 1138.0
2.057 2.091 2.125 2.158 2.190
1900 1920 1940 1960 1980
2127 2152 2177 2202 2227
3.218 3.238 3.258 3.277 3.296
600 610 620 630 640
607.0 617.5 628.1 638.6 649.2
1.211 1.238 1.264 1.289 1.314
1100 1120 1140 1160 1180
1161.1 1184.3 1207.6 1230.9 1254.3
2.223 2.254 2.285 2.316 2.346
2000 2050 2100 2150 2200
2252 2315 2377 2440 2504
3.215 3.362 3.408 3.453 3.496
*Values rounded off from Chappell and Cockshutt, Nat. Res. Counc. Can. Rep. NRC LR 759 (NRC No. 14300), 1974. This source tabulates values of seven thermodynamic functions at 1-K increments from 200 to 2200 K in SI units and at other increments for two other unit systems. An earlier report (NRC LR 381, 1963) gives a more detailed description of an earlier fitting from 200 to 1400 K. In the above table h = specific enthalpy, kJ/kg, and Ψ2 − Ψ1 = log10 (P2 /P1)s for an isentrope. In terms of the Keenan and Kaye function φ, Ψ = (log10 e/R) ⋅ φ.
THERMODYNAMIC PROPERTIES
FIG. 2-5 Temperature-entropy diagram for air. [Landsbaum, Dadds, Stevens, et al., Am. Inst. Chem. Eng. J., 1(3), 303 (1955). Reproduced by permission of the authors and of the editor, American Institute of Chemical Engineers.]
2-213
2-214
PHYSICAL AND CHEMICAL DATA
TABLE 2-231
Air
Other tables include Stewart, R. B., S. G. Penoncello, et al., University of Idaho CATS report, 85-5, 1985 (0.1–700 bar, 85–750 K), and a revision is in process of publication. Tables including reactions with hydrocarbons include Gordon, S., NASA Techn. Paper 1907, 4 vols., 1982. See also Gupta, R. N., K-P. Lee, et al., NASA RP 1232, 1990 (89 pp.) and RP 1260, 1991 (75 pp.). Analytic expressions for high temperatures were given by Matsuzaki, R., Jap. J. Appl. Phys., 21, 7 (1982): 1009–1013 and Japanese National Aerospace Laboratory report NAL TR 671, 1981 (45 pp.). Functions from 1500 to 15000 K were tabulated by Hilsenrath, J. and M. Klein, AEDC-TR-65-58 = AD 612 301, 1965 (333 pp.). Tables from 10000 to 10,000,000 K were authored by Gilmore, F. R., Lockheed rept. 3-27-67-1, vol 1., 1967 (340 pp.), also published as Radiative Properties of Air, IFI/Plenum, New York, 1969 (648 pp.). Saturation and superheat tables and a chart to 7000 psia, 660°R appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity see Thermophysical Properties of Refrigerants, ASHRAE, 1993. AIR, MOIST An ASHRAE publication, Thermodynamic Properties of Dry Air and Water and S. I. Psychrometric Charts, 1983 (360 pp.), extensively reviews moist air properties. Gandiduson, P., Chem. Eng., Oct. 29, 1984 gives on page 118 a nomograph from 50 to 120°F, while equations in SI units were given by Nelson, B., Chem. Eng. Progr. 76, 5 (May 1980): 83–85. Liley, P. E., 2000 Solved Problems in M.E. Thermodynamics, McGraw-Hill, New York, 1989, gives four simple equations with which most calculations can be made. Devres, Y.O., Appl. Energy 48 (1994): 1–18 gives equations with which three known properties can be used to determine four others. Klappert, M. T. and G. F. Schilling, Rand RM-4244-PR = AD 604 856, 1984 (40 pp.) gives tables from 100 to 270 K, while programs from −60 to 2°F are given by Sando, F. A., ASHRAE Trans., 96, 2 (1990): 299–308. Viscosity references include Kestin, J. and J. H. Whitelaw, Int. J. Ht. Mass Transf. 7, 11 (1964): 1245–1255; Studnokov, E. L., Inz.-Fiz. Zhur. 19, 2 (1970): 338–340; Hochramer, D. and F. Munczak, Setzb. Ost. Acad. Wiss II 175, 10 (1966): 540–550. For thermal conductivity see, for instance, Mason, E. A. and L. Monchick, Humidity and Moisture Control in Science and Industry, Reinhold, New York, 1965 (257–272).
TABLE 2-232
Saturated Ammonia*
P, bar
vf, m3/kg
vg, m3/kg
h f, kJ/kg
h g, kJ/kg
s f, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
c pf, kJ/(kg⋅K)
µ f, 10−4 Pa⋅s
k f, W/(m⋅K)
195.5t 200 210 220 230
0.0608 0.0865 0.1775 0.3381 0.6044
1.327.–3 1.372.–3 1.394.–3 1.417.–3 1.442.–3
15.648 11.237 5.729 3.135 1.822
−1110.1 −1088.8 −1044.1 −1000.6 −957.0
380.1 388.5 406.7 424.1 440.7
4.203 4.311 4.529 4.731 4.925
11.827 11.698 11.438 11.207 11.002
4.73 4.61 4.38 4.35 4.38
4.25 4.07 3.69 3.34 3.02
0.715 0.709 0.685 0.661 0.638
240 250 260 270 280
1.0226 1.6496 2.5529 3.8100 5.5077
1.468.–3 1.495.–3 1.524.–3 1.551.–3 1.589.–3
1.115 0.712 0.472 0.324 0.228
−912.9 −868.2 −823.1 −777.3 −730.9
456.2 470.6 483.8 495.6 506.0
5.113 5.294 5.471 5.643 5.811
10.817 10.650 10.498 10.358 10.228
4.43 4.48 4.54 4.60 4.66
2.73 2.45 2.20 1.97 1.76
0.615 0.592 0.569 0.546 0.523
T, K
290 300 310 320 330
7.741 10.61 14.24 18.72 24.20
1.626.–3 1.666.–3 1.710.–3 1.760.–3 1.815.–3
0.165 0.121 0.091 0.069 0.053
−683.8 −636.0 −587.2 −537.5 −486.7
514.7 521.5 526.1 528.2 527.5
5.975 6.135 6.293 6.448 6.602
10.108 9.994 9.885 9.779 9.675
4.73 4.82 4.91 5.02 5.17
1.58 1.41 1.26 1.13 1.02
0.500 0.477 0.454 0.431 0.408
340 350 360 370 380
30.79 38.64 47.90 58.74 71.35
1.878.–3 1.952.–3 2.039.–3 2.148.–3 2.291.–3
0.0410 0.0319 0.0249 0.0194 0.0149
−434.3 −380.0 −323.2 −262.6 −196.5
523.3 515.1 501.8 481.9 452.7
6.755 6.908 7.063 7.222 7.391
9.571 9.465 9.354 9.235 9.100
5.37 5.64 6.04 6.68 7.80
0.92 0.83 0.75 0.69 0.61
0.385 0.361 0.337 0.313 0.286
85.98 103.0 113.0
2.499.–3 2.882.–3 4.255.–3
0.0113 0.0077 0.0043
−120.9 −23.5 142.7
408.1 329.0 142.7
7.578 7.813 8.216
8.935 8.694 8.216
0.50 0.39 0.25
0.254 0.21 ∞
390 400 405.4c
10.3 21. ∞
*P, v, h, and s values condensed from ASHRAE Handbook, 1981: Fundamentals. Copyright 1981 by the American Society of Heating, Refrigerating and Air Conditioning Engineers, Inc., and reproduced by permission of the copyright owner. cp, µ, and k values are interpolated and converted from Thermophysical Properties of Refrigerants, ASHRAE, New York, 1976. t = triple point; c = critical point. The notation 1.327.–3 signifies 1.327 × 10−3. At 195.5 K, the viscosity of the saturated liquid is 4.25 × 10−4 Pa⋅s. Most recent tabulations of ammonia properties are based upon the extensive tabulation to 5000 bar, 750 K of Haar, L. and J. S. Gallagher, J. Phys. Chem. Ref. Data, 7, 3 (1978): 635–792, which does, however, neglect dissociation. For tables to 70,000 psia, 920°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). A chart in fps units corresponding with these tables appears on page 17.34 of the ASHRAE 1989 Fundamentals Handbook. Simmons, A. L., C. E. Miller III, et al., Tables and Charts of Equilibrium Thermodynamic Properties of Ammonia for Temperatures from 500 to 50000 K, NASA SP 3099, 1976 (255 pp.), tabulates ρ, h, s, cp, cv, Z, and so on, from 0.01 to 400 bar and also 18 species of decomposition products. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the ITS 90 scale for saturation temperatures from −77.66 to 132.22 °C. The same diagram reproduced here appears in that source.
THERMODYNAMIC PROPERTIES
FIG. 2-6 Enthalpy–log-pressure diagram for ammonia. 1 MPa = 10 bar. (Copyright 1981 by the American Society of Heating, Refrigerating and Air-Conditioning Engineers and reproduced by permission of the copyright owner.)
2-215
2-216
FIG. 2-7 Enthalpy-concentration diagram for aqueous ammonia. From Thermodynamic and Physical Properties NH3 –H2O, Int. Inst. Refrigeration, Paris, France, 1994 (88 pp.). Reproduced by permission. In order to determine equilibrium compositions, draw a vertical from any liquid composition on any boiling line (the lowest plots) to intersect the appropriate auxiliarly curve (the intermediate curves). A horizontal then drawn from this point to the appropriate dew line (the upper curves) will establish the vapor composition. The Int. Inst. Refrigeration publication also gives extensive P-v-x tables from −50 to 316°C. Other sources include Park, Y. M. and Sonntag, R. E., ASHRAE Trans., 96, 1 (1990): 150–159 (x, h, s, tables, 360 to 640 K); Ibrahim, O. M. and S. A. Klein, ASHRAE Trans., 99, 1 (1993): 1495–1502 (Eqs., 0.2 to 110 bar, 293 to 413 K); Smolen, T. M., D. B. Manley, et al., J. Chem. Eng. Data, 36 (1991): 202–208 (p-x correlation, 0.9 to 450 psia, 293–413 K); Ruiter, J. P., Int. J. Refrig., 13 (1990): 223–236 gives ten subroutines for computer calculations.
TABLE 2-233 T, K
Saturated Argon (R740)* vf, m3/kg
P, bar
10 20 30 40 50
vg, m3/kg
5.646.–4 5.666.–4 5.707.–4 5.763.–4 5.831.–4
hf, kJ/kg
hg, kJ/kg
0.20 2.20 6.12 11.30 17.26
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
0.0266 0.1559 0.3129 0.4610 0.5937
µf, 10−4 Pa⋅s
kf, W/(m⋅K)
0.083 0.306 0.466 0.560 0.627
60 70 80 83.8t 83.8t
0.082 0.406 0.687 0.687
5.912.–4 6.008.–4 6.125.–4 6.178.–4 7.068.–4
2.1800 0.3918 0.2434 0.2434
23.85 31.08 39.07 42.34 71.88
229.08 232.88 235.06 235.06
0.7138 0.8250 0.9316 0.9720 1.333
3.415 3.364 3.280 3.280
0.687 0.752 0.836 0.877 1.050
2.93
0.134
85 87.3 90 95 100
0.790 1.013 1.338 2.137 3.247
7.107.–4 7.174.–4 7.269.–4 7.440.–4 7.628.–4
0.2145 0.1710 0.1327 0.0864 0.0588
73.16 75.61 78.55 84.15 89.85
235.55 236.39 237.37 238.91 240.20
1.348 1.375 1.403 1.462 1.520
3.258 3.216 3.168 3.091 3.023
1.058 1.073 1.091 1.124 1.158
2.81 2.60 2.40 2.08 1.82
0.132 0.128 0.124 0.116 0.109
110 115 120 125 130
6.665 9.107 12.13 15.81 20.23
8.064.–4 8.322.–4 8.618.–4 8.965.–4 9.620.–4
0.0299 0.0221 0.0166 0.0126 0.0096
101.83 108.11 114.62 121.50 128.79
241.66 241.78 241.33 240.30 238.41
1.632 1.685 1.738 1.792 1.846
2.903 2.848 2.794 2.743 2.690
1.229 1.274 1.336 1.427 1.550
1.46 1.32 1.21 1.12 1.01
0.096 0.090 0.084 0.078 0.072
135 140 145 150 150.9
25.49 31.68 38.93 47.39 48.98
9.906.–4 1.061.–3 1.172.–3 1.468.–3 1.867.–3
0.0074 0.0056 0.0041 0.0026 0.0019
136.76 145.58 155.73 174.64 189.94
234.60 230.74 223.09 204.35 189.94
1.902 1.961 2.026 2.133 2.201
2.633 2.570 2.490 2.331 2.201
1.752
0.89 0.75 0.60 0.45 0.28
0.066 0.060 0.054 ∞
*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This source contains values for the compressed state for pressures up to 1000 bar, etc. t = triple point. Above the solid line the condensed phase is solid; below it, it is liquid. The notation 5.646.–4 signifies 5.646 × 10−4. At 83.8 K, the viscosity of the saturated liquid is 2.93 × 10−4 Pa⋅s = 0.000293 Ns/m2. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.). TABLE 2-234
Thermodynamic Properties of Compressed Argon* Pressure, bar
T, K
1
100
200
300
400
500
600
700
800
900
v 100 h s
0.2035 243.4 3.299
7.420.–4 93.6 1.494
7.255.–4 97.9 1.464
7.120.–4 102.5 1.438
7.006.–4 107.2 1.414
6.907.–4 112.0 1.393
6.819.–4 116.8 1.372
6.050.–4 91.1 1.037
6.009.–4 96.1 1.026
5.976.–4 101.0 1.016
5.935.–4 106.0 1.007
1000
v 200 h s
0.4151 296.4 3.667
2.96.–3 250.2 2.538
1.430.–5 217.1 2.276
1.159.–3 209.1 2.173
1.045.–3 207.9 2.112
9.778.–4 209.2 2.068
9.312.–4 211.9 2.033
8.962.–4 215.1 2.004
8.683.–4 218.9 1.979
8.454.–4 223.0 1.957
8.260.–4 227.4 1.936
v 300 h s
0.6241 348.6 3.879
5.96.–3 330.9 2.872
2.976.–3 316.3 2.686
2.071.–3 306.6 2.572
1.666.–3 301.4 2.493
1.443.–3 299.3 2.435
1.304.–3 299.2 2.389
1.207.–3 300.5 2.352
1.136.–3 302.7 2.320
1.081.–3 305.6 2.293
1.037.–3 310.0 2.269
v 400 h s
0.8326 400.7 4.028
8.37.–3 391.3 3.048
4.279.–3 383.6 2.881
2.957.–3 378.4 2.780
2.322.–3 375.2 2.707
1.955.–3 373.8 2.651
1.719.–3 373.8 2.603
1.557.–3 374.8 2.565
1.435.–3 376.6 2.533
1.344.–3 379.2 2.505
1.271.–3 382.0 3.480
v 500 h s
1.0409 452.8 4.145
1.062.–2 447.7 3.174
5.464.–3 444.3 3.018
3.772.–3 442.0 2.924
2.940.–3 440.9 2.854
2.448.–3 440.6 2.801
2.124.–3 441.4 2.755
1.899.–3 422.9 2.718
1.730.–3 444.7 2.685
1.607.–3 447.1 2.658
1.506.–3 449.9 2.633
v 600 h s
1.2489 504.9 4.240
1.280.–2 502.4 3.274
6.589.–3 501.6 3.122
4.539.–3 501.4 3.031
3.525.–3 501.8 2.966
2.922.–3 503.0 2.914
2.522.–3 504.6 2.870
2.238.–3 506.6 2.834
2.023.–3 508.7 2.801
1.866.–3 511.2 2.774
1.736.–3 513.9 2.750
v 700 h s
1.4569 556.9 4.320
1.495.–2 556.5 3.356
7.686.–3 556.9 3.207
5.281.–3 558.0 3.118
4.088.–3 559.8 3.054
3.377.–3 561.8 3.005
2.906.–3 564.2 2.963
2.570.–3 566.9 2.928
2.317.–3 569.6 2.897
2.123.–3 527.5 2.870
1.966.–3 575.3 2.845
v 800 h s
1.6659 609.9 4.389
1.708.–2 609.8 3.427
8.768.–3 611.0 3.279
6.011.–3 612.9 3.191
4.640.–3 615.2 3.129
3.822.–3 618.1 3.081
3.280.–3 621.2 3.039
2.893.–3 624.5 3.005
2.603.–3 627.8 2.975
2.376.–3 631.3 2.948
2.196.–3 634.8 2.924
v 900 h s
1.8739 661.0 4.451
1.920.–2 662.7 3.490
9.841.–3 664.6 3.342
6.732.–3 667.2 3.255
5.183.–3 670.1 3.193
4.259.–3 673.3 3.145
3.646.–3 676.8 3.105
3.209.–3 680.7 3.071
2.881.–3 684.4 3.042
2.626.–3 688.3 3.016
2.423.–3 692.3 2.992
v 1000 h s
2.0819 713.1 4.506
2.131.–2 715.4 3.545
1.091.–2 717.9 3.398
7.448.–3 720.9 3.312
5.723.–3 724.3 3.250
4.692.–3 727.8 3.203
4.008.–3 731.5 3.163
3.520.–3 735.6 3.129
3.156.–3 739.8 3.100
2.872.–3 744.1 3.074
2.645.–3 748.5 3.051
*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K). This source contains an exhaustive tabulation of values. The notation 7.420.–4 signifies 7.420 × 10−4. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.). The 1993 ASHRAE Handbook—Fundamentals (SI ed.) has a thermodynamic chart for pressures from 1 to 2000 bar, temperatures from 90 to 700 K. Saturation and superheat tables and a chart to 50,000 psia, 1220 °R appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity see Thermophysical Properties of Refrigerants, ASHRAE, 1993. Extensive tables for 10 properties from 0.9–100 bar, 86–400 K are given by Jacques, A., Fermi Accelerator Lab., Batavia, IL, rept TM 1517, 1988 (201 pp.). In Hilsenrath, J., C. G. Messina, et al., AEDC-TR-66-248 = AD 644 081, 1966 (121 pp.), thermodynamic properties and chemical composition from 2400 to 35,000 K are tabulated. See also Drellishak, K. S. et al., AEDC-TDR-63-146, 1963; AEDC-TDR-64-12 = AD 427839, 1964. 2-217
2-218
PHYSICAL AND CHEMICAL DATA
TABLE 2-235
Liquid-Vapor Equilibrium Data for the Argon-Nitrogen-Oxygen System*
Liquid mole fraction N2 /N2 + O2
Vapor mole fraction Ar
N2
Ar
O2
Temperature,°R
Relative volatility N2 /Ar
N2 /O2
Ar/O2
Pressure activity coefficient
Enthalpy, Btu/ (lb·mol)
Heat capacity, Btu/(lb·mol·°R)
N2
Ar
O2
Liquid
Vapor
Liquid
Vapor
Pressure, 1 atm 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0.0154 0.0306 0.0456 0.0603 0.0748 0.1031 0.1439 0.2687 0.4796 0.6605 0.8293 0.9136
1.0000 0.9845 0.9694 0.9544 0.9397 0.9253 0.8970 0.8561 0.7313 0.5204 0.3395 0.1707 0.0865
162.4 162.3 162.2 162.1 162.0 161.9 161.7 161.5 160.7 159.4 158.5 157.7 157.5
2.575 2.581 2.586 2.592 2.597 2.602 2.613 2.629 2.682 2.786 2.888 2.991 3.042
4.010 4.007 4.004 4.001 3.998 3.995 3.989 3.979 3.941 3.852 3.746 3.632 3.572
1.557 1.553 1.548 1.544 1.540 1.535 1.526 1.513 1.469 1.382 1.297 1.214 1.174
1.118 1.117 1.115 1.113 1.112 1.110 1.107 1.103 1.091 1.076 1.075 1.087 1.099
1.165 1.161 1.158 1.155 1.151 1.148 1.142 1.132 1.104 1.058 1.026 1.008 1.003
0.999 1.000 1.000 1.000 1.001 1.001 1.002 1.003 1.010 1.034 1.072 1.127 1.162
−1841. −1844. −1847. −1850. −1852. −1855. −1860. −1868. −1893. −1938. −1978. −2015. −2032.
1093. 1087. 1082. 1076. 1071. 1066. 1056. 1041. 997. 924. 862. 807. 779.
13.2 13.1 13.1 13.1 13.0 13.0 12.9 12.9 12.6 11.9 11.3 10.7 10.4
7.406 7.374 7.342 7.311 7.281 7.251 7.192 7.107 6.847 6.406 6.026 5.669 5.491
0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10
0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90
0.3135 0.3095 0.3056 0.3017 0.2978 0.2939 0.2863 0.2752 0.2399 0.1759 0.1169 0.0595 0.0303
0. 0.0119 0.0237 0.0354 0.0470 0.0585 0.0812 0.1145 0.2207 0.4170 0.6036 0.7933 0.8937
0.6865 0.6786 0.6707 0.6630 0.6553 0.6476 0.6325 0.6103 0.5394 0.4072 0.2795 0.1471 0.0762
157.7 157.6 157.6 157.6 157.6 157.5 157.5 157.4 157.2 157.0 156.9 156.9 157.1
2.621 2.626 2.631 2.636 2.641 2.645 2.655 2.669 2.717 2.812 2.906 3.001 3.048
4.111 4.106 4.100 4.095 4.090 4.085 4.074 4.058 4.003 3.887 3.766 3.640 3.576
1.568 1.563 1.558 1.554 1.549 1.544 1.534 1.520 1.473 1.382 1.296 1.213 1.173
1.103 1.102 1.100 1.099 1.098 1.096 1.094 1.090 1.080 1.070 1.072 1.086 1.099
1.168 1.164 1.161 1.157 1.154 1.151 1.144 1.135 1.106 1.061 1.029 1.009 1.004
1.012 1.012 1.012 1.013 1.013 1.013 1.014 1.015 1.022 1.045 1.082 1.134 1.166
−1834. −1837. −1839. −1842. −1844. −1846. −1851. −1858. −1882. −1926. −1969. −2009. −2029.
1060. 1057. 1053. 1049. 1045. 1042. 1034. 1024. 990. 928. 871. 813. 783.
13.2 13.1 13.1 13.1 13.1 13.0 13.0 12.9 12.5 11.9 11.3 10.7 10.4
7.410 7.386 7.361 7.337 7.313 7.289 7.242 7.173 6.951 6.540 6.147 5.746 5.534
0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20
0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90
0.5095 0.5042 0.4990 0.4938 0.4886 0.4834 0.4732 0.4580 0.4083 0.3123 0.2162 0.1144 0.0593
0. 0.0096 0.0192 0.0288 0.0383 0.0477 0.0666 0.0946 0.1866 0.3680 0.5550 0.7602 0.8744
0.4905 0.4861 0.4818 0.4775 0.4731 0.4688 0.4602 0.4474 0.4051 0.3197 0.2288 0.1254 0.0663
154.0 154.0 154.0 154.1 154.1 154.1 154.1 154.2 154.3 154.8 155.4 156.2 156.7
2.641 2.646 2.651 2.655 2.660 2.665 2.674 2.688 2.735 2.829 2.921 3.009 3.052
4.155 4.149 4.143 4.137 4.131 4.125 4.112 4.094 4.032 3.907 3.779 3.647 3.580
1.573 1.568 1.563 1.558 1.553 1.548 1.538 1.523 1.474 1.381 1.294 1.212 1.173
1.085 1.084 1.083 1.082 1.081 1.080 1.078 1.075 1.068 1.062 1.068 1.086 1.099
1.171 1.168 1.164 1.161 1.158 1.154 1.148 1.139 1.110 1.064 1.032 1.011 1.006
1.026 1.026 1.026 1.027 1.027 1.027 1.028 1.030 1.036 1.058 1.093 1.140 1.169
−1814. −1816. −1819. −1821. −1824. −1826. −1831. −1839. −1863. −1911. −1958. −2004. −2026.
1035. 1032. 1029. 1027. 1024. 1021. 1016. 1008. 981. 930. 877. 819. 787.
13.2 13.2 13.1 13.1 13.1 13.0 13.0 12.9 12.6 11.9 11.3 10.7 10.4
7.422 7.402 7.382 7.362 7.342 7.322 7.283 7.224 7.031 6.648 6.252 5.817 5.574
0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40
0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90
0.7333 0.7279 0.7226 0.7172 0.7118 0.7064 0.6956 0.6794 0.6244 0.5075 0.3743 0.2121 0.1141
0. 0.0070 0.0140 0.0210 0.0280 0.0350 0.0491 0.0703 0.1426 0.2977 0.4776 0.7010 0.8382
0.2667 0.2651 0.2635 0.2619 0.2602 0.2586 0.2553 0.2503 0.2331 0.1948 0.1482 0.0870 0.0477
148.7 148.8 148.8 148.9 148.9 149.0 149.1 149.3 149.9 151.2 152.8 154.8 155.9
2.629 2.634 2.640 2.645 2.650 2.656 2.667 2.683 2.737 2.841 2.939 3.025 3.063
4.124 4.119 4.114 4.108 4.103 4.098 4.087 4.072 4.018 3.907 3.788 3.657 3.586
1.569 1.564 1.558 1.553 1.548 1.543 1.533 1.517 1.468 1.375 1.289 1.209 1.171
1.050 1.049 1.049 1.048 1.048 1.047 1.047 1.046 1.044 1.048 1.062 1.084 1.099
1.187 1.183 1.179 1.176 1.172 1.169 1.162 1.152 1.122 1.074 1.039 1.016 1.008
1.065 1.065 1.065 1.066 1.066 1.066 1.066 1.067 1.071 1.088 1.116 1.154 1.177
−1748. −1751. −1754. −1757. −1760. −1763. −1770. −1779. −1810. −1871. −1932. −1991. −2020.
997. 996. 994. 992. 991. 989. 986. 981. 964. 928. 885. 829. 794.
13.3 13.3 13.2 13.2 13.2 13.1 13.1 13.0 12.6 11.9 11.3 10.7 10.4
7.452 7.437 7.422 7.407 7.392 7.377 7.347 7.301 7.145 6.811 6.423 5.944 5.651
0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60
0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90
0.8569 0.8521 0.8472 0.8424 0.8375 0.8326 0.8227 0.8076 0.7557 0.6391 0.4938 0.2961 0.1647
0. 0.0056 0.0111 0.0167 0.0224 0.0280 0.0394 0.0566 0.1163 0.2507 0.4191 0.6500 0.8047
0.1431 0.1424 0.1416 0.1409 0.1402 0.1395 0.1380 0.1357 0.1280 0.1102 0.0871 0.0539 0.0306
144.9 145.0 145.1 145.1 145.2 145.3 145.5 145.7 146.5 148.4 150.6 153.5 155.2
2.575 2.582 2.589 2.595 2.602 2.609 2.622 2.642 2.707 2.833 2.945 3.037 3.071
3.993 3.991 3.988 3.985 3.983 3.980 3.975 3.966 3.937 3.867 3.777 3.662 3.592
1.551 1.546 1.541 1.536 1.531 1.526 1.516 1.501 1.454 1.365 1.283 1.206 1.170
1.024 1.024 1.024 1.024 1.024 1.023 1.023 1.023 1.025 1.036 1.056 1.083 1.098
1.218 1.214 1.210 1.206 1.202 1.198 1.190 1.179 1.145 1.090 1.049 1.021 1.010
1.126 1.125 1.125 1.124 1.124 1.123 1.122 1.121 1.120 1.126 1.143 1.169 1.185
−1663. −1667. −1672. −1676. −1680. −1684. −1692. −1704. −1744. −1824. −1903. −1978. −2014.
970. 969. 968. 967. 966. 965. 963. 960. 948. 923. 888. 837. 801.
13.6 13.5 13.5 13.4 13.4 13.4 13.3 13.2 12.8 12.0 11.3 10.7 10.4
7.483 7.471 7.459 7.446 7.434 7.421 7.396 7.357 7.224 6.926 6.556 6.055 5.723
0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80
0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90
0.9384 0.9340 0.9296 0.9252 0.9207 0.9162 0.9071 0.8934 0.8452 0.7339 0.5869 0.3690 0.2117
0. 0.0047 0.0094 0.0142 0.0189 0.0237 0.0334 0.0481 0.0994 0.2177 0.3740 0.6059 0.7736
0.0616 0.0613 0.0610 0.0607 0.0604 0.0601 0.0595 0.0586 0.0554 0.0483 0.0391 0.0252 0.0147
142.0 142.0 142.1 142.2 142.3 142.4 142.6 142.8 143.8 146.0 148.7 152.2 154.5
2.501 2.509 2.517 2.525 2.533 2.540 2.556 2.580 2.658 2.809 2.943 3.045 3.078
3.811 3.811 3.812 3.812 3.813 3.813 3.814 3.814 3.814 3.797 3.751 3.662 3.595
1.524 1.519 1.515 1.510 1.506 1.501 1.492 1.478 1.435 1.352 1.275 1.202 1.168
1.013 1.013 1.013 1.013 1.013 1.013 1.012 1.013 1.015 1.028 1.051 1.082 1.098
1.273 1.268 1.263 1.257 1.258 1.247 1.238 1.224 1.181 1.112 1.062 1.026 1.013
1.214 1.212 1.210 1.209 1.207 1.205 1.202 1.197 1.185 1.173 1.173 1.184 1.193
−1570. −1575. −1580. −1585. −1590. −1595. −1606. −1621. −1671. −1772. −1870. −1964. −2007.
949. 948. 947. 947. 946. 945. 944. 942. 934. 916. 889. 843. 806.
14.0 13.9 13.9 13.8 13.8 13.7 13.6 13.5 13.0 12.2 11.4 10.7 10.3
7.514 7.503 7.492 7.481 7.470 7.459 7.437 7.403 7.284 7.013 6.662 6.153 5.790
*Calculated values, from Wilson, Silverberg, and Zellner, USAF Aero Propulsion Laboratory. Rep. APL TDR 64-64 (AD 603 151), 1964. Relative volatility = α i − j = (yi /xi)(x j /yj), where x = liquid composition, y = vapor composition. Pressure activity coefficient = yi P/x i p0, where p0 = vapor pressure. These data were confirmed by the analyses of Bender, Cryogenics, 13, 11 (1973); and by Elshayal and Lu, J. Chem. Eng. Data, 16, 31 (1971). See also Armstrong, G. T., J. M. Goldstein, et al., J. Res. N.B.S., 55, 5 (1955): 265–277; Bender, E., Cryogenics, 13, 1 (1973): 11–18; Elshayal, I. M. and B. C-y Lu, J. Chem. Eng. Data, 16, 1 (1971): 31–37; Funada, I., S. Yoshimura, et al., Advan. Cryog. Engng., 7 (1982): 893–901; and Hwang, S-C., Fluid Phase Equila., 37 (1987): 153–167.
THERMODYNAMIC PROPERTIES TABLE 2-235
Liquid-Vapor Equilibrium Data for the Argon-Nitrogen-Oxygen System (Continued )
Liquid mole fraction N2 /N2 + O2
2-219
Vapor mole fraction Ar
N2
Ar
O2
Temperature,°R
Relative volatility N2 /Ar
N2 /O2
Pressure activity coefficient
Ar/O2
Enthalpy, Btu/ (lb·mol)
Heat capacity, Btu/(lb·mol·°R)
N2
Ar
O2
Liquid
Vapor
Liquid
Vapor
Pressure, 1 atm (Cont.) 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90
0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90
0.9709 0.9667 0.9624 0.9581 0.9538 0.9494 0.9407 0.9274 0.8808 0.7723 0.6262 0.4018 0.2339
0. 0.0044 0.0088 0.0133 0.0177 0.0222 0.0312 0.0450 0.0931 0.2048 0.3552 0.5860 0.7589
0.0291 0.0289 0.0288 0.0286 0.0285 0.0284 0.0281 0.0276 0.0261 0.0229 0.0186 0.0122 0.0072
140.6 140.7 140.8 140.9 141.0 141.1 141.3 141.6 142.6 144.9 147.8 151.7 154.2
2.459 2.468 2.476 2.485 2.493 2.502 2.519 2.545 2.629 2.793 2.938 3.048 3.082
3.710 3.712 3.714 3.716 3.718 3.720 3.724 3.729 3.743 3.755 3.733 3.660 3.597
1.509 1.504 1.500 1.496 1.491 1.487 1.478 1.465 1.424 1.344 1.270 1.201 1.167
1.015 1.014 1.014 1.013 1.013 1.013 1.013 1.012 1.014 1.026 1.050 1.081 1.098
1.311 1.305 1.299 1.293 1.287 1.281 1.270 1.254 1.204 1.126 1.069 1.029 1.014
1.271 1.268 1.265 1.263 1.260 1.257 1.252 1.245 1.226 1.200 1.190 1.192 1.197
−1522. −1527. −1533. −1538. −1544. −1550. −1561. −1578. −1633. −1745. −1853. −1956. −2004.
939. 938. 938. 937. 937. 936. 935. 934. 928. 912. 889. 846. 809.
14.2 14.2 14.1 14.1 14.0 14.0 13.8 13.7 13.2 12.3 11.4 10.7 10.3
7.530 7.519 7.509 7.498 7.488 7.477 7.456 7.423 7.310 7.050 6.708 6.197 5.822
0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97
0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90
0.9916 0.9874 0.9832 0.9790 0.9748 0.9705 0.9619 0.9488 0.9032 0.7965 0.6513 0.4236 0.2489
0. 0.0042 0.0085 0.0127 0.0170 0.0213 0.0300 0.0432 0.0893 0.1969 0.3433 0.5728 0.7489
0.0084 0.0084 0.0083 0.0083 0.0083 0.0082 0.0081 0.0080 0.0076 0.0066 0.0054 0.0036 0.0021
139.8 139.9 140.0 140.1 140.2 140.3 140.5 140.8 141.8 144.2 147.2 151.3 153.9
2.429 2.438 2.447 2.456 2.465 2.474 2.492 2.519 2.608 2.780 2.934 3.050 3.084
3.638 3.641 3.644 3.647 3.650 3.653 3.658 3.667 3.691 3.722 3.718 3.658 3.597
1.498 1.494 1.489 1.485 1.481 1.477 1.468 1.456 1.415 1.339 1.267 1.199 1.167
1.018 1.018 1.017 1.017 1.016 1.016 1.015 1.014 1.014 1.025 1.049 1.081 1.098
1.342 1.335 1.329 1.322 1.315 1.309 1.296 1.278 1.224 1.137 1.075 1.031 1.015
1.316 1.313 1.309 1.306 1.302 1.299 1.293 1.284 1.257 1.220 1.202 1.198 1.200
−1488. −1494. −1500. −1505. −1511. −1517. −1529. −1547. −1606. −1725. −1841. −1951. −2002.
933. 932. 932. 931. 931. 930. 929. 928. 923. 910. 889. 847. 810.
14.4 14.4 14.3 14.2 14.2 14.1 14.0 13.9 13.3 12.3 11.4 10.7 10.3
7.541 7.531 7.520 7.510 7.500 7.489 7.469 7.437 7.327 7.073 6.737 6.226 5.844
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90
1.0000 0.9959 0.9917 0.9875 0.9833 0.9791 0.9705 0.9576 0.9122 0.8063 0.6616 0.4327 0.2553
0. 0.0041 0.0083 0.0125 0.0167 0.0209 0.0295 0.0424 0.0878 0.1937 0.3384 0.5674 0.7447
0. 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0. 0. 0.
139.4 139.5 139.6 139.7 139.8 139.9 140.1 140.4 141.5 143.9 147.0 151.1 153.8
2.416 2.425 2.434 2.443 2.452 2.462 2.480 2.507 2.598 2.774 2.932 3.050 3.085
3.607 3.611 3.614 3.617 3.621 3.624 3.630 3.640 3.668 3.708 3.712 3.657 3.598
1.493 1.489 1.485 1.480 1.476 1.472 1.464 1.452 1.412 1.337 1.266 1.199 1.166
1.021 1.020 1.019 1.019 1.018 1.018 1.017 1.016 1.015 1.025 1.048 1.080 1.098
1.357 1.350 1.343 1.336 1.329 1.322 1.309 1.290 1.232 1.142 1.077 1.032 1.016
1.338 1.334 1.330 1.326 1.322 1.318 1.311 1.301 1.271 1.230 1.208 1.201 1.201
−1473. −1479. −1485. −1491. −1497. −1503. −1516. −1534. −1595. −1716. −1836. −1949. −2001.
930. 929. 929. 929. 928. 928. 927. 926. 921. 909. 888. 848. 811.
14.5 14.4 14.4 14.3 14.3 14.2 14.1 13.9 13.4 12.4 11.4 10.7 10.3
7.546 7.535 7.525 7.515 7.505 7.495 7.474 7.443 7.333 7.082 6.749 6.838 5.853
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0.0137 0.0272 0.0405 0.0537 0.0668 0.0924 0.1299 0.2471 0.4548 0.6410 0.8190 0.9084
1.0000 0.9863 0.9728 0.9595 0.9463 0.9332 0.9076 0.8701 0.7529 0.5452 0.3590 0.1811 0.0916
190.6 190.5 190.4 190.3 190.2 190.1 189.9 189.6 188.8 187.5 186.5 185.8 185.5
2.020 2.023 2.026 2.030 2.033 2.037 2.043 2.053 2.086 2.148 2.207 2.264 2.292
2.776 2.775 2.773 2.772 2.770 2.768 2.765 2.759 2.738 2.688 2.627 2.560 2.524
1.375 1.372 1.369 1.366 1.362 1.359 1.353 1.344 1.313 1.251 1.190 1.131 1.101
0.987 0.986 0.985 0.985 0.984 0.983 0.982 0.980 0.974 0.967 0.967 0.976 0.983
1.111 1.109 1.106 1.104 1.102 1.100 1.096 1.089 1.070 1.038 1.016 1.003 1.000
1.000 1.001 1.001 1.002 1.002 1.003 1.004 1.005 1.013 1.034 1.066 1.110 1.137
−1466. −1470. −1473. −1477. −1480. −1484. −1490. −1500. −1533. −1593. −1647. −1698. −1723.
1224. 1218. 1212. 1207. 1201. 1196. 1185. 1169. 1121. 1037. 964. 896. 862.
13.4 13.4 13.4 13.4 13.4 13.3 13.3 13.3 13.1 12.7 12.3 11.9 11.8
8.122 8.094 8.065 8.037 8.010 7.982 7.929 7.850 7.605 7.166 6.771 6.389 6.196
0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10
0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90
0.2393 0.2363 0.2334 0.2305 0.2277 0.2248 0.2192 0.2109 0.1843 0.1353 0.0896 0.0452 0.0229
0. 0.0116 0.0230 0.0344 0.0457 0.0569 0.0792 0.1120 0.2174 0.4150 0.6045 0.7961 0.8957
0.7607 0.7521 0.7436 0.7351 0.7267 0.7183 0.7017 0.6772 0.5983 0.4497 0.3058 0.1588 0.0814
186.2 186.1 186.1 186.0 186.0 186.0 185.9 185.8 185.5 185.2 185.0 185.1 185.2
2.063 2.066 2.069 2.072 2.075 2.078 2.083 2.092 2.119 2.173 2.224 2.273 2.296
2.831 2.828 2.825 2.822 2.820 2.817 2.811 2.802 2.772 2.707 2.638 2.564 2.526
1.372 1.369 1.365 1.362 1.359 1.356 1.349 1.340 1.308 1.246 1.186 1.128 1.100
0.994 0.994 0.993 0.992 0.991 0.990 0.988 0.986 0.979 0.971 0.971 0.978 0.985
1.120 1.117 1.115 1.113 1.110 1.108 1.104 1.097 1.077 1.044 1.020 1.006 1.002
1.020 1.021 1.021 1.021 1.022 1.022 1.023 1.024 1.030 1.049 1.078 1.117 1.141
−1452. −1455. −1459. −1462. −1465. −1469. −1475. −1485. −1517. −1578. −1637. −1693. −1720.
1193. 1188. 1184. 1180. 1175. 1171. 1163. 1150. 1110. 1037. 968. 900. 865.
13.7 13.6 13.6 13.6 13.6 13.5 13.5 13.4 13.2 12.8 12.4 12.0 11.8
8.150 8.126 8.102 8.078 8.054 8.031 7.984 7.915 7.691 7.269 6.860 6.444 6.226
0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20
0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90
0.4170 0.4126 0.4082 0.4038 0.3994 0.3951 0.3864 0.3735 0.3316 0.2506 0.1706 0.0883 0.0452
0. 0.0099 0.0198 0.0297 0.0395 0.0493 0.0688 0.0979 0.1932 0.3808 0.5714 0.7742 0.8834
0.5831 0.5776 0.5721 0.5666 0.5611 0.5557 0.5448 0.5286 0.4752 0.3686 0.2580 0.1376 0.0715
182.4 182.4 182.4 182.4 182.4 182.4 182.4 182.5 182.6 183.0 183.6 184.3 184.8
2.094 2.097 2.100 2.102 2.105 2.107 2.112 2.120 2.145 2.194 2.239 2.281 2.301
2.861 2.857 2.854 2.851 2.847 2.844 2.837 2.827 2.792 2.720 2.645 2.568 2.528
1.366 1.363 1.359 1.356 1.353 1.349 1.343 1.333 1.301 1.240 1.181 1.126 1.099
0.997 0.996 0.995 0.994 0.993 0.992 0.990 0.988 0.982 0.974 0.974 0.980 0.986
1.128 1.125 1.123 1.120 1.118 1.116 1.111 1.104 1.083 1.049 1.024 1.008 1.003
1.041 1.041 1.042 1.042 1.042 1.042 1.043 1.044 1.049 1.065 1.090 1.124 1.145
−1431. −1434. −1438. −1441. −1445. −1448. −1455. −1465. −1498. −1562. −1625. −1687. −1717.
1166. 1162. 1159. 1155. 1152. 1149. 1142. 1132. 1099. 1035. 971. 904. 867.
13.8 13.8 13.8 13.8 13.7 13.7 13.7 13.6 13.3 12.9 12.4 12.0 11.8
8.189 8.167 8.146 8.125 8.104 8.083 8.041 7.978 7.771 7.363 6.944 6.497 6.255
Pressure, 4 atm
2-220
PHYSICAL AND CHEMICAL DATA
TABLE 2-235
Liquid-Vapor Equilibrium Data for the Argon-Nitrogen-Oxygen System (Concluded )
Liquid mole fraction N2 /N2 + O2
Vapor mole fraction Ar
N2
Ar
O2
Temperature,°R
Relative volatility N2 /Ar
N2 /O2
Pressure activity coefficient
Ar/O2
Enthalpy, Btu/ (lb·mol)
Heat capacity, Btu/(lb·mol·°R)
N2
Ar
O2
Liquid
Vapor
Liquid
Vapor
Pressure, 4 atm (Cont.) 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40
0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90
0.6560 0.6506 0.6453 0.6400 0.6347 0.6293 0.6186 0.6025 0.5482 0.4348 0.3104 0.1684 0.0882
0. 0.0077 0.0155 0.0232 0.0310 0.0387 0.0543 0.0778 0.1575 0.3259 0.5141 0.7334 0.8595
0.3441 0.3417 0.3393 0.3369 0.3345 0.3320 0.3271 0.3197 0.2943 0.2393 0.1756 0.0982 0.0523
176.3 176.4 176.4 176.5 176.6 176.6 176.8 177.0 177.7 179.2 180.9 182.9 184.1
2.124 2.127 2.129 2.132 2.135 2.137 2.143 2.151 2.176 2.223 2.264 2.297 2.309
2.859 2.856 2.853 2.850 2.846 2.843 2.837 2.827 2.794 2.725 2.652 2.573 2.531
1.346 1.343 1.340 1.337 1.333 1.330 1.324 1.315 1.284 1.226 1.171 1.120 1.096
0.992 0.991 0.991 0.990 0.990 0.989 0.988 0.986 0.982 0.978 0.979 0.985 0.989
1.146 1.143 1.140 1.138 1.135 1.133 1.128 1.120 1.098 1.061 1.033 1.013 1.006
1.090 1.090 1.090 1.090 1.090 1.090 1.090 1.090 1.091 1.100 1.116 1.139 1.153
−1372. −1376. −1380. −1384. −1387. −1391. −1399. −1410. −1449. −1525. −1600. −1674. −1710.
1121. 1119. 1117. 1115. 1112. 1110. 1106. 1099. 1077. 1029. 975. 910. 872.
14.1 14.0 14.0 14.0 13.9 13.9 13.9 13.8 13.5 13.0 12.5 12.0 11.8
8.277 8.260 8.242 8.224 8.206 8.188 8.153 8.098 7.915 7.527 7.095 6.597 6.312
0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60
0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90
0.8076 0.8023 0.7971 0.7918 0.7865 0.7812 0.7704 0.7542 0.6980 0.5739 0.4261 0.2413 0.1293
0. 0.0064 0.0127 0.0192 0.0256 0.0321 0.0451 0.0649 0.1332 0.2848 0.4667 0.6963 0.8367
0.1924 0.1913 0.1902 0.1891 0.1879 0.1868 0.1845 0.1810 0.1688 0.1413 0.1073 0.0625 0.0340
171.7 171.8 171.9 172.0 172.1 172.2 172.4 172.6 173.7 175.9 178.5 181.6 183.4
2.120 2.123 2.127 2.130 2.133 2.137 2.143 2.153 2.184 2.239 2.283 2.311 2.317
2.798 2.796 2.794 2.792 2.790 2.788 2.784 2.778 2.757 2.708 2.648 2.576 2.533
1.320 1.317 1.314 1.311 1.308 1.305 1.299 1.290 1.262 1.209 1.160 1.115 1.093
0.986 0.986 0.985 0.985 0.985 0.984 0.984 0.983 0.981 0.980 0.984 0.989 0.992
1.173 1.170 1.167 1.164 1.161 1.159 1.153 1.144 1.119 1.076 1.043 1.019 1.009
1.154 1.153 1.152 1.152 1.151 1.150 1.149 1.148 1.143 1.141 1.145 1.155 1.161
−1296. −1301. −1305. −1310. −1315. −1319. −1329. −1343. −1389. −1482. −1573. −1661. −1704.
1086. 1084. 1083. 1082. 1080. 1079. 1076. 1072. 1056. 1021. 976. 916. 876.
14.2 14.2 14.1 14.1 14.1 14.0 14.0 13.9 13.6 13.0 12.5 12.0 11.8
8.369 8.354 8.338 8.322 8.306 8.289 8.257 8.208 8.038 7.666 7.228 6.690 6.367
0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80
0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90
0.9152 0.9102 0.9052 0.9001 0.8950 0.8899 0.8795 0.8638 0.8087 0.6826 0.5231 0.3077 0.1684
0. 0.0055 0.0110 0.0165 0.0221 0.0277 0.0390 0.0562 0.1162 0.2535 0.4274 0.6624 0.8150
0.0848 0.0843 0.0839 0.0834 0.0829 0.0825 0.0815 0.0801 0.0751 0.0638 0.0496 0.0299 0.0166
168.0 168.1 168.2 168.3 168.4 168.5 168.7 169.1 170.3 173.1 176.4 180.3 182.7
2.095 2.099 2.103 2.107 2.112 2.116 2.124 2.136 2.175 2.244 2.295 2.323 2.325
2.699 2.699 2.699 2.698 2.698 2.698 2.698 2.697 2.693 2.674 2.636 2.576 2.535
1.288 1.286 1.283 1.280 1.278 1.275 1.270 1.263 1.238 1.192 1.149 1.109 1.090
0.986 0.986 0.986 0.985 0.985 0.984 0.984 0.983 0.981 0.983 0.988 0.993 0.994
1.216 1.212 1.209 1.205 1.201 1.198 1.190 1.180 1.148 1.095 1.055 1.024 1.012
1.239 1.237 1.235 1.234 1.232 1.230 1.227 1.222 1.208 1.188 1.176 1.171 1.169
−1209. −1215. −1220. −1226. −1232. −1237. −1249. −1266. −1322. −1434. −1543. −1647. −1698.
1057. 1056. 1055. 1054. 1053. 1052. 1050. 1047. 1037. 1012. 976. 920. 880.
14.3 14.2 14.2 14.2 14.1 14.1 14.0 14.0 13.7 13.1 12.5 12.0 11.8
8.462 8.447 8.432 8.417 8.402 8.387 8.356 8.309 8.148 7.786 7.347 6.777 6.420
0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90
0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90
0.9596 0.9547 0.9497 0.9448 0.9397 0.9347 0.9245 0.9090 0.8546 0.7287 0.5659 0.3387 0.1873
0. 0.0051 0.0103 0.0155 0.0208 0.0260 0.0367 0.0529 0.1096 0.2407 0.4102 0.6467 0.8045
0.0404 0.0402 0.0399 0.0397 0.0395 0.0393 0.0388 0.0381 0.0358 0.0305 0.0239 0.0146 0.0082
166.3 166.4 166.5 166.7 166.8 166.9 167.2 167.5 168.8 171.8 175.3 179.7 182.4
2.077 2.081 2.086 2.091 2.095 2.100 2.109 2.122 2.166 2.242 2.299 2.328 2.329
2.641 2.641 2.642 2.643 2.644 2.645 2.646 2.649 2.653 2.651 2.627 2.575 2.536
1.271 1.269 1.267 1.264 1.262 1.260 1.255 1.248 1.225 1.182 1.143 1.106 1.089
0.990 0.990 0.989 0.989 0.988 0.988 0.987 0.986 0.984 0.985 0.990 0.995 0.995
1.245 1.241 1.236 1.232 1.228 1.224 1.215 1.203 1.167 1.107 1.062 1.027 1.013
1.292 1.289 1.287 1.284 1.281 1.279 1.274 1.267 1.246 1.215 1.193 1.179 1.173
−1163. −1169. −1175. −1181. −1188. −1194. −1206. −1225. −1287. −1409. −1527. −1640. −1694.
1044. 1043. 1042. 1042. 1041. 1040. 1039. 1036. 1028. 1007. 975. 922. 882.
14.3 14.3 14.2 14.2 14.2 14.1 14.1 14.0 13.7 13.1 12.5 12.0 11.8
8.509 8.494 8.479 8.464 8.449 8.434 8.404 8.358 8.198 7.840 7.400 6.818 6.446
0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97
0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.10 0.20 0.40 0.60 0.80 0.90
0.9882 0.9834 0.9784 0.9735 0.9685 0.9635 0.9534 0.9380 0.9380 0.8840 0.7584 0.5940 0.3597 0.2003
0. 0.0050 0.0099 0.0149 0.0200 0.0250 0.0353 0.0509 0.0509 0.1056 0.2327 0.3990 0.6361 0.7972
0.0118 0.0117 0.0116 0.0116 0.0115 0.0114 0.0113 0.0111 0.0111 0.0104 0.0089 0.0070 0.0043 0.0024
165.2 165.3 165.5 165.6 165.7 165.8 166.1 166.5 166.5 167.9 170.9 174.6 179.3 182.1
2.062 2.067 2.072 2.077 2.082 2.087 2.097 2.111 2.111 2.158 2.240 2.302 2.332 2.331
2.598 2.599 2.601 2.602 2.604 2.605 2.608 2.612 2.612 2.624 2.633 2.620 2.574 2.537
1.260 1.257 1.255 1.253 1.251 1.248 1.244 1.237 1.237 1.216 1.176 1.138 1.104 1.088
0.995 0.994 0.994 0.993 0.992 0.992 0.991 0.989 0.989 0.987 0.986 0.991 0.996 0.996
1.269 1.264 1.259 1.254 1.250 1.245 1.236 1.222 1.222 1.182 1.116 1.067 1.030 1.014
1.334 1.330 1.327 1.324 1.321 1.317 1.311 1.302 1.302 1.276 1.235 1.206 1.185 1.176
−1130. −1136. −1143. −1149. −1156. −1163. −1176. −1195. −1195. −1261. −1390. −1516. −1635. −1692.
1035. 1035. 1034. 1033. 1033. 1032. 1031. 1029. 1029. 1022. 1003. 974. 923. 883.
14.3 14.3 14.2 14.2 14.2 14.2 14.1 14.0 14.0 13.7 13.1 12.5 12.0 11.8
8.541 8.527 8.512 8.497 8.482 8.467 8.437 8.391 8.391 8.233 7.877 7.436 6.846 6.464
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
0. 0.01 0.02 0.03 0.04 0.05 0.07 0.10 0.20 0.40 0.60 0.80 0.90
1.0000 0.9951 0.9902 0.9853 0.9803 0.9753 0.9653 0.9499 0.8960 0.7706 0.6055 0.3684 0.2058
0. 0.0049 0.0098 0.0147 0.0197 0.0247 0.0347 0.0501 0.1040 0.2294 0.3945 0.6316 0.7942
0. 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0. 0. 0.
164.7 164.9 165.0 165.1 165.3 165.4 165.7 166.1 167.4 170.6 174.4 179.1 182.0
2.057 2.061 2.066 2.071 2.076 2.081 2.091 2.106 2.154 2.239 2.303 2.333 2.332
2.580 2.581 2.583 2.584 2.586 2.588 2.591 2.596 2.610 2.626 2.617 2.574 2.537
1.254 1.252 1.250 1.248 1.246 1.243 1.239 1.233 1.212 1.173 1.136 1.103 1.088
0.997 0.997 0.996 0.995 0.995 0.994 0.993 0.991 0.988 0.987 0.992 0.996 0.996
1.280 1.275 1.270 1.265 1.260 1.255 1.245 1.231 1.189 1.120 1.069 1.030 1.015
1.352 1.349 1.346 1.342 1.338 1.335 1.328 1.319 1.289 1.244 1.211 1.188 1.177
−1115. −1122. −1129. −1136. −1142. −1149. −1163. −1183. −1250. −1382. −1511. −1633. −1691.
1032. 1031. 1031. 1030. 1029. 1029. 1028. 1026. 1019. 1002. 973. 923. 884.
14.3 14.3 14.3 14.2 14.2 14.2 14.1 14.0 13.7 13.1 12.5 12.0 11.8
8.555 8.541 8.526 8.511 8.496 8.481 8.451 8.405 8.247 7.892 7.451 6.858 6.471
THERMODYNAMIC PROPERTIES TABLE 2-236
2-221
Thermodynamic Properties of the International Standard Atmosphere* T, K
P, bar
ρ, kg/m3
g, m/s2
M
a, m/s
µ, Pa⋅s
k, W/(m⋅K)
λ, m
0 1,000 2,000 3,000 4,000
288.15 281.65 275.15 268.66 262.17
1.01325 0.89876 0.79501 0.70121 0.61660
1.2250 1.1117 1.0066 0.90925 0.81935
9.80665 9.8036 9.8005 9.7974 9.7943
28.964 28.964 28.964 28.964 28.964
340.29 336.43 332.53 328.58 324.59
1.79.−5 1.76.−5 1.73.−5 1.69.−5 1.66.−5
2.54.−5 2.49.−5 2.43.−5 2.38.−5 2.33.−5
6.63.−8 7.31.−8 8.07.−8 8.94.−8 9.92.−8
0 1,000 2,999 2,999 3,997
5,000 6,000 7,000 8,000 9,000
255.68 249.19 242.70 236.22 229.73
0.54048 0.47217 0.41105 0.35651 0.30800
0.73643 0.66011 0.59002 0.52579 0.46706
9.7912 9.7882 9.7851 9.7820 9.7789
28.964 28.964 28.964 28.964 28.964
320.55 316.45 312.31 308.11 303.85
1.63.−5 1.59.−5 1.56.−5 1.53.−5 1.49.−5
2.28.−5 2.22.−5 2.17.−5 2.12.−5 2.06.−5
1.10.−7 1.23.−7 1.38.−7 1.55.−7 1.74.−7
4,996 5,994 6,992 7,990 8,987
10,000 15,000 20,000 25,000 30,000
223.25 216.65 216.65 221.55 226.51
0.26499 0.12111 0.05529 0.02549 0.01197
0.41351 0.19476 0.08891 0.04008 0.01841
9.7759 9.7605 9.7452 9.7300 9.7147
28.964 28.964 28.964 28.964 28.964
299.53 295.07 295.07 298.39 301.71
1.46.−5 1.42.−5 1.42.−5 1.45.−5 1.48.−5
2.01.−5 1.95.−5 1.95.−5 1.99.−5 2.04.−5
1.97.−7 4.17.−7 9.14.−7 2.03.−6 4.42.−6
9,984 14,965 19,937 24,902 29,859
40,000 50,000 60,000 70,000 80,000
250.35 270.65 247.02 219.59 198.64
2.87.−3 8.00.−4 2.20.−4 5.22.−5 1.05.−5
4.00.−3 1.03.−3 3.10.−4 8.28.−5 1.85.−5
9.6844 9.6542 9.6241 9.5942 9.5644
28.964 28.964 28.964 28.964 28.964
317.19 329.80 315.07 297.06 282.54
1.60.−5 1.70.−5 1.58.−5 1.44.−5 1.32.−5
2.23.−5 2.40.−5 2.21.−5 1.98.−5 1.80.−5
2.03.−5 7.91.−5 2.62.−4 9.81.−4 4.40.−3
39,750 49,610 59,439 69,238 79,006
90,000 100,000 150,000 200,000 250,000
186.87 195.08 634.39 854.56 941.33
1.84.−6 3.20.−7 4.54.−9 8.47.−10 2.48.−10
3.43.−6 5.60.−7 2.08.−9 2.54.−10 6.07.−11
9.5348 9.5052 9.3597 9.2175 9.0785
28.95 28.40 24.10 21.30 19.19
2.37.−2 0.142 33 240 890
88,744 98,451 146,542 193,899 240,540
300,000 400,000 500,000 600,000 800,000
976.01 995.83 999.24 999.85 999.99
8.77.−11 1.45.−11 3.02.−12 8.21.−13 1.70.−13
1.92.−11 2.80.−12 5.22.−13 1.14.−13 1.14.−14
8.9427 8.6799 8.4286 8.1880 7.7368
17.73 15.98 14.33 11.51 5.54
2600 1.6.+4 7.7.+4 2.8.+5 1.4.+6
286,480 376,320 463,540 548,252 710,574
1,000,000
1000.00
7.51.−14
3.56.−15
7.3218
3.94
3.1.+6
864,071
Z, m
H, m
*Extracted from U.S. Standard Atmosphere, 1976, National Oceanic and Atmospheric Administration, National Aeronautics and Space Administration and the U.S. Air Force, Washington, 1976. Z = geometric altitude, T = temperature, P = pressure, g = acceleration of gravity, M = molecular weight, a = velocity of sound, µ = viscosity, k = thermal conductivity, λ = mean free path, ρ = density, and H = geopotential altitude. The notation 1.79.−5 signifies 1.79 × 10−5. TABLE 2-237 T, K
Saturated Benzene*
P, bar
vf, m3/kg –3
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−4 Pa⋅s
kf, W/(m⋅K)
290 300 310 320 330
0.0860 0.1382 0.2139 0.3206 0.4665
1.133.−10 1.147.−10–3 1.162.−10–3 1.176.−10–3 1.192.−10–3
3.569.−10 2.292.−10 1.525.−10 1.046.−10 7.379.−10–1
371.1 388.3 405.9 423.8 442.1
810.3 820.4 830.8 841.5 852.4
2.172 2.229 2.286 2.344 2.400
3.686 3.670 3.657 3.650 3.643
1.719 1.746 1.774 1.804 1.836
6.75 5.80 5.14 4.52 3.95
0.147 0.144 0.141 0.138 0.135
340 350 360 370 380
0.6615 0.9162 1.2419 1.6517 2.1588
1.207.−10–3 1.224.−10–3 1.241.−10–3 1.259.−10–3 1.277.−10–3
5.332.−10–1 3.938.−10–1 2.965.−10–1 2.233.−10–1 1.767.−10–1
460.8 479.6 498.7 518.1 537.7
863.6 875.0 886.7 898.6 910.6
2.455 2.510 2.564 2.617 2.669
3.641 3.641 3.642 3.646 3.651
1.868 1.890 1.920 1.950 1.989
3.55 3.23 2.99 2.72 2.46
0.132 0.129 0.126 0.123 0.120
390 400 410 420 430
2.7774 3.5228 4.4091 5.4540 6.6739
1.297.−10–3 1.318.−10–3 1.340.−10–3 1.363.−10–3 1.388.−10–3
1.393.−10–1 1.112.−10–1 8.972.−10–2 7.309.−10–2 6.003.−10–2
557.6 577.9 598.6 619.7 641.3
922.9 935.2 947.8 960.4 973.0
2.592 2.644 2.823 2.873 2.924
3.657 3.665 3.674 3.684 3.695
2.030 2.070 2.110 2.160 2.210
2.24 2.05 1.88 1.73 1.60
0.117 0.114 0.111 0.107 0.104
440 450 460 470 480
8.0861 9.7088 11.451 13.660 16.028
1.415.−10–3 1.444.−10–3 1.475.−10–3 1.510.−10–3 1.548.−10–3
4.965.−10–2 4.131.−10–2 3.455.−10–2 2.901.−10–2 2.441.−10–2
663.5 686.3 709.7 733.8 758.6
985.6 998.2 1010.7 1022.9 1034.9
2.974 3.025 3.075 3.126 3.179
3.706 3.718 3.730 3.742 3.753
2.260 2.320 2.380 2.450 2.519
1.48 1.37 1.28 1.10 1.12
0.101 0.098 0.095 0.092 0.089
490 500 510 520 530
18.685 21.651 24.952 28.613 32.669
1.591.−10–3 1.640.−10–3 1.697.−10–3 1.765.−10–3 1.849.−10–3
2.059.−10–2 1.736.−10–2 1.462.−10–2 1.226.−10–2 1.020.−10–2
784.3 810.9 838.5 867.2 897.2
1046.4 1057.3 1067.5 1076.6 1084.3
3.230 3.284 3.336 3.391 3.446
3.765 3.777 3.785 3.794 3.800
2.590 2.670 2.750 2.839 2.941
1.05 0.98 0.91 0.84 0.77
0.086 0.083
540 550 560 562.2
37.161 42.144 47.696 48.979
2.126.−10–3 2.258.−10–3 2.512.−10–3 3.290.−10–3
8.349.−10–3 6.616.−10–3 4.696.−10–3 3.290.−10–3
928.8 963.2 1007.3 1043.0
1089.5 1090.4 1077.6 1043.0
3.504 3.565 3.642 3.706
3.802 3.797 3.769 3.706
0.70 0.65 0.60
*Converted from a tabulation by Counsell, Lawrenson, and Lees, Nat. Phys. Lab. Teddington (U.K.) Rep. Chem. 52, 1976. Another tabulation by Kesselman et al., in Vargaftik (ed.), Tables on the Thermophysical Properties of Liquids and Gases, Hemisphere, Washington and London, 1975, shows some differences. The notation 1.133.−6 signifies 1.133 × 10−6. Other tables are given by Goodwin, R. D., J. Phys. Chem. Ref. Data, 17, 4 (1988): 1541–1636.
2-222
PHYSICAL AND CHEMICAL DATA
TABLE 2-238
Saturated Bromine* vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−4 Pa⋅s
kf, W/(m⋅K)
260 280 300 320 340
0.042 0.124 0.310 0.680 1.330
3.106.−4 3.168.−4 3.232.−4 3.311.−4 3.385.−4
3.195 1.169 0.5002 0.2425 0.1309
−147.2 −138.9 −131.6 −124.2 −112.3
51.8 56.2 60.6 64.8 71.1
0.903 0.933 0.956 0.978 1.004
1.669 1.629 1.597 1.570 1.539
0.486 0.479 0.475 0.473 0.471
13.4 11.5 9.3 7.8 6.7
0.131 0.127 0.122 0.118 0.114
360 380 400 420 440
2.384 4.010 6.390 9.730 14.25
3.464.−4 3.550.−4 3.647.−4 3.752.−4 3.885.−4
0.0767 0.0477 0.0311 0.0211 0.0148
−108.6 −100.6 −93.4 −85.8 −77.7
73.1 76.9 80.6 84.0 87.1
1.026 1.048 1.063 1.084 1.103
1.531 1.515 1.501 1.488 1.477
0.470 0.471 0.475 0.480 0.489
5.7 5.0 4.5 4.0 3.7
0.109 0.104 0.099 0.094 0.089
460 480 500 520 540
20.17 27.75 37.21 48.81 62.80
4.023.−4 4.179.−4 4.378.−4 4.623.−4 4.938.−4
0.0107 0.00786 0.00589 0.00445 0.00337
−69.0 −59.7 −49.3 −37.7 −24.0
89.9 92.2 94.0 95.0 94.8
1.122 1.142 1.161 1.183 1.207
1.467 1.457 1.448 1.438 1.428
0.503 0.527 0.595 0.710 0.860
3.3 3.1 2.8 2.6 2.5
0.084 0.079 0.073 0.066 0.059
79.41 98.90 103.4
5.368.−4 6.250.−4 8.475.−4
0.00251 0.00167 0.00085
−7.1 18.8 64.8
92.5 82.5 64.8
1.237 1.280 1.356
1.414 1.390 1.356
1.063 2.31 ∞
2.3 2.2 2.1
0.050 0.035 ∞
T, K
P, bar
560 580 584.2c
*Reproduced or converted from a tabulation by Seshadri, Viswanath, and Kuloor, Ind. J. Technol., 6 (1970): 191–198. c = critical point.
TABLE 2-239
Saturated Normal Butane (R600)* hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−4 Pa⋅s
kf, W/(m⋅K)
0.00 9.95 29.44 49.10 68.94
494.21 499.96 511.39 523.13 535.16
2.3056 2.3778 2.5121 2.6389 2.7592
5.9702 5.8779 5.7251 5.6016 5.5017
1.946 1.953 1.970 1.985 2.001
15.8 14.4 12.0 9.94 8.26
0.181 0.179 0.175 0.171 0.167
88.97 109.22 129.71 150.45 171.49
547.48 560.07 572.93 586.06 599.42
2.8738 2.9835 3.0887 3.1900 3.2879
5.4211 5.3564 5.3048 5.2643 5.2331
2.018 2.035 2.055 2.077 2.101
6.87 5.71 4.83 4.15 3.61
0.163 0.160 0.156 0.152 0.148
2.31 1.40 0.893 0.592 0.406
192.83 214.50 236.52 258.92 281.72
613.02 626.83 640.82 654.97 669.24
3.3828 3.4749 3.5647 3.6523 3.7380
5.2097 5.1929 5.1818 5.1755 5.1732
2.128 2.158 2.192 2.231 2.274
3.18 2.83 2.55 2.31 2.10
0.144 0.140 0.136 0.132 0.128
1.686.−3 1.718.−3 1.752.−3 1.790.−3 1.830.−3
0.286 0.207 0.1533 0.1156 0.0885
309.94 328.62 352.77 377.46 402.71
683.60 697.99 712.36 726.67 740.84
3.8220 3.9046 3.9860 4.0663 4.1458
5.1744 5.1783 5.1846 5.1928 5.2025
2.323 2.377 2.437 2.503 2.577
1.93 1.77 1.62 1.47 1.34
0.124 0.120 0.116 0.113 0.109
5.9179 7.5354 9.4573 11.72 14.35
1.874.−3 1.923.−3 1.978.−3 2.041.−3 2.114.−3
0.0687 0.0539 0.0427 0.0340 0.0272
428.61 455.25 482.74 511.22 540.88
754.80 768.49 781.79 794.60 806.72
4.2248 4.3035 4.3822 4.4613 4.5412
5.2132 5.2248 5.2367 5.2485 5.2597
2.657 2.746 2.842 2.947 3.062
1.21 1.08 0.97 0.87 0.78
0.105 0.101 0.097 0.093 0.089
380 390 400 410 420
17.40 20.90 24.92 29.54 34.86
2.200.−3 2.307.−3 2.447.−3 2.652.−3 3.048.−3
0.0218 0.0174 0.0138 0.0106 0.0075
571.94 604.76 639.85 678.30 723.89
817.86 827.56 834.95 838.10 830.34
4.6225 4.7058 4.7922 4.8842 4.9903
5.2696 5.2771 5.2800 5.2740 5.2437
3.20 3.34 3.50 3.69 3.84
0.69 0.62 0.55 0.49 0.44
0.085 0.081 0.077 0.074 0.072
425.2c
37.96
4.405.−3
0.0044
783.50
783.50
5.1290
5.1290
∞
P, bar
vf, m3/kg
134.9 140 150 160 170
6.7.−6 1.7.−5 8.7.−5 3.5.−4 1.17.−3
1.360.−3 1.369.−3 1.387.−3 1.405.−3 1.424.−3
180 190 200 210 220
3.37.−3 8.53.−3 1.94.−2 4.05.−2 7.81.−2
1.443.−3 1.463.−3 1.484.−3 1.505.−3 1.528.−3
230 240 250 260 270
0.1411 0.2408 0.3915 0.6100 0.9155
1.551.−3 1.575.−3 1.601.−3 1.628.−3 1.656.−3
280 290 300 310 320
1.3297 1.8765 2.5811 3.4706 4.5731
330 340 350 360 370
T, K t
vg, m3/kg 28630 11635 2470 654 207 76.4 31.8 14.7 7.39 4.00
∞
*Values rounded and reproduced or converted from Goodwin, NBSIR 79-1621, 1979. t = triple point; c = critical point. The notation 6.7.−6 signifies 6.7 × 10−6.
THERMODYNAMIC PROPERTIES TABLE 2-240
2-223
Superheated Normal Butane* Temperature, K
P, bar
150
200
250
300
350
400
450
500
600
700
v 1.013 h s
0.00139 29.6 2.512
0.00148 129.8 3.088
0.00160 236.6 3.564
0.4106 718.9 5.334
0.4847 810.7 5.616
0.5575 913.1 5.889
0.6297 1026.0 6.155
0.7013 1149.0 6.414
0.8440 1423 6.913
0.9861 1730 7.386
v 5h s
0.00139 30.0 2.511
0.00148 130.2 3.088
0.00160 237.0 3.563
0.00175 352.9 3.985
0.0909 798.5 5.363
0.1078 904.3 5.645
0.1238 1019.3 5.916
0.1393 1143.7 6.178
0.1693 1420 6.680
0.1988 1728 7.155
v 10 h s
0.00139 30.6 2.510
0.00148 130.8 3.087
0.00160 237.4 3.562
0.00175 353.3 3.983
0.00198 482.7 4.382
0.0502 891.9 5.524
0.0593 1010.3 5.803
0.0677 1136.8 6.069
0.0835 1415 6.575
0.0987 1725 7.052
v 20 h s
0.00138 31.7 2.509
0.00148 131.8 3.085
0.00160 238.4 3.560
0.00174 354.0 3.980
0.00196 482.6 4.376
0.0205 860.0 5.364
0.0268 990.1 5.670
0.0318 1122.0 5.948
0.0406 1406 6.464
0.0487 1718 6.945
v 30 h s
0.00138 32.8 2.507
0.00148 132.9 3.082
0.00159 239.3 3.557
0.00174 354.7 3.976
0.00195 482.6 4.370
0.00240 637.3 4.783
0.0156 965.5 5.570
0.0198 1105.9 5.866
0.0263 1396 6.394
0.0320 1711 6.880
v 40 h s
0.00138 33.9 2.505
0.00148 134.0 3.080
0.00159 240.3 3.555
0.00173 355.4 3.973
0.00194 482.7 4.365
0.00234 633.6 4.768
0.0097 932.2 5.468
0.0137 1088.1 5.797
0.0192 1387 6.341
0.0237 1705 6.832
v 50 h s
0.00138 35.0 2.503
0.00148 135.0 3.078
0.00159 241.3 3.552
0.00173 356.2 3.970
0.00193 428.8 4.360
0.00229 631.0 4.755
0.00549 877.0 5.329
0.0101 1068.2 5.734
0.0149 1377 6.297
0.0188 1699 6.792
v 60 h s
0.00138 36.2 2.501
0.00148 136.1 3.076
0.00159 242.3 3.550
0.00172 356.9 3.967
0.00192 483.1 4.355
0.00255 629.1 4.745
0.00352 825.1 5.204
0.00764 1046.4 5.673
0.0121 1367 6.258
0.0155 1692 6.759
v 80 h s
0.00138 38.4 2.498
0.00147 138.3 3.072
0.00158 244.2 3.545
0.00172 358.5 3.960
0.00190 483.7 4.346
0.00219 626.5 4.727
0.00286 798.1 5.130
0.00482 1001.5 5.559
0.00868 1347 6.191
0.0114 1680 6.704
v 100 h s
0.00138 40.6 2.495
0.00147 140.4 3.069
0.00158 246.2 3.540
0.00171 360.1 3.954
0.00188 484.5 4.337
0.00214 624.9 4.712
0.00264 787.9 5.095
0.00368 971.3 5.310
0.00669 1329 6.134
0.00901 1668 6.658
v 200 h s
0.00137 51.9 2.478
0.00146 151.3 3.049
0.00156 257.9 3.518
0.00167 368.8 3.927
0.00178 490.3 4.301
0.00200 624.4 4.660
0.00225 773.3 5.010
0.00258 933.7 5.348
0.00349 1270 5.960
0.00460 1623 6.849
v 300 h s
0.00136 63.2 2.462
0.00145 162.2 3.032
0.00154 266.7 3.498
0.00164 378.3 3.903
0.00176 498.0 4.273
0.00191 629.2 4.623
0.00209 773.4 4.962
0.00231 928.4 5.288
0.00284 1255 5.884
0.00345 1603 6.419
v 400 h s
0.00136 74.5 2.447
0.00144 173.3 3.015
0.00152 277.4 3.479
0.00162 388.2 3.882
0.00173 506.8 4.248
0.00185 636.2 4.593
0.00200 778.0 4.927
0.00217 930.2 5.247
0.00255 1253 5.836
0.00298 1600 6.366
v 500 h s
0.00136 85.8 2.432
0.00143 184.4 2.999
0.00151 288.1 3.461
0.00160 398.4 3.863
0.00170 516.3 4.226
0.00181 644.5 4.569
0.00193 784.8 4.898
0.00207 935.3 5.215
0.00240 1256 5.799
0.00272 1599 6.328
*Converted and rounded from tables of Goodwin, NBSIR 79-1621, 1979. Saturation and superheat tables and a diagram to 100 bar, 580 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For material to 10,000 psia, 640°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.
2-224
PHYSICAL AND CHEMICAL DATA
TABLE 2-241
Saturated Carbon Dioxide*
T, K
P, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
216.6 220 225 230 235
5.180 5.996 7.357 8.935 10.75
8.484.−4 8.574.−4 8.710.−4 8.856.−4 9.011.−4
0.0712 0.0624 0.0515 0.0428 0.0357
386.3 392.6 401.8 411.1 420.5
731.5 733.1 735.1 736.7 737.9
2.656 2.684 2.723 2.763 2.802
4.250 4.232 4.204 4.178 4.152
1.707 1.761
240 245 250 255 260
12.83 15.19 17.86 20.85 24.19
9.178.−4 9.358.−4 9.554.−4 9.768.−4 1.000.−3
0.0300 0.0253 0.0214 0.0182 0.0155
430.2 440.1 450.3 460.8 471.6
738.9 739.4 739.6 739.4 738.7
2.842 2.882 2.923 2.964 3.005
4.128 4.103 4.079 4.056 4.032
270 275 280 290 300
32.03 36.59 41.60 53.15 67.10
1.056.−3 1.091.−3 1.130.−3 1.241.−3 1.470.−3
0.0113 0.0097 0.0082 0.0058 0.0037
494.4 506.5 519.2 547.6 585.4
735.6 732.8 729.1 716.9 690.2
3.089 3.132 3.176 3.271 3.393
3.981 3.954 3.925 3.854 3.742
304.2c
73.83
2.145.−3
0.0021
636.6
636.6
3.558
3.558
1.879
µf, 10−4 Pa⋅s
1.64
kf, W/(m⋅K) 0.182 0.178 0.171 0.164 0.160
1.933
1.45
1.992
1.28
0.156 0.148 0.140 0.134 0.128
2.125
1.14
2.410
1.02
2.887 3.724
0.91 0.79 0.60
0.116 0.109 0.102 0.088 0.074
∞
0.31
∞
*c = critical point. The notation 8.484.−4 signifies 8.484 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the LPTS 68 scale for saturation temperatures from −56.57 to 30.98 degrees Celsius. The thermodynamic diagram from 4 to 1000 bar extends to 420°C. Saturation and superheat tables and a chart to 15,000 psia, 840°F appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see ASHRAE Thermophysical Properties of Refrigerants, 1993. Saturation and superheat tables and a diagram to 200 bar, 1000 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Holste, J. C., D. M. Bailey, et al., Energy Progr., 6, 2 (1986): 125–130, give properties mainly in the range 0–100 bar, 200–450 K for the superheated vapor. Compare these with Angus, S., B. Armstrong, et al., International Tables of the Fluid State—Carbon Dioxide, Pergamon, Oxford, 1976 (377 pp.). In Miller, C. E. III and S. E. Wilder, NASA SP 3097, 1976 (489 pp.), many properties and decomposition products are tabulated for pressures from 10−7 to 104 atm., 100–25,000 K. For the range to 50 kb, 400–2100 K, see Bottinga, Y. and P. Richet, Amer. J. Sci., 281 (1981): 615–660.
THERMODYNAMIC PROPERTIES TABLE 2-242
2-225
Superheated Carbon Dioxide* Temperature, K
P, bar
300
350
400
450
500
600
700
800
900
1000
v 1h s
0.5639 809.3 4.860
0.6595 853.1 4.996
0.7543 899.1 5.118
0.8494 947.1 5.231
0.9439 997.0 5.337
1.1333 1102 5.527
1.3324 1212 5.697
1.5115 1327 5.850
1.7005 1445 5.990
1.8894 1567 6.120
v 5h s
0.1106 805.5 4.548
0.1304 850.3 4.686
0.1498 897.0 4.810
0.1691 945.5 4.925
0.1882 995.8 50.31
0.2264 1101 5.222
0.2645 1211 5.392
0.3024 1326 5.546
0.3403 1445 5.685
0.3782 1567 5.814
v 10 h s
0.0539 800.7 4.405
0.0642 846.9 4.548
0.0742 894.4 4.674
0.0841 943.5 4.790
0.0938 994.1 4.897
0.1131 1100 5.089
0.1322 1211 5.260
0.1513 1326 5.414
0.1703 1445 5.555
0.1893 1567 5.683
v 20 h s
0.0255 790.2 4.249
0.0311 839.8 4.402
0.0364 889.3 4.534
0.0416 939.4 4.653
0.0466 990.8 4.762
0.0564 1098 4.955
0.0661 1209 5.127
0.0757 1325 5.282
0.0853 1444 5.423
0.0948 1567 5.551
v 30 h s
0.0159 778.5 4.144
0.0201 832.4 4.341
0.0238 883.8 4.447
0.0274 935.2 4.569
0.0309 987.3 4.679
0.0375 1096 4.876
0.0441 1208 5.049
0.0505 1324 5.204
0.0570 1444 5.346
0.0633 1566 5.474
v 40 h s
0.0110 764.9 4.055
0.0146 824.6 4.239
0.0175 878.3 4.380
0.0203 931.1 4.507
0.0230 984.3 4.619
0.0281 1094 4.818
0.0331 1205 4.993
0.0379 1323 5.148
0.0428 1443 5.291
0.0476 1566 5.419
v 50 h s
0.0080 748.2 3.968
0.0112 816.3 4.179
0.0138 872.6 4.330
0.0161 926.9 4.457
0.0183 981.1 4.572
0.0224 1091 4.773
0.0265 1205 4.948
0.0304 1322 5.104
0.0343 1443 5.247
0.0382 1566 5.377
v 60 h s
0.0058 726.9 3.878
0.0090 807.7 4.126
0.0113 866.9 4.314
0.0133 922.7 4.416
0.0151 977.8 4.532
0.0187 1089 4.736
0.0221 1204 4.912
0.0254 1321 5.069
0.0286 1442 5.212
0.0318 1565 5.341
v 80 h s
0.0062 788.4 4.029
0.0081 855.1 4.208
0.0097 914.2 4.347
0.0112 971.3 4.468
0.0140 1085 4.675
0.0166 1201 4.854
0.0191 1320 5.011
0.0216 1441 5.155
0.0240 1565 5.286
v 100 h s
0.0045 766.2 3.936
0.0062 843.0 4.144
0.0076 905.7 4.290
0.0089 964.9 4.417
0.0111 1081 4.627
0.0133 1198 4.808
0.0153 1318 4.967
0.0173 1440 5.111
0.0193 1564 5.241
v 150 h s
0.0023 704.5 3.716
0.0038 811.9 4.005
0.0049 884.8 4.177
0.0058 949.4 4.313
0.0074 1072 4.536
0.0089 1192 4.722
0.0103 1314 4.884
0.0117 1437 5.030
0.0130 1562 5.162
v 200 h s
0.0017 670.0 3.591
0.0027 783.2 3.894
0.0035 865.2 4.088
0.0043 934.9 4.234
0.0056 1063 4.468
0.0067 1186 4.668
0.0078 1310 4.824
0.0088 1435 4.970
0.0099 1561 5.104
v 300 h s
0.0017 745.3 3.747
0.0023 834.0 3.956
0.0029 910.6 4.118
0.0038 1047 4.367
0.0046 1176 4.573
0.0053 1303 4.743
0.0060 1431 4.886
0.0067 1559 5.021
v 400 h s
0.0015 728.1 3.663
0.0018 814.6 3.867
0.0022 893.3 4.033
0.0029 1035 4.292
0.0035 1168 4.497
0.0041 1298 4.671
0.0047 1428 4.824
0.0052 1558 4.960
0.0016 803.5 3.805
0.0018 881.9 3.970
0.0024 1027 4.234
0.0029 1162 4.443
0.0034 1294 4.620
0.0038 1426 4.774
0.0043 1557 4.913
v 500 h s
*Interpolated and rounded from Vukalovich and Altunin, Thermophysical Properties of Carbon Dioxide, Atomizdat, Moscow, 1965; and Collett, England, 1968. TABLE 2-243
Saturated Carbon Monoxide*
T, K
P, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
81.62 83.36 88.25 96.16 101.51
1.01 1.52 2.03 4.05 6.08
1.268.−3 1.295.−3 1.317.−3 1.385.−3 1.440.−3
0.0666 0.0631 0.0606 0.0547 0.0513
150.25 158.56 165.00 182.76 195.0
365.30 368.07 370.00 374.21 375.98
3.005 3.104 3.178 3.368 3.489
5.640 5.559 5.501 5.359 5.271
105.69 109.17 116.08 121.48 125.97
8.12 10.13 15.20 20.27 25.33
1.489.−3 1.535.−3 1.651.−3 1.778.−3 1.936.−3
0.0318 0.0253 0.0163 0.0116 0.0085
204.8 213.2 231.0 246.3 261.2
376.6 376.6 374.5 370.2 363.6
3.580 3.656 3.807 3.918 4.041
5.206 5.152 5.043 4.948 4.854
129.84 132.91c
30.40 34.96
2.168.−3 3.337.−3
0.0063 0.0033
277.6
313.15
4.161
4.747
*Pressure and volume values converted, and enthalpy and entropy values reproduced, from Hust and Stewart, NBS Tech. Note 202, 1963. This source gives values at and above 72.373 K at closer pressure intervals. c = critical point. The notation 1.268.−3 signifies 1.268 × 10−3. Goodwin, R. D., J. Phys. Chem. Ref. Data, 14, 4 (1985): 849–932, gives properties to 1000 bar, 68–1000 K.
2-226
PHYSICAL AND CHEMICAL DATA
FIG. 2-8 Temperature-entropy diagram for carbon monoxide. Pressure P, in atmospheres; density ρ, in grams per cubic centimeter; enthalpy H, in joules per gram. (From Hust and Stewart, NBS Tech. Note 202, 1963.)
THERMODYNAMIC PROPERTIES TABLE 2-244
2-227
Thermophysical Properties of Saturated Carbon Tetrachloride hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf , 10−6 Pa·s
kf , W/(m⋅K)
Pr
280 290 300 310 320
0.064 0.105 0.165 0.251 0.370
0.000 0.000 0.000 0.000 0.000
619 625 633 641 649
2.414 1.495 0.971 0.669 0.463
205.5 212.9 220.9 228.8 236.9
420.7 425.7 430.9 436.1 441.3
1.018 1.042 1.068 1.095 1.121
1.787 1.775 1.768 1.764 1.760
0.835 0.844 0.853 0.863 0.874
1042 892 774 679 603
0.1043 0.1020 0.0998 0.0975 0.0952
8.34 7.38 6.62 6.01 5.54
330 340 350 360 370
0.531 0.743 1.017 1.361 1.795
0.000 0.000 0.000 0.000 0.000
657 666 674 684 694
0.3306 0.2407 0.1802 0.1370 0.1053
246.0 254.5 263.1 271.8 280.8
446.4 451.5 456.6 461.7 466.6
1.149 1.174 1.199 1.224 1.248
1.756 1.754 1.752 1.751 1.751
0.885 0.897 0.910 0.924 0.939
539 486 441 402 368
0.0930 0.0907 0.0884 0.0861 0.0839
5.13 4.81 4.54 4.31 4.12
380 390 400 410 420
2.327 2.970 3.735 4.642 5.700
0.000 0.000 0.000 0.000 0.000
704 715 727 739 753
0.0820 0.0651 0.0525 0.0426 0.0350
289.7 298.1 307.9 317.1 326.0
471.5 475.8 481.2 485.8 490.4
1.272 1.295 1.319 1.341 1.363
1.750 1.751 1.752 1.753 1.754
0.954 0.970 0.987 1.010 1.034
338 311 287 265 246
0.0816 0.0794 0.0771 0.0749 0.0726
3.95 3.80 3.67 3.57 3.50
430 440 450 460 470
6.927 8.342 9.958 11.792 13.869
0.000 0.000 0.000 0.000 0.000
766 780 796 801 834
0.02899 0.02413 0.02020 0.01692 0.01425
335.2 344.3 353.6 363.1 372.8
494.9 499.2 503.4 507.3 511.1
1.384 1.405 1.426 1.446 1.467
1.756 1.757 1.759 1.760 1.761
1.060 1.094 1.141 1.207 1.240
227 211 195 180 167
0.0704 0.0682 0.0660 0.0638 0.0666
3.42 3.38 3.37 3.36 3.36
480 490 500 510 520
16.21 18.83 21.77 25.02 28.68
0.000 0.000 0.000 0.000 0.000
856 880 858 945 987
0.01205 0.01011 0.00858 0.00722 0.00607
382.6 392.0 402.5 412.9 424.3
514.6 517.5 520.2 522.6 524.2
1.487 1.507 1.526 1.546 1.568
1.762 1.763 1.762 1.761 1.760
1.278 1.320 1.375 1.44 1.52
156 145 133
0.0594 0.0511 0.0549
3.36 3.35 3.35
530 540 550 556.4c
32.71 37.18 44.12 45.60
0.001 0.001 0.001 0.001
041 121 248 792
0.00500 0.00400 0.00309 0.00179
436.4 448.3 463.4 494.4
524.5 522.7 518.2 494.4
1.590 1.614 1.638 1.692
1.756 1.749 1.738 1.692
T, K
vf, m3/kg
P, bar
vg, m3/kg
c = critical point. Base points: hf = 200 at 273.15 K = 0°C = hA − 300 kJ/kg; sf = 1.000 at 273.15 K = 0°C = sA − 4.000 kJ/(kg⋅K). Values mostly rounded and converted from Altunin, V. V., V. Z. Geller, et al., Thermophysical Properties of Freons, vol. 9, Hemisphere, Washington, DC, 1987 (243 pp.). Some irregularities exist in these data.
TABLE 2-245
Saturated Carbon Tetrafluoride (R14)*
P, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−4 Pa⋅s
kf, W/(m⋅K)
100 110 120 130 140
0.0089 0.0286 0.0924 0.2986 0.6901
5.370.−4 5.515.−4 5.668.−4 5.834.−4 6.018.−4
10.77 3.648 1.228 0.4051 0.1855
495.8 502.7 510.4 518.8 527.7
648.4 652.9 657.1 661.1 664.8
5.487 5.556 5.624 5.691 5.757
7.003 6.919 6.847 6.786 6.736
0.887 0.887 0.890 0.896 0.904
3.56
0.136 0.128 0.119 0.111 0.104
150 160 170 180 190
1.4074 2.598 4.426 7.067 10.702
6.225.−4 6.460.−4 6.733.−4 7.055.−4 7.449.−4
0.0951 0.0532 0.0318 0.0200 0.0131
537.2 549.4 557.6 568.2 579.3
668.3 671.4 674.0 676.1 677.4
5.822 5.885 5.947 6.007 6.066
6.696 6.662 6.629 6.607 6.583
0.922 0.975 1.031 1.104 1.203
3.28 3.03 2.80 2.59 2.39
0.097 0.089 0.081 0.072 0.064
200 210 220 227.5c
15.531 21.794 29.269 37.45
7.957.−4 8.674.−4 9.931.−4 1.598.−3
0.0087 0.0058 0.0036 0.0016
591.0 603.5 618.5 646.9
677.8 676.4 671.4 646.9
6.124 6.182 6.233 6.371
6.558 6.536 6.490 6.371
1.334 1.506 1.73 ∞
2.19 2.01 1.85
0.057 0.049 0.042 ∞
T, K
*P, v, h, and s values interpolated, extrapolated, and converted from Oguchi, Reito, 52 (1977): 869–889. c = critical point. The notation 5.370.−4 signifies 5.370 × 10−4. Equations and constants approximated to ASHRAE tables are given by Mecaryk, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2–3 (1991). The 1993 ASHRAE Handbook—Fundamentals (S.I. ed.) contains a saturation table from −140 to −45.65°C and an enthalpy–log-pressure diagram from 0.1 to 300 bar, −140 to 300°C. For properties to 1000 bar from 90 to 420 K, see Rublo, R. G., J. A. Zollweg, et al., J. Chem. Eng. Data, 36 (1991): 171–184. Saturation and superheat tables and a diagram to 80 bar, 600 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Chari, Ph.D. thesis, University of Michigan, 1960, presents saturation-temperature tables in fps units for 1°F increments from −270 to −51°F. Thermodynamic and transport properties, equations, and computer code and tables at constant entropy from 89 to 845 K are given by Hunt, J. L. and Boney, L. R., NASA TN D-7181, 1973 (105 pp.), largely based upon the Chari data.
2-228
PHYSICAL AND CHEMICAL DATA
TABLE 2-246 T, K 301.6m 400 500 600 700 800 900 1000 1200 1500
Saturated Cesium* P, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
2.66.−9 3.83.−6 3.11.−4 5.65.−3 0.0440
5.444.−4 5.615.−4 5.800.−4 5.999.−4 6.215.−4
7.01.+7 6.54.+4 1001 65.63 9.671
74.6 98.5 122.0 144.9 167.0
637.6 651.9 666.1 678.4 688.9
0.696 0.765 0.817 0.859 0.893
2.563 2.148 1.905 1.748 1.638
0.245 0.240 0.232 0.224 0.219
0.2029 0.6620 1.693 6.790 27.6
6.443.−4 6.689.−4 6.954.−4 7.628.−4 8.84.−4
188.7 210.6 233.2 281.1 358.8
698.3 707.3 716.4 736.1 772.2
0.922 0.975 0.972 1.015 1.072
1.559 1.500 1.455 1.394 1.345
0.217 0.222 0.231 0.248 0.275
2.353 0.796 0.335 0.097 0.029
*Converted from tables in Vargaftik, Tables of the Thermophysical Properties of Liquids and Gases, Nauka, Moscow, 1972, and Hemisphere, Washington, 1975. m = melting point. The notation 2.66.−9 signifies 2.66 × 109. Many of the Vargaftik values also appear in Ohse, R. W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Sci. Pubs., Oxford, 1985 (1020 pp.). This source contains superheat data. Saturation and superheat tables and a diagram to 30 bar, 1550 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram from 0.1 to 327 psia, 1300−2700°R, see Weatherford, W. D., J. C. Tyler, et al., WADD-TR-61-96, 1961. An extensive review of properties of the solid and the saturated liquid was given by Alcock, C. B., M. W. Chase, et al., J. Phys. Chem. Ref. Data, 23, 3 (1994): 385–497.
TABLE 2-247
Thermophysical Properties of Saturated Chlorine
T, °C
P, bar
vf , m3/kg
vg , m3/kg
hf , kJ/ kg
hg , kJ/kg
sf , kJ/(kg·K)
sg , kJ/(kg·K)
cpf , kJ/(kg·K)
cpg , kJ/(kg·K)
µf , 10−6 Pa· s
µg , 10−6 Pa·s
kf ,W/(m ·K)
kg,W/(m· K)
Prf
Prg
−50 −40 −30 −20 −10
0.475 0.773 1.203 1.802 2.608
0.000 0.000 0.000 0.000 0.000
623 634 645 656 668
0.5448 0.3481 0.2314 0.1593 0.1134
221.5 231.0 240.6 250.3 260.0
518.2 522.2 526.1 529.9 533.9
1.7650 1.8074 1.8480 1.8869 1.9243
3.0946 3.0562 3.0223 2.9921 2.9649
0.9454 0.9474 0.9496 0.9520 0.9547
0.476 0.484 0.497 0.513 0.532
565 520 483 452 422
10.3 10.8 11.4 11.9 12.4
0.1684 0.1650 0.1613 0.1573 0.1527
0.0061 0.0065 0.0069 0.0074 0.0078
3.17 2.99 2.85 2.74 2.64
0.809 0.815 0.820 0.826 0.841
0 10 20 30 40
3.664 5.014 6.702 8.774 11.27
0.000 0.000 0.000 0.000 0.000
681 695 710 726 744
0.0829 0.0619 0.0471 0.0364 0.0286
269.7 279.4 289.2 299.0 308.8
537.4 540.5 543.3 545.7 548.0
1.9604 1.9953 2.0291 2.0622 2.0946
2.9402 2.9177 2.8924 2.8777 2.8593
0.9579 0.9618 0.9667 0.9728 0.9816
0.554 0.579 0.607 0.638 0.674
393 368 348 333 318
13.0 13.5 14.1 14.7 15.2
0.1478 0.1427 0.1378 0.1327 0.1282
0.0083 0.0088 0.0093 0.0099 0.0104
2.55 2.48 2.45 2.44 2.43
0.864 0.888 0.918 0.950 0.985
50 60 70 80 90
14.25 17.76 21.85 26.65 32.17
0.000 0.000 0.000 0.000 0.000
763 784 808 834 865
0.02276 0.01827 0.01481 0.01202 0.00972
318.6 329.1 340.0 351.4 364.1
549.8 551.2 552.1 552.5 552.4
2.1264 2.1578 2.1892 2.2207 2.2528
2.8417 2.8245 2.8074 2.7900 2.7714
0.9968 1.022 1.054 1.124 1.253
0.720 0.786 0.885 1.017 1.205
304 290 278 267 256
15.8 16.4 17.1 17.9 18.7
0.1230 0.1171 0.1122 0.1050 0.0986
0.0110 0.0117 0.0126 0.0137 0.0149
2.46 2.53 2.61 2.85 3.26
1.034 1.107 1.201 1.331 1.510
100 110 120 130 140
38.44 45.54 53.57 62.68 72.84
0.000 0.000 0.001 0.001 0.001
901 956 016 121 335
0.00789 0.00639 0.00508 0.00392 0.00282
377.8 391.3 407.1 426.1 451.1
551.0 548.8 543.7 535.0 517.3
2.2860 2.3207 2.3590 2.4032 2.4595
2.7502 2.7317 2.7064 2.6733 2.6198
1.418 1.632 1.891
1.434 1.696 1.960
247 238 230
19.5 20.6 22.2
0.0916 0.0850 0.0775
0.0163 0.0178 0.0195
3.82 4.57 5.61
1.700 1.96 2.23
144c
77.10
0.001 77
0.00177
483.1
483.1
2.5365
2.5365
c = critical point. Values interpolated and converted from Martin, J. J., 1977 (private communication), and from Heat Exchanger Design Handbook, vol. 5, Hemisphere, Washington, DC, 1983. Values of Ziegler, Chem.-Ing.Tech., 22 (1950): 229, apparently were also used in Landolt-Bornstein, IVa, (1967): 238–239, and in Ullmans Enzyklopädie der technische Chemie, 9, Verlag Chemie, Weinheim, 1975 (317–372). 2-229
PHYSICAL AND CHEMICAL DATA 10 Chlorine
8 6
200°C 180
4
2.
6
160
140 2. 7
2 120
100°C 2. 8
10 8 80 6
9
2.
0 3.
4
Satu rate d va por
Pressure, bar
2-230
2
kJ
/kg
•
K
60
0
3. t En
p ro
y
40
20
1
3.
1 2
3.
0.8 0°C 0.6 –20
3.3
0.4
–40 3.4
0.2 –60
0.1
FIG. 2-9
500 Enthalpy–log-pressure diagram for chlorine.
3.5
550 Enthalpy, kJ/kg
600
THERMODYNAMIC PROPERTIES TABLE 2-248 T, K
2-231
Saturated Chloroform (R20) vf, m3/kg
P, bar
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg·K)
sg, kJ/(kg·K)
280 300 320 340 360
0.115 0.293 0.620 1.224 2.255
0.000 0.000 0.000 0.000 0.000
660 678 695 715 739
1.689 0.714 0.358 0.190 0.107
−46.0 −32.6 −13.4 5.2 23.3
219.5 230.6 241.1 252.1 263.0
−0.165 −0.105 −0.041 0.015 0.065
0.798 0.773 0.754 0.741 0.731
380 400 420 440 460
3.830 6.039 9.058 13.39 18.80
0.000 0.000 0.000 0.000 0.000
765 795 822 871 921
0.0653 0.0425 0.0288 0.0195 0.0137
41.7 61.4 82.8 106.1 131.6
273.7 284.2 294.2 303.6 311.2
0.114 0.165 0.217 0.270 0.325
480 500 520 530 536.6c
26.00 34.66 44.68 50.44 54.72
0.000 0.001 0.001 0.001 0.002
980 059 193 328 00
0.00962 0.00673 0.00467 0.00359 0.00200
157.4 186.2 219.6 242.7 284.1
316.5 320.8 321.3 315.7 284.1
0.380 0.436 0.499 0.540 0.602
cpf, kJ/(kg·K)
µf, 10−6 Pa·s
kf, W/(m·K)
Prf
1.03
748 587 468 381 319
0.120 0.114 0.109 0.103 0.095
3.35
0.725 0.722 0.721 0.719 0.716
1.07 1.11 1.15 1.21 1.32
273 237 206 177 155
0.0921 0.0863 0.0808 0.0750 0.0694
3.17 3.04 2.93 2.86 2.95
0.711 0.706 0.694 0.678 0.602
1.43 1.59
129.6 105.5 81.2 67.7
0.0641 0.0584 0.0518 0.0461
2.89 2.87
c = critical point. hf = sf = 0 at n.b.p., 334.5 K. P, v, h, and s interpolated from Altunin, V. V., V. Z. Geller, et al., Thermophysical Properties of Freons, U.S.S.R. N.S.R.D.S. series, vol. 9., Hemisphere.
TABLE 2-249
Saturated Decane*
P, bar
vf, m3/kg
243.5 260 280 300 320
0.00001 0.00006 0.00042 0.00197 0.00720
1.319.−3 1.334.−3 1.356.−3 1.381.−3 1.410.−3
340 360 380 400 420
0.02155 0.05522 0.1248 0.2549 0.4789
1.442.−3 1.478.−3 1.515.−3 1.552.−3 1.591.−3
440 447.3 460 480 500
0.8387 1.0133 1.3852 2.1745 3.2690
T, K m
vg, m3/kg
µf, 10−4 Pa·s
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg·K)
sg, kJ/(kg·K)
cpf, kJ/(kg·K)
418.1 452.7 495.3 539.0 584.0
812.5 836.3 866.9 899.2 933.2
2.561 2.699 2.856 3.007 3.153
4.092 4.120 4.158 4.200 4.246
2.119 2.109 2.155 2.217 2.286
8.883 3.763 1.750 0.892 0.490
631.1 680.1 730.7 782.0 835.6
968.9 1006.2 1045.0 1085.0 1126.2
3.303 3.443 3.581 3.712 3.842
4.296 4.350 4.408 4.469 4.534
5.2 4.16 3.52 2.98 2.54
1.632.−3 1.650.−3 1.682.−3 1.735.−3 1.797.−3
0.290 0.243 0.178 0.115 0.0759
889.6 909.4 944.5 1002.6 1062.7
1168.4 1184.0 1211.4 1255.2 1299.4
3.968 4.014 4.089 4.213 4.335
4.602 4.627 4.670 4.739 4.808
2.23 2.09
20750 3300. 443. 88.74 22.73
520 540 560 580 600
4.733 6.633 9.062 12.16 16.12
1.868.−3 1.952.−3 2.067.−3 2.255.−3 2.588.−3
0.0525 0.0369 0.0248 0.0154 0.0093
1124.5 1190.1 1256.1 1318.5 1384.5
1344.4 1389.5 1432.2 1468.1 1495.6
4.456 4.573 4.698 4.802 4.913
4.879 4.949 5.011 5.060 5.098
617.5c
20.97
4.238.−3
0.0042
1483.2
1483.2
5.073
5.073
kf, W/(m · K)
25.0 16.6 11.3 8.2 6.5
*Values converted from Das and Kuloor, Ind. J. Technol., 5 (1967): 75. m = melting point; c = critical point. The notation 1.319.−3 signifies 1.319 × 10−3.
0.149 0.144 0.139 0.134 0.129 0.124 0.119 0.116 0.110
2-232
PHYSICAL AND CHEMICAL DATA TABLE 2-250
Saturated Normal Deuterium*
T, K
P, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
18.71 19 20 21 22
0.1709 0.1944 0.2944 0.4297 0.6072
0.005752 0.005771 0.005840 0.005914 0.005993
2.232 1.988 1.365 0.968 0.705
−161.1 −160.0 −152.8 −145.9 −138.7
158.6 159.1 163.9 167.6 170.6
4.54 4.68 4.97 5.30 5.63
21.62 21.48 20.81 20.23 19.69
23 24 25 26 27
0.8344 1.1192 1.4694 1.8932 2.3989
0.00608 0.00617 0.00627 0.00638 0.00650
0.5256 0.3995 0.3088 0.2421 0.1921
−131.4 −123.8 −116.1 −108.2 −100.2
173.0 174.6 175.5 175.7 175.1
5.95 6.26 6.57 6.87 7.16
19.18 18.70 18.23 17.79 17.36
28 29 30 31 32
2.995 3.690 4.493 5.412 6.457
0.00663 0.00678 0.00694 0.00713 0.00735
0.1540 0.1246 0.1015 0.0831 0.0683
−92.0 −83.6 −74.9 −65.9 −56.5
173.8 171.7 168.7 165.0 160.3
7.44 7.72 8.00 8.27 8.54
16.94 16.52 16.12 15.72 15.32
33 34 35 36 37
7.455 8.962 10.44 12.09 13.91
0.00761 0.00793 0.00834 0.00890 0.00976
0.0563 0.0465 0.0382 0.0311 0.0249
−46.4 −35.5 −23.2 −8.6 10.0
154.7 148.0 140.0 130.1 117.1
8.83 9.12 9.45 9.82 10.28
14.92 14.52 14.11 13.67 13.17
38 38.34c
15.92 16.65
0.01158 0.01433
0.0185 0.0143
39.7 69.2
95.0 69.2
11.01 11.76
12.47 11.76
*Condensed and converted from tables of Prydz, NBS Rep. 9276, 1967. c = critical point. For equations and T-s and Z charts from 0.1 to 100 atm, 20–300 K, see also Prydz, R. and K. D. Timmerhaus, Advan. Cryog. Eng., 13 (1968): 384–396. TABLE 2-251
Saturated Deuterium Oxide*
P, bar
vf, m3/kg
vg, m3/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
277.0 278.2 283.2 288.2 293.2
0.00668 0.00720 0.01030 0.01449 0.02011
9.047.−4 9.045.−4 9.042.−4 9.043.−4 9.047.−4
172.2 160.4 114.1 82.48 60.45
0.0 5.0 25.9 46.9 67.8
2320.9 2322.5 2330.9 2339.3 2347.6
0.000 0.0188 0.0920 0.166 0.239
8.380 8.351 8.233 8.122 8.016
298.2 303.2 308.2 313.2 318.2
0.02758 0.03730 0.04990 0.06598 0.08638
9.054.−4 9.063.−4 9.075.−4 9.091.−4 9.108.−4
44.88 33.71 25.59 19.66 15.24
88.7 109.6 130.5 151.5 172.4
2356.0 2364.0 2372.3 2380.7 2388.6
0.311 0.382 0.450 0.518 0.585
7.915 7.818 7.725 7.637 7.550
323.2 333.2 353.2 373.2 398.2
0.1120 0.1831 0.4439 0.9646 2.2427
9.127.−4 9.170.−4 9.274.−4 9.403.−4 9.599.−4
11.93 7.52 3.27 1.58 0.72
193.3 234.7 318.4 402.0 507.5
2396.6 2413.3 2445.1 2474.8 2509.6
0.650 0.776 1.020 1.253 1.527
7.468 7.315 7.042 6.807 6.555
T, K t
423.2 448.2 473.2 498.2 523.2 548.2 573.2 598.2 623.2 644.7c
4.653 8.806 15.46 25.52 39.99 60.04 86.97 122.4 168.3 218.4
hf, kJ/kg
9.835.−4 1.012.−3 1.044.−3 1.082.−3 1.133.−3
0.362 0.198 0.115 0.0704 0.0447
612.5 718.8 826.8 938.5 1055.2
2541.8 2569.4 2585.7 2597.0 2598.7
1.781 2.020 2.256 2.483 2.707
6.341 6.149 5.973 5.812 5.658
1.200.−3 1.276.−3 1.392.−3 1.596.−3 2.950.−3
0.0290 0.0191 0.0124 0.0075 0.0030
1177.4 1306.7 1445.6 1607.1
2587.0 2555.6 2492.4 2366.5
2.930 3.153 3.356 3.631
5.501 5.332 5.132 4.850
*Extracted or converted from values in Kazavchinskii, Kesselman, et al., Thermophysical Properties of Heavy Water, Moscow and Leningrad, 1963; NBS-NSF transl. 70-50094, 1971. t = triple point; c = critical point. The notation 9.047.−4 signifies 9.047 × 10−4. Hill, P. G., MacMillan, R. D. and others give extensive tables for 0–1000 bar, 4–800°C in Atomic Energy of Canada, Chalk River rept. AECL-7531, 1981 (196 pp.). See also J. Phys. Chem. Ref. Data, 11, 1 (1982): 1–14; 19, 5 (1990): 1233–1274. TABLE 2-252
Deuterium Oxide Gas at 1-kg/cm3 Pressure
T, K
400
450
500
550
600
650
700
750
v, m3/ kg h, kJ/ kg s, kJ/(kg⋅K)
1.676 2525 6.931
1.895 2619 7.151
2.112 2712 7.349
2.322 2807 7.529
2.535 2904 7.697
2.747 3002 7.855
2.960 3102 8.003
3.172 3205 8.153
THERMODYNAMIC PROPERTIES TABLE 2-253
2-233
Saturated Diphenyl* hg, kJ/ kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−4 Pa⋅s
kf, W/(m⋅K)
0.0 13.0 30.0 47.2 65.0
444.2 444.2 446.7 449.7 454.5
0.000 0.036 0.084 0.130 0.178
1.298 1.266 1.236 1.213 1.200
1.760 1.782 1.813 1.844 1.875
15.0 13.5 11.7 10.3 9.1
0.139 0.138 0.136 0.135 0.133
82.7 99.3 139.9 180.3 222.7
462.7 461.2 499.0 532.4 569.7
0.224 0.273 0.358 0.451 0.545
1.194 1.202 1.228 1.267 1.378
1.906 1.936 1.998 2.060 2.122
8.1 7.3 6.0 5.0 4.3
0.132 0.130 0.127 0.125 0.122
0.9594 0.4452 0.3652 0.2261 0.1447
267.6 314.9 361.5 404.5 457.2
611.6 651.8 687.8 723.8 762.7
0.652 0.746 0.824 0.915 1.032
1.367 1.424 1.477 1.529 1.582
2.184 2.246 2.308 2.370 2.432
3.7 3.3 2.7 2.4 2.2
0.119 0.116 0.113 0.110 0.107
1.258.−3 1.291.−3 1.326.−3 1.366.−3 1.412.−3
0.0977 0.0685 0.0504 0.0381 0.0301
522.3 563.7 630.4 689.1 745.9
801.7 842.4 886.4 930.9 977.1
1.125 1.223 1.316 1.375 1.457
1.635 1.688 1.740 1.748 1.791
2.494 2.556 2.618 2.680 2.741
1.90 1.71 1.54 1.39 1.24
0.105 0.102 0.099 0.096 0.093
2.803 2.865 2.93 3.00
1.10 0.97
0.090 0.087
T, K
P, bar
vf, m3/kg
vg, m3/kg
343 350 360 370 380
0.0010 0.0016 0.0029 0.0049 0.0064
1.010.−3 1.014.−3 1.021.−3 1.030.−3 1.037.−3
252.5 156.1 85.0 49.9 29.9
390 400 420 440 460
0.0129 0.0200 0.0432 0.0879 0.1694
1.046.−3 1.054.−3 1.072.−3 1.092.−3 1.112.−3
480 500 520 540 560
0.3112 0.5218 0.8375 1.290 1.941
1.132.−3 1.154.−3 1.177.−3 1.204.−3 1.230.−3
580 600 620 640 660
2.818 3.926 5.408 7.328 9.572
18.3 11.7 5.84 3.021 1.652
hf, kJ/kg
680 700 720 740 760
12.05 15.21 19.14 23.93 28.71
1.465.−3 1.529.−3 1.56.−3 1.70.−3 1.95.−3
0.0236 0.0186 0.0147 0.0113 0.0085
802.8 860.1 917.5 975.2 1033.1
1024.9 1073.1 1116.7 1152.8 1182.5
1.585 1.663 1.746 1.822 1.901
1.856 1.951 2.003 2.058 2.099
780 800
34.83 42.46
2.16.−3 3.18.−3
0.0058 0.0032
1091.2 1148.4
1163.0 1148.4
1.977 2.047
2.107 2.047
*Interpolated by P. E. Liley from the Landolt-Börnstein band IVa, p. 557, 1967 tables based on Technical Data on Fuel, British National Committee, World Energy Conference, London.
TABLE 2-254
Saturated Ethane (R170)* hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−4 Pa⋅s
kf, W/(m⋅K)
176.8 198.7 221.5 244.4 267.4
769.4 782.4 795.0 807.2 819.3
2.560 2.790 3.008 3.207 3.391
9.113 8.627 8.222 7.897 7.637
2.260 2.274 2.284 2.292 2.302
14.19 9.37 6.57 4.89 3.81
0.215 0.208 0.201 0.194 0.187
10.08 4.263 2.039 1.075 0.6139
290.5 313.7 337.2 360.9 384.9
831.4 843.5 855.6 867.6 879.4
3.562 3.722 3.873 4.017 4.154
7.426 7.254 7.113 6.998 6.901
2.316 2.333 2.355 2.383 2.417
3.07 2.55 2.17 1.88 1.65
0.180 0.174 0.167 0.160 0.153
1.862 1.908 1.958 2.014 2.076
0.3738 0.2395 0.1602 0.1109 0.0789
409.3 434.2 459.7 485.9 512.8
890.8 901.7 911.9 921.4 929.6
4.285 4.412 4.535 4.655 4.773
6.819 6.750 6.689 6.635 6.585
2.458 2.508 2.568 2.640 2.730
1.47 1.33 1.21 1.11 1.03
0.147 0.140 0.133 0.126 0.119
9.670 13.01 17.12 22.10 28.06
2.148 2.231 2.330 2.452 2.613
0.0573 0.0423 0.0316 0.0237 0.0177
540.8 569.9 600.7 633.6 669.3
936.6 941.9 945.4 946.4 943.6
4.890 5.006 5.123 5.233 5.370
6.539 6.493 6.449 6.392 6.350
2.843 2.991 3.214 3.511 4.011
0.96 0.82 0.73 0.64 0.55
0.112 0.106 0.099 0.092 0.085
35.14 43.54 48.71
2.847 3.295 4.891
0.0129 0.0087 0.0048
709.8 761.6 841.2
934.7 910.8 841.2
5.502 5.669 5.919
6.278 6.166 5.919
5.089 9.919 ∞
0.44 0.31
0.078 0.067
P, bar
vf, m3/kg
90.4 100 110 120 130
1.131.−5 1.110.−4 7.467.−3 3.545.−3 1.291.−2
1.534.−3 1.546 1.573 1.615 1.644
140 150 160 170 180
3.831.−2 9.672.−2 0.2146 0.4290 0.7874
1.675 1.708 1.743 1.780 1.819
190 200 210 220 230
1.347 2.174 3.340 4.922 7.004
240 250 260 270 280 290 300 305.3c
T, K t
vg, m3/kg 21945 2484.5 407.0 93.61 27.83
*Values reproduced or converted from Goodwin, Roder, and Straty, NBS Tech. Note 684, 1976. t = triple point; c = critical point. The notation 1.131.−5 signifies 1.131 × 10−5.
2-234
PHYSICAL AND CHEMICAL DATA
TABLE 2-255
Superheated Ethane* Temperature, K
P, bar
100
150
200
250
300
350
400
450
500
600
700
v 1.013 h s
0.00156 198.9 2.790
0.00171 313.8 3.722
0.5310 909.3 6.993
0.6725 984.7 7.330
0.8118 1068.3 7.634
0.9500 1161.5 7.921
1.0877 1265.3 8.198
1.2250 1379.8 8.467
1.3622 1504.6 8.730
1.6360 1783 9.237
1.9096 2097 9.720
v 5h s
0.00156 199.4 2.789
0.00171 314.3 3.720
0.00191 434.5 4.411
0.1288 973.3 6.858
0.1595 1060.3 7.175
0.1890 1155.6 7.468
0.2178 1260.7 7.748
0.2464 1376.1 8.020
0.2747 1501.5 8.284
0.3308 1781 8.793
0.3867 2096 9.227
v 10 h s
0.00156 200.0 2.788
0.00171 314.9 3.719
0.00190 435.0 4.408
0.0590 956.5 6.618
0.0765 1050.0 6.959
0.0923 1148.2 7.262
0.1073 1255.0 7.547
0.1220 1371.5 7.821
0.1365 1497.9 8.087
0.1650 1777 8.598
0.1933 2094 9.083
v 20 h s
0.00156 201.3 2.785
0.00170 316.1 3.715
0.00190 435.9 4.404
0.00222 569.8 4.999
0.0346 1026.1 6.710
0.0438 1132.3 7.038
0.0521 1243.3 7.334
0.0599 1362.4 7.614
0.0674 1490.5 7.884
0.0822 1774 8.399
0.0966 2090 8.886
v 40 h s
0.00155 203.9 2.780
0.00170 318.5 3.709
0.00189 437.9 4.394
0.00219 569.9 4.982
0.0118 947.9 6.309
0.0193 1096.2 6.770
0.0244 1218.6 7.097
0.0288 1343.8 7.391
0.0329 1475.9 7.670
0.0407 1764 8.194
0.0482 2083 8.686
v 60 h s
0.00155 206.5 2.775
0.00170 321.0 3.702
0.00188 439.8 4.385
0.00217 570.3 4.966
0.00290 738.1 5.574
0.0109 1050.9 6.557
0.0132 1192.0 6.934
0.0185 1324.8 7.247
0.0215 1461.2 7.535
0.0270 1754 8.068
0.0321 2077 8.564
v 80 h s
0.00155 209.1 2.769
0.00169 323.4 3.696
0.00188 441.9 4.377
0.00215 570.9 4.951
0.00273 728.1 5.522
0.00667 993.8 6.345
0.0106 1163.6 6.800
0.0134 1305.5 7.135
0.0158 1446.7 7.432
0.0201 1745 7.975
0.0459 2070 8.476
v 100 h s
0.00155 211.7 2.764
0.00169 325.8 3.690
0.00187 443.9 4.368
0.00213 571.8 4.938
0.00263 722.7 5.486
0.00465 924.4 6.166
0.00791 1134.7 6.682
0.0104 1286.3 7.040
0.0124 1432.4 7.348
0.0160 1736 7.900
0.0193 2064 8.406
v 150 h s
0.00155 218.1 2.752
0.00168 332.0 3.674
0.00185 449.2 4.348
0.00209 574.6 4.907
0.00247 716.4 5.423
0.00328 887.4 5.955
0.00488 1075.2 6.457
0.00655 1242.3 6.851
0.00805 1399.3 7.182
0.0107 1715 7.758
0.0130 2050 8.274
v 200 h s
0.00154 224.6 2.738
0.00167 338.2 3.660
0.00184 454.7 4.329
0.00205 578.2 4.880
0.00237 714.8 5.377
0.00291 870.5 5.863
0.00383 1041.7 6.320
0.00495 1210.2 6.717
0.00605 1327.3 7.059
0.00806 1697 7.651
0.00986 2038 8.176
v 300 h s
0.00153 237.6 2.715
0.00166 350.6 3.632
0.00181 465.9 4.294
0.00200 586.8 4.833
0.00225 715.9 5.309
0.00259 860.9 5.757
0.00307 1014.9 6.168
0.00367 1175.5 6.547
0.00433 1338.7 6.891
0.00563 1671 7.496
0.00686 2019 8.032
v 400 h s
0.00153 250.6 2.692
0.00165 363.2 3.605
0.00179 477.6 4.262
0.00195 596.6 4.793
0.00216 723.7 5.257
0.00244 861.6 5.688
0.00276 1008.3 6.080
0.00316 62.5 6.443
0.00361 1322.7 6.780
0.00454 1657 7.388
0.00545 2008 7.930
v 500 h s
0.00152 263.5 2.670
0.00163 375.8 3.580
0.00176 489.3 4.234
0.00192 607.1 4.758
0.00210 732.0 5.213
0.00232 866.5 5.634
0.00258 1009.3 6.015
0.00288 1159.3 6.369
0.00322 1316.9 6.00
0.00392 1650 7.306
0.00465 2003 7.851
*Converted and rounded off from the tables of Goodwin, Roder, and Straty, NBS Tech. Note 684, 1976. v = specific volume, m3/kg; h = specific enthalpy, kJ/ kg; s = specific entropy, kJ/(kg⋅K). Saturation and superheat tables and a diagram to 300 bar, 580 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Saturation and superheat tables and a chart to 10,000 psia, 640°F appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) contains a thermodynamic diagram from 0.1 to 700 bar for temperatures to 600 K.
THERMODYNAMIC PROPERTIES TABLE 2-256 T, K
P, bar
2-235
Saturated Ethanol vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−6 Pa·s
kf, W/(m·K)
Prf
250 260 270 280 290
0.0027 0.0059 0.0128 0.025 0.048
0.001 0.001 0.001 0.001 0.001
184 196 208 220 233
2.113 2.167 2.227 2.294 2.369
295 229 193 156 127
0.177 0.175 0.173 0.171 0.170
35.2 28.4 24.8 20.9 17.7
300 310 320 330 340
0.088 0.151 0.253 0.406 0.632
0.001 0.001 0.001 0.001 0.001
246 260 274 288 304
2.45 2.54 2.64 2.75 2.86
104 86 72 61 52
0.168 0.165 0.162 0.159 0.157
15.2 13.2 11.7 10.6 9.5
350 360 370 380 390
0.956 1.409 2.023 2.837 3.897
0.001 0.001 0.001 0.001 0.001
318 337 357 379 403
0.7656 0.5052 0.3555 0.2556 0.1873
199.9 230.1 262.2 295.1 329.1
1161.9 1178.4 1193.9 1208.4 1221.5
2.99 3.12 3.27 3.42 3.58
45.0 39.0 34.2 30.0 26.1
0.155 0.153 0.151 0.149 0.147
8.7 8.0 7.4 6.9 6.3
400 410 420 430 440
5.251 6.954 9.063 11.64 14.72
0.001 0.001 0.001 0.001 0.001
430 461 495 532 574
0.1398 0.1058 0.0812 0.0631 0.0493
364.2 400.8 435.7 472.2 512.7
1233.6 1244.2 1254.2 1262.3 1269.2
3.74 3.99 4.26 4.55 4.88
22.7 20.0 17.6 15.3 13.9
0.145 0.144 0.142 0.140 0.139
5.9 5.5 5.3 5.0 4.9
450 460 470 480 490
18.33 22.61 27.66 33.55 40.39
0.001 0.001 0.001 0.001 0.001
623 682 752 832 950
0.0389 0.0308 0.0243 0.0193 0.0148
557.2 605.0 653.7 704.5 757.7
1274.2 1275.5 1271.1 1262.3 1250.2
5.23
12.5
0.137
4.8
500 510 516.3c
48.28 57.32 63.90
0.002 091
0.0110
818.9
1232.7
c = critical point. Values interpolated and converted from Heat Exchanger Design Handbook, vol. 5, Hemisphere, Washington, DC, 1983, and from various literature sources.
FIG. 2-10 Enthalpy-concentration diagram for aqueous ethyl alcohol. Reference states: Enthalpies of liquid water and ethyl alcohol at 0°C are zero. NOTE: In order to interpolate equilibrium compositions, a vertical may be erected from any liquid composition on the boiling line and its intersection with the auxiliary line determined. A horizontal from this intersection will establish the equilibrium vapor composition on the dew line. (Bosnjakovic, Technische Thermodynamik, T. Steinkopff, Leipzig, 1935.)
2-236
PHYSICAL AND CHEMICAL DATA
TABLE 2-257
Saturated Ethylene (Ethene—R1150)
Temperature, K
Pressure, bar
104.0t 110 120 130 140
0.00123 0.00334 0.01380 0.04456 0.1191
0.001 0.001 0.001 0.001 0.001
527 545 576 609 644
150 160 170 180 190
0.2747 0.5636 1.0526 1.8207 2.9574
0.001 0.001 0.001 0.001 0.001
681 721 763 810 861
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
251.36 97.57 25.75 8.62 3.46
−323.81 −309.54 −284.17 −259.13 −234.80
244.36 251.47 263.23 274.87 286.28
−1.9901 −1.8571 −1.6362 −1.4358 −1.2554
3.4730 3.2431 2.9255 2.6717 2.4663
2.497 2.500 2.539 2.465 2.405
1.5977 0.8232 0.4625 0.2784 0.1770
−210.90 −187.12 −163.23 −139.05 −114.46
297.37 308.00 318.04 327.35 335.79
−1.0908 −0.9378 −0.7935 −0.6559 −0.5244
2.2977 2.1566 2.0375 1.9352 1.7812
2.377 2.377 2.395 2.427 2.472
200 210 220 230 240
4.560 6.730 9.575 13.206 17.742
0.001 0.001 0.002 0.002 0.002
918 981 054 139 241
0.1177 0.0810 0.0573 0.0413 0.0302
−89.33 −63.52 −36.84 −9.04 20.23
343.21 349.41 354.18 357.17 357.90
−0.3967 −0.2730 −0.1515 −0.0314 0.0088
1.7659 1.6932 1.6258 1.5609 1.4957
2.531 2.608 2.711 2.852 3.055
250 260 270 280 282.3c
23.307 30.046 38.132 47.834 50.403
0.002 0.002 0.002 0.003 0.004
369 541 804 442 669
0.02222 0.01624 0.01152 0.00720 0.00467
51.55 85.91 125.79 183.40 234.55
355.37 348.68 333.71 292.83 234.55
0.2114 0.3397 0.4819 0.6803 0.8585
1.4276 1.3503 1.3054 1.0711 0.8585
3.372 3.945 5.40 20.0
t = triple point; c = critical point. hf = sf = 0 at 233.15 K = −40°C. Converted from Jacobsen, R. T., M. Jahangiri, et al., Ethylene—Intl. Thermodyn. Tables of the Fluid State—10, Blackwell Sci. Publ., Oxford, U.K., 1988 (299 pp.). Saturation and superheat tables and a diagram to 100 bar, 460 K are given by Reynolds, W. C., Thermodynamic properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Saturation and superheat tables and a chart to 6000 psia, 360°F appear in Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) has a thermodynamic chart for pressures from 0.1 to 400 bar and temperatures up to 460 K.
THERMODYNAMIC PROPERTIES TABLE 2-258
Compressed Ethylene Temperature, K
Pressure, bar
110
125
150
175
200
225
250
275
v (m3/kg) 1 h (kJ/kg) s (kJ/kg⋅K)
0.001 545 −309.4 −1.858
0.001 592 −271.4 −1.534
0.001 681 −210.8 −1.091
0.5036 324.8 2.091
0.5814 357.0 2.264
0.6580 389.9 2.419
0.7337 424.0 2.562
0.8091 459.7 2.698
v (m3/kg) 5 h (kJ/kg) s (kJ/kg⋅K)
0.001 544 −308.9 −1.859
0.001 591 −271.0 −1.535
0.001 680 −210.4 −1.093
0.001 785 −150.8 −0.726
0.001 917 −89.3 −0.397
0.1240 378.4 1.907
0.1407 415.0 2.061
0.1569 452.3 2.203
v (m3/kg) 10 h (kJ/kg) s (kJ/kg⋅K)
0.001 543 −308.3 −1.860
0.001 591 −270.4 −1.537
0.001 679 −209.8 −1.095
0.001 783 −150.3 −0.728
0.001 914 −89.0 −0.400
0.05643 361.2 1.646
0.06672 402.4 1.820
0.07525 442.3 1.973
v (m3/kg) 20 h (kJ/kg) s (kJ/kg⋅K)
0.001 542 −307.1 −1.863
0.001 589 −269.2 −1.540
0.001 676 −208.7 −1.098
0.001 780 −149.4 −0.733
0.001 908 −88.2 −0.406
0.002 084 −23.0 −0.099
0.02810 370.3 1.520
0.03405 419.7 1.708
v (m3/kg) 30 h (kJ/kg) s (kJ/kg⋅K)
0.001 541 −305.9 −1.866
0.001 588 −268.0 −1.543
0.001 674 −207.6 −1.102
0.001 776 −148.4 −0.737
0.001 903 −87.5 −0.412
0.002 072 −22.8 −0.107
0.002 347 50.5 0.201
0.01978 390.7 1.508
v (m3/kg) 40 h (kJ/kg) s (kJ/kg⋅K)
0.001 540 −304.7 −1.869
0.001 587 −266.8 −1.546
0.001 672 −206.5 −1.106
0.001 773 −147.4 −0.741
0.001 897 −86.7 −0.418
0.002 062 −22.5 −0.115
0.002 318 49.1 0.186
0.01163 344.7 1.284
v (m3/kg) 50 h (kJ/kg) s (kJ/kg⋅K)
0.001 539 −303.5 −1.872
0.001 585 −265.7 −1.550
0.001 670 −205.4 −1.110
0.001 770 −146.4 −0.746
0.001 892 −85.9 −0.423
0.002 052 −22.2 −0.123
0.002 293 48.1 0.173
0.002 846 139.8 0.521
v (m3/kg) 60 h (kJ/kg) s (kJ/kg⋅K)
0.001 538 −302.3 −1.875
0.001 584 −264.5 −1.553
0.001 668 −204.2 −1.113
0.001 767 −145.4 −0.750
0.001 887 −85.1 −0.428
0.002 043 −21.8 −0.130
0.002 270 47.4 0.161
0.002 723 132.3 0.484
v (m3/kg) 80 h (kJ/kg) s (kJ/kg⋅K)
0.001 535 −299.8 −1.881
0.001 581 −262.1 −1.559
0.001 664 −202.0 −1.120
0.001 761 −143.4 −0.759
0.001 877 −83.5 −0.439
0.002 025 −20.9 −0.145
0.002 232 46.5 0.139
0.002 585 124.1 0.434
v (m3/kg) 100 h (kJ/kg) s (kJ/kg⋅K)
0.001 533 −297.4 −1.887
0.001 579 −259.7 −1.565
0.001 660 −199.7 −1.127
0.001 754 −141.2 −0.767
0.001 867 −81.8 −0.449
0.002 009 −19.9 −0.158
0.002 199 46.1 0.120
0.002 495 119.6 0.400
v (m3/kg) 150 h (kJ/kg) s (kJ/kg⋅K)
0.001 528 −291.3 −1.901
0.001 571 −253.7 −1.580
0.001 650 −194.0 −1.145
0.001 740 −136.0 −0.787
0.001 846 −77.3 −0.473
0.001 973 −16.7 −0.188
0.002 136 46.6 0.079
0.002 356 114.4 0.337
v (m3/kg) 200 h (kJ/kg) s (kJ/kg⋅K)
0.001 522 −285.3 −1.914
0.001 565 −247.7 −1.595
0.001 641 −188.3 −1.161
0.001 727 −130.7 −0.806
0.001 826 −72.5 −0.495
0.001 943 −13.0 −0.215
0.002 086 48.6 0.045
0.002 268 113.2 0.291
v (m3/kg) 250 h (kJ/kg) s (kJ/kg⋅K)
0.001 517 −279.2 −1.928
0.001 559 −241.7 −1.610
0.001 633 −182.5 −1.177
0.001 715 −125.2 −0.824
0.001 809 −67.6 −0.516
0.001 918 −8.9 −0.240
0.002 046 51.4 0.015
0.002 203 113.9 0.253
v (m3/kg) 300 h (kJ/kg) s (kJ/kg⋅K)
0.001 512 −273.0 −1.942
0.001 552 −235.7 −1.623
0.001 625 −174.5 −1.192
0.001 704 −119.6 −0.841
0.001 793 −62.5 −0.536
0.001 895 −4.4 −0.262
0.002 012 54.9 −0.012
0.002 151 115.9 0.220
v (m3/kg) 400 h (kJ/kg) s (kJ/kg⋅K)
0.001 503 −260.8 −1.968
0.001 542 −223.6 −1.650
0.001 609 −164.8 −1.221
0.001 683 −108.3 −0.873
0.001 765 −51.9 −0.572
0.001 855 5.1 −0.303
0.001 957 63.0 −0.059
0.002 072 122.1 0.166
v (m3/kg) 500 h (kJ/kg) s (kJ/kg⋅K)
0.001 499 −246.9 −1.978
0.001 531 −211.4 −1.676
0.001 596 −152.9 −1.249
0.001 665 −96.8 −0.906
0.001 740 −40.9 −0.605
0.001 823 15.3 −0.339
0.001 913 72.3 −0.099
0.002 01 130.1 0.121
Converted from Jacobsen, R. T., M. Jahangiri, et al., Ethylene—Intl. Thermodyn. Tables of the Fluid State—10, Blackwell Sci. Publ., Oxford, 1988 (299 pp.). sf = hf = 0 at 233.15 K = −40°C.
2-237
2-238
PHYSICAL AND CHEMICAL DATA TABLE 2-258
Compressed Ethylene (Concluded ) Temperature, K
Pressure, bar
300
325
350
375
400
425
450
v (m3/kg) 1 h (kJ/kg) s (kJ/kg⋅K)
0.8842 497.3 2.829
0.9591 536.9 2.956
1.0339 578.8 3.079
1.1084 622.8 3.201
1.1830 668.9 3.320
1.2575 717.2 3.437
1.3319 767.9 3.553
v (m3/kg) 5 h (kJ/kg) s (kJ/kg⋅K)
0.1728 491.0 2.338
0.1884 531.5 2.467
0.2039 574.1 2.593
0.2193 618.6 2.716
0.2346 665.2 2.836
0.2499 713.9 2.954
0.2650 764.9 3.071
v (m3/kg) 10 h (kJ/kg) s (kJ/kg⋅K)
0.08380 482.8 2.113
0.09207 542.5 2.247
0.1002 568.0 2.375
0.1081 613.3 2.500
0.1160 660.5 2.622
0.1238 709.7 2.742
0.1316 761.1 2.859
v (m3/kg) 20 h (kJ/kg) s (kJ/kg⋅K)
0.03914 465.0 1.866
0.04379 509.8 2.009
0.04823 555.5 2.144
0.05257 602.4 2.274
0.05675 650.9 2.399
0.06088 701.2 2.521
0.06491 753.5 2.640
v (m3/kg) 30 h (kJ/kg) s (kJ/kg⋅K)
0.02404 444.7 1.696
0.02763 493.8 1.853
0.03090 542.3 1.996
0.03400 591.2 2.131
0.03700 641.2 2.261
0.03990 692.6 2.387
0.04270 745.9 2.508
v (m3/kg) 40 h (kJ/kg) s (kJ/kg⋅K)
0.01630 420.6 1.550
0.01947 476.3 1.728
0.02220 528.3 1.882
0.02473 579.5 2.023
0.02710 631.2 2.157
0.02938 688.9 2.286
0.03160 738.2 2.409
v (m3/kg) 50 h (kJ/kg) s (kJ/kg⋅K)
0.01140 390.4 1.404
0.01451 456.9 1.617
0.01697 513.4 1.784
0.01916 567.5 1.933
0.02119 621.1 2.072
0.02311 675.1 2.207
0.02495 730.5 2.330
v (m3/kg) 60 h (kJ/kg) s (kJ/kg⋅K)
0.007 757 347.8 1.230
0.01116 435.1 1.510
0.01347 497.7 1.696
0.01546 555.2 1.854
0.01725 610.9 1.999
0.01892 666.3 2.135
0.02052 722.9 2.263
v (m3/kg) 80 h (kJ/kg) s (kJ/kg⋅K)
0.003 672 238.7 0.832
0.006 864 382.8 1.295
0.009 136 463.5 1.534
0.01085 529.4 1.717
0.01237 590.1 1.874
0.01374 648.5 2.016
0.01502 707.6 2.151
v (m3/kg) 100 h (kJ/kg) s (kJ/kg⋅K)
0.003 094 210.2 0.715
0.004 698 330.2 1.098
0.006 596 427.5 1.387
0.008 163 503.1 1.596
0.009 492 569.3 1.768
0.01068 630.9 1.918
0.01177 692.7 2.059
v (m3/kg) 150 h (kJ/kg) s (kJ/kg⋅K)
0.002 684 188.8 0.596
0.003 223 272.7 0.864
0.004 040 361.4 1.126
0.004 983 445.5 1.359
0.005 914 521.6 1.556
0.006 765 592.1 1.722
0.007 578 658.1 1.878
v (m3/kg) 200 h (kJ/kg) s (kJ/kg⋅K)
0.002 508 181.7 0.529
0.002 840 255.0 0.763
0.003 292 332.4 0.992
0.003 838 410.6 1.208
0.004 445 487.0 1.406
0.005 058 560.1 1.580
0.005 664 629.5 1.742
v (m3/kg) 250 h (kJ/kg) s (kJ/kg⋅K)
0.002 397 179.2 0.480
0.002 644 247.7 0.698
0.003 024 319.1 0.910
0.003 327 392.1 1.111
0.003 743 465.5 1.301
0.004 190 538.2 1.476
0.004 648 608.5 1.639
v (m3/kg) 300 h (kJ/kg) s (kJ/kg⋅K)
0.002 317 179.1 0.440
0.002 517 244.6 0.670
0.002 578 312.6 0.850
0.003 037 382.2 1.043
0.003 351 452.9 1.226
0.003 690 524.1 1.398
0.004 042 593.9 1.558
v (m3/kg) 400 h (kJ/kg) s (kJ/kg⋅K)
0.002 203 182.7 0.377
0.002 352 244.9 0.576
0.002 522 308.8 0.764
0.002 711 374.3 0.946
0.002 919 441.3 1.119
0.003 413 509.8 1.285
0.003 382 578.3 1.442
v (m3/kg) 500 h (kJ/kg) s (kJ/kg⋅K)
0.002 122 189.1 0.326
0.002 245 250.7 0.523
0.002 379 312.8 0.707
0.002 524 376.1 0.882
0.002 678 440.4 1.048
0.002 847 505.8 1.206
0.003 022 572.0 1.358
THERMODYNAMIC PROPERTIES TABLE 2-259
2-239
Saturated Fluorine* hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−4 Pa⋅s
kf, W/(m⋅K)
−158.6 −153.5 −149.1 −141.8 −134.4
40.9 42.0 45.8 49.6 53.2
1.602 1.642 1.768 1.885 1.995
5.314 5.235 5.004 4.816 4.666
1.446 1.442 1.437 1.442 1.450
8.8 8.0 6.0 4.7 3.8
0.186 0.184 0.177 0.170 0.162
0.583 0.309 0.176 0.108 0.069
−127.0 −119.5 −111.9 −104.3 −96.5
56.8 60.1 63.3 66.1 68.6
2.097 2.194 2.285 2.372 2.455
4.540 4.433 4.342 4.262 4.191
1.460 1.474 1.498 1.535 1.555
3.19 2.71 2.33 2.00 1.76
0.154 0.146 0.137 0.129 0.120
7.193.−4 7.412.−4 7.659.−4 7.948.−4 8.283.−4
0.0466 0.0323 0.0231 0.0168 0.0125
−88.6 −80.5 −72.2 −63.6 −54.5
70.7 72.4 73.6 74.1 73.9
2.535 2.612 2.688 2.763 2.837
4.127 4.068 4.012 3.959 3.906
1.585 1.630 1.692 1.782 1.888
1.53 1.36 1.21 1.08 0.96
0.112 0.103 0.095 0.087 0.080
8.696.−4 9.223.−4 9.963.−4 1.119.−3 1.743.−3
0.0093 0.0069 0.0051 0.0036 0.0017
−44.9 −34.5 −22.7 −8.4 23.9
72.7 70.2 65.6 56.9 23.9
2.912 2.989 3.073 3.170 3.388
3.864 3.795 3.727 3.636 3.388
2.05 2.33 2.90 3.64 ∞
0.86 0.74 0.63 0.49
0.073 0.066 0.070 0.105 ∞
T, K
P, bar
vf, m3/kg
vg, m3/kg
53.5t 55 60 65 70
0.0025 0.0041 0.0155 0.0477 0.1230
5.866.−4 5.898.−4 6.005.−4 6.119.−4 6.240.−4
46.2 17.1 8.46 2.93 1.24
75 80 85 90 95
0.276 0.555 1.019 1.740 2.802
6.369.−4 6.508.−4 6.657.−4 6.819.−4 6.997.−4
100 105 110 115 120
4.280 6.280 8.885 12.20 16.33
125 130 135 140 144.3c
21.37 27.48 34.72 43.47 52.15
*Values reproduced or converted from Prydz and Straty, NBS Tech. Note 392, rev., September 1973. t = triple point; c = critical point. The notation 5.866.−4 signifies 5.866 × 10−4. TABLE 2-260
Fluorine Gas at Atmospheric Pressure*
T, K
84.95
90
100
120
140
160
180
200
220
240
260
280
300
v, m3/kg h, kJ/kg s, kJ/(kg⋅K)
0.1776 63.22 4.342
0.1892 67.30 4.390
0.2118 75.27 4.474
0.2562 90.96 4.616
0.3002 106.53 4.737
0.3439 122.06 4.840
0.3874 137.62 4.932
0.4309 153.2 5.014
0.4744 169.0 5.090
0.5176 184.9 5.158
0.5610 201.0 5.221
0.6043 217.2 5.282
0.6476 233.7 5.340
*Extracted from Prydz and Straty, NBS Tech. Note 392, 1970. This source is recommended for other pressures and temperatures. Other information is contained in J. Chem. Phys., 53 (1970): 2359; and J. Res. NBS, 74A (1970): 499, 661, 747. TABLE 2-261
TABLE 2-262
Flutec
Proprietary name for a series of fluorocarbons produced by the Imperial Smelting Corp., Avonmouth, Bristol, UK. Bulletins of thermodynamic properties include PP1 (C6F14), PP2 (C7F14), PP3 (C8F16), PP5 (C10F18), PP9 (C11F20), and PP50, usually for 0.1–100 kg/m2, 0–500°C. See also Green, S. W., Chem. & Ind. (1969): 63–67.
TABLE 2-263
Halon
A series of fire-extinguishing fluids. Halon 1211 is produced by ICI, and Halon 1301, by duPont, the latter issuing a bulletin with thermodynamic properties and a diagram for the range 0.6–600 psia, −160–460°F.
Saturated Helium3*
T, K
P, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
1.0 1.1 1.2 1.3 1.4
0.0122 0.0182 0.0274 0.0370 0.0517
0.01222 0.01224 0.01227 0.01231 0.01236
1.72 1.33 1.02 0.805 0.649
−5.69 −5.49 −5.26 −5.01 −4.75
6.75 7.40 8.03 8.65 9.27
2.28 2.65 2.95 3.20 3.40
14.72 14.34 14.02 13.70 13.41
1.5 1.6 1.7 1.8 1.9
0.0659 0.0871 0.107 0.137 0.163
0.01241 0.01247 0.01254 0.01262 0.01271
0.526 0.437 0.363 0.308 0.260
−4.47 −4.17 −3.84 −3.47 −3.07
9.88 10.46 11.04 11.60 12.15
3.60 3.80 3.91 4.01 4.13
13.13 12.88 12.53 12.38 12.14
2.0 2.1 2.2 2.3 2.4
0.202 0.237 0.284 0.326 0.385
0.01282 0.01294 0.01308 0.01324 0.01343
0.222 0.189 0.164 0.142 0.124
−2.64 −2.17 −1.55 −0.99 −0.34
12.68 13.19 13.67 14.13 14.57
4.26 4.40 4.55 4.71 4.87
11.91 11.69 11.47 11.25 11.04
2.5 2.6 2.7 2.8 2.9
0.438 0.508 0.576 0.653 0.732
0.01365 0.01390 0.01419 0.01456 0.01497
0.109 0.096 0.085 0.074 0.064
0.36 1.16 2.01 2.96 4.01
14.98 15.37 15.89 16.40 16.37
5.03 5.20 5.38 5.57 5.77
10.84 10.64 10.41 10.17 9.92
3.0 3.1 3.2 3.3 3.32c
0.803 0.907 1.023 1.128 1.165
0.01549 0.01614 0.01720 0.01902 0.02394
0.055 0.047 0.039 0.028 0.024
5.28 6.70 8.44 10.66 13.25
16.32 16.20 15.98 14.50 13.25
6.00 6.24 6.54 6.96 7.50
9.66 9.34 8.90 8.35 7.50
*Converted and smoothed from a tabulation of Gibbons and Nathan, USAF Rep. AFML-TR-67-175, 1967. c = critical point. Kelly, D. P. and W. K. Haubach, in AEC R&D rept. MLM 1161, 1963 (56 pp.), give a comprehensive graphical comparison of the properties of He3 and He4.
2-240
PHYSICAL AND CHEMICAL DATA
TABLE 2-264
Saturated Helium4
Temperature, K
Pressure, bar
vf, m3/kg
0.8 0.9 1.0 1.1 1.2
1.475.−5 5.379.−5 1.557.−4 3.800.−4 8.148.−4
0.00689 0.00689 0.00689 0.00689 0.00689
1.3 1.4 1.5 1.6 1.7
0.00158 0.00282 0.00472 0.00746 0.01128
0.00689 0.00689 0.00689 0.00688 0.00688
1.8 1.9 2.0 2.1 2.2
0.01638 0.02299 0.03129 0.04141 0.05335
0.00688 0.00687 0.00686 0.00685 0.00684
2.3 2.4 2.5 2.6 2.7
0.06730 0.08354 0.01023 0.1237 0.1481
2.8 2.9 3.0 3.1 3.2
vg, m3/kg
hf, kJ/kg
h g, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf , kJ/(kg⋅K)
cpg, kJ/(kg⋅K)
0.0019 0.0054 0.0127 0.0268 0.0518
19.42 19.94 20.44 20.95 21.44
0.0047 0.0087 0.0163 0.0296 0.0510
23.94 21.86 20.44 18.82 17.67
0.022 0.050 0.100 0.185 0.318
5.210 5.230 5.262 5.305 5.360
0.0932 0.1579 0.2543 0.3923 0.5836
21.92 22.40 22.87 23.32 23.77
0.0836 0.1308 0.1962 0.2839 0.3981
16.70 15.86 15.13 14.49 13.93
0.511 0.780 1.138 1.602 2.193
5.424 5.496 5.574 5.654 5.736
2.1993 1.6420 1.2601 0.9921 0.7994
0.8422 1.186 1.642 2.261 3.090
24.20 24.63 25.04 25.45 25.85
0.5437 0.7270 0.9578 1.256 1.638
13.43 12.98 12.58 12.23 11.92
2.938 3.893 5.187 7.244 4.222
5.818 5.898 5.975 6.046 6.111
0.00685 0.00687 0.00690 0.00693 0.00695
0.6566 0.5470 0.4608 0.3923 0.3367
3.418 3.678 3.922 4.161 4.408
26.24 26.63 27.00 27.37 27.72
1.780 1.886 1.980 2.068 2.155
11.65 11.40 11.17 10.96 10.76
2.685 2.375 2.284 2.320 2.351
6.170 6.228 6.285 6.344 6.406
0.1755 0.2063 0.2405 0.2784 0.3201
0.00699 0.00703 0.00707 0.00713 0.00717
0.2913 0.2537 0.2223 0.1958 0.1728
4.662 4.923 5.195 5.483 5.787
28.06 28.38 28.69 28.98 29.26
2.240 2.324 2.408 2.494 2.581
10.57 10.39 10.22 10.05 9.90
2.403 2.486 2.597 2.740 2.896
6.470 6.540 6.616 6.700 6.792
3.3 3.4 3.5 3.6 3.7
0.3659 0.4159 0.4704 0.5296 0.5935
0.00723 0.00728 0.00735 0.00742 0.00749
0.1542 0.1376 0.1232 0.1107 0.0997
6.108 6.448 6.806 7.184 7.581
29.52 29.76 29.97 30.17 30.34
2.670 2.780 2.852 2.946 3.042
9.747 9.600 9.458 9.318 9.181
3.061 3.273 3.413 3.601 3.801
6.897 7.015 7.150 7.305 7.484
3.8 3.9 4.0 4.1 4.2
0.6625 0.7366 0.8162 0.9014 0.9923
0.00758 0.00766 0.00776 0.00786 0.00797
0.0900 0.0814 0.0738 0.0669 0.0606
7.998 8.437 8.899 9.387 9.901
30.48 30.60 30.68 30.73 30.74
3.140 3.239 3.341 3.444 3.551
9.046 8.911 8.776 8.641 8.504
4.017 4.254 4.519 4.820 5.170
7.694 7.942 8.238 8.641 9.033
4.3 4.4 4.5 4.6 4.7
1.089 1.193 1.303 1.419 1.543
0.00810 0.00824 0.00841 0.00860 0.00881
0.0550 0.0499 0.0452 0.0408 0.0367
10.45 11.02 11.64 12.31 13.04
30.71 30.62 30.47 30.24 29.91
3.661 3.775 3.893 4.018 4.151
8.363 8.218 8.067 7.906 7.732
5.587 6.097 6.742 7.590 8.763
4.8 4.9 5.0 5.1 5.195c
1.674 1.813 1.960 2.116 2.275
0.00907 0.00941 0.00986 0.01056 0.01436
0.0329 0.0291 0.0252 0.0207 0.0145
13.85 14.76 15.85 17.26
29.45 28.80 27.83 26.08
4.296 4.458 4.649 4.898
7.539 7.327 7.041 6.624
1125.9 347.1 133.0 59.8 30.4 16.93 10.17 6.49 4.35 3.04
c = critical pt. From Arp, V. D. and R. D. McCarty, N.I.S.T. TN 1334, 1989 (142 pp.).
10.51 13.38 19.02 34.60
9.58 10.29 11.22 12.50 14.37 17.32 22.64 34.93 95.84
THERMODYNAMIC PROPERTIES TABLE 2-265
Superheated Helium*
P, bars
0
2-241
Temp., °C 1 v h s
5.677 0.327 0.0116
100
200
300
400
500
600
800
1000
7.754 519.6 1.620
9.831 1039 2.853
11.908 1558 3.849
13.985 2078 4.684
16.063 2597 5.403
18.140 3116 6.035
22.294 4155 7.106
26.448 5193 7.993
5 v h s
1.138 1.636 −3.343
1.553 520.9 −1.723
1.968 1040 −0.490
2.384 1560 0.506
2.799 2079 1.341
3.215 2598 2.060
3.630 3117 2.692
4.461 4156 3.763
5.291 5194 4.650
10 v h s
0.5704 3.272 −4.782
0.780 522.5 −3.162
0.986 1042 −1.929
1.193 1561 −0.934
1.401 2080 −0.098
1.609 2600 0.621
1.816 3119 1.252
2.232 4157 2.323
2.647 5196 3.211
20 v h s
0.2867 6.544 −6.221
0.3904 525.8 −4.601
0.4942 1045 −3.368
0.5979 1564 −2.373
0.7017 2083 −1.537
0.9093 2603 −0.818
1.1169 3122 −0.187
1.3245 4160 0.884
1.8435 5199 1.771
50 v h s
0.1164 16.360 −8.121
0.1579 535.5 −6.501
0.1993 1055 −5.268
0.2408 1574 −4.273
0.2822 2093 −3.438
0.3257 2612 −2.719
0.3652 3131 −2.088
0.4481 4169 −1.017
0.5311 5207 −0.130
100 v h s
0.0597 37.720 −9.555
0.0803 551.7 −7.936
0.1010 1071 −6.703
0.1217 1590 −5.709
0.1424 2108 −4.874
0.1631 2627 −4.155
0.1838 3146 −3.524
0.2252 4184 −2.454
0.2666 5222 −1.567
150 v h s
0.0407 49.080 −10.391
0.0545 567.9 −8.773
0.0682 1087 −7.541
0.0820 1605 −6.546
0.0958 2124 −5.712
0.1095 2643 −4.994
0.1233 3161 −4.363
0.1509 4199 −3.293
0.1785 5236 −2.407
200 v h s
0.0312 65.440 −10.983
0.0416 584.1 −9.635
0.0518 1103 −8.134
0.0622 1621 −7.139
0.0725 2140 −6.306
0.0828 2658 −5.588
0.0931 3176 −4.957
0.1137 4213 −3.888
0.1344 5250 −3.002
*Extracted from Tsederberg, Popov, et al., Thermodynamic and Thermophysical Properties of Helium, Atomizdat, Moscow, 1969, and NBS-NSF TT 50096, 1971. Copyright material. Reproduced by permission. This source contains entries for many more temperatures and pressures than can be reproduced here. v = volume, m3/kg; h = enthalpy, kJ/ kg; s = entropy, kJ/(kg·K). The 1993 ASHRAE Handbook—Fundamentals (SI ed.) has a thermodynamic chart for pressures from 0.1 to 50 bar and temperatures from 2.5 to 15 K. Saturation and superheat tables to 9000 psia, 800°R; and a chart to 700 psia, 40°R appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. A useful compilation of properties is given by Betts, D. S., Cryogenics, 16, 1 (1976): 3–16. A 32-term equation of state for the range up to 20,000 bar, 2–1500 K is given by McCarty, R. D. and V. D. Arp, Advan. Cryog. Engng., 35 (1990): 1465–1475.
TABLE 2-266
Helium4 Gas at Atmospheric Pressure*
T, K
4.224
5
10
20
30
40
50
75
100
200
300
400
500
600
800
1000
v, m3/kg h, kJ/kg s, kJ/(kg⋅K)
0.0591 30.30 8.327
0.0834 36.18 9.614
0.1612 64.91 13.369
0.4094 117.95 17.321
0.6161 170.24 19.442
0.8218 222.4 20.94
1.0273 274.4 22.10
1.5403 404.4 24.21
2.053 534.2 25.71
4.102 1054 29.30
6.154 1573 31.41
8.191 2092 32.90
10.24 2612 34.06
12.31 3131 35.01
16.40 4170 36.50
20.50 5208 37.66
*From McCarty, NBS Rep. 9762, 1970. Reproduced by permission. The source contains values for further temperatures and for other functions, usually to additional significant figures.
2-242
PHYSICAL AND CHEMICAL DATA
TABLE 2-267
Saturated n-Heptane*
P, bar
vf, m3/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µ f, 10−4 Pa⋅s
k f, W/(m⋅K)
182.6t 200 220 240 250
0.00002 0.00019 0.00133 0.00303
1.292.−3 1.316.−3 1.344.−3 1.374.−3 1.389.−3
284.1 319.4 359.7 400.5 421.3
722.6 757.1 791.4 808.3
2.260 2.441 2.636 2.814 2.899
4.457 4.442 4.443 4.447
2.025 2.011 2.026 2.063 2.088
39.4 21.0 12.6 8.52 7.23
0.150 0.148 0.145 0.142 0.140
260 270 280 290 300
0.00635 0.01316 0.02347 0.03997 0.06674
1.405.−3 1.422.−3 1.440.−3 1.457.−3 1.475.−3
3.744
442.3 463.6 485.2 507.2 529.6
824.9 841.2 857.8 874.8 891.9
2.981 3.061 3.140 3.217 3.293
4.453 4.460 4.471 4.485 4.501
2.117 2.147 2.180 2.216 2.252
6.52 5.46 4.83 4.29 3.85
0.137 0.135 0.132 0.129 0.126
310 320 330 340 350
0.1070 0.1656 0.2461 0.3614 0.5130
1.494.−3 1.514.−3 1.534.−3 1.555.−3 1.578.−3
2.412 1.596 1.101 0.7650 0.5510
552.3 575.4 598.8 622.8 647.0
908.9 926.0 943.3 961.2 979.1
3.367 3.441 3.513 3.584 3.655
4.517 4.537 4.557 4.579 4.604
2.291 2.329 2.370 2.412 2.454
3.48 3.17 2.89 2.66 2.45
0.123 0.121 0.119 0.116 0.114
360 370 371.6 380 390
0.712 0.967 1.013 1.289 1.689
1.601.−3 1.625.−3 1.629.−3 1.651.−3 1.678.−3
0.4058 0.3036 0.2904 0.2308 0.1781
671.9 697.1 701.9 723.9 750.4
997.5 1016.1 1019.8 1035.4 1054.2
3.725 3.794 3.805 3.864 3.932
4.629 4.656 4.660 4.684 4.711
2.500 2.548 2.556 2.60 2.65
2.24 2.04 2.01 1.86 1.71
0.111 0.109 0.108 0.107 0.105
400 420 440 460 480
2.180 3.471 5.268 7.691 10.92
1.708.−3 1.775.−3 1.853.−3 1.954.−3 2.065.−3
0.1388 0.0734 0.0576 0.0389 0.0265
777.2
1073.2
4.000
4.740
2.70 2.81 2.93 3.05 3.19
1.58 1.35 1.15 0.97 0.82
0.103 0.099 0.095 0.091 0.087
500 520 540.1c
15.10 20.43 27.35
2.235.−3 2.52.−3 4.3.−3
3.38 3.7
0.67
0.080
0.0043
T, K
vg, m3/kg
h f, kJ/kg
0.0178
*Values of P and v interpolated and converted from tables in Vargaftik, Handbook of Thermophysical Properties of Gases and Liquids, Hemisphere, Washington, and McGraw-Hill, New York, 1975. Values of h and s calculated from API tables published by the Thermodynamics Research Center, Texas A&M University, College Station. t = triple point; c = critical point. Saturation and superheat tables and a diagram to 200 bar, 680 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.).
TABLE 2-268
Hexane
Saturation and superheat tables and a diagram to 100 bar, 680 K, are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.).
THERMODYNAMIC PROPERTIES TABLE 2-269
2-243
Saturated Hydrazine
Temperature, K
vf, m3/kg
vg, m3/kg
h f, kJ/kg
hg, kJ/kg
s f, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
1.013 1.135 1.560 2.102 2.786
0.001 053 0.001 060 0.001 081 0.001 104 0.001 127
0.9833 0.8850 0.6579 0.4994 0.3850
−105.9 −104.8 −101.4 −97.6 −93.9
65.4 66.0 68.2 70.6 73.0
0.5994 0.6029 0.6120 0.6211 0.6300
1.0426 1.0409 1.0360 1.0314 1.0275
4.732 7.610 11.76 17.42 29.59
0.001 178 0.001 235 0.001 299 0.001 374 0.001 460
0.2355 0.1500 0.1005 0.0690 0.0407
−86.1 −76.9 −67.1 −57.3 −47.8
77.6 82.1 86.6 90.8 94.6
0.6492 0.6707 0.6916 0.7124 0.7320
1.0212 1.0163 1.0118 1.0086 1.0058
Pressure, bar
386.6 390 400 410 420 440 460 480 500 520 540 560 580 600 620
34.75 47.09 62.44 81.17 102.7
0.001 563 0.001 681 0.001 835 0.002 045 0.002 320
0.0353 0.0263 0.0196 0.0142 0.0106
−36.0 −25.2 −12.4 5.2 23.2
97.7 101.2 103.6 104.2 103.6
0.7566 0.7762 0.8002 0.8335 0.8671
1.0042 1.0020 1.0002 0.9988 0.9967
640 653 c
128.1 146.9
0.002 86 0.004 33
0.0074 0.0043
45.9 83.7
98.1 83.7
0.9035 0.9715
0.9906 0.9715
Converted from E. F. Fricke, Republic Aviation Co. rept. F-5028-101. c = critical point.
TABLE 2-270
Saturated n-Hydrogen*
T, K
P, bar
vf, m3/kg
v g, m3/kg
hf, kJ/kg
h g, kJ/kg
s f, kJ/(kg⋅K)
s g, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µ f, 10−4 Pa⋅s
k f, W/(m⋅K)
13.95t 14 15 16 17
0.072 0.074 0.127 0.204 0.314
0.01298 0.01301 0.01316 0.01332 0.01348
7.974 7.205 4.488 2.954 2.032
218.3 219.6 226.4 233.8 241.6
667.4 669.3 678.2 686.7 694.7
14.079 14.173 14.640 15.104 15.568
46.635 46.301 44.763 43.418 42.227
6.36 6.47 6.91 7.36 7.88
0.255 0.248 0.218 0.194 0.175
0.073 0.075 0.083 0.089 0.093
18 19 20 21 22
0.461 0.654 0.901 1.208 1.585
0.01366 0.01387 0.01407 0.01430 0.01455
1.449 1.064 0.8017 0.6177 0.4828
249.9 258.8 268.3 278.4 289.2
702.1 708.8 714.8 720.2 724.4
16.032 16.498 16.966 17.440 17.919
41.158 40.188 39.299 38.485 37.710
8.42 8.93 9.45 10.13 10.82
0.159 0.146 0.135 0.125 0.116
0.095 0.097 0.098 0.100 0.101
23 24 25 26 27
2.039 2.579 3.213 3.950 4.800
0.01483 0.01515 0.01551 0.01592 0.01639
0.3829 0.3072 0.2489 0.2032 0.1667
300.8 313.3 326.7 341.2 357.0
727.6 729.8 730.7 730.2 728.0
18.405 18.901 19.408 19.929 20.473
36.973 36.266 35.579 34.900 34.221
11.69 12.52 13.44 14.80 16.17
0.108 0.101 0.094 0.088 0.082
0.101 0.101 0.100 0.098 0.096
28 29 30 31 32
5.770 6.872 8.116 9.510 11.07
0.01696 0.01765 0.01854 0.01977 0.02174
0.1370 0.1125 0.0919 0.0738 0.0571
374.3 393.6 415.4 441.3 474.7
723.7 716.6 705.9 689.7 663.2
21.041 21.650 22.315 23.075 24.032
33.524 32.795 32.002 31.091 29.926
18.48 22.05 26.59 36.55 65.37
0.076 0.070 0.065 0.058 0.051
0.094 0.091 0.087 0.086 0.092
33.18c
13.13
0.03182
0.0318
565.4
565.4
26.680
26.680
∞
*Values extracted and occasionally rounded off from McCarty, Hord, and Roder, NBS Monogr. 168, 1981. t = triple point; c = critical point.
∞
2-244
PHYSICAL AND CHEMICAL DATA
TABLE 2-271
Compressed n-Hydrogen* Temperature, K
Pressure, bar
15
20
30
40
50
60
80
100
150
200
6.076 679.2 46.02
8.176 731.6 49.04
12.333 835.5 53.25
16.473 938.9 56.23
20.606 1042.3 58.53
24.736 1146 60.43
32.991 1356 63.45
41.244 1575 65.89
61.870 2172 70.68
82.495 2826 74.46
1 v h s
0.0131 227.3 14.62
0.0141 268.3 16.96
1.196 826.0 43.56
1.625 932.7 46.63
2.046 1037.9 48.98
2.463 1143 50.89
3.295 1354 53.93
4.123 1574 56.38
6.190 2172 61.17
8.254 2826 64.96
5 v h s
0.0131 231.7 14.57
0.0140 272.1 16.88
0.2006 775.0 35.80
0.3039 903.4 39.52
0.3958 1017.6 42.07
0.4839 1128 44.07
0.6553 1345 47.20
0.8238 1568 49.68
1.241 2170 54.66
1.655 2826 58.31
10 v h s
0.0130 237.2 14.50
0.0138 277.0 16.77
0.0181 412.1 22.09
0.1376 861.8 35.95
0.1895 991.1 38.85
0.2366 1109 40.99
0.3255 1334 44.23
0.4116 1560 46.75
0.6221 2167 51.63
0.8303 2826 55.44
20 v h s
0.0129 248.2 14.37
0.0136 286.9 16.58
0.0167 406.5 21.33
0.0521 752.0 31.07
0.0866 934.7 35.19
0.1135 1070 37.67
0.1611 1312 41.15
0.2057 1546 43.76
0.3129 2163 48.71
0.4179 2826 52.55
40 v h s
0.0133 307.3 16.26
0.0155 413.5 20.50
0.0216 589.3 25.49
0.0376 823.5 30.73
0.0533 997 33.91
0.0796 1271 37.87
0.1033 1521 40.65
0.1586 2155 45.75
0.2119 2826 49.64
60 v h s
0.0130 328.0 15.98
0.0147 427.2 19.95
0.0182 570.1 24.03
0.0254 757.0 28.19
0.0351 940 31.54
0.0532 1237 35.82
0.0697 1499 38.76
0.1073 2149 43.99
0.1433 2828 47.92
80 v h s
0.0127 348.9 15.74
0.0142 443.5 19.53
0.0167 572.3 23.21
0.0211 732.8 26.78
0.0273 905 29.93
0.0406 1210 34.34
0.0531 1482 37.37
0.0818 2146 42.72
0.1090 2831 46.69
100 v h s
0.0125 369.8 15.53
0.0138 461.1 19.19
0.0158 581.5 22.63
0.0190 727.4 25.88
0.0233 888 28.80
0.0335 1192 33.19
0.0434 1469 36.28
0.0666 2144 41.73
0.0885 2835 45.73
200 v h s
0.0117 474.4 14.71
0.0125 556.1 17.99
0.0136 658.7 20.93
0.0150 776.9 23.56
0.0167 908 25.94
0.0207 1182 29.88
0.0253 1458 32.97
0.0368 2156 38.59
0.0480 2869 42.72
400 v h s
0.0113 751.0 16.59
0.0119 841.9 19.20
0.0126 945.4 21.50
0.0134 1059 23.58
0.0151 1303 27.07
0.0171 1560 29.94
0.0225 2249 35.48
0.0279 2973 39.67
600 v h s
0.0106 941.5 15.68
0.0110 1027 18.14
0.0115 1124 20.29
0.0120 1231 22.24
0.0131 1463 25.57
0.0144 1709 28.31
0.0178 2385 33.74
0.0214 3107 37.92
800 v h s
0.0104 1209 17.35
0.0107 1302 19.43
0.0111 1405 21.30
0.0120 1628 24.50
0.0130 1870 27.20
0.0155 2535 32.54
0.0181 3255 36.70
1000 v h s
0.0099 1387 16.72
0.0102 1478 18.75
0.0106 1578 20.58
0.0112 1796 23.70
0.0120 2032 26.33
0.0140 2692 31.63
0.0160 3403 35.75
0.1 v h s
*Values extracted and sometimes rounded off from the tables of McCarty, Hord, and Roder, NBS Monogr. 168, 1981. This source contains an exhaustive tabulation of property values for both the normal and the para forms of hydrogen. v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K). The 1993 ASHRAE Handbook—Fundamentals (SI ed.) has a thermodynamic diagram for 0.1 to 500 bar for temperatures up to 100 K. Tables and a Mollier chart from 10−4 to 1000 atm, 300–20,000 K are given by Kubin, R. F. and L. L. Presley, NASA SP 3002, 1964. Liebenberg, D. H., R. L. Mills, and others, in LA-6645-MS, 1977 (26 pp.), give properties from 75 to 307 K for pressures from 2 to 20 kbar. See also Baker, J. R. and H. F. Swift, J. Appl. Phys., 43, 3 (1972): 950–953. An extensive collection of data for H2, D2, T2, and so on below 30 K is given by Souers, P. C., UCRL 52628, 1979 (91 pp.); and for temperatures below 40 K by Roder, H. M., G. E. Childs, et al., NBS TN 641, 1973 (114 pp.). Saturation and superheat tables to 10,000 psia, 900°R and a chart to 180°R appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For viscosity, thermal conductivity, and specific heat, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.
THERMODYNAMIC PROPERTIES
2-245
Temperature, K 250
300
350
400
450
500
600
700
800
900
1000
103.12 3517 77.53
123.23 4227 80.13
144.35 4945 82.34
164.97 5668 84.27
185.60 6393 85.98
206.22 7118 87.51
247.46 8571 90.15
288.70 10028 92.40
329.94 11493 94.36
371.18 12969 96.10
412.43 14458 97.66
10.32 3517 68.03
12.38 4227 70.63
14.44 4946 72.85
16.50 5669 74.78
18.57 6393 76.48
20.63 7118 78.01
24.75 8571 80.66
28.88 10029 82.91
33.00 11494 84.86
37.13 12969 86.60
41.25 14459 88.17
2.069 3518 61.39
2.482 4229 63.99
2.895 4948 66.21
3.307 5671 68.14
3.720 6396 69.84
4.132 7121 71.37
4.957 8574 74.02
5.782 10032 76.27
6.607 11497 78.23
7.432 12973 79.96
8.257 14462 81.53
1.038 3519 58.52
1.245 4231 61.12
1.451 4951 63.34
1.658 5674 65.28
1.864 6399 66.98
2.070 7125 68.51
2.483 8578 71.16
2.896 10036 73.41
3.308 11501 75.37
3.720 12977 77.10
4.133 14467 78.67
0.522 3522 55.65
0.6259 4235 58.26
0.7294 4956 60.48
0.8328 5680 62.41
0.9361 6406 64.12
1.040 7132 65.65
1.246 8586 68.30
1.452 10044 70.55
1.658 11509 72.51
1.865 12985 74.24
2.071 14475 75.81
0.2644 3527 52.76
0.3166 4244 55.38
0.3685 4967 57.61
0.4204 5692 59.55
0.4721 6419 61.26
0.5238 7146 62.79
0.6271 8601 65.44
0.7303 10059 67.69
0.8335 11525 69.65
0.9366 13002 71.39
1.040 14492 72.95
0.1786 3533 51.05
0.2136 4253 53.69
0.2483 4978 55.92
0.2829 5705 57.86
0.3174 6432 59.58
0.3519 7160 61.11
0.4209 8616 63.76
0.4897 10075 66.02
0.5585 11542 67.97
0.6273 13018 70.51
0.6961 14508 71.28
0.1357 3540 49.84
0.1621 4263 52.49
0.1882 4989 54.73
0.2142 5718 56.67
0.2401 6446 58.39
0.2660 7174 59.92
0.3177 8631 62.57
0.3694 10091 64.83
0.4120 11558 66.79
0.4726 13035 68.52
0.5242 14525 70.09
0.1099 3547 48.89
0.1312 4273 51.55
0.1521 5001 53.79
0.1730 5731 55.74
0.1937 6460 57.46
0.2145 7189 59.00
0.2559 8647 61.65
0.2972 10107 63.90
0.3385 11574 65.87
0.3798 13051 67.60
0.4211 14542 69.17
0.0588 3594 45.94
0.0695 4329 48.62
0.0801 5064 50.89
0.0905 5798 52.85
0.1001 6531 54.58
0.1114 7263 56.12
0.1321 8724 58.78
0.1528 10187 61.04
0.1734 11656 63.00
0.1941 13134 64.74
0.2147 14625 66.31
0.0334 3716 42.98
0.0388 4458 45.68
0.0441 5202 47.97
0.0493 5943 49.95
0.0545 6681 51.69
0.0597 7416 53.24
0.0701 8883 55.91
0.0804 10349 58.17
0.0908 11820 60.14
0.1011 13300 61.88
0.1114 14792 63.45
0.0249 3854 41.24
0.0285 4600 43.95
0.0321 5349 46.26
0.0355 6095 48.26
0.0390 6836 50.00
0.0425 7574 51.56
0.0494 9045 54.24
0.0562 10513 56.50
0.0631 11985 58.47
0.0700 13466 60.21
0.0768 14958 61.78
0.0207 4003 40.03
0.0234 4748 42.73
0.0260 5501 45.05
0.0286 6249 47.05
0.0312 6993 48.81
0.0338 7734 50.37
0.0390 9207 53.05
0.0441 10677 55.32
0.0492 12150 57.29
0.0543 13631 59.03
0.0594 15124 60.60
0.0181 4156 39.10
0.0202 4898 41.79
0.0223 5654 44.12
0.0244 6405 46.12
0.0265 7151 47.88
0.0286 7893 49.45
0.0327 9370 52.14
0.0367 10842 54.41
0.0408 12316 56.38
0.0449 13797 58.12
0.0490 15289 59.69
2-246
PHYSICAL AND CHEMICAL DATA
TABLE 2-272
Saturated para-Hydrogen*
T, K
P, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf , kJ/(kg⋅K)
µf, 10−4 Pa⋅s
kf, W/(m⋅K)
13.8t 14 15 16 17
0.070 0.079 0.134 0.216 0.329
0.0130 0.0130 0.0132 0.0133 0.0135
7.97 7.20 4.49 2.96 2.03
−308.9 −307.6 −300.9 −293.4 −285.6
140.3 142.1 151.1 159.6 167.6
4.97 5.06 5.53 5.99 6.45
37.52 37.19 36.65 34.31 33.11
6.37 6.47 6.91 7.36 7.88
0.255 0.248 0.218 0.194 0.175
0.073 0.075 0.082 0.089 0.092
18 19 20 21 22
0.482 0.682 0.935 1.250 1.634
0.0137 0.0139 0.0141 0.0143 0.0146
1.45 1.07 0.802 0.618 0.483
−277.3 −268.4 −258.9 −248.8 −237.9
175.0 181.7 187.7 193.0 197.3
6.92 7.38 7.85 8.32 8.80
32.05 31.08 30.19 29.37 28.60
8.42 8.93 9.45 10.13 10.82
0.159 0.146 0.135 0.125 0.116
0.095 0.097 0.098 0.100 0.101
23 24 25 26 27
2.096 2.645 3.288 4.035 4.892
0.0148 0.0152 0.0155 0.0159 0.0164
0.383 0.307 0.249 0.203 0.167
−226.3 −213.9 −200.4 −185.9 −170.2
200.5 202.7 203.6 203.1 200.9
9.29 9.78 10.29 10.81 11.36
27.86 27.15 26.46 25.79 25.11
11.69 12.52 13.44 14.81 16.18
0.108 0.101 0.094 0.088 0.082
0.101 0.100 0.099 0.098 0.096
0.076 0.070 0.065 0.058 0.051
0.094 0.091 0.087 0.088 0.092
28 29 30 31 32
5.88 6.98 8.23 9.63 11.20
0.0170 0.0177 0.0185 0.0198 0.0217
0.137 0.113 0.092 0.074 0.057
−152.9 −133.6 −111.7 −85.8 −52.4
196.5 189.5 178.8 162.6 136.1
11.93 12.54 13.20 13.96 14.92
24.41 23.68 22.89 21.98 20.81
18.5 22.1 26.6 36.6 65.4
33c
12.93
0.0318
0.032
38.3
38.3
17.56
17.56
∞
∞
*Values extracted and occasionally rounded off from McCarty, Hord, and Roder, NBS Monogr. 168, 1981. t = triple point; c = critical point. Saturation and superheat tables to 12,000 psia, 900°R and a chart to 180°R appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) has a thermodynamic chart for pressures from 0.1 to 1000 bar for temperatures up to 100 K.
TABLE 2-273 T, K 273 300 350 400 450
Saturated Hydrogen Peroxide*
P, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−4 Pa⋅s
kf, W/(m⋅K)
0.0004 0.0031 0.0564 0.4521 2.143
0.00068 0.00069 0.00072 0.00076 0.00081
1672 235 15.1 2.12 0.487
−5577 −5510 −5376 −5238 −5091
−4027 −3995 −3933 −3878 −3820
2.990 3.224 3.631 4.032 4.346
8.662 8.269 7.758 7.440 7.172
1.45 1.48 1.54 1.61 1.68
18.0 11.3 4.3 2.2 1.3
0.483 0.481 0.474 0.464 0.453
1.75 1.82 1.90
500 550 600 650 700
7.126 18.56 40.75 79.27 141.7
0.00088 0.00095 0.00107 0.00125 0.00171
0.155 0.0605 0.0268 0.0125 0.0048
−4945 −4794 −4635 −4463 −4195
−3777 −3745 −3731 −3746 −3860
4.656 4.941 5.209 5.485 5.682
6.992 6.846 6.720 6.582 6.339
708.5c
155.3
0.00284
0.0028
−4012
−4012
5.732
5.732
0.89 0.65 0.50
0.443 0.431 0.416
*Values reproduced or converted from a tabulation by Tsykalo and Tabachnikov in V. A. Rabinovich (ed.), Thermophysical Properties of Gases and Liquids, Standartov, Moscow, 1968; NBS-NSF transl. TT 69-55091, 1970. The reader may be reminded that very pure hydrogen peroxide is very difficult to obtain owing to its decomposition or instability. c = critical point. The FMC Corp., Philadelphia, PA tech. bull. 67, 1969 (100 pp.) contains an enthalpy-pressure diagram to 3000 psia, 1100 K.
TABLE 2-274
Hydrogen Sulfide
West, J. R., Chem. Eng. Progr., 44, 4 (1948): 207–292 gives tables and a chart for the range 1–90 atm., −76 to 1300°F while properties from 10 to 330 bar, 300 to 500 K were tabulated by Lui, C-H., D. M. Bailey, et al., Hydroc. Proc., 65, 7 ( July 1986): 41–43.
THERMODYNAMIC PROPERTIES
Enthalpy-concentration diagram for aqueous hydrogen chloride at 1 atm. Reference states: enthalpy of liquid water at 0°C is zero; enthalpy of pure saturated HCl vapor at 1 atm (−85.03°C) is 8000 kcal/mol. NOTE: It should be observed that the weight basis includes the vapor, which is particularly important in the two-phase region. Saturation values may be read at the ends of the tie lines. [Van Nuys, Trans. Am. Inst. Chem. Eng., 39, 663 (1943).]
FIG. 2-11
2-247
2-248
PHYSICAL AND CHEMICAL DATA
TABLE 2-275
Saturated Isobutane (R600a)* µf , 10−4 Pa⋅s
kf, W/(m⋅K)
1.78 1.87 1.93 1.99
9.46
0.163 0.158 0.149
4.984 4.916 4.878 4.863 4.861
2.05 2.12 2.19 2.28 2.33
6.06 4.21 3.11 2.40 2.14
0.142 0.134 0.127 0.120 0.117
3.617 3.700 3.783 3.865 3.946
4.863 4.867 4.874 4.882 4.891
2.39 2.46 2.53 2.61 2.70
1.93 1.75 1.59 1.46 1.35
0.113 0.110 0.106 0.102 0.099
T, K
P, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
113.6t 120 140 160 180
1.9.−7 9.3.−7 4.8.−5 8.2.−4 0.0070
1.349.−3 1.360.−3 1.396.−3 1.435.−3 1.476.−3
8.60.+6 1.84.+6 4210 278.2 36.66
0.0 11.0 46.0 82.1 119.5
485.3 491.1 510.1 530.8 533.0
1.863 1.957 2.226 2.467 2.688
6.136 5.957 5.541 5.272 5.097
200 220 240 260 270
0.0369 0.1374 0.3989 0.9600 1.4081
1.520.−3 1.568.−3 1.621.−3 1.680.−3 1.712.−3
7.723 2.265 0.8432 0.3738 0.2617
158.5 199.0 241.4 285.8 308.8
576.7 601.5 627.4 654.2 667.7
2.893 3.086 3.270 3.446 3.532
280 290 300 310 320
2.0020 2.7686 3.7365 4.934 6.392
1.746.−3 1.784.−3 1.824.−3 1.868.−3 1.916.−3
0.1882 0.1385 0.1040 0.0794 0.0614
332.3 356.4 381.1 406.4 432.4
681.3 694.9 708.4 721.7 734.8
330 340 350 360 370
8.140 10.21 12.64 15.46 18.72
1.971.−3 2.032.−3 2.103.−3 2.187.−3 2.289.−3
0.0481 0.0380 0.0301 0.0240 0.0190
459.2 486.9 515.7 545.6 577.1
747.7 760.0 771.8 782.7 792.3
4.028 4.109 4.191 4.273 4.357
4.902 4.912 4.923 4.932 4.939
2.81 2.92 3.04 3.17 3.31
1.25 1.15 1.05 0.95 0.85
0.095 0.092 0.088 0.083 0.080
380 390 400 408.0c
22.48 26.82 31.86 36.55
2.420.−3 2.604.−3 2.920.−3 4.464.−3
0.0150 0.0115 0.0083 0.0045
610.6 647.1 689.6 752.5
799.8 803.7 799.6 752.5
4.444 4.536 4.639 4.791
4.942 4.937 4.915 4.791
3.45 3.62 3.85 ∞
0.75 0.63 0.51
0.076 0.071 0.065 ∞
*Values reproduced or converted from Goodwin, NBSIR 79-1612, 1979. t = triple point; c = critical point. The notation 1.9.−7 signifies 1.9 × 10−7. Slightly different values for the range 0.5 to 34.5 bar, 250–404 K appear in Waxman, M. and J. S. Gallagher, J. Chem. Eng. Data, 28, (1983): 224–241. This source also contains superheat tables for 1–400 bar, 250–600 K. Saturation and superheat tables and a diagram to 200 bar, 600 K are given by Reynolds, W. C., Thermodynamic properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Saturation and superheat tables and a chart to 10,000 psia, 640°F appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. Equations and data for thermal conductivity and viscosity are given by Nieuwoldt, J. C., B. LeNeindre, et al., J. Chem. Eng. Data, 32, (1987): 1–8.
TABLE 2-276 T, K
Saturated Krypton*
P, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−4 Pa⋅s
kf, W/(m⋅K)
10 20 30 40 50
3.235.−4 3.246.−4 3.265.−4 3.288.−4 3.313.−4
0.22 1.59 3.84 6.49 9.37
0.0256 0.1141 0.2034 0.2791 0.3431
0.070 0.188 0.247 0.276 0.295
60 70 80 90 100
3.341.−4 3.372.−4 3.407.−4 3.446.−4 3.492.−4
12.40 15.57 18.97 22.58 26.42
0.3982 0.4471 0.4925 0.5353 0.5765
0.311 0.327 0.345 0.366 0.389
110 115.76 115.76 119.76 120
0.732 1.013 1.032
3.544.−4 3.579.−4 4.090.−4 4.143.−4 4.146.−4
0.1529 0.1136 0.1116
30.52 33.18 52.78 54.99 55.09
161.8 162.6 162.6
0.6165 0.6390 0.8095 0.8279 0.8291
1.751 1.726 1.724
0.414 0.427 0.547 0.545 0.544
3.72
0.0900
130 140 150 160 170
2.112 3.878 6.552 10.37 15.57
4.284.−4 4.440.−4 4.619.−4 4.831.−4 5.091.−4
0.0578 0.0330 0.0201 0.0130 0.0086
60.55 66.02 71.58 77.34 83.48
164.1 165.3 166.1 166.4 166.0
0.8724 0.9124 0.9499 0.9859 1.022
1.669 1.622 1.580 1.543 1.507
0.542 0.546 0.559 0.587 0.641
3.16 2.64 2.20 1.87 1.54
0.0828 0.0756 0.0688 0.0625 0.0558
180 190 200 209.39
22.41 31.20 42.23 54.96
5.423.−4 5.882.−4 6.641.−4 1.098.−3
0.0059 0.0040 0.0026 0.0011
90.26 98.19 108.40 133.90
164.6 161.8 156.0 133.9
1.058 1.098 1.147 1.262
1.472 1.433 1.386 1.262
0.734 0.905 1.515 ∞
1.28 1.05 0.80
0.0494 0.0433 0.0348 ∞
*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This source contains values for the compressed state for pressures up to 1000 bar, etc. The notation 3.235.−4 signifies 3.235 × 10−4. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).
THERMODYNAMIC PROPERTIES TABLE 2-277 Temperature, K
2-249
Compressed Krypton* Pressure, bar 1
10
20
40
60
80
100
200
400
600
800
1000
100
v h s
3.49.−4 26.42 0.5765
3.49.−4 26.69 0.5760
3.49.−4 26.99 0.5755
3.48.−4 27.59 0.5745
3.48.−4 28.18 0.5735
3.47.−4 28.78 0.5724
3.47.−4 29.38 0.5714
3.45.−4 32.38 0.5667
3.42.−4 38.42 0.5580
3.39.−4 44.47 0.5503
3.36.−4 50.52 0.5432
3.33.−4 56.57 0.5366
200
v h s
0.1971 183.1 1.859
0.0184 179.3 1.618
8.39.−3 174.5 1.533
3.00.−3 159.4 1.405
6.19.−4 105.6 1.129
5.94.−4 104.4 1.116
5.76.−4 103.6 1.106
5.27.−4 102.7 1.073
4.83.−4 105.3 1.037
4.58.−4 109.7 1.013
4.41.−4 114.9 0.993
4.28.−4 120.3 0.977
v 300 h s
0.2971 208.1 1.961
0.0292 206.3 1.728
0.0143 204.2 1.654
6.84.−3 200.0 1.575
4.37.−3 195.7 1.525
3.14.−3 191.2 1.485
2.41.−3 186.6 1.451
1.09.−3 166.8 1.333
6.92.−4 155.1 1.239
5.94.−4 155.2 1.196
5.44.−4 158.2 1.169
5.13.−4 162.3 1.149
400
v h s
0.3966 233.0 2.032
0.0394 231.9 1.802
0.0196 230.7 1.730
9.67.−3 228.3 1.657
6.37.−3 225.9 1.612
4.73.−3 223.6 1.579
3.75.−3 221.3 1.552
1.85.−3 211.4 1.463
1.01.−3 199.7 1.368
7.79.−4 196.8 1.317
6.76.−4 197.9 1.284
6.14.−4 200.8 1.259
500
v h s
0.4960 257.8 2.088
0.0495 257.1 1.858
0.0247 256.3 1.788
0.0123 254.9 1.716
8.20.−3 253.3 1.673
6.15.−3 251.9 1.642
4.91.−3 250.5 1.617
2.49.−3 244.5 1.537
1.33.−3 236.9 1.451
9.81.−4 234.2 1.400
8.22.−4 234.7 1.365
7.29.−4 237.4 1.340
600
v h s
0.5953 282.7 2.133
0.0596 282.2 1.904
0.0298 281.7 1.834
0.0149 280.7 1.763
9.96.−3 279.7 1.721
7.49.−3 278.8 1.691
6.01.−3 277.9 1.667
3.07.−3 274.2 1.591
1.64.−3 269.6 1.511
1.18.−3 268.1 1.462
9.67.−4 269.1 1.428
8.44.−4 271.7 1.403
700
v h s
0.6946 307.5 2.171
0.0696 307.2 1.942
0.0348 306.9 1.873
0.0175 306.2 1.803
0.0117 305.6 1.761
8.80.−3 305.1 1.732
7.07.−3 304.5 1.708
3.62.−3 302.2 1.634
1.93.−3 299.8 1.557
1.38.−3 299.6 1.511
1.11.−3 301.1 1.478
9.56.−4 304.0 1.453
800
v h s
0.7939 332.3 2.204
0.0795 332.2 1.975
0.0399 331.9 1.906
0.0200 331.6 1.837
0.0134 331.2 1.795
0.0101 330.9 1.766
8.11.−3 330.5 1.743
4.16.−3 329.3 1.671
2.21.−3 328.6 1.596
1.57.−3 329.4 1.551
1.25.−3 331.5 1.518
1.07.−3 334.6 1.494
900
v h s
0.8931 357.1 2.233
0.0895 357.0 2.005
0.0448 356.9 1.936
0.0225 356.8 1.866
0.0151 356.6 1.825
0.0114 356.4 1.796
9.13.−3 356.3 1.773
4.68.−3 355.8 1.702
2.48.−3 356.3 1.628
1.75.−3 358.0 1.584
1.39.−3 360.7 1.553
1.18.−3 364.2 1.528
1000
v h s
0.9924 381.9 2.260
0.0994 381.9 2.031
0.0498 381.9 1.962
0.0250 381.8 1.893
0.0168 381.8 1.852
0.0126 381.8 1.823
0.0102 381.8 1.800
5.20.−3 381.9 1.729
2.74.−3 383.4 1.657
1.93.−3 385.9 1.614
1.53.−3 389.0 1.583
1.29.−3 392.8 1.559
*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This source contains an exhaustive tabulation of values. v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K). The notation 3.49.−4 signifies 3.49 × 10−4. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).
TABLE 2-278 T, K 453.7m 500 600 700 800 900 1000 1200 1400 1500
Saturated Lithium*
P, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
1.78.−13 8.21.−12 4.18.−9 3.51.−7 9.57.−6
1.912.−3 1.946.−3 1.988.−3 2.028.−3 2.070.−3
2.40.+7 9.94.+5
1703 1905 2334 2697 3174
24259 24390 24674 24869 25162
6.776 7.199 7.983 8.633 9.192
56.492 52.169 45.216 40.307 36.678
4.30 4.34 4.23 4.19 4.17
1.24.−4 9.60.−4 0.0204 0.1794 0.4269
2.114.−3 2.160.−3 2.262.−3 2.370.−3 2.433.−3
8.55.+4 1.22.+4 669.3 86.06 38.17
3590 4006 4835 5668 6088
25341 25477 25654 25778 25845
9.682 10.120 10.876 11.518 11.808
33.850 31.591 28.225 25.882 24.979
4.16 4.16 4.14 4.19 4.20
*Converted from tables in Vargaftik, Tables of the Thermophysical Properties of Liquids and Gases, Nauka, Moscow, 1972, and Hemisphere, Washington, 1975. m = melting point. The notation 1.78.−13 signifies 1.78 × 10−13. Many of the Vargaftik values also appear in Ohse, R. W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Sci. Pubs., Oxford, 1985 (1020 pp.). This source contains superheat data. Saturation and superheat tables and a diagram to 14 bar, 2200 K are given by Reynolds, W. C., Thermodynamic properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram from 0.1 to 140 psia, 2100–3600°R, see Weatherford, P. M., J. C. Tyler, et al., WADD-TR-61-96, 1961. An extensive review of properties of the solid and the saturated liquid was given by Alcock, C. B., M. W. Chase, et al., J. Phys. Chem. Ref. Data, 23, 3 (1994): 385–497.
TABLE 2-279
Lithium Bromide—Water Solutions
Ruiter, J. P., Rev. Int. Froid = Int. J. Refrig., 13 (1990): 223–236 gives subroutines for computer calculations. See also ASHRAE Handbook—Fundamentals.
2-250
PHYSICAL AND CHEMICAL DATA
TABLE 2-280
Saturated Mercury*
T, K
P, bar
vf × 105, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
hfg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
203.15 213.15 223.15 233.15 243.15
2.298 ⋅ 10−11 1.288 ⋅ 10−10 6.169 ⋅ 10−10 2.580 ⋅ 10−9 9.573 ⋅ 10−9
7.26239 7.27570 7.28900 7.30231 7.31563
3.665 ⋅ 109 6.862 ⋅ 108 1.499 ⋅ 108 3.746 ⋅ 107 1.053 ⋅ 107
33.131 34.567 35.997 37.422 38.842
342.637 343.674 344.710 345.746 346.782
309.506 309.107 308.713 308.324 307.940
0.32434 0.33124 0.33780 0.34404 0.35001
1.84787 1.78142 1.72123 1.66647 1.61647
253.15 263.15 273.15 283.15 293.15
3.198 ⋅ 10−8 9.736 ⋅ 10−8 2.728 ⋅ 10−7 7.101 ⋅ 10−7 1.729 ⋅ 10−6
7.32896 7.34229 7.35563 7.36898 7.38234
3.281 ⋅ 106 1.120 ⋅ 106 4.150 ⋅ 105 1.653 ⋅ 105 7.026 ⋅ 104
40.258 41.668 43.074 44.476 45.874
347.819 348.855 349.891 350.927 351.964
307.561 307.187 306.817 306.451 306.090
0.35571 0.36118 0.36642 0.37146 0.37631
1.57065 1.52852 1.48967 1.45375 1.42045
303.15 313.15 323.15 333.15 343.15
3.968 ⋅ 10−6 8.626 ⋅ 10−6 1.786 ⋅ 10−5 3.356 ⋅ 10−5 6.724 ⋅ 10−5
7.39572 7.40911 7.42252 7.43594 7.44938
3.167 ⋅ 104 1.505 ⋅ 104 7.501 ⋅ 103 3.905 ⋅ 103 2.115 ⋅ 103
47.268 48.659 50.046 51.430 52.810
353.000 354.036 355.072 356.108 357.145
305.732 305.377 305.026 304.678 304.335
0.38099 0.38550 0.38986 0.39408 0.39816
1.38951 1.36068 1.33378 1.30862 1.28505
353.15 363.15 373.15 383.15
1.232 ⋅ 10−4 2.182 ⋅ 10−4 3.745 ⋅ 10−4 6.247 ⋅ 10−4
7.46285 7.47633 7.48984 7.50337
1.188 ⋅ 103 6.899 ⋅ 102 413.0 254.2
54.188 55.563 56.936 58.306
358.181 359.217 360.253 361.289
303.993 303.654 303.317 302.983
0.40212 0.40596 0.40969 0.41331
1.26292 1.24213 1.22255 1.20408
393.15 403.15 413.15 423.15 433.15
1.015 ⋅ 10−3 1.608 ⋅ 10−3 2.491 ⋅ 10−3 3.778 ⋅ 10−3 5.618 ⋅ 10−3
7.51693 7.53052 7.55415 7.55780 7.57148
153.6 103.9 68.75 46.43 31.96
59.674 61.039 62.403 63.765 65.125
362.326 363.362 364.397 365.433 366.469
302.652 302.323 301.994 301.668 301.344
0.41684 0.42027 0.42361 0.42687 0.43004
1.18665 1.17017 1.15456 1.13978 1.12575
443.15 453.15 463.15 473.15 483.15
8.204 ⋅ 10−3 1.178 ⋅ 10−2 1.664 ⋅ 10−2 2.315 ⋅ 10−2 3.177 ⋅ 10−2
7.58520 7.59897 7.61277 7.62662 7.64051
66.484 67.842 69.198 70.553 71.908
367.504 368.539 369.574 370.609 371.642
301.020 300.697 300.376 300.056 299.734
0.43314 0.43617 0.43913 0.44203 0.44486
1.11242 1.09975 1.08768 1.07619 1.06524
493.15 503.15 513.15 523.15 533.15
4.304 ⋅ 10−2 5.758 ⋅ 10−2 7.614 ⋅ 10−2 9.959 ⋅ 10−2 0.12892
7.65444 7.66843 7.68247 7.69656 7.71071
4.748 3.621 2.793 2.176 1.7132
73.261 74.614 75.967 77.319 78.671
372.676 373.708 374.740 375.771 376.800
299.415 299.094 298.773 298.452 298.129
0.44763 0.45035 0.45301 0.45562 0.45818
1.05478 1.04479 1.03525 1.02611 1.01737
543.15 553.15 563.15 573.15 583.15
0.16527 0.20993 0.26435 0.33015 0.40910
7.72491 7.73918 7.75351 7.7679 7.7823
1.3613 1.0912 0.88213 0.71874 0.59002
80.023 81.375 82.728 84.080 85.434
377.829 378.855 379.880 380.904 381.925
297.806 297.480 297.152 296.824 296.491
0.46069 0.46316 0.46558 0.46796 0.47030
1.00899 1.00095 0.99324 0.98584 0.97893
593.15 603.15 613.15 623.15 633.15
0.50320 0.61460 0.74567 0.89896 1.0772
7.7969 7.8115 7.8262 7.8409 7.8558
0.48779 0.40600 0.34008 0.28660 0.24291
86.788 88.143 89.499 90.856 92.215
382.944 383.960 384.973 385.984 386.991
296.156 295.817 295.474 295.128 294.776
0.47260 0.47487 0.47709 0.47929 0.48145
0.97190 0.96532 0.95899 0.95289 0.94702
643.15 653.15 663.15 673.15 683.15
1.2834 1.5207 1.9725 2.1024 2.454
7.8707 7.8858 7.9008 7.9160 7.9313
0.20702 0.17735 0.15269 0.13207 0.11476
93.575 94.937 96.300 97.666 99.033
387.994 388.994 389.989 390.980 391.966
294.419 294.057 293.689 293.314 292.933
0.48358 0.48568 0.48774 0.48978 0.49180
0.94135 0.93589 0.93061 0.92552 0.92059
693.15 703.15 713.15 723.15 733.15
2.852 3.299 3.801 4.362 4.986
7.9467 7.9622 7.9778 7.9935 8.0094
0.10014 0.08775 0.07719 0.06815 0.06039
100.403 101.775 103.150 104.528 105.908
392.947 393.923 394.893 395.858 396.816
292.544 292.148 291.743 291.330 290.908
0.49378 0.49574 0.49768 0.49959 0.50148
0.91583 0.91123 0.90677 0.90245 0.89827
743.15 753.15 763.15 773.15 783.15
5.679 6.446 7.292 8.222 9.242
8.0252 8.0413 8.0574 8.074 8.090
0.05369 0.04789 0.04285 0.03846 0.03462
107.292 108.679 110.069 111.463 112.861
397.767 398.711 399.649 400.579 401.501
290.475 290.032 289.580 289.116 288.640
0.50335 0.50519 0.50702 0.50882 0.51061
0.89422 0.89029 0.88647 0.88277 0.87917
793.15 803.15 813.15 823.15 833.15
10.358 11.576 12.901 14.340 15.899
8.106 8.123 8.140 8.157 8.174
0.03124 0.02827 0.02565 0.02333 0.02126
114.262 115.668 117.078 118.492 119.911
402.415 403.321 404.218 405.106 405.985
288.153 287.653 287.140 286.614 286.074
0.51238 0.51412 0.51586 0.51757 0.51927
0.87568 0.87228 0.86898 0.86576 0.86263
22.39 15.95 11.54 8.469 6.301
*From Vukalovich, Ivanov, Fokin, and Yakovlev, Thermophysical Properties of Mercury, Standartov, Moscow, 1971. For the saturated liquid the specific volume at 203.15 K is 7.26239 × 10−5 m3/kg, etc. All the tabular values for 203.15 K, 213.15 K, 223.15 K, and 233.15 K represent a metastable equilibrium between the subcooled liquid and the saturated vapor. Saturation and superheat tables and a diagram to 100 bar, 1600 K are given by Reynolds, W. C., Thermodynamic properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram from 1 to 8200 psia and 2700°R, see Weatherford, W. D., J. C. Tyler, et al., WADD-TR-61-96, 1961.
THERMODYNAMIC PROPERTIES
2-251
Saturated Mercury* (Concluded )
TABLE 2-280 T, K
P, bar
vf × 105, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
hfg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
843.15 853.15 863.15 873.15 883.15
17.584 19.403 21.36 23.46 25.72
8.191 8.209 8.226 8.244 8.262
0.019426 0.017785 0.016317 0.015000 0.013815
121.335 122.763 124.197 125.636 127.080
406.855 407.715 408.565 409.405 410.235
285.520 284.952 284.368 283.769 283.155
0.52095 0.52262 0.52427 0.52591 0.52753
0.85959 0.85662 0.85372 0.85090 0.84815
893.15 903.15 913.15 923.15 933.15
28.14 30.72 33.47 36.41 39.53
8.280 8.298 8.316 8.335 8.353
0.012748 0.011784 0.010911 0.010120 0.009401
128.530 129.986 131.448 132.915 134.389
411.054 411.861 412.658 413.444 414.218
282.524 281.875 281.210 280.529 279.829
0.52914 0.53074 0.53232 0.53389 0.53545
0.84546 0.84284 0.84028 0.83777 0.83533
943.15 953.15 963.15 973.15 983.15
42.85 46.36 50.09 54.03 58.20
8.372 8.391 8.410 8.430 8.450
0.008746 0.008150 0.007604 0.007105 0.006648
135.869 137.356 138.850 140.350 141.858
414.980 415.731 416.469 417.195 417.909
279.111 278.375 277.619 276.845 276.051
0.53700 0.53854 0.54006 0.54158 0.54308
0.83294 0.83060 0.82831 0.82606 0.82387
993.15 1003.15 1013.15 1023.15 1033.15
62.59 67.22 72.10 77.22 82.60
8.468 8.488 8.508 8.529 8.550
0.006228 0.005842 0.005487 0.005159 0.004856
143.372 144.894 146.424 147.961 149.506
418.610 419.298 419.974 420.636 421.286
275.238 274.404 273.550 272.675 271.780
0.54458 0.54607 0.54754 0.54901 0.55047
0.82172 0.81961 0.81754 0.81552 0.81353
1043.15 1053.15 1063.15 1073.15
88.25 94.17 100.37 106.85
8.570 8.590 8.612 8.632
0.004576 0.004317 0.004077 0.003854
151.059 152.619 154.188 155.766
421.923 422.546 423.156 423.752
270.864 269.927 268.968 267.986
0.55192 0.55336 0.55479 0.55621
0.81158 0.80966 0.80778 0.80593
TABLE 2-281
Saturated Methane*
P, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−4 Pa⋅s
kf, W/(m⋅K)
90.7 95 100 105 110
0.117 0.198 0.345 0.565 0.884
2.215.−3 2.244.−3 2.278.−3 2.316.−3 2.353.−3
3.976 2.463 1.479 0.940 0.625
216.4 232.5 246.3 263.2 280.1
759.9 769.0 776.9 785.7 794.5
4.231 4.406 4.556 4.719 4.882
10.225 10.034 9.862 9.710 9.558
3.288 3.318 3.369 3.425 3.478
2.02 1.71 1.56 1.33 1.22
0.225 0.215 0.206 0.197 0.189
115 120 125 130 135
1.325 1.919 2.693 3.681 4.912
2.396.−3 2.438.−3 2.487.−3 2.536.−3 2.594.−3
0.430 0.306 0.223 0.167 0.127
297.7 315.3 333.5 351.7 370.6
802.5 810.4 817.3 824.1 829.5
5.035 5.188 5.332 5.476 5.614
9.436 9.314 9.062 8.810 8.871
3.525 3.570 3.620 3.679 3.755
1.09 0.98 0.89 0.81 0.73
0.181 0.173 0.165 0.158 0.150
140 145 150 155 160
6.422 8.246 10.41 12.97 15.94
2.652.−3 2.722.−3 2.792.−3 2.882.−3 2.971.−3
0.098 0.077 0.061 0.049 0.039
389.5 409.5 429.4 450.8 472.1
834.8 844.4 853.9 848.5 843.0
5.751 5.885 6.019 6.151 6.283
8.932 8.891 8.849 8.725 8.601
3.849 3.965 4.101 4.27 4.47
0.66 0.61 0.56 0.51 0.46
0.143 0.136 0.129 0.122 0.115
165 170 175 180 185
19.39 23.81 27.81 32.86 38.59
3.095.−3 3.218.−3 3.419.−3 3.619.−3 3.979.−3
0.032 0.026 0.020 0.016 0.012
495.4 518.6 545.8 572.9 605.4
840.0 837.0 827.6 818.1 797.7
6.417 6.551 6.697 6.843 7.017
8.513 8.424 8.315 8.205 8.049
4.75 5.16 5.89 7.27 11.1
0.42 0.38 0.34 0.30 0.25
0.108 0.101 0.094 0.088 0.085
190 190.6c
45.20 45.99
4.900.−3 6.233.−3
0.008 0.006
661.6 704.4
750.7 704.4
7.293 7.516
7.762 7.516
70. ∞
0.19 0.17
0.090 ∞
T, K t
*Values reproduced or converted from Goodwin, NBS Tech. Note 653, 1974. t = triple point; c = critical point. The notation 2.215.−3 signifies 2.215 × 10−3.
2-252
PHYSICAL AND CHEMICAL DATA
FIG. 2-12
2-280.)
Enthalpy-log-pressure diagram for mercury. (Drawn from tabular data in footnote reference to Table
THERMODYNAMIC PROPERTIES TABLE 2-282
2-253
Superheated Methane* Temperature, K
P, bar
100
150
200
250
300
350
400
450
500
1
v h s
0.00228 246.4 4.555
0.7661 879.0 10.152
1.0299 984.3 10.757
1.2915 1090.4 11.230
1.5521 1199.8 11.629
1.8122 1314.8 11.983
2.0719 1437.4 12.310
2.3669 1568.8 12.618
2.5911 1708.9 12.914
5
v h s
0.00228 247.0 4.553
0.1434 865.0 9.256
0.2006 976.1 9.896
0.2549 1084.7 10.381
0.3083 1195.5 10.785
0.3611 1311.5 11.142
0.4136 1434.7 11.471
0.4657 1566.6 11.781
0.5181 1706.9 12.066
10
v h s
0.00227 247.8 4.549
0.0643 843.6 8.797
0.0968 965.5 9.501
0.1254 1077.9 10.002
0.1528 1190.6 10.414
0.1798 1307.9 10.775
0.2063 1432.0 11.106
0.2327 1564.1 11.417
0.2590 1705.3 11.715
20
v h s
0.00227 249.4 4.542
0.00277 429.8 6.003
0.0446 941.9 9.059
0.0606 1063.6 9.603
0.0751 1180.7 10.030
0.0891 1300.6 10.400
0.1027 1426.5 10.736
0.1162 1560.3 11.050
0.1295 1702.1 11.349
40
v h s
0.00226 252.5 4.528
0.00274 430.8 5.973
0.0176 879.3 8.465
0.0281 1032.9 9.155
0.0363 1160.5 9.621
0.0438 1286.0 10.008
0.0510 1415.7 10.354
0.0579 1552.1 10.674
0.0648 1696.0 10.978
60
v h s
0.00226 255.7 4.515
0.00271 432.2 5.946
0.00615 734.0 7.623
0.0173 999.8 8.847
0.0234 1140.0 9.359
0.0287 1271.7 9.765
0.0338 1405.1 10.121
0.0386 1544.2 10.440
0.0432 1690.0 10.756
80
v h s
0.00225 258.9 4.502
0.00268 433.8 5.920
0.00411 660.5 7.209
0.0119 964.4 8.590
0.0171 1119.7 9.158
0.0213 1257.7 9.584
0.0252 1394.9 9.951
0.0289 1536.6 10.283
0.0324 1684.4 10.595
100
v h s
0.00224 262.1 4.489
0.00266 435.5 5.897
0.00375 644.5 7.090
0.00888 928.5 8.364
0.0133 1099.6 8.991
0.0169 1244.2 9.437
0.0201 1385.2 9.814
0.0231 1529.4 10.153
0.0260 1679.0 10.469
150
v h s
0.00223 270.2 4.458
0.00261 440.7 5.843
0.00337 630.2 6.930
0.00555 860.0 7.953
0.00852 1054.1 8.664
0.0111 1213.1 9.155
0.0134 1362.8 9.555
0.0155 1513.0 9.907
0.0175 1667.0 10.233
200
v h s
0.00221 278.3 4.429
0.00256 446.5 5.796
0.00318 626.5 6.829
0.00447 825.0 7.719
0.00644 1019.8 8.426
0.00837 1187.2 8.944
0.0101 1343.8 9.362
0.0118 1498.9 9.727
0.0133 1656.9 10.060
300
v h s
0.00218 294.7 4.373
0.00249 459.6 5.714
0.00296 629.2 6.690
0.00369 804.4 7.471
0.00474 982.9 8.122
0.00593 1153.6 8.649
0.00708 1316.8 9.085
0.00818 1478.5 9.465
0.00924 1642.2 9.811
400
v h s
0.00244 473.8 5.645
0.00282 637.7 6.588
0.00336 802.4 7.323
0.00406 970.1 7.935
0.00486 1137.8 8.451
0.00569 1303.0 8.893
0.00560 1467.7 9.280
0.00729 1634.7 9.633
500
v h s
0.00239 488.8 5.584
0.00272 648.9 6.507
0.00315 807.7 7.215
0.00368 969.0 7.802
0.00428 1132.8 8.307
0.00492 1297.8 8.748
0.00555 1464.2 9.139
0.00616 1633.2 9.496
*Converted and rounded off from the tables of Goodwin, NBS Tech. Note 654, 1974. v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K). For a thermodynamic diagram from 0.1 to 400 bar and 620°C, see the 1993 ASHRAE Handbook—Fundamentals (SI ed.). Saturation and superheat tables and a chart to 6000 psia, 680°F appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. See also Friend, D. G., J. F. Ely, et al., J. Phys. Chem. Ref. Data. 18, 2 (1989): 583–638.
2-254
PHYSICAL AND CHEMICAL DATA
TABLE 2-283 Pressure, bar
Thermophysical Properties of Saturated Methanol kf, W/(m⋅K)
Prf
2.531 2.554 2.669 2.777
625 525 401 329
0.204 0.196 0.193 0.189
7.75 6.84 5.55 4.83
7.5379 7.4836 7.4398 7.4051 7.3474
2.845 2.894 2.946 2.984 3.050
288 268 242 227 204
0.186 0.184 0.182 0.181 0.179
4.41 4.22 3.92 3.74 3.48
4.7307 4.7836 4.8678 4.9366 5.0708
7.2992 7.2624 7.1988 7.1471 7.0461
3.117 3.176 3.265 3.349 3.540
187 174 156 141 117
0.178 0.177 0.175 0.173 0.171
3.27 3.12 2.91 2.73 2.42
1553.8 1486.4 1474.7 1450.1 1423.2
5.1744 5.2605 5.3355 5.4650 5.5793
6.9677 6.9017 6.8435 6.7409 6.6475
3.72 3.91 4.12 4.67 5.55
102 92 84 72 63
0.169 0.167 0.165 0.160 0.154
2.25 2.15 2.10 2.10 2.27
1391.8 1318.7 1186.8
5.6889 5.8803 6.0979
6.5543 6.3791 6.0979
hf, kJ/kg
hg, kJ/kg
0.001 0.001 0.001 0.001 0.001
057 257 276 307 336
1700000 7.309 3.801 1.599 0.819
0.0 261.0 293.9 345.0 391.7
1303.1 1440.3 1455.4 1476.2 1492.1
2.8114 3.9383 4.0493 4.2117 4.3516
10.2328 8.0281 7.9032 7.7386 7.6104
1.5 2.0 2.5 3.0 4.0
348.0 356.0 362.5 368.0 377.1
0.001 0.001 0.001 0.001 0.001
356 371 385 396 417
0.5632 0.4276 0.3443 0.2893 0.2188
421.0 444.2 463.6 479.8 507.8
1500.3 1505.8 1509.8 1512.4 1515.9
4.4361 4.5014 4.5536 4.5992 4.6728
5 6 8 10 15
384.5 390.8 401.3 409.8 426.3
0.001 0.001 0.001 0.001 0.001
434 450 479 504 560
0.17569 0.14683 0.11015 0.08783 0.05761
529.7 549.6 582.7 610.3 665.8
1517.4 1518.4 1518.0 1516.1 1507.9
20 25 30 40 50
438.9 449.3 458.2 472.9 484.9
0.001 0.001 0.001 0.001 0.001
611 666 710 814 934
0.04224 0.03290 0.02661 0.01863 0.01373
710.5 749.0 783.8 846.7 905.2
60 80 80.95c
495.1 508.1 512.6
0.002 086 0.002 507 0.003 715
0.01032 0.00642 0.00372
963.3 1065.3 1186.8
t
Temp., K
vf, m3/kg
µf, 10−6 Pa·s
vg, m3/kg
175.6 288.4 301.7 320.7 337.7
4 × 10−6 0.1 0.2 0.5 1.013
sf, kJ/(kg⋅K) sg, kJ/(kg⋅K) cpf, kJ/(kg⋅K)
t = triple point; c = critical point. v, h, s, and cp interpolated and converted from Goodwin, R. D., J. Phys. Chem. Ref. Data, 16, 4 (1987): 799–891.
THERMODYNAMIC PROPERTIES TABLE 2-284
2-255
Thermodynamic Properties of Compressed Methanol Temperature, K
Pressure, bar
200
250
300
350
v (m3/ kg) 0.1 h (kJ/ kg) s (kJ/ kg·K)
0.001137 57.5 3.096
400
450
500
550
600
0.001203 169.8 3.597
7.630 1456.7 8.081
8.942 1529.5 8.305
10.23 1607.5 8.514
11.56 1691.5 8.711
12.84 1781.7 8.901
1878.1 9.085
15.45 1980.2 9.263
v (m3/ kg) 0.5 h (kJ/ kg) s (kJ/ kg·K)
0.001137 57.5 3.096
0.001202 169.9 3.597
0.001274 290.5 4.038
1.764 1522.7 7.877
2.033 1603.0 8.091
2.296 1687.9 8.291
2.558 1778.9 8.482
2.818 1875.3 8.666
3.078 1977.4 8.844
v (m3/ kg) 1.013 h (kJ/ kg) s (kJ/ kg·K)
0.001137 57.6 3.096
0.001202 169.9 3.597
0.001274 290.5 4.038
0.8560 1514.0 7.675
0.9958 1598.7 7.902
1.1283 1685.1 8.105
1.2843 1795.4 8.117
1.3870 1873.5 8.482
1.5157 1975.8 8.660
v (m3/ kg) 10 h (kJ/ kg) s (kJ/ kg·K)
0.001136 58.4 3.095
0.001201 170.7 3.596
0.001272 291.2 4.036
0.001357 427.4 4.451
0.001474 578.8 4.857
0.1068 1638.1 7.427
0.1236 1751.5 7.667
0.1381 1858.0 7.870
0.1519 1965.2 8.056
v (m3/ kg) 15 h (kJ/ kg) s (kJ/ kg·K)
0.001136 58.8 3.094
0.001201 171.1 3.595
0.001272 291.6 4.035
0.001356 427.7 4.450
0.001472 578.9 4.856
0.0673 1601.9 7.253
0.0806 1735.6 7.536
0.0911 1849.4 7.752
0.1007 1960.2 7.946
v (m3/ kg) 20 h (kJ/ kg) s (kJ/ kg·K)
0.001135 59.2 3.094
0.001200 171.6 3.595
0.001271 292.0 4.035
0.001355 428.1 4.449
0.001469 579.0 4.854
0.0466 1565.3 7.087
0.0589 1717.7 7.431
0.0675 1840.0 7.664
0.0751 1954.8 7.864
v (m3/ kg) 30 h (kJ/ kg) s (kJ/ kg·K)
0.001134 60.1 3.092
0.001199 172.4 3.593
0.001269 292.9 4.036
0.001355 428.8 4.447
0.001465 579.4 4.851
0.001659 751.3 5.264
0.0367 1675.4 7.253
0.0436 1818.7 7.526
0.0492 1942.7 7.743
v (m3/ kg) 40 h (kJ/ kg) s (kJ/ kg·K)
0.001133 61.0 3.091
0.001198 173.3 3.592
0.001268 293.7 4.032
0.001350 429.5 4.445
0.001461 579.8 4.849
0.001649 750.4 5.258
0.0251 1623.0 7.088
0.0314 1794.2 7.414
0.0361 1928.8 7.650
v (m3/ kg) 50 h (kJ/ kg) s (kJ/ kg·K)
0.001133 61.9 3.090
0.001197 174.2 3.591
0.001266 294.5 4.030
0.001348 430.2 4.443
0.001457 580.2 4.846
0.001637 749.7 5.579
0.0176 1556.4 6.912
0.0239 1766.7 7.314
0.0282 1913.4 7.570
v (m3/ kg) 60 h (kJ/ kg) s (kJ/ kg·K)
0.001131 62.8 3.089
0.001196 175.0 3.589
0.001265 295.3 4.029
0.001346 430.9 4.442
0.001453 580.6 4.843
0.001628 749.1 5.248
0.0120 1461.8 6.692
0.0188 1736.1 7.220
0.0228 1896.6 7.500
v (m3/ kg) 75 h (kJ/ kg) s (kJ/ kg·K)
0.001130 64.1 3.087
0.001194 176.3 3.587
0.001263 296.6 4.027
0.001343 431.9 4.439
0.001448 581.2 4.839
0.001614 748.3 5.241
0.002084 982.1 5.718
0.01359 1683.9 7.081
0.0174 1869.1 7.405
v (m3/ kg) 100 h (kJ/ kg) s (kJ/ kg·K)
0.001128 66.3 3.084
0.001191 178.5 3.584
0.001259 298.6 4.023
0.001337 433.8 4.435
0.001439 582.4 4.833
0.001595 747.5 5.230
0.001952 964.8 5.673
0. 1572.9 6.829
0.01188 1818.8 7.261
v (m3/ kg) 150 h (kJ/ kg) s (kJ/ kg·K)
0.001125 70.7 3.078
0.001186 182.8 3.578
0.001252 302.8 4.016
0.001328 437.4 4.426
0.001423 584.9 4.822
0.001562 746.8 5.211
0.001825 948.4 5.622
1248.8 6.302
0.006513 1704.3 6.997
v (m3/ kg) 200 h (kJ/ kg) s (kJ/ kg·K)
0.001121 75.1 3.071
0.001182 187.2 3.571
0.001246 307.0 4.009
0.001317 441.2 4.418
0.001408 587.8 4.811
0.001535 747.0 5.194
0.001751 939.9 5.587
0.002314 1223.5 6.125
0.004091 1583.5 6.752
v (m3/ kg) 300 h (kJ/ kg) s (kJ/ kg·K)
0.001113 83.9 3.060
0.001172 195.9 3.559
0.001234 315.4 3.996
0.001302 448.9 4.403
0.001384 593.8 4.791
0.001492 749.4 5.166
0.001656 932.0 5.537
0.001957 1173.4 5.996
0.002600 1443.5 6.466
v (m3/ kg) 400 h (kJ/ kg) s (kJ/ kg·K)
0.001107 92.7 3.048
0.001164 204.7 3.548
0.001223 324.0 3.983
0.001288 456.9 4.388
0.001363 600.5 4.774
0.001459 753.4 5.142
0.001593 929.6 5.500
0.001808 1154.4 5.926
0.002182 1388.1 6.335
v (m3/ kg) 500 h (kJ/ kg) s (kJ/ kg·K)
0.001101 101.5 3.037
0.001156 213.4 3.536
0.001213 332.6 3.971
0.001274 465.1 4.375
0.001345 607.7 4.757
0.001431 758.4 5.121
0.001546 930.4 5.470
0.001716 1145.6 5.880
0.001980 1360.6 6.254
Converted and interpolated from Goodwin, R. D., J. Phys. Chem. Ref. Data, 16, 4 (1987): 799–891. These extensive tables extend to 700 bar for temperatures from 175.6 to 800 K. Another extensive compilation is by deReuck, K. M. and R. J. B. Craven, Methanol, C.R.C. Press, 1993 (320 pp.). Equations and diagrams to 30 bar, 200°C are given by Eicholz, H. D., S. Schulz, et al., Kalte u Klim., no. 9, (1981) 322–331. For pressures to 1040 bar, 298–489 K, see Machado, J. R. S. and W. B. Street, J.Chem.Eng.Data, 28 (1983): 218–223; to 2800 bar from 273 to 333 K, see Sun, T., S. N. Biswas, et al., J. Chem. Eng. Data, 33 (1988): 395–398. Dissociation was considered by Yerlett, T. K. and C. J. Wormald, J. Chem. Thermo., 18 (1986): 719–726, and by Kazarnovskii, Ya. S. and E. V. Pavlova, Russ. J. Phys. Chem., 56, 6 (1982): 847–851.
TABLE 2-285
Saturated Methyl Chloride* µf, 10−4 Pa⋅s
kf, W/(m⋅K)
1.483 1.486 1.489 1.492 1.496
4.44 4.27 4.11 3.96 3.82
0.241 0.236 0.232 0.228 0.224
5.896 5.866 5.845 5.822 5.786
1.500 1.504 1.508 1.513 1.518
3.69 3.57 3.46 3.35 3.25
0.219 0.215 0.211 0.207 0.202
4.050 4.080 4.110 4.139 4.168
5.762 5.740 5.720 5.699 5.680
1.523 1.528 1.533 1.539 1.546
3.16 3.08 3.00 2.92 2.85
0.198 0.194 0.190 0.186 0.182
824.4 826.8 829.0 831.2 833.2
4.197 4.225 4.253 4.280 4.308
5.662 5.644 5.628 5.612 5.597
1.554 1.565 1.574 1.583 1.594
2.78 2.72 2.66 2.61 2.56
0.177 0.173 0.169 0.165 0.160
461.2 469.3 477.4 485.6 493.8
835.2 837.0 838.8 840.5 841.9
4.334 4.361 4.388 4.414 4.440
5.581 5.567 5.553 5.540 5.527
1.605 1.617 1.631 1.644 1.658
2.51 2.46 2.42 2.37 2.33
0.156 0.152 0.148 0.143 0.139
0.0386 0.0343 0.0282 0.0228 0.0186
502.1 510.4 518.8 538.3 562.9
843.3 844.5 846.4 847.5 847.6
4.465 4.491 4.542 4.592 4.643
5.516 5.504 5.481 5.457 5.434
2.30 2.27 2.12 1.99 1.87
0.135 0.131 0.124 0.117 0.110
13.47.−4 14.11.−4 14.67.−4 15.66.−4 16.48.−4
0.0151 0.0117 0.0096 0.0075 0.0063
581.6 602.8 622.9 643.6 663.2
845.9 842.6 837.4 826.4 819.1
4.694 4.747 4.805 4.870 4.904
5.398 5.382 5.358 5.323 5.289
1.77 1.67 1.59 1.51
0.103 0.095 0.086 0.075
17.97.−4 21.10.−4 27.40.−4
0.0052 0.0038 0.0027
677.3 714.1 749.3
807.1 778.6 749.3
4.954 5.025 5.116
5.256 5.200 5.116
T, K
P, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
175 180 185 190 195
0.0117 0.0165 0.0233 0.0327 0.0462
8.84.−4 8.91.−4 8.97.−4 9.04.−4 9.10.−4
27.90 19.85 14.12 10.12 7.208
274.5 280.9 287.5 294.5 301.7
764.3 767.7 771.0 774.3 777.5
3.529 3.570 3.603 3.647 3.684
6.328 6.274 6.222 6.172 6.124
1.469 1.472 1.475 1.477 1.480
200 205 210 215 220
0.0653 0.0919 0.1315 0.181 0.243
9.17.−4 9.25.−4 9.33.−4 9.40.−4 9.48.−4
5.137 3.835 2.656 1.975 1.505
309.0 316.3 323.7 331.0 338.4
780.7 783.9 787.0 790.1 793.2
3.722 3.756 3.791 3.825 3.859
6.080 6.038 5.998 5.961 5.928
225 230 235 240 245
0.319 0.417 0.539 0.688 0.866
9.56.−4 9.65.−4 9.73.−4 9.81.−4 9.89.−4
1.168 0.911 0.718 0.572 0.462
345.7 353.1 360.5 368.0 375.6
796.3 799.3 802.3 805.3 808.2
3.892 3.925 3.957 3.988 4.019
250 255 260 265 270
1.076 1.328 1.627 1.970 2.364
9.98.−4 10.08.−4 10.18.−4 10.27.−4 10.36.−4
0.377 0.311 0.257 0.215 0.1807
383.2 390.7 398.3 406.0 413.7
811.1 814.0 816.8 819.4 822.0
275 280 285 290 295
2.830 3.347 3.936 4.612 5.361
10.46.−4 10.57.−4 10.68.−4 10.79.−4 10.91.−4
0.1524 0.1301 0.1115 0.0960 0.0830
421.5 429.4 437.3 445.2 453.2
300 305 310 315 320
6.189 7.110 8.111 9.243 10.47
11.03.−4 11.15.−4 11.27.−4 11.40.−4 11.55.−4
0.0723 0.0632 0.0556 0.0489 0.0433
325 330 340 350 360
11.78 13.27 16.52 20.53 25.29
11.70.−4 11.86.−4 12.17.−4 12.54.−4 12.97.−4
370 380 390 400 405
30.74 36.99 44.05 52.29 56.6
410 415 416c
61.5 67.4 69.0
*Interpolated by P. E. Liley from the Landolt-Börnstein band IVa, p. 677, 1967 tables by Steinle/Dienemann. c = critical point. The notation 8.84.−4 signifies 8.84 × 10−4. TABLE 2-286
Saturated Neon*
P, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−4 Pa⋅s
kf, W/(m⋅K)
10 20 24.6m 24.6m 26
0.434 0.718
6.654.−4 6.823.−4 6.696.−4 8.012.−4 8.172.−4
0.2266 0.1429
0.75 6.78 11.96 28.22 30.90
117.0 118.1
0.0992 0.4906 0.7257 1.388 1.494
5.006 4.846
0.278 0.945 1.345 1.802 1.868
1.57 1.37
0.146 0.132
28 30 32 34 36
1.321 2.238 3.552 5.352 7.728
8.413.−4 8.687.−4 9.001.−4 9.370.−4 9.820.−4
0.0817 0.0501 0.0323 0.0217 0.0149
34.75 38.80 43.06 47.57 52.34
119.3 120.1 120.6 120.6 119.9
1.634 1.771 1.905 2.036 2.166
4.653 4.483 4.329 4.184 4.043
1.955 2.052 2.163 2.302 2.506
1.16 1.00 0.84 0.71 0.59
0.124 0.115 0.106 0.097 0.088
1.039.−3 1.116.−3 1.232.−3 1.538.−3 2.070.−3
0.0104 0.0073 0.0050 0.0031 0.0021
57.52 63.33 69.82 80.83 92.50
118.4 115.8 111.8 103.0 92.5
2.297 2.435 2.582 2.812 3.062
3.900 3.749 3.582 3.316 3.062
2.825 3.436 5.26 25.0 ∞
0.48 0.38 0.31 0.25
0.078 0.069 0.059
T, K
38 40 42 44 44.4c
10.78 14.62 19.39 25.22 26.53
∞
*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. m = melting point; c = critical point. The notation 6.654.−4 signifies 6.654 × 10−4. This source contains values for the compressed state up to 1000 bar, etc. This book was published in English translation by Hemisphere, New York 1988 (604 pp.). Saturation and superheat tables and a diagram to 200 bar, 320 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Saturation and superheat tables to 60,000 psia, 900°R and a chart to 4000 psia, 560°R appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. 2-256
THERMODYNAMIC PROPERTIES TABLE 2-287
2-257
Compressed Neon* Pressure, bar
Temperature, K
1
10
20
40
60
80
100
200
400
600
800
1000
v 100 h s
0.4117 195.4 6.129
0.0410 194.0 5.168
0.0204 192.4 4.869
0.0102 189.4 4.556
6.76.−3 186.6 4.363
5.08.−3 184.0 4.221
4.09.−3 181.6 4.106
2.22.−3 174.1 3.739
1.42.−3 173.3 3.386
1.18.−3 180.0 3.197
1.06.−3 189.2 3.066
9.74.−4 199.2 2.964
v 200 h s
0.8243 298.5 6.844
0.0828 298.4 5.893
0.0416 298.4 5.605
0.0210 298.2 5.315
0.0142 298.2 5.143
0.0107 298.2 5.020
8.69.−3 298.3 4.924
4.61.−3 299.4 4.620
2.61.−3 304.6 4.308
1.95.−3 312.4 4.124
1.63.−3 321.8 3.994
1.43.−3 332.1 3.893
v 300 h s
1.236 401.6 7.262
0.1241 401.8 6.312
0.0624 402.2 6.026
0.0315 402.8 5.739
0.0212 403.5 5.570
0.0160 404.1 5.450
0.0129 404.9 5.357
6.77.−3 408.8 5.065
3.71.−3 417.8 4.769
2.69.−3 427.8 4.593
2.18.−3 438.5 4.469
1.87.−3 449.7 4.372
v 400 h s
1.648 504.6 7.558
0.1654 505.0 6.609
0.0830 505.4 6.323
0.0418 506.4 6.037
0.0281 507.4 5.896
0.0212 508.3 5.750
0.0171 509.3 5.657
8.88.−3 514.4 5.369
4.77.−3 525.3 5.078
3.40.−3 536.7 4.907
2.72.−3 548.4 4.785
2.30.−3 560.2 4.690
v 500 h s
2.060 607.6 7.788
0.2066 608.1 6.839
0.1036 608.6 6.553
0.0521 609.7 6.267
0.0350 610.8 6.100
0.0264 611.9 5.981
0.0213 613.0 5.889
0.0110 618.8 5.601
5.82.−3 630.7 5.313
4.10.−3 642.9 5.144
3.24.−3 655.2 5.023
2.73.−3 667.5 4.929
v 600 h s
2.472 710.6 7.975
0.2478 711.1 7.027
0.1242 711.7 6.741
0.0625 712.9 6.455
0.0419 714.1 6.288
0.0316 715.3 6.169
0.0254 716.5 6.077
0.0130 722.5 5.791
6.85.−3 735.0 5.504
4.80.−3 747.8 5.335
3.77.−3 760.5 5.215
3.15.−3 773.2 5.122
v 700 h s
2.884 813.5 8.134
0.2890 814.1 7.186
0.1449 814.7 6.900
0.0728 816.0 6.614
0.0487 817.2 6.447
0.0367 818.5 6.328
0.0295 819.7 6.236
0.0151 826.0 5.950
7.89.−3 838.9 5.664
5.49.−3 851.9 5.496
4.29.−3 865.0 5.376
3.57.−3 878.1 5.284
v 800 h s
3.296 916.5 8.272
0.3302 917.1 7.323
0.1655 917.7 7.038
0.0831 919.0 6.752
0.0556 920.3 6.585
0.0419 921.6 6.466
0.0336 922.9 6.374
0.0172 929.3 6.088
8.92.−3 942.4 5.802
6.18.−3 955.7 5.634
4.81.−3 969.0 5.515
3.98.−3 982.2 5.423
v 900 h s
3.708 1020 8.393
0.3714 1020 7.444
0.1861 1021 7.159
0.0934 1022 6.873
0.0625 1023 6.706
0.0470 1025 6.588
0.0378 1026 6.496
0.0192 1033 6.210
9.96.−3 1046 5.924
6.87.−3 1059 5.756
5.32.−3 1073 5.637
4.40.−3 1086 5.545
v 1000 h s
4.120 1123 8.502
0.4126 1123 7.553
0.2067 1124 7.267
0.1037 1125 6.982
0.0693 1126 6.815
0.0522 1128 6.696
0.0419 1129 6.604
0.0213 1136 6.318
0.0110 1149 6.032
7.56.−3 1163 5.856
5.84.−3 1176 5.746
4.81.−3 1190 5.654
*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This source contains an exhaustive tabulation of values. v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K). The notation 6.76.−3 signifies 6.76 × 10−3. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.). TABLE 2-288
Saturated Nitrogen (R728)*
T, K
P, bar
vf, 10–3 m3/kg
63.15t 65 70 75 77.35
0.1253 0.1743 0.3859 0.7609 1.0133
1.155 1.165 1.193 1.224 1.239
80 85 90 95 100
1.369 2.287 3.600 5.398 7.775
1.258 1.297 1.340 1.390 1.447
vg, m3/kg 1477 1091 525.6 281.8 216.9 164.0 101.7 66.28 44.87 31.26
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−4 Pa⋅s
kf, W/(m⋅K)
−148.5 −144.9 −135.2 −125.4 −120.8
64.1 65.8 70.5 74.9 76.8
2.459 2.516 2.657 2.789 2.849
5.826 5.757 5.595 5.460 5.404
1.928 1.930 1.937 1.948 1.955
2.74 2.17 1.77 1.60
0.170 0.160 0.151 0.141 0.136
−115.6 −105.7 −95.6 −85.2 −74.5
78.9 82.3 85.0 86.8 87.7
2.913 3.032 3.147 3.256 3.363
5.345 5.244 5.152 5.067 4.985
1.964 1.989 2.028 2.086 2.176
1.48 1.27 1.10 0.97 0.87
0.132 0.123 0.114 0.105 0.097
0.79 0.71 0.60 0.48 0.32
0.088 0.080 0.071 0.063 0.052
105 110 115 120 125
10.83 14.67 19.40 25.15 32.05
1.514 1.597 1.714 1.892 2.324
22.23 15.98 11.47 8.031 5.016
−63.8 −51.4 −38.1 −21.4 5.1
87.4 85.6 81.8 74.3 57.2
3.469 3.575 3.687 3.821 4.024
4.904 4.820 4.729 4.619 4.444
2.319 2.566 3.063
126.25c
33.96
3.289
3.289
34.8
34.8
4.252
4.252
∞
∞
*Reproduced and converted from Vasserman and Rabinovich, Thermophysical Properties of Liquid Air and Its Components, Standartov, Moscow, 1968; and Israel Program for Scientific Translations, TT 69-55092, 1970. t = triple point; c = critical point. Other extensive tables are given by Angus, S., International Thermodynamic Tables of the Fluid State—6. Nitrogen, Pergamon, 1977 (244 pp.); Hanley, H. J. M., R. D. McCarty, et al., J. Phys. Chem. Ref. Data, 3 (1974): 979–1019. Saturation and superheat tables to 30,000 psia and a chart to 10,000 psia, all to 860°R, appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) has a thermodynamic chart for pressures from 0.1 to 800 bar and temperatures from 80 to 500 K.
2-258
FIG. 2-13
4729.)
PHYSICAL AND CHEMICAL DATA
Temperature-entropy diagram for nitrogen. Section of T-S diagram for nitrogen by E. S. Burnett, 1950. (Reprinted from U.S. Bur. Mines Rep. Invest.
TABLE 2-289
Thermophysical Properties of Nitrogen (R728) at Atmospheric Pressure
T (K) v (m3/ kg) h (kJ/ kg) s (kJ/ kg·K) cp (kJ/ kg·K) Z vs (m/s) η (10−6 Pa·s) k (W/m·K) NPr
77.4b 0.2164 76.7 5.403 1.341 0.9545 172 5.0 0.0074 0.913
80 0.2252 80.0 5.446 1.196 0.9610 177 5.2 0.0077 0.811
100 0.2871 101.9 5.690 1.067 0.9801 202 6.7 0.0098 0.728
120 0.3474 123.1 5.884 1.056 0.9883 222 8.0 0.0117 0.727
140 0.4071 144.2 6.046 1.050 0.9927 240 9.3 0.0136 0.723
160 0.4664 165.2 6.186 1.047 0.9952 257 10.6 0.0154 0.721
180 0.5255 186.1 6.309 1.045 0.9967 273 11.8 0.0171 0.720
200 0.5845 207.0 6.419 1.043 0.9977 288 12.9 0.0187 0.719
220 0.6434 227.8 6.519 1.043 0.9984 302 14.0 0.0203 0.718
240 0.7023 248.7 6.609 1.042 0.9990 316 15.0 0.0218 0.717
T (K) v (m3/ kg) h (kJ/ kg) s (kJ/ kg·K) cp (kJ/ kg·K) Z vs (m/s) η (10−6 Pa·s) k (W/m·K) NPr
260 0.7611 269.5 6.693 1.042 0.9994 329 16.0 0.0232 0.717
280 0.8199 290.3 6.770 1.041 0.9997 341 17.0 0.0247 0.716
300 0.8786 311.2 6.842 1.041 0.9998 359 17.9 0.0260 0.716
320 0.9371 332.0 6.909 1.042 0.9999 365 18.8 0.0273 0.717
340 0.9960 352.8 6.972 1.042 1.0000 376 19.7 0.0286 0.717
360 1.0546 373.7 7.032 1.043 1.0001 387 20.5 0.0299 0.717
380 1.1134 394.5 7.088 1.044 1.0002 397 21.4 0.0311 0.717
400 1.1719 411.5 7.142 1.045 1.0002 408 22.2 0.0324 0.717
420 1.2305 436.3 7.193 1.047 1.0002 417 23.0 0.0336 0.717
440 1.2892 457.3 7.242 1.048 1.0003 427 23.8 0.0347 0.717
T (K) v (m3/ kg) h (kJ/ kg) s (kJ/ kg·K) cp (kJ/ kg·K) Z vs (m/s) η (10−6 Pa·s) k (W/m·K) NPr
460 1.3481 478.3 7.288 1.051 1.0003 437 24.5 0.0359 0.718
480 1.4065 499.3 7.333 1.053 1.0004 446 25.3 0.0371 0.718
500 1.4654 520.4 7.376 1.056 1.0004 455 26.0 0.0383 0.718
600 1.758 626.9 7.570 1.075 1.000 496 29.5 0.0440 0.722
700 2.052 735.6 7.738 1.098 1.000 534 32.8 0.0496 0.726
800 2.344 846.6 7.886 1.122 1.000 568 35.9 0.0551 0.730
900 2.636 960.0 8.019 1.146 1.000 601 38.8 0.0606 0.734
1000 2.931 1075.7 8.141 1.167 1.001 631 41.6 0.658 0.737
1500 4.396 1680.5 8.630 1.244 1.001 765
2000 5.862 2313.5 8.995 1.284 1.001 879
b = normal boiling point. TABLE 2-290
Saturated Nitrogen Tetroxide vg, m3/kg
Mf
Mg
299.32 309.57 326.66 337.43 345.45
0.000 0.000 0.000 0.000 0.000
694 711 733 749 762
0.2996 0.1630 0.0876 0.0608 0.0469
91.857 91.886 91.766 91.625 91.488
79.157 76.503 73.538 71.748 70.480
10 15 20 30 40
351.88 364.09 373.17 386.57 396.52
0.000 0.000 0.000 0.000 0.000
774 800 822 863 903
0.0382 0.0262 0.0199 0.0133 0.0098
91.346 90.979 90.601 89.823 89.018
69.483 67.742 66.547 64.997 64.099
50 60 80 100
404.50 411.20 422.07 430.76
0.000 0.000 0.001 0.001
945 993 129 577
0.00761 0.00607 0.00394 0.00209
88.191 87.344 85.602 83.817
63.532 63.181 62.959 63.366
Pressure, bar
Temperature, K
1.0133 2 4 6 8
vf, m3/kg
Condensed from McCarty, R. D., H-U. Steurer, et al., NBS IR 86 - 3054, 1986 (106 pp.). M = mol wt for the reaction N2O4 A 2NO2 A 2NO + O2. No derived thermodynamic functions were tabulated due to unduly large differences in literature values, but 92 references are given. TABLE 2-291
Saturated Nitrous Oxide
Temp., °F
Pressure, psia
vf, ft3/lbm
vg, ft3/lbm
h f, Btu/lbm
h g, Btu/lbm
s f, Btu/lbm°R
sg, Btu/lbm°R
−127.2 −100 −80 −60 −40
14.70 33.68 56.79 90.29 136.68
0.01310 0.01358 0.01398 0.01444 0.01495
5.069 2.374 1.463 0.939 0.648
0.0 11.7 20.8 30.2 40.3
161.7 165.9 168.8 171.5 173.7
0.0000 0.0304 0.0534 0.0782 0.1044
0.4864 0.4591 0.4433 0.4315 0.4222
−20 0 20 40 60
198.62 278.97 380.88 507.51 662.69
0.01555 0.01625 0.01711 0.01819 0.01968
0.450 0.316 0.227 0.164 0.117
50.6 60.5 70.2 80.3 91.9
175.3 176.2 176.2 175.0 172.1
0.1296 0.1518 0.1718 0.1920 0.2145
0.4133 0.4036 0.3928 0.3815 0.3687
0.0222 0.0247 0.0354
0.0792 0.0611 0.0354
105.7 114.7 136.4
165.0 157.5 136.4
0.2382 0.2523 0.2890
0.3480 0.3302 0.2890
80 90 97.6c
851.5 961.0 1052.2
Rounded and condensed from Couch, E. J. and K. A. Kobe, Univ. Texas Rep., Cont. DAI-23-072-ORD-685, June 1, 1956. c = critical point. 2-259
2-260
PHYSICAL AND CHEMICAL DATA
Mollier diagram for nitrous oxide. (Fig. 9, Univ. Texas Rep., Cont. DAI-23-072-ORD685, June 1, 1956, by Couch and Kobe. Reproduced by permission.) Some irregularity in the compressibility factors from 80 to 160 atm, 50 to 100°C exists (Couch, private communication, 1967). See Couch et al., J. Chem. Eng. Data, 6, (1961) for PVT data.
FIG. 2-14
THERMODYNAMIC PROPERTIES TABLE 2-292
2-261
Nonane* h g, kJ/kg
s f, kJ/(kg⋅K)
s g, kJ/(kg⋅K)
c pf, kJ/(kg⋅K)
µ f, 10−4 Pa⋅s
k f, W/(m⋅K)
358.4 359.2 400.6 442.2 484.8
828.7 859.4
2.424 2.427 2.607 2.774 2.932
4.210 4.243
2.07 2.07 2.08 2.10 2.16
33.5 33.0 17.9 12.1 8.7
0.150 0.150 0.145 0.140 0.134
30.35 10.19 4.00 1.80 0.894
528.6 573.8 622.0 671.3 722.5
891.7 925.6 961.1 998.2 1036.5
3.083 3.229 3.370 3.511 3.650
4.282 4.324 4.368 4.419 4.476
1.591.−3 1.637.−3 1.690.−3 1.748.−3 1.815.−3
0.485 0.286 0.161 0.104 0.069
776.7 833.3 890.2 950.3 1012.1
1076.0 1116.6 1157.1 1199.2 1241.3
3.788 3.927 4.053 4.186 4.316
4.536 4.601 4.660 4.727 4.794
5.309 7.437 10.20 13.76 18.02
1.895.−3 2.00.−3 2.13.−3 2.35.−3 2.78.−3
0.045 0.030 0.021 0.013 0.008
1076.2 1141.3 1207.7 1275.4 1342.9
1282.9 1324.5 1363.8 1338.7 1318.1
4.444 4.569 4.691 4.811 4.927
4.857 4.921 3.980 5.029 5.056
22.90
4.23.−3
0.004
1305.2
1305.2
5.032
5.032
vf, m3/kg
T, K
P, bar
219.7t 220 240 260 280
2.6.−6 2.7.−6 3.74.−5 2.97.−4 1.61.−3
300 320 340 360 380
6.40.−3 0.0203 0.0547 0.1279 0.2678
1.404.−3 1.436.−3 1.471.−3 1.508.−3 1.548.−3
400 420 440 460 480
0.513 0.911 1.521 2.401 3.639
500 520 540 560 580 594.6c
vg, m3/kg
h f, kJ/kg
2.22 2.30
6.53 5.13 4.16 3.44 2.91
0.129 0.123 0.118 0.112 0.107
2.50 2.18
0.101 0.096 0.092 0.089 0.085 0.082
*Values of p and v interpolated and converted from tables in Vargaftik, Handbook of Thermophysical Properties of Gases and Liquids, Hemisphere, Washington, and McGraw-Hill, New York, 1975. Values of h and s calculated from API tables published by Texas A&M University, College Station. t = triple point; c = critical point.
TABLE 2-293
Octane* h g, kJ/kg
s f, kJ/(kg⋅K)
s g, kJ/(kg⋅K)
c pf, kJ/(kg⋅K)
µ f, 10−3 Pa⋅s
k f, W/(m⋅K)
365.9 373.2 414.1 455.8 498.4
811.4 842.1 873.5
2.487 2.520 2.698 2.865 3.023
4.207 4.259 4.312
2.033 2.035 2.059 2.105 2.165
2.25 2.01 1.24 0.87 0.65
0.149 0.148 0.143 0.138 0.133
10.7 4.01 1.752 0.844 0.448
542.4 589.8 637.9 687.1 737.7
906.2 939.8 974.6 1010.4 1047.3
3.175 3.325 3.471 3.611 3.747
4.366 4.419 4.461 4.509 4.562
2.231
0.504 0.405 0.334 0.282 0.244
0.128 0.123 0.118 0.112 0.107
1.632.−3 1.685.−3 1.747.−3 1.818.−3 1.904.−3
0.252 0.155 0.100 0.066 0.045
790.1 843.1 897.5 954.8 1013.5
1084.8 1123.6 1162.5 1202.0 1241.8
3.881 4.010 4.137 4.264 4.388
4.617 4.677 4.740 4.802 4.864
0.200 0.167 0.143 0.121 0.103
0.102 0.099 0.095 0.091 0.087
2.013.−3 2.16.−3 2.37.−3 2.81.−3 4.26.−3
0.031 0.021 0.014 0.008 0.004
1072.8 1136.0 1201.5 1276.7 1331.7
1281.2 1318.6 1352.4 1370.4 1331.7
4.508 4.629 4.749 4.880 4.977
4.924 4.980 5.028 5.048 4.977
0.086 0.072 0.058 0.044
0.083
vf, m3/kg
T, K
P, bar
216.4t 220 240 260 280
1.49.−5 2.41.−5 2.18.−4 0.0014 0.0061
1.353.−3 1.368.−3 1.384.−3
300 320 340 360 380
0.0207 0.0575 0.1384 0.3000 0.5856
1.420.−3 1.457.−3 1.495.−3 1.536.−3 1.582.−3
400 420 440 460 480
1.0507 1.758 2.797 4.246 6.201
500 520 540 560 568.8c
8.785 12.15 16.46 21.98 24.97
vg, m3/kg
700 125 31.9
h f, kJ/kg
*Values of p and v interpolated and converted from tables in Vargaftik, Handbook of Thermophysical Properties of Gases and Liquids, Hemisphere, Washington, and McGraw-Hill, New York, 1975. Values of h and s calculated from API tables published by Texas A&M University, College Station. t = triple point; c = critical point. Saturation and superheat tables and a diagram to 100 bar, 680 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.).
2-262
PHYSICAL AND CHEMICAL DATA
TABLE 2-294
Saturated Oxygen (R732)*
T, K
P, bar
vf, 10–3 m3/kg
54.35 t 55 60 65 70
0.0015 0.0018 0.0073 0.0233 0.0624
0.776 0.778 0.790 0.802 0.816
75 80 85 90 90.18
0.1448 0.3003 0.5677 0.9943 1.0133
0.827 0.845 0.862 0.880 0.881
1.634 2.547 3.794 5.443 7.559
0.899 0.920 0.944 0.970 0.998
95 100 105 110 115
vg, 10–3 m3/kg 93980 77920 21240 7200 2894 1330 680.7 379.7 227.1 223.2 143.9 95.46 65.81 46.81 34.15
cpf, kJ/(kg⋅K)
µ f, 10−4 Pa⋅s
kf , W/(m⋅K)
5.642 5.510 5.397 5.301 5.297
1.570 1.589 1.607 1.625 1.626
3.04 2.54 2.16 1.88 1.87
0.170 0.164 0.157 0.151 0.151
3.045 3.113 3.196 3.276 3.354
5.216 5.141 5.073 5.009 4.950
1.645 1.672 1.706 1.752 1.814
1.66 1.51 1.34 1.20 1.07
0.144 0.138 0.131 0.125 0.118
h f, kJ/kg
hg, kJ/kg
s f, kJ/(kg⋅K)
s g, kJ/(kg⋅K)
−189.8 −188.9 −181.1 −173.3 −165.5
48.9 49.5 53.8 58.1 62.4
2.156 2.172 2.308 2.432 2.545
6.548 6.507 6.223 5.992 5.801
−159.2 −149.7 −141.7 −133.7 −133.4
66.6 70.8 74.9 78.8 78.9
2.631 2.754 2.849 2.940 2.943
−125.4 −117.1 −108.6 −99.9 −90.0
82.4 85.7 88.5 90.8 92.6
120 125 130 135 140
10.21 13.48 17.44 22.19 27.82
1.031 1.070 1.116 1.170 1.237
25.42 19.21 14.67 11.25 8.612
−81.6 −71.8 −61.5 −50.6 −38.9
93.6 93.9 93.3 91.6 88.4
3.432 3.510 3.588 3.667 3.748
4.892 4.836 4.779 4.720 4.657
1.896 2.004 2.148 2.341 2.629
0.97 0.86 0.78 0.70 0.60
0.111 0.103 0.096 0.088 0.080
145 150 154.77c
34.45 42.23 50.87
1.332 1.487 2.464
6.499 4.705 2.464
−25.9 −10.8 35.2
82.9 73.1 35.2
3.833 3.928 4.219
4.583 4.487 4.219
3.141 3.935 ∞
0.52
0.072 ∞
*Reproduced and converted from Vasserman and Rabinovich, Thermophysical Properties of Liquid Air and Its Components, Standartov, Moscow, 1968; and Israel Program for Scientific Translations, TT 69-55092, 1970. t = triple point; c = critical point. Other tables are given by Sytchev, V. V., A. A. Vasserman, et al., Thermodynamic Properties of Oxygen, Hemisphere, New York, 1987 (307 pp.); Stewart, R. B., R. T. Jacobsen, et al., J. Phys. Chem. Ref. Data, 20, 5 (1991): 917–1021; For fps units, see Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. See also Roder, H. M., Transport Properties of Oxygen, NASA Ref. Publ. 1102, 1983 (87 pp.); Laesecke, A., K. Krauss, et al., J. Phys. Chem. Ref. Data, 19, 5 (1990): 1089–1122.
THERMODYNAMIC PROPERTIES
FIG. 2-15 Temperature-entropy chart for oxygen. Pressure P, atm; density ρ, (g⋅mol)/L; temperature, K; enthalpy H, J/(g⋅mol); entropy, J/(g⋅mol⋅K). (NBS Chart D-56. Reproduced by permission.)
2-263
2-264
PHYSICAL AND CHEMICAL DATA
Enthalpy-concentration diagram for oxygen-nitrogen mixture at 1 atm. Reference states: Enthalpies of liquid oxygen and liquid nitrogen at the normal boiling point of nitrogen are zero. (Dodge, Chemical Engineering Thermodynamics, McGraw-Hill, New York, 1944.) Wilson, Silverberg, and Zellner, AFAPL TDR 64-64 (AD 603151), 1964, p. 314, present extensive vapor-liquid equilibrium data for the three-component system argon-nitrogen-oxygen as well as for binary systems including oxygen-nitrogen.
FIG. 2-16
TABLE 2-295
Pentane
Canjar and Manning (Thermodynamic Properties and Reduced Correlations for Gases, Gulf, Houston, 1967) give extensive tables and an enthalpy–log-pressure diagram, based upon Brydon, Walen, and Canjar [Chem. Eng. Prog. Symp. Ser., 49, 7, (1951): 151–157]. For isopentane, Arnold, Liou, and Eldridge [ J. Chem. Eng. Data, 10, 88 (1965)] used the Benedict-WebbRubin equation to generate information to 600°F and 60 atm. Das and Kuloor used the same equation in Ind. J. Technol., 5, 46 (1967) to calculate information up to 1500 K and 1000 atm. Saturation and superheat tables and a diagram to 200 bar, 600 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For equations, see Grigoryev, B. A., Yu. L. Rastorguyev, et al., Int. J. Thermophys., 11, 3 (1990): 487–502.
TABLE 2-296 T, K m
336.4 400 500 600 700 800 1000 1200 1400 1500
Saturated Potassium*
P, bar
vf, m3/kg
1.37.−9 1.84.−7 3.13.−5 9.26.−4 0.01022
0.001208 0.001229 0.001266 0.001304 0.001346
0.06116 0.7322 3.913 12.44 20.0
0.001389 0.001488 0.001605 0.001742 0.001816
vg, m3/kg
h f, kJ/kg
h g, kJ/kg
sf, kJ/(kg⋅K)
s g, kJ/(kg⋅K)
c pf, kJ/(kg⋅K)
4.64.+6 3.39.+4 3164 142.3
93.8 145.5 225.1 302.7 379.4
2327 2342 2390 2433 2468
1.928 2.068 2.246 2.388 2.506
8.567 7.559 6.576 5.937 5.490
0.822 0.805 0.785 0.771 0.762
455.5 609.7 773.5 948.0 1040.0
2498 2552 2610 2679 2718
2.608 2.780 2.929 3.063 3.123
5.161 4.722 4.459 4.299 4.209
0.761 0.792 0.846 0.899 0.924
26.75 2.691 0.584 0.207 0.132
*Converted from tables in Vargaftik, Tables of the Thermophysical Properties of Liquids and Gases, Nauka, Moscow, 1972; and Hemisphere, Washington, 1975. m = melting point. The notation 1.37.−9 signifies 1.37 × 10−9. Many of the Vargaftik values also appear in Ohse, R. W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Sci. Pubs., Oxford, 1985 (1020 pp.). This source contains superheat data. Saturation and superheat tables and a diagram to 30 bar, 1650 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram from 0.1 to 250 psia, 1300 to 2700°R, see Weatherford, W. D., J. C. Tyler, et al., WADD-TR61-96, 1961. An extensive review of properties of the solid and the saturated liquid is given by Alcock, C. B., M. W. Chase, et al., J. Phys. Chem. Ref. Data, 23, 3 (1994): 385–497.
THERMODYNAMIC PROPERTIES
2-265
FIG. 2-17 Mollier diagram for potassium. Basis: enthalpy = 0.0 cal/g atom at 298 K; entropy = 15.8 cal/(g atom·K) at 298 K. (Aerojet-General Rep. AGN8194, vol. 2, 1967. Reproduced by permission.)
2-266
PHYSICAL AND CHEMICAL DATA
TABLE 2-297
Saturated Propane (R290)* vg, m3/kg
µ f, 10−4 Pa⋅s
k f, W/(m⋅K)
1.96 1.98 2.00 2.02 2.04
6.61 5.54 4.67
0.191 0.183 0.175
5.9862 5.9114 5.8502 5.8005 5.7603
2.07 2.10 2.13 2.16 2.20
3.97 3.27 2.98 2.65 2.36
0.166 0.158 0.150 0.143 0.136
3.8631 3.9605 4.0563 4.1505 4.2433
5.7280 5.7022 5.6817 5.6656 5.6528
2.25 2.29 2.34 2.41 2.48
2.07 1.86 1.69 1.53 1.40
0.129 0.123 0.117 0.111 0.106
906.03 916.54 926.41 935.45 943.38
4.3349 4.4257 4.5160 4.6062 4.6971
5.6426 5.6343 5.6270 5.6200 5.6124
2.56 2.65 2.76 2.89 3.06
1.29 1.19 1.10 0.93 0.82
0.100 0.096 0.091 0.086 0.082
949.79 953.92 954.23 946.56 879.20
4.7896 4.8850 4.9861 5.0997 5.3300
5.6030 5.5896 5.5681 5.5277 5.3300
3.28 3.62 4.23 5.98 ∞
0.72 0.62 0.52 0.40 0.29
0.078 0.073 0.069 0.066 ∞
T, K
P, bar
v f, m3/kg
h f, kJ/kg
h g, kJ/kg
s f, kJ/(kg⋅K)
s g, kJ/(kg⋅K)
c pf, kJ/(kg⋅K)
85.5t 90 100 110 120
3.0.−9 1.5.−8 3.2.−7 3.9.−6 3.1.−5
1.364.−3 1.373.−3 1.392.−3 1.412.−3 1.432.−3
5.37.+7 1.12.+7 5.85.+5 53275 7350
124.92 133.56 152.74 172.03 191.46
690.02 693.58 702.23 711.71 721.78
1.8738 1.9723 2.1743 2.3581 2.5271
8.3548 8.0953 7.6163 7.2377 6.9343
1.92 1.92 1.93 1.94 1.95
130 140 150 160 170
1.8.−4 7.7.−4 2.74.−3 8.22.−3 0.0214
1.453.−3 1.475.−3 1.497.−3 1.521.−3 1.545.−3
1400 344 103 36.8 15.0
211.03 230.77 250.67 270.78 291.10
732.27 743.07 754.12 765.37 776.80
2.6838 2.8300 2.9674 3.0971 3.2202
6.6885 6.4881 6.3237 6.1886 6.0775
180 190 200 210 220
0.0495 0.1035 0.1993 0.3574 0.6031
1.570.−3 1.597.−3 1.625.−3 1.654.−3 1.686.−3
6.84 3.43 1.868 1.087 0.669
311.66 332.48 353.61 375.07 396.90
788.40 800.15 812.03 824.01 836.04
3.3377 3.4503 3.5586 3.6631 3.7645
230 240 250 260 270
0.9661 1.4800 2.1819 3.1118 4.3120
1.719.−3 1.754.−3 1.792.−3 1.833.−3 1.878.−3
0.432 0.290 0.2020 0.1445 0.1059
419.16 442.07 465.58 489.70 514.45
848.08 860.07 871.94 883.62 895.02
280 290 300 310 320
5.8278 7.7063 9.9973 12.75 16.03
1.927.−3 1.982.−3 2.044.−3 2.115.−3 2.200.−3
0.0791 0.0600 0.0461 0.0357 0.0279
539.88 566.06 593.11 621.18 650.49
330 340 350 360 369.8c
19.88 24.36 29.56 35.55 42.42
2.301.−3 2.430.−3 2.607.−3 2.896.−3 4.566.−3
0.0218 0.0170 0.0130 0.0095 0.0046
681.37 714.38 750.52 792.50 879.20
*Values converted and mostly rounded off from those of Goodwin, NBSIR 77-860, 1977. t = triple point; c = critical point. The notation 3.0.−9 signifies 3.0 × 10−9. Later tables for the same temperature range for saturation and for the superheat state from 0.1 to 1000 bar, 85.5 to 600 K, were published by Younglove, B. A. and J. F. Ely, J. Phys. Chem. Ref. Data, 16, 4 (1987): 685–721, but the lower temperature saturation tables contain some errors. Saturation and superheat tables and a chart to 10,000 psia, 800°F appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For thermodynamic properties for 0.1 to 1000 bar, 100 to 700 K, see Sychev, V. V., A. A. Vasserman, et al., Thermodynamic Properties of Propane, Hemisphere, New York, NY, 1991 (275 pp.). Saturation and superheat tables and a diagram to 200 bar, 600 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.
THERMODYNAMIC PROPERTIES TABLE 2-298
2-267
Saturated Propylene (Propene, R1270)
T, K
P, bar
v f, m3/kg
vg, m3/kg
h f, kJ/kg
hg, kJ/ kg
s f, kJ/(kg⋅K)
s g, kJ/(kg⋅K)
c pf, kJ/(kg⋅K)
87.9t 90 100 110 120
9.54.−9 2.05.−8 4.81.−7 6.08.−6 4.88.−5
0.001 301 0.001 305 0.001 325 0.001 346 0.001 367
1.82.+7 8.66.+6 411 165 35 753 4 856
−290.1 −285.1 −265.4 −247.7 −229.8
279.2 281.1 290.2 299.6 309.3
−1.923 −1.867 −1.659 −1.490 −1.335
4.554 4.424 3.897 3.488 3.158
1.695 1.760 1.820
130 140 150 160 170
2.77.−4 1.20.−3 4.17.−3 0.0122 0.0309
0.001 389 0.001 411 0.001 434 0.001 458 0.001 483
927.0 230.91 71.043 25.903 10.842
−211.4 −192.4 −172.9 −153.1 −133.1
319.3 329.4 339.8 350.4 361.2
−1.187 −1.046 −0.912 −0.784 −0.663
2.895 2.681 2.506 2.363 2.245
180 190 200 210 220
0.0697 0.1425 0.2686 0.4727 0.7849
0.001 508 0.001 535 0.001 563 0.001 593 0.001 624
5.080 2.613 1.452 0.860 0.538
−112.7 −92.2 −71.4 −50.3 −28.8
372.1 383.1 394.2 405.3 416.3
−0.547 −0.436 −0.329 −0.226 −0.127
225.5 b 230 240 250 260
1.0133 1.2401 1.8775 2.7401 3.8737
0.001 642 0.001 657 0.001 693 0.001 732 0.001 774
0.4241 0.3515 0.2388 0.1674 0.1206
−16.9 −7.0 15.3 38.0 61.3
422.2 427.1 437.8 448.2 458.2
270 280 290 300 310
5.3269 7.1499 9.3954 12.12 15.38
0.001 820 0.001 872 0.001 929 0.001 995 0.002 071
0.0888 0.0666 0.0507 0.0390 0.0303
85.2 109.9 135.3 161.6 189.0
320 330 340 350 360
19.23 23.75 29.01 35.12 42.20
0.002 162 0.002 273 0.002 418 0.002 628 0.003 038
0.0236 0.0184 0.0142 0.0107 0.0075
365.6 c
46.65
0.004 476
0.0045
µ f, 10−6 Pa·s
k f, W/(m⋅K)
Pr
2017 1526 1185
0.214 0.209 0.204
15.98 12.85 10.57
1.875 1.923 1.964 1.996 2.020
941 735 587 478 397
0.198 0.193 0.188 0.183 0.178
8.91 7.32 6.13 5.21 4.50
2.147 2.066 1.999 1.943 1.896
2.044 2.067 2.094 2.128 2.162
334.5 286.1 244.9 212.7 187.0
0.173 0.168 0.162 0.157 0.152
3.95 3.52 3.17 2.88 2.66
−0.073 −0.030 0.064 0.157 0.247
1.874 1.857 1.825 1.797 1.774
2.182 2.199 2.243 2.298 2.369
175.0 166.2 149.2 135.0 123.0
0.149 0.147 0.142 0.137 0.131
2.56 2.49 2.36 2.26 2.22
467.8 476.9 485.3 492.8 499.3
0.336 0.425 0.512 0.600 0.688
1.753 1.735 1.719 1.704 1.688
2.418 2.494 2.584 2.693 2.842
112.9 106.6 100.0 93.0 85.6
0.126 0.121 0.116 0.112 0.109
2.17 2.20 2.23 2.24 2.23
217.7 248.2 280.9 317.6 364.1
504.3 507.4 507.6 502.8 486.0
0.776 0.867 0.961 1.062 1.188
1.672 1.652 1.627 1.592 1.527
3.007 3.335 3.723 4.669
77.8 69.6 61.0
0.104 0.097 0.090 0.082
2.25 2.39 2.52
433.3
433.3
1.374
1.374
t = triple point; b = normal boiling point; c = critical point. The notation 9.54.−9 signifies 9.54 × 10−9. hf = sf = 0 at 233.15 K = −40°C. Converted from Angus, S., B. Armstrong, et al., Intnl. Thermodynamic Properties of the Fluid State—7. Propylene (Propene) R1270, Pergamon Press, Oxford, 1980 (401 pp.).
2-268
PHYSICAL AND CHEMICAL DATA
TABLE 2-299
Compressed Propylene (Propene, R1270) Temperature, K
Pressure, bar
225
250
275
300
325
350
375
400
425
450
v (m3/ kg) 1 h (kJ/ kg) s (kJ/ kg⋅K)
0.00164 −17.9 −0.0779
0.4817 455.5 2.0169
0.5334 491.2 2.1530
0.5846 529.1 2.2847
0.6354 569.1 2.4128
0.6858 611.4 2.5380
0.7361 656.0 2.6610
0.7861 702.8 2.7821
0.8361 751.9 2.9010
0.8859 803.2 3.0183
v (m3/ kg) 10 h (kJ/ kg) s (kJ/ kg⋅K)
0.00164 −17.2 −0.0810
0.00173 38.5 0.1535
0.00184 97.6 0.3788
0.04986 501.2 1.7631
0.05670 547.0 1.9653
0.06295 593.2 2.0466
0.06889 640.8 2.1774
0.07460 689.7 2.3042
0.08014 740.6 2.4273
0.08557 793.3 2.5480
v (m3/ kg) 20 h (kJ/ kg) s (kJ/ kg⋅K)
0.00163 −16.3 −0.0841
0.00172 39.3 0.1497
0.00184 98.1 0.3736
0.00198 161.5 0.5941
0.02324 512.7 1.6920
0.02773 568.2 1.8567
0.03149 621.5 2.0024
0.03489 673.5 2.1381
0.03803 726.5 2.2674
0.04104 781.5 2.3923
v (m3/ kg) 40 h (kJ/ kg) s (kJ/ kg⋅K)
0.00163 −14.4 −0.0908
0.00172 40.8 0.1419
0.00182 99.0 0.3638
0.00196 161.3 0.5806
0.00216 230.0 0.8001
0.00256 313.3 1.0466
0.01465 633.6 1.9256
0.01682 695.4 2.0758
0.01872 755.6 2.2131
v (m3/ kg) 60 h (kJ/ kg) s (kJ/ kg⋅K)
0.00162 −12.6 −0.0970
0.00171 42.3 0.1345
0.00181 100.1 0.3546
0.00194 161.5 0.5684
0.00211 228.2 0.7816
0.00240 303.8 1.0055
0.00743 575.4 1.7272
0.00944 656.7 1.9250
0.01126 726.1 2.0832
v (m3/ kg) 80 h (kJ/ kg) s (kJ/ kg⋅K)
0.00162 −10.7 −0.1031
0.00170 44.0 0.1274
0.00180 101.3 0.3458
0.00192 162.0 0.5570
0.00208 227.2 0.7657
0.00231 299.0 0.9781
0.00402 499.7 1.5107
0.00605 607.7 1.7795
0.00757 693.3 1.9693
v (m3/ kg) 100 h (kJ/ kg) s (kJ/ kg⋅K)
0.00161 −8.8 −0.1091
0.00169 45.6 0.1202
0.00179 102.6 0.3374
0.00190 162.7 0.5466
0.00204 226.7 0.7514
0.00224 296.1 0.9570
0.00256 373.5 1.1704
0.00316 464.9 1.4061
0.00426 567.8 1.6456
0.00551 660.1 1.8669
v (m3/ kg) 150 h (kJ/ kg) s (kJ/ kg⋅K)
0.00160 −4.1 −0.1236
0.00167 49.9 0.1038
0.00176 106.1 0.3180
0.00186 165.0 0.5228
0.00198 227.0 0.7214
0.00214 292.8 0.9163
0.00234 362.9 1.1100
0.00262 438.5 1.3049
0.00300 519.5 1.5021
0.00354 604.6 1.6958
v (m3/ kg) 200 h (kJ/ kg) s (kJ/ kg⋅K)
0.00159 0.8 −0.1371
0.00166 54.4 0.0884
0.00174 110.0 0.3004
0.00183 167.9 0.5021
0.00194 228.7 0.6963
0.00207 292.3 0.8852
0.00222 359.2 1.0701
0.00242 429.9 1.2521
0.00266 504.4 1.4319
0.00296 581.4 1.6086
v (m3/ kg) 300 h (kJ/ kg) s (kJ/ kg⋅K)
0.00157 10.9 −0.1625
0.00163 63.8 0.0601
0.00170 118.5 0.2688
0.00178 175.2 0.4660
0.00187 234.2 0.6549
0.00197 295.5 0.8367
0.00208 359.3 1.0127
0.00221 425.7 1.1839
0.00236 494.4 1.3507
0.00253 565.6 1.5133
v (m3/ kg) 400 h (kJ/ kg) s (kJ/ kg⋅K)
0.00155 21.2 −0.1863
0.00161 73.6 0.0347
0.00167 127.7 0.2407
0.00174 183.6 0.4351
0.00182 241.5 0.6207
0.00190 301.5 0.7985
0.00199 363.7 0.9705
0.00209 428.0 1.1362
0.00220 494.4 1.2972
0.00232 562.9 1.4536
v (m3/ kg) 500 h (kJ/ kg) s (kJ/ kg⋅K)
0.00153 31.6 −0.2082
0.00159 83.7 0.0112
0.00165 137.3 0.2155
0.00171 192.6 0.4080
0.00178 249.8 0.5910
0.00185 309.0 0.7664
0.00193 370.1 0.9351
0.00201 433.3 1.0981
0.00210 498.7 1.2559
0.00220 565.4 1.4092
Converted and interpolated from Angus, S., B. Armstrong, et al., International Thermodynamic Tables of the Fluid State—7. Propylene, Pergamon, Oxford, 1980 (401 pp.). The 1993 ASHRAE Handbook—Fundamentals (SI ed.) has a thermodynamic chart for pressures from 0.1 to 1000 bar for temperatures up to 580 K. Saturation and superheat tables and a diagram to 30,000 psia, 580°F appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.
THERMODYNAMIC PROPERTIES TABLE 2-300
2-269
Saturated Refrigerant 11*
P, bar
vf, m3/kg
vg, m3/kg
h f, kJ/kg
h g, kJ/kg
s f, kJ/(kg⋅K)
s g, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µ f, 10−4 Pa⋅s
k f, W/(m⋅K)
200 220 240 260 270
0.0043 0.0417 0.0768 0.2215 0.3514
5.901.−4 6.061.−4 6.225.−4 6.398.−4 6.491.−4
28.06 6.272 1.882 0.703 0.458
−14.37 −8.20 4.97 21.01 29.53
186.30 195.89 205.85 216.06 221.23
−0.0651 −0.0361 0.0210 0.0851 0.1172
0.9431 0.8925 0.8581 0.8353 0.8272
0.815 0.828 0.842 0.856 0.863
1.674 1.142 0.831 0.635 0.563
0.115 0.110 0.104 0.098 0.095
280 290 300 310 320
0.5364 0.7917 1.1341 1.5821 2.1556
6.587.−4 6.688.−4 6.794.−4 6.908.−4 7.027.−4
0.309 0.216 0.154 0.113 0.0847
38.25 47.10 56.06 65.10 74.22
226.40 231.58 236.73 241.83 246.88
0.1489 0.1799 0.2102 0.2397 0.2686
0.8209 0.8160 0.8124 0.8099 0.8081
0.870 0.878 0.887 0.897 0.907
0.504 0.454 0.413 0.377 0.346
0.093 0.090 0.087 0.084 0.081
330 340 350 360 380
2.876 3.764 4.845 6.142 9.487
7.156.−4 7.293.−4 7.442.−4 7.603.−4 7.974.−4
0.0645 0.0500 0.0392 0.0311 0.0201
83.42 92.72 102.12 111.64 131.12
251.84 256.69 261.40 265.95 274.40
0.2967 0.3243 0.3513 0.3778 0.4298
0.8071 0.8065 0.8064 0.8065 0.8069
0.917 0.928 0.939 0.950 0.975
0.320 0.297 0.276 0.259 0.229
0.079 0.076 0.073 0.070 0.065
8.435.−4 9.042.−4 9.930.−4 1.167.−3 1.799.−3
0.0134 0.0090 0.0059 0.0036 0.0018
151.38 172.76 196.01 223.85 258.70
281.69 287.20 289.72 285.36 258.70
0.4808 0.5317 0.5840 0.6435 0.7162
0.8066 0.8041 0.7970 0.7773 0.7162
1.004 1.04 1.09 1.19 ∞
0.203 0.169 0.131 0.084 0.033
0.059 0.053 0.048 0.037 ∞
T, K
400 420 440 460 471.2c
14.02 19.98 27.65 37.36 44.09
*Values reproduced or converted from Table 1, p. 17.75, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and AirConditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 5.901.−4 signifies 5.901 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the ITS 90 scale. For experimental isochores for the compressed liquid from 12 to 301 bar, 254 to 453 K, see Blanke, W. and R. Weiss, PTB Bericht W 30, Braunschweig, Germany, 1992 (54 pp.). Equations and constants approximated to 1985 ASHRAE tables are given by Mecarik, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. For tables and a chart to 3000 psia, 460°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For similar material to 80 bar, 650 K, see Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For specific heat at constant pressure, thermal conductivity, and viscosity in both SI and fps units, see Liley, P. E., Thermophysical Properties of Refrigerants, ASHRAE, Atlanta, GA, 1993.
FIG. 2-18 Enthalpy–log-pressure diagram for Refrigerant 11. 1 MPa = 10 bar. (Copyright 1981 by the American Society of Heating, Refrigerating and Air-Conditioning Engineers and reproduced by permission of the copyright owner.) This chart, redrawn with a different zero point and temperatures in Celsius, appears in ASHRAE Handbook—Fundamentals (SI ed.), Atlanta, GA, 1993.
2-270
PHYSICAL AND CHEMICAL DATA
TABLE 2-301
Saturated Refrigerant 12* h f, kJ/kg
h g, kJ/kg
s f, kJ/(kg⋅K)
s g, kJ/(kg⋅K)
c pf, kJ/(kg⋅K)
µ f, 10−4 Pa⋅s
k f, W/(m⋅K)
294.6 302.3 310.3 318.3 326.5
496.0 500.2 504.5 508.9 513.5
3.492 3.543 3.591 3.637 3.681
4.835 4.780 4.734 4.696 4.665
0.808 0.817 0.827 0.836 0.845
18.9 15.1 12.1 9.69 7.94
0.123 0.119 0.116 0.113 0.109
1.370 0.7589 0.4476 0.2784 0.1811
334.8 343.2 351.8 360.6 369.5
518.1 522.7 527.4 531.1 536.8
3.724 3.765 3.805 3.844 3.881
4.640 4.620 4.603 4.590 4.579
0.855 0.864 0.873 0.882 0.891
6.64 5.65 4.88 4.26 3.77
0.105 0.102 0.098 0.094 0.090
6.810.−4 6.970.−4 7.112.−4 7.282.−4 7.470.−4
0.1225 0.08559 0.06147 0.04543 0.03888
378.0 387.7 397.0 406.5 416.1
541.5 546.1 550.7 555.1 559.4
3.918 3.954 3.989 4.023 4.057
4.570 4.563 4.558 4.554 4.551
0.902 0.913 0.926 0.942 0.959
3.37 3.03 2.75 2.52 2.31
0.087 0.083 0.080 0.076 0.072
T, K
P, bar
vf, m3/kg
v g, m3/kg
150 160 170 180 190
0.00091 0.00305 0.00871 0.02178 0.04877
5.767.−4 5.849.−4 5.926.−4 6.024.−4 6.118.−4
179.12 36.05 13.40 5.666 2.665
200 210 220 230 240
0.0996 0.1879 0.3317 0.5531 0.8781
6.217.−4 6.139.−4 6.431.−4 6.549.−4 6.675.−4
250 260 270 280 290
1.3359 1.959 2.784 3.825 5.184
300 310 320 330 340
6.840 8.860 11.29 14.17 17.58
7.678.−4 7.912.−4 8.173.−4 8.478.−4 8.840.−4
0.02582 0.01992 0.01553 0.01218 0.00957
426.0 436.0 446.2 456.8 467.8
563.5 567.3 570.9 574.0 576.5
4.090 4.122 4.154 4.186 4.218
4.548 4.546 4.543 4.541 4.538
0.979 1.005 1.041 1.093 1.166
2.14 2.00 1.86 1.74 1.60
0.069 0.065 0.061 0.058 0.054
350 360 370 380 385c
21.57 26.19 31.56 37.76 41.31
9.286.−4 9.868.−4 1.072.−3 1.237.−3 1.876.−3
0.00750 0.00582 0.00439 0.00305 0.00188
479.4 492.1 506.4 524.7 551.1
578.2 578.7 577.2 571.2 551.1
4.250 4.285 4.322 4.369 4.437
4.533 4.525 4.514 4.900 4.437
1.264 1.39 1.55
1.45 1.28 1.06 0.75 0.31
0.050 0.046 0.041
∞
∞
*P, v, h, and s data interpolated from Perelshteyn (ed.), Tables and Diagrams of the Thermodynamic Properties of Refrigerants 12, 13, and 22, Moscow, 1971. cp, µ, and k data interpolated and converted from Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1976. c = critical point. The notation 5.767.−4 signifies 5.767 × 10−4.
TABLE 2-302
Saturated Refrigerant 13*
P, bar
vf, m3/kg
91 100 110 120 130
3.817.−6 3.418.−5 2.563.−4 0.00137 0.00571
5.367.−4 5.448.−4 5.538.−4 5.635.−4 5.739.−4
140 150 160 170 180
0.01895 0.05250 0.1258 0.2680 0.5186
5.850.−4 5.969.−4 6.095.−4 6.231.−4 6.380.−4
190 200 210 220 230
0.9269 1.5507 2.456 3.712 5.396
T, K
vg, m3/kg
cpf, kJ/(kg⋅K)
µf, 10−4 Pa⋅s
kf, W/(m⋅K)
4.610 4.558 4.516 4.482 4.454
0.826 0.845 0.865 0.884
6.83 5.60 4.59 3.83
0.114 0.109 0.104 0.099
3.642 3.688 3.732 3.777 3.820
4.431 4.413 4.397 4.385 4.374
0.898 0.910 0.924 0.943 0.972
3.26 2.82 2.48 2.20 1.97
0.093 0.088 0.083 0.078 0.072
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
238.1 243.8 251.1 258.2 265.8
424.9 429.0 433.1 437.2 441.3
3.080 3.140 3.205 3.267 3.327
5.133 4.990 4.860 4.759 4.677
5.865 2.2617 1.0019 0.4962 0.2689
273.7 281.7 290.0 298.4 307.1
455.3 449.3 453.5 457.6 461.8
3.385 3.441 3.494 3.545 3.594
6.536.−4 6.709.−4 6.899.−4 7.110.−4 7.346.−4
0.1567 9.69.−2 6.28.−2 4.24.−2 2.95.−2
315.9 325.0 334.3 343.8 353.6
465.9 469.9 473.8 477.5 481.0
19557 2392 347.0 70.25 18.15
240 250 260 270 280
7.589 10.37 13.85 18.13 23.32
7.615.−4 7.928.−4 8.302.−4 8.769.−4 9.320.−4
2.11.−2 1.53.−2 1.13.−2 8.28.−3 6.10.−3
363.5 373.9 384.7 396.2 408.8
484.1 486.1 489.1 490.5 490.6
3.862 3.903 3.944 3.986 4.029
4.364 4.355 4.346 4.336 4.323
1.014 1.072 1.151 1.255 1.386
1.79 1.63 1.50 1.34 1.14
0.067 0.062 0.057 0.051 0.045
290 300 302.0c
29.57 37.05 38.70
1.035.−3 1.284.−3 1.808.−3
4.34.−3 2.60.−3 1.81.−3
423.6 445.3 463.1
488.3 477.5 463.1
4.080 4.151 4.209
4.303 4.257 4.209
1.549 1.75 ∞
0.87 0.52 0.29
0.038 ∞
*P, v, h, and s data interpolated from Perelshteyn (ed.), Tables and Diagrams of the Thermodynamic Properties of Refrigerants 12, 13 and 22, Moscow, 1971. cp, µ, and k data interpolated and converted from Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1976. c = critical point. The notation 3.817.−6 signifies 3.817 × 10−6. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) contains a table at closer temperature increments and also an enthalpy–log-pressure diagram from 0.1 to 70 bar, −100 to 240°C. Equations and constants approximated to 1985 ASHRAE tables are given by Mecarik, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. Saturation and superheat tables and a diagram to 60 bar, 600 K are given by Reynolds, W. C., Thermodynamic Properties in SI, Stanford Univ. publ., 1979 (173 pp.). For tables and a chart to 1000 psia, 520°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.
THERMODYNAMIC PROPERTIES
2-271
Enthalpy–log-pressure diagram for Refrigerant 12. 1 MPa = 10 bar. (Copyright 1981 by the American Society of Heating, Refrigerating and AirConditioning Engineers and reproduced by permission of the copyright owner.) This chart, redrawn for integral Celsius temperatures with a different zero point, appears on p. 17.4 of the 1993 ASHRAE Handbook—Fundamentals (SI ed.). This handbook gives material for integral degrees Celsius with temperatures on the ITS 90 scale. For experimental isochores for the compressed liquid from 10 to 302 bar, 122 to 462 K, see Blanke, W. and R. Weiss, PTB Bericht W30, Braunschweig, Germany, 1992 (54 pp.). Equations and constants approximated to 1985 ASHRAE tables are given by Mecarik, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. Tables at 2°C increments to 240°C, 50 bar are given by Watson, J. T. R., Thermophysical Properties of Refrigerant 12, H.M.S.O., Edinburgh, Scotland, 1975 (183 pp.). Saturation and superheat tables and a diagram to 40 bar, 620 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ., 1979 (173 pp.). Tables and a chart to 1100 psia, 480°F are given by Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.
FIG. 2-19
2-272
PHYSICAL AND CHEMICAL DATA
TABLE 2-303
Saturated Refrigerant 13B1*
P, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−4 Pa⋅s
kf, W/(m⋅K)
170 180 190 200 210
0.059 0.127 0.250 0.455 0.777
4.594.−4 4.677.−4 4.765.−4 4.860.−4 4.961.−4
1.6015 0.7840 0.4190 0.2407 0.1467
−40.90 −34.75 −28.51 −22.17 −15.68
90.95 94.37 97.83 101.32 104.82
−0.2033 −0.1682 −0.1345 −0.1020 −0.0704
0.5723 0.5491 0.5305 0.5154 0.5033
0.597 0.618 0.634 0.648 0.663
9.54 7.60 6.20 5.13 4.33
0.101 0.096 0.091 0.086 0.082
215.4 220 230 240 250
1.013 1.254 1.933 2.863 4.096
5.020.−4 5.071.−4 5.190.−4 5.321.−4 5.466.−4
0.1147 0.0940 0.0628 0.0433 0.0308
−12.09 −9.02 −2.19 4.83 12.03
106.70 108.28 111.68 114.99 118.16
−0.0536 −0.0396 −0.0094 0.0202 0.0494
0.4978 0.4936 0.4857 0.4793 0.4739
0.670 0.676 0.690 0.703 0.721
3.97 3.71 3.22 2.83 2.51
0.079 0.077 0.073 0.068 0.063
260 270 280 290 300
5.690 7.703 10.20 13.25 16.91
5.627.−4 5.809.−4 6.018.−4 6.264.−4 6.562.−4
0.0224 0.0166 0.0124 0.0094 0.0072
19.44 27.06 34.94 43.11 51.68
121.16 123.93 126.41 128.51 130.09
0.0781 0.1064 0.1345 0.1625 0.1908
0.4693 0.4652 0.4612 0.4570 0.4522
0.742 0.767 0.800 0.842 0.891
2.25 2.04 1.84 1.69 1.57
0.059 0.054 0.049 0.045 0.040
310 320 330 340.2c
21.28 26.44 32.48 39.64
6.940.−4 7.458.−4 8.295.−4 1.344.−3
0.0055 0.0041 0.0030 0.0013
60.81 70.80 82.42 108.70
130.97 130.76 128.59 108.70
0.2197 0.2503 0.2845 0.3605
0.4460 0.4376 0.4245 0.3605
0.951 1.09 1.29 ∞
1.45 1.26 0.99 0.35
0.035 0.030 0.026 ∞
T, K
*Values reproduced or converted from Table 4, p. 17.83, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 4.594.−4 signifies 4.594 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) contains a table at closer temperature increments and also an enthalpy–log-pressure diagram from 0.1 to 35 bar, −80 to 220°C. For tables and a chart to 500 psia, 480°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.
Refrigerant 14 (tetrafluoromethane) See Carbon Tetrafluoride (Table 2-245).
TABLE 2-304
Refrigerant 20 See Chloroform (Table 2-248).
Saturated Refrigerant 21
Temperature, K
Pressure, bar
250 260 270 280 290
0.2415 0.3953 0.6200 0.9364 1.3682
0.000 0.000 0.000 0.000 0.000
300 310 320 330 340
1.9417 2.6849 3.6279 4.8022 6.2409
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
677 687 698 709 722
0.8292 0.5247 0.3455 0.2355 0.1654
16.6 26.5 36.6 46.7 57.1
274.8 279.9 284.9 290.0 295.0
0.0687 0.1076 0.1454 0.1824 0.2186
1.1015 1.0820 1.0653 1.0511 1.0389
0.000 0.000 0.000 0.000 0.000
735 748 763 778 794
0.1192 0.0879 0.0661 0.0505 0.0391
67.7 78.4 89.5 100.7 112.3
300.0 304.8 309.5 314.1 318.4
0.2543 0.2894 0.3242 0.3586 0.3927
1.0286 1.0196 1.0119 1.0051 0.9989
sg, kJ⋅(kg⋅K)
350 360 370 380 390
7.978 10.049 12.489 15.337 18.630
0.000 0.000 0.000 0.000 0.000
812 830 850 870 893
0.0307 0.0243 0.0194 0.0155 0.0125
124.1 136.2 148.6 161.2 173.9
322.4 326.1 329.3 331.9 333.8
0.4266 0.4602 0.4935 0.5264 0.5587
0.9932 0.9877 0.9820 0.9758 0.9688
400 410 420 430 440
22.41 26.72 31.60 37.10 43.26
0.000 0.000 0.000 0.001 0.001
918 944 972 002 034
0.01011 0.00820 0.00672 0.00564 0.00491
186.4 198.3 208.7 216.4 221.1
334.8 334.7 333.7 332.4 332.3
0.5896 0.6180 0.6418 0.6587 0.6682
0.9605 0.9506 0.9394 0.9286 0.9208
Reproduced and rounded from unpublished Center for Applied Thermodynamic Studies, Moscow ID report, 1981. For a thermodynamic diagram to 350 bar, 370°C, see Rombusch, U. K., Allgem. Warme., 11, 3 (1962).
TABLE 2-305
Saturated Refrigerant 22*
P, bar
vf, m3/kg
vg, m3/kg
150 160 170 180 190
0.0017 0.0054 0.0150 0.0369 0.0821
6.209.−4 6.293.−4 6.381.−4 6.474.−4 6.573.−4
83.40 28.20 10.85 4.673 2.225
200 210 220 230 240
0.1662 0.3116 0.5470 0.9076 1.4346
6.680.−4 6.794.−4 6.917.−4 7.050.−4 7.195.−4
250 260 270 280 290
2.174 3.177 4.497 6.192 8.324
300 310 320 330 340 350 360 369.3c
T, K
µf, 10−4 Pa⋅s
kf, W/(m⋅K)
1.059 1.058 1.057 1.058 1.060
0.770 0.647 0.554
0.161 0.156 0.151 0.146 0.141
4.921 4.885 4.854 4.828 4.805
1.065 1.071 1.080 1.091 1.105
0.481 0.424 0.378 0.340 0.309
0.136 0.131 0.126 0.121 0.117
3.898 3.942 3.986 4.029 4.071
4.785 4.768 4.752 4.738 4.725
1.122 1.143 1.169 1.193 1.220
0.282 0.260 0.241 0.225 0.211
0.112 0.107 0.102 0.097 0.092
612.8 615.1 616.7 617.3 616.5
5.113 4.153 4.194 4.235 4.278
4.713 4.701 4.688 4.674 4.658
1.257 1.305 1.372 1.460 1.573
0.198 0.186 0.176 0.167 0.151
0.087 0.082 0.077 0.072 0.067
613.3 605.5 570.0
4.324 4.378 4.501
4.637 4.605 4.501
1.718 1.897 ∞
0.130 0.106
0.062
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
268.2 278.2 288.3 298.7 308.6
547.3 552.1 557.0 561.9 566.8
3.355 3.430 3.494 3.551 3.605
5.215 5.141 5.075 5.013 4.963
1.145 0.6370 0.3772 0.2352 0.1532
318.8 329.1 339.7 350.6 361.7
571.6 576.5 581.2 585.9 590.5
3.657 3.707 3.756 3.804 3.852
7.351.−4 7.523.−4 7.733.−4 7.923.−4 8.158.−4
0.1037 0.07237 0.05187 0.03803 0.02838
373.0 384.5 396.3 408.2 420.4
594.9 599.0 603.0 606.6 610.0
10.956 14.17 18.02 22.61 28.03
8.426.−4 8.734.−4 9.096.−4 9.535.−4 1.010.−3
0.02148 0.01643 0.01265 9.753.−3 7.479.−3
432.7 445.5 458.6 472.4 487.2
34.41 41.86 49.89
1.086.−3 1.212.−3 2.015.−3
5.613.−3 4.036.−3 2.015.−3
503.7 523.7 570.0
*P, v, h, and s data interpolated from Perelshteyn (ed.), Tables and Diagrams of the Thermodynamic Properties of Refrigerants 12, 13 and 22, Moscow, 1971. cp, µ, and k data interpolated and converted from Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1976. c = critical point. The notation 6.209.−4 signifies 6.209 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives a saturation table from −150 to 96.14°C and an enthalpy–log-pressure diagram from 0.1 to 150 bar, −60 to 200°C. For experimental isochores for the compressed liquid from 12 to 297 bar, 120 to 378 K, see Blanke, W. and R. Weiss, PTB Bericht W 30, Braunschweig, Germany, 1992 (54 pp.). Equations and constants approximated to 1985 ASHRAE tables are given by Mecarik, K. and M. Masaryk, Heat Recovery Systems and CHP, 11 2/3 (1991): 193–197. Saturation and superheat tables and a diagram to 100 bar, 620 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (179 pp.). For tables and a chart to 2000 psia, 480°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.
FIG. 2-20 Enthalpy–log-pressure diagram for Refrigerant 22. 1 MPa = 10 bar. (Copyright 1981 by the American Society of Heating, Refrigerating and Air-Conditioning Engineers and reproduced by permission of the copyright owner.)
2-273
2-274
PHYSICAL AND CHEMICAL DATA
TABLE 2-306
Thermophysical Properties of Compressed R22 Temperature, K
Pressure, bar
Property
275
300
325
350
375
400
425
450
1
cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr
0.639 11.8 0.0091 0.829
0.653 12.8 0.0106 0.793
0.689 13.9 0.0121 0.787
0.714 14.9 0.0136 0.782
0.739 15.8 0.0151 0.773
0.758 16.7 0.0166 0.762
0.781 17.7 0.0181 0.765
0.806 18.7 0.0196 0.769
5
cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr
0.725 11.8 0.0096 0.887
0.728 12.8 0.0107 0.871
0.744 13.8 0.0123 0.852
0.759 15.0 0.0138 0.839
0.766 16.2 0.0153 0.803
0.775 17.0 0.0170 0.775
0.791 18.0 0.0184 0.773
0.816 18.8 0.0199 0.771
10
cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr
1.166 211 0.0954 2.58
0.847 13.7 0.0121 0.959
0.810 14.4 0.0128 0.901
0.799 15.1 0.0144 0.838
0.797 16.1 0.0160 0.802
0.803 17.1 0.0175 0.785
0.814 18.1 0.0190 0.775
0.828 19.0 0.0205 0.767
20
cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr
0.164 211 0.0963 2.55
1.237 159 0.0849 2.32
0.949 16.5 0.0157 0.997
0.889 17.3 0.0172 0.894
0.865 18.0 0.0184 0.846
0.858 18.8 0.0199 0.811
0.859 19.6 0.0214 0.787
40
cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr
1.152 218 0.0980 2.56
1.217 164 0.0872 2.29
1.359 123 0.0767 2.18
1.373 20.7 0.0219 1.30
1.089 20.5 0.0210 1.063
0.996 20.7 0.0217 0.950
0.956 21.2 0.0233 0.870
60
cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr
1.142 221 0.0993 2.54
1.191 170 0.0889 2.28
1.311 128 0.0786 2.14
1.460 94.6
1.767 24.7 0.0305 1.431
1.221 24.2 0.0287 1.030
1.089 23.9 0.0268 0.971
80
cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr
1.132 226 0.1003 2.55
1.177 175 0.0904 2.28
1.277 133 0.0803 2.12
1.444 101 0.0690 2.12
1.861 73.6 0.0523 2.62
1.396 29.9 0.0374 1.12
1.262 27.7 0.0337 1.04
100
cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m·K) Pr
1.122 230 0.1013 2.55
1.154 179 0.0916 2.26
1.247 138 0.0817 2.11
1.361 108 0.0716 2.04
1.564 83.2 0.0607 2.14
1.923 37.6 0.0421 1.72
1.471 32.4 0.0378 1.26
2.073 55.5 0.0504 2.28
Some values are approximate as significant differences exist in the literature.
TABLE 2-307
Saturated Refrigerant 23
Temp., K
Pressure, bar
180 190 191.1b 200 210
0.510 0.950 1.013 1.652 2.709
0.000 0.000 0.000 0.000 0.000
220 230 240 250 260
4.298 6.312 9.091 12.69 17.25
270 280 290 299.1c
22.94 29.98 38.68 48.36
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
678 693 695 710 729
0.4088 0.2279 0.2139 0.1353 0.0845
−66.0 −54.4 −53.1 −42.6 −30.3
181.1 185.3 185.7 189.1 192.4
−0.3179 −0.2554 −0.2485 −0.1948 −0.1353
1.0549 1.0062 1.0011 0.9635 0.9254
0.000 0.000 0.000 0.000 0.000
751 777 807 844 889
0.0551 0.0372 0.0259 0.0183 0.0132
−17.5 −4.3 9.4 23.6 38.1
195.4 197.8 199.6 200.7 200.9
−0.0764 −0.0182 0.0392 0.0957 0.1512
0.8913 0.8602 0.8314 0.8042 0.7773
0.000 0.001 0.001 0.001
948 031 169 905
0.0095 0.0068 0.0046 0.0019
53.5 70.5 92.0 143.0
199.8 196.4 188.1 143.0
0.2071 0.2665 0.3387 0.5062
0.7493 0.7162 0.6698 0.5062
cpf, kJ/(kg⋅K)
µf, 10−6 Pa·s
kf, W/(m⋅K)
Pr
0.710 1.043 1.289 1.497
170.1 150.4 131.2 113.0
0.105 0.098 0.091 0.084
1.15 1.56 1.85 2.00
b = normal boiling point; c = critical point. hf = sf = 0 at 233.15 K = −40°C. Interpolated and converted from ASHRAE Handbook—Fundamentals, 1993. Experimental P−ρ−T data from 95 to 413 K reported in J. Phys. Chem., 89 (1985): 4637–4646 were used by Rubio, R. G., J. A. Zollweg, et al., J. Chem. Eng. Data, 36, (1991): 171–184, to calculate properties up to 1000 bar from 126 to 332° K. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives a saturation table from −100 to 25.92°C and an enthalpy–log-pressure diagram from 0.1 to 80 bar, −100 to 280°C. Equations and constants approximated to the 1985 ASHRAE tables are given by Mecaryk, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. For an enthalpy–log-pressure diagram from 0.005 to 200 bar, −140 to 180°C, see Morsy, T. E., Kaltetechnik—Klimat., 18, 9 (1966): 347–349. Saturation and superheat tables and a diagram to 100 bar, 600 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For tables and a chart to 1000 psia, 560°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (21 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.
TABLE 2-308 Temp., K
Thermophysical Properties of Saturated Difluoromethane (R32)
Pressure, bar
vf , m3/kg
vg , m3/kg
hf , kJ/kg
hg , kJ/kg
sf , kJ/(kg⋅K)
sg , kJ/(kg⋅K)
cpf , kJ/(kg⋅K)
cpg , kJ/(kg⋅K)
µf , 10−6 Pa·s
µg , 10−6 Pa·s
200 210 220 230 240
0.2960 0.5440 0.9384 1.5345 2.3963
7.845.−4 8.025.−4 8.208.−4 8.402.−4 8.611.−4
1.0580 0.5990 0.3593 0.2261 0.1483
−52.340 −37.750 −20.690 −4.984 10.947
351.160 356.880 361.970 366.773 371.129
−0.2418 −0.1652 −0.0909 −0.0215 0.0459
1.7757 1.7098 1.6485 1.5948 1.5468
1.557 1.580 1.613
0.799 0.839 0.894
283.8 247.8 220.4
10.30 10.37 10.46
250 260 270 280 290
3.5966 5.2160 7.3423 10.070 13.502
8.842.−4 9.096.−4 9.376.−4 9.696.−4 1.006.−3
0.1005 0.07020 0.05009 0.03643 0.02687
374.971 378.224 380.786 382.525 383.262
0.1117 0.1763 0.2397 0.3029 0.3654
1.5030 1.4624 1.4247 1.3886 1.3534
1.642 1.682 1.730 1.786 1.863
0.963 1.043 1.138 1.244 1.375
198.1 177.9 159.1 141.8 126.1
300 310 320 330 340
17.749 22.931 29.186 36.675 45.603
1.049.−3 1.100.−3 1.166.−3 1.243.−3 1.394.−3
0.02001 0.01497 0.01117 0.00822 0.00581
115.754 135.801 157.212 180.724 208.262
382.737 380.576 376.163 368.357 354.460
0.4283 0.4919 0.5574 0.6264 0.7047
1.3182 1.2815 1.2415 1.1950 1.1347
1.955 2.084 2.282 2.620 3.560
1.560 1.810 2.16 2.62 4.21
112.0 98.8 86.1 75.1 65.4
350 351.4c
56.336 57.927
0.00237
0.00317 0.00237
274.640 286.675
337.933 286.675
0.8927 0.9269
1.0735 0.9269
27.1778 43.786 60.849 78.456 96.713
kf , W/(m⋅K)
kg , W/(m⋅K)
Prf
Prg
10.66 10.95 11.31 11.70 12.21
0.1646 0.1562 0.1487 0.1403 0.1308
0.0097 0.0106 0.0115 0.0125 0.0136
1.98 1.92 1.85 1.81 1.80
1.06 1.08 1.12 1.16 1.23
12.82 13.71 14.4 15.3 17.3
0.1228 0.1155 0.1073 0.0990
0.0149 0.0165 0.0184 0.0205 0.0236
1.78 1.78 1.83 1.99
1.34 1.50 1.69 1.96 3.10
c = critical point. The notation 7.845.−4 signifies 7.845 × 10−4. P, v, T, h, s, and cp converted and extrapolated from Defibaugh, D. R., G. Morrison, et al., J. Chem. Eng. Data, 39 (1994): 333–340. Saturated liquid and vapor viscosities from smooth curve fits of Oliveira, C. M. B. P. and W. A. Wakeham, Int. J. Thermophys., 14, 6 (1993): 1131–1143. Thermal conductivity values based upon papers by Geller, V. Z. and M. E. Perlaitis, and by Gross, Proc. 10th Symp. Thermophys. Props., Boulder, CO, 1994. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives a saturation table to 78.41°C and a diagram to 200 bar, 200°C.
2-275
2-276
PHYSICAL AND CHEMICAL DATA
TABLE 2-309
Specific Heat at Constant Pressure, Thermal Conductivity, Viscosity, and Prandtl Number of R32 Gas P, bar
Temp., K 250
260
270
280
290
300
5
P, bar
Property
1
10
cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr
0.805 10.55 0.0094 0.908
cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr
0.810 11.00 0.0100 0.890
1.025 10.96 0.0104 1.080
cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr
0.818 11.42 0.0107 0.873
0.991 11.37 0.0111 1.015
cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr
0.825 11.82 0.0116 0.860
0.969 11.77 0.0118 0.967
1.238 11.68 0.0125 1.157
cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr
0.837 12.28 0.0121 0.849
0.959 12.22 0.0125 0.938
1.161 12.17 0.0131 1.079
cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr
0.849 12.70 0.0128 0.842
0.951 12.69 0.0132 0.914
1.118 12.66 0.0138 1.026
15
Temp., K 310
320
330
340
350
1
5
10
15
20
0.861 13.12 0.0135 0.878
0.945 13.10 0.0139 0.891
1.084 13.08 0.0144 0.985
1.279 13.07 0.0150 1.114
1.560 13.09 0.0159 1.284
25
30
40
0.873 13.54 0.0142 0.836
0.944 13.54 0.0146 0.875
1.059 13.55 0.0150 0.957
1.207 13.56 0.0156 1.049
1.400 13.60 0.0164 1.161
1.704 13.75 0.0173 1.354
0.885 13.96 0.0148 0.834
0.942 13.96 0.0152 0.865
1.038 13.98 0.0156 0.930
1.158 14.01 0.0162 1.001
1.301 14.15 0.0169 1.089
1.508 14.28 0.0177 1.217
1.837 14.52 0.0187 1.426
0.897 14.38 0.0155 0.832
0.937 14.40 0.0159 0.849
1.020 14.43 0.0163 0.903
1.135 14.47 0.0168 0.978
1.242 14.53 0.0175 1.031
1.388 14.65 0.0182 1.117
1.612 14.85 0.0190 1.260
2.488 16.00 0.0217 1.834
0.910 14.80 0.0162 0.831
0.934 14.82 0.0165 0.839
1.004 14.84 0.0169 0.882
1.118 14.87 0.0174 0.955
1.200 14.92 0.0180 0.995
1.308 15.06 0.0186 1.060
1.440 15.21 0.0194 1.130
1.914 16.16 0.0216 1.432
1.370 12.62 0.0144 1.201
Some values read from charts may be approximate. cp values interpolated and converted from Thermodynamic Properties of KLEA 32, I.C.I., 1993 (47 pp.). Viscosity interpolated from Takahashi, M., C. Yokoyama, et al., Proc. 14th Symp. Thermophys. Props., Japan, 1993 (pp. 427–430). Thermal conductivities are taken from Geller, V. Z. and M. E. Perlaitis, and from Gross, Proc. 10th Symp. Thermophys. Props., Boulder, CO, 1994.
TABLE 2-310 Temp., °C
Saturated SUVA MP 39
Pf, bar
Pg, bar
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−6 Pa·s
kf , W/(m·K)
Prf
−40 −30 −20 −10 0
0.733 1.155 1.748 2.553 3.615
0.533 0.871 1.361 2.043 2.965
0.000 0.000 0.000 0.000 0.000
712 728 744 762 781
0.3778 0.2391 0.1576 0.1075 0.0755
154.0 164.9 176.2 188.6 200.0
385.0 390.6 396.3 401.8 407.3
0.8188 0.8647 0.9099 0.9577 1.0000
1.8244 1.8059 1.7907 1.7781 1.7675
1.078 1.109 1.137 1.165 1.197
351 323 291 266 241
0.1209 0.1154 0.1107 0.1057 0.1012
3.13 3.06 2.99 2.93 2.85
10 20 30 40 50
4.984 6.712 8.857 11.475 14.628
4.177 5.733 7.697 10.133 13.112
0.000 0.000 0.000 0.000 0.000
803 826 851 878 909
0.0544 0.0399 0.0298 0.0225 0.0172
212.7 225.3 238.3 252.0 266.4
412.6 417.6 422.2 426.5 430.1
1.0454 1.0884 1.1316 1.1752 1.2194
1.7587 1.7510 1.7439 1.7372 1.7304
1.233 1.277 1.329 1.392 1.468
221 202 186 170 157
0.0967 0.0922 0.0877 0.0830 0.0781
2.82 2.80 2.83 2.85 2.95
60 70 80 90 100
18.378 22.79 27.92 33.83 40.53
16.711 21.01 26.12 32.13 39.22
0.000 0.000 0.001 0.001 0.001
944 988 028 084 140
0.01313 0.01005 0.00764 0.00570 0.00403
281.6 297.9 315.9 336.2 361.4
433.0 434.9 435.4 433.5 426.9
1.2647 1.3118 1.3616 1.4163 1.4820
1.7228 1.7138 1.7022 1.6858 1.6584
1.564 1.652 1.802 1.958 2.16
143 131 122 115 110
0.0737 0.0684 0.0631 0.0577 0.0533
3.04 3.16 3.48 3.90 4.46
108.0c
46.04
46.04
0.001 96
0.00196
397
397
vf, m3/kg
c = critical point. SUVA MP 39 = R401A = CHClF2 (R22) 53% wt + CH3CHF2 (R 152a) 13% wt + CHClFCF3 (R124) 34% wt, near-azeotropic blend. Some values read from charts are approximate. Material used by permission of DuPont Fluoroproducts.
TABLE 2-311
SUVA MP 39 at Atmospheric Pressure
Temp., °C
−27.01
−20
0
20
40
60
80
100
120
140
v (m3/ kg) h (kJ/ kg) s (kJ/ kg⋅K) cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr Z
0.2102 351.7 1.8009 0.648 10.17 0.00878 0.750 0.9829
0.2167 396.9 1.8193 0.669 10.43 0.00921 0.758 0.9852
0.2351 410.4 1.8706 0.698 11.18 0.01041 0.750 0.9906
0.2534 424.5 1.9204 0.727 11.93 0.01161 0.749 0.9949
0.2715 439.2 1.9689 0.757 12.68 0.01282 0.749 0.9979
0.2896 454.4 2.0161 0.787 13.42 0.01404 0.748 1.0005
0.3076 470.3 2.0623 0.811 14.17 0.01536 0.748 1.0025
0.3256 486.6 2.1073 0.836 14.89 0.01668 0.748 1.0043
0.3435 503.5 2.1513 0.859 15.61 0.01796 0.747 1.0056
0.3613 521.2 2.1943 0.883 16.32 0.01929 0.747 1.0060
For composition see footnote to Table 2-310. Some values read from charts are approximate. Material used by permission of DuPont Fluoroproducts.
50 R 32
20
1.
40
30
0
1.3
350 20
g
.40
•
340
K
y1
p tro
0.95
0.90
k kJ/
330 320
En
310 300 K
0
1.5
Satu rated vapor
Pressure, bar
10 8
6
290
1.6
0
280 270
4
0
1.7
260
2
0 1.8 250 240 230
1
FIG. 2-21
360
380
400 Enthalpy, kJ/kg
420
440
Enthalpy–log-pressure diagram for Refrigerant 32. 2-277
2-278
PHYSICAL AND CHEMICAL DATA TABLE 2-312 Pressure, bar
Thermodynamic Properties of Saturated KLEA 60 Tf , K
Tg , K
vg , m3/kg
hf , kJ/kg
hg , kJ/kg
227.3 236.1 242.8 248.3 253.0
234.0 242.5 249.1 254.5 259.1
0.000 0.000 0.000 0.000 0.000
7118 7263 7381 7483 7573
0.2097 0.1433 0.1093 0.0885 0.0744
−7.80 3.92 12.89 20.27 26.57
229.64 235.02 239.07 242.35 245.08
0.9965 0.9833 0.9744 0.9679 0.9629
4 5 6 8 10
260.7 267.3 272.9 282.1 289.8
266.8 273.1 278.5 287.5 295.0
0.000 0.000 0.000 0.000 0.000
7735 7880 8012 8254 8480
0.0564 0.0442 0.0384 0.0286 0.0228
37.23 46.12 53.84 67.02 78.23
249.54 253.07 255.24 260.70 263.86
0.9552 0.9496 0.9450 0.9378 0.9318
12.5 15 17.5 20 22.5
297.9 304.8 311.0 316.5 321.4
302.8 309.5 315.4 320.7 325.5
0.000 0.000 0.000 0.000 0.000
8750 9017 9290 9613 9884
0.01802 0.01481 0.01247 0.01069 0.00928
90.50 101.51 111.64 121.18 130.31
266.95 269.12 270.58 271.46 271.79
0.9257 0.9190 0.9128 0.9065 0.8999
25 27.5 30
326.1 330.4 334.5
329.8 333.9 337.6
0.001 023 0.001 063 0.001 115
0.00828 0.00717 0.00635
139.17 147.89 156.58
271.63 270.97 269.81
0.8927 0.8850 0.8765
1 1.5 2 2.5 3
vf , m3/kg
sf , kJ/(kg⋅K)
sg , kJ/(kg⋅K)
hf = sf = 0 at 233.15 K = −40°C. Converted and interpolated from Thermodynamic Properties of Klea 60 (British units, 20 pp.), copyright ICI Chemicals and Polymers Limited, 1993. Reproduced by permission. Tf = bubble point temperature; Tg = dew point temperature.
TABLE 2-313 Pressure, bar
Thermodynamic Properties of Saturated KLEA 61 vf , m3/kg
vg , m3/kg
hf , kJ/kg
hg , kJ/kg
6852 6994 7110 7211 7301
0.1800 0.1230 0.0937 0.0758 0.0637
−9.45 2.52 9.72 16.59 22.47
191.64 196.90 200.88 204.10 206.80
0.8433 0.8341 0.8282 0.8245 0.8215
0.000 0.000 0.000 0.000 0.000
7463 7607 7740 7985 8214
0.04831 0.03888 0.03249 0.02435 0.01936
32.43 40.76 48.00 59.82 70.98
211.22 214.74 217.65 222.21 225.63
0.8172 0.8141 0.8123 0.8080 0.8048
298.7 305.5 311.4 316.7 321.5
0.000 0.000 0.000 0.000 0.000
8491 8768 9053 9353 9680
0.01528 0.01251 0.01049 0.00896 0.00774
82.59 93.02 102.67 111.79 120.55
228.80 231.08 232.64 233.60 233.99
0.8010 0.7971 0.7929 0.7882 0.7829
325.9 330.0 333.7
0.001 005 0.001 048 0.001 102
0.00674 0.00590 0.00518
129.11 137.62 146.21
233.85 233.16 231.84
0.7769 0.7700 0.7619
Tf , K
Tg , K
225.6 234.3 241.8 246.4 251.1
230.0 238.5 245.0 250.4 254.9
0.000 0.000 0.000 0.000 0.000
4 5 6 8 10
258.9 265.4 270.9 280.2 287.8
262.6 269.0 274.4 283.4 290.9
12.5 15 17.5 20 22.5
295.8 302.8 308.8 314.3 319.3
25 27.5 30
323.9 328.1 332.1
1 1.5 2 2.5 3
sf , kJ/(kg⋅K)
sg , kJ/(kg⋅K)
Converted and interpolated from Thermodynamic Properties of Klea 61 (British units, 20 pp.), copyright ICI Chemicals and Polymers Limited, 1993. Reproduced by permission. Tf = bubble-point temperature; Tg = dew-point temperature. hf = sf = 0 at 233.15 K = −40 °C.
THERMODYNAMIC PROPERTIES
2-279
30 0.9 0
5
0.70 0.75
0 .9
20
R 60
0.80
0
425
1.0
15
0.90
400
1.1 0
10
1.0
5
kJ/ kg
•
K
0.85
8
5
375
350
1.2
6
0
1.1
Pressure, bar
0.95
1.3
0
325
1.2
5
4
1.
35
300 K
40
2
5
1.
275
1
FIG. 2-22
1.4
1.5
250
Enthalpy–log-pressure diagram for KLEA 60.
300 Enthalpy (h), kJ/kg
350
400
2-280
PHYSICAL AND CHEMICAL DATA
30 0.80
0.70 0.75
0.8 5
0.80
20
450
0.9 0
0.85 0.90 15
0.9
5
0.95
425
1.0 0k J/k g
•
K
400
10
5 1.0
rated
0
375
350
1.1
6
Satu
Pressure, bar
vapo
r
8
1.1
5
4
1.2
0
325
1.2
5
300
30
2
1.3
5
1.
275 1.5 250 K
1 200 FIG. 2-23
250 Enthalpy–log-pressure diagram for KLEA 61.
300 Enthalpy (h), kJ/kg
350
THERMODYNAMIC PROPERTIES TABLE 2-314 Temp., °C
2-281
Saturated SUVA HP 62
Pf, bar
Pg, bar
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−6 Pa·s
kf, W/(m⋅K)
−50 −40 −30 −20 −10
0.852 1.367 2.095 3.087 4.404
0.821 1.325 2.041 3.018 4.321
0.000 0.000 0.000 0.000 0.000
761 779 799 820 843
0.2244 0.1434 0.0953 0.0656 0.0463
133.1 145.6 159.9 172.8 186.1
337.3 343.8 350.3 356.5 362.6
0.7318 0.7862 0.8460 0.8975 0.9487
1.6487 1.6380 1.6301 1.6245 1.6202
0.0970
1.220 1.260 1.302
370 318 276 238 207
0.0868 0.0834 0.0801
3.88 3.60 3.37
0 10 20 30 40
6.111 8.278 10.977 14.287 18.292
6.013 8.165 10.851 14.150 18.148
0.000 0.000 0.000 0.000 0.001
868 898 933 977 037
0.03338 0.02444 0.01809 0.01348 0.01003
200.0 214.5 229.9 246.2 263.8
368.3 373.6 378.3 382.2 385.0
1.0000 1.0515 1.1038 1.1574 1.2130
1.6188 1.6138 1.6106 1.6065 1.6005
1.351 1.412 1.489 1.592 1.753
181 158 138 122 106
0.0767 0.0733 0.0698 0.0663 0.0624
3.19 3.04 2.94 2.93 2.98
50 60 70 72.1c
23.08 28.75 35.58 37.32
22.94 28.63
0.001 122 0.001 261
386.1 384.2 375.9 361
1.5910 1.5742
2.09
91 76 61
0.0583 0.0535
3.26
0.002 06
283.2 305.8 339.8 361
1.2723 1.3389
37.32
0.00739 0.00527 0.00285 0.00206
vf, m3/kg
Prf
c = critical point. SUVA HP 62 = CHF2CF3 (R125) 44% wt + CH3CF3 (R143a) 52% wt + CH2FCF3 (R134a) 4% wt, near-azeotropic blend. Material used by permission of DuPont Fluoroproducts. Some values read from charts may be approximate.
TABLE 2-315
SUVA HP 62 at Atmospheric Pressure
Temp., °C
−45.63
−40
−20
0
20
40
60
80
100
120
v (m3/ kg) h (kJ/ kg) s (kJ/ kg⋅K) cp (kJ/ kg⋅K) µ (10−6 Pa⋅s) k (W/m⋅K) Pr Z
0.1866 336.0 1.6599 0.732 9.47 0.00860 0.806 0.9755
0.1921 344.4 1.6636 0.738 9.68 0.00932 0.767 0.9800
0.2100 359.9 1.7274 0.781 10.45 0.01059 0.771 0.9867
0.2278 376.2 1.7891 0.821 11.22 0.01186 0.777 0.9919
0.2455 393.1 1.8491 0.860 11.99 0.01313 0.785 0.9961
0.2630 410.9 1.9076 0.897 12.76 0.01440 0.795 0.9989
0.2805 429.3 1.9646 0.933 13.53 0.01568 0.805 1.0014
0.2980 448.4 2.0203 0.967 14.30 0.01695 0.816 1.0037
0.3153 468.2 2.0747 1.000 15.07 0.01827 0.827 1.0050
0.3325 488.7 2.1278 1.032 15.84 0.01949 0.839 1.0060
v,h, and s from DuPont bull. T—HP62—SI, June 1993 (17 pp.). cp and k from DuPont bull. ART 18, June 1993 (37 pp.). Some values read from charts may be approximate. Material used by permission of DuPont Fluoroproducts.
TABLE 2-316
Thermodynamic Properties of Saturated KLEA 66
Pressure, bar
Tf, K
Tg, K
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
0.69 1 1.5 2 2.5
221.46 228.89 237.69 244.45 249.99
228.77 236.05 244.69 251.33 256.76
0.000 0.000 0.000 0.000 0.000
7122 7237 7382 7501 7600
0.31325 0.22131 0.15140 0.11537 0.09104
−16.16 −5.89 6.22 15.51 23.12
241.25 245.91 251.38 255.52 258.95
1.0729 1.0580 1.0430 1.0330 1.0258
3 4 5 6 8
254.36 262.60 269.12 274.70 284.03
261.39 269.14 275.51 280.98 290.08
0.000 0.000 0.000 0.000 0.000
7695 7857 8001 8133 8375
0.07855 0.05964 0.04806 0.04021 0.03022
29.62 40.60 49.74 57.69 71.22
261.63 266.16 269.74 272.68 277.25
1.0201 1.0114 1.0055 0.9993 0.9913
10 12.5 15 17.5 20
291.74 299.87 306.87 313.05 318.60
297.56 305.44 312.18 318.10 323.40
0.000 0.000 0.000 0.000 0.000
8599 8867 9131 9400 9680
0.02410 0.01910 0.01571 0.01324 0.01137
82.73 95.32 106.59 116.97 126.73
280.64 283.74 285.93 287.28 288.26
0.9834 0.9770 0.9701 0.9633 0.9564
22.5 25 27.5 30
323.7 328.3 332.7 336.7
328.2 332.5 336.6 340.4
0.000 0.001 0.001 0.001
9981 032 072 125
0.00988 0.00883 0.00766 0.00703
136.0 145.1 153.9 162.7
288.6 288.4 287.8 286.6
0.9493 0.9418 0.9338 0.9251
vf, m3/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
Converted and interpolated from Thermodynamic Properties of Klea 66 (British units, 22 pp.), copyright ICI Chemicals and Polymers Limited, 1993. Reproduced by permission. Tf = bubble-point temperature; Tg = dew-point temperature. hf = sf = 0 at 233.15 K = −40°C.
2-282
PHYSICAL AND CHEMICAL DATA
0 1 .2
5 1 .1
0
450 1 .1
1.0 5
0.9
5
0.75
1.0 0
30
K
0.80 20
1.2 5
15
kJ
425
0.90
/kg
•
0.85
Satu ra
8
1 .3
375
1. 35
Pressure, bar
400 K ted v apor
10
0
Qua
lity
0.95
6
350
py En
tro
325
1.
40
4
1.4
5
300
2 275
1.5
0
1.5 250 K 1 250 FIG. 2-24
300 Enthalpy–log-pressure diagram for KLEA 66.
350 Enthalpy (h), kJ/kg
400
THERMODYNAMIC PROPERTIES TABLE 2-317 Temp., °C
2-283
Saturated SUVA MP 66
Pf, bar
Pg, bar
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−6 Pa·s
kf, W/(m⋅K)
Prf
−40 −30 −20 −10 0
0.788 1.239 1.872 2.726 3.850
0.585 0.952 1.479 2.212 3.198
0.000 0.000 0.000 0.000 0.000
710 725 740 758 778
0.3498 0.2224 0.1471 0.1008 0.0710
153.8 164.8 176.0 188.6 200.0
386.0 391.6 397.1 402.6 407.8
0.8184 0.8643 0.9095 0.9577 1.0000
1.8291 1.8100 1.7940 1.7807 1.7694
1.078 1.109 1.137 1.165 1.197
349 313 282 257 236
0.1209 0.1154 0.1106 0.1057 0.1012
3.11 3.01 2.90 2.83 2.79
10 20 30 40 50
5.297 7.120 9.379 12.133 15.444
4.491 6.146 8.229 10.808 13.955
0.000 0.000 0.000 0.000 0.000
801 827 858 895 939
0.05124 0.03771 0.02818 0.02131 0.01625
212.6 225.1 238.2 251.9 266.3
412.9 417.7 422.1 426.1 429.4
1.0450 1.0879 1.1311 1.1747 1.2190
1.7598 1.7512 1.7433 1.7357 1.7278
1.233 1.277 1.329 1.392 1.468
217 198 181 168 151
0.0967 0.0922 0.0877 0.0830 0.0781
2.77 2.74 2.74 2.82 2.84
60 70 80 90 100
19.378 24.00 29.37 35.55 42.30
17.750 22.28 27.64 33.96
0.000 0.001 0.001 0.001
994 066 164 313
0.01244 0.00951 0.00721 0.00534
281.6 298.1 316.3 337.2
431.9 433.4 433.2 430.4
1.2645 1.3120 1.3625 1.4187
1.7191 1.7088 1.6956 1.6768
1.564 1.652 1.802
139 127 116
0.0737 0.0684 0.0631 0.0577 0.0533
2.95 3.07 3.31
106.1c
46.82
46.82
0.001 95
0.00195
389
389
vf, m3/kg
c = critical point. SUVA MP 66 = R401 = CHClF2 (R22) 61% wt + CH3CHF2 (R152a) 11% wt + CHClFCF3 (R124) 28% wt, near-azeotropic blend. Material used by permission of DuPont Fluoroproducts. Some values read from charts are approximate.
TABLE 2-318
SUVA MP 66 at Atmospheric Pressure
Temp., °C
−28.63b
−20
0
20
40
60
80
100
120
140
v (m3/ kg) h (kJ/ kg) s (kJ/ kg⋅K) cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr Z
0.2086 392.2 1.8081 0.641 9.78 0.00817 0.767 0.9652
0.2177 397.9 1.8299 0.652 10.43 0.00921 0.738 0.9730
0.2362 411.2 1.8804 0.688 11.18 0.01041 0.737 0.9783
0.2545 425.1 1.9295 0.716 11.93 0.01161 0.736 0.9822
0.2727 439.6 1.9772 0.744 12.68 0.01282 0.735 0.9852
0.2908 454.6 2.0237 0.771 13.42 0.01404 0.735 0.9876
0.3089 470.1 2.0690 0.796 14.17 0.01536 0.734 0.9896
0.3269 486.2 2.1132 0.822 14.89 0.01668 0.734 0.9912
0.3449 502.7 2.1564 0.844 15.61 0.01796 0.733 0.9925
0.3629 519.4 2.1986 0.866 16.32 0.01929 0.733 0.9937
v, h, and s from DuPont bull. T—MP 66—SI, Jan. 1993 (17 pp.). cp, µ, and k from DuPont bull. ART 10, Jan. 1993 (27 pp.). Some values read from charts may be approximate. Material used by permission of DuPont Fluoroproducts. b = normal boiling point.
TABLE 2-319 Temp., °C
Saturated SUVA HP 80 vf, m3/kg
cpf, kJ/(kg⋅K)
µf, 10−6 Pa·s
kf, W/(m⋅K)
1.6327 1.6206 1.6110 1.6034 1.5972
0.0970
1.193 1.217 1.236
377 317 283 247 215
0.0880 0.0849 0.0813
3.84 3.54 3.27
1.0000 1.0461 1.0927 1.1403 1.1897
1.5919 1.5870 1.5820 1.5762 1.5690
1.253 1.286 1.340 1.412 1.512
188 165 146 128 113
0.0778 0.0743 0.0708 0.0672 0.0634
3.03 2.86 2.76 2.69 2.70
1.2420 1.2998
1.5589 1.5433
1.64 1.81
98 83 68
0.0593 0.0551
2.71 2.79
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
679 695 713 733 757
0.2033 0.1303 0.0869 0.0598 0.0423
139.6 150.8 163.1 174.9 187.6
334.1 339.9 345.6 351.1 356.4
0.7578 0.8070 0.8584 0.9053 0.9541
785 819 860 911 977
0.03060 0.02248 0.01671 0.01250 0.00936
200.0 213.0 226.7 241.2 256.8
361.3 365.9 369.8 373.1 375.4
24.04 29.97
0.001 070 0.001 212
0.00696 0.00505
273.9 293.6
376.2 374.6
41.35
0.001 850
0.00185
340
340
Pf, bar
Pg, bar
−50 −40 −30 −20 −10
0.962 1.520 2.305 3.370 4.776
0.872 1.403 2.156 3.188 4.560
0.000 0.000 0.000 0.000 0.000
0 10 20 30 40
6.588 8.877 11.720 15.195 19.388
6.336 8.592 11.404 14.855 19.034
0.000 0.000 0.000 0.000 0.000
50 60 70 75.5c
24.39 30.30 41.35
Prf
c = critical point. SUVA HP 80 = R402 = CHF2CF3 (R125) 60% wt + CH3CH2CH3 (R290) 2% wt + CHClF2 (R22) 38% wt, near-azeotropic blend. Material used by permission of DuPont Fluoroproducts. Some values, read from charts, may be approximate.
2-284
PHYSICAL AND CHEMICAL DATA
TABLE 2-320
SUVA HP 80 at Atmospheric Pressure
Temp., °C
−46.95b
−40
−20
0
20
40
60
80
100
120
v (m3/ kg) h (kJ/ kg) s (kJ/ kg⋅K) cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr Z
0.1768 335.9 1.6286 0.648 9.42 0.00888 0.687 0.9673
0.1827 340.5 1.6490 0.654 9.69 0.00932 0.680 0.9697
0.1996 354.3 1.7055 0.687 10.45 0.01059 0.678 0.9758
0.2164 368.6 1.7599 0.721 11.22 0.01186 0.681 0.9804
0.2331 383.5 1.8124 0.749 11.99 0.01313 0.685 0.9840
0.2497 398.7 1.8633 0.779 12.75 0.01440 0.690 0.9868
0.2663 414.9 1.9128 0.807 13.52 0.01568 0.696 0.9892
0.2828 431.4 1.9610 0.836 14.29 0.01695 0.703 0.9910
0.2992 448.5 2.0081 0.863 15.06 0.01822 0.713 0.9923
0.3155 466.1 2.0541 0.890 15.82 0.01949 0.722 0.9932
b = normal boiling pt. v, h, and s from DuPont bull. T—HP 80—SI, Jan. 1993 (17 pp.). cp, µ, and k from DuPont bull. ART 18, June 1993 (37 pp.). Some values read from charts may be approximate. Material used by permission of DuPont Fluoroproducts.
TABLE 2-321 Temp., °C
Saturated SUVA HP 81 vf, m3/kg
cpf, kJ/(kg⋅K)
µf, 10−6 Pa·s
kf, W/(m⋅K)
Prf
1.7122 1.6957 1.6820 1.6706 1.6611
1.178 1.191 1.204
383 333 290 253 223
0.1031 0.0983 0.0941 0.0900 0.0863
3.63 3.35 3.11
1.0000 1.0450 1.0905 1.1367 1.1842 1.2339
1.6528 1.6451 1.6376 1.6299 1.6211 1.6104
1.221 1.288 1.313 1.37 1.75 2.07
195 173 151 137 122 106
0.0818 0.0790 0.0753 0.0715 0.0676 0.0633
2.91 2.82 2.63 2.49 3.16 3.47
1.2873 1.3164
1.5961 1.5866
91 75
0.0586 0.0544
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
687 702 719 739 761
0.2425 0.1548 0.1028 0.0707 0.0499
140.3 151.4 163.3 174.9 187.8
351.7 357.2 362.7 368.0 373.0
0.7606 0.8092 0.8589 0.9054 0.9550
787 817 854 899 955 030
0.03610 0.02656 0.01980 0.01490 0.01125 0.00848
200.0 212.7 226.0 240.1 255.1 271.4
377.8 382.2 386.0 389.3 391.5 392.8
28.03 34.60
0.001 136 0.001 307
0.00632 0.00456
289.5 299.6
392.2 390.9
44.45
0.001 88
0.00188
351
351
Pf, bar
Pg, bar
−50 −40 −30 −20 −10
0.883 1.403 2.135 3.132 4.451
0.787 1.273 1.967 2.923 4.198
0.000 0.000 0.000 0.000 0.000
0 10 20 30 40 50
6.153 8.307 10.984 14.261 18.216 22.93
5.852 7.959 10.591 13.827 17.750 22.45
0.000 0.000 0.000 0.000 0.000 0.001
60 70 80 82.6c
28.50 35.01 44.45
c = critical point. SUVA HP 81 = R402 (38/2/60) = CHF2CF3 (R125) 38% wt + CH3CH2CH3 (R290) 2% wt + CHClF2 (R22) 60% wt, near-azeotropic blend. Material used by permission of DuPont Fluoroproducts. Some values read from charts may be approximate.
TABLE 2-322
SUVA HP 81 at Atmospheric Pressure
Temp., °C
−44.87b
−40
−20
0
20
40
60
80
100
120
v (m3/ kg) h (kJ/ kg) s (kJ/ kg⋅K) cp (kJ/ kg⋅K) µ (10−6 Pa·s) k (W/m⋅K) Pr Z
0.1903 354.7 1.7032 1.187 10.16 0.00739 1.632 0.9622
0.1960 357.7 1.7169 1.177 10.33 0.00768 1.583 0.9703
0.2142 370.8 1.7711 1.169 11.10 0.00902 1.439 0.9766
0.2322 384.6 1.8232 1.159 11.86 0.01036 1.327 0.9811
0.2500 398.8 1.8735 1.149 12.62 0.01170 1.239 0.9843
0.2678 413.6 1.9222 1.143 13.39 0.01304 1.174 0.9870
0.2856 428.9 1.9696 1.134 14.15 0.01438 1.124 0.9894
0.3032 444.7 2.0158 1.128 14.78 0.01572 1.061 0.9909
0.3209 461.0 2.0607 1.124 15.54 0.01706 1.024 0.9926
0.3386 477.7 2.1047 1.120 16.30 0.01840 0.992 0.9940
b = normal boiling point. v, h, and s from DuPont bull. T—HP 81—SI, Jan. 1993 (17 pp.). cp, µ, and k from DuPont bull. ART 18, June 1993 (37 pp.). Some values, read from charts, may be approximate. Material used by permission of DuPont Fluoroproducts.
THERMODYNAMIC PROPERTIES TABLE 2-323
2-285
Saturated Refrigerant 113*
P, bar
v f, m3/kg
vg, m3/kg
h f, kJ/kg
hg, kJ/kg
s f, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
c pf, kJ/(kg⋅K)
µ f, 10−4 Pa⋅s
k f, W/(m⋅K)
240 250 260 270 280
0.0233 0.0435 0.0767 0.1290 0.2076
5.908.−4 5.986.−4 6.066.−4 6.150.−4 6.237.−4
4.548 2.537 1.492 0.9189 0.5893
5.70 14.19 22.83 31.65 40.63
171.97 178.06 184.22 190.46 196.75
0.0241 0.0587 0.0926 0.1259 0.1585
0.7169 0.7142 0.7134 0.7141 0.7161
0.845 0.877 0.895 0.916 0.933
17.9 14.8 12.3 10.4 8.9
0.087 0.084 0.083 0.081 0.079
290 300 310 320 330
0.3217 0.4817 0.6999 0.9897 1.3657
6.328.−4 6.422.−4 6.522.−4 6.626.−4 6.737.−4
0.3917 0.2687 0.1895 0.1370 0.1012
49.77 59.07 68.51 78.09 87.80
203.08 209.44 215.80 222.17 228.53
0.1906 0.2221 0.2530 0.2833 0.3131
0.7192 0.7233 0.7281 0.7336 0.7396
0.946 0.958 0.971 0.983 0.992
7.6 6.6 5.9 5.2 4.7
0.077 0.075 0.073 0.071 0.069
340 350 360 370 380
1.8347 2.4406 3.174 4.062 5.123
6.854.−4 6.979.−4 7.112.−4 7.255.−4 7.411.−4
0.0762 0.0584 0.0454 0.0357 0.0284
97.64 107.58 117.65 127.82 138.11
234.86 241.16 247.41 253.59 259.70
0.3424 0.3711 0.3993 0.4270 0.4542
0.7460 0.7528 0.7598 0.7669 0.7742
1.000 1.013 1.029 1.042 1.059
4.2 3.8 3.4 3.2 2.9
0.066 0.065 0.062 0.060 0.058
T, K
390 400 410 420 430
6.379 7.849 9.556 11.52 13.78
7.580.−4 7.767.−4 7.975.−4 8.211.−4 8.483.−4
0.0229 0.0185 0.0151 0.0124 0.0102
148.52 159.07 169.78 180.69 191.85
265.71 271.59 277.31 282.83 288.09
0.4810 0.5075 0.5336 0.5595 0.5853
0.7815 0.7888 0.7958 0.8027 0.8091
1.084 1.109 1.14 1.18 1.22
2.7 2.46 2.28 2.10 1.93
0.056 0.054 0.052 0.050 0.047
440 450 460 470 480
16.35 19.26 22.56 26.29 30.52
8.806.−4 9.201.−4 9.713.−4 1.044.−3 1.174.−3
0.0083 0.0068 0.0055 0.0044 0.0032
203.35 215.31 227.97 241.79 258.16
292.98 297.38 301.03 303.41 303.00
0.6112 0.6375 0.6645 0.6933 0.7264
0.8149 0.8198 0.8234 0.8244 0.8198
1.27 1.32 1.38 1.45 1.54
1.75 1.58 1.33 1.07 0.77
0.045 0.042 0.039 0.035 0.031
487.5c
34.11
1.754.−3
0.0018
288.10
288.10
0.7828
0.7828
∞
0.30
∞
*Values reproduced or converted from Table 8, p. 17.91, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and AirConditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 5.908.−4 signifies 5.908 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives a saturation table from −30 to 214.4°C and an enthalpy–log-pressure diagram from 0.1 to 60 bar, 0 to 260°C. Equations and constants approximated to the 1985 ASHRAE tables were given by Mecaryk, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. For experimental isochores for the compressed liquid from 21 to 304 bar, 266 to 453 K, see Blanke, W. and R. Weiss, PTB Bericht W 30, Braunschweig, Germany, 1992 (54 pp.). For tables to 300 bar, 460 K, see Geller, V. Z. and V. A. Rabinovich (ed.), Thermophysical Properties of Substances and Materials, Standartov, Moscow, 7 (1973): 135–154. Mastroianni, M. J., R. F. Stahl, et al., J. Chem. Eng. Data, 23, 2 (1978): 113–118 give a diagram to 1000 psia, 600°F. Tables and a diagram to 800 psia, 520°F are given by Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.
2-286
PHYSICAL AND CHEMICAL DATA
TABLE 2-324
Saturated Refrigerant 114*
P, bar
vf, m3/kg
vg, m3/kg
h f, kJ/kg
h g, kJ/kg
s f, kJ/(kg⋅K)
s g, kJ/(kg⋅K)
c pf, kJ/(kg⋅K)
µ f , 10−4 Pa⋅s
k f, W/(m⋅K)
190 200 210 220 230
0.0058 0.0137 0.029 0.059 0.109
6.326.−4 6.344.−4 6.366.−4 6.391.−4 6.421.−4
15.823 7.094 3.465 1.822 1.021
−42.58 −31.87 −21.48 −11.37 −1.50
125.78 131.01 136.41 141.95 147.61
−0.2091 −0.1542 −0.1035 −0.0565 −0.0126
0.6794 0.6648 0.6541 0.6466 0.6419
0.765 0.787 0.810 0.831 0.854
23.9 18.2 14.3 11.5 9.4
0.093 0.090 0.088 0.085 0.082
240 250 260 270 280
0.190 0.317 0.505 0.773 1.143
6.457.−4 6.500.−4 6.554.−4 6.619.−4 6.700.−4
0.604 0.375 0.2431 0.1633 0.1132
8.18 17.74 27.22 36.71 46.27
153.36 159.18 165.05 170.95 176.85
0.0286 0.0676 0.1047 0.1405 0.1751
0.6393 0.6387 0.6396 0.6418 0.6452
0.877 0.900 0.923 0.946 0.967
7.9 6.61 5.66 4.96 4.30
0.080 0.077 0.075 0.072 0.069
290 300 310 320 330
1.636 2.279 3.096 4.116 5.366
6.799.−4 6.918.−4 7.060.−4 7.224.−4 7.412.−4
0.0807 0.0590 0.0440 0.0334 0.0257
55.95 65.79 75.79 85.92 96.16
182.75 188.61 194.44 200.19 205.84
0.2090 0.2422 0.2748 0.3067 0.3379
0.6494 0.6543 0.6598 0.6657 0.6719
0.991 1.015 1.038 1.062 1.087
3.80 3.35 3.02 2.69 2.48
0.067 0.064 0.061 0.059 0.056
340 350 360 370 380
6.877 8.683 10.82 13.32 16.24
7.624.−4 7.863.−4 8.135.−4 8.453.−4 8.836.−4
0.0201 0.0158 0.0125 0.0099 0.0079
106.49 116.96 127.63 138.60 149.99
211.37 216.71 221.82 226.57 230.84
0.3685 0.3984 0.4280 0.4575 0.4872
0.6781 0.6843 0.6903 0.6957 0.7002
1.111 1.136 1.160 1.185 1.210
2.27 2.07 1.91 1.76 1.59
0.054 0.051 0.048 0.045 0.042
390 400 410 419.0c
19.62 23.52 28.00 32.61
9.324.−4 1.001.−3 1.118.−3 1.795.−3
0.0062 0.0048 0.0035 0.0018
162.01 175.03 190.13 219.90
234.36 236.61 236.20 219.90
0.5176 0.5496 0.5857 0.6559
0.7032 0.7036 0.6980 0.6559
1.236 1.261 1.5 ∞
1.39 1.17 0.87 0.34
0.038 0.034 0.030 ∞
T, K
*Values reproduced or converted from Table 9, p. 17.93, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and AirConditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 6.326.−4 signifies 6.326 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives a saturation table from −80 to 145.88°C and an enthalpy–log-pressure diagram from 0.1 to 100 bar, −20 to 220°C. Equations and constants approximated to the 1985 ASHRAE tables were given by Mecaryk, K. and M. Masaryk, Heat Recovery Systems and CHP., 11, 2/3 (1991): 193–197. Saturation and superheat tables and a diagram to 60 bar, 540 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (179 pp.). Tables and a chart to 1500 psia, 480°F are given by Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.
TABLE 2-325 Temp., °F −100 −80 −60 −40 −20
Saturated Refrigerant 115* Volume, ft3/lb
Enthalpy, Btu/lb
Entropy, Btu/(lb)(°F)
Pressure, lb/in2 abs.
Liquid
Vapor
Liquid
Vapor
Liquid
Vapor
2.327 4.573 8.306 14.13 22.74
0.00966 0.00986 0.01009 0.01033 0.01060
10.57 5.624 3.218 1.953 1.245
−13.07 −8.78 −4.43 0.00 4.50
45.83 48.39 50.96 53.53 56.07
−0.0335 −0.0219 −0.0108 0.0000 0.0104
0.1302 0.1286 0.1278 0.1275 0.1277
0 20 40 60 80
34.94 51.59 73.65 102.1 138.1
0.01090 0.01123 0.01161 0.01204 0.01255
0.8257 0.5657 0.3979 0.2857 0.2081
9.09 13.76 18.54 23.45 28.54
58.56 61.00 63.35 65.60 67.71
0.0206 0.0305 0.0401 0.0496 0.0591
0.1282 0.1290 0.1298 0.1308 0.1317
100 120 140 160 170
182.7 237.3 303.2 382.0 427.0
0.01316 0.01393 0.01496 0.01664 0.01838
0.1530 0.1125 0.0817 0.0567 0.0444
33.85 39.50 45.67 52.76 56.56
69.63 71.24 72.36 72.42 71.33
0.0686 0.0782 0.0884 0.0996 0.1055
0.1325 0.1330 0.1329 0.1314 0.1290
175.89 c
457.6
0.0261
0.0261
64.30
64.30
0.1175
0.1175
*Unpublished data of General Chemicals Division, Allied Chemical Company. Used by permission. c = critical temperature. No material in SI units appears in the 1993 ASHRAE Handbook—Fundamentals (SI ed.). Tables and a chart to 50 ata, 200°C are given by Mathias, H. and H. J. Loffler, Techn. Univ. Berlin rept., 1966 (42 pp.). A chart to 1500 psia, 500°F was given by Mears, W. H., E. Rosenthal, et al., J. Chem. Eng. Data, 11, 3 (1966): 338–343.
THERMODYNAMIC PROPERTIES TABLE 2-326
2-287
Thermodynamic Properties of Refrigerant 123
Pressure, bar
Temp., K
vg, m3/kg
h f, kJ/kg
h g, kJ/kg
s f, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
c pf, kJ/(kg⋅K)
k f, W/(m⋅K)
µ f, 10−6 Pa·s
Prf
0.1 0.5 1.0 1.013 1.5
249.49 282.87 300.62 300.99 312.25
0.000 0.000 0.000 0.000 0.000
6315 6664 6862 6868 7008
1.3430 0.2993 0.1567 0.1546 0.1070
13.25 41.72 58.62 58.99 70.51
198.51 218.53 229.20 229.43 236.52
0.0548 0.1610 0.2195 0.2208 0.2582
0.7977 0.7863 0.7869 0.7870 0.7892
0.849 0.923 1.000 1.001 1.038
0.0908 0.0811 0.0759 0.0758 0.0726
798.7 503.7 409.8 408.1 361.5
7.46 5.73 5.40 5.39 5.17
2.0 2.5 3.0 4.0 5.0
321.18 328.50 334.79 345.29 353.95
0.000 0.000 0.000 0.000 0.000
7126 7230 7323 7490 7640
0.08139 0.06546 0.05525 0.03836 0.03358
79.90 87.76 94.59 106.16 115.83
241.76 246.20 249.96 256.17 261.17
0.2877 0.3118 0.3323 0.3661 0.3935
0.7917 0.7942 0.7965 0.8006 0.8042
1.063 1.079 1.091 1.108 1.120
0.0696 0.0678 0.0660 0.0630 0.0605
329.6 306.1 287.4 259.7 239.1
5.03 4.87 4.75 4.57 4.43
6 8 10 15 20
361.41 373.92 384.19 404.54 420.30
0.000 0.000 0.000 0.000 0.000
7779 8038 8280 8874 9512
0.02799 0.02090 0.01675 0.01062 0.00751
124.23 138.48 150.35 174.49 194.19
265.34 272.04 277.18 286.01 291.01
0.4168 0.4551 0.4860 0.5462 0.5928
0.8073 0.8124 0.8162 0.8218 0.8232
1.130 1.148 1.168 1.234 1.345
25 30 36.68*
433.33 444.10 456.83
0.001 030 0.001 136 0.001 818
0.00549 0.00408 0.00182
212.00 228.26 264.54
293.05 291.27 264.54
0.6334 0.6692 0.7393
0.8203 0.8112 0.7393
1.559 2.005
vf, m3/kg
h f = s f = 0 at −40°C = 233.15 K. s f, s g, cp,f units: kJ/kg⋅K. Interpolated and converted from 1993 ASHRAE Handbook—Fundamentals (SI ed.) saturation table from −40 to 183.68°C. This source also contains an enthalpy–log-pressure diagram from 0.1 to 200 bar, −40 to 320°C.
TABLE 2-327 Temp., °C
Saturated Refrigerant 124 Pressure, bar
vf, m3/kg
vg, m3/kg
h f, kJ/kg
h g, kJ/kg
s f, kJ/(kg⋅K)
s g, kJ/(kg⋅K)
−40 −30 −20 −10 0
0.2680 0.4499 0.7197 1.1044 1.6348
0.000 644 0.000 655 0.000 668 0.000 681 0.000 696
0.5173 0.3185 0.2049 0.1369 0.0945
159.1 169.3 179.5 189.7 200.0
334.9 340.6 346.2 351.8 357.4
0.8384 0.8813 0.9222 0.9616 1.0000
1.5927 1.5856 1.5808 1.5777 1.5762
10 20 30 40 50
2.3447 3.2710 4.4529 5.9320 7.7521
0.000 711 0.000 728 0.000 747 0.000 768 0.000 791
0.06703 0.04867 0.03604 0.02713 0.02069
210.5 221.3 282.3 243.7 255.4
363.0 368.5 373.9 379.2 384.4
1.0376 1.0747 1.1115 1.1480 1.1843
1.5760 1.5768 1.5785 1.5808 1.5836
60 70 80 90 100
9.9599 12.605 15.742 19.432 23.749
0.000 818 0.000 849 0.000 887 0.000 935 0.000 999
0.01594 0.01236 0.00961 0.00744 0.00569
267.5 280.1 293.2 307.0 321.9
389.3 393.9 398.0 401.3 403.4
1.2207 1.2572 1.2942 1.3318 1.3710
1.5864 1.5890 1.5909 1.5915 1.5894
110 120 122.5c
28.787 34.702 36.340
0.001 098 0.001 338 0.001 810
0.00420 0.00269 0.00181
338.4 360.6 378.5
403.0 394.9 378.5
1.4133 1.4685
1.5820 1.5558
c = critical point. Bull. T—124—SI, Jan. 1993 (28 pp.). Used by permission of DuPont Fluoroproducts. The 1993 ASHRAE Handbook— Fundamentals (SI ed.) gives a saturation table to 122.47°C and a diagram to 200 bar, 320°C.
FIG. 2-25
2-288
Enthalpy–log-pressure diagram for Refrigerant 123.
THERMODYNAMIC PROPERTIES TABLE 2-328
2-289
Thermophysical Properties of Saturated Refrigerant 125 µf , 10−6 Pa·s
Temp., K
Pressure, bar
vf, m3/kg
vg , m3/kg
h f, kJ/kg
h g , kJ/kg
s f, kJ/(kg⋅K)
s g, kJ/(kg·K)
172.5* 180 190 200 210
0.035 0.064 0.133 0.257 0.465
0.000591 0.000599 0.000611 0.000624 0.000638
3.48 1.958 0.986 0.5312 0.3057
−30.5 −23.3
140.8 146.7
−0.1386 −0.1024
0.7183 0.7067
220 224.9† 230 240 250
0.794 1.013 1.290 2.005 3.000
0.000653 0.000660 0.000669 0.000686 0.000705
0.1854 0.1475 0.1175 0.0775 0.0527
−13.9 −8.8 −3.4 7.7 19.3
152.6 155.5 158.4 164.2 170.0
−0.0604 −0.0386 −0.0147 0.0324 0.0800
0.6981 0.6948 0.6919 0.6875 0.6847
1.077 1.139 1.184
445.8 411.2 379.6 326.6 282.8
260 270 280 290 300
4.336 6.078 8.298 11.068 14.476
0.000725 0.000749 0.000776 0.000809 0.000848
0.0369 0.0264 0.0193 0.0143 0.0106
31.3 43.7 56.5 69.7 83.5
175.3 180.5 185.4 189.9 194.2
0.1274 0.1743 0.2206 0.2666 0.3126
0.6831 0.6822 0.6819 0.6815 0.6805
1.221 1.257 1.299 1.356 1.437
245.8 213.6 185.3 159.7 136.3
310 320 330 339.4‡
18.62 23.63 29.65 35.95
0.000898 0.000969 0.001088 0.00175
0.0079 0.0059 0.0041 0.0018
98.1 113.9 132.2 169.0
196.9 198.5 197.4 169.0
0.3597 0.4079 0.4621 0.5699
0.6774 0.6726 0.6639 0.5699
1.57 1.82
115.0 95.4
cpf, kJ/(kg⋅K)
644 531
* = triple point; † = normal boiling point; ‡ = critical point. Converted, extrapolated and interpolated from 1993 ASHRAE Handbook—Fundamentals (SI ed.) hf = sf = 0 at 233.15 K = −40°C. This source also contains an enthalpy–log-pressure diagram from 0.3 to 100 bar, −65 to 175°C. An apparently identical diagram but a different saturation table is contained in Duarte-Garza, H.A., Hwang, C.A. et al., ASHRAE Trans., 99, 2 (1993): 649–664. R124: The 1993 ASHRAE Handbook—Fundamentals (SI ed.) contains a saturation table from −60 to 122.47°C.
TABLE 2-329
Thermophysical Properties of Refrigerant 134a Temp., K
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
s f, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf , kJ/(kg·K)
0.0039t 0.5 0.6 0.8 1.0
169.85 232.69 236.22 242.04 246.80
0.0006285 0.0007062 0.0007113 0.0007199 0.0007272
35.263 0.3692 0.3015 0.2375 0.1924
−76.68 −0.57 3.85 11.15 17.14
186.50 225.27 227.52 231.19 234.15
−0.3830 −0.0025 0.0161 0.0467 0.0713
1.1665 0.9669 0.9636 0.9560 0.9507
1.147 1.242 1.248 1.258 1.267
1.013 1.5 2.0 2.5 3.0
247.03 256.03 263.09 268.88 273.82
0.0007276 0.0007421 0.0007543 0.0007648 0.0007743
0.1902 0.1312 0.0999 0.0806 0.0677
17.50 28.96 38.13 45.75 52.33
234.33 239.86 244.14 247.60 250.50
0.0728 0.1181 0.1533 0.1819 0.2059
0.9503 0.9419 0.9364 0.9326 0.9297
1.268 1.288 1.306 1.322 1.337
4.0 5 6 8 10
282.08 288.89 294.72 304.47 312.53
0.0007912 0.0008063 0.0008203 0.0008460 0.0008703
0.0512 0.04116 0.03434 0.02565 0.02035
63.50 72.87 81.04 95.00 106.86
255.22 258.99 262.09 267.01 270.74
0.2458 0.2784 0.3062 0.3522 0.3901
0.9256 0.9232 0.9208 0.9171 0.9144
12 14 16 18 20
319.47 325.57 330.11 336.04 340.63
0.0008938 0.0009170 0.0009362 0.0009555 0.0009894
0.01675 0.01414 0.01247 0.01059 0.00931
117.34 126.80 134.00 143.68 151.39
273.65 275.92 277.40 279.01 279.95
0.4227 0.4515 0.4729 0.5013 0.5236
25 30 35 40 40.56c
350.73 359.37 366.89 373.50 374.18
0.0010585 0.001144 0.001270 0.001606 0.001948
0.00695 0.00528 0.00399 0.00255 0.00195
169.30 185.05 203.19 229.24 241.22
280.64 278.32 273.52 257.12 241.22
0.5738 0.6212 0.6657 0.7292 0.7620
Pressure, bar
µ f, 10−6 Pa·s
k f, W/(m⋅K)
Prf
0.1121 0.1105 0.1078 0.1056
5.61 5.42 5.12 4.90
406 358.7 326.6 303.2 285.1
0.1054 0.1013 0.0980 0.0954 0.0931
4.89 4.56 4.35 4.20 4.09
1.363 1.387 1.410 1.454 1.497
257.7 237.5 221.6 197.6 179.5
0.0893 0.0861 0.0835 0.0790 0.0753
3.93 3.83 3.74 3.64 3.57
0.9120 0.9095 0.9073 0.9041 0.9010
1.541 1.589 1.631 1.698 1.764
165.1 153.0 144.3 133.2 124.8
0.0721 0.0693 0.0672 0.0645 0.0623
3.53 3.51 3.50 3.51 3.53
0.8913 0.8807 0.8574 0.8038 0.7620
1.987 2.418
106.6 90.4
0.0577 0.0538
3.67 4.06
2187 506 480 438 408
t = triple point, c = critical point. hf = sf = 0 at −40°C = 233.15 K. T, v, h, and s interpolated and converted from Refrigerant 134a—Thermodynamic and Physical Properties, Int. Inst. Refrig., Paris, France, 1992 (28 pp.). Other properties from this source and from Oliveira, C. M. B. P. and W. A. Wakeham, Int. J. Thermophys., 14, 1 (1993): 33–44; Krauss, R., J. Luettmer-Strathmann, et al., Int. J. Thermophys., 14, 4 (1993): 951–988; ASHRAE Handbook—Fundamentals, Atlanta, GA, 1993; ICI KLEA 134a bulletin, 1993 (43 pp.); and R134a— Thermodynamic and Physical Properties, Int. Inst. Refrig., Paris, France, 1992 (28 pp.). Papers giving polynomial curve fits and similar simple equations include Cleland, A. C., Rev. Int. Froid = Int. J. Refrig., 17, 4 (1994): 245–249; Dobrokhotov, A., A. Grebenkov, et al., Proc. 14th Japan Symp. Thermophys. Props., (1993): 271–274; Huber, M. L. and J. F. Ely, Rev. Int. Froid = Int. J. Refrig., 17, 1 (1994): 18–31 (includes extensive list of vapor pressure and liquid density sources for many refrigerants); Modic, J., Proc. 11th Int. Symp. Htg, Refrig and Air-Condg., Zagreb, (1991): 174–185; and Kabelac, S., Int. J. Refrig., 14, (1991): 217–222. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives saturation data for integral degrees Celsius with temperatures on the ITS 90 scale from −103.03°C to 101.03°C. The thermodynamic diagram from 0.1 to 200 bar extends to 320°C.
FIG. 2-26
2-290
Enthalpy–log-pressure diagram for Refrigerant 125.
THERMODYNAMIC PROPERTIES TABLE 2-330
2-291
Thermophysical Properties of Compressed Gaseous Refrigerant 134a Pressure, bar
Temp., K
0
1
2
3
4
5
6
cp (kJ/ kg⋅K) µ (10−6 Pa·s) 230 k (W/m⋅K) Pr
— — — —
— — — —
— — — —
— — — —
— — — —
— — — —
cp (kJ/ kg⋅K) −6 240 µ (10 Pa·s) k (W/m⋅K) Pr
— — — —
— — — —
— — — —
— — — —
— — — —
— — — —
cp (kJ/ kg⋅K) −6 250 µ (10 Pa·s) k (W/m⋅K) Pr
0.7437 10.11 0.0096 0.783
0.7953 10.15 0.0097 0.797
— — — —
— — — —
— — — —
— — — —
— — — —
cp (kJ/ kg⋅K) −6 260 µ (10 Pa·s) k (W/m⋅K) Pr
0.7627 10.47 0.0105 0.761
0.8048 10.51 0.0107 0.790
— — — —
— — — —
— — — —
— — — —
— — — —
cp (kJ/ kg) −6 270 µ (10 Pa·s) k (W/m⋅K) Pr
0.7813 10.84 0.0117 0.724
0.8158 10.88 0.0118 0.761
0.8557 10.94 0.0118 0.793
— — — —
— — — —
— — — —
— — — —
cp (kJ/ kg⋅K) −6 280 µ (10 Pa·s) k (W/m⋅K) Pr
0.7996 11.22 0.0122 0.735
0.8283 11.26 0.0123 0.757
0.8604 11.29 0.0123 0.790
— — — —
— — — —
— — — —
cp (kJ/ kg⋅K) −6 290 µ (10 Pa·s) k (W/m⋅K) Pr
0.8176 11.62 0.0130 0.731
0.8412 11.65 0.0131 0.748
0.8673 11.68 0.0131 0.773
0.8938 11.71 0.0133 0.787
0.9335 11.74
cp (kJ/ kg⋅K) −6 300 µ (10 Pa·s) k (W/m⋅K) Pr
0.8354 12.05 0.0139 0.730
0.8556 12.06 0.0139 0.742
0.8771 12.08 0.0140 0.757
0.8972 12.10 0.0141 0.770
0.9277 12.15 0.0142 0.792
0.0143 0.816
cp (kJ/ kg⋅K) −6 310 µ (10 Pa·s) k (W/m⋅K) Pr
0.8530 12.44 0.0145 0.730
0.8703 12.45 0.0146 0.742
0.8875 12.47 0.0147 0.753
0.9046 12.49 0.0148 0.763
0.9292 12.52 0.0149 0.781
0.9546 12.54 0.0150 0.798
cp (kJ/ kg⋅K) −6 320 µ (10 Pa·s) k (W/m⋅K) Pr
0.8703 12.83 0.0153 0.730
0.8843 12.84 0.0153 0.740
0.8993 12.86 0.0154 0.751
0.9163 12.88 0.0155 0.761
0.9356 12.90 0.0156 0.774
0.9548 12.93 0.0157 0.786
0.9750 12.97 0.0158 0.800
cp (kJ/ kg⋅K) −6 330 µ (10 Pa·s) k (W/m⋅K) Pr
0.8874 13.22 0.0160 0.729
0.8996 13.23 0.0161 0.739
0.9114 13.25 0.0161 0.750
0.9268 13.27 0.0162 0.759
0.9398 13.29 0.0163 0.766
0.9569 13.32 0.0164 0.777
0.9750 13.35 0.0165 0.789
cp (kJ/ kg⋅K) −6 340 µ (10 Pa·s) k (W/m⋅K) Pr
0.9042 13.61 0.0169 0.728
0.9152 13.62 0.0169 0.738
0.9262 13.64 0.0169 0.748
0.9372 13.66 0.0170 0.755
0.9502 13.68 0.0170 0.765
0.9632 13.70 0.0171 0.772
0.9770 13.73 0.0171 0.780
cp (kJ/ kg⋅K) −6 350 µ (10 Pa·s) k (W/m⋅K) Pr
0.9208 13.98 0.0175 0.730
0.9307 13.99 0.0176 0.740
0.9406 14.01 0.0176 0.749
0.9505 14.03 0.0177 0.754
0.9607 14.05 0.0177 0.763
0.9695 14.07 0.0178 0.767
0.9830 14.10 0.0179 0.774
— — — — 0.9606
0.9976
0.9827 0.0152
Dashes indicate unavailable states; blanks indicate no data. Note that “profound differences” presently exist in the transport properties of R134a according to Chemistry International, 16(6), 233, Nov. 1994, and 18(2) 44–47, 1996.
2-292
PHYSICAL AND CHEMICAL DATA
TABLE 2-330
Thermophysical Properties of Compressed Gaseous R134a (Concluded ) Pressure, bar
Temp., K
8
10
12.5
15
17.5
20
22.5
cp (kJ/ kg⋅K) 300 µ (10−6 Pa·s) k (W/m⋅K) Pr
— — — —
— — — —
— — — —
— — — —
— — — —
— — — —
— — — —
cp (kJ/ kg⋅K) 310 µ (10−6 Pa·s) k (W/m⋅K) Pr
1.053
— — — —
— — — —
— — — —
— — — —
— — — —
— — — —
cp (kJ/ kg⋅K) 320 µ (10−6 Pa·s) k (W/m⋅K) Pr
1.028 13.05 0.0161 0.833
1.097 13.13
— — — —
— — — —
— — — —
— — — —
— — — —
cp (kJ/ kg⋅K) 330 µ (10−6 Pa·s) k (W/m⋅K) Pr
1.015 13.41 0.0168 0.810
1.065 13.49 0.0171 0.840
1.151 13.64 0.0177 0.887
1.276 13.86 0.0184 0.961
— — — —
— — — —
— — — —
cp (kJ/ kg⋅K) 340 µ (10−6 Pa·s) k (W/m⋅K) Pr
1.008 13.79 0.0174 0.799
1.049 13.86 0.0177 0.821
1.107 13.98 0.181 0.855
1.187 14.17 0.0187 0.899
1.319
— — — —
— — — —
cp (kJ/ kg⋅K) 350 µ (10−6 Pa·s) k (W/m⋅K) Pr
1.008 14.15 0.0181 0.788
1.040 14.22 0.0183 0.828
1.086 14.34 0.0186 0.837
1.148 14.49 0.0192 0.866
1.225
1.340 14.97 0.0205
0.0155
TABLE 2-331
0.0198
Refrigerant 141b
The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives saturation data to 150°C and a diagram to 20 bar, 150°C. For equation of state including decomposition, see Weber, L. A., paper 69, Proc. 18th Int. Congr. Refrig., Montreal, 1991.
1.525 0.0215
THERMODYNAMIC PROPERTIES
2-293
40 410
5
5
0.
6
400 K
1 .7
0.
0.
8
Q ua
lit
y
1 .8
=
20
0
0.
7
25
1 .7
30
0
0.4
1. 25
390
90
0
10 9 8 7
1.
1.
6 95
py
5
tro
1.
En
Pressure (P ), bar
0. 9
1.
kJ /k g
•
85
K
15
4
00
2.
380
3
370 360
2.5 340 330
2 310 1.5
290 260
1
270
320
05
2.
300 K
280 10
2.
250 400
450 Enthalpy (h), kJ/kg
FIG. 2-27
350 K
Enthalpy–log-pressure diagram for Refrigerant 134a.
500
2-294
PHYSICAL AND CHEMICAL DATA
TABLE 2-332
Refrigerant 142b* µ f, 10−4 Pa⋅s
k f, W/(m⋅K)
1.15 1.17 1.18 1.19
0.517
0.123 0.118 0.114 0.111 0.109
0.9341 0.9264 0.9204 0.9155 0.9116
1.21 1.22 1.24 1.26 1.28
0.466 0.422 0.385 0.355 0.329
0.103 0.099 0.095 0.091 0.088
1.30 1.32 1.34
0.305 0.285 0.267 0.241 0.216
0.084 0.080 0.075 0.072 0.068
0.192
0.064 0.060 0.056 0.052 0.048
T, K
P, bar
vf, m3/kg
vg, m3/kg
h f, kJ/kg
h g, kJ/kg
s f, kJ/(kg⋅K)
s g, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
200 210 220 230 240
0.0380 0.0728 0.1314 0.2252 0.3691
7.505.−4 7.626.−4 7.751.−4 7.883.−4 8.019.−4
4.337 2.374 1.373 0.833 0.527
−24.36 −17.48 −10.21 −2.52 9.92
200.49 206.82 213.28 219.82 229.74
−0.1123 −0.0788 −0.0450 −0.0109 0.0414
1.0119 0.9893 0.9708 0.9558 0.9387
250 260 270 280 290
0.5815 0.8846 1.3046 1.8714 2.6184
8.164.−4 8.317.−4 8.480.−4 8.653.−4 8.843.−4
0.346 0.234 0.162 0.115 0.0838
14.32 23.54 33.32 43.68 54.60
233.05 239.66 246.18 252.57 258.77
0.0592 0.0952 0.1320 0.1695 0.2076
300 310 320 330 340
3.583 4.803 6.324 8.187 10.44
9.047.−4 9.273.−4 9.525.−4 9.810.−4 1.014.−3
0.0619 0.0464 0.0353 0.0271 0.0210
66.07 78.07 90.55 103.45 116.71
264.69 270.26 275.40 280.01 283.99
0.2462 0.2851 0.3243 0.3634 0.4024
0.9082 0.9051 0.9020 0.8985 0.8943
350 360 370 380 390
13.13 16.30 20.01 24.29 29.20
1.052.−3 1.099.−3 1.157.−3 1.235.−3 1.348.−3
0.0164 0.0129 0.0102 0.0080 0.0062
130.30 144.18 158.45 173.45 190.16
287.23 289.61 291.01 291.22 289.77
0.4409 0.4791 0.5170 0.5557 0.5974
0.8893 0.8831 0.8753 0.8656 0.8528
400 410c
34.78 41.5
1.541.−3 2.300.−3
0.0046 0.0023
212.57 255.00
284.04 255.00
0.6521
0.8307
0.044 ∞
*Values reproduced and converted from Table 10, p. 17.95, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and AirConditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 7.505.−4 signifies 7.505 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the IPTS 68 scale from −50 to 125°C. The thermodynamic diagram from 0.1 to 35 bar extends to 180°C. For experimental isochores for the compressed liquid from 6 to 298 bar, 147 to 432 K, see Blanke, W. and R. Weiss, PTB Bericht W 30, Braunschweig, Germany, 1992 (54 pp.). Tables and a diagram to 500 psia, 400°F are given in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.
TABLE 2-333 Temp., K t
161.82 170 180 190 200 210 220 225.92 230 240 250 260 270 280 290 300 310 320 330 340 346.75c
Saturated Refrigerant R143a*
Pressure, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
h g, kJ/kg
s f , kJ/(kg⋅K)
s g, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µ f , 10−4 Pa⋅s
k f, W/(m⋅K)
0.01124 0.02497 0.05914 0.126 0.2458 0.4455 0.7586 1.01325 1.225 1.89 2.806 4.027 5.613 7.629 10.14 13.23 16.98 21.48 26.85 33.25 38.32
0.000752 0.000764 0.000778 0.000793 0.000809 0.000826 0.000845 0.000856 0.000865 0.000886 0.000910 0.000936 0.000966 0.000999 0.001038 0.001084 0.001140 0.001214 0.001321 0.001514 0.002311
14.22 6.709 2.991 1.474 0.7898 0.4532 0.2754 0.2098 0.1755 0.1164 0.07975 0.05617 0.04045 0.02964 0.02200 0.01646 0.01234 0.009182 0.006678 0.004520 0.002311
53.2 63.0 75.3 87.7 100.3 113.1 126.1 133.9 139.3 152.8 166.6 180.8 195.3 210.3 225.9 242.1 259.2 277.4 297.5 321.8 360.6
320.5 325.5 331.8 338.1 344.5 350.8 357.1 360.8 363.3 369.4 375.3 380.9 386.2 391.1 395.4 399.0 401.6 402.7 401.0 393.4 360.6
0.3181 0.3774 0.4474 0.5147 0.5792 0.6415 0.7018 0.7367 0.7604 0.8176 0.8736 0.9287 0.983 1.037 1.091 1.144 1.199 1.255 1.315 1.385 1.471
1.970 1.922 1.872 1.832 1.800 1.774 1.752 1.741 1.734 1.720 1.708 1.698 1.690 1.682 1.675 1.668 1.659 1.647 1.629 1.595 1.471
1.188 1.215 1.235 1.252 1.268 1.287 1.308 1.323 1.333 1.362 1.394 1.431 1.475 1.527 1.593 1.679 1.804 2.006 2.421 4.021 —
6.011 5.366 4.692 4.121 3.636 3.221 2.864 2.676 2.556 2.288 2.055 1.852 1.673 1.496 1.334 1.192 1.067 0.9569 0.8321 0.6560 —
0.1416 0.1403 0.1372 0.1331 0.1281 0.1226 0.1168 0.1133 0.1108 0.1047 0.09862 0.09272 0.08682 0.08098 0.07521 0.06951 0.06381 0.05803 0.05202 0.04480 —
*Values calculated from NIST Thermodynamic Properties of Refrigerants and Refrigerant Mixtures Database (REFPROP, Version 5). Thermodynamic properties are from 32-term MBWR equation of state; transport properties are from extended corresponding states model. t = triple point; c = critical point.
THERMODYNAMIC PROPERTIES TABLE 2-334
2-295
Saturated Refrigerant R152a*
Temp., K
Pressure, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
h g, kJ/kg
s f , kJ/(kg⋅K)
s g, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µ f , 10−4 Pa⋅s
k f, W/(m⋅K)
154.56t 160 170 180 190 200 210 220 230 240 249.12 250 260 270 280 290 300 310 320 330 340 350 360 370 380 386.41c
0.000641 0.001297 0.004145 0.01141 0.02775 0.06088 0.1224 0.2284 0.4004 0.6647 1.01325 1.053 1.603 2.354 3.354 4.650 6.297 8.351 10.87 13.92 17.57 21.90 27.00 32.97 39.97 45.17
0.000839 0.000846 0.000859 0.000873 0.000887 0.000902 0.000918 0.000935 0.000952 0.000971 0.000989 0.000991 0.001012 0.001035 0.001060 0.001087 0.001118 0.001152 0.001190 0.001235 0.001289 0.001355 0.001440 0.001563 0.001785 0.002717
303.6 155.2 51.59 19.82 8.588 4.110 2.138 1.193 0.7064 0.4397 0.2961 0.2855 0.1922 0.1334 0.09500 0.06916 0.05126 0.03857 0.02935 0.02251 0.01735 0.01335 0.01020 0.007603 0.005274 0.002717
14.0 22.2 37.2 52.4 67.6 82.9 98.3 113.8 129.5 145.5 160.2 161.6 178.0 194.7 211.7 229.1 246.9 265.2 284.1 303.7 324.2 345.7 368.8 394.3 425.4 477.3
419.8 423.5 430.6 437.9 445.3 452.8 460.4 468.0 475.6 483.1 489.8 490.5 497.7 504.7 511.4 517.8 523.8 529.3 534.1 538.2 541.3 542.9 542.5 538.5 526.2 477.3
0.1130 0.1647 0.2560 0.3425 0.4247 0.5032 0.5784 0.6507 0.7205 0.7881 0.8481 0.8538 0.9178 0.9805 1.042 1.103 1.162 1.222 1.281 1.340 1.400 1.460 1.523 1.591 1.671 1.778
2.738 2.673 2.570 2.484 2.413 2.353 2.303 2.261 2.225 2.195 2.171 2.169 2.147 2.129 2.112 2.098 2.085 2.073 2.062 2.051 2.038 2.024 2.006 1.980 1.936 1.778
1.492 1.500 1.510 1.517 1.525 1.535 1.547 1.562 1.580 1.600 1.622 1.624 1.651 1.681 1.716 1.756 1.803 1.859 1.928 2.015 2.131 2.299 2.573 3.143 5.407 —
10.85 9.614 8.058 6.940 6.012 5.236 4.582 4.028 3.556 3.153 2.834 2.805 2.500 2.231 1.995 1.789 1.607 1.447 1.305 1.180 1.069 1.005 0.9225 0.8191 0.6638 —
0.1932 0.1894 0.1822 0.1753 0.1685 0.1618 0.1552 0.1487 0.1423 0.1361 0.1304 0.1299 0.1239 0.1179 0.1121 0.1064 0.1008 0.09526 0.08986 0.08457 0.07939 0.07336 0.06666 0.06032 0.05176 —
*Values calculated from NIST Thermodynamic Properties of Refrigerants and Refrigerant Mixtures Database (REFPROP, Version 5). Thermodynamic properties are from 32-term MBWR equation of state; transport properties are from extended corresponding states model. t = triple point; c = critical point.
TABLE 2-335 Temp., °F
Saturated Refrigerant 216* Pressure, lb/in2 abs.
Volume, ft3/lb
Enthalpy, Btu/lb
Entropy, Btu/(lb)(°F)
Liquid
Vapor
Liquid
Vapor
Liquid
Vapor
−40 −20 0 20 40
0.339 0.713 1.382 2.497 4.247
0.00927 0.00942 0.00958 0.00974 0.00992
59.957 29.749 15.986 9.184 5.582
0.000 4.778 9.541 14.298 19.056
62.415 65.276 68.208 71.199 74.239
0.0000 0.0111 0.0217 0.0318 0.0415
0.1487 0.1487 0.1493 0.1504 0.1520
60 80 100 120 140
6.862 10.612 15.797 22.753 31.845
0.01010 0.01030 0.01050 0.01073 0.01097
3.558 2.361 1.6215 1.1462 0.8304
23.821 28.598 33.391 38.205 43.049
77.319 80.429 83.559 86.701 89.845
0.0509 0.0599 0.0686 0.0770 0.0852
0.1538 0.1559 0.1582 0.1607 0.1632
160 180 200 220 240
43.468 58.046 76.033 97.913 124.21
0.01124 0.01153 0.01186 0.01223 0.01266
0.6142 0.4623 0.3529 0.2725 0.2121
47.930 52.861 57.857 62.939 68.132
92.981 96.099 99.186 102.225 105.196
0.0931 0.1009 0.1085 0.1161 0.1235
0.1658 0.1685 0.1712 0.1739 0.1765
260 280 300 320 340
155.50 192.40 235.63 286.03 344.81
0.01317 0.01378 0.01458 0.01570 0.01764
0.1660 0.1300 0.1013 0.0776 0.0565
73.474 79.015 84.835 91.089 98.234
108.066 110.789 113.282 115.373 116.538
0.1309 0.1384 0.1460 0.1539 0.1628
0.1790 0.1813 0.1834 0.1851 0.1856
355.98c
399.45
0.02771
0.0277
110.248
110.248
0.1773
0.1773
*From published data, Chemicals Division, Union Carbide Corporation. Used by permission. The paper describing these data is by Shank, ASHRAE J., 7 (1965): 94–101. c = critical temperature. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the ITS 90 scale from −118.59 to 113.26°C. The thermodynamic diagram from 0.1 to 30 bar extends to 180°C. For tables and a diagram to 400 psia, 360°F, see Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993. Thermal conductivity data as a function of pressure and temperature are reported by Krauss, R. and K. Stephan, Proc. 12th Symp. Thermophys. Props., Boulder, CO, 1994.
2-296
PHYSICAL AND CHEMICAL DATA TABLE 2-336
Saturated Refrigerant 245*
P, bar
v f, m3/kg
vg, m3/kg
172 180 190 200 210
0.0034 0.0076 0.0190 0.0425 0.0870
6.46.−4 6.57.−4 6.70.−4 6.83.−4 6.97.−4
31.49 14.63 6.20 2.91 1.48
220 230 240 250 260
0.1654 0.2946 0.4958 0.7946 1.2204
7.11.−4 7.25.−4 7.40.−4 7.55.−4 7.72.−4
270 280 290 300 310
1.806 2.584 3.600 4.888 6.491
320 330 340 350 360 370 375 380.1c
T, K
h f, kJ/kg
h g, kJ/kg
s f, kJ/(kg⋅K)
s g, kJ/(kg⋅K)
−63.4 −55.9 −46.2 −36.0 −25.7
133.8 138.7 145.1 151.7 158.5
−0.3131 −0.2707 −0.2182 −0.1666 −0.1157
0.8327 0.8099 0.7885 0.7725 0.7612
0.822 0.475 0.292 0.192 0.125
−14.8 −3.6 8.0 19.9 32.3
165.4 172.5 179.6 186.8 194.0
−0.0654 −0.0156 0.0337 0.0824 0.1305
0.7539 0.7500 0.7487 0.7497 0.7525
7.89.−4 8.08.−4 8.30.−4 8.53.−4 8.80.−4
0.0862 0.0611 0.0443 0.0327 0.0246
44.9 57.9 71.1 84.6 98.4
201.1 208.3 215.3 222.2 228.9
0.1781 0.2249 0.2711 0.3161 0.3614
0.7567 0.7621 0.7683 0.7751 0.7822
8.456 10.83 13.67 17.04 21.02
9.11.−4 9.48.−4 9.93.−4 0.00105 0.00113
0.0186 0.0143 0.0111 0.0084 0.0063
112.6 127.1 142.1 157.2 174.7
235.3 241.4 246.9 251.5 254.8
0.4057 0.4497 0.4937 0.5382 0.5844
0.7893 0.7960 0.8018 0.8060 0.8071
25.71 28.46 31.37
0.00125 0.00137 0.00204
0.0045 0.0036 0.0020
193.6 205.2 231.8
255.2 252.5 231.8
0.6349 0.6649 0.7341
0.8013 0.7953 0.7341
*Values converted from tables of Shank, Thermodynamic Properties of UCON 245 Refrigerant, Union Carbide Corporation, New York, 1966. See also Shank, J. Chem. Eng. Data, 12, 474–480 (1967). c = critical point. The notation 6.46.−4 signifies 6.46 × 10−4.
TABLE 2-337
Refrigerant C 318*
P, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−4 Pa⋅s
kf, W/(m⋅K)
200 210 220 230 240
0.0216 0.0449 0.0875 0.1608 0.2810
5.507.−4 5.593.−4 5.683.−4 5.778.−4 5.879.−4
3.810 1.931 1.038 0.588 0.349
353.5 361.0 369.2 377.6 386.4
498.0 500.1 502.2 504.4 510.9
3.909 3.947 3.984 4.022 4.060
4.560 4.564 4.569 4.574 4.578
0.98 1.00
11.7 9.55
0.088 0.085
250 260 270 280 290
0.466 0.741 1.133 1.672 2.392
5.988.−4 6.106.−4 6.234.−4 6.375.−4 6.529.−4
0.2166 0.1401 0.0938 0.0647 0.0458
395.6 405.2 415.1 425.8 436.2
517.4 524.0 530.7 537.3 543.9
4.097 4.133 4.172 4.210 4.247
4.584 4.592 4.599 4.609 4.618
1.02 1.03 1.05 1.07 1.09
7.90 6.63 5.64 4.85 4.22
0.082 0.078 0.075 0.071 0.068
300 310 320 330 340
3.325 4.522 6.007 7.826 10.018
6.694.−4 6.893.−4 7.115.−4 7.365.−4 7.666.−4
0.0332 0.0245 0.0184 0.0139 0.0106
447.3 458.7 470.5 482.7 495.2
550.4 556.9 563.3 569.4 575.4
4.284 4.322 4.359 4.396 4.433
4.626 4.638 4.648 4.659 4.669
1.12 1.15 1.18 1.23 1.27
3.70 3.20 2.94 2.66 2.33
0.065 0.061 0.058 0.054 0.051
350 360 370 380 388.5c
12.632 15.71 19.33 23.59 27.83
8.034.−4 8.508.−4 9.172.−4 1.031.−3 1.613.−3
0.0082 0.0062 0.0047 0.0033 0.0016
508.1 521.5 535.6 551.4 577.2
581.0 585.8 589.9 591.5 577.2
4.469 4.507 4.544 4.585 4.651
4.678 4.685 4.691 4.691 4.651
1.32 1.39
2.00
0.048
T, K
*Values of P, v, h, and s interpolated, extrapolated, and converted from tables of Oguchi, Reito, 52 (1977): 869–889. Values of cp, µ, and k interpolated and converted from tables in Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1976. c = critical point. Saturation and superheat tables and a diagram to 80 bar, 580 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For equations, see Cipollone, R., ASHRAE Trans., 97, 2 (1991): 262–267.
THERMODYNAMIC PROPERTIES TABLE 2-338
2-297
Saturated Refrigerant 500*
P, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−4 Pa⋅s
kf, W/(m⋅K)
200 210 220 230 240
0.1219 0.2258 0.3936 0.6511 1.0291
6.966.−4 7.090.−4 7.222.−4 7.361.−4 7.509.−4
1.360 0.766 0.457 0.286 0.187
−29.56 −21.03 −12.17 −2.97 6.58
185.87 191.25 196.63 201.96 207.23
−0.1363 −0.0948 −0.0536 −0.0130 0.0277
0.9408 0.9161 0.8955 0.8782 0.8638
1.044 1.018 0.997 0.987 0.987
6.11 5.15 4.42 3.85 3.42
0.113 0.109 0.106 0.102 0.098
250 260 270 280 290
1.5632 2.2932 3.2624 4.5172 6.1064
7.668.−4 7.839.−4 8.024.−4 8.226.−4 8.450.−4
0.1261 0.0879 0.0628 0.0459 0.0342
16.50 26.78 37.44 48.48 59.91
212.40 217.45 222.35 227.06 231.56
0.0680 0.1082 0.1481 0.1878 0.2275
0.8517 0.8415 0.8329 0.8257 0.8194
0.997 1.017 1.048 1.089 1.140
3.04 2.74 2.48 2.26 2.08
0.094 0.090 0.086 0.082 0.078
300 310 320 330 340
8.0809 10.49 13.40 16.86 20.93
8.699.−4 8.981.−4 9.306.−4 9.690.−4 1.016.−3
0.0259 0.0198 0.0154 0.0119 0.0093
71.76 84.05 96.83 110.17 124.20
235.79 239.69 243.19 246.14 248.36
0.2671 0.3067 0.3464 0.3864 0.4271
0.8139 0.8088 0.8038 0.7985 0.7922
1.201 1.273 1.355 1.447 1.550
1.92 1.77 1.63 1.48 1.34
0.074 0.070 0.066 0.062 0.058
350 360 370 378.6c
25.70 31.25 37.72 44.26
1.077.−3 1.162.−3 1.307.−3 2.012.−3
0.0072 0.0055 0.0040 0.0020
139.18 155.66 175.59 219.50
249.47 248.71 244.26 219.50
0.4689 0.5135 0.5650 0.6729
0.7841 0.7721 0.7509 0.6729
1.663 1.919 2.07 ∞
T, K
*Values reproduced and converted from Table 12, p. 17.99, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 6.966.−4 signifies 6.966 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the IPTS 68 scale from −70 to 105.60°C. The thermodynamic diagram from 0.1 to 70 bar extends to 240°C. Equations and constants approximated to the 1985 ASHRAE tables were given by Mecaryk, K. and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. Saturation and superheat tables and a diagram to 80 bar, 560 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Tables and a chart to 1000 psia, 480°F are given by Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). Specific heat and viscosity appear in Thermophysical Properties of Refrigerants, ASHRAE, 1993.
TABLE 2-339
Saturated Refrigerant 502*
P, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−4 Pa⋅s
kf, W/(m⋅K)
200 210 220 230 240
0.2274 0.4098 0.6965 1.1251 1.7392
6.381.−4 6.507.−4 6.640.−4 6.783.−4 6.938.−4
0.646 0.374 0.228 0.146 0.0969
−29.04 −20.83 −12.15 −2.99 6.66
153.34 158.42 163.49 168.50 173.42
−0.1337 −0.0937 −0.0534 −0.0128 0.0280
0.7782 0.7599 0.7449 0.7328 0.7228
1.018 1.036 1.055 1.075 1.097
5.72 4.88 4.23 3.71 3.28
0.103 0.099 0.095 0.091 0.087
250 260 270 280 290
2.5867 3.7188 5.1893 7.0530 9.3660
7.105.−4 7.289.−4 7.492.−4 7.720.−4 7.979.−4
0.0665 0.0470 0.0340 0.0251 0.0188
16.78 27.36 38.36 49.77 61.55
178.20 182.81 187.21 191.35 195.16
0.0691 0.1102 0.1514 0.1923 0.2330
0.7148 0.7082 0.7027 0.6980 0.6937
1.120 1.144 1.170 1.197 1.225
2.94 2.65 2.41 2.18 1.99
0.083 0.079 0.075 0.072 0.068
T, K
300 310 320 330 340
12.19 15.57 19.60 24.35 29.95
8.280.−4 8.637.−4 9.081.−4 9.666.−4 1.053.−3
0.0143 0.0109 0.0084 0.0064 0.0048
73.68 86.17 99.06 112.53 127.13
198.56 201.43 203.57 204.62 203.71
0.2734 0.3134 0.3532 0.3933 0.4351
0.6896 0.6852 0.6798 0.6723 0.6604
1.254 1.285 1.317 1.351 1.386
1.79 1.59 1.40 1.23 1.07
0.064 0.060 0.056 0.052 0.048
350 355.3c
36.62 40.75
1.220.−3 1.786.−3
0.0033 0.0018
145.44 174.00
197.82 174.00
0.4859 0.5634
0.6355 0.5634
1.422
0.93
0.044
*Values reproduced and converted from Table 13, p. 17.101, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and AirConditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. c = critical point. The notation 6.381.−4 signifies 6.381 × 10−4. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the IPTS 68 scale from −70 to 82.2°C. The thermodynamic diagram from 0.1 to 80 bar extends to 180°C. Equations and constants approximated to 1985 ASHRAE tables are given by Mecaryk, K., and M. Masaryk, Heat Recovery Systems and CHP, 11, 2/3 (1991): 193–197. Saturation and superheat tables and a diagram to 20 bar, 515 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Tables and a chart to 1000 psia, 400°F appear in Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat and viscosity, see Thermophysical Properties of Refrigerants, ASHRAE, 1993.
2-298
PHYSICAL AND CHEMICAL DATA
TABLE 2-340
Saturated Refrigerant 503*
P, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−4 Pa⋅s
kf, W/(m⋅K)
150 160 170 180 190
0.0750 0.1798 0.3828 0.7395 1.3187
6.384.−4 6.478.−4 6.585.−4 6.700.−4 6.850.−4
1.894 0.837 0.414 0.224 0.130
−89.60 −79.73 −69.55 −59.08 −48.36
111.02 115.40 119.70 123.84 127.77
−0.4694 −0.4057 −0.3441 −0.2844 −0.2267
0.8681 0.8139 0.7691 0.7318 0.7003
0.482 0.554 0.620 0.682 0.747
6.12 5.05 4.16 3.43 2.94
0.128 0.123 0.116 0.111 0.105
200 210 220 230 240
2.1999 3.4713 5.2281 7.5713 10.61
7.014.−4 7.204.−4 7.426.−4 7.687.−4 8.001.−4
0.0803 0.0520 0.0350 0.0242 0.0172
−37.45 −26.36 −15.10 − 3.65 8.07
131.45 134.84 137.87 140.49 142.58
−0.1710 −0.1173 −0.0656 −0.0155 0.0334
0.6735 0.6503 0.6298 0.6112 0.5939
0.817 0.896 0.988 1.017 1.227
2.56 2.25 1.98 1.73 1.52
0.099 0.094 0.088 0.082 0.076
250 260 270 280 290
14.46 19.25 25.13 32.27 40.87
8.386.−4 8.874.−4 9.526.−4 1.050.−3 1.264.−3
0.0124 0.0090 0.0064 0.0045 0.0028
20.22 33.10 47.22 63.64 86.41
143.98 144.38 143.23 139.25 127.51
0.0817 0.1305 0.1816 0.2384 0.3131
0.5767 0.5585 0.5373 0.5085 0.4548
1.382 1.57 1.79 2.03 2.35
1.33 1.17 1.03 0.91
0.070 0.065 0.059 0.054
292.6c
43.57
1.773.−3
0.0018
110.20
110.20
0.3864
0.3864
∞
T, K
∞
*P, v, h, and s values reproduced and converted from Table 14, p. 17.103, ASHRAE Handbook, 1981: Fundamentals, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, 1981. Copyright material. Reproduced by permission of the copyright owner. cp, µ, and k values interpolated and converted from Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1976. c = critical point. The notation 6.384.−4 signifies 6.384 × 10−4. Saturation and superheat tables and a diagram to 80 bar, 600 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). Tables and a chart to 1000 psia, 460°F are given by Stewart, R. B., R. T. Jacobsen, et al., Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat and viscosity see Thermophysical Properties of Refrigerants, ASHRAE, 1993. The 1993 ASHRAE Handbook—Fundamentals (SI ed.) gives material for integral degrees Celsius with temperatures on the IPTS 68 scale for saturation conditions from −125 to 19.50°C. The thermodynamic diagram from 0.1 to 80 bar extends to 220°C. TABLE 2-341
Saturated Refrigerant 504* Volume, ft3/lb
Enthalpy, Btu/lb
Entropy, Btu/(lb)(°F)
Temp., °F
Pressure, lb/in2 abs.
Liquid
Vapor
Liquid
Vapor
Liquid
Vapor
−120 −100 −80 −60 −40
2.964 6.042 11.34 19.85 32.76
0.01095 0.01119 0.01146 0.01175 0.01206
15.31 7.874 4.372 2.585 1.609
−21.48 −16.39 −11.12 −5.65 0.00
86.69 89.31 91.84 94.25 96.50
−0.0565 −0.0420 −0.0277 −0.0137 0.0000
0.2609 0.2519 0.2435 0.2362 0.2299
−20 0 20 40 60
51.44 77.41 112.3 158.0 216.2
0.01242 0.01282 0.01328 0.01379 0.01443
1.045 0.7029 0.4859 0.3431 0.2458
5.85 11.91 18.22 24.81 31.78
98.58 100.45 102.09 103.44 104.41
0.0135 0.0269 0.0401 0.0533 0.0667
0.2244 0.2195 0.2150 0.2107 0.2065
80 100 120 140 150
289.2 379.1 488.3 618.1 692.2
0.01522 0.01629 0.01783 0.02083 0.02597
0.1773 0.1274 0.0893 0.0578 0.0394
39.25 47.43 56.78 69.97 76.96
104.85 104.49 102.72 97.70 89.76
0.0804 0.0948 0.1107 0.1322 0.1432
0.2020 0.1968 0.1899 0.1784 0.1642
*Unpublished data of Allied Chemical Company, 1970. Used by permission. TABLE 2-342
Thermodynamic Properties of Refrigerant 507*
Temp., K
Pressure, bar
230.5 240 250 260 270 280
1.013 1.59 2.42 3.54 4.95 6.70
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
0.000 0.000 0.000 0.000 0.000 0.000
574 602 627 658 695 738
0.1280 0.0826 0.0546 0.0377 0.0270 0.0198
−3.1 10.3 22.6 37.6 51.6 64.7
143.3 150.2 154.5 159.0 163.8 169.0
−0.015 0.042 0.092 0.149 0.202 0.250
0.620 0.623 0.619 0.617 0.618 0.620
290 300 310 320 330
8.85 11.52 14.74 18.76 23.65
0.000 0.000 0.000 0.001 0.001
787 839 903 006 221
0.0148 0.0112 0.0084 0.0062 0.0042
77.2 89.4 101.6 115.7 135.5
174.6 180.3 185.4 188.6 189.3
0.295 0.336 0.378 0.422 0.481
0.634 0.640 0.648 0.649 0.641
340 341.5c
29.57 32.67
0.001 618 0.001 97
0.0025 0.0020
161.7 172.7
179.9 172.7
0.557 0.590
0.611 0.590
*Azeotropic mixture of R152a and R218. hf = sf = 0 at 233.15 K = −40°C. Interpolated, extrapolated and converted from Lavrenchenko, G. K., M. G. Khmelnuk, et al., Int. J. Refrig., 17, 7 (1994): 461. Some values are tentative. This source also gives a ln P–h diagram from 0.6 to 30 bar, −50 to 70°C. Differences exist between the published diagram and tables. c = critical point.
THERMODYNAMIC PROPERTIES TABLE 2-343
Saturated Rubidium*
T, K
P, bar
vf, m3/kg
312.7m 400 500 600 700
2.46.−9 1.69.−6 1.73.−4 0.0037 0.0317
6.75.−4 6.98.−4 7.22.−4 7.46.−4 7.73.−4
0.1584 1.467 6.466 18.6 28.5
8.10.−4 8.65.−4 9.40.−4 1.03.−3 1.08.−3
800 1000 1200 1400 1500
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
2.3.+5 2790 156.6 20.75
118.7 151.6 188.8 225.4 261.3
1036 1057 1078 1096 1111
0.998 1.091 1.174 1.241 1.296
3.932 3.355 2.953 2.692 2.511
0.379 0.375 0.369 0.362 0.357
4.662 0.605 0.159
296.8 367.6 440.1
1124 1150 1179
1.343 1.422 1.490
2.378 2.205 2.104
0.353 0.360 0.385
*Converted from tables in Vargaftik, Tables of the Thermophysical Properties of Liquids and Gases, Nauka, Moscow, 1972, and Hemisphere, Washington, 1975. m = melting point. The notation 2.46.−9 signifies 2.46 × 10−9. Many of the Vargaftik values also appear in Ohse, R. W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Sci. Pubs., Oxford, 1985 (1020 pp.). This source contains superheat data. Saturation and superheat tables and a diagram to 40 bar, 1600 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram from 0.1 to 320 psia, 1200 to 2700°R, see Weatherford, W. D., J. C. Tyler, et al., WADD-TR-61-96, 1961. An extensive review of properties of the solid and the saturated liquid was given by Alcock, C. B., M. W. Chase, et al., J. Phys. Chem. Ref. Data, 23, 3 (1994): 385–497.
TABLE 2-344 Temp., °C
Thermophysical Properties of Saturated Seawater
Pressure, bar
v, (m3/kg)103
cp, kJ/(kg⋅K)
µ, Ns/m2
k, W/(m⋅K)
NPr
105κ, 1/bar
0 1 2 3 4
0.005993 0.006438 0.006916 0.007427 0.007970
1.000158 1.000099 1.000057 1.000033 1.000025
4.000 4.000 4.000 4.000 4.001
0.001884 0.001827 0.001772 0.001720 0.001669
0.560 0.563 0.565 0.567 0.569
13.46 12.98 12.55 12.13 11.74
5.06 5.02 4.98 4.95 4.92
5 6 7 8 9
0.008548 0.009163 0.009816 0.010511 0.011248
1.000033 1.000057 1.000096 1.000149 1.000261
4.001 4.001 4.002 4.002 4.002
0.001620 0.001574 0.001529 0.001486 0.001445
0.571 0.574 0.576 0.578 0.580
11.35 10.97 10.62 10.29 9.97
4.89 4.86 4.83 4.80 4.78
10 11 12 13 14
0.01203 0.01286 0.01374 0.01467 0.01566
1.000298 1.000392 1.000500 1.000620 1.000727
4.003 4.003 4.003 4.004 4.004
0.001405 0.001367 0.001330 0.001294 0.001259
0.582 0.584 0.586 0.588 0.590
9.70 9.37 9.09 8.81 8.54
4.76 4.74 4.72 4.70 4.68
15 16 17 18 19
0.01671 0.01781 0.01898 0.02022 0.02153
1.000899 1.001055 1.001224 1.001404 1.001595
4.005 4.005 4.006 4.006 4.007
0.001226 0.001195 0.001165 0.001136 0.001107
0.592 0.594 0.595 0.597 0.599
8.29 8.06 7.82 7.62 7.41
4.66 4.65 4.63 4.62 4.60
20 21 22 23 24
0.02291 0.02437 0.02591 0.02753 0.02924
1.001796 1.002009 1.002232 1.002465 1.002708
4.007 4.007 4.008 4.008 4.009
0.001080 0.001054 0.001029 0.001005 0.000981
0.600 0.602 0.604 0.605 0.607
7.21 7.02 6.82 6.66 6.48
4.59 4.57 4.56 4.55 4.54
25 26 27 28 29
0.03104 0.03294 0.03494 0.03705 0.03926
1.002961 1.003224 1.003496 1.003778 1.004069
4.009 4.009 4.010 4.010 4.011
0.000958 0.000936 0.000915 0.000895 0.000875
0.608 0.609 0.611 0.612 0.614
6.31 6.16 6.01 5.86 5.72
4.53 4.52 4.51 4.50 4.49
30
0.04159
1.004369
4.011
0.000855
0.615
5.58
4.48
κ = (−1/V)(∂v/∂p)T ⋅ 105. Thus, at 0°C, the compressibility is 5.06 × 10−5/bar. For further information see, for instance, Bromley, LeR. A., J. Chem. Eng. Data, 12, 2 (1967): 202–206; 13, 1 (1968): 60–62 and 13, 3: 399–402; 15, 2 (1970): 246–253; and A.I.Ch.E.J., 20, 2 (1974): 326–335. Thermal conductivity data sources include Castelli, V. J., E. M. Stanley, et al., Deep Sea Res., 211 (1974): 311–318; Levy, F. L., Int. J. Refrig., 5, 3 (1982): 155–159. For velocity of sound, see, for instance, U.S. Naval Oceanographic Office SP 58, 1962 (50 pp.). More recent information is contained in UNESCO technical papers. See Marine Science No. 38, 1981 (6 pp.) and No. 44, 1983 (53 pp.). For sea ice properties, see Fukusako, S., Int. J. Thermophys., 11, 2 (1990): 353–372.
2-299
TABLE 2-345 Saturated Sodium Temp., K
Pressure, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
cpg, kJ/(kg⋅K)
µf, 10−6 Pa·s
µg, 10−6 Pa·s
kf, W/(m⋅K)
kg, W/(m⋅K)
Prg
1.59.−10 1.80.−9 8.99.−7 5.57.−5 0.00105
0.001 0.001 0.001 0.001 0.001
078 088 115 144 174
8.54.+9 8.08.+8 1.99.+6 38022 2320
207 247 382 514 642
4739 4757 4817 4872 4921
2.259 2.920 3.222 3.462 3.661
14.475 14.195 12.092 10.745 10.631
1.383 1.372 1.334 1.301 1.277
0.86 1.25 1.80 2.28
688 599 415 321 264
800 900 1000 1100 1154.7
0.00941 0.05147 0.1995 0.6016 1.013
0.001 0.001 0.001 0.001 0.001
208 242 280 323 347
291.5 58.8 16.6 5.95 3.89
769 895 1020 1146 1215
4966 5007 5044 5079 5097
3.830 3.978 4.110 4.230 4.290
9.076 8.547 8.134 7.805 7.652
1.260 1.252 1.252 1.261 1.271
2.59 2.72 2.70 2.62 2.56
227 201 181 166 159
19.6 20.6 23.0 25.3 26.5
62.9 58.3 54.2 50.5 48.7
0.0343 0.0406 0.0455 0.0492 0.0522
0.0045 0.0043 0.0042 0.0042 0.0041
1.48 1.38 1.36 1.35 1.30
27.5 29.9 32.2 34.6 37.1
47.2 44.0 41.1 38.2 35.4
0.0547 0.0570 0.0592
0.0041 0.0042 0.0044 0.0046 0.0050
1.26 1.27 1.30
1200 1300 1400 1500 1600
1.50 3.26 6.30 11.13 18.28
0.001 0.001 0.001 0.001 0.001
366 416 471 531 597
2.54 1.24 0.676 0.400 0.253
1273 1402 1534 1671 1812
5111 5140 5168 5193 5217
4.340 4.444 4.542 4.636 4.727
7.538 7.319 7.138 6.984 6.855
1.279 1.305 1.340 1.384 1.437
2.51 2.43 2.39 2.36 2.34
153 143 135 128 122
1700 1800 1900 2000 2100
28.28 41.61 58.70 79.91 105.5
0.001 0.001 0.001 0.001 0.002
675 761 862 984 174
0.168 0.117 0.084 0.063 0.0472
1959 2113 2274 2444 2625
5238 5256 5268 5273 5265
4.816 4.904 4.992 5.079
6.745 6.650 6.568 6.494
1.500 1.574 1.661 1.764 1.926
2.41 2.46 2.53 2.66 2.91
117 112 108 104
2200 2300 2400 2500 2503.7c
135.7 170.6 210.3 254.7 256.4
0.002 0.002 0.002 0.004 0.004
320 584 985 19 57
0.0361 0.0275 0.0203 0.0098 0.0046
2822 3047 3331 3965 4294
5241 5188 5078 4617 4294
2.190 2.690 4.012 39.3
3.40 4.47 8.03 417.
89.4 87.2 80.1 73.7 68.0
Prf
371 400 500 600 700
32.6 29.7 26.6 23.2
0.0106 0.0094 0.0069 0.0057 0.0050
0.0054 0.0059 0.0067 0.0079
c = critical point. sf values converted from Cordfunke, E. H. P. and R. J. M. Konings, Thermochemical Data for Reactor Materials and Fission Products, North Holland Elsevier, NY, 1990. sg determined as sf + (hg − hf)/T. µg and kg values estimated by P. E. Liley. All other values are from Fink, J. K. and L. Leibowitz, Argonne Nat. Lab Rept. ANL/RE-95-2, 1995. The Fink and Leibowitz work also appeared in High Temp. Materials Sci., 35, 65–103, 1996. Saturation and superheat tables and a diagram to 14 bar, 1700 K are given by Reynolds, W. C., Thermodynamic Properties in S.I., Stanford Univ. publ., 1979 (173 pp.). For a Mollier diagram for 0.1–150 psia, 1500–2700°R, see Weatherford, P. M., J. C. Tyler, et al., WADD-TR-61-96, 1961.
2-300
THERMODYNAMIC PROPERTIES
2-301
2500 K 2400 6500
2300 2200 2100 2000 1900 1800
ar
1700
5
10 b
1600
1 ba
r
2
h, kJ/kg
6000
0.5
1500
0.2
1400
0.1
1300
5500
0.0
5
1200
0.0
2
1100
0.0 1
1000
5000 7.0
800
r
8.0
9.0 Entropy (s), kJ/kg • K
bar
vapo
01
te d
0.0
tura
0.0 02
Sa
0.0 05
900
700
10.0
FIG. 2-28 Mollier Diagram for Sodium. Drawn from the Vargaftik et al. values in Ohse, R. W., Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Sci. Pubs., Oxford, UK, 1985. These values are identical with those of Vargaftik, N. B., Handbook of Thermophysical Properties of Gases and Liquids, Moscow, 1972, and the Hemisphere translation, pp. 19. An apparent discontinuity exists between the superheat values and the saturation values, not reproduced here. For a Mollier diagram in f.p.s. units from 0.1 to 150 psia, 1500 to 2700°R, see Fig. 3-36, p. 3-232 of the 6th edition of this handbook. An extensive review of properties of the solid and the saturated liquid was given by Alcock, C. B., Chase, M. W. et al., J. Phys. Chem. Ref. Data, 23(3), 385–497, 1994.
2-302
PHYSICAL AND CHEMICAL DATA
FIG. 2-29 Enthalpy-concentration diagram for aqueous sodium hydroxide at 1 atm. Reference states: enthalpy of liquid water at 32°F and vapor pressure is zero; partial molal enthalpy of infinitely dilute NaOH solution at 64°F and 1 atm is zero. [McCabe, Trans. Am. Inst. Chem. Eng., 31, 129 (1935).]
TABLE 2-346
Saturated Sulfur Dioxide*
P, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−4 Pa⋅s
200 210 220 230 240
0.02056 0.04569 0.09997 0.1844 0.3202
6.189.−4 6.284.−4 6.384.−4 6.488.−4 6.596.−4
12.602 5.946 2.876 1.605 0.9602
7.4 9.1 28.6 43.5 56.5
433.3 446.1 453.8 459.5 464.5
0.033 0.041 0.123 0.198 0.254
2.212 2.159 2.075 2.001 1.952
1.280 1.284 1.288 1.293 1.299
12.3 10.6 8.37 7.03 5.97
250 260 270 280 290
0.5430 0.8778 1.3634 2.0402 2.9574
6.707.−4 6.819.−4 6.938.−4 7.057.−4 7.184.−4
0.5864 0.3745 0.2479 0.1699 0.1197
70.0 85.1 99.8 114.8 129.2
469.7 474.5 479.3 484.3 488.5
0.308 0.363 0.425 0.473 0.523
1.906 1.865 1.827 1.793 1.763
1.308 1.317 1.328 1.343 1.363
5.11 4.39 3.78 3.30 2.87
0.262 0.243 0.224 0.206 0.190
300 310 320 330 340
4.1675 5.7372 7.8226 10.301 13.229
7.312.−4 7.447.−4 7.590.−4 7.847.−4 8.066.−4
0.08647 0.06366 0.04707 0.03572 0.02792
143.1 157.1 170.1 183.0 196.0
492.5 496.3 498.9 501.2 502.5
0.568 0.612 0.649 0.690 0.731
1.732 1.706 1.678 1.654 1.633
1.389 1.422 1.459 1.499 1.546
2.51 2.19 1.91 1.67 1.46
0.174 0.162 0.151 0.139 0.128
350 360 370 380 390
16.759 21.01 26.01 31.92 38.76
8.303.−4 8.571.−4 8.877.−4 9.236.−4 9.671.−4
0.02209 0.01755 0.01399 0.01110 0.00877
211.2 223.7 239.9 257.9 277.7
502.9 503.1 502.9 502.7 500.7
0.781 0.817 0.862 0.910 0.962
1.614 1.593 1.573 1.555 1.534
1.603 1.68 1.75 1.84 1.97
1.27 1.11 0.96 0.84 0.73
0.117 0.108 0.098 0.089 0.081
400 410 420 425.1c
46.67 55.80 66.19 78.81
1.023.−3 1.098.−3 1.235.−3 1.906.−3
0.00685 0.00559 0.00387 0.00191
300.2 326.2 355.6 423.6
496.7 489.5 474.1 423.6
1.020 1.083 1.155 1.304
1.511 1.481 1.436 1.304
2.12
0.63 0.53 0.44
0.072 0.064 0.055
T, K
kf, W/(m⋅K)
*Values interpolated and converted from tables of Kang, McKetta, et al., Bur. Eng. Res. Repr. 59, University of Texas, Austin, 1961. See also J. Chem. Eng. Data, 6 (1961): 220–227; and Am. Inst. Chem. Eng. J., 7 (1961): 418. c = critical point. The notation 6.189.−4 signifies 6.189 × 10−4. The AIChE publication contains a Mollier diagram to 4500 psia, 480°F, while the reprint contains saturation and superheat tables.
THERMODYNAMIC PROPERTIES TABLE 2-347
2-303
Thermodynamic Properties of Saturated Sulfur Hexafluoride (SF6)*
Temp., K
Pressure, bar
vf, m3/kg
vg, m3/kg
hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
222.4 225 230 235 240 245 250 255 260 265 270 275 280 285 290 295 300 305 310 315 318.7
2.200 2.470 3.045 3.710 4.475 5.346 6.332 7.442 8.684 10.07 11.60 13.30 15.18 17.25 19.52 22.01 24.75 27.76 31.05 34.67 37.79
0.0005389 0.0005429 0.0005507 0.0005588 0.0005675 0.0005768 0.0005866 0.0005971 0.0006085 0.0006207 0.0006341 0.0006488 0.0006652 0.0006836 0.0007047 0.0007295 0.0007594 0.000798 0.000851 0.000949 0.001372
0.05428 0.04861 0.03978 0.03286 0.02737 0.02296 0.01939 0.01647 0.01406 0.01205 0.01035 0.00892 0.00769 0.00663 0.00571 0.00490 0.00418 0.00352 0.00291 0.00228 0.00137
−57.55 −54.41 −48.51 −42.67 −36.87 −31.13 −25.47 −19.87 −14.33 −8.85 −3.41 2.00 7.42 12.88 18.45 24.20 30.22 36.75 44.05 53.98 71.74
59.08 60.49 63.12 65.68 68.18 70.61 72.96 75.22 77.39 79.44 81.38 83.19 84.84 86.30 87.52 88.45 89.00 88.97 88.06 85.22 71.74
−0.2310 −0.2171 −0.1913 −0.1663 −0.1421 −0.1186 −0.0960 −0.0741 −0.0528 −0.0323 −0.0123 0.0071 0.0262 0.0451 0.0639 0.0829 0.1025 0.1233 0.1462 0.1769 0.2317
0.2935 0.2936 0.2940 0.2947 0.2956 0.2966 0.2977 0.2988 0.2999 0.3009 0.3017 0.3024 0.3027 0.3027 0.3021 0.3008 0.2984 0.2945 0.2881 0.2761 0.2317
cpf, kJ/(kg⋅K)
cpg, kJ/(kg⋅K)
0.409 0.631 0.870 1.17 1.63 2.48 ∞
0.579 0.583 0.592 0.602 0.613 0.626 0.640 0.656 0.674 0.695 0.720 0.748 0.783 0.827 0.882 0.941 1.070 1.26 1.63 2.40 ∞
*See also Oda, A., M. Uematsu, et al., Bull. JSME, 26, 219 (1983): 1590–1596. Ulybin, S.A., Thermodynamic Properties of Sulfur Hexafluoride, Moscow, 1977 (53 pp.). For thermal conductivity to 500 bar, see Rastorguev, Yu L., B. A. Grigorev, et al., Teploenergetika 24, 6 (1977): 78–81 and Bakulin, S. S. and S. A. Ulybin, Teplofiz. Vysok. Temp., 16, 1 (1978): 59–66. For viscosity to 400 bar, see Grigorev, B. A., A. S. Keramidi, et al., Teploenergetika, 24, 9 (1977): 85–87; and Ulybin, S. A. and V. I. Makanushkin, Teplofiz. Vysok. Temp., 15, 6 (1977): 1195–1201.
FIG. 2-30 Enthalpy-concentration diagram for aqueous sulfuric acid at 1 atm. Reference states: enthalpies of pure-liquid components at 32°F and vapor pressures are zero. NOTE: It should be observed that the weight basis includes the vapor, which is particularly important in the two-phase region. The upper ends of the tie lines in this region are assumed to be pure water. (Hougen and Watson, Chemical Process Principles, part I, Wiley, New York, 1943.)
FIG. 2-31
Enthalpy–log-pressure diagram for sulfur hexafluoride.
TABLE 2-348
Saturated SUVA AC 9000
DuPont bulletin T–AC–9000–SI, 1994 (16 pp.) gives tables and a chart to 100 bar, 235°C. With a stated composition of 23% wt CH2F2 (R23), 25% wt CHF2CF3 (R125), and 52% wt CH2FCH3 (R134a) this is apparently identical to KLEA 66, to which the reader is referred. TABLE 2-349
Saturated Toluene* hf, kJ/kg
hg, kJ/kg
sf, kJ/(kg⋅K)
sg, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µf, 10−4 Pa⋅s
kf, W/(m⋅K)
316.7 333.0 349.6 366.5 383.7
745.7 756.1 766.8 777.8 789.2
2.236 2.295 2.353 2.410 2.467
3.825 3.806 3.792 3.782 3.776
1.64 1.66 1.68 1.71 1.74
8.02 6.96 6.10 5.41 4.83
0.141 0.138 0.136 0.133 0.131
2.67 1.80 1.25 0.891 0.698
401.3 419.6 437.4 456.0 475.1
800.9 812.9 825.2 837.8 850.7
2.522 2.577 2.632 2.686 2.739
3.771 3.771 3.772 3.777 3.783
1.78 1.81 1.84 1.88 1.92
4.34 3.93 3.58 3.28 3.01
0.128 0.126 0.124 0.121 0.119
1.261.−3 1.277.−3 1.294.−3 1.312.−3 1.350.−3
0.481 0.364 0.279 0.218 0.137
494.6 514.4 534.7 555.4 598.1
863.8 877.2 890.9 904.8 933.1
2.792 2.846 2.898 2.950 3.054
3.791 3.801 3.811 3.824 3.852
1.96 2.01 2.05 2.09 2.17
2.78 2.56 2.37 2.19 1.89
0.117 0.114 0.112 0.110 0.105
1.64
0.101 0.096 0.091 0.086 0.082
P, bar
vf, m3/kg
vg, m3/kg
270 280 290 300 310
0.0076 0.0139 0.0246 0.0418 0.0682
1.127.−3 1.138.−3 1.150.−3 1.162.−3 1.175.−3
34.9 19.1 10.6 6.46 4.08
320 330 340 350 360
0.1072 0.1633 0.2416 0.3480 0.4894
1.188.−3 1.201.−3 1.215.−3 1.230.−3 1.245.−3
370 380 390 400 420
0.6736 0.9090 1.2049 1.5713 2.5589
T, K
440 460 480 500 520
3.965 5.892 8.451 11.76 15.96
1.393.−3 1.443.−3 1.499.−3 1.567.−3 1.651.−3
9.00.−2 6.11.−2 4.26.−2 3.03.−2 2.19.−2
642.3 688.1 735.5 784.4 834.9
962.0 991.3 1021.1 1051.3 1081.4
3.156 3.258 3.358 3.457 3.554
3.883 3.917 3.953 3.989 4.027
2.24 2.31 2.38 2.45 2.53
540 560 580 590 591.8c
21.99 27.65 35.56 40.16 41.04
1.761.−3 1.919.−3 2.213.−3 2.650.−3 3.432.−3
1.58.−2 1.13.−2 7.59.−3 5.28.−3 3.43.−3
887.3 942.8 1005.6 1050.2 1084.9
1109.6 1132.1 1142.3 1128.1 1084.9
3.651 3.750 3.857 3.932 3.989
4.062 4.088 4.093 4.063 3.989
2.65 2.82
0.078 0.074
*Values converted and mostly rounded off from the tables of Counsell, Lawrenson, and Lees, Nat. Phys. Lab., Teddington (U.K.) Rep. Chem. 52, 1976. c = critical point. The notation 1.127.−6 signifies 1.127 × 10−6. For other tables, see Goodwin, R. D., J. Phys. Chem. Ref. Data, 18, 4 (1989): 1565–1636. TABLE 2-350
Saturated Solid/Vapor Water* Volume, ft3/lb
Enthalpy, Btu/lb
Entropy, Btu/(lb)(°F)
Temp., °F
Pressure, lb/in2 abs.
Solid
Vapor
Solid
Vapor
Solid
Vapor
−160 −150 −140 −130 −120
4.949.−8 1.620.−7 4.928.−7 1.403.−6 3.757.−6
0.01722 0.01723 0.01724 0.01725 0.01726
3.607.+9 1.139.+9 3.864.+8 1.400.+8 5.386.+7
−222.05 −218.82 −215.49 −212.08 −208.58
990.38 994.80 999.21 1003.63 1008.05
−0.4907 −0.4801 −0.4695 −0.4590 −0.4485
3.5549 3.4387 3.3301 3.2284 3.1330
−110 −100 −90 −80 −70
9.517.−6 2.291.−5 5.260.−5 1.157.−4 2.443.−4
0.01728 0.01729 0.01730 0.01731 0.01732
2.189.+7 9.352.+6 4.186.+6 1.955.+6 9.501.+5
−204.98 −201.28 −197.49 −193.60 −189.61
1012.47 1016.89 1021.31 1025.73 1030.15
−0.4381 −0.4277 −0.4173 −0.4069 −0.3965
3.0434 2.9591 2.8796 2.8045 2.7336
−60 −50 −45 −40 −35
4.972.−4 9.776.−4 1.354.−3 1.861.−3 2.540.−3
0.01734 0.01735 0.01736 0.01737 0.01737
4.788.+5 2.496.+5 1.824.+5 1.343.+5 9.961.+4
−185.52 −181.34 −179.21 −177.06 −174.88
1034.58 1039.00 1041.21 1043.42 1045.63
−0.3862 −0.3758 −0.3707 −0.3655 −0.3604
2.6664 2.6028 2.5723 2.5425 2.5135
−30 −25 −20 −15 −10
3.440.−3 4.627.−3 6.181.−3 8.204.−3 1.082.−2
0.01738 0.01739 0.01739 0.01740 0.01741
7.441.+4 5.596.+4 4.237.+4 3.228.+4 2.475.+4
−172.68 −170.46 −168.21 −165.94 −163.65
1047.84 1050.05 1052.26 1054.47 1056.67
−0.3552 −0.3501 −0.3449 −0.3398 −0.3347
2.4853 2.4577 2.4308 2.4046 2.3791
−5 0 5 10 15
1.419.−2 1.849.−2 2.396.−2 3.087.−2 3.957.−2
0.01741 0.01742 0.01743 0.01744 0.01744
1.909.+4 1.481.+4 1.155.+4 9.060.+3 7.144.+3
−161.33 −158.98 −156.61 −154.22 −151.80
1058.88 1061.09 1063.29 1065.50 1067.70
−0.3295 −0.3244 −0.3193 −0.3142 −0.3090
2.3541 2.3297 2.3039 2.2827 2.2600
16 18 20 22 24
4.156.−2 4.581.−2 5.045.−2 5.552.−2 6.105.−2
0.01745 0.01745 0.01745 0.01746 0.01746
6.817.+3 6.210.+3 5.662.+3 5.166.+3 4.717.+3
−151.32 −150.34 −149.36 −148.38 −147.39
1068.14 1069.02 1069.90 1070.38 1071.66
−0.3080 −0.3060 −0.3039 −0.3019 −0.2998
2.2555 2.2466 2.2378 2.2291 2.2205
26 28 30 31 32
6.708.−2 7.365.−2 8.080.−2 8.461.−2 8.858.−2
0.01746 0.01746 0.01747 0.01747 0.01747
4.311.+3 3.943.+3 3.608.+3 3.453.+3 3.305.+3
−146.40 −145.40 −144.40 −143.90 −143.40
1072.53 1073.41 1074.29 1074.73 1075.16
−0.2978 −0.2957 −0.2937 −0.2927 −0.2916
2.2119 2.2034 2.1950 2.1908 2.1867
*Condensed from Fundamentals, American Society of Heating, Refrigerating and Air-Conditioning Engineers, 1967 and 1972. Reproduced by permission. The validity of many standard reference tables has been critically reviewed by Jancso, Pupezin, and van Hook, J. Phys. Chem., 74 (1970): 2984. This source is recommended for further study. The notation 4.949.−8, 3.607.+9, etc., means 4.949 × 10−8, 3.607 × 109, etc. 2-304
THERMODYNAMIC PROPERTIES TABLE 2-351 Temp., °F
Saturated Water Substance—Temperature (fps units)* Pressure, lb/in2 abs.
Volume, ft3/lb Liquid
Enthalpy, Btu/lb Vapor
Entropy, Btu/(lb)(°F)
Liquid
Vapor
Liquid
Vapor
32.018 35 40 45 50
0.08865 0.09991 0.12163 0.14744 0.17796
0.016022 0.016020 0.016019 0.016020 0.016023
3302.4 2948.1 2445.8 2037.8 1704.8
0.000 3.002 8.027 13.044 18.054
1075.5 1076.8 1079.0 1081.2 1083.4
0.0000 0.0061 0.0162 0.0262 0.0361
2.1872 2.1767 2.1594 2.1426 2.1262
55 60 65 70 75
0.21392 0.25611 0.30545 0.36292 0.42964
0.016027 0.016033 0.016041 0.016050 0.016060
1432.0 1207.6 1022.1 868.4 740.3
23.059 28.060 33.057 38.052 43.045
1085.6 1087.7 1089.9 1092.1 1094.3
0.0458 0.0555 0.0651 0.0745 0.0839
2.1102 2.0946 2.0794 2.0645 2.0500
80 85 90 95 100
0.50683 0.59583 0.69813 0.81534 0.94294
0.016072 0.016085 0.016099 0.016114 0.016130
633.3 543.6 468.1 404.4 350.4
48.037 53.027 58.018 63.008 67.999
1096.4 1098.6 1100.8 1102.9 1105.1
0.0932 0.1024 0.1115 0.1206 0.1295
2.0359 2.0221 2.0086 1.9954 1.9825
110 120 130 140 150
1.2750 1.6927 2.2230 2.8892 3.7184
0.016165 0.016204 0.016247 0.016293 0.016343
265.39 203.26 157.33 122.98 97.07
77.98 87.97 97.96 107.89 117.95
1109.3 1113.6 1117.8 1122.0 1126.1
0.1472 0.1646 0.1817 0.1985 0.2150
1.9577 1.9339 1.9112 1.8895 1.8686
160 170 180 190 200
4.7414 5.9926 7.5110 9.340 11.526
0.016395 0.016451 0.016510 0.016572 0.016637
77.27 62.06 50.225 40.957 33.639
127.96 137.97 148.00 158.04 168.09
1130.2 1134.2 1138.2 1142.1 1146.0
0.2313 0.2473 0.2631 0.2787 0.2940
1.8487 1.8295 1.8111 1.7934 1.7764
210 212 220 230 240
14.123 14.696 17.186 20.779 24.968
0.016705 0.016719 0.016775 0.016849 0.016926
27.816 26.799 23.148 19.381 16.321
178.15 180.17 188.23 198.33 208.45
1149.7 1150.5 1153.4 1157.1 1160.6
0.3091 0.3121 0.3241 0.3388 0.3533
1.7600 1.7568 1.7442 1.7290 1.7142
250 260 270 280 290
29.825 35.427 41.856 49.200 57.550
0.017066 0.017089 0.017175 0.017264 0.01736
13.819 11.762 10.060 8.644 7.4603
218.59 228.76 238.95 249.17 259.4
1164.0 1167.4 1170.6 1173.8 1167.8
0.3677 0.3819 0.3960 0.4098 0.4236
1.7000 1.6862 1.6729 1.6599 1.6473
300 320 340 360 380
67.005 89.643 117.992 153.01 195.73
0.01745 0.01766 0.01787 0.01811 0.01836
6.4658 4.9138 3.7878 2.9573 2.3353
269.7 290.4 311.3 332.3 353.6
1179.7 1185.2 1190.1 1194.4 1198.0
0.4372 0.4640 0.4902 0.5161 0.5416
1.6351 1.6116 1.5892 1.5678 1.5473
400 420 440 460 480
247.26 308.78 381.54 466.87 566.15
0.01864 0.01894 0.01926 0.01961 0.02000
1.8630 1.4997 1.2169 0.99424 0.81717
375.1 396.9 419.0 441.5 464.5
1201.0 1203.1 1204.4 1204.8 1204.1
0.5667 0.5915 0.6161 0.6405 0.6648
1.5274 1.5080 1.4890 1.4704 1.4518
500 520 540 560 580
680.86 812.53 962.79 1133.38 1326.17
0.02043 0.02091 0.02146 0.02207 0.02279
0.67492 0.55956 0.46513 0.38714 0.32216
487.9 512.0 536.8 562.4 589.1
1202.2 1199.0 1194.3 1187.7 1179.0
0.6890 0.7133 0.7378 0.7625 0.7876
1.4333 1.4146 1.3954 1.3757 1.3550
600 620 640 660 680
1543.2 1786.9 2059.9 2365.7 2708.6
0.02364 0.02466 0.02595 0.02768 0.03037
0.26747 0.22081 0.18021 0.14431 0.11117
617.1 646.9 679.1 714.9 758.5
1167.7 1153.2 1133.7 1107.0 1068.5
0.8134 0.8403 0.8686 0.8995 0.9365
1.3330 1.3092 1.2821 1.2498 1.2086
700 702 704 705.47
3094.3 3135.5 3177.2 3208.2
0.03662 0.03824 0.04108 0.05078
0.07519 0.06997 0.06300 0.05078
825.2 835.0 854.2 906.0
991.7 979.7 956.2 906.0
0.9924 1.0006 1.0169 1.0612
1.1359 1.1210 1.1046 1.0612
*Extracted and condensed from 1967 ASME Steam Tables. Copyright reserved. Reproduced by permission.
2-305
TABLE 2-352
Temp., K
Saturated Water Substance—Temperature (SI units)
Pressure, bar*
Volume, m3/ kg Vapor
Entropy, kJ/(kg⋅K)
Condensed† Vapor Condensed†
Specific heat, Cp, kJ/(kg⋅K) Vapor
Viscosity, Ns/m2 Condensed†
Vapor
Thermal conductivity, W/(m⋅K) Condensed†
Vapor
Prandtl no. Condensed†
Vapor
Surface tension, N/m Temp., Condensed† K
Vapor
Condensed†
150
6.30.−11
1.073.−3
9.55.+9
−539.6
2273
−2.187
16.54
1.155
3.73
150
160 170 180 190 200
7.72.−10 7.29.−9 5.38.−8 3.23.−7 1.62.−6
1.074.−3 1.076.−3 1.077.−3 1.078.−3 1.079.−3
9.62.+8 1.08.+8 1.55.+7 2.72.+6 5.69.+5
−525.7 −511.7 −497.8 −483.8 −467.5
2291 2310 2328 2347 2366
−2.106 −2.026 −1.947 −1.868 −1.789
15.49 14.57 13.76 13.03 12.38
1.233 1.311 1.389 1.467 1.545
3.52 3.34 3.18 3.04 2.91
160 170 180 190 200
210 220 230 240 250
7.01.−6 2.65.−5 8.91.−5 3.72.−4 7.59.−4
1.081.−3 1.082.−3 1.084.−3 1.085.−3 1.087.−3
1.39.+5 3.83.+4 1.18.+4 4.07.+3 1.52.+3
−451.2 −435.0 −416.3 −400.1 −381.5
2384 2403 2421 2440 2459
−1.711 −1.633 −1.555 −1.478 −1.400
11.79 11.20 10.79 10.35 9.954
1.623 1.701 1.779 1.857 1.935
2.79 2.69 2.59 2.50 2.42
210 220 230 240 250
255 260 265 270 273.15
1.23.−3 1.96.−3 3.06.−3 4.69.−3 6.11.−3
1.087.−3 1.088.−3 1.089.−3 1.090.−3 1.091.−3
956.4 612.2 400.4 265.4 206.3
−369.8 −360.5 −351.2 −339.6 −333.5
2468 2477 2486 2496 2502
−1.361 −1.323 −1.281 −1.296 −1.221
9.768 9.590 9.461 9.255 9.158
1.974 2.013 2.052 2.091 2.116
2.38 2.35 2.31 2.27 2.26
255 260 265 270 273.15
273.15 275 280 285 290
0.00611 0.00697 0.00990 0.01387 0.01917
1.000.−3 1.000.−3 1.000.−3 1.000.−3 1.001.−3
206.3 181.7 130.4 99.4 69.7
0.0 7.8 28.8 49.8 70.7
2502 2505 2514 2523 2532
0.000 0.028 0.104 0.178 0.251
9.158 9.109 8.980 8.857 8.740
4.217 4.211 4.198 4.189 4.184
1.854 1.855 1.858 1.861 1.864
1750.−6 1652.−6 1422.−6 1225.−6 1080.−6
8.02.−6 8.09.−6 8.29.−6 8.49.−6 8.69.−6
0.569 0.574 0.582 0.590 0.598
0.0182 0.0183 0.0186 0.0189 0.0193
12.99 12.22 10.26 8.81 7.56
0.815 0.817 0.825 0.833 0.841
0.0755 0.0753 0.0748 0.0743 0.0737
273.15 275 280 285 290
295 300 305 310 315
0.02617 0.03531 0.04712 0.06221 0.08132
1.002.−3 1.003.−3 1.005.−3 1.007.−3 1.009.−3
51.94 39.13 27.90 22.93 17.82
91.6 112.5 133.4 154.3 175.2
2541 2550 2559 2568 2577
0.323 0.393 0.462 0.530 0.597
8.627 8.520 8.417 8.318 8.224
4.181 4.179 4.178 4.178 4.179
1.868 1.872 1.877 1.882 1.888
959.−6 855.−6 769.−6 695.−6 631.−6
8.89.−6 9.09.−6 9.29.−6 9.49.−6 9.69.−6
0.606 0.613 0.620 0.628 0.634
0.0195 0.0196 0.0201 0.0204 0.0207
6.62 5.83 5.20 4.62 4.16
0.849 0.857 0.865 0.873 0.883
0.0727 0.0717 0.0709 0.0700 0.0692
295 300 305 310 315
320 325 330 335 340
0.1053 0.1351 0.1719 0.2167 0.2713
1.011.−3 1.013.−3 1.016.−3 1.018.−3 1.021.−3
13.98 11.06 8.82 7.09 5.74
196.1 217.0 237.9 258.8 279.8
2586 2595 2604 2613 2622
0.649 0.727 0.791 0.854 0.916
8.151 8.046 7.962 7.881 7.804
4.180 4.182 4.184 4.186 4.188
1.895 1.903 1.911 1.920 1.930
577.−6 528.−6 489.−6 453.−6 420.−6
9.89.−6 10.09.−6 10.29.−6 10.49.−6 10.69.−6
0.640 0.645 0.650 0.655 0.660
0.0210 0.0213 0.0217 0.0220 0.0223
3.77 3.42 3.15 2.88 2.66
0.894 0.901 0.908 0.916 0.925
0.0683 0.0675 0.0666 0.0658 0.0649
320 325 330 335 340
345 350 355 360 365
0.3372 0.4163 0.5100 0.6209 0.7514
1.024.−3 1.027.−3 1.030.−3 1.034.−3 1.038.−3
4.683 3.846 3.180 2.645 2.212
300.7 321.7 342.7 363.7 384.7
2630 2639 2647 2655 2663
0.977 1.038 1.097 1.156 1.214
7.729 7.657 7.588 7.521 7.456
4.191 4.195 4.199 4.203 4.209
1.941 1.954 1.968 1.983 1.999
389.−6 365.−6 343.−6 324.−6 306.−6
10.89.−6 11.09.−6 11.29.−6 11.49.−6 11.69.−6
0.665 0.668 0.671 0.674 0.677
0.0226 0.0230 0.0233 0.0237 0.0241
2.45 2.29 2.14 2.02 1.91
0.933 0.942 0.951 0.960 0.969
0.0641 0.0632 0.0623 0.0614 0.0605
345 350 355 360 365
370 373.15 375 380 385
0.9040 1.0133 1.0815 1.2869 1.5233
1.041.−3 1.044.−3 1.045.−3 1.049.−3 1.053.−3
1.861 1.679 1.574 1.337 1.142
405.8 419.1 426.8 448.0 469.2
2671 2676 2679 2687 2694
1.271 1.307 1.328 1.384 1.439
7.394 7.356 7.333 7.275 7.218
4.214 4.217 4.220 4.226 4.232
2.017 2.029 2.036 2.057 2.080
289.−6 279.−6 274.−6 260.−6 248.−6
11.89.−6 12.02.−6 12.09.−6 12.29.−6 12.49.−6
0.679 0.680 0.681 0.683 0.685
0.0245 0.0248 0.0249 0.0254 0.0258
1.80 1.76 1.70 1.61 1.53
0.978 0.984 0.987 0.995 1.004
0.0595 0.0589 0.0586 0.0576 0.0566
370 373.15 375 380 385
390 400 410 420 430
1.794 2.455 3.302 4.370 5.699
1.058.−3 1.067.−3 1.077.−3 1.088.−3 1.099.−3
0.980 0.731 0.553 0.425 0.331
490.4 532.9 575.6 618.6 661.8
2702 2716 2729 2742 2753
1.494 1.605 1.708 1.810 1.911
7.163 7.058 6.959 6.865 6.775
4.239 4.256 4.278 4.302 4.331
2.104 2.158 2.221 2.291 2.369
237.−6 217.−6 200.−6 185.−6 173.−6
12.69.−6 13.05.−6 13.42.−6 13.79.−6 14.14.−6
0.686 0.688 0.688 0.688 0.685
0.0263 0.0272 0.0282 0.0293 0.0304
1.47 1.34 1.24 1.16 1.09
1.013 1.033 1.054 1.075 1.10
0.0556 0.0536 0.0515 0.0494 0.0472
390 400 410 420 430
2-306
Condensed†
Enthalpy, kJ/kg
440 450 460 470 480
7.333 9.319 11.71 14.55 17.90
1.110.−3 1.123.−3 1.137.−3 1.152.−3 1.167.−3
0.261 0.208 0.167 0.136 0.111
705.3 749.2 793.5 838.2 883.4
2764 2773 2782 2789 2795
2.011 2.109 2.205 2.301 2.395
6.689 6.607 6.528 6.451 6.377
4.36 4.40 4.44 4.48 4.53
2.46 2.56 2.68 2.79 2.94
162.−6 152.−6 143.−6 136.−6 129.−6
14.50.−6 14.85.−6 15.19.−6 15.54.−6 15.88.−6
0.682 0.678 0.673 0.667 0.660
0.0317 0.0331 0.0346 0.0363 0.0381
1.04 0.99 0.95 0.92 0.89
1.12 1.14 1.17 1.20 1.23
0.0451 0.0429 0.0407 0.0385 0.0362
440 450 460 470 480
490 500 510 520 530
21.83 26.40 31.66 37.70 44.58
1.184.−3 1.203.−3 1.222.−3 1.244.−3 1.268.−3
0.0922 0.0766 0.0631 0.0525 0.0445
929.1 975.6 1023 1071 1119
2799 2801 2802 2801 2798
2.479 2.581 2.673 2.765 2.856
6.312 6.233 6.163 6.093 6.023
4.59 4.66 4.74 4.84 4.95
3.10 3.27 3.47 3.70 3.96
124.−6 118.−6 113.−6 108.−6 104.−6
16.23.−6 16.59.−6 16.95.−6 17.33.−6 17.72.−6
0.651 0.642 0.631 0.621 0.608
0.0401 0.0423 0.0447 0.0475 0.0506
0.87 0.86 0.85 0.84 0.85
1.25 1.28 1.31 1.35 1.39
0.0339 0.0316 0.0293 0.0269 0.0245
490 500 510 520 530
540 550 560 570 580
52.38 61.19 71.08 82.16 94.51
1.294.−3 1.323.−3 1.355.−3 1.392.−3 1.433.−3
0.0375 0.0317 0.0269 0.0228 0.0193
1170 1220 1273 1328 1384
2792 2784 2772 2757 2737
2.948 3.039 3.132 3.225 3.321
5.953 5.882 5.808 5.733 5.654
5.08 5.24 5.43 5.68 6.00
4.27 4.64 5.09 5.67 6.40
101.−6 97.−6 94.−6 91.−6 88.−6
18.1.−6 18.6.−6 19.1.−6 19.7.−6 20.4.−6
0.594 0.580 0.563 0.548 0.528
0.0540 0.0583 0.0637 0.0698 0.0767
0.86 0.87 0.90 0.94 0.99
1.43 1.47 1.52 1.59 1.68
0.0221 0.0197 0.0173 0.0150 0.0128
540 550 560 570 580
1.05 1.14 1.30 1.52 1.65
1.84 2.15 2.60 3.46 4.20
0.0105 0.0084 0.0063 0.0045 0.0035
590 600 610 620 625
0.0026 0.0015 0.0008 0.0001 0.0000
630 635 640 645 647.3‡
590 600 610 620 625
108.3 123.5 137.3 159.1 169.1
1.482.−3 1.541.−3 1.612.−3 1.705.−3 1.778.−3
0.0163 0.0137 0.0115 0.0094 0.0085
1443 1506 1573 1647 1697
2717 2682 2641 2588 2555
3.419 3.520 3.627 3.741 3.805
5.569 5.480 5.318 5.259 5.191
6.41 7.00 7.85 9.35 10.6
7.35 8.75 11.1 15.4 18.3
84.−6 81.−6 77.−6 72.−6 70.−6
21.5.−6 22.7.−6 24.1.−6 25.9.−6 27.0.−6
0.513 0.497 0.467 0.444 0.430
0.0841 0.0929 0.103 0.114 0.121
630 635 640 645 647.3‡
179.7 190.9 202.7 215.2 221.2
1.856.−3 1.935.−3 2.075.−3 2.351.−3 3.170.−3
0.0075 0.0066 0.0057 0.0045 0.0032
1734 1783 1841 1931 2107
2515 2466 2401 2292 2107
3.875 3.950 4.037 4.223 4.443
5.115 5.025 4.912 4.732 4.443
12.6 16.4 26 90 ∞
22.1 27.6 42
67.−6 64.−6 59.−6 54.−6 45.−6
28.0.−6 30.0.−6 32.0.−6 37.0.−6 45.0.−6
0.412 0.392 0.367 0.331 0.238
0.130 0.141 0.155 0.178 0.238
∞
2.0 2.7 4.2 12 ∞
4.8 6.0 9.6 26 ∞
*1 bar = 105 N/m2. †Above the solid line, the condensed phase is solid; below it, liquid. ‡Critical temperature. −11 −3 9 NOTE: The notations 6.30.−11, 1.073.−3, 9.55.+9, etc. signify 6.30 × 10 , 1.073 × 10 , 955 × 10 , etc. Tables 2-351 and 2-352 are provided for general use. Tables to higher precision are available over certain ranges and for various properties. The most current internationally accepted tables are found in Haar, L., J. S. Gallagher, and G. S. Kell, NBS/NRC Steam Tables, Hemisphere, Washington, DC, 1984 (320 pp.). These do not tabulate certain properties at saturation states. A revised release on the IAPWS Skeleton Tables 1985 for the thermodynamic properties of ordinary water substance, Sept. 1993 (15 pp), is apparently the latest international publication. In J. Phys. Chem. Ref. Data 17, 4 (1988): 1439–1540, H. Sato, M. Uematsu, and others review existing steam tables and present the 1985 formulation of skeleton tables. Property codes and programs include Cheng, S. C. and C. Nguyen, Modeling and Simulation on Microcomputers 1989 (R. W. Allen, ed.), S.C.S. Intl., San Diego, 1989 (pp. 138–141); Garland, W. J. and B. J. Hand, Nucl. Engng. & Des., 113, (1989): 21–34; Dickey, D. S., Chem. Eng. 98, 9 (1991): 207–8 and 98, 11: 235–6; Muneer, T. and S. M. Scott, Proc. Inst. Mech. Eng., 205, (1991): 25–29; and Energy Convsn. Mgmt., 31, 4 (1991): 315–325. Useful pictorial representations of 20 properties as a function of both temperature (to 800°C) and pressure (to 1000 bar) are given by Grigull, U., J. Bach, et al., Warme- u. Stoff., 1 (1968): 202–213. Property equations for the saturated liquid for the range 0–300°C are given by Charters, W. W. S. and H. A. Sadafi, Rev. Int. Froid, 10, (Mar. 1987): 105–6. Gordon, S., NASA Tech. Paper 1906, 1982 gives detailed tables for ice from 0 K. Ice and snow properties are reviewed by Fukusako, S., Int. J. Thermophys., 11, 2 (1990): 353–372. See also Wagner, W., A. Saul, et al., J. Phys. Chem. Ref. Data, 23, 3 (1994): 515–525, and Table 2-358.
2-307
2-308
PHYSICAL AND CHEMICAL DATA
TABLE 2-353
Saturated Liquid Water—Miscellaneous Properties
Temperature, °C
104 β
104 kT / bar
104 ks / bar
vs, m/s
µf , 10−6 Pa⋅s
cp, kJ/kg⋅K
k, W/m⋅K
Pr, bar
0 1 2 3 4
−0.681 −0.501 −0.327 −0.160 0.003
0.50885 0.50509 0.50151 0.49808 0.49481
0.50855 0.50493 0.50143 0.49806 0.49481
1402.4 1407.4 1412.2 1417.0 1421.6
1.793 1.732 1.675 1.621 1.569
4.2176 4.2140 4.2107 4.2077 4.2048
0.567 0.569 0.570 0.572 0.573
13.32 12.83 12.37 11.93 11.51
σ, N/m 0.07565 0.07551 0.07537 0.07522 0.07508
5 6 7 8 9
0.160 0.312 0.460 0.604 0.744
0.49169 0.48871 0.48587 0.48315 0.48056
0.49167 0.48865 0.48573 0.48291 0.48019
1426.2 1430.6 1434.9 1439.1 1443.3
1.520 1.474 1.429 1.387 1.346
4.2022 4.1999 4.1977 4.1956 4.1938
0.575 0.577 0.578 0.580 0.581
11.11 10.73 10.38 10.04 9.72
0.07494 0.07480 0.07465 0.07451 0.07436
10 11 12 13 14
0.880 1.012 1.141 1.267 1.389
0.47809 0.47573 0.47347 0.47133 0.46928
0.47757 0.47504 0.47260 0.47024 0.46797
1447.3 1451.2 1455.0 1458.7 1462.4
1.308 1.271 1.236 1.202 1.170
4.1921 4.1906 4.1892 4.1879 4.1867
0.5828 0.5844 0.5859 0.5875 0.5891
9.41 9.11 8.84 8.57 8.32
0.07422 0.07407 0.07393 0.07378 0.07364
15 16 17 18 19
1.509 1.626 1.740 1.852 1.961
0.46733 0.46548 0.46371 0.46203 0.46043
0.46578 0.46366 0.46162 0.45966 0.45776
1465.9 1469.4 1472.7 1476.0 1479.2
1.139 1.110 1.081 1.054 1.028
4.1856 4.1847 4.1838 4.1830 4.1823
0.5906 0.5922 0.5937 0.5953 0.5968
8.07 7.84 7.62 7.41 7.20
0.07349 0.07334 0.07319 0.07304 0.07289
20 21 22 23 24
2.068 2.173 2.275 2.376 2.475
0.45892 0.45748 0.45612 0.45484 0.45362
0.45593 0.45417 0.45248 0.45084 0.44927
1482.3 1485.3 1488.3 1491.2 1493.9
1.003 0.979 0.955 0.933 0.911
4.1817 4.1812 4.1807 4.1802 4.1798
0.5983 0.5999 0.6014 0.6029 0.6044
7.01 6.82 6.64 6.47 6.30
0.07274 0.07259 0.07244 0.07228 0.07213
25 26 27 28 30
2.572 2.667 2.761 2.852 3.032
0.45247 0.45139 0.45038 0.44943 0.44771
0.44776 0.44630 0.44490 0.44355 0.44102
1496.7 1499.3 1501.9 1504.3 1509.1
0.891 0.871 0.852 0.833 0.798
4.1795 4.1792 4.1790 4.1788 4.1785
0.6059 0.6074 0.6089 0.6104 0.6133
6.15 5.99 5.85 5.70 5.44
0.07198 0.07182 0.07167 0.07151 0.07120
32 34 36 38 40
3.206 3.375 3.539 3.698 3.853
0.44622 0.44496 0.44390 0.44305 0.44239
0.43869 0.43655 0.43459 0.43280 0.43118
1513.6 1517.8 1521.7 1525.4 1528.9
0.765 0.734 0.705 0.679 0.653
4.1783 4.1782 4.1783 4.1784 4.1786
0.6162 0.6190 0.6218 0.6246 0.6273
5.19 4.95 4.74 4.54 4.35
0.07089 0.07058 0.07025 0.06992 0.06960
42 44 46 48 50
4.004 4.152 4.296 4.438 4.576
0.44192 0.44162 0.44149 0.44153 0.44173
0.42972 0.42842 0.42726 0.42624 0.42535
1532.1 1535.0 1537.7 1540.3 1542.6
0.629 0.607 0.586 0.566 0.547
4.1789 4.1792 4.1797 4.1801 4.1807
0.6299 0.6315 0.6351 0.6375 0.6400
4.17 4.02 3.86 3.71 3.57
0.06927 0.06894 0.06861 0.06828 0.06795
55 60 65 70 75
4.910 5.231 5.539 5.837 6.128
0.44290 0.44496 0.44788 0.45162 0.45614
0.42370 0.42281 0.42262 0.42309 0.42418
1547.4 1551.0 1553.4 1554.8 1555.1
0.5043 0.4668 0.4338 0.4045 0.3784
4.1824 4.1844 4.1869 4.1897 4.1929
0.6457 0.6511 0.6561 0.6607 0.6649
3.267 3.000 2.768 2.565 2.386
0.06710 0.06624 0.06537 0.06449 0.06359
80 85 90 95 100
6.411 6.689 6.962 7.233 7.501
0.46143 0.46748 0.47429 0.48185 0.49019
0.42587 0.42812 0.43093 0.43429 0.43819
1554.4 1552.9 1550.5 1547.2 1543.1
0.3550 0.3340 0.3150 0.2979 0.2823
4.1965 4.2005 4.2050 4.2102 4.2164
0.6686 0.6721 0.6753 0.6779 0.6800
2.228 2.088 1.962 1.850 1.756
0.06268 0.06176 0.06083 0.05988 0.05892
Values mostly from Aleksandrov, A. A. and M. S. Trakhtenhertz, Thermophysical Properties of Water at Atmospheric Pressure, Standartov, Moscow, 1977 (99 pp.).
THERMODYNAMIC PROPERTIES TABLE 2-354 Temperature, K
2-309
Thermodynamic Properties of Compressed Steam* Pressure, bar 0.1
0.5
1
5
10
20
40
60
80
350
v h s
16.12 2644 8.327
1.027.−3 321.7 1.037
1.027.−3 231.8 1.037
1.027.−3 322.1 1.037
1.027.−3 322.5 1.037
1.026.−3 323.3 1.036
1.025.−3 324.9 1.035
1.024.−3 326.4 1.034
1.023.−3 328.1 1.032
1.023.−3 329.7 1.031
400
v h s
18.44 2739 8.581
3.67 2735 7.831
1.827 2730 7.502
1.067.−3 533.1 1.601
1.067.−3 533.4 1.600
1.066.−3 534.1 1.599
1.065.−3 535.4 1.597
1.064.−3 536.8 1.595
1.063.−3 538.2 1.593
1.061.−3 539.6 1.592
450
v h s
20.75 2835 8.811
4.14 2833 8.061
2.063 2830 7.736
0.410 2804 6.949
1.124.−3 749.0 2.110
1.123.−3 749.8 2.107
1.121.−3 750.8 2.105
1.119.−3 751.9 2.102
1.118.−3 753.0 2.099
1.116.−3 754.1 2.097
500
v h s
23.07 2932 9.012
4.61 2931 8.261
2.298 2929 7.944
0.452 2912.4 7.177
0.221 2891.2 6.823
0.104 2839.4 6.422
1.201.−3 975.9 2.578
1.198.−3 976.3 2.575
1.196.−3 976.8 2.571
1.193.−3 977.3 2.567
600
v h s
27.7 3131 9.374
5.53 3130 8.630
2.76 3129 8.309
0.548 3120 7.560
0.271 3109 7.223
0.133 3087 6.875
0.0630 3036 6.590
0.0396 2976 6.224
0.0276 2906 5.997
0.0201 2820 5.775
700
v h s
32.3 3335 9.692
6.46 3335 8.946
3.23 3334 8.625
0.643 3328 7.877
0.319 3322 7.550
0.158 3307 7.215
0.0769 3278 6.864
0.0500 3247 6.644
0.0346 3214 6.431
0.0283 3179 6.334
800
v h s
36.9 3547 9.971
7.38 3546 9.228
3.69 3546 8.908
0.736 3542 8.161
0.367 3537 7.837
0.182 3526 7.507
0.0889 3506 7.151
0.0589 3485 6.965
0.0436 3464 6.809
0.0343 3442 6.685
900
v h s
41.5 3765 10.228
8.31 3765 9.485
4.15 3764 9.165
0.829 3761 8.420
0.414 3757 8.097
0.206 3750 7.770
0.102 3737 7.462
0.0674 3719 7.237
0.0501 3704 7.092
0.0398 3688 6.975
1000
v h s
46.2 3990 10.466
9.23 3990 9.723
4.615 3990 9.402
0.921 3987 8.659
0.460 3984 8.336
0.229 3978 8.011
0.114 3967 7.682
0.0758 3955 7.486
0.0564 3944 7.345
0.0449 3935 7.233
1500
v h s
69.2 5231 11.47
13.9 5228 10.77
6.92 5227 10.40
1.385 5225 9.66
0.692 5224 9.34
0.341 5221 9.015
0.1730 5217 8.693
0.1153 5212 8.503
0.0865 5207 8.368
0.0692 5203 8.262
2000
v h s
93.0 6832 12.38
18.6 6734 11.58
9.26 6706 11.25
1.850 6662 10.48
0.925 6649 10.15
0.462 6639 9.828
0.231 6629 9.503
0.1543 6623 9.313
0.1157 6619 9.178
0.0926 6616 9.073
2500
v h s
123.7 10417 13.95
24.0 9330 12.73
11.90 9046 12.28
2.35 8621 11.35
1.171 8504 10.80
0.583 8413 10.62
0.291 8342 10.26
0.1942 8307 10.06
0.1457 8285 9.920
0.1166 8269 9.810
*v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K). The notation 1.027.−3 signifies 1.027 × 10−3.
100
2-310
PHYSICAL AND CHEMICAL DATA
TABLE 2-354
Thermodynamic Properties of Compressed Steam (Concluded ) Pressure, bar
Temperature, K
150
200
250
300
400
500
600
700
800
900
1000
v 350 h s
1.020.−3 333.7 1.028
1.018.−3 337.7 1.025
1.016.−3 341.7 1.022
1.014.−3 344.7 1.019
1.009.−3 353.8 1.013
1.005.−3 361.8 1.007
1.002.−3 369.7 1.001
9.977.−4 377.7 0.996
9.937.−4 385.7 0.991
9.900.−4 393.7 0.985
9.865.−4 401.7 0.979
400
v h s
1.059.−3 543.1 1.587
1.056.−3 546.5 1.583
1.053.−3 550.1 1.578
1.050.−3 553.5 1.574
1.045.−3 560.6 1.565
1.041.−3 567.8 1.557
1.035.−3 574.9 1.549
1.031.−3 582.1 1.541
1.027.−3 589.3 1.533
1.022.−3 596.5 1.526
1.018.−3 603.8 1.518
450
v h s
1.112.−3 756.8 2.088
1.108.−3 759.5 2.082
1.105.−3 762.3 2.076
1.101.−3 765.2 2.070
1.094.−3 771.0 2.060
1.088.−3 776.9 2.049
1.082.−3 783.0 2.039
1.076.−3 789.6 2.029
1.070.−3 795.3 2.019
1.065.−3 801.6 2.010
1.059.−3 807.9 2.002
500
v h s
1.187.−3 978.8 2.558
1.181.−3 980.3 2.549
1.175.−3 981.9 2.541
1.170.−3 983.7 2.533
1.160.−3 987.4 2.517
1.151.−3 991.5 2.502
1.142.−3 995.9 2.488
1.134.−3 1000.5 2.474
1.126.−3 1005.3 2.461
1.119.−3 1010.3 2.449
1.112.−3 1015.4 2.437
600
v h s
1.519.−3 1499 3.501
1.483.−3 1489 3.469
1.454.−3 1479 3.443
1.428.−3 1472 3.419
1.392.−3 1462 3.379
1.362.−3 1456 3.346
1.337.−3 1452 3.316
1.315.−3 1449 3.290
1.296.−3 1447 3.266
1.280.−3 1447 3.244
1.265.−3 1447 3.223
700
v h s
1.724.−2 3082 6.037
1.157.−2 2965 5.770
7.986.−3 2821 5.494
5.416.−3 2635 5.179
2.630.−3 2233 4.554
2.038.−3 2084 4.308
1.831.−3 2021 4.192
1.716.−3 1986 4.116
1.639.−3 1962 4.058
1.589.−3 1946 4.012
1.536.−3 1931 3.972
800
v h s
2.195.−2 3386 6.444
1.575.−2 3325 6.252
1.201.−2 3261 6.086
9.512.−3 3193 5.934
6.391.−3 3047 5.654
4.576.−3 2895 5.397
3.496.−3 2734 5.175
2.866.−3 2648 4.998
2.484.−3 2567 4.864
2.239.−3 2508 4.761
2.072.−3 2465 4.701
900
v h s
2.590.−2 3649 6.755
1.899.−2 3609 6.587
1.483.−2 3568 6.449
1.207.−2 3526 6.327
8.619.−3 3440 6.119
6.581.−3 3354 5.940
5.257.−3 3269 5.780
4.348.−3 3188 5.637
3.704.−3 3113 5.510
3.454.−3 3049 5.399
2.907.−3 2995 5.305
1000
v h s
2.954.−2 3904 7.023
2.186.−2 3874 6.867
1.726.−2 3845 6.741
1.420.−2 3816 6.633
1.038.−2 3756 6.453
8.102.−3 3697 6.302
6.605.−3 3640 6.172
5.557.−3 3584 6.055
4.792.−3 3532 5.951
4.212.−3 3482 5.856
3.763.−3 3435 5.727
1500
v h s
0.0461 5202 8.074
0.0346 5198 7.936
0.0277 5186 7.827
0.0231 5180 7.738
0.0173 5171 7.597
0.0139 5157 7.484
0.0116 5144 7.391
0.00993 5133 7.310
0.00871 5120 7.239
0.00776 5108 7.176
0.00700 5095 7.118
2000
v h s
0.0619 6613 8.883
0.0465 6610 8.748
0.0372 6608 8.642
0.0311 6605 8.555
0.0234 6599 8.418
0.0188 6595 8.310
0.0157 6590 8.222
0.0135 6585 8.147
0.0119 6581 8.082
0.0106 6577 8.024
0.0096 6574 7.971
2500
v h s
0.0778 8269 9.610
0.0584 8269 9.468
0.0468 8269 9.358
0.0391 8268 9.270
0.0294 8267 9.129
0.0236 8265 9.020
0.0197 8261 8.930
0.0170 2856 8.854
0.0149 8250 8.788
0.0133 8244 8.730
0.0120 8240 8.677
TABLE 2-355 Density, Specific Heats at Constant Pressure and at Constant Volume and Velocity of Sound for Compressed Water, 1–1000 bar, 0–150°C 0°C (ITS-90)
10°C (ITS-90)
20°C (ITS-90)
30°C (ITS-90)
P, bar
density, kg/m3
Cp, kJ/ (kg⋅K)
Cv, kJ/ (kg⋅K)
w, m/s
density, kg/m3
Cp, kJ/(kg⋅K)
Cv, kJ/(kg⋅K)
w, m/s
density, kg/m3
Cp, kJ/(kg⋅K)
Cv, kJ/(kg⋅K)
w, m/s
density, kg/m3
Cp, kJ/(kg⋅K)
Cv, kJ/(kg⋅K)
w, m/s
1 50 100 150 200 250 300 400 500 600 800 1000
999.702 1002.03 1004.38 1006.71 1009.01 1011.28 1013.53 1017.97 1022.31 1026.57 1034.85 1042.83
4.1923 4.174 4.156 4.139 4.123 4.108 4.093 4.065 4.040 4.018 3.979 3.948
4.1877 4.168 4.149 4.130 4.112 4.095 4.078 4.046 4.016 3.988 3.937 3.892
1447.3 1455 1464 1472 1480 1489 1497 1513 1529 1545 1577 1609
998.207 1000.44 1002.69 1004.93 1007.13 1009.32 1011.48 1015.74 1019.92 1024.02 1031.99 1039.68
4.1812 4.166 4.151 4.137 4.124 4.110 4.098 4.074 4.052 4.032 3.996 3.967
4.1538 4.137 4.119 4.103 4.087 4.071 4.056 4.027 3.999 3.974 3.926 3.884
1482.3 1491 1499 1507 1516 1524 1532 1548 1565 1581 1613 1644
995.650 997.82 1000.02 1002.19 1004.34 1006.47 1008.57 1012.72 1016.79 1020.79 1028.56 1036.06
4.1774 4.164 4.151 4.139 4.127 4.115 4.104 4.083 4.063 4.044 4.011 3.982
4.1148 4.099 4.084 4.069 4.055 4.041 4.027 4.001 3.976 3.952 3.908 3.869
1509.1 1517 1526 1534 1543 1551 1559 1576 1592 1608 1640 1671
992.217 994.36 996.52 998.66 1000.77 1002.87 1004.94 1009.03 1013.03 1016.97 1024.62 1032.00
4.1775 4.166 4.154 4.142 4.131 4.121 4.110 4.091 4.072 4.055 4.023 3.995
4.0715 4.058 4.044 4.031 4.018 4.005 3.993 3.969 3.946 3.924 3.884 3.847
1528.9 1537 1546 1554 1563 1571 1579 1596 1612 1628 1660 1692
P, bar
density, kg/m3
Cp, kJ/ (kg⋅K)
Cv, kJ/ (kg⋅K)
w, m/s
density, kg/m3
Cp, kJ/(kg⋅K)
Cv, kJ/(kg⋅K)
w, m/s
density, kg/m3
Cp, kJ/(kg⋅K)
Cv, kJ/(kg⋅K)
w, m/s
density, kg/m3
Cp, kJ/ (kg⋅K)
Cv, kJ/(kg⋅K)
w, m/s
1 50 100 150 200 250 300 400 500 600 800 1000
988.036 990.16 992.31 994.44 996.54 998.62 1000.68 1004.74 1008.72 1012.62 1020.21 1027.53
4.1799 4.169 4.158 4.147 4.137 4.126 4.117 4.098 4.080 4.063 4.033 4.005
4.0248 4.012 4.000 3.988 3.976 3.965 3.954 3.932 3.911 3.892 3.854 3.820
1542.6 1551 1560 1568 1577 1585 1594 1610 1627 1643 1676 1707
983.197 985.33 987.48 989.61 991.71 993.80 995.86 999.92 1003.90 1007.80 1015.38 1022.69
4.1840 4.173 4.163 4.152 4.142 4.132 4.123 4.105 4.087 4.071 4.041 4.013
3.9755 3.964 3.953 3.943 3.932 3.922 3.911 3.892 3.873 3.855 3.821 3.789
1551.0 1560 1568 1577 1586 1594 1603 1620 1637 1653 1686 1718
977.766 979.92 982.09 984.23 986.36 988.46 990.53 994.62 998.62 1002.54 1010.15 1017.48
4.1896 4.179 4.169 4.158 4.148 4.139 4.129 4.111 4.094 4.078 4.048 4.020
3.9246 3.915 3.905 3.895 3.885 3.876 3.867 3.849 3.832 3.815 3.784 3.754
1554.8 1564 1573 1582 1590 1599 1608 1625 1642 1659 1693 1726
971.791 973.98 976.18 978.35 980.51 982.63 984.74 988.87 992.92 996.88 1004.56 1011.94
4.1967 4.186 4.176 4.165 4.155 4.146 4.136 4.118 4.101 4.085 4.054 4.027
3.8727 3.864 3.855 3.846 3.838 3.829 3.821 3.805 3.789 3.774 3.745 3.717
1554.5 1564 1573 1582 1591 1600 1609 1627 1644 1662 1696 1730
50°C (ITS-90)
40°C (ITS-90)
70°C (ITS-90)
60°C (ITS-90)
Prepared by H. Sato, Keio University, Japan, Oct. 1994. Based upon “An equation of state for the thermodynamic properties of water in the liquid phase including the metastable state,” from Properties of Water and Steam,” Proc. 11th Int. Conf. Props. Steam (M. Pichal and O. Sifner, eds.), Hemisphere, New York, 1990 (551 pp.).
2-311
TABLE 2-355 Density, Specific Heats at Constant Pressure and at Constant Volume and Velocity of Sound for Compressed Water, 1–1000 bar, 0–150°C (Concluded ) 80°C (ITS-90)
90°C (ITS-90)
100°C (ITS-90)
110°C (ITS-90)
P, bar
density, kg/m3
Cp, kJ/ (kg⋅K)
Cv, kJ/ (kg⋅K)
w, m/s
density, kg/m3
Cp, kJ/(kg⋅K)
Cv, kJ/(kg⋅K)
w, m/s
density, kg/m3
Cp, kJ/(kg⋅K)
Cv, kJ/(kg⋅K)
w, m/s
density, kg/m3
Cp, kJ/(kg⋅K)
Cv, kJ/(kg⋅K)
w, m/s
1 50 100 150 200 250 300 400 500 600 800 1000
965.309 967.54 969.79 972.00 974.20 976.36 978.50 982.71 986.82 990.83 998.62 1006.08
4.2056 4.195 4.184 4.174 4.164 4.154 4.144 4.126 4.108 4.092 4.061 4.033
3.8206 3.813 3.805 3.797 3.789 3.782 3.774 3.759 3.745 3.731 3.704 3.678
1550.5 1560 1569 1579 1588 1597 1607 1625 1643 1661 1696 1731
958.348 960.64 962.94 965.21 967.45 969.67 971.85 976.15 980.34 984.43 992.34 999.92
4.2164 4.205 4.194 4.184 4.173 4.163 4.153 4.135 4.117 4.100 4.068 4.039
3.7689 3.762 3.755 3.748 3.741 3.734 3.727 3.714 3.701 3.688 3.663 3.639
1543.1 1553 1563 1572 1582 1592 1601 1620 1639 1658 1694 1730
950.927 953.28 955.65 957.99 960.30 962.57 964.82 969.21 973.50 977.69 985.76 993.47
4.2296 4.218 4.206 4.195 4.184 4.174 4.164 4.144 4.126 4.108 4.076 4.046
3.7181 3.712 3.706 3.699 3.693 3.687 3.681 3.669 3.657 3.645 3.621 3.598
1532.5 1543 1553 1563 1573 1583 1593 1613 1632 1652 1690 1727
943.059 945.50 947.95 950.36 952.74 955.09 957.40 961.92 966.32 970.61 978.87 986.73
4.2453 4.233 4.221 4.209 4.198 4.187 4.176 4.155 4.136 4.118 4.084 4.053
3.6684 3.663 3.657 3.652 3.646 3.641 3.635 3.624 3.613 3.602 3.579 3.556
1519.0 1530 1540 1551 1561 1572 1582 1603 1623 1644 1683 1723
P, bar
density, kg/m3
Cp, kJ/ (kg⋅K)
Cv, kJ/ (kg⋅K)
w, m/s
density, kg/m3
Cp, kJ/(kg⋅K)
Cv, kJ/(kg⋅K)
w, m/s
density, kg/m3
Cp, kJ/(kg⋅K)
Cv, kJ/(kg⋅K)
w, m/s
density, kg/m3
Cp, kJ/(kg⋅K)
Cv, kJ/(kg⋅K)
w, m/s
1 50 100 150 200 250 300 400 500 600 800 1000
934.749 937.28 939.83 942.33 944.79 947.22 949.61 954.27 958.81 963.22 971.68 979.72
4.2639 4.251 4.238 4.225 4.213 4.201 4.190 4.168 4.148 4.128 4.092 4.059
3.6201 3.615 3.610 3.605 3.600 3.595 3.590 3.580 3.569 3.558 3.536 3.512
1502.8 1514 1525 1536 1547 1558 1569 1591 1612 1634 1676 1717
925.997 928.64 931.29 933.90 936.46 938.97 941.45 946.28 950.96 955.51 964.20 972.44
4.2859 4.271 4.257 4.244 4.231 4.218 4.206 4.183 4.161 4.140 4.101 4.066
3.5733 3.569 3.564 3.560 3.555 3.550 3.545 3.535 3.525 3.514 3.491 3.466
1484.1 1496 1508 1519 1531 1543 1554 1577 1600 1623 1667 1710
916.797 919.57 922.34 925.06 927.73 930.35 932.92 937.94 942.79 947.48 956.43 964.88
4.3114 4.296 4.280 4.266 4.251 4.238 4.224 4.199 4.176 4.153 4.112 4.073
3.5279 3.524 3.520 3.516 3.511 3.506 3.501 3.491 3.480 3.469 3.444 3.417
1463.0 1475 1488 1501 1513 1525 1538 1562 1586 1610 1658 1704
907.143 910.06 912.97 915.82 918.61 921.34 924.03 929.24 934.27 939.13 948.36 957.04
4.3408 4.324 4.307 4.291 4.276 4.261 4.246 4.219 4.194 4.169 4.124 4.081
3.4848 3.481 3.477 3.473 3.469 3.464 3.459 3.448 3.436 3.423 3.395 3.364
1439.8 1453 1467 1480 1493 1507 1520 1546 1572 1598 1648 1698
120°C (ITS-90)
2-312
140°C (ITS-90)
130°C (ITS-90)
150°C (ITS-90)
TABLE 2-356
Specific Heat and Other Thermophysical Properties of Water Substance* Temperature, K
Pressure, bar
300
350
400
450
500
600
700
800
900
1000
1200
1400
1600
1800
µ 1 cp k Pr
8.57.−4 4.18 0.614 5.81
3.70.−4 4.19 0.668 2.32
1.32.−5 1.99 0.0268 0.980
1.52.−5 1.97 0.0311 0.967
1.73.−5 1.98 0.0358 0.955
2.15.−5 2.02 0.0464 0.936
2.57.−5 2.09 0.0581 0.920
2.98.−5 2.15 0.0710 0.906
3.39.−5 2.22 0.0843 0.891
3.78.−5 2.29 0.0981 0.881
4.48.−5 2.43 0.13 0.83
5.06.−5 2.58 0.16 0.80
5.65.−5 2.73 0.21 0.75
6.19.−5 3.02 0.33 0.57
6.70.−5 3.79 0.57 0.45
5
µ cp k Pr
8.57.−4 4.18 0.614 5.82
3.70.−4 4.19 0.668 2.32
2.17.−4 4.26 0.689 1.34
1.49.−5 2.21 0.0335 0.983
1.72.−5 2.10 0.0369 0.973
2.15.−5 2.07 0.0469 0.947
2.57.−5 2.11 0.0585 0.925
2.98.−5 2.16 0.0713 0.907
3.39.−5 2.23 0.0846 0.892
3.78.−5 2.29 0.0984 0.881
4.45.−5 2.43 0.13 0.83
5.06.−5 2.58 0.16 0.81
5.65.−5 2.73 0.20 0.77
6.19.−5 2.98 0.28 0.65
6.70.−5 3.40 0.43 0.53
10
µ cp k Pr
8.57.−4 4.18 0.615 5.82
3.70.−4 4.19 0.668 2.32
2.17.−4 4.25 0.689 1.34
1.51.−4 4.39 0.677 0.981
1.71.−5 2.29 0.0380 1.028
2.15.−5 2.13 0.0474 0.963
2.58.−5 2.13 0.0590 0.931
2.99.−5 2.18 0.0717 0.908
3.39.−5 2.24 0.0851 0.892
3.78.−5 2.30 0.0988 0.881
4.45.−5 2.44 0.13 0.84
5.06.−5 2.58 0.16 0.82
5.65.−5 2.73 0.20 0.78
6.19.−5 2.95 0.26 0.70
6.70.−5 3.29 0.39 0.57
20
µ cp k Pr
8.56.−4 4.17 0.616 5.80
3.71.−4 4.19 0.669 2.32
2.18.−4 4.25 0.689 1.34
1.51.−4 4.39 0.679 0.979
1.68.−5 2.84 0.0402 1.19
2.15.−5 2.26 0.0485 0.999
2.59.−5 2.19 0.0599 0.946
3.00.−5 2.21 0.0726 0.912
3.40.−5 2.26 0.0859 0.893
3.79.−5 2.32 0.0996 0.881
4.46.−5 2.45 0.13 0.84
5.06.−5 2.59 0.16 0.82
5.65.−5 2.73 0.20 0.79
6.19.−5 2.92 0.25 0.72
6.70.−5 3.21 0.36 0.60
40
µ cp k Pr
8.55.−4 4.17 0.617 5.78
3.71.−4 4.19 0.671 2.31
2.18.−4 4.25 0.690 1.34
1.52.−4 4.38 0.680 0.977
1.19.−4 4.65 0.644 0.862
2.15.−5 2.60 0.516 1.08
2.61.−5 2.32 0.0620 0.975
3.02.−5 2.28 0.0744 0.924
3.42.−5 2.30 0.0877 0.895
3.80.−5 2.34 0.101 0.881
4.47.−5 2.46 0.13 0.84
5.07.−5 2.59 0.16 0.82
5.65.−5 2.73 0.19 0.80
6.19.−5 2.90 0.24 0.73
6.70.−5 3.14 0.33 0.63
60
µ cp k Pr
8.54.−4 4.16 0.619 5.74
3.72.−4 4.18 0.672 2.31
2.19.−4 4.24 0.692 1.34
1.53.−4 4.37 0.682 0.976
1.20.−4 4.63 0.646 0.859
2.14.−5 3.11 0.0561 1.19
2.63.−5 2.47 0.0645 1.008
3.04.−5 2.35 0.0764 0.934
3.43.−5 2.34 0.0895 0.899
3.82.−5 2.37 0.103 0.879
4.48.−5 2.48 0.13 0.84
5.07.−5 2.60 0.16 0.82
5.66.−5 2.73 0.19 0.81
6.19.−5 2.89 0.24 0.74
6.70.−5 3.11 0.32 0.65
80
µ cp k Pr
8.53.−4 4.16 0.620 5.72
3.72.−4 4.18 0.674 2.31
2.19.−4 4.24 0.693 1.34
1.53.−4 4.36 0.684 0.975
1.20.−4 4.62 0.648 0.856
2.14.−5 3.88 0.0628 1.33
2.66.−5 2.65 0.0672 1.046
3.06.−5 2.43 0.0785 0.946
3.45.−5 2.39 0.0914 0.902
3.83.−5 2.40 0.105 0.877
4.48.−5 2.49 0.13 0.84
5.08.−5 2.61 0.16 0.83
5.66.−5 2.73 0.10 0.81
6.19.−5 2.88 0.24 0.74
6.70.−5 3.09 0.31 0.66
100
µ cp k Pr
8.52.−4 4.15 0.622 5.69
3.73.−4 4.17 0.675 2.31
2.20.−4 4.23 0.694 1.34
1.53.−4 4.35 0.685 0.975
1.21.−4 4.60 0.651 0.853
2.14.−5 5.22 0.0730 1.74
2.69.−5 2.85 0.0704 1.088
3.08.−5 2.52 0.0807 0.960
3.47.−5 2.44 0.0934 0.905
3.85.−5 2.44 0.107 0.876
4.49.−5 2.50 0.13 0.84
5.08.−5 2.62 0.16 0.83
5.66.−5 2.73 0.19 0.81
6.19.−5 2.88 0.24 0.74
6.70.−5 3.08 0.31 0.67
µ cp k Pr
8.51.−4 4.14 0.624 5.64
3.74.−4 4.16 0.678 2.30
2.22.−4 4.22 0.699 1.34
1.56.−4 4.34 0.693 0.974
1.22.−4 4.54 0.657 0.842
8.22.−5
150
2.72.−5 3.55 0.079 1.22
3.12.−5 2.74 0.086 0.994
3.51.−5 2.57 0.098 0.916
3.89.−5 2.53 0.110 0.891
4.52.−5 2.54 0.14 0.84
5.09.−5 2.65 0.16 0.83
5.67.−5 2.75 0.19 0.82
6.19.−5 2.88 0.23 0.76
6.70.−5 3.06
µ cp k Pr
8.50.−4 4.12 0.626 5.59
3.75.−4 4.15 0.681 2.29
2.24.−4 4.21 0.702 1.34
1.57.−4 4.32 0.697 0.974
1.23.−4 4.51 0.661 0.833
2.80.−5 4.67 0.095 1.38
3.17.−5 3.04 0.095 1.014
3.54.−5 2.71 0.104 0.925
3.93.−5 2.62 0.113 0.903
4.54.−5 2.57 0.14 0.84
5.11.−5 2.67 0.16 0.83
5.67.−5 2.76 0.19 0.82
2.88
3.05
250
µ cp k Pr
8.49.−4 4.12 0.627 5.57
3.76.−4 4.14 0.683 2.28
2.26.−4 4.20 0.705 1.34
1.59.−4 4.30 0.701 0.974
1.23.−4 4.49 0.672 0.826
8.41.−5 5.90 0.537 0.924
2.89.−5 6.16 0.112 1.590
3.24.−5 3.40 0.103 1.070
3.59.−5 2.86 0.110 0.940
3.98.−5 2.71 0.119 0.910
4.56.−5 2.61 0.136 0.85
5.12.−5 2.69 0.16 0.84
5.68.−5 2.77
2.89
3.04
300
µ cp k Pr
8.49.−4 4.10 0.629 5.53
3.77.−4 4.13 0.685 2.27
2.28.−4 4.19 0.708 1.34
1.60.−4 4.29 0.704 0.973
1.24.−4 4.44 0.675 0.820
8.50.−5 5.60 0.548 0.859
3.7.−5 10.20 0.173 2.18
3.4.−5 3.82 0.113 1.149
3.64.−5 3.03 0.113 0.976
4.02.−5 2.81 0.123 0.917
4.59.−5 2.65 0.14 0.87
5.14.−5 2.72
5.68.−5 2.78
2.90
3.04
400
µ cp k Pr
8.49.−4 4.08 0.631 5.49
3.80.−4 4.12 0.689 2.26
2.30.−4 4.16 0.714 1.34
1.62.−4 4.26 0.710 0.971
1.26.−4 4.42 0.676 0.817
8.64.−5 5.31 0.567 0.799
5.3.−5 13.20 0.327 2.14
3.6.−5 4.86 0.145 1.207
3.8.−5 3.39 0.129 0.999
4.1.−5 3.01 0.134 0.926
4.6.−5 2.70 0.15
5.17.−5 2.77
2.81
2.91
3.04
500
µ cp k Pr
8.50.−4 4.06 0.634 5.44
3.82.−4 4.10 0.695 2.25
2.31.−4 4.15 0.719 1.33
1.64.−4 4.23 0.717 0.971
1.28.−4 4.38 0.693 0.814
8.83.−5 5.08 0.583 0.773
5.8.−5 8.44 0.378 1.30
4.0.−5 5.70 0.186 1.225
4.0.−5 3.90 0.147 1.061
4.2.−5 3.21 0.145 0.932
4.7.−5 2.77
2.81
2.84
2.92
3.04
200
0.520 8.32.−5 0.525
2000
2-313
TABLE 2-356
Specific Heat and Other Thermophysical Properties of Water Substance (Concluded ) Temperature, K
Pressure, bar
300
350
400
450
500
600
700
800
900
1000
µ 600 cp k Pr
8.51.−4 4.04 0.639 5.38
3.85.−4 4.08 0.699 2.24
2.32.−4 4.13 0.725 1.32
1.66.−4 4.20 0.725 0.970
1.30.−4 4.33 0.700 0.812
9.17.−5 4.92 0.597 0.755
6.5.−5 6.93 0.420 1.073
4.4.−5 6.83 0.239 1.175
4.2.−5 4.19 0.170 1.035
4.4.−5 3.38 0.159 0.935
2.87
2.86
2.86
2.92
3.04
700
µ cp k Pr
8.52.−4 4.01 0.644 5.33
3.87.−4 4.07 0.706 2.23
2.33.−4 4.12 0.730 1.32
1.69.−4 4.17 0.732 0.970
1.33.−4 4.29 0.707 0.810
9.50.−5 4.78 0.614 0.739
6.9.−5 6.12 0.442 1.047
4.9.−5 6.26 0.279 1.098
4.5.−5 4.62 0.198 1.010
4.6.−5 3.59 0.177 0.935
2.94
2.91
2.88
2.93
3.05
800
µ cp k Pr
8.53.−4 3.99 0.648 5.28
3.90.−4 4.05 0.709 2.23
2.34.−4 4.10 0.735 1.31
1.72.−4 4.15 0.736 0.970
1.36.−4 4.26 0.714 0.808
9.82.−5 4.67 0.625 0.725
7.3.−5 5.60 0.478 0.855
5.4.−5 6.09 0.320 1.028
4.8.−5 4.77 0.228 1.003
4.8.−5 3.75 0.193 0.933
3.01
2.96
2.91
2.95
3.05
900
µ cp k Pr
8.54.−4 3.98 0.651 5.23
3.93.−4 4.03 0.713 2.22
2.35.−4 4.08 0.738 1.30
1.74.−4 4.13 0.742 0.969
1.38.−4 4.23 0.724 0.806
1.00.−4 4.57 0.636 0.712
7.6.−5 5.29 0.496 0.810
5.8.−5 5.86 0.351 0.968
5.1.−5 4.85 0.260 0.950
5.0.−5 3.86 0.210 0.919
3.08
3.00
2.94
2.97
3.06
1000
µ cp k Pr
8.56.−4 3.97 0.653 5.19
3.96.−4 4.02 0.717 2.22
2.36.−4 4.06 0.743 1.30
1.76.−4 4.11 0.747 0.968
1.40.−4 4.20 0.731 0.804
1.02.−4 4.47 0.650 0.701
7.9.−5 5.08 0.516 0.778
6.2.−5 5.51 0.372 0.918
5.4.−5 4.88 0.288 0.900
5.1.−5 3.96 0.228 0.886
3.16
3.05
2.97
2.98
3.07
*µ = viscosity, Ns/m2; cp = specific heat at constant pressure, kJ/(kg⋅K); k = thermal conductivity, W/(m⋅K); Pr = Prandtl number.
2-314
1200
1400
1600
1800
2000
THERMODYNAMIC PROPERTIES TABLE 2-357
Thermodynamic Properties of Water Substance along the Melting Line T, °C
103 v f , m3/kg
h f , kJ/kg
s f, kJ/kg⋅K
cpf, kJ/kg⋅K
cmelt, kJ/kg⋅K
106α f, K−1
106K f,T bar−1
6.117.−5t 1.01325 50 100 150
0.0100 0.0026 −0.3618 −0.7410 −1.1249
1.00021 1.00016 0.99770 0.99523 0.99278
0 0.0719 3.5140 6.9794 10.3964
0 −0.0001 −0.0054 −0.0110 −0.0167
4.219 4.218 4.196 4.174 4.152
3.969 3.970 3.997 4.023 4.047
−67.42 −67.17 −54.92 −42.52 −30.24
50.90 50.88 50.30 49.73 49.17
200 250 300 400 500
−1.5166 −1.9151 −2.3206 −3.1532 −4.0156
0.99037 0.98798 0.98562 0.98098 0.97643
13.7648 17.0843 20.3547 26.7472 32.9403
−0.0225 −0.0285 −0.0347 −0.0474 −0.0607
4.132 4.112 4.092 4.056 4.022
4.070 4.092 4.113 4.150 4.184
−18.05 −5.93 6.12 30.09 53.97
48.63 48.11 47.59 46.61 45.68
600 800 1000
−4.909 −6.790 −8.803
0.97196 0.96326 0.95493
38.932 50.300 60.836
−0.0747 −0.1046 −0.1371
3.992 3.937 3.893
4.215 4.270 4.320
77.87 126.18 175.98
44.80 43.19 41.74
P, bar
2-315
Condensed from U. Grigull, Private communication, January 18, 1995. Materials prepared at Technical University München, Germany by U. Grigull and S. Marek. For a table as a function of temperature, see Grigull, U. and S. Marek, Warme u. Stoff., 30 (1994): 1–8. t = the triple point (at 6.117 × 10−5 bar, 0.01°C); vf = 0.0010021 m3/kg: α f = −67.42 × 10−6/K. Other equations for properties are given by Jones, F. E. and G. L. Harris, J. Res. N.I.S.T., 97, 3 (1992): 335–340, and by Wagner, W. and A. Pruss, J. Phys. Chem. Ref. Data, 22, 3 (1993): 783–787. Steam tables include Walker, W. A., U.S. Naval Ordn. Lab. rept. NOLTR NOLTR-66-217 = AD 651105 (0–1000 bar, 0–150°C), 1967 (72 pp.); Grigull, U., J. Straub, et al., Steam Tables in S.I. Units (0.01–1000 bar, 0–1000°C), Springer-Verlag, Berlin, 1990 (133 pp.); Tseng, C. M., T. A. Hamp, et al., Atomic Energy of Canada rept. (30 props, sat liq & vap., 1–220 bar), AECL-5910 1977 (90 pp.). For dissociation, see e.g., Knonicek, V., Rozpr. Cesko Acad Ved., Rada techn ved (0.01–100 bar, 1000–5000 K). 77, 1 (1967). The proceedings of the 10th international conference on the properties of steam were edited by Sytchev, V. V. and A. A. Aleksandrov, Plenum, NY, 1984; and for the 11th conference by Pichal, M. and O. Sifner, Hemisphere, 1989 (550 pp.). For electrical conductivity, see e.g., Marshall, W. L., J. Chem. Eng. Data, 32 (1987): 221–226.
TABLE 2-358 T, K
Saturated Xenon*
P, bar
v f, m3/kg
vg, m3/kg
h f, kJ/kg
hg, kJ/kg
s f, kJ/(kg⋅K)
s g, kJ/(kg⋅K)
cpf, kJ/(kg⋅K)
µ f, 10−4 Pa⋅s
k f, W/(m⋅K)
10 20 30 40 50
2.642.−4 2.650.−4 2.661.−4 2.675.−4 2.689.−4
0.19 1.21 2.74 4.47 6.31
0.0236 0.0901 0.1510 0.2003 0.2410
0.058 0.133 0.164 0.178 0.186
60 80 100 120 140
2.704.−4 2.737.−4 2.776.−4 2.820.−4 2.874.−4
8.21 12.14 16.30 20.81 25.67
0.2755 0.3319 0.3783 0.4197 0.4581
0.191 0.202 0.214 0.231 0.251
160 161.4m 161.4m 170 180
0.816 1.336 2.218
2.941.−4 2.946.−4 3.372.−4 3.439.−4 3.523.−4
0.1219 0.0776 0.0487
30.94 31.30 48.98 52.01 55.52
145.5 146.5 147.5
0.4946 0.4969 0.6072 0.6253 0.6452
1.206 1.181 1.156
0.270 0.271 0.350 0.349 0.349
4.50 3.99
0.0707 0.0663
190 200 210 220 230
3.480 5.212 7.504 10.45 14.16
3.615.−4 3.715.−4 3.828.−4 3.955.−4 4.100.−4
0.0321 0.0220 0.0156 0.0113 0.0084
59.04 62.61 66.25 70.00 73.91
148.3 148.9 149.2 149.4 149.2
0.6641 0.6820 0.6994 0.7163 0.7330
1.134 1.113 1.095 1.077 1.060
0.352 0.357 0.365 0.379 0.400
3.51 3.09 2.71 2.39 2.09
0.0622 0.0582 0.0542 0.0506 0.0468
240 250 260 270 280
18.72 24.25 30.87 38.69 47.86
4.271.−4 4.476.−4 4.730.−4 5.079.−4 5.689.−4
0.0063 0.0047 0.0036 0.0027 0.0019
78.05 82.54 87.52 93.30 100.6
148.5 147.5 145.7 142.8 138.0
0.7498 0.7671 0.7855 0.8058 0.8308
1.044 1.027 1.009 0.989 0.964
0.432 0.482 0.560 0.685 0.995
1.83 1.60 1.38 1.18 0.95
0.0429 0.0393 0.0355 0.0313 0.0275
289.7 c
58.21
9.091.−4
0.0009
120.0
120.0
0.8962
0.896
∞
∞
*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon, Standards Press, Moscow, 1976. This source contains values for the compressed state for pressures up to 1000 bar, etc. m = melting point; c = critical point. The notation 2.642.−4 signifies 2.642 × 10−4. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).
2-316
PHYSICAL AND CHEMICAL DATA
TABLE 2-359
Compressed Xenon* Pressure, bar
T, K 100 v h s
1 2.776.−4 16.30 0.3783
100
200
300
400
500
600
700
2.764.−4 18.84 0.3762
2.752.−4 21.40 0.3742
2.742.−4 23.95 0.3723
2.731.−4 26.50 0.3704
2.721.−4 29.05 0.3686
2.711.−4 31.59 0.3669
2.702.−4 34.13 0.3652
2.693.−4 36.67 0.3636
800
2.684.−4 39.21 0.3621
900
2.675.−4 41.74 0.3802
1000
200 v h s
0.1245 151.8 1.228
3.623.−4 64.22 0.6727
3.547.−4 66.14 0.6643
3.484.−4 68.19 0.6570
3.430.−4 70.34 0.6505
3.383.−4 72.56 0.6446
3.342.−4 74.83 0.6391
3.304.−4 77.13 0.6340
3.270.−4 79.46 0.6292
3.240.−4 81.81 0.6247
3.211.−4 84.18 0.6204
300 v h s
0.1890 168.0 1.294
5.729.−4 106.4 0.8401
4.769.−4 101.6 0.8073
4.431.−4 101.3 0.7908
4.220.−4 102.0 0.7789
4.068.−4 103.3 0.7691
3.955.−4 104.9 0.7608
3.862.−4 106.7 0.7540
3.783.−4 108.5 0.7477
3.716.−4 110.6 0.7424
3.657.−4 112.8 0.7370
400 v h s
0.2527 183.9 1.340
1.998.−3 164.2 1.012
8.759.−4 145.4 0.9330
6.452.−4 137.4 0.8945
5.604.−4 134.7 0.8730
5.141.−4 134.1 0.8581
4.839.−4 134.5 0.8467
4.622.−4 135.5 0.8373
4.457.−4 136.8 0.8292
4.325.−4 138.3 0.8220
4.217.−4 140.0 0.8162
500 v h s
0.3163 199.8 1.375
2.899.−3 187.8 1.065
1.389.−3 177.1 1.004
9.449.−4 169.4 0.9664
7.577.−4 165.1 0.9409
6.593.−4 163.0 0.9228
5.986.−4 162.3 0.9088
5.570.−4 162.4 0.8975
5.268.−4 163.1 0.8881
5.038.−4 164.3 0.8801
4.859.−4 165.7 0.8731
600 v h s
0.3798 215.7 1.404
3.673.−3 207.4 1.101
1.823.−3 200.3 1.047
1.240.−3 194.8 1.013
9.699.−4 191.1 0.9885
8.206.−4 188.9 0.9700
7.273.−4 187.9 0.9555
6.636.−4 187.6 0.9435
6.172.−4 188.0 0.9334
5.820.−4 188.8 0.9247
5.545.−4 189.9 0.9172
700 v h s
0.4432 231.5 1.428
4.397.−3 225.6 1.129
2.217.−3 220.6 1.078
1.513.−3 216.7 1.047
1.175.−3 213.8 1.023
9.815.−4 212.2 1.006
8.583.−4 211.3 0.9916
7.734.−4 211.1 0.9797
7.115.−4 211.3 0.9695
6.642.−4 212.0 0.9606
6.268.−4 213.1 0.9528
800 v h s
0.5066 247.4 1.450
5.093.−3 243.0 1.152
2.587.−3 239.5 1.103
1.769.−3 236.7 1.073
1.370.−3 234.8 1.052
1.137.−3 233.6 1.035
9.870.−4 233.0 1.021
8.824.−4 232.9 1.009
8.057.−4 233.3 0.9988
7.469.−4 234.0 0.9901
7.005.−4 235.0 0.9823
900 v h s
0.5700 263.2 1.468
5.773.−3 260.1 1.172
2.944.−3 257.5 1.125
2.014.−3 255.7 1.096
1.557.−3 254.4 1.075
1.288.−3 253.6 1.058
1.112.−3 253.4 1.045
9.893.−4 253.7 1.033
8.989.−4 254.2 1.023
8.289.−4 254.9 1.015
7.737.−4 256.1 1.007
1000 v h s
0.6333 279.1 1.485
6.441.−3 276.8 1.190
3.291.−3 275.1 1.143
2.252.−3 273.9 1.115
1.738.−3 273.2 1.095
1.435.−3 272.9 1.079
1.235.−3 273.0 1.065
1.094.−3 273.4 1.054
9.899.−4 274.1 1.044
9.097.−4 275.1 1.036
8.461.−4 276.2 1.028
*Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, Argon, Krypton and Xenon. Standards Press, Moscow, 1976. This source contains an exhaustive tabulation of values. v = specific volume, m3/kg; h = specific enthalpy, kJ/kg; s = specific entropy, kJ/(kg⋅K). The notation 2.776.−4 signifies 2.776 × 10−4. This book was published in English translation by Hemisphere, New York, 1988 (604 pp.).
THERMODYNAMIC PROPERTIES
2-317
TABLE 2-360
Surface Tension (N/m) of Saturated Liquid Refrigerants*
R no.
−50
−25
0
25
50
75
100
125
150
0.0279 0.0188 0.0092 0.0197 0.0115
0.0244 0.0152 0.0056 0.0156 0.0065
0.0210 0.0118 0.0025 0.0117 0.0025
0.0178 0.0085 0.0002 0.0081
0.0146 0.0055 — 0.0047 —
0.0116 0.0029 — 0.0018 —
0.0087 0.0007 — — —
0.0060 — — — —
0.0036 — — — —
—
0.0231 0.0154
0.0069 0.0172 0.0109 0.0047 0.0082
0.0032 0.0144 0.0082 0.0022 0.0050
0.0002 0.0118 0.0056
0.0192
0.0201 0.0138 0.0075 0.0117
0.0021
— 0.0092 0.0033 — 0.0000
— 0.0067 0.0012 — —
— 0.0045 — — —
142b 152a 170 290 C318
0.0213 0.0201 0.0100
0.0178 0.0166 0.0051
—
0.0143
0.0145 0.0132 0.0032 0.0101 0.0113
0.0113 0.0100 0.0005 0.0082 0.0085
0.0083 0.0068 — 0.0041 0.0048
0.0055 0.0038 — 0.0016 0.0033
0.0029 0.0011 — — 0.0011
— — — —
— — — — —
502 503 600 600a 718
0.0159 0.0094
0.0121 0.0053 0.0180
0.0054 — 0.0122 0.0101 0.0720
0.0026 — 0.0094 0.0073 0.0680
— 0.0068 0.0047 0.0636
— — 0.0043 0.0024 0.0590
— — 0.0020 0.0005 0.0540
— — 0.0001 — 0.0488
0.0005
—
—
—
—
—
0.0070
0.0041
0.0014
Temperature, °C 11 12 13 22 23 32 113 114 115 134a
744 1150 1270
—
—
0.0086 0.0018 0.0150 0.0132 0.0755
0.0100 0.0171
0.0096 0.0055 0.0136
0.0044 0.0013 0.0102
*Dashes indicate inaccessible states; blanks indicate no available data. Values and equations were given by Srinivasan, K., Can. J. Chem. Eng. (27 liquids), 68 (1990): 493; Lielmezs, J. and T. A. Herrick, Chem. Eng. J. (34 liquids), 32 (1986): 165–169. Somayajulu, G. R., Int. J. Thermophys. (64 liquids); 9, 4 (1988): 559–566; Ibrahim, N. and S. Murad, Chem. Eng. Commun. (29 polar liquids), 79 (1979): 165–174; Yaws, C. L.; Morachevsky, A. G. and I. B. Sladkov, Physico-Chemical Properties of Molecular Inorganic Compounds (200 compounds), Khimiya, Leningrad, 1987, Jasper, J., J. Phys. Chem. Ref. Data (2200 compounds), 1, (1972): 841–1009; and Vargaftik, N. B., B. N. Volkov, et al., J. Phys. Chem. Ref. Data (water), 12, 3 (1983): 817–820. See also Escobedo, J. and Mansoori, G. R., AIChE J., 42(5), May 1996: 1425–1433.
TABLE 2-361
Velocity of Sound (m/s) in Gaseous Refrigerants at Atmospheric Pressure* Temperature, °C
−50
−25
0
25
50
75
100
125
150
11 12 13 14 22
— — 142 158 —
— 136 150 166 166
— 143 157 173 174
141 150 164 180 182
147 156 170 187 189
153 162 176 194 196
158 168 182 200 202
163 173 188 206 208
168 179 193 212 215
23 32 113 114 134a
179
188
197
205
212
220
227
234
240
— —
— — 146
— — 154
— 120 162
121 126 169
126 131 175
131 136 180
135 141 186
140 146 192
170 290 600 600a 718
272 — — — —
286 227 — — —
299 238 200 201 —
311 249 210 211 —
323 258 220 221 —
334 268 228 229 —
344 277 237 237 473
355 286 245 246 490
364 294 252 253 505
290 —
248 305 235
258 318 246
269 330 257
279 341 267
288 352 277
297 363 286
307 373 295
316 384 303
R. no.
744 1150 1270
*Dashes indicate inaccessible states; blanks indicate no available data.
2-318
PHYSICAL AND CHEMICAL DATA TABLE 3-362
Velocity of Sound (m/s) in Saturated Liquid Refrigerants* Temperature, °C
R. no.
−50
−25
0
25
50
75
100
125
150
11 12 13 14 22
933 829 602 182 899
843 695 444 — 790
772 564 302 — 682
705 434
639
569
493
— 571
— — 446
— — 319
— — —
408 — — — —
323 — — — —
538
348
191
—
—
— —
— —
— —
— 853
871 726
786 623 454
700 540 346
633 453 255
371
284 —
183 —
— —
858 1210 1290 1205 —
743 982 1163 1078 —
626 884 1031 947 1402
517 719 896 812 1495
387 551 759 661 1542
262 367 609 528 1554
105 — 477 378 1543
— — 325 208 1514
— — 142 — 1468
751 644 1022
525 372 859
272
—
—
—
—
—
874 1184
694
524
335
23 32 113 114 115 134a 290 600 600a 718 744 1150 1270
*Dashes indicate inaccessible states; blanks indicate no available data.
TRANSPORT PROPERTIES INTRODUCTION Extensive tables of the viscosity and thermal conductivity of air and of water or steam for various pressures and temperatures are given with the thermodynamic-property tables. The thermal conductivity and the viscosity for the saturated-liquid state are also tabulated for many fluids along with the thermodynamic-property tables earlier in this section. UNITS CONVERSIONS For this subsection the following units conversions are applicable: Diffusivity: to convert square centimeters per second to square feet per hour, multiply by 3.8750; to convert square meters per second to square feet per hour, multiply by 38,750. Pressure: to convert bars to pounds-force per square inch, multiply by 14.504. Temperature: °F = 9⁄5 °C + 32; °R = 9⁄5 K. Thermal conductivity: to convert watts per meter-kelvin to British thermal unit–feet per hour–square foot–degree Fahrenheit, multiply by 0.57779; and to convert British thermal unit–feet per hour–square foot–degree Fahrenheit to watts per meter-kelvin, multiply by 1.7307. Viscosity: to convert pascal-seconds to centipoises, multiply by 1000. ADDITIONAL REFERENCES An extensive coverage of the general pressure and temperature variation of thermal conductivity is given in the monograph by Vargaftik,
Filippov, Tarzimanov, and Totskiy, Thermal Conductivity of Liquids and Gases (in Russian), Standartov, Moscow, 1978, now published in English translation by CRC Press, Miami, FL. For a similar work on viscosity, see Stephan and Lucas, Viscosity of Dense Fluids, Plenum, New York and London, 1979. Tables and polynomial fits for refrigerants in both the gaseous and the liquid state are contained in ASHRAE Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Ventilating Engineers, Atlanta, GA, 1993. Other sources for viscosity include Fischer & Porter Co. catalog 10-A-94, “Fluid densities and viscosities,” 1953 (200 industrial fluids in 48 pp.) and van Velzen, D., R. L. Cardozo et al., EURATOM Ispra, Italy rept. 4735 e, 1972 (160 pp.). Liquid viscosity, 314 cpds, is summarized in I&EC Fundtls., 11 (1972): 20–26. Five hundred forty-nine binary and ternary systems are discussed in Skubla, P., Coll. Czech. Chem. Commun., 46 (1981): 303–339. See also Duhne, C. R., Chem. Eng. (NY), 86, 15 (July 16, 1979): 83–91 (equations and 326 liquids); and Rao, K. V. K., Chem. Eng. (NY), 90, 11 (May 30, 1983): 90–91 (nomograph, 87 liquids). For rheology, non-Newtonian behavior, and the like, see, for instance, Barnes, H., The Chem. Engr. (UK), (June 24, 1993): 17–23; Hyman, W. A., I&EC Fundtls., 16 (1976): 215–218; and Ferguson, J. and Z. Kemblowski, Applied Fluid Rheology, Elsevier, 1991 (325 pp.). Other sources for thermal conductivity include Ho, C. Y., R. W. Powell et al., J. Phys. Chem. Ref. Data, 1 (1972) and 3, suppl. 1 (1974); Childs, Ericks et al., N.B.S. Monogr. 131, 1973; Jamieson, D. T., J. B. Irving et al., Liquid Thermal Conductivity, H.M.S.O., Edinburgh, Scotland, 1975 (220 pp.).
TRANSPORT PROPERTIES TABLE 2-363
2-319
Transport Properties of Selected Gases at Atmospheric Pressure* Viscosity, 10−4 Pa⋅s Temperature, K
Thermal conductivity, W/(m⋅K) Temperature, K Substance
250
300
400
500
Acetone Acetylene Ammonia Argon Benzene
0.0080 0.0162 0.0197 0.0152 0.0077
0.0115 0.0213 0.0246 0.0177 0.0104
0.0201 0.0332 0.0364 0.0223 0.0195
0.0310 0.0452 0.0506 0.0264 0.0335
600
Bromine Butane CO2 CCl4 Chlorine
0.0038 0.0117 0.0129 0.0053 0.0071
0.0048 0.0160 0.0166 0.0067 0.0089
0.0067 0.0264 0.0244 0.0099 0.0124
0.0377 0.0323 0.0126 0.0156
Deuterium Ethane Ethylene Helium Heptane
0.122 0.0156 0.0152 0.134 0.0082
0.141 0.0218 0.0214 0.150 0.0120
0.176 0.0360 0.0342 0.180 0.0214
0.0516 0.0491 0.211 0.0325
0.0685 0.0653 0.247 0.0447
Hydrogen Methane Nitrogen Oxygen Pentane
0.156 0.0277 0.0222 0.0225 0.0107
0.182 0.0343 0.0260 0.0267 0.0152
0.221 0.0484 0.0325 0.0343 0.0250
0.256 0.0671 0.0386 0.0412 0.0362
0.291 0.0948 0.0441 0.0480
Propane Propylene R 11 R 12 R 13
0.0129 0.0114
0.0295 0.0226 0.0119 0.0151 0.0185
0.0417 0.0430
0.0072 0.0091
0.0183 0.0168 0.0078 0.0097 0.0121
0.0208 0.0248
R 21 R 22 SO2
0.0080 0.0078
0.0088 0.0109 0.0096
0.0135 0.0170 0.0143
0.0181 0.0230 0.0200
250
0.0561 0.0656 0.0301 0.0524
0.085 0.195
0.0403
0.126
0.0190
0.0580
0.0290 0.0256
Prandtl number, dimensionless Temperature, K
300
400
500
600
0.077 0.104 0.102 0.229 0.076
0.101 0.135 0.139 0.289 0.101
0.128 0.164 0.175 0.343 0.127
0.156
250
300
400
500
0.211 0.390 0.154
0.669
0.91 0.668
0.87 0.666
0.86 0.663
0.076 0.150 0.101 0.136
0.203 0.101 0.196 0.131 0.178
0.260 0.125 0.239 0.162 0.218
0.291 0.151 0.278 0.191 0.259
0.793
0.805 0.778
0.820 0.752
0.734
0.817 0.812 0.671
0.773 0.796 0.668
0.746 0.769 0.663
0.746 0.750 0.661
0.71 0.742 0.721
0.71 0.739 0.714
0.71 0.737 0.708
0.71 0.736 0.707
0.810 0.860
0.788 0.762 0.761 0.745 0.759
0.826
0.827 0.796
0.774 0.797 0.814 0.781 0.766
0.820
0.779 0.771
0.773 0.760
0.111 0.079 0.087 0.176
0.126 0.094 0.103 0.199
0.153 0.123 0.135 0.243 0.080
0.178 0.148 0.162 0.284 0.099
0.201 0.171 0.187 0.322 1.116
0.080 0.095 0.156 0.179
0.090 0.112 0.180 0.207
0.109 0.142 0.223 0.258
0.126 0.170 0.261 0.306
0.143 0.195 0.295 0.348
0.069 0.073 0.094 0.108 0.123
0.082 0.087 0.110 0.126 0.145
0.108 0.115 0.144 0.162 0.190
0.131 0.141
0.100 0.109
0.115 0.129 0.129
0.154 0.168 0.175
0.217
0.708 0.757
0.256
*An approximate interpolation scheme is to plot the logarithm of the viscosity or the thermal conductivity versus the logarithm of the absolute temperature. At 250 K the viscosity of gaseous argon is to be read as 1.95 × 10−5 Pa⋅s = 0.0000195 Ns/m2.
2-320
PHYSICAL AND CHEMICAL DATA
TABLE 2-364
Viscosities of Gases: Coordinates for Use with Fig. 2-32* µ × 107 p
Gas
X
Y
Acetic acid Acetone Acetylene Air Ammonia Amylene (β) Argon Arsine Benzene Bromine Butane (n) Butane (iso) Butyl acetate (iso) Butylene (α) Butylene (β) Butylene (iso) Butyl formate (iso) Cadmium Carbon dioxide Carbon disulfide Carbon monoxide Carbon oxysulfide Carbon tetrachloride Chlorine Chloroform Cyanogen Cyclohexane Cyclopropane Deuterium Diethyl ether Dimethyl ether Diphenyl ether Diphenyl methane Ethane Ethanol Ethyl acetate Ethyl chloride Ethylene Ethyl propionate Fluorine Freon-11 Freon-12 Freon-14 Freon-21 Freon-22 Freon-113 Freon-114 Helium Heptane (n) Hexane (n) Hydrogen Hydrogen-helium 10% H2, 90% He 25% H2, 75% He 40% H2, 60% He 60% H2, 40% He 81% H2, 19% He
7.0 8.4 9.3 10.4 8.4 8.6 9.7 8.6 8.7 8.8 8.6 8.6 5.7 8.4 8.7 8.3 6.6 7.8 8.9 8.5 10.5 8.2 8.0 8.8 8.8 8.2 9.0 8.3 11.0 8.8 9.0 8.6 8.0 9.0 8.2 8.4 8.5 9.5 12.0 7.3 8.6 9.0 9.5 9.0 9.0 11.0 9.4 11.3 8.6 8.4 11.3
14.6 13.2 15.5 20.4 16.0 12.2 22.6 20.0 13.2 19.4 13.2 13.2 16.3 13.5 13.1 13.9 16.0 22.5 19.1 15.8 20.0 17.9 15.3 18.3 15.7 16.2 12.2 14.7 16.2 12.7 15.0 10.4 10.3 14.5 14.5 13.4 15.6 15.2 12.4 23.8 16.2 17.4 20.4 16.7 17.7 14.0 16.4 20.8 10.6 12.0 12.4
825 (50°C) 735 1017 1812 1000 676 2215 1576 746 1495 735 744 778 761 746 786 840 5690 (500) 1463 990 1749 1220 966 1335 1000 1002 701 870 1240 730 925 610 (50) 605 (50) 915 835 743 978 1010 890 2250 1298 (93) 1496 (93) 1716 1389 (93) 1554 (93) 1166 (93) 1364 (93) 1946 618 (50) 644 880
11.0 11.0 10.7 10.8 10.5
20.5 19.4 18.4 16.7 15.0
1780 (0) 1603 (0) 1431 (0) 1227 (0) 1016 (0)
Ref.
Gas
X
Y
µ × 107 p
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 3 5 3 3 3 3 1 1 1 1 1
Hydrogen–sulfur dioxide 10% H2, 90% SO2 20% H2, 80% SO2 50% H2, 50% SO2 80% H2, 20% SO2 Hydrogen bromide Hydrogen chloride Hydrogen cyanide Hydrogen iodide Hydrogen sulfide Iodine Krypton Mercury Mercuric bromide Mercuric chloride Mercuric iodide Mesitylene Methane Methane (deuterated) Methanol Methyl acetate Methyl acetylene 3-Methyl-1-butene Methyl butyrate (iso) Methyl bromide Methyl chloride 3-Methylene-1-butene Methylene chloride Methyl formate Neon Nitric oxide Nitrogen Nitrous oxide Nonane (n) Octane (n) Oxygen Pentane (n) Pentane (iso) Phosphene Propane Propanol (n) Propanol (iso) Propyl acetate Propylene Pyridine Silane Stannic chloride Stannic bromide Sulfur dioxide Thiazole Thiophene Toluene 2,2,3-Trimethylbutane Trimethylethane Water Xenon Zinc
8.7 8.6 8.9 9.7 8.4 8.5 7.1 8.5 8.4 8.7 9.4 7.4 8.5 7.7 8.4 9.5 9.5 9.5 8.3 8.4 8.9 8.0 6.6 8.1 8.5 8.0 8.5 5.1 11.1 10.4 10.6 9.0 9.2 8.8 10.2 8.5 8.9 8.8 8.9 8.4 8.4 8.0 8.5 8.6 9.0 9.1 9.0 8.4 10.0 8.3 8.6 10.0 8.0 8.0 9.3 8.0
18.1 18.2 18.3 17.7 21.6 19.2 14.5 21.5 18.0 18.7 24.0 24.9 19.0 18.7 18.0 10.2 15.8 17.6 15.6 14.0 14.3 13.3 15.8 18.7 16.5 13.3 15.8 18.0 25.8 20.8 20.0 19.0 8.9 9.8 21.6 12.3 12.1 17.0 13.5 13.5 13.6 14.3 14.4 13.3 16.8 16.0 16.7 18.2 14.4 14.2 12.5 10.4 13.0 16.0 23.0 22.0
1259 (17) 1277 (17) 1332 (17) 1306 (17) 1843 1425 737 1830 1265 1730 (100) 2480 4500 (200) 2253 2200 (200) 2045 (200) 660 (50) 1092 1290 935 870 (50) 867 716 824 1327 1062 716 989 923 3113 1899 1766 1460 554 (50) 586 (50) 2026 668 685 1150 800 770 774 797 840 830 (50) 1148 1330 (100) 142 (100) 1250 958 901 (50) 686 691 (50) 686 1250 (100) 2255 5250 (500)
Ref. 4
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
*Viscosity at 20°C unless otherwise indicated. From Beerman, Meas. Control (June 1982): 154–157. References: 1. I. F. Golubev, Viscosity of Gases and Gas Mixtures, Moscow 1959; transl. U.S. Department of Commerce, Clearinghouse for Federal Scientific and Technical Information, Springfield, Va., TT 70-50022, ISPT Cat. No. 5680, Table 4, Jerusalem 1970. 2. R. H. Perry and C. H. Chilton, Chemical Engineers’ Handbook, 5th ed., McGraw-Hill, New York, 1973, pp. 3-210, 3-211. 3. Ibid., Table 3-282, p. 3-210. 4. By interpolation of data in Ref. 1. 5. Thermophysical Properties of Refrigerants, American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York. 6. N. A. Lange, Handbook of Chemistry, 4th ed., Handbook Publishers, Sandusky, Ohio, 1941. For another alignment chart for 165 hydrocarbons from −100 to 500°C, see Sastry, R. C. and A. Satyanarayan, Chem. Industry Devs. (July 1978): 11–14.
TRANSPORT PROPERTIES
Nomograph for determining (a) absolute viscosity of a gas as a function of temperature near ambient pressure and (b) relative viscosity of a gas compared with air. For coordinates see Table 2-364. To convert poises to pascal-seconds, multiply by 0.1. [From Beerman, Meas. Control, 154–157 (June 1982).]
FIG. 2-32
2-321
2-322
PHYSICAL AND CHEMICAL DATA
TABLE 2-365
Viscosities of Liquids: Coordinates for Use with Fig. 2-33
Liquid
X
Y
Liquid
X
Y
Acetaldehyde Acetic acid, 100% Acetic acid, 70% Acetic anhydride Acetone, 100% Acetone, 35% Acetonitrile Acrylic acid Allyl alcohol Allyl bromide Allyl iodide Ammonia, 100% Ammonia, 26% Amyl acetate Amyl alcohol Aniline Anisole Arsenic trichloride Benzene Brine, CaCl2, 25% Brine, NaCl, 25% Bromine Bromotoluene Butyl acetate Butyl acrylate Butyl alcohol Butyric acid Carbon dioxide Carbon disulfide Carbon tetrachloride Chlorobenzene Chloroform Chlorosulfonic acid Chlorotoluene, ortho Chlorotoluene, meta Chlorotoluene, para Cresol, meta Cyclohexanol Cyclohexane Dibromomethane Dichloroethane Dichloromethane Diethyl ketone Diethyl oxalate Diethylene glycol Diphenyl Dipropyl ether Dipropyl oxalate Ethyl acetate Ethyl acrylate Ethyl alcohol, 100% Ethyl alcohol, 95% Ethyl alcohol, 40% Ethyl benzene Ethyl bromide 2-Ethyl butyl acrylate Ethyl chloride Ethyl ether Ethyl formate 2-Ethyl hexyl acrylate Ethyl iodide Ethyl propionate Ethyl propyl ether Ethyl sulfide Ethylene bromide Ethylene chloride Ethylene glycol Ethylidene chloride Fluorobenzene Formic acid Freon-11 Freon-12 Freon-21 Freon-22
15.2 12.1 9.5 12.7 14.5 7.9 14.4 12.3 10.2 14.4 14.0 12.6 10.1 11.8 7.5 8.1 12.3 13.9 12.5 6.6 10.2 14.2 20.0 12.3 11.5 8.6 12.1 11.6 16.1 12.7 12.3 14.4 11.2 13.0 13.3 13.3 2.5 2.9 9.8 12.7 13.2 14.6 13.5 11.0 5.0 12.0 13.2 10.3 13.7 12.7 10.5 9.8 6.5 13.2 14.5 11.2 14.8 14.5 14.2 9.0 14.7 13.2 14.0 13.8 11.9 12.7 6.0 14.1 13.7 10.7 14.4 16.8 15.7 17.2
4.8 14.2 17.0 12.8 7.2 15.0 7.4 13.9 14.3 9.6 11.7 2.0 13.9 12.5 18.4 18.7 13.5 14.5 10.9 15.9 16.6 13.2 15.9 11.0 12.6 17.2 15.3 0.3 7.5 13.1 12.4 10.2 18.1 13.3 12.5 12.5 20.8 24.3 12.9 15.8 12.2 8.9 9.2 16.4 24.7 18.3 8.6 17.7 9.1 10.4 13.8 14.3 16.6 11.5 8.1 14.0 6.0 5.3 8.4 15.0 10.3 9.9 7.0 8.9 15.7 12.2 23.6 8.7 10.4 15.8 9.0 5.6 7.5 4.7
Freon-113 Glycerol, 100% Glycerol, 50% Heptane Hexane Hydrochloric acid, 31.5% Iodobenzene Isobutyl alcohol Isobutyric acid Isopropyl alcohol Isopropyl bromide Isopropyl chloride Isopropyl iodide Kerosene Linseed oil, raw Mercury Methanol, 100% Methanol, 90% Methanol, 40% Methyl acetate Methyl acrylate Methyl i-butyrate Methyl n-butyrate Methyl chloride Methyl ethyl ketone Methyl formate Methyl iodide Methyl propionate Methyl propyl ketone Methyl sulfide Napthalene Nitric acid, 95% Nitric acid, 60% Nitrobenzene Nitrogen dioxide Nitrotoluene Octane Octyl alcohol Pentachloroethane Pentane Phenol Phosphorus tribromide Phosphorus trichloride Propionic acid Propyl acetate Propyl alcohol Propyl bromide Propyl chloride Propyl formate Propyl iodide Sodium Sodium hydroxide, 50% Stannic chloride Succinonitrile Sulfur dioxide Sulfuric acid, 110% Sulfuric acid, 100% Sulfuric acid, 98% Sulfuric acid, 60% Sulfuryl chloride Tetrachloroethane Thiophene Titanium tetrachloride Toluene Trichloroethylene Triethylene glycol Turpentine Vinyl acetate Vinyl toluene Water Xylene, ortho Xylene, meta Xylene, para
12.5 2.0 6.9 14.1 14.7 13.0 12.8 7.1 12.2 8.2 14.1 13.9 13.7 10.2 7.5 18.4 12.4 12.3 7.8 14.2 13.0 12.3 13.2 15.0 13.9 14.2 14.3 13.5 14.3 15.3 7.9 12.8 10.8 10.6 12.9 11.0 13.7 6.6 10.9 14.9 6.9 13.8 16.2 12.8 13.1 9.1 14.5 14.4 13.1 14.1 16.4 3.2 13.5 10.1 15.2 7.2 8.0 7.0 10.2 15.2 11.9 13.2 14.4 13.7 14.8 4.7 11.5 14.0 13.4 10.2 13.5 13.9 13.9
11.4 30.0 19.6 8.4 7.0 16.6 15.9 18.0 14.4 16.0 9.2 7.1 11.2 16.9 27.2 16.4 10.5 11.8 15.5 8.2 9.5 9.7 10.3 3.8 8.6 7.5 9.3 9.0 9.5 6.4 18.1 13.8 17.0 16.2 8.6 17.0 10.0 21.1 17.3 5.2 20.8 16.7 10.9 13.8 10.3 16.5 9.6 7.5 9.7 11.6 13.9 25.8 12.8 20.8 7.1 27.4 25.1 24.8 21.3 12.4 15.7 11.0 12.3 10.4 10.5 24.8 14.9 8.8 12.0 13.0 12.1 10.6 10.9
TRANSPORT PROPERTIES
Nomograph for viscosities of liquids at 1 atm. For coordinates see Table 2-365. To convert centipoises to pascalseconds, multiply by 0.001.
FIG. 2-33
2-323
2-324
PHYSICAL AND CHEMICAL DATA
TABLE 2-367
TABLE 2-366
Viscosity of Sucrose Solutions* Viscosity in centipoises
Percentage sucrose by weight
Temp., °C
20
40
0 5 10 15 20 25 30 35 40 45
3.818 3.166 2.662 2.275 1.967 1.710 1.510 1.336 1.197 1.074
14.82 11.60 9.830 7.496 6.223 5.206 4.398 3.776 3.261 2.858
60
Temp., °C
113.9 74.9 56.7 44.02 34.01 26.62 21.30 17.24
50 55 60 65 70 75 80 85 90 95
Percentage sucrose by weight 20
40
60
0.974 0.887 0.811 0.745 0.688 0.637 0.592 0.552
2.506 2.227 1.989 1.785 1.614 1.467 1.339 1.226 1.127 1.041
14.06 11.71 9.87 8.37 7.18 6.22 5.42 4.75 4.17 3.73
*International Critical Tables, vol. 5, p. 23. Bingham and Jackson, Bur. Standards Bull. 14 (1919): 59.
FIG. 2-34 and TABLE 2-367
Nomograph (right) for thermal conductivity of organic liquids.
TRANSPORT PROPERTIES TABLE 2-368
2-325
Prandtl Number of Air* Pressure, bar
Temperature, K
1
5
10
20
30
40
50
60
70
80
90
100
80 90 100 120 140
mix 0.796 0.786 0.773 0.763
2.31 1.76 0.872 0.813 0.782
2.32 1.77 1.54 0.89 0.82
2.35 1.78 1.53 1.44 0.94
2.37 1.79 1.53 1.65 1.20
2.40 1.81 1.53 1.54 1.59
2.42 1.82 1.53 1.48 2.14
2.45 1.83 1.53 1.43 2.43
2.48 1.85 1.53 1.40 2.07
2.51 1.87 1.54 1.38 1.78
2.54 1.89 1.54 1.36 1.62
2.57 1.91 1.55 1.34 1.52
160 180 200 240 280
0.754 0.745 0.738 0.724 0.710
0.765 0.754 0.743 0.727 0.711
0.78 0.763 0.749 0.729 0.713
0.84 0.792 0.766 0.737 0.717
0.92 0.830 0.788 0.746 0.721
1.03 0.876 0.812 0.756 0.726
1.13 0.932 0.841 0.767 0.731
1.25 1.00 0.87 0.78 0.737
1.37 1.07 0.90 0.80 0.742
1.65 1.14 0.95 0.81 0.75
1.83 1.20 0.97 0.81 0.75
1.72 1.25 1.00 0.82 0.76
300 350 400 450 500
0.705 0.699 0.694 0.691 0.689
0.707 0.699 0.694 0.691 0.689
0.708 0.699 0.694 0.691 0.689
0.712 0.701 0.695 0.691 0.689
0.715 0.703 0.696 0.692 0.689
0.717 0.705 0.697 0.692 0.690
0.721 0.707 0.698 0.693 0.690
0.725 0.709 0.699 0.693 0.690
0.728 0.711 0.700 0.694 0.690
0.732 0.712 0.701 0.695 0.691
0.737 0.714 0.703 0.695 0.691
0.742 0.716 0.704 0.696 0.691
600 700 800 900 1000
0.690 0.696 0.705 0.709 0.711
0.690 0.696 0.704 0.709 0.711
0.690 0.695 0.704 0.708 0.711
0.689 0.695 0.704 0.708 0.711
0.689 0.695 0.704 0.708 0.711
0.689 0.695 0.703 0.708 0.710
0.689 0.695 0.703 0.708 0.710
0.689 0.695 0.703 0.708 0.710
0.689 0.695 0.703 0.708 0.710
0.690 0.695 0.702 0.708 0.709
0.690 0.695 0.702 0.708 0.709
0.690 0.695 0.702 0.708 0.709
*Compiled by P. E. Liley from tables of specific heat at constant pressure, thermal conductivity, and viscosity given in SI units for integral kelvin temperatures and pressures in bars by Vasserman. Thermophysical Properties of Air and Its Components and Thermophysical Properties of Liquid Air and Its Components. Nauka, Moscow, and in translated form by the National Bureau of Standards, Washington. The number of significant figures given above reflects the similar numbers appearing for the constituent properties in the source references. While reasonable agreement occurs for atmospheric pressure with some other works, the fragmentary data available for the saturated, etc., states show large deviations.
TABLE 2-369
Prandtl Number of Liquid Refrigerants* Temperature, K Refrigerant
Trichlorofluoromethane Dichlorodifluoromethane Chlorotrifluoromethane Bromotrifluoromethane Dichlorofluoromethane Chlorodifluoromethane Methyl chloride Trichlorotrifluoroethane Dichlorotetrafluoroethane Chloropentafluoroethane Ethane Propane Octafluorocyclobutane Dichlorodifluoromethane/difluoroethane Chlorodifluoromethane/chloropentafluoroethane Trifluoromethane/chlorotrifluoromethane Methylene fluoride/chloropentafluoroethane Butane Isobutane (2-methyl propane) Ammonia Water Ethylene Propylene
No. 11 12 13 13B1 21 22 40 113 114 115 170 290 C318 500 502 503 504 600 600a 717 718 1150 1270
180
200
220
240
260
280
300
11.9 5.25 2.96 3.75
8.64 4.27 2.67 3.27 5.72
6.73 3.65 2.69 2.94 4.50
5.33 3.27 3.05 2.83 3.87
4.74 3.08 3.57 3.03 3.48
4.18 3.04 3.61 3.25
2.93 2.42
2.79 2.40
2.77 2.45
2.87 2.60
— 15.13 7.85
3.23 2.53 — 11.18 6.16
8.59 5.21
6.94 4.67
5.77 4.40
5.06 4.46
2.55 5.28 —
2.29 4.46 — 5.78 5.73
2.22 3.88 — 4.23 4.71
2.40 3.44 11.2 3.40 4.13
2.70 3.16 8.74 3.13 3.81
—
—
3.02 7.35 3.01
3.16 6.37 3.13
5.87 3.35
5.96 3.72
2.10
2.09 4.90 6.19 8.26
2.24 3.60 5.20 6.36
2.43 3.04 4.44 5.18 1.97
2.89 2.79 3.83 4.49 1.76
2.69 3.44 3.93 1.54
— 2.85 3.22 3.66 1.40
— 3.30 3.07 3.53 1.29
— — 3.02 3.53 1.24
— 1.74 2.24
— 1.78 1.88
— 2.07 1.71
— 2.70 1.71
10.3 4.4 1.88
5.69 — 2.24
3.65 — 3.91
2.60 — 4.73
7.00 4.80 4.68 — 25.7 —
8.35
3.76
— — 1.85 3.80
320
340
360
380
3.19 — 4.52 3.16
3.44 —
4.00 — —
— —
3.18 2.85 7.04 4.78 4.90
3.54
3.17
6.23 4.82
— 5.61
5.18
—
—
—
—
—
— — —
— —
— —
3.77 1.25
4.68 1.34
1.99 —
1.59 — —
*Dashes indicate inaccessible states. Average uncertainty is about 20 percent. Values derived from formulations for thermal conductivity, specific heat at constant pressure, and viscosity contained in Thermophysical Properties of Refrigerants. American Society of Heating, Refrigerating and Air-Conditioning Engineers, New York, 1976. For further details see M. W. Johnson, M.S.M.E. thesis, Purdue University, West Lafayette, Ind., 1976.
2-326
PHYSICAL AND CHEMICAL DATA
TABLE 2-370
Thermophysical Properties of Miscellaneous Saturated Liquids Temperature, °C
Substance
Property
Acetaldehyde ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr Acetic acid
−50
−40
−30
−20
−10
0
863 2.05 460 0.211 4.47
852 2.08 404 0.206 4.08
840 2.11 358 0.200 3.78
828 2.14 321 0.195 3.52
816 2.17 290 0.189 3.33
804 2.20 263 0.184 3.14
10
20
794 2.24 241 0.182 2.97
783 2.28 222 0.180 2.81
ρ (kg/m3) cp (kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr
1049 2.031 1210 0.173 14.2
30
40
50
60
1039
1028
1018
1006
1102 0.170
1010 0.168
795 0.167
600 0.165
70
80
90
100
995
984
972
960
0.163
0.161
Aniline
ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr
— — — — —
— — — — —
— — — — —
— — — — —
— — — — —
1039 2.024 10200 0.186 111
1030 2.047 6500 0.184 72
1022 2.071 4400 0.182 50
1013 2.093 3160 0.180 36.7
1005 2.113 2370 0.177 28.3
996 2.132 1850 0.174 22.7
987 2.17 1510 0.171 19.2
978 2.20 1270 0.169 16.5
969 2.23 1090 0.168 14.5
960 2.27 935 0.167 12.7
951 2.32 825 0.167 11.5
Butanol
ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr
845 1.947 34700 0.175 3860
841 1.996 22400 0.174 2570
837 2.046 14700 0.173 1740
833 2.100 10300 0.172 1260
829 2.153 7400 0.171 930
825 2.202 5190 0.170 670
817 2.262 3870 0.168 120
810 2.345 2950 0.167 41
803 2.437 2300 0.166 33.8
797 2.524 1780 0.165 27.2
791 2.621 1410 0.164 22.5
784
776
768
760
753
1140 0.163
930 0.162
760 630 535 0.161 0.160 0.159
ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr
1362 0.988 630 0.194 3.21
1348 0.989 580 0.190 3.02
1334 0.990 535 0.186 2.85
1320 0.991 496 0.182 2.70
1306 0.993 463 0.178 2.58
1292 0.996 435 0.174 2.49
1278 1.004 405 0.170 2.39
1263 1.017 375 0.166 2.30
350 0.161
330 0.158
0.156
0.154
0.152
0.150
ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr
— — — — —
— — — — —
— — — — —
— — — — —
— — — — —
— — — — —
789 2.068 1175 0.122 19.9
779 2.081 980 0.120 17.0
769 2.094 820 0.119 14.4
759 2.106 710 0.118 12.7
750 2.119 605 0.117 11.0
740
731
721
540 0.116
0.114
0.112
ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr
2.01 6400 0.188 68.4
2.04 4790 0.186 52.5
2.08 3650 0.184 41.3
2.13 2825 0.181 33.2
2.19 2220 0.179 27.2
806 2.27 1770 0.177 22.7
798 2.35 1470 0.175 19.7
789 2.43 1200 0.173 16.9
781 2.52 1000 0.171 14.7
776 2.62 835 0.168 13.0
763 2.73 700 0.165 11.6
754 2.83 590 0.162 10.3
745 2.93 500 0.159 9.2
735 3.03 435 0.156 8.4
725 3.19 370 0.153 7.7
716 3.30 314 0.151 6.9
947
935
924
912
888
876
863
851
838
825
811
797
580
510
901 2.01 455 0.145 6.3
400 0.142
370 0.139
345 0.136
310 0.133
280 0.130
250 230 220 0.127 0.123 0.119
Carbon disulfide
Cyclohexane
Ethanol
Ethyl acetate
ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) 1090 k (W/m⋅K) Pr
Ethylamine
ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr
761 2.95 580 0.204 8.39
750 2.97 500 0.201 7.39
739 2.98 435 0.199 6.51
729 3.00 390 0.196 5.97
718 3.01 350 0.194 5.43
707 3.03 320 0.191 5.08
695
683
671
658
646
633
620
607
Ethyl ether
ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr
790 2.135 550 0.159 7.39
780 2.156 470 0.155 6.54
769 2.179 410 0.151 5.92
758 2.205 365 0.147 5.48
747 2.233 330 0.144 5.12
736 2.265 290 0.140 4.69
725 2.299 265 0.139 4.38
714 2.332 233 0.134 4.05
702 2.36 214 0.129 3.92
689 2.39 197 0.125 3.77
676 2.43 181 0.120 3.67
666 2.47 166 0.116 3.54
653 2.51 153 0.112 3.43
640
625
611
140
129
118
Ethyl iodide
ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr
0.656 0.663 0.670 0.677 730 0.092 5.37
0.684 655 0.090 4.98
0.691 590 0.088 4.63
0.698 539 0.086 4.30
0.705 495 0.085 4.11
0.712 455 0.083 3.90
0.718 420 0.081 3.72
0.724 390 0.080 3.53
Ethylene glycol
ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr
1127 2.272 57000 0.254 510
1120 2.327 33300 0.255 305
1113 2.381 20200 0.256 190
1106 2.431 13400 0.258 126
1099 2.484 9100 0.259 87.3
1092 2.536 7070 0.260 69.0
1085 2.586 4000
1077 2.636 3450
1070 1063 1056 2.685 2.734 2.779 3000 2440 2000
Formic acid
ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr
1241
1231
1220
1209
1196
1184
1170
1156
1140
0.265
2260 0.261
1800 0.257
1470 0.257
1220 0.253
1030 0.250
890 0.246
780 0.243
680 615 550 0.240 0.236 0.232
1124
1108
TRANSPORT PROPERTIES TABLE 2-370
2-327
Thermophysical Properties of Miscellaneous Saturated Liquids (Concluded ) Temperature, °C
Substance Gasoline
Glycerol
Kerosine
Property
−50
ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) 1710 k (W/m⋅K) 0.131 Pr ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr
—
−40
1400 0.128 —
−30
−20
784 1.88 1170 990 0.125 0.123 15.1 —
—
−10
0
775 1.92 850 0.121 13.5 —
10
20
30
40
50
60
70
80
90
100
767 1.97 735 0.120 12.1
759 2.02 645 0.118 11.0
751 2.06 530 0.116 9.41
743 2.11 464 0.114 8.59
735 2.15 410 0.112 7.87
721 2.20 367 0.110 7.34
717 2.25 330 0.108 6.88
708 2.30 298 0.106 6.47
699 2.35 270 0.104 6.10
690 2.41 246 0.102 5.81
681 2.46 225 0.100 5.54
1276
1270
1260 2.393 1.5.+6 0.284 12650
1254 2.406
1248 2.457
1242 2.504
2.548
2.588
2.625 2.657 2.686
0.285
0.287
0.288
0.289
0.291
0.293 0.294 0.295
2.28 73
2.32 66
2.35 60
2.38 55
1.2.+7 4.0.+6
ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) 1150 k (W/m⋅K) Pr
725
500
360
275
781 1.91 215 0.140 2.93
774 1.96 173 0.139 2.44
767 2.02 149 0.139 2.17
760 2.07 126 0.138 1.89
754 2.13 108 0.138 1.67
748 2.18 95 0.137 1.51
742 2.23 83 0.137 1.35
ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k(W/m⋅K) Pr
2.30 2305 0.225 23.6
2.32 1800 0.222 18.8
2.35 1410 0.219 15.1
2.37 1170 0.216 12.9
2.40 975 0.212 11.0
2.42 820 0.209 9.53
2.45 692 0.206 8.23
2.47 590 0.203 7.18
783 2.49 510 0.199 6.38
774 2.52 455 0.195 5.88
766 2.55 400 0.192 5.31
756 2.65 355 0.189 4.98
746 2.78 315 0.187 4.68
736 2.94 271 0.184 4.34
725 3.13 240 0.182 4.13
711 3.30 218 0.180 3.99
Methyl formate
ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr
1069 1.84 830 0.217 7.04
1056 1.86 711 0.213 6.21
1043 1.88 618 0.209 5.56
1030 1.90 544 0.205 5.04
1017 1.92 481 0.200 4.62
1003 1.95 430 0.195 4.30
989 1.99 380 0.191 3.96
975 2.03 345 0.186 3.77
960 2.08 315 0.180 3.64
944
929
913
897
880
863
845
Oil, castor
ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr
Methanol
Oil, olive
Pentane
Propanol
Sulfuric acid
2,420,000 986,000 451,000 231,000 125,000 74,000 43,000 0.182 0.181 0.180 0.179 0.178 0.177 0.176 0.175 0.174 0.17
ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr
138,000 0.170
914 1.633 84,000 0.169 810
52,000 0.168
36,300 0.167
24,500 0.166
17,000 12,400 0.166 0.165 0.165 0.164 0.164
616
606
596
585
574
562
209 0.115
190 0.112
175 0.108
161 0.105
148 0.101
137 124 113 0.098 0.095 0.091
ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr
693 2.060 489 0.142 7.14
665 2.137 339 0.132 5.49
656 2.167 307 0.128 5.20
646 2.206 279 0.125 4.92
636 2.239 254 0.122 4.66
ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr
849 1.955 20,200 13,500 9500 6900 0.167 0.166 0.165 236
819 2.219 3900
811
814
796
788
779
770
761
752
5110
2900
2245
1720 0.171
1400 0.169
1130 0.168
921 0.167
760 0.165
630 508 447 0.164 0.163 0.162
1834 1.382 25,400
15,700
11,500
8820
7220
6090
5190
829 1.80 380 0.124 5.5
820 1.83 355 0.122 5.3
810 1.87 325 0.119 5.1
820
730
675
684 2.084 428 0.139 6.42
674 2.110 379 0.136 5.88
ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr
Toluene
ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr
Turpentine
ρ (kg/m3) cp(kJ/ kg⋅K) µ (10−6Pa⋅s) k (W/m⋅K) Pr
48,400 35,200 0.314 932 1.514 2120 0.152 21.1
923 1.535 1670 0.149 17.8
913 1.556 1345 0.147 14.2
904 1.579 1100 0.144 12.1
895 1.602 915 0.142 10.3
626 2.273 234 0.119 4.47
886 1.633 770 0.139 9.0
876 1.652 670 0.137 8.1
867 1.675 590 0.134 7.4
858 1.701 520 0.132 6.7
848 1.73 470 0.129 6.3
839 1.76 420 0.126 5.9
1.72 2250 0.130 29.8
1.76 1780 0.129 24.3
1.80 1490 0.128 20.9
1270 0.127 18.4
1070 0.126 16.1
1.93 925 0.125 14.3
550
747
800 1.92 295 0.117 4.8
538
743
790 1.97 270 0.114 4.7
2-328
PHYSICAL AND CHEMICAL DATA
TABLE 2-371
Diffusivities of Pairs of Gases and Vapors (1 atm) Dv in cm2/s
Substance Acetic acid Acetone n-Amyl alcohol sec-Amyl alcohol Amyl butyrate Amyl formate i-Amyl formate Amyl isobutyrate Amyl propionate Aniline Anthracene Argon Benzene Benzidine Benzyl chloride n-Butyl acetate i-Butyl acetate n-Butyl alcohol i-Butyl alcohol Butyl amine i-Butyl amine i-Butyl butyrate i-Butyl formate i-Butyl isobutyrate i-Butyl proprionate i-Butyl valerate Butyric acid i-Butyric acid Cadmium Caproic acid i-Caproic acid Carbon dioxide
Carbon disulfide Carbon monoxide Carbon tetrachloride Chlorobenzene Chloroform Chloropicrin m-Chlorotoluene o-Chlorotoluene p-Chlorotoluene Cyanogen chloride Cyclohexane n-Decane Diethylamine 2,3-Dimethyl butane Diphenyl n-Dodecane Ethane Ethanol Ether (diethyl) Ethyl acetate Ethyl alcohol Ethyl benzene Ethyl n-butyrate Ethyl i-butyrate Ethylene Ethyl formate Ethyl propionate Ethyl valerate Eugenol Formic acid Helium n-Heptane n-Hexane Hexyl alcohol Hydrogen
Temp., °C 0 0 0 30 0 0 0 0 0 0 30 0 20 0 0 0 0 0 0 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 25 500‡ 0 0 450‡ 0 30 0 25 0 0 0 0 15 45 90 0 15 0 126 0 0 0 0 30 0 0 0 0 0 0 0 0 0 0 0 20 38 15 0 0 25 500
Air
A
H2
0.1064 .109 .0589 .072 .040 .0543 .058 .0419 .046 .0610 .075 .0421
0.416 .361 .235
.077 .0298 .066 .058 .0612 .0703 .088 .0727 .0821 .0853 .0468 .0705 .0457 .0529 .0424 .067 .0679
.306
O2
N2
CO2
N2O
CH4
C2H6
C2H4
n-C4H10
i-C4H10
0.0716 .0422
.171 .1914
.0347
0.194 0.0797
.0528
.2364 .2716
.0425 .0476
.2771
.0483
.185
.0327
.191 .203 .173 .264 .271
.0364 .0366 .0308 .0476 .0471 .17
.050 .0513 .138
.550
.139
0.096
0.153
.163 .0996*
.00215†
.9 .0892
.369 .651 .293
.185 1.0 0.0636
.319
.0744
.063 .137
0.116
.075 .091 .088 .054 .059 .051 .111 0.0719
.0760
.086 .306
.0841
.0884 .0657
.301
.0753
.0751
.0610 .308 .459 .377 .298 .273
.0778 .0715 .089 .102 .0658 .0579 .0591 .0840 .068 .0512 .0377 .1308
.0813 .0686 .0546 .0487
.375
.0685
.224 .229 .486 .337 .236 .205
.0407 .0413 .0573 .0450 .0367
.510
Ref. 8 6, 16 8 5 8 8 8 8 8 8 5 8 18 8, 15 8 8 8 8 8 5 8 8 8 8 8 8 8 8 8 8 13 8 8 8 19 1, 9 18 8 8 18 16, 17 5 6 10 8 8 8 10 3 6 3 8 3 8 3 8 20 7, 8 8 5 8 8 8 8 8 8 4, 8 8 8 8 8 19
.0874
.641 .705 .066§ .0663 .0499 .611
.290 .200
.0753 .697 4.2
.0757 .674
.0351 .550 .646
.535
.625
0.459 .537
0.486 .726
0.272
0.277
3 8 8 2 18
TRANSPORT PROPERTIES TABLE 2-371
2-329
Diffusivities of Pairs of Gases and Vapors (1 atm) (Concluded ) Dv in cm2/s
Substance
Temp., °C
Air
Hydrogen cyanide Hydrogen peroxide Iodine Mercury Mesitylene Methane Methyl acetate Methyl alcohol Methyl butyrate Methyl i-butyrate Methyl cyclopentane Methyl formate Methyl propionate Methyl valerate Naphthalene Nitrogen
0 60 0 0 0 500 0 0 0 0 15 0 0 0 0 0 25 0 0 30 0 0 0 0 0 0 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 25 0 30 0
.173 .188 .07 .112 .056
Nitrous oxide n-Octane Oxygen Phosgene Propionic acid Propyl acetate n-Propyl alcohol i-Propyl alcohol n-Propyl benzene i-Propyl benzene n-Propyl bromide i-Propyl bromide Propyl butyrate Propyl formate n-Propyl iodide i-Propyl iodide n-Propyl isobutyrate i-Propyl isobutyrate Propyl propionate Propyl valerate Safrol i-Safrol Sulfur hexafluoride Toluene Trimethyl carbinol 2,2,4-Trimethyl pentane 2,2,3-Trimethyl heptane n-Valeric acid i-Valeric acid Water
H2
O2
N2
CO2
N2O
CH4
C2H6
C2H4
n-C4H10
i-C4H10
.070 .13
.53 1.1
.084 .132 .0633 .0639 .0731 .0872 .0735 0.0569 .0513
.333 .506 .242 .257 .318
.0567 .0879 .0446 .0451 0.0742
.0758
.295
.0528 0.181 0.165 .096
0.535 .0505 0.0642 .178 .095 .0829 .067 .085 .0818 .101 .0481 .0489 .085 .0902 .0530 .0712 .079 .0802 .0549 .059 .057 .0466 .0434 .0455
.271 .697
.0705
0.0710 .181
.139
.330
.0588
.315
.0577
.206 .281
.0364 .0490
.212
.0388
.212 .189
.0395 .0341
.418 .076 .088 .087
30 90 0 0 0 450
A
.071
.0618
.288
.0688
.270 .050 .0544 .220
.212 .75
0.148
0.163
0.0960
0.0908
Ref. 10 11 8, 12, 14 8, 12, 13 8 18 8 8 8 8 3 8 8 8 8 8 2 8 8 3 8 10 8 8 8 8 5 8 8 8 8 8 8 8 8 8 8 8 8 8 8 2 4, 8 5 8
.0705
3
.0684
3 8 8 8, 20 18
.0376 .138 1.3
* 320 mm Hg. † 40 atm. ‡ Also at other temperatures. § Strong function of concentration. References 1 Amdur, Irvine, Mason, and Ross, J. Chem. Phys., 20, 436 (1952). 2 Boyd, Stein, Steingrimsson, and Rumpel, J. Chem. Phys., 19, 548 (1951). 3 Cummings and Ubbelohde, J. Chem. Soc. (London), 1953, p. 3751. 4 Fairbanks and Wilke, Ind. Eng. Chem., 42, 471 (1950). 5 Gilliland, Ind. Eng. Chem., 26, 681 (1934). 6 Gorynnova and Kuvskinskii, Zhur. Tekh. Fiz., 18, 1421 (1948). 7 Hansen, Dissertation, Jena, 1907. 8 “International Critical Tables,” vol. 5, p. 62. 9 Jeffries and Drickamer, J. Chem. Phys., 22, 436 (1954). 10 Klotz and Miller, J. Am. Chem. Soc., 69, 2557 (1947). 11 McMurtrie and Keyes, J. Am. Chem. Soc., 70, 3755 (1948). 12 Mullaly and Jacques, Phil. Mag., 48, 6, 1105 (1924). 13 Spier, Physica, 6 (1939): 453; 7, 381 (1940). 14 Topley and Whytlaw-Gray, Phil. Mag., 4, 873 (1927). 15 Trautz and Ludwig, Ann. Physik, 5, 5, 887 (1930). 16 Trautz and Muller, Ann. Physik, 22, 353 (1935). 17 Trautz and Ries, Ann. Physik, 8, 163 (1931). 18 Walker and Westenberg, J. Chem. Phys., 32, 136 (1960). 19 Westenberg and Walker, J. Chem. Phys., 26, 1753 (1957). 20 Winkelmann, Wied. Ann., 22, 152 (1884); 23, 203 (1884); 26, 105 (1885); 33, 445 (1888); 36, 92 (1889).
2-330
PHYSICAL AND CHEMICAL DATA
In this table are a representative selection of diffusion coefficients. The subsection “Prediction and Correlation of Physical Properties” should be consulted for estimation techniques. As general references, the works by Hirschfelder, Curtiss, and Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1964; Chapman and Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge, New York, 1970; Reid and Sherwood, The Properties of Gases and Liquids, TABLE 2-372
McGraw-Hill, New York, 1964; and Bretsznajder, Prediction of Transport and Other Physical Properties of Fluids, Pergamon, New York, 1971, may be found useful. The most exhaustive recent compilation for gases is by Mason and Marrero, J. Phys. Chem. Ref. Data, 1 (1972). Unfortunately, the Mason and Marrero work cites only equations and equation constants and not direct tabulations. For these, the LandoltBörnstein series is suggested.
Diffusivities in Liquids (25°C)
Dilute solutions and 1 atm unless otherwise noted; use DLµ/T = constant to estimate effect of temperature; * indicates that reference gives effect of concentration.
Solute
Solvent
DL × 105, sq cm/sec
Acetal* Acetamide* Acetamide* Acetic acid Acetic acid Acetic acid Acetic acid Acetic acid Acetic acid* Acetonitrile Acetylene Allyl alcohol* Allyl alcohol Ammonia* i-Amyl alcohol* i-Amyl alcohol Benzene Benzene (50 mole %) Benzene (50 mole %) Benzene (50 mole %) Benzene (50 mole %) Benzene (50 mole %) Benzene (50 mole %) Benzoic acid Benzoic acid Benzoic acid Benzoic acid Benzoic acid Bromine Bromine Bromine Bromobenzene Bromoform* Bromoform Bromoform Bromoform* Bromoform Bromoform n-Butanol Caffeine Carbon dioxide Carbon dioxide Carbon disulfide (50 mole %, 200 atm.) Carbon disulfide (50 mole %, 200 atm.) Carbon disulfide (50 mole %, 218 atm.) Carbon disulfide (50 mole %, 200 atm.) Carbon disulfide (50 mole %, 100 atm.) Carbon disulfide (50 mole %, 50 atm.) Carbon disulfide (50 mole %, 200 atm.) Carbon disulfide (50 mole %) Carbon tetrachloride Carbon tetrachloride* Carbon tetrachloride Carbon tetrachloride Carbon tetrachloride* Carbon tetrachloride Carbon tetrachloride Carbon tetrachloride Carbon tetrachloride Carbon tetrachloride Chloral* Chloral hydrate
Ethanol Ethanol Water Acetone Benzene Carbon tetrachloride Ethylene glycol Toluene Water Water Water Ethanol Water Water Ethanol Water Carbon tetrachloride n-Decane 2,4-Dimethyl pentane n-Dodecane n-Heptane n-Hexadecane n-Octadecane Acetone Benzene Carbon tetrachloride Ethylene glycol Toluene Benzene Carbon disulfide Water Benzene Acetone i-Amyl alcohol Ethanol Ethyl ether Methanol n-Propanol Water Water Ethanol Water n-Butanol i-Butanol Chlorobenzene 2,4-Dimethyl pentane n-Heptane Methyl cyclohexane n-Octane Toluene Benzene Cyclohexane Decalin Dioxane Ethanol n-Heptane Kerosene Methanol i-Octane Tetralin Ethanol Water
1.25 0.68 1.19 3.31 2.11 1.49 0.13 2.26 1.24 1.66 1.78, 2.11 1.06 1.19 1.7, 2.0, 2.3 0.87 1.0 1.53 1.72 2.49 1.40 2.47 0.96 0.86 2.62 1.38 0.91 0.043 1.49 2.7 4.1 1.3 2.30 2.90 0.53 1.08 3.62 2.20 0.94 0.96 0.63 4.0 1.96 3.57 2.42 3.00 3.63 3.0 3.5 3.10 2.06 2.04 1.49 0.776 1.02 1.50 3.17 0.961 2.30 2.57 0.735 0.68 0.77
Estimated possible, error, %1 5 5 3
3 5 5 6 5 8
5
5 6 6 1
3 2 2 2 2 2 2 2 2 2 5 7
Ref. 11 11 11 4 1, 4 4 4 4 11 11 1, 24 11 11 1, 11 11 11, 25 7 26 26 26 26 26 26 4 4 4 4 4 11 11 11 25 11 11 11 11 23 11 1, 11, 18, 25 11 11 1, 3, 5, 20, 24, 28 14 14 14 14 14 14 14 14 7, 9 9, 10* 9 9 9, 10* 9 9 9 9 9 11 11
TRANSPORT PROPERTIES TABLE 2-372
Diffusivities in Liquids (25°C) (Continued )
Dilute solutions and 1 atm unless otherwise noted; use DL µ /T = constant to estimate effect of temperature; * indicates that reference gives effect of concentration.
Solute Chlorine Chlorobenzene Chloroform Chloroform Cinnamic acid Cinnamic acid Cinnamic acid Cinnamic acid 1,1′-Dichloropropanol Dicyanodiamide* Diethyl ether Diethyl ether 2,4-Dimethyl pentane (50 mole %) 2,4-Dimethyl pentane (50 mole %) Ethanol* Ethyl acetate Ethylene dichloride Formic acid Formic acid Formic acid Formic acid Formic acid Formic acid Glucose Glycerol Glycerol Glycerol* n-Heptane (50 mole %) n-Heptane (50 mole %) n-Heptane (50 mole %) n-Heptane (50 mole %) Hexamethylene tetramine Hydrogen chloride* Hydrogen Hydrogen sulfide Hydroquinone* Hydroquinone* Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine* Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodine Iodobenzene Lactose* Maltose* Mannitol* Methanol Nicotine* Nitric acid* Nitrobenzene Nitrogen Nitrous oxide Oxalic acid*
Solvent
DL × 105, sq cm/sec
Water Benzene Benzene Ethanol Acetone Benzene Carbon tetrachloride Toluene Water Water Benzene Water n-Dodecane n-Hexadecane Water Ethyl benzoate Benzene Acetone Benzene Carbon tetrachloride Ethylene glycol Toluene Water Water i-Amyl alcohol Ethanol Water n-Dodecane n-Hexadecane n-Octadecane n-Tetradecane Water Water Water Water Ethanol Water Acetic acid Anisole Benzene Bromobenzene Carbon disulfide Carbon tetrachloride Chloroform Cyclohexane Dioxane Ethanol Ethyl acetate Ethyl ether Ethylene bromide n-Heptane n-Hexane Mesitylene Methanol Methyl cyclohexane n-Octane Tetrabromoethane n-Tetradecane Toluene m-Xylene Ethanol Water Water Water Water Water Water Carbon tetrachloride Water Water Water
1.44 2.66 2.50 1.38 2.41 1.12 0.76 2.41 1.0 1.18 2.73 0.85 1.44 0.88 1.28 0.94 2.8 3.77 2.28 1.89 0.094 2.65 1.37 0.69 0.12 0.56 0.94 1.58 1.00 0.92 1.29 0.67 3.10 5.85 (4.4) 1.61 0.53 0.88, 1.12 1.13 1.25 1.98 1.25 3.2 1.45 2.30 1.80 1.07 1.30 2.2 3.61 0.93 3.4, 2.5 4.15 1.49 1.74 2.1 2.76 2.0 0.96 2.1 1.82 1.09 0.49 0.48 0.65 1.6 0.60 2.98 1.00 1.9 1.8 1.61
Estimated possible, error, %1 4 6 3
6 4
4
10 6 6
3 5
10 8 3
3 5 5 5 8 2
2
Ref. 1, 28 25 1, 25 11 4 4 4 4 11 11 25 2 26 26 1, 7, 9,* 11,* 22 6 1, 25 4 4 4 4 11 11 11 11 1, 11* 26 26 26 26 11 4, 11,* 12* 1, 11, 24(?) 1 11 2, 11* 11 11 9, 19, 23 4, 11, 19 11, 19, 23 9, 11, 19 11, 23 4 9 4, 11* 11, 19 11 11 9, 11, 19 4, 9 9 19 4 4 11 4 11 9, 11 11 11 11 11 1, 7, 11 11 11 7 1, 24 1, 11 11
2-331
2-332
PHYSICAL AND CHEMICAL DATA TABLE 2-372
Diffusivities in Liquids (25°C) (Concluded )
Dilute solutions and 1 atm unless otherwise noted; use DL µ /T = constant to estimate effect of temperature; * indicates that reference gives effect of concentration.
Solute Oxygen Oxygen Oxygen Pentaerythritol* Phenol Phenol Phenol Phenol Phenol Phenol n-Propanol Pyridine* Pyridine Pyrogallol Raffinose* Resorcinol* Resorcinol* Saccharose* Stearic acid* Succinic acid* Sucrose Sulfur dioxide Sulfuric acid* Tartaric acid* 1,1,2,2-Tetrabromoethane Toluene Toluene Toluene Toluene Toluene Urea Urea Urethane Water
Solvent Glycerol*-water (106 poise) Sucrose*-water (125 poise) Water Water i-Amyl alcohol Benzene Carbon disulfide Chloroform Ethanol Ethyl ether Water Ethanol Water Water Water Ethanol Water Water Ethanol Water Water Water Water Water 1,1,2,2-Tetrachloroethane n-Decane n-Dodecane n-Heptane n-Hexane n-Tetradecane Ethanol Water Water Glycerol
DL × 105, sq cm/sec
Estimated possible, error, %1
Ref.
0.24
13
0.25
13
2.5 0.77 0.2 1.68 3.7 2.0 0.89 3.9 1.1 1.24 0.76 0.74 0.41 0.46 0.87 0.49 0.65 0.94 0.56 1.7 1.97 0.80 0.61 2.09 1.38 3.72 4.21 1.02 0.73 1.37 1.06 0.021
References 1 Arnold, J. Am. Chem. Soc., 52, 3937 (1930). 2 Calvet, J. Chim. Phys., 44, 47 (1947). 3 Carlson, J. Am. Chem. Soc., 33, 1027 (1911). 4 Chang and Wilke, J. Phys. Chem., 59, 592 (1955). 5 Davidson and Cullen, Trans. Inst. Chem. Eng., 35, 51 (1957). 6 Dummer, Z. Anorg. Chem., 109, 31 (1949). 7 Gerlach, Ann. Phys. (Leipzig), 10, 437 (1931). 8 Gosting and Akeley, J. Am. Chem. Soc., 74, 2058 (1952). 9 Hammond and Stokes, Trans. Faraday Soc., 49, 890 (1953); 49, 886 (1953). 10 Hammond and Stokes, Trans. Faraday Soc., 52, 781 (1956). 11 International Critical Tables, vol. 5, p. 63. 12 James, Hollingshead, and Gordon, J. Chem. Phys., 7, 89 (1939); 7, 836 (1939). 13 Jordon, Ackermann, and Berger, J. Am. Chem. Soc., 78, 2979 (1956). 14 Koeller and Drickamer, J. Chem. Phys., 21, 575 (1953). 15 Kolthoff and Miller, J. Am. Chem. Soc., 63, 1013 (1941).
20 4
3 7 7 4 5 4 4 5 6 3 10 4
2
1, 3, 15, 21, 24 11 11 1 11 11 11 11 1, 7, 11 11 11 11 11 11 11 11 11 11 2, 27 15, 17 11 11 11 4 4 4 4 4 11 8, 11 11, 25 16
TRANSPORT PROPERTIES TABLE 2-373
2-333
Thermal Conductivities of Some Building and Insulating Materials* k = Btu/(h⋅ft2)(°F/ft)
Material
Apparent density ρ, lb/ft3 at room temperature
Aerogel, silica, opacified
8.5
Asbestos-cement boards Asbestos sheets Asbestos slate
120 55.5 112 112 29.3 29.3 36 36 36 36 43.5 43.5 0.2
Asbestos
Aluminum foil (7 air spaces per 2.5 in.) Ashes, wood Asphalt Boiler scale (Note 1) Bricks: Alumina (92–99% Al2O3 by wt.) fused Alumina (64–65% Al2O3 by wt.) (See also Bricks, fire clay) Building brick work Carbon Chrome brick (32% Cr2O3 by wt.) Diatomaceous earth, natural, across strata (Note 2) Diatomaceous, natural, parallel to strata (Note 2) Diatomaceous earth, molded and fired (Note 2) Diatomaceous earth and clay, molded and fired (Note 2) Diatomaceous earth, high burn, large pores (Note 3)
132
Kaolin insulating firebrick (Note 4) Magnesite (86.8% MgO, 6.3% Fe2O3, 3% CaO, 2.6% SiO2 by wt.)
Silicon carbide brick, recrystallized (Note 3)
Calcium carbonate, natural White marble Chalk Calcium sulfate (4H2O), artificial plaster (artificial) (building) Cambric (varnished) Carbon, gas Carbon stock Cardboard, corrugated
k
120 290 20 51 0 60 −200 0 0 100 200 400 −200 0 38 177 0–100 20
0.013 .026 .43 .096 .087 .114 .043 .090 .087 .111 .120 .129 .090 .135 .025 .038 .041 .43
427 1315 800 1100 20
96.7 200 200 200
200 650 1315
1.8 2.7 0.62 .63 .4 3.0 .67 .85 1.0
27.7 27.7
204 871
0.051 .077
27.7 27.7 38 38
204 871 204 871
.081 .106 .14 .18
42.3 42.3
204 871
.14 .19
27 27 19 19
200 1000 200 600 1000 1400 500 1150 200 760
.13 .34 .58 .85 .95 1.02 0.15 .26 .050 .113
158 158 158
204 650 1200
2.2 1.6 1.1
129 129 129 129 129 162
600 800 1000 1200 1400 30
10.7 9.2 8.0 7.0 6.3 1.3 1.7 0.4 .22 .43 .25 .091 2.0 0.55 3.6 0.037
115 115
37 37
Fire clay (Missouri)
Kaolin insulating brick (Note 3)
t, °C
96 84.6 132 77.9 94
40 75 25 38 0–100 −184 0
Material Cotton wool Cork board Cork (regranulated) (ground) Diatomaceous earth powder, coarse (Note 2) fine (Note 2) molded pipe covering (Note 2) 4 vol. calcined earth and 1 vol. cement, poured and fired (Note 2) Dolomite Ebonite Enamel, silicate Felt, wool Fiber insulating board Fiber, red (with binder, baked) Gas carbon Glass Borosilicate type Window glass Soda glass Granite Graphite, longitudinal powdered, through 100 mesh Gypsum (molded and dry) Hair felt (perpendicular to fibers) Ice Infusorial earth, see diatomaceous earth Kapok Lampblack Lava Leather, sole Limestone (15.3 vol. % H2O) Linen Magnesia (powdered) Magnesia (light carbonate) Magnesium oxide (compressed) Marble Mica (perpendicular to planes) Mill shavings Mineral wool
Apparent density ρ, lb/ft3 at room temperature
t, °C
k
5 10 8.1 9.4 20.0 20.0 17.2 17.2 26.0 26.0
30 30 30 30 38 871 204 871 204 871
0.024 .025 .026 .025 .036 .082 .040 .074 .051 .088
61.8 61.8 167
204 871 50
38 20.6 14.8 80.5
139
30 78 17 57.5 0.88 10 62.4 103
20 40
0 21 21 20 40 21
0.020 .038 .49 .092 .54 .05 .35 0.034 .32 1.2–1.7 0.25 0.033–0.05 0.0225 .024 .075 .14 3.4 2.9 0.88 .17 .14 .075 .087 .109 0.075–0.092 0.19 1.06 0.03
38 24–127 30 94 0 100 21 21 30 30
.026 .096 .064 .022 .86 .27 0.09–0.097 0.16 .028 .04 .034
49.7 13 49.9
24 30 47 21 20
9.4 19.7
30 30
50
Paper Paraffin wax Petroleum coke Porcelain Portland cement, see concrete Pumice stone Pyroxylin plastics Rubber (hard) (para) (soft) Sand (dry) Sandstone Sawdust Scale (Note 1) Silk varnished Slag, blast furnace Slag wool Slate Snow Sulfur (monoclinic) (rhombic) Wall board, insulating type Wall board, stiff paste board Wood shavings
.16 .23 1.0 0.10 0.5–0.75 30 0.03 21 .028 20 .27 20–97 .097 0–100 2.0 0.2–0.73 30–75 0.63 0.3–0.61 0.3–0.44 1.0–2.3 20 95 40 0.104 20 .25 30 .021 0 1.3
0 100 500 200 90 21–66 74.8 94.6 140 12 6.3 12 34.7 14.8 43 8.8
2-334
PHYSICAL AND CHEMICAL DATA
TABLE 2-373
Thermal Conductivities of Some Building and Insulating Materials* (Concluded ) k = Btu/(h⋅ft2)(°F/ft) Apparent density ρ, lb/ft3 at room temperature
Material Celluloid Charcoal flakes
87.3 11.9 15
Clinker (granular) Coke, petroleum Coke, petroleum (20–100 mesh) Coke (powdered) Concrete (cinder) (stone) (1:4 dry)
62
t, °C
k
30 80 80 0–700 100 500 400 0–100
.12 .043 .051 .27 3.4 2.9 0.55 .11 .20 .54 .44
Material Wood (across grain): Balsa Oak Maple Pine, white Teak White fir Wood (parallel to grain): Pine Wool, animal
Apparent density ρ, lb/ft3 at room temperature
t, °C
k
7–8 51.5 44.7 34.0 40.0 28.1
30 15 50 15 15 60
0.025–0.03 0.12 .11 .087 .10 .062
34.4 6.9
21 30
.20 .021
*Marks, Mechanical Engineers’ Handbook, 4th ed., McGraw-Hill, New York, 1941. International Critical Tables, McGraw-Hill, 1929, and other sources. Note 1: B. Kamp [Z. tech. Physik, 12, 30 (1931)] shows the effect of increased porosity in decreasing thermal conductivity of boiler scale. Partridge [University of Michigan, Eng. Research Bull., 15, 1930] has published a 170-page treatise on Formation and Properties of Boiler Scale. Note 2: Townshend and Williams, Chem. & Met., 39, 219 (1932). Note 3: Norton, Refractories, 2d ed., McGraw-Hill, New York, 1942. Note 4: Norton, private communication. TABLE 2-374
Thermal-Conductivity-Temperature Table for Metals* Thermal conductivities tabulated in watts per meter-kelvin Temperature, K
Substance
10
20
40
60
80
100
200
300
400
500
7 38,000 470 47 240
32 13,500 230 196 100
121 2,300 110 810 45
174 850 80 1,400 31
160 380 60 1,650 24
125 300 48 1,490 22
55 237 32 480 18
36 273 26 272 16
26 240 22 196 14
20 237 20 146 12
165 900 400 250 4
305 250 570 450 9
400 150 450 380 16
327 120 250 250 18
230 110 180 190 19
170 110 158 160 20
45 105 111 120 23
25 104 90 100 25
15 101 87 85 27
Copper Gallium Gold Graphite† Graphite‡
19,000 2,200 2,800 27 81
10,700 640 1,500 108 420
2,100 250 520 135 1,630
850 200 380 81 2,980
570 170 350 54 4,290
483 140 345 39 4,980
413 100 327 15 3,250
398 85 315 10 2,000
Hastelloy Inconel Iridium Iron Lead
1 2 1,300 710 175
3 4 1,900 1,000 57
4 8 750 560 43
5 10 360 270 42
6 11 230 170 41
7 11 172 132 40
9 14 147 94 37
10 15 145 80 35
11
13
143 69 34
140 61 33
Magnesium Magnesium oxide Manganese Manganin Mercury
1,200 1,100 2 2 54
1,300 3,100 2 4 40
620 2,200 4 9 35
290 950 5 11 33
190 460 5 13 33
169 260 6 13 32
159 75 7 17 32
156 48 8 22 8
153 36 9 28 10
Molybdenum Nickel Nylon Palladium Platinum
150 2,600 0.04 1,200 1,200
280 1,700 0.10 610 490
350 570 0.17 160 130
250 290 0.20 100 92
210 200 0.23 88 82
179 158 0.25 80 79
143 106 0.28 78 75
138 91 0.30 78 73
PTFE§ Pyrex Quartz Rhodium Rubber
0.94 0.12 1,200 2,900
1.43 0.20 480 3,900
1.94 0.33 82 1,000 0.13
2.1 0.42 40 370 0.15
2.15 0.51 30 250 0.16
2.16 0.57
2.20 0.88
2.25 1.1
190 0.17
160 0.20
Alumina Aluminum Antimony Beryllium oxide Bismuth Boron Cadmium Chromium Cobalt Constantan
Selenium (axis) Silica Silver Tantalum Tellurium Tin Titanium Tungsten Uranium Zinc Zirconium
1200
1400
16 232
10 220
8 93
7 99
6 105
111
70
47
33
25
12 99 85 70 30
81
71
65
62
61
392
388
383
371
357
342
312 7 1,460
309 5 1,140
304 4 930
292 3 680
278 3 530
262 2 440
2 370
55 31
43 19
33 22
28 24
31 26
151 27 9 34 11
149 21
146 13
84 10
98 8
112 7
40 12
13
14
134 80
130 72
126 66
118 67
112 72
105 76
100 80
78 72
80 72
72
73
78
78
81
358 61
62
2.3 1.6
2.5 2.1
150 0.22
145 0.24
140 0.25
140
57
25
15
10
8
6
16,500 108 300
5,200 146 93
1,100 88 29
630 68 17
500 62 13
430 59 11
425 58 6
4 1.34 424 57 4
3 1.52 420 58 3
2 1.70 413 58 3
14
320 28
130 39 880
100
110
59
101 37 330 20 150 42
90 33 310 22 135 38
84 31 280 23 130 34
72 26 190 26 123 25
67 21 180 28 120 23
62 20 170 30 116 22
60 20 150 32 110 21
600
1.87 405 59
800
2.22 389 59
1000
2.60 374 60
19 140 110 21
* Especially at low temperatures, the thermal conductivity can often be markedly reduced by even small traces of impurities. This table, for the highest-purity specimens available, should thus be used with caution in applications with commercial materials. From Perry, Engineering Manual, 3d ed., McGraw-Hill, New York, 1976. A more detailed table appears as Section 5.5.6 in the Heat Exchanger Design Handbook, Hemisphere Pub. Corp., Washington, DC, 1983. † Parallel to basal plane. ‡ Perpendicular to basal plane. § Also known as Teflon, etc.
TRANSPORT PROPERTIES TABLE 2-375
Thermal Conductivity of Chromium Alloys* k = Btu/(h⋅ft2)(°F/ft)
American Iron and Steel Institute Type No.
k at 212°F
k at 932°F
9.4 8.8 8.0 9.3 14.4 15.1 12.5 21.2
12.4 12.5 10.8 12.8 16.6 15.2 14.2 19.5
301, 302, 302B, 303, 304, 316† 308 309, 310 321, 347 403, 406, 410, 414, 416† 430, 430F† 442 501, 502†
* Table 3-322 is based on information from manufacturers. † Shelton and Swanger (National Bureau of Standards), Trans. Am. Soc. Steel Treat., 21, 1061–1078 (1933).
TABLE 2-376 Thermal Conductivity of Some Alloys at High Temperature*
TABLE 2-377 Thermal Conductivities of Some Materials for Refrigeration and Building Insulation* k = Btu/(h⋅ft2)(°F/ft) at approximately room temperature
Material Soft flexible materials in sheet form: Chemically treated wood fiber Eel grass between paper Felted cattle hair Flax fibers between paper Hair and asbestos fibers, felted Insulating hair, and jute Jute and asbestos fibers, felted Loose materials: Cork, regranulated, fine particles Charcoal, 6 mesh Diatomaceous earth, powdered Glass wool, curled Gypsum in powdered form Mineral wool, fibrous
Thermal conductivity, Btu/(ft)(hr)(°R) °R
Kovar
500 600 700 800 900
2-335
Advance
Monel
Hastelloy A
Inconel
Nichrome V
7.8 8.3 8.6 8.7 8.7
11.4 12.6 13.9 15.1
9.0 10.2 11.2 12.3 13.4
5.6 6.2 6.8 7.3 7.8
6.0 6.5 7.0 7.6 8.1
5.5 6.1 6.7 7.3 7.8
Sawdust Wood shavings, from planer Semiflexible materials in sheet form: Flax fiber Semirigid materials in board form: Corkboard
1000 1100 1200 1300 1400
8.9 9.2 9.5 9.8 10.2
16.4 17.6 18.8 20.0 21.2
14.4 15.4 16.5 17.6 18.7
8.4 9.0 9.5 10.1 10.7
8.6 9.1 9.7 10.2 10.8
8.4 9.0 9.5 10.1 10.7
Mineral wool, block, with binder Stiff fibrous materials in sheet form: Wood pulp Sugar-cane fiber Cellular gypsum
1500 1600 1700 1800 1900
10.5 10.8 11.1 11.3 11.5
22.5 23.8 25.0 26.2 27.4
19.8 20.8 21.9 23.0 24.0
11.3 11.8 12.3 12.9 13.4
11.3 11.8 12.4 13.0 13.6
11.3 11.9 12.4 13.0 13.5
2000 2100 2200
11.8 12.1 12.3
28.7 30.0
25.1 26.1 27.2
14.0 14.6 15.1
14.0 14.5 15.0
14.1 14.7 15.3
Apparent density, lb/cu ft room temp.
k
2.2 3.4–4.6 11–13 4.9 7.8 6.1–6.3 10.0
0.023 0.021–0.022 0.022 .023 .023 0.022–0.023 0.031
8–9 15.2 10.6 4–10 26–34 6 10 14 18 12 8.8
.025 .031 .026 .024 0.043–0.05 0.0217 .0225 .0233 .0242 .034 .034
13.0
.026
7.0 10.6 16.7
.0225 .025 .031
16.2–16.9 13.2–14.8 8 12 18 24 30
.028 .028 .029 .037 .049 .064 .083
*Abstracted from U.S. Bur. Standards Letter Circ. 227, Apr. 19, 1927.
*Silverman, J. Metals, 5, 631 (1953). Copyright American Institute of Mining, Metallurgical and Petroleum Engineers, Inc.
TABLE 2-378
Thermal Conductivities of Insulating Materials at High Temperatures* k = Btu/(h⋅ft2)(°F/ft)
Material Laminated asbestos felt (approx. 40 laminations per in) Laminated asbestos felt (approx. 20 laminations per in) Corrugated asbestos (4 plies per in) 85% magnesia (density, 13 lb/ft3) Diatomaceous earth, asbestos and bonding material Diatomaceous earth brick Diatomaceous earth brick Diatomaceous earth brick Diatomaceous earth powder (density, 18 lb/ft3) Rock wool
For temperatures, °F up to 700 500 300 600 1600 1600 2000 2500
Mean temperatures, °F 100
200
300
400
500
0.033 .045 .050 .034 .045 .054 .127 .128 .039 .030
0.037 .050 .058 .036 .047 .056 .130 .131 .042 .034
0.040 .055 .069 .038 .049 .058 .133 .135 .044 .039
0.044 .060
0.048 .065
.040 .050 .060 .137 .139 .048 .044
.053 .063 .140 .143 .051 .050
Asbestos cement, 1.2; 85% magnesia cement, 0.05; asbestos and rock wool cement, 0.075 approx. *Marks, “Mechanical Engineers’ Handbook,” 4th ed., McGraw-Hill, New York, 1941.
600
800
1000
0.055 .065 .143 .148 .054 .057
0.060 .069 .150 .155 .061
0.065 .073 .158 .163 .068
1500
2000
0.176 .183
0.203
2-336
PHYSICAL AND CHEMICAL DATA
TABLE 2-379 Thermal Conductivities of Insulating Materials at Moderate Temperatures (Nusselt)* k = Btu/(h⋅ft2)(°F/ft) Weight, lb/cu ft
Material Asbestos Burned infusorial earth for pipe coverings Insulating composition (loose) Cotton Silk hair Silk Wool Pulverized cork Infusorial earth (loose)
TABLE 2-380 Thermal Conductivities of Insulating Materials at Low Temperatures (Gröber)* k = Btu/(h⋅ft2)(°F/ft)
Temperatures, °F 32
100
200
300
400
600
800
36.0
0.087 0.097 0.110 0.117 0.121 0.125 0.130
12.5
.043 .046 .052 .057 .062 .073 .085
25.0 5.0 9.1 6.3 8.5 10.0 22.0
.040 .032 .026 .025 .022 .021 .035
.046 .035 .030 .028 .027 .026 .039
.050 .053 .055 .039 .034 .034 .033 .032 .045 .047 .050 .053
Temperatures, °F
Material
Weight, lb/cu ft
32
−50
−100
−200
−300
Asbestos Asbestos Cotton Silk
44.0 29.0 5.0 6.3
0.135 .0894 .0325 .0290
0.132 .0860 .0302 .0256
0.130 .0820 .0276 .0235
0.125 .0720 .0235 .0196
0.100 .0545 .0198 .0155
*Marks, Mechanical Engineers’ Handbook, 4th ed., McGraw-Hill, New York, 1941.
*Marks, Mechanical Engineers’ Handbook, 4th ed., McGraw-Hill, New York, 1941.
TABLE 2-381
Thermal Diffusivity (m2/s) of Selected Elements* Temperature, K
Element
20
40
60
80
100
Aluminum Beryllium Chromium Copper Gold
0.50
200
400
600
800
1000
0.012
0.0014
4.4. − 4
0.038 0.16 0.005
0.0037 0.0040 4.5. − 4
5.9. − 4 6.9. − 4 2.3. − 4
2.0. − 4 3.1. − 4 1.8. − 4
2.3. − 4 0.0036 1.2. − 4 2.2. − 4 1.5. − 4
1.1. − 4 1.5. − 4 4.1. − 5 1.3. − 4 1.3. − 4
9.4. − 5 4.0. − 5 2.6. − 5 1.1. − 4 1.2. − 4
8.4. − 5 2.6. − 5 2.0. − 5 1.0. − 4 1.2. − 4
7.4. − 5 2.1. − 5 1.7. − 5 9.0. − 5 1.1. − 4
6.6. − 5 1.7. − 5 1.4. − 5 9.0. − 5 9.8. − 5
Iridium Iron Lead Molybdenum Nickel
0.046 0.043 9.3. − 5 0.0095 0.033
3.2. − 3 3.9. − 5 0.0014 0.0017
4.9. − 4 3.3. − 5 4.0. − 4 3.1. − 4
1.6. − 4 3.1. − 5 2.0. − 4 1.3. − 4
8.4. − 5 8.2. − 5 2.9. − 5 1.3. − 4 8.0. − 5
5.6. − 5 3.1. − 5 2.6. − 5 6.3. − 5 3.1. − 5
4.8. − 5 1.8. − 5 2.3. − 5 5.1. − 5 1.9. − 5
4.4. − 5 1.3. − 5 2.0. − 5 4.5. − 5 1.3. − 5
4.1. − 5 1.1. − 5 1.3. − 5 4.2. − 5 1.4. − 5
3.5. − 5 1.0. − 5 1.5. − 5 3.8. − 5 1.5. − 5
Platinum Silver Zinc
0.0029 0.031 0.0046
1.6. − 4 0.0013 3.1. − 4
6.3. − 5 4.5. − 4 1.0. − 4
4.3. − 5 2.8. − 4 7.0. − 5
3.6. − 5 2.3. − 4 5.5. − 5
2.7. − 5 1.8. − 4 4.7. − 5
2.5. − 5 1.7. − 4 3.9. − 5
2.5. − 5 1.6. − 4 3.4. − 5
2.5. − 5 1.5. − 4 1.8. − 5
2.5. − 5 1.4. − 4 2.2. − 5
*Tables for up to 24 temperatures for 47 elements appear in the Handbook of Heat Transfer, 2d ed., McGraw-Hill, New York, 1984. The notation 3.2. − 4 signifies 2.3 × 10−4.
PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-382
2-337
Thermophysical Properties of Selected Nonmetallic Solid Substances Density, kg/m3
Material Alumina Asphalt Bakelite Beryllia Brick
3975 2110 1300 3000 1925
Brick, fireclay Carbon, amorphous Clay Coal Cotton
2640 1950 1460 1350 80
Diamond Granite Hardboard Magnesite Magnesia
3500 2630 1000 3025 3635
Emissivity
Specific heat, kJ/(kg⋅K)
Thermal conductivity, W/(m⋅K)
0.82 0.93
0.765 0.920 1.465 1.030 0.835
36 0.06 1.4 270 0.72
11.9 0.03 0.74 88 0.45
0.960 0.724 0.880 1.26 1.30
1.0 1.6 1.3 0.26 0.06
0.39 1.13 1.01 0.15 0.58
0.509 0.775 1.38 1.13 0.943
2300 2.79 0.15 4.0 48
1290 1.37 0.11 1.2 14
0.93 0.86 0.91 0.80
0.38 0.72
Oak Paper Pine Plaster board Plywood
770 930 525 800 540
0.90 0.83 0.84 0.91
2.38 1.34 2.75
Pyrex Rubber Rubber, foam Salt Sandstone
2250 1150 70
0.835 2.00
2150
0.92 0.92 0.90 0.34 0.59
Silica Sapphire Silicon carbide Soil
3975 3160 2050
Teflon Thoria Urethane foam Vermiculite
2200 4160 70 120
Thermal diffusivity, m2/s × 106
0.18 0.011 0.12 0.17 0.12
0.10 0.01 0.54
0.74 0.09
0.854 0.745
1.4 0.2 0.03 7.1 2.9
0.79 0.48 0.86 0.38
0.743 0.765 0.675 1.84
1.3 46 490 0.52
15 230 0.14
0.92 0.28
0.35 0.71 1.05 0.84
0.26 14 0.03 0.06
0.34 4.7 0.36 0.60
1.22
0.18
1.8
NOTE: Difficulties of accurately characterizing many of the specimens mean that many of the values presented here must be regarded as being of order of magnitude only. For some materials, actual measurement may be the only way to obtain data of the required accuracy. To convert kilograms per cubic meter to pounds per cubic foot, multiply by 0.062428; to convert kilojoules per kilogram-kelvin to British thermal units per pound-degree Fahrenheit, multiply by 0.23885.
PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES INTRODUCTION In the absence of reliable experimental data, the methods presented here provide physical property estimates that are sufficiently accurate for many engineering applications. These techniques have been selected on the basis of accuracy, generality, and, in most cases, simplicity; they are divided into 11 categories: (1) pure component constants: critical properties, normal freezing and boiling temperatures, acentric factor, radius of gyration, dipole moment, and van der Waals area and volume; (2) vapor pressure; (3) ideal gas thermal properties: heat capacity and enthalpy, Gibbs energy, and entropy of formation; (4) enthalpy of vaporization and fusion; (5) solid and liquid heat capacity; (6) vapor, liquid, and solid density; (7) vapor and liquid viscosity; (8) vapor and liquid thermal conductivity; (9) vapor and liquid diffusivity; (10) surface tension; and (11) flammability properties: flash point, flammability limits, and autoignition temperature. The definition of the property and limitations and accuracy of each method of correla-
tion or prediction are given for each property. Numerical examples are included for many of the methods. Equation symbols are listed under “Nomenclature,” and literature citations, indicated by superscript numbers, follow the nomenclature under “References.” Essentially all of the methods are derived from work on the American Petroleum Institute Technical Data Book 23 (hydrocarbon compounds and their mixtures), the AIChE Design Institute for Physical Property Data (DIPPR) Data Prediction Manual 22 (nonhydrocarbon compounds and their mixtures), and the DIPPR Data Compilation Project 24. UNITS Applicable dimensional units are shown individually with each equation. The International Metric System (SI) is used when feasible; otherwise commonly used U.S. engineering units are employed. The reader is referred to Sec. 1 for unit conversion factors.
2-338
PHYSICAL AND CHEMICAL DATA Nomenclature Symbol
Definition
SI units
U.S. customary units
B Cp Cv d D F k K ln M MeABP N NA P P sat [P] rel den R R S T V x y Z
Second virial coefficient Heat capacity at constant pressure Heat capacity at constant volume Mass density Diffusivity Conductivity factor in Eq. (2-136) Thermal conductivity Watson/UOP characterization factor = (1.8Tb)1/3/rel den Denotes natural logarithm Molecular weight Mean average boiling point Carbon number Avogadro’s number Pressure Vapor pressure Parachor Relative density at 15°C and 0.1 MPa Universal gas constant; e.g., 8314 Pa·m3/kmol K Radius of gyration Absolute entropy Absolute temperature Molar volume Mole fraction of component in liquid phase Mole fraction of component in vapor phase Compressibility factor = PV/RT
m3/kmol J/(kmol·K) kg/m3 m2/s
ft3/lbmol Btu/(lbm °F) lbm/ft3 ft2/s
W/(m·K)
Btu/(h·ft °F)
Pa Pa
lbf/in2 lbf/in2
J/kg K m3/kmol
Btu/lbm R ft3/lbmole
∆Gf ∆Hf ∆Hfus ∆HV ∆Sf ∆Sfus ∆ZV
J/kg J/kg J/kg J/kg J/(kg·K) J/(kg·K)
Btu/lbm Btu/lbm Btu/lbm Btu/lbm Btu/lbm Btu/lbm
Θ λ µ ν ρ σ φ ω
Gibbs energy of formation Enthalpy (heat) of formation Enthalpy (heat) of fusion Enthalpy (heat) of vaporization Entropy of formation Entropy of fusion Difference of vapor and liquid compressibility factors defined in Eq. (2-55) Tb /Tc Dipole moment Absolute viscosity Kinematic viscosity Molar density Surface tension Volume fraction Acentric factor
Pa·s m2/s kmol/m3 N/m
lbm/(ft·s) ft2/s lb·mole/ft3 dyne/cm
r (0) (1) ° ′
At reference condition Simple spherical molecule, corresponding states Correction factor, corresponding states Of the ideal gas At atmospheric pressure
b bp c G HI i j L m mc mlt o pc r RA sat S V w
At normal boiling temperature At bubble point At critical point Of the gas/vapor Upper limit Component index, of the ith component Component index, of the jth component Of the liquid Of the mixture Mixture correspondence At melting temperature Of organic component Pseudocritical quantity Reduced quantity Rackett parameter Saturated Of the solid Of the vapor Of water
Greek symbols
Superscripts
Subscripts
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2-339
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Basic pure component constants required to characterize components or mixtures for calculation of other properties include the melting point, normal boiling point, critical temperature, critical pressure, critical volume, critical compressibility factor, acentric factor, and several other characterization properties. This section details for each property the method of calculation for an accurate technique of prediction for each category of compound, and it references other accurate techniques for which space is not available for inclusion. Critical Temperature The critical temperature of a compound is the temperature above which a liquid phase cannot be formed, no matter what the pressure on the system. The critical temperature is important in determining the phase boundaries of any compound and is a required input parameter for most phase equilibrium thermal property or volumetric property calculations using analytic equations of state or the theorem of corresponding states. Critical temperatures are predicted by various empirical methods according to the type of compound or mixture being considered. For pure hydrocarbons, the method of Ambrose2 is the most accurate and will also be useful for predicting critical pressure and volume. Equation (2-1) requires only the normal boiling point, Tb, and the molecular structure of the compound.
log Tc = A + B log10 (rel den) + C log Tb
(2-1)
Tc and Tb are the critical and normal boiling temperatures, respectively, expressed in kelvins. Values of ∆T from Table 2-383 are summed for each part of the molecule to yield ∆T (e.g., for isobutane, 3 × CH3 + 1 × CH). ∆(Platt no.) is equal to the Platt number of any alkyl chains in the molecule minus the Platt number of the n-alkane with the same number of carbon atoms. The Platt number is defined as the number of pairs of carbon atoms that are separated by three carbon-carbon bonds and is an indicator of branching (e.g., for 2,2,3trimethylpentane, the Platt number is 5). The Platt number of an n-alkane is the number of carbon atoms minus three (e.g., for n-octane, the Platt number is five). Errors in Tc average about 4 K for paraffins to C20 and other hydrocarbons to C14.
(2-2)
Tc and Tb are the critical and normal boiling temperatures, respectively, expressed in kelvins. The relative density (rel den) of the liquid at 15°C is 0.1 MPa. The regression constants A, B, and C are tabulated by family in Table 2-384. Errors average about 3 K. For pure nonhydrocarbon organics, the most accurate method for prediction of critical temperature for all compound groups is also the Ambrose2 method. Equation (2-1) applies to all nonhydrocarbon compounds except perfluorocarbons, where the constant 1.242 is replaced by 1.570. For compounds containing any of C, H, O, N, S, or halogens up to C13 and ranging in critical temperature from 228– 790 K, the average error is about 6 K. Alternate methods for nonhydrocarbon organics are the first order method of Lydersen63 with an average error of 9 K although the method of Ambrose is considerably better for alcohols and ketones. Equation (2-3) is the Lydersen equation for critical temperature and requires only the normal boiling point and the molecular structure for solution. Tb Tc = [0.567 + ∆T − ( ∆T)2]
(2-3)
Contributions to ∆T are given in Table 2-385. For pure inorganic compounds, the method of Gambill31 was modified to yield Eq. (2-4) and only requires the normal boiling point as input. Tc = 1.64Tb
PURE COMPONENT CONSTANTS
1 Tc = Tb 1 + 1.242 + ∆T − 0.023∆(Platt no.)
Equation (2-2), another somewhat simpler method for estimating the critical temperature of pure hydrocarbons only, is the method of Nokay76 and requires the normal boiling point, the relative density, and the compound family.
(2-4)
Although this equation was tested with available experimental data (38 compounds), it can only be considered a rough approximation. Inorganic-organic and inorganic-halide compounds are predicted better by replacing the constant 1.64 in Eq. (2-4) by 1.55. For both hydrocarbons and nonhydrocarbon organic defined mixtures, the method of Li60 is used with a relatively simple volumetric average mixing rule as shown in Eq. (2-5) to calculate the true critical temperature. yjVc j Tcm = Tc j (2-5) j
yiVc i
i
Tcm is the mixture critical temperature in K. Vc is the critical volume of a component, m3/kmole. The mole fraction of a component is y. The mixture contains i components. For hydrocarbon systems, the average error is about 3 K, while for systems containing nonhydrocarbons, the average error is 15 K with the highest errors occurring where simple gases are present. The method of Chueh and Prausnitz18 yields average errors slightly lower than the method of Li, but it is computationally more complex. Critical Pressure The critical pressure of a compound is the vapor pressure of the compound at the critical temperature. Below the critical temperature, any compound above its vapor pressure will be a liquid. The critical pressure is required for calculations discussed in the part of the section on critical temperature. For pure hydrocarbons, the method of Ambrose2 is the most accurate. Equation (2-6) requires only the molecular weight (M) and the molecular structure of the compound. 0.101325M Pc = [0.339 + ∆P − 0.026∆(Platt no.)]2
(2-6)
Pc is the critical pressure, MPa. Values of ∆P from Table 2-383 are summed for each part of the molecule to yield ∆P. Calculation of the Platt number is discussed under “Critical Temperature.” Errors in Pc average 0.07 MPa and are less reliable for compounds with 12 or more carbon atoms.
TABLE 2-383
Group Increments for the Ambrose Method ∆T
∆P
∆V
Group description
CH3
0.138
0.2260
55.1
Aromatic Compounds (Cont.)
CH2
0.138
0.2260
55.1
CH
0.095
0.2200
47.1
C
0.018
0.1960
38.1
CH2
0.113
0.1935
45.1
CH
0.113
0.1935
45.1
C
0.070
0.1875
37.1
C
0.088
0.1610
35.1
CH
0.038
0.1410
35.1
C
0.038
0.1410
35.1
Group description
Ring Increments CH2
0.090
0.1820
44.5
CH
0.090
0.1820
44.5
CH
in fused ring *
0.030
0.1820
44.5
C
0.090
0.1820
44.5
CH
0.075
0.1495
37.0
C
0.075
0.1495
37.0
C
0.060
0.1170
29.5
Aromatic Compounds 0.458
0.448
0.488
0.9240
0.8940
0.9440
222
222
222
0.488
0.9440
222
0.438
0.8640
222
0.478
0.9140
0.8340
∆P
∆V
0.468
0.8840
222
0.468
0.8840
222
0.418
0.8040
222
0.368
0.7240
222
0.220
0.5150
148
in fused ring* Nonring Increments OH
Use Eq. (a), below Use Eq. (b), below
O
0.138
0.160
CO
0.220
0.282
0.220
0.220
0.578
0.450
1.156
0.900
CHO COOH CO
O
CO
O
OC
0.330
0.470
NO2
0.370
0.420
NH2
0.208
0.095
NH
0.208
0.135
N
0.088
0.170
CN
0.423
0.360
S
0.105
0.270
SH
0.090
0.270
Si
0.138
0.461
SiH
0.371
0.507
SiH3
0.195
—
0.159
0.725
0.131
0.663
222 Si
0.428
∆T
222
Si
O
O
cyclic
2-341
2-342
PHYSICAL AND CHEMICAL DATA
TABLE 2-383
Group Increments for the Ambrose Method (Concluded ) ∆T
∆P
−0.050 −0.200 0.180 (1st) 0.055 0.110 (1st) 0.055 0.110 (1st) 0.055
−0.065 −0.170 0.223 0.318 0.500
0.448 0.448 0.198 0.220 0.080
0.924 0.850 −0.025 0.515 0.183
Group description Double Bond Triple Bond F Cl Br Aromatic Compounds Corrections benzene pyridine OH C4H4 (fused ring) F *Group contributions for
CH and
∆V
∆T
∆P
0.080 0.080
0.318 0.600 0.850
0.010
0
0.030
0.020
−0.040 −0.080
−0.050 0
Group description Cl Br I The single or first substituent on an aromatic ring The second or subsequent ring substituents Each pair of ring substituents in ortho positions with respect to each other If one of the ortho pair is OH
∆V
in fused rings have been calculated from minimal data and may be less reliable than the other values.
Substituent increments do not apply to halogens. Eq. (a): ∆T (OH) = 0.87 − 0.11n + 0.003n2 Eq. (b): ∆p (OH) = 0.100 − 0.013n n = carbon number of compound. For branched alcohols, an effective carbon number can be determined by interpolation between the normal alcohol of the same carbon number and the immediately lower normal alcohol using normal boiling points. (See Example 1.)
TABLE 2-384
∆T = 2(.138) + .138 + .095 + .558 = 1.067 1 Tc = 372.7 1 + 1.242 + 1.067 − 0
Nokay Equation
Type compound
A
B
C
Paraffin Naphthene Olefin Acetylene Diolefin Aromatic
1.35940 0.65812 1.09534 0.74673 0.1384 1.0615
0.43684 −0.07165 0.27749 0.30381 −0.39618 0.22732
0.56224 0.81196 0.65563 0.79987 0.99481 0.66929
Tc = 534.1 K An accurate experimental value is 536.05 K.
∆P = 2(.226) + .226 + .220 + .0597 = 0.9577 .101325(74.12) Pc = 2 (.339 + .9577 − 0)
Example 1 Estimate the critical temperature and critical pressure of 2-butanol using the Ambrose method, Eqs. (2-1) and (2-6). The experimental normal boiling point is 372.7 K. 1 Tc = Tb 1 + 1.242 + ∆ T − 0.023∆ (Platt no.)
0.101325M Pc = [0.339 + ∆P − 0.026∆ (Platt no.)]2 Determine group contributions from Table 2-383 for a structure CH3CHOHCH2CH3: Number of groups
∆T
∆P
CH3
2
0.138
0.2260
CH2
1
0.138
0.2260
CH–
1
0.095
0.2200
OH
1
(a)
(b)
Group
Pc = 4.467 MPa An accurate experimental value is 4.179 MPa. For pure nonhydrocarbon organics, the simplest accurate method for prediction of critical pressure is the method of Lydersen.63 Equation (2-7) requires the molecular weight (M) and the molecular structure of the compound. 0.101325M Pc = (2-7) (0.34 + ∆P)2 Pc is the critical pressure, MPa. Values of ∆P from Table 2-385 are summed to yield ∆P. The average error in Pc is about 0.2 MPa when tested on compounds ranging in carbon number from C1 to C10.
Example 2 Estimate the critical temperature and critical pressure of 2-butanol, which has an experimental normal boiling point of 372.7 K. Use the Lydersen method, Eqs. (2-3) and (2-7). The structure of 2-butanol is CH3CHOHCH2CH3. Determine group contributions from Table 2-385. Group
Number of groups
∆T
∆P
CH3
2
0.020
0.227
CH2
1
0.020
0.227
To determine n, the normal boiling points of 1-propanol (370.3 K) and 1-butanol (390.9 K) are required:
CH
1
0.012
0.210
372.7 − 370.3 n = 3 + (4 − 3) = 3.1 390.9 − 370.3
OH
1
0.082
0.06
∆T°(OH) = 0.87 − 0.11(n) + 0.003 n = 0.558 ∆p°(OH) = 0.100 − 0.013(n) = 0.0597 2
The Platt number of the compound is determined by substituting a CH3 group for the OH. CH3CH(CH3) − CH2CH3, Platt no. = 2 The Platt number of the n-alkane of the same carbon number is 5 − 3 = 2. Thus, ∆(Platt no.) = 2 − 2 = 0. Therefore:
0.154 Tb 372.7 Tc = 2 = 2 0.567 + ∆T − ( ∆T) .567 + .154 − (.154) Tc = 534.5 K
0.951
PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-385
Group Increments for the Lydersen Method Incremental contributions
Group description Nonring Elements CH3, CH2 CH
∆T
∆P
∆V
Incremental contributions ∆P
∆V
0.085
(0.4)
0.080
0.047
0.47
0.080
(0.02)
(0.12)
(0.011)
0.018 0.017 0.010 0.012
0.224 0.320 (0.50) (0.83)
0.018 0.049 (0.070) (0.095)
0.031 0.031
0.095 0.135
0.028 (0.037)
(0.024)
(0.09)
(0.027)
0.014
0.17
(0.042)
(0.007)
(0.13)
(0.032)
CN
(0.060)
(0.36)
(0.080)
NO2
(0.055)
(0.42)
(0.078)
0.015 0.015 (0.008) (0.003)
0.27 0.27 (0.24) (0.24)
0.055 0.055 (0.045) (0.047)
Si
0.026
0.468
—
SiH
0.040
0.513
—
SiH3
0.027
—
—
0.025
0.730
—
0.027
0.668
—
Group description
C
0.020
0.227
0.055
0.012
0.210
0.051
0.00
C CH,
0.210
0.041
CH
0.018
0.198
0.045
C
0.00
0.198
0.036
0.005
0.153
(0.036)
0.013
0.184
0.0445
, C
Ring Increments CH2
∆T
Oxygen Increments (Cont.) COOH (acid) COO
CH2,
2-343
(ester)
O (except for combinations above) Halogen Increments F Cl Br I Nitrogen Increments NH2 NH (nonring) NH (ring)
0.012
CH
0.192
0.046 N
(−0.007)
C
0.154
(0.031) N
C
,
C
CH
0.011
0.154
0.036
0.011
0.154
0.037
0.066
0.924
Organometallic Increments
OH (alcohols)
0.082
0.06
(0.018)
OH (phenols)
0.031
(−0.02)
(0.003)
O
(nonring)
0.021
0.16
0.020
O
(ring)
(0.014)
(0.12)
(0.008)
O (nonring)
C
O (ring)
HC
O (aldehyde)
(ring)
Sulfur Increments SH S (nonring) S (ring) 苷S
Oxygen Increments
C
(nonring)
0.040
0.29
0.060
(0.033)
(0.2)
(0.050)
0.048
0.33
0.073
Values in parentheses are based on too few experimental points to be reliable.
Si
O
Si
O
cyclic
2-344
PHYSICAL AND CHEMICAL DATA TABLE 2-386 Atomic and Structural Contributions for the Fedors Method
The accurate experimental critical temperature is 536.05 K. 0.101325M (0.101325)(74.12) = Pc = (0.34 + ∆P)2 (0.34 + 0.951)2
Atomic increments
Pc = 4.506 MPa The accurate experimental critical pressure is 4.179 MPa. No known method is available to predict the critical pressure of inorganic compounds. For both hydrocarbon and nonhydrocarbon organic defined mixtures, the method of Kreglewski and Kay52 is recommended. The critical temperature, critical pressure, and acentric factor of each compound and the critical temperature of the mixture must be known or predicted from the methods of this section.
Tcm − Tpc
x ω T
Pcm = Ppc + Ppc 5.808 + 4.93
n
i
i
i=1
∆V
Feature
∆V
C H O O (alcohols) N N (amines) F
0.034426 0.009172 0.020291 0.018000 0.048855 0.047422 0.022242
Si Sisiloxane Sicyclic siloxane
0.086174 0.126483 0.126483
Cl Br I S 3-membered ring 4-membered ring 5-membered ring 6-membered ring double bond triple bond ring attached directly to another ring
0.052801 0.071774 0.096402 0.050866 −0.105824 −0.017247 −0.039126 −0.039508 +0.005028 +0.000797 +0.035524
(2-8)
pc
Use of Eq. (2-8) requires the pseudocritical properties defined by Eqs. (2-9) and (2-10)
Structural increments
Atom
n
Tpc = xiTc i
(2-9)
i=1 n
Ppc = xiPc i
(2-10)
i=1
Each component i of the mixture must have available its Tc, Pc, and ω. Tcm can be predicted from Eq. (2-5). For hydrocarbon systems, average errors in predicted critical pressures are about 0.2 MPa, except when organic gases are present and errors are unacceptably large. Errors for nonhydrocarbon organics not including inorganic gases average 0.5 MPa.
Critical Volume The critical volume of a compound is the volume occupied by a set mass of a compound at its critical temperature and pressure. While useful in itself, the critical volume is extensively used in equations for estimating volumetric fractions. For pure hydrocarbons, two methods are quite accurate. The Ambrose2 method used for Tc and Pc is also used for critical volume. Eq. (2-11) only requires the molecular structure of the compound. Vc = 10−3(40 + ∆V)
(2-11)
Vc is the critical volume, m3/kmole. Values of ∆V are given in Table 2-383. The average error for hydrocarbons of twelve or less carbon atoms is about 0.01 m3/kmole. The Riedel method90 requires the critical temperature (Tc), critical pressure (Pc), and acentric factor (ω) of the compound as given by Eqs. (2-12) and (2-13). If the gas constant is in Pa·m3/kmole·K, the critical volume will be in m3/kmole. RTc Vc = (2-12) Pc[3.72 + 0.26(α − 7.00)] α = 5.811 + 4.919ω
(2-13)
The average error for paraffins up to C18 and other hydrocarbon families up to C11 is about 0.015 m3/kmole. If estimated values of Tc and Pc are used, errors may be higher. For pure nonhydrocarbon organics, the method of Fedors,29 which requires only molecular structure, is the most accurate. Equation (2-14) shows the method to depend only on the molecular structure. The resulting Vc will be in m3/kmole. Vc = 0.0266 + ∆V
(2-14)
Values for ∆V are given in Table 2-386. The average error for compounds up to C7 is about 0.007 m3/kmole, although a maximum error of 0.03 m3/kmole has been noted. No experimental data above C7 are available for comparison. Example 3 Estimate the critical volume of 2-butanol. Use the method of Fedors, Eq. (2-14). Vc = 0.0266 + ∆V The molecular formula is C4H10O. Using the atomic contribution values from Table 2-386.
∆V = 4(.034426) + 10(.009172) + 1(.018000)
∆V = 0.2474 Vc = 0.0266 + 0.2474 = 0.2740 m3/kmole The accurate experimental critical volume is 0.2690 m3/kmole.
The method of Lydersen63 may also be used for prediction of critical volume, but it is not so accurate as the method of Fedors. Equation (2-15) depends only on molecular structure and gives a critical volume in m3/kmole. Vc = 0.040 + ∆V
(2-15)
Group contributions for ∆V are given in Table 2-385. Errors average about 0.01 m3/kmole. There is no known method for predicting the critical volume of inorganic compounds. For both hydrocarbon and nonhydrocarbon organic defined mixtures, the method of Chueh and Prausnitz19 is useful. For hydrocarbon systems, the mixing rule is shown by Eq. (2-16) for binaries and by Eq. (2-17) for multicomponents. Equations (2-18) through (2-20) give the input parameters. The mixture critical volume Vcm is a function of the pure component critical volumes. The constant C is zero for hydrocarbon systems and 0.1559 for systems containing a nonhydrocarbon gas. Vcm = φ1Vc1 + φ2Vc2 + 2φ1φ2ν12 n
n
i
j
Vcm = φiφ jνi j
(i ≠ j)
2/3 cj
(2-16) (2-17)
xjV φj = n
xiV 2/3ci
(2-18)
Vij(Vci + Vcj) νij = 2.0
(2-19)
i=1
Vci − Vcj (2-20) Vij = −1.4684 + C Vci + Vcj Errors average about 10 percent for systems containing hydrocarbons. Systems containing only organics or organics and gases may give very high errors. A specialized modification of the method (Chueh and Prausnitz18) is available for binary mixtures containing organics but may give errors over 20 percent.
Critical Compressibility Factor The critical compressibility factor of a compound is calculated from the experimental or predicted values of the critical properties by the definition, Eq. (2-21). PcVc Zc = (2-21) RTc Critical compressibility factors are used as characterization parameters in corresponding states methods (especially those of Lydersen) to predict volumetric and thermal properties. The factor varies from about 0.23 for water to 0.26–0.28 for most hydrocarbons to slightly above 0.30 for light gases. Normal Freezing Temperature (Melting Point) The melting point is the temperature at which melting occurs at atmospheric pressure. In most cases, measurements are made in air, making values slightly lower than if the measurements were made in vacuum. Impurities can cause a substantial decrease in the measured melting point. The melting point is very slightly higher than the triple point temperature—the temperature at which equilibrium exists between solid, liquid, and vapor—for a pure compound. For practical purposes, the two temperatures are equal. Reliable methods for predicting melting points have not until recently been advanced. Constantinou and
PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Gani20 derived a group contribution method that shows promise. Initial evaluations show average errors of 5 to 10 percent (10–30 K) on a wide variety of compounds, but larger errors can occur. It is recommended that several compounds of known melting point in the same or a similar family be predicted in order to estimate the probable error. Normal Boiling Temperature The normal boiling temperature (point) is the temperature at which the vapor pressure equals exactly 101,325 Pa (1 atmosphere). Caution should be taken in using values from older references, where the temperature may be reported for the prevailing pressure (0.95–0.97 atm) rather than at 1 atmosphere. If at least two values of vapor pressure very close to 1 atmosphere are available, the normal boiling point can be interpolated or extrapolated on a plot of log P sat vs. 1/T. The section on vapor pressure discusses this in more detail. Various methods are available for estimation of the normal boiling point of organic compounds. Lyman et al.64 review and give calculational procedures for the methods of Meissner, Miller, and Lydersen/ Forman-Thodos. A more recent method that has been determined to be more accurate is the method of Pailhes,80 which requires one experimental vapor pressure point and Lydersen group contributions for critical temperature and critical pressure (Table 2-385). log Pc + (1 − Θ) log (1/p) Tb = T (2-22) log Pc where T and p = the one low pressure vapor pressure point, in K and atm, respectively Θ = Tc /Tb calculated from Eq. (2-3) using Lydersen contributions Pc = critical pressure calculated from Eq. (2-7) using Lydersen contributions A recent study of the method on a wide variety of complex organics shows an overall average error of less than 2 percent (~10 K). If no vapor pressure point is available, the new group contribution method of Constantinou and Gani20 discussed under the section on melting point gives an overall average error of about 4 percent (~20 K) and may be useful. The method of Miller (Lyman et al.64), which requires only the molecular structure, has also been found to be relatively accurate for organics. Example 4 Estimate the normal boiling point of 2-butanol. One vapor pressure point of 0.802 psia at 100°F is available. Use the Pailhes method, Eq. (2-22). log Pc + (1 − θ) log (1/p) TB = T log Pc From Example 2, Pc = 4.506 MPa; θ = Tc /Tb = 1.434; T = 100°F = 310.9 K; and p = 0.802 psia = 5528 Pa. log (4.506 × 106) + (1 − 1.434) log (1/5528) Tb = (310.9) log (4.506 × 106) Tb = (310.9)(1.244) = 386.8 K An accepted experimental normal boiling point is 372.7 K. Note the error here is 3.8 percent, a value above the average. If the vapor pressure point available would have been closer to one atmosphere, the error would have been much lower.
Acentric Factor The acentric factor of a compound (ω) is primarily a measure of the shape of a molecule, though it also measures a molecule’s polarity. It is calculated from the reduced vapor pressure (Prsat ) at a reduced temperature of 0.7 by the definition, Eq. (2-23). ω = −log (Prsat)Tr = 0.7 − 1.000
(2-23)
Critical temperature and pressure are required and can be estimated from the methods of this section. Vapor pressure is predicted by the methods of the next section. Experimental values should be used if available. The acentric factor is used as a third parameter with Tc and Pc in Pitzer-type corresponding states methods to predict volumetric properties and in cubic equations of state such as the Redlich-KwongSoave and Peng-Robinson equations. For simple spherical molecules, the acentric factor is essentially zero, rising as branching and molecu-
2-345
lar weight increases. For compounds of similar size and shape the acentric factor increases slightly with increasing polarity. For mixtures, the acentric factor is usually taken as a simple molar average value of the n components of the mixture. n
ω = xiω i
(2-24)
i=1
Miscellaneous Characterizing Constants The radius of gyration (R ) is a simultaneous size-shape factor varying with the manner in which mass is distributed about the center of gravity of the molecule. For planar molecules, the radius of gyration is R=
(AB) N
M 1/2
A
(2-25)
For three-dimensional molecules, it is R=
(ABC) 2πN
M 1/3
A
(2-26)
AB and ABC are the products of the principal moments of inertia. Moments of inertia are calculated from bond angles and bond lengths. Many values are given by Landolt-Bornstein.53 NA is Avogadro’s number, and M is the molecular weight of the molecule. Stuper et al.105 give a computerized method for prediction of the radius of gyration. The dipole moment (λ) of a molecule is the first moment of the electric charge density of a molecule. Paraffins have dipole moments of zero, while dipole moments of almost all hydrocarbons are small. McClellan68 lists many dipole moments. The computer method of Dixon and Jurs27 is the most useful method for predicting dipole moments. Lyman et al.64 give other methods of calculation. The van der Waals volume and area are characterizing parameters relating molecular configurations. Bondi8 describes group contribution methods for their calculation. VAPOR PRESSURE Vapor pressure is the most important of the basic thermodynamic properties affecting liquids and vapors. The vapor pressure is the pressure exerted by a pure component at equilibrium at any temperature when both liquid and vapor phases exist and thus extends from a minimum at the triple point temperature to a maximum at the critical temperature, the critical pressure. This section briefly reviews methods for both correlating vapor pressure data and for predicting vapor pressure of pure compounds. Except at very high total pressures (above about 10 MPa), there is no effect of total pressure on vapor pressure. If such an effect is present, a correction, the Poynting correction, can be applied. The pressure exerted above a solid-vapor mixture may also be called vapor pressure but is normally only available as experimental data for common compounds that sublime. Correlation Methods Vapor pressure is correlated as a function of temperature by numerous methods mainly derived from the Clapeyron equation discussed in the section on enthalpy of vaporization. The classic simple equation used for correlation of low to moderate vapor pressures is the Antoine4 equation (2-27). B (2-27) ln Psat = A + T+C A, B, and C are regression constants for the specific compound. The Antoine equation does not fit data accurately much above the normal boiling point. Thus, as regression by computer is now standard, more accurate expressions applicable to the critical point have become usable. The entire DIPPR Compilation24 is regressed with the modified Riedel89 equation (2-28) with constants available for over 1500 compounds. B ln P sat = A + + C ln T + DT E (2-28) T A, B, C, and D are regression constants and E is an exponent equal to 1, 2, or 6 depending on which regression gives the most accurate fit of the data.
2-346
PHYSICAL AND CHEMICAL DATA
For purposes of the API Technical Data Book (Daubert and Danner23), another modified Riedel equation (2-29) was chosen and found to fit hydrocarbon data well over the entire pressure range. Coefficients are given for several hundred hydrocarbons. B E ln P sat = A + + C ln T + DT 2 + 2 (2-29) T T Both equations (2-28) and (2-29) are also extrapolatable above the critical temperature where necessary for thermodynamic calculations. The other modern equation used for correlation is the modified and linearized Wagner124 equation (2-30), which has the advantage that it will match critical data exactly, although it cannot be extrapolated above the critical point. The equation is also included with coefficients to several hundred compounds in the Technical Data Book—Petroleum Refining. ln Pr = aX1 + bX2 + cX3 + dX4 where
(2-30)
1−T (1 − Tr)1.5 (1 − Tr)2.6 (1 − Tr)5 X1 = r , X2 = , X3 = , X4 = Tr Tr Tr Tr Pr = P /Pc sat
Tr = T / Tc sat
(2-31)
(ln P ) = 5.92714 − 6.09648/Tr − 1.28862 ln Tr + 0.169347T 6r (2-32) sat (0) r
(1) (ln P sat = 15.2518 − 15.6875/Tr − 13.4721 ln Tr + 0.43577T 6r r ) (2-33)
The method is applicable at reduced temperatures above 0.30 or the freezing point, whichever is higher, and below the critical point. The method is most reliable when 0.5 < Tr < 0.95, where errors in prediction average 3.5 percent when experimental critical properties are known. Errors are higher for predicted criticals. The method is useful when solved iteratively with Eq. (2-23) to predict the acentric factor. Example 5 Estimate the vapor pressure of 1-butene at 98°C. Use Eq. (2-31): (1) sat (0) ln P sat + ω(ln P sat r = (ln P r ) r )
Pure component properties of 1-butene are Tc = 146.4°C, Pc = 4.02 MPa, and ω = 0.1867. 371.1 Tr = = 0.885 419.5 From Eq. (2-32):
(ln Prsat)(0) = −0.7227
From Eq. (2-33):
(ln Prsat)(1) = −0.6190
3000.538X − 6.761560 log Psat = 43X − 0.987672
ln Prsat = −.7227 + (.1867)(−0.6190) ln Prsat = −0.8383 Prsat = 0.4325 P sat = PrsatPc = (.4325)(4.02) = 1.74 MPa An experimental value is 1.72 MPa. When criticals cannot be estimated with reasonable accuracy, the method of Maxwell and Bonnell67 is recommended. The normal boiling point and the specific gravity at 60°F (15.5°C) are required inputs. According to what vapor pressure range is expected, the vapor pressure is calculated from Eqs. (2-34), (2-35), or (2-36). If the wrong range is selected, the procedure will need to be repeated.
(2-34)
For 0.0013 ≤ X ≤ 0.0022 (2 mm Hg ≤ P ≤ 760 mm Hg): sat
2663.129X − 5.994296 log Psat = 95.76X − 0.972546
(2-35)
For X < 0.0013 (P sat > 760 mm Hg): 2770.085X − 6.412631 log Psat = 36X − 0.989679
(2-36)
X is calculated from Eq. (2-37) and T′b is calculated from Eq. (2-38). Iterative calculation may be required. T ′b − 0.0002867(T′b) T X = 748.1 − 0.2145(T′b) (2-37) Psat Tb − T′b = 2.5f(K − 12) log 760 where
Both Riedel and Wagner regressions usually fit data within a few tenths of a percent over the entire range between the triple point and the critical point. Prediction Methods Two methods have gained almost universal acceptance for prediction of the vapor pressure of pure hydrocarbons. The method of Lee and Kesler55 is the preferred method if the critical temperature and the critical pressure of the hydrocarbon is known or can be reasonably predicted by the methods of the first section. The corresponding states method is shown in equation (2-31) with the simple fluid and correction terms to be calculated from equations (2-32) and (2-33), respectively, for any Tr. (1) sat (0) ln P sat + ω(ln P sat r = (ln P r ) r )
For X > 0.0022 (Psat < 2 mm Hg):
(2-38)
Tb = normal boiling point, °R T ′b = normal boiling point corrected to K = 12, °R T = absolute temperature, °R f = correction factor. For all subatmospheric vapor pressures and for all substances having normal boiling points greater than 400°F, f = 1. For substances having normal boiling points less than 200°F, f = 0. For superatmospheric vapor pressures of substances having normal boiling points between 200°F and 400°F, f is given by (Tb − 659.7)/200 K = Watson characterization factor, T b1/3/sp gr
Evaluation of the method for pure hydrocarbons shows errors averaging 8 percent for vapor pressures above 1 mm Hg and 30 percent below 1 mm Hg. The method is also usable for narrow boiling (range up to 50°F) undefined hydrocarbon mixtures with the only change being that the mean average boiling point replaces the normal boiling point in all calculations.
Example 6 Estimate the vapor pressure of tetralin at 150°C (302°F). Its normal boiling point is 207.6°C (405.7°F) and its Watson characterization factor is 9.78. Use the Maxwell-Bonnell method, Eq. (2-35). 2663.129X − 5.994296 log Psat = 95.76X − 0.972546 At 150°C with a normal boiling point of 207.6°C, a vapor pressure between 2 and 760 mm Hg would be expected. Assume T b′ = Tb = 405.7°F as a first trial. From Eq. (2-37): (865.7/762) − 0.0002867(865.7) X = 748.1 − 0.2145(865.7) X = 0.001579 2663.129(0.001579) − 5.994296 log Psat = = 2.179 95.76(0.001579) − 0.972546 Psat = 151.0 mm Hg Use Eq. (2-38) to calculate the correction. 151 Psat Tb − T′b = 2.5f(K − 12) log = 2.5(1)(9.48 − 12) log 760 760 Tb − T′b = 4.4° Thus, for the second trial, T′b = 405.7 − 4.4 = 401.3°F. Using Eq. (2-37), recalculate X. (861.3/762) − 0.0002867(861.3) X = = 0.001568 748.1 − 0.2145(861.3) From Eq. (2-35), log Psat = 2.214; Psat = 163.7 mm Hg. Use Eq. (2-38) to recalculate the correction. 163.7 Tb − T′b = 2.5(1)(9.48 − 12) log = 4.2° 760 Thus, T′b = 401.5°F. If greater accuracy is desired, carry out a third trial. (An experimental vapor pressure is 161.8 mm Hg.) For nonhydrocarbon organics, vapor pressures above 15 kPa for com-
PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES pounds of known or estimable normal boiling point are predicted using the method of Riedel89 given by Eq. (2-39). log P°r = φ(Tr) − (α − 7)ψ(Tr)
Use Eq. (2-44), determining parameters from Eqs. (2-45), (2-46), and (2-47). log Psat = m log Pwsat + C
(2-39)
at T1 = 256.55 K, P1sat = 2.00 × 104 Pa, log P1sat = 4.3010
Correlation functions φ(Tr), ψ(Tr), and ζ(Tr) are given by Eqs. (2-40), (2-41), and (2-42), respectively.
at T2 = 294.95 K, P2sat = 1.067 × 105 Pa, log P2sat = 5.0282 Use Eq. (2-45) to calculate the vapor pressure of water at T1 and T2.
φ(Tr) = 0.118ζ(Tr) − 7 log10 Tr
(2-40)
ψ(Tr) = 0.0364ζ(Tr) − log10 Tr
(2-41)
log Pωsat1 = 2.2282,
Pωsat1 = 169.12 Pa
ζ(Tr) = 36/Tr + 96.7 log10 Tr − 35 − Tr6
(2-42)
log Pωsat2 = 3.4213,
Pωsat2 = 2637.9 Pa
From Eq. (2-46): log P1sat − log P sat 4.3010 − 5.0282 2 m = = = 0.6093 log P wsat1 − log P wsat2 2.2282 − 3.4213 From Eq. (2-47):
The Riedel α is calculated from Eq. (2-43). 0.136ζ(Tr ) + log10 Pc − 5.01 α = 0.0364ζ(Tr ) − log10 Tr b
b
(2-43)
b
Critical properties, if not available, can be estimated from the methods of the previous section. Tr is the reduced temperature at the temperature of interest, while Tr is the reduced temperature at the normal boiling point. The method is accurate within 2 to 3 percent above 15 kPa, while errors increase to 10–30 percent at lower pressures. Care should be taken not to use the method below the freezing point temperature.
2-347
C = log P1sat − m log P wsat1 = 4.3010 − 0.6093(2.2282) C = 2.9434
b
T = 273.15 K (0°C) 3129.8 log P = 31.51 − − 7.1385 log 273.15 + 1.757 × 10−6(273.15)2 273.15 log P sat w = 2.7907
at
sat w
Example 7 Estimate the vapor pressure of thiophene at 500 K. Pure component properties are Tc = 579.4 K, Pc = 5.694 MPa, and Tb = 357.5 K. Use the Riedel method, Eq. (2-39).
log Psat = (0.6093)(2.7907) + 2.9434 log Psat = 4.6438
log P sat r = −φ(Tr) − (α − 7) ψ (Tr) 500 357.5 Tr = = 0.8629 Tr = = 0.6170 579.4 579.4 From Eq. (2-42): (36) ζ(Tr ) = + 96.7 log (0.6170) − 35 − (0.6170)6 = 3.01 (0.6170) From Eq. (2-43): 0.136(3.01) + log (5.694 × 106) − 5.01 α = (0.0364)(3.01) − log (0.6170)
Psat = 44,030 Pa = 44.0 kPa
b
IDEAL GAS THERMAL PROPERTIES
b
α = 6.749 From Eq. (2-42), calculate ζ(Tr) and then calculate φ(Tr) and ψ(Tr) from Eqs. (2-40) and (2-41). log Prsat = −0.461 − (6.749 − 7)(0.068) log Prsat = −0.444 Prsat = 0.3598 Psat = PrsatPc = (.3598)(5.694) = 2.049 MPa An experimental value is 2.037 MPa. For nonhydrocarbon organics for which normal boiling points are unknown or expected vapor pressures are below 15 kPa, the reference substance method of Othmer and Yu78 as given by Eq. (2-44) is recommended. log Psat = m log P wsat + C
(2-44)
The vapor pressure of water P wsat may be calculated by Eq. (2-45). 3.1298 × 103 log Pwsat = 31.51 − − 7.1385 log T + 1.757 × 10−6T 2 (2-45) T with temperatures in K and vapor pressures in Pa. Values of the compound specific constants m and c were originally derived by Othmer et al. and greatly expanded to over 600 common organics by Danner and Daubert.22 If constants are not available but any two vapor pressure data points are available, the constants m and C can be calculated using Eqs. (2-46) and (2-47). sat log P sat 1 − log P 2 m = (2-46) log P wsat1 − log P wsat2 sat C = log P sat 1 − m log P w1
(2-47)
where the subscripts 1 and 2 refer to the two reference temperatures T1 and T2. Average errors at low pressures for compounds with tabulated m and C are within a few percent. When values of m and C are calculated from only two vapor pressure points, the method should be used only for interpolation and limited extrapolation. The method is usable from about 220 K (so long as it is above the freezing point of the compound) to the critical point of water (about 647 K).
Example 8 Estimate the vapor pressure of acetaldehyde at 0°C. Two vapor pressure points are 20.0 kPa at 256.55 K and 107.6 kPa at 294.85 K.
A substance is in the ideal gas state when the volume of its molecules is a zero fraction of the total volume taken up by the substance and when the individual molecules are far enough apart from each other so that there is no interaction between them. Although this only occurs at infinite volume and zero pressure, in practice, ideal gas properties can be used for gases up to a pressure of two atmospheres with little loss of accuracy. Thermal properties of ideal gas mixtures may be obtained by mole-fraction averaging the pure component values. Heat Capacity, Cpo Heat capacity is defined as the amount of energy required to change the temperature of a unit mass or mole one degree; typical units are J/kg·K or J/kmol·K. There are many sources of ideal gas heat capacities in the literature; e.g., Daubert et al.,24 Daubert and Danner,23 JANAF thermochemical tables,15 TRC thermodynamic tables,115,116 and Stull et al.104 If Cpo values are not in the preceding sources, there are several estimation techniques that require only the molecular structure. The methods of Thinh et al.113 and Benson et al.6,7 are the most accurate but are also somewhat complicated to use. The equation of Harrison and Seaton36 for Cpo between 300 and 1500 K is almost as accurate and easy to use: Cpo = a1 + a2C + a3H + a4O + a5N + a6S + a7F + a8Cl + a9I + a10Br + a11Si + a12Al + a13B + a14P + a15E where
(2-48)
C = ideal gas heat capacity, J/mol K a1–a15 = constant parameters obtained from Table 2-387 as a function of temperature C = number of carbon atoms in the molecule H = number of hydrogen atoms in the molecule O = number of oxygen atoms in the molecule N = number of nitrogen atoms in the molecule S = number of sulfur atoms in the molecule F = number of fluorine atoms in the molecule Cl = number of chlorine atoms in the molecule I = number of iodine atoms in the molecule Br = number of bromine atoms in the molecule Si = number of silicon atoms in the molecule Al = number of aluminum atoms in the molecule B = number of boron atoms in the molecule P = number of phosphorus atoms in the molecule E = number of atoms in the molecule excluding the 13 atom-types listed above o p
2-348
PHYSICAL AND CHEMICAL DATA
TABLE 2-387
Values of the Constant Parameters a1–a15 in Eq. (2-48) at Different Temperatures
Temp., K
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
300 400 500 600 800 1000 1500
4.86 0.864 −1.85 −4.61 −7.49 −8.53 −7.37
9.04 12.6 15.5 17.5 20.1 21.6 23.9
5.69 7.37 8.89 10.5 13.1 15.2 17.9
11.4 13.9 15.7 17.5 19.4 20.4 20.6
11.9 14.0 16.0 17.3 19.4 20.4 21.1
15.3 17.0 19.4 20.3 22.3 22.9 22.5
12.7 16.2 17.9 20.1 21.5 22.4 22.1
16.8 18.9 20.2 21.4 22.4 22.8 22.6
18.7 20.5 22.1 23.3 25.0 25.4 24.6
17.8 19.9 21.2 22.4 23.4 23.8 23.0
14.6 17.5 19.6 20.9 23.2 23.9 24.1
15.8 18.3 20.0 21.1 22.3 22.8 23.2
11.5 14.7 17.0 18.3 20.8 22.2 24.2
18.0 20.9 21.6 22.8 23.0 23.4 24.2
19.5 20.8 21.7 22.1 23.0 23.3 23.3
Parameter a
n = number of different atomic groups contained in the molecule Ni = number of atomic groups i contained in the molecule ∆Gi = numeric value of atomic group i obtained from Table 2-388
Results and parameters may be interpolated between temperatures. Average errors are between 2 and 6 percent, with the higher errors at the lower temperatures. Example 9 Using Eq. 2-48 to estimate the ideal gas heat capacity of acetone (C3H6O) at 600 K: Cp° = −4.61 + (17.5)(3) + (10.5)(6) + (17.5)(1) = 128.39 J/mol K
Average errors of 8 to 9 kJ/mol may be expected.
Daubert et al.24 report a value of 121.8 J/mol K.
Enthalpy of Formation The ideal gas standard enthalpy (heat) of formation (∆H°f 298) of a chemical compound is the increment of enthalpy associated with the reaction of forming that compound in the ideal gas state from the constituent elements in their standard states, defined as the existing phase at a temperature of 298.15 K and one atmosphere (101.3 kPa). Sources for data are Refs. 15, 23, 24, 104, 115, and 116. The most accurate, but again complicated, estimation method is that of Benson et al.6,7 A compromise between complexity and accuracy is based on the additive atomic group-contribution scheme of Joback44; his original units of kcal/mol have been converted to kJ/mol by the conversion 1 kcal/mol = 4.1868 kJ/mol:
Example 11 The ∆G°f 298 of phenol is estimated using Table 2-388. The molecular groups are
(2-49)
i=1
where ∆H°f 298 = enthalpy of formation at 298.15 K, kJ/mol n = number of different atomic groups contained in the molecule Ni = number of atomic groups i contained in the molecule ∆Hi = numeric value of atomic group i obtained from Table 2-388. Average expected errors are about 9 kJ/mol. For other temperatures: ∆H°f T = ∆H°f 298 +
T
C°p dT
(2-50)
298
C
1
OH (phenol)
(both ring)
∆G°f 298 = 53.88 + 5(11.30) + (54.05) + (−197.37) = −32.94 kJ/mol
∆G°f T = ∆H°f T − T ∆S°f T
where
(2-52)
∆G°f T = Gibbs energy of formation at T, kJ/mol ∆H°f T = enthalpy of formation at T, kJ/mol (see above) ∆S°f T = entropy of formation at T, kJ/mol K (see below)
Entropy of Formation The ideal gas standard entropy of formation (∆S°f 298) of a chemical compound is the increment of entropy associated with the reaction of forming that compound in the ideal gas state from the constituent elements in their standard state defined as the existing phase at a temperature of 298.15 K and one atmosphere (101.325 kPa). Thus: n
r ∆S°f 298 = S°compound − Ni Selement i
Example 10 The ∆H°f 298 of 2-butanol is estimated using Table 2-388. The molecular groups are 2CH3, 1CH2, 1CH (all nonring), and 1OH (alcohol). Therefore: ∆H°f 298 = 68.29 + 2(−76.45) + (−20.64) + (29.89) + (−208.04) = −283.40 kJ/mol The value from Daubert et al.24 is −292.9 kJ/mol.
Gibbs Energy of Formation The ideal gas standard Gibbs energy of formation (∆G°f 298) of a chemical compound is the increment of Gibbs energy associated with the reaction of forming that compound in the ideal gas state from the constituent elements in their standard state defined as the existing phase at a temperature of 298.15 K and one atmosphere (101.325 kPa). Refs. 15, 23, 24, 104, 115, and 116 are good sources of data. The additive atomic groupcontribution scheme of Joback44 may be used to estimate ∆G°f 298; his original units of kcal/mol have been converted to kJ/mol by the conversion 1 kcal/mol = 4.1868 kJ/mol: n
(2-51)
i=1
where
1
The value from Daubert et al.24 is −32.64 kJ/mol. For other temperatures, the exact Eq. (2-52) may be used at temperature T (K):
See above for discussion of the ideal gas heat capacity (C°p ).
∆G°f 298 = 53.88 + Ni ∆Gi
CH
Therefore,
n
∆H°f 298 = 68.29 + Ni ∆Hi
5
∆G°f 298 = Gibbs energy of formation at 298.15 K, kJ/mol
(2-53)
i=1
where
∆S°f 298 = entropy of formation at 298.15 K and 1 atm, J/mol K S°compound = ideal gas absolute entropy of the compound at 298.15 K and 1 atm, J/mol K n = number of different elements contained in the compound Ni = moles of element i contained in one mole of compound r Selement i = absolute entropy of element i in its standard state at 298.15 K and 1 atm, J/mol K.
Ideal gas absolute entropies of many compounds may be found in Daubert et al.,24 Daubert and Danner,23 JANAF Thermochemical Tables,15 TRC Thermodynamic Tables,115,116 and Stull et al.104 Otherwise, the estimation method of Benson et al.6,7 is reasonably accurate, with average errors of 1–2 J/mol K. Elemental standard-state absolute entropies may be found in Cox et al.21 Values from this source for some common elements are listed in Table 2-389. ∆S°f 298 may also be calculated from Eq. (2-52) if values for ∆H°f 298 and ∆G°f 298 are known.
PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-388
Atomic Group Contributions to Estimate ⌬H f°298 and ⌬G f°298 ∆H
Nonring Increments CH3
∆G
−76.45
−43.96
CH2
−20.64
8.42
CH
29.89
58.36
∆H
∆G
Oxygen Increments (Cont.) CHO (aldehyde)
−162.03
−143.48
COOH (acid)
−426.72
−387.87
COO
−337.92
−301.95
−247.61
−250.83
−22.02
14.07
NH (nonring)
53.47
89.39
NH (ring)
31.65
75.61
(ester)
O (except for above) C
82.23
116.02
CH2
−9.63
3.77
CH
37.97
48.53
C
83.99
92.36
C CH C Ring Increments CH2 CH
C
123.34
163.16
(nonring)
23.61
—
79.30
77.71
N
(ring)
55.52
79.93
115.51
109.82
NH
93.70
119.66
−26.80
−3.68
CN
88.43
89.22
NO2
−66.57
−16.83
8.67
40.99
Sulfur Increments SH S (nonring) S (ring)
−17.33 41.87 39.10
−22.99 33.12 27.76
−251.92 −71.55 −29.48 21.06
−247.19 −64.31 −38.06 5.74
87.88
46.43
54.05
−208.04
−189.20
OH (phenol)
−221.65
−197.37
O
(nonring)
−132.22
−105.00
O
(ring)
−138.16
−98.22
C
O (nonring)
−133.22
−120.50
C
O (ring)
−164.50
−126.27
Halogen Increments F Cl Br I
where
ENTHALPY OF VAPORIZATION AND FUSION Enthalpy of Vaporization The enthalpy (heat) of vaporization ∆HV is defined as the difference of the enthalpies of a unit mole or mass of a saturated vapor and saturated liquid of a pure component; i.e., at a temperature (below the critical temperature) and corresponding vapor pressure. ∆HV is related to vapor pressure by the thermodynamically exact Clausius-Clapeyron equation: d ln Psat ∆Hv = −R ∆ZV d (1/T)
(nonring)
N
11.30
(aromatic or cyclic olefin)
N 136.70
2.09
Oxygen Increments OH (alcohol)
Nitrogen Increments NH2
142.14
79.72
CH (aromatic or cyclic olefin) C
2-349
(2-54)
R = gas constant in energy units ∆ZV = ZG − ZL ZG = compressibility factor of the saturated vapor ZL = compressibility factor of the saturated liquid Psat = vapor pressure T = absolute temperature
(2-55)
If accurate ZG and ZL data are available, excellent ∆HV values can be obtained by differentiating a vapor pressure correlation and using Eq. (2-54). If not, ∆ZV may be esimated by Haggenmacher’s equation34:
2-350
PHYSICAL AND CHEMICAL DATA where Tr 1,2 = reduced temperature = T1 /Tc or T2 /Tc T1,2 = temperature, K Tc = critical temperature, K
TABLE 2-389 Standard-State Entropy of Elements at 298.15 K and 1 Atmosphere Element
State
Absolute entropy J/mol K
C H2 O2 N2 S F2 Cl2 Br2 I2
crystal (graphite) gas gas gas crystal (rhombic) gas gas liquid crystal
5.74 130.571 205.043 191.500 32.054 202.682 222.972 152.21 116.14
P 1/2 ∆Zv = 1 − r Tr where Pr = reduced pressure = P/Pc Tr = reduced temperature = T/Tc 3
Equation (2-59) works best between the normal boiling and critical temperatures, producing values of engineering accuracy.
Example 14 Estimate ∆Hv of ethyl acetate at 450 K, using the normal boiling point values as a basis (see Example 13). ∆Hv1 = 32.23 kJ/mol, Tr1 = 0.6692, and Tr2 = 450.0/523.3 = 0.8599. Substituting in Eq. (2-59): 1 − 0.8599 0.38 ∆Hv(450) = 32.23 = 23.25 kJ/mol 1 − 0.6692 A value of 23.16 kJ/mol is obtained from Daubert et al.24
(2-56)
However, Eq. (2-56) should be used only near or below the normal boiling point; even then, the accuracy of the resulting ∆HV is significantly reduced. The corresponding states approach suggested by Pitzer et al.82 requires only the critical temperature and acentric factor of the compound. For a close approximation, an analytical representation of this method proposed by Reid et al.86 for 0.6 < Tr < 1.0 is: ∆HV /RTc = 7.08(1 − Tr)0.354 + 10.95ω(1 − Tr)0.456
(2-57)
where ∆HV = enthalpy of vaporization, kJ/mol R = gas constant = 0.008314 kJ/mol K Tc = critical temperature, K Tr = reduced temperature, T/Tc T = temperature, K ω = acentric factor
Enthalpy of Fusion The enthalpy (heat) of fusion ∆Hfus is defined as the difference of the enthalpies of a unit mole or mass of a solid and liquid at its melting temperature and one atmosphere pressure of a pure component. There are no generally applicable estimation techniques that are very accurate. However, if the melting temperature is known, the atomic group contribution method of Chickos et al.16 yields approximate results: ∆Hfus = Tmlt∆Sfus where
a = 35.19 NR + 4.289 (NCH2 − 3 NR) where
Example 12 Estimate ∆HV of Propionaldehyde at 350 K. The required properties from Daubert et al.24 are Tc = 504.4 K and ω = 0.2559. Tr = 350.0/504.4 = 0.6939. Substituting in Eq. (2-57): ∆HV /RTc = (7.08)(1 − 0.6939)0.354 + (10.95)(0.2559)(1 − 0.6939)0.456 = 6.289 ∆HV = (6.289)(0.008314)(504.4) = 26.37 kJ/mol The reported value is 26.85 kJ/mol.24 The enthalpy of vaporization at the normal boiling temperature ∆Hvb (kJ/mol) can be estimated by an equation suggested by Riedel90: (ln [Pc /101.325] − 1) (2-58) ∆Hvb = 1.093 RTc Tbr 0.930 − Tbr where R = gas constant = 0.008314 kJ/mol K. Tc = critical temperature, K Tbr = reduced normal boiling temperature = Tb /Tr Tb = normal boiling temperature, K Pc = critical pressure, kPa
Average errors are about 2 percent.
Example 13 Estimate ∆HVb of Ethyl Acetate. The required properties for ethyl acetate are from Daubert et al.24: Tc = 523.3 K, Tb = 350.2 K, and Pc = 3880.0 kPa. Tbr = 350.2/523.3 = 0.6692. Substituting in Eq. (2-58): (ln [3880.0/101.325] − 1) ∆HVb = (1.093)(0.008314)(523.3) (0.6692) 0.930 − 0.6692
= 32.28 kJ/mol The value from Daubert et al.24 is 32.23 kJ/mol. The enthalpy of vaporization decreases with temperature and is zero at the critical point. If the value of an enthalpy of vaporization ∆Hv1 is known at temperature T1, this temperature dependency can be represented by the Watson relation127 to calculate another enthalpy of vaporization ∆Hv2 at any other temperature T2: 1 − Tr 0.38 (2-59) ∆Hv2 = ∆Hv1 2 1 − Tr1
(2-60)
∆Hfus = enthalpy of fusion at the melting temperature, J/mol Tmlt = melting temperature, K ∆Sfus = a + b = entropy of fusion at the melting temperature, J/mol K.
It should be noted that the methodology for a and b results in a ∆Sfus associated with the phase change from a solid at 0 K to the liquid at Tmlt. No entropy changes resulting from solid transitions are taken into account, and ∆Sfus for a substance that undergoes such a transition will be overestimated by this technique.
Maximum errors are in the order of 8 percent.
(2-61)
NR = number of nonaromatic rings NCH2 = number of CH2 atomic groups in nonaromatic ring(s) required to form a cyclic paraffin of the same ring size(s) as contained in the molecule of interest.
Example: For
O
, NCH 2 = 5; a = 0 if there are no nonaromatic rings
in the molecule of interest. If a nonaromatic ring in fact contains a CH2 atomic group, then no consideration of that group in the b term in Eq. (2-62) is required. ng
ns
nf
i=1
j=1
k=1
b = (Ng)i(∆s)i + (Ns)j(Cs)j(∆s)j + (Nf )k(Ct)k(∆s)k where
(2-62)
ng = number of different nonring or aromatic C-H atomic groups bonded to other carbon atoms in the molecule of interest (Ng)i = number of C-H atomic groups i bonded to other carbon atoms in the molecule of interest ns = number of different nonring or aromatic C-H atomic groups bonded to at least one functional group or atom in the molecule of interest (Ns)j = number of C-H atomic groups j bonded to at least one functional group or atom in the molecule of interest nf = number of different functional groups or atoms in the molecule of interest (Nf)k = number of functional groups or atoms k in the molecule of interest (Cs)j = coefficient for C-H atomic group j bonded to at least one functional group or atom in the molecule of interest; numeric values for C-H atomic groups are found in Table 2-390 Ct = coefficient for the functional group or atom k in the molecule of interest, where t = the total number of functional groups or atoms in the molecule of interest. Exception: Molecules containing any number of fluorine atoms are treated as having only one functional fluorine atom. Numeric values of C1–C4 are in Table 2-391 for functional groups or atoms (∆s)i, j, k = contribution of the atomic group or atom i, j, or k to the entropy of fusion, J/mol K. Numeric values for C-H atomic groups are in Table 2-390; values for functional groups or atoms are in Table 2-391
PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-390 Cs and ⌬s Values for C-H Atomic Groups to Estimate ⌬Hfus Cs
∆s
CH3
1.0
18.33
CH2
1.0
9.41
CH
0.69
−16.19
C
0.67
−38.70
CH2
1.0
14.56
CH
3.23
4.85
C
1.0
−11.38
CH
1.0
10.88
C
1.0
2.18
1.0
6.44
Nonring
2-351
There are no reliable prediction methods for solid heat capacity as a function of temperature. However, the atomic element contribution method of Hurst and Harrison,41 which is a modification of Kopp’s Rule,50 provides estimations at 298.15 K and is easy to use: n
Aromatic CH C
(bonded to paraffinic C)
1.0
−10.33
C
(bonded to olefinic C or non-C)
1.0
−4.27
CpS = Ni ∆Ei
(2-63)
i=1
where CpS = solid heat capacity at 298.15 K, J/mol K n = number of different atomic elements in the compound Ni = number of atomic elements i in the compound ∆Ei = numeric value of the contribution of atomic element i found in Table 2-393 Average errors are in the 9–10 percent range. Example 15 Estimate solid heat capacity of dibenzothiophene, C12H8S. The required atomic element contributions from Table 2-393 are: C = 10.89, H = 7.56, and S = 12.36. Substituting in Eq. (2-63): CpS = (12)(10.89) + (8)(7.56) + (1)(12.36) = 203.52 J/mol K Daubert et al.24 report a value of 198.5 J/mol K.
Liquid Heat Capacity The two commonly used liquid heat capacities are either at constant pressure or at saturated conditions. There is negligible difference between them for most compounds up to a reduced temperature (temperature/critical temperature) of 0.7. Liquid heat capacity increases with increasing temperature, although a minimum occurs near the triple point for many compounds. There are a number of reliable estimating techniques for obtaining pure-component liquid heat capacity as a function of temperature, including Ruzicka and Dolmalski,93,94 Tarakad and Danner,112 and Lee and Kesler.55 These methods are somewhat complicated. The relatively simple atomic group contribution approach of Chueh and Swanson17 for liquid heat capacity at 293.15 K is presented here: n
CpL = Ni ∆cpi + 18.83m
(2-64)
i=1
1.0
−2.51
CH
0.76
−15.98
C
1.0
−32.97
CH
0.62
−4.35
C
0.86
−11.72
1.0
−5.36
C
(bonded to acetylenic C)
Ring
C or
C
where CpL = liquid heat capacity at 293.15 K, J/mol K. n = number of different atomic groups in the compound Ni = number of atomic groups i in the compound ∆cpi = numeric value of the contribution of atomic element i found in Table 2-394. The original units of cal/mol K have been converted to J/mol K by the conversion 1 cal/mol K = 4.184 J/mol K m = number of carbon groups requiring an additional contribution, which are those that are joined by a single bond to a carbon group, which in turn is connected to a third carbon group by a double or triple bond. If a carbon group meets this criterion in more than one way, m should be increased by one for each of the ways. Exceptions: CH3 groups or carbon groups in a ring never require an additional contribution; and the first additional contribution for a CH2 group is 10.46 J/mol K rather than 18.83 J/mol K. However, if the CH2 group meets the criterion in a second way, the second additional contribution reverts to the 18.83 J/mol K value (see Example 17, below).
Chickos et al.16 report an average error of 2050 J/mol for monofunctional molecules and 3180 J/mol for multifunctional molecules when using their method to estimate ∆Hfus. Four example estimations are shown in Table 2-392.
Errors should be less than 6 percent for all compounds except for acids, amines, and halides.
SOLID AND LIQUID HEAT CAPACITY
Example 16 Estimate the liquid heat capacity at 293.15 K of 2-butanol. The atomic groups are:
The heat capacity is defined as the amount of energy required to change the temperature of a unit mass or mole one degree; typical units are J/kg·K or J/kmol·K. Solid Heat Capacity Solid heat capacity increases with increasing temperature, with steep rises near the triple point for many compounds. When experimental data are available, a simple polynomial equation in temperature is often used to correlate the data. It should be noted that step changes in heat capacity occur if the compound undergoes crystalline state changes at different temperatures.
2
CH3
1
CH2
1 CHOH Substituting in Eq. (2-64) the atomic group contributions from Table 2-394 with m = 0: CpL = (2)(36.82) + (1)(30.38) + (1)(76.15) = 180.17 J/mol K
The value from Daubert et al.24 is 190.3 J/mol K.
TABLE 2-391 Ct and Ds Values for Functional Groups and Atoms to Estimate DHfus ∆s
C1
C2
C3
C4
OH (alcohol)
1.0
12.6
18.9
26.4
1.13
OH (phenol)
1.0
1.0
1.0
1.0
16.57
O
(ether, nonring)
1.0
1.0
1.0
1.0
1.09
O
(ether, ring)
1.0
1.0
1.0
1.0
1.34
C
O (ketone, nonring)
1.0
1.0
3.14
C
O (ketone, ring)
1.0
1.0
−1.88
CHO (aldehyde)
1.0
COOH (acid)
1.0
1.83
1.88
1.72
14.90
COO
(ester)
1.0
1.0
1.0
1.0
3.68
NH2 (aliphatic)
1.0
1.82
16.23
NH2 (aromatic)
1.0
1.0
15.48
NH (nonring)
1.0
1.0
−2.18
NH (ring)
1.0
1.84
N
(nonring)
1.0
−15.90
N
(ring)
1.0
1.0
−17.07
N
(ring)
1.0
1.0
1.67
N
(aromatic)
1.0
1.0
CN (nitrile)
1.0
1.4
NO2
1.0
1.0
1.0
1.0
26.19
1.0
1.0
−0.42
1.0
1.0
17.99
19.66
1.0
7.32 9.62
1.0
17.36
O C NH2 O C NH SH S S
(nonring)
1.0
(ring)
1.0
–
2-352
0.36 1.0
7.20 2.18
SO2 (nonring)
1.0
F (on
C)
1.0
1.0
1.0
1.0
14.73
F (on
C)
1.0
1.0
1.0
1.0
13.01
F (on ring C)
1.0
1.0
1.0
1.0
15.90
Cl
1.0
2.0
2.0
1.93
8.37
Br
1.0
1.0
1.0
0.82
17.95
I
1.0
1.0
3.26
16.95
PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-392
2-353
Examples of Estimations of DHfus, J/mol Melting temp., K
Molecule
Atomic group cyclopentane
(35.19)(1) + (4.289) [5 − (3)(1)]
=
43.77
(1)(0.76)(−15.98)
=
−12.14
(1)(1.0)(17.99) ∆Sfus
= =
17.99 49.62
∆Hfus = (49.62)(155.4) ∆Hfus = (experimental)16
= =
(35.19) (1) + (4.289)[5 − (3)(1)]
=
43.77
CH (aromatic)
(4)(6.44)
=
25.76
C
(ring)
(1)(−11.72)
=
−11.72
C
(ring)
(1)(0.86)(−11.72)
=
−10.08
CH (ring)
(1)(−4.35)
=
−4.35
CH (ring)
(1)(0.62)(−4.35)
=
−2.69
(1)(1.0)(2.18) ∆Sfus
= =
2.18 42.87
∆Hfus = (42.87)(304.5) ∆Hfus = (experimental)16
= =
CH (aromatic)
(4)(6.44)
=
25.76
C
(aromatic)
(1)(1.0)(−4.27)
=
−4.27
C)
(1)(1.0)(13.01)
=
13.01
(1)(−10.33)
=
−10.33
CH
(1)(0.69)(−16.19)
=
−11.17
COOH
(1)(1.88)(14.90)
=
28.01
OH (alcohol)
(1)(18.9)(1.13) ∆Sfus
= =
21.36 62.37
∆Hfus = (62.37)(363.0) ∆Hfus = (experimental)16
= =
CH2
(2)(1.0) 9.41
=
18.82
CH
(2)(0.69)(−16.19)
=
−22.34
OH (alcohol)
(2)(26.4)(1.13)
=
59.66
Br
(2)(0.82)(17.95) ∆Sfus
= =
29.44 85.58
∆Hfus = (85.58)(338.2) ∆Hfus = (experimental)16
= =
CH SH
155.4
Contribution
(ring)
SH
(t = 1) cyclopentane
7710.9 7802
S 304.5 (t = 1)
S
(ring)
13054 11823
F CH
363.0
COOH
F (on
OH C
(aromatic)
(t = 3)
HO
CH2
H
Br
C
C
Br H (t = 4)
338.2 CH2
OH
22640 20959
28943 29291
2-354
PHYSICAL AND CHEMICAL DATA
TABLE 2-393 Atomic Element Contributions to Estimate Solid Heat Capacity at 298.15 K Atomic element
∆E
Atomic element
∆E
Atomic element
∆E
C H O N S F Cl Br I Al B
10.89 7.56 13.42 18.74 12.36 26.16 24.69 25.36 25.29 18.07 10.10
Ba Be Ca Co Cu Fe Hg K Li Mg Mn
32.37 12.47 28.25 25.71 26.92 29.08 27.87 28.78 23.25 22.69 28.06
Mo Na Ni Pb Si Sr Ti V W Zr All other
29.44 26.19 25.46 31.60 17.00 28.41 27.24 29.36 30.87 26.82 26.63
TABLE 2-394
Example 17 Estimate liquid heat capacity at 293.15 K of 1,4-pentadiene, CH2苷CHCH2CH苷CH2. The atomic groups are: 2
CH2
2
CH
1
CH2
The CH2 group is twice joined by a single bond to a carbon group, which in turn is connected to a third carbon group by a double bond, and m = 2. However, by the second exception, the first additional contribution is 10.46 J/mol K rather than 18.83 J/mol K. Substituting in Eq. (2-64) the atomic group contributions from Table 2-394: CpL = (2)(21.76) + (2)(21.34) + (1)(30.38) + 10.46 + 18.83 = 145.87 J/mol K
Atomic Group Contributions to Estimate Liquid Heat Capacity at 293.15 K ∆cp
Nonring Increments CH3
36.82
CH2
30.38
CH
20.92
∆cp Oxygen Increments (Cont.) CH2OH CHOH
7.36
CH2
21.76
CH
21.34
C
15.90
CH C Ring Increments CH2 CH
OH (except for above) ONO2 Nitrogen Increments NH2
C
CH
43.93
24.69
N
31.38
24.69
N
25.94
CN
(ring)
Sulfur Increments SH
12.13
Halogen Increments F
18.83 58.16
44.77 33.47
16.74
Cl (first or second on a carbon)
35.98
Cl (third or fourth on a carbon)
25.10
Br
37.66
I
35.98
22.18
Oxygen Increments O C
58.58
NH
18.41
or
44.77 119.24
S C
76.15
111.29
COH C
73.22
O
35.15 52.97
CHO (aldehyde)
52.97
COOH (acid)
79.91
COO
60.67
(ester)
Hydrogen Increment H (for formic acid, formates, hydrogen cyanide, etc.)
14.64
PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Daubert et al.24 report a value of 145.6 J/mol K. For liquid mixtures, the values of the pure components can be molefraction-averaged. This procedure neglects any heat of mixing effects.
DENSITY Density is defined as the mass of a substance contained in a unit volume. In the SI system of units, the ratio of the density of a substance to the density of water at 15°C is known as its relative density, while the older term specific gravity is the ratio relative to water at 60°F. Various units of density, such as kg/m3, lb-mass/ft3, and g/cm3, are commonly used. In addition, molar densities, or the density divided by the molecular weight, is often specified. This section briefly discusses methods of correlation of density as a function of temperature and presents the most common accurate methods for prediction of vapor, liquid, and solid density. Correlation Methods Vapor densities are not correlated as functions of temperature alone, as pressure and temperature are both important. At high temperatures and very low pressures, the ideal gas law can be applied; while at moderate temperature and low pressure, vapor density is usually correlated by the virial equation. Both methods will be discussed later. Molar liquid density (ρ) is best correlated by an equation adopted from the Rackett predictor. The equation has the form of Eq. (2-65): A ρ = D (2-65) T B 1 − C The regression constants A, B, and D are determined from the nonlinear regression of available data, while C is usually taken as the critical temperature. The liquid density decreases approximately linearly from the triple point to the normal boiling point and then nonlinearly to the critical density (the reciprocal of the critical volume). A few compounds such as water cannot be fit with this equation over the entire range of temperature. Liquid density data to be regressed should be at atmospheric pressure up to the normal boiling point, above which saturated liquid data should be used. Constants for 1500 compounds are given in the DIPPR compilation.24 Solid density data are sparse and usually only available over a narrow temperature range, for which the general decrease in density with temperature is approximately linear. Vapor Density Prediction A myriad of methods exist for prediction of vapor density as a function of temperature and pressure. This section will only present the most accurate and generally used methods. For simple molecules at temperatures above the critical and at pressures no more than a few atmospheres, the ideal gas law, Eq. (2-66), may be used to estimate vapor density.
1 P ρ== V RT
(2-66)
At slightly higher pressures up to a reduced pressure of about 0.4, the truncated virial equation, Eq. (2-67), is commonly used for all types of organic fluids. PV B Z==1+ (2-67a) RT V P Z = = 1 + Bρ ρRT
(2-67b)
Second virial coefficients, B, are a function of temperature and are available for about 1500 compounds in the DIPPR compilation.24 The second virial coefficient can be regressed from experimental PVT data or can be reasonably and accurately predicted. Tsonopoulos117 proposed a prediction method for nonpolar compounds that requires the critical temperature, critical pressure, and acentric factor. Equations (2-68) through (2-70) describe the method. BPc (2-68) = B0 + ωB1 RTc
2-355
0.330 0.1385 0.0121 0.000607 B0 = 0.1445 − − − − Tr T r2 T 3r T r5
(2-69)
0.0331 0.423 0.008 − − (2-70) B1 = 0.0637 + T r2 T 3r T 8r For non-hydrogen-bonding polar compounds such as carbonyls and ethers, Tsonopoulos117 recommends that Eq. (2-68) be expanded to a third term that is a function of the reduced dipole moment (µr) as described by Eqs. (2-71) through (2-73): BPc (2-71) = B0 + ωB1 + B2 RTc B2 = −0.0002410λr − 4.308 × 10−21λ8r (2-72) 105λ2p Pc λr = (2-73) T c2 The dipole moment λp in Eq. (2-73) is in debyes, while Pc is in atm and Tc is in K. Units must be watched carefully. For hydrogen-bonding molecules, Eq. (2-71) can be used with a value of B2 calculated by Eq. (2-74). a b B2 = 6 − 8 (2-74) Tr Tr Variables a and b are specific constants reported by Tsonopoulos117 for some alcohols and water (e.g., methanol: a = 0.0878, b = 0.0560; and water: a = 0.0279, b = 0.0229). Tsonopoulos also gives specific prediction methods for haloalkanes118 and water pollutants.119 Example 18 Estimate the molar volume of isobutane at 155°C and 1.0 MPa pressure. Properties of isobutane are Tc = 135.0°C, Pc = 3.647 MPa, and ω = 0.1170. 1.0 155 + 273.1 Tr = = 1.05 Pr = = 0.274 135 + 273.1 3.647 Since reduced pressure is below 0.4, use virial equation (2-67a). Calculate B by the Tsonopoulos method, Eq. (2-68). BPc = B0 + ωB1 RTc Using Eq. (2-69):
0.330 0.1385 0.0121 0.000607 − − B0 = 0.1445 − − 1.05 1.052 1.053 1.055
B0 = 0.1445 − 0.3143 − 0.1256 − 0.0105 − 0.0005 = −0.3064 Using Eq. (2-70):
0.0331 0.423 0.008 − − B1 = 0.0637 + 1.052 1.053 1.058
B1 = 0.0637 + 0.0300 − 0.3654 − 0.0054 = −0.2771 [−0.3064 + (0.1770)(−0.2771)](8314)(408.1) B = = −0.3307 (3.647 × 106) B PV =1+ RT V 106 V 0.3307 = 1 − (8314)(428.1) V Trial and error or the quadratic formula can be used for the solution. If you opt for trial and error, start with the ideal gas value (B = 0) where V = 3.559 m3/ kmole [Eq. (2-66)]. Solving, V = 3.190 m3/kmole. For prediction of vapor density of pure hydrocarbon and nonpolar gases, the corresponding states method of Pitzer et al.82 is the most accurate method, with errors of less than 1 percent except in the critical region where errors of up to 30 percent can occur. The method correlates the compressibility factor by Eq. (2-75), after which the density can be calculated by Eq. (2-75): Z = Z(0) + ωZ(1) (2-75) P 1 ρ== (2-76) ZRT V Z(0) is the compressibility factor for the simple fluid, while Z(1) is the correction term for molecular acentricity, both of which are functions of Tr and Pr. Both
2-356
PHYSICAL AND CHEMICAL DATA
plots and detailed tabulations of the functions are available in the Technical Data Book.23 Critical temperature and pressure and the acentric factor from tabulations or as predicted are required. For hydrogen, Tc and Pc should be taken as 41.7 K and 2100 kPa, respectively. For approximate calculations, Figs. 2-35 and 2-36 should be used for calculating Z(0) and Z(1), respectively, for superheated vapors with 0.2 ≤ Pr ≤ 10. Z(0) will approach 1, and Z(1) will approach 0 for Pr < 0.2. More accurately, Eq. (2-77) can be used for Pr between 0 and 0.2.
FIG. 2-35
P Z = 1 + r [(0.1445 + 0.073ω) − (0.330 − 0.46ω)T r−1 Tr − (0.1385 + 0.50ω)T r−2 − (0.0121 + 0.097ω)T r−3 − 0.0073ωT r−8]
(2-77)
Extension of the pressure range to Pr = 14 is available in the Technical Data Book. For saturated vapor densities, the values of Z(0) and Z(1) are tabulated as a
Generalized compressibility factors—Pitzer Method, simple fluid term.
PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES function of reduced pressure in Table 2-395. If the saturation temperature rather than the saturation pressure is known, the vapor pressure of the compound can be determined either from data or the vapor pressure prediction methods discussed earlier.
As high pressure, use Eq. (2-75) to calculate Z and then Eq. (2-76) to estimate the molar volume. The properties of isobutane necessary are Tc = 135.0°C, Pc = 3.647 MPa, and ω = 0.1770.
Example 19 Estimate the molar volume of isobutane at 155°C and 8.6 MPa pressure.
FIG. 2-36
2-357
Generalized compressibility factors—Pitzer Method, correction term.
155 + 273.1 Tr = = 1.05 135 + 273.1
8.6 Pr = = 2.36 3.647
2-358
PHYSICAL AND CHEMICAL DATA
TABLE 2-395
Saturated Vapor Density Parameters
Pr
Z (0)
Z (1)
Pr
Z (0)
Z (1)
1.00 0.99 0.98 0.97 0.96 0.95 0.94 0.92 0.90 0.85 0.80 0.75 0.70
0.291 0.35 0.38 0.40 0.41 0.42 0.43 0.45 0.47 0.50 0.53 0.56 0.59
−0.080 −0.083 −0.085 −0.087 −0.088 −0.089 −0.089 −0.090 −0.091 −0.090 −0.087 −0.081 −0.075
0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05
0.615 0.64 0.665 0.688 0.711 0.734 0.758 0.783 0.809 0.835 0.864 0.896 0.935
−0.069 −0.063 −0.056 −0.049 −0.041 −0.033 −0.025 −0.018 −0.012 −0.008 −0.005 −0.002 0.000
Using Fig. 2-35, Z(0) = 0.385. Using Fig. 2-36, Z(1) = −0.063. Z = Z(0) + ω Z(1) Z = 0.385 + 0.1770(−0.063) = 0.374 ZRT (0.374)(8314)(428.1) V = = P 8.6 × 106 V = 0.155 m3/kmol An experimental value for Z is 0.377. Note that use of the Lee-Kesler fit [Eq. (2-78)] would give a slightly more accurate answer than the graphical method, and this fit is used for any computer applications. Lee and Kesler55 fit the entire Pitzer method to equations, rewriting the basic Eq. (2-75) with respect to a heavy reference fluid (n-octane) as shown by Eq. (2-78). ω (h) Z = Z(0) + (Z − Z(0)) ω(h)
(2-78)
where h specifies the heavy reference fluid with an acentric factor of 0.3978. The parameters in the equation are calculated for the simple fluid and the heavy reference fluid with an acentric factor of 0.3978. The parameters in the equation are calculated for the simple fluid and the heavy reference fluid from Eq. (2-79) PrVr B C D c4 γ −γ Z(i) = = 1 + + 2 + 5 + β + 2 exp 2 Tr Vr V r Vr T r3 V 2r Vr Vr
(2-79)
b2 b3 b4 B = b1 − − 2 − 3 Tr Tr Tr
c2 c3 C = c1 − − 3 Tr Tr
d2 D = d1 + Tr where
Z(i) = Z(0) when the constants in the equation correspond to the simple fluid and Z(h) when the constants in the equation correspond to the heavy reference fluid P = pressure, kPa Pc = critical pressure of the compound whose density is sought, kPa V = molar volume of the simple fluid or the heavy reference fluid, as the case may be, in m3/kmole. R = gas constant = 8.3140 m3 kPa/k mole K Tc = critical temperature of the compound whose density is sought, K T = temperature, K
Constant
Simple fluid
Heavy reference fluid
b1 b2 b3 b4 c1 c2 c3 c4 d1 × 104 d2 × 104 β γ
0.1181193 0.265728 0.154790 0.030323 0.0236744 0.0186984 0.0 0.042724 0.155488 0.623689 0.65392 0.060167
0.2026579 0.331511 0.027655 0.203488 0.0313385 0.0503618 0.016901 0.041577 0.48736 0.0740336 1.226 0.03754
For hydrocarbon and nonpolar gas mixtures, the Pitzer pure component method can be used to predict vapor density by replacing the true critical properties with pseudocritical properties defined in Eqs. (2-80) and (2-81) by Kay.47 n
Tpc = xiTci
(2-80)
i=1 n
Ppc = xiPci
(2-81)
i=1
The mixture acentric factor, Eq. (2-82), can also be used. n
ω = xiωi
(2-82)
i=1
Errors in compressibility factors tabulated for over 6500 data points rarely exceed 2 percent except in the critical region, where 15 percent errors may be expected and 50 percent errors can occur. For mixtures near the critical point, special techniques are available as discussed in the sixth chapter of the Technical Data Book. For pure organic vapors, the Lydersen et al.63 corresponding states method is the most accurate technique for predicting compressibility factors and, hence, vapor densities. Critical temperature, critical pressure, and critical compressibility factor defined by Eq. (2-21) are used as input parameters. Figure 2-37 is used to predict the compressibility factor at Zc = 0.27, and the result is corrected to the Zc of the desired fluid using Eq. (2-83). Z = Z@ Zc = 0.27 + Di(Zc − 0.27)
(2-83)
Di is equal to Da read from Fig. 2-38 if Zc > 0.27; and Di is equal to Db read from Fig. 2-39 if Zc < 0.27. At reduced temperatures less than 0.9, Di can be taken as 0. The density is then calculated from Eq. (2-76). All families of organic compounds except mercaptans and carboxylic acids are predicted within an average deviation of 5 percent. No specific mixing rules have been tested for predicting compressibility factors for defined organic mixtures. However, the Lydersen method using pseudocritical properties as defined in Eqs. (2-80), (2-81), and (2-82) in place of true critical properties will give a reasonable estimate of the compressibility factor and hence the vapor density. Vapor densities for pure compounds can also be predicted by cubic equations of state. For hydrocarbons, relatively accurate RedlichKwong-type equations such as the Soave98 and Peng-Robinson81 equations are often used. Both require only Tc, Pc, and ω as inputs. For organic compounds, the Lee-Erbar-Edmister54 equation (which requires the same input parameters) has been used with errors essentially equivalent to those determined for the Lydersen method. While analytical equations of state are not often used when only densities are required, values from equations of state are used as inputs to equation of state formulations for thermal and equilibrium properties. Liquid Density Prediction Methods for the prediction of pure saturated hydrocarbons and nonhydrocarbon organics, compressed hydrocarbon liquids, and defined and undefined hydrocarbon mixtures were evaluated. Only the most accurate and convenient methods are included here. The most convenient method for predicting the saturated liquid density of both pure hydrocarbons and pure organic liquids is the method of Rackett84 as modified by Spencer and Danner.99 Equation (2-84) is used to calculate the saturated liquid molar density at any temperature using input parameters of Tc, Pc, and ZRA. ZRA is a parameter regressed from experimental data. Values for some common substances are given in Table 2-396. Extensive tabulations are given in the Technical Data Book23 for hydrocarbons and nonhydrocarbons as well as organic and inorganic gases. Additional values are given in the Data Prediction Manual22 for nonhydrocarbons. RTc 1 (2-84) = Vsat = Z nRA ρsat Pc
n = 1.0 + (1.0 − Tr)2/7 Errors for hydrocarbons between the triple and critical points average about 0.7 percent, with organics averaging about 1.2 percent. The cor-
PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES
FIG. 2-37
2-359
Generalized compressibility factor—Lydersen Method, Zc = 0.27.
relation is especially sensitive to the value of ZRA near the critical point. If no value of ZRA is available or derivable, the critical compressibility factor can be used in Eq. (2-84) as originally proposed by Rackett. Use of Zc increases the average error to about 3.0 percent.
Example 20 Estimate the density of saturated liquid propane at 0°C. Use Eq. (2-84).
1 Z
Pc ρsat = RTc
n RA
n = 1.0 + (1.0 − Tr)2/7
2-360
PHYSICAL AND CHEMICAL DATA Pure component properties of propane are M = 44.1, Tc = 96.7°C, and Pc = 4.246 MPa. From Table 2-388, ZRA = 0.2763. 273.1 Tr = = 0.739, n = 1.0 + (1.0 − 0.739)2/7 = 1.6813 369.8 n = 1.6813 (4.246 × 106) 1 ρsat = (8314)(369.8) 0.27631.6813
ρsat = 1.201 kmole/m = 529.6 kg/m3 3
An accepted experimental value is 531 kg/m3. An alternate method with approximately the same accuracy as the Rackett method is the COSTALD method of Hankinson and Thomson.35 The critical temperature, a characteristic volume near the critical volume, and an acentric factor optimized for vapor pressure prediction by the Soave98 equation of state are required input parameters. The method is detailed in the Technical Data Book.23 Prediction of the density of compressed pure liquid hydrocarbons and their defined mixtures are readily and accurately predicted by the method of Lu.62 One value of low pressure liquid density and the critical temperature and pressure are required to predict the density at higher pressures from Eq. (2-85).
C ρ2 = 2 ρ1 C1
(2-85)
The constants C1 and C2 are both obtained from Fig. 2-40: C1, usually from the saturated liquid line; and C2, at the higher pressure. Errors should be less than 1 percent for pure hydrocarbons except at reduced temperatures above 0.95 where errors of up to 10 percent may occur. The method can be used for defined mixtures substituting pseudocritical properties for critical properties. For mixtures, the Technical Data Book—Petroleum Refining gives a more complex and accurate mixing rule than merely using the pseudocritical properties. The saturated low pressure value should be obtained from experiment or from prediction procedures discussed in this section for both pure and mixed liquids. FIG. 2-38
Compressibility factor correction for Lydersen Method, Zc > 0.27.
Example 21 Estimate the liquid density of n-nonane at 104.5°C and 6.893 MPa pressure. A tabulated value of liquid density at 60°F (15.5°C) and 1 atm is 719.8 kg/m3. Pure component properties are Tc = 321.5°C and Pc = 2.288 MPa. Use Eq. (2-85) to correct a low pressure density. C ρ2 = 2 ρ1 C1 where ρ1 is a reference density. C2 and C1 are functions of Tr and Pr. 288.6 377.6 Tr2 = = 0.635 Tr1 = = 0.485 594.6 594.6
0.1013 6.893 Pr1 = = 0.0443 Pr2 = = 3.013 2.288 2.288 From Fig. 2-40, C1 = 1.08 and C2 = 0.998.
0.998 ρ2 = (719.8) = 665.1 kg/m3 1.08 An accepted experimental value at this temperature is 658.5 kg/m3, a 1 percent difference. An analytical method for the prediction of compressed liquid densities was proposed by Thomson et al.114 The method requires the saturated liquid density at the temperature of interest, the critical temperature, the critical pressure, an acentric factor (preferably the one optimized for vapor pressure data), and the vapor pressure at the temperature of interest. All properties not known experimentally may be estimated. Errors range from about 1 percent for hydrocarbons to 2 percent for nonhydrocarbons. For prediction of the densities of a defined liquid mixture at its bubble point (ρbp), the method of Spencer and Danner100 is the simplest. The density is calculated from Eq. (2-86) using inputs from Eqs. (2-87) and (2-88). For hydrocarbons, Tmc is calculated by Eqs. (2-89) through (2-92) if high accuracy is desired or by Eq. (2-93) for a less accurate answer. 1 =R ρbp
x P Z n
Tc
i
i
i=1
ci
[1 + (1 − Tr)2/7] RAm
(2-86)
n
ZRAm = xi ZRAi
(2-87)
i=1
T Tr = Tmc n
FIG. 2-39
Compressibility factor correction for Lydersen Method, Zc < 0.27.
(2-88) n
Tmc = φiφjTcij i=1 j=i
(2-89)
PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-396
2-361
The Modified Rackett Equation Input Parameters for Calculating Pure Saturated Liquid Densities
Liquid
ZRA
Hydrocarbons Methane Ethane Propane n-Butane 2-Methylpropane (isobutane) n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane n-Dodecane n-Tetradecane n-Hexadecane n-Octadecane n-Eicosane Cyclopentane Methylcyclopentane Cyclohexane Methylcyclohexane Ethene (ethylene) Propene (propylene) 1-Butene cis-2-Butene trans-2-Butene 2-Methylpropene (isobutylene) 1-Hexene 1,3-Butadiene 2-Methyl-1,3-butadiene Ethyne (acetylene) Propyne (methylacetylene) Benzene Methylbenzene (toluene) Ethylbenzene 1,2-Dimethylbenzene (o-xylene) 1,3-Dimethylbenzene (m-xylene) 1,4-Dimethylbenzene (p-xylene) Isopropylbenzene (cumene) Biphenyl Naphthalene
Liquid
0.2880 0.2819 0.2763 0.2730 0.2760 0.2685 0.2637 0.2610 0.2569 0.2555 0.2527 0.2471 0.2270 0.2386 0.2292 0.2281 0.2709 0.2712 0.2729 0.2702 0.2813 0.2783 0.2735 0.2705 0.2722 0.2727 0.2654 0.2713 0.2680 0.2707 0.2703 0.2696 0.2645 0.2619 0.2626 0.2594 0.2590 0.2616 0.2746 0.2611
xiVci φi = n
xjVc j
(2-90)
j=1
Tcij = T c Tc (1 − kij)
(2-91)
j
i
1/3 1/3 V V ci cj
kij = 1.0 − (Vc1/3 + Vc1/3 )/2 i j
3
(2-92)
n
Tmc = xi Tci
(2-93)
i=1
Errors for binary hydrocarbon systems average about 2.5 percent except near the critical, where errors can approach 20 percent. If inorganic gases are included, errors average 4 percent except at high concentrations of carbon dioxide or hydrogen, where higher errors would be expected. If the simplified method for predicting Tmc is used, average errors of 5 to 7 percent should be expected for both binary hydrocarbon and nonhydrocarbon systems. No data are available to test the method for systems with more than two components. A similarly accurate but slightly more complex method for prediction of densities of defined liquid hydrocarbon mixtures at their bubble points was published by Hankinson and Thomson35 and was previously cited for prediction of pure liquid hydrocarbons. For undefined hydrocarbon mixtures, the liquid density may be predicted at any temperature (T) from the mean average boiling point (MeABP) and the specific gravity (sp gr) by Eq. (2-94), adopted from Ritter et al.92
P = 62.3636 (sp gr)2 (1.2655)(sp gr) − 0.5098 + 8.011 × 10−5MeABP(T − 519.67) 1/2 − (2-94) MeABP 3 The density is calculated in lbm /ft if the temperatures are both in °R. Errors
ZRA
Organics Acetic acid Methanol Ethanol 2-Propanol Acetaldehyde Acetone Methyl ethyl ketone Methyl isobutyl ketone Ethylamine Aniline Methyl formate Methyl acetate Ethyl acetate Ethyl acrylate Methyl-n-butyl ether Diethyl ether Diisopropyl ether
0.2242 0.2340 0.2523 0.2508 0.2387 0.2448 0.2524 0.2589 0.2640 0.2607 0.2581 0.2553 0.2538 0.2583 0.2655 0.2643 0.2699
Halogen Compounds Methyl chloride Dichloromethane Chloroform Tetrachloromethane Chlorobenzene Propionitrile
0.2679 0.2619 0.2751 0.2721 0.2650 0.2156
Inorganics Ammonia Argon Carbon dioxide Carbon disulfide Carbon monoxide Chlorine Hydrogen Hydrogen chloride Hydrogen sulfide Nitrogen Oxygen Sulfur dioxide Sulfur trioxide
0.2466 0.2933 0.2729 0.2850 0.2898 0.2781 0.3218 0.2673 0.2818 0.2893 0.2890 0.2667 0.2513
average about 0.3 percent at atmospheric pressure. At high pressures, the liquid density of undefined hydrocarbon mixtures can be predicted from the low pressure value by the method of Wright131 fully outlined in the Technical Data Book.23
Solid Density Prediction The prediction of solid density is an inexact science and sometimes is taken as the liquid density at the triple point, although the solid density normally is higher than this value with a discontinuity at the triple point. Based on solid density data reviewed for the DIPPR compilation,24 the solid density at the triple point can be estimated for organic compounds as 1.17 times the liquid density at the triple point. As liquid density at low temperatures varies little with temperature, the density of the liquid at the lowest estimable point above the triple point can be used with little degradation of the result. As solid density only decreases very slightly with increasing temperature and very little data on solid density as a function of temperature exist, no methods have been developed for predicting the solid density vs. temperature. VISCOSITY Viscosity is defined as the shear stress per unit area at any point in a confined fluid divided by the velocity gradient in the direction perpendicular to the direction of flow. If this ratio is constant with time at a given temperature and pressure for any species, the fluid is called a Newtonian fluid. This section is limited to Newtonian fluids, which include all gases and most nonpolymeric liquids and their mixtures. Most polymers, pastes, slurries, waxy oils, and some silicate esters are examples of non-Newtonian fluids. The absolute viscosity (µ) is defined as the sheer stress at a point
2-362
PHYSICAL AND CHEMICAL DATA
FIG. 2-40
Densities of compressed pure liquid hydrocarbons and their defined mixtures.
divided by the velocity gradient at that point. The most common unit is the poise (1 g/cm sec). The SI unit is the Pa·sec (1 kg/m sec). As many common fluids have viscosities in the hundredths of a poise the centipoise (cp) is often used. One centipoise is then equal to one mPa sec. The kinematic viscosity (ν) is defined as the ratio of the absolute viscosity to density at the same temperature and pressure. The most common unit corresponding to the poise is the stoke (1 cm2/sec). The SI unit would be m2/sec. Correlation Methods This section briefly discusses methods for correlating viscosities as a function of temperature and presents the most common accurate methods for prediction of vapor and liquid viscosity. Vapor viscosity is accurately correlated as a function of temperature by Eq. (2-95). AT B µv = (2-95) C D 1 + + 2 T T
If data are available over a wide range, all four regression constants (A, B, C, and D) are usually used. Over narrow temperature ranges, only constants A and B are necessary. Liquid viscosity is accurately correlated as a function of temperature by the modified Riedel equation previously discussed for correlation of vapor pressure and shown by Eq. (2-96). B µᐉ = exp A + + C ln (T) + DT E (2-96) T For most systems, only the first three terms are used. Only the first two terms are used for narrow ranges. If data are available in a wide range extending far above the normal boiling point, all four terms are used, with values of E varying in integers from −10 to 10 (excluding 0 and −1). Constants for about 1500 compounds for both viscosities are available in the DIPPR compilation.24 Vapor Viscosity Methods for prediction of vapor viscosity abound such that only the most accurate and generally used methods are included.
PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES For prediction of the vapor viscosity of pure hydrocarbons at low pressure (below Tr of 0.6), the method of Stiel and Thodos101 is the most accurate. Only the molecular weight, the critical temperature, and the critical pressure are required. Equation (2-97) with values of N from Eqs. (2-98) and (2-99) is used. NM1/2P c2/3 µv = 4.60 × 10−4 T c1/6 0.94 N = 0.0003400Tr for Tr ≤ 1.5 N = 0.0001778(4.58Tr − 1.67)0.625
(2-97) (2-98) for Tr > 1.5
(2-99)
The resultant viscosity is in centipoise (mPa·sec) if Tc and Pc are given in K and Pa, respectively. This method can also be used for light nonhydrocarbon gases except for hydrogen where, special N’s are required. For hydrocarbons below ten carbon atoms, average errors of about 3 percent can be expected, with errors increasing to 5–10 percent for heavier hydrocarbons. Example 22 Estimate the vapor viscosity of propane at 101.3 kPa and 80°C. Use Eq. (2-97).
Tc = 96.7°C,
Z = Z(0) + ωZ(1) 398.1 Tr = = 1.62 245.5
Pc = 4.246 MPa,
and M = 44.1.
To determine whether Eq. (2-98) or (2-99) should be used to calculate N, calculate Tr. 80 + 273.1 Tr = = 0.955 96.7 + 273.1 Use Eq. (2-98):
Example 23 Estimate the vapor viscosity of a mixture of propane and methane. Assume 60 mole percent methane and 40 mole percent propane at 125°C and 10.34 MPa total pressure. The low pressure viscosity is 0.0123 cp. Use Eq. (2-102): M1/2P 2/3 c µ − µo = 5.0 × 10−8 [exp (1.439ρr) − exp (−1.11ρ1.858 )] r T1/6 c Properties of the pure components are: Methane. M = 16.04, Tc = −110.4°C, Pc = 4.593 MPa Propane. M = 44.10, Tc = 96.7°C, Pc = 4.246 MPa For the mixture: Tpc = (0.60)(−110.4 + 273.1) + (0.40)(96.7 + 273.1) = 245.5 K Ppc = (0.60)(4.593) + (0.40)(4.246) = 4.454 MPa Mm = (0.60)(16.04) + (0.40)(44.10) = 27.26 To calculate ρr, use Eq. (2-76) to calculate ρ using Pitzer corresponding states first to calculate Z. Then calculate ρr = ρ/ρc = ρVpc Additional properties required are: Methane ω = 0.0115, Vc = 0.0986 m3/kmole Propane ω = 0.1523, Vc = 0.2002 m3/kmole For the mixture: ωm = (0.60)(0.0115) + (0.40)(0.1523) = 0.0678 Vpc = (0.60)(0.0986) + (0.40)(0.2002) = 0.1392 m3/kmole Using Eq. (2-58):
M1/2Pc2/3 µv = 4.60 × 10−4N T c1/6
x 1 + Qij j xi j=1 j≠1
µi 1/2 Mj 1/4 2 Mi µj Qij = Mi 1/2 8 1 + Mj 1+
(2-100)
P (10.34) ρ = = ZRT (0.88)(8.314 × 10−3)(398.1)
µ − µo = 0.0037 µ = 0.0037 + 0.0123 = 0.0160 cp An experimental value of 0.0167 cp compares favorably. For pure nonhydrocarbon polar gases at low pressures, the viscosity can be estimated by the method of Reichenberg85 given by Eq. (2-103). ATr µ = (2-103) [1 + 0.36Tr(Tr − 1)]1/6 For organic compounds: M1/2Tc A=
niCi
(2-101)
Errors, when tested against binary and multicomponent mixtures of both hydrocarbons and nonhydrocarbon gas mixtures, average about 3 percent. For prediction of the vapor viscosity of gaseous hydrocarbons and mixtures of hydrocarbons at high pressures (not applicable to nonhydrocarbon gases) above a Tr of 0.6, low pressure values are calculated from Eq. (2-97) and/or (2-100) and then corrected for pressure by the method of Dean and Stiel25 given by Eq. (2-102). M1/2P c2/3 [exp (1.439ρr) − exp (−1.11ρ1.858 )] µ − µo = 5.0 × 10−8 r T c1/6
Z = 0.87 + (0.0678)(.20) = 0.88
(5.08 × 10−8)(5.22)(27,085) µ − µo = [2.0358 − 0.7413] (2.503)
An experimental value of 0.0095 cp compares favorably. For prediction of the vapor viscosity of gaseous mixtures of hydrocarbons and nonhydrocarbon gases at low pressures below a Tr of 0.6, the method of Bromley and Wilke13 is recommended. The mixing rule is given by Eq. (2-100) with the interaction parameter Q for each pair of components defined by Eq. (2-101). µi
Z(0) 0.20
(5.0 × 10−8)(27.26)1/2(4.454 × 106)2/3 1.439(.494) −1.11(.494)1.858 µ − µo = [e −e ] (245.5)1/6
µv = 0.0097 cp
n
From Fig. 2-36:
ρ = 3.55 kmole/m3
(4.60 × 10−4)(3.255 × 10−4)(44.1)1/2(4.246 × 106)2/3 µv = (369.8)1/6
n
Z(0) = 0.87
10.34 Pr = = 2.32 4.454
ρr = ρVpc = (3.55)(.1392) = 0.494
Thus, N = 3.255 × 10−4.
i=1
From Fig. 2-35:
Hence:
N = 0.0003400Tr0.94
µm =
2-363
(2-102)
If critical pressure and critical temperature are given in Pa and K, respectively, viscosities in centipoise result. The variable µo is either the low pressure pure component or mixture viscosity according to whether a pure component or mixture is being considered. For mixtures, simple molar average pseudocritical temperature (Kay’s rule), pressure, and density, and molar average molecular weight are used. The vapor density can be predicted by the methods previously discussed. Errors of above 5 percent are common for hydrocarbons and their mixtures. Experimental densities will reduce the errors slightly.
For inorganic gases:
(2-104)
M1/2P c2/3 (2-105) A = 1.6104 × 10−10 T c1/6 Viscosities are calculated in Pa sec (103 cp = 1 Pa sec) with Tc in K and Pc in Pa. Group contributions based on atomic structure for organic compounds necessary for Eq. (2-104) are tabulated in Table 2-397. Errors average about 5 percent for most organics, with slightly higher errors for inorganic gases. For pure nonhydrocarbon nonpolar gases, an alternate method is the method of Yoon and Thodos,132 which requires the same input parameters as the Reichenberg method. The method, when evaluated for the Technical Data Manual, showed errors of about 3 percent for compounds with low dipole moments and requires special correlations for hydrogen and helium.
Example 24 Estimate the vapor viscosity of isopropyl alcohol at 251°C and atmospheric pressure. Use Eq. (2-103) with A determined from Eq. (2-104). ATr µ = [1 + 0.36Tr(Tr − 1)]1/6 M1/2Tc A=
niCi
2-364
PHYSICAL AND CHEMICAL DATA
TABLE 2-397
Group Contribution Values for Eq. (2-104) Contribution Ci × 10−8 m⋅s⋅K/kg
Group CH3
0.904
CH2 (nonring)
0.647
CH
0.267
Pure component properties are Tc = 508.3 K and M = 60.1. Isopropyl alcohol group contributions for niCi are calculated using Table 2-397.
2 1 1
C
(nonring)
CH2
0.768
CH
0.553
(nonring)
C
(nonring)
0.178
CH
0.741 (nonring)
C
0.524
CH2 (ring)
0.691
group = (1)(.267 × 108) +
OH group = (1)(.796 × 108) = 2.871 × 108
60.11/2(508.3) A = × 108 = 1.373 × 10−5 Pa sec 2.871 (1.373 × 10−5)(1.031) µ = = 1.413 × 10−5 Pa sec [1 + 0.36(1.031)(.031)]1/6 (An experimental value of 1.380 × 10−5 Pa sec compares favorably.) For pure nonhydrocarbon polar and nonpolar gases at high pressure, a method of prediction attributed to Stiel and Thodos102 depending on the reduced density as a corrector to the low pressure gas viscosity (µo) takes various forms for polar gases according to the reduced density (ρr) as shown in Eqs. (2-106) through (2-109). ρr ≤ 0.1
(µ − µo)B = 1.656 × 10−7ρ1.111 r
(2-106)
0.1 ≤ ρr ≤ 0.9
(µ − µo)B = 6.07 × 10−9(9.045ρr + 0.63)1.739
(2-107)
0.9 ≤ ρr < 2.2
log {4 − log10 [(µ − µo)107B]} = 0.6439 − 0.1005ρr
(2-108)
2.2 ≤ ρr < 2.6
log {4 − log10 [(µ − µo)107B]} = 0.6439 − 0.1005ρr B = 2173.424 T
C
(ring)
CH C
0.023
(ring)
0.590
(ring)
0.359
F
0.446
Cl
1.006
Br
1.283
OH (alcohols)
0.796
O (nonring)
0.359
C
O (nonring)
1.202
CHO (aldehydes)
1.402
COOH (acids)
1.865
COO
(esters) or HCOO
(formates)
1.341 0.971
NH2 NH (nonring)
0.368
N
0.497
(ring)
− 4.75 × 10−4(ρ3r − 10.65)2
(2-109)
M−1/2 P c−2/3
(2-110)
0.116
(ring)
CH
CH
524.1 Tr = = 1.031 508.3
−0.153
(nonring)
CH3 groups = (2)(.904 × 108) +
CN
1.813
S (ring)
0.886
1/6 c
For nonpolar gases, Jossi et al.45 extended the method as shown in Eq. (2-111) for 0.1 < ρr < 3.0. [(µ − µo)B107 + 1]1/4 = 1.0230 + 0.23364ρr + 0.58533ρr2 − 0.40758ρr3 + 0.093324ρr4 (2-111) In all cases viscosities are in Pa sec with Tc in K and Pc in Pa. For nonpolars, errors are very small; while for polars, average errors reach 11 percent.
Example 25 Estimate the vapor viscosity of carbon dioxide at 350 K and a total pressure of 20 MPa. An experimental low pressure viscosity at 350 K is 1.7386 × 10−5 Pa sec. Use Eq. (2-111) with B calculated by Eq. (2-110). The pure component properties necessary are Tc = 304.19 K, Pc = 7.3815 MPa, Vc = 0.094 m3/kmole, and M = 44.01. Using the Lee-Kesler form of the Pitzer method [Eq. (2-78)], Z = 0.4983. (0.094)(2 × 107) V Vc P ρr = c = = = 1.2965 V ZRT (0.4983)(8314)(350) (2173.424)(304.19)1/6 2173.424T1/6 c B = = M1/2P 2/3 (44.01)1/2(7.3815 × 106)2/3 c B = 2.2411 × 10−2 [(µ − µo)B107 + 1]1/4 = 1.0230 + 0.23364(1.2965) + 0.58533(1.2965)2 − 0.40758(1.2965)3 + 0.093324(1.2965)4 = 1.6852 (µ − 1.7386 × 10−5)(2.2411 × 10−2)(107) + 1 = 8.0659 µ − 1.7386 × 10−5 = 3.1529 × 10−5 µ = 4.89 × 10−5 Pa sec An experimental value of 4.73 × 10−5 Pa sec compares favorably. For both polar and nonpolar nonhydrocarbon gaseous mixtures at low pressures, the most accurate viscosity prediction method is the method of Brokaw.10,11 The method is quite accurate but requires the dipole moment and the Stockmayer energy parameter (ε /k) for polar components as well as pure component viscosities, molecular weights, the normal boiling point, and the liquid molar volume at the normal boiling point. The Technical Data Manual should be consulted for the full method. For nonpolar, nonhydrocarbon vapor mixtures at high pressures, the method of Dean and Stiel25 [Eq. (2-102)] discussed earlier can be used. The accuracy of the method is excellent and dependent on the pure component viscosity values used as input parameters.
PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Liquid Viscosity The viscosity of both pure hydrocarbon and pure nonhydrocarbon liquids are most accurately predicted by the method of van Velzen et al.122 The basic equation (2-112) depends on group contributions which are dependent on structure for the calculation of compound-specific constants B and To. 1 1 log µ = B − − 3.0 (2-112) T To Resultant viscosities are in Pa sec. If the −3.0 on the right is deleted,
TABLE 2-398
2-365
answers are in cp. To is calculated by Eq. (2-114) or (2-115) according to the value of an adjusted carbon number N* calculated by Eq. (2-113) using the actual carbon number N and group contributions from Table 2-398. N* = N + ∆Ni (2-113) i
N* ≤ 20
To = 28.86 + 37.439N* − 1.3547N*2 + 0.02076N*3
(2-114)
N* > 20
To = 8.164N* + 238.59
(2-115)
Group Contribution Values for Liquid Viscosity Prediction
Structures or functional group n-Alkanes Isoalkanes Saturated hydrocarbons with two methyl groups in isoposition n-Alkanes n-Alkadienes Isoalkanes Isoalkadienes Hydrocarbon with one double bond and two methyl groups in isoposition Hydrocarbon with two double bonds and two methyl groups in isoposition Cyclopentanes Cyclohexanes Alkyl benzenes Polyphenols Alcohols Primary Secondary Tertiary Diols (correction) Phenols (correction) OH on side chain to aromatic ring (correction) Acids Isoacids Acids with aromatic nucleus in structure (correction) Esters Esters with aromatic nucleus in structure (correction) Ketones Ketones with aromatic nucleus in structure (correction) Ethers Aromatic ethers Amines Primary Primary amine in side chain of aromatic compound (correction) Secondary Tertiary Primary amines with NH2 group on aromatic nucleus Secondary or tertiary amine with at least one aromatic group attached to amino nitrogen Nitro compounds 1-Nitro 2-Nitro 3-Nitro 4-Nitro; 5-Nitro Aromatic nitrocompounds Nitrile
∆ Ni
∆Bi
0 1.389 − 0.238N 2.319 − 0.238N
0 15.51 15.51
−0.152 − 0.042N −0.304 − 0.084N 1.237 − 0.280N 1.085 − 0.322N 2.626 − 0.518N
−44.94 + 5.410N* −44.94 + 5.410N* −36.01 + 5.410N* −36.01 + 5.410N* −36.01 + 5.410N*
2.474 − 0.560N
−36.01 + 5.410N*
0.205 + 0.069N 3.971 − 0.172N 1.48 6.517 − 0.311N 0.60 3.055 − 0.161N −5.340 + 0.815N
−45.96 + 2.224N* −339.67 + 23.135N* −272.85 + 25.041N* −272.85 + 25.041N* −140.04 + 13.869N* −140.04 + 13.869N* −188.40 + 9.558N*
10.606 − 0.276N 11.200 − 0.605N 11.200 − 0.605N See remarks 16.17 − N −0.16
−589.44 + 70.519N* 497.58 928.83 557.77 213.68 213.68
b
6.795 + 0.365N 10.71 See remarks
−249.12 + 22.449N* −249.12 + 22.449N* −249.12 + 22.449N*
4.81
−188.40 + 9.558N*
N < 11, not recommended for N = 1,2 N ≥ 11 Calculate ∆B as for straight-chain acid; calculate ∆N for straight-chain acid but reduce ∆N by 0.24 for each methyl group in isoposition
Remarks
For any additional CH3 groups in isoposition, increase ∆N by 1.389 − 0.238N For any additional CH3 groups in isoposition, increase ∆N by 1.389 − 0.238N N ≤ 16; not recommended for N = 5,6 N ≥ 16 N < 17; not recommended for N = 6,7 N ≥ 17 N < 16; not recommended for N = 6,7a,e,f N ≥ 16a,e, f
a
b b
For ∆N, use alcohol contributions and add N − 2.50
a,c,d
4.337 − 0.230N −1.174 + 0.376N
−149.13 + 18.695N* −140.04 + 13.869N*
If hydrocarbon groups have isoconfiguration, see e Add to values of ∆N, ∆B calculated for ester
3.265 − 0.122N 2.70
−117.21 + 15.781N* −760.65 + 50.478N*
If hydrocarbon groups have isoconfiguration, see e Add to values of ∆N, ∆B calculated for ketone
0.298 + 0.209N 11.5 − N
−9.39 + 2.848N* −140.04 + 13.869N*
If hydrocarbon groups have isoconfiguration, see e The ∆N value is not a correction to regular ether value, but the ∆B value is a correction to regular etherc
3.581 + 0.325N −0.16
25.39 + 8.744N* 0
If hydrocarbon groups have isoconfiguration, see e Corrections to be added to amine calculatione
1.390 − 0.461N 3.27 15.04 − N
25.39 + 8.744N* 25.39 + 8.744N* 0
If hydrocarbon groups have isoconfiguration, see e If hydrocarbon groups have isoconfiguration, see e The ∆N value is not a correction to regular amine value; to find ∆B, use primary amine valuea,c
f
7.812 − 0.236N 5.84 5.56 5.36 7.182 − 0.236N 4.039 − 0.0103N
f
−213.14 + 18.330N* −213.4 + 18.330N* −338.01 + 25.086N* −338.01 + 25.086N* −213.14 + 18.330N* −241.66 + 27.937N*
Note alkene contribution is necessary For aromatic correction, see f
2-366
PHYSICAL AND CHEMICAL DATA
TABLE 2-398
Group Contribution Values for Liquid Viscosity Prediction (Concluded ) ∆ Ni
∆Bi
−0.7228 + 0.1755N 2.321 − 0.2357N 10.452 − 1.1276N
286.26 − 31.009N* −26.063 − 11.516N* 3599.9 − 199.96N*
Structures or functional group Amines (Cont.) Isomethyl on nitrile Aromatic nitrile Dinitrile Halogenated compounds Fluoride Chloride Bromide Iodide Special configurations (corrections) C(Cl)x CClCCl C(Br)x CBrCBr CF3 In alcohols In other compounds Aldehydes Aldehydes with an aromatic nucleus in structure (correction) Anhydrides Anhydrides with an aromatic nucleus in structure (correction) Amides Amides with an aromatic nucleus in structure (correction) Sulfide Isomethyl on sulfide a
1.43 3.21 4.39 5.76
Remarks
5.75 −17.03 −101.97 + 5.954N* −85.32
1.91 − 1.459x 0.96 0.50 1.60
e, f e, f e, f
−26.38 0 81.34 − 86.850x −57.73
−3.93 −3.93 3.38 2.70
341.68 25.55 146.45 − 25.11N* −760.65 + 50.478N*
7.97 − 0.50N 2.70
−33.50 −760.65 + 50.478N*
13.12 + 1.49N 2.70
524.63 − 20.72N* −760.65 + 50.478N*
3.9965 − 0.1861N 0.1601
−76.676 + 8.1403N* −25.026
For substitutions on an aromatic nucleus in more than one position, additional corrections are required: Ortho
∆N = 0.51
Meta Para
∆N = 0.11 ∆N = −0.04
∆B = −571.94 (with OH) 54.84 (without OH) ∆B = 27.25 ∆B = 17.57
For alcohols, if there is a methyl group in the isoposition, increase ∆ N by 0.24 and ∆ B by 94.23. If the compound has an aromatic OH or NH2, or if there is an aromatic ether, use ∆N contribution in table but neglect other substituents on the ring such as halogen, CH3, NO2, and the like. For the calculation of ∆ B, however, such substituents must be taken into account. d For aromatic alcohols and compounds with an OH on a side chain, the alcohol contribution (primary, etc.) must be included. For example, o-chlorophenol: ∆ B = ∆B (primary alcohol) + ∆ B (chlorine) + ∆ B (phenol) + ∆ B (ortho correction—see footnote a) With N* = 16.17 (see footnote c): ∆B = (−589.44 + 70.519 × 16.17) + (−17.03) + (213.68) + (−571.94) = 175.56 Ba = 745.94 B = Ba + ∆B = 921.50 2-Phenylethanol: N = 8; ∆N = ∆N (primary alcohol) + ∆N (correction) = [10.606 − (0.276)(8)] + (−0.16) = 8.24 N* = N + ∆N = 8 + 8.24 = 16.24 ∆ B = ∆B (primary alcohol) + ∆ B (correction) = [−589.44 + (70.519)(16.24)] + 213.68 = 769.47 Ba = 747.43 B = Ba + ∆B = 1516.9 e For esters, alkylbenzenes, halogenated hydrocarbons, and ketones: If the hydrocarbon chain has a methyl group in an isoposition, decrease ∆N by 0.24 and increase ∆ B by 8.93 for each such grouping. For ethers and amines, decrease ∆ N by 0.50 and increase ∆ B by 8.93 for each isogroup. f For alkylbenzenes, nitrobenzenes, halogenated benzenes and for secondary or tertiary amines where at least one aromatic group is connected to an amino nitrogen, add the following corrections for each aromatic nucleus. If N < 16, increase ∆N by 0.60; if N ≥ 16, increase ∆ N by 3.055 − 0.161N for each aromatic group. For any N, increase ∆B by (−140.04 + 13.869N*). From van Velzen et al. (122). b c
B is calculated by Eq. (2-116) using values of Ba calculated from Eq. (2-117) or (2-118) according to the value of N* and group contributions from Table 2-398. B = Ba + ∆Bi
(2-116)
N* ≤ 20
Ba = 24.79 + 66.885N* − 1.3173N*2 − 0.00377N*3
(2-117)
N* > 20
Ba = 530.59 + 13.740N*
(2-118)
i
The method should not be used for the first member of a homologous series or for temperatures much above the normal boiling point (Tr ≈ 0.75). Errors for both hydrocarbons and nonhydrocarbons average 15 percent for a wide variety of compounds. Higher errors are noted for amines, diols, ethers, and fluorides. Table 2-398 gives ∆N and ∆B contributions for most common groups. Space prohibits examples for
each type of compound or inclusion of specialized cases that are more fully discussed in the Technical Data Manual. Example 26 Estimate the liquid viscosity of cis-1,4-dimethylcyclohexane at 0°C. Use Eq. (2-112) with N* calculated from Eq. (2-113) and B calculated from Eq. (2-116). Determine group contributions from Table 2-398.
Cyclohexanes n-Alkanes
(1) (2)
∆Ni
∆Bi
1.48 0
−272.85 + 25.041 N* 0
N* = N + ni∆Ni i
N* = 8 + (1)(1.48) + (2)(0) = 9.48
PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES Use Eq. (2-114) to calculate To: To = 28.86 + 37.439(9.48) − 1.3547(9.48)2 + (.02076)(9.48)3 To = 279.72 B = Ba + ni∆Bi i
Use Eq. (2-117) to calculate Ba: Ba = 24.79 + 66.885(9.48) − 1.3173(9.48)2 − 0.00377(9.48)3 Ba = 537.26
1 1 log µ = B − − 3.0 T To
1 1 log µ = 501.80 − − 3.0 273.15 279.72 log µ = −2.9569 µ = 0.001104 Pa sec = 1.104 cp An experimental value at 0°C is 1.224 cp. A mixing rule developed by Kendall and Monroe48 is useful for determining the liquid viscosity of defined hydrocarbon mixtures. Equation (2-119) depends only on the pure component viscosities at the given temperature and pressure and the mixture composition. µm =
xµ 1/3 i
i
(2-119)
For mixtures of the same chemical family, errors average less than 3 percent, while errors overall average 5–6 percent, with errors of mixed families averaging from 10–15 percent. For estimating the liquid viscosity of defined nonhydrocarbon mixtures, a mixing rule shown by Eq. (2-120) was recommended by the Technical Data Manual. ln µm = xi ln µi
(2-120)
i
Errors average near 15 percent. For interpolating viscosities of hydrocarbon mixtures within a limited range knowing viscosities at two temperatures, ASTM Procedure D341-89, including both charts and equations, is recommended. Several recommended methods for predicting the viscosity of undefined hydrocarbon mixtures, such as petroleum fractions and coal liquids, are presented and evaluated in the Technical Data Book—Petroleum Refining. In addition, several methods for determining the liquid viscosity of blends of hydrocarbon mixtures and a method for liquid viscosity of pure hydrocarbons blended with undefined mixtures are given. The most accurate method for predicting the liquid viscosity of hydrocarbons containing fewer than 20 carbon atoms at high pressure is a corresponding states method developed by Graboski and included in the Technical Data Book—Petroleum Refining. Critical temperature, critical pressure, the acentric factor, and knowing or being able to calculate at least one viscosity at a reference temperature and pressure are required. Errors average about 5 percent. For high molecular weight hydrocarbons at high pressure, low pressure viscosities can be converted instead of using the method of Kouzel,51 which requires a low pressure liquid viscosity as input. Compounds of more than 20 carbon atoms and their mixtures are treated in this way. For predicting the liquid viscosity of pure hydrocarbon mixtures at high temperatures, the method of Letsou and Stiel58 is available. Error analyses with only a small amount of data shows errors averaging 34 percent in the reduced temperature range of 0.76 to 0.98. Equation (2-121) defines the method with inputs of Eqs. (2-122) and (2-123). T1/6 c µ 2173.424 = µ(0) + ωµ(1) (2-121) M1/2P 2/3 c
Cp may be assumed to be the ideal gas heat capacity, Cpo. Average errors can be expected to be less than 5 percent.
3
n
i=1
µ = 1.5174 × 10 − 2.135 × 10−5Tr + 7.5 × 10−6T r2
(2-122)
µ(1) = 4.2552 × 10−5 − 7.674 × 10−5Tr + 3.4 × 10−5T r2
(2-123)
(0)
Gases For pure component, low pressure (<350 kPa) hydrocarbon gases, Misic and Thodos70,71 recommend the following equations. For methane and cyclic compounds below reduced temperatures of 1.0: C kG = 4.45 × 10−7 Tr p (2-124) λ For these hydrocarbons above reduced temperatures of 1.0 and for other hydrocarbons at any temperature: C kG = 10−7 (14.52 Tr − 5.14)2/3 p (2-125) λ In these equations, 2/3 1/2 101.325 λ = T 1/6 (2-126) c M Pc where kG = vapor thermal conductivity, W/m K Tr = reduced temperature, T/Tc T = temperature, K Tc = critical temperature, K Cp = heat capacity at constant pressure, J/kmol K M = molecular weight Pc = critical pressure, kPa
B = 537.26 + (1)[−272.85 + 25.041 (9.48)] B = 501.80
2-367
−5
Results are in Pa sec with inputs of Tc in K and Pc in Pa.
VAPOR AND LIQUID THERMAL CONDUCTIVITY Thermal conductivity describes the ease with which conductive heat can flow through a vapor, liquid, or solid layer of a substance. It is defined as the proportionality constant in Fourier’s law of heat conduction in units of energy ⋅ length/time ⋅ area ⋅ temperature; e.g., W/m K.
Example 27 Estimate thermal conductivity for n-hexane. For n-hexane at 373.15 K and low pressure, the required properties from Daubert et al.24 are: Tc = 507.6 K, M = 86.18, Pc = 3025.0 kPa, and Cpo = 1.721 × 105. Tr = 373.15/ 507.6 = 0.7351. Using Eq. (2-126): 101.325 2/3 λ = (507.6)1/6 (86.18)1/2 3025.0
λ = 2.724 Substituting into Eq. (2-125): 1.721 × 105 kG = 10−7 [(14.52)(0.7351) − 5.14]2/3 2.724
kG = 0.01977 W/m K The reported value is 0.02025 W/m K.24 For pure nonhydrocarbon gases at low pressures (up through 1 atm), the following equations may be used at temperature T (K): ηGCv Monatomic gases12: kG = 2.5 (2-127) M Linear molecules12:
ηG 2928.8 kG = 1.30 Cv + 14644.0 − M Tr
ηG kG = (1.15 Cv + 16903.36) M kG = vapor thermal conductivity, W/m K ηG = vapor viscosity, Pa·s Cv = heat capacity at constant volume, J/kmol K M = molecular weight Tr = reduced temperature, T/Tc Tc = critical temperature, K
Nonlinear molecules102: where
(2-128) (2-129)
Cv may be calculated as Cpo − R, where Cpo is the ideal gas heat capacity in J/kmol K and R is the gas constant, 8314 J/kmol K. Average errors are in the 8–10 percent range but may be higher for polar compounds. This method should not be used for molecules that associate; e.g., organic acids.
Example 28 Estimate thermal conductivity of carbon dioxide at 370 K and low pressure. For carbon dioxide (a linear molecule) at 370.0 K and low pressure, the required properties from Daubert et al.24 are: Tc = 304.2 K, ηG = 1.828 × 10−5 Pa·s, M = 44.01, and Cpo = 40520 J/kmol K. Tr = 370.0/304.2 = 1.2163, and Cv = 40520 − 8314 = 32206 J/kmol K. Substituting in Eq. (2-128): 1.828 × 10−5 2928.8 kG = (1.30) (32206) + 14644.0 − 44.01 1.2163
kG = 0.02247 W/m K A value of 0.0220 W/m K is obtained from Vargaftik et al.124
2-368
PHYSICAL AND CHEMICAL DATA
For pure gases above atmospheric pressure, the method of Stiel and Thodos102 may be used: A × 10−4(eBρr + C) kG = k′G + (2-130) Tc1/ 6 M1/2 5 Zc 2/3 Pc
ρr < 0.5 0.5 < ρr > 2.0 2.0 < ρr > 2.8
A = 2.702 A = 2.528 A = 0.574
B = 0.535 B = 0.670 B = 1.155
373.15 K and low pressure. The required pure component properties from Daubert et al.24 are: k1 = 0.02504 W/m K, k2 = 0.01587 W/m K, µ1 = 1.161 × 10−5 Pa·s, µ2 = 1.361 × 10−5 Pa·s, M1 = 46.07, M2 = 50.49, Tb1 = 248.3 K, and Tb2 = 248.9 K. By Eq. (2-134), S1 = (1.5)(248.3) = 372.45, and S2 = (1.5)(248.9) = 373.35. By Eq. (2-133), S12 = S21 = (1.0)[(372.45)(373.35)]1/2 = 372.90. Substituting in Eq. (2-132): A11 = A22 = 1.0
C = −1.000 C = −1.069 C = 2.016
where kG = vapor thermal conductivity at the temperature T (K) and pressure P of interest, W/m K k′G = vapor thermal conductivity at T and atmospheric pressure, W/m K ρr = reduced density = Vc /V Vc = critical molar volume, m3/kmol V = molar volume at T and P, m3/kmol Tc = critical temperature, K M = molecular weight Pc = critical pressure, MPa Zc = critical compressibility factor = PcVc /RTc R = gas constant = 0.008314 MPa m3/kmol K Errors in the range of 5–6 percent are typical with this method but may be higher for branched compounds.
Example 29 Estimate the thermal conductivity of carbon dioxide at
1.161 × 10−5
373.15 + 372.45 1.361 × 10 46.07 373.15 + 373.35
1 A12 = 1 + 4
50.49
3/4
1/2 2
−5
373.15 + 372.90 = 0.956 373.15 + 372.45
373.15 + 373.35 1.361 × 10 1.161 × 10 50.49 373.15 + 372.45
1 A21 = 1 + 4
−5
46.07
3/4
1/2 2
−5
373.15 + 372.90 = 1.047 373.15 + 373.35
Using Eq. (2-131): (0.23)(0.02504) (0.77)(0.01587) km = + = 0.01805 W/m K (0.23)(1.0) + (0.77)(0.956) (0.23)(1.047) + (0.77)(1.0) The experimental value is 0.01778 W/m K.66
24
370 K and 10 MPa pressure. The required properties from Daubert et al. are: Tc = 304.2 K, M = 44.01, Pc = 7.383 MPa, and Vc = 0.0940 m3/kmol; Zc = (7.383) × (0.0940)/(0.008314)(304.2) = 0.274. k′G = 0.0220 W/m K123, and V = 0.22809 m3/ kmol3; ρr = 0.0940/0.22809 = 0.4121. Substituting in Eq. (2-130) using the equation constants A, B, and C for ρr < 0.5: (2.702 × 10−4)(e(0.535)(0.4121) − 1.000) kG = 0.0220 + = 0.0315 W/m K (304.2)1/6(44.01)1/2 (0.274)5 (7.383)2/3
124
Vargaftik et al.
report a value of 0.0308 W/m K.
The thermal conductivity of low pressure (1 atm or less) gas mixtures can be determined from the relation of Wassiljewa126: n yi ki km = (2-131) n i=1
yj Aij j−1
where km = mixture thermal conductivity, W/m K n = number of components yi, j = mole fraction of component i or j in the vapor mixture ki = thermal conductivity of pure component i at the temperature of interest The binary interaction parameter Aij is obtained by the method of Lindsay and Bromley61: 1 µi Mj 3/4 T + Si 1/2 2 T + Sij Aij = 1 + (2-132) 4 µj Mi T + Sj T + Si where µi, j = vapor viscosity of pure component i or j at the temperature T of interest and low pressure, Pa·s Mi, j = molecular weight of pure component i or j T = temperature, K Sij = Sji; see Eq. (2-133) C = 1.0 except when either or both components i and j are very polar; then C = 0.73 Si, j = 79 K for helium, hydrogen, and neon; for all others, see Eq. (2-134) Tbi, j = normal boiling temperature of pure component i or j, K
Sji = C(SiSj)1/2 Si,j = 1.5Tbi, j
(2-133) (2-134)
Expected errors for this method are 4–5 percent. At higher pressures, a pressure correction using Eq. (2-130) may be used. The mixture is treated as a hypothetical pure component with mixture critical properties obtained via Eqs. (2-5), (2-8), and (2-17) and with the molecular weight being mole-averaged. Example 30 Estimate thermal conductivity of a mixture of 0.23 mole fraction dimethylether (1) and 0.77 mole fraction methyl chloride (2) at
Liquids For pure component hydrocarbon liquids at reduced temperatures between 0.25 and 0.8 and at pressures below 3.4 MPa, an equation based on the methods of Pachaiyappan et al.79 and Riedel88 may be used: 3 + 20(1 − Tr)2/3 kL = CρM n (2-135) 293.15 2/3 3 + 20 1 − Tc
where kL = liquid thermal conductivity, W/m K M = molecular weight ρ = molar density at 293.15 K, kmol/m3 Tr = reduced temperature, T/Tc T = temperature, K Tc = critical temperature, K For unbranched, straight chain hydrocarbons, n = 1.001 and C = 1.811 × 10−4. For branched and cyclic hydrocarbons, n = 0.7717 and C = 4.407 × 10−4. Average errors are 5 percent when this equation is used. For pressures greater than 3.4 MPa, the thermal conductivity from Eq. (2-135) may be corrected by the technique suggested by Lenoir.57 The correction factor is the ratio of conductivity factors F/F′, where F is at the desired temperature and higher pressure, and F′ is at the same temperature and lower pressure (usually atmospheric). The conductivity factors are calculated from: 2.054T 2r F = 17.77 + 0.065Pr − 7.764Tr − (2-136) e0.2 Pr where Tr = reduced temperature as in Eq. (2-135) Pr = reduced pressure, P/Pc P = pressure, MPa Pc = critical pressure, MPa The average error in the pressure correction alone is typically 3 percent. Example 31 Estimate thermal conductivity of n-octane at 373.15 K and pressures of 0.1 MPa and 20.0 MPa. The required properties from Daubert et 24 al. are: ρ at 293.15 K = 6.155 kmol/m3, M = 114.2, Tc = 568.7 K, and Pc = 2.490 MPa. Tr = 373.15/568.7 = 0.6561, n = 1.001, and C = 1.811 × 10−4. Substituting in Eq. (2-135) for the thermal conductivity at 0.1 MPa:
3 + (20)(1 − 0.6561)2/3 kL = (1.811 × 10−4)(6.155)(114.2)1.001 293.15 2/3 3 + (20) 1 − 568.7 = 0.107 W/m K
PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES The reported value is 0.111 W/m K.123 To correct to 20 MPa, Pr = 20.0/2.490 = 8.032, and P′r = 0.1/2.490 = 0.04016. Substituting in Eq. (2-136) to calculate F and F′: (2.054)(0.6561)2 F = 17.77 + (0.065)(8.032) − (7.764)(0.6561) − e(0.2)(8.032) F = 13.02 (2.054)(0.6561)2 F′ = 17.77 + (0.065)(0.04016) − (7.764)(0.6561) − e(0.2)(0.04016) F′ = 11.80 Thus, the thermal conductivity of n-octane at 373.15 K and 20.0 MPa is estimated to be (0.111)(13.02/11.80) = 0.122 W/m K as compared with an experimental value of 0.121 W/m K.123 For pure component hydrocarbon liquids above the normal boiling point and all pressures, the method of Kanitkar and Thodos46 is recommended: α eβρr P < 10,000 kPa (2-137) kL = λ βρ 2.596 × 10−4 P 1.6 r + α e r kL = P > 10,000 kPa (2-138) λ where kL = liquid thermal conductivity at the temperature T (K) and pressure P (kPa) of interest, W/m K α = 0.0112 β−3.322 (2-139) β = 0.40 + 0.986e−0.64λ (2-140)
101.325 2/3 1/2 λ = T 1/6 c M Pc ρr = reduced density = Vc /V Vc = critical molar volume, m3/kmol V = molar volume at T and P, m3/kmol Tc = critical temperature, K M = molecular weight Pc = critical pressure, kPa Pr = reduced pressure, P/Pc
(2-141)
β = 0.40 + 0.986e−(0.64)(3.639)
= 0.496
α = (0.0112)(0.496)
= 0.115
−3.322
Using Eq. (2-138): (2.596 × 10−4)(6.024)1.6 + 0.115e(0.496)(2.478) kL = = 0.109 W/m K 3.639 The reported value is 0.106 W/m K.123 The thermal conductivity of pure component nonhydrocarbon liquids may be estimated by the method of Baroncini et al.,5 with a modification by Myers75 for silicon compounds, at reduced temperatures between 0.3 and 0.8 and at pressures below 3.5 MPa: abc (1 − Tr)0.38 kL = (2-142) T 1/6 m r where kL = liquid thermal conductivity, W/m K Tr = reduced temperature, T/Tc T = temperature, K Tc = critical temperature, K a = constant parameter b = constant parameter = function of normal boiling temperature Tb, K c = constant paramerer = function of critical temperature Tc, K m = constant parameter = function of molecular weight M
Compound class
a
b
c
m
Acids* Alcohols, phenols Esters† Ethers Halides‡ Refrigerants R20–R23 Ketones Alkoxysilanes Alkyl-(aryl)-chlorosilanes
0.00319 0.00339 0.0415 0.0385 0.494 0.562 0.00383 0.00482 0.6510
T b6/5 T b6/5 T b6/5 T b6/5 1.0 1.0 T b6/5 T b6/5 1.0
T c−1/6 T c−1/6 T c−1/6 T c−1/6 T 1/6 c T 1/6 c T c−1/6 T c−1/6 T 1/6 c
M 1/2 M 1/2 M M M 1/2 M 1/2 M 1/2 M 1/2 M 1/2
*Do not use for formic, myristic, or oleic acids. †Do not use for butyl stearate. ‡Do not use for refrigerants R20–R23 (CHCl3, CHFCl2, CHClF2, or CHF3). 390.8 K. Tr = 360.0/563.0 = 0.6394. From Table 2-399, a = 0.00339, b = (Tb)6/5 = (390.8)6/5 = 1289.3, c = (Tc)−1/6 = (563.0)−1/6 = 0.3480, and m = M1/2 = (74.12)1/2 = 8.609. Substituting in Eq. (2-142): (0.00339)(1289.3)(0.3480) (1 − 0.6394)0.38 kL = = 0.1292 W/m K 8.609 (0.6394)1/6 123 The reported value is 0.1429 W/m K. For pure component nonhydrocarbon liquids for which Eq. (2-142) is not applicable, the method of Missenard72,73 may be used at temperature T (K) and below pressures of 3.5 MPa: 3 + 20 (1 − Tr)2/3 kL = kLr (2-143) 273.15 2/3 3 + 20 1 − Tc
(Tbρr)1/2 Cpr kLr = 2.656 × 10−7 M1/2 N1/4
Example 32 Estimate thermal conductivity of n-octane at 473.15 K and 15,000 kPa. The required properties from Daubert et al.24 are: Tc = 568.7 K, M = 114.2, Pc = 2490.0 kPa, and Vc = 0.4860 m3/kmol. The specific volume at 473.15 K and 15,000 kPa is 0.001717 m3/kg123; V = (0.001717)(114.2) = 0.1961 m3/kmol. Thus, Pr = 15000/2490 = 6.024, and ρr = 0.4860/0.1961 = 2.478. Substituting in Eqs. (2-141), (2-140), and (2-139): 101.325 2/3 λ = (568.7)1/6 (114.2)1/2 = 3.639 2490
TABLE 2-399 Values of Constant Parameters in Eq. (2-142) for Various Compound Classes
Average errors can be expected to be in the order of 10 percent.
Values of a, b, c, and m for various compound classes are found in Table 2-399. Average errors are about 8 percent.
Example 33 Estimate thermal conductivity of n-butanol. The properties required to estimate the liquid thermal conductivity of n-butanol at 360.0 K and 0.1 MPa from Daubert et al.24 are: Tc = 563.0 K, M = 74.12, and Tb =
2-369
where
(2-144)
kL = liquid thermal conductivity, W/m K kLr = liquid thermal conductivity at 273.15 K, W/m K Tr = reduced temperature, T/Tc Tc = critical temperature, K Tb = normal boiling temperature, K ρr = molar density at 273.15 K, kmol/m3 Cpr = molar heat capacity at 273.15 K, J/kmol K M = molecular weight N = number of atoms in the molecule
Errors in the order of 8 percent can be expected.
Example 34 Estimate the thermal conductivity of n-propionaldehyde (CH3CH2CHO) at 318.15 K and low pressure (0.1 MPa); the necessary properties from Daubert et al.24 are: Tc = 504.4 K, Tb = 321.1 K, ρr = 14.11 kmol/m3, C pr = 1.309 × 105 J/kmol K, and M = 58.08. N = 10, and Tr = 318.15/504.4 = 0.6307. Substituting in Eq. (2-144): [(321.1) (14.11)]1/2 (1.309 × 105) kLr = (2.656 × 10−7) = 0.1727 W/m K (58.08)1/2 (10)1/4 Using Eq. (2-143): 3 + 20 (1 − 0.6307)2/3 = 0.1542 W/m K kL = 0.1727 3 + 20 (1 − 273.15/504.4)2/3
A value from the literature is 0.1541 W/m K.24 For pressures greater than 3.5 MPa, the correction factor suggested by Missenard74 may be used to obtain the thermal conductivity of pure component nonhydrocarbon liquids. Thus: Pr kL = k′L 0.98 + 0.0079 PrT r1.4 + 0.63 T 1.2 (2-145) r 30 + Pr where kL = liquid thermal conductivity at the desired temperature T (K) and pressure P (MPa), W/m K k′L = liquid thermal conductivity at T and pressure of 0.1 MPa, W/m K Pr = reduced pressure, P/Pc Pc = critical pressure, MPa Tr = reduced temperature, T/Tc Tc = critical temperature, K
Average errors are in the range of 5–20 percent.
Example 35 Estimate thermal conductivity of n-butanol. The required properties at 360 K and 15 MPa from Daubert et al.24 are: Tc = 563.0 K and
2-370
PHYSICAL AND CHEMICAL DATA
Pc = 4.423 MPa. Tr = 360.0/563.0 = 0.6394 and Pr = 15.0/4.423 = 3.391. k′L = 0.1429 W/m K.123 Substituting in Eq. (2-145): 3.391 kL = 0.1429 0.98 + (0.0079) (3.391) (0.6394)1.4 + (0.63) (0.6394)1.2 30 + 3.391
= 0.1474 W/m K Vargaftik123 reports a value of 0.1494 W/m K. For both aqueous and nonaqueous liquid mixtures, the method of Li59 is suggested for pressures below 3.5 MPa: n
km =
n
φ φ k i
j
ij
(2-146)
i=1 j=1
xi Vi φi = n
xj Vj
(2-147)
Gas Diffusivity For prediction of the gas diffusivity of binary hydrocarbon-hydrocarbon gas systems at low pressures (below about 500 psia [3.5 MPa]) the method of Gilliland32 given by Eq. (2-150) is recommended. 1 1 0.5 0.1014T 1.5 + M1 M2 (2-150) D12 = 1/3 2 P(V 1/3 1 + V2 )
Component 1 is the solute, while component 2 is the solvent. Units of T, P, and V are °R, psia, and cm3/gmole, respectively. Diffusivity is then in ft2/hr. The molar volumes V1 and V2 at the normal boiling point are estimated by Tyn and Calus,120 Eq. (2-151). Vi = 0.285Vc1.048 i
j=1
2 kij = (2-148) (1/ki) + (1/kj) where km = mixture liquid thermal conductivity at temperature T (K), W/m K n = number of components xi, j = mole fraction of component i or j in the liquid mixture Vi, j = liquid molar volume of pure component i or j at temperature T, m3/kmol ki, j = liquid thermal conductivity of pure component i or j at temperature T, W/m K Expected errors are in the 4–6 percent range. At pressures greater than 3.5 MPa, a pressure correction using Eq. (2-145) may be used. The mixture is treated as a hypothetical pure component with mixture critical properties obtained via Eqs. (2-5) and (2-8).
Example 36 Estimate thermal conductivity of a mixture of 0.302 mole fraction diethyl ether (1) and 0.698 mole fraction methanol (2) at 273.15 K and 0.1 MPa. The required properties from Daubert et al.24 are: V1 = 0.1007 m3/kmol, V2 = 0.03942 m3/kmol, k1 = 0.1383 W/m K, and k2 = 0.2069 W/m K. Substituting in Eq. (2-148): 2 k12 = = 0.1658 1/0.1383 + 1/0.2069 Then, using Eq. (2-147) to obtain φ1 and φ2: (0.302)(0.1007) 0.03041 φ1 = = = 0.525 (0.302)(0.1007) + (0.698)(0.03942) 0.05793
(2-151)
The method gives average errors of less than 4 percent. For prediction of the gas diffusivity of binary air-hydrocarbon or nonhydrocarbon gas mixtures at low pressures, the method of Fuller et al.30 given by Eq. (2-152) is recommended. 1 1 0.5 0.1013T 1.75 + M1 M2 D12 = (2-152) P[( v1)1/3 + ( v2)1/3]2
Units of T and P are K and Pa, respectively, with the resulting diffusivity in m2/sec. All vi are group contribution values for the subscript component summed over atoms, groups, and structural features given in Table 2-400. For air-hydrocarbon systems, average deviations do not exceed 9 percent. For general nonhydrocarbon gas systems, the average deviation is about 6 percent. Example 37 Estimate the diffusivity of benzene vapor diffusing into air at 30°C and 96.5 kPa total pressure. Use Eq. (2-152). 1 1 0.5 0.1013T1.75 + M M 1 2 D12 = P[( v1)1/3 + ( v2)1/3]2
T = 303.1 K, M1 = 78.1, M2 = 28.86, and P = 96,500 Pa.
(0.698)(0.03942) φ2 = = 0.475 0.05793 Substituting in Eq. (2-146):
Now consult Table 2-400. For benzene (C6H6):
km = (0.525)2(0.1383) + 2(0.525)(0.475)(0.1658) + (0.475)2(0.2069) = 0.167 W/m K
For air, v2 = 20.1:
v1 = 6(16.5) + 6(1.98) − 20.2
v1 = 90.68
1 1 0.5 (.01013)303.11.75 + 78.1 28.86 D12 = (96,500)(90.681/3 + 20.11/3)2
Jamieson and Hastings42 report a value of 0.173 W/m K for this mixture.
DIFFUSIVITY Diffusion is the molecular transport of mass without flow. The diffusivity (D) or diffusion coefficient is the proportionality constant between the diffusion and the concentration gradient causing diffusion. It is usually defined by Fick’s first law for one-dimensional, binary component diffusion for molecular transport without turbulence shown by Eq. (2-149) dC1 N1 (2-149) = −D12 A dL The molar flow of species 1(N1) per unit area (A) is directly proportional to the change in concentration of species 1 (C1) per distance diffused (L). The usual units of diffusivity are m2/sec. In chemical engineering, the primary application of the diffusivity is to calculate the Schmidt number (µ/ρD) used to correlate mass transfer properties. This number is also used in reaction rate calculations involving transport to and away from catalyst surfaces. Experimental diffusion coefficients are scarce and not highly accurate, especially in the liquid phase, leading to prediction methods with marginal accuracy. However, use of the values predicted are generally suitable for engineering calculations. At concentrations above about 10 mole percent, predicted values should be used with caution. Diffusivities in liquids are 104–105 times lower than those in gases.
D12 = 9.68 × 10−6 m2/sec This value is within 1 percent of an available experimental value. TABLE 2-400
Atomic Diffusion Volumes for Use in Eq. (2-153)
Atomic and structural diffusion-volume increments ν C H O N
16.5 1.98 5.481 (5.69)
Cl S Aromatic ring Heterocyclic ring
(19.5) (17.0) −20.2 −20.2
Diffusion volumes for simple molecules H2 D2 He N2 O2 Air Ar Kr Xe
7.07 6.70 2.88 17.9 16.6 20.1 16.1 22.8 (37.9)
CO CO2 N2O NH3 H2O CCl2F2 SF6 CI2 Br2 SO2
18.9 26.9 35.9 14.9 12.7 (114.8) (69.7) (37.7) (67.2) (41.1)
Parentheses indicate that the value listed is based only on a few data points.
PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-401
Parameters for Eq. (2-153)
Pr
(D 12 P)R
A
B
C
0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 4.0 5.0
1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.02 1.02 1.02 1.02 1.03 1.03 1.04 1.05 1.06 1.07
0.38042 0.067433 0.098371 0.137610 0.175081 0.216376 0.314051 0.385736 0.514553 0.599184 0.557725 0.593007 0.696001 0.790770 0.502100 0.837452 0.890390
1.52267 2.16794 2.42910 2.77605 2.98256 3.11384 0.50264 3.07773 3.54744 3.61216 3.41882 3.18415 3.3760 3.27984 2.39031 3.23513 3.13001
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.141211 0.278407 0.372683 0.504894 0.678469 0.0665702 0.0 0.602907 0.0 0.0
E
13.45454 14.00000 10.00900 8.57519 10.37483 11.21674
D′12 D12 = 1.013 × 105 (D12 P)R(1 − AT r−B)(1 − CT −E r ) P
6.19043
(2-153)
Example 38 Estimate the diffusivity of hydrogen (1) in nitrogen (2) at 60°C and 17.23 MPa. A value of the low pressure diffusivity obtained using Eq. (2-152) is D′12 = 9.2 × 10−5 m2/sec. Use Eq. (2-153): D′12 D12 = 1.013 × 105 (D12P)R(1 − AT r−B)(1 − CT −E r ) P For nitrogen, Tc = 126.2 K and Pc = 3.394 MPa. 17.23 333.2 Pr = = 5.077 Tr = = 2.640 126.2 3.394 From Table 2-401: (D12P)R = 1.07, A = 0.89039, B = 3.1300, C = 0 9.2 × 10−5 D12 = 1.013 × 105 7 (1.07)[1 − .89039(2.64)−3.13] 1.723 × 10
D12 = 5.54 × 10 m /sec
y D j
(1.1728 × 10−16)(298.15)(1.0 × 112.56)1/2 D12 = (7.548 × 10−4)(0.0745)0.6 D12 = 2.33 × 10−9 m2/sec
Liquid Diffusivity Liquid diffusivities are in general not as accurately predicted as vapor diffusivities, and specialized methods have been developed. References to each method determined to be accurate are given, but only the most common methods will be presented. For predicting liquid diffusivities of binary nonpolar liquid systems at high solute dilution, Umesi120 developed a method that only depends on the viscosity of the solvent (2) and the radius of gyration of the solvent (2) and the solute (1). The Technical Data Book— Petroleum Refining gives the method and values of the radii of gyration for common hydrocarbons. Errors average 16 percent but reach 30 percent at times. For predicting diffusivities in binary polar or associating liquid systems at high solute dilution, the method of Wilke and Chang129 defined in Eq. (2-156) can be utilized. The Tyn and Calus equation (2-152) can be used to determine the molar volume of the solute at the normal boiling point. Errors average 20 percent, with occasional errors of 35 percent. The method is not considered to be accurate above a solute concentration of 5 mole percent. For concentrated binary nonpolar liquid systems (more than 5 mole percent solute), the diffusivity can be estimated by a molar average mixing rule developed by Caldwell and Babb,14 Eq. (2-156). D1m = x1D21 + (1 − x1)D12
(2-156)
D21 and D12 are dilute solution binary diffusivities. Errors depend on the procedure used to determine the dilute solution diffusivities. For multicomponent nonpolar liquid systems, Leffler and Cullinan56 developed a mixing rule, Eq. (2-157). 2
An available experimental value is 4.89 × 10−7 m2/sec. For prediction of gas phase diffusion coefficients in multicomponent hydrocarbon/nonhydrocarbon gas systems, the method of Wilke128 shown in Eq. (2-154) is used. 1 − yi
For chlorobenzene, M2 = 112.56, χ2 = 1.0, µ2 @ 25°C = 7.548 × 10−4 Pa sec, and V1 = 0.0745 m3/kmole.
D1mµm = (D12µ2)x (D13µ3)x . . .
2
Dim =
T(χ2M2)1/2 D12 = 1.1728 × 10−16 µ2V 10.6
An experimental value of 2.77 × 10−9 m2/sec is available.
D′12 is the low pressure diffusivity at the temperature of interest. (D12P)R is a reduced diffusivity pressure product at infinite reduced temperature; and A, B, C, and E are constants. All are a function of Pr tabulated in Table 2-401. Component 1 is the diffusing species, while component 2 is the concentrated species. Critical properties are for the solvent. The pressure is given in Pa. The diffusivity is in m2/sec. Errors from evaluation average near 15 percent.
−7
molecular weight are required input parameters. For the common solvents, χ decreases to 1 as polarity decreases, with values of 2.6 for water, 1.9 for methanol, 1.5 for ethanol, and 1.0 for less polar solvents. When tested with both hydrocarbon and nonhydrocarbon systems, average errors are about 25 percent—not excessive, considering the magnitude of the diffusivity.
Example 39 Estimate the infinite dilution diffusivity of propane (1) in chlorobenzene (2) at 25°C. Use Eq. (2-113):
An alternate method for gas diffusivity of binary gas mixtures at low pressures is the method of Hirschfelder et al.40 The method requires several molecular parameters and, when evaluated, gives an average absolute error of about 10 percent. The method is discussed in detail in the Data Prediction Manual. For predicting the diffusivity of binary gas mixtures at high pressures, the method of Takahashi,109 Eq. (2-153), applies.
2-371
(2-154)
ij
j j≠1
This mixing rule is used to determine the diffusivity of any component in a j + 1 component mixture and requires binary diffusivities of component i with all other components. It has been estimated that errors are about 5 percent greater than the greatest error in the binary diffusivities. Fairbanks and Wilke,28 using the same Eq. (2-154), made the same recommendation with essentially the same errors. For prediction of the diffusivity of a dilute dissolved gas (hydrocarbon or nonhydrocarbon) in a liquid, the standard method is that of Wilke and Chang129 shown by Eq. (2-155). T(χ2M2)1/2 (2-155) D12 = 1.1728 × 10−16 µ2V 0.6 1 Component 1 is the diffusing gas, while component 2 is the solvent. The solvent viscosity µ2 in Pa sec, the solute molar volume at the normal boiling point V1 in m3/kmole, and the solvent association parameter χ2 multiplied by the solvent
(2-157)
3
The diffusivity of solute 1 in the mixture is related to the binary infinite dilution diffusivities for each of the other components calculated from Eq. (2-155) or the Umesi method. The viscosities are calculated by the methods in the previous section. Errors are not quantifiable, as little experimental data exist, although these errors would be related to those assumed for the binary pairs. For concentrated binary liquid nonhydrocarbon systems, the method of Caldwell and Babb,14 Eq. (2-156) has been modified by introduction of a thermodynamic correction term as shown in Eq. (2-158). d ln γ1 D1mµm = [xiD21 + (1 − x1)D12] 1 + d ln x1
(2-158)
The activity coefficient (γ) based corrector is calculated using any applicable activity correlating equation such as the van Laar (slightly polar) or Wilson (more polar) equations. The average absolute error is 20 percent. An alternate method for binary concentrated liquid systems where activity coefficients are not available or estimable is the method of Leffler and Cullinan56 previously given in Eq. (2-156). Absolute errors average 25 percent. For estimating the diffusivity of the dilute solute (10 mole percent) in water, the method of Hayduk and Laudie,37 Eq. (2-159), applies.
2-372
PHYSICAL AND CHEMICAL DATA 8.621 × 10−14 D12 = µ1.14 V 0.589 2 1
Component 1 is the solute, while component 2 is water. The molar volume of the solute in m3/kmole is at the solute normal boiling point, while the viscosity of water in Pa sec is at the temperature of the system resulting in a diffusivity in m2/sec. The average error is about 9 percent when tested on 36 experimental systems. For estimating the diffusivity of a dilute solute (<10 mole percent) in any solvent except water, the method of King et al.,49 Eq. (2-160), applies. D12 = 4.4 × 10
−15
λ2 λ1
T V2 µ2 V1
1/6
1/2 1/2 ln (D1m µ1/2 m ) = x2 ln (D* 12 µ 2 ) + x3 ln (D* 13 µ 3 )
σ = a − bT
(2-162)
where a and b are listed for most of the substances. To obtain values at a higher temperature than the upper temperature limit indicated by Jasper, the following expression may be used: σ = d(1 − Tr)e
(2-163)
where d and e are determined such that σ and dσ/dT at the upper temperature limit THI have the same values when calculated from both Eqs. (2-162) and (2-163): b(Tc − THI) e = (2-164) a − b THI d = (a − b THI)(1 − THI)−e
σ = 4.601 × 10−4(5.49 × 106)2/3(499.15)1/3(0.6170)(1 − 0.6073)11/9 σ = 22.36 mN/m The reported experimental value is 22.68 mN/m.43 For hydrocarbon and polar chemicals, the approach originally suggested by Macleod65 as further developed by Sugden106,107 can be used:
(2-165)
where Tc = critical temperature, °C Tr = reduced temperature = (T + 273.15)/(Tc + 273.15) Eq. (2-163) correctly predicts that the surface tension becomes zero at the critical point.87 For nonpolar, nonhydrocarbon chemicals not found in Jasper,43 use can be made of the corresponding states approach of Brock and Bird9 as modified by Miller69 at temperature T (K): σ = 4.601 × 10−4 P c2/3 T c1/3 Q (1 − Tr)11/9
(2-166)
Tbr(ln Pc − 11.5261) Q = 0.1207 1 + − 0.281 1 − Tbr
(2-167)
[P] σ = (ρL − ρG) 1000
(2-161)
The molecules in a gas-liquid interface are in tension and tend to contract to a minimum surface area. This tension may be quantified by the surface tension, which is defined as the force in the plane of the surface per unit length. Jasper43 has made a critical evaluation of experimental surface tension data for approximately 2200 pure chemicals. He correlates surface tension σ (mN/m = dyn/cm) with temperature T (°C) over a specified temperature range as
σ = surface tension, mN/m Pc = critical pressure, Pa
0.6173 (ln 5.49 × 106 − 11.5261) Q = 0.1207 1 + − 0.281 1 − 0.6173 Q = 0.6170
SURFACE TENSION
where
Example 40 Estimate surface tension for ethyl mercaptan. The required properties from Daubert et al.24 are Pc = 5.49 × 106 Pa, Tc = 499.15 K, and Tb = 308.15 K. For T = 303.15 K, Tr = 303.15/499.15 = 0.6073, and Tbr = 308.15/ 499.15 = 0.6173. Substituting into Eqs. (2-167) and (2-166):
(2-160)
The solute 1 is dissolved in a solvent pair of 2 and 3. D* are infinite dilution binary diffusivities estimated by the proper method discussed previously. The mixture viscosity can be predicted by methods of the previous section. The average absolute error when tested on 40 systems is 25 percent. The method gives higher errors if the solute is gaseous.
Errors are usually less than 5 percent.
1/2
Component 1 is the solute, while component 2 is the solvent. The latent heats, λ, are at the normal boiling point, as are the molar volumes. Using T in K and µ in Pa sec yields a diffusivity in m2/sec. The average error is 21 percent when tested on 237 experimental systems. For prediction of the liquid diffusivity of a solute in a pair of mixed solvents, the method of Tang and Himmelblau,111 Eq. (2-161), is recommended.
Tc = critical temperature, K Tr = reduced temperature, T/Tc Tb = normal boiling temperature, K Tbr = reduced normal boiling temperature, Tb /Tc
(2-159)
where
4
(2-168)
σ = surface tension, mN/m ρL = liquid density, kmol/m3 ρG = vapor density, kmol/m3
The temperature-independent parachor [P] may be calculated by the additive scheme proposed by Quale.83 The atomic group contributions for this method, with contributions for silicon, boron, and aluminum from Myers,75 are shown in Table 2-402. At low pressures, where ρL >> ρG, the vapor density term may be neglected. Errors using Eq. (2-168) are normally less than 5 to 10 percent.
Example 41 Estimate surface tension for isobutyric acid. For isobutyric acid, the liquid density from Daubert et al.24 is 10.77 kmol/m3 at 293.15 K. [P] is determined from Table 2-402: O
CH3 CH
C
CH3
OH
is made up of the two groups CH3CH(CH3) and COOH. Therefore, [P] = 133.3 + 73.8 = 207.1. With Eq. (2-168), neglecting the vapor density,
= 24.75 mN/m
207.1 σ = (10.77) 1000
4
Jasper43 quotes a value of 25.04 mN/m at 293.15 K. In general, the surface tension of a liquid mixture is not a simple function of the pure component surface tensions because the composition of the mixture surface is not the same as the bulk. For nonaqueous solutions of n components, the method of Winterfeld, Scriven, and Davis130 is applicable: n n xi xi (2-169) σm = ρ2 (σiσj)1/2 ρLi ρLi i=1 j=1
n
1 xi (2-170) = ρ i = 1 ρLi where σm = mixture surface tension, mN/m xi, j = mole fraction of component i or j in the liquid mixture ρLi, j = pure component liquid density of component i or j, kmol/m3 σi, j = pure component surface tension of component i or j, mN/m Accuracies of 3–4 percent average deviation are typical when using this method.
Example 42 Estimate surface tension of a mixture. At 298.15 K, Daubert et al.24 report the liquid density of n-pentane to be 8.617 kmol/m3 and its surface tension to be 15.47 mN/m. From the same source, the corresponding values for dichloromethane are 15.52 kmol/m3 and 27.22 mN/m. Using Eqs. (2-170) and (2-169) for a mixture of 0.1606 mole fraction n-pentane and 0.8394 mole fraction dichloromethane: 1 0.1606 0.8394 =+ ρ 8.617 15.52 ρ = 13.75 kmol/m3
PREDICTION AND CORRELATION OF PHYSICAL PROPERTIES TABLE 2-402
2-373
Atomic Group Contributions for Calculation of the Parachor [P ]
Atomic group
[P]
Carbon-hydrogen C H CH3 (CH2)n n = 1 − 12 n > 12 CH3CH(CH3) CH3CH2 CH(CH3) CH3CH2CH2CH(CH3) CH3CH(CH3)CH2 CH3CH2CH(C2H5) CH3C(CH3)2 CH3CH2C(CH3)2 CH3CH(CH3)CH(CH3) CH3CH(CH3)C(CH3)2 C6H5
Atomic group
9.0 15.5 55.5 40.0 40.3 133.3 171.9 211.7 173.3 209.5 170.4 207.5 207.9 243.5 189.6
Special Groups H in OH H in HN O OH O2 in acids, esters COO COOH N NH2 S P Si Si (silanes)
10.0 12.5 19.8 29.8 54.8 63.8 73.8 17.5 42.5 49.1 40.5 30.3 43.3
[P]
Special Groups (Cont.) B Al F Cl Br I
13.2 34.9 26.1 55.2 68.0 90.3
Ethylenic Bond Terminal* 2,3-position 3,4-position†
19.1 17.7 16.3
Triple Bond
40.6
Ring Closure 3-membered 4-membered 5-membered 6-membered 7-membered
12.5 6.0 3.0 0.8 4.0
ⴝ0 (ketone) 3 carbon atoms 4 carbon atoms 5 carbon atoms 6 carbon atoms 7 carbon atoms 8 carbon atoms 9 carbon atoms 10 carbon atoms 11 carbon atoms
22.3 20.0 18.5 17.3 17.3 15.1 14.1 13.0 12.6
* Use the value for double bonds in cyclic compound. Assume 3 double bonds for the aromatic ring. † Use 16.3 for double bonds in the 3, 4 or higher positions.
(15.47) 0.1606 0.8394 + 2(13.75) [(15.47)(27.22)] 8.617 15.52 0.8394 + (13.75) (27.22) 15.52
0.1606 σm = (13.75)2 8.617
2
2
1/2
σm = 23.89 mN/m De Soria et al.26 give an experimental value of 24.24 mN/m for this mixture. Surface tensions for aqueous solutions are more difficult to predict than those for nonaqueous mixtures because of the nonlinear dependence on mole fraction. Small concentrations of the organic material may significantly affect the mixture surface tension value. For many binary organic-water mixtures, the method of Tamura, Kurata, and Odani110 may be used: where
(2-172)
ψw is defined by the relation:
Chloroacetic acid
V(chloroacetic acid) q = 2 V(acetic acid) Expected errors are less than 10 percent when q is less than 5 and within 20 percent when q is greater than 5.
Example 43 Estimate surface tension of a water-methanol mixture. Equation (2-171) can be used with a water-methanol mixture at 303.15 K when the methanol mole fraction is 0.122. From Jasper,43 σw = 71.40 mN/m, and σo = 21.73 mN/m. The density of water (per Ref. 24) is 55.16 kmol/m3; Vw = 0.01813 m3/kmol. The density of methanol is 24.49 kmol/m3 (Ref. 24); Vo = 0.04083 m3/kmol. For methanol, q = 1. Using Eq. (2-173) to obtain ψw: ψw (0.878)(0.01813) log10 = log10 1 − ψw (0.122)(0.04083)
(ψw)q (xwVw)q log10 = log10 (xwVw + xoVo)1 − q (1 − ψw) xoVo
q σoV 2/3 o + 44.1 − σwV w2/3 T q where
Example Acetic acid: q = 2 Acetone: q = 2
(2-171)
σm = mixture surface tension, mN/m σw = surface tension of pure water, mN/m σo = surface tension of pure organic component, mN/m ψo = 1 − ψw
q Number of carbon atoms One less than the number of carbon atoms Number of carbon atoms times the ratio of the molar volume of the halogen derivative to the parent fatty acid
Halogen derivatives of fatty acids
2
2
1/4 σ m1/4 = ψwσ 1/4 w + ψoσ o
Organic component Fatty acids, alcohols Ketones
(2-173)
xw = bulk mole fraction of pure water xo = bulk mole fraction of pure organic component Vw = molar volume of pure water, m3/kmol Vo = molar volume of pure organic component, m3/kmol T = temperature, K q = constant depending upon the size and type of the organic component; see table:
44.1 + [(21.73)(0.04083)2/3 − (71.40)(0.01813)2/3] 303.15 = 0.505 − 0.342 = 0.163 ψw = 100.163 = 1.455 1 − ψw 1.455 ψw = = 0.593 2.455
2-374
PHYSICAL AND CHEMICAL DATA
Using Eq. (2-172): ψo = 1 − 0.593 = 0.407. Substituting into Eq. (2-171): σ m1/4 = (0.593)(71.40)1/4 + (0.407)(21.73)1/4 σ m1/4 = 2.603 σm = 45.91 mN/m The reported experimental value is 46.1 mN/m.110
FLAMMABILITY PROPERTIES Flash points, lower and upper flammability limits, and autoignition temperatures are the three properties used to indicate safe operating limits of temperature when processing organic materials. Prediction methods are somewhat erratic, but, together with comparisons with reliable experimental values for families or similar compounds, they are valuable in setting a conservative value for each of the properties. The DIPPR compilation includes evaluated values for over 1000 common organics. Detailed examples of most of the methods discussed are available in Danner and Daubert.22 The flash point is the lowest temperature at which a liquid gives off sufficient vapor to form an ignitable mixture with air near the surface of the liquid or within the vessel used. ASTM test methods include procedures using a closed cup (ASTM D56, ASTM D93, and ASTM D3828), which is preferred, and an open cup (ASTM D92 and ASTM D1310). When several values are available, the lowest temperature is usually taken in order to assure safe operation of the process. The method of Shebeko et al.96 is the preferred flash point prediction method. The formula of the compound, the system pressure, and vapor pressure data for the compound must be available or estimable. Equation (2-174) is the basic equation. P Psat = = 0 1 + 4.76(2β − 1)
(2-174)
(NH − NX) NO β = NC + NS + − 4 2 N’s are the numbers of atoms of carbon (C), sulfur (S), hydrogen (H), halogens (X), and oxygen (O) in the molecule. P is the total system pressure. Psat is the vapor pressure of the compound at the flash point temperature. If Psat is available as a function of temperature, Eq. (2-174) can be solved directly for the flash point temperature. Otherwise, trial and error with a table of Psat vs. T is required. Errors average about 5°C but may be as much as 15°C. An alternate method for flash point prediction is the method of Gmehling and Rasmussen33 and depends on the lower flammability limit (discussed later). Vapor pressure as a function of temperature is also required. The method is generally not as accurate as the preceding method as flammability limit errors are propagated. The authors have also extended the method to defined mixtures of organics. The upper and lower flammability limits are the boundary-line mixtures of vapor or gas with air, which, if ignited, will just propagate flame and are given in terms of percent by volume of gas or vapor in the air. Each of these limits also has a temperature at which the flammability limits are reached. The temperature corresponding to the lower-limit partial vapor pressure should equal the flash point. The
temperature corresponding to the upper-limit partial vapor pressure is somewhat above the lower limit and is usually considerably below the autoignition temperature. Flammability limits are calculated at one atmosphere total pressure and normally are considered synonymous with explosive limits. Limits in oxygen rather than air are sometimes measured and available. Limits are generally reported at 298 K and 1 atm. If temperature or pressure are increased, the lower limit will decrease while the upper limit will increase, giving a wider range of compositions over which flame will propagate. The most generally applicable method for prediction of the property is the method of Seaton,95 which depends only on the molecular structure of the molecule and utilizes second order (Benson-type) groups to construct the molecule. Equation (2-175) sums the groups’ number of each type group (ni) to get both the upper and lower limits. zu or
(ni fi ) zl = nf
ii gi
(2-175)
Two sets of fi and gi are given in the article for each second-order group to cover both upper (u) and lower (l) limits (z) in volume percent units. A study of this method for about 80 organic compounds in 14 families shows absolute errors of 0.15 percent and 2.3 percent for the lower and upper limits, respectively. The upper limit prediction should not be used for ethers. Alternate group contribution methods dependent only on molecular structure are the method of Shebeko et al.97 modified by High and Siegel39 for lower flammability limit and the method of High and Danner38 for upper flammability limit. Both methods are detailed by Danner and Daubert.22 A study comparing these methods with the Seaton method shows slightly higher absolute errors of 0.23 percent and 2.9 percent for the lower and upper limits, respectively. The upper limit prediction should not be used for ethers. Both methods are recommended to be used only for qualitative guidance. Lower flammability limits can also be back-calculated from a known flash point by the method of Gmehling and Rasmussen33 discussed earlier. The autoignition temperature is the minimum temperature for a substance to initiate self-combustion in air in the absence of a spark or flame. The temperature is no lower than and is generally considerably higher than the temperature corresponding to the upper flammability limit. Large differences can occur in reported values determined by different procedures. The lowest reasonable value should be accepted in order to assure safety. Values are also sometimes given in oxygen rather than in air. Values for hydrocarbons other than alkynes and alkadienes can be predicted by the method of Suzuki et al.108 The best model includes the descriptors Tc, Pc, the parachor, the molecular surface area (which can be approximated by the van der Waals area), and the zero-order connectivity index. Excluding alkynes and alkadienes, a study for 58 alkanes, aromatics, and cycloalkanes showed an average deviation from experimental values of about 30 K. Another method of estimating autoignition temperatures is to compare values for a compound with other members of its homologous series on a plot vs. carbon number as the temperature decreases and carbon number increases. Affens1 gives a formal procedure for such estimation.
Section 3
Mathematics*
Bruce A. Finlayson, Ph.D., Rehnberg Professor and Chair, Department of Chemical Engineering, University of Washington: Member, National Academy of Engineering (Numerical methods and all general material; section editor) James F. Davis, Ph.D., Professor of Chemical Engineering, Ohio State University (Intelligent Systems) Arthur W. Westerberg, Ph.D., Swearingen University Professor of Chemical Engineering, Carnegie Mellon University: Member, National Academy of Engineering (Optimization) Yoshiyuki Yamashita, Ph.D., Associate Professor of Chemical Engineering, Tohoku University, Sendai, Japan (Intelligent Systems)
MATHEMATICS General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miscellaneous Mathematical Constants. . . . . . . . . . . . . . . . . . . . . . . . . . The Real-Number System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebraic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-7 3-8 3-8 3-9 3-9
MENSURATION FORMULAS Plane Geometric Figures with Straight Boundaries . . . . . . . . . . . . . . . . Plane Geometric Figures with Curved Boundaries . . . . . . . . . . . . . . . . Solid Geometric Figures with Plane Boundaries . . . . . . . . . . . . . . . . . . Solids Bounded by Curved Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miscellaneous Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Irregular Areas and Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-10 3-10 3-11 3-11 3-12 3-12
ELEMENTARY ALGEBRA Operations on Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Permutations, Combinations, and Probability. . . . . . . . . . . . . . . . . . . . . Theory of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-12 3-13 3-13 3-14 3-14
ANALYTIC GEOMETRY Plane Analytic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solid Analytic Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-16 3-18
PLANE TRIGONOMETRY Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions of Circular Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inverse Trigonometric Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relations between Angles and Sides of Triangles . . . . . . . . . . . . . . . . . . Hyperbolic Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximations for Trigonometric Functions . . . . . . . . . . . . . . . . . . . . .
3-20 3-20 3-21 3-22 3-22 3-23
DIFFERENTIAL AND INTEGRAL CALCULUS Differential Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multivariable Calculus Applied to Thermodynamics . . . . . . . . . . . . . . . Integral Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-23 3-26 3-27
INFINITE SERIES Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operations with Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tests for Convergence and Divergence. . . . . . . . . . . . . . . . . . . . . . . . . . Series Summation and Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-30 3-31 3-31 3-32
COMPLEX VARIABLES Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trigonometric Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Powers and Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elementary Complex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Functions (Analytic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-33 3-33 3-33 3-33 3-33 3-34
* The contributions of William F. Ames (retired), Georgia Institute of Technology; Arthur E. Hoerl (deceased), University of Delaware; and M. Zuhair Nashed, University of Delaware, to material that was used from the sixth edition is gratefully acknowledged. 3-1
3-2
MATHEMATICS
DIFFERENTIAL EQUATIONS Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ordinary Differential Equations of the First Order . . . . . . . . . . . . . . . . Ordinary Differential Equations of Higher Order . . . . . . . . . . . . . . . . . Special Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-35 3-36 3-36 3-37 3-38
DIFFERENCE EQUATIONS Elements of the Calculus of Finite Differences . . . . . . . . . . . . . . . . . . . Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-41 3-41
INTEGRAL EQUATIONS Classification of Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relation to Differential Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methods of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-42 3-43 3-43
INTEGRAL TRANSFORMS (OPERATIONAL METHODS) Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convolution Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourier Cosine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-44 3-45 3-45 3-46 3-46
MATRIX ALGEBRA AND MATRIX COMPUTATIONS Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matrix Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-47 3-48
NUMERICAL APPROXIMATIONS TO SOME EXPRESSIONS Approximation Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
NUMERICAL ANALYSIS AND APPROXIMATE METHODS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Solution of Linear Equations. . . . . . . . . . . . . . . . . . . . . . . . . Numerical Solution of Nonlinear Equations in One Variable . . . . . . . . Interpolation and Finite Differences. . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Integration (Quadrature) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Solution of Ordinary Differential Equations as Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ordinary Differential Equations-Boundary Value Problems . . . . . . . . . Numerical Solution of Integral Equations. . . . . . . . . . . . . . . . . . . . . . . . Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Solution of Partial Differential Equations. . . . . . . . . . . . . . . Spline Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-54 3-57 3-60 3-60 3-61 3-64 3-64
OPTIMIZATION Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conditions for Optimality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strategies of Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-65 3-66 3-67
STATISTICS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enumeration Data and Probability Distributions . . . . . . . . . . . . . . . . . . Measurement Data and Sampling Densities. . . . . . . . . . . . . . . . . . . . . . Tests of Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error Analysis of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Factorial Design of Experiments and Analysis of Variance . . . . . . . . . .
3-69 3-71 3-72 3-76 3-83 3-87 3-87
DIMENSIONAL ANALYSIS PROCESS SIMULATION
3-49
3-49 3-50 3-50 3-51 3-53 3-53
INTELLIGENT SYSTEMS IN PROCESS ENGINEERING
GENERAL REFERENCES: The list of references for this section is selected to provide a broad perspective on classical and modern mathematical methods that are useful in chemical engineering. The references supplement and extend the treatment given in this section. Also included are selected references to important areas of mathematics that are not covered in the Handbook but that may be useful for certain areas of chemical engineering, e.g., additional topics in numerical analysis and software, optimal control and system theory, linear operators, and functional-analysis methods. Readers interested in brief summaries of theory, together with many detailed examples and solved problems on various topics of college mathematics and mathematical methods for engineers, are referred to the Schaum’s Outline Series in Mathematics, published by the McGraw-Hill Book Comapny.
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1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.
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237. Rall, L. B. Computational Solution of Nonlinear Operator Equations, Wiley, New York (1969) and Dover, New York (1981). 238. Ralston, A., and A. Rabinowitz. A First Course in Numerical Analysis, 2d ed., McGraw-Hill, New York (1978). 239. Ramirez, W. F. Computational Methods for Process Simulations, Butterworths, Boston (1989). 240. Rauch, J. Partial Differential Equations, Springer-Verlag (1991). 241. Reddy, J. N., and D. K. Gartling. The Finite Element Method in Heat Transfer and Fluid Dynamics, CRC Press (1994). 242. Reklaitis, G. V. Introduction to Material and Energy Balances, Wiley (1983). 243. Rekalitis, G. V., and H. D. Spriggs. Proceedings of the First International Conference on Foundations of Computer-Aided Operations, Elsevier Science Publishers, Inc., New York (1987). 244. Reklaitis, G. V., A. Ravindran, and K. M. Ragsdell. Engineering Optimization Methods and Applications, Wiley, New York (1983). 245. Rhee, H.-K., R. Aris, and N. R. Amundson. First-Order Partial Differential Equations, vol. I., Prentice Hall, Englewood Cliffs, NJ (1986). 246. ———. Matrix Computations and Mathematical Software, McGrawHill, New York (1981). 247. ———. Numerical Methods, Software, and Analysis, 2d ed., Academic, New York (1993). 248. Rich, E., and K. Kevin. Artificial Intelligence, 2d ed., McGraw-Hill, New York (1991). 249. Riggs, J. B. An Introduction to Numerical Methods for Chemical Engineers, Texas Tech Univ. Press, Lubbock, TX (1994). 250. Rippin, D. W. T., J. C. Hale, and J. F. Davis (ed.). Proceedings of the Second International Conference on Foundations of Computer-Aided Operations, CACHE Corporation, Austin, TX (1994). 251. Ritchmyer, R., and K. Morton. Difference Methods for Initial-Value Problems, 2d ed., Interscience, New York (1967). 252. Saaty, T. L., and J. Bram. Nonlinear Mathematics, McGraw-Hill, New York (1964) and Dover, New York (1981). 253. Schiesser, W. E. The Numerical Method of Lines, Academic Press (1991). 254. Schittkowski, K. Num. Math. 38 (1982): 83. 255. Seader, J. D. Computer Modeling of Chemical Processes, AIChE Monog. Ser. No. 15 (1985). 256. Seborg, D. E., T. F. Edgar, and D. A. Mellichamp. Process Dynamics and Control, Wiley, New York (1989). 257. Shampine, L. Numerical Solution of Ordinary Differential Equations, Chapman & Hall (1994). 258. Shapiro, S. C., D. Eckroth et. al (ed.). Encyclopedia of Artificial Intelligence, Wiley, New York (1987). 259. Shanno, D. F. Math. Comp. 24 (1970): 647. 260. Shenk, A. Calculus and Analytic Geometry, Goodyear Publishing Company, Santa Monica, CA (1977). 261. Shockley, J. E. Calculus and Analytic Geometry, Saunders, Philadelphia (1982). 262. Siirola, J. J., I. E. Grossmann, and G. Stephanopoulos. Proceedings of the Second International Conference on Foundations of Computer-Aided Design, Elsevier Science Publishers, Inc., New York (1990). 263. Simmons, G. F. Differential Equations, McGraw-Hill, New York (1972). 264. Simmonds, J. G. A Brief on Tensor Analysis, Springer-Verlag (1994). 265. Sincich, T., and Mendenhall, W. Statistics for Engineering and the Sciences, 4th ed., Prentice Hall, Englewood Cliffs, NJ (1995). 266. Sincovec, R. F. Math. Comp. 26 (1972): 893–895. 267. Smith, I. M., J. L. Siemienivich, and I. Gladwell. “A Comparison of Old and New Methods for Large Systems of Ordinary Differential Equations Arising from Parabolic Partial Differential Equations,” Num. Anal. Rep. Department of Engineering, no. 13, University of Manchester, England (1975). 268. Smith, W. K. Analytic Geometry, Macmillan (1972). 269. Sobel, M. A., and N. Lerner. College Algebra, Prentice Hall, Englewood Cliffs, NJ (1983). 270. Sod, G. A. Numerical Methods in Fluid Dynamics, Cambridge Univ. Press (1985). 271. Sokolnikoff, I. S., and Sokolnikoff, E. S. Higher Mathematics for Engineers and Physicists, McGraw-Hill, New York (1941). 272. Spiegel, M. R. Applied Differential Equations, 3d ed., Prentice Hall, Englewood Cliffs, NJ (1981). 273. Stakgold, I. Green’s Functions and Boundary Value Problems, Interscience, New York (1979). 274. Stein, S. K. Calculus and Analytic Geometry, 3d ed., McGraw-Hill, New York (1982). 275. Stephanopoulos, G., and J. F. Davis (eds.). Artificial Intelligence in Process Engineering, CACHE Monograph Series, CACHE, Austin (1990–1992).
276. Stephanopoulos, G., and H. Chonghun. “Intelligent Systems in Process Engineering: A Review,” Proceedings of PSE ’94, Korea (1994). 277. Stillwell, J. C. Elements of Algebra, CRC Press, New York (1994). 278. Stoer, J., and R. Bulirsch. Introduction to Numerical Analysis, Springer, New York (1993). 279. Strang, G. Linear Algebra and Its Applications, 2d ed., Academic, New York (1980). 280. Strang, G. Introduction to Linear Algebra, Wellesley-Cambridge, Cambridge, MA (1993). 281. ——— and G. Fix. An Analysis of the Finite Element Method, Prentice Hall, Englewood Cliffs, NJ (1973). 282. Swokowski, E. W. Calculus with Analytic Geometry, 2d ed., Prindle, Weber & Schmidt, Boston (1981). 283. Taylor, A. E., and D. C. Lay. Introduction to Functional Analysis, 2d ed., Wiley, New York (1980). 284. Umeda, T., and A. Ichikawa. I&EC Proc. Design Develop. 10 (1971): 229. 285. Vasantharajan, S., and L. T. Biegler. Computers and Chemical Engineering 12 (1988): 1087. 286. Vemuri, V., and W. Karplus. Digital Computer Treatment of Partial Differential Equations, Prentice Hall, Englewood Cliffs, NJ (1981). 287. Vichnevetsky, R. Computer Methods for Partial Differential Equations, vols. 1 and 2, Prentice Hall, Englewood Cliffs, NJ (1981, 1982). 288. Villadsen, J. V., and M. L. Michelsen. Solution of Differential Equation Models by Polynomial Approximation, Prentice Hall, Englewood Cliffs, NJ (1978). 289. Villadsen, J., and W. E. Stewart. Chem. Eng. Sci. 22 (1967): 1483–1501. 290. Walas, S. M. Modeling with Differential Equations in Chemical Engineering, Butterworth-Heinemann, Stoneham, MA (1991). 291. Weisberg, S. Applied Linear Regression, 2d ed., Wiley, New York (1985). 292. Weld, D. S., and J. de Kleer (ed.). Readings in Qualitative Reasoning About Physical Systems, Morgan Kaufman, San Mateo, CA (1990). 293. Westerberg, A. W., H. P. Hutchison, R. L. Motard, and P. Winter. Process Flowsheeting, Cambridge University Press, London (1979). 294. Westerberg, A. W., and H. H. Chien (ed.). Proceedings of the Second International Conference on Foundations of Computer-Aided Design, CACHE Corporation, Austin, TX (1984). 295. Westerberg, A. W. “Optimization” in A. K. Sunol, D. W. T. Rippin, G. V. Reklaitis, O. Hortacsu (eds.), Batch Processing Systems Engineering: Current Status and Future Directions, vol. 143, NATO ASI Series F, Springer, Berlin (1995). 296. Whipkey, K. L., and M. N. Whipkey. The Power of Calculus, 3d ed., Wiley, New York (1979). 297. Wilkinson, J. H. The Algebraic Eigenvalue Problem, Clarendon Press, Oxford (1988). 298. Williams, G. Computational Linear Algebra with Models, 2d ed., Allyn and Bacon, Boston (1981). 299. Wolfram, S. Mathematica, Addison-Wesley, New York (1988). 300. Wouk, A. A Course of Applied Functional Analysis, Interscience, New York (1979). 301. Wylie, C. R. Advanced Engineering Mathematics, 5th ed., McGraw-Hill, New York (1982). 302. Young, D. M. Iterative Solution for Large Linear Systems, Academic, New York (1971). 303. Zienkiewicz, O. C., and R. L. Taylor. The Finite Element Method, McGraw-Hill, London (1989). 304. ——— and K. Morgan. Finite Elements and Approximations, Wiley, New York (1983). REFERENCES FOR GENERAL AND SPECIFIC TOPICS Advanced engineering mathematics: Upper undergraduate level, 19, 80, 127, 144, 156, 173, 194, 203, 209, 301. Graduate level, 79, 127, 212, 273. Mathematical tables, mathematical dictionaries, and handbooks of mathematical functions and formulas, 1, 28, 48, 57, 118, 126, 188, 208, 217, 222, 234. Mathematical modeling of physical phenomena, 17, 19, 31, 44, 98, 132, 194, 273. Mathematical theory of reaction, diffusion, and transport processes, 10, 16, 19, 88, 223. Mathematical methods in chemical engineering, 13, 15, 61, 85, 104, 106, 138, 239, 249, 288. Inequalities, 28, 126, 139, 290. Vector and tensor analysis, 18, 163, 173, 264. Special functions in physics and engineering, 190, 197, 235. Green’s functions and applications, 75, 127, 273. Perturbation and asymptotic methods in applied mathematics, 170, 215, 216. Approximation theory and interpolation, 86, 87. Functional analysis; linear operators, 129, 147, 162, 174, 196, 216, 226, 283, 300. Generalized inverses and least-squares problems, 33, 128, 189.
MATHEMATICS
3-7
MATHEMATICS GENERAL The basic problems of the sciences and engineering fall broadly into three categories: 1. Steady state problems. In such problems the configuration of the system is to be determined. This solution does not change with time but continues indefinitely in the same pattern, hence the name “steady state.” Typical chemical engineering examples include steady temperature distributions in heat conduction, equilibrium in chemical reactions, and steady diffusion problems. 2. Eigenvalue problems. These are extensions of equilibrium problems in which critical values of certain parameters are to be determined in addition to the corresponding steady-state configurations. The determination of eigenvalues may also arise in propagation problems. Typical chemical engineering problems include those in heat transfer and resonance in which certain boundary conditions are prescribed. 3. Propagation problems. These problems are concerned with predicting the subsequent behavior of a system from a knowledge of the initial state. For this reason they are often called the transient (time-varying) or unsteady-state phenomena. Chemical engineering examples include the transient state of chemical reactions (kinetics), the propagation of pressure waves in a fluid, transient behavior of an adsorption column, and the rate of approach to equilibrium of a packed distillation column. The mathematical treatment of engineering problems involves four basic steps: 1. Formulation. The expression of the problem in mathematical language. That translation is based on the appropriate physical laws governing the process. 2. Solution. Appropriate mathematical operations are accomplished so that logical deductions may be drawn from the mathematical model. 3. Interpretation. Development of relations between the mathematical results and their meaning in the physical world. 4. Refinement. The recycling of the procedure to obtain better predictions as indicated by experimental checks. Steps 1 and 2 are of primary interest here. The actual details are left to the various subsections, and only general approaches will be discussed. The formulation step may result in algebraic equations, difference equations, differential equations, integral equations, or combinations of these. In any event these mathematical models usually arise from statements of physical laws such as the laws of mass and energy conservation in the form. Input of conserved quantity − output of conserved quantity + conserved quantity produced = accumulation of conserved quantity Rate of input of conserved quantity − rate of output of conserved quantity + rate of conserved quantity produced = rate of accumulation of conserved quantity These statements may be abbreviated by the statement Input − output + production = accumulation
FIG. 3-1
Boundary conditions.
is called a boundary-value problem (Fig. 3-1). Schematically, the problem is characterized by a differential equation plus an open region in which the equation holds and, on the boundaries of the region, by certain conditions (boundary conditions) that are dictated by the physical problem. The solution of the equation must satisfy the differential equation inside the region and the prescribed conditions on the boundary. In mathematical language, the propagation problem is known as an initial-value problem (Fig. 3-2). Schematically, the problem is characterized by a differential equation plus an open region in which the equation holds. The solution of the differential equation must satisfy the initial conditions plus any “side” boundary conditions. The description of phenomena in a “continuous” medium such as a gas or a fluid often leads to partial differential equations. In particular, phenomena of “wave” propagation are described by a class of partial differential equations called “hyperbolic,” and these are essentially different in their properties from other classes such as those that describe equilibrium (“elliptic”) or diffusion and heat transfer (“parabolic”). Prototypes are: 1. Elliptic. Laplace’s equation ∂2u ∂2u 2 + 2 = 0 ∂x ∂y Poisson’s equation
∂2u ∂2u 2 + 2 = g(x,y) ∂x ∂y These do not contain the variable t (time) explicitly; accordingly, their solutions represent equilibrium configurations. Laplace’s equation corresponds to a “natural” equilibrium, while Poisson’s equation corresponds to an equilibrium under the influence of an external force of density proportional to g(x, y). 2. Parabolic. The heat equation ∂u ∂2u ∂2u = 2 + 2 ∂t ∂x ∂y describes nonequilibrium or propagation states of diffusion as well as heat transfer. 3. Hyperbolic. The wave equation ∂2u ∂2u ∂2u = 2 + 2 ∂t2 ∂x ∂y describes wave propagation of all types when the assumption is made that the wave amplitude is small and that interactions are linear. The solution phase has been characterized in the past by a concentration on methods to obtain analytic solutions to the mathematical
When the basic physical laws are expressed in this form, the formulation is greatly facilitated. These expressions are quite often given the names, “material balance,” “energy balance,” and so forth. To be a little more specific, one could write the law of conservation of energy in the steady state as Rate of energy in − rate of energy out + rate of energy produced = 0 Many general laws of the physical universe are expressible by differential equations. Specific phenomena are then singled out from the infinity of solutions of these equations by assigning the individual initial or boundary conditions which characterize the given problem. In mathematical language one such problem, the equilibrium problem,
FIG. 3-2
Propagation problem.
3-8
MATHEMATICS
equations. These efforts have been most fruitful in the area of the linear equations such as those just given. However, many natural phenomena are nonlinear. While there are a few nonlinear problems that can be solved analytically, most cannot. In those cases, numerical methods are used. Due to the widespread availability of software for computers, the engineer has quite good tools available. Numerical methods almost never fail to provide an answer to any particular situation, but they can never furnish a general solution of any problem. The mathematical details outlined here include both analytic and numerical techniques useful in obtaining solutions to problems. Our discussion to this point has been confined to those areas in which the governing laws are well known. However, in many areas, information on the governing laws is lacking. Interest in the application of statistical methods to all types of problems has grown rapidly since World War II. Broadly speaking, statistical methods may be of use whenever conclusions are to be drawn or decisions made on the basis of experimental evidence. Since statistics could be defined as the technology of the scientific method, it is primarily concerned with the first two aspects of the method, namely, the performance of experiments and the drawing of conclusions from experiments. Traditionally the field is divided into two areas: 1. Design of experiments. When conclusions are to be drawn or decisions made on the basis of experimental evidence, statistical techniques are most useful when experimental data are subject to errors. The design of experiments may then often be carried out in such a fashion as to avoid some of the sources of experimental error and make the necessary allowances for that portion which is unavoidable. Second, the results can be presented in terms of probability statements which express the reliability of the results. Third, a statistical approach frequently forces a more thorough evaluation of the experimental aims and leads to a more definitive experiment than would otherwise have been performed. 2. Statistical inference. The broad problem of statistical inference is to provide measures of the uncertainty of conclusions drawn from experimental data. This area uses the theory of probability, enabling scientists to assess the reliability of their conclusions in terms of probability statements. Both of these areas, the mathematical and the statistical, are intimately intertwined when applied to any given situation. The methods of one are often combined with the other. And both in order to be successfully used must result in the numerical answer to a problem—that is, they constitute the means to an end. Increasingly the numerical answer is being obtained from the mathematics with the aid of computers. MISCELLANEOUS MATHEMATICAL CONSTANTS Numerical values of the constants that follow are approximate to the number of significant digits given. π = 3.1415926536 e = 2.7182818285 γ = 0.5772156649 ln π = 1.1447298858 log π = 0.4971498727 Radian = 57.2957795131° Degree = 0.0174532925 rad Minute = 0.0002908882 rad Second = 0.0000048481 rad
Pi Napierian (natural) logarithm base Euler’s constant Napierian (natural) logarithm of pi, base e Briggsian (common logarithm of pi, base 10
THE REAL-NUMBER SYSTEM The natural numbers, or counting numbers, are the positive integers: 1, 2, 3, 4, 5, . . . . The negative integers are −1, −2, −3, . . . . A number in the form a/b, where a and b are integers, b ≠ 0, is a rational number. A real number that cannot be written as the quotient of two integers is called an irrational number, e.g., 2, 3, 5, π, 3 e, 2. There is a one-to-one correspondence between the set of real numbers and the set of points on an infinite line (coordinate line).
Order among Real Numbers; Inequalities a > b means that a − b is a positive real number. If a < b and b < c, then a < c. If a < b, then a c < b c for any real number c. If a < b and c > 0, then ac < bc. If a < b and c < 0, then ac > bc. If a < b and c < d, then a + c < b + d. If 0 < a < b and 0 < c < d, then ac < bd. If a < b and ab > 0, then 1/a > 1/b. If a < b and ab < 0, then 1/a < 1/b. Absolute Value For any real number x, |x| = x −x
if x ≥ 0 if x < 0
Properties If |x| = a, where a > 0, then x = a or x = −a. |x| = |−x|; −|x| ≤ x ≤ |x|; |xy| = |x| |y|. If |x| < c, then −c < x < c, where c > 0. ||x| − |y|| ≤ |x + y| ≤ |x| + |y|. x2 = |x|. a = c , then a + b = c + d , a − b = c − d , Proportions If b d d b d b a−b c−d = . a+b c+d Indeterminants Form
Example
(∞)(0) 00 ∞0 1∞
xe−x xx (tan x)cos x (1 + x)1/x
x→∞ x → 0+ − x→aπ x → 0+
∞−∞ 0 0 ∞ ∞
1 − x− 1 x+ sin x x ex x
x→∞ x→0 x→∞
Limits of the type 0/∞, ∞/0, 0∞, ∞ ⋅ ∞, (+∞) + (+∞), and (−∞) + (−∞) are not indeterminate forms. Integral Exponents (Powers and Roots) If m and n are positive integers and a, b are numbers or functions, then the following properties hold: a−n = 1/an a≠0 (ab)n = anbn (an)m = anm, n
a = a1/n
anam = an + m if
a>0
mn
= a, a > 0 a m
n
n
am a > 0 am/n = (am)1/n = , a0 = 1 (a ≠ 0) 0a = 0 (a ≠ 0) Infinity (∞) is not a real number. It is possible to extend the realnumber system by adjointing to it “∞” and “−∞,” and within the extended system, certain operations involving +∞ or −∞ are possible. For example, if 0 < a < 1, then a∞ = lim x→∞ ax = 0, whereas if a > 1, then a∞ = ∞, ∞a = ∞ (a > 0), ∞a = 0 (a < 0). Care should be taken in the case of roots and fractional powers of a product; e.g., xy ≠ xy if x and y are negative. This rule applies if one is careful about the domain of the functions involved; so xy = xy if x > 0 and y > 0. Given any number b > 0, there is a unique function f(x) defined for all real numbers x such that (1) f(x) = bx for all rational x; (2) f is increasing if b > 1, constant if b = 1, and decreasing if 0 < b < 1. This function is called the exponential function bx. For any b > 0, f(x) = bx is
MATHEMATICS a continuous function. Also with a,b > 0 and x,y any real numbers, we have (ab)x = axbx bx by = bx + y (bx)y = bxy The exponential function with base b can also be defined as the inverse of the logarithmic function. The most common exponential function in applications corresponds to choosing b the transcendental number e. Logarithms log ab = log a + log b, a > 0, b > 0 log an = n log a log (a/b) = log a − log b n log a = (1/n) log a The common logarithm (base 10) is denoted log a or log10a. The natural logarithm (base e) is denoted ln a (or in some texts loge a). Roots If a is a real number, n is a positive integer, then x is called the nth root of a if xn = a. The number of nth roots is n, but not all of them are necessarily real. The principal nth root means the following: (1) if a > 0 the principal nth root is the unique positive root, (2) if a < 0, and n odd, it is the unique negative root, and (3) if a < 0 and n even, it is any of the complex roots. In cases (1) and (2), the root can be found on a calculator by taking y = ln a/n and then x = e y. In case (3), see the section on complex variables. PROGRESSIONS Arithmetic Progression n−1
1
n
(a + kd) = na + 2 n(n − 1)d = 2 (a + ᐉ)
k=0
where ᐉ is the last term, ᐉ = a + (n − 1)d. Geometric Progression n a(rn − 1) ar k − 1 = (r ≠ 1) r−1 k=1 Arithmetic-Geometric Progression n−1 a − [a + (n − 1)d]r n dr(1 − r n − 1) (a + kd)r k = + 1−r (1 − r)2 k=0 n 1 2 n (n + 1)2(2n2 + 2n − 1) k5 = 12 k=1 2
k=1 n
(2k − 1)
3
2
n
n
n
k k
k
k=1
2
k
k=1
2
k=1
The equality holds if, and only if, the vectors a, b are linearly dependent (i.e., one vector is scalar times the other vector). Minkowski’s Inequality Let a1, a2, . . . , an and b1, b2, . . . , bn be any two sets of complex numbers. Then for any real number p > 1,
|a + b | ≤ |a | + |b | 1/p
n
k
k
1/p
n
p
k
k=1
1/p
n
p
k
k=1
p
k=1
Hölder’s Inequality Let a1, a2, . . . , an and b1, b2, . . . , bn be any two sets of complex numbers, and let p and q be positive numbers with 1/p + 1/q = 1. Then
a b ≤ |a | |b | n
1/p
n
k k
k
k=1
1/q
n
p
q
k
k=1
k=1
The equality holds if, and only if, the sequences |a1|p, |a2|p, . . . , |an|p and |b1|q, |b2|q, . . . , |bn|q are proportional and the argument (angle) of the complex numbers akb k is independent of k. This last condition is of course automatically satisfied if a1, . . . , an and b1, . . . , bn are positive numbers. Lagrange’s Inequality Let a1, a2, . . . , an and b1, b2, . . . , bn be real numbers. Then 2
n
n
2 k
k=1
2 k
k=1
(akbj − aj bk)2
1≤k≤j≤n
Example Two chemical engineers, John and Mary, purchase stock in the same company at times t1, t2, . . . , tn, when the price per share is respectively p1, p2, . . . , pn. Their methods of investment are different, however: John purchases x shares each time, whereas Mary invests P dollars each time (fractional shares can be purchased). Who is doing better? While one can argue intuitively that the average cost per share for Mary does not exceed that for John, we illustrate a mathematical proof using inequalities. The average cost per share for John is equal to n
x pi 1 n Total money invested i=1 = = pi Number of shares purchased n i=1 nx
nP n = n n P 1 i i = 1 pi = 1 pi
= n2(2n2 − 1)
1 γ = lim − ln n = 0.577215 n→∞ m m=1 ALGEBRAIC INEQUALITIES Arithmetic-Geometric Inequality Let An and Gn denote respectively the arithmetic and the geometric means of a set of positive numbers a1, a2, . . . , an. The An ≥ Gn, i.e., a1 + a2 + ⋅⋅⋅ + an ≥ (a1a2 ⋅⋅⋅ an)1/n n The equality holds only if all of the numbers ai are equal. Carleman’s Inequality The arithmetic and geometric means just defined satisfy the inequality n
G
r
r=1
a b ≤ |a | |b |
The average cost per share for Mary is
1 = n(4n2 − 1) 3
k=1
n
⋅⋅⋅ ar)1/r ≤ neAn
where e is the best possible constant in this inequality. Cauchy-Schwarz Inequality Let a = (a1, a2, . . . , an), b = (b1, b2, . . . , bn), where the ai’s and bi’s are real or complex numbers. Then
k=1
k=1
2
1 2
r=1
k k
n
n
n
(a a
n
(2k − 1) = n (2k − 1)
or, equivalently,
a b = a b −
(r ≠ 1)
3-9
≤ neAn
Thus the average cost per share for John is the arithmetic mean of p1, p2, . . . , pn, whereas that for Mary is the harmonic mean of these n numbers. Since the harmonic mean is less than or equal to the arithmetic mean for any set of positive numbers and the two means are equal only if p1 = p2 = ⋅⋅⋅ = pn, we conclude that the average cost per share for Mary is less than that for John if two of the prices pi are distinct. One can also give a proof based on the Cauchy-Schwarz inequality. To this end, define the vectors a = (p1−1/2, p2−1/2, . . . , pn−1/2)
b = (p11/2, p21/2, . . . , pn1/2)
Then a ⋅ b = 1 + ⋅⋅⋅ + 1 = n, and so by the Cauchy-Schwarz inequality n 1 (a ⋅ b)2 = n2 ≤ i = 1 pi
n
p
i
i=1
with the equality holding only if p1 = p2 = ⋅⋅⋅ = pn. Therefore n
pi n i=1 n ≤ 1 n i = 1 pi
3-10
MATHEMATICS
MENSURATION FORMULAS Let A denote areas and V, volumes, in the following.
Radius R of Circumscribed Circle abc R = 4 s( s −a )( s − b )( s −c)
PLANE GEOMETRIC FIGURES WITH STRAIGHT BOUNDARIES Triangles (see also “Plane Trigonometry”) A = a bh where b = base, h = altitude. Rectangle A = ab where a and b are the lengths of the sides. Parallelogram (opposite sides parallel) A = ah = ab sin α where a, b are the lengths of the sides, h the height, and α the angle between the sides. See Fig. 3-3. Rhombus (equilateral parallelogram) A = aab where a, b are the lengths of the diagonals. Trapezoid (four sides, two parallel) A = a(a + b)h where the lengths of the parallel sides are a and b, and h = height. Quadrilateral (four-sided) A = aab sin θ where a, b are the lengths of the diagonals and the acute angle between them is θ. Regular Polygon of n Sides See Fig. 3-4. 1 180° A = nl 2 cot where l = length of each side 4 n l 180° R = csc where R is the radius of the circumscribed circle 2 n l 180° r = cot where r is the radius of the inscribed circle 2 n 360° β= n (n − 2)180° θ = n β β l = 2r tan = 2R sin 2 2 Inscribed and Circumscribed Circles with Regular Polygon of n Sides Let l = length of one side.
Figure
n
Area
Radius of circumscribed circle
Equilateral triangle Square Pentagon Hexagon Octagon Decagon Dodecagon
3 4 5 6 8 10 12
0.4330 l2 1.0000 l2 1.7205 l2 2.5981 l2 4.8284 l2 7.6942 l2 11.1962 l2
0.5774 l 0.7071 l 0.8507 l 1.0000 l 1.3065 l 1.6180 l 1.8660 l
Radius of inscribed circle 0.2887 l 0.5000 l 0.6882 l 0.8660 l 1.2071 l 1.5388 l 1.9318 l
Radius r of Circle Inscribed in Triangle with Sides a, b, c r=
FIG. 3-3
(s − a)(s − b)(s − c) s
Parallelogram.
where s = a(a + b + c)
FIG. 3-4
Regular polygon.
Area of Regular Polygon of n Sides Inscribed in a Circle of Radius r A = (nr 2/2) sin (360°/n) Perimeter of Inscribed Regular Polygon P = 2nr sin (180°/n) Area of Regular Polygon Circumscribed about a Circle of Radius r A = nr 2 tan (180°/n) Perimeter of Circumscribed Regular Polygon 180° P = 2nr tan n PLANE GEOMETRIC FIGURES WITH CURVED BOUNDARIES Circle (Fig. 3-5) Let C = circumference r = radius D = diameter A = area S = arc length subtended by θ l = chord length subtended by θ H = maximum rise of arc above chord, r − H = d θ = central angle (rad) subtended by arc S C = 2πr = πD (π = 3.14159 . . .) S = rθ = aDθ l = 2 r2 − d2 = 2r sin (θ/2) = 2d tan (θ/2) 1 1 θ d = 4 r2 −l2 = l cot 2 2 2 S −1 d −1 l θ = = 2 cos = 2 sin r r D A (circle) = πr2 = dπD2 A (sector) = arS = ar 2θ A (segment) = A (sector) − A (triangle) = ar 2(θ − sin θ) r−H = r2 cos−1 − (r − H) 2rH − H2 r Ring (area between two circles of radii r1 and r2 ) The circles need not be concentric, but one of the circles must enclose the other. A = π(r1 + r2)(r1 − r2)
FIG. 3-5
Circle.
r1 > r2
MENSURATION FORMULAS
3-11
Volume and Surface Area of Regular Polyhedra with Edge l
FIG. 3-6
Ellipse.
Ellipse (Fig. 3-6)
FIG. 3-7
Parabola.
Let the semiaxes of the ellipse be a and b A = πab C = 4aE(k)
where e2 = 1 − b2/a2 and E(e) is the complete elliptic integral of the second kind, π 1 2 E(e) = 1 − e2 + ⋅⋅⋅ 2 2
2 2+ b ]. [an approximation for the circumference C = 2π (a )/2 Parabola (Fig. 3-7) y2 x2 +y2 2x + 4 Length of arc EFG = 4 x2 +y2 + ln 2x y
4 Area of section EFG = xy 3 Catenary (the curve formed by a cord of uniform weight suspended freely between two points A, B; Fig. 3-8) y = a cosh (x/a) Length of arc between points A and B is equal to 2a sinh (L/a). Sag of the cord is D = a cosh (L/a) − 1. SOLID GEOMETRIC FIGURES WITH PLANE BOUNDARIES Cube Volume = a3; total surface area = 6a2; diagonal = a3, where a = length of one side of the cube. Rectangular Parallelepiped Volume = abc; surface area = 2 2(ab + ac + bc); diagonal = a2 + b +c2, where a, b, c are the lengths of the sides. Prism Volume = (area of base) × (altitude); lateral surface area = (perimeter of right section) × (lateral edge). Pyramid Volume = s (area of base) × (altitude); lateral area of regular pyramid = a (perimeter of base) × (slant height) = a (number of sides) (length of one side) (slant height). Frustum of Pyramid (formed from the pyramid by cutting off the top with a plane
1⋅A 2)h V = s (A1 + A2 + A where h = altitude and A1, A2 are the areas of the base; lateral area of a regular figure = a (sum of the perimeters of base) × (slant height).
FIG. 3-8
Catenary.
Type of surface
Name
Volume
Surface area
4 equilateral triangles 6 squares 8 equilateral triangles 12 pentagons 20 equilateral triangles
Tetrahedron Hexahedron (cube) Octahedron Dodecahedron Icosahedron
0.1179 l3 1.0000 l3 0.4714 l3 7.6631 l3 2.1817 l3
1.7321 l2 6.0000 l2 3.4641 l2 20.6458 l2 8.6603 l2
SOLIDS BOUNDED BY CURVED SURFACES Cylinders (Fig. 3-9) V = (area of base) × (altitude); lateral surface area = (perimeter of right section) × (lateral edge). Right Circular Cylinder V = π (radius)2 × (altitude); lateral surface area = 2π (radius) × (altitude). Truncated Right Circular Cylinder V = πr 2h; lateral area = 2πrh h = a (h1 + h2) Hollow Cylinders Volume = πh(R2 − r 2), where r and R are the internal and external radii and h is the height of the cylinder. Sphere (Fig. 3-10) V (sphere) = 4⁄ 3πR3, jπD3 V (spherical sector) = wπR2h = jπh1(3r 22 + h12) V (spherical segment of one base) = jπh1(3r 22 + h12) V (spherical segment of two bases) = jπh 2(3r 12 + 3r 22 + h 22 ) A (sphere) = 4πR2 = πD2 A (zone) = 2πRh = πDh A (lune on the surface included between two great circles, the inclination of which is θ radians) = 2R2θ. Cone V = s (area of base) × (altitude). Right Circular Cone V = (π/3) r 2h, where h is the altitude and r is the radius of the base; curved surface area = πr r2 + h2, curved sur2 face of the frustum of a right cone = π(r1 + r2) h2 +( r1 −r 2), where r1, r2 are the radii of the base and top, respectively, and h is the altitude; volume of the frustum of a right cone = π(h/3)(r 21 + r1r2 + r 22) = h/3(A1 + A2 + A 1A2), where A1 = area of base and A2 = area of top. Ellipsoid V = (4 ⁄3) πabc, where a, b, c are the lengths of the semiaxes. Torus (obtained by rotating a circle of radius r about a line whose distance is R > r from the center of the circle) V = 2π2Rr 2
Surface area = 4π2Rr
Prolate Spheroid (formed by rotating an ellipse about its major axis [2a]) Surface area = 2πb2 + 2π(ab/e) sin−1 e
V = 4 ⁄3πab2
where a, b are the major and minor axes and e = eccentricity (e < 1). Oblate Spheroid (formed by the rotation of an ellipse about its minor axis [2b]) Data as given previously.
FIG. 3-9
Cylinder.
FIG. 3-10
Sphere.
3-12
MATHEMATICS b2 1+e Surface area = 2πa2 + π ln e 1−e
V = 4 ⁄3πa2b
MISCELLANEOUS FORMULAS See also “Differential and Integral Calculus.” Volume of a Solid Revolution (the solid generated by rotating a plane area about the x axis) V=π
[ f(x)] dx b
2
a
where y = f(x) is the equation of the plane curve and a ≤ x ≤ b. Area of a Surface of Revolution
y ds
IRREGULAR AREAS AND VOLUMES
b
S = 2π
a
y/d x) 2 dx and y = f(x) is the equation of the plane where ds = 1 +(d curve rotated about the x axis to generate the surface. Area Bounded by f(x), the x Axis, and the Lines x = a, x = b A=
1. If a plane area is revolved about a line which lies in its plane but does not intersect the area, then the volume generated is equal to the product of the area and the distance traveled by the area’s center of gravity. 2. If an arc of a plane curve is revolved about a line that lies in its plane but does not intersect the arc, then the surface area generated by the arc is equal to the product of the length of the arc and the distance traveled by its center of gravity. These theorems are useful for determining volumes V and surface areas S of solids of revolution if the centers of gravity are known. If S and V are known, the centers of gravity may be determined.
f(x) dx b
[ f(x) ≥ 0]
Irregular Areas Let y0, y1, . . . , yn be the lengths of a series of equally spaced parallel chords and h be their distance apart. The area of the figure is given approximately by any of the following: AT = (h/2)[(y0 + yn) + 2(y1 + y2 + ⋅⋅⋅ + yn − 1)] As = (h/3)[(y0 + yn) + 4(y1 + y3 + y5 + ⋅⋅⋅ + yn − 1) + 2(y2 + y4 + ⋅⋅⋅ + yn − 2)]
a
Length of Arc of a Plane Curve If y = f(x), Length of arc s =
(n even, Simpson’s rule)
The greater the value of n, the greater the accuracy of approximation. Irregular Volumes To find the volume, replace the y’s by crosssectional areas Aj and use the results in the preceding equations.
dy 1 + dx dx b
(trapezoidal rule)
2
a
If x = g(y), Length of arc s =
dx 1 + dy dy 2
d
c
If x = f(t), y = g(t), Length of arc s =
dy dx + dt dt dt t1
2
2
t0
In general, (ds)2 = (dx)2 + (dy)2. Theorems of Pappus (for volumes and areas of surfaces of revolution)
FIG. 3-11
Irregular area.
ELEMENTARY ALGEBRA REFERENCES:
20, 102, 108, 191, 269, 277.
Dividend Divisor ex + 1 | 3e2x + ex + 1 3ex − 2 quotient 3e2x + 3ex
OPERATIONS ON ALGEBRAIC EXPRESSIONS
−2ex + 1 −2ex − 2
An algebraic expression will here be denoted as a combination of letters and numbers such as 3ax − 3xy + 7x2 + 7x 3/ 2 − 2.8xy Addition and Subtraction Only like terms can be added or subtracted in two algebraic expressions. Example (3x + 4xy − x ) + (3x + 2x − 8xy) = 5x − 4xy + 2x . 2
2
2
Example (2x + 3xy − 4x1/2) + (3x + 6x − 8xy) = 2x + 3x + 6x − 5xy − 4x1/2. Multiplication Multiplication of algebraic expressions is term by term, and corresponding terms are combined. Example (2x + 3y − 2xy)(3 + 3y) = 6x + 9y + 9y − 6xy . 2
2
Division This operation is analogous to that in arithmetic. Example Divide 3e2x + ex + 1 by ex + 1.
+ 3 (remainder) Therefore, 3e + e + 1 = (e + 1)(3e − 2) + 3. 2x
x
x
x
Operations with Zero All numerical computations (except division) can be done with zero: a + 0 = 0 + a = a; a − 0 = a; 0 − a = −a; (a)(0) = 0; a0 = 1 if a ≠ 0; 0/a = 0, a ≠ 0. a/0 and 0/0 have no meaning. Fractional Operations x −x x −x x −x x ax − = − = = ; = ; = , if a ≠ 0. y −y −y y y −y y ay
x z xz = ; y y y
y t = yt ; x
z
xz
xt x/y x t = = z/t y z yz
Factoring That process of analysis consisting of reducing a given expression into the product of two or more simpler expressions called factors. Some of the more common expressions are factored here: (1) (x2 − y2) = (x − y)(x + y)
ELEMENTARY ALGEBRA s (n − 1)d 2s a = l − (n − 1)d = − = − l n 2 n
(2) x2 + 2xy + y2 = (x + y)2 (3) x + ax + b = (x + c)(x + d) where c + d = a, cd = b 2
l−a 2(s − an) 2(nl − s) d=== n−1 n(n − 1) n(n − 1)
(4) by2 + cy + d = (ey + f)(gy + h) where eg = b, fg + eh = c, fh = d (5) x2 + y2 + z2 + 2yz + 2xz + 2xy = (x + y + z)2
2l + d + (2 l + d )2 −8 ds 2s l−a n = + 1 = = 2d d l+a
(6) x2 − y2 − z2 − 2yz = (x − y − z)(x + y + z) (7) x2 + y2 + z2 − 2xy − 2xz + 2yz = (x − y − z)2 (8) x3 − y3 = (x − y)(x2 + xy + y2) (9) (x + y ) = (x + y)(x − xy + y ) 3
3
2
2
(10) (x4 − y4) = (x − y)(x + y)(x2 + y2) (11) x5 + y5 = (x + y)(x4 − x3y + x2y2 − xy3 + y4) (12) xn − yn = (x − y)(xn − 1 + xn − 2y + xn − 3y2 + ⋅⋅⋅ + yn − 1) Laws of Exponents (an)m = anm; an + m = an ⋅ am; an/m = (an)1/m; an − m = an/am; a1/m = ma; a1/2 = a; x2 = |x| (absolute value of x). For x > 0, y > 0, xy = x n y; for x > 0 xm = xm/n; 1 /x = 1/x
The arithmetic mean or average of two numbers a, b is (a + b)/2; of n numbers a1, . . . , an is (a1 + a2 + ⋅⋅⋅ + an)/n. A geometric progression is a succession of terms such that each term, except the first, is derivable from the preceding by the multiplication of a quantity r called the common ratio. All such progressions have the form a, ar, ar 2, . . . , ar n − 1. With a = first term, l = last term, r = ratio, n = number of terms, s = sum of the terms, the following relations hold: [a + (r − 1)s] (r − 1)sr n − 1 l = ar n − 1 = = r rn − 1 a(r n − 1) a(1 − r n) rl − a lr n − l s==== r−1 1−r r − 1 rn − rn − 1
THE BINOMIAL THEOREM
l (r − 1)s s−a log l − log a a= = r = log r = rn − l rn − 1 s−l n−1
If n is a positive integer, n(n − 1) (a + b)n = an + nan − 1b + an − 2 b2 2! n n(n − 1)(n − 2) n n−j j + an − 3b3 + ⋅⋅⋅ + bn = a b j 3! j=0
where
n! = number of combinations of n things taken j at nj = j!(n − j)!
log l − log a log[a + (r − 1)s] − log a n = + 1 = log r log r ; of n The geometric mean of two nonnegative numbers a, b is ab numbers is (a1a2 . . . an)1/n. Example Find the sum of 1 + a + d + ⋅⋅⋅ + 1⁄64. Here a = 1, r = a, n = 7. Thus a(1⁄64) − 1 s = = 127/64 a−1
a time. n! = 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅⋅⋅ n, 0! = 1. Example Find the sixth term of (x + 2y) . The sixth term is obtained by setting j = 5. It is 12 12 − 5 x (2y)5 = 792x7(2y)5 5
a ar n s = a + ar + ar 2 + ⋅⋅⋅ + ar n − 1 = − 1−r 1−r
12
j = (1 + 1) 14
Example
n
14
3-13
a lim s = 1−r which is called the sum of the infinite geometric progression. If |r| < 1,
then
n→∞
Example The present worth (PW) of a series of cash flows Ck at the end
= 214.
of year k is
j=0
PROGRESSIONS
n Ck PW = k k = 1 (1 + i) where i is an assumed interest rate. (Thus the present worth always requires specification of an interest rate.) If all the payments are the same, Ck = R, the present worth is n 1 PW = R k k = 1 (1 + i) This can be rewritten as n R 1 R n−1 1 = PW = 1 + i k = 1 (1 + i)k − 1 1 + i j = 0 (1 + i) j
An arithmetic progression is a succession of terms such that each term, except the first, is derivable from the preceding by the addition of a quantity d called the common difference. All arithmetic progressions have the form a, a + d, a + 2d, a + 3d, . . . . With a = first term, l = last term, d = common difference, n = number of terms, and s = sum of the terms, the following relations hold:
This is a geometric series with r = 1/(1 + i) and a = R/(1 + i). The formulas above give R (1 + i)n − 1 PW (=s) = (1 + i)n i The same formula applies to the value of an annuity (PW) now, to provide for equal payments R at the end of each of n years, with interest rate i.
If n is not a positive integer, the sum formula no longer applies and an infinite series results for (a + b)n. The coefficients are obtained from the first formulas in this case. Example (1 + x)1/2 = 1 + ax − a ⋅ dx2 + a ⋅ d ⋅ 3⁄ 6 x3 ⋅⋅⋅ (convergent for
x2 < 1).
Additional discussion is under “Infinite Series.”
d l = a + (n − 1)d = − + 2
2 d s + a − d2 2
s (n − 1) =+d n 2 n n n s = [2a + (n − 1)d] = (a + l) = [2l − (n − 1)d] 2 2 2
A progression of the form a, (a + d )r, (a + 2d )r 2, (a + 3d )r 3, etc., is a combined arithmetic and geometric progression. The sum of n such terms is a − [a + (n − 1)d]r n rd(1 − r n − 1) s = + I−r (1 − r)2 a If |r| < 1, lim s = + rd/(1 − r)2. n→∞ 1−r
3-14
MATHEMATICS
The non-zero numbers a, b, c, etc., form a harmonic progression if their reciprocals 1/a, 1/b, 1/c, etc., form an arithmetic progression. Example The progression 1, s, 1⁄5, 1⁄7, . . . , 1⁄31 is harmonic since 1, 3, 5, 7, . . . , 31 form an arithmetic progression. The harmonic mean of two numbers a, b is 2ab/(a + b). PERMUTATIONS, COMBINATIONS, AND PROBABILITY Each separate arrangement of all or a part of a set of things is called a permutation. The number of permutations of n things taken r at a time, written n! P(n, r) = = n(n − 1)(n − 2) ⋅⋅⋅ (n − r + 1) (n − r)! Example The permutations of a, b, c two at a time are ab, ac, ba, ca, cb, and bc. The formula is P(3,2) = 3!/1! = 6. The permutations of a, b, c three at a time are abc, bac, cab, acb, bca, and cba. Each separate selection of objects that is possible irrespective of the order in which they are arranged is called a combination. The number of combinations of n things taken r at a time, written C(n, r) = n!/ [r!(n − r)!].
has three complex roots. If the coefficients are real numbers, then at least one of the roots must be real. The cubic equation x3 + bx2 + cx + d = 0 may be reduced by the substitution x = y − (b/3) to the form y3 + py + q = 0, where p = s(3c − b2), q = 1⁄27(27d − 9bc + 2b3). This equation has the solutions y1 = A + B, y2 = −a(A + B) + (i3/2)(A − B), 3 y3 = −a(A + B) − (i3/2)(A − B), where i2 = −1, A = − q /2 + R, 3 3 2 B = − q /2 − R, and R = (p/3) + (q/2) . If b, c, d are all real and if R > 0, there are one real root and two conjugate complex roots; if R = 0, there are three real roots, of which at least two are equal; if R < 0, there are three real unequal roots. If R < 0, these formulas are impractical. In this case, the roots are given by yk = 2 − p /3 cos [(φ/3) + 120k], k = 0, 1, 2 where φ = cos−1
Example Two dice may be thrown in 36 separate ways. What is the probability of throwing such that their sum is 7? Seven may arise in 6 ways: 1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, 6 and 1. The probability of shooting 7 is j.
2
3
and the upper sign applies if q > 0, the lower if q < 0. Example x3 + 3x2 + 9x + 9 = 0 reduces to y3 + 6y + 2 = 0 under x = y − 1. 3 3 4. The desired roots in y are Here p = 6, q = 2, R = 9. Hence A = 2, B = − 3 3 3 3 3 3 4 and −a(2 − 4) (i3/2)(2 + 4). The roots in x are x = 2 − − y − 1. Example y3 − 7y + 7 = 0. p = −7, q = 7, R < 0. Hence 28 φ cos + 120k 3 3 27 φ φ = , = 3°37′52″. 28 3
xk = −
Example The combinations of a, b, c taken 2 at a time are ab, ac, bc; taken 3 at a time is abc. An important relation is r! C(n, r) = P(n, r). If an event can occur in p ways and fail to occur in q ways, all ways being equally likely, the probability of its occurrence is p/(p + q), and that of its failure q/(p + q).
q /4 −p /27
where
The roots are approximately −3.048916, 1.692020, and 1.356897.
Example Many equations of state involve solving cubic equations for the compressibility factor Z. For example, the Redlich-Kwong-Soave equation of state requires solving Z 3 − Z 2 + cZ + d = 0,
d<0
THEORY OF EQUATIONS
where c and d depend on critical constants of the chemical species. In this case, only positive solutions, Z > 0, are desired.
Linear Equations A linear equation is one of the first degree (i.e., only the first powers of the variables are involved), and the process of obtaining definite values for the unknown is called solving the equation. Every linear equation in one variable is written Ax + B = 0 or x = −B/A. Linear equations in n variables have the form
Quartic Equations See Ref. 118. General Polynomials of the nth Degree Denote the general polynomial equation of degree n by
a11 x1 + a12 x2 + ⋅⋅⋅ + a1n xn = b1 a21 x1 + a22 x2 + ⋅⋅⋅ + a2n xn = b2 ⯗ am1 x1 + am2 x2 + ⋅⋅⋅ + amn xn = bm The solution of the system may then be found by elimination or matrix methods if a solution exists (see “Matrix Algebra and Matrix Computations”). Quadratic Equations Every quadratic equation in one variable is expressible in the form ax 2 + bx + c = 0. a ≠ 0. This equation has two solutions, say, x1, x2, given by
4ac −b x1 b2− = x2 2a
If a, b, c are real, the discriminant b − 4ac gives the character of the roots. If b2 − 4ac > 0, the roots are real and unequal. If b2 − 4ac < 0, the roots are complex conjugates. If b2 − 4ac = 0 the roots are real and equal. Two quadratic equations in two variables can in general be solved only by numerical methods (see “Numerical Analysis and Approximate Methods”). If one equation is of the first degree, the other of the second degree, a solution may be obtained by solving the first for one unknown. This result is substituted in the second equation and the resulting quadratic equation solved. Cubic Equations A cubic equation, in one variable, has the form x3 + bx2 + cx + d = 0. Every cubic equation having complex coefficients 2
P(x) = a0 x n + a1 x n − 1 + ⋅⋅⋅ + an − 1 x + an = 0 If n > 4, there is no formula which gives the roots of the general equation. For fourth and higher order (even third order), the roots can be found numerically (see “Numerical Analysis and Approximate Methods”). However, there are some general theorems that may prove useful. Remainder Theorems When P(x) is a polynomial and P(x) is divided by x − a until a remainder independent of x is obtained, this remainder is equal to P(a). Example P(x) = 2x4 − 3x2 + 7x − 2 when divided by x + 1 (here a = −1) results in P(x) = (x + 1)(2x3 − 2x2 − x + 8) − 10 where −10 is the remainder. It is easy to see that P(−1) = −10. Factor Theorem If P(a) is zero, the polynomial P(x) has the factor x − a. In other words, if a is a root of P(x) = 0, then x − a is a factor of P(x). If a number a is found to be a root of P(x) = 0, the division of P(x) by (x − a) leaves a polynomial of degree one less than that of the original equation, i.e., P(x) = Q(x)(x − a). Roots of Q(x) = 0 are clearly roots of P(x) = 0. Example P(x) = x3 − 6x2 + 11x − 6 = 0 has the root + 3. Then P(x) = (x − 3)(x2 − 3x + 2). The roots of x2 − 3x + 2 = 0 are 1 and 2. The roots of P(x) are therefore 1, 2, 3. Fundamental Theorem of Algebra Every polynomial of degree n has exactly n real or complex roots, counting multiplicities.
ELEMENTARY ALGEBRA Every polynomial equation a0 x n + a1 x n − 1 + ⋅⋅⋅ + an = 0 with rational coefficients may be rewritten as a polynomial, of the same degree, with integral coefficients by multiplying each coefficient by the least common multiple of the denominators of the coefficients. Example The coefficients of 3⁄2 x4 + 7⁄3 x3 − 5⁄6 x2 + 2x − j = 0 are rational
numbers. The least common multiple of the denominators is 2 × 3 = 6. Therefore, the equation is equivalent to 9x4 + 14x3 − 5x2 + 12x − 1 = 0.
Upper Bound for the Real Roots Any number that exceeds all the roots is called an upper bound to the real roots. If the coefficients of a polynomial equation are all of like sign, there is no positive root. Such equations are excluded here since zero is the upper bound to the real roots. If the coefficient of the highest power of P(x) = 0 is negative, replace the equation by −P(x) = 0. If in a polynomial P(x) = c0 x n + c1 x n − 1 + ⋅⋅⋅ + cn − 1 x + cn = 0, with c0 > 0, the first negative coefficient is preceded by k coefficients which are positive or zero, and if G denotes the greatest of the numerical values of the negative coefficients, then each real root is less than k 1 + G /c0. A lower bound to the negative roots of P(x) = 0 may be found by applying the rule to P(−x) = 0. Example P(x) = x + 2x + 4x − 8x − 32 = 0. Here3 k = 5 (since 2 coefficients are zero), G = 32, c0 = 1. The upper bound is 1 + 32 = 3. P(−x) = −x7 − 5 2x + 4x4 − 8x2 − 32 = 0. −P(−x) = x73+ 2x5 − 4x4 + 8x2 + 32 = 0. Here k = 3, G = 4, c0 = 1. The lower bound is −(1 + 4) ≈ −2.587. Thus all real roots r lie in the range −2.587 < r < 3. 7
5
4
2
Descartes Rule of Signs The number of positive real roots of a polynomial equation with real coefficients either is equal to the number v of its variations in sign or is less than v by a positive even integer. The number of negative roots of P(x) = 0 either is equal to the number of variations of sign of P(−x) or is less than that number by a positive even integer. Example P(x) = x4 + 3x3 + x − 1 = 0. v = 1; so P(x) has one positive root. P(−x) = x4 − 3x3 − x − 1. Here v = 1; so P(x) has one negative root. The other two roots are complex conjugates. Example P(x) = x4 − x2 + 10x − 4 = 0. v = 3; so P(x) has three or one positive roots. P(−x) = x4 − x2 − 10x − 4. v = 1; so P(x) has exactly one negative root. Numerical methods are often used to find the roots of polynomials. A detailed discussion of these techniques is given under “Numerical Analysis and Approximate Methods.” Determinants Consider the system of two linear equations a11x1 + a12x2 = b1 a21x1 + a22x2 = b2 If the first equation is multiplied by a22 and the second by −a12 and the results added, we obtain (a11a22 − a21a12)x1 = b1a22 − b2a12 The expression a11a22 − a21a12 may be represented by the symbol a11
a11 a12 a a22 |A| = 21 ⯗ an1 an2
a13 ⋅⋅⋅ a1n ⋅⋅⋅⋅⋅ a2n
an3 ⋅⋅⋅ ann
is called a determinant. The n2 quantities aij are called the elements of the determinant. In the determinant |A| let the ith row and jth column be deleted and a new determinant be formed having n − 1 rows and columns. This new determinant is called the minor of aij denoted Mij.
Example
a11 a12 a13 a21 a22 a23 = a31A31 + a32A32 + a33A33 a31 a32 a33 = a31
aa
a13 a a a a − a32 11 13 + a33 11 12 a23 a21 a23 a21 a22
12 22
In general, Aij will be determinants of order n − 1, but they may in turn be expanded by the rule. Also, n
a
j=1
n
ji
A jk = a ij A jk = |A| 0 j=1
i=k i≠k
Fundamental Properties of Determinants 1. The value of a determinant |A| is not changed if the rows and columns are interchanged. 2. If the elements of one row (or one column) of a determinant are all zero, the value of |A| is zero. 3. If the elements of one row (or column) of a determinant are multiplied by the same constant factor, the value of the determinant is multiplied by this factor. 4. If one determinant is obtained from another by interchanging any two rows (or columns), the value of either is the negative of the value of the other. 5. If two rows (or columns) of a determinant are identical, the value of the determinant is zero. 6. If two determinants are identical except for one row (or column), the sum of their values is given by a single determinant obtained by adding corresponding elements of dissimilar rows (or columns) and leaving unchanged the remaining elements. Example
31 25 + 47 25 = 13 + 6 = 19
Directly
78 25 = 35 − 16 = 19
By rule 6
7. The value of a determinant is not changed if to the elements of any row (or column) are added a constant multiple of the corresponding elements of any other row (or column). 8. If all elements but one in a row (or column) are zero, the value of the determinant is the product of that element times its cofactor. The evaluation of determinants using the definition is quite laborious. The labor can be reduced by applying the fundamental properties just outlined. The solution of n linear equations (not all bi zero) a11 x1 + a12x2 + ⋅⋅⋅ + a1n xn = b1
21
The cofactor Aij of the element aij is the signed minor of aij determined by the rule Aij = (−1) i + jMij. The value of |A| is obtained by forming any of the n n equivalent expressions j = 1 aij Aij , t = 1 aij Aij, where the elements aij must be taken from a single row or a single column of A.
a12 = a11a22 − a21a12 a22 This symbol is called a determinant of second order. The value of the square array of n2 quantities aij, where i = 1, . . . , n is the row index, j = 1, . . . , n the column index, written in the form
a
a11 a12 a13 a a a21 a22 a23 The minor of a23 is M23 = 11 12 a31 a32 a31 a32 a33
Example
3-15
a21 x1 + a22x2 + ⋅⋅⋅ + a2n xn = b2 ⯗⯗⯗ ⯗ an1x1 + an2x2 + ⋅⋅⋅ + annxn = bn a11 ⋅⋅⋅ a1n a21 ⋅⋅⋅ a2n where |A| = ⯗ ≠0 an1 ⋅⋅⋅ ann
has a unique solution given by x1 = |B1|/ |A|, x2 = |B2|/ |A|, . . . , xn = |Bn|/ |A|, where Bk is the determinant obtained from A by replacing its kth column by b1, b2, . . . , bn. This technique is called Cramer’s rule. It requires more labor than the method of elimination and should not be used for computations.
3-16
MATHEMATICS
ANALYTIC GEOMETRY REFERENCES: 108, 188, 193, 260, 261, 268, 274, 282. Analytic geometry uses algebraic equations and methods to study geometric problems. It also permits one to visualize algebraic equations in terms of geometric curves, which frequently clarifies abstract concepts.
PLANE ANALYTIC GEOMETRY Coordinate Systems The basic concept of analytic geometry is the establishment of a one-to-one correspondence between the points of the plane and number pairs (x, y). This correspondence may be done in a number of ways. The rectangular or cartesian coordinate system consists of two straight lines intersecting at right angles (Fig. 3-12). A point is designated by (x, y), where x (the abscissa) is the distance of the point from the y axis measured parallel to the x axis, positive if to the right, negative to the left. y (ordinate) is the distance of the point from the x axis, measured parallel to the y axis, positive if above, negative if below the x axis. The quadrants are labeled 1, 2, 3, 4 in the drawing, the coordinates of points in the various quadrants having the depicted signs. Another common coordinate system is the polar coordinate system (Fig. 3-13). In this system the position of a point is designated by the pair (r, θ), r = x2 +y2 being the distance to the origin 0(0,0) and θ being the angle the line r makes with the positive x axis (polar axis). To change from polar to rectangular coordinates, use x = r cos θ and y = r sin θ. To change from rectangular to polar coordinates, use r = x2 +y2 and θ = tan−1 (y/x) if x ≠ 0; θ = π/2 if x = 0. The distance between two points (x1, y1), (x2, y2) is defined 2 2 by d = (x1 −x +(y1 −y 2) 2) in rectangular coordinates or by d = 2 2 r 1 +r −2 r1 r2cos (θ1 − θ 2 2) in polar coordinates. Other coordinate systems are sometimes used. For example, on the surface of a sphere latitude and longitude prove useful. The Straight Line (Fig. 3-14) The slope m of a straight line is the tangent of the inclination angle θ made with the positive x axis. If (x1, y1) and (x2, y2) are any two points on the line, slope = m = (y2 − y1)/ (x2 − x1). The slope of a line parallel to the x axis is zero; parallel to the y axis, it is undefined. Two lines are parallel if and only if they have the same slope. Two lines are perpendicular if and only if the product of their slopes is −1 (the exception being that case when the lines are parallel to the coordinate axes). Every equation of the type Ax + By + C = 0 represents a straight line, and every straight line has an equation of this form. A straight line is determined by a variety of conditions:
origin are given (see Fig. 3-15). Let p = length of the perpendicular and α the angle that the perpendicular makes with the positive x axis. The equation of the line is x cos + y sin = p. The equation of a line perpendicular to a given line of slope m and passing through a point (x1, y1) is y − y1 = −(1/m) (x − x1). The distance from a point (x1, y1) to a line with equation Ax + by + C = 0 is |Ax1 + By1 + C| d = A2 + B2 Example If it is known that centigrade C and Fahrenheit F are linearly related and when C = 0°, F = 32°; C = 100°, F = 212°, find the equation relating C and F and that point where C = F. By using the two-point form, the equation is 212 − 32 F − 32 = (C − 0) 100 − 0 or F = 9⁄ 5C + 32. Equivalently 100 − 0 C − 0 = (F − 32) 212 − 32 or C = 5⁄ 9(F − 32). Letting C = F, we have from either equation F = C = −40.
Occasionally some nonlinear algebraic equations can be reduced to linear equations under suitable substitutions or changes of variables. In other words, certain curves become the graphs of lines if the scales or coordinate axes are appropriately transformed. Example Consider y = bxn. B = log b. Taking logarithms log y = n log x + log b. Let Y = log y, X = log x, B = log b. The equation then has the form Y = nX + B, which is a linear equation. Consider k = k0 exp (−E/RT), taking logarithms loge k = loge k0 − E/(RT). Let Y = loge k, B = loge k0, and m = −E/R, X = 1/T, and the result is Y = mX + B. Next consider y = a + bxn. If the substitution t = x n is made, then the graph of y is a straight line versus t. Asymptotes The limiting position of the tangent to a curve as the point of contact tends to an infinite distance from the origin is called an asymptote. If the equation of a given curve can be expanded in a Laurent power series such that n n b f(x) = ak x k + kk k=0 k=0 x n
lim f(x) = akxk
and Given conditions (1) (2) (3) (4) (5)
Parallel to x axis Parallel y axis Point (x1, y1) and slope m Intercept on y axis (0, b), m Intercept on x axis (a, 0), m
(6)
Two points (x1, y1), (x2, y2)
(7)
Two intercepts (a, 0), (0, b)
Equation of line y = constant x = constant y − y1 = m(x − x1) y = mx + b y = m(x − a) y2 − y1 y − y1 = (x − x1) x2 − x1 x/a + y/b = 1
The angle β a line with slope m1 makes with a line having slope m2 is given by tan β = (m2 − m1)/(m1m2 + 1). A line is determined if the length and direction of the perpendicular to it (the normal) from the
FIG. 3-12
Rectangular coordinates.
x→∞
FIG. 3-13
Polar coordinates.
k=0
then the equation of the asymptote is y = k = 0 ak x k. If n = 1, then the asymptote is (in general oblique) a line. In this case, the equation of the asymptote may be written as n
y = mx + b
m = lim f ′(x) x→∞
b = lim [ f(x) − xf ′(x)] x→∞
Geometric Properties of a Curve When the Equation Is Given The analysis of the properties of an equation is facilitated by the investigation of the equation by using the following techniques: 1. Points of maximum, minimum, and inflection. These may be investigated by means of the calculus.
FIG. 3-14
Straight line.
FIG. 3-15
Determination of line.
ANALYTIC GEOMETRY 2.
Symmetry. Let F(x, y) = 0 be the equation of the curve.
The table characterizes the curve represented by the equation.
Condition on F(x, y)
Symmetry
B2 − 4AC < 0
F(x, y) = F(−x, y) F(x, y) = F(x, −y) F(x, y) = F(−x, −y) F(x, y) = F(y, x)
With respect to y axis With respect to x axis With respect to origin With respect to the line y = x
A < 0 A ≠ C, an ellipse A < 0 A = C, a circle A > 0, no locus
3. Extent. Only real values of x and y are considered in obtaining the points (x, y) whose coordinates satisfy the equation. The extent of them may be limited by the condition that negative numbers do not have real square roots. 4. Intercepts. Find those points where the curves of the function cross the coordinate axes. 5. Asymptotes. See preceding discussion. 6. Direction at a point. This may be found from the derivative of the function at a point. This concept is useful for distinguishing among a family of similar curves. Example y2 = (x2 + 1)/(x2 − 1) is symmetric with respect to the x and y axis, the origin, and the line y = x. It has the vertical asymptotes x = 1. When x = 0, y2 = −1; so there are no y intercepts. If y = 0, (x2 + 1)/(x2 − 1) = 0; so there are no x intercepts. If |x| < 1, y2 is negative; so |x| > 1. From x2 = ( y2 + 1)/(y2 − 1), y = 1 are horizontal asymptotes and |y| > 1. As x → 1+, y → + ∞; as x → + ∞, y → + 1. The graph is given in Fig. 3-16. Conic Sections The curves included in this group are obtained from plane sections of the cone. They include the circle, ellipse, parabola, hyperbola, and degeneratively the point and straight line. A conic is the locus of a point whose distance from a fixed point called the focus is in a constant ratio to its distance from a fixed line, called the directrix. This ratio is the eccentricity e. If e = 0, the conic is a circle; if 0 < e < 1, the conic is an ellipse; if e = 1, the conic is a parabola; if e > 1, the conic is a hyperbola. Every conic section is representable by an equation of second degree. Conversely, every equation of second degree in two variables represents a conic. The general equation of the second degree is Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. Let be defined as the determinant
2A = B D
FIG. 3-16
B 2C E
D E 2F
Graph of y2 = (x2 + 1)/(x2 − 1)
≠0
=0
B2 − 4AC = 0
B2 − 4AC > 0
Parabola
Hyperbola
2 parallel lines if Q = D2 + E2 − 4(A + C)F > 0 1 straight line if Q = 0, no locus if Q < 0
Point
2 intersecting straight lines
Example 3x2 + 4xy − 2y2 + 3x − 2y + 7 = 0.
6 = 4 3
4 −4 −2
3 −2 = −596 ≠ 0, B2 − 4AC = 40 > 0 14
The curve is therefore a hyperbola.
To translate the axes to a new origin at (h, k), substitute for x and y in the original equation x + h and y + k. Translation of the axes can always be accomplished to eliminate the linear terms in the seconddegree equation in two variables having no xy term. Example x2 + y2 + 2x − 4y + 2 = 0. Rewrite this as x2 + 2x + 1 + y2 − 4y + 4 − 5 + 2 = 0 or (x + 1)2 + (y − 2)2 = 3. Let u = x + 1, v = y − 2. Then u2 + v2 = 3. The axis has been translated to the new origin (−1, 2). The type of curve determined by a specific equation of the second degree can also be easily determined by reducing it to a standard form by translation and/or rotation. In the case in which the equation has no xy term, the procedure is merely to complete the squares of the terms in x and y separately. To rotate the axes through an angle α, substitute for x the quantity x cos α − y sin α and for y the quantity x sin α + y cos α. A rotation of the axes through α = a cot−1 (A − C)/B will eliminate the crossproduct term in the general second-degree equation. Example Consider 3x2 + 2xy + y2 − 2x + 3y = 7. A rotation of axes through α = a cot−1 1 = 22a° eliminates the cross-product term. The following tabulation gives the form of the more common equations. Polar equation
Type of curve
(1) r = a (2) r = 2a cos θ (3) r = 2a sin θ (4) r2 − 2br cos (θ − β) + b2 − a2 = 0
Circle Circle Circle Circle at (b, β), radius a
ke (5) r = 1 − e cos θ
e = 1 parabola 0 < e < 1 ellipse e > 1 hyperbola
Some common equations in parametric form are given below. (1) (x − h)2 + (y − k)2 = a2 (x − h)2 (y − k)2 (2) + =1 a2 b2
(3) z2 + y2 = a2 x (4) y = a cosh a (5) Cycloid
3-17
x = h + a cos θ y = k + a sin θ x = h + a cos φ y = k + a sin φ −at x= t2 +1 a y= t2 + 1
Circle (Fig. 3-23) Parameter is angle θ.
s x = a sinh−1 a y2 = a2 + s2 x = a(φ − sin φ) y = a(1 − cos φ)
Catenary (Fig. 3-24; such as hanging cable under gravity) Parameter s = arc length from (0, a) to (x, y). See Fig. 3-24.
Ellipse (Fig. 3-20) Parameter is angle φ. dy Circle Parameter is t = = slope of tangent at (x, y). dx
3-18
MATHEMATICS
FIG. 3-17
Circle center (0,0) r = a.
FIG. 3-18
Circle center (a,0) r = 2a cos θ.
FIG. 3-19
Circle center (0,a) r = 2a sin θ.
FIG. 3-20
Ellipse, 0 < e < 1.
FIG. 3-21
Hyperbola, e > 1, r = ke/(1 − e cos θ).
FIG. 3-22
Parabola, e = 1.
Circle at (b, β), radius a: r2 − 2br cos (θ − β) b2 − a2 = 0.
Graphs of Polar Equations The equation r = 0 corresponds to x = 0, y = 0 regardless of θ. The same point may be represented in several different ways; thus the point (2, π/3) or (2, 60°) has the following representations: (2, 60°), (2, −300°). These are summarized in (2, 60° + n 360°), n = 0, 1, 2, or in radian measure [2, (π/3) + 2nπ], n = 0, 1, 2. Plotting of polar equations can be facilitated by the following steps: 1. Find those points where r is a maximum or minimum. 2. Find those values of θ where r = 0, if any. 3. Symmetry: The curve is symmetric about the origin if the equation is unchanged when θ is replaced by θ π, symmetric about the x axis if the equation is unchanged when θ is replaced by −θ, and symmetric about the y axis if the equation is unchanged when θ is replaced by π − θ. Parametric Equations It is frequently useful to write the equations of a curve in terms of an auxiliary variable called a parameter. For example, a circle of radius a, center at (0, 0), can be written in the equivalent form x = a cos θ, y = a sin φ where θ is the parameter. Similarly, x = a cos φ, y = b sin φ are the parametric equations of the ellipse x2/a2 + y2/b2 = 1 with parameter φ.
FIG. 3-23
Circle.
FIG. 3-24
Cycloid.
SOLID ANALYTIC GEOMETRY Coordinate Systems The commonly used coordinate systems are three in number. Others may be used in specific problems (see Ref. 212). The rectangular (cartesian) system (Fig. 3-25) consists of mutually orthogonal axes x, y, z. A triple of numbers (x, y, z) is used to represent each point. The cylindrical coordinate system (r, θ, z; Fig. 3-26) is frequently used to locate a point in space. These are essentially the polar coordinates (r, θ) coupled with the z coordinate. As
FIG. 3-25
Cartesian coordinates.
FIG. 3-26
Cylindrical coordinates.
ANALYTIC GEOMETRY before, x = r cos θ, y = r sin θ, z = z and r2 = x2 + y2, y/x = tan θ. If r is held constant and θ and z are allowed to vary, the locus of (r, θ, z) is a right circular cylinder of radius r along the z axis. The locus of r = C is a circle, and θ = constant is a plane containing the z axis and making an angle θ with the xz plane. Cylindrical coordinates are convenient to use when the problem has an axis of symmetry. The spherical coordinate system is convenient if there is a point of symmetry in the system. This point is taken as the origin and the coordinates (ρ, φ, θ) illustrated in Fig. 3-27. The relations are x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ, and r = ρ sin φ. θ = constant is a plane containing the z axis and making an angle θ with the xz plane. φ = constant is a cone with vertex at 0. ρ = constant is the surface of a sphere of radius ρ, center at the origin 0. Every point in the space may be given spherical coordinates restricted to the ranges 0 ≤ φ ≤ π, ρ ≥ 0, 0 ≤ θ < 2π. Lines and Planes The distance between two points (x1, y1, z1), 2 2 2 (x2, y2, z2) is d = (x −x +(y −y +( z1 −z 1 2) 1 2) 2). There is nothing in the geometry of three dimensions quite analogous to the slope of a line in the plane case. Instead of specifying the direction of a line by a trigonometric function evaluated for one angle, a trigonometric function evaluated for three angles is used. The angles α, β, γ that a line segment makes with the positive x, y, and z axes, respectively, are called the direction angles of the line, and cos α, cos β, cos γ are called the direction cosines. Let (x1, y1, z1), (x2, y2, z2) be on the line. Then cos α = (x2 − x1)/d, cos β = (y2 − y1)/d, cos γ = (z2 − z1)/d, where d = the distance between the two points. Clearly cos2 α + cos2 β + cos2 γ = 1. If two lines are specified by the direction cosines (cos α1, cos β1, cos γ1), (cos α2, cos β2, cos γ2), then the angle θ between the lines is cos θ = cos α1 cos α2 + cos β1 cos β2 + cos γ1 cos γ2. Thus the lines are perpendicular if and only if θ = 90° or cos α1 cos α2 + cos β1 cos β2 + cos γ1 cos γ2 = 0. The equation of a line with direction cosines (cos α, cos β, cos γ) passing through (x1, y1, z1) is (x − x1)/cos α = (y − y1)/cos β = (z − z1)/cos γ. The equation of every plane is of the form Ax + By + Cz + D = 0. The numbers A B C , , 2 2 2 2 2 2 A + B + C2 A + B + C2 A + B + C2
FIG. 3-27
Spherical coordinates.
FIG. 3-29
y2 z2 x2 Ellipsoid. 2 + 2 + 2 = 1 (sphere if a = b = c) b c a
FIG. 3-30
y2 z2 x2 Hyperboloid of one sheet. 2 + 2 − 2 = 1 b c a
FIG. 3-31
y2 z2 x2 Hyperboloid of two sheets. 2 + 2 − 2 = −1 b c a
FIG. 3-28
Parabolic cylinder.
are direction cosines of the normal lines to the plane. The plane through the point (x1, y1, z1) whose normals have these as direction cosines is A(x − x1) + B(y − y1) + C(z − z1) = 0. Example Find the equation of the plane through (1, 5, −2) perpendicular to the line (x + 9)/7 = (y − 3)/−1 = z/8. The numbers (7, −1, 8) are called direction numbers. They are a constant multiple of the direction cosines. cos α = 7/114, cos β = −1/114, cos γ = 8/114. The plane has the equation 7(x − 1) − 1(y − 5) + 8(z + 2) = 0 or 7x − y + 8z + 14 = 0. The distance from the point (x1, y1, z1) to the plane Ax + By + Cz + D = 0 is |Ax1 + By1 + Cz1 + D| d = 2+ B2+ C2 A
Space Curves Space curves are usually specified as the set of points whose coordinates are given parametrically by a system of equations x = f(t), y = g(t), z = h(t) in the parameter t. Example The equation of a straight line in space is (x − x1)/a = (y − y1)/b = (z − z1)/c. Since all these quantities must be equal (say, to t), we may write x = x1 + at, y = y1 + bt, z = z1 + ct, which represent the parametric equations of the line. Example The equations z = a cos βt, y = a sin βt, z = bt, a, β, b positive constants, represent a circular helix. Surfaces The locus of points (x, y, z) satisfying f(x, y, z) = 0, broadly speaking, may be interpreted as a surface. The simplest surface is the plane. The next simplest is a cylinder, which is a surface generated by a straight line moving parallel to a given line and passing through a given curve. Example The parabolic cylinder y = x2 (Fig. 3-28) is generated by a straight line parallel to the z axis passing through y = x2 in the plane z = 0.
3-19
3-20
MATHEMATICS
A surface whose equation is a quadratic in the variables x, y, and z is called a quadric surface. Some of the more common such surfaces are tabulated and pictured in Figs. 3-29 to 3-37.
Elliptic paraboloid.
FIG. 3-33
y2 z2 x2 FIG. 3-32 Cone. 2 + 2 + 2 = 0 b c a
y2 x2 2 + 2 + 2z = 0 b a
FIG. 3-35
y2 x2 Elliptic cylinder. 2 + 2 = 1 b a
FIG. 3-36
Hyperbolic cylinder. y2 x2 2 − 2 = 1 b a
y2 x2 FIG. 3-34 Hyperbolic paraboloid. 2 − 2 + 2z = 0 b a
FIG. 3-37
Parabolic cylinder. y2 + 2ax = 0
PLANE TRIGONOMETRY REFERENCES:
20, 108, 131, 158, 166, 202.
ANGLES An angle is generated by the rotation of a line about a fixed center from some initial position to some terminal position. If the rotation is clockwise, the angle is negative; if it is counterclockwise, the angle is positive. Angle size is unlimited. If α, β are two angles such that α + β = 90°, they are complementary; they are supplementary if α + β = 180°. Angles are most commonly measured in the sexagesimal system or by radian measure. In the first system there are 360 degrees in one complete revolution; one degree = 1⁄90 of a right angle. The degree is subdivided into 60 minutes; the minute is subdivided into 60 seconds. In the radian system one radian is the angle at the center of a circle subtended by an arc whose length is equal to the radius of the circle. Thus 2 rad = 360°; 1 rad = 57.29578°; 1° = 0.01745 rad; 1 min = 0.00029089 rad. The advantage of radian measure is that it is dimensionless. The quadrants are conventionally labeled as Fig. 3-38 shows.
FIG. 3-38
Quadrants.
FUNCTIONS OF CIRCULAR TRIGONOMETRY The trigonometric functions of angles are the ratios between the various sides of the reference triangles shown in Fig. 3-39 for the various quadrants. Clearly r = x2 +y2 ≥ 0. The fundamental functions (see Figs. 3-40, 3-41, 3-42) are
FIG. 3-39
Triangles.
FIG. 3-40
Graph of y = sin x.
PLANE TRIGONOMETRY
3-21
Relations between Functions of a Single Angle sec θ = 1/ cos θ; csc θ = 1/sin θ, tan θ = sin θ/cos θ = sec θ/csc θ = 1/cot θ; sin2 θ + cos2 θ = 1; 1 + tan2 θ = sec2 θ; 1 + cot2 θ = csc2 θ. For 0 ≤ θ ≤ 90° the following results hold: sin θ = cos θ/cot θ = 1 −co s2 θ = cos θ tan θ
FIG. 3-41
tan θ 1 θ θ = = = 2 sin cos +t an2 θ 1 +co t2 θ 1 2 2
Graph of y = cos x.
and
1 2θ = cos θ = 1 −sin 2 θ +t an 1 cot θ sin θ θ θ = = = cos2 − sin2 2 2 +co t2 θ tan θ 1
The cofunction property is very important. cos θ = sin (90° − θ), sin θ = cos (90° − θ), tan θ = cot (90° − θ), cot θ = tan (90° − θ), etc. Functions of Negative Angles sin (−θ) = −sin θ, cos (−θ) = cos θ, tan (−θ) = −tan θ, sec (−θ) = sec θ, csc (−θ) = −csc θ, cot (−θ) = −cot θ. FIG. 3-42
Graph of y = tan x.
Plane Trigonometry Sine of θ = sin θ = y/r Cosine of θ = cos θ = x/r Tangent of θ = tan θ = y/x
Secant of θ = sec θ = r/x Cosecant of θ = csc θ = r/y Cotangent of θ = cot θ = x/y
Magnitude and Sign of Trigonometric Functions 0 ≤ θ ≤ 360° Function
0° to 90°
90° to 180°
180° to 270°
270° to 360°
sin θ csc θ cos θ sec θ tan θ cot θ
+0 to +1 +∞ to +1 +1 to 0 +1 to +∞ +0 to +∞ +∞ to +0
+1 to +0 +1 to +∞ −0 to −1 −∞ to −1 −∞ to −0 −0 to −∞
−0 to −1 −∞ to −1 −1 to −0 −1 to −∞ +0 to +∞ +∞ to +0
−1 to −0 −1 to −∞ +0 to +1 +∞ to +1 −∞ to −0 −0 to −∞
Identities Sum and Difference Formulas Let x, y be two angles. sin (x y) = sin x cos y cos x sin y; cos (x y) = cos x cos y sin x sin y; tan (x y) = (tan x tan y)/(1 tan x tan y); sin x sin y = 2 sin a(x y) cos a(x y); cos x + cos y = 2 cos a(x + y) cos a(x − y); cos x − cos y = −2 sin a(x + y) sin a(x − y); tan x tan y = [sin (x y)]/(cos x cos y); sin2 x − sin2 y = cos2 y − cos2 x = sin (x + y) sin (x − y); cos2 x − sin2 y = cos2 y − sin2 x = cos (x + y) cos (x − y); sin (45° + x) = cos (45° − x); sin (45° − x) = cos (45° + x); tan (45° x) = cot (45° x). A cos x + 2 2 B sin x = A + B2 sin (α + x) = A + B2 cos (β − x) where tan α = A/B, tan β = B/A; both α and β are positive acute angles. Multiple and Half Angle Identities Let x = angle, sin 2x = 2 sin x cos x; sin x = 2 sin ax cos ax; cos 2x = cos2 x − sin2x = 1 − 2 sin2x = 2 cos2x − 1. tan 2x = (2 tan x)/(1 − tan2 x); sin 3x = 3 sin x − 4 sin3x; cos 3x = 4 cos3 x − 3 cos x. tan 3x = (3 tan x − tan3 x)/(1 − 3 tan2 x); sin 4x = 4 sin x cos x − 8 sin3 x cos x; cos 4x = 8 cos4 x − 8 cos2 x + 1. x sin = a (1 −co s x) 2
x (1 cos = a +o csx) 2
Values of the Trigonometric Functions for Common Angles θ°
θ, rad
sin θ
cos θ
tan θ
0 30 45 60 90
0 π/6 π/4 π/3 π/2
0 1/2 2/2 3/2 1
1 3/2 2/2 1/2 0
0 3/3 1 3 +∞
If 90° ≤ θ ≤ 180°, sin θ = sin (180° − θ); cos θ = −cos (180° − θ); tan θ = −tan (180° − θ). If 180° ≤ θ ≤ 270°, sin θ = −sin (270° − θ); cos θ = −cos (270° − θ); tan θ = tan (270° − θ). If 270° ≤ θ ≤ 360°, sin θ = −sin (360° − θ); cos θ = cos (360° − θ); tan θ = −tan (360° − θ). The reciprocal properties may be used to find the values of the other functions. If it is desired to find the angle when a function of it is given, the procedure is as follows: There will in general be two angles between 0° and 360° corresponding to the given value of the function.
1 − cos x sin x 1 − cos x == 1 + cos x 1 + cos x sin x
x tan = 2
Relations between Three Angles Whose Sum Is 180° Let x, y, z be the angles. x y z six x + sin y + sin z = 4 cos cos cos 2 2 2
x y z cos x + cos y + cos z = 4 sin sin sin + 1 2 2 2
x y z sin x + sin y − sin z = 4 sin sin cos 2 2 2
sin x + sin y + sin z = 2 cos x cos y cos z + 2; tan x + tan y + tan z = tan x tan y tan z; sin 2x + sin 2y + sin 2z = 4 sin x sin y sin z. 2
2
2
INVERSE TRIGONOMETRIC FUNCTIONS Given (a > 0)
Find an acute angle θ0 such that
Required angles are
sin θ = +a cos θ = +a tan θ = +a sin θ = −a cos θ = −a tan θ = −a
sin θ0 = a cos θ0 = a tan θ0 = a sin θ0 = a cos θ0 = a tan θ0 = a
θ0 and (180° − θ0) θ0 and (360° − θ0) θ0 and (180° + θ0) 180° + θ0 and 360° − θ0 180° − θ0 and 180° + θ0 180° − θ0 and 360° − θ0
y = sin −1 x = arcsin x is the angle y whose sine is x. Example y = sin−1 a, y is 30°. The complete solution of the equation x = sin y is y = (−1)n sin−1 x + n(180°), −π/2 ≤ sin−1 x ≤ π/2 where sin−1 x is the principal value of the angle whose sine is x. The range of principal values of the cos−1 x is 0 ≤ cos−1 x ≤ π and −π/2 ≤ tan−1 x ≤ π/2. If these restrictions are allowed to hold, the following formulas result:
3-22
MATHEMATICS
x −x2 1 sin−1 x = cos−1 1 −x2 = tan−1 2 = cot−1 x −x 1 1 1 π = sec−1 2 = csc−1 = − cos−1 x x 2 −x 1 −x2 1 −x2 = tan−1 cos−1 x = sin−1 1 x FIG. 3-44
x 1 = cot−1 2 = sec−1 x −x 1
Right triangle.
a = (c + b)( c− b) = c sin α = b tan α
x 1 tan−1 x = sin−1 2 = cos−1 2 −x +x 1 1 +x2 1 1 = cot−1 = sec−1 1 +x2 = csc−1 x x RELATIONS BETWEEN ANGLES AND SIDES OF TRIANGLES Solutions of Triangles (Fig. 3-43) Let a, b, c denote the sides and α, β, γ the angles opposite the sides in the triangle. Let 2s = a + b + c, A = area, r = radius of the inscribed circle, R = radius of the circumscribed circle, and h = altitude. In any triangle α + β + γ = 180°.
+a)( c− a) = c cos α = a cot α b = (c a b a 2 2 a b, sin α = , cos α = , tan α = , β = 90° − α c = c c b
1 a2 b2 tan α c2 sin 2α A = ab = = = 2 2 tan α 2 4 Oblique Triangles (Fig. 3-45) There are four possible cases. 1. Given b, c and the included angles α, b−c 1 1 1 1 tan (β + γ) (β + γ) = 90° − α; tan (β − γ) = b +c 2 2 2 2 1 1 1 1 b sin α β = (β + γ) + (β − γ); γ = (β + γ) − (β − γ); a = 2 2 2 2 sin β 2. Given the three sides a, b, c, s = a (a + b + c); r=
Triangle.
Law of Sines sin α/a = sin β/ b = sin γ/ c. Law of Tangents a + b tan a(α + β) b + c tan a(β + γ) a + c tan a(α + γ) = ; = ; = a − b tan a(α − β) b − c tan a(β − γ) a − c tan a(α − γ) Law of Cosines a2 = b2 + c2 − 2bc cos α; b2 = a2 + c2 − 2ac cos β; c = a2 + b2 − 2ab cos γ. Other Relations In this subsection, where appropriate, two more formulas can be generated by replacing a by b, b by c, c by a, α by β, β by γ, and γ by α. cos α = (b2 + c2 − a2)/2bc; a = b cos γ + c = cos β; sin α = (2/bc) s( s −a )( s − b )( s −c) ; 2
α sin = 2
(s − b)(s − c) α ; cos = bc 2
s(s − a) 1 ; A = bh bc 2
1 a2 sin β sin γ = ab sin γ = = s( s− a)( s− b)( s− c) = rs 2 2 sin α where r =
Oblique triangle.
Right Triangle (Fig. 3-44) Given one side and any acute angle α or any two sides, the remaining parts can be obtained from the following formulas:
1 π = csc−1 2 = − sin−1 x −x 2 1
FIG. 3-43
FIG. 3-45
(s − a)(s − b)(s − c) s
R = a/(2 sin α) = abc/4A; h = c sin a = a sin γ = 2rs/b. Example a = 5, b = 4, α = 30°. Use the law of sines. 0.5/5 = sin β/4, sin β = 2⁄ 5, β = 23°35′, γ = 126°25′. So c = sin 126°25′/ 1⁄10 = 10(.8047) = 8.05. The relations given here suffice to solve any triangle. One method for each triangle is given.
(s − a)(s − b)(s − c) s
1 r 1 r 1 r tan α = ; tan β = ; tan γ = 2 s−a 2 s−b 2 s−c 3. Given any two sides a, c and an angle opposite one of them α, sin γ = (c sin α)/a; β = 180° − a − γ; b = (a sin β)/(sin α). There may be two solutions here. γ may have two values γ1, γ2; γ1 < 90°, γ2 = 180° − γ1 > 90°. If α + γ2 > 180°, use only γ1. This case may be impossible if sin γ > 1. 4. Given any side c and two angles α and β, γ = 180° − α − β; a = (c sin α)/(sin γ); b = (c sin β)/(sin γ). HYPERBOLIC TRIGONOMETRY The hyperbolic functions are certain combinations of exponentials ex and e−x. ex + e−x ex − e−x sinh x ex − e−x cosh x = ; sinh x = ; tanh x = = 2 2 cosh x ex + e−x ex + e−x 1 cosh x 1 2 coth x = = = ; sech x = = ; ex − e−x tanh x sinh x cosh x ex + e−x 2 1 csch x = = sinh x ex − e−x Fundamental Relationships sinh x + cosh x = ex; cosh x − sinh x = e−x; cosh2 x − sinh2 x = 1; sech2 x + tanh2 x = 1; coth2 x − csch2 x = 1; sinh 2x = 2 sinh x cosh x; cosh 2x = cosh2 x + sinh2 x = 1 + 2 sinh2 x = 2 cosh2 x − 1. tanh 2x = (2 tanh x)/(1 + tanh2 x); sinh (x y) = sinh x cosh y cosh x sinh y; cosh (x y) = cosh x cosh y sinh x sinh y; 2 sinh2 x/2 = cosh x − 1; 2 cosh2 x/2 = cosh x + 1; sinh (−x) = −sinh x; cosh (−x) = cosh x; tanh (−x) = −tanh x. When u = a cosh x, v = a sinh x, then u2 − v2 = a2; which is the equation for a hyperbola. In other words, the hyperbolic functions in the parametric equations u = a cosh x, v = a sinh x have the same relation to the hyperbola u2 − v2 = a2 that the equations u = a cos θ, v = a sin θ have to the circle u2 + v2 = a2.
DIFFERENTIAL AND INTEGRAL CALCULUS Inverse Hyperbolic Functions If x = sinh y, then y is the inverse hyperbolic sine of x written y = sinh−1 x or arcsinh x. sinh−1 x = loge (x + x2 +1) 1 1+x −1 x2 +1 ); tanh −1 x = loge ; cosh x = loge (x + 2 1−x 1 x+1 −x2 1 + 1 coth−1 x = loge ; sech−1 x = loge ; 2 x−1 x
1 + 1 +x2 csch−1 = loge x Magnitude of the Hyperbolic Functions cosh x ≥ 1 with equality only for x = 0; −∞ < sinh x < ∞; −1 < tanh x < 1. cosh x ∼ ex/2 as x → ∞; sinh x → ex/2 as x → ∞.
3-23
APPROXIMATIONS FOR TRIGONOMETRIC FUNCTIONS For small values of θ (θ measured in radians) sin θ ≈ θ, tan θ ≈ θ; cos θ ≈ 1 − (θ2/2). The following relations actually hold: sin θ < θ < tan θ; cos θ < sin θ/θ < 1; θ 1 −θ2 < sin θ < θ; cos θ < θ/tan θ < 1; θ θ2 θ 1 − < sin θ < θ and θ < tan θ < 2 − θ 1 2
The behavior ratio of the functions as θ → 0 is given by the following: lim sin θ/θ = 1; sin θ/tan θ = 1. θ→0
DIFFERENTIAL AND INTEGRAL CALCULUS REFERENCES: 114, 158, 260, 261, 274, 282, 296. See also “General References: References for General and Specific Topics—Advanced Calculus.” For computer evaluations of the calculus described here, see Refs. 68, 299.
Limits The limit of function f(x) as x approaches a (a is finite or else x is said to increase without bound) is the number N. lim f(x) = N x→a
DIFFERENTIAL CALCULUS An Example of Functional Notation Suppose that a storage warehouse of 16,000 ft3 is required. The construction costs per square foot are $10, $3, and $2 for walls, roof, and floor respectively. What are the minimum cost dimensions? Thus, with h = height, x = width, and y = length, the respective costs are Walls = 2 × 10hy + 2 × 10hx = 20h(y + x) Roof = 3xy Floor = 2xy Total cost = 2xy + 3xy + 20h(x + y) = 5xy + 20h(x + y)
This states that f(x) can be calculated as close to N as desirable by making x sufficiently close to a. This does not put any restriction on f(x) when x = a. Alternatively, for any given positive number ε, a number δ can be found such that 0 < |a − x| < δ implies that |N − f(x)| < ε. The following operations with limits (when they exist) are valid: lim bf(x) = b lim f(x) x→a
x→a
(3-1)
x→a
x→a
lim f(x) x→a f(x) lim = lim g(x) x→a g(x)
(3-2)
x→a
Solving for h from Eq. (3-2), h = volume/xy = 16,000/xy 320,000 Cost = 5xy + (y + x) = 5xy + 320,000 xy In this form it can be shown that the minimum x = y; therefore Cost = 5x2 + 640,000 (1/x)
x→a
(3-3)
x + y 1
1
(3-4)
cost will occur for
By evaluation, the smallest cost will occur when x = 40. Cost = 5(1600) + 640,000/40 = $24,000 The dimensions are then x = 40 ft, y = 40 ft, h = 16,000/(40 × 40) = 10 ft. Symbolically, the original cost relationship is written Cost = f(x, y, h) = 5xy + 20h(y + x) and the volume relation Volume = g(x, y, h) = xyh = 16,000 In terms of the derived general relationships (3-1) and (3-2), x, y, and h are independent variables—cost and volume, dependent variables. That is, the cost and volume become fixed with the specification of dimensions. However, corresponding to the given restriction of the problem, relative to volume, the function g(x, y, z) = xyh becomes a constraint function. In place of three independent and two dependent variables the problem reduces to two independent (volume has been constrained) and two dependent as in functions (3-3) and (3-4). Further, the requirement of minimum cost reduces the problem to three dependent variables (x, y, h) and no degrees of freedom, that is, freedom of independent selection.
x→a
lim [ f(x)g(x)] = lim f(x) ⋅ lim g(x)
and the restriction Total volume = xyh
x→a
lim [ f(x) + g(x)] = lim f(x) + lim g(x) x→a
if
lim g(x) ≠ 0 x→a
Continuity A function f(x) is continuous at the point x = a if lim [ f(a + h) − f(a)] = 0 h→0
Rigorously, it is stated f(x) is continuous at x = a if for any positive ε there exists a δ > 0 such that |f(a + h) − f(a)| < ε for all x with |x − a| < δ. For example, the function (sin x)/x is not continuous at x = 0 and therefore is said to be discontinuous. Discontinuities are classified into three types: 1. Removable 2. Infinite 3. Jump
y = sin x/x at x = 0 y = 1/x at x = 0 y = 10/(1 + e1/x) at x = 0+ y = 0+ x=0 y=0 x = 0− y = 10
Derivative The function f(x) has a derivative at x = a, which can be denoted as f ′(a), if f(a + h) − f(a) lim h→0 h exists. This implies continuity at x = a. Conversely, a function may be continuous but not have a derivative. The derivative function is df f(x + h) − f(x) f ′(x) = = lim dx h→0 h Differentiation Define ∆y = f(x + ∆x) − f(x). Then dividing by ∆x ∆y f(x + ∆x) − f(x) = ∆x ∆x
3-24
MATHEMATICS ∆y dy lim = ∆x dx
Call
∆x→0
Example Find the derivative of y = sin x. dy sin (x + ∆x) − sin(x) = lim dx ∆x→0 ∆x sin x cos ∆x + sin ∆x cos x − sin x = lim ∆x→0 ∆x sin ∆x cos x sin x(cos ∆x − 1) = lim + lim ∆x→0 ∆x→0 ∆x ∆x sin ∆x = cos x since lim = 1 ∆x→0 ∆x
Differential Operations The following differential operations are valid: f, g, . . . are differentiable functions of x, c is a constant; e is the base of the natural logarithms. dc (3-5) =0 dx dx =1 dx
(3-6)
d df dg (f + g) = + dx dx dx
(3-7)
d dg df (f × g) = f + g dx dx dx dy 1 = dx dx/dy
if
(3-8) dx ≠0 dy
d df f n = nf n − 1 dx dx d f g(df/dx) − f(dg/dx) = dx g g2
df df dv =× dx dv dx
d tan x = sec x dx
(3-20)
d cot x = −csc2 x dx
(3-21)
d sec x = tan x sec x dx
(3-22)
d csc x = −cot x csc x dx
(3-23)
d sin−1 x = (1 − x2)−1/2 dx
(3-24)
d cos−1x = −(1 − x2)−1/2 dx
(3-25)
d tan−1 x = (1 + x2)−1 dx
(3-26)
d cot−1 x = −(1 + x2)−1 dx
(3-27)
d sec−1 x = x−1(x2 − 1)−1/2 dx
(3-28)
d csc−1 x = −x−1(x2 − 1)−1/2 dx
(3-29)
d sinh x = cosh x dx
(3-30)
d cosh x = sinh x dx
(3-31)
d tanh x = sech2 x dx
(3-32)
d coth x = −csch2 x dx
(3-33)
d sech x = −sech x tanh x dx
(3-34)
d csch x = −csch x coth x dx
(3-35)
d sinh x = (x + 1)
(3-36)
−1
(3-9)
−1/2
2
dx
d cosh−1 = (x2 − 1)−1/2 dx
(3-37)
d tanh−1 x = (1 − x2)−1 dx
(3-38)
d coth−1 x = −(x2 − 1)−1 dx
(3-39)
d sech−1 x = −(1/x)(1 − x2)−1/2 dx
(3-40)
d csch x = −x (x + 1)
(3-41)
−1
−1
−1/2
2
dx
Example Find dy/dx for y = x cos (1 − x ). 2
(3-10)
dy d d = x cos (1 − x2) + cos (1 − x2) x dx dx dx
(3-11)
d d cos (1 − x2) = −sin (1 − x2) (1 − x2) dx dx
Using (3-8) (3-19)
= −sin (1 − x )(0 − 2x) 2
(chain rule)
(3-12)
g
df df dg = gf g − 1 + f g ln f dx dx dx
(3-13)
dax = (ln a) ax dx
(3-14)
(3-5), (3-10)
dx 1 = x−1/2 dx 2
(3-10)
1 dy = 2x3/2 sin (1 − x2) + x−1/2 cos (1 − x2) dx 2
Example Find the derivative of tan x with respect to sin x. v = sin x
Example Derive dy/dx for x2 + y3 = x + xy + A. Here
(3-19)
2
f(x + ∆x) − f(x) dy lim = ∆x→0 dx ∆x
then
d cos x = −sin x dx
y = tan x
d d d d d x2 + y3 = x + xy + A dx dx dx dx dx
d tan x dy dy dx == d sin x dv dx dv 1 d tan x = d sin x dx dx = sec2 x/cos x
dy dy 2x + 3y2 = 1 + y + x + 0 dx dx by rules (3-10), (3-10), (3-6), (3-8), and (3-5) respectively. dy 2x − 1 − y Thus = dx x − 3y2
Differentials dex = ex dx
(3-15a)
d(ax) = ax log a dx
(3-15b)
d ln x = (1/x) dx
(3-16)
d log x = (log e/x)dx
(3-17)
d sin x = cos x dx
(3-18)
Using (3-12) (3-9) (3-18), (3-20)
Very often in experimental sciences and engineering functions and their derivatives are available only through their numerical values. In particular, through measurements we may know the values of a function and its derivative only at certain points. In such cases the preceding operational rules for derivatives, including the chain rule, can be applied numerically. Example Given the following table of values for differentiable functions f and g; evaluate the following quantities:
DIFFERENTIAL AND INTEGRAL CALCULUS x
f(x)
f′(x)
g(x)
g′(x)
1 3 4
3 0 −2
1 2 10
4 4 3
−4 7 6
d [ f(x) + g(x)]| x = 4 = f ′(4) + g′(4) = 10 + 6 = 16 dx f ′
f ′(1)g(1) − f(1)g′(1)
1 ⋅ 4 − 3(−4)
= = = 1 g (1) = [g(1)] (−4) 16 2
16
Example Given f(x) = 3x3 + 2x + 1, calculate all derivative values at x = 3. df(x) = 9x2 + 2 dx
x = 3, f′(3) = 9(9) + 2 = 83
d2f(x) = 18x dx2
x = 3, f″(3) = 18(3) = 54
d3f(x) = 18 dx3
x = 3, f ″(3) = 18
dnf(x) =0 dxn
for n ≥ 4
If f ′(x) > 0 on (a, b), then f is increasing on (a, b). If f ′(x) < 0 on (a, b), then f is decreasing on (a, b). The graph of a function y = f(x) is concave up if f ′ is increasing on (a, b); it is concave down if f ′ is decreasing on (a, b). If f ″(x) exists on (a, b) and if f ″(x) > 0, then f is concave up on (a, b). If f ″(x) < 0, then f is concave down on (a, b). An inflection point is a point at which a function changes the direction of its concavity. Indeterminate Forms: L’Hospital’s Theorem Forms of the type 0/0, ∞/∞, 0 × ∞, etc., are called indeterminates. To find the limiting values that the corresponding functions approach, L’Hospital’s theorem is useful: If two functions f(x) and g(x) both become zero at x = a, then the limit of their quotient is equal to the limit of the quotient of their separate derivatives, if the limit exists or is + ∞ or − ∞. sin x x sin x d sin x cos x lim = lim = lim = 1 x→0 x→0 x→0 x dx 1
Example Find lim . n→0 Here
(1.1) x
Example Find lim . 1000 x→∞ (1.1)x d(1.1)x (ln 1.1)(1.1)x = lim = lim lim x→∞ x1000 x→∞ dx1000 x→∞ 1000x999 1.1x Obviously lim = ∞ since repeated application of the rule will reduce the x→∞ x1000 denominator to a finite number 1000! while the numerator remains infinitely large.
x3 6 lim x3 e−x = lim x = lim x = 0 x→∞ x→∞ e x→∞ e
∂e ∂z ∂x = y + ey ∂x ∂x ∂x
∂ey ∂z 2 ∂y = ex + x ∂y ∂y ∂y
= 2xyex + e y
= ex + xey
2
2
Order of Differentiation It is generally true that the order of differentiation is immaterial for any number of differentiations or variables provided the function and the appropriate derivatives are continuous. For z = f(x, y) it follows: ∂3f ∂3f ∂3f == ∂y2 ∂x ∂y ∂x ∂y ∂x ∂y2 General Form for Partial Differentiation 1. Given f(x, y) = 0 and x = g(t), y = h(t). df ∂f dx ∂f dy Then = + dt ∂x dt ∂y dt d 2f ∂2f dx 2 = 2 dt ∂x dt
∂2
∂2
f dy f ∂f d x + + +2 ∂x ∂y dt dt ∂y dt ∂x dt 2
dx
dy
y = (1 − x)
1/x
ln y = (1/x) ln (1 − x) ln(1 − x) lim (ln y) = lim = −1 x→0 x→0 x lim y = e−1
2
2
2
2
∂f d 2y + ∂y dt2 Example Find df/dt for f = xy, x = ρ sin t, y = ρ cos t. df ∂(xy) d ρ sin t ∂(xy) d ρ cos t = + dt ∂x dt ∂y dt = y(ρ cos t) + x(−ρ sin t) = ρ2 cos2 t − ρ2 sin2 t
Given f(x, y) = 0 and x = g(t, s), y = h(t, s). ∂f ∂f ∂x ∂f ∂y Then =+ ∂t ∂x ∂t ∂y ∂t 2.
∂f ∂f ∂x ∂f ∂y =+ ∂s ∂x ∂s ∂y ∂x Differentiation of Composite Function dy ∂f/∂x ∂f Rule 1. Given f(x, y) = 0, then = − ≠ 0 . dx ∂f/∂y ∂y Rule 2. Given f(u) = 0 where u = g(x), then df du = f ′(u) dx dx du + f ′(u) dx
d 2f du 2 = f ″(u) dx dx
2
2
2
df d sin2 u d1 −x2 = dx du dx
1 = 2 sin u cos u (−2x)(1 − x2)−1/2 2
Example Find lim (1 − x)1/x. x→0
x→0
2
x2
Example Find df/dx for f = sin2 u and u = 1−x2
Example Find lim x3 e−x. x→∞
Therefore,
Example Find ∂z/∂x and ∂z/∂y for z = ye x + xey.
x
Let
Partial Derivative The abbreviation z = f(x, y) means that z is a function of the two variables x and y. The derivative of z with respect to x, treating y as a constant, is called the partial derivative with respect to x and is usually denoted as ∂z/∂x or ∂f(x, y)/∂x or simply fx. Partial differentiation, like full differentiation, is quite simple to apply. Conversely, the solution of partial differential equations is appreciably more difficult than that of differential equations.
2
Higher Differentials The first derivative of f(x) with respect to x is denoted by f′ or df/dx. The derivative of the first derivative is called the second derivative of f(x) with respect to x and is denoted by f ″, f (2), or d 2 f/dx 2; and similarly for the higher-order derivatives.
3-25
− u 1 = −2 sin u cos u u 2
Rule 3. Given f(u) = 0 where u = g(x,y), then ∂f ∂u ∂f ∂u = f ′(u) + = f ′(u) ∂x ∂x ∂y ∂y
3-26
MATHEMATICS ∂2f ∂u 2 = f ″ ∂x ∂x
∂ + f′ ∂x 2
2
u
V is specified once T and p are specified; it is therefore a state function. All applications are for closed systems with constant mass. If a process is reversible and only p-V work is done, the first law and differentials can be expressed as follows.
2
∂2f ∂u ∂u ∂2u = f ″ + f′ ∂x ∂y ∂x ∂y ∂x ∂y ∂2f ∂u 2 = f ″ ∂y ∂y
dU = T dS − p dV dH = T dS + V dp dA = −S dT − p dV dG = −S dT + V dp
∂2u
+ f′ ∂y 2
2
MULTIVARIABLE CALCULUS APPLIED TO THERMODYNAMICS Many of the functional relationships needed in thermodynamics are direct applications of the rules of multivariable calculus. This section reviews those rules in the context of the needs of themodynamics. These ideas were expounded in one of the classic books on chemical engineering thermodynamics.151 State Functions State functions depend only on the state of the system, not on past history or how one got there. If z is a function of two variables, x and y, then z(x,y) is a state function, since z is known once x and y are specified. The differential of z is dz = M dx + N dy The line integral
(M dx + N dy)
is independent of the path in x-y space if and only if ∂M ∂N = ∂y ∂x The total differential can be written as ∂z ∂z dz = dx + dy ∂x y ∂y x and the following condition guarantees path independence. ∂ ∂z ∂ ∂z = ∂y ∂x y ∂x ∂y x
∂2z ∂2z = ∂y ∂x ∂x ∂y
or
(3-42)
(3-43)
x
z
∂z ∂z ∂y (∂y/∂x)z (3-45) =− =− ∂x y ∂x z ∂y x (∂y/∂z)x Alternatively, divide Eq. (3-43) by dy when holding some other variable w constant to obtain ∂z ∂z ∂x ∂z (3-46) = + ∂y w ∂x v ∂y w ∂y x Also divide both numerator and denominator of a partial derivative by dw while holding a variable y constant to get
(∂z/∂w) ∂z ∂w = ∂z∂x = (∂x/∂w) ∂w ∂x y y
y
∂T ∂p = − ∂V ∂S
(3-47)
y
Themodynamic State Functions In thermodynamics, the state functions include the internal energy, U; enthalpy, H; and Helmholtz and Gibbs free energies, A and G, respectively, defined as follows: H=U+pV A = U − TS G = H − TS = U + pV − TS = A + pV S is the entropy, T the absolute temperature, p the pressure, and V the volume. These are also state functions, in that the entropy is specified once two variables (like T and p) are specified, for example. Likewise,
V
This is one of the Maxwell relations, and the other Maxwell relations can be derived in a similar fashion by applying Eq. (3-44). ∂T ∂V = ∂p S ∂S p ∂S ∂p = ∂V T ∂T V
∂S ∂V = − ∂p ∂T
p
In process simulation it is necessary to calculate enthalpy as a function of state variables. This is done using the following formulas, derived from the above relations by considering S and H as functions of T and p. ∂V dH = Cp dT + V − T dp ∂T p Enthalpy differences are then given by the following formula. T2 p 2 ∂V H(T2, p2) − H(T1, p1) = Cp(T, p1) dT + V−T dp T1 p1 ∂T p T2,p The same manipulations can be done for internal energy as a function of T and V. (∂V/∂T) dU = CV dT − p + T p dV (∂V/∂p)T
Rearrangement gives
y
T
(3-44)
dy
S
Example Suppose z is constant and apply Eq. (3-43). ∂z ∂z 0 = dx + ∂x y ∂y
This is
C
Alternatively, if the internal energy is considered a function of S and V, then the differential is: ∂U ∂U dU = dS + dV ∂S V ∂V S This is the equivalent of Eq. (3-43) and gives the following definitions. ∂U ∂U T= , p=− ∂S V ∂V S Since the internal energy is a state function, then Eq. (3-44) must be satisfied. ∂2U ∂2U = ∂V ∂S ∂S ∂V
Partial Derivatives of All Thermodynamic Functions The various partial derivatives of the thermodynamic functions can be classified into six groups. In the general formulas below, the variables U, H, A, G or S are denoted by Greek letters, while the variables V, T, or p are denoted by Latin letters. Type I (3 possibilities plus reciprocals) ∂a ∂p General: ; Specific: ∂b c ∂T V Eq. (3-45) gives ∂p ∂V ∂p (∂V/∂T)p =− =− ∂T V ∂T p ∂V T (∂V/∂p)T
Type II
(30 possibilities) ∂α ∂G General: ; Specific: ∂b c ∂T
V
DIFFERENTIAL AND INTEGRAL CALCULUS The differential for G gives ∂G ∂p = −S + V ∂T V ∂T V Using the other equations for U, H, A, or S gives the other possibilities. Type III (15 possibilities plus reciprocals) ∂a ∂V General: ; Specific: ∂b α ∂T S First expand the derivative using Eq. (3-45). (∂S/∂T)V ∂V ∂S ∂V =− =− ∂T S ∂T V ∂S T (∂S/∂V)T Then evaluate the numerator and denominator as type II derivatives. CV ∂V T ∂V CV ∂p T = − ∂V ∂p = ∂T S T ∂V − ∂T p ∂V T ∂T p
f ′(x) dx = f(x) + c where c is an arbitrary constant to be determined by the problem. By virtue of the known formulas for differentiation the following relationships hold (a is a constant):
(du + dv + dw) = du + dv + dw a dv = a dv v v dv = + c (n ≠ −1) n+1
n
dv = ln |v| + c v
∂G
∂G
v
∂T
= = (∂A/∂T) ∂A ∂T ∂A p
p
(∂G/∂T)p
p
p
This operation has created two type II derivatives; by substitution we obtain ∂G S = ∂A p S + p (∂V/∂T)p Type V (60 possibilities) ∂α ∂G General: ; Specific: ∂b β ∂p A Start from the differential for dG. Then we get ∂G ∂T = −S + V ∂p A ∂p A The derivative is type III and can be evaluated by using Eq. (3-45). ∂G (∂A/∂p)T = S + V ∂p A (∂A/∂T)p The two type II derivatives are then evaluated. ∂G Sp (∂V/∂p)T = + V ∂p A S + p (∂V/∂T)p These derivatives are also of interest for free expansions or isentropic changes. Type VI (30 possibilities plus reciprocals) ∂α ∂G General: ; Specific: ∂β γ ∂A H We use Eq. (3-47) to obtain two type V derivatives. (∂G/∂T)H ∂G = ∂A H (∂A/∂T)H These can then be evaluated using the procedures for Type V derivatives.
(3-54) (3-55) (3-56)
2
(3-57) (3-58) (3-59) (3-60)
2
−1
2
2
v +c a
(3-61)
dv 1 v−a = ln + c v −a 2a v+a 2
(3-62)
2
dv = ln |v + v a| + c v a sec v dv = ln (sec v + tan v) + c csc v dv = ln (csc v − cot v) + c
(3-53)
2
dv = sin a −v
2
2
v
2
(3-52)
−1
(3-50)
v
v
(3-49)
(3-51)
a a dv = +c ln a e dv = e + c sin v dv = −cos v + c cos v dv = sin v + c sec v dv = tan v + c csc v dv = −cot v + c sec v tan v dv = sec v + c csc v cot v dv = −csc v + c dv 1 v = tan + c v +a a a
These derivatives are of importance for reversible, adiabatic processes (such as in an ideal turbine or compressor), since then the entropy is constant. An example is the Joule-Thomson coefficient. ∂T 1 ∂V = −V + T ∂p H Cp ∂T p Type IV (30 possibilities plus reciprocals) ∂α ∂G General: ; Specific: ∂β c ∂A p Use Eq. (3-47) to introduce a new variable.
(3-48)
n+1
3-27
2
2
(3-63) (3-64) (3-65)
Example Derive ∫ av dv = (av/ln a) + c. By reference to the differentiation formula dav/dv = av ln a, or in the more usable form d(av/ln a) = av dv, let f ′ = av dv; then f = av/ln a and hence ∫ av dv = (av/ln a) + c. Example Find ∫ (3x2 + ex − 10) dx using Eq. (3-48). ∫ (3x2 + ex − 10) dx = 3 ∫ x2 dx + ∫ ex dx − 10 ∫ dx = x3 + ex − 10x + c (by Eqs. 3-50, 3-53). 7x dx . Let v = 2 − 3x ; dv = −6x dx 2 − 3x 7x dx x dx 7 −6x dx =7 =− 2 − 3x 2 − 3x 6 2 − 3x 7 dv =− 6 v
Example Find Thus
2
2
2
2
2
INTEGRAL CALCULUS
7 = − ln |v| + c 6
Indefinite Integral If f ′(x) is the derivative of f(x), an antiderivative of f ′(x) is f(x). Symbolically, the indefinite integral of f ′(x) is
7 = − ln |2 − 3x2| + c 6
3-28
MATHEMATICS
Example—Constant of Integration By definition the derivative of x3 is 3x2, and x3 is therefore the integral of 3x2. However, if f = x3 + 10, it follows that f′ = 3x2, and x3 + 10 is therefore also the integral of 3x2. For this reason the constant c in ∫ 3x2 dx = x3 + c must be determined by the problem conditions, i.e., the value of f for a specified x. Methods of Integration In practice it is rare when generally encountered functions can be directly integrated. For example, the integrand in ∫ si nx dx which appears quite simple has no elementary function whose derivative is si nx. In general, there is no explicit way of determining whether a particular function can be integrated into an elementary form. As a whole, integration is a trial-and-error proposition which depends on the effort and ingenuity of the practitioner. The following are general procedures which can be used to find the elementary forms of the integral when they exist. When they do not exist or cannot be found either from tabled integration formulas or directly, the only recourse is series expansion as illustrated later. Indefinite integrals cannot be solved numerically unless they are redefined as definite integrals (see “Definite Integral”), i.e., F(x) = ∫ f(x) dx, indefinite, whereas F(x) = ∫ ax f(t) dt, definite. Direct Formula Many integrals can be solved by transformation in the integrand to one of the forms given previously. Example Find ∫ x 3 x + 10 dx. Let v = 3x + 10 for which dv = 9x dx. 2
Thus
3
3
x 3x +10 dx = (3x + 10) 2
3
3
1/2
(x2 dx)
(3x + 10)
1 = 9
v
1/2
x dx = (3 + 4x) 1/4
1 = 4
1/2
(9x2 dx)
dv
3/2
1 v =+c 9 3⁄ 2
General The number of possible transformations one might use are unlimited. No specific overall rules can be given. Success in handling integration problems depends primarily upon experience and ingenuity. The following example illustrates the extent to which alternative approaches are possible.
e dx− 1
Example Find . Let ex = y; then ex dx = dy or dx = 1/y dy. x dx (1/y) dy e −2 dy y−1 = = = ln = ln e −1 y−1 y −y e y x
x
A B C =++ x+2 x−2 x−1 A(x − 2)(x − 1) + B(x + 2)(x − 1) + C(x + 2)(x − 2) = (x + 2)(x − 2)(x − 1) x2(A + B + C) + x(−3A + B) + (2A − 2B − 4C) = (x + 2)(x − 2)(x − 1) Equate coefficients and solve for A, B, and C. A+B+C=0 −3A + B = 0 2A − 2B − 4C = 1 A = 1⁄12, B = d, C = −s 1 1 1 1 =+− x3 − x2 − 4x + 4 12(x + 2) 4(x − 2) 3(x − 1)
x +a Let x = a tan θ 2
a2 −x2 Let x = a sin θ − 9 x 4 2 2 dx. Let x = sin θ; then dx = cos θ dθ. x 3 3 2
2
(2 /3 ) −x θ 2 2/31 −sin dx = 3 cos θ dθ x (2/3) sin θ 3 2
2
2
2
2
=3
cos θ dθ sin θ
=3
cot θ dθ
Hence
dx dx dx dx = + − x − x − 4x + 4 12(x + 2) 4(x − 2) 3(x − 1)
2
3
2
d(uv) = u dv + v du
2
= −3 cot θ − 3θ + c by trigonometric transform − 9 x2 3 4 = − − 3 sin−1 x + c in terms of x x 2
Algebraic Substitution Functions containing elements of the type (a + bx)1/n are best handled by the algebraic transformation yn = a + bx. Example Find . Let 3 + 4x = y4; then 4dx = 4y3 dy and 1/4
2
Parts An extremely useful formula for integration is the relation
2
(3 +x dx4x)
x
2
Partial Fractions Rational functions are of the type f(x)/g(x) where f(x) and g(x) are polynomial expressions of degrees m and n respectively. If the degree of f is higher than g, perform the algebraic division—the remainder will then be at least one degree less than the denominator. Consider the following types: Type 1 Reducible denominator to linear unequal factors. For example, 1 1 = x3 − x2 − 4x + 4 (x + 2)(x − 2)(x − 1)
x2 −a2 Let x = a sec θ
3
4
1 1 = (3 + 4x)7/4 − (3 + 4x)3/4 + c 28 4
[by Eq. (3-50)]
Trigonometric Substitution This technique is particularly well adapted to integrands in the form of radicals. For these the function is transformed into a trigonometric form. In the latter form they may be more easily recognizable relative to the identity formulas. These functions and their transformations are
Example Find
2
1 y7 3 y3 =−+c 4 7 4 3
2 = (3x3 + 10)3/2 + c 27
2
y (y − 3) dy
2
1 = 9
3
y4 − 3 y3 dy 4 y
and or
u dv + v du u dv = uv − v du uv =
No general rule for breaking an integrand can be given. Experience alone limits the use of this technique. It is particularly useful for trigonometric and exponential functions. Example Find xex dx. Let u=x du = dx
and
dv = ex dx v = ex
DIFFERENTIAL AND INTEGRAL CALCULUS
xe dx = xe − e dx x
Therefore
x
Techniques for determining when integration is valid under these conditions are available in the references. However, the following simplified rules will, in general, serve as a guide for most practical applications. Rule 1 For the integral ∞ φ(x) dx 0 xn
x
= xex − ex + c
Example Find ex sin x dx. Let u = ex
dv = sin x dx v = −cos x
du = ex dx
e sin x dx = −e cos x + e cos x dx x
x
x
u = ex
Again
dv = cos x dx v = sin x
du = ex dx
e sin x dx = −e cos x + e sin x − e sin x dx + c x
x
x
x
c = (ex/2)(sin x − cos x) + 2
Series Expansion When an explicit function cannot be found, the integration can sometimes be carried out by a series expansion. Example Find e− x dx. Since 2
4
6
x x 2 e− x = 1 − x2 + − + ⋅⋅⋅ 2! 3!
e
−x2
dx =
dx − x dx + 2!x dx − 3!x dx + ⋅⋅⋅ 4
if φ(x) is bounded, the integral will converge for n < 1 and diverge for n ≥ 1. Thus 1 1 dx x 0
will converge (exist) since a = n < 1. Properties The fundamental theorem of calculus states
f(x) dx = F(b) − F(a)
6
for all x
b
a
dF(x)/dx = f(x)
where
Definite Integral The concept and derivation of the definite integral are completely different from those for the indefinite integral. These are by definition different types of operations. However, the formal operation ∫ as it turns out treats the integrand in the same way for both. Consider the function f(x) = 10 − 10e−2x. Define x1 = a and xn = b, and suppose it is desirable to compute the area between the curve and the coordinate axis y = 0 and bounded by x1 = a, xn = b. Obviously, by a sufficiently large number of rectangles this area could be approximated as closely as desired by the formula
if φ(x) is bounded, the integral will converge for n > 1 and not converge for n ≤ 1. ∞ It is easily seen that ∫ 0 e−x dx converges by noting 1/x2 > 1/ex > 0 for large x. Rule 2 For the integral b φ(x) n dx, a (a − x)
2
x3 x5 x7 = x − + − + ⋅⋅⋅ 3 5.2! 7.3!
n−1
Other properties of the definite integral are
c[ f(x) dx] = c f(x) dx b
[ f (x) + f (x)] dx = f (x) dx + f (x) dx
a
i
2
a
a
f(x) dx = − f(x) dx b
a
a
b
f(x) dx = b
c
a
f(x) dx +
a
f(x) dx b
c
f(x) dx = (b − a) f(ξ) for some ξ in (a, b)
xi − 1 ≤ ξi − 1 ≤ xi
b
i+1
− xi)
i=1
f(x) dx indefinite integral where dF/dx = f(x) F(a, b) = f(x) dx definite integral F(α) = f(x, α) dx
∂ ∂b
f(x) dx = f(b)
∂ ∂a
f(x) dx = −f(a)
dF(α) = dα
F(x) =
a b
a
b
a
∂f(x, α) dx if a and b are constant ∂α b
a
b
d
a
when F(x) =
d
c
c
b
a
b(x)
f(x, y) dy
a(x)
a
There are certain restrictions of the integration definition, “The function f(x) must be continuous in the finite interval (a, b) with at most a finite number of finite discontinuities,” which must be observed before integration formulas can be generally applied. Two of these restrictions give rise to so-called improper integrals and require special handling. These occur when ∞ 1. The limits of integration are not both finite, i.e., ∫ 0 e−x dx. 2. The function becomes infinite within the interval of integration, i.e., 1 1 dx x 0
b
dx f(x, α) dα = dα f(x, α) dx
b
b
1
a
where the points x1, x2, . . . , xn are equally spaced. For a rigorous definition of the definite integral the references should be consulted. Thus, the value of a definite integral depends on the limits a, b, and any selected variable coefficients in the function but not on the dummy variable of integration x. Symbolically
or
2
a
n
n→∞
b
1
The definite integral of f(x) is defined as
f(x) dx = lim f(ξ )(x
a
b
i=1
+ ⋅⋅⋅ + f(ξn − 1)(b − xn − 1)
b
a
f(ξi)(xi + 1 − xi) = f(ξ1)(x2 − a) + f(ξ2)(x3 − x2)
b
3-29
the Leibniz rule gives dF db da = f [x, b(x)] − f [x, a(x)] + dx dx dx
b(x)
a(x)
∂f dy ∂x
π/2
Example Find
sin x dx.
0
π/2
0
since
π sin x dx = [−cos x]0π/2 = − cos − cos 0 = 1 2
−d cos x/dx = sin x
(3-66)
3-30
MATHEMATICS dx . Direct application of the formula would yield (x − 1) 2
Example Find the incorrect value
1 dx = − (x − 1) x−1 2
2
0
c = π/2 φ(α) = −tan−1 α + π/2
2
= −2
0
It should be noted that f(x) = 1/(x − 1) becomes unbounded as x → 1 and by Rule 2 the integral diverges and hence is said not to exist. Methods of Integration All the methods of integration available for the indefinite integral can be used for definite integrals. In addition, several others are available for the latter integrals and are indicated below. Change of Variable This substitution is basically the same as previously indicated for indefinite integrals. However, for definite integrals, the limits of integration must also be changed: i.e., for x = φ(t), 2
f (x) dx = b
Integration It is sometimes useful to generate a double integral to solve a problem. By this approach, the fundamental theorem indicated by Eq. (3-66) can be used.
0
x 1
α
Consider
0
dα x b
α
dα x a
2
π/2
0
cos2 θ dθ = 16[aθ + d sin 2θ]0π/2 = 4π
∞
e−αx sin x dx (α > 0) x Since this is a continuous function of α, it may be differentiated under the integral sign ∞ dφ = − e−αx sin x dx 0 dα φ(α) =
b
dα b+1 = ln α+1 a+1
= −1/(1 + α ) φ(α) = −tan−1 α + c 2
1
b
α
1
dα =
a
0
b+1 xb − xα dx = ln ln x a+1
xb − xα dx ln x
Complex Variable Certain definite integrals can be evaluated by the technique of complex variable integration. This is described in the references for “Complex Variables.” Numerical Because of the property of definite integrals another method for obtaining their solution is available which cannot be applied to indefinite integrals. This involves a numerical approximation based on the previously outlined summation definition:
0
dx x 0
Therefore
Differentiation Here the application of the general rules for differentiating under the integral sign may be useful. Example Find
a
dx =
0
1
x = 4 sin θ (x = 0, θ = 0) dx = 4 cos θ dθ (x = 4, θ = π/2) 0
1
α
16−x dx. Let 2
0
dx =
0
But also b
16−x dx = 16
1
a
t0
4
Then
1 dx = (α > −1) α+1
Then multiplying both sides by dα and integrating between a and b,
4
0
xb − xα dx ln x
1
f [φ(t)]φ′(t) dt
t = t0 when x = a t = t1 when x = b
Example Find
Example Find
t1
a
where
and since φ(α) → 0 as α → ∞,
2
0
n−1
lim f(ξi)(xi + 1 − xi) =
n→∞
where x1 = a
and
1
f(x) dx b
a
xn = b
Examples of this procedure are given in the subsection “Numerical Analysis and Approximate Methods.”
INFINITE SERIES REFERENCES: 53, 126, 127, 163. For asymptotic series and asymptotic methods, see Refs. 51, 127.
DEFINITIONS
while valid for any finite value of r and n now takes on a different interpretation. In this sense it is necessary to consider the limit of Sn as n increases indefinitely: S = lim Sn
A succession of numbers or terms that are formed according to some definite rule is called a sequence. The indicated sum of the terms of a sequence is called a series. A series of the form a0 + a1(x − c) + a2(x − c)2 + ⋅⋅⋅ + an(x − c)n + ⋅⋅⋅ is called a power series. Consider the sum of a finite number of terms in the geometric series (a special case of a power series). Sn = a + ar + ar 2 + ar 3 + ⋅⋅⋅ + ar n − 1
(3-67)
For any number of terms n, the sum equals 1 − rn Sn = a 1−r In this form, the geometric series is assumed finite. In the form of Eq. (3-67), it can further be defined that the terms in the series be nonending and therefore an infinite series. S = a + ar + ar 2 + ⋅⋅⋅ + ar n + ⋅⋅⋅ However, the defined sum of the terms [Eq. (3-67)] 1 − rn Sn = a r≠1 1−r
(3-68)
n→∞
1 − rn = a lim n→∞ 1 − r For this, it is stated the infinite series converges if the limit of Sn approaches a fixed finite value as n approaches infinity. Otherwise, the series is divergent. On this basis an analysis of 1 − rn S = a lim n→∞ 1 − r shows that if r is less than 1 but greater than −1, the infinite series is convergent. For values outside of the range −1 < r < 1, the series is divergent because the sum is not defined. The range −1 < r < 1 is called the region of convergence. (We assume a ≠ 0.) Consider the divergence of Eq. (3-68) when r = −1 and +1. For the former case r = −1, S = a + a(−1) + a(−1)2 + a(−1)3 + ⋅⋅⋅ + a(−1)n + ⋅⋅⋅ = a − a + a − a + a − ⋅⋅⋅
INFINITE SERIES and for which 1 − rn S = a lim n→∞ 1 − r 1 − (−1)n = a lim undefined limit (if a ≠ 0) n→∞ 1+1 Since the limit sum does not exist, the series is divergent. This is defined as a bounded or oscillating divergent series. Similarly for the value r = +1, S = a + a(1) + a(1)2 + a(1)3 + ⋅⋅⋅ + a(1)n + ⋅⋅⋅ S = a + a + a + a + ⋅⋅⋅ + a + ⋅⋅⋅
The series is also divergent but defined as an unbounded divergent series. There are also two types of convergent series. Consider the new series 1 1 1 1 S = 1 − + − + ⋅⋅⋅ + (−1)n + 1 + ⋅⋅⋅ (3-69) 2 3 4 n It can be shown that the series (3-69) does converge to the value S = log 2. However, if each term is replaced by its absolute value, the series becomes unbounded and therefore divergent (unbounded divergent): 1 1 1 1 S = 1 + + + + + ⋅⋅⋅ (3-70) 2 3 4 5 In this case the series (3-69) is defined as a conditionally convergent series. If the replacement series of absolute values also converges, the series is defined to converge absolutely. Series (3-69) is further defined as an alternating series, while series (3-70) is referred to as a positive series. OPERATIONS WITH INFINITE SERIES 1. The convergence or divergence of an infinite series is unaffected by the removal of a finite number of finite terms. This is a trivial theorem but useful to remember, especially when using the comparison test to be described in the subsection “Tests for Convergence and Divergence.” 2. If a series is conditionally convergent, its sums can be made to have any arbitrary value by a suitable rearrangement of the series; it can in fact be made divergent or oscillatory (Riemann’s theorem). This seemingly paradoxical theorem can be illustrated by the following example. 1 1 1 1 1 2 3 4 5 6 The series is rearranged so that each positive term is followed by two negative terms: 1 1 1 1 1 1 1 1 t = 1 − − + − − + − − + ⋅⋅⋅ 2 4 3 6 8 5 10 12 Define t3n for the first 3n terms in the series
Example S = 1 − + − + − + ⋅⋅⋅
1 1 1 1 1 1 1 1 t3n = 1 − − + − − + ⋅⋅⋅ + − − 2 4 3 6 8 2n − 1 4n − 2 4n 1 1 1 1 1 1 = − + − + ⋅⋅⋅ + − 2 4 6 8 4n − 2 4n
1 1 1 1 1 1 = 1 − + − + ⋅⋅⋅ + − 2 2 3 4 2n − 1 2n
3. A series of positive terms, if convergent, has a sum independent of the order of its terms; but if divergent, it remains divergent however its terms are rearranged. 4. An oscillatory series can always be made to converge by grouping the terms in brackets. Example Consider the series 1 2 3 4 5 1 − + − + − + ⋅⋅⋅ 2 3 4 5 6 which oscillates between the values 0.306 and 1.306. However, the series 1 2 3 4 5 1 1 1 1 1 − + − + − + ⋅⋅⋅ = − − − − ⋅⋅⋅ 0.306 ⋅⋅⋅ 2 3 4 5 6 2 12 30 56
(a ≠ 0)
3-31
and
1 1 2 3 4 5 6 1 1 1 − − − − − − + ⋅⋅⋅ = 1 + + + + ⋅⋅⋅ = 1.306 ⋅⋅⋅ 2 3 4 5 6 7 6 20 42
5. A power series can be inverted, provided the first-degree term is not zero. Given y = b1 x + b2 x2 + b3 x3 + b4 x4 + b5 x5 + b6 x6 + b7 x7 + ⋅⋅⋅ then
x = B1 y + B2 y2 + B3 y3 + B4 y4 + B5 y5 + B6 y6 + B7 y7 + ⋅⋅⋅
where B1 = 1/b1 B2 = −b2 /b13 B3 = (1/b15 ) (2b22 − b1b3 ) B4 = (1/b17 )(5b1b2 b3 − b12 b4 − 5b23 ) Additional coefficients are available in the references. 6. Two series may be added or subtracted term by term provided each is a convergent series. The joint sum is equal to the sum (or difference) of the individuals. 7. The sum of two divergent series can be convergent. Similarly, the sum of a convergent series and a divergent series must be divergent. Example Given 1+n
∞
= + + + + ⋅⋅⋅ n 1 4 9 16
n=1
2
3
1−n
∞
5
4
(a divergent series)
2
= − − − + ⋅⋅⋅ n 4 9 16
n=1
1
2
3
(a divergent series)
2
1+n
1−n
1+n+1−n
+ = n n n
However,
2
2
2
1 = 2 2 n
(convergent)
8. A power series may be integrated term by term to represent the integral of the function within an interval of the region of convergence. If f(x) = a0 + a1x + a2x2 + ⋅⋅⋅, then
x2
x1
f(x) dx =
x2
x1
a0 dx +
x2
x1
a1x dx +
x2
a2 x2 dx + ⋅⋅⋅
x1
9. A power series may be differentiated term by term and represents the function df(x)/dx within the same region of convergence as f(x). TESTS FOR CONVERGENCE AND DIVERGENCE
1 = S2n 2 where S2n is the sum of the first 2n terms of the original series. Thus 1 lim t3n = lim S2n n→∞ n→∞ 2 1 t=S 2 and since lim t3n + 2 = lim t3n + 1 = lim t3n, it follows the sum of the series t is (a) S. Hence a rearrangement of the terms of an alternating series alters the sum of the series.
In general, the problem of determining whether a given series will converge or not can require a great deal of ingenuity and resourcefulness. There is no all-inclusive test which can be applied to all series. As the only alternative, it is necessary to apply one or more of the developed theorems in an attempt to ascertain the convergence or divergence of the series under study. The following defined tests are given in relative order of effectiveness. For examples, see references on advanced calculus. 1. Comparison Test. A series will converge if the absolute value of each term (with or without a finite number of terms) is less than the corresponding term of a known convergent series. Similarly, a positive series is divergent if it is termwise larger than a known divergent series of positive terms.
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MATHEMATICS
2. nth-Term Test. A series is divergent if the nth term of the series does not approach zero as n becomes increasingly large. 3. Ratio Test. If the absolute ratio of the (n + 1) term divided by the nth term as n becomes unbounded approaches a. A number less than 1, the series is absolutely convergent b. A number greater than 1, the series is divergent c. A number equal to 1, the test is inconclusive 4. Alternating-Series Leibniz Test. If the terms of a series are alternately positive and negative and never increase in value, the absolute series will converge, provided that the terms tend to zero as a limit. 5. Cauchy’s Root Test. If the nth root of the absolute value of the nth term, as n becomes unbounded, approaches a. A number less than 1, the series is absolutely convergent b. A number greater than 1, the series is divergent c. A number equal to 1, the test is inconclusive 6. Maclaurin’s Integral Test. Suppose an is a series of positive terms and f is a continuous decreasing function such that f(x) ≥ 0 for 1∞ ≤ x < ∞ and f(n) = an. Then the series and the improper integral ∫ 1 f(x) dx either both converge or both diverge. SERIES SUMMATION AND IDENTITIES Sums for the First n Numbers to Integer Powers n
j=1 n
j=1 n
j=1 n
j=1
n(n + 1) j = = 1 + 2 + 3 + 4 + ⋅⋅⋅ + n 2 n(n + 1)(2n + 1) j 2 = = 12 + 22 + 32 + 42 + ⋅⋅⋅ + n2 6
Taylor’s Series x2 x3 f(x + h) = f(h) + xf ′(h) + f ″(h) + f ′′′(h) + ⋅⋅⋅ 2! 3! f ″(x0) f ′′′(x0) or f(x) = f(x0) + f ′(x0) (x − x0) + (x − x0)2 + (x − x0)3 + ⋅⋅⋅ 2! 3! Example Find a series expansion for f(x) = ln (1 + x) about x0 = 0. f′(x) = (1 + x)−1, thus
f(0) = 0,
f″(x) = −(1 + x)−2,
f′(0) = 1,
f ″(0) = −1,
which converges for −1 < x ≤ 1.
Maclaurin’s Series x2 x3 f(x) = f(0) + xf ′(0) + f ″(0) + f ′′′(0) + ⋅⋅⋅ 2! 3! This is simply a special case of Taylor’s series when h is set to zero. Exponential Series x2 x3 xn ex = 1 + x + + + ⋅⋅⋅ + + ⋅⋅⋅ − ∞ < x < ∞ 2! 3! n! Logarithmic Series
n2(n + 1)2 j = = 13 + 23 + 33 + ⋅⋅⋅ + n3 4
x−1 1 x−1 ln x = + x 2 x
n(n + 1)(2n + 1)(3n2 + 3n − 1) j 4 = = 14 + 24 + 34 + ⋅⋅⋅ + n4 30
ln x = 2
n
[a + (k − 1)d] = a + (a + d ) + (a + 2d )
+ (a + 3d ) + ⋅⋅⋅ + [a + (n − 1)]d
k=1
ar
j−1
3
1
x−1 + + ⋅⋅⋅
x + 1 3 x + 1 3
1
(x > a)
(x > 0)
−∞ < x < ∞ −∞ < x < ∞
x3 1 3 x5 1 3 5 x7 sin−1 x = x + + ⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅⋅⋅ 6 2 4 5 2 4 6 7
= a + ar + ar 2 + ar 3 + ⋅⋅⋅ + ar n − 1 1 − rn =a 1−r
2
x2 x4 x 6 cos x = 1 − + − + ⋅⋅⋅ 2! 4! 6!
Geometric Progression j=1
x−1
x−1 + ⋅⋅⋅ + 3 x
Trigonometric Series* x3 x5 x7 sin x = x − + − + ⋅⋅⋅ 3! 5! 7!
1 = na + n(n − 1)d 2
n
f ′′′(1) = 2, etc.
x2 x3 x4 xn ln (x + 1) = x − + − + ⋅⋅⋅ + (−1)n + 1 + ⋅⋅⋅ 2 3 4 n
3
Arithmetic Progression
f′′′(x) = 2(1 + x)−3, etc.
r≠1
1 1 1 tan−1 x = x − x3 + x5 − x7 + ⋅⋅⋅ 3 5 7
∂f f(x, y) = f(x0, y0) + ∂x
n
1 1 1 1 1 1 1 + + + + ⋅⋅⋅ + = a + a + d a + 2d a + 3d a + 4d a + nd k = 0 a + kd
1 ∂2f + 2 2! ∂x
The reciprocals of the terms of the arithmetic-progression series are called harmonic progression. No general summation formulas are available for this series. n(n − 1) (x + y)n = xn + nx n − 1y + x n − 2 y2 2! n(n − 1)(n − 2) n − 3 3 n! + x y + ⋅⋅⋅ + x n − ryr + ⋅⋅⋅ + yn 3! (n − r)!r! n(n − 1) n(n − 1)(n − 2) (1 x)n = 1 nx + x2 x3 + ⋅⋅⋅ (x2 < 1) 2! 3!
(x2 < 1)
Taylor Series The Taylor series for a function of two variables, expanded about the point (x0, y0), is
Harmonic Progression
Binomial Series
(x2 < 1)
x 0, y0
∂f (x − x0) + ∂y
∂2f (x − x0)2 + 2 x 0, y0 ∂x∂y
(y − y0) x 0, y0
(x − x0)(y − y0) x 0 , y0
∂2f + 2 ∂y
(y − y0)2 + ⋅⋅⋅ x 0, y0
Partial Sums of Infinite Series, and How They Grow Calculus textbooks devote much space to tests for convergence and divergence of series that are of little practical value, since a convergent
* tan x series has awkward coefficients and should be computed as sin x (sign) . 2 −sin x 1
COMPLEX VARIABLES series either converges rapidly, in which case almost any test (among those presented in the preceding subsections) will do; or it converges slowly, in which case it is not going to be of much use unless there is
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some way to get at its sum without adding up an unreasonable number of terms. To find out, as accurately as possible, how fast a convergent series converges and how fast a divergent series diverges, see Ref. 34.
COMPLEX VARIABLES REFERENCES: General. 73, 163, 172, 179. Applied and computational complex analysis. 141, 146, 179.
Numbers of the form z = x + iy, where x and y are real, i = −1, are called complex numbers. The numbers z = x + iy are representable in the plane as shown in Fig. 3-46. The following definitions and terminology are used: 1. Distance OP = r = modulus of z written |z|. |z| = x2 + y2. 2. x is the real part of z. 3. y is the imaginary part of z. 4. The angle θ, 0 ≤ θ < 2π, measured counterclockwise from the positive x axis to OP is the argument of z. θ = arctan y/x = arcsin y/r = arccos x/r if x ≠ 0, θ = π/2 if x = 0 and y > 0. 5. The numbers r, θ are the polar coordinates of z. 6. z = x − iy is the complex conjugate of z. 2
ALGEBRA Let z1 = x1 + iy1, z2 = x2 + iy2. Equality z1 = z2 if and only if x1 = x2 and y1 = y2. Addition z1 + z2 = (x1 + x2) + i(y1 + y2). Subtraction z1 − z2 = (x1 − x2) + i(y1 − y2). Multiplication z1 ⋅ z2 = (x1x2 − y1y2) + i(x1y2 + x2y1). x2y1 − x1y2 x1x2 + y1y2 Division z1 /z2 = + i , z2 ≠ 0. x 22 + y22 x 22 + y22 SPECIAL OPERATIONS zz = x2 + y2 = |z|2; z1z2 = z1 z2; z1 = z1; z1z2 = z1z2; |z1 ⋅ z2| = |z1| ⋅ |z2|; arg (z1 ⋅ z2) = arg z1 + arg z2; arg (z1 /z2) = arg z1 − arg z2; i4n = 1 for n any integer; i2n = −1 where n is any odd integer; z + z = 2x; z − z = 2iy. Every complex quantity can be expressed in the form x + iy. TRIGONOMETRIC REPRESENTATION By referring to Fig. 3-46, there results x = r cos θ, y = r sin θ so that z = x + iy = r (cos θ + i sin θ), which is called the polar form of the complex number. cos θ + i sin θ = e iθ. Hence z = x + iy = re iθ. z = x − iy = re−iθ. Two important results from this are cos θ = (eiθ + e−iθ)/2 and sin θ = (eiθ − e−iθ)/2i. Let z1 = r1e iθ1, z2 = r2e iθ2. This form is convenient for multiplication for z1z2 = r1 r2 e i(θ1 + θ2) and for division for z1 /z2 = (r1 /r2)ei(θ1 − θ2), z2 ≠ 0. POWERS AND ROOTS If n is a positive integer, z = (re ) = r e If n is a positive integer, n
FIG. 3-46
Complex plane.
iθ n
n inθ
= r (cos nθ + i sin nθ). n
θ + 2kπ θ + 2kπ z1/ n = r 1/ nei[(θ + 2kπ)/n] = r 1/n cos + i sin n n
and selecting values of k = 0, 1, 2, 3, . . . , n − 1 give the n distinct values of z1/n. The n roots of a complex quantity are uniformly spaced around a circle, with radius r 1/n, in the complex plane in a symmetric fashion. Example Find the three cube roots of −8. Here r = 8, θ = π. The roots are z0 = 2(cos π/3 + i sin π/3) = 1 + i 3, z1 = 2(cos π + i sin π) = −2, z2 = 2(cos 5π/3 + i sin 5π/3) = 1 − i 3. ELEMENTARY COMPLEX FUNCTIONS Polynomials A polynomial in z, anzn + an − 1zn − 1 + ⋅⋅⋅ + a0, where n is a positive integer, is simply a sum of complex numbers times integral powers of z which have already been defined. Every polynomial of degree n has precisely n complex roots provided each multiple root of multiplicity m is counted m times. Exponential Functions The exponential function ez is defined by the equation ez = ex + iy = ex ⋅ eiy = ez(cos y + i sin y). Properties: e0 = 1; ez1 ⋅ ez2 = ez1 + z2; ez1/ez2 = ez1 − z2; ez + 2kπi = ez. Trigonometric Functions sin z = (eiz − e−iz)/2i; cos z = (eiz + e−iz)/2; tan z = sin z/cos z; cot z = cos z/sin z; sec z = 1/cos z; csc z = 1/sin z. Fundamental identities for these functions are the same as their real counterparts. Thus cos2 z + sin2 z = 1, cos (z1 z2) = cos z1 cos z2 sin z1 sin z2, sin (z1 z2) = sin z1 cos z2 cos z1 sin z2. The sine and cosine of z are periodic functions of period 2π; thus sin (z + 2π) = sin z. For computation purposes sin z = sin (x + iy) = sin x cosh y + i cos x sinh y, where sin x, cosh y, etc., are the real trigonometric and hyperbolic functions. Similarly, cos z = cos x cosh y − i sin x sinh y. If x = 0 in the results given, cos iy = cosh y, sin iy = i sinh y. Example Find all solutions of sin z = 3. From previous data sin z = sin x cosh y + i cos x sinh y = 3. Equating real and imaginary parts sin x cosh y = 3, cos x sinh y = 0. The second equation can hold for y = 0 or for x = π/2, 3π/2, . . . . If y = 0, cosh 0 = 1 and sin x = 3 is impossible for real x. Therefore, x = π/2, 3π/2, . . . (2n + 1)π/2, n = 0, 1, 2, . . . . However, sin 3π/2 = −1 and cosh y ≥ 1. Hence x = π/2, 5π/2, . . . . The solution is z = [(4n + 1)π]/2 + i cosh−13, n = 0, 1, 2, 3, . . . . Example Find all solutions of ez = −i. ez = ex(cos y + i sin y) = −i. Equating real and imaginary parts gives ex cos y = 0, ex sin y = −1. From the first y = π/2, 3π/2, . . . . But ex > 0. Therefore, y = 3π/2, 7π/2, −π/2, . . . . Then x = 0. The solution is z = i[(4n + 3)π]/2. Two important facets of these functions should be recognized. First, the sin z is unbounded; and, second, ez takes all complex values except 0. Hyperbolic Functions sinh z = (ez − e−z)/2; cosh z = (ez + e−z)/2; tanh z = sinh z/cosh z; coth z = cosh z/sinh z; csch z = 1/sinh z; sech z = 1/cosh z. Identities are: cosh2 z − sinh2 z = 1; sinh (z1 + z2) = sinh z1 cosh z2 + cosh z1 sinh z2; cosh (z1 + z2) = cosh z1 cosh z2 + sinh z1 sinh z2; cosh z + sinh z = ez; cosh z − sinh z = e−z. The hyperbolic sine and hyperbolic cosine are periodic functions with the imaginary period 2πi. That is, sinh (z + 2πi) = sinh z. Logarithms The logarithm of z, log z = log |z| + i(θ + 2nπ), where log |z| is taken to the base e and θ is the principal argument of z, that is, the particular argument lying in the interval 0 ≤ θ < 2π. The logarithm of z is infinitely many valued. If n = 0, the resulting logarithm is called the principal value. The familiar laws log z1 z2 = log z1 + log z2, log z1 /z2 = log z1 − log z2, log zn = n log z hold for the principal value.
3-34
MATHEMATICS
π4
Example log (1 + i) = log 2 + i + 2nπ . General powers of z are defined by zα = eα log z. Since log z is infinitely many valued, so too is zα unless α is a rational number. DeMoivre’s formula can be derived from properties of ez. zn = rn (cos θ + i sin θ)n = rn (cos nθ + i sin nθ) (cos θ + i sin θ)n = cos nθ + i sin nθ
Thus
Example ii = eilog i = ei[log|i| + i(π/2 + 2nπ)] = e−(π/2 + 2nπ). Thus i i is real with principal value (n = 0) = e−π/2. Example (2)1 + i = e(1 + i) log 2 = elog 2. ei log 2 = 2. (cos log 2 + i sin log 2) = 2[cos (0.3466) + i sin (0.3466)]. Inverse Trigonmetric Functions cos−1 z = −i log (z z2 −1); i + z i −1 2 −1 sin z = −i log (iz 1 −z); tan z = log . These functions 2 i−z are infinitely many valued. Inverse Hyperbolic Functions cosh−1 z = log (z z2−1); 1+z 1 z2 +1); tanh−1 z = log . sinh−1 z = log (z 2 1−z
COMPLEX FUNCTIONS (ANALYTIC) In the real-number system a greater than b(a > b) and b less than c(b < c) define an order relation. These relations have no meaning for complex numbers. The absolute value is used for ordering. Some important relations follow: |z| ≥ x; |z| ≥ y; |z1 z2| ≤ |z1| + |z2|; |z1 − z2| ≥ ||z1| − |z2||; |z| ≥ (|x| + |y|)/2. Parts of the complex plane, commonly called regions or domains, are described by using inequalities. Example |z − 3| ≤ 5. This is equivalent to (x −3 )2 +y2 ≤ 5, which is the set of all points within and on the circle, centered at x = 3, y = 0 of radius 5.
Example |z − 1| ≤ x represents the set of all points inside and on the parabola 2x = y2 + 1 or, equivalently, 2x ≥ y2 + 1. Functions of a Complex Variable If z = x + iy, w = u + iv and if for each value of z in some region of the complex plane one or more values of w are defined, then w is said to be a function of z, w = f(z). Some of these functions have already been discussed, e.g., sin z, log z. All functions are reducible to the form w = u(x, y) + iv(x, y), where u, v are real functions of the real variables x and y.
∂v/∂x = −∂u/∂y. The last two equations are called the CauchyRiemann equations. The derivative dw ∂u ∂v ∂v ∂u =+i=−i dz ∂x ∂x ∂y ∂y If f(z) possesses a derivative at zo and at every point in some neighborhood of z0, then f(z) is said to be analytic at z0. If the Cauchy-Riemann equations are satisfied and ∂u ∂u ∂v ∂v u, v, , , , ∂x ∂y ∂x ∂y are continuous in a region of the complex plane, then f(z) is analytic in that region. Example w = zz = x2 + y2. Here u = x2 + y2, v = 0. ∂u/∂x = 2x, ∂u/∂y = 2y, ∂v/∂x = ∂v/∂y = 0. These are continuous everywhere, but the Cauchy-Riemann equations hold only at the origin. Therefore, w is nowhere analytic, but it is differentiable at z = 0 only. Example w = ez = ex cos y + iex sin y. u = ex cos y, v = ex sin y. ∂u/∂x = ex cos y, ∂u/∂y = −ex sin y, ∂v/∂x = ex sin y, ∂v/∂y = ex cos y. The continuity and CauchyRiemann requirements are satisfied for all finite z. Hence ez is analytic (except at ∞) and dw/∂z = ∂u/∂x + i(∂v/∂x) = ez. 1 z
x − iy x +y
x x +y
y x +y
Example w = = = −i 2 2 2 2 2 2 It is easy to see that dw/dz exists except at z = 0. Thus 1/z is analytic except at z = 0.
Singular Points If f(z) is analytic in a region except at certain points, those points are called singular points. Example 1/z has a singular point at zero. Example tan z has singular points at z = (2n + 1)(π/2), n = 0, 1, 2, . . . . The derivatives of the common functions, given earlier, are the same as their real counterparts. Example (d/dz)(log z) = 1/z, (d/dz)(sin z) = cos z. Harmonic Functions Both the real and the imaginary parts of any analytic function f = u + iv satisfy Laplace’s equation ∂2φ/∂x2 + ∂2φ/∂y2 = 0. A function which possesses continuous second partial derivatives and satisfies Laplace’s equation is called a harmonic function.
Example z3 = (x + iy)3 = x3 + 3x2(iy) + 3x(iy)2 + (iy)3 = (x3 − 3xy2) +
Example ez = ex cos y + iex sin y. u = ex cos y, ∂u/∂x = ex cos y, ∂2u/∂x2 = ex cos y, ∂u/∂y = −ex sin y, ∂2u/∂y2 = −ex cos y. Clearly ∂2u/∂x2 + ∂2u/∂y2 = 0. Similarly, v = ex sin y is also harmonic.
Example cos z = cos x cosh y − i sin x sinh y.
If w = u + iv is analytic, the curves u(x, y) = c and v(x, y) = k intersect at right angles, if wi(z) ≠ 0.
i(3x2y − y3).
Differentiation The derivative of w = f(z) is dw f(z + ∆z) − f(z) lim = ∆z→0 dz ∆z and for the derivative to exist the limit must be the same no matter how ∆z approaches zero. If w1, w2 are differentiable functions of z, the following rules apply: dw2 d(w1w2) dw1 dw2 d(w1 w2) dw1 = w2 + w1 = dz dz dz dz dz dz d(w1/w2) w2(dw1/dz) − w1(dw2/dz) = dz w22 dw1n dw1 = nw1n − 1 dz dz For w = f(z) to be differentiable, it is necessary that ∂u/∂x = ∂v/∂y and and
Example z3 = (x3 − 3xy2) + i(3x2y − y3). Set u = x3 − 3xy2 = c, v = 3x2y − y3 = k. By implicit differentiation there results, respectively, dy/dx = (x2 − y2)/2xy, dy/dx = 2xy/(y2 − x2), which are clearly negative reciprocals, the condition for perpendicularity. Integration In much of the work with complex variables a simple extension of integration called line or curvilinear integration is of fundamental importance. Since any complex line integral can be expressed in terms of real line integrals, we define only real line integrals. Let F(x,y) be a real, continuous function of x and y and c be any continuous curve of finite length joining the points A and B (Fig. 3-47). F(x,y) is not related to the curve c. Divide c up into n segments, ∆si, whose projection on the x axis is ∆xi and on the y axis is ∆yi. Let (εi, ηi) be the coordinates of an arbitrary point on ∆si. The limits of the sums
F(ε , η ) ∆s = F(x, y) ds n
lim
∆si→0
i
i=1
i
i
c
DIFFERENTIAL EQUATIONS
Conformal transformation.
FIG. 3-48
Line integral.
FIG. 3-47
grals are the same as those for ordinary integrals. That is, ∫c [ f(z) g(z)] dz = ∫c f(z) dz ∫c g(z) dz; ∫c kf(z) dz = k ∫c f(z) dz for any constant k, etc.
F(ε , η ) ∆x = F(x, y) dx n
lim
∆si→0
i
i
i
c
i=1
F(ε , η ) ∆ y = F(x, y) dy
lim
∆si→0
i
i
c
2
are known as line integrals. Much of the initial strangeness of these integrals will vanish if it be observed that the ordinary definite integral b ∫a f(x) dx is just a line integral in which the curve c is a line segment on the x axis and F(x, y) is a function of x alone. The evaluation of line integrals can be reduced to evaluation of ordinary integrals. Example ∫c y(1 + x) dy, where c: y = 1 − x21 from (−1, 0) to (1, 0). Clearly y = 1 − x2, dy = −2x dx. Thus ∫c y(1 + x) dy = −2 ∫ −1 (1 − x2)(1 + x)x dx = −8⁄15. Example ∫c x2y ds, c is the square whose vertices are (0, 0), (1, 0), (1, 1), x2 + dy2. When dx = 0, ds = dy. From (0, 0) to (1, 0), y = 0, dy = (0, 1). ds = d 0. Similar arguments for the other sides give
x y ds = 0.x dx + y dy + x dx + 0.y dy = a − s = ⁄ 1
2
1
0
2
0
0
2
0
16
1
2
c
i
i=1
(x + iy) dz along c: y = x, 0 to 1 + i. This becomes (x + iy) dz = (x dx − y dy) + i (y dx + x dy) = x dx − x dx + i x dx + i x dx = − ⁄
Example
n
c
3-35
1
Let f(z) be any function of z, analytic or not, and c any curve as above. The complex integral is calculated as ∫c f(z) dz = ∫c (u dx − v dy) + i ∫c (v dx + u dy), where f(z) = u(x, y) + iv(x, y). Properties of line inte-
2
c
c
1
2
c
1
1
1
2
0
2
0
0
16
+ 5i/6
0
Conformal Mapping Every function of a complex variable w = f(z) = u(x, y) + iv(x, y) transforms the x, y plane into the u, v plane in some manner. A conformal transformation is one in which angles between curves are preserved in magnitude and sense. Every analytic function, except at those points where f ′(z) = 0, is a conformal transformation. See Fig. 3-48. Example w = z2. u + iv = (x2 − y2) + 2ixy or u = x2 − y2, v = 2xy. These are the transformation equations between the (x, y) and (u, v) planes. Lines parallel to the x axis, y = c1 map into curves in the u, v plane with parametric equations u = x2 − c12, v = 2c1x. Eliminating x, u = (v2/4c12) − c12, which represents a family of parabolas with the origin of the w plane as focus, the line v = 0 as axis and opening to the right. Similar arguments apply to x = c2. The principles of complex variables are useful in the solution of a variety of applied problems. See the references for additional information.
DIFFERENTIAL EQUATIONS REFERENCES: Ordinary Differential Equations: Elementary level, 41, 44, 62, 81, 204, 236, 263. Intermediate level, 30, 43, 144. Theory and Advanced topics, 252. Applications, 9, 263. Partial Differential Equations: Elementary level and solution methods, 9, 41, 61, 72, 144, 156, 229. Theory and advanced level, 79, 220, 240. See also “Numerical Analysis and Approximate Methods” and “General References: References for General and Specific Topics—Advanced Engineering Mathematics” for additional references on topics in ordinary and partial differential equations.
The natural laws in any scientific or technological field are not regarded as precise and definitive until they have been expressed in mathematical form. Such a form, often an equation, is a relation between the quantity of interest, say, product yield, and independent variables such as time and temperature upon which yield depends. When it happens that this equation involves, besides the function itself, one or more of its derivatives it is called a differential equation. Example The homogeneous bimolecular reaction A + B → C is characterized by the differential equation dx/dt = k(a − x)(b − x), where a = initial concentration of A, b = initial concentration of B, and x = x(t) = concentration of C as a function of time t. k
Example The differential equation of heat conduction in a moving fluid with velocity components vx, vy is ∂u ∂u ∂u K ∂2u ∂2u + vx + vy = 2 + 2 ∂t ∂x ∂y ρcp ∂x ∂y
where u = u(x, y, t) = temperature, K = thermal conductivity, ρ = density, and cp = specific heat at constant pressure.
ORDINARY DIFFERENTIAL EQUATIONS When the function involved in the equation depends upon only one variable, its derivatives are ordinary derivatives and the differential equation is called an ordinary differential equation. When the function depends upon several independent variables, then the equation is called a partial differential equation. The theories of ordinary and partial differential equations are quite different. In almost every respect the latter is more difficult. Whichever the type, a differential equation is said to be of nth order if it involves derivatives of order n but no higher. The equation in the first example is of first order and that in the second example of second order. The degree of a differential equation is the power to which the derivative of the highest order is raised after the equation has been cleared of fractions and radicals in the dependent variable and its derivatives. A relation between the variables, involving no derivatives, is called a solution of the differential equation if this relation, when substituted in the equation, satisfies the equation. A solution of an ordinary differential equation which includes the maximum possible number of “arbitrary” constants is called the general solution. The maximum number of “arbitrary” constants is exactly equal to the order of the dif-
3-36
MATHEMATICS
ferential equation. If any set of specific values of the constants is chosen, the result is called a particular solution.
are distinct real roots, r1 and r2, say, the solution is y = Aer1x + Ber2x, where A and B are arbitrary constants.
Example The general solution of (d2x/dt2) + k2x = 0 is x = A cos kt + B sin kt, where A, B are arbitrary constants. A particular solution is x = a cos kt + 3 sin kt.
Example y″ + 4y′ + 3 = 0. The characteristic equation is m2 + 4m + 3 = 0. The roots are −3 and −1, and the general solution is y = Ae−3x + Be−x.
In the case of some equations still other solutions exist called singular solutions. A singular solution is any solution of the differential equation which is not included in the general solution. Example y = x(dy/dx) − d(dy/dx) has the general solution y = cx − dc , where c is an arbitrary constant; y = x2 is a singular solution, as is easily verified. 2
2
ORDINARY DIFFERENTIAL EQUATIONS OF THE FIRST ORDER Equations with Separable Variables Every differential equation of the first order and of the first degree can be written in the form M(x, y) dx + N(x, y) dy = 0. If the equation can be transformed so that M does not involve y and N does not involve x, then the variables are said to be separated. The solution can then be obtained by quadrature, which means that y = ∫ f(x) dx + c, which may or may not be expressible in simpler form. Example Two liquids A and B are boiling together in a vessel. Experimentally it is found that the ratio of the rates at which A and B are evaporating at any time is proportional to the ratio of the amount of A (say, x) to the amount of B (say, y) still in the liquid state. This physical law is expressible as (dy/dt)/(dx/dt) = ky/x or dy/dx = ky/x, where k is a proportionality constant. This equation may be written dy/y = k(dx/x), in which the variables are separated. The solution is ln y = k ln x + ln c or y = cxk. Exact Equations The equation M(x, y) dx + N(x, y) dy = 0 is exact if and only if ∂M/∂y = ∂N/∂x. In this case there exists a function w = f(x, y) such that ∂f /∂x = M, ∂f /∂y = N, and f(x, y) = C is the required solution. f(x, y) is found as follows: treat y as though it were constant and evaluate ∫ M(x, y) dx. Then treat x as though it were constant and evaluate ∫ N(x, y) dy. The sum of all unlike terms in these two integrals (including no repetitions) is f(x, y). Example (2xy − cos x) dx + (x2 − 1) dy = 0 is exact for ∂M/∂y = 2x, ∂N/∂x = 2x. ∫ M dx = ∫ (2xy − cos x) dx = x2y − sin x, ∫ N dy = ∫ (x2 − 1) dy = x2y − y. The solution is x2y − sin x − y = C, as may easily be verified. Linear Equations A differential equation is said to be linear when it is of first degree in the dependent variable and its derivatives. The general linear first-order differential equation has the form dy/dx + P(x)y = Q(x). Its general solution is y = e−∫ P dx
Qe
∫ P dx
dx + C
Example A tank initially holds 200 gal of a salt solution in which 100 lb is dissolved. Six gallons of brine containing 4 lb of salt run into the tank per minute. If mixing is perfect and the output rate is 4 gal/min, what is the amount A of salt in the tank at time t? The differential equation of A is dA/dt + [1/(100 + t)]A = 4. Its general solution is A = 2(100 + t) + C/(100 + t). At t = 0, ′A = 100; so the particular solution is A = 2(100 + t) − 104/(100 + t). ORDINARY DIFFERENTIAL EQUATIONS OF HIGHER ORDER The higher-order differential equations, especially those of order 2, are of great importance because of physical situations describable by them. Equation y(n) = f(x) Such a differential equation can be solved by n integrations. The solution will contain n arbitrary constants. Linear Differential Equations with Constant Coefficients and Right-Hand Member Zero (Homogeneous) The solution of y″ + ay′ + by = 0 depends upon the nature of the roots of the characteristic equation m2 + am + b = 0 obtained by substituting the trial solution y = emx in the equation. Distinct Real Roots If the roots of the characteristic equation
Multiple Real Roots If r1 = r2, the solution of the differential equation is y = e r1x(A + Bx). Example y″ + 4y + 4 = 0. The characteristic equation is m2 + 4m + 4 = 0 with roots −2 and −2. The solution is y = e−2x(A + Bx). Complex Roots If the characteristic roots are p iq, then the solution is y = epx(A cos qx + B sin qx). Example The differential equation My″ + Ay′ + ky = 0 represents the vibration of a linear system of mass M, spring constant k, and damping constant , the roots of the characteristic equation A. If A < 2 kM A Mm2 + Am + k = 0 are complex − i 2M and the solution is y = e−(At/2M)
k A − M 2M
2
A − t + c sin − t c cos M 2M M 2M k
A
1
2
k
2
2
This solution is oscillatory, representing undercritical damping.
All these results generalize to homogeneous linear differential equations with constant coefficients of order higher than 2. These equations (especially of order 2) have been much used because of the ease of solution. Oscillations, electric circuits, diffusion processes, and heat-flow problems are a few examples for which such equations are useful. Second-Order Equations: Dependent Variable Missing Such an equation is of the form dy d 2y F x, , 2 = 0 dx dx It can be reduced to a first-order equation by substituting p = dy/dx and dp/dx = d 2y/dx2. Second-Order Equations: Independent Variable Missing Such an equation is of the form dy d 2y F y, , 2 = 0 dx dx
dy d 2y dp = p, 2 = p dx dx dy The result is a first-order equation in p, dp F y, p, p = 0 dy Set
Example The capillary curve for one vertical plate is given by
dx
d 2y 4y 1+ 2 = dx c2 Its solution by this technique is c 2 c2 −y2 − c2 − h x + 0 = 2 where c, h0 are physical constants.
dy
2 3/2
cosh
−1
c c − cosh−1 y h0
Example The equation governing chemical reaction in a porous catalyst in plane geometry of thickness L is dc d 2c D 2 = k f(c), (0) = 0, c(L) = c0 dx dx where D is a diffusion coefficient, k is a reaction rate parameter, c is the concentration, k f(c) is the rate of reaction, and c0 is the concentration at the boundary. Making the substitution gives k dp p = f(c) dc D
DIFFERENTIAL EQUATIONS
c p2 k f(c) dc = 2 D c(0) If the reaction is very fast, c(0) ≈ 0 and the average reaction rate is related to p(L). See Ref. 106. This variable is given by 1/2 2k c0 p(L) = f(c) dc D 0 Thus, the average reaction rate can be calculated without solving the complete problem.
Integrating gives
Linear Nonhomogeneous Differential Equations Linear Differential Equations Right-Hand Member f(x) ≠ 0 Again the specific remarks for y″ + ay′ + by = f(x) apply to differential equations of similar type but higher order. We shall discuss two general methods. Method of Undetermined Coefficients Use of this method is limited to equations exhibiting both constant coefficients and particular forms of the function f(x). In most cases f(x) will be a sum or product of functions of the type constant, xn (n a positive integer), emx, cos kx, sin kx. When this is the case, the solution of the equation is y = H(x) + P(x), where H(x) is a solution of the homogeneous equations found by the method of the preceding subsection and P(x) is a particular integral found by using the following table subject to these conditions: (1) When f(x) consists of the sum of several terms, the appropriate form of P(x) is the sum of the particular integrals corresponding to these terms individually. (2) When a term in any of the trial integrals listed is already a part of the homogeneous solution, the indicated form of the particular integral is multiplied by x. Form of Particular Integral
If f(x) is
Then P(x) is
a (constant) axn aerx c cos kx d sin kx
A (constant) An xn + An − 1 xn − 1 + ⋅⋅⋅A1 x + A0 Berx
gxnerx cos kx hxnerx sin kx
A cos kx + B sin kx
(Anxn + ⋅⋅⋅ + A0)erx cos kx + (Bn xn + ⋅⋅⋅ + B0)erx sin kx
d 2y
1 dy
dx
x dx
Example (1 − x2) 2 − = x. The homogeneous equation d 2y 1 dy (1 − x2) 2 − = 0 x dx dx dp dx = p x(1 − x2) when we set dy/dx = p. Upon integrating twice, y = c1 x2 − 1 + c2 is the homogeneous solution. Now assume that the particular solution has the form y = u x2 −1 + v. The equations for u and v become reduces to
u′ = du/dx = x2 −1 dv 2 v′ = = 1 − x dx so that 1 x2 −1 − ln (x + x2 −1)] and v = x − x3/3. u = [x 2 The complete solution is x x3 1 x2 −1 + c2 + − − x2 −1 ln (x + x2 −1). y = c1 2 6 2
Perturbation Methods If the ordinary differential equation has a parameter that is small and is not multiplying the highest derivative, perturbation methods can give solutions for small values of the parameter. Example Consider the differential equation for reaction and diffusion in a catalyst; the reaction is second order: c″ = ac2, c′(0) = 0, c(1) = 1. The solution is expanded in the following Taylor series in a. c(x, a) = c0(x) + ac1(x) + a2c2(x) + … The goal is to find equations governing the functions {ci(x)} and solve them. Substitution into the equations gives the following equations: c0″(x) + a c″1(x) + a2c″2(x) + … = a[c0(x) + ac1(x) + a2c2(x) + …]2 c′0(0) + ac′1(0) + a2c′2(0) + … = 0 c0(1) + ac1(1) + a2c2(1) + … = 1 Like terms in powers of a are collected to form the individual problems. c″0 = 0,
Since the form of the particular integral is known, the constants may be evaluated by substitution in the differential equation. Example y″ + 2y′ + y = 3e − cos x + x . The characteristic equation is (m + 1)2 = 0 so that the homogeneous solution is y = (c1 + c2x)e−x. To find a particular solution we use the trial solution from the table, y = a1e2x + a2 cos x + a3 sin x + a4x3 + a5x2 + a6x + a7. Substituting this in the differential equation collecting and equating like terms, there results a1 = s, a2 = 0, a3 = −a, a4 = 1, a5 = −6, a6 = 18, and a7 = −24. The solution is y = (c1 + c2x)e−x + se2x − a sin x + x3 − 6x2 + 18x − 24. 2x
3
Method of Variation of Parameters This method is applicable to any linear equation. The technique is developed for a second-order equation but immediately extends to higher order. Let the equation be y″ + a(x)y′ + b(x)y = R(x) and let the solution of the homogeneous equation, found by some method, be y = c1 f1(x) + c2 f2(x). It is now assumed that a particular integral of the differential equation is of the form P(x) = uf1 + vf2 where u, v are functions of x to be determined by two equations. One equation results from the requirement that uf1 + vf2 satisfy the differential equation, and the other is a degree of freedom open to the analyst. The best choice proves to be u′f1 + v′f2 = 0
and
u′f 1′ + v′f′2 = R(x)
f2 du Then u′ = = − R(x) dx f1 f 2′ − f2 f 1′ f1 dv v′ = = R(x) dx f1 f 2′ − f2 f 1′ and since f1, f2, and R are known u, v may be found by direct integration.
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c″1 = c 02,
c′0(0) = 0, c′1(0) = 0,
c″2 = 2c0 c1,
c0(1) = 1 c1(1) = 0
c′2(0) = 0,
c2(1) = 0
The solution proceeds in turn. c0(x) = 1,
(x2 − 1) c1(x) = , 2
5 − 6x2 + x4 c2(x) = 12
SPECIAL DIFFERENTIAL EQUATIONS (SEE REF. 1) Euler’s Equation The linear equation x ny(n) + a1x n − 1y(n − 1) + ⋅⋅⋅ + an − 1xy′ + any = R(x) can be reduced to a linear equation with constant coefficients by the change of variable x = et. To solve the homogeneous equation substitute y = x r into it, cancel the powers of x, which are the same for all terms, and solve the resulting polynomial for r. In case of multiple or complex roots there results the form y = x r(log x)r and y = x α[cos (β log x) + i sin (β log x)]. Example Solve x2y″ − 2y = 0. By setting y = xr, xr[r(r − 1) − 2] = 0. The roots of r 2 − r − 2 = 0 are r = 2, −1. The general solution is y = Ax2 + B/x. The equation (ax + b)ny(n) + a1(ax + b)n − 1y(n − 1) + ⋅⋅⋅ + an y = R(x) can be reduced to the Euler form by the substitution ax + b = z. It may be treated without change of variable, the homogeneous equation having solutions of the form y = (ax + b)r. Bessel’s Equation The linear equation x2(d 2y/dx2) + (1 − 2α) x(dy/dx) + [β 2 γ 2 x 2γ + (α2 − p2γ 2)]y = 0 is the general Bessel equation. By series methods, not to be discussed here, this equation can be shown to have the solution y = AxαJp(βxγ) + BxαJ−p(βxγ)
p not an integer or zero
y = AxαJp(βxγ) + BxαYp(βxγ)
p an integer
3-38 where
MATHEMATICS ∞
PARTIAL DIFFERENTIAL EQUATIONS
k!Γ(p + k + 1)
x Jp(x) = 2
p
The analysis of situations involving two or more independent variables frequently results in a partial differential equation.
k=0
Γ(n) = x
x J−p(x) = 2
(−1)k(x/2)2k
–p
∞
k=0
(−1)k(x/2)2k k!Γ(k + 1 − p)
∞
n − 1 −x
e dx
p not an integer
n>0
0
is the gamma function. For p an integer x p ∞ (−1)k(x/2)2k Jp(x) = 2 k = 0 k!(p + k)!
(Bessel function of the first kind of order p) [ Jp(x) cos (pπ) − J−p(x)] Yp(x) = sin (pπ) (replace right-hand side by limiting value if P is an integer or zero). The series converge for all x. Much of the importance of Bessel’s equation and Bessel functions lies in the fact that the solutions of numerous linear differential equations can be expressed in terms of them. Example d 2y/dx2 + [9x − (63/4x2)]y = 0. In general form this is x2(d 2y/dx2) + (9x3 − 63⁄ 4)y = 0. Thus α = a, γ = 3⁄ 2; β = 2, p = 8⁄ 3. The solution is (since p integer) y = Ax1/2J8/3(2x3/2) + Bx1/2J–8/3(2x3/2). Tables are available for the evaluation of many of these functions. Example The heat flow through a wedge-shaped fin is characterized by the equation x2(d 2y/dx2) + x(dy/dx) − a2xy = 0, where y = T − Tair, α is a combination of physical constants, and x = distance from fin end. By comparing this with the standard equation, there results α = 0, p = 0, γ = a, β 2 = −4a2 or β = 2ai. The solution is y = AJ0(2ai x) + BY0(2ai x). Legendre’s Equation The Legendre equation (1 − x2)y″ − 2xy′ + n(n + 1)y = 0, n ≥ 0, has the solution y = Aun(x) + Bvn(x) for n not an integer where n(n + 1) n(n − 2)(n + 1)(n + 3) un(x) = 1 − x2 + x4 2! 4! n(n − 2)(n − 4)(n + 1)(n + 3)(n + 5) 6 − x + ⋅⋅⋅ 6! (n − 1)(n + 2) 3 (n − 1)(n − 3)(n + 2)(n + 4) 5 vn(x) = x − x + x − ⋅⋅⋅⋅ 3! 5! If n is an even integer or zero, un is a polynomial in x. If n is an odd integer, then vn is a polynomial. The interval of convergence for the series is −1 < x < 1. If n is an integer, set un(x) vn(x) Pn(x) = (n even or zero), Pn = (n odd) un(1) vn(1) The polynomials Pn are the so-called Legendre polynomials, P0(x) = 1, P1(x) = x, P2(x) = a(3x2 − 1), P3(x) = a(5x3 − 3x), . . . . Laguerre’s Equation The Laguerre equation x(d 2y/dx2) + (c − x) (dy/dx) − ay = 0 is satisfied by the confluent hypergeometric function. See Refs. 1 and 173. Hermite’s Equation The Hermite equation y″ − 2xy′ + 2ny = 0 is satisfied by the Hermite polynomial of degree n, y = AHn(x) if n is a positive integer or zero. H0(x) = 1, H1(x) = 2x, H2(x) = 4x2 − 2, H3(x) = 8x3 − 12x, H4(x) = 16x4 − 48x2 + 12, Hr + 1(x) = 2xHr(x) − 2rHr − 1(x). Example y″ − 2xy′ + 6y = 0. Here n = 3; so y = AH3 = A(8x3 − 12x) is a solution.
Chebyshev’s Equation The equation (1 − x2)y″ − xy′ + n2y = 0 for n a positive integer or zero is satisfied by the nth Chebyshev polynomial y = ATn(x). T0(x) = 1, T1(x) = x, T2(x) = 2x2 − 1, T3(x) = 4x3 − 3x, T4(x) = 8x4 − 8x2 + 1; Tr + 1(x) = 2xTr(x) − Tr − 1(x). Example (1 − x2)y″ − xy′ + 36y = 0. Here n = 6. A solution is y = T6(x) = 2xT5(x) − T4(x) = 2x(2xT4 − T3) − T4 = 32x6 − 48x4 + 18x2 − 1. Further details on these special equations and others can be found in the literature.
Example The equation ∂T/∂t = K(∂2T/∂x2) represents the unsteady onedimensional conduction of heat. Example The equation for the unsteady transverse motion of a uniform beam clamped at the ends is ∂4y ρ ∂2y =0 4 + ∂x EI ∂t2 Example The expansion of a gas behind a piston is characterized by the simultaneous equations ∂u ∂u c2 ∂ρ ∂ρ ∂ρ ∂u + u + = 0 and + u + ρ = 0 ∂t ∂x ρ ∂x ∂t ∂x ∂x Example The heating of a diathermanous solid is characterized by the equation α(∂2θ/∂x2) + βe−γz = ∂θ/∂t. The partial differential equation ∂2f/∂x ∂y = 0 can be solved by two integrations yielding the solution f = g(x) + h(y), where g(x) and h(y) are arbitrary differentiable functions. This result is an example of the fact that the general solution of partial differential equations involves arbitrary functions in contrast to the solution of ordinary differential equations, which involve only arbitrary constants. A number of methods are available for finding the general solution of a partial differential equation. In most applications of partial differential equations the general solution is of limited use. In such applications the solution of a partial differential equation must satisfy both the equation and certain auxiliary conditions called initial and/or boundary conditions, which are dictated by the problem. Examples of these include those in which the wall temperature is a fixed constant T(x0) = T0, there is no diffusion across a nonpermeable wall, and the like. In ordinary differential equations these auxiliary conditions allow definite numbers to be assigned to the constants of integration. In partial differential equations the boundary conditions demand that the arbitrary functions resulting from integration assume specific forms. Except for a few cases (some first-order equations, D’Alembert’s solution of the wave equation, and others) a procedure which first determines the arbitrary functions and then specializes them to fit the boundary conditions is usually not feasible. A more fruitful attack is to determine directly a set of particular solutions and then combine them so that the boundary conditions are satisfied. The only area in which much analysis has been accomplished is for linear homogeneous partial differential equations. Such equations have the property that if f1, f2, . . . , ∞ fn, . . . are individually solutions, then the function f = i=1 fi is also a solution, provided the series converges and is differentiable up to the order (termwise) of the equation. Partial Differential Equations of Second and Higher Order Many of the applications to scientific problems fall naturally into partial differential equations of second order, although there are important exceptions in elasticity, vibration theory, and elsewhere. A second-order differential equation can be written as ∂2u ∂2u ∂2u a 2 + b + c 2 = f ∂x ∂x∂y ∂y where a, b, c, and f depend upon x, y, u, ∂u/∂x, and ∂u/∂y. This equation is hyperbolic, parabolic, or elliptic, depending on whether the discriminant b2 − 4ac is >0, =0, or <0, respectively. Since a, b, c, and f depend on the solution, the type of equation can be different at different x and y locations. If the equation is hyperbolic, discontinuities can be propagated. See Refs. 11, 79, 105, 159, and 192. Phenomena of propagation such as vibrations are characterized by equations of “hyperbolic” type which are essentially different in their properties from other classes such as those which describe equilibrium (elliptic) or unsteady diffusion and heat transfer (parabolic). Prototypes are as follows: Elliptic Laplace’s equation ∂2u/∂x2 + ∂2u/∂y2 = 0 and Poisson’s equation ∂2u/∂x2 + ∂2u/∂y2 = g(x, y) do not contain the variable time
DIFFERENTIAL EQUATIONS explicitly and consequently represent equilibrium configurations. Laplace’s equation is satisfied by static electric or magnetic potential at points free from electric charges or magnetic poles. Other important functions satisfying Laplace’s equation are the velocity potential of the irrotational motion of an incompressible fluid, used in hydrodynamics; the steady temperature at points in a homogeneous solid, and the steady state of diffusion through a homogeneous body. The gravitational potential V at points occupied by mass of density d satisfies Poisson’s equation ∂2 V/∂x2 + ∂2 V/∂y2 + ∂2 V/∂z2 = −4πd. Parabolic The heat equation ∂T/∂t = ∂ 2T/∂x2 + ∂2T/∂y2 represents nonequilibrium or unsteady states of heat conduction and diffusion. Hyperbolic The wave equation ∂2u/∂t2 = c2(∂2u/∂x2 + ∂2u/∂y2) represents wave propagation of many varied types. Quasilinear first-order differential equations are like ∂u ∂u a+b=f ∂x ∂y where a, b, and f depend on x, y, and u, with a2 + b2 ≠ 0. This equation can be solved using the method of characteristics, which writes the solution in terms of a parameter s, which defines a path for the characteristic. dx dy du = a, = b, = f ds ds ds These equations are integrated from some initial conditions. For a specified value of s, the value of x and y shows the location where the solution is u. The equation is semilinear if a and b depend just on x and y (and not u), and the equation is linear if a, b, and f all depend on x and y, but not u. Such equations give rise to shock propagation, and conditions have been derived to deduce the presence of shocks, Ref. 245. For further information, see Refs. 79, 159, 192, and 245. An example of a linear hyperbolic equation is the advection equation for flow of contaminants when the x and y velocity components are u and v, respectively. ∂c ∂c ∂c +u+v=0 ∂t ∂x ∂y The equations for flow and adsorption in a packed bed or chromatography column give a quasilinear equation. ∂c ∂c df ∂c φ + φ u + (1 − φ) = 0 ∂t ∂x dc ∂t Here n = f(c) is the relation between concentration on the adsorbent and fluid concentration. The solution of problems involving partial differential equations often revolves about an attempt to reduce the partial differential equation to one or more ordinary differential equations. The solutions of the ordinary differential equations are then combined (if possible) so that the boundary conditions as well as the original partial differential equation are simultaneously satisfied. Three of these techniques are illustrated. Similarity Variables The physical meaning of the term “similarity” relates to internal similitude, or self-similitude. Thus, similar solutions in boundary-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, “separation of variables” (not the classical concept) and the use of “continuous transformation groups.” The basic theory is available in Ames (see the references). Example The equation ∂θ/∂x = (A/y)(∂2θ/∂y2) with the boundary conditions θ = 0 at x = 0, y > 0; θ = 0 at y = ∞, x > 0; θ = 1 at y = 0, x > 0 represents the nondimensional temperature θ of a fluid moving past an infinitely wide flat plate immersed in the fluid. Turbulent transfer is neglected, as is molecular transport except in the y direction. It is now assumed that the equation and the boundary conditions can be satisfied by a solution of the form θ = f(y/xn) = f(u), where θ =
3-39
0 at u = ∞ and θ = 1 at u = 0. The purpose here is to replace the independent variables x and y by the single variable u when it is hoped that a value of n exists which will allow x and y to be completely eliminated in the equation. In this case since u = y/xn, there results after some calculation ∂θ/∂x = −(nu/x)(dθ/du), ∂2θ/∂y2 = (1/x2n)(d2θ/du2), and when these are substituted in the equation, −(1/x)nu(dθ/du) = (1/x3n)(A/u)(d2θ/du2). For this to be a function of u only, choose n = s. There results (d2θ/du2) + (u2/3A)(dθ/du) = 0. Two integrations and use of the boundary conditions for this ordinary differential equation give the solution θ=
∞
u
exp (−u3/9A) du
∞
exp (−u3/9A) du
0
Group Method The type of transformation can be deduced using group theory. For a complete exposition, see Refs. 9, 12, and 145; a shortened version is in Ref. 106. Basically, a similarity transformation should be considered when one of the independent variables has no physical scale (perhaps it goes to infinity). The boundary conditions must also simplify (and combine) since each transformation leads to a differential equation with one fewer independent variable. Example A similarity variable is found for the problem ∂c ∂ ∂c = D(c) , c(0,t) = 1, c(∞,t) = 0, c(x,0) = 0 ∂t ∂x ∂x Note that the length dimension goes to infinity, so that there is no length scale in the problem statement; this is a clue to try a similarity transformation. The transformation examined here is
t = a α t,
x = a β x, c = a γ c With this substitution, the equation becomes ∂c ∂ ∂c aα − γ = a2β − γ D(a−γ c) ∂t ∂x ∂x Group theory says a system is conformally invariant if it has the same form in the new variables; here, that is
γ = 0, α − γ = 2β − γ,
or α = 2β
The invariants are β δ= α
x η = δ , t and the solution is
c(x, t) = f(η)tγ/α We can take γ = 0 and δ = β/α = a. Note that the boundary conditions combine because the point x = ∞ and t = 0 give the same value of η and the conditions on c at x = ∞ and t = 0 are the same. We thus make the transformation x η = , c(x, t) = f(η) 4 D 0t The use of the 4 and D0 makes the analysis below simpler. The result is
df d df D(c) + 2η = 0, dη dη dη
f(0) = 1,
f(∞) = 0
Thus, we solve a two-point boundary value problem instead of a partial differential equation. When the diffusivity is constant, the solution is the error function, a tabulated function. c(x,t) = 1 − erf η = erfc η erf η =
η
0
e−ξ dξ 2
∞
2
e−ξ dξ
0
Separation of Variables This is a powerful, well-utilized method which is applicable in certain circumstances. It consists of assuming that the solution for a partial differential equation has the form U = f(x)g(y). If it is then possible to obtain an ordinary differential equation on one side of the equation depending only on x and on the other side only on y, the partial differential equation is said to be separable in the variables x, y. If this is the case, one side of the equation is a function of x alone and the other of y alone. The two can be equal only if each is a constant, say λ. Thus the problem has again been reduced to the solution of ordinary differential equations. Example Laplace’s equation ∂2V/∂x2 + ∂2V/∂y2 = 0 plus the boundary conditions V(0, y) = 0, V(l, y) = 0, V(x, ∞) = 0, V(x, 0) = f(x) represents the steadystate potential in a thin plate (in z direction) of infinite extent in the y direction
3-40
MATHEMATICS
and of width l in the x direction. A potential f(x) is impressed (at y = 0) from x = 0 to x = 1, and the sides are grounded. To obtain a solution of this boundaryvalue problem assume V(x, y) = f(x)g(y). Substitution in the differential equation yields f ″(x)g(y) + f(x)g″(y) = 0, or g″(y)/g(y) = −f″(x)/f(x) = λ2 (say). This system becomes g″(y) − λ2g(y) = 0 and f ″(x) + λ2f(x) = 0. The solutions of these ordinary differential equations are respectively g(y) = Aeλy + Be−λy, f(x) = C sin λx + D cos λx. Then f(x)g(y) = (Aeλy + Be−λy) (C sin λx + D cos λx). Now V(0, y) = 0 so that f(0)g(y) = (Aeλy + Be−λy) D 0 for all y. Hence D = 0. The solution then has the form sin λx (Aeλy + Be−λy) where the multiplicative constant C has been eliminated. Since V(l, y) = 0, sin λl(Aeλy + Be−λy) 0. Clearly the bracketed function of y is not zero, for the solution would then be the identically zero solution. Hence sin λl = 0 or λn = nπ/l, n = 1, 2, . . . where λn = nth eigenvalue. The solution now has the form sin (nπx/l)(Aenπy/l + Be−nπy/l). Since V(x, ∞) = 0, A must be taken to be zero because ey becomes arbitrarily large as y → ∞. The solution then reads Bn sin (nπx/l)e−nπy/l, where Bn is the multiplicative constant. The differential equation is linear and homogeneous so that n∞= 1 Bne−nπy/l sin (nπx/l) is also a solution. Satisfaction of the last boundary condition is ensured by taking 2 Bn = l
f(x) sin (nπx/l) dx = Fourier sine coefficients of f(x) l
0
Further, convergence and differentiability of this series are established quite easily. Thus the solution is ∞ nπx V(x, y) = Bne−nπy/l sin l n=1
Example The diffusion problem ∂c ∂ ∂c = D(c) , c(0, t) = 1, c(∞, t) = 0, c(x, 0) = 0 ∂t ∂x ∂x can be solved by separation of variables. First transform the problem so that the boundary conditions are homogeneous (having zeroes on the right-hand side). Let c(x, t) = 1 − x + u(x, t) Then u(x, t) satisfies
∂u ∂ 2u = D 2 , ∂t ∂x
u(x, 0) = x − 1,
u(0, t) = 0,
u(1, t) = 0
Assume a solution of the form u(x, t) = X(x) T(t), which gives dT d 2X X=DT dt dx2 Since both sides are constant, this gives the following ordinary differential equations to solve. 1 d 2X 1 dT = −λ = −λ, D T dt X dx2 The solution of these is T=Ae
−λDt
,
x + E sin λ x X = B cos λ
The combined solution for u(x,t) is
x + E sin λ x) e−λDt u = A (B cos λ Apply the boundary condition that u(0,t) = 0 to give B = 0. Then the solution is
x)e−λDt u = A (sin λ where the multiplicative constant E has been eliminated. Apply the boundary condition at x = L.
L)e−λDt 0 = A (sin λ This can be satisfied by choosing A = 0, which gives no solution. However, it can also be satisfied by choosing λ such that
L = 0, λ L=nπ sin λ n2π2 Thus λ= L2 The combined solution can now be written as
sin nπx 2 2 2 u = A e−n π Dt/L L
Since the initial condition must be satisfied, we use an infinite series of these functions. ∞ sin nπx 2 2 2 u = An e−n π Dt/L L n=1 At t = 0, we satisfy the initial condition. ∞ sin nπx x − 1 = An L n=1 This is done by multiplying the equation by sin mπx L and integrating over x: 0 → L. (This is the same as minimizing the mean-square error of the initial condition.) This gives L AmL = (x − 1) sin mπx dx 0 2 which completes the solution.
Integral-Transform Method A number of integral transforms are used in the solution of differential equations. Only one, the Laplace transform, will be discussed here [for others, see “Integral Transforms (Operational Methods)”]. The one-sided Laplace transform indicated ∞ by L[ f(t)] is defined by the equation L[ f(t)] = ∫0 f(t)e−st dt. It has numerous important properties. The ones of interest here are L[ f′(t)] = sL[f(t)] − f(0); L[ f″(t)] = s2L[f(t)] − sf(0) − f′(0); L[ f (n)(t)] = snL[ f(t)] − sn − 1f(0) − sn − 2f′(0) − ⋅⋅⋅ − f (n − 1)(0) for ordinary derivatives. For partial derivatives an indication of which variable is being transformed avoids confusion. Thus, if ∂y y = y(x, t), Lt = sL[y(x, t)] − y(x, 0) ∂t
∂y dLt[y(x, t)] Lt = ∂x dx since L[y(x, t)] is “really” only a function of x. Otherwise the results are similar. These facts coupled with the linearity of the transform, i.e., L[af(t) + bg(t)] = aL[ f(t)] + bL[g(t)], make it a useful device in solving some linear differential equations. Its use reduces the solution of ordinary differential equations to the solution of algebraic equations for L[y]. The solution of partial differential equations is reduced to the solution of ordinary differential equations. In both situations the inverse transform must be obtained either from tables, of which there are several, or by use of complex inversion methods.
whereas
Example The equation ∂c/∂t = D(∂2c/∂x2) represents the diffusion in a semi-infinite medium, x ≥ 0. Under the boundary conditions c(0, t) = c0, c(x, 0) = 0 find a solution of the diffusion equation. By taking the Laplace transform of both sides with respect to t, ∞ ∂2c 1 ∞ −st ∂c e−st 2 dt = e dt 0 ∂x D 0 ∂t
sF d2F = (1/D)sF − c(x, 0) = dx2 D where F(x, s) = Lt[c(x, t)]. Hence d 2F s − F=0 dx2 D The other boundary condition transforms into F(0, s) = c0 /s. Finally the solution of the ordinary differential equation for F subject to F(0, s) = c0 /s and F remains x. Reference to a table shows that the funcfinite as x → ∞ is F(x, s) = (c0 /s)e−s/D tion having this as its Laplace transform is or
2 c(x, t) = c0 1 − π
t x/2D
0
e−u du 2
Matched-Asymptotic Expansions Sometimes the coefficient in front of the highest derivative is a small number. Special perturbation techniques can then be used, provided the proper scaling laws are found. See Refs. 32, 170, and 180.
DIFFERENCE EQUATIONS
3-41
DIFFERENCE EQUATIONS REFERENCES:
30, 43.
Certain situations are such that the independent variable does not vary continuously but has meaning only for discrete values. Typical illustrations occur in the stagewise processes found in chemical engineering such as distillation, staged extraction systems, and absorption columns. In each of these the operation is characterized by a finite between-stage change of the dependent variable in which the independent variable is the integral number of the stage. The importance of difference equations is twofold: (1) to analyze problems of the type described and (2) to obtain approximate solutions of problems which lead, in their formulation, to differential equations. In this subsection only problems of analysis are considered; the application to approximate solutions is considered under “Numerical Analysis and Approximate Methods.” ELEMENTS OF THE CALCULUS OF FINITE DIFFERENCES Let y = f(x) be defined for discrete equidistant values of x, which will be denoted by xn. The corresponding value of y will be written yn = f(xn). The first forward difference of f(x) denoted by ∆f(x) = f(x + h) − f(x) where h = xn − xn − 1 = interval length. Example Let f(x) = x2. Then ∆f(x) = (x + h)2 − x2 = 2hx + h2. The second forward difference is obtained by taking the difference of the first; thus ∆∆f(x) = ∆2f(x) = ∆f(x + h) − ∆f(x) = f(x + 2h) − 2f(x + h) + f(x). Example f(x) = x2, ∆2f(x) = ∆[∆f(x)] = ∆2hx + ∆h2 = 2h(x + h) − 2hx + h2 −
h2 = 2h2.
Similarly the nth forward difference is defined by the relation ∆nf(x) = ∆[∆n − 1f(x)]. Other difference relations are also quite useful. Some of these are ∇f(x) = f(x) − f(x − h), which is called the backward difference, and δf(x) = f [x + (h/2)] − f [x − (h/2)], called the central difference. Some properties of the operator ∆ are quite important. If C is any constant, ∆C = 0; if f(x) is any function of period h, ∆f(x) = 0 (in fact, periodic functions of period h play the same role here as constants do in the differential calculus); ∆[ f(x) + g(x)] = ∆f(x) + ∆g(x); ∆m[∆nf(x)] = ∆m + nf(x); ∆[f(x)g(x)] = f(x) ∆g(x) + g(x + h) ∆f(x) g(x) ∆f(x) − f(x) ∆g(x) f(x) ∆ = g(x) g(x)g(x + h)
Example ∆(x sin x) = x∆ sin x + sin (x + h) ∆x = 2x sin (h/2) cos [x + (h/2)] +
h sin (x + h).
DIFFERENCE EQUATIONS A difference equation is a relation between the differences and the independent variable, φ(∆ny, ∆n − 1y, . . . , ∆y, y, x) = 0, where φ is some given function. The general case in which the interval between the successive points is any real number h, instead of 1, can be reduced to that with interval size 1 by the substitution x = hx′. Hence all further difference-equation work will assume the interval size between successive points is 1. Example f(x + 1) − (α + 1)f(x) + αf(x − 1) = 0. Common notation usually is yx = f(x). This equation is then written yx + 1 − (α + 1)yx + αyx − 1 = 0. Example yx + 2 + 2yx yx + 1 + yx = x2. Example yx + 1 − yx = 2x. The order of the difference equation is the difference between the largest and smallest arguments when written in the form of the second example. The first and second examples are both of order 2, while the third example is of order 1. A linear difference equation involves no
products or other nonlinear functions of the dependent variable and its differences. The first and third examples are linear, while the second example is nonlinear. A solution of a difference equation is a relation between the variables which satisfies the equation. If the difference equation is of order n, the general solution involves n arbitrary constants. The techniques for solving difference equations resemble techniques used for differential equations. Equation Dny = a The solution of ∆ny = a, where a is a constant, is a polynomial of degree n plus an arbitrary periodic function of period 1. That is, y = (axn/n!) + c1xn − 1 + c2xn − 2 + ⋅⋅⋅ + cn + f(x), where f(x + 1) = f(x). Example ∆3y = 6. The solution is y = x3 + c1x2 + c2x + c3 + f(x); c1, c2, c3 are arbitrary constants, and f(x) is an arbitrary periodic function of period 1. Equation yx + 1 - yx = φ(x) This equation states that the first difference of the unknown function is equal to the given function φ(x). The solution by analogy with solving the differential equation dy/dx = φ(x) by integration is obtained by “finite integration” or summation. When there are only a finite number of data points, this is easily accomplished by writing yx = y0 + tx = 1 φ(t − 1), where the data points are numbered from 1 to x. This is the only situation considered here. Examples If φ(x) = 1, yx = x. If φ(x) = x, yx = [x(x − 1)]/2. If φ(x) = ax, a ≠ 0, yx = ax/(a − 1). In all cases y0 = 0. Other examples may be evaluated by using summation, that is, y2 = y1 + φ(1), y3 = y2 + φ(2) = y1 + φ(1) + φ(2), y4 = y3 + φ(3) = y1 + φ(1) + φ(2) + φ(3), . . . , yx = y1 + tx =− 11 φ(t). Example yx + 1 − ryx = 1, r constant, x > 0 and y0 = 1. y1 = 1 + r, y2 = 1 + r + r2, . . . , yx = 1 + r + ⋅⋅⋅ + rx = (1 − rx + 1)/(1 − r) for r ≠ 1 and yx = 1 + x for r = 1. Linear Difference Equations The linear difference equation of order n has the form Pnyx + n + Pn − 1yx + n − 1 + ⋅⋅⋅ + P1yx + 1 + P0yx = Q(x) with Pn ≠ 0 and P0 ≠ 0 and Pj ; j = 0, . . . , n are functions of x. Constant Coefficient and Q(x) = 0 (Homogeneous) The solution is obtained by trying a solution of the form yx = cβ x. When this trial solution is substituted in the difference equation, a polynomial of degree n results for β. If the solutions of this polynomial are denoted by β1, β2, . . . , βn then the following cases result: (1) if all the βj’s are n real and unequal, the solution is yx = j = 1 cj β jx, where the c1, . . . , cn are arbitrary constants; (2) if the roots are real and repeated, say, βj has multiplicity m, then the partial solution corresponding to βj is β jx(c1 + c2 x + ⋅⋅⋅ + cm xm − 1); (3) if the roots are complex conjugates, say, a + ib = peiθ and a − ib = pe−iθ, the partial solution corresponding to this pair is px(c1 cos θx + c2 sin θx); and (4) if the roots are multiple complex conjugates, say, a + ib = peiθ and a − ib = pe−iθ are m-fold, then the partial solution corresponding to these is px[(c1 + c2 x + ⋅⋅⋅ + cm x m − 1) cos θx + (d1 + d2 x + ⋅⋅⋅ + dm x m − 1) sin θx]. Example The equation yx + 1 − (α + 1)yx + αyx − 1 = 0, y0 = c0 and ym + 1 = xm + 1/k represents the steady-state composition of transferable material in the raffinate stream of a staged countercurrent liquid-liquid extraction system. Clearly y is a function of the stage number x. α is a combination of system constants. By using the trial solution yx = cβx, there results β2 − (α + 1)β + α = 0, so that β1 = 1, β2 = α. The general solution is yx = c1 + c2αx. By using the side conditions, c1 = c0 − c2, c2 = (ym + 1 − c0)/(αm + 1 − 1). The desired solution is (yx − c0)/(ym + 1 − c0) = (αx − 1)/(αm + 1 − 1). Example yx + 3 − 3yx + 2 + 4yx = 0. By setting yx = cβ x, there results β3 − 3β2 + 4 = 0 or β1 = −1, β2 = 2, β3 = 2. The general solution is yx = c1(−1)x + 2x(c2 + c3x). Example yx + 1 − 2yx + 2yx − 1 = 0. β1 = 1 + i, β2 = 1 − i. p = 1 + 1 = 2, θ = π/4. The solution is yx = 2 x/2 [c1 cos (xπ/4) + c2 sin (xπ/4)]. Constant Coefficients and Q(x) ≠ 0 (Nonhomogeneous) In this case the general solution is found by first obtaining the homoge-
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MATHEMATICS
neous solution, say, yHx and adding to it any particular solution with Q(x) ≠ 0, say, y xP. There are several means of obtaining the particular solution. Method of Undetermined Coefficients If Q(x) is a product or linear combination of products of the functions ebx, a x, x p (p a positive integer or zero) cos cx and sin cx, this method may be used. The “families” [ax], [ebx], [sin cx, cos cx] and [x p, x p − 1, . . . , x, 1] are defined for each of the above functions in the following way: The family of a term fx is the set of all functions of which fx and all operations of the form a x + y, cos c(x + y), sin c(x + y), (x + y)p on fx and their linear combinations result in. The technique involves the following steps: (1) Solve the homogeneous system. (2) Construct the family of each term. (3) If the family has no representative in the homogeneous solution, assume y Px is a linear combination of the families of each term and determine the constants so that the equation is satisfied. (4) If a family has a representative in the homogeneous solution, multiply each member of the family by the smallest integral power of x for which all such representatives are removed and revert to step 3. Example yx + 1 − 3yx + 2yx − 1 = 1 + ax. a ≠ 0. The homogeneous solution is yxH = c1 + c2 2x. The family of 1 is 1 and of ax is ax. However, 1 is a solution of the homogeneous system. Therefore, try yxP = Ax + Bax. Substituting in the equation there results a yx = c1 + c2 2x − x + axa ≠ 1, a ≠ 2 (a − 1)(a − 2) If a = 1, yx = c1 + c22x − 2x. If a = 2, yx = c1 + c22x − x + x2x.
Example The family of x23x is [x23x, x3x, 3x]. Method of Variation of Parameters This technique is applicable to general linear difference equations. It is illustrated for the second-order system yx + 2 + Ayx + 1 + Byx = φ(x). Assume that the homogeneous solution has been found by some technique and write yxH = c1ux + c2vx. Assume that a particular solution yxP = Dxux + Exvx. Ex and Dx can be found by solving the equations: ux + 1φ(x) Ex + 1 − Ex = ux + 1vx + 2 − ux + 2vx + 1 vx + 1φ(x) Dx + 1 − Dx = vx + 1ux + 2 − vx + 2ux + 1 by summation. The general solution is then yx = yxP + yxH. Variable Coefficients The method of variation of parameters applies equally well to the linear difference equation with variable coefficients. Techniques are therefore needed to solve the homogeneous system with variable coefficients. Equation yx + 1 - axyx = 0 By assuming that this equation is valid for x ≥ 0 and y0 = c, the solution is yx = c xn = 1 an − 1. x+2 x+1
Example yx + 1 + yx = 0. The solution is x x+1 n+1 2 3 yx = c − = c(−1)x ⋅ ⋅⋅⋅ = (−1)xc(x + 1) n 1 2 x n=1
Example yx + 1 − xyx = 0. The solution is yx = c(x − 1)!
Reduction of Order If one homogeneous solution, say, ux, can be found by inspection or otherwise, an equation of lower order can be obtained by the substitution vx = yx /ux. The resultant equation must be satisfied by vx = constant or ∆vx = 0. Thus the equation will be of reduced order if the new variable Ux = ∆(yx /ux) is introduced. Example (x + 2)yx + 2 − (x + 3)yx + 1 + yx = 0. By observation ux = 1 is a solution. Set Ux = ∆yx = yx + 1 − yx. There results (x + 2)Ux + 1 − Ux = 0, which is of degree one lower than the original equation. The complete solution for yx is finally x 1 yx = c0 + c1 n = 0 n! Factorization If the difference equation can be factored, then the general solution can be obtained by solving two or more successive equations of lower order. Consider yx + 2 + Axyx + 1 + Bxyx = φ(x). If there exists ax, bx such that ax + bx = −Ax and axbx = Bx, then the difference equation may be written yx + 2 − (ax + bx) yx + 1 + axbxyx = φ(x). First solve Ux + 1 − bxUx = φ(x) and then yx + 1 − axyx = Ux. Example yx + 2 − (2x + 1)yx + 1 + (x2 + x)yx = 0. Set ax = x, bx = x + 1. Solve ux + 1 − (x + 1)ux = 0 and then yx + 1 − xyx = ux. Substitution If it is possible to rearrange a difference equation so that it takes the form afx + 2yx + 2 + bfx + 1yx + 1 + cfxyx = φ(x) with a, b, c constants, then the substitution ux = fxyx reduces the equation to one with constant coefficients. Example (x + 2)2yx + 2 − 3(x + 1)2yx + 1 + 2x2yx = 0. Set ux = x2yx. The equation becomes ux + 2 − 3ux + 1 + 2ux = 0, which is linear and easily solved by previous methods. The substitution ux = yx / fx reduces afx fx + 1yx + z + bfx fx + 2yx + 1 + cfx + 1 fx + 2yx = φ(x) to an equation with constant coefficients. Example x(x + 1)yx + 2 + 3x(x + 2)yx + 1 − 4(x + 1)(x + 2)yx = x. Set ux = yx /fx = yx /x. Then yx = xux, yx + 1 = (x + 1)ux + 1 and yx + 2 = (x + 2)ux + 2. Substitution in the equation yields x(x + 1)(x + 2)ux + 2 + 3x(x + 2)(x + 1)uu + 1 − 4x(x + 1)(x + 2) ux = x or ux + 2 + 3ux + 1 − 4ux = 1/(x + 1)(x + 2), which is a linear equation with constant coefficients. Nonlinear Difference Equations: Riccati Difference Equation The Riccati equation yx + 1yx + ayx + 1 + byx + c = 0 is a nonlinear difference equation which can be solved by reduction to linear form. Set y = z + h. The equation becomes zx + 1zx + (h + a)zx + 1 + (h + b)zx + h2 + (a + b)h + c = 0. If h is selected as a root of h2 + (a + b)h + c = 0 and the equation is divided by zx + 1zx there results [(h + b)/zx + 1] + [(h + a)/zx] + 1 = 0. This is a linear equation with constant coefficients. The solution is 1 yx = h + a+h x 1 c − − b+h (a + h) + (b + h)
Example This equation is obtained in distillation problems, among others, in which the number of theoretical plates is required. If the relative volatility is assumed to be constant, the plates are theoretically perfect, and the molal liquid and vapor rates are constant, then a material balance around the nth plate of the enriching section yields a Riccati difference equation.
INTEGRAL EQUATIONS REFERENCES: 75, 79, 105, 195, 273. See also “Numerical Analysis and Approximate Methods.”
An integral equation is any equation in which the unknown function appears under the sign of integration and possibly outside the sign of integration. If derivatives of the dependent variable appear elsewhere in the equation, the equation is said to be integrodifferential.
CLASSIFICATION OF INTEGRAL EQUATIONS Volterra integral equations have an integral with a variable limit. The Volterra equation of the second kind is u(x) = f(x) + λ
K(x, t)u(t) dt x
a
INTEGRAL EQUATIONS whereas a Volterra equation of the first kind is u(x) = λ
METHODS OF SOLUTION
K(x, t)u(t) dt x
a
Equations of the first kind are very sensitive to solution errors so that they present severe numerical problems. Volterra equations are similar to initial value problems. A Fredholm equation of the second kind is u(x) = f(x) + λ
K(x, t)u(t) dt b
a
whereas a Fredholm equation of the first kind is u(x) =
b
a
The limits of integration are fixed, and these problems are analogous to boundary value problems. An eigenvalue problem is a homogeneous equation of the second kind, and solutions exist only for certain λ.
K(x, t)u(t) dt b
a
See Refs. 105 and 195 for further information and existence proofs. If the unknown function u appears in the equation in any way except to the first power, the integral equation is said to be nonlinear. b The equation u(x) = f(x) + ∫a K(x, t)[u(t)]3/2 dt is nonlinear. The differential equation du/dx = g(x, u) is equivalent to the nonlinear integral x equation u(x) = c + ∫a g[t, u(t)] dt. An integral equation is said to be singular when either one or both of the limits of integration become infinite or if K(x, t) becomes infinite for one or more points of the interval under discussion.
∞
cos (xt)u(t) dt and f(x) =
The Leibniz rule (see “Integral Calculus”) can be used to show the equivalence of the initial-value problem consisting of the secondorder differential equation d 2y/dx2 + A(x)(dy/dx) + B(x)y = f(x) together with the prescribed initial conditions y(a) = y0, y′(a) = y′0 to the integral equation. y(x) =
K(x, t)y(t) dt + F(x) x
and
(x − t)f(t) dt + [A(a)y + y′](x − a) + y x
0
0
0
a
This integral equation is a Volterra equation of the second kind. Thus the initial-value problem is equivalent to a Volterra integral equation of the second kind. Example d 2y/dx2 + x2(dy/dx) + xy = x, y(0) = 1, y′(0) x= 0. Here A(x) = x2, B(x) = x, f(x) = x. The equivalent integral equation is y(x) = ∫ 0 K(x, t)y(t) dt + F(x) x where K(x, t) = t(x − t) − t2 and F(x) = ∫ 0 (x − t)t dt + 1 = x3/6 + 1. Combining these x y(x) = ∫ 0 t[x − 2t]y(t) dt + x3/6 + 1. Eigenvalue problems can also be related. For example, the problem (d y/dx2) + λy = 0 with y(0) = 0, y(a) = 0 is equivalent to the integral a equation y(x) = λ ∫ 0 K(x, t)y(t) dt, where K(x, t) = (t/a)(a − x) when t < x and K(x, t) = (x/a)(a − t) when t > x. The differential equation may be recovered from the integral equation by differentiating the integral equation by using the Leibniz rule.
Method of Successive Approximations Consider the equation b y(x) = f(x) + λ ∫ a K(x, t)y(t) dt. In this method a unique solution is obtained in sequence form as follows: Substitute in the right-hand member of the equation y0(t) for y(t). Upon integration there results b y1(t) = f(x) + λ ∫ a K(x, t)y0(t) dt. Continue in like manner by replacing y0 by y1, y1 by y2, etc. A series of functions y0(x), y1(x), y2(x), . . . are obtained which satisfy the equations yn(x) = f(x) + λ
K(x, t)y b
(t) dt
n−1
a
Then yn(x) = f(x) + λ K(x, t)f(t) dt + λ K(x, t) ∫ a K(t, t1)f(t1) dt1 dt + b λ3 ∫ ab K(x, t) ∫ a K(t, t ) K(t1, t2)f(t2) dt2 dt1 dt + ⋅⋅⋅ + Rn, where Rn is the remainder, and max. y0 |Rn| ≤ |λn| Mn(b − a)n a≤x≤b 2 b ∫a
∫ ab b 1 ∫a
b
where M = maximum value of |K| in the rectangle a ≤ t ≤ b, a ≤ x ≤ b. If |λ|M(b − a) < 1, lim Rn = 0. Then yn(x) → y(x), which is the unique n→∞ solution. 1
Example Consider the equation y(x) = 1 + λ ∫ 0 (1 − 3xt)y(t) dt.
(1 − 3xt) dt + λ (1 − 3xt) (1 − 3tt ) dt dt + ⋅⋅⋅ 1
1
1
2
1
0
K(x, t) = (t − x)[B(t) − A′(t)] − A(t) F(x) =
L[1] 1/p p E(t) = L−1 = L−1 2 = L−1 = cosh λt 1 − λL[K(t)] 1 − λ/p p2 − λ
y(x) = 1 + λ
a
where
Example In a certain linear system, the effect E(t) due to a cause C = λE at time τ is a function only of the elapsed time t − τ. If the system has the activity level 1 at time t < 0, thet cause λE and effect (E) relation is given by the integralt equation E(t) = 1 + λ ∫ 0 K(t − τ)E(τ) dτ. Let K(t − τ) = t − τ. Then E(t) = 1 + λ ∫ 0 (t − τ)E(τ) dτ. By using the transform method
x
RELATION TO DIFFERENTIAL EQUATIONS
L[ f(x)] L[ f(x)] L[u(x)] = , u(x) = L−1 1 − λL[K(x)] 1 − λL[K(x)] Equations of the type considered here occur quite frequently in practice in what can be called “cause-and-effect” systems.
u(t) dt are both 0 0 x − t singular. The kernel of the first equation is cos (xt), and that of the second is (x − t)−1.
Example u(x) = x +
In general, the solution of integral equations is not easy, and a few exact and approximate methods are given here. Often numerical methods must be employed, as discussed in “Numerical Solution of Integral Equations.” Equations of Convolution Type The equation u(x) = f(x) + x λ ∫ 0 K(x − t)u(t) dt is a special case of the linear integral equation of the second kind of Volterra type. The integral part is the convolution integral discussed under “Integral Transforms (Operational Methods)”; so the solution can be accomplished by Laplace transforms; L[u(x)] = L[ f(x)] + λL[u(x)]L[K(x)] or
K(x, t)u(t) dt
u(x) = λ
3-43
0
1
0
3 1 1 3 λ4 1 3 = 1 + λ 1 − x + λ2 + λ3 1 − x + + λ5 1 − x + ⋅⋅⋅ 2 4 4 2 16 16 2
λ λ 3 = 1 + + + ⋅⋅⋅ 1 + λ 1 − x 4 16 2
2
4
1 + λ(1 − ⁄ x) = , 1 − dλ2 32
|λ|<2
Example dy/dx = x2 x+ y. x0 = 0, y0 = 1. This problem is equivalent to the integral equation y = 1 + ∫ 0 (x2 + y) dx. Let the initial approximation for y be 1. Then x x3 y(1) = 1 + (x2 + 1) dx = 1 + x + 0 3
2
y(2) = 1 +
[x + y x
2
0
] dx = 1 +
(1)
x + 1 + x + x3 dx x
3
2
0
x4 x2 x3 = 1 + x + + + , etc. 2 3 12
3-44
MATHEMATICS
INTEGRAL TRANSFORMS (OPERATIONAL METHODS) REFERENCES:
63, 64, 71, 72, 97, 137, 217.
The term “operational method” implies a procedure of solving differential and difference equations by which the boundary or initial conditions are automatically satisfied in the course of the solution. The technique offers a very powerful tool in the applications of mathematics, but it is limited to linear problems. Most integral transforms are special cases of the equation g(s) = b ∫ a f(t)K(s, t) dt in which g(s) is said to be the transform of f(t) and K(s, t) is called the kernel of the transform. A tabulation of the more important kernels and the interval (a, b) of applicability follows. The first three transforms are considered here.
Properties of the Laplace Transform 1. The Laplace transform is a linear operator: L{af(t) + bg(t)} = aL{ f(t)} + bL{g(t)} for any constants a, b and any two functions f and g whose Laplace transforms exist. 2. The Laplace transform of a real-valued function is real for real s. If f(t) is a complex-valued function, f(t) = u(t) + iv(t), where u and v are real, then L{ f(t)} = L{u(t)} + iL{v(t)}. Thus L{u(t)} is the real part of L{ f(t)}, and L{v(t)} is the imaginary part of L{ f(t)}. 3. The Laplace transform of a function in the class Λ has derivatives of all orders, and L{tkf(t)} = (−1)kd kF(s)/ds k, k = 1, 2, 3, . . . .
Name of transform
(a, b)
Laplace
(0, ∞)
Fourier
(−∞, ∞)
Fourier cosine
(0, ∞)
Fourier sine
(0, ∞)
Mellin
(0, ∞)
Hankel
(0, ∞)
K(s, t) e−st 1 e−ist 2π
π2 cos st π2 sin st
2as a e−st sin at dt = , s > 0. By property 3, = (s2 + a2)2 s2 + a2 e−st t sin at dt = L{t sin at}.
Example
0
0
Example By applying property 3 with f(t) = 1 and using the preceding results, we obtain k! dk 1 L{tk} = (−1)k k = ds s sk + 1
provided Re s > 0; k = 1, 2, . . . . Similarly, we obtain
t
tJν (st), ν ≥ −a
The Laplace transform of a function f(t) is defined by F(s) = L{f(t)} = ∞ ∫ 0 e−stf(t) dt, where s is a complex variable. Note that the transform is an improper integral and therefore may not exist for all continuous functions and all values of s. We restrict consideration to those values of s and those functions f for which this improper integral converges. The function L[f(t)] = g(s) is called the direct transform, and L−1[g(s)] = f(t) is called the inverse transform. Both the direct and the inverse transforms are tabulated for many often-occurring functions. In general, 1 + i∞ st L−1[g(s)] = e g(s) ds 2πi − i∞ and to evaluate this integral requires a knowledge of complex variables, the theory of residues, and contour integration. A function is said to be piecewise continuous on an interval if it has only a finite number of finite (or jump) discontinuities. A function f on 0 < t < ∞ is said to be of exponential growth at infinity if there exist constants M and α such that |f(t)| ≤ Meαt for sufficiently large t. Sufficient Conditions for the Existence of Laplace Transform Suppose f is a function which is (1) piecewise continuous on every finite interval 0 < t < T, (2) of exponential growth at infinity, and (3) ∫ 0δ |f(t)| dt exist (finite) for every finite δ > 0. Then the Laplace transform of f exists for all complex numbers s with sufficiently large real part. Note that condition 3 is automatically satisfied if f is assumed to be piecewise continuous on every finite interval 0 ≤ t < T. The function f(t) = t−1/2 is not piecewise continuous on 0 ≤ t ≤ T but satisfies conditions 1 to 3. Let Λ denote the class of all functions on 0 < t < ∞ which satisfy conditions 1 to 3.
Example Let f(t) be the Heaviside step function at t = t0; i.e., f(t) = 0 for t ≤ t0, and f(t) = 1 for t > t0. Then ∞ T 1 e−st0 e−st dt = lim e−st dt = lim (e−st0 − e−sT) = L{ f(t)} = T→∞ t0 T→∞ s t0 s provided s > 0.
Example Let f(t) = eat, t ≥ 0, where a is a real number. Then L{eat} = ∞ ∫ 0 e−(s − a) dt = 1/(s − a), provided Re s > a.
dk 1 k! L{tkeat} = (−1)k k = ds s − a (s − a)k + 1
s−1
LAPLACE TRANSFORM
∞
∞
4. Frequency-shift property (or, equivalently, the transform of an exponentially modulated function). If F(s) is the Laplace transform of a function f(t) in the class Λ, then for any constant a, L{eat f(t)} = F(s − a). 1 (s + a)
Example L{te−at} = 2 , s > 0. 5. Time-shift property. Let u(t − a) be the unit step function at t = a. Then L{f (t − a)u(t − a)} = e−asF(s). 6. Transform of a derivative. Let f be a differentiable function such that both f and f ′ belong to the class Λ. Then L{ f ′(t)} = sF(s) − f(0). 7. Transform of a higher-order derivative. Let f be a function which has continuous derivatives up to order n on (0, ∞), and suppose that f and its derivatives up to order n belong to the class Λ. Then L{ f ( j)(t)} = s jF(s) − s j − 1f(0) − s j − 2 f ′(0) − ⋅ ⋅ ⋅ − sf ( j − 2)(0) − f ( j − 1)(0) for j = 1, 2, . . . , k. Example L{ f″(t)} = s2L{ f(t)} − sf(0) − f′(0) L{ f″′(t)} = s3L{ f(t)} − s2f(0) − sf ′(0) − f″(0)
Example Solve y″ + y = 2et, y(0) = y′(0) = 2. L[y″] = −y′(0) − sy(0) + s2L[y] = −2 − 2s + s2L[y]. Thus 2 −2 − 2s + s2L[y] + L[y] = 2L[et] = s−1 1 1 s 2s2 L[y] = =++ (s − 1)(s2 + 1) s − 1 s2 + 1 s2 + 1 Hence y = et + cos t + sin t.
A short table (Table 3-1) of very common Laplace transforms and inverse transforms follows. The references include more detailed ∞ tables. NOTE: Γ(n + 1) = ∫ 0 xne−x dx (gamma function); Jn(t) = Bessel function of the first kind of order n. t 1 1 0 8. L f(t) dt = L[ f(t)] + f(t) dt a s s a
1 s
s −1 a
Example Find f(t) if L[ f(t)] = 2 2 2 Therefore f(t) =
sinh at 1a sinh at dt dt = a1 − t a t
t
0
0
2
1 1 L sinh at = a s2 − a2
INTEGRAL TRANSFORMS (OPERATIONAL METHODS) TABLE 3-1
Laplace Transforms
f(t)
g(s)
f(t)
g(s)
1
1/s
e−at(1 − at)
s 2 (s + a)
tn, (n a + integer)
n! sn + 1
t sin at 2a
s (s2 + a2)2
tn, n ≠ + integer
Γ(n + 1) sn + 1
1 2 sin at sinh at 2a
s s4 + 4a4
cos at
s s2 + a2
cos at cosh at
s2 s4 + 4a4
a s2 + a2
sin at
2
s s4 − a4
1 (sinh at + sin at) 2a
cosh at
s s2 − a2
a (cosh at + cos at)
s2 s4 − a4
sinh at
a s2 − a2
sin at t
a tan−1 s
e−at
1 s+a
J0(at)
1 s2 +a2
e−bt cos at
s+b (s + b)2 + a2
Jn(at) nan t
1 s2 +a2 − sn
e−bt sin a
a (s + b)2 + a2
) J0 (2 at
1 e−a/s s
3-45
I0 = zero-order Bessel function of an imaginary argument. For large u, In(u) ∼ eu/2π u . Hence for large n, exp [−(θ − n )2] y(n, θ) ∼ 2π1/2(nθ)1/4 or for sufficiently large n, the peak concentration occurs near θ = n.
Other applications of Laplace transforms are given under “Differential Equations.” CONVOLUTION INTEGRAL The convolution integral (faltung) of two functions f(t), r(t) is x(t) = f(t)°r(t) = ∫ 0t f(τ)r(t − τ) dτ. Example t° sin t =
τ sin (t − τ) dτ = t − sin t. t
0
L[ f(t)]L[h(t)] = L[ f(t)°h(t)] Z-TRANSFORM
See Refs. 198, 218, and 256. The z-transform is useful when data is available at only discrete points. Let f *(t) = f(tk) be the value of f at the sample points tk = k ∆t,
k = 0, 1, 2, . . .
Then the function f *(t) is ∞
9.
f(t) L = t
⋅⋅⋅
∞
f(t) L = tk
g(s) ds
s
∞
s
f *(t) = f(tk) δ(t − tk) k=0
∞
k
g(s)(ds)
Take the Laplace transform of this.
s
sin at Example L = t
10.
∞
L[sin at] ds =
s
s
a ds s = cot−1 s2 + a2 a
01
ta
∞0
k=0
∞
k=0
L[u(t − a)] = e−as/s
at t = a elsewhere
L[u′(t − a)] = e−as
12. L−1[e−asg(s)] = f(t − a)u(t − a) (second shift theorem). 13. If f(t) is periodic of period b, i.e., f(t + b) = f(t), then b 1 L[f(t)] = e−stf(t) dt 1 − e−bs 0
k=0
g*(z) = f(tk) z−k
The unit impulse function is δ(a) = u′(t − a) =
∞
For convenience, replace e−s∆t by z and call g*(z) the z-transform of f *(t).
The unit step function u(t − a) =
11.
∞
∞
g*(s) = L[ f *(t)] = f(tk) e−stk = f(tk) e−s∆tk
k integrals
Example The partial differential equations relating gas composition to position and time in a gas chromatograph are ∂y/∂n + ∂x/∂θ = 0, ∂y/∂n = x − y, where x = mx′, n = (kGaP/Gm)h, θ = (mkGaP/ρB)t and GM = molar velocity, y = mole fraction of the component in the gas phase, ρB = bulk density, h = distance from the entrance, P = pressure, kG = mass-transfer coefficient, and m = slope of the equilibrium line. These equations are equivalent to ∂2y/∂n ∂θ + ∂y/∂n + ∂y/∂θ = 0, where the boundary conditions considered here are y(0, θ) = 0 and x(n, 0) = y(n, 0) + (∂y/∂n) (n, 0) = δ(0) (see property 11). The problem is conveniently solved by using the Laplace transform of y with respect to n; write ∞ g(s, θ) = ∫ 0 e−nsy(n, θ) dn. Operating on the partial differential equation gives s(dg/dθ) − (∂y/∂θ) (0, θ) + sg − y(0, θ) + dg/dθ = 0 or (s + 1) (dg/dθ) + sg = (∂y/∂θ) (0, θ) + y(0, θ) = 0. The second boundary condition gives g(s, 0) + sg(s, 0) − y(0, 0) = 1 or g(s, 0) + sg(s, 0) = 1 (L[δ(0)] = 1). A solution of the ordinary differential equation for g consistent with this second condition is 1 g(s, θ) = e−sθ /(s + 1) s+1
θ) where Inversion of this transform gives the solution y(n, θ) = e−(n + θ) I0(2 n
The z-transform is used in process control when the signals are at intervals of ∆t. A brief table (Table 3-2) is provided here. The z-transform can also be used to solve difference equations, just like the Laplace transform can be used to solve differential equations. TABLE 3-2
z-Transforms
f(k)
g*(z)
1(k)
1 1 − z−1
k ∆t
∆t z−1 (1 − z−1)2
(k ∆t)n − 1
∂n − 1 1 lim (−1)n − 1 a→0 ∂an − 1 1 − e−a∆tz−1
sin a k ∆t
z−1 sin a ∆t (1 − 2 z−1 cos a ∆t + z−2)
cos a k ∆t
1 − z−1 cos a ∆t (1 − 2 z−1 cos a ∆t + z−2)
e−ak∆t
1 1 − e−a∆tz−1
e−bk∆t cos a k ∆t
1 − z−1 e−b∆t cos a ∆t 1 − 2 z−1 e−b∆t cos a ∆t + z−2 e−2b∆t
1 e−bk∆t sin a k ∆t b
1 z−1 e−b∆t sin a ∆t b 1 − 2 z−1 e−b∆t cos a ∆t + z−2 e−2b∆t
3-46
MATHEMATICS
Example The difference equation for y(k) is
and its inverse by
y(k) + a1 y(k − 1) + a2 y(k − 2) = b1u(k)
F c−1[g(s)] = f(t) =
Take the z-transform −2
The inverse transform must be found, usually from a table of inverse transforms.
The Fourier transform is given by ∞ 1 F[ f(t)] = f(t)e−ist dt = g(s) 2 π −∞ and its inverse by ∞ 1 g(s)eist dt = f(t) F−1[g(s)] = π −∞ 2
1−a≤t≤a has F[ f(t)] = 0 elsewhere a a a 2 sin sa eist dt + e−ist dt = 2 cos st dt = 0 0 0 s
a
−a
Example Find F[e−a|t|] by property 7. e−a|t| is even. So L[e−at] = 1/(s + a). Therefore, F[e−a|t|] = 1/(is + a) + 1/(−is + a) = 2a/(s2 + a2). Tables of this transform may be found in Higher Transcendental Functions, vols. I, II, and III, A. Erdelyi, et al., McGraw-Hill, New York, 1953–1955. FOURIER COSINE TRANSFORM The Fourier cosine transform is given by
π2
a
cos st dt =
0
4. Fc[t−2nf(t)] = (−1)n (d 2ng/ds2n). 5. Fc[t2n + 1f(t)] = (−1)n (d 2n + 1/ds2n + 1) Fs[ f(t)]. A short table (Table 3-3) of Fourier cosine transforms follows.
e−ist dt =
Properties of the Fourier Transform Let F [ f (t)] = g(s); F−1[g(s)] = f(t). 1. F[ f (n)(t)] = (is)nF[ f(t)]. 2. F[af(t) + bh(t)] = aF[ f(t)] + bF [h(t)]. 3. F[ f(−t)] = g(−s). 1 s 4. F[ f(at)] = g , a > 0. a a 5. F[e−iwt f(t)] = g(s + w). 6. F[ f(t + t1)] = eist1g(s). 7. F [ f(t)] = G(is) + G(−is) if f(t) = f(−t) ( f even) F[ f(t)] = G(is) − G(−is) if f(t) = −f(−t) ( f odd) where G(s) = L[f(t)]. This result allows the use of the Laplacetransform tables to obtain the Fourier transforms.
Fc[f(t)] = g(s) =
π2
1 s+b s−b Fc[ f(at) cos bt] = g + g , a, b > 0 2a a a
3.
In brief, the condition for the Fourier transform to exist is that ∞ ∫ -∞ |f(t)| dt < ∞, although certain functions may have a Fourier transform even if this is violated.
0
10
Properties of the Fourier Cosine Transform Fc[ f(t)] = g(s). 1. Fc[af(t) + bh(t)] = aFc[ f(t)] + bFc[h(t)]. 2. Fc[ f(at)] = (1/a)g(s/a).
Example Fc[ f(t)], f(t) = 2 sin as π s
FOURIER TRANSFORM
Example The function f(t) =
g(s) cos st ds
The Fourier sine transform Fs is obtainable by replacing the cosine by the sine in these integrals.
u*(z) y*(z) = 1 + a1z−1 + a2z−2
Then
∞
0
(1 + a1z + a2z ) y*(z) = u*(z) −1
π2
∞
f(t) cos st dt
0
TABLE 3-3
Fourier Transforms g(s) 2/π
f(t) t 2−t 0 t−1/2 0 (t − a)−1/2 (t2 + a2)−1 e−at
0
1 2 [2 cos s − 1 − cos 2s] s π1/2(2s)−1/2
0
π1/2(2s)−1/2 [cos as − sin as]
a>0
a πa−1e−as
e−at
2
a>0
a s2 + a2 2 aπ1/2a−1/2e−s /4a
sin at t
a>0
π/2 π/4 0
sa
Example The temperature in the semi-infinite rod 0 ≤ x < ∞ is determined by the differential equation ∂ ∂t = k(∂2 ∂x2) and the condition = 0 when t = 0, x ≥ 0; ∂ ∂x = −µ = constant when x = 0, t > 0. By using the Fourier cosine transform a solution may be found as 2µ
(x, t) = π
∞
0
cos px (1 − e−kp2t ) dp. p
MATRIX ALGEBRA AND MATRIX COMPUTATIONS REFERENCES: General (textbooks that cover at an introductory level a variety of topics that constitute a core of numerical methods for practicing engineers), 2, 3, 4, 22, 56, 59, 70, 77, 133, 135, 143, 150, 155, 219. Numerical solution of nonlinear equations, 153, 171, 237, 302. Numerical solution of ordinary differential equations, 76, 117, 127, 185, 257. Numerical solution of integral equa-
tions, 23, 26, 129, 162. Numerical solution of partial differential equations, 11, 76, 127, 133, 155, 210, 251, 286, 287, 213, 233, 253. Spline functions and applications, 38, 56, 70, 230. Finite elements and applications, 5, 29, 83, 130, 164, 210, 241, 281, 287, 303, 304. Fast Fourier transforms, 47, 56, 135, 238. Software, 187, 231.
MATRIX ALGEBRA AND MATRIX COMPUTATIONS MATRIX ALGEBRA Matrices A rectangular array of rows and n columns a11 a A = (aij) = 21 ⯗ am1
mn quantities, arranged in m
… …
a1n a2n
…
amn
is called a matrix. The elements aij may be real or complex. The notation aij means the element in the ith row and jth column, i is called the row index, j the column index. If m = n the matrix is said to be square and of order n. A matrix, even if it is square, does not have a numerical value, as a determinant does. However, if the matrix A is square, a determinant can be formed which has the same elements as the matrix A. This is called the determinant of the matrix and is written det (A) or |A|. If A is square and det (A) ≠ 0, A is said to be nonsingular; if det (A) = 0, A is said to be singular. A matrix A has rank r if and only if it has a nonvanishing determinant of order r and no nonvanishing determinant of order > r. Equality of Matrices Let A = (aij), B = (bij). Two matrices A and B are equal (=) if and only if they are identical; that is, they have the same number of rows and the same number of columns and equal corresponding elements (aij = bij for all i and j). Addition and Subtraction The operations of addition (+) and subtraction (−) of two or more matrices are possible if and only if they have the same number of rows and columns. Thus A B = (aij bij); i.e., addition and subtraction are of corresponding elements. Transposition The matrix obtained from A by interchanging the rows and columns of A is called the transpose of A, written A′ or AT.
1 2 Example A = 12 31 46 , AT = 3 1 4 6
Note that (AT )T = A.
Multiplication Let A = (aij), i = 1, . . . , m1; j = 1, . . . , m2. B = (bij), i = 1, . . . , n1, j = 1, . . . , n2. The product AB is defined if and only if the number of columns of A (m2) equals the number of rows of B(n1), i.e., n1 = m2. For two such matrices the product P = AB is defined by summing the element by element products of a row of A by a column of B. This is the row by column rule. Thus n1
pij = aikbkj k=1
The resulting matrix has m1 rows and n2 columns. Example
3 1 5
2 1 4
0 −2
1 0
5 1
6 = 3
−4 −2 −8
3 1 5
17 6 29
24 9 42
It is helpful to remember that the element pij is formed from the ith row of the first matrix and the jth column of the second matrix. The matrix product is not commutative. That is, AB ≠ BA in general. Inverse of a Matrix A square matrix A is said to have an inverse if there exists a matrix B such that AB = BA = I, where I is the identity matrix of order n. 1 0.... 0 0 1.. ⯗ 1 0 0 ....0 1
−1
The inverse B is a square matrix of the order of A, designated by A . Thus AA−1 = A−1A = I. A square matrix A has an inverse if and only if A is nonsingular. Certain relations are important: (1)
(AB)−1 = B−1A−1
(2)
(AB)T = BTAT
(3)
(A−1)T = (AT )−1
(4)
(ABC)−1 = C−1B−1A−1
3-47
Scalar Multiplication Let c be any real or complex number. Then cA = (caij). Adjugate Matrix of a Matrix Let Aij denote the cofactor of the element aij in the determinant of the matrix A. The matrix BT where B = (Aij) is called the adjugate matrix of A written adj A = BT. The elements bij are calculated by taking the matrix A, deleting the ith row and jth column, and calculating the determinant of the remaining matrix times (−1)i + j. Then A−1 = adj A/ |A|. This definition may be used to calculate A−1. However, it is very laborious and the inversion is usually accomplished by numerical techniques shown under “Numerical Analysis and Approximate Methods.”
3
0
3
6
Example Let A = −1 2
adj A = BT =
0 6 −12
−1 1 3
Form B = (A i j), B =
0 −6 2
6 12 −2
−12 −18 6
−6 2 12 −2 ; |A| = 12 −18 6
−a 1 −3 ⁄ 2
0 adj A A−1 = = a |A| −1
1 ⁄6 −1⁄6 a
Linear Equations in Matrix Form Every set of n nonhomogeneous linear equations in n unknowns a11 x1 + a12 x2 + ⋅⋅⋅ + a1n xn = b1 a21 x1 + a22 x2 + ⋅⋅⋅ + a2n xn = b2 an1 x1 + an2 x2 + ⋅⋅⋅ + ann xn = bn can be written in matrix form as AX = B, where A = (aij), X T = [x1 ⋅⋅⋅ xn], and BT = [b1 ⋅⋅⋅ bn]. The solution for the unknowns is X = A−1B. Special Square Matrices 1. A triangular matrix is a matrix all of whose elements above or below the main diagonal (set of elements a11, . . . , ann) are zero. If A is triangular, det (A) = a11. a22 . . . ann. 2. A diagonal matrix is one such that all elements both above and below the main diagonal are zero (i.e., aij = 0 for all i ≠ j). If all diagonal elements are equal, the matrix is called scalar. If A is diagonal, A = (aij), A−1 = (1/aij). 3. If aij = aji for all i and j (i.e., A = AT ), the matrix is symmetric. 4. If aij = −aji for i ≠ j but the aij are not all zero, the matrix is skew. 5. If aij = −aji for all i and j (i.e., aii = 0), the matrix is skew symmetric. 6. If AT = A−1, the matrix A is orthogonal. 7. If the matrix A* = (aij)T, aij = complex conjugate of aij, A* is the hermitian conjugate of A. 8. If A = A−1, A is involutory. 9. If A = A*, A is hermitian. 10. If A = −A*, A is skew hermitian. 11. If A−1 = A*, A is unitary. If A is any matrix, then AAT and ATA are square symmetric matrices, usually of different order.
5
Example Let A = 3
35 AAT = 22 8
2
1 3 4 1 −2 0
22 51 3
8 3 , 9
3 4 1 5
2 −2 0 1
38 13 18 17
13 21 7 18
18 7 10 5
5 0 1 5 , AT = 3 1 0
ATA =
17 18 5 26
Using a program such as MATLAB, these are easily calculated.
Matrix Calculus Differentiation Let the elements of A = [aij(t)] be differentiable dA daij(t) functions of t. Then = . dt dt
3-48
MATHEMATICS
interchange two rows of A to produce a (nonzero) pivot, such as if we start with 0 2 A= 1 6
sin t cos t dA cos t −sin t Example A = −cos , = . t sin t sin t cos t dt
Integration The integral ∫ A dt = [∫ aij(t) dt].
2 t 2t Example A = t2 e2t , ∫ A dt = tt3/2 . /3 et
The matrix B = A − λI is called the characteristic (eigen) matrix of A. Here A is square of order n, λ is a scalar parameter, and I is the n × n identity. det B = det (A − λI) = 0 is the characteristic (eigen) equation for A. The characteristic equation is always of the same degree as the order of A. The roots of the characteristic equation are called the eigenvalues of A.
Example A = 13 28 , B = 13 28 − λ0 λ0 = 1−λ3 8−λ2
is the characteristic matrix and f(λ) = det (B) = det (A − λI) = (1 − λ)(8 − λ) − 6 = 2 − 9λ + λ2 = 0 is the characteristic equation. The eigenvalues of A are the roots of λ2 − 9λ + 2 = 0, which are (9 73)/2.
A nonzero matrix Xi, which has one column and n rows, called a column vector satisfying the equation (A − λI)Xi = 0 and associated with the ith characteristic root λi is called an eigenvector. Vector and Matrix Norms To carry out error analysis for approximate and iterative methods for the solutions of linear systems, one needs notions for vectors in Rn and for matrices that are analogous to the notion of length of a geometric vector. Let Rn denote the set of all vectors with n components, x = (x1, . . . , xn). In dealing with matrices it is convenient to treat vectors in Rn as columns, and so x = (x1, . . . , xn)T; however, we shall here write them simply as row vectors. A norm on Rn is a real-valued function f defined on Rn with the following properties: 1. f(x) ≥ 0 for all x Rn. 2. f(x) = 0 if and only if x = (0, 0, . . . , 0). 3. f(ax) = |a| f(x) for all real numbers a and x Rn. 4. f(x + y) f(x) + f(y) for all x, y Rn. The usual notation for a norm is f(x) = x. The norm of a matrix is κ(A) A A−1 where
If A is a square matrix and if principal submatrices of A are all nonsingular, then we may choose P as the identity in the preceding factorization and obtain A = LU. This factorization is unique if L is normalized (as assumed previously), so that it has unit elements on the main diagonal. Solution of Ax = b by Using LU Factorization Suppose that the indicated system is compatible and that A = LU (the case PA = LU is similarly handled and amounts to rearranging the equations). Let z = Ux. Then Ax = LUx = b implies that Lz = b. Thus to solve Ax = b we first solve Lz = b for z and then solve Ux = z for x. This procedure does not require that A be invertible and can be used to determine all solutions of a compatible system Ax = b. Note that the systems Lz = b and Ux = z are both in triangular forms and thus can be easily solved. The LU decomposition is essentially a Gaussian elimination, arranged for maximum efficiency (Ref. 112). The chief reason for doing an LU decomposition is that it takes fewer multiplications than would be needed to find an inverse. Also, once the LU decomposition has been found, it is possible to solve for multiple right-hand sides with little increase in work. The multiplication count for an n × n matrix and m right-hand sides is 1 1 operation count = n3 − n + mn2 3 3 If an inverse is desired, it can be calculated by solving for the LU decomposition and then solving n problems with right-hand sides consisting of all zeroes except one entry. Thus 4n2/3 − n/3 multiplications are required for the inverse. The determinant is given by n
Det A = aii(i) i=1
where aii(i) are the diagonal elements obtained in the LU decomposition. A tridiagonal matrix is one in which the only nonzero entries lie on the main diagonal and the diagonal just above and just below the main diagonal. The set of equations can be written as ai xi − 1 + bi xi + ci xi + 1 = di The LU decomposition is b1 = b1 for k=2,n do ak ak a′k = , b′k = bk − ck − 1 b′k − 1 b′k − 1 enddo d′1 = d1 for k=2,n do d′k = dk − a′k d′k − 1 enddo xn = d n′ / b′n for k=n−1,1 do d k′ − ck xk + 1 xk = b′k enddo
n A x A = supx ≠ 0 = maxk |ajk| j=1 x
The norm is useful when doing numerical calculations. If the computer’s floating-point precision is 10−6, then κ = 106 indicates an illconditioned matrix. If the floating-point precision is 10−12 (double precision), then a matrix with κ = 1012 may be ill-conditioned. Two other measures are useful and are more easily calculated: (k) |det A| maxk |akk | Ratio = V = , αi = (a 2i1 + a 2i2 + . . . a 2in)1/2 (k) , mink |akk | α1 α2 . . . αn where akk(k) are the diagonal elements of the LU decomposition. MATRIX COMPUTATIONS The principal topics in linear algebra involve systems of linear equations, matrices, vector spaces, linear transformations, eigenvalues and eigenvectors, and least-squares problems. The calculations are routinely done on a computer. LU Factorization of a Matrix To every m × n matrix A there exists a permutation matrix P, a lower triangular matrix L with unit diagonal elements, and an m × n (upper triangular) echelon matrix U such that PA = LU. The Gauss elimination is in essence an algorithm to determine U, P, and L. The permutation matrix P may be needed since it may be necessary in carrying out the Gauss elimination to
The operation count for an n × n matrix with m right-hand sides is 2(n − 1) + m(3n − 2) If
|bi| > |ai| + |ci |
no pivoting is necessary, and this is true for many boundary-value problems and partial-differential equations. Sparse matrices are ones in which the majority of the elements are
NUMERICAL ANALYSIS AND APPROXIMATE METHODS zero. If the structure of the matrix is exploited, the solution time on a computer is greatly reduced. See Refs. 27, 55, 95, 96, 101, and 246. The conjugate gradient method is one method for solving sparse matrix problems, since it only involves multiplication of a matrix times a vector. Thus the sparseness of the matrix is easy to exploit. The conjugate gradient method is an iterative method that converges for sure in n iterations where the matrix is an n × n matrix. See Refs. 142 and 206. The singular value decomposition is useful when the matrix is singular or nearly singular (see Ref. 231). Matrix methods, in particular finding the rank of the matrix, can be used to find the number of independent reactions in a reaction set. If the stoichiometric numbers for the reactions and molecules are put in the form of a matrix, the rank of the matrix gives the number of independent reactions (see Ref. 13). Pivoting in Gauss Elimination It might seem that the Gauss elimination completely disposes of the problem of finding solutions of linear systems, and theoretically it does. In practice, however, things are not so simple. Example Assume three-decimal floating arithmetic (i.e., only the three most significant digits of any number are retained), and solve the following system by Gauss elimination: 0.000100x1 + 1.00x2 = 1.00 1.00x1 + 1.00x2 = 2.00 We obtain 0.100 × 10−3x1 + 0.100 × 101x2 = 0.100 × 101 −0.100 × 105x2 = −0.100 × 105 so that x2 = 1.00 and x1 = 0.00.
3-49
We check our solution by computing the residual vector r = b − Ax: r1 = 0.100 × 101 − 0.100 × 10−3x1 − 0.100 × 101x2 = 0.00 r2 = 0.200 × 101 − 0.100 × 101x1 − 0.100 × 101x2 = 0.100 × 101 The fact that r2 = 1 indicates that our “solution” is not very good. Indeed the exact solution of the system is x1 = 1.00010 and x2 = 0.99990, so the result computed by Gauss elimination is pretty bad. Now reverse the order of the equations (that is, pivot) and solve 0.100 × 101x1 + 0.100 × 101x2 = 0.200 × 101 0.100 × 101x2 = 0.100 × 101 so that x2 = 1.00 and x1 = 1.00. In this case the residual vector is r1 = 0.00 and r2 = 0.100 × 10−3, a considerable improvement over the previous result. In fact, the solution is as good as one could hope for by using three-digit arithmetic.
The moral of the preceding example is that the order of equations can make a large difference in how good an answer is obtained. It should be clear that the poor results in the first case are caused by having the large multiplier (0.100 × 101)/(0.100 × 10−3), which resulted from dividing by a relatively small a11. It is not enough just to avoid zero “pivots”; one must also avoid using pivots that are relatively small. This magnification of errors can be reduced if we arrange that the pivot at any stage is larger in magnitude than any remaining element in the column. If this is done, the multipliers will then be less than or equal to 1 in magnitude. Gauss elimination modified in this manner is called pivotal condensation or partial pivoting. This is routinely done by computer programs.
NUMERICAL APPROXIMATIONS TO SOME EXPRESSIONS APPROXIMATION IDENTITIES For the following relationships the sign means approximately equal to, when X is small: Approximation 1 1X 1X 1+Y 1+YX 1X
Approximation X X1 1 2 −n
(1 X) 1 nX
Approximation
Approximation
(1 X)n 1 nX
X (1 X)−1/2 1 2
(a X)2 a2 2aX
ez 1 + X
sin X X(X rad)
tan X X
2Y + X (Y + X) Y 2
X2 X 2 + X2 Y + small Y 2Y Y
NUMERICAL ANALYSIS AND APPROXIMATE METHODS REFERENCES: General (textbooks that cover at an introductory level a variety of topics that constitute a core of numerical methods for practicing engineers), 2, 3, 4, 22, 56, 59, 70, 77, 133, 135, 143, 150, 155, 219. Numerical solution of nonlinear equations, 153, 171, 237, 302. Numerical solution of ordinary differential equations, 76, 117, 127, 185, 257. Numerical solution of integral equations, 23, 26, 129, 162. Numerical solution of partial differential equations, 11, 76, 127, 133, 155, 210, 251, 286, 287, 213, 233, 253. Spline functions and applications, 38, 56, 70, 230. Finite elements and applications, 5, 29, 83, 130, 164, 210, 241, 281, 287, 303, 304. Fast Fourier transforms, 47, 56, 135, 238. Software, 187, 231.
INTRODUCTION The goal of approximate and numerical methods is to provide convenient techniques for obtaining useful information from mathematical formulations of physical problems. Often this mathematical statement is not solvable by analytical means. Or perhaps analytic solutions are available but in a form that is inconvenient for direct interpretation
numerically. In the first case it is necessary either to attempt to approximate the problem satisfactorily by one which will be amenable to analysis, to obtain an approximate solution to the original problem by numerical means, or to use the two techniques in combination. Numerical techniques therefore do not yield exact results in the sense of the mathematician. Since most numerical calculations are inexact, the concept of error is an important feature. The error associated with an approximate value is defined as True value = approximate value + error The four sources of error are as follows: 1. Gross errors. These result from unpredictable human, mechanical, or electrical mistakes. 2. Round-off errors. These are the consequence of using a number specified by m correct digits to approximate a number which requires more than m digits for its exact specification. For example, approximate the irrational number 2 by 1.414. Such errors are often
3-50
MATHEMATICS
present in experimental data, in which case they may be called inherent errors, due either to empiricism or to the fact that the computer dictates the number of digits. Such errors may be especially damaging in areas such as matrix inversion or the numerical solution of partial differential equations when the number of algebraic operations is extremely large. 3. Truncation errors. These errors arise from the substitution of a finite number of steps for an infinite sequence of steps which would yield the exact result. To illustrate this error consider the infinite series for e−x ⋅ e−x = 1 − x + x2/2 − x3/6 + ET(x), where ET is the truncation error, ET = (1/24)e−εx4, 0 < ε < x. If x is positive, ε is also positive. Hence e−ε < 1. The approximation e−x ≈ 1 − x + x2/2 − x3/6 is in error by a positive amount smaller than (1/24)x4. 4. Inherited errors. These arise as a result of errors occurring in the previous steps of the computational algorithm. The study of errors in a computation is related to the theory of probability. In what follows a relation for the error will be given in certain instances. NUMERICAL SOLUTION OF LINEAR EQUATIONS See the section entitled “Matrix Algebra and Matrix Computation.” NUMERICAL SOLUTION OF NONLINEAR EQUATIONS IN ONE VARIABLE Special Methods for Polynomials Consider a polynomial equation of degree n: P(x) = a0 xn + a1xn − 1 + a2 xn − 2 + ⋅⋅⋅ + an − 1x + an = 0
(3-71)
with real coefficients. P(x) has exactly n roots, which may be real or complex. If all the coefficients of P(x) are integers, then any rational root, say, r/s (r, s integers, having no common divisors) of P(x), must be such that r is an integral divisor of an and s is an integral divisor of a0. Further, any polynomial with rational coefficients may be converted into one with integral coefficients by multiplying by the lowest common multiple of the denominators of the coefficients. Example 3x4 − 5⁄ 3 x2 + 1⁄5 x − 2 = 0. The lowest common multiple of the denominator is 15. Thus multiplying by 15 (which does not change the roots) gives 45x4 − 25x2 + 3x − 30 = 0. The only possible rational roots r/s are such that r may have the values 30, 15, 10, 6, 5, 3, 2, 1. s may have the values 45, 15, 9, 5, 3, 1. The possible rational roots may then be formed from all possible quotients, having no common factor. In addition to these results, one can obtain an upper and lower bound for the real roots by the following device: If a0 > 0 in Eq. (3-71) and if in Eq. (3-71) the first negative coefficient is preceded by k coefficients which are positive or zero, and if G is the greatest of the absolute values of the negative coefficients, then each real root is less k than 1 + G/a 0. Example P(x) = x5 + 3x4 − 7x2 − 40x + 2 = 0. Here a03 = 1, G = 40, and k = 3
Example f(x) = x4 − 13x2 + 4x − 2 = 0 has three changes in sign; therefore, there are either three or one positive roots. f(−x) = x4 − 13x2 − 4x − 2 has one change in sign. Therefore, there is one negative root. General Methods for Nonlinear Equations in One Variable Successive Substitutions Let f(x) = 0 be the nonlinear equation to be solved. If this is rewritten as x = F(x), then an iterative scheme can be set up in the form xk + 1 = F(xk). To start the iteration an initial guess must be obtained graphically or otherwise. The convergence or divergence of the procedure depends upon the method of writing x = F(x), of which there will usually be several forms. However, if a is a root of f(x) = 0, and if |F′(a)| < 1, then for any initial approximation sufficiently close to a, the method converges to a. This process is called first order because the error in xk + 1 is proportional to the first power of the error in xk for large k. Example f(x) = x3 − x − 1 = 0. A rough plot shows a real root of approximately 1.3. The equation can be written in the form x = F(x) in several ways such as x = x3 − 1, x = 1/(x2 − 1), and x = (1 + x)1/3. In the first case F′(x) = 3x2 = 5.07 at x = 1.3, in the second F(1.3) = −5.46, and only in the third case is F′(1.3) < 1. Hence only the third iterative process has a chance to converge. This is illustrated in the following table. Step k
x = x3 − 1
x = 1/(x2 − 1)
x = (1 + x)1/3
0 1 2 3 4
1.3 1.197 0.7150 −0.6345
1.3 1.4493 0.9088 −5.742
1.3 1.32 1.3238 1.3245 1.3247
Another way of writing the equation is xk + 1 = xk + β f(xk). The choice of β is made such that |1 + β df/dx(a)| < 1. Convergence is guaranteed by the theorem given for simultaneous equations.
Methods of Perturbation Let f(x) = 0 be the equation. In general, the iterative relation is xk + 1 = xk − [ f(xk)/ak] where the iteration begins with x0 as an initial approximation and αk as some functional. Newton-Raphson Procedure This variant chooses αk = f ′(xk) where f ′ = df/dx and geometrically consists of replacing the graph of f(x) by the tangent line at x = xk in each successive step. If f′(x) and f ″(x) have the same sign throughout an interval a ≤ x ≤ b containing the solution, with f(a), f(b) of opposite signs, then the process converges starting from any x0 in the interval a ≤ x ≤ b. The process is second order. (0.5)x − 0.5 0.3
Example f(x) = x − 1 + f′ (x) = 1 − 2.3105[0.5]x An approximate root (obtained graphically) is 2.
since we must supply 0 as the coefficient for x . Thus 1 + 40 ≈ 4.42 is an upper bound for the real roots.
Step k
xk
f(xk)
f′(xk)
A lower bound to the real roots may be found by applying the criterion to the equation P(−x).
0 1 2
2 1.6054 1.4632
0.1667 0.0342 0.0055
0.4224 0.2407 0.1620
3
Example P(−x) = −x5 + 3x4 − 7x2 + 40x + 2 = 0, which is equivalent to x5 − 3x4 + 7x2 − 40x − 2 = 0 since a0 must be +. Then a0 = 1, G = 40, and k = 1. Hence −(1 + 40) = −41 is a lower bound. Thus all real roots −41 < r < 4.42. One last result is helpful in getting an estimate of how many positive and negative real roots there are. Descartes Rule The number of positive real roots of a polynomial with real coefficients is either equal to the number of changes in sign v or is less than v by a positive even integer. The number of negative roots of f(x) is either equal to the number of variations of sign of f(−x) or is less than this by a positive even integer.
Method of False Position This variant is commenced by finding x0 and x1 such that f(x0), f(x1) are of opposite signs. Then α1 = slope of secant line joining [x0, f(x0)] and [x1, f(x1)] so that x1 − x0 x2 = x1 − f(x1) f(x1) − f(x0) In each following step αk is the slope of the line joining [xk, f(xk)] to the most recently determined point where f(xj) has the opposite sign from that of f(xk). This method is of first order. If one uses the most recently determined point (regardless of sign), the method is a secant method.
NUMERICAL ANALYSIS AND APPROXIMATE METHODS Method of Wegstein This is a variant of the method of successive substitutions which forces and/or accelerates convergence. The iterative procedure xk + 1 = F(xk) is revised by setting ˆxk + 1 = F(xk) and then taking xk + 1 = qxk + (1 − q)ˆxk + 1, where q is a suitably chosen number which may be taken as constant throughout or may be adjusted at each step. Wegstein found that suitable q’s are: Behavior of successive substitution process
Range of optimum q
Oscillatory convergence Oscillatory divergence Monotonic convergence Monotonic divergence
0
At each step q may be calculated to give a locally optimum value by setting xˆ k + 1 − xˆ k q = xˆ k + 1 − 2 xˆ k + 1 + xˆ k− 1 The Wegstein method is a secant method applied to g(x) x − F(x). Numerical Solution of Simultaneous Nonlinear Equations The techniques illustrated here will be demonstrated for two simultaneous equations f(x, y) = 0, g(x, y) = 0. They immediately generalize to more than two simultaneous equations. Method of Successive Substitutions Write a system of equations as αi = fi(α),
or α = f(α)
The following theorem guarantees convergence. Let α be the solution to αi = fi(α). Assume that given h > 0, there exists a number 0 < µ < 1 such that n ∂fi ≤ µ for |xi − αi| < h, i = 1, . . . , n ∂x j=1 j
x Then
k+1 i
= fi(x ik)
x ik → αi
Step k
xk
yk
f(xk, yk)
g(xk, yk)
0 1 2 3
0.4 0.43673 0.42672 0.42925
0.3 0.24184 0.25573 0.24943
−0.26 0.078 −0.0170 0.0077
0.07 0.0175 −0.007 0.0010
Method of Continuity (Homotopy) In the case of n equations in n unknowns, when n is large, determining the approximate solution may involve considerable effort. In such a case the method of continuity is admirably suited for use on digital computers. It consists basically of the introduction of an extra variable into the n equations fi(x1, x2, . . . , xn) = 0
i = 1, . . . , n
(3-72)
and replacing them by fi(x1, x2, . . . , xn, λ) = 0
i = 1, . . . , n
(3-73)
where λ is introduced in such a way that the functions (3-73) depend in a simple way upon λ and reduce to an easily solvable system for λ = 0 and to the original equations (3-72) for λ = 1. A system of ordinary differential equations, with independent variable λ, is then constructed by differentiating Eqs. (3-73) with respect to λ. There results n ∂fi dxf ∂fi (3-74) +=0 ∂λ j = 1 ∂xj dλ where x1, . . . , xn are considered as functions of λ. Equations (3-74) are integrated, with initial conditions obtained from Eqs. (3-73) with λ = 0, from λ = 0 to λ = 1. If the solution can be continued to λ = 1, the values of x1, . . . , xn for λ = 1 will be a solution of the original equations. If the integration becomes infinite, the parameter λ must be introduced in a different fashion. Integration of the differential equations (which are usually nonlinear in λ) may be accomplished by using techniques described under “Numerical Solution of Ordinary Differential Equations.” Other Methods Other methods can be found in the literature. See Ref. 66. INTERPOLATION AND FINITE DIFFERENCES
as k increases (see Ref. 106). Newton-Raphson Method To solve the set of equations Fi(x1, x2, . . . , xn) = 0,
or Fi({xj}) = 0,
or Fi(x) = 0
one uses a truncated Taylor series to give n ∂F 0 = Fi({x k}) + i j = 1 ∂xj
xk
(x jk + 1 − x jk)
Linear Interpolation If a function f(x) is approximately linear in a certain range, then the ratio f(x1) − f(x0) = f [x0, x1] x1 − x0 is approximately independent of x0, x1 in the range. The linear approximation to the function f(x), x0 < x < x1 then leads to the interpolation formula f(x) ≈ f(x0) + (x − x0)f [x0, x1] x − x0 ≈ f(x0) + [ f(x1) − f(x0)] x1 − x0 1 ≈ [(x1 − x) f(x0) − (x0 − x)f(x1)] x1 − x0
Thus one solves iteratively from one point to another. n
A
k ij
(x
k+1 j
− x ) = −Fi({x }) k
k j
j=1
where
3-51
∂F A kij = i ∂xj
xk
This method requires solution of sets of linear equations until the functions are zero to some tolerance or the changes of the solution between iterations is small enough. Convergence is guaranteed provided the norm of the matrix A is bounded, F(x) is bounded for the initial guess, and the second derivative of F(x) with respect to all variables is bounded. See Refs. 106 and 155. Example f(x, y) = 4x2 + 6x − 4xy + 2y2 − 3 g(x, y) = 2x2 − 4xy + y2
Example Find cosh 0.83 by linear interpolation given cosh 0.8 and cosh 0.9. xi
f(xi)
xi − 0.83
0.8 0.9
1.33743 1.43309
−0.03 +0.07
f(0.83) ≈ 1/0.10[(0.07)(1.33743) − (−0.03)(1.43309)] f(0.83) ≈ 1.36613
By plotting one of the approximate roots is found to be x0 = 0.4, y0 = 0.3. At this point there results ∂f/∂x = 8, ∂f/∂y = −0.4, ∂g/∂x = 0.4, and ∂g/∂y = −1.
Since the true five-place value is 1.36468, it is seen that here linear interpolation gives three significant figures.
8(xk + 1 − xk) − 0.4( yk + 1 − yk) = +0.26
Divided Differences of Higher Order and Higher-Order Interpolation The first-order divided difference f [x0, x1] was defined previously. Divided differences of second and higher order are defined iteratively by
0.4(xk + 1 − xk) − 1( yk + 1 − yk) = −0.07 The first few iteration steps are as follows:
3-52
MATHEMATICS f [x1, x2] − f[x0, x1] f [x0, x1, x2] = x2 − x0 ⯗ f [x1, . . . , xk] − f[x0, x1, . . . , xk − 1] f [x0, x1, . . . , xk] = xk − x0
and a convenient form for computational purposes is k′ f(xj) f [x0, x1, . . . , xk] = j = 0 (xj − x0)(xj − x1) ⋅⋅⋅ (xj − xk) for any k ≥ 0, where the ′ means that the term (xj − xj) is omitted in the denominator. For example, f(x0) f(x1) f(x2) f [x0, x1, x2] = + + (x0 − x1)(x0 − x2) (x1 − x0)(x1 − x2) (x2 − x0)(x2 − x1) If the accuracy afforded by a linear approximation is inadequate, a generally more accurate result may be based upon the assumption that f(x) may be approximated by a polynomial of degree 2 or higher over certain ranges. This assumption leads to Newton’s fundamental interpolation formula with divided differences f(x) ≈ f(x0) + (x − x0) f[x0, x1] + (x − x0)(x − x1) f [x0, x1, x2] + ⋅⋅⋅ + (x − x0)(x − x1) ⋅⋅⋅ (x − xn − 1) f [x0, x1, . . . , xn] + En(x) 1 En(x) = error = f n + 1(ε)π(x) (n + 1)! where minimum (x0, . . . , x) < ε < maximum (x0, x1, . . . , xn, x) and π(x) = (x − x0)(x − x1) ⋅⋅⋅ (x − xn). In order to use the previous equation most effectively one may first form a divided-difference table. For example, for third-order interpolation the difference table is
where
and the finite interpolation formula through the points y0, y1, . . . , yn is written as follows: α (α − 1) α (α − 1) ⋅⋅⋅ (α − n + 1) yα = y0 + α ∆y0 + ∆2y0 + ⋅⋅⋅ + ∆ny0 2! n! Keeping only the first two terms gives a straight line through (x0, y0)–(x1, y1); keeping the first three terms gives a quadratic function of position going through those points plus (x2, y2). The value α = 0 gives x = x0; α = 1 gives x = x1, and so on. Equally Spaced Backward Differences Backward differences are defined by ∇yn = yn − yn − 1 ∇ 2yn = ∇yn − ∇yn − 1 = yn − 2 yn − 1 + yn − 2 The interpolation polynomial of order n through the points y0, y−1, . . . , y−n is α (α + 1) α (α + 1) ⋅⋅⋅ (α + n − 1) yα = y0 + α ∇y0 + ∇2y0 + ⋅⋅⋅ + ∇ny0 2! n! The value of α = 0 gives x = x0; α = −1 gives x = x−1, and so on. Alternatively, the interpolation polynomial of order n through the points y1, y0, y−1, . . . , y−n is α (α − 1) yα = y1 + (α − 1) ∇y1 + ∇ 2y1 2! (α − 1) α (α + 1) ⋅⋅⋅ (α + n − 2) + ⋅⋅⋅ + ∇ny1 n! Now α = 1 gives x = x1; α = 0 gives x = x0. Central Differences The central difference denoted by h h δf(x) = f x + − f x − 2 2
h h δ2f(x) = δn − 1 f x + − δn − 1 f x − 2 2 is useful for calculating at the interior points of tabulated data. Lagrange Interpolation Formulas A global polynomial is defined over the entire region of space m
where each entry is given by taking the difference between diagonally adjacent entries to the left, divided by the abscissas corresponding to the ordinates intercepted by the diagonals passing through the calculated entry. Equally Spaced Forward Differences If the ordinates are equally spaced, i.e., xj − xj − 1 = ∆x for all j, then the first differences are denoted by ∆f(x0) = f(x1) − f(x0) or ∆y0 = y1 − y0, where y = f(x). The differences of these first differences, called second differences, are denoted by ∆2y0, ∆2y1, . . . , ∆2yn. Thus ∆2y0 = ∆y1 − ∆y0 = y2 − y1 − y1 + y0 = y2 − 2y1 + y0 And in general j
∆ y0 = (−1) j
n
n=0
j y n j−n
j! j where = = binomial coefficients. n n!( j − n)! If the ordinates are equally spaced, xn + 1 − xn = ∆x yn = y(xn) then the first and second differences are denoted by ∆yn = yn + 1 − yn ∆2yn = ∆yn + 1 − ∆yn = yn + 2 − 2yn + 1 + yn A new variable is defined xα − x0 α= ∆x
Pm(x) = cj x j j=0
This polynomial is of degree m (highest power is x m) and order m + 1 (m + 1 parameters {cj}). If we are given a set of m + 1 points y1 = f(x1), y2 = f(x2), . . . , ym + 1 = f(xm + 1) then Lagrange’s formula gives a polynomial of degree m that goes through the m + 1 points: (x − x2) (x − x3) ⋅⋅⋅ (x − xm + 1) Pm(x) = y1 (x1 − x2) (x1 − x3) ⋅⋅⋅ (x1 − xm + 1) (x − x1) (x − x3) ⋅⋅⋅ (x − xm + 1) + y2 + ⋅⋅⋅ (x2 − x1) (x2 − x3) ⋅⋅⋅ (x2 − xm + 1) (x − x1) (x − x2) ⋅⋅⋅ (x − xm + 1) + ym + 1 (xm + 1 − x1) (xm + 1 − x2) ⋅⋅⋅ (xm + 1 − xm) Note that each coefficient of yj is a polynomial of degree m that vanishes at the points {xj} (except for one value of j) and takes the value of 1.0 at that point: Pm(xj) = yj,
j = 1, 2, . . . , m + 1
If the function f(x) is known, the error in the approximation is, per Ref. 14, |xm + 1 − x1|m + 1 |error(x)| ≤ maxx1 ≤ x ≤ xm + 1 | f (m + 2)(x)| (m + 2)! The evaluation of Pm(x) at a point other than at the defining points can be made with Neville’s algorithm (Ref. 231). Let P1 be the value at x of the unique function passing through the point (x1, y1); or P1 = y1. Let
NUMERICAL ANALYSIS AND APPROXIMATE METHODS P12 be the value at x of the unique polynomial passing through the points x1 and x2. Likewise, Pijk . . . r is the unique polynomial passing through the points xi, xj, xk, . . . xr. Then use the table x1 y1 = P1 x2 y2 = P2 x3 y3 = P3 x4 y4 = P4
3-53
f 0′ = 1/10h [−2f−2 − f−1 + f1 + 2f2] If derivatives are required at end points, with all points and ordinates to one side, the derivatives are f0′ = 1/20h [−21 f0 + 13 f1 + 17 f2 − 9 f3]
P12
P123
P23
P234
P34
f0′ = 1/20h [21 f0 − 13 f−1 − 17 f−2 + 9 f−1] f−1 ′ = 1/20h [ 11 f0 − 3 f−1 − 7 f−2 − f−1]
These entries are defined using (x − xi + m) Pi(i + 1) . . . (i + m − 1) + (xi − x) P(i + 1)(i + 2) . . . (i + m) Pi(i + 1) . . . (i + m) = xi − xi + m For example, consider P1234. The terms on the right-hand side involve P123 and P234. The “parents,” P123 and P234, already agree at points 2 and 3. Here i = 1, m = 3; thus, the parents agree at xi + 1, . . . , xi + m − 1 already. The formula makes Pi(i + 1) . . . (i + m) agree with the function at the additional points xi + m and xi. Thus, Pi(i + 1) . . . (i + m) agrees with the function at all the points {xi, xi + 1, . . . xi + m}. NUMERICAL DIFFERENTIATION Numerical differentiation should be avoided whenever possible, particularly when data are empirical and subject to appreciable observation errors. Errors in data can affect numerical derivatives quite strongly; i.e., differentiation is a roughening process. When such a calculation must be made, it is usually desirable first to smooth the data to a certain extent. Use of Interpolation Formula If the data are given over equidistant values of the independent variable x, an interpolation formula such as the Newton formula (see Refs. 143 and 185) may be used and the resulting formula differentiated analytically. If the independent variable is not at equidistant values, then Lagrange’s formulas must be used. By differentiating three- and five-point Lagrange interpolation formulas the following differentiation formulas result for equally spaced tabular points: Three-Point Formulas Let x0, x1, x2 be the three points. 1 h2 f ′(x0) = [−3f(x0) + 4f(x1) − f(x2)] + f ′″(ε) 2h 3 1 h2 f ′(x1) = [−f(x0) + f(x2)] − f ′″(ε) 2h 6 h2 1 f ′(x2) = [ f(x0) − 4f(x1) + 3f(x2)] + f ′″(ε) 2h 3 where the last term is an error term min. xj < ε < max. xj. j
f1′ = 1/20h [−11 f0 + 3 f1 + 7 f2 + f3]
P1234
j
Smoothing Techniques These techniques involve the approximation of the tabular data by a least-squares fit of the data by using some known functional form, usually a polynomial (for the concept of least squares see “Statistics”). In place of approximating f(x) by a single least-squares polynomial of degree n over the entire range of the tabulation, it is often desirable to replace each tabulated value by the value taken on by a least-squares polynomial of degree n relevant to a subrange of 2M + 1 points centered, when possible, at the point for which the entry is to be modified. Thus each smoothed value replaces a tabulated value. Let fj = f(xj) be the tabular points and yj = smoothed values. First-Degree Least Squares with Three Points y0 = j[5f0 + 2 f1 − f2] y1 = s[ f0 + f1 + f2] y2 = j[−f0 + 2 f1 + 5f2] Second-Degree Least Squares with Five Points For five evenly spaced points x−2, x−1, x0, x1, and x2 (separated by distance h) and their ordinates f−2, f−1, f0, f1, and f2, assume a parabola is fit by least squares. Then the derivative at the center point is
Numerical Derivatives The results given above can be used to obtain numerical derivatives when solving problems on the computer, in particular for the Newton-Raphson method and homotopy methods. Suppose one has a program, subroutine, or other function evaluation device that will calculate f given x. One can estimate the value of the first derivative at x0 using f [x0(1 + ε)] − f [x0] df ≈ dx x0 ε (a first-order formula) or f [x0(1 + ε)] − f [x0(1 − ε)] df ≈ dx x0 2ε
(a second-order formula). The value of ε is important; a value of 10−6 is typical, but smaller or larger values may be necessary depending on the computer precision and the application. One must also be sure that the value of x0 is not zero and use a different increment in that case. NUMERICAL INTEGRATION (QUADRATURE) A multitude of formulas have been developed to accomplish numerical integration, which consists of computing the value of a definite integral from a set of numerical values of the integrand. Newton-Cotes Integration Formulas (Equally Spaced Ordinates) for Functions of One Variable The definite integral b ∫ a f(x) dx is to be evaluated. Trapezoidal Rule This formula consists of subdividing the interval a ≤ x ≤ b into n subintervals a to a + h, a + h to a + 2h, . . . and replacing the graph of f(x) by the result of joining the ends of adjacent ordinates by line segments. If fj = f(xj) = f(a + jh), f0 = f(a), fn = f(b), the integration formula is b h f(x) dx = [ f0 + 2f1 + 2f2 + ⋅⋅⋅ + 2fn − 1 + fn] + En a 2
nh3 (b − a)3 |En| = |f″ (ε)| = | f ″(ε)| a<ε
nh5 (b − a)5 (IV) |En| = | f (IV)(ε)| = | f (ε)| a<ε
x = a + (b − a)u,
dx = (b − a)du
f(x) dx = (b − a) f(u) du b
a
1
0
3-54
MATHEMATICS
f(u) du = W f(u ) 1
m
i
0
i
i=1
The quadrature is exact when f is a polynomial of degree 2m − 1 in x. Because there are m weights and m Gauss points, we have 2m parameters that are chosen to exactly represent a polynomial of degree 2m − 1, which has 2m parameters. The Gauss points and weights are given in the table. Gaussian Quadrature Points and Weights m
ui
Wi
2
0.21132 48654 0.78867 51346 0.11270 16654 0.50000 00000 0.88729 83346 0.06943 18442 0.33000 94783 0.66999 05218 0.93056 81558 0.04691 00771 0.23076 53450 0.50000 00000 0.76923 46551 0.95308 99230
0.50000 00000 0.50000 00000 0.27777 77778 0.44444 44445 0.27777 77778 0.17392 74226 0.32607 25774 0.32607 25774 0.17392 74226 0.11846 34425 0.23931 43353 0.28444 44444 0.23931 43353 0.11846 34425
3
4
5
Example Evaluate the same integral (3-75) using the trapezoid rule and then apply the Romberg method. To achieve four-digit accuracy, any result from J2 through L1 are suitable, even though the base results (I1–I4) are not that close. I1 = 0.967058363
I2 = 0.923704741 J1 = 0.909253534
f(x) dx = b
1/a
a
1/b
b
g2(x)
sin x dx
f(x, y) dx dy =
g1(x)
(3-75)
Using the Gaussian quadrature formulas gives the following values for various values of m. Clearly, three internal points, requiring evaluation of the integrand at only three points, gives excellent results.
1 2 3 4 5
0.908185 0.910089 0.909336367 0.909330666 0.909330674
Romberg’s Method Romberg’s method uses extrapolation techniques to improve the answer (Ref. 231). If we let I1 be the value of the integral obtained using interval size h = ∆x, and I2 be the value of I obtained when using interval size h/2, and I0 the true value of I, then the error in a method is approximately hm, or I1 ≈ I0 + chm h I2 ≈ I0 + c 2
m
Replacing the ≈ by an equality (an approximation) and solving for c and I0 gives 2mI2 − I1 I0 = 2m − 1 This process can also be used to obtain I1, I2, . . . , by halving h each time, and then calculating new estimates from each pair, calling them J1, J2, . . . ; that is, in the formula above, replace I0 with J1. The formulas are reapplied for each pair of J to obtain K1, K2, . . . The process continues until the required tolerance is obtained. I1
I2 I3 I4 J1 J2 J3 K1 K2 L1
Romberg’s method is most useful for a low-order method (small m) because significant improvement is then possible.
b
g2(x)
f(x, y) dy
Gaussian quadrature can also be used in two dimensions, provided the integration is on a square or can be transformed to one. (Domain transformations might be used to convert the domain to a square.)
f(x, y) dx dy = W 1
I
G(x) dx g1(x)
0
m
ab > 0
a
G(x) =
1
−x
1 1 2 f du u u
Two-Dimensional Formula Two-dimensional integrals can be calculated by breaking down the integral into one-dimensional integrals. a
e
I4 = 0.910227902 J3 = 0.909330366 K2 = 0.909331898 L1 = 0.909325916
Singularities When the integrand has singularities, a variety of techniques can be tried. The integral may be divided into one part that can be integrated analytically near the singularity and another part that is integrated numerically. Sometimes a change of argument allows analytical integration. Series expansion might be helpful, too. When the domain is infinite, it is possible to use Gauss-Legendre or Gauss-Hermite quadrature. Also a transformation can be made (Ref. 26). For example, let u = 1/x and then
Example Calculate the value of the following integral. I=
I3 = 0.912920511 J2 = 0.909325768 K1 = 0.909349846
1
mx
xi
0
0
i=1
my
W
yj
f(xi, yj)
j=1
NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS AS INITIAL VALUE PROBLEMS A differential equation for a function that depends on only one variable, often time, is called an ordinary differential equation. The general solution to the differential equation includes many possibilities; the boundary or initial conditions are needed to specify which of those are desired. If all conditions are at one point, then the problem is an initial value problem and can be integrated from that point on. If some of the conditions are available at one point and others at another point, then the ordinary differential equations become two-point boundary value problems, which are treated in the next section. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and in all models where there are no spatial gradients in the unknowns. A higher-order differential equation y(n) + F(y(n − 1), y(n − 2), . . . , y′, y) = 0 with initial conditions Gi(y(n − 1)(0), y(n − 2)(0), . . . , y(0), y(0)) = 0,
i = 1, . . . , n
can be converted into a set of first-order equations using d(i − 1)y d dyi − 1 = y(i − 2) = yi y(i − 1) = dt(i − 1) dt dt The higher-order equation can be written as a set of first-order equations. dy1 = y2 dt dy2 = y3 dt
NUMERICAL ANALYSIS AND APPROXIMATE METHODS dy3 = y4 dt ... dyn = −F(yn − 1, yn − 2, . . . , y2, y1) dt The initial conditions would have to be specified for variables y1(0), . . . , yn(0), or equivalently y(0), . . . , y(n − 1)(0). The set of equations is then written as dy = f(y, t) dt All the methods in this section are described for a single equation; the methods apply to multiple equations. See Refs. 106 and 185 for more details. Euler’s method is first-order. yn + 1 = yn + ∆t f(yn) and errors are proportional to ∆t. The second-order Adams-Bashforth method is ∆t yn + 1 = yn + [3 f(yn) − f(yn − 1)] 2 Errors are proportional to ∆ t2, and high-order methods are available. Notice that the higher-order explicit methods require knowing the solution (or the right-hand side) evaluated at times in the past. Since these were calculated to get to the current time, this presents no problem except for starting the problem. Then it may be necessary to use Euler’s method with a very small step size for several steps in order to generate starting values at a succession of time points. The error terms, order of the method, function evaluations per step, and stability limitations are listed in Ref. 106. The advantage of the high-order Adams-Bashforth method is that it uses only one function evaluation per step yet achieves high-order accuracy. The disadvantage is the necessity of using another method to start. Runge-Kutta methods are explicit methods that use several function evaluations for each time step. Runge-Kutta methods are traditionally written for f(t, y). The first-order Runge-Kutta method is Euler’s method. A second-order Runge-Kutta method is ∆t yn + 1 = yn + [ f n + f(tn + ∆t, yn + ∆t f n)] 2 while the midpoint scheme is also a second-order Runge-Kutta method. ∆t ∆t yn + 1 = yn + ∆t f tn + , yn + f n 2 2 A popular fourth-order Runge-Kutta method is the Runge-KuttaFeldberg formulas (Ref. 111), which have the property that the method is fourth-order but achieves fifth-order accuracy. The popular integration package RKF45 is based on this method.
k1 = ∆t f(tn, yn) ∆t k1 k2 = ∆t f tn + , yn + 4 4
12 1932 7200 7296 k4 = ∆t f tn + ∆t, yn + k1 − k2 + k3 13 2197 2197 2197
439 3680 845 k5 = ∆t f tn + ∆t, yn + k1 − 8k2 + k3 − k4 216 513 4104
∆t 8 3544 1859 11 k6 = ∆t f tn + , yn − k1 + 2k2 − k3 + k4 − k5 2 27 2565 4104 40
1 25 1408 2197 yn + 1 = yn + k1 + k3 + k4 − k5 216 2565 4104 5
2 16 6656 28561 9 zn + 1 = yn + k1 + k3 + k4 − k5 + k6 135 12825 56430 50 55 The value of yn + 1 − zn + 1 is an estimate of the error in yn + 1 and can be used in step-size control schemes. Usually one would use a high-order method to achieve high accuracy. The Runge-Kutta-Feldberg method is popular because it is high order and does not require a starting method (as does an AdamsBashforth method). However, it does require four function evaluations per time step, or four times as many as a fourth-order Adams-Bashforth method. For problems in which the function evaluations are a significant portion of the calculation time, this might be important. Given the speed of present-day computers and the widespread availability of microcomputers (which can be run while you are doing something else, if need be), the efficiency of the methods is most important only for very large problems that are going to be solved many times. For other problems, the most important criterion for choosing a method is probably the time the user spends setting up the problem. The stability limits for the explicit methods are based on the largest eigenvalue of the linearized system of equations (see Ref. 106). For linear problems, the eigenvalues do not change, so that the stability and oscillation limits must be satisfied for every eigenvalue of the matrix A. When solving nonlinear problems, the equations are linearized about the solution at the local time, and the analysis applies for small changes in time, after which a new analysis about the new solution must be made. Thus, for nonlinear problems, the eigenvalues keep changing, and the largest stable time step changes, too. The stability limits are: Euler method, λ ∆t ≤ 2 Runge-Kutta, 2nd order, λ ∆t < 2 Runge-Kutta-Feldberg, λ ∆t < 3.0 Richardson extrapolation can be used to improve the accuracy of a method. Suppose we step forward one step ∆t with a pth-order method. Then redo the problem, this time stepping forward from the same initial point, but in two steps of length ∆t/2, thus ending at the same point. Call the solution of the one-step calculation y1 and the solution of the two-step calculation y2. Then an improved solution at the new time is given by 2 py2 − y1 y= 2p − 1 This gives a good estimate provided ∆t is small enough that the method is truly convergent with order p. This process can also be repeated in the same way Romberg’s method was used for quadrature. The error term in the various methods can be used to deduce a step size that will give a user-specified accuracy. Most packages today are based on a user-specified tolerance; the step-size is changed during the calculation to achieve that accuracy. The accuracy itself is not guaranteed, but it improves as the tolerance is decreased. Implicit Methods By using different interpolation formulas involving yn + 1, it is possible to derive implicit integration methods. Implicit methods result in a nonlinear equation to be solved for yn + 1 so that iterative methods must be used. The backward Euler method is a first-order method. yn + 1 = yn + ∆t f(yn + 1)
3 3 9 k3 = ∆t f t + ∆t, yn + k1 + k2 8 32 32 n
3-55
Errors are proportional to ∆t for small ∆t. The trapezoid rule is a second-order method. ∆t yn + 1 = yn + [ f(yn) + f(yn + 1)] 2 Errors are proportional to ∆ t2 for small ∆ t. When the trapezoid rule is used with the finite difference method for solving partial differential equations, it is called the Crank-Nicolson method. The implicit methods are stable for any step size but do require the solution of a set of nonlinear equations, which must be solved iteratively. The set of equations can be solved using the successive substitution method or Newton-Raphson method. See Ref. 36 for an application to dynamic distillation problems. The best packages for stiff equations (see below) use Gear’s back-
3-56
MATHEMATICS
ward difference formulas. The formulas of various orders are, per Refs. 59 and 117, (1) yn + 1 = yn + ∆t f(yn + 1) 4 1 2 (2) yn + 1 = yn − yn − 1 + ∆t f(yn + 1) 3 3 3 18 9 2 6 (3) yn + 1 = yn − yn − 1 + yn − 2 + ∆t f(yn + 1) 11 11 11 11 48 36 16 3 12 (4) yn + 1 = yn − yn − 1 + yn − 2 − yn − 3 + ∆t f(yn + 1) 25 25 25 25 25 300 300 200 75 12 (5) yn + 1 = yn − yn − 1 + yn − 2 − yn − 3 + yn − 4 137 137 137 137 137 60 + ∆t f(yn + 1) 137 Stiffness The concept of stiffness is described for a system of linear equations. dy =Ay dt Let λi be the eigenvalues of the matrix A (Ref. 267). Then, per Ref. 181, the stiffness ratio is defined as maxi |Re (λi)| SR = mini |Re (λi)| SR = 20 is not stiff, SR = 103 is stiff, and SR = 106 is very stiff. If the problem is nonlinear, then the solution is expanded about the current state. n dyi ∂fi = fi [y(tn)] + [yj − yj(tn)] dt j = 1 ∂yj The question of stiffness then depends on the solution at the current time. Consequently nonlinear problems can be stiff during one time period and not stiff during another. While the chemical engineer may not actually calculate the eigenvalues, it is useful to know that they determine the stability and accuracy of the numerical scheme and the step size used. Problems are stiff when the time constants for different phenomena have very different magnitudes. Consider flow through a packed bed reactor. The time constants for different phenomena are: 1. Time for device flow-through L φAL tflow = = u Q where Q is the volumetric flow rate, A is the cross sectional area, L is the length of the packed bed, and φ is the void fraction; 2. Time for reaction 1 tr × n = k where k is a rate constant (time−1); 3. Time for diffusion inside the catalyst εR2 tinternal diffusion = De where ε is the porosity of the catalyst, R is the catalyst radius, and De is the effective diffusion coefficient inside the catalyst; 4. Time for heat transfer is R2 ρsCsR2 tinternal heat transfer = = α ke where ρs is the catalyst density, Cs is the catalyst heat capacity per unit mass, ke is the effective thermal conductivity of the catalyst, and α is the thermal diffusivity. For example, in the model of a catalytic converter for an automobile (Ref. 103), the time constants for internal diffusion was 0.3 seconds; internal heat transfer, 21 seconds; and device flow-through, 0.003 seconds. The device flow-through is so fast that it might as well be instantaneous. The stiffness is approximately 7000.
Implicit methods must be used to integrate the equations. Alternatively, a quasistate model can be developed (Ref. 239). Differential-Algebraic Systems Sometimes models involve ordinary differential equations subject to some algebraic constraints. For example, the equations governing one equilibrium stage (as in a distillation column) are dxn M = Vn + 1yn + 1 − Lnxn − V nyn + Ln − 1xn − 1 dt xn − 1 − xn = En(xn − 1 − x*,n) N
x =1 i
i=1
where x and y are the mole fraction in the liquid and vapor, respectively; L and V are liquid and vapor flow rates, respectively; M is the holdup; and the superscript is the stage number. The efficiency is E, and the concentration in equilibrium with the vapor is x*. The first equation is an ordinary differential equation for the mass of one component on the stage, while the third equation represents a constraint that the mass fractions add to one. This is a differential-algebraic system of equations. Differential-algebraic equations can be written in the general notation dy F t, y, = 0 dt
To solve the general problem using the backward Euler method, replace the nonlinear differential equation with the nonlinear algebraic equation for one step. yn + 1 − yn F t, yn + 1, = 0 ∆t This equation must be solved for yn + 1. The Newton-Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, the higher-order backward-difference Gear methods are used in DASSL (Ref. 224). Differential-algebraic systems are more complicated than differential systems because the solution may not always be defined. Pontelides et al. (Ref. 226) introduced the term index to identify the possible problems. The index is defined as the minimum number of times the equations need to be differentiated with respect to time to convert the system to a set of ordinary differential equations. These higher derivatives may not exist, and the process places limits on which variables can be given initial values. Sometimes the initial values must be constrained by the algebraic equations (Ref. 226). For a differential-algebraic system modeling a distillation tower, Ref. 226 shows that the index depends on the specification of pressure for the column. Byrne and Ponzi (Ref. 58) also list several chemical engineering examples of differential-algebraic systems and solve one involving two-phase flow. Computer Software Efficient computer packages are available for solving ordinary differential equations as initial value problems. The packages are widely available and good enough that most chemical engineers use them and do not write their own. Here we discuss three of them: RKF45, LSODE, and EPISODE. In each of the packages, the user specifies the differential equation to be solved and a desired error criterion. The package then integrates in time and adjusts the step size to achieve the error criterion within the limitations imposed by stability. A popular explicit, Runge-Kutta package is RKF45, developed by Forsythe et al. (Ref. 111). The method is based on the Runge-KuttaFeldberg formulas. Notice there that an estimate of the truncation error at each step is available. Then the step size can be reduced until this estimate is below the user-specified tolerance. The method is thus automatic, and the user is assured of the results. Note, however, that the tolerance is set on the local truncation error, namely from one step to another, whereas the user is usually interested in the global truncation error, or the error after several steps. The global error is generally made smaller by making the tolerance smaller, but the absolute accuracy is not the same as the tolerance. If the problem is stiff, then very
NUMERICAL ANALYSIS AND APPROXIMATE METHODS small step sizes are used; the computation becomes very lengthy. The RKF45 code discovers this and returns control to the user with a message indicating the problem is too hard to solve with RKF45. A popular implicit package is LSODE, a version of Gear’s method (Ref. 117) written by Alan Hindmarsh at Lawrence Livermore Laboratory (Ref. 148). In this package, the user specifies the differential equation to be solved and the tolerance desired. Now the method is implicit and therefore stable for any step size. The accuracy may not be acceptable, however, and sets of nonlinear equations must be solved. Thus, in practice the step size is limited but not nearly so much as in the Runge-Kutta methods. In these packages, both the step size and order of the method are adjusted by the package. Suppose we are calculating with a kth order method. The truncation error is determined by the (k + 1)th order derivative. This is estimated using difference formulas and the values of the right-hand sides at previous times. An estimate is also made for the kth and (k + 2)th derivative. Then it is possible to estimate the error in a (k − 1)th order method, a kth order method, and a (k + 1)th order method. Furthermore, the step size needed to satisfy the tolerance with each of these methods can be determined. Then we can choose the method and step size for the next step that achieves the biggest step, with appropriate adjustments due to the different work required for each order. The package generally starts with a very small step size and a first-order method, the backward Euler method. Then it integrates along, adjusting the order up (and later down) depending on the error estimates. The user is thus assured that the local truncation error meets the tolerance. There is a further difficulty, since the set of nonlinear equations must be solved. Usually a good guess of the solution is available, since the solution is evolving in time and past history can be extrapolated. Thus, the Newton-Raphson method will usually converge. The package protects itself, though, by only doing three iterations. If convergence is not reached within this many iterations, then the step size is reduced and the calculation is redone for that time step. The convergence theorem for the Newton-Raphson method (p. 3-50) indicates that the method will converge if the step size is small enough. Thus the method is guaranteed to work. Further economies are possible. The Jacobian needed in the Newton-Raphson method can be fixed over several time steps. Then, if the iteration does not converge, the Jacobian can be reevaluated at the current time-step. If the iteration still does not converge, then the step-size is reduced and a new Jacobian is evaluated. Also the successive substitution method can be used, which is even faster, except that it may not converge. However, it, too, will converge if the time step is small enough. Comparisons of the methods and additional details are provided for chemical engineering problems by Refs. 59 and 106. Generally, the Runge-Kutta methods give extremely good accuracy, especially when the step size is kept small for stability reasons. When the computation time is comparable for LSODE and RKF45, the RKF45 package generally gives much more accurate results. The RKF45 package is unsuitable, however, for many chemical reactor problems because they are so stiff. Generally, though, standard packages must have a highorder explicit method (usually a version of Runge-Kutta) and a multistep, implicit method (usually a version of GEAR, EPISODE, or LSODE). The package DASSL (Ref. 224) uses similar principles to solve the differential-algebraic systems. The software described here is available by electronic mail over the Internet. Sending the message send index to [email protected]
will retrieve an index and descriptions of how to obtain the software. Stability, Bifurcations, Limit Cycles Some aspects of this subject involve the solution of nonlinear equations; other aspects involve the integration of ordinary differential equations; applications include chaos and fractals as well as unusual operation of some chemical engineering equipment. Ref. 176 gives an excellent introduction to the subject and the details needed to apply the methods. Ref. 66 gives more details of the algorithms. A concise survey with some chemical engineering examples is given in Ref. 91. Bifurcation results are closely connected with stability of the steady states, which is essentially a transient phenomenon.
3-57
Sensitivity Analysis When solving differential equations, it is frequently necessary to know the solution as well as the sensitivity of the solution to the value of a parameter. Such information is useful when doing parameter estimation (to find the best set of parameters for a model) and for deciding if a parameter needs to be measured accurately. See Ref. 105. ORDINARY DIFFERENTIAL EQUATIONS-BOUNDARY VALUE PROBLEMS Diffusion problems in one dimension lead to boundary value problems. The boundary conditions are applied at two different spatial locations: at one side the concentration may be fixed and at the other side the flux may be fixed. Because the conditions are specified at two different locations, the problems are not initial value in character. It is not possible to begin at one position and integrate directly because at least one of the conditions is specified somewhere else and there are not enough conditions to begin the calculation. Thus, methods have been developed especially for boundary value problems. Shooting Methods The first method is one that utilizes the techniques for initial value problems but allows for an iterative calculation to satisfy all the boundary conditions. Consider the nonlinear boundary value problem d 2y dy 2 = f x, y, , y(0) = α, y(1) = β dx dx Convert this second-order equation into two first-order equations along with the boundary conditions written to include a parameter s to represent the unknown value of v(0) = dy/dx(0). dy dv = v, = f(x, y, v), y(0) = α, v(0) = s dx dx The parameter s is chosen so that the last boundary condition is satisfied: y(1) = β. Define the function
χ(s) = y(1, s) − β and iterate on s to make χ(s) = 0. Note that the condition at x = 0 is satisfied for any s, the differential equation is satisfied by the integration routine, and only the last boundary condition is yet to be satisfied. Both successive substitution and the Newton-Raphson methods can be used. The technique can be used when the boundary conditions are more general and convergence can be proved (see Refs. 106 and 167). Computer software exists: the IMSL program DTPTB uses DVERK, which employs Runge-Kutta integration to integrate the ordinary differential equations (Ref. 55). Finite Difference Method To apply the finite difference method, we first spread grid points through the domain. Figure 3-49 shows a uniform mesh of n points (nonuniform meshes are possible, too). The unknown, here c(x), at a grid point xi is assigned the symbol ci = c(xi). The finite difference method can be derived easily by using a Taylor expansion of the solution about this point. Expressions for the derivatives are: dc dc ci + 1 − ci d 2c ∆x ci − ci − 1 d 2c ∆x = − 2 + ⋅⋅⋅ , = + 2 + ⋅⋅⋅ dx i ∆x dx i 2 dx i ∆x dx i 2
dc ci + 1 − ci − 1 d c ∆x = − 3 dx i 2∆x dx i 3! The truncation error in the first two expressions is proportional to ∆x, and the methods are said to be first-order. The truncation error in the third expression is proportional to ∆x2, and the method is said to be second-order. Usually the last equation is used to insure the best accuracy. The finite difference representation of the second derivative is:
FIG. 3-49
3
Finite difference mesh; ∆x uniform.
2
3-58
MATHEMATICS
d 2c ci + 1 − 2ci + ci − 1 d4c 2∆x2 − 4 + ⋅⋅⋅ 2 = dx i ∆x2 dx i 4! The truncation error is proportional to ∆x2. To solve a differential equation, it is evaluated at a point i and then these expressions are inserted for the derivatives.
FIG. 3-50
Example Consider the equation for convection, diffusion, and reaction in
Finite difference grid with variable spacing.
a tubular reactor. 1 d 2c dc 2 − = Da R(c) Pe dx dx The finite difference representation is 1 ci + 1 − 2ci + ci − 1 ci + 1 − ci − 1 − = Da R(ci) Pe ∆x2 2∆x This equation is written for i = 2 to n − 1, or the internal points. The equations would then be coupled but would also involve the values of c1 and cn, as well. These are determined from the boundary conditions. If the boundary condition involves a derivative, it is important that the derivatives be evaluated using points that exist. Three possibilities exist: dc c2 − c1 = dx 1 ∆x
dc −3c1 + 4c2 − c3 = dx 1 2∆x The third alternative is to add a false point, outside the domain, as c0 = c(x = −∆x). c2 − c0 dc = dx 1 2∆x Since this equation introduces a new variable, c0, another equation is needed and is obtained by writing the finite difference equation for i = 1, too. The sets of equations can be solved using the Newton-Raphson method. The first form of the derivative gives a tridiagonal system of equations, and the standard routines for solving tridiagonal equations suffice. For the other two options, some manipulation is necessary to put them into a tridiagonal form (see Ref. 105). Frequently, the transport coefficients, such as diffusion coefficient or thermal conductivity, depend on the dependent variable, concentration, or temperature, respectively. Then the differential equation might look like dc d D(c) = 0 dx dx This could be written as two equations. dJ dc −=0 J = −D(c) dx dx Because the coefficient depends on c, the equations are more complicated. A finite difference method can be written in terms of the fluxes at the midpoints, i + 1/2. ci + 1 − ci Ji + 1/2 − Ji − 1/2 − =0 Ji +1/2 = −D(ci + 1/2) ∆x ∆x These are combined to give the complete equation. D(ci + 1/2) (ci + 1 − ci) − D(ci − 1/2) (ci − ci − 1) =0 ∆x2 This represents a set of nonlinear algebraic equations that can be solved with the Newton-Raphson method. However, in this case, a viable iterative strategy is to evaluate the transport coefficients at the last value and then solve D(cki + 1/2) (cik++11 − c ik + 1) − D(cik− 1/2) (cik + 1 − c ik−+11) =0 ∆x2 The advantage of this approach is that it is easier to program than a full NewtonRaphson method. If the transport coefficients do not vary radically, then the method converges. If the method does not converge, then it may be necessary to use the full Newton-Raphson method. There are three common ways to evaluate the transport coefficient at the midpoint. The first one uses the transport coefficient evaluated at the average value of the solutions on either side. 1 D (ci + 1/2) ≈ D (ci + 1 + ci) 2 The truncation error of this approach is ∆x2 (Ref. 106). The second approach uses the average of the transport coefficients on either side. 1 D (ci + 1/2) ≈ [D(ci + 1) + D(ci)] 2
The truncation error of this approach is also ∆x2 (Ref. 106). The third approach uses an “upstream” transport coefficient. D (ci + 1/2) ≈ D(ci + 1),
when D (ci + 1) > D(ci)
D (ci + 1/2) ≈ D(ci),
when D (ci + 1) < D(ci)
This approach is used when the transport coefficients vary over several orders of magnitude, and the “upstream” direction is defined as the one in which the transport coefficient is larger. The truncation error of this approach is only ∆x (Refs. 106 and 107), but this approach is useful if the numerical solutions show unrealistic oscillations. If the grid spacing is not uniform, the formulas must be revised. The notation is shown in Fig. 3-50. The finite-difference form of the equations is then Ji + 1/2 − Ji − 1/2 ci + 1 − ci ci − ci − 1 − = 0 Ji + 1/2 = −Di + 1/2 , Ji − 1/2 = −Di − 1/2 a(∆xi + ∆xi + 1) ∆xi + 1 ∆xi If average diffusion coefficients are used, then the finite difference equation is as follows.
1 1 1 (Di + 1 + Di) (ci + 1 − ci) − (Di + Di − 1) (ci − ci − 1) = 0 ∆xi + 1 + ∆xi ∆xi + 1 ∆xi Rigorous error bounds are discussed for linear ordinary differential equations solved with the finite difference method by Isaacson and Keller (Ref. 107). Computer software exists to solve two-point boundary value problems. The IMSL routine DVCPR uses the finite difference method with a variable step size (Ref. 247). Finlayson (Ref. 106) gives FDRXN for reaction problems.
Example A reaction diffusion problem is solved with the finite difference method. d 2c 2 = φ2c, dx
dc (0) = 0, dx
c(1) = 1
The solution is derived for φ = 2. It is solved several times, first with two intervals and three points (at x = 0, 0.5, 1), then with four intervals, then with eight intervals. The reason is that when an exact solution is not known, one must use several ∆x and see that the solution converges as ∆x approaches zero. With two intervals, the equations are as follows. The points are x1 = 0, x2 = 0.5, and x3 = 1.0; and the solution at those points are c1, c2, and c3, respectively. A false boundary is used at x0 = −0.5. c0 − c2 = 0, 2∆x
c0 − 2c1 + c2 − φ2c1 = 0, ∆x2
c1 − 2c2 + c3 − φ2c2 = 0, ∆x2
c3 = 1
The solution is c1 = 0.2857, c2 = 0.4286, and c3 = 1.0. Since the solution is only an approximation and approaches the exact solution only as ∆x approaches zero, it is necessary to find out if ∆x is small enough to be considered zero. This is done by solving the problem again with more grid points. The value of concentration at x = 0 takes the following values for different ∆x. These values are extrapolated using the Richardson extrapolation technique to give c(0) = 0.265826. Using this value as the best estimate of the exact solution, the errors in the solution are tabulated versus ∆x. Clearly the errors go as ∆x2 (decreasing by a factor of 4 when ∆x decreases by a factor of 2), thus validating the solution. The exact solution (given below) is 0.265802. n−1
∆x
c(0)
2 4 8
0.5 0.25 0.125
0.285714 0.271043 0.267131
n−1
∆x
Error in c(0)
2 4 8
0.5 0.25 0.125
0.01989 0.00521 0.00130
Finite Difference Methods Solved with Spreadsheets A convenient way to solve the finite difference equations for simple
NUMERICAL ANALYSIS AND APPROXIMATE METHODS
3-59
The residual is then made orthogonal to the set of basis functions. db b (x) a − φ R a b (x) dx = 0 dx 1
2
NT
j
i
0
FIG. 3-51
Finite difference method using spreadsheets.
problems is to use a computer spreadsheet. The equations for the problem solved in the example can be cast into the following form 2c2 c1 = 2 + φ2∆x2
i=1
NT
i
2
i i
2
j = 1, . . . , NT
i=1
This is the process that makes the method a Galerkin method. The basis for the orthogonality condition is that a function that is made orthogonal to each member of a complete set is then zero. The residual is being made orthogonal, and if the basis functions are complete and you use infinitely many of them, then the residual is zero. Once the residual is zero, the problem is solved. This equation is integrated by parts to give the following equation NT
− i=1
1
0
b (x)R a b (x) dx
dbj dbi dxai = φ2 dx dx
1
NT
j
i i
0
i=1
j = 1, . . . , NT − 1
ci + 1 + ci − 1 ci = 2 + φ2∆x2 cn + 1 = 1 Let us solve the problem using 6 nodes, or 5 intervals. Then the connection between the cell in the spreadsheet and the nodal value is shown in Fig. 3-51. The following equations are placed into the various cells. A1: = 2*B1/(2.+(phi*dx)**2) B1: = (A1 + C1)/(2.+(phi*dx)**2) F1: = 1.
(3-76)
This equation defines the Galerkin method and a solution that satisfies this equation (for all j = 1, . . . , ∞) is called a weak solution. For an approximate solution, the equation is written once for each member of the trial function, j = 1, . . . , NT − 1, and the boundary condition is applied. NT
a b (1) = c i
i
B
i=1
The equation in cell B1 is copied into cells C1 though E1. Then turn on the iteration scheme in the spreadsheet and watch the solution converge. Whether or not convergence is achieved can depend on how you write the equations, so some experimentation may be necessary. Theorems for convergence of the successive substitution method are useful in this regard. Orthogonal Collocation The orthogonal collocation method has found widespread application in chemical engineering, particularly for chemical reaction engineering. In the collocation method, the dependent variable is expanded in a series of orthogonal polynomials, and the differential equation is evaluated at certain collocation points. The collocation points are the roots to an orthogonal polynomial, as first used by Lanczos (Refs. 182 and 183). A major improvement was proposed by Villadsen and Stewart (Refs. 288 and 289), who proposed that the entire solution process be done in terms of the solution at the collocation points rather than the coefficients in the expansion. This method is especially useful for reaction-diffusion problems that frequently arise when modeling chemical reactors. It is highly efficient when the solution is smooth, but the finite difference method is preferred when the solution changes steeply in some region of space. See Ref. 105 for comparisons. Galerkin Finite Element Method In the finite element method, the domain is divided into elements and an expansion is made for the solution on each finite element. In the Galerkin finite element method an additional idea is introduced: the Galerkin method is used to solve the equation. The Galerkin method is explained before the finite element basis set is introduced, using the equations for reaction and diffusion in a porous catalyst pellet. d 2c 2 = φ2R(c) dx dc (0) = 0, c(1) = 1 dx The unknown solution is expanded in a series of known functions {bi(x)} with unknown coefficients {ai}. NT
c(x) = ai bi(x) i=1
The trial solution is substituted into the differential equation to obtain the residual. NT NT d 2bi Residual = ai − φ2R aibi(x) 2 dx i=1 i=1
The Galerkin finite element method results when the Galerkin method is combined with a finite element trial function. The domain is divided into elements separated by nodes, as in the finite difference method. The solution is approximated by a linear (or sometimes quadratic) function of position within the element. These approximations are substituted into Eq. (3-76) to provide the Galerkin finite element equations. The element integrals are defined as 1 e BJI = − ∆xe
1
0
dNJ dNI du, du du
F Je = φ2∆xe
N (u)R c N (u) du 1
NP
e I
J
0
I
I=1
and the entire method can be written in the following compact notation:
B
c = FJe
e e JI I
e
e
The matrices for various terms are given in the table. This equation can also be written in the form AAc = f where the matrix AA is sparse; if linear elements are used, the matrix is tridiagonal. Once the solution is found, the solution at any point can be recovered from ce(u) = c eI = 1(1 − u) + c eI = 2u for linear elements. Element Matrices for Galerkin Method with Linear Shape Functions
dN dN N1 = 1 − u, N2 = u, 1 = −1, 2 = 1 du du
−a a du = N −a a du N N du = sj js, N du = aa, N u du = js 1
0
dNJ dNI 1 −1 , du = −1 1 du du
1
dNI
J
0
1
J
0
1
1
I
J
0
j
0
Example Solve the specified problem when φ = 2, the rate expression is linear, R(c) = c, and the boundary condition is 1.0. The Galerkin finite element method is used with ∆x = 0.33333. The element nodes are at x = 0, 0.3333, 0.6667, and 1.0. The solution at x = 1.0 is c4 = 1.0. The Galerkin equations for one element are obtained from the table.
−11 −11 = φ ∆x s⁄ s⁄
−
2
16
2
16
3-60
MATHEMATICS
When these are summed over all elements the result is
−1 1 0 0
1 −1 − 1 1 0
0 1 −1 − 1 0
0 0 1 1
1
c1 ⁄6 s 0 1 4 1⁄6 s + s c2 ⁄6 = 1 c3 0 ⁄ 6 s + s 9 0 0 c4 0
0 0 ⁄6 0
1
c1 0 0 c2 + 0 c3 1 c4
u(x) = f(x) + λ(b − a)
After rearrangement this is
31⁄27
−25⁄27 0
−25⁄27 62⁄27 −25⁄27
0
−25⁄27 62⁄27
c1 c2 = c3
25
⁄27
Cubic B-Splines Cubic B-splines can also be used to solve differential equations (Refs. 105 and 266). Adaptive Meshes In many two-point boundary value problems, the difficulty in the problem is the formation of a boundary layer region, or a region in which the solution changes very dramatically. In such cases, it is prudent to use small mesh spacing there, either with the finite difference method or the finite element method. If the region is known a priori, small mesh spacings can be assumed at the boundary layer. If the region is not known, though, other techniques must be used. These techniques are known as adaptive mesh techniques. The mesh size is made small where some property of the solution is large. For example, if the truncation error of the method is nth order, then the nth-order derivative of the solution is evaluated and a small mesh is used where it is large. Alternatively, the residual (the differential equation with the numerical solution substituted into it) can be used as a criterion. See Refs. 21 and 107. It is also possible to define the error that is expected from a method one order higher and one order lower. Then a decision about whether to increase or decrease the order of the method can be made, taking into account the relative work of the different orders. This provides a method of adjusting both the mesh spacing (∆x, or sometimes called h) and the degree of polynomial (p). Such methods are called h-p methods. Singular Problems and Infinite Domains If the solution being sought has a singularity, it may be difficult to find a good numerical solution. Sometimes even the location of the singularity may not be known (Ref. 11). One method of solving such problems is to refine the mesh near the singularity, relying on the better approximation due to a smaller ∆x. Another approach is to incorporate the singular trial function into the approximation. Thus, if the solution approaches f(x) as x goes to zero and f(x) becomes infinite, one may define a new variable u(x) = y(x) − f(x) and derive an equation for u. The differential equation is more complicated, but the solution is better near the singularity. See Refs. 39 and 231. Sometimes the domain is semi-infinite, as in boundary layer flow. The domain can be transformed from the x domain (0–∞) to the η domain (1–0) using the transformation η = exp (−x). Another approach is to use a variable mesh, perhaps with the same transformation. For example, use η = exp (−βx) and a constant mesh size in η; the value of β is found experimentally. Still another approach is to solve on a finite mesh in which the last point is far enough away that its location does not influence the solution (Ref. 59). A location that is far enough away must be found by trial and error. NUMERICAL SOLUTION OF INTEGRAL EQUATIONS
b
a
for u(x)
n
i
i
(3-78)
i
where t1, . . . , tn are points of subdivision of the t axis, a ≤ t ≤ b, and the c’s are coefficients whose values depend upon the type of numerical integration formula used. Now Eq. (3-78) must hold for all values of x, a ≤ x ≤ b; so it must hold for x = t1, x = t2, . . . , x = tn. Substituting for x successively t1, t2, . . . , tn and setting u(ti) = ui, f(ti) = fi, we get n linear algebraic equations for the n unknowns u1, . . . , un. That is, ui = fi + (b − a)[c1k(ti, t1)u1 + c2 k(ti, t2)u2 + ⋅⋅⋅ + cnk(ti, tn)un]
i = 1, 2, . . . , n
These uj may be solved for by the methods under “Numerical Solution of Linear Equations and Associated Problems” and substituted into Eq. (3-78) to yield an approximate solution for Eq. (3-77). Example Solve numerically u(x) = x + s ∫ 01 (t + x)u(t) dt. In this example a = 0, b = 1. Take n = 3, t1 = 0, t2 = a, t3 = 1. Then Eq. (3-78) takes the form (for which we have used the parabolic rule) a u(x) = x + (s) [(t1 + x)u(t1) + 4(t2 + x)u(t2) + (t3 + x)u(t3)] 3 = x + (1/18)[(t1 + x)u(t1) + 4(t2 + x)u(t2) + (t3 + x)u(t3)] This must hold for all x, 0 ≤ x ≤ 1. Here t1 = 0, t2 = a, and t3 = 1. Evaluate at x = ti. u(t1) = t1 + 1⁄18[2t1u(t1) + 4(t2 + t1)u(t2) + (t3 + t1)u(t3)] u(t2) = t2 + 1⁄18[(t1 + t2)u(t1) + 4(2t2)u(t2) + (t3 + t2)u(t3)] u(t3) = t3 + 1⁄18[(t1 + t3)u(t1) + 4(t2 + t3)u(t2) + 2t3u(t3)] By setting in the values of t1, t2, t3 and u(ti) = ui, 18u1 − 2u2 − u3 = 0 −u1 + 28u2 − 3u3 = 18 −u1 − 6u2 + 16u3 = 18 with the solution u1 = 12⁄ 71, u2 = 57⁄71, u3 = 102⁄71. Thus u(x) = x + 1⁄18[x12⁄71 + 4(a + x)57⁄71 + (1 + x)102⁄71] = 90⁄ 71x + 12⁄71
Because of the work involved in solving large systems of simultaneous linear equations it is desirable that only a small number of u’s be computed. Thus the gaussian integration formulas are useful because of the economy they offer. See references on numerical solutions of integral equations. Solutions for Volterra equations are done in a similar fashion, except that the solution can proceed point by point, or in small groups of points depending on the quadrature scheme. See Refs. 105 and 195. There are methods that are analogous to the usual methods for integrating differential equations (Runge-Kutta, predictor-corrector, Adams methods, etc.). Explicit methods are fast and efficient until the time step is very small to meet the stability requirements. Then implicit methods are used, even though sets of simultaneous algebraic equations must be solved. The major part of the calculation is the evaluation of integrals, however, so that the added time to solve the algebraic equations is not excessive. Thus, implicit methods tend to be preferred (Ref. 195). Volterra equations of the first kind are not well posed, and small errors in the solution can have disastrous consequences. The boundary element method uses Green’s functions and integral equations to solve differential equations (Refs. 45 and 200). MONTE CARLO SIMULATIONS
In this subsection is considered a method of solving numerically the Fredholm integral equation of the second kind:
k(x, t)u(t) dt
c k(x, t )u(t ) i=1
0 0
The solution is c1 = 0.2560, c2 = 0.3174, c3 = 0.5312, and c4 = 1. The exact solution is derived using the section entitled “Ordinary Differential Equations: Linear Differential Equations with Constant Coefficients.” e2x + e−2x cosh (2x) c= = e2 + e−2 cosh (2) The values of the exact solution at the same finite element nodes are c1 = 0.2658, c2 = 0.3271, c3 = 0.5392, and c4 = 1, indicating that the three-element finite element solution is accurate within 3 percent. When the exact solution is not known, the problem must be solved several times, each with a different number of elements, so that convergence is seen as the number of elements increases.
u(x) = f(x) + λ
The method discussed arises because a definite integral can be closely approximated by any of several numerical integration formulas (each of which arises by approximating the function by some polynomial over an interval). Thus the definite integral in Eq. (3-77) can be replaced by an integration formula, and Eq. (3-77) may be written
(3-77)
Some physical problems, such as those involving interaction of molecules, are usually formulated as integral equations. Monte Carlo methods are especially well-suited to their solution. This section cannot give a comprehensive treatment of such methods, but their use in
NUMERICAL ANALYSIS AND APPROXIMATE METHODS
3-61
calculating the value of an integral will be illustrated. Suppose we wish to calculate the integral G=
Ω0
g(x)f(x) dx
where the distribution function f(x) satisfies: f(x) ≥ 0,
Ω0
f(x) dx = 1
The distribution function f(x) can be taken as constant; for example, 1/Ω0. We choose variables x1, x2, . . . , xN randomly from f(x) and form the arithmetic mean 1 GN = g(xi) N i The quantity GN is an estimation of G, and the fundamental theorem of Monte Carlo guarantees that the expected value of GN is G, if G exists (Ref. 161). The error in the calculation is given by σ1 ε= N1/2 where σ 21 is calculated from σ 21 =
Ω0
g2(x) f(x) dx − G2
Thus the number of terms needed to achieve a specified accuracy can be calculated once an estimate of σ 21 is known. σ12 N= ε Various methods, such as influence sampling, can be used to reduce the number of calculations needed (Ref. 161). NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS Parabolic Equations in One Dimension By combining the techniques applied to initial value problems and boundary value problems it is possible to easily solve parabolic equations in one dimension. The method is often called the method of lines. It is illustrated here using the finite difference method, but the Galerkin finite element method and the orthogonal collocation method can also be combined with initial value methods in similar ways. The analysis is done by example. Example Consider the diffusion equation, with boundary and initial conditions.
∂c ∂2c = D 2 ∂t ∂x c(x, 0) = 0 c(0, t) = 1, c(1, t) = 0
We denote by ci the value of c(xi, t) at any time. Thus, ci is a function of time, and differential equations in ci are ordinary differential equations. By evaluating the diffusion equation at the ith node and replacing the derivative with a finite difference equation, the following working equation is derived for each node i, i = 2, . . . , n (see Fig. 3-52). dci ci + 1 − 2ci + ci − 1 = D dt ∆x2 This can be written in the general form of a set of ordinary differential equations by defining the matrix AA. dc = AAc dt This set of ordinary differential equations can be solved using any of the standard methods, and the stability of the integration of these equations is governed by the largest eigenvalue of AA. If Euler’s method is used for integration, the time step is limited by 2 ∆t ≤ |λ|max
FIG. 3-52
Computational molecules. h = ∆x = ∆y.
whereas, if the Runge-Kutta-Feldberg method is used, the 2 in the numerator is replaced by 3.0. The largest eigenvalue of AA is bounded by Gerschgorin’s Theorem (Ref. 155, p. 135). n 4D |λ|max ≤ max 2 < j < n |AA ji | = 2 ∆x i=2 This gives the well-known stability limit D 1 ∆t 2 ≤ ∆x 2
3-62
MATHEMATICS
The smallest eigenvalue is independent of ∆x (it is Dπ 2/L2) so that the ratio of largest to smallest eigenvalue is proportional to 1/∆x2. Thus, the problem becomes stiff as ∆x approaches zero (Ref. 106). Another way to study the stability of explicit equations is to use the positivity theorem. For Euler’s method, the equations can be written in the form c in + 1 − cin cni+ 1 − 2c ni + cni− 1 = D ∆t ∆x2 n n where c i = c(xi, t ). Then the new value is given by
D∆t n D∆t n D∆t n c in + 1 = ci + 1 + 1 − 2 ci + ci − 1 ∆x2 ∆x2 ∆x2 Theorem. If c ni + 1 = Acni+ 1 + Bc in + Ccni− 1 and A, B, and C are positive and A + B + C ≤ 1, then the scheme is stable and the errors die out. Here the theorem requires D∆t 1−2 >0 ∆x2 which gives the same stability condition (Ref. 106). Implicit methods can also be used. Write a finite difference form for the time derivative and average the right-hand sides, evaluated at the old and new time. c in + 1 − cin cni+ 1 − 2cni + cni− 1 cin++11 − 2c in + 1 + cni −+11 + Dθ = D(1 − θ) 2 ∆t ∆x ∆x2 Now the equations are of the form
D∆tθ n + 1 D∆tθ n + 1 D∆tθ n + 1 − ci + 1 + 1 + 2 ci − ci − 1 ∆x2 ∆x2 ∆x2 D∆t(1 − θ) n = c in + (ci + 1 − 2c in + cni− 1) ∆x2 and require solving a set of simultaneous equations, which have a tridiagonal structure. Using θ = 0 gives the Euler method (as above), θ = 0.5 gives the Crank-Nicolson method, and θ = 1 gives the backward Euler method. The Crank-Nicolson method is also the same as applying the trapezoid rule to do the integration. The stability limit is given by D∆t 0.5 ≤ ∆x2 1 − 2θ If the ∆t satisfies the following equation, then the solution will not oscillate from node to node (a numerical artifact). See Ref. 106. 0.25 D∆t ≤ ∆x2 1−θ Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements (see Ref. 106). Spectral methods employ Chebyshev polynomials and the Fast Fourier Transform and are quite useful for hyperbolic or parabolic problems on rectangular domains (Ref. 125). Packages exist that use various discretizations in the spatial direction and an integration routine in the time variable. PDECOL uses B-splines for the spatial direction and various GEAR methods in time (Ref. 247). PDEPACK and DSS (Ref. 247) use finite differences in the spatial direction and GEARB in time (Ref. 66). REACOL (Ref. 106) uses orthogonal collocation in the radial direction and LSODE in the axial direction, while REACFD uses finite difference in the radial direction; both codes are restricted to modeling chemical reactors. Elliptic Equations Elliptic equations can be solved with both finite difference and finite element methods. One-dimensional elliptic problems are two-point boundary value problems. Two- and threedimensional elliptic problems are often solved with iterative methods when the finite difference method is used and direct methods when the finite element method is used. So there are two aspects to consider: how the equations are discretized to form sets of algebraic equations and how the algebraic equations are then solved.
The prototype elliptic problem is steady-state heat conduction or diffusion, ∂2T ∂2T + =Q k ∂x2 ∂y2
possibly with a heat generation term per unit volume, Q. The boundary conditions taken here are T = f(x, y) on the boundary (S) with f a known function. Illustrations are given for constant thermal conductivity k while Q is a known function of position. The finite difference formulation is given using the following nomenclature: Ti, j = T(i∆x, j∆y) The finite difference formulation is then (see Fig. 3-52) Ti + 1, j − 2Ti, j + Ti − 1, j Ti, j + 1 − 2Ti, j + Ti, j − 1 + = Qi, j ∆x2 ∆y2
(3-79)
Ti, j = f(xi, yj) on S If the boundary is parallel to a coordinate axis any derivative is evaluated as in the section on boundary value problems, using either a onesided, centered difference or a false boundary. If the boundary is more irregular and not parallel to a coordinate line then more complicated expressions are needed and the finite element method may be the better method. Equation (3-79) is rewritten in the form ∆ x2 ∆x2 Qi, j 2 1 + 2 Ti, j = Ti + 1, j + Ti − 1, j + 2 (Ti, j + 1 + Ti, j − 1) − ∆x2 ∆y ∆y k
The relaxation method solves this equation iteratively. Qi, j ∆ x2 ∆ x2 2 1 + 2 T*i, j = Tis+ 1, j + T si −+ 1,1 j + 2 (T i,s j + 1 + T i,s +j −1 1) − ∆ x2 ∆y ∆y k
T si,+j 1 = Ti,s j + β(T*i, j − Ti,s j) If β = 1, this is the Gauss-Seidel method. If β > 1, it is overrelaxation; if β < 1 it is underrelaxation. The value of β may be chosen empirically, 0 < β < 2, but it can be selected theoretically for simple problems like this (Refs. 106 and 221). In particular, these equations can be programmed in a spreadsheet and solved using the iteration feature, provided the boundaries are all rectangular. The alternating direction method can be used for elliptic problems by using sequences of iteration parameters (Refs. 106 and 221). The method is well suited to transient problems as well. These are the classical iterative techniques. Recently preconditioned conjugate gradient methods have been developed (see Ref. 100). In these methods, a series of matrix multiplications are done iteration by iteration; and the steps lend themselves to the efficiency available in parallel computers. In the multigrid method, the problem is solved on several grids, each more refined than the previous one. As one iterates between the solutions on the different grids, one converges to the solution of the algebraic equations. See Juncu and Mihail (Ref. 68) for a chemical engineering application. The Galerkin finite element method (FEM) is useful for solving elliptic problems and is particularly effective when the domain or geometry is irregular. As an example, cover the domain with triangles and define a trial function on each triangle. The trial function takes the value 1.0 at one corner and 0.0 at the other corners and is linear in between. See Fig. 3-53. These trial functions on each triangle are pieced together to give a trial function on the whole domain. General treatments of the finite element method are available (see references). The steps in the solution method are similar to those described for boundary value problems, except now the problems are much bigger so that the numerical analysis must be done very carefully to be efficient. Most engineers, though, just use a finite element program without generating it. There are three major caveats that must be addressed. The first one is that the solution is dependent on the mesh laid down, and the only way to assess the accuracy of the solution is to solve the problem with a more refined mesh. The second concern is that the solution obeys the shape of the trial function inside
NUMERICAL ANALYSIS AND APPROXIMATE METHODS the element. Thus, if linear functions are used on triangles, a threedimensional view of the solution, plotting the solution versus x and y, consists of a series of triangular planes joined together at the edges, as in a geodesic dome. The third caveat is that the Galerkin finite element method is applied to both the differential equations and the boundary conditions. Computer programs are usually quite general and may allow the user to specify boundary conditions that are not realistic. Also, natural boundary conditions are satisfied if no other boundary condition (ones involving derivatives) is set at a node. Thus, the user of finite element codes must be very clear what boundary conditions and differential equations are built into the computer code. When the problem is nonlinear, the Newton-Raphson method is used to iterate from an initial guess. Nonlinear problems lead to complicated integrals to evaluate, and they are usually evaluated using Gaussian quadrature. One nice feature of the finite element method is the use of natural boundary conditions. It may be possible to solve the problem on a domain that is shorter than needed to reach some limiting condition (such as at an outflow boundary). The externally applied flux is still applied at the shorter domain, and the solution inside the truncated domain is still valid. Examples are given in Refs. 67 and 107. The effect of this is to allow solutions in domains that are smaller, thus saving computation time and permitting the solution in semi-infinite domains. A general purpose package for general two-dimensional domains and rectangular three-dimensional rectangular domains is ELLPACK (Ref. 247). This package allows choice of a variety of methods: finite difference, Hermite collocation, spline Galerkin, collocation, as well as others. Comparisons of the various methods are available (Ref. 154). The program FISHPAK solves the Helmholtz equation in multiple dimensions when the domain is separable (since fast methods like FFT are used). See Ref. 247. Hyperbolic Equations The most common situation yielding hyperbolic equations involves unsteady phenomena with convection. Two typical equations are the convective diffusive equation ∂c ∂2c ∂c + u = D 2 ∂t ∂x ∂x and the chromatography equation (Ref. 245) ∂c ∂c df ∂c φ + φu + (1 − φ) = 0 ∂t ∂x dc ∂t where φ is the void fraction and f(c) gives the equilibrium relation between the concentration in the fluid phase and the concentration in the solid phase. If the diffusion coefficient is zero, the convective diffusion equation is hyperbolic. If D is small, the phenomenon may be essentially hyperbolic, even though the equations are parabolic. Thus the numerical methods for hyperbolic equations may be useful even for parabolic equations. Equations for several methods are given here, as taken from the book by Finlayson (Ref. 107). If the convective term is treated with a centered difference expression, the solution exhibits oscillations from node to node, and these only go away if a very fine grid is used. The
FIG. 3-53 Trial functions for Galerkin finite element method: linear polynomial on triangle.
3-63
simplest way to avoid the oscillations with a hyperbolic equation is to use upstream derivatives. If the flow is from left to right, this would give dci ci − ci − 1 ci + 1 − 2ci + ci − 1 + u = D dt ∆x ∆x2 d ci − ci − 1 [ φci + (1 − φ)f(ci)] + φui = 0 dt ∆x (See Ref. 227 for the reason the equation is written in this form.) The effect of using upstream derivatives is to add artificial or numerical diffusion to the model. This can be ascertained by rearranging the finite difference form of the convective diffusion equation dci ci + 1 − ci − 1 u∆x ci + 1 − 2ci + ci − 1 + u = D + dt 2∆x 2 ∆x2 Thus the diffusion coefficient has been changed from u∆x D to D + 2 Another method often used for hyperbolic equations is the MacCormack method. This method has two steps, and it is written here for the convective diffusion equation. u∆t D∆t n c*i n + 1 = cin − (cni+ 1 − c ni ) + (c i + 1 − 2c ni + cni − 1) ∆x ∆x2
1 u∆t c in + 1 = (cin + c*i n + 1) − (c*i n + 1 − c*i −n 1+ 1) 2 2∆x D∆t + 2 (c*i +n 1+ 1 − 2c*i n + 1 + c*i −n 1+ 1) 2∆x The concentration profile is steeper for the MacCormack method than for the upstream derivatives, but oscillations can still be present. The flux-corrected transport method can be added to the MacCormack method. A solution is obtained both with the upstream algorithm and the MacCormack method and then they are combined to add just enough diffusion to eliminate the oscillations without smoothing the solution too much. The algorithm is complicated and lengthy but well worth the effort (Refs. 37, 107, and 270). Stability conditions can be constructed in terms of Co = u∆t/∆x and r = D∆t/∆x2 by using Fourier analysis (Ref. 107). All the methods require u∆t Co = ≤ 1 ∆x where Co is the Courant number. How much Co should be less than one depends on the method and on r = D∆t/∆x2. For example, the upstream method requires Co ≤ 1 − 2r. The MacCormack method depends less on r and is stable for most Co as long as r ≤ 0.5. Each of these methods is trying to avoid oscillations that would disappear if the mesh were fine enough. For the steady convective diffusion equation, these oscillations do not occur provided u∆x ≤1 2D For large velocity u, the ∆x must be small to meet this condition. An alternative is to use a small ∆x in regions where the solution changes drastically. Since these regions change in time, it is necessary that the elements or grid points move. The criteria to move the grid points can be quite complicated, and typical methods are reviewed in Ref. 107. Similar considerations apply to the nonlinear chromatography problem (Ref. 227). See especially Ref. 192. Parabolic Equations in Two or Three Dimensions Computations become much more lengthy when there are two or more spatial dimensions. For example, we may have the unsteady heat conduction equation ∂T ∂2T ∂2T ρCp = k + −Q ∂t ∂x2 ∂y2
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MATHEMATICS
In the finite difference method an explicit technique would evaluate the right-hand side at the nth time level. T i,n j+ 1 − T i,nj k n ρCp = 2 (T in+ 1, j − 2T i, j + T in− 1, j) ∆t ∆x k + 2 (T ni, j + 1 − 2T i,n j + Ti,nj − 1) − Q ∆y When Q = 0 and ∆x = ∆y, the time step limit can be found using the positivity rule. ∆x2ρC ∆x2 ∆t ≤ p or 4k 4D These time steps are smaller than for one-dimensional problems. For three dimensions, the limit is ∆x2 ∆t ≤ 6D To avoid such small time steps, which become smaller as ∆x decreases, an implicit method could be used. This leads to large, sparse matrices rather than convenient tridiagonal matrices. These can be solved, but the alternating direction method is also useful (Ref. 221). This reduces a problem on an n n grid to a series of 2n one-dimensional problems on an n grid. SPLINE FUNCTIONS Splines are functions that match given values at the points x1, . . . , xNT and have continuous derivatives up to some order at the knots, or the points x2, . . . , xNT − 1. Cubic splines are most common; see Ref. 38. The function is represented by a cubic polynomial within each interval (xi, xi + 1) and has continuous first and second derivatives at the knots. Two more conditions can be specified arbitrarily. These are usually the second derivatives at the two end points, which are commonly taken as zero; this gives the natural cubic splines. Take yi = y(xi) at each of the points xi, and let ∆xi = xi + 1 − xi. Then, in the interval (xi, xi + 1), the function is represented as a cubic polynomial.
FAST FOURIER TRANSFORM (REF. 231) Suppose a signal y(t) is sampled at equal intervals
n = . . . , −2, −1, 0, 1, 2, . . .
∆ = sampling rate (e.g., number of samples per second) The Fourier transform and inverse transform are Y(ω) =
∞
−∞
y(t)eiωt dt
1 ∞ Y(ω)e−iωt dt y(t) = 2π −∞ The Nyquist critical frequency or critical angular frequency is 1 π fc = , ωc = 2∆ ∆ If a function y(t) is bandwidth-limited to frequencies smaller than fc, such as Y(ω) = 0
for ω > ωc
then the function is completely determined by its samples yn. Thus, the entire information content of a signal can be recorded by sampling at a rate ∆−1 = 2fc. If the function is not bandwidth-limited, then aliasing occurs. Once a sample rate ∆ is chosen, information corresponding to frequencies greater than fc is simply aliased into that range. The way to detect this in a Fourier transform is to see if the transform approaches zero at fc; if not, aliasing has occurred, and a higher sampling rate is needed. Next, suppose we have N samples, where N is even yk = y(tk)
tk = k∆
k = 0,1,2, . . . , N − 1
and the sample rate is ∆. With only N values {yk}, it is not possible to determine the complete Fourier transform Y(ω). We calculate the value Y(ωn) at the discrete points 2πn N N ωn = , n = − , . . . , 0, . . . , N∆ 2 2 Yn =
N−1
ye k
2πikn/N
k=0
Y(ωn) = ∆Yn
Ci(x) = a0i + a1i x + a2i x2 + a3i x3 The interpolating function takes on specified values at the knots and has continuous first and second derivatives at the knots. Within the ith interval, the function is (x − xi) Ci(x) = Ci(xi) + C′i (xi)(x − xi) + C″i (xi) 2 (x − xi)3 + [C″i (xi + 1) − C″i (xi)] 6∆xi where Ci(xi) = yi. The second derivative C″i (xi) = y″i is found by solving the following tridiagonal system of equations: yi − yi − 1 yi + 1 − yi y″i − 1∆xi − 1 + y″i 2(∆xi − 1 + ∆xi) + y″i + 1∆xi = 6 − ∆xi − 1 ∆xi Since the continuity conditions apply only for i = 2, . . . , NT − 1, we have only NT − 2 conditions for the NT values of y″i . Two additional conditions are needed, and these are usually taken as the value of the second derivative at each end of the domain, y″1 , y″NT. If these values are zero, we get the natural cubic splines; they can also be set to achieve some other purpose, such as making the first derivative match some desired condition at the two ends. With these values taken as zero in the natural cubic spline, we have a NT − 2 system of tridiagonal equations, which is easily solved. Once the second derivatives are known at each of the knots, the first derivatives are given by ∆x yi + 1 − yi ∆x y′i = − y″i i − y″i + 1 i ∆xi 3 6 The function itself is then known within each element.
yn = y(n∆),
The discrete inverse Fourier transform is 1 N−1 yk = Yne−2πikn/N N n=0 The fast Fourier transform (FFT) is used to calculate the Fourier transform as well as the inverse Fourier transform. A discrete Fourier transform of length N can be written as the sum of two discrete Fourier transforms, each of length N/2. Yk = Yke + W k Yko Here Yk is the kth component of the Fourier transform of y, and Yke is the kth component of the Fourier transform of the even components of {yj} and is of length N/2. Similarly, Yko is the kth component of the Fourier transform of the odd components of {yj} and is of length N/2. W is a constant, which is taken to the kth power. W = e2πi/N Since Yk has N components, while Y ek and Yko have N/2 components, Y ek and Yko are repeated once to give N components in the calculation of Yk. This decomposition can be used recursively. Thus, Y ek is split into even and odd terms of length N/4. Yke = Ykee + W k Ykeo Yko = Ykoe + W k Ykoo This process is continued until there is only one component. For this reason, the number N is taken as a power of 2. The vector {yj } is filled with zeroes, if need be, to make N = 2 p for some p. For the computer program, see Ref. 26. The standard Fourier transform takes N 2 operations to calculation, whereas the fast Fourier transform takes only N log2 N. For large N, the difference is significant; at N = 100 it is a factor of 15, but for N = 1000 it is a factor of 100.
OPTIMIZATION The discrete Fourier transform can also be used for differentiating a function, and this is used in the spectral method for solving differential equations. Suppose we have a grid of equidistant points L xn = n∆x, n = 0, 1, 2, . . . , 2N − 1, ∆x = 2N The solution is known at each of these grid points {Y(xn)}. First the Fourier transform is taken. 1 2N − 1 yk = Y(xn)e−2ikπx n /L 2N n = 0 The inverse transformation is 1 N Y(x) = yke2ikπx/L L k = −N Differentiate this to get dY 1 N 2πik = yk e2ikπx/L dx L k = −N L
3-65
Thus at the grid points
dY 1 N 2πik = yk e2ikπx n /L dx n L k = −N L The process works as follows. From the solution at all grid points the Fourier transform is obtained using FFT, {yk}. Then this is multiplied by 2πik/L to obtain the Fourier transform of the derivative. 2πik yk = yk L Then the inverse Fourier transform is taken using FFT, giving the value of the derivative at each of the grid points.
dY 1 N = yke2ikπx n /L dx n L k = −N
OPTIMIZATION INTRODUCTION Optimization should be viewed as a tool to aid in decision making. Its purpose is to aid in the selection of better values for the decisions that can be made by a person in solving a problem. To formulate an optimization problem, one must resolve three issues. First, one must have a representation of the artifact that can be used to determine how the artifact performs in response to the decisions one makes. This representation may be a mathematical model or the artifact itself. Second, one must have a way to evaluate the performance—an objective function—which is used to compare alternative solutions. Third, one must have a method to search for the improvement. This section concentrates on the third issue, the methods one might use. The first two items are difficult ones, but discussing them at length is outside the scope of this section. Example optimization problems are: (1) determining the optimal thickness of pipe insulation; (2) finding the best equipment sizes and operating schedules for the design of a new batch process to make a given slate of products; (3) choosing the best set of operating conditions for a set of experiments to determine the constants in a kinetic model for a given reaction; (4) finding the amounts of a given set of ingredients one should use for making a carbon rod to be used as an electrode in an arc welder. For the first problem, one will usually write a mathematical model of how insulation of varying thicknesses restricts the loss of heat from a pipe. Evaluation requires that one develop a cost model for the insulation (a capital cost in dollars) and the heat that is lost (an operating cost in dollars/year). Some method is required to permit these two costs to be compared, such as a present worth analysis. Finally, if the model is simple enough, the method one can use is to set the derivative of the evaluation function to zero with respect to wall thickness to find candidate points for its optimal thickness. For the second problem, selecting a best operating schedule involves discrete decisions, which will generally require models that have integer variables. It may not be possible to develop a mathematical model for the fourth problem if not enough is known to characterize the performance of a rod versus the amounts of the various ingredients used in its manufacture. The rods may have to be manufactured and judged by ranking the rods relative to each other, perhaps based partially or totally on opinions. Pattern search methods have been devised to attack problems in this class. In this section assume a mathematical model is possible for the problem to be solved. The model may be encoded in a subroutine and be known only implicitly, or the equations may be known explicitly. A general form for such an optimization problem is min F = F(z), such that h(z) = 0 and g(z) ≤ 0
where F represents a specified objective function that is to be minimized. Functions h and g represent equality and inequality constraints that must be satisfied at the final problem solution. Variables z are used to model such things as flows, mole fractions, physical properties, temperatures, and sizes. The objective function F is generally assumed to be a scalar function, one which represents such things as cost, net present value, safety, or flexibility. Sometimes several objective functions are specified (e.g., minimizing cost while maximizing reliability); these are commonly combined into one function, or else one is selected for the optimization while the others are specified as constraints. Equations h(z) = 0 are typically algebraic equations, linear or nonlinear, when modeling steady-state processes, or algebraic coupled with ordinary and/or partial differential equations when optimizing time-varying processes. Inequalities g(z) ≤ 0 put limits on the values variables can take, such as a minimum and maximum temperature, or they restrict one pressure to be greater than another. An important issue is how to solve large problems that occur in distributed systems. The optimization of distributed systems is discussed in Refs. 52, 120, 244, and 285. For further reading on optimization, readers are directed to Refs. 120 and 244 as well as introductory texts on optimization applied to chemical engineering (Refs. 99 and 225). The material in this section is part of a more advanced treatment (Ref. 295). Packages There are a number of packages available for optimization, some of which are listed here. 1. Frameworks • GAMS. This framework is commercially available. It provides a uniform language to access several different optimization packages, many of them listed below. It will convert the model as expressed in “GAMS” into the form needed to run the package chosen. • AMPL. This framework is by Fourier and coworkers (Ref. 113) at Northwestern University. It is well suited for constructing complex models. • ASCEND. This framework is by Westerberg and coworkers (Ref. 295) at Carnegie-Mellon University. It features an objectoriented modeling language and is well suited for constructing complex models. 2. Algebraic optimization with equality and inequality constraints • SQP. A package by Biegler at Carnegie-Mellon University. • MINOS5.4. A package available from Stanford Research Institute (affiliated with Stanford University). This package is the state of the art for mildly nonlinear programming problems. • GRG. A package from Lasdon at the University of Texas, Dept. of Management Science. 3. Linear programming. Most current commercial codes for lin-
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MATHEMATICS
ear programming extend the Simplex algorithm, and they can typically handle problems with up to 15,000 constraints. • MPSX. From IBM • SCICONIC. From the company of that name • MINOS5.4 • Cplex. A package by R. Bixby at Rice University and Cplx, Inc. CONDITIONS FOR OPTIMALITY
du dF
i
∆h = 0
= 0,
i = 1, 2, . . . , r
Equality Constrained Problems—Lagrange Multipliers Form a scalar function, called the Lagrange function, by adding each of the equality constraints multiplied by an arbitrary multiplier to the objective function. m
Local Minimum Point for Unconstrained Problems Consider the following unconstrained optimization problem: Min {F(u) | u Rr} u
If F is continuous and has continuous first and second derivatives, it is necessary that F is stationary with respect to all variations in the independent variables u at a point û, which is proposed as a minimum to F; that is, ∂F or ∇u F = 0 at u = û (3-80) = 0, i = 1, 2, . . . , r ∂ui These are only necessary conditions, as point û may be a minimum, maximum, or saddle point. Sufficient conditions are that any local move away from the optimal point û gives rise to an increase in the objective function. Expand F in a Taylor series locally around the candidate point û up to second-order terms: 1 2 F(u) = F(û) + ∇u F T û (u − û) + (u − û)T∇uu F û (u − û) + … 2 If û satisfies necessary conditions [Eq. (3-80)], the second term disappears in this last line. Sufficient conditions for the point to be a local 2 minimum are that the matrix of second partial derivatives ∇uu F is positive definite. This matrix is symmetric, so all of its eigenvalues are real; to be positive definite, they must all be greater than zero. Constrained Derivatives—Equality Constrained Problems Consider minimizing the objective function F written in terms of n variables z and subject to m equality constraints h(z) = 0, or Mzin {F(z) | h(z) = 0, z Rn, h:Rn → Rm}
(3-81)
The point zˆ is tested to see if it could be a minimum point. It is necessary that F be stationary for all infinitesimal moves for z that satisfy the equality constraints. Linearize the m equality constraints around zˆ, getting h(ˆz + ∆z) = h(ˆz) + ∇z hT zˆ ∆z
(3-82)
where ∆z = z − zˆ . There are m constraints here, so m of the variables are dependent, leaving r = n − m independent variables. Partition the variables ∆z into a set of m dependent variables ∆x and r = n − m independent variables ∆u. Equation (3-82), rearranged and then rewritten in terms of these variables, becomes
L(x, u, λ) = F(x, u) + λi hi(x, u) = F(x, u) + λT h(x, u) i=1
At any point where the functions h(z) are zero, the Lagrange function equals the objective function. Next differentiate L with respect to variables x, u, and λ. ∇x LT zˆ = ∇x F T zˆ + λT∇h xT zˆ
= 0T
(3-83)
∇u L zˆ = ∇u F zˆ + λ ∇h
=0
(3-84)
∇λ LT zˆ = hT(x, u)
=0
T
T
T
T u zˆ
T T
Solve Eq. (3-83) for the Lagrange multipliers λT = −∇x F T[∇h Tx ]−1
(3-85)
and then eliminate these multipliers from Eq. (3-84). ∇u LT = ∇u F T − ∇x F T[∇hTx ]∇h Tu = 0 T ∇uL is equal to the constrained derivatives for the problem, which should be zero at the solution to the problem. Also, these stationarity conditions very neatly provide the necessary conditions for optimality of an equality-constrained problem. Lagrange multipliers are often referred to as shadow prices, adjoint variables, or dual variables, depending on the context. Suppose the variables are at an optimum point for the problem. Perturb the variables such that only constraint hi changes. We can write ∆L = ∆F + λi ∆hi = 0 which is zero because, as just shown, the Lagrange function is at a stationary point at the optimum. Solving for the change in the objective function: ∆F = −λi∆hi The multiplier tells how the optimal value of the objective function changes for this small change in the value of a constraint while holding all the other constraints at zero. It is for this reason that they are often called shadow prices. Equality- and Inequality-Constrained Problems—KuhnTucker Multipliers Next a point is tested to see if it is an optimum one when there are inequality constraints. The problem is Mzin {F(z) | h(z) = 0, g(z) ≤ 0, z Rn, F:Rn → R1, h:Rn → Rm, g:Rn → Rp}
The Lagrange function here is similar to that used above.
∆h = ∇x hT zˆ ∆x + ∇uhT zˆ ∆u = 0
L(z, λ, µ) F(z) + λTh(z) + µTg(z)
This enables the solution for ∆x. Linearize the objective function F(z) in terms of the partitioned variables
Each of the inequality constraints gi(z) multiplied by what is called a Kuhn-Tucker multiplier µi is added to form the Lagrange function. The necessary conditions for optimality, called the Karush-KuhnTucker conditions for inequality-constrained optimization problems, are
∆F = ∇x F T zˆ ∆x + ∇u F T zˆ ∆u and substitute for ∆x. ∆F = {∇x F T − ∇u F T[∇x hT]−1 ∇u hT}zˆ ∆u
dF = du
T ∆h = 0
r
dF ∆u = dui i=1
∆h = 0
∆ui
There is one term for each ∆ui in the row vector which is in the curly braces {}. These terms are called constrained derivatives, which tells how the object function changes when the independent variables ui are changed while keeping the constraints satisfied (by varying the dependent variables xi). Necessary conditions for optimality are that these constrained derivatives are zero; that is,
∇z L zˆ = ∇z F zˆ + ∇z h zˆ λ + ∇z g zˆ µ = 0 ∇λL = h(z) = 0 g(z) ≤ 0 µi gi(z) = 0 ,
i = 1, 2, . . . , p
µi ≥ 0 ,
i = 1, 2, . . . , p
(3-86)
Conditions in Eq. (3-86), called complementary slackness conditions, state that either the constraint gi(z) = 0 and/or its corresponding multiplier µi is zero. If constraint gi(z) is zero, it is behaving like an equality constraint, and its multiplier µi is exactly the same as a Lagrange multiplier for an equality constraint. If the constraint is
OPTIMIZATION away from zero, it is not a part of the problem and should not affect it. Setting its multiplier to zero removes it from the problem. As the goal is to minimize the objective function, releasing the constraint into the feasible region must not decrease the objective function. Using the shadow price argument above, it is evident that the multiplier must be nonnegative (Ref. 177). Sufficiency conditions to assure that a Kuhn-Tucker point is a local minimum point require one to prove that the objective function will increase for any feasible move away from such a point. To carry out such a test, one has to generate the matrix of second derivatives of the Lagrange function with respect to all the variables z evaluated at zˆ . The test is seldom done, as it requires too much work. STRATEGIES OF OPTIMIZATION The theory just covered tells if a candidate point is or is not the optimum point, but how is the candidate point found? The simplest strategy is to place a grid of points throughout the feasible space, evaluating the objective function at every grid point. If the grid is fine enough, then the point yielding the highest value for the objective function can be selected as the optimum. Twenty variables gridded over only ten points would take place over 1020 points in our grid, and, at one nanosecond per evaluation, it would take in excess of four thousand years to carry out these evaluations. Most strategies limit themselves to finding a local minimum point in the vicinity of the starting point for the search. Such a strategy will find the global optimum only if the problem has a single minimum point or a set of “connected” minimum points. A “convex” problem has only a global optimum. Pattern Search Suppose the optimization problem is to find the right mix of a given set of ingredients and the proper baking temperature and time to make the best cake possible. A panel of judges can be formed to judge the cakes; assume they are only asked to rank the cakes and that they can do that task in a consistent manner. Our approach will be to bake several cakes and ask the judges to rank them. For this type of problem, pattern-search methods can be used to find the better conditions for manufacturing the product. We shall only describe the ideas behind this approach. Details on implementing it can be found in Ref. 284. The complex method is one such pattern search method (see Fig. 3-54). First, form a “complex” of at least r + 1 (r = 2 and 4 points are used in Fig. 3-54) different points at which to bake the cakes by picking a range of suitable values for the r independent variables for the baking process. Bake the cakes and then ask the judges to identify the worst cake. For each independent variable, form the average value at which it was run in the complex. Draw a line from the coordinates of the worst cake through the average point—called the centroid—and continue on that line a distance that is twice that between these two points. This point will be the next test point. First decide if it is feasible. If so, bake the cake and discover if it leads to a cake that is better than the worst cake from the last set of cakes. If it is not feasible or it is not better, then return half the distance toward the average values from the last
FIG. 3-54
Complex method, a pattern search optimization method.
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test and try again. If it is better, toss out the worst point of the last test and replace it with this new one. Again, ask the judges to find the worst cake. Continue as above until the cakes are all the same quality in the most recent test. It might pay to restart at this point, stopping finally if the restart leads to no improvement. The method takes large steps if the steps are being successful in improving the recipe. It collapses onto a set of points quite close to each other otherwise. The method works reasonably well, but it requires one to bake lots of cakes. The following strategies are all examples of Generalized Reduced Gradient (GRG) methods. Optimization of Unconstrained Objective Assume the objective function F is a function of independent variables ui, i = 1 ⋅⋅⋅ r. A computer program, given the values for the independent variables, can calculate F and its derivatives with respect to each ui. Assume that F is well approximated as an as-yet-unknown quadratic function in u. 1 F ≈ a + bTu + uTQu 2 where a is a scalar; b, a vector; and Q, an r × r symmetric positive definite matrix. The gradient of the approximate function is ∇u F = b + Qu Setting the gradient to zero allows an estimate for its minimum. u = −Q−1b
(3-87)
Initially, Q and b are not known and the calculation proceeds as follows: b contains r unknown coefficients and Q another r(r + 1)/2. To estimate b and Q, the computer code is used repeatedly, getting r equations each time—namely (∇u F)(1) = b + Qu(1) (∇u F)(2) = b + Qu(2) ... (∇u F )(t) = b + Qu(t)
(3-88)
As soon as there are as many independent equations as there are unknown coefficients, these linear equations are solved for b and Q. A proper choice of the points u(i) guarantees getting independent equations to solve here. Given b and Q, Eq. (3-87) provides a new estimate for u as a candidate minimum point. The subroutine is used again to obtain the gradient of F at this point. If the gradient is essentially zero, the calculations stop, since a point has been found that satisfies the necessary conditions for optimality. If not, the equations are written in the form of Eq. (3-88) for this new point, adding them to the set while removing the oldest set of equations. The new set of equations for b and Q are solved, and the calculations continue until a minimum point is found. If removal of the oldest equations from the set in Eq. (3-88) leads to a singular set of equations, then different equations have to be selected for removal. Alternatively, all the older equations can be kept, with the new ones added to the top of the list. Pivoting can be done by proceeding down the list until a nonsingular set of equations is found. Then the older equations are used only if necessary. Also, since only one set of equations is being replaced, clever methods are available to find the solution to the equations with much less work than is required to solve the set of equations the first time (Refs. 89 and 259). Quadratic Fit for the Equality Constrained Case Next consider solving a problem of the form of Eq. (3-82). For each iteration k: 1. Enter with values provided for variables u(k). 2. Given values for u(k), solve equations h(x, u) = 0 for x(k). These will be m equations in m unknowns. If the equations are nonlinear, solving can be done using a variant of the Newton-Raphson method. 3. Use Eq. (3-85) to solve for the Lagrange multipliers λ(k). If the Newton-Raphson method (or any or several variants to it) is used to solve the equations, the jacobian matrix ∇ xT h| z (k) and its LU factors are already known so solving Eq. (3-85) requires very little effort. 4. Substitute λ(k) into Eq. (3-84), which in general will not be zero. The gradient ∇u L(k) computed will be the constrained derivatives of F with respect to the independent variables u(k). 5. Return.
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MATHEMATICS
The calculations begin with given values for the independent variables u and exit with the (constrained) derivatives of the objective function with respect to them. Use the routine described above for the unconstrained problem where a succession of quadratic fits is used to move toward the optimal point for an unconstrained problem. This approach is a form of the generalized reduced gradient (GRG) approach to optimizing, one of the better ways to carry out optimization numerically. Inequality Constrained Problems To solve inequality constrained problems, a strategy is needed that can decide which of the inequality constraints should be treated as equalities. Once that question is decided, a GRG type of approach can be used to solve the resulting equality constrained problem. Solving can be split into two phases: phase 1, where the goal is to find a point that is feasible with respect to the inequality constraints; and phase 2, where one seeks the optimum while maintaining feasibility. Phase 1 is often accomplished by ignoring the objective function and using instead p
F=
i=1
g 2i (z) if gi(z) > 0) 0 otherwise
until all the inequality constraints are satisfied. Then at each point, check which of the inequality constraints are active, or exactly equal to zero. These can be placed into the active set and treated as equalities. The remaining can be put aside to be used only for testing. A step can then be proposed using the GRG algorithm. If it does not cause one to violate any of the inactive inequality constraints, the step is taken. Otherwise one can add the closest inactive inequality constraint to the active set. Finding the closet inactive equality will almost certainly require a line search in the direction proposed by the GRG algorithm. When one comes to a stationary point, one has to test the active inequality constraints at that point to see if they should remain active. This test is done by examining the sign (they should be nonnegative if they are to remain active) of their respective Kuhn-Tucker multipliers. If any should be released, it has to be done carefully as the release of a constraint changes the multipliers for all the constraints. One can find oneself cycling through the testing to decide whether to release the constraints. A correct approach is to add slack variables s to the problem to convert the inequality constraints to equalities and then require the slack variables to remain positive. The multipliers associated with the inequalities s ≥ 0 all behave independently, and their sign tells one directly to keep or release the constraints. In other words, simultaneously release all the slack variables that have multipliers strictly less than zero. If released, the slack variables must be treated as a part of the set of independent variables until one is well away from the associated constraints for this approach to work. Successive Quadratic Programming (SQP) The above approach to finding the optimum is called a feasible path method, as it attempts at all times to remain feasible with respect to the equality and inequality constraints as it moves to the optimum. A quite different method exists called the Successive Quadratic Programming (SQP) method, which only requires one be feasible at the final solution. Tests that compare the GRG and SQP methods generally favor the SQP method so it has the reputation of being one of the best methods known for nonlinear optimization for the type of problems considered here. Assume certain inequality constraints will be active at the final solution. The necessary conditions for optimality are ∇z L(z, µ, λ) = ∇F + ∇g A µ + ∇hλ = 0,
gA(z) = 0,
h(z) = 0
Then one can apply Newton’s method to the necessary conditions for optimality, which are a set of simultaneous (non)linear equations. The Newton equations one would write are
∇zzL[z(i), u(i), λ(i)]
∇gA[z(i)]
∇h[z(i)]
∇gA[z(i)]T
0
0
∇h[z(i)]T
0
0
∆z(i)
∇zL[z(i), µ(i), λ(i)]
∆µ(i) = −
gA[z(i)]
∆λ(i)
h[z(i)]
A sufficient condition for a unique Newton direction is that the matrix of constraint derivatives is of full rank (linear independence
of constraints) and the Hessian matrix of the Lagrange function [∇zz L(z, µ, λ)] projected into the space of the linearized constraints is positive definite. The linearized system actually represents the solution of the following quadratic programming problem: 1 Min ∇F [z(i)]T∆z + ∆ zT ∇zz L[z(i), µ(i), λ(i)]∆z ∆z 2 subject to gA[z(i)] + ∇g A[z(i)]T∆z = 0
and
h[z(i)] + ∇h[z(i)]T∆z = 0
Reformulating the necessary conditions as a linear quadratic program has an interesting side effect. We can simply add linearizations of the inactive inequalities to the problem and let the active set be selected by the algorithm used to solve the linear quadratic program. Problems with calculating second derivatives as well as maintaining positive definiteness of the Hessian matrix can be avoided by approximating this matrix by B(i) using a quasi-Newton formula such as BFGS (Refs. 50, 84, 109, 110, 122, and 259). One maintains positive definiteness by skipping the update if it causes the matrix to lose this property. Here gradients of the Lagrange function are used to calculate the update formula (Refs. 136 and 228). The resulting quadratic program, which generates the search direction at each iteration i, becomes: 1 Min ∇F[z(i)]T∆z + ∆zTB(i)∆z ∆z 2 subject to
g[z(i)] + ∇g[z(i)]T∆z ≤ 0 h[z(i)] + ∇h[z(i)]T∆z = 0
This linear quadratic program will have a unique solution if B(i) is kept positive definite. Efficient solution methods exist for solving it (Refs. 119 and 123). Finally, to ensure convergence of this algorithm from poor starting points, a step size α is chosen along the search direction so that the point at the next iteration (zi + 1 = zi + αd) is closer to the solution of the NLP (Refs. 65, 136, and 254). These problems get very large as the Lagrange function involves all the variables in the problem. If one has a problem with 5000 variables z and the problem has only 10 degrees of freedom (i.e., the partitioning will select 4990 variables x and only 10 variables u), one is still faced with maintaining a matrix B that is 5000 × 5000. See Westerberg (Ref. 40) for references to this case. Interior Point Algorithms for Linear Programming Problems There has been considerable excitement in the popular press about so-called interior point algorithms (Ref. 23) for solving extremely large linear programming problems. Computational demands for these algorithms grow less rapidly than for the Simplex algorithm, with a breakeven point being a few thousand constraints. A key idea for an interior method is that one heads across the feasible region to locate the solution rather than around its edges as one does for the Simplex algorithm. This move is found by computing the direction of steepest descent for the objective function with respect to changing the slack variables. Variables u are computed in terms of the slack variables by using the inequality constraints. The direction of steepest descent is a function of the scaling of the variables used for the problem. See Refs. 6, 124, 199, and 295. Linear Programming The combined term linear programming is given to any method for finding where a given linear function of several variables takes on an extreme value, and what that value is, when the variables are nonnegative and are constrained by linear equalities or inequalities. A very general problem consists of maximizing f = nf = 1 cj zj subject to the constraints zj ≥ 0 ( j = 1, 2, . . . , n) n and j = 1 aij zj ≤ bi (i = 1, 2, . . . , m). With S the set of all points whose coordinates zj satisfy all the constraints, we must ask three questions: (1) Are the constraints consistent? If not, S is empty and there is no solution. (2) If S is not empty, does the function f become unbounded on S? If so, the problem has no solution. If not, then there is a point P of S that is optimal in the sense that if Q is any point of S then f(Q) ≤ f(P). (3) How can we find P? The simplex algorithm, in a sense, prepares the problem before cal-
STATISTICS culation in such a way that favorable answers to these questions are tentatively assumed for the given problem and can be guaranteed for the prepared problem. The calculations then reveal whether or not those assumptions are justified for the given problem. The simplex algorithm terminates automatically, yielding full information on the given problem and so-called dual problem. The dual of the general problem of linear programming is to minimizem d(µ1, . . . , µm) = m i = 1 µi bi subject to µi ≥ 0 (i = 1, 2, . . . , m) and i = 1 µ i aif ≥ cj( j = 1, 2, . . . , n). Let A be the matrix [aij], c = [cj], U = [µi] be row vectors, and B = [bi]T, Z = [zj]T be column vectors. In matrix form the original (primal) problem is to maximize f(Z) = CZ subject to Z ≥ 0, AZ ≤ B. The dual is to minimize d(U) = UB subject to U ≥ 0, UA ≥ C. Example Maximize 3z1 + 4z2 subject to the constraints z1 ≥ 0, z2 ≥ 0, 2z1 + 4z2 ≤ 8, and 4z1 + 3z2 ≤ 10. The dual problem is to minimize 8µ1 + 10µ2 subject to the constraints µ1 ≥ 0, µ2 ≥ 0, 2µ1 + 4µ2 ≥ 3, and 5µ1 + 3µ2 ≥ 4. Simplex Method 1. Original problem. Let the column vector [zj]T = z ( j = 1, 2, n . . . , n) and the row vector [cj] = c. To maximize f(z) = j = 1 cj zj = cTz subject to then n constraints zj ≥ 0 ( j = 1, . . . , n) and m further constraints hi: j = 1 aij zj° ibi (i = 1, 2, . . . , m) where °i can be ≥ = or ≤. If any bi ≤ 0, multiply hi by − 1; thus we may assume bi ≥ 0. We suppose the m constraints have been arranged so that °i is ≥ for i = 1, . . . , g; °i is = for i = g + 1, . . . , g + e; °i is ≤ for i = g + e + 1, . . . , g + e + l = m. 2. Adjusted original problem. Introduce m + g further variables
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with associated constraints and coefficients for use in f. Thus, replacing j by j + m, f becomes f(x) = mj =+mn + 1 cj zj and constraints zj ≥ 0 and hi: mj =+mn + 1 aij zj° i bi, i = 1, . . . , m. 3. Prepared problem. For i = 1, . . . , g replace hi by Hi: zi + mj =+mn + 1 aij zj zm + n + i = bi, define ci = −M(M > 0 and “large”) and Cm + n + i = 0, and add the constraints zi ≥ 0, zm + n + i ≥ 0. For i = g + e + 1, . . . , m replace hi by Hi. xi + mj =+mn + 1 aij zj = bi, define ci = 0, and adjoin zi ≥ 0. Let J run from 1 to N = n + m + g; put Z = [zj]T and j = m + 1, . . . , m + n. The new function to be maximized is f(Z) = N g=e vm + n J = 1 cj z j. Actually this is f(Z) = −M i = 1 zi + j = m + 1 cjzj, for all other coefficients are zero. Thus for J = g + e + 1, . . . , m and m + n + 1, . . . , N the variables z j make no contribution to f. They are called slack variables, since they take up the slack permitted by the inequalities (≤ and ≥) in hi. Any variable zj, i = 1, . . . , g + e whose value is not zero gives rise to a large negative term −Mzi. Such a term will keep f(Z) less than it would be with that zi = 0. The effect of ci = −M(i = 1, . . . , g + e) is to make it likely that the optimal solution will have the artificial variables zi = 0(i = 1, . . . , g + e). N The prepared problem now has the form—maximize f(Z) = j = 1 cjzJ N subject to zJ ≥ 0 and Hi: j = 1 aijzj = (i = 1, . . . , m), where bi ≥ 0, aiβ = δiβ =
i≠β (β = 1, . . . , m), ai,m + n + β = −δiβ (β = 1, . . . , g) i≠β
l
0
and aij came from hi. The set of feasible points SP (points satisfying all constraints) for the prepared problem is not empty, and f(Z) is bounded above on SP.
STATISTICS GENERAL REFERENCES:
69, 93, 134, 169, 186, 211, 265, 291.
INTRODUCTION Statistics represents a body of knowledge which enables one to deal with quantitative data reflecting any degree of uncertainty. There are six basic aspects of applied statistics. These are: 1. Type of data 2. Random variables 3. Models 4. Parameters 5. Sample statistics 6. Characterization of chance occurrences From these can be developed strategies and procedures for dealing with (1) estimation and (2) inferential statistics. The following has been directed more toward inferential statistics because of its broader utility. Detailed illustrations and examples are used throughout to develop basic statistical methodology for dealing with a broad area of applications. However, in addition to this material, there are many specialized topics as well as some very subtle areas which have not been discussed. The references should be used for more detailed information. Type of Data In general, statistics deals with two types of data: counts and measurements. Counts represent the number of discrete outcomes, such as the number of defective parts in a shipment, the number of lost-time accidents, and so forth. Measurement data are treated as a continuum. For example, the tensile strength of a synthetic yarn theoretically could be measured to any degree of precision. A subtle aspect associated with count and measurement data is that some types of count data can be dealt with through the application of techniques which have been developed for measurement data alone. This ability is due to the fact that some simplified measurement statistics serve as an excellent approximation for the more tedious count statistics. Random Variables Applied statistics deals with quantitative data. In tossing a fair coin the successive outcomes would tend to be
different, with heads and tails occurring randomly over a period of time. Given a long strand of synthetic fiber, the tensile strength of successive samples would tend to vary significantly from sample to sample. Counts and measurements are characterized as random variables, that is, observations which are susceptible to chance. Virtually all quantitative data are susceptible to chance in one way or another. Models Part of the foundation of statistics consists of the mathematical models which characterize an experiment. The models themselves are mathematical ways of describing the probability, or relative likelihood, of observing specified values of random variables. For example, in tossing a coin once, a random variable x could be defined by assigning to x the value 1 for a head and 0 for a tail. Given a fair coin, the probability of observing a head on a toss would be a .5, and similarly for a tail. Therefore, the mathematical model governing this experiment can be written as x
P(x)
0 1
.5 .5
where P(x) stands for what is called a probability function. This term is reserved for count data, in that probabilities can be defined for particular outcomes. The probability function that has been displayed is a very special case of the more general case, which is called the binomial probability distribution. For measurement data which are considered continuous, the term probability density is used. For example, consider a spinner wheel which conceptually can be thought of as being marked off on the circumference infinitely precisely from 0 up to, but not including, 1. In spinning the wheel, the probability of the wheel’s stopping at a specified marking point at any particular x value, where 0 ≤ x < 1, is zero, for example, stopping at the value x = .5. For the spinning wheel, the probability density function would be defined by f(x) = 1 for 0 ≤ x < 1. Graphically, this is shown in Fig. 3-55. The relative-probability
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MATHEMATICS 1 n x = xi with n = 6 n i=1 = (1/6)(.1009 + .3754 + ⋅⋅⋅ + .6606) = .3899 The sample standard deviation s is defined by the computation s= Density function.
FIG. 3-55
=
concept refers to the fact that density reflects the relative likelihood of occurrence; in this case, each number between 0 and 1 is equally likely. For measurement data, probability is defined by the area under the curve between specified limits. A density function always must have a total area of 1. P [0 ≤ x ≤ .4] = .4 P [.2 ≤ x ≤ .9] = .7 P[.6 ≤ x < 1] = .4 and so forth. Since the probability associated with any particular point value is zero, it makes no difference whether the limit point is defined by a closed interval (≤ or ≥) or an open interval (< or >).
Many different types of models are used as the foundation for statistical analysis. These models are also referred to as populations. Parameters As a way of characterizing probability functions and densities, certain types of quantities called parameters can be defined. For example, the center of gravity of the distribution is defined to be the population mean, which is designated as µ. For the coin toss µ = .5, which corresponds to the average value of x; i.e., for half of the time x will take on a value 0 and for the other half a value 1. The average would be .5. For the spinning wheel, the average value would also be .5. Another parameter is called the standard deviation, which is designated as σ. The square of the standard deviation is used frequently and is called the popular variance, σ2. Basically, the standard deviation is a quantity which measures the spread or dispersion of the distribution from its mean µ. If the spread is broad, then the standard deviation will be larger than if it were more constrained. For specified probability and density functions, the respective means and variances are defined by the following: Probability functions
Probability density functions E(x) = µ =
Var(x) = σ = (x − µ) P(x) 2
x f(x) dx x
x
Var(x) = σ =
2
2
(x − µ) f(x) dx 2
x
x
(xi − x)2 n−1
n x 2i − ( xi)2 n(n − 1)
In effect, this represents the root of a statistical average of the squares. The divisor quantity (n − 1) will be referred to as the degrees of freedom. The value of n − 1 is used in the denominator because the deviations from the sample average must total zero, or
(x − x) = 0 i
Thus knowing n − 1 values of xi − x permits calculation of the nth value of xi − x. The sample value of the standard deviation for the data given is .3686. The following is a tabulation of the deviations (xi − xj) for the data:
Example For the density of Fig. 3-55 the
E(x) = µ = x P(x)
where E(x) is defined to be the expected or average value of x. Sample Statistics Many types of sample statistics will be defined. Two very special types are the sample mean, designated as x, and the sample standard deviation, designated as s. These are, by definition, random variables. Parameters like µ and σ are not random variables; they are fixed constants. Example In an experiment, six random numbers (rounded to four decimal places) were observed from the uniform distribution f(x) = 1 for 0 ≤ x < 1: .1009 .3754 .0842 .9901 .1280 .6606 The sample mean corresponds to the arithmetic average of the observations, which will be designated as x1 through x6, where
x .1009 .3754 .0842 .9901 .1280 .6606
x − x −.2890 −.0145 −.3057 .6002 −.2619 .2707
x = .3899
s = .3686
In effect, the standard deviation quantifies the relative magnitude of the deviation numbers, i.e., a special type of “average” of the distance of points from their center. In statistical theory, it turns out that the corresponding variance quantities s2 have remarkable properties which make possible broad generalities for sample statistics and therefore also their counterparts, the standard deviations. For the corresponding population, the parameter values are µ = .50 and σ = .2887. If, instead of using individual observations only, averages of six were reported, then the corresponding population parameter values would be µ = .50 and σx = σ/6 = .1179. The corresponding variance for an average will be written occasionally as Var (x) = var (x)/n. In effect, the variance of an average is inversely proportional to the sample size n, which reflects the fact that sample averages will tend to cluster about µ much more closely than individual observations. This is illustrated in greater detail under “Measurement Data and Sampling Densities.” Characterization of Chance Occurrences To deal with a broad area of statistical applications, it is necessary to characterize the way in which random variables will vary by chance alone. The basic foundation for this characteristic is laid through a density called the gaussian, or normal, distribution. Determining the area under the normal curve is a very tedious procedure. However, by standardizing a random variable that is normally distributed, it is possible to relate all normally distributed random variables to one table. The standardization is defined by the identity z = (x − µ)/σ, where z is called the unit normal. Further, it is possible to standardize the sampling distribution of averages x by the identity z = (x − µ)/(σ/ n). A remarkable property of the normal distribution is that, almost regardless of the distribution of x, sample averages x will approach the gaussian distribution as n gets large. Even for relatively small values of n, of about 10, the approximation in most cases is quite close. For example, sample averages of size 10 from the uniform distribution will have essentially a gaussian distribution. Also, in many applications involving count data, the normal distribution can be used as a close approximation. In particular, the approximation is quite close for the binomial distribution within certain guidelines.
STATISTICS ENUMERATION DATA AND PROBABILITY DISTRIBUTIONS Introduction Many types of statistical applications are characterized by enumeration data in the form of counts. Examples are the number of lost-time accidents in a plant, the number of defective items in a sample, and the number of items in a sample that fall within several specified categories. The sampling distribution of count data can be characterized through probability distributions. In many cases, count data are appropriately interpreted through their corresponding distributions. However, in other situations analysis is greatly facilitated through distributions which have been developed for measurement data. Examples of each will be illustrated in the following subsections. Binomial Probability Distribution Nature Consider an experiment in which each outcome is classified into one of two categories, one of which will be defined as a success and the other as a failure. Given that the probability of success p is constant from trial to trial, then the probability of observing a specified number of successes x in n trials is defined by the binomial distribution. The sequence of outcomes is called a Bernoulli process, Nomenclature n = total number of trials x = number of successes in n trials p = probability of observing a success on any one trial pˆ = x/n, the proportion of successes in n trials Probability Law x n x P(x) = P = p (1 − p)n − x x = 0, 1, 2, . . . , n x n
where
x = x!(n − x)! n
n!
Properties E(x) = np E( p ˆ)=p
5. The number of spinning machines that break down during 1 day at a large plant. Nomenclature x = total number of arrivals in a total length L or total period T a = average rate of arrivals for a unit length or unit time λ = aL = expected or average number of arrivals for the total length L λ = aT = expected or average number of arrivals for the total time T Probability Law Given that a is constant for the total length L or period T, the probability of observing x arrivals in some period L or T is given by λx P(x) = e−λ x = 0, 1, 2, . . . x! Properties E(x) = λ
Var (x) = λ
Example The number of broken filaments in a threadline has been averaging .015 per yard. What is the probability of observing exactly two broken filaments in the next 100 yd? In this example, a = .015/yd and L = 100 yd; therefore λ = (.015)(100) = 1.5: (1.5)2 P(x = 2) = e−1.5 = .2510 2! Example A commercial item is sold in a retail outlet as a unit product. In the past, sales have averaged 10 units per month with no seasonal variation. The retail outlet must order replacement items 2 months in advance. If the outlet starts the next 2-month period with 25 items on hand, what is the probability that it will stock out before the end of the second month? Given a = 10/month, then λ = 10 × 2 = 20 for the total period of 2 months: ∞
25
26
0
P(x ≥ 26) = P(x) = 1 − P(x)
2025 20x −20 −20 20 202 e =e 1 + + + ⋅⋅⋅ + 0 x! 1 2! 25! 25
= .887815
Var(x) = np(1 − p) Var( p ˆ ) = p(1 − p)/n
Geometric Probability Distribution Nature Consider an experiment in which each outcome is classified into one of two categories, one of which will be defined as a success and the other as a failure. Given that the probability of success p is constant from trial to trial, then the probability of observing the first success on the xth trial is defined by the geometric distribution. Nomenclature p = probability of observing a success on any one trial x = the number of trials to obtain the first success Probability Law P(x) = p(1 − p)x − 1
x = 1, 2, 3, . . .
Properties E(x) = 1/p
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Var (x) = (1 − p)/p2
Poisson Probability Distribution Nature In monitoring a moving threadline, one criterion of quality would be the frequency of broken filaments. These can be identified as they occur through the threadline by a broken-filament detector mounted adjacent to the threadline. In this context, the random occurrences of broken filaments can be modeled by the Poisson distribution. This is called a Poisson process and corresponds to a probabilistic description of the frequency of defects or, in general, what are called arrivals at points on a continuous line or in time. Other examples include: 1. The number of cars (arrivals) that pass a point on a high-speed highway between 10:00 and 11:00 A.M. on Wednesdays 2. The number of customers arriving at a bank between 10:00 and 10:10 A.M. 3. The number of telephone calls received through a switchboard between 9:00 and 10:00 A.M. 4. The number of insurance claims that are filed each week
Therefore P(x ≥ 26) = .112185 or roughly an 11 percent chance of a stockout.
Hypergeometric Probability Distribution Nature In an experiment in which one samples from a relatively small group of items, each of which is classified in one of two categories, A or B, the hypergeometric distribution can be defined. One example is the probability of drawing two red and two black cards from a deck of cards. The hypergeometric distribution is the analog of the binomial distribution when successive trials are not independent, i.e., when the total group of items is not infinite. This happens when the drawn items are not replaced. Nomenclature N = total group size n = sample group size X = number of items in the total group with a specified attribute A N − X = number of items in the total group with the other attribute B x = number of items in the sample with a specified attribute A n − x = number of items in the sample with the other attribute B
Category A Category B Total
Probability Law P(x) =
Population
Sample
X N−X N
x n−x n
Nn −− Xx XxNn
nX E(x) = N N−n var (x) = nP(1 − P) N−1
3-72
MATHEMATICS
Example What is the probability that an appointed special committee of 4 has no female members when the members are randomly selected from a candidate group of 10 males and 7 females? 10 7 4 0 P(x = 0) = = .0882 17 4
Example A bin contains 300 items, of which 240 are good and 60 are defective. In a sample of 6 what is the probability of selecting 4 good and 2 defective items by chance?
4 2 = .2478 300 6 240
P(x) =
60
Multinomial Distribution Nature For an experiment in which successive outcomes can be classified into two or more categories and the probabilities associated with the respective outcomes remain constant, then the experiment can be characterized through the multinomial distribution. Nomenclature n = total number of trials k = total number of distinct categories pj = probability of observing category j on any one trial, j = 1, 2, . . . , k xj = total number of occurrences in category j in n trials Probability Law n! P(x1, x2, . . . , xk) = p1x1 p2x2 ⋅⋅⋅ pkx k x1! x2! . . . xk! Example In tossing a die 12 times, what is the probability that each face value will occur exactly twice? 12! 1 2 1 2 1 2 1 2 1 2 1 2 p(2, 2, 2, 2, 2, 2) = = .003438 2!2!2!2!2!2! 6 6 6 6 6 6
Monday 16
Tuesday 17
Wednesday 18
Thursday 19
Friday 20
Groups of 25 pooled
37.68 36.38 38.43 39.07 33.06 x = 36.92 s = 2.38
35.97 35.92 36.51 33.89 36.01 35.66 1.02
33.71 32.34 33.29 32.81 37.13 33.86 1.90
35.61 37.13 31.37 35.89 36.33 35.27 2.25
36.65 37.91 42.18 39.25 33.32 37.86 3.27
35.91 2.52
Pooled sample of 100: x = 35.16
Even if the process were at steady state, tensile strength, a key property would still reflect some variation. Steady state, or stable operation of any process, has associated with it a characteristic variation. Superimposed on this is the testing method, which is itself a process with its own characteristic variation. The observed variation is a composite of these two variations. Assume that the table represents “typical” production-line performance. The numbers themselves have been generated on a computer and represent random observations from a population with µ = 35 and a population standard deviation σ = 2.45. The sample values reflect the way in which tensile strength can vary by chance alone. In practice, a production supervisor unschooled in statistics but interested in high tensile performance would be despondent on the eighth day and exuberant on the twentieth day. If the supervisor were more concerned with uniformity, the lowest and highest points would have been on the eleventh and seventeenth days. An objective of statistical analysis is to serve as a guide in decision making in the context of normal variation. In the case of the production supervisor, it is to make a decision, with a high probability of being correct, that something has in fact changed the operation. Suppose that an engineering change has been made in the process and five new tensile samples have been tested with the results: 36.81 38.34 34.87 39.58 36.12
MEASUREMENT DATA AND SAMPLING DENSITIES Introduction The following example data are used throughout this subsection to illustrate concepts. Consider, for the purpose of illustration, that five synthetic-yarn samples have been selected randomly from a production line and tested for tensile strength on each of 20 production days. For this, assume that each group of five corresponds to a day, Monday through Friday, for a period of 4 weeks: Monday 1
Tuesday 2
Wednesday 3
Thursday 4
Friday 5
36.48 35.33 35.92 32.28 31.61 x = 34.32 s = 2.22
38.06 31.86 33.81 30.30 35.27 33.86 3.01
35.28 36.58 38.81 33.31 33.88 35.57 2.22
36.34 36.25 30.46 37.37 37.52 35.59 2.92
36.73 37.17 33.07 34.27 36.94 35.64 1.85
6
7
8
9
10
38.67 32.08 33.79 32.85 35.22 x = 34.52 s = 2.60
36.62 33.05 35.43 36.63 31.46 34.64 2.30
35.03 36.22 32.71 32.52 27.23 32.74 3.46
35.80 33.16 35.19 32.91 35.44 34.50 1.36
36.82 36.49 32.83 32.43 34.16 34.54 2.03
11
12
13
14
15
39.63 34.38 36.51 30.00 39.64 x = 36.03 s = 4.04
34.52 37.39 34.16 35.76 37.63 35.89 1.59
36.05 35.36 35.00 33.61 36.98 35.40 1.25
36.64 31.18 36.13 37.51 39.05 36.10 2.96
31.57 36.21 33.84 35.01 34.95 34.32 1.75
Groups of 25 pooled
34.19 2.35
x = 37.14 s = 1.85
In this situation, management would inquire whether the product has been improved by increased tensile strength. To answer this question, in addition to a variety of analogous questions, it is necessary to have some type of scientific basis upon which to draw a conclusion. A scientific basis for the evaluation and interpretation of data is contained in the accompanying table descriptions. These tables characterize the way in which sample values will vary by chance alone in the context of individual observations, averages, and variances.
Table number
35.00 2.40
s = 2.47
Designated symbol
Variable
Sampling distribution of
3-4
z
x−µ σ
3-4
z
x − µ σ/ n
Averages
3-5
t
x − µ s/ n
Averages when σ is unknown*
3-6
χ2
(s2/σ2)(df)
3-7
F
s12 /s 22
Observations*
Variances* Ratio of two independent sample variances*
*When sampling from a gaussian distribution.
35.55 2.42
Normal Distribution of Observations Many types of data follow what is called the gaussian, or bell-shaped, curve; this is especially true of averages. Basically, the gaussian curve is a purely mathematical function which has very special properties. However, owing to some mathematically intractable aspects primary use of the function is restricted to tabulated values.
STATISTICS
3-73
0 and 2 is .4772; therefore, the area between 1 and 2 is .4772 − .3413 = .1359. Also, since the normal curve is symmetric, areas to the left can be determined in exactly the same way. For example, the area between −2 and +1 would include the area between −2 and 0, .4772 (the same as 0 to 2), plus the area between 0 and 1, .3413, or a total area of .8185. Any types of observation which are applicable to the normal curve can be transformed to Z values by the relationship z = (x − µ)/σ and, conversely, Z values to x values by x = µ + σz, as shown in Fig. 3-56. For example, for tensile strength, with µ = 35 and σ = 2.45, this would dictate z = (x − 35)/2.45 and x = 35 + 2.45z. Example What proportion of tensile values will fall between 34 and 36? FIG. 3-56
z1 = (34 − 35)/2.45 = −.41
Transformation of z values.
z2 = (36 − 35)/2.45 = .41
P[−.41 ≤ z ≤ .41] = .3182, or roughly 32 percent
Basically, the tabled values represent area (proportions or probability) associated with a scaling variable designated by Z in Fig. 3-56. The normal curve is centered at 0, and for particular values of Z, designated as z, the tabulated numbers represent the corresponding area under the curve between 0 and z. For example, between 0 and 1 the area is .3413. (Get this number from Table 3-4. The value of A includes the area on both sides of zero. Thus we want A/2. For z = 1, A = 0.6827, A/2 = 0.3413. For z = 2, A/2 = 0.4772.) The area between
TABLE 3-4
The value 0.3182 is interpolated from Table 3-4 using z = 0.40, A = 0.3108, and z = 0.45, A = 0.3473.
Example What midrange of tensile values will include 95 percent of the sample values? Since P[−1.96 ≤ z ≤ 1.96] = .95, the corresponding values of x are x1 = 35 − 1.96(2.45) = 30.2 x2 = 35 + 1.96(2.45) = 39.8 P[30.2 ≤ x ≤ 39.8] = .95
or
Ordinates and Areas between Abscissa Values -z and +z of the Normal Distribution Curve
z
X
Y
A
1−A
z
X
0 0.05 .10 .15 .20
µ µ 0.05σ µ .10σ µ .15σ µ .20σ
0.399 .398 .397 .394 .391
0.0000 .0399 .0797 .1192 .1585
1.0000 0.9601 .9203 .8808 .8415
1.50 1.55 1.60 1.65 1.70
µ 1.50σ µ 1.55σ µ 1.60σ µ 1.65σ µ 1.70σ
0.1295 .1200 .1109 .1023 .0940
0.8664 .8789 .8904 .9011 .9109
0.1336 .1211 .1096 .0989 .0891
.25 .30 .35 .40 .45
µ µ µ µ µ
.25σ .30σ .35σ .40σ .45σ
.387 .381 .375 .368 .361
.1974 .2358 .2737 .3108 .3473
.8026 .7642 .7263 .6892 .6527
1.75 1.80 1.85 1.90 1.95
µ 1.75σ µ 1.80σ µ 1.85σ µ 1.90σ µ 1.95σ
.0863 .0790 .0721 .0656 .0596
.9199 .9281 .9357 .9446 .9488
.0801 .0719 .0643 .0574 .0512
.50 .55 .60 .65 .70
µ µ+ µ µ µ
.50σ .55σ .60σ .65σ .70σ
.352 .343 .333 .323 .312
.3829 .4177 .4515 .4843 .5161
.6171 .5823 .5485 .5157 .4839
2.00 2.05 2.10 2.15 2.20
µ 2.00σ µ 2.05σ µ 2.10σ µ 2.15σ µ 2.20σ
.0540 .0488 .0440 .0396 .0335
.9545 .9596 .9643 .9684 .9722
.0455 .0404 .0357 .0316 .0278
.75 .80 .85 .90 .95
µ µ µ µ µ
.75σ .80σ .85σ .90σ .95σ
.301 .290 .278 .266 .254
.5467 .5763 .6047 .6319 .6579
.4533 .4237 .3953 .3681 .3421
2.25 2.30 2.35 2.40 2.45
µ 2.25σ µ 2.30σ µ 2.35σ µ 2.40σ µ 2.45σ
.0317 .0283 .0252 .0224 .0198
.9756 .9786 .9812 .9836 .9857
.0244 .0214 .0188 .0164 .0143
1.00 1.05 1.10 1.15 1.20
µ 1.00σ µ 1.05σ µ 1.10σ µ 1.15σ µ 1.20σ
.242 .230 .218 .206 .194
.6827 .7063 .7287 .7499 .7699
.3173 .2937 .2713 .2501 .2301
2.50 2.55 2.60 2.65 2.70
µ 2.50σ µ 2.55σ µ 2.60σ µ 2.65σ µ 2.70σ
.0175 .0154 .0136 .0119 .0104
.9876 .9892 .9907 .9920 .9931
.0124 .0108 .0093 .0080 .0069
1.25 1.30 1.35 1.40 1.45
µ 1.25σ µ 1.30σ µ 1.35σ µ 1.40σ µ 1.45σ
.183 .171 .160 .150 .139
.7887 .8064 .8230 .8385 .8529
.2113 .1936 .1770 .1615 .1471
2.75 2.80 2.85 2.90 2.95
µ 2.75σ µ 2.80σ µ 2.85σ µ 2.90σ µ 2.95σ
.0091 .0079 .0069 .0060 .0051
.9940 .9949 .9956 .9963 .9968
.0060 .0051 .0044 .0037 .0032
1.50
µ 1.50σ
.130
.8664
.1336
3.00 4.00 5.00
µ 3.00σ µ 4.00σ µ 5.00s
.0044 .0001 .000001
.9973 .99994 .9999994
.0027 .00006 .0000006
0.000 .126 .253 .385 .524 .674 .842
µ µ 0.126σ µ .253σ µ .385σ µ .524σ µ .674σ µ .842σ
0.3989 .3958 .3863 .3704 .3477 .3178 .2800
.0000 .1000 .2000 .3000 .4000 .5000 .6000
1.0000 0.9000 .8000 .7000 .6000 .5000 .4000
1.036 1.282 1.645 1.960 2.576 3.291 3.891
µ 1.036σ µ 1.282σ µ 1.645σ µ 1.960σ µ 2.576σ µ 3.291σ µ 3.891σ
Y
0.2331 .1755 .1031 .0584 .0145 .0018 .0002
A
0.7000 .8000 .9000 .9500 .9900 .9990 .9999
1−A
0.3000 .2000 .1000 .0500 .0100 .0010 .0001
3-74
MATHEMATICS
Normal Distribution of Averages An examination of the tensile-strength data previously tabulated would show that the range (largest minus the smallest) of tensile strength within days averages 5.72. The average range in x values within each week is 2.37, while the range in the four weekly averages is 1.72. This reflects the fact that averages tend to be less variable in a predictable way. Given that the variance of x is var (x) = σ2, then the variance of x based on n observations is var (x) = σ2/n. For averages of n observations, the corresponding relationship for the Z-scale relationship is σ z = (x − µ/σ/ n) or z x = µ + n Example What proportion of daily tensile averages will fall between 34 and 36? z1 = (34 − 35)/(2.45/5) = −.91
z2 = (36 − 35)/(2.45/5) = .91
P[−.91 ≤ z ≤ .91] = .6372, or roughly 64 percent
Example What midrange of daily tensile averages will include 95 percent of the sample values? x1 = 35 − 1.96(2.45/5) = 32.85 x2 = 35 + 1.96(2.45/5) = 37.15 P[32.85 ≤ x ≤ 37.15] = .95
or
Example What proportion of weekly tensile averages will fall between 34 and 36? z1 = (34 − 35)/(2.45/ 25) = −2.04 z2 = (36 − 35)/(2.45/ 25) = 2.04 P[−2.04 ≤ z ≤ 2.04] = .9586, or roughly 96 percent
Distribution of Averages The normal curve relies on a knowledge of σ, or in special cases, when it is unknown, s can be used with the normal curve as an approximation when n > 30. For example, with n > 30 the intervals x s and x 2s will include roughly 68 and 95 percent of the sample values respectively when the distribution is normal. In applications sample sizes are usually small and σ unknown. In these cases, the t distribution can be used where t = (x − µ)/(s/n ) or x = µ + ts/n The t distribution is also symmetric and centered at zero. It is said to be robust in the sense that even when the individual observations x are not normally distributed, sample averages of x have distributions which tend toward normality as n gets large. Even for small n of 5 through 10, the approximation is usually relatively accurate. In reference to the tensile-strength table, consider the summary statistics x and s by days. For each day, the t statistic could be computed. If this were repeated over an extensive simulation and the resultant t quantities plotted in a frequency distribution, they would match the corresponding distribution of t values summarized in Table 3-5. Since the t distribution relies on the sample standard deviation s, the resultant distribution will differ according to the sample size n. To designate this difference, the respective distributions are classified according to what are called the degrees of freedom and abbreviated as df. In simple problems, the df are just the sample size minus 1. In more complicated applications the df can be different. In general, degrees of freedom are the number of quantities minus the number of constraints. For example, four numbers in a square which must have row and column sums equal to zero have only one df, i.e., four numbers minus three constraints (the fourth constraint is redundant).
TABLE 3-5 df
t.30
t.20
t.10
t.05
t.025
t.01
t.005
1 2 3 4 5
0.325 .289 .277 .271 .267
0.727 .617 .584 .569 .559
1.376 1.061 0.978 .941 .920
3.078 1.886 1.638 1.533 1.476
6.314 2.920 2.353 2.132 2.015
12.706 4.303 3.182 2.776 2.571
31.821 6.965 4.541 3.747 3.365
63.657 9.925 5.841 4.604 4.032
6 7 8 9 10
.265 .263 .262 .261 .260
.553 .549 .546 .543 .542
.906 .896 .889 .883 .879
1.440 1.415 1.397 1.383 1.372
1.943 1.895 1.860 1.833 1.812
2.447 2.365 2.306 2.262 2.228
3.143 2.998 2.896 2.821 2.764
3.707 3.499 3.355 3.250 3.169
11 12 13 14 15
.260 .259 .259 .258 .258
.540 .539 .538 .537 .536
.876 .873 .870 .868 .866
1.363 1.356 1.350 1.345 1.341
1.796 1.782 1.771 1.761 1.753
2.201 2.179 2.160 2.145 2.131
2.718 2.681 2.650 2.624 2.602
3.106 3.055 3.012 2.977 2.947
16 17 18 19 20
.258 .257 .257 .257 .257
.535 .534 .534 .533 .533
.865 .863 .862 .861 .860
1.337 1.333 1.330 1.328 1.325
1.746 1.740 1.734 1.729 1.725
2.120 2.110 2.101 2.093 2.086
2.583 2.567 2.552 2.539 2.528
2.921 2.898 2.878 2.861 2.845
21 22 23 24 25
.257 .256 .256 .256 .256
.532 .532 .532 .531 .531
.859 .858 .858 .857 .856
1.323 1.321 1.319 1.318 1.316
1.721 1.717 1.714 1.711 1.708
2.080 2.074 2.069 2.064 2.060
2.518 2.508 2.500 2.492 2.485
2.831 2.819 2.807 2.797 2.787
26 27 28 29 30
.256 .256 .256 .256 .256
.531 .531 .530 .530 .530
.856 .855 .855 .854 .854
1.315 1.314 1.313 1.311 1.310
1.706 1.703 1.701 1.699 1.697
2.056 2.052 2.048 2.045 2.042
2.479 2.473 2.467 2.462 2.457
2.779 2.771 2.763 2.756 2.750
40 60 120 ∞
.255 .254 .254 .253
.529 .527 .526 .524
.851 .848 .845 .842
1.303 1.296 1.289 1.282
1.684 1.671 1.658 1.645
2.021 2.000 1.980 1.960
2.423 2.390 2.358 2.326
2.704 2.660 2.617 2.576
Above values refer to a single tail outside the indicated limit of t. For example, for 95 percent of the area to be between −t and +t in a two-tailed t distribution, use the values for t0.025 or 2.5 percent for each tail.
Example What is the sample value of t for the first day of tensile data? Sample t = (34.32 − 35)/(2.22/5) = −.68 Note that on the average 90 percent of all such sample values would be expected to fall within the interval 2.132.
t Distribution for the Difference in Two Sample Means Population Variances Are Equal The t distribution can be readily extended to the difference in two sample means when the respective populations have the same variance σ2: (x1 − x2) − (µ1 − µ2) t = sp1 /n +1 /n2 1 where s 2p is a pooled variance defined by (n1 − 1)s12 + (n2 − 1)s22 s 2p = (n1 − 1) + (n2 − 1) In this application, the t distribution has (n1 + n2 − 2) df. Population Variances Are Unequal When population variances are unequal, an approximate t quantity can be used:
Example For a sample size n = 5, what values of t define a midarea of 90 percent. For 4 df the tabled value of t corresponding to a midarea of 90 percent is 2.132; i.e., P[−2.132 ≤ t ≤ 2.132] = .90. Example For a sample size n = 25, what values of t define a midarea of 95 percent? For 24 df the tabled value of t corresponding to a midarea of 95 percent is 2.064; i.e., P[−2.064 ≤ t ≤ 2.064] = .95.
Values of t
t.40
with and
(x1 − x2) − (µ1 − µ2) t = a + b a = s12 / n1 b = s22 / n2 (a + b)2 df = a2/(n1 − 1) + b2/(n2 − 1)
STATISTICS Chi-Square Distribution For some industrial applications, product uniformity is of primary importance. The sample standard deviation s is most often used to characterize uniformity. In dealing with this problem, the chi-square distribution can be used where χ 2 = (s2/σ2) (df). The chi-square distribution is a family of distributions which are defined by the degrees of freedom associated with the sample variance. For most applications, df is equal to the sample size minus 1. The probability distribution function is −(df )2 p(y) = y0ydf − 2 exp 2 where y0 is chosen such that the integral of p(y) over all y is one. In terms of the tensile-strength table previously given, the respective chi-square sample values for the daily, weekly, and monthly figures could be computed. The corresponding df would be 4, 24, and 99 respectively. These numbers would represent sample values from the respective distributions which are summarized in Table 3-6. In a manner similar to the use of the t distribution, chi square can be interpreted in a direct probabilistic sense corresponding to a midarea of (1 − α):
P[χ 21 ≤ (s2/σ2)(df) ≤ χ 22 ] = 1 − α where χ 21 corresponds to a lower-tail area of α/2 and χ 22 an upper-tail area of α/2. The basic underlying assumption for the mathematical derivation of chi square is that a random sample was selected from a normal distribution with variance σ2. When the population is not normal but skewed, square probabilities could be substantially in error. Example On the basis of a sample size n = 5, what midrange of values will include the sample ratio s/σ with a probability of 90 percent? Use Table 3-6 for 4 df and read χ 21 = 0.484 for a lower tail area of 0.05/2, 2.5 percent, and read χ 22 = 11.1 for an upper tail area of 97.5 percent.
TABLE 3-6
3-75
P[.484 ≤ (s2/σ2)(4) ≤ 11.1] = .90 P[.35 ≤ s/σ ≤ 1.66] = .90
or
Example On the basis of a sample size n = 25, what midrange of values will include the sample ratio s/σ with a probability of 90 percent? P[12.4 ≤ (s2/σ2)(24) ≤ 39.4] = .90 P[.72 ≤ s/σ ≤ 1.28] = .90
or
This states that the sample standard deviation will be at least 72 percent and not more than 128 percent of the population variance 90 percent of the time. Conversely, 10 percent of the time the standard deviation will underestimate or overestimate the population standard deviation by the corresponding amount. Even for samples as large as 25, the relative reliability of a sample standard deviation is poor.
The chi-square distribution can be applied to other types of application which are of an entirely different nature. These include applications which are discussed under “Goodness-of-Fit Test” and “Two-Way Test for Independence of Count Data.” In these applications, the mathematical formulation and context are entirely different, but they do result in the same table of values. F Distribution In reference to the tensile-strength table, the successive pairs of daily standard deviations could be ratioed and squared. These ratios of variance would represent a sample from a distribution called the F distribution or F ratio. In general, the F ratio is defined by the identity F(γ1, γ 2) = s12 / s22 where γ1 and γ2 correspond to the respective df’s for the sample variances. In statistical applications, it turns out that the primary area of interest is found when the ratios are greater than 1. For this reason, most tabled values are defined for an upper-tail area. However, defining F2 to be that value corresponding to an upper-tail area of α/2, then F1 for a lower-tail area of α/2 can be determined through the identity
Percentiles of the c2 Distribution Percent
df
10
90
95
97.5
0.000039 .0100 .0717 .207 .412
0.00016 .0201 .115 .297 .554
0.00098 .0506 .216 .484 .831
0.0039 .1026 .352 .711 1.15
0.0158 .2107 .584 1.064 1.61
2.71 4.61 6.25 7.78 9.24
3.84 5.99 7.81 9.49 11.07
5.02 7.38 9.35 11.14 12.83
6.63 9.21 11.34 13.28 15.09
7.88 10.60 12.84 14.86 16.75
6 7 8 9 10
.676 .989 1.34 1.73 2.16
.872 1.24 1.65 2.09 2.56
1.24 1.69 2.18 2.70 3.25
1.64 2.17 2.73 3.33 3.94
2.20 2.83 3.49 4.17 4.87
10.64 12.02 13.36 14.68 15.99
12.59 14.07 15.51 16.92 18.31
14.45 16.01 17.53 19.02 20.48
16.81 18.48 20.09 21.67 23.21
18.55 20.28 21.96 23.59 25.19
11 12 13 14 15
2.60 3.07 3.57 4.07 4.60
3.05 3.57 4.11 4.66 5.23
3.82 4.40 5.01 5.63 6.26
4.57 5.23 5.89 6.57 7.26
5.58 6.30 7.04 7.79 8.55
17.28 18.55 19.81 21.06 22.31
19.68 21.03 22.36 23.68 25.00
21.92 23.34 24.74 26.12 27.49
24.73 26.22 27.69 29.14 30.58
26.76 28.30 29.82 31.32 32.80
16 18 20 24 30
5.14 6.26 7.43 9.89 13.79
5.81 7.01 8.26 10.86 14.95
6.91 8.23 9.59 12.40 16.79
7.96 9.39 10.85 13.85 18.49
9.31 10.86 12.44 15.66 20.60
23.54 25.99 28.41 33.20 40.26
26.30 28.87 31.41 36.42 43.77
28.85 31.53 34.17 39.36 46.98
32.00 34.81 37.57 42.98 50.89
34.27 37.16 40.00 45.56 53.67
40 60 120
20.71 35.53 83.85
22.16 37.48 86.92
24.43 40.48 91.58
26.51 43.19 95.70
29.05 46.46 100.62
51.81 74.40 140.23
55.76 79.08 146.57
59.34 83.30 152.21
63.69 88.38 158.95
66.77 91.95 163.64
1 2 3 4 5
0.5
1
2.5
5
99
99.5
For large values of degrees of freedom the approximate formula
2 χ 2a = n 1 − + za 9n
2 9n
3
where za is the normal deviate and n is the number of degrees of freedom, may be used. For example, χ.299 = 60[1 − 0.00370 + 2.326(0.06086)]3 = 60(1.1379)3 = 88.4 for the 99th percentile for 60 degrees of freedom.
3-76
MATHEMATICS F1(γ1, γ 2) = 1/F2(γ 2, γ1)
With 103 df and α = .05, t = 1.96 using t.025 in Table 3-5. Therefore
The F distribution, similar to the chi square, is sensitive to the basic assumption that sample values were selected randomly from a normal distribution. Example For two sample variances with 4 df each, what limits will bracket their ratio with a midarea probability of 90 percent? Use Table 3-7 with 4 df in the numerator and denominator and upper 5 percent points (to get both sides totaling 10 percent). The entry is 6.39. Thus: P[1/6.39 ≤ s 12 /s 22 ≤ 6.39] = .90 P[.40 ≤ s1 /s2 ≤ 2.53] = .90
or
Confidence Interval for a Mean For the daily sample tensilestrength data with 4 df it is known that P[−2.132 ≤ t ≤ 2.132] = .90. This states that 90 percent of all samples will have sample t values which fall within the specified limits. In fact, for the 20 daily samples exactly 16 do fall within the specified limits (note that the binomial with n = 20 and p = .90 would describe the likelihood of exactly none through 20 falling within the prescribed limits—the sample of 20 is only a sample). Consider the new daily sample (with n = 5, x = 37.14, and s = 1.85) which was observed after a process change. In this case, the same probability holds. However, in this instance the sample value of t cannot be computed, since the new µ, under the process change, is not known. Therefore P[−2.132 ≤ (37.14 − µ)/(1.85/5) ≤ 2.132] = .90. In effect, this identity limits the magnitude of possible values for µ. The magnitude of µ can be only large enough to retain the t quantity above −2.132 and small enough to retain the t quantity below +2.132. This can be found by rearranging the quantities within the bracket; i.e., P[35.78 ≤ µ ≤ 38.90] = .90. This states that we are 90 percent sure that the interval from 35.78 to 38.90 includes the unknown parameter µ. In general, s s P x − t ≤ µ ≤ x + t = 1 − α n n
where t is defined for an upper-tail area of α/2 with (n − 1) df. In this application, the interval limits (x + t s/n ) are random variables which will cover the unknown parameter µ with probability (1 − α). The converse, that we are 100 (1 − α) percent sure that the parameter value is within the interval, is not correct. This statement defines a probability for the parameter rather than the probability for the interval. Example What values of t define the midarea of 95 percent for weekly samples of size 25, and what is the sample value of t for the second week? P[−2.064 ≤ t ≤ 2.064] = .95 (34.19 − 35)/(2.35/25) = 1.72.
and
Example For the composite sample of 100 tensile strengths, what is the 90 percent confidence interval for µ? Use Table 3-5 for t.05 with df ≈ ∞.
2.47 2.47 P 35.16 − 1.645 < µ < 35.16 + 1.645 = .90 100 100 or
P[34.75 ≤ µ ≤ 35.57] = .90
Confidence Interval for the Difference in Two Population Means The confidence interval for a mean can be extended to include the difference between two population means. This interval is based on the assumption that the respective populations have the same variance σ2:
1/n 2 ≤ µ1 − µ2 ≤ (x1 − x2) + tsp1/n 2 (x1 − x2) − tsp1/n 1/n 1 + 1 + Example Compute the 95 percent confidence interval based on the original 100-point sample and the subsequent 5-point sample: 99(2.47)2 + 4(1.85)2 s p2 = = 5.997 103 or
sp = 2.45
00 = −1.98 2.20 +1/5 (35.16 − 37.14) 1.96(2.45) 1/1 or
−4.18 ≤ (µ1 − µ2) ≤ .22
Note that if the respective samples had been based on 52 observations each rather than 100 and 5, the uncertainty factor would have been .94 rather than the observed 2.20. The interval width tends to be minimum when n1 = n2.
Confidence Interval for a Variance The chi-square distribution can be used to derive a confidence interval for a population variance σ2 when the parent population is normally distributed. For a 100(1 − α) percent confidence interval (df )s2 (df )s2 ≤ σ2 ≤ 2 χ2 χ12 where χ 12 corresponds to a lower-tail area of α/2 and χ 22 to an upper-tail area of α/2. Example For the first week of tensile-strength samples compute the 90 percent confidence interval for σ2 (df = 24, corresponding to n = 25, using 5 percent and 95 percent in Table 3-6): 24(2.40)2 24(2.40)2 ≤ σ2 ≤ 36.4 13.8 3.80 ≤ σ2 ≤ 10.02 or
1.95 ≤ σ ≤ 3.17
TESTS OF HYPOTHESIS General Nature of Tests The general nature of tests can be illustrated with a simple example. In a court of law, when a defendant is charged with a crime, the judge instructs the jury initially to presume that the defendant is innocent of the crime. The jurors are then presented with evidence and counterargument as to the defendant’s guilt or innocence. If the evidence suggests beyond a reasonable doubt that the defendant did, in fact, commit the crime, they have been instructed to find the defendant guilty; otherwise, not guilty. The burden of proof is on the prosecution. Jury trials represent a form of decision making. In statistics, an analogous procedure for making decisions falls into an area of statistical inference called hypothesis testing. Suppose that a company has been using a certain supplier of raw materials in one of its chemical processes. A new supplier approaches the company and states that its material, at the same cost, will increase the process yield. If the new supplier has a good reputation, the company might be willing to run a limited test. On the basis of the test results it would then make a decision to change suppliers or not. Good management would dictate that an improvement must be demonstrated (beyond a reasonable doubt) for the new material. That is, the burden of proof is tied to the new material. In setting up a test of hypothesis for this application, the initial assumption would be defined as a null hypothesis and symbolized as H0. The null hypothesis would state that yield for the new material is no greater than for the conventional material. The symbol µ0 would be used to designate the known current level of yield for the standard material and µ for the unknown population yield for the new material. Thus, the null hypothesis can be symbolized as H0: µ ≤ µ0. The alternative to H0 is called the alternative hypothesis and is symbolized as H1: µ > µ0. Given a series of tests with the new material, the average yield x would be compared with µ0. If x < µ0, the new supplier would be dismissed. If x > µ0, the question would be: Is it sufficiently greater in the light of its corresponding reliability, i.e., beyond a reasonable doubt? If the confidence interval for µ included µ0, the answer would be no, but if it did not include µ0, the answer would be yes. In this simple application, the formal test of hypothesis would result in the same conclusion as that derived from the confidence interval. However, the utility of tests of hypothesis lies in their generality, whereas confidence intervals are restricted to a few special cases.
TABLE 3-7 F Distribution Upper 5% Points (F.95)
Degrees of freedom for denominator
Degrees of freedom for numerator 1
2
3
4
5
6
7
8
9
10
12
15
20
24
30
40
60
120
∞
1 2 3 4 5
161 18.5 10.1 7.71 6.61
200 19.0 9.55 6.94 5.79
216 19.2 9.28 6.59 5.41
225 19.2 9.12 6.39 5.19
230 19.3 9.01 6.26 5.05
234 19.3 8.94 6.16 4.95
237 19.4 8.89 6.09 4.88
239 19.4 8.85 6.04 4.82
241 19.4 8.81 6.00 4.77
242 19.4 8.79 5.96 4.74
244 19.4 8.74 5.91 4.68
246 19.4 8.70 5.86 4.62
248 19.4 8.66 5.80 4.56
249 19.5 8.64 5.77 4.53
250 19.5 8.62 5.75 4.50
251 19.5 8.59 5.72 4.46
252 19.5 8.57 5.69 4.43
253 19.5 8.55 5.66 4.40
254 19.5 8.53 5.63 4.37
6 7 8 9 10
5.99 5.59 5.32 5.12 4.96
5.14 4.74 4.46 4.26 4.10
4.76 4.35 4.07 3.86 3.71
4.53 4.12 3.84 3.63 3.48
4.39 3.97 3.69 3.48 3.33
4.28 3.87 3.58 3.37 3.22
4.21 3.79 3.50 3.29 3.14
4.15 3.73 3.44 3.23 3.07
4.10 3.68 3.39 3.18 3.02
4.06 3.64 3.35 3.14 2.98
4.00 3.57 3.28 3.07 2.91
3.94 3.51 3.22 3.01 2.85
3.87 3.44 3.15 2.94 2.77
3.84 3.41 3.12 2.90 2.74
3.81 3.38 3.08 2.86 2.70
3.77 3.34 3.04 2.83 2.66
3.74 3.30 3.01 2.79 2.62
3.70 3.27 2.97 2.75 2.58
3.67 3.23 2.93 2.71 2.54
11 12 13 14 15
4.84 4.75 4.67 4.60 4.54
3.98 3.89 3.81 3.74 3.68
3.59 3.49 3.41 3.34 3.29
3.36 3.26 3.18 3.11 3.06
3.20 3.11 3.03 2.96 2.90
3.09 3.00 2.92 2.85 2.79
3.01 2.91 2.83 2.76 2.71
2.95 2.85 2.77 2.70 2.64
2.90 2.80 2.71 2.65 2.59
2.85 2.75 2.67 2.60 2.54
2.79 2.69 2.60 2.53 2.48
2.72 2.62 2.53 2.46 2.40
2.65 2.54 2.46 2.39 2.33
2.61 2.51 2.42 2.35 2.29
2.57 2.47 2.38 2.31 2.25
2.53 2.43 2.34 2.27 2.20
2.49 2.38 2.30 2.22 2.16
2.45 2.34 2.25 2.18 2.11
2.40 2.30 2.21 2.13 2.07
16 17 18 19 20
4.49 4.45 4.41 4.38 4.35
3.63 3.59 3.55 3.52 3.49
3.24 3.20 3.16 3.13 3.10
3.01 2.96 2.93 2.90 2.87
2.85 2.81 2.77 2.74 2.71
2.74 2.70 2.66 2.63 2.60
2.66 2.61 2.58 2.54 2.51
2.59 2.55 2.51 2.48 2.45
2.54 2.49 2.46 2.42 2.39
2.49 2.45 2.41 2.38 2.35
2.42 2.38 2.34 2.31 2.28
2.35 2.31 2.27 2.23 2.20
2.28 2.23 2.19 2.16 2.12
2.24 2.19 2.15 2.11 2.08
2.19 2.15 2.11 2.07 2.04
2.15 2.10 2.06 2.03 1.99
2.11 2.06 2.02 1.98 1.95
2.06 2.01 1.97 1.93 1.90
2.01 1.96 1.92 1.88 1.84
21 22 23 24 25
4.32 4.30 4.28 4.26 4.24
3.47 3.44 3.42 3.40 3.39
3.07 3.05 3.03 3.01 2.99
2.84 2.82 2.80 2.78 2.76
2.68 2.66 2.64 2.62 2.60
2.57 2.55 2.53 2.51 2.49
2.49 2.46 2.44 2.42 2.40
2.42 2.40 2.37 2.36 2.34
2.37 2.34 2.32 2.30 2.28
2.32 2.30 2.27 2.25 2.24
2.25 2.23 2.20 2.18 2.16
2.18 2.15 2.13 2.11 2.09
2.10 2.07 2.05 2.03 2.01
2.05 2.03 2.01 1.98 1.96
2.01 1.98 1.96 1.94 1.92
1.96 1.94 1.91 1.89 1.87
1.92 1.89 1.86 1.84 1.82
1.87 1.84 1.81 1.79 1.77
1.81 1.78 1.76 1.73 1.71
30 40 60 120 ∞
4.17 4.08 4.00 3.92 3.84
3.32 3.23 3.15 3.07 3.00
2.92 2.84 2.76 2.68 2.60
2.69 2.61 2.53 2.45 2.37
2.53 2.45 2.37 2.29 2.21
2.42 2.34 2.25 2.18 2.10
2.33 2.25 2.17 2.09 2.01
2.27 2.18 2.10 2.02 1.94
2.21 2.12 2.04 1.96 1.88
2.16 2.08 1.99 1.91 1.83
2.09 2.00 1.92 1.83 1.75
2.01 1.92 1.84 1.75 1.67
1.93 1.84 1.75 1.66 1.57
1.89 1.79 1.70 1.61 1.52
1.84 1.74 1.65 1.55 1.46
1.79 1.69 1.59 1.50 1.39
1.74 1.64 1.53 1.43 1.32
1.68 1.58 1.47 1.35 1.22
1.62 1.51 1.39 1.25 1.00
Upper 1% Points (F.99)
Degrees of freedom for denominator
Degrees of freedom for numerator 1
2
3
4
5
6
7
8
9
10
12
15
20
24
30
40
60
120
∞
1 2 3 4 5
4052 98.5 34.1 21.2 16.3
5000 99.0 30.8 18.0 13.3
5403 99.2 29.5 16.7 12.1
5625 99.2 28.7 16.0 11.4
5764 99.3 28.2 15.5 11.0
5859 99.3 27.9 15.2 10.7
5928 99.4 27.7 15.0 10.5
5982 99.4 27.5 14.8 10.3
6023 99.4 27.3 14.7 10.2
6056 99.4 27.2 14.5 10.1
6106 99.4 27.1 14.4 9.89
6157 99.4 26.9 14.2 9.72
6209 99.4 26.7 14.0 9.55
6235 99.5 26.6 13.9 9.47
6261 99.5 26.5 13.8 9.38
6287 99.5 26.4 13.7 9.29
6313 99.5 26.3 13.7 9.20
6339 99.5 26.2 13.6 9.11
6366 99.5 26.1 13.5 9.02
6 7 8 9 10
13.7 12.2 11.3 10.6 10.0
10.9 9.55 8.65 8.02 7.56
9.78 8.45 7.59 6.99 6.55
9.15 7.85 7.01 6.42 5.99
8.75 7.46 6.63 6.06 5.64
8.47 7.19 6.37 5.80 5.39
8.26 6.99 6.18 5.61 5.20
8.10 6.84 6.03 5.47 5.06
7.98 6.72 5.91 5.35 4.94
7.87 6.62 5.81 5.26 4.85
7.72 6.47 5.67 5.11 4.71
7.56 6.31 5.52 4.96 4.56
7.40 6.16 5.36 4.81 4.41
7.31 6.07 5.28 4.73 4.33
7.23 5.99 5.20 4.65 4.25
7.14 5.91 5.12 4.57 4.17
7.06 5.82 5.03 4.48 4.08
6.97 5.74 4.95 4.40 4.00
6.88 5.65 4.86 4.31 3.91
11 12 13 14 15
9.65 9.33 9.07 8.86 8.68
7.21 6.93 6.70 6.51 6.36
6.22 5.95 5.74 5.56 5.42
5.67 5.41 5.21 5.04 4.89
5.32 5.06 4.86 4.70 4.56
5.07 4.82 4.62 4.46 4.32
4.89 4.64 4.44 4.28 4.14
4.74 4.50 4.30 4.14 4.00
4.63 4.39 4.19 4.03 3.89
4.54 4.30 4.10 3.94 3.80
4.40 4.16 3.96 3.80 3.67
4.25 4.01 3.82 3.66 3.52
4.10 3.86 3.66 3.51 3.37
4.02 3.78 3.59 3.43 3.29
3.94 3.70 3.51 3.35 3.21
3.86 3.62 3.43 3.27 3.13
3.78 3.54 3.34 3.18 3.05
3.69 3.45 3.25 3.09 2.96
3.60 3.36 3.17 3.00 2.87
16 17 18 19 20
8.53 8.40 8.29 8.19 8.10
6.23 6.11 6.01 5.93 5.85
5.29 5.19 5.09 5.01 4.94
4.77 4.67 4.58 4.50 4.43
4.44 4.34 4.25 4.17 4.10
4.20 4.10 4.01 3.94 3.87
4.03 3.93 3.84 3.77 3.70
3.89 3.79 3.71 3.63 3.56
3.78 3.68 3.60 3.52 3.46
3.69 3.59 3.51 3.43 3.37
3.55 3.46 3.37 3.30 3.23
3.41 3.31 3.23 3.15 3.09
3.26 3.16 3.08 3.00 2.94
3.18 3.08 3.00 2.92 2.86
3.10 3.00 2.92 2.84 2.78
3.02 2.92 2.84 2.76 2.69
2.93 2.83 2.75 2.67 2.61
2.84 2.75 2.66 2.58 2.52
2.75 2.65 2.57 2.49 2.42
21 22 23 24 25
8.02 7.95 7.88 7.82 7.77
5.78 5.72 5.66 5.61 5.57
4.87 4.82 4.76 4.72 4.68
4.37 4.31 4.26 4.22 4.18
4.04 3.99 3.94 3.90 3.86
3.81 3.76 3.71 3.67 3.63
3.64 3.59 3.54 3.50 3.46
3.51 3.45 3.41 3.36 3.32
3.40 3.35 3.30 3.26 3.22
3.31 3.26 3.21 3.17 3.13
3.17 3.12 3.07 3.03 2.99
3.03 2.98 2.93 2.89 2.85
2.88 2.83 2.78 2.74 2.70
2.80 2.75 2.70 2.66 2.62
2.72 2.67 2.62 2.58 2.53
2.64 2.58 2.54 2.49 2.45
2.55 2.50 2.45 2.40 2.36
2.46 2.40 2.35 2.31 2.27
2.36 2.31 2.26 2.21 2.17
30 40 60 120 ∞
7.56 7.31 7.08 6.85 6.63
5.39 5.18 4.98 4.79 4.61
4.51 4.31 4.13 3.95 3.78
4.02 3.83 3.65 3.48 3.32
3.70 3.51 3.34 3.17 3.02
3.47 3.29 3.12 2.96 2.80
3.30 3.12 2.95 2.79 2.64
3.17 2.99 2.82 2.66 2.51
3.07 2.89 2.72 2.56 2.41
2.98 2.80 2.63 2.47 2.32
2.84 2.66 2.50 2.34 2.18
2.70 2.52 2.35 2.19 2.04
2.55 2.37 2.20 2.03 1.88
2.47 2.29 2.12 1.95 1.79
2.39 2.20 2.03 1.86 1.70
2.30 2.11 1.94 1.76 1.59
2.21 2.02 1.84 1.66 1.47
2.11 1.92 1.73 1.53 1.32
2.01 1.80 1.60 1.38 1.00
Interpolation should be performed using reciprocals of the degrees of freedom. 3-77
3-78
MATHEMATICS
Test of Hypothesis for a Mean Procedure Nomenclature µ = mean of the population from which the sample has been drawn σ = standard deviation of the population from which the sample has been drawn µ0 = base or reference level H0 = null hypothesis H1 = alternative hypothesis α = significance level, usually set at .10, .05, or .01 t = tabled t value corresponding to the significance level α. For a two-tailed test, each corresponding tail would have an area of α/2, and for a one-tailed test, one tail area would be equal to α. If σ2 is known, then z would be used rather than the t. t = (x − µ0)/(s/ n) = sample value of the test statistic. Assumptions 1. The n observations x1, x2, . . . , xn have been selected randomly. 2. The population from which the observations were obtained is normally distributed with an unknown mean µ and standard deviation σ. In actual practice, this is a robust test, in the sense that in most types of problems it is not sensitive to the normality assumption when the sample size is 10 or greater. Test of Hypothesis 1. Under the null hypothesis, it is assumed that the sample came from a population whose mean µ is equivalent to some base or reference designated by µ0. This can take one of three forms: Form 1
Form 2
Form 3
H0: µ = µ0 H1: µ ≠ µ0 Two-tailed test
H0: µ ≤ µ0 H1: µ > µ0 Upper-tailed test
H0: µ ≥ µ0 H1: µ < µ0 Lower-tailed test
2. If the null hypothesis is assumed to be true, say, in the case of a two-sided test, form 1, then the distribution of the test statistic t is known. Given a random sample, one can predict how far its sample value of t might be expected to deviate from zero (the midvalue of t) by chance alone. If the sample value of t does, in fact, deviate too far from zero, then this is defined to be sufficient evidence to refute the assumption of the null hypothesis. It is consequently rejected, and the converse or alternative hypothesis is accepted. 3. The rule for accepting H0 is specified by selection of the α level as indicated in Fig. 3-57. For forms 2 and 3 the α area is defined to be in the upper or the lower tail respectively. 4. The decision rules for each of the three forms are defined as follows: If the sample t falls within the acceptance region, accept H0 for lack of contrary evidence. If the sample t falls in the critical region, reject H0 at a significance level of 100α percent. Example Application. In the past, the yield for a chemical process has been established at 89.6 percent with a standard deviation of 3.4 percent. A new supplier of raw materials will be used and tested for 7 days. Procedure 1. The standard of reference is µ0 = 89.6 with a known σ = 3.4. 2. It is of interest to demonstrate whether an increase in yield is achieved with the new material; H0 says it has not; therefore, H0: µ ≤ 89.6
H1: µ > 89.6
3. Select α = .05, and since σ is known (the new material would not affect the day-to-day variability in yield), the test statistic would be z with a corresponding critical value cv(z) = 1.645 (Table 3-5, df = ∞). 4. The decision rule: Accept H0 if sample z < 1.645 Reject H0 if sample z > 1.645 5. A 7-day test was carried out, and daily yields averaged 91.6 percent with a sample standard deviation s = 3.6 (this is not needed for the test of hypothesis). 6. For the data sample z = (91.6 − 89.6)/(3.4/7) = 1.56. 7. Since the sample z < cv(z), accept the null hypothesis for lack of contrary evidence; i.e., an improvement has not been demonstrated beyond a reasonable doubt.
Example Application. In the past, the break strength of a synthetic yarn has averaged 34.6 lb. The first-stage draw ratio of the spinning machines has been increased. Production management wants to determine whether the break strength has changed under the new condition. Procedure 1. The standard of reference is µ0 = 34.6. 2. It is of interest to demonstrate whether a change has occurred; therefore, H0: µ = 34.6
Accept H0 if −2.086 < sample t < 2.086 Reject H0 if sample t < −2.086 or > 2.086 6. For the data sample t = (35.55 − 34.6)/(2.041/21) = 2.133. 7. Since 2.133 > 2.086, reject H0 and accept H1. It has been demonstrated that an improvement in break strength has been achieved.
Two-Population Test of Hypothesis for Means Nature Two samples were selected from different locations in a plastic-film sheet and measured for thickness. The thickness of the respective samples was measured at 10 close but equally spaced points in each of the samples. It was of interest to compare the average thickness of the respective samples to detect whether they were significantly different. That is, was there a significant variation in thickness between locations? From a modeling standpoint statisticians would define this problem as a two-population test of hypothesis. They would define the respective sample sheets as two populations from which 10 sample thickness determinations were measured for each. In order to compare populations based on their respective samples, it is necessary to have some basis of comparison. This basis is predicated on the distribution of the t statistic. In effect, the t statistic characterizes the way in which two sample means from two separate populations will tend to vary by chance alone when the population means and variances are equal. Consider the following: Population 1
Acceptance region.
Population 2
Normal
Sample 1
Normal
Sample 2
µ1
n1 x1 s12
µ2
n2 x2 s22
σ
FIG. 3-57
H1: µ ≠ 34.6
3. Select α = .05, and since with the change in draw ratio the uniformity might change, the sample standard deviation would be used, and therefore t would be the appropriate test statistic. 4. A sample of 21 ends was selected randomly and tested on an Instron with the results x = 35.55 and s = 2.041. 5. For 20 df and a two-tailed α level of 5 percent, the critical values of t are given by 2.086 with a decision rule (Table 3-5, t.025, df = 20):
2 1
σ
2 2
Consider the hypothesis µ1 = µ2. If, in fact, the hypothesis is correct, i.e., µ1 = µ2 (under the condition σ 12 = σ 22), then the sampling distribution of (x1 − x2) is predictable through the t distribution. The observed sample values then can be compared with the corresponding t distribution. If the sample values are reasonably close (as reflected through the α level), that is, x1 and x2 are not “too different” from each other on the basis of the t distribution, the null hypothesis would be accepted. Conversely, if they deviate from each other “too much” and the deviation is therefore not ascribable to chance, the conjecture would be questioned and the null hypothesis rejected.
STATISTICS Example Application. Two samples were selected from different locations in a plastic-film sheet. The thickness of the respective samples was measured at 10 close but equally spaced points. Procedure 1. Demonstrate whether the thicknesses of the respective sample locations are significantly different from each other; therefore, H0: µ1 = µ2 2. 3.
H1: µ1 ≠ µ2
Select α = .05. Summarize the statistics for the respective samples: Sample 1
Sample 2
1.473 1.484 1.484 1.425 1.448
1.367 1.276 1.485 1.462 1.439
1.474 1.501 1.485 1.435 1.348
1.417 1.448 1.469 1.474 1.452
x1 = 1.434
s1 = .0664
x2 = 1.450
s2 = .0435
4. As a first step, the assumption for the standard t test, that σ 12 = σ 22, can be tested through the F distribution. For this hypothesis, H0: σ 12 = σ 22 would be tested against H1: σ 12 ≠ σ 22. Since this is a two-tailed test and conventionally only the upper tail for F is published, the procedure is to use the largest ratio and the corresponding ordered degrees of freedom. This achieves the same end result through one table. However, since the largest ratio is arbitrary, it is necessary to define the true α level as twice the value of the tabled value. Therefore, by using Table 3-7 with α = .05 the corresponding critical value for F(9,9) = 3.18 would be for a true α = .10. For the sample,
Test of Hypothesis for Matched Pairs: Procedure Nomenclature di = sample difference between the ith pair of observations s = sample standard deviation of differences µ = population mean of differences σ = population standard deviation of differences µ0 = base or reference level of comparison H0 = null hypothesis H1 = alternative hypothesis α = significance level t = tabled value with (n − 1) df t = (d n), the sample value of t − µ0)/(s/ Assumptions 1. The n pairs of samples have been selected and assigned for testing in a random way. 2. The population of differences is normally distributed with a mean µ and variance σ2. As in the previous application of the t distribution, this is a robust procedure, i.e., not sensitive to the normality assumption if the sample size is 10 or greater in most situations. Test of Hypothesis 1. Under the null hypothesis, it is assumed that the sample came from a population whose mean µ is equivalent to some base or reference level designated by µ0. For most applications of this type, the value of µ0 is defined to be zero; that is, it is of interest generally to demonstrate a difference not equal to zero. The hypothesis can take one of three forms:
Sample F = (.0664/.0435)2 = 2.33
Form 1
Form 2
Therefore, the ratio of sample variances is no larger than one might expect to observe when in fact σ 12 = σ 22. There is not sufficient evidence to reject the null hypothesis that σ 12 = σ 22. 5. For 18 df and a two-tailed α level of 5 percent the critical values of t are given by 2.101 (Table 3-5, t0.025, df = 18). 6. The decision rule:
H0: µ = µ0 H1: µ ≠ µ0 Two-tailed test
H0: µ ≤ µ0 H1: µ > µ0 Upper-tailed test
Accept H0 if −2.101 ≤ sample t ≤ 2.101 Reject H0 otherwise 7.
For the sample the pooled variance estimate is given by 9(.0664)2 + 9(.0435)2 (.0664)2 + (.0435)2 s p2 = = = .00315 9+9 2 sp = .056
or 8.
The sample statistic value of t is 1.434 − 1.450 Sample t = = −.64 0 0 .0561/1 + 1/1
9. Since the sample value of t falls within the acceptance region, accept H0 for lack of contrary evidence; i.e., there is insufficient evidence to demonstrate that thickness differs between the two selected locations.
Test of Hypothesis for Paired Observations Nature In some types of applications, associated pairs of observations are defined. For example, (1) pairs of samples from two populations are treated in the same way, or (2) two types of measurements are made on the same unit. For applications of this type, it is not only more effective but necessary to define the random variable as the difference between the pairs of observations. The difference numbers can then be tested by the standard t distribution. Examples of the two types of applications are as follows: 1. Sample treatment a. Two types of metal specimens buried in the ground together in a variety of soil types to determine corrosion resistance b. Wear-rate test with two different types of tractor tires mounted in pairs on n tractors for a defined period of time 2. Same unit a. Blood-pressure measurements made on the same individual before and after the administration of a stimulus b. Smoothness determinations on the same film samples at two different testing laboratories
3-79
Form 3 H0: µ ≥ µ0 H1: µ < µ0 Lower-tailed test
2. If the null hypothesis is assumed to be true, say, in the case of a lower-tailed test, form 3, then the distribution of the test statistic t is known under the null hypothesis that limits µ = µ0. Given a random sample, one can predict how far its sample value of t might be expected to deviate from zero by chance alone when µ = µ0. If the sample value of t is too small, as in the case of a negative value, then this would be defined as sufficient evidence to reject the null hypothesis. 3. Select α. 4. The critical values or value of t would be defined by the tabled value of t with (n − 1) df corresponding to a tail area of α. For a twotailed test, each tail area would be α/2, and for a one-tailed test there would be an upper-tail or a lower-tail area of α corresponding to forms 2 and 3 respectively. 5. The decision rule for each of the three forms would be to reject the null hypothesis if the sample value of t fell in that area of the t distribution defined by α, which is called the critical region. Otherwise, the alternative hypothesis would be accepted for lack of contrary evidence. Example Application. Pairs of pipes have been buried in 11 different locations to determine corrosion on nonbituminous pipe coatings for underground use. One type includes a lead-coated steel pipe and the other a bare steel pipe. Procedure 1. The standard of reference is taken as µ0 = 0, corresponding to no difference in the two types. 2. It is of interest to demonstrate whether either type of pipe has a greater corrosion resistance than the other. Therefore, H0: µ = 0
H1: µ ≠ 0
3. Select α = .05. Therefore, with n = 11 the critical values of t with 10 df are defined by t = 2.228 (Table 3.5, t.025). 4. The decision rule: Accept H0 if −2.228 ≤ sample t ≤ 2.228 Reject H0 otherwise 5. The sample of 11 pairs of corrosion determinations and their differences are as follows:
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MATHEMATICS
Soil type
Lead-coated steel pipe
Bare steel pipe
d = difference
A B C D E F
27.3 18.4 11.9 11.3 14.8 20.8
41.4 18.9 21.7 16.8 9.0 19.3
−14.1 −0.5 −9.8 −5.5 5.8 1.5
G H I J K
17.9 7.8 14.7 19.0 65.3
32.1 7.4 20.7 34.4 76.2
−14.2 0.4 −6.0 −15.4 −10.9
6.
The sample statistics: 11 d 2 − ( d)2 s2 = = 52.56 11 × 10
d = −6.245
s = 7.25
or
Sample t = (−6.245 − 0)/(7.25/11) = −2.86 7. Since the sample t of −2.86 < tabled t of −2.228, reject H0 and accept H1; that is, it has been demonstrated that, on the basis of the evidence, lead-coated steel pipe has a greater corrosion resistance than bare steel pipe.
Example Application. A stimulus was tested for its effect on blood pressure. Ten men were selected randomly, and their blood pressure was measured before and after the stimulus was administered. It was of interest to determine whether the stimulus had caused a significant increase in the blood pressure. Procedure 1. The standard of reference was taken as µ0 ≤ 0, corresponding to no increase. 2. It was of interest to demonstrate an increase in blood pressure if in fact an increase did occur. Therefore, H0: µ0 ≤ 0
H1: µ0 > 0
3. Select α = .05. Therefore, with n = 10 the critical value of t with 9 df is defined by t = 1.833 (Table 3-5, t.05, one-sided). 4. The decision rule: Accept H0 if sample t < 1.833 Reject H0 if sample t > 1.833 5. The sample of 10 pairs of blood pressure and their differences were as follows:
proportion of workers in a plant who are out sick, (2) lost-time worker accidents per month, (3) defective items in a shipment lot, and (4) preference in consumer surveys. The procedure for testing the significance of a sample proportion follows that for a sample mean. In this case, however, owing to the nature of the problem the appropriate test statistic is Z. This follows from the fact that the null hypothesis requires the specification of the goal or reference quantity p0, and since the distribution is a binomial proportion, the associated variance is [p0(1 − p0)]n under the null hypothesis. The primary requirement is that the sample size n satisfy normal approximation criteria for a binomial proportion, roughly np > 5 and n(1 − p) > 5. Test of Hypothesis for a Proportion: Procedure Nomenclature p = mean proportion of the population from which the sample has been drawn p0 = base or reference proportion [p0(1 − p0)]/n = base or reference variance p ˆ = x/n = sample proportion, where x refers to the number of observations out of n which have the specified attribute H0 = assumption or null hypothesis regarding the population proportion H1 = alternative hypothesis α = significance level, usually set at .10, .05, or .01 z = Tabled Z value corresponding to the significance level α. The sample sizes required for the z approximation according to the magnitude of p0 are given in Table 3-5. z = (p ˆ − p0)/ p 1 − p /n, the sample value of the test 0( 0) statistic Assumptions 1. The n observations have been selected randomly. 2. The sample size n is sufficiently large to meet the requirement for the Z approximation. Test of Hypothesis 1. Under the null hypothesis, it is assumed that the sample came from a population with a proportion p0 of items having the specified attribute. For example, in tossing a coin the population could be thought of as having an unbounded number of potential tosses. If it is assumed that the coin is fair, this would dictate p0 = 1/2 for the proportional number of heads in the population. The null hypothesis can take one of three forms:
Individual
Before
After
d = difference
Form 1
Form 2
Form 3
1 2 3 4 5
138 116 124 128 155
146 118 120 136 174
8 2 −4 8 19
H0: p = p0 H1: p ≠ p0 Two-tailed test
H0: p ≤ p0 H1: p > p0 Upper-tailed test
H0: p ≥ p0 H1: p < p0 Lower-tailed test
6 7 8 9 10
129 130 148 143 159
133 129 155 148 155
4 −1 7 5 −4
2. If the null hypothesis is assumed to be true, then the sampling distribution of the test statistic Z is known. Given a random sample, it is possible to predict how far the sample proportion x/n might deviate from its assumed population proportion p0 through the Z distribution. When the sample proportion deviates too far, as defined by the significance level α, this serves as the justification for rejecting the assumption, that is, rejecting the null hypothesis. 3. The decision rule is given by Form 1: Accept H0 if lower critical z < sample z < upper critical z Reject H0 otherwise Form 2: Accept H0 if sample z < upper critical z Reject H0 otherwise Form 3: Accept H0 if lower critical z < sample z Reject H0 otherwise
6.
The sample statistics: d = 4.4
s = 6.85
Sample t = (4.4 − 0)/(6.85/10) = 2.03 7. Since the sample t = 2.03 > critical t = 1.833, reject the null hypothesis. It has been demonstrated that the population of men from which the sample was drawn tend, as a whole, to have an increase in blood pressure after the stimulus has been given. The distribution of differences d seems to indicate that the degree of response varies by individuals.
Example Test of Hypothesis for a Proportion Nature Some types of statistical applications deal with counts and proportions rather than measurements. Examples are (1) the
Application. A company has received a very large shipment of rivets. One product specification required that no more than 2 percent of the rivets have diameters greater than 14.28 mm. Any rivet with a diameter greater than this would be classified as defective. A random sample of 600 was selected and
STATISTICS tested with a go–no-go gauge. Of these, 16 rivets were found to be defective. Is this sufficient evidence to conclude that the shipment contains more than 2 percent defective rivets? Procedure 1. The quality goal is p ≤ .02. It would be assumed initially that the shipment meets this standard; i.e., H0: p ≤ .02. 2. The assumption in step 1 would first be tested by obtaining a random sample. Under the assumption that p ≤ .02, the distribution for a sample proportion would be defined by the z distribution. This distribution would define an upper bound corresponding to the upper critical value for the sample proportion. It would be unlikely that the sample proportion would rise above that value if, in fact, p ≤ .02. If the observed sample proportion exceeds that limit, corresponding to what would be a very unlikely chance outcome, this would lead one to question the assumption that p ≤ .02. That is, one would conclude that the null hypothesis is false. To test, set H0: p ≤ .02
H1: p > .02
3. Select α = .05. 4. With α = .05, the upper critical value of Z = 1.645 (Table 3-5, t.05, df = ∞, one-sided). 5. The decision rule:
Test of Hypothesis 1. Under the null hypothesis, it is assumed that the respective two samples have come from populations with equal proportions p1 = p2. Under this hypothesis, the sampling distribution of the corresponding Z statistic is known. On the basis of the observed data, if the resultant sample value of Z represents an unusual outcome, that is, if it falls within the critical region, this would cast doubt on the assumption of equal proportions. Therefore, it will have been demonstrated statistically that the population proportions are in fact not equal. The various hypotheses can be stated: Form 1
Form 2
Form 3
H0: p1 = p2 H1: p1 ≠ p2 Two-tailed test
H0: p1 ≤ p2 H1: p1 > p2 Upper-tailed test
H0: p1 ≥ p2 H1: p1 < p2 Lower-tailed test
2.
Accept H0 if sample z < 1.645 Reject H0 if sample z > 1.645 6.
The sample z is given by (16/600) − .02 Sample z = 02)( .9 8)/ 600 (. = 1.17
7. Since the sample z < 1.645, accept H0 for lack of contrary evidence; there is not sufficient evidence to demonstrate that the defect proportion in the shipment is greater than 2 percent.
Test of Hypothesis for Two Proportions Nature In some types of engineering and management-science problems, we may be concerned with a random variable which represents a proportion, for example, the proportional number of defective items per day. The method described previously relates to a single proportion. In this subsection two proportions will be considered. A certain change in a manufacturing procedure for producing component parts is being considered. Samples are taken by using both the existing and the new procedures in order to determine whether the new procedure results in an improvement. In this application, it is of interest to demonstrate statistically whether the population proportion p2 for the new procedure is less than the population proportion p1 for the old procedure on the basis of a sample of data. Test of Hypothesis for Two Proportions: Procedure Nomenclature p1 = population 1 proportion p2 = population 2 proportion n1 = sample size from population 1 n2 = sample size from population 2 x1 = number of observations out of n1 that have the designated attribute x2 = number of observations out of n2 that have the designated attribute p ˆ 1 = x1/n1, the sample proportion from population 1 p ˆ 2 = x2/n2, the sample proportion from population 2 α = significance level H0 = null hypothesis H1 = alternative hypothesis z = tabled Z value corresponding to the stated significance level α ˆ2 p ˆ1−p z = , the sample value of Z pˆ 1(1 − pˆ 1)/n1 + pˆ 2(1 − pˆ 2)/n2 Assumptions 1. The respective two samples of n1 and n2 observations have been selected randomly. 2. The sample sizes n1 and n2 are sufficiently large to meet the requirement for the Z approximation; i.e., x1 > 5, x2 > 5.
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The decision rule for form 1 is given by Accept H0 if lower critical z < sample z < upper critical z Reject H0 otherwise
Example Application. A change was made in a manufacturing procedure for component parts. Samples were taken during the last week of operations with the old procedure and during the first week of operations with the new procedure. Determine whether the proportional numbers of defects for the respective populations differ on the basis of the sample information. Procedure 1. The hypotheses are H0: p1 = p2
H1: p1 ≠ p2
2. Select α = .05. Therefore, the critical values of z are 1.96 (Table 3-4, A = 0.9500). 3. For the samples, 75 out of 1720 parts from the previous procedure and 80 out of 2780 parts under the new procedure were found to be defective; therefore, p ˆ 1 = 75/1720 = .0436 4.
p ˆ 2 = 80/2780 = .0288
The decision rule: Accept H0 if −1.96 ≤ sample Z ≤ 1.96 Reject H0 otherwise
5.
The sample statistic: .0436 − .0288 Sample z = 0436)( .9 564)/ 1720 288)( .9 712)/ 2780 +(.0 (. = 2.53
6. Since the sample z of 2.53 > tabled z of 1.96, reject H0 and conclude that the new procedure has resulted in a reduced defect rate.
Goodness-of-Fit Test Nature A standard die has six sides numbered from 1 to 6. If one were really interested in determining whether a particular die was well balanced, one would have to carry out an experiment. To do this, it might be decided to count the frequencies of outcomes, 1 through 6, in tossing the die N times. On the assumption that the die is perfectly balanced, one would expect to observe N/6 occurrences each for 1, 2, 3, 4, 5, and 6. However, chance dictates that exactly N/6 occurrences each will not be observed. For example, given a perfectly balanced die, the probability is only 1 chance in 65 that one will observe 1 outcome each, for 1 through 6, in tossing the die 6 times. Therefore, an outcome different from 1 occurrence each can be expected. Conversely, an outcome of six 3s would seem to be too unusual to have occurred by chance alone. Some industrial applications involve the concept outlined here. The basic idea is to test whether or not a group of observations follows a preconceived distribution. In the case cited, the distribution is uniform; i.e., each face value should tend to occur with the same frequency. Goodness-of-Fit Test: Procedure Nomenclature Each experimental observation can be classified into one of r possible categories or cells.
3-82
MATHEMATICS
r = total number of cells Oj = number of observations occurring in cell j Ej = expected number of observations for cell j based on the preconceived distribution N = total number of observations f = degrees of freedom for the test. In general, this will be equal to (r − 1) minus the number of statistical quantities on which the Ej’s are based (see the examples which follow for details). Assumptions 1. The observations represent a sample selected randomly from a population which has been specified. 2. The number of expectation counts Ej within each category should be roughly 5 or more. If an Ej count is significantly less than 5, that cell should be pooled with an adjacent cell. Computation for Ej On the basis of the specified population, the probability of observing a count in cell j is defined by pj. For a sample of size N, corresponding to N total counts, the expected frequency is given by Ej = Npj. Test Statistics: Chi Square r (Oj − Ej)2 χ2 = with f df Ej j=1 Test of Hypothesis 1. H0: The sample came from the specified theoretical distribution H1: The sample did not come from the specified theoretical distribution 2. For a stated level of α, Reject H0 if sample χ2 > tabled χ2 Accept H0 if sample χ2 < tabled χ2 Example Application A production-line product is rejected if one of its characteristics does not fall within specified limits. The standard goal is that no more than 2 percent of the production should be rejected. Computation 1. Of 950 units produced during the day, 28 units were rejected. 2. The hypotheses: H0: the process is in control H1: the process is not in control 3. Assume that α = .05; therefore, the critical value of χ2(1) = 3.84 (Table 3-6, 95 percent, df = 1). One degree of freedom is defined since (r − 1) = 1, and no statistical quantities have been computed for the data. 4. The decision rule: Reject H0 if sample χ2 > 3.84 Accept H0 otherwise 5. Since it is assumed that p = .02, this would dictate that in a sample of 950 there would be on the average (.02)(950) = 19 defective items and 931 acceptable items:
Category
Observed Oj
Expectation Ej = 950pj
Acceptable Not acceptable Total
922 28 950
931 19 950
(922 − 931)2 (28 − 19)2 Sample χ = + 931 19 = 4.35 with critical χ2 = 3.84 2
6. Conclusion. Since the sample value exceeds the critical value, it would be concluded that the process is not in control.
Example Application A frequency count of workers was tabulated according to the number of defective items that they produced. An unresolved question is whether the observed distribution is a Poisson distribution. That is, do observed and expected frequencies agree within chance variation? Computation 1. The hypotheses: H0: there are no significant differences, in number of defective units, between workers H1: there are significant differences
2. 3.
Assume that α = .05. Test statistic: No. of defective units 0 1 2 3 4 5 6 7 8 9 ≥10 Sum
Oj 8 7 9 12 9 6 3 2 0 1 0 52
Ej
10
2.06 6.64 10.73 11.55 9.33 6.03
8.70 pool
3.24 1.50 .60 .22 .10 52
6
5.66 pool
The expectation numbers Ej were computed as follows: For the Poisson distribution, λ = E(x); therefore, an estimate of λ is the average number of defective units per worker, i.e., λ = (1/52)(0 × 3 + 1 × 7 + ⋅⋅⋅ + 9 × 1) = 3.23. Given this approximation, the probability of no defective units for a worker would be (3.23)0/0!)e−3.23 = .0396. For the 52 workers, the number of workers producing no defective units would have an expectation E = 52(0.0396) = 2.06, and so forth. The sample chi-square value is computed from (10 − 8.70)2 (9 − 10.73)2 (6 − 5.66)2 χ2 = + + ⋅⋅⋅ + 8.70 10.73 5.66 = .53 4. The critical value of χ2 would be based on four degrees of freedom. This corresponds to (r − 1) − 1, since one statistical quantity λ was computed from the sample and used to derive the expectation numbers. 5. The critical value of χ2(4) = 9.49 (Table 3-6) with α = .05; therefore, accept H0.
Two-Way Test for Independence for Count Data Nature When individuals or items are observed and classified according to two different criteria, the resultant counts can be statistically analyzed. For example, a market survey may examine whether a new product is preferred and if it is preferred due to a particular characteristic. Count data, based on a random selection of individuals or items which are classified according to two different criteria, can be statistically analyzed through the χ2 distribution. The purpose of this analysis is to determine whether the respective criteria are dependent. That is, is the product preferred because of a particular characteristic? Two-Way Test for Independence for Count Data: Procedure Nomenclature 1. Each observation is classified into each of two categories: a. The first one into 2, 3, . . . , or r categories b. The second one into 2, 3, . . . , or c categories 2. Oij = number of observations (observed counts) in cell (i, j) with i = 1, 2, . . . , r j = 1, 2, . . . , c 3. N = total number of observations 4. Eij = computed number for cell (i,j) which is an expectation based on the assumption that the two characteristics are independent 5. Ri = subtotal of counts in row i 6. Cj = subtotal of counts in column j 7. α = significance level 8. H0 = null hypothesis 9. H1 = alternative hypothesis 10. χ2 = critical value of χ2 corresponding to the significance level α and (r − 1)(c − 1) df c,r (Oij − Eij)2 11. Sample χ2 = Eij i, j Assumptions 1. The observations represent a sample selected randomly from a large total population.
STATISTICS 2. The number of expectation counts Eij within each cell should be approximately 2 or more for arrays 3 × 3 or larger. If any cell contains a number smaller than 2, appropriate rows or columns should be combined to increase the magnitude of the expectation count. For arrays 2 × 2, approximately 4 or more are required. If the number is less than 4, the exact Fisher test should be used. Test of Hypothesis Under the null hypothesis, the classification criteria are assumed to be independent, i.e., H0: the criteria are independent H1: the criteria are not independent For the stated level of α, Reject H0 if sample χ2 > tabled χ2 Accept H0 otherwise Computation for Eij Compute Eij across rows or down columns by using either of the following identities: R Eij = Cj i across rows N
Cj Eij = Ri down columns N Sample c Value 2
(Oij − Eij)2 χ2 = Eij i, j
In the special case of r = 2 and c = 2, a more accurate and simplified formula which does not require the direct computation of Eij can be used: [|O11O22 − O12O21| − aN]2N χ2 = R1R2C1C2 Example Application A market research study was carried out to relate the subjective “feel” of a consumer product to consumer preference. In other words, is the consumer’s preference for the product associated with the feel of the product, or is the preference independent of the product feel? Procedure 1. It was of interest to demonstrate whether an association exists between feel and preference; therefore, assume H0: feel and preference are independent H1: they are not independent 2. A sample of 200 people was asked to classify the product according to two criteria: a. Liking for this product b. Liking for the feel of the product
127/200 = 63.5 percent suggest further that there are other attributes of the product which tend to nullify the beneficial feel of the product.
LEAST SQUARES When experimental data is to be fit with a mathematical model, it is necessary to allow for the fact that the data has errors. The engineer is interested in finding the parameters in the model as well as the uncertainty in their determination. In the simplest case, the model is a linear equation with only two parameters, and they are found by a least-squares minimization of the errors in fitting the data. Multiple regression is just linear least squares applied with more terms. Nonlinear regression allows the parameters of the model to enter in a nonlinear fashion. The following description of maximum likelihood applies to both linear and nonlinear least squares (Ref. 231). If each measurement point yi has a measurement error ∆yi that is independently random and distributed with a normal distribution about the true model y(x) with standard deviation σi, then the probability of a data set is N 1 yi − y(xi) 2 P = exp − ∆y 2 σi i=1
Yes
No
Ri
114 55 169
13 18 31
= 127 = 73 200
3. Select α = .05; therefore, with (r − 1)(c − 1) = 1 df, the critical value of χ2 is 3.84 (Table 3-6, 95 percent). 4. The decision rule: Accept H0 if sample χ2 < 3.84 Reject H0 otherwise 5.
∞
ta − 1e−t dt
0
Like feel Yes No Cj
Here, yi is the measured value, σi is the standard deviation of the ith measurement, and ∆y is needed to say a measured value ∆y has a certain probability. Given a set of parameters (maximizing this function), the probability that this data set plus or minus ∆y could have occurred is P. This probability is maximized (giving the maximum likelihood) if the negative of the logarithm is minimized. N yi − y(xi) 2 − N log ∆y 2 σi i=1 Since N, σi, and ∆y are constants, this is the same as minimizing χ2. N yi − y(xi; a1, . . . , aM) 2 χ2 = σi i=1 with respect to the parameters {aj}. Note that the standard deviations {σi} of the measurements are expected to be known. The goodness of fit is related to the number of degrees of freedom, ν = N − M. The probability that χ2 would exceed a particular value (χ0)2 is ν 1 P = 1 − Q , χ 20 2 2 where Q(a, x) is the incomplete gamma function x 1 Q(a, x) = e−t ta − 1 dt (a > 0) Γ(a) 0 and Γ(a) is the gamma function Γ(a) =
Like product
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The sample value of χ2 by using the special formula is [|114 × 18 − 13 × 55| − 100]2200 Sample χ2 = (169)(31)(127)(73) = 6.30
6. Since the sample χ2 of 6.30 > tabled χ2 of 3.84, reject H0 and accept H1. The relative proportionality of E11 = 169(127/200) = 107.3 to the observed 114 compared with E22 = 31(73/200) = 11.3 to the observed 18 suggests that when the consumer likes the feel, the consumer tends to like the product, and conversely for not liking the feel. The proportions 169/200 = 84.5 percent and
Both functions are tabulated in mathematical handbooks (Ref. 1). The function P gives the goodness of fit. Call χ02 the value of χ2 at the minimum. Then P > 0.1 represents a believable fit; if Q > 0.001, it might be an acceptable fit; smaller values of Q indicate the model may be in error (or the σi are really larger.) A “typical” value of χ2 for a moderately good fit is χ2 ∼ ν. Asymptotically for large ν, the statistic χ2 becomes normally distributed with a mean ν and a standard deviation (2 ν) (Ref. 231). If values σi are not known in advance, assume σi = σ (so that its value does not affect the minimization of χ2). Find the parameters by minimizing χ2 and compute: N [yi − y(xi)]2 σ2 = N i=1 This gives some information about the errors (i.e., the variance and standard deviation of each data point), although the goodness of fit, P, cannot be calculated. The minimization of χ2 requires yi − y(xi) ∂y(xi; a1, . . . , aM)
= 0, σ ∂a N
i=1
2 i
k
k = 1, . . . , M
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MATHEMATICS
Linear Least Squares When the model is a straight line yi − a − bxi χ2(a, b) = σi i=1 N
Define
N 1 S = 2 , i = 1 σi N
N xi Sx = , 2 i = 1 σi
2 i
y = a + bx + ε where x = strip-method determination y = magnetic-method determination
N y Sy = i2 i = 1 σi
N
x Sxx = 2 , i = 1 σi
Nomenclature. The calibration between the magnetic and the stripping methods can be determined through the model
2
xiyi Sxy = , 2 i = 1 σi
Sample data
Thickness, 10−5 ln
N
1 S ti = xi − x , σi S
Stt = t 2i
Stripping method, x
Magnetic method, y
1 σ b2 = Stt
104 114 116 129 132 139
85 115 105 127 120 121
We thus get the values of a and b with maximum likelihood as well as the variances of a and b. Using the value of χ2 for this a and b, we can also calculate the goodness of fit, P. In addition, the linear correlation coefficient r is related by
174 312 338 465 720
155 250 310 443 630
Then
1 N tiyi b= , Stt i = 1 σi
Sy − Sxb a= , S
Sx Cov (a, b) = − , SStt
i=1
1 S 2x σ a2 = 1 + , S SStt Cov (a, b) rab = σaσb
N
χ2 = (1 − r2) (yi − y)2 i=1
(xi − x)(yi − y) σ2i i=1 r = N N (xi − x )2 (yi − y)2 2 σi σ 2i i=1 i=1
Computations. The normal equations are defined by na + ( x)b = y
N
Here
Values of r near 1 indicate a positive correlation; r near −1 means a negative correlation and r near zero means no correlation. The form of the equations here is given to provide good accuracy when many terms are used and to provide the variances of the parameters. Another form of the equations for a and b is simpler, but is sometimes inaccurate unless many significant digits are kept in the calculations. The minimization of χ2 when σi is the same for all i gives the following equations for a and b. N
N
i=1
i=1
aN + b xi = yi N
N
N
a xi + b x = yi xi 2 i
i=1
i=1
i=1
N
The solution is
N
N
N yi xi − xi yi i=1 i=1 i=1 b = N N 2 2 N x i − xi
i=1
N
N
y = yi N, i=1
i=1
( x)a + ( x2)b = xy
For the sample
11a + 2743b = 2461 2743a + 1,067,143b = 952,517 with y2 = 852,419. The solution to the normal equations is given by a = 3.19960
χ2 = y2 − a y − b xy if a sufficient number of significant digits is retained (usually six or seven digits are sufficient). Here χ2 = 2175.14 If the normalized method is used in addition, the value of Stt is 3.8314 × 105/σ2, where σ2 is the variance of the measurement of y. The values of a and b are, of course, the same. The variances of a and b are σ a2 = 0.2532σ2, σ b2 = 2.610 × 10−6σ2. The correlation coefficient is 0.996390, which indicates that there is a positive correlation between x and y. The small value of the variance for b indicates that this parameter is determined very well by the data. The residuals show no particular pattern, and the predictions are plotted along with the data in Fig. 3-58. If the variance of the measurements of y is known through repeated measurements, then the variance of the parameters can be made absolute.
Multiple Regression A general linear model is one expressed as M
x = xi N i=1
a = y − bx
The value of χ2 can be calculated from the formula N
N
N
i=1
i=1
i=1
χ2 = y 2i − a yi − b yi xi It is usually advisable to plot the observed pairs of yi versus xi to support the linearity assumption and to detect potential outliers. Suspected outliers can be omitted from the least-squares “fit” and then subsequently tested on the basis of the least-squares fit.
y(x) = akXk(x) k=1
where the parameters are {ak}, and the expression is linear with respect to them, and Xk(x) can be any (nonlinear) functions of x, not depending on the parameters {ak}. Then: N M 1 2 yi − aj Xj(xi) Xk(xi) = 0, k = 1, . . . , M i = 1 σi j=1 This is rewritten as M N N 1 yi 2 Xj(xi)Xk(xi) aj = 2 Xk(xi) j = 1 i = 1 σi i = 1 σi
M
Example Application. Brenner (Magnetic Method for Measuring the Thickness of Non-magnetic Coatings on Iron and Steel, National Bureau of Standards, RP1081, March 1938) suggests an alternative way of measuring the thickness of nonmagnetic coatings of galvanized zinc on iron and steel. This procedure is based on a nondestructive magnetic method as a substitute for the standard destructive stripping method. A random sample of 11 pieces was selected and measured by both methods.
b = .884362
The error sum of squares can be computed from the formula
or as
α
a = βk
kj j
j=1
Solving this set of equations gives the parameters {aj}, which maximize the likelihood. The variance of aj is σ2(aj) = Cj j where Cj k = α−1 j k, or C is the inverse of α. The covariance of aj and ak
STATISTICS
FIG. 3-58
3-85
Plot of data and correlating line. 4.5βˆ 0 + 2.85βˆ 1 + 2.025βˆ 2 = 14.08957
is given by Cj k. If rounding errors affect the result, then we try to make the functions orthogonal. For example, using Xk(x) = xk − 1
2.85βˆ 0 + 2.025βˆ 1 + 1.5333βˆ 2 = 8.828813 The algebraic solution to the simultaneous equations is βˆ = 3.19513 βˆ = .4425 βˆ = −.7653
will cause rounding errors for a smaller M than
0
Xk(x) = Pk − 1(x)
1
2
The inverse of the product matrix
where Pk − 1 are orthogonal polynomials. If necessary, a singular value decomposition can be used. Various global and piecewise polynomials can be used to fit the data. Most approximations are to be used with M < N. One can sometimes use more and more terms, and calculating the value of χ2 for each solution. Then stop increasing M when the value of χ2 no longer increases with increasing M. Example Application. Merriman (“The Method of Least Squares Applied to a Hydraulic Problem,” J. Franklin Inst., 233–241, October 1877) reported on a study of stream velocity as a function of relative depth of the stream. Sample data
α=
α−1 =
is
10 4.5 2.85
4.5 2.85 2.025
.6182 −2.5909 2.2727
2.85 2.025 1.5333
−2.5909 16.5530 −17.0455
2.2727 −17.0455 18.9394
The variances are then the diagonal elements of the inverse of matrix α (0.6182, 16.5530, 18.9394) times the variance of the measurement of y, σ y2. The value of χ2 is 5.751 × 10−5, the correlation coefficient r = 0.99964, and σ = 0.002398. t values. A sample t value can be computed for each regression coefficient j through the identity tj = βˆ j /(ˆσ cjj), where cjj is the ( j,j) element in the inverse. For the two variables x1 and x2,
Velocity, y, ft/s
Coefficient
cjj
Sample t value
.1 .2 .3 .4
3.1950 3.2299 3.2532 3.2611 3.2516
.4425 −.7653
16.55 18.94
45.3 −73.3
.5 .6 .7 .8 .9
3.2282 3.1807 3.1266 3.0594 2.9759
Depth* 0
*As a fraction of total depth. Model. Owing to the curvature of velocity with depth, a quadratic model was specified:
Computational note. From a computational standpoint, it is usually advisable to define the variables in deviation units. For example, in the problem presented, let x1 = depth − dep th = depth − .45 For expansion terms such as a square, define x2 = x21 − x¯21 ( x¯21 = .0825) For the previous sample data, Deviation units
Velocity = β0 + β1x1 + β2x2 where x2 = x 12 . Normal equations. The three normal equations are defined by (n)βˆ + ( x )βˆ + ( x )βˆ = y 0
1
1
2
2
( x1)βˆ 0 + ( x12)βˆ 1 + ( x1x2)βˆ 2 = x1 y ( x2)βˆ 0 + ( x1x2)βˆ 1 + ( x22)βˆ 2 = x2 y For the sample data, the normal equations are 10βˆ + 4.5βˆ + 2.85 βˆ = 31.7616 0
1
2
x1
x2
x1
x2
−.45 −.35 −.25 −.25 −.15 −.05
.12 .04 −.02 −.02 −.06 −.08
.05 .15 .25 .35 .45
−.08 −.06 −.02 .04 .12
The resultant analysis-of-variance tables will remain exactly the same. However,
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MATHEMATICS
the corresponding coefficient t value for the linear coefficient will usually be improved. This is an idiosyncrasy of regression modeling. With the coded data presented, the least-squares solution is given by Yˆ = 3.17616 − .2462x1 − .7653x2 with a corresponding t value for βˆ 1 = −.2462 of t = −63.63. When expansion terms are used but not expanded about the mean, the corresponding t values for the generating terms should not be used. For example, if x3 = x1x2 is used rather than the correct expansion (x1 − x1)(x2 − x2), then the corresponding t values for x1 and x2 should not be used.
Nonlinear Least Squares There are no analytic methods for determining the most appropriate model for a particular set of data. In many cases, however, the engineer has some basis for a model. If the parameters occur in a nonlinear fashion, then the analysis becomes more difficult. For example, in relating the temperature to the elapsed time of a fluid cooling in the atmosphere, a model that has an asymptotic property would be the appropriate model (temp = a + b exp(−c time), where a represents the asymptotic temperature corresponding to t → ∞. In this case, the parameter c appears nonlinearly. The usual practice is to concentrate on model development and computation rather than on statistical aspects. In general, nonlinear regression should be applied only to problems in which there is a welldefined, clear association between the two variables; therefore, a test of hypothesis on the significance of the fit would be somewhat ludicrous. In addition, the generalization of the theory for the associate confidence intervals for nonlinear coefficients is not well developed. The Levenberg-Marquardt method is used when the parameters of the model appear nonlinearly (Ref. 231). We still define N yi − y(xi; a) χ2(a) = σ 2i i=1
2
1. Choose a and calculate χ2(a). 2. Choose λ, say λ = 0.001. 3. Solve Eq. (3-91) for ak + 1 and evaluate χ2(ak + 1). 4. If χ2(ak + 1) ≥ χ2(ak) then increase λ by a factor of, say, 10 and go back to step 3. This makes the step more like a steepest descent. 5. If χ2(ak + 1) < χ2(ak) then update a, i.e., use a = ak + 1, decrease λ by a factor of 10, and go back to step 3. 6. Stop the iteration when the decrease in χ2 from one step to another is not statistically meaningful, i.e., less than 0.1 or 0.01 or 0.001. 7. Set λ = 0 and compute the estimated covariance matrix: C = α−1. This gives the standard errors in the fitted parameters a. For normally distributed errors the parameter region in which χ2 = constant can give boundaries of the confidence limits. The value of a obtained in the Marquardt method gives the minimum χ2min. If we set χ2 = χ2min + ∆χ for some ∆χ and then look at contours in parameter space where χ12 = constant then we have confidence boundaries at the probability associated with χ 12. For example, in a chemical reactor with radial dispersion the heat transfer coefficient and radial effective heat conductivity are closely connected: decreasing one and increasing the other can still give a good fit. Thus, the confidence boundaries may look something like Fig. 3-59. The ellipse defined by ∆χ2 = 2.3 contains 68.3 percent of the normally distributed data. The curve defined by ∆χ2 = 6.17 contains 95.4 percent of the data. Example Application. Data were collected on the cooling of water in the atmosphere as a function of time. Sample data
and near the optimum represent χ2 by 1 χ2(a) = χ02 − dT ⋅ a + aT ⋅ D ⋅ a 2 where d is an M × 1 vector and D is an M × M matrix. We then calculate iteratively D ⋅ (ak + 1 − ak) = −∇χ2(ak)
(3-89)
The notation alk means the lth component of a evaluated on the kth iteration. If ak is a poor approximation to the optimum, we might use steepest descent instead. ak + 1 − ak = −constant × ∇χ2(ak)
(3-90)
and choose the constant somehow to decrease χ2 as much as possible. The gradient of χ2 is N ∂χ2 yi − y(xi; a) ∂y(xi; a) = −2 ∂ak σ 2i ∂ak i=1
k = 1, 2, . . . , M
The second derivative (in D) is N ∂2χ2 1 ∂y(xi; a) ∂y(xi; a) ∂2y(xi; a) = 2 2 − [yi − y(xi; a)] ∂ak∂al ∂ak ∂al ∂ak∂al i=1 σi
Time x
Temperature y
0 1 2 3 5
92.0 85.5 79.5 74.5 67.0
7 10 15 20
60.5 53.5 45.0 39.5
Model form. On the basis of the nature of the data, an exponential model was selected initially to represent the trend y = a + becx. In this example, the resultant temperature would approach as an asymptotic (a with c negative) the wet-bulb temperature of the surrounding atmosphere. Unfortunately, this temperature was not reported. Using a computer package in MATLAB gives the following results: a = 33.54, b = 57.89, c = 0.11. The value of χ2 is 1.83. An alternative form of model is y = a + b/(c + x). For this model the results were a = 9.872, b = 925.7, c = 11.27, and the value of χ2 is 0.19. Since this model had a smaller value of χ2, it might be the chosen one, but it is only a fit of the specified data and may not be generalized beyond that. Both curve fits give an equivalent plot. The second form is shown in Fig. 3-60.
Both Eq. (3-89) and Eq. (3-90) are included if we write M
α′
kl
(a kl + 1 − a lk ) = βk
(3-91)
l=1
where
N 1 ∂y(xi; a) ∂y(xi; a) α′kl = 2 ∂ak ∂al i=1 σi N 1 ∂y(xi; a) α′kk = 2 ∂ak i=1 σi
k≠1
(1 + λ) 2
N yi − y(xi; a) ∂y(xi; a) β k = σ 2i ∂ak i=1
The second term in the second derivative is dropped because it is usually small [remember that yi will be close to y(xi, a)]. The LevenbergMarquardt method then iterates as follows
FIG. 3-59
Parameter estimation for heat transfer.
STATISTICS
FIG. 3-60
Data in nonlinear regression example.
ERROR ANALYSIS OF EXPERIMENTS Consider the problem of assessing the accuracy of a series of measurements. If measurements are for independent, identically distributed observations, then the errors are independent and uncorrelated. Then y, the experimentally determined mean, varies about E(y), the true mean, with variance σ2/n, where n is the number of observations in y. Thus, if one measures something several times today, and each day, and the measurements have the same distribution, then the variance of the means decreases with the number of samples in each day’s measurement, n. Of course, other factors (weather, weekends) may make the observations on different days not distributed identically. Consider next the problem of estimating the error in a variable that cannot be measured directly but must be calculated based on results of other measurements. Suppose the computed value Y is a linear combination of the measured variables {yi}, Y = α1y1 + α2y2 + … . Let the random variables y1, y2, . . . have means E(y1), E(y2), . . . and variances σ2(y1), σ2(y2), . . . . The variable Y has mean E(Y) = α1E(y1) + α2 E(y2) + … and variance (Ref. 82) n
n
σ2(Y) = α2i σ2(yi) + 2 i=1
n
αi αj Cov (yi, yj)
i=1 j=i+1
If the variables are uncorrelated and have the same variance, then σ2(Y) =
3-87
α σ n
2 i
2
i=1
Next suppose the model relating Y to {yi} is nonlinear, but the errors are small and independent of one another. Then a change in Y is related to changes in yi by ∂Y ∂Y dY = dy1 + dy2 + … ∂y1 ∂y2 If the changes are indeed small, then the partial derivatives are constant among all the samples. Then the expected value of the change, E(dY), is zero. The variances are given by the following equation (Refs. 25 and 40): N ∂Y 2 σ2(dY) = σ i2 ∂yi i=1 Thus, the variance of the desired quantity Y can be found. This gives an independent estimate of the errors in measuring the quantity Y from the errors in measuring each variable it depends upon.
Example Suppose one wants to measure the thermal conductivity of a solid (k). To do this, one needs to measure the heat flux (q), the thickness of the
sample (d), and the temperature difference across the sample (∆T). Each measurement has some error. The heat flux (q) may be the rate of electrical heat ˙ divided by the area (A), and both quantities are measured to some tolinput (Q) erance. The thickness of the sample is measured with some accuracy, and the temperatures are probably measured with a thermocouple to some accuracy. These measurements are combined, however, to obtain the thermal conductivity, and it is desired to know the error in the thermal conductivity. The formula is d ˙ k=Q A∆T The variance in the thermal conductivity is then k 2 k 2 k 2 2 k 2 σ k2 = σ d2 + σ Q˙ + σ A2 + σ 2∆T ˙ Q d A ∆T
FACTORIAL DESIGN OF EXPERIMENTS AND ANALYSIS OF VARIANCE Statistically designed experiments consider, of course, the effect of primary variables, but they also consider the effect of extraneous variables and the interactions between variables, and they include a measure of the random error. Primary variables are those whose effect you wish to determine. These variables can be quantitative or qualitative. The quantitative variables are ones you may fit to a model in order to determine the model parameters (see the section “Least Squares”). Qualitative variables are ones you wish to know the effect of, but you do not try to quantify that effect other than to assign possible errors or magnitudes. Qualitative variables can be further subdivided into Type I variables, whose effect you wish to determine directly, and Type II variables, which contribute to the performance variability and whose effect you wish to average out. For example, if you are studying the effect of several catalysts on yield in a chemical reactor, each different type of catalyst would be a Type I variable because you would like to know the effect of each. However, each time the catalyst is prepared, the results are slightly different due to random variations; thus, you may have several batches of what purports to be the same catalyst. The variability between batches is a Type II variable. Since the ultimate use will require using different batches, you would like to know the overall effect including that variation, since knowing precisely the results from one batch of one catalyst might not be representative of the results obtained from all batches of the same catalyst. A randomized block design, incomplete block design, or Latin square design (Ref. 40), for example, all keep the effect of experimental error in the blocked variables from influencing the effect of the primary variables. Other uncontrolled variables are accounted for by introducing randomization in parts of the experimental design. To study all variables and their interaction requires a factorial design, involving all possible
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MATHEMATICS
combinations of each variable, or a fractional factorial design, involving only a selected set. Statistical techniques are then used to determine which are the important variables, what are the important interactions, and what the error is in estimating these effects. The discussion here is only a brief overview of the excellent Ref. 40. Suppose we have two methods of preparing some product and we wish to see which treatment is best. When there are only two treatments, then the sampling analysis discussed in the section “TwoPopulation Test of Hypothesis for Means” can be used to deduce if the means of the two treatments differ significantly. When there are more treatments, the analysis is more detailed. Suppose the experimental results are arranged as shown in the table: several measurements for each treatment. The goal is to see if the treatments differ significantly from each other; that is, whether their means are different when the samples have the same variance. The hypothesis is that the treatments are all the same, and the null hypothesis is that they are different. The statistical validity of the hypothesis is determined by an analysis of variance. Estimating the Effect of Four Treatments Treatment
Treatment average Grand average
1
2
3
4
— — —
— — — —
— — — — —
— — — — — — —
—
— —
—
The data for k = 4 treatments is arranged in the table. For each treatment, there are nt experiments and the outcome of the ith experiment with treatment t is called yti. Compute the treatment average nt
y
Basically the test for whether the hypothesis is true or not hinges on a comparison of the within-treatment estimate sR2 (with νR = N − k degrees of freedom) with the between-treatment estimate s2T (with νT = k − 1 degrees of freedom). The test is made based on the F distribution for νR and νT degrees of freedom (Table 3-7). Next consider the case that uses randomized blocking to eliminate the effect of some variable whose effect is of no interest, such as the batch-to-batch variation of the catalysts in the chemical reactor example. Suppose there are k treatments and n experiments in each treatment. The results from nk experiments can be arranged as shown in the block design table; within each block, the various treatments are applied in a random order. Compute the block average, the treatment average, as well as the grand average as before. Block Design with Four Treatments and Five Blocks Treatment
1
2
3
4
Block average
Block 1 Block 2 Block 3 Block 4 Block 5
— — — — —
— — — — —
— — — — —
— — — — —
— — — — —
The following quantities are needed for the analysis of variance table. Name
Formula
dof
average
2 SA = nky
blocks
2 SB = k i = 1 (y i − y)
n−1
treatments
ST = n t = 1 (y t − y)
k−1
residuals
SR = t = 1 i = 1 (yti − yi − yt + y)2
(n − 1)(k − 1)
total
S = t = 1 i = 1 y
N = nk
1
n
k
2
k
k
n
n
2 ti
ti
i=1
The key test is again a statistical one, based on the value of
yt = nt
sT2 , sR2
Also compute the grand average k
n y
t t
k
N = nt
t=1
y = , N
t=1
Next compute the sum of squares of deviations from the average within the tth treatment nt
St = (yti − yt)2 i=1
Since each treatment has nt experiments, the number of degrees of freedom is nt − 1. Then the sample variances are St s t2 = nt − 1 The within-treatment sum of squares is k
SR = St t=1
and the within-treatment sample variance is SR sR2 = N−k Now, if there is no difference between treatments, a second estimate of σ2 could be obtained by calculating the variation of the treatment averages about the grand average. Thus compute the betweentreatment mean square k ST 2 sT2 = , ST = nt(y t − y) k−1 t=1
ST sT2 = , k−1
SR s R2 = (n − 1)(k − 1)
and the F distribution for νR and νT degrees of freedom (Table 3-7). The assumption behind the analysis is that the variations are linear (Ref. 40). There are ways to test this assumption as well as transformations to make if it is not true. Reference 40 also gives an excellent example of how the observations are broken down into a grand average, a block deviation, a treatment deviation, and a residual. For twoway factorial design in which the second variable is a real one rather than one you would like to block out, see Ref. 40. To measure the effects of variables on a single outcome a factorial design is appropriate. In a two-level factorial design, each variable is considered at two levels only, a high and low value, often designated as a + and −. The two-level factorial design is useful for indicating trends, showing interactions, and it is also the basis for a fractional factorial design. As an example, consider a 23 factorial design with 3 variables and 2 levels for each. The experiments are indicated in the factorial design table. Two-Level Factorial Design with Three Variables Variable Run
1
2
3
1 2 3 4 5 6 7 8
− + − + − + − +
− − + + − − + +
− − − − + + + +
DIMENSIONAL ANALYSIS The main effects are calculated by calculating the difference between results from all high values of a variable and all low values of a variable; the result is divided by the number of experiments at each level. For example, for the first variable: [(y2 + y4 + y6 + y8) − (y1 + y3 + y5 + y7)] Effect of variable 1 = 4 Note that all observations are being used to supply information on each of the main effects and each effect is determined with the precision of a fourfold replicated difference. The advantage of a one-at-atime experiment is the gain in precision if the variables are additive and the measure of nonadditivity if it occurs (Ref. 40). Interaction effects between variables 1 and 2 are obtained by calculating the difference between the results obtained with the high and low value of 1 at the low value of 2 compared with the results obtained with the high and low value 1 at the high value of 2. The 12-interaction is [(y4 − y3 + y8 − y7) − (y2 − y1 + y6 − y5)] 12-interaction = 2
3-89
The key step is to determine the errors associated with the effect of each variable and each interaction so that the significance can be determined. Thus, standard errors need to be assigned. This can be done by repeating the experiments, but it can also be done by using higher-order interactions (such as 123 interactions in a 24 factorial design). These are assumed negligible in their effect on the mean but can be used to estimate the standard error (see Ref. 40). Then, calculated effects that are large compared with the standard error are considered important, while those that are small compared with the standard error are considered to be due to random variations and are unimportant. In a fractional factorial design one does only part of the possible experiments. When there are k variables, a factorial design requires 2k experiments. When k is large, the number of experiments can be large; for k = 5, 25 = 32. For a k this large, Box et al. (Ref. 82, p. 376) do a fractional factorial design. In the fractional factorial design with k = 5, only 16 experiments are done. Cropley (Ref. 82) gives an example of how to combine heuristics and statistical arguments in application to kinetics mechanisms in chemical engineering.
DIMENSIONAL ANALYSIS Dimensional analysis allows the engineer to reduce the number of variables that must be considered to model experiments or correlate data. Consider a simple example in which two variables F1 and F2 have the units of force and two additional variables L1 and L2 have the units of length. Rather than having to deduce the relation of one variable on the other three, F1 = fn (F2, L1, L2), dimensional analysis can be used to show that the relation must be of the form F1 /F2 = fn (L1 /L2). Thus considerable experimentation is saved. Historically, dimensional analysis can be done using the Rayleigh method or the Buckingham pi method. This brief discussion is equivalent to the Buckingham pi method but uses concepts from linear algebra; see Ref. 13 for further information. The general problem is posed as finding the minimum number of variables necessary to define the relationship between n variables. Let {Qi} represent a set of fundamental units, like length, time, force, and so on. Let [Pi] represent the dimensions of a physical quantity Pi; there are n physical quantities. Then form the matrix αij
Q1 Q2 … Qm
[P1]
[P2]
…
[Pn]
α11 α21
α12 α22
… …
α1n α2n
αm1
αm2
…
αmn
in which the entries are the number of times each fundamental unit appears in the dimensions [Pi]. The dimensions can then be expressed as follows.
Fluid thermal conductivity = k = (F/θT) Fluid specific heat = cp = (FL/MT) Dimensional constant = gc = (ML/Fθ2) The matrix α in this case is as follows. [Pi]
Qj
F M L θ T
Example: Buckingham Pi Method—Heat-Transfer Film Coefficient It is desired to determine a complete set of dimensionless groups with which to correlate experimental data on the film coefficient of heat transfer between the walls of a straight conduit with circular cross section and a fluid flowing in that conduit. The variables and the dimensional constant believed to be involved and their dimensions in the engineering system are given below: Film coefficient = h = (F/LθT) Conduit internal diameter = D = (L) Fluid linear velocity = V = (L/θ) Fluid density = ρ = (M/L3) Fluid absolute viscosity = µ = (M/Lθ)
D
V
ρ
µ
k
Cp
gc
1 0 −1 −1 −1
0 0 1 0 0
0 0 1 −1 0
0 1 −3 0 0
0 1 −1 −1 0
1 0 0 −1 −1
1 −1 1 0 −1
−1 1 1 −2 0
Here m ≤ 5, n = 8, p ≥ 3. Choose D, V, µ, k, and gc as the primary variables. By examining the 5 × 5 matrix associated with those variables, we can see that its determinant is not zero, so the rank of the matrix is m = 5; thus, p = 3. These variables are thus a possible basis set. The dimensions of the other three variables h, ρ, and Cp must be defined in terms of the primary variables. This can be done by inspection, although linear algebra can be used, too. hD h [h] = D−1k+1; thus = is a dimensionless group D−1 k k ρVD ρ [ρ] = µ1V −1D −1; thus = is a dimensionless group µ1V−1D−1 µ Cpµ Cp [Cp] = k+1µ−1; thus = is a dimensionless group k+1 µ−1 k Thus, the dimensionless groups are [Pi] hD ρVD Cp µ : , , Q1α1i Q2α2i⋅⋅⋅Qmαmi k µ k
[Pi] = Q1α1i Q2α2i⋅⋅⋅Qmαmi Let m be the rank of the α matrix. Then p = n − m is the number of dimensionless groups that can be formed. One can choose m variables {Pi} to be the basis and express the other p variables in terms of them, giving p dimensionless quantities.
h
The dimensionless group hD/k is called the Nusselt number, NNu, and the group Cp µ /k is the Prandtl number, NPr. The group DVρ/µ is the familiar Reynolds number, NRe , encountered in fluid-friction problems. These three dimensionless groups are frequently used in heat-transfer-film-coefficient correlations. Functionally, their relation may be expressed as φ(NNu, NPr, NRe) = 0 or as
(3-91)
NNu = φ1(NPr, NRe)
It has been found that these dimensionless groups may be correlated well by an equation of the type hD/k = K(cpµ/k)a(DVρ/µ)b in which K, a, and b are experimentally determined dimensionless constants. However, any other type of algebraic expression or perhaps simply a graphical relation among these three groups that accurately fits the experimental data would be an equally valid manner of expressing Eq. (3-91).
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MATHEMATICS
Naturally, other dimensionless groups might have been obtained in the example by employing a different set of five repeating quantities that would not form a dimensionless group among themselves. Some of these groups may be found among those presented in Table 3-8. Such a complete set of three dimensionless groups might consist of Stanton, Reynolds, and Prandtl numbers or of Stanton, Peclet, and Prandtl numbers. Also, such a complete set different from that obtained in the preceding example will result from a multiplication of appropriate powers of the Nusselt, Prandtl, and Reynolds numbers. For such a set to be complete, however, it must satisfy the condition that each of the three dimensionless groups be independent of the other two.
TABLE 3-8 Dimensionless Groups in the Engineering System of Dimensions Biot number Condensation number Number used in condensation of vapors Euler number Fourier number Froude number Graetz number Grashof number Mach number Nusselt number Peclet number Prandtl number Reynolds number Schmidt number Stanton number Weber number
NBi NCo NCv NEu NFo NFr NGz NGr NMa NNu NPe NPr NRe NSc NSt NWe
hL/k (h/k)(µ2/ρ2g)1/3 L3ρ2gλ/kµ∆t gc(−dp)/ρV2 kθ/ρcL2 V2/Lg wc/kL L3ρ2βg∆t/µ2 V/Va hD/k DVρc/k cµ/k DVρ/µ µ/ρDυ h/cVρ LV2ρ/σgc
PROCESS SIMULATION Classification Process simulation refers to the activity in which mathematical models of chemical processes and refineries are modeled with equations, usually on the computer. The usual distinction must be made between steady-state models and transient models, following the ideas presented in the introduction to this section. In a chemical process, of course, the process is nearly always in a transient mode, at some level of precision, but when the time-dependent fluctuations are below some value, a steady-state model can be formulated. This subsection presents briefly the ideas behind steady-state process simulation (also called flowsheeting), which are embodied in commercial codes. The transient simulations are important for designing startup of plants and are especially useful for the operating of chemical plants. Process Modules The usual first step in process simulation is to perform a mass and energy balance for a chosen process. The most important aspect of the simulation is that the thermodynamic data of the chemicals be modeled correctly. The computer results of vaporliquid equilibria, for example, must be checked against experimental data to insure their validity before using the data in more complicated computer calculations. At this first level of detail, it is not necessary to know the internal parameters for all the units, since what is desired is just the overall performance. For example, in a heat exchanger design, it suffices to know the heat duty, the total area, and the temperatures of the output streams; the details like the percentage baffle cut, tube layout, or baffle spacing can be specified later when the details of the proposed plant are better defined. Each unit operation is modeled by a subroutine, which is governed by equations (presented throughout this book). Some of the inputs to the units are known, some are specified by the user as design variables, and some are to be found using the simulation. It is important to know the number of degrees of freedom for each option of the unit operation, because at least that many parameters must be specified in order for the simulation to be able to calculate unit outputs. Sometimes the quantities the user would like to specify are targets, and parameters in the unit operation are to be changed to meet that target. This is not always possible, and the designer will have to adjust the parameters of the unit operation to achieve the desired target, possibly using the convergence tools discussed below. For example, in a reaction/separation system, if there is an impurity that must be purged, a common objective is to set the purge fraction so that the impurity concentration into the reactor is kept at some moderate value. Yet the solution techniques do not readily lend themselves to this connection, so convergence strategies must be employed.
Solution Strategies Consider a chemical process consisting of a series of units, such as distillation towers, reactors, and so forth. If the feed to the process is known and the operating parameters of the unit operations are specified by the user, then one can begin with the first unit, take the process input, calculate the unit output, carry that output to the input of the next unit, and continue the process. In this way, one can simulate the entire process. However, if the process involves a recycle stream, as nearly all chemical processes do, then when the calculation is begun, it is discovered that the recycle stream is unknown. Thus the calculation cannot begin. This situation leads to the need for an iterative process: the flow rates, temperature, and pressure of the unknown recycle stream are guessed and the calculations proceed as before. When one reaches the end of the process, where the recycle stream is formed to return to the inlet, it is necessary to check to see if the recycle stream is the same as assumed. If not, an iterative procedure must be used to cause convergence. The techniques like Wegstein (see “Numerical Solution of Nonlinear Equations in One Variable”) can be used to accelerate the convergence. When doing these iterations, it is useful to analyze the process using precedence ordering and tearing to minimize the number of recycle loops (Refs. 201, 242, 255, and 293). When the recycle loops interact with one another the iterations may not lead to a convergent solution. The designer usually wants to specify stream flow rates or parameters in the process, but these may not be directly accessible. For example, the desired separation may be known for a distillation tower, but the simulation program requires the specification of the number of trays. It is left up to the designer to choose the number of trays that lead to the desired separation. In the example of the purge stream/ reactor impurity, a controller module may be used to adjust the purge rate to achieve the desired reactor impurity. This further complicates the iteration process. An alternative method of solving the equations is to solve them as simultaneous equations. In that case, one can specify the design variables and the desired specifications and let the computer figure out the process parameters that will achieve those objectives. It is possible to overspecify the system or give impossible conditions. However, the biggest drawback to this method of simulation is that large sets (10,000s) of algebraic equations must be solved simultaneously. As computers become faster, this is less of an impediment. For further information, see Refs. 90, 175, 255, and 293. For information on computer software, see the Annual CEP Software Directory (Ref. 8) and other articles (Refs. 7 and 175).
INTELLIGENT SYSTEMS IN PROCESS ENGINEERING
3-91
INTELLIGENT SYSTEMS IN PROCESS ENGINEERING REFERENCES: General, 232, 248, 258, 275, 276. Knowledge-Based Systems, 49, 232, 275. Neural Networks, 54, 140. Qualitative Simulation, 178, 292. Fuzzy Logic, 94. Genetic Algorithms, 121. Applications, 15, 24, 205, 232, 250, 262, 294.
Intelligent system is a term that refers to computer-based systems that include knowledge-based systems, neural networks, fuzzy logic and fuzzy control, qualitative simulation, genetic algorithms, natural language understanding, and others. The term is often associated with a variety of computer programming languages and/or features that are used as implementation media, although this is an imprecise use. Examples include object-oriented languages, rule-based languages, prolog, and lisp. The term intelligent system is preferred over the term artificial intelligence. The three intelligent-system technologies currently seeing the greatest amount of industrial application are knowledge-based systems, fuzzy logic, and artificial neural networks. These technologies are components of distributed systems. Mathematical models, conventional numeric and statistical approaches, neural networks, knowledge-based systems, and the like, all have their place in practical implementation and allow automation of tasks not welltreated by numerical algorithms. Fundamentally, intelligent-system techniques are modeling techniques. They allow the encoding of qualitative models that draw upon experience and expertise, thereby extending modeling capacity beyond mathematical description. An important capability of intelligent system techniques is that they can be used not only to model physical behaviors but also decision-making processes. Decision processes reflect the selection, application, and interpretation of highly relevant pieces of information to draw conclusions about complex situations. Activity-specific decision processes can be expressed at a functional level, such as diagnosis, design, planning, and scheduling, or as their generic components, such as classification, abduction, and simulation. Decision process models address how information is organized and structured and then assimilated into active decisions. Knowledge-Based Systems Knowledge-based system (KBS) approaches capture the structural and information processing features of qualitative problem solving associated with sequential consideration, selection, and search. These technologies not only provide the means of capturing decision-making knowledge but also offer a medium for exploiting efficient strategies used by experts. KBSs, then, are computer programs that model specific ways of organizing problem-specific fragments of knowledge and then searching through them by establishing appropriate relationships to reach correct conclusions. Deliberation is a general label for the algorithmic process for sorting through the knowledge fragments. The basic components of KBSs are knowledge representation (structure) and search. They are the programming mechanisms that facilitate the use and application of the problem-specific knowledge appropriate to solving the problem. Together they are used to form conclusions, decisions, or interpretations in a symbolic form. See Refs. 49, 232, and 275. Qualitative simulation is a specific KBS model of physical processes that are not understood well enough to develop a physics-based numeric model. Corrosion, fouling, mechanical wear, equipment failure, and fatigue are not easily modeled, but decisions about them can be based on qualitative reasoning. See Refs. 178 and 292. Qualitative description of physical behaviors require that each continuous variable space be quantized. Quantization is typically based on landmark values that are boundary points separating qualitatively distinct regions of continuous values. By using these qualitative quantity descriptions, dynamic relations between variables can be modeled as qualitative equations that represent the structure of the system. The
solution to the equations represents the possible sequences of qualitative states as well as the explanations for changes in behaviors. Building and explaining a complex model requires a unified view called an ontology. Methods of qualitative reasoning can be based on different viewpoints; the dominant viewpoints are device, process, and constraints. Behavior generation is handled with two approaches: (1) simulating successive states from one or more initial states, and (2) determining all possible state-to-state transitions once all possible states are determined. Fuzzy Logic Fuzzy logic is a formalism for mapping between numerical values and qualitative or linguistic interpretations. This is useful when it is difficult to define precisely such terms as “high” and “low,” since there may be no fixed threshold. Fuzzy sets use the concept of degree of membership to overcome this problem. Degree of membership allows a descriptor to be associated with a range of numeric values but in varying degrees. A fuzzy set is explicitly defined by a degree of membership for each linguistic variable that is applicable, mA(x) where mA is the degree of membership for linguistic variable A. For fuzzy sets, logical operators, such as complement (NOT), intersection (AND), and union (OR) are defined. The following are typical definitions. NOT: mNOT A(x) = 1 − mA(x) AND: mA AND B(x) = min [mA(x), mB(x)] OR: mA OR B(x) = max [mA(x), mB(x)] Using these operators, fuzzy inference mechanisms are then developed to manipulate rules that include fuzzy values. The largest difference between fuzzy inference and ordinary inference is that fuzzy inference allows “partial match” of input and produces an “interpolated” output. This technology is useful in control also. See Ref. 94. Artificial Neural Networks An artificial neural network (ANN) is a collection of computational units that are interconnected in a network. Knowledge is captured in the form of weights, and input-output mappings are produced by the interactions of the weights and the computational units. Each computational unit combines weighted inputs and generates an output base on an activation function. Typical activation functions are (1) specified limit, (2) sigmoid, and (3) gaussian. ANNs can be feedforward, with multiple layers of intermediate units, or feedback (sometimes called recurrent networks). The ability to generalize on given data is one of the most important performance characteristics. With appropriate selection of training examples, an optimal network architecture, and appropriate training, the network can map a relationship between input and output that is complete but bounded by the coverage of the training data. Applications of neural networks can be broadly classified into three categories: 1. Numeric-to-numeric transformations are used as empirical mathematical models where the adaptive characteristics of neural networks learn to map between numeric sets of input-output data. In these modeling applications, neural networks are used as an alternative to traditional data regression schemes based on regression of plant data. Backpropagation networks have been widely used for this purpose. 2. Numeric-to-symbolic transformations are used in patternrecognition problems where the network is used to classify input data vectors into specific labeled classes. Pattern recognition problems include data interpretation, feature identification, and diagnosis. 3. Symbolic-to-symbolic transformations are used in various symbolic manipulations, including natural language processing and rulebased system implementation. See Refs. 54 and 140.
Section 4
Thermodynamics
Hendrick C. Van Ness, D.Eng., Howard P. Isermann Department of Chemical Engineering, Rensselaer Polytechnic Institute; Fellow, American Institute of Chemical Engineers; Member, American Chemical Society Michael M. Abbott, Ph.D., Howard P. Isermann Department of Chemical Engineering, Rensselaer Polytechnic Institute; Member, American Institute of Chemical Engineers
INTRODUCTION Postulate 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postulate 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postulate 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postulate 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postulate 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-3 4-3 4-4 4-4 4-4
VARIABLES, DEFINITIONS, AND RELATIONSHIPS Constant-Composition Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy and Entropy as Functions of T and P. . . . . . . . . . . . . . . . . . Internal Energy and Entropy as Functions of T and V. . . . . . . . . . . . Heat-Capacity Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Systems of Variable Composition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Molar Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gibbs/Duhem Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Molar Gibbs Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Ideal Gas State and the Compressibility Factor . . . . . . . . . . . . . Residual Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-5 4-5 4-6 4-6 4-6 4-7 4-7 4-7 4-8 4-8 4-8
SOLUTION THERMODYNAMICS Ideal Gas Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fugacity and Fugacity Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental Residual-Property Relation. . . . . . . . . . . . . . . . . . . . . . . . The Ideal Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental Excess-Property Relation . . . . . . . . . . . . . . . . . . . . . . . . . Summary of Fundamental Property Relations . . . . . . . . . . . . . . . . . . . . Property Changes of Mixing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Behavior of Binary Liquid Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-8 4-9 4-9 4-10 4-10 4-11 4-11 4-12
EVALUATION OF PROPERTIES Residual-Property Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid/Vapor Phase Transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-14 4-15
Liquid-Phase Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties from PVT Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pitzer’s Corresponding-States Correlation . . . . . . . . . . . . . . . . . . . . . Alternative Property Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . Virial Equations of State. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalized Correlation for the Second Virial Coefficient. . . . . . . . . Cubic Equations of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Benedict/Webb/Rubin Equation of State . . . . . . . . . . . . . . . . . . . . . . Expressions for the Excess Gibbs Energy . . . . . . . . . . . . . . . . . . . . . . . .
4-15 4-15 4-16 4-16 4-19 4-20 4-20 4-21 4-22
EQUILIBRIUM Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Phase Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Application of the Phase Rule . . . . . . . . . . . . . . . . . . . . . Vapor/Liquid Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gamma/Phi Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solute/Solvent Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K-Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equation-of-State Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid/Liquid and Vapor/Liquid/Liquid Equilibria . . . . . . . . . . . . . . . . Chemical-Reaction Stoichiometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical-Reaction Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standard Property Changes of Reaction . . . . . . . . . . . . . . . . . . . . . . . Equilibrium Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2: Single-Reaction Equilibrium . . . . . . . . . . . . . . . . . . . . . . Complex Chemical-Reaction Equilibria . . . . . . . . . . . . . . . . . . . . . . . Example 3: Minimization of Gibbs Energy . . . . . . . . . . . . . . . . . . . . .
4-24 4-24 4-25 4-25 4-25 4-26 4-27 4-28 4-28 4-30 4-31 4-31 4-31 4-32 4-32 4-33 4-34
THERMODYNAMIC ANALYSIS OF PROCESSES Calculation of Ideal Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lost Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of Steady-State, Steady-Flow Processes . . . . . . . . . . . . . . . . . . Example 4: Lost-Work Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-34 4-35 4-35 4-36
4-1
4-2
THERMODYNAMICS
Nomenclature and Units Symbols are omitted that are correlation- or application-specific. Symbol A âi B C D B′ C′ D′ Bij Cijk CP CV EK EP fi fˆi G g g H Ki Kj ki M
Mi Mi ∆M ∆M°j m m ˙ n ni P Pc Pisat pi Q Q˙ R
U.S. customary units
Definition
SI units
Helmholtz energy Activity of species i in solution 2d virial coefficient, density expansion 3d virial coefficient, density expansion 4th virial coefficient, density expansion 2d virial coefficient, pressure expansion 3d virial coefficient, pressure expansion 4th virial coefficient, pressure expansion Interaction 2d virial coefficient Interaction 3d virial coefficient Heat capacity at constant pressure Heat capacity at constant volume Kinetic energy Gravitational potential energy Fugacity of pure species i Fugacity of species i in solution Molar or unit-mass Gibbs energy Acceleration of gravity GE/RT Molar or unit-mass enthalpy
J Dimensionless
Equilibrium K-value, yi/xi Equilibrium constant for chemical reaction j Henry’s constant Molar or unit-mass value of any extensive thermodynamic property of a solution Molar or unit-mass value of any extensive property of pure species i Partial molar property of species i in solution Property change of mixing Standard property change of reaction j Mass Mass flow rate Number of moles Number of moles of species i Absolute pressure Critical pressure Saturation or vapor pressure of species i Partial pressure of species i in gas mixture (yiP) Heat Rate of heat transfer Universal gas constant
Dimensionless Dimensionless
Btu/lb mol or Btu/lbm Dimensionless Dimensionless
kPa
psi
cm3/mol 6
Btu Dimensionless cm3/mol
2
cm /mol
cm6/mol2
cm9/mol3
cm9/mol3
kPa−1
kPa−1
kPa−2
kPa−2
−3
kPa
kPa−3
cm3/mol
cm3/mol
cm6/mol2
cm6/mol2
J/(mol⭈K)
Btu/(lb mol⭈R)
J/(mol⭈K)
Btu/(lb mol⭈R)
J J kPa kPa
Btu Btu psi psi
J/mol or J/kg
Btu/lb mol or Btu/lbm ft/s2
m/s2 J/mol or J/kg
Symbol
Definition
SI units
S
Molar or unit-mass entropy
T Tc U
Absolute temperature Critical temperature, Molar or unit-mass internal energy Velocity Molar or unit-mass volume
u V W Ws ˙s W xi
yi Z z
Work Shaft work for flow process Shaft power for flow process Mole fraction in general or liquid-phase mole fraction of species i in solution Vapor-phase mole fraction of species i in solution Compressibility factor Elevation above a datum level
J/(mol⭈K) or J/(kg⭈K) K K J/mol or J/kg
U.S. customary units
m/s m3/mol or m3/kg J J J/s Dimensionless
Btu/(lb mol⭈R) or Btu/(lb⭈R) R R Btu/lb mol or Btu/lbm ft/s ft3/lb mol or ft3/lbm Btu Btu Btu/s Dimensionless
Dimensionless
Dimensionless
Dimensionless m
Dimensionless ft
Superscripts E id ig l lv R t v ∞
Denotes excess thermodynamic property Denotes value for an ideal solution Denotes value for an ideal gas Denotes liquid phase Denotes phase transition from liquid to vapor Denotes residual thermodynamic property Denotes a total value of a thermodynamic property Denotes vapor phase Denotes a value at infinite dilution
C c H r rev
Denotes a value for a colder heat reservoir Denotes a value for the critical state Denotes a value for a hotter heat reservoir Denotes a reduced value Denotes a reversible process
α,β
As superscripts, identify phases Volume expansivity, species i Reaction coordinate for reaction j Defined by Eq. (4-72) Heat-capacity ratio, CP/CV Activity coefficient of species i in solution Chemical potential of species i Stoichiometric number of species i in reaction j Molar density As a subscript, denotes a heat reservoir Defined by Eq. (4-283) Fugacity coefficient of pure species i Fugacity coefficient of species i in solution Acentric factor
Subscripts
Greek letters
βi εj Γi(T) γ γi
kg kg/s
lbm lbm/s
kPa kPa kPa
psi psi psi
ρ σ
kPa
psi
Φi φi
J J/s J/(mol⭈K)
Btu Btu/s Btu/(lb mol⭈R)
µi νi, j
φˆ i ω
K−1 mol
R−1 lb mol
J/mol Dimensionless Dimensionless
Btu/lb mol Dimensionless Dimensionless
J/mol Dimensionless
Btu/lb mol Dimensionless
mols/m3
lb moles/ft3
Dimensionless Dimensionless
Dimensionless Dimensionless
Dimensionless
Dimensionless
Dimensionless
Dimensionless
GENERAL REFERENCES: Abbott, M.M., and H.C. Van Ness, Schaum’s Outline of Theory and Problems of Thermodynamics, 2d ed., McGraw-Hill, New York, 1989. Tester, J.W. and M. Modell, Thermodynamics and its Applications, 3d ed., Prentice-Hall, Englewood Cliffs, N.J., 1996. Prausnitz, J.M., R.N. Lichtenthaler, and E.G. de Azevedo, Molecular Thermodynamics of Fluid-Phase Equilibria, 2d ed., Prentice-Hall, Englewood Cliffs, N.J., 1986. Reid, R.C., J.M. Prausnitz, and B.E. Poling, The Properties of Gases and Liquids, 4th ed.,
McGraw-Hill, New York, 1987. Sandler, S.I., Chemical and Engineering Thermodynamics, 2d ed., Wiley, New York, 1989. Smith, J.M., H.C. Van Ness, and M.M. Abbott, Introduction to Chemical Engineering Thermodynamics, 5th ed., McGraw-Hill, New York, 1996. Van Ness, H.C., and M.M. Abbott, Classical Thermodynamics of Nonelectrolyte Solutions: With Applications to Phase Equilibria, McGraw-Hill, New York, 1982.
INTRODUCTION Thermodynamics is the branch of science that embodies the principles of energy transformation in macroscopic systems. The general restrictions which experience has shown to apply to all such transformations are known as the laws of thermodynamics. These laws are primitive; they cannot be derived from anything more basic. The first law of thermodynamics states that energy is conserved; that, although it can be altered in form and transferred from one place to another, the total quantity remains constant. Thus, the first law of thermodynamics depends on the concept of energy; but, conversely, energy is an essential thermodynamic function because it allows the first law to be formulated. This coupling is characteristic of the primitive concepts of thermodynamics. The words system and surroundings are similarly coupled. A system is taken to be any object, any quantity of matter, any region, and so on, selected for study and set apart (mentally) from everything else, which is called the surroundings. The imaginary envelope which encloses the system and separates it from its surroundings is called the boundary of the system. Attributed to this boundary are special properties which may serve either (1) to isolate the system from its surroundings, or (2) to provide for interaction in specific ways between system and surroundings. An isolated system exchanges neither matter nor energy with its surroundings. If a system is not isolated, its boundaries may permit exchange of matter or energy or both with its surroundings. If the exchange of matter is allowed, the system is said to be open; if only energy and not matter may be exchanged, the system is closed (but not isolated), and its mass is constant. When a system is isolated, it cannot be affected by its surroundings. Nevertheless, changes may occur within the system that are detectable with such measuring instruments as thermometers, pressure gauges, and so on. However, such changes cannot continue indefinitely, and the system must eventually reach a final static condition of internal equilibrium. For a closed system which interacts with its surroundings, a final static condition may likewise be reached such that the system is not only internally at equilibrium but also in external equilibrium with its surroundings. The concept of equilibrium is central in thermodynamics, for associated with the condition of internal equilibrium is the concept of state. A system has an identifiable, reproducible state when all its properties, such as temperature T, pressure P, and molar volume V, are fixed. The concepts of state and property are again coupled. One can equally well say that the properties of a system are fixed by its state. Although the properties T, P, and V may be detected with measuring instruments, the existence of the primitive thermodynamic properties (see Postulates 1 and 3 following) is recognized much more indirectly. The number of properties for which values must be specified in order to fix the state of a system depends on the nature of the system and is ultimately determined from experience. When a system is displaced from an equilibrium state, it undergoes a process, a change of state, which continues until its properties attain new equilibrium values. During such a process the system may be
caused to interact with its surroundings so as to interchange energy in the forms of heat and work and so to produce in the system changes considered desirable for one reason or another. A process that proceeds so that the system is never displaced more than differentially from an equilibrium state is said to be reversible, because such a process can be reversed at any point by an infinitesimal change in external conditions, causing it to retrace the initial path in the opposite direction. Thermodynamics finds its origin in experience and experiment, from which are formulated a few postulates that form the foundation of the subject. The first two deal with energy: POSTULATE 1 There exists a form of energy, known as internal energy, which for systems at internal equilibrium is an intrinsic property of the system, functionally related to its characteristic coordinates. POSTULATE 2 (FIRST LAW OF THERMODYNAMICS) The total energy of any system and its surroundings is conserved. Internal energy is quite distinct from such external forms as the kinetic and potential energies of macroscopic bodies. Although a macroscopic property characterized by the macroscopic coordinates T and P, internal energy finds its origin in the kinetic and potential energies of molecules and submolecular particles. In applications of the first law of thermodynamics, all forms of energy must be considered, including the internal energy. It is therefore clear that Postulate 2 depends on Postulate 1. For an isolated system, the first law requires that its energy be constant. For a closed (but not isolated) system, the first law requires that energy changes of the system be exactly compensated by energy changes in the surroundings. Energy is exchanged between such a system and its surroundings in two forms: heat and work. Heat is energy crossing the system boundary under the influence of a temperature difference or gradient. A quantity of heat Q represents an amount of energy in transit between a system and its surroundings, and is not a property of the system. The convention with respect to sign makes numerical values of Q positive when heat is added to the system and negative when heat leaves the system. Work is again energy in transit between a system and its surroundings, but resulting from the displacement of an external force acting on the system. Like heat, a quantity of work W represents an amount of energy, and is not a property of the system. The sign convention, analogous to that for heat, makes numerical values of W positive when work is done on the system by the surroundings and negative when work is done on the surroundings by the system. When applied to closed (constant-mass) systems for which the only form of energy that changes is the internal energy, the first law of thermodynamics is expressed mathematically as dU t = dQ + dW (4-1) 4-3
4-4
THERMODYNAMICS
where U t is the total internal energy of the system. Note that dQ and dW, differential quantities representing energy exchanges between the system and its surroundings, serve to account for the energy change of the surroundings. On the other hand, dU t is directly the differential change in internal energy of the system. Integration of Eq. (4-1) gives for a finite process ∆Ut = Q + W
(4-2)
where ∆U is the finite change given by the difference between the final and initial values of U t. The heat Q and work W are finite quantities of heat and work; they are not properties of the system nor functions of the thermodynamic coordinates that characterize the system.
encountered in chemical technology is one for which the primary characteristic variables are temperature T, pressure P, molar volume V, and composition, not all of which are necessarily independent. Such systems are usually made up of fluids (liquid or gas) and are called PVT systems. For closed systems of this kind, the work of a reversible process may always be calculated from dWrev = −P dV t
t
POSTULATE 3 There exists a property called entropy, which for systems at internal equilibrium is an intrinsic property of the system, functionally related to the measurable coordinates which characterize the system. For reversible processes, changes in this property may be calculated by the equation: dS t = dQrev /T
(4-3)
where St is the total entropy of the system and T is the absolute temperature of the system. POSTULATE 4 (SECOND LAW OF THERMODYNAMICS) The entropy change of any system and its surroundings, considered together, resulting from any real process is positive, approaching zero when the process approaches reversibility. In the same way that the first law of thermodynamics cannot be formulated without the prior recognition of internal energy as a property, so also the second law can have no complete and quantitative expression without a prior assertion of the existence of entropy as a property. The second law requires that the entropy of an isolated system either increase or, in the limit, where the system has reached an equilibrium state, remain constant. For a closed (but not isolated) system it requires that any entropy decrease in either the system or its surroundings be more than compensated by an entropy increase in the other part or that in the limit, where the process is reversible, the total entropy of the system plus its surroundings be constant. The fundamental thermodynamic properties that arise in connection with the first and second laws of thermodynamics are internal energy and entropy. These properties, together with the two laws for which they are essential, apply to all types of systems. However, different types of systems are characterized by different sets of measurable coordinates or variables. The type of system most commonly
(4-4)
t
where P is the absolute pressure and V is the total volume of the system. This equation follows directly from the definition of mechanical work. POSTULATE 5 The macroscopic properties of homogeneous PVT systems at internal equilibrium can be expressed as functions of temperature, pressure, and composition only. This postulate imposes an idealization, and is the basis for all subsequent property relations for PVT systems. The PVT system serves as a satisfactory model in an enormous number of practical applications. In accepting this model one assumes that the effects of fields (e.g., electric, magnetic, or gravitational) are negligible and that surface and viscous-shear effects are unimportant. Temperature, pressure, and composition are thermodynamic coordinates representing conditions imposed upon or exhibited by the system, and the functional dependence of the thermodynamic properties on these conditions is determined by experiment. This is quite direct for molar or specific volume V, which can be measured, and leads immediately to the conclusion that there exists an equation of state relating molar volume to temperature, pressure, and composition for any particular homogeneous PVT system. The equation of state is a primary tool in applications of thermodynamics. Postulate 5 affirms that the other molar or specific thermodynamic properties of PVT systems, such as internal energy U and entropy S, are also functions of temperature, pressure, and composition. These molar or unit-mass properties, represented by the plain symbols V, U, and S, are independent of system size and are called intensive. Temperature, pressure, and the composition variables, such as mole fraction, are also intensive. Total-system properties (V t, U t, St ) do depend on system size, and are extensive. For a system containing n moles of fluid, M t = nM, where M is a molar property. Applications of the thermodynamic postulates necessarily involve the abstract quantities internal energy and entropy. The solution of any problem in applied thermodynamics is therefore found through these quantities.
VARIABLES, DEFINITIONS, AND RELATIONSHIPS Consider a single-phase closed system in which there are no chemical reactions. Under these restrictions the composition is fixed. If such a system undergoes a differential, reversible process, then by Eq. (4-1) dU t = dQrev + dWrev
where i is an index identifying the chemical species present. When U, S, and V represent specific (unit-mass) properties, n is replaced by m. Equation (4-5) shows that for the single-phase, nonreacting, closed system specified, nU = u(nS, nV)
Substitution for dQrev and dWrev by Eqs. (4-3) and (4-4) gives dU t = T dS t − P dV t
Then
Although derived for a reversible process, this equation relates properties only and is valid for any change between equilibrium states in a closed system. It may equally well be written d(nU) = T d(nS) − P d(nV)
(4-5)
where n is the number of moles of fluid in the system and is constant for the special case of a closed, nonreacting system. Note that n n1 + n2 + n3 + ⋅⋅⋅ = ni i
∂(nU) d(nU) = ᎏ ∂(nS)
nV,n
∂(nU) d(nS) + ᎏ ∂(nV)
d(nV) nS,n
where the subscript n indicates that all mole numbers ni (and hence n) are held constant. Comparison with Eq. (4-5) shows that ∂(nU) =T (4-6) ᎏ ∂(nS) nV,n
∂(nU) ᎏ ∂(nV)
= −P nS,n
(4-7)
VARIABLES, DEFINITIONS, AND RELATIONSHIPS Consider now an open system consisting of a single phase and assume that nU = (nS, nV, n1, n2, n3, . . .) Then ∂(nU) ∂(nU) ∂(nU) d(nS) + ᎏ d(nV) + ᎏ dni d(nU) = ᎏ ∂(nS) nV,n ∂(nV) nS,n ∂ni nS,nV,n i where the summation is over all species present in the system and subscript nj indicates that all mole numbers are held constant except the ith. Let ∂(nU) µi ᎏ ∂ni nS,nV,nj
j
Together with Eqs. (4-6) and (4-7), this definition allows elimination of all the partial differential coefficients from the preceding equation: d(nU) = T d(nS) − P d(nV) + µi dni
(4-8)
i
Equation (4-8) is the fundamental property relation for singlephase PVT systems, from which all other equations connecting properties of such systems are derived. The quantity µi is called the chemical potential of species i, and it plays a vital role in the thermodynamics of phase and chemical equilibria. Additional property relations follow directly from Eq. (4-8). Since ni = xin, where xi is the mole fraction of species i, this equation may be rewritten: d(nU) − T d(nS) + P d(nV) − µi d(xin) = 0 i
Upon expansion of the differentials and collection of like terms, this becomes
dU − T dS + P dV − µi dxi n + U − TS + PV − xiµi dn = 0 i
i
Since n and dn are independent and arbitrary, the terms in brackets must separately be zero. Then dU = T dS − P dV + µi dxi
(4-9)
i
U = TS − PV + xiµi
(4-10)
i
Equations (4-8) and (4-9) are similar, but there is an important difference. Equation (4-8) applies to a system of n moles where n may vary; whereas Eq. (4-9) applies to a system in which n is unity and invariant. Thus Eq. (4-9) is subject to the constraint that i xi = 1 or that i dxi = 0. In this equation the xi are not independent variables, whereas the ni in Eq. (4-8) are. Equation (4-10) dictates the possible combinations of terms that may be defined as additional primary functions. Those in common use are: Enthalpy Helmholtz energy Gibbs energy
H U + PV A U − TS G U + PV − TS = H − TS
(4-11) (4-12) (4-13)
Additional thermodynamic properties are related to these and arise by arbitrary definition. Multiplication of Eq. (4-11) by n and differentiation yields the general expression: d(nH) = d(nU) + P d(nV) + nV dP d(nH) = T d(nS) + nV dP + µi dni
and so on, as a function of a particular set of independent variables; these are the canonical variables for the property. The choice of which equation to use in a particular application is dictated by convenience. However, the Gibbs energy G is special, because of its unique functional relation to T, P, and the ni, which are the variables of primary interest in chemical processing. A similar set of equations is developed from Eq. (4-9). This set also follows from the preceding set when n = 1 and ni = xi. The two sets are related exactly as Eq. (4-8) is related to Eq. (4-9). The equations written for n = 1 are, of course, less general. Furthermore, the interdependence of the xi precludes those mathematical operations which depend on independence of these variables. CONSTANT-COMPOSITION SYSTEMS For 1 mole of a homogeneous fluid of constant composition Eqs. (4-8) and (4-14) through (4-16) simplify to: dU = T dS − P dV dH = T dS + V dP dA = −S dT − P dV dG = −S dT + V dP Implicit in these are the following: ∂U ∂H T= ᎏ = ᎏ ∂S V ∂S
∂U ∂A −P = ᎏ = ᎏ ∂V ∂V ∂H ∂G V = ᎏ = ᎏ ∂P ∂P
(4-14)
i
The total differentials of nA and nG are obtained similarly: d(nA) = −nS dT − P d(nV) + µi dni
(4-15)
d(nG) = −nS dT + nV dP + µi dni
(4-16)
i
i
Equations (4-8) and (4-14) through (4-16) are equivalent forms of the fundamental property relation. Each expresses a property nU, nH,
(4-17) (4-18) (4-19) (4-20)
(4-21)
P
S
T
S
T
(4-22) (4-23)
∂A ∂G (4-24) −S = ᎏ = ᎏ ∂T V ∂T P In addition, the common Maxwell equations result from application of the reciprocity relation for exact differentials: ∂T ∂P (4-25) ᎏ =− ᎏ ∂V S ∂S V
∂T ∂V = ᎏ ᎏ ∂P ∂S S
∂P ∂S = ᎏ ᎏ ∂T ∂V V
(4-26) P
(4-27) T
∂V ∂S = − ᎏ ᎏ ∂T ∂P P
(4-28)
T
In all these equations the partial derivatives are taken with composition held constant. Enthalpy and Entropy as Functions of T and P At constant composition the molar thermodynamic properties are functions of temperature and pressure (Postulate 5). Thus ∂H ∂H dH = ᎏ dT + ᎏ dP (4-29) ∂T P ∂P T
∂S ∂S (4-30) dS = ᎏ dT + ᎏ dP ∂T P ∂P T The obvious next step is to eliminate the partial-differential coefficients in favor of measurable quantities. The heat capacity at constant pressure is defined for this purpose: ∂H CP ᎏ (4-31) ∂T P It is a property of the material and a function of temperature, pressure, and composition. Equation (4-18) may first be divided by dT and restricted to constant pressure, and then be divided by dP and restricted to constant temperature, yielding the two equations:
Substitution for d(nU) by Eq. (4-8) reduces this result to:
4-5
4-6
THERMODYNAMICS ∂H
∂S = T ᎏ ᎏ ∂T ∂T P
P
∂H ∂S ᎏ =T ᎏ +V ∂P T ∂P T In view of Eq. (4-31), the first of these becomes ∂S CP (4-32) ᎏ =ᎏ ∂T P T and in view of Eq. (4-28), the second becomes ∂H ∂V (4-33) ᎏ =V−T ᎏ ∂P T ∂T P Combination of Eqs. (4-29), (4-31), and (4-33) gives ∂V dH = CP dT + V − T ᎏ dP (4-34) ∂T P and in combination Eqs. (4-30), (4-32), and (4-28) yield CP ∂V dS = ᎏ dT − ᎏ dP (4-35) T ∂T P Equations (4-34) and (4-35) are general expressions for the enthalpy and entropy of homogeneous fluids at constant composition as functions of T and P. The coefficients of dT and dP are expressed in terms of measurable quantities. Internal Energy and Entropy as Functions of T and V Because V is related to T and P through an equation of state, V rather than P can serve as an independent variable. In this case the internal energy and entropy are the properties of choice; whence ∂U ∂U dU = ᎏ dT + ᎏ dV (4-36) ∂T V ∂V T
∂S ∂S (4-37) dS = ᎏ dT + ᎏ dV ∂T V ∂V T The procedure now is analogous to that of the preceding section. Define the heat capacity at constant volume by ∂U CV ᎏ (4-38) ∂T V It is a property of the material and a function of temperature, pressure, and composition. Two relations follow immediately from Eq. (4-17): ∂U ∂S ᎏ =T ᎏ ∂T V ∂T V
∂U ∂S = T ᎏ − P ᎏ ∂V ∂V T
T
As a result of Eq. (4-38) the first of these becomes CV ∂S (4-39) ᎏ =ᎏ ∂T V T and as a result of Eq. (4-27), the second becomes ∂U ∂P (4-40) ᎏ =T ᎏ −P ∂V T ∂T V Combination of Eqs. (4-36), (4-38), and (4-40) gives ∂P dU = CV dT + T ᎏ − P dV (4-41) ∂T V and Eqs. (4-37), (4-39), and (4-27) together yield ∂P CV dS = ᎏ dT + ᎏ dV (4-42) T ∂T V Equations (4-41) and (4-42) are general expressions for the internal energy and entropy of homogeneous fluids at constant composition as functions of temperature and molar volume. The coefficients of dT and dV are expressed in terms of measurable quantities.
Heat-Capacity Relations In Eqs. (4-34) and (4-41) both dH and dU are exact differentials, and application of the reciprocity relation leads to ∂CP ∂2V (4-43) ᎏ = −T ᎏ2 ∂P T ∂T P
∂CV
∂2P
= T ᎏ ᎏ ∂V ∂T 2
T
(4-44)
V
Thus, the pressure or volume dependence of the heat capacities may be determined from PVT data. The temperature dependence of the heat capacities is, however, determined empirically and is often given by equations such as CP = α + βT + γ T 2 Equations (4-35) and (4-42) both provide expressions for dS, which must be equal for the same change of state. Equating them and solving for dT gives T ∂V T ∂P dT = ᎏ ᎏ dP + ᎏ ᎏ dV CP − CV ∂T P CP − CV ∂T V However, at constant composition T = T(P,V), and ∂T ∂T dT = ᎏ dP + ᎏ dV ∂P V ∂V P Equating coefficients of either dP or dV in these two expressions for dT gives ∂V ∂P CP − CV = T ᎏ ᎏ (4-45) ∂T P ∂T V Thus the difference between the two heat capacities may be determined from PVT data. Division of Eq. (4-32) by Eq. (4-39) yields the ratio of these heat capacities: CP (∂S/∂T)P (∂S/∂V)P(∂V/∂T)P ᎏ = ᎏ = ᎏᎏ CV (∂S/∂T)V (∂S/∂P)V(∂P/∂T)V
Replacement of each of the four partial derivatives through the appropriate Maxwell relation gives finally C ∂V ∂P γ ᎏP = ᎏ (4-46) ᎏ CV ∂P T ∂V S where γ is the symbol conventionally used to represent the heatcapacity ratio. The Ideal Gas The simplest equation of state is the ideal gas equation: PV = RT
where R is a universal constant, values of which are given in Table 1-9. The following partial derivatives are obtained from the ideal gas equation: ∂P R P ∂2P ᎏ =ᎏ=ᎏ ᎏ2 = 0 ∂T V V T ∂T V
∂V R V ∂V =ᎏ=ᎏ =0 ᎏ ᎏ ∂T P T ∂T 2
2
P
P
∂P P ᎏ =−ᎏ ∂V T V The general equations for constant-composition fluids derived in the preceding subsections reduce to very simple forms when the relations for an ideal gas are substituted into them:
∂U ∂H = ᎏ =0 ᎏ ∂V ∂P T
T
∂S R ∂S R =−ᎏ =ᎏ ᎏ ᎏ ∂P P ∂V V T
dU = CV dT
T
VARIABLES, DEFINITIONS, AND RELATIONSHIPS dH = CP dT CV R dS = ᎏ dT + ᎏ dV T V
Combining these expressions with Eq. (4-48) and collecting like terms gives ∂M ∂M dM − ᎏ dT − ᎏ dP − Mi dxi n + M − Mi xi dn = 0 ∂T P,x ∂P T,x i i
CP R dS = ᎏ dT − ᎏ dP T P ∂CV
∂CP
∂ ln P C γ ᎏP = − ᎏ CV ∂ ln V
and
SYSTEMS OF VARIABLE COMPOSITION The composition of a system may vary because the system is open or because of chemical reactions even in a closed system. The equations developed here apply regardless of the cause of composition changes. Partial Molar Properties Consider a homogeneous fluid solution comprised of any number of chemical species. For such a PVT system let the symbol M represent the molar (or unit-mass) value of any extensive thermodynamic property of the solution, where M may stand in turn for U, H, S, and so on. A total-system property is then nM, where n = ini and i is the index identifying chemical species. One might expect the solution property M to be related solely to the properties Mi of the pure chemical species which comprise the solution. However, no such generally valid relation is known, and the connection must be established experimentally for every specific system. Although the chemical species which make up a solution do not in fact have separate properties of their own, a solution property may be arbitrarily apportioned among the individual species. Once an apportioning recipe is adopted, then the assigned property values are quite logically treated as though they were indeed properties of the species in solution, and reasoning on this basis leads to valid conclusions. For a homogeneous PVT system, Postulate 5 requires that nM = (T, P, n1, n2, n3 , . . .) The total differential of nM is therefore
P,n
∂(nM) dT + ᎏ ∂P
T,n
∂(nM) dP + ᎏ ∂ni i
dni T,P,nj
where subscript n indicates that all mole numbers ni are held constant, and subscript nj signifies that all mole numbers are held constant except the ith. This equation may also be written ∂M d(nM) = n ᎏ ∂T
P,x
∂M dT + n ᎏ ∂P
T,x
∂(nM) dP + ᎏ ∂ni i
dni T,P,nj
where subscript x indicates that all mole fractions are held constant. The derivatives in the summation are called partial molar properties M i; by definition, ∂(nM) M (4-47) i ᎏ ∂ni T,P,nj
The basis for calculation of partial properties from solution properties is provided by this equation. Moreover, the preceding equation becomes ∂M ∂M d(nM) = n ᎏ dT + n ᎏ dP + M (4-48) i dni ∂T P,x ∂P T,x i
Important equations follow from this result through the relations: d(nM) = n dM + M dn dni = d(xin) = xi dn + n dxi
and arbitrary, the terms in brackets ∂M
ᎏ ∂P
T,x
dP + M i dxi
(4-49)
i
M = xiM i
(4-50)
i
S
These equations clearly show that for an ideal gas U, H, CP, and CV are functions of temperature only and are independent of P and V. The entropy of an ideal gas, however, is a function of both T and P or of both T and V.
∂(nM) d(nM) = ᎏ ∂T
T
CP − CV = R
Since n and dn are independent must separately be zero; whence ∂M dM = ᎏ dT + ∂T P,x
= ᎏ =0 ᎏ ∂V ∂P T
4-7
Equation (4-49) is merely a special case of Eq. (4-48); however, Eq. (4-50) is a vital new relation. Known as the summability equation, it provides for the calculation of solution properties from partial properties. Thus, a solution property apportioned according to the recipe of Eq. (4-47) may be recovered simply by adding the properties attributed to the individual species, each weighted by its mole fraction in solution. The equations for partial molar properties are also valid for partial specific properties, in which case m replaces n and the xi are mass fractions. Equation (4-47) applied to the definitions of Eqs. (4-11) through (4-13) yields the partial-property relations: Hi = Ui + PV i A Ui − TS i i = G Hi − TS i i = Pertinent examples on partial molar properties are presented in Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., Sec. 10.3, McGraw-Hill, New York, 1996). Gibbs/Duhem Equation Differentiation of Eq. (4-50) yields dM = xi dM i + M i dxi i
i
Since this equation and Eq. (4-49) are both valid in general, their right-hand sides can be equated, yielding ∂M ∂M (4-51) ᎏ dT + ᎏ dP − xi dM i = 0 ∂T P,x ∂P T,x i
This general result, the Gibbs/Duhem equation, imposes a constraint on how the partial molar properties of any phase may vary with temperature, pressure, and composition. For the special case where T and P are constant: =0 x dM i
(constant T, P)
i
(4-52)
i
Symbol M may represent the molar value of any extensive thermodynamic property; for example, V, U, H, S, or G. When M H, the derivatives (∂H/∂T)P and (∂H/∂P)T are given by Eqs. (4-31) and (4-33). Equations (4-49), (4-50), and (4-51) then become ∂V dH = CP dT + V − T ᎏ dP + H (4-53) i dxi ∂T P,x i
H = xiH i
(4-54)
i
∂V CP dT + V − T ᎏ dP − xi dH (4-55) i = 0 ∂T P,x i Similar equations are readily derived when M takes on other identities. Equation (4-47), which defines a partial molar property, provides a general means by which partial property values may be determined. However, for a binary solution an alternative method is useful. Equation (4-50) for a binary solution is
M = x1 M 1 + x2 M 2
(4-56)
Moreover, the Gibbs/Duhem equation for a solution at given T and P, Eq. (4-52), becomes x1 dM 1 + x2 dM 2 = 0
(4-57)
4-8
THERMODYNAMICS
These two equations can be combined to give dM M 1 = M + x2 ᎏ dx1
The following equation is a mathematical identity:
dM M2 = M − x1 ᎏ dx1
nG 1 nG d ᎏ ᎏ d(nG) − ᎏ2 dT RT RT RT
(4-58b)
Substitution for d(nG) by Eq. (4-16) and for G by H − TS (Eq. [4-13]) gives, after algebraic reduction,
Thus for a binary solution, the partial properties are given directly as functions of composition for given T and P. For multicomponent solutions such calculations are complex, and direct use of Eq. (4-47) is appropriate. Partial Molar Gibbs Energy Implicit in Eq. (4-16) is the relation ∂(nG) µi = ᎏ ∂ni T,P,nj
In view of Eq. (4-47), the chemical potential and the partial molar Gibbs energy are therefore identical: µi = Gi
(4-59)
The reciprocity relation for an exact differential applied to Eq. (416) produces not only the Maxwell relation, Eq. (4-28), but also two other useful equations: ∂µi ∂(nV) = Vi (4-60) ᎏ = ᎏ ∂P T,n ∂ni T,P,nj
∂µ ∂(nS) = − ᎏ ᎏ ∂T ∂n i
P,n
i
T,P,nj
= −S i
(4-61)
In a solution of constant composition, µi = µ(T,P); whence ∂µi ∂µi dµi dG i = ᎏ dT + ᎏ dP ∂T P,n ∂P T,n
or
dG i = −S i dT + V i dP
(4-62)
Comparison with Eq. (4-20) provides an example of the parallelism that exists between the equations for a constant-composition solution and those for the corresponding partial properties. This parallelism exists whenever the solution properties in the parent equation are related linearly (in the algebraic sense). Thus, in view of Eqs. (4-17), (4-18), and (4-19): dU i = T dS i − P dV i Vi dP dH i = T dS i + dA i = −S i dT − P dV i
(4-58a)
(4-63) (4-64) (4-65)
nG nV nH µi d ᎏ = ᎏ dP − ᎏ2 dT + ᎏ dni RT RT RT RT i
(4-66)
Equation (4-66) is a useful alternative to the fundamental property relation given by Eq. (4-16). All terms in this equation have the units of moles; moreover, the enthalpy rather than the entropy appears on the right-hand side. The Ideal Gas State and the Compressibility Factor The simplest equation of state for a PVT system is the ideal gas equation: PV ig = RT ig
where V is the ideal-gas–state molar volume. Similarly, H ig, S ig, and G ig are ideal gas–state values; that is, the molar enthalpy, entropy, and Gibbs energy values that a PVT system would have were the ideal gas equation the correct equation of state. These quantities provide reference values to which actual values may be compared. For example, the compressibility factor Z compares the true molar volume to the ideal gas molar volume as a ratio: V V PV Z=ᎏ =ᎏ=ᎏ V ig RT/P RT Generalized correlations for the compressibility factor are treated in Sec. 2. Residual Properties These quantities compare true and ideal gas properties through differences: MR M − M ig
(4-67)
where M is the molar value of an extensive thermodynamic property of a fluid in its actual state and Mig is the corresponding value for the ideal gas state of the fluid at the same T, P, and composition. Residual properties depend on interactions between molecules and not on characteristics of individual molecules. Since the ideal gas state presumes the absence of molecular interactions, residual properties reflect deviations from ideality. Most commonly used of the residual properties are: V R V − V ig HR H − H ig SR S − S ig GR G − G ig
Residual volume Residual enthalpy Residual entropy Residual Gibbs energy
Note that these equations hold only for species in a constantcomposition solution.
SOLUTION THERMODYNAMICS M iig(T, P) = Miig(T, pi)
IDEAL GAS MIXTURES An ideal gas is a model gas comprising imaginary molecules of zero volume that do not interact. Each chemical species in an ideal gas mixture therefore has its own private properties, uninfluenced by the presence of other species. The partial pressure of species i in a gas mixture is defined as pi = xiP
The partial molar property, other than the volume, of a constituent species in an ideal gas mixture is equal to the corresponding molar property of the species as a pure ideal gas at the mixture temperature but at a pressure equal to its partial pressure in the mixture.
This is expressed mathematically for generic partial property M iig by the equation
(4-68)
Miig = Miig ig
where Mi is evaluated at the mixture T and P. Thus, for the enthalpy,
(i = 1, 2, . . . , N)
where xi is the mole fraction of species i. The sum of the partial pressures clearly equals the total pressure. Gibbs’ theorem for a mixture of ideal gases may be stated as follows:
(M ≠ V)
For those properties of an ideal gas that are independent of P, for example, U, H, and CP, this becomes simply
Hiig = Hiig The entropy of an ideal gas does depend on pressure: dSiig = −R d ln P
(constant T )
Integration from pi to P gives P P Siig(T, P) − Siig(T, pi) = −R ln ᎏ = −R ln ᎏ = R ln xi pi xi P Whence
Siig(T, pi) = Siig(T, P) − R ln xi
(4-69)
SOLUTION THERMODYNAMICS Substituting this result into Eq. (4-68) written for the entropy gives Sl = S − R ln xi ig
ig i
(4-70)
where Siig is evaluated at the mixture T and P. For the Gibbs energy of an ideal gas mixture, Gig = Hig − TSig; the parallel relation for partial properties is Giig = Hiig − TSiig In combination with Eqs. (4-69) and (4-70), this becomes G Hiig − TSiig + RT ln xi iig = or
µiig Giig = Giig + RT ln xi
(4-72)
where Γi(T), the integration constant for a given temperature, is a function of temperature only. Equation (4-71) now becomes µigi = Γi(T) + RT ln xi P
(4-73)
FUGACITY AND FUGACITY COEFFICIENT The chemical potential µi plays a vital role in both phase and chemicalreaction equilibria. However, the chemical potential exhibits certain unfortunate characteristics which discourage its use in the solution of practical problems. The Gibbs energy, and hence µi, is defined in relation to the internal energy and entropy, both primitive quantities for which absolute values are unknown. Moreover, µi approaches negative infinity when either P or xi approaches zero. While these characteristics do not preclude the use of chemical potentials, the application of equilibrium criteria is facilitated by introduction of the fugacity, a quantity that takes the place of µi but which does not exhibit its less desirable characteristics. The origin of the fugacity concept resides in Eq. (4-72), an equation valid only for pure species i in the ideal gas state. For a real fluid, an analogous equation is written: Gi Γi(T) + RT ln f i
(4-74)
in which a new property f i replaces the pressure P. This equation serves as a partial definition of the fugacity f i. Subtraction of Eq. (4-72) from Eq. (4-74), both written for the same temperature and pressure, gives f Gi − Giig = RT ln ᎏi P According to the definition of Eq. (4-67), Gi − Giig is the residual Gibbs energy, GiR. The dimensionless ratio fi /P is another new property called the fugacity coefficient φi. Thus, (4-75) G = RT ln φi fi where φi ᎏ (4-76) P The definition of fugacity is completed by setting the ideal-gas–state fugacity of pure species i equal to its pressure: R i
f iig = P Thus, for the special case of an ideal gas, GiR = 0, φi = 1, and Eq. (4-72) is recovered from Eq. (4-74). The definition of the fugacity of a species in solution is parallel to the definition of the pure-species fugacity. An equation analogous to the ideal gas expression, Eq. (4-73), is written for species i in a fluid mixture: µ Γ (T) + RT ln fˆ (4-77) i
i
i in solution. Since it is not a partial molar property, it is identified by a circumflex rather than an overbar. Subtracting Eq. (4-73) from Eq. (4-77), both written for the same temperature, pressure, and composition, yields fˆi µi − µiig = RT ln ᎏ xiP Analogous to the defining equation for the residual Gibbs energy of a mixture, GR G − Gig, is the definition of a partial molar residual Gibbs energy: G Gi − Gigi = µi − µigi iR
(4-71)
Elimination of Giig from this equation is accomplished by Eq. (4-20), written for pure species i as: RT dGiig = Viig dP = ᎏ dP = RT d ln P (constant T) P Integration gives Giig = Γi(T) + RT ln P
4-9
i
where the partial pressure xi P is replaced by fˆi , the fugacity of species
GiR = RT ln φˆ i (4-78) ˆ f i where by definition φˆ i ᎏ (4-79) xiP The dimensionless ratio φˆ i is called the fugacity coefficient of species i in solution. Eq. (4-78) is the analog of Eq. (4-75), which relates φi to GiR. For an ideal gas, GiR is necessarily 0; therefore φˆ iig = 1, and fˆ ig = x P Therefore
i
i
Thus, the fugacity of species i in an ideal gas mixture is equal to its partial pressure. Pertinent examples are given in Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., Secs. 10.5–10.7, McGraw-Hill, New York, 1996). FUNDAMENTAL RESIDUAL-PROPERTY RELATION In view of Eq. (4-59), the fundamental property relation given by Eq. (4-66) may be written Gi nH nG nV d ᎏ = ᎏ dP − ᎏ2 dT + ᎏ dni (4-80) RT RT RT RT i
This equation is general, and may be written for the special case of an ideal gas: G nGig nVig nHig iig d ᎏ = ᎏ dP − ᎏ2 dT + ᎏ dni RT RT RT RT i
Subtraction of this equation from Eq. (4-80) gives GiR nGR nVR nHR d ᎏ = ᎏ dP − ᎏ2 dT + ᎏ dni RT RT RT RT i
(4-81)
where the definitions GR G − Gig and G iR G i − G iig have been imposed. Equation (4-81) is the fundamental residual-property relation. An alternative form follows by introduction of the fugacity coefficient as given by Eq. (4-78): nGR nVR nHR d ᎏ = ᎏ dP − ᎏ2 dT + ln φˆ i dni (4-82) RT RT RT i
These equations are of such generality that for practical application they are used only in restricted forms. Division of Eq. (4-82) by dP and restriction to constant T and composition leads to: VR ∂(GR/RT) ᎏ = ᎏᎏ RT ∂P
(4-83)
T,x
Similarly, division by dT and restriction to constant P and composition gives HR ∂(GR/RT) ᎏ = −T ᎏᎏ RT ∂T
(4-84)
P,x
Also implicit in Eq. (4-82) is the relation ∂(nGR/RT) ln φˆ i = ᎏᎏ (4-85) T,P,nj ∂ni This equation demonstrates that ln φˆ i is a partial property with respect to GR/RT. The partial-property analogs of Eqs. (4-83) and (4-84) are therefore:
4-10
THERMODYNAMICS ∂ ln φ ᎏ ∂P ∂ ln φˆ ᎏ ∂T ˆi
ViR =ᎏ T,x RT
(4-86)
H iR = − ᎏ2 (4-87) P,x RT The partial-property relationship of ln φˆ i to GR/RT also means that the summability relation applies; thus GR (4-88) ᎏ = xi ln φˆ i RT i i
THE IDEAL SOLUTION The ideal gas is a useful model of the behavior of gases and serves as a standard to which real gas behavior can be compared. This is formalized by the introduction of residual properties. Another useful model is the ideal solution, which serves as a standard to which real solution behavior can be compared. This is formalized by introduction of excess properties. The partial molar Gibbs energy of species i in an ideal gas mixture is given by Eq. (4-71). This equation takes on new meaning when Giig, the Gibbs energy of pure species i in the ideal gas state, is replaced by Gi, the Gibbs energy of pure species i as it actually exists at the mixture T and P and in the same physical state (real gas, liquid, or solid) as the mixture. It then becomes applicable to species in real solutions; indeed, to liquids and solids as well as to gases. The ideal solution is therefore defined as one for which G iid Gi + RT ln xi
(4-89)
where superscript id denotes an ideal-solution property. This equation is the basis for development of expressions for all other thermodynamic properties of an ideal solution. Equations (4-60) and (4-61), applied to an ideal solution with µi replaced by Gi, can be written id id ∂G ∂G i i V and Siid = − ᎏ iid = ᎏ ∂P T,x ∂T P,x Appropriate differentiation of Eq. (4-89) in combination with these relations and Eqs. (4-23) and (4-24) yields
id i
Thus, the fugacity coefficient of species i in an ideal solution is equal to the fugacity coefficient of pure species i in the same physical state as the solution and at the same T and P. Ideal solution behavior is often approximated by solutions comprised of molecules not too different in size and of the same chemical nature. Thus, a mixture of isomers conforms very closely to ideal solution behavior. So do mixtures of adjacent members of a homologous series. FUNDAMENTAL EXCESS-PROPERTY RELATION The residual Gibbs energy and the fugacity coefficient are useful where experimental PVT data can be adequately correlated by equations of state. Indeed, if convenient treatment of all fluids by means of equations of state were possible, the thermodynamic-property relations already presented would suffice. However, liquid solutions are often more easily dealt with through properties that measure their deviations from ideal solution behavior, not from ideal gas behavior. Thus, the mathematical formalism of excess properties is analogous to that of the residual properties. If M represents the molar (or unit-mass) value of any extensive thermodynamic property (e.g., V, U, H, S, G, and so on), then an excess property M E is defined as the difference between the actual property value of a solution and the value it would have as an ideal solution at the same temperature, pressure, and composition. Thus, M E M − Mid
(4-99)
This definition is analogous to the definition of a residual property as given by Eq. (4-67). However, excess properties have no meaning for pure species, whereas residual properties exist for pure species as well as for mixtures. In addition, analogous to Eq. (4-99) is the partialproperty relation, MiE = Mi − Miid (4-100)
Gid = xiGi + RT xi ln xi
(4-93)
Vid = xiVi
(4-94)
where M iE is a partial excess property. The fundamental excessproperty relation is derived in exactly the same way as the fundamental residual-property relation and leads to analogous results. Equation (4-80), written for the special case of an ideal solution, is subtracted from Eq. (4-80) itself, yielding: GiE nGE nV E nH E d ᎏ = ᎏ dP − ᎏ2 dT + ᎏ dni (4-101) RT RT RT RT i This is the fundamental excess-property relation, analogous to Eq. (4-81), the fundamental residual-property relation. The excess Gibbs energy is of particular interest. Equation (4-77) may be written: G = Γ (T ) + RT ln fˆ
S = xiSi − R xi ln xi
(4-95)
In accord with Eq. (4-97) for an ideal solution, this becomes
Hid = xiHi
(4-96)
Viid = Vi
(4-90)
S iid = Si − R ln xi
(4-91)
G + TS Since H i = , substitutions by Eqs. (4-89) and (4-91) yield id
same T and P. Division of both sides of Eq. (4-97) by xiP and substitution of φˆ iid for fˆiid/xiP (Eq. [4-79]) and of φi for fi /P (Eq. [4-76]) gives an alternative form: φˆ iid = φi (4-98)
id i
Hiid = Hi
(4-92)
The summability relation, Eq. (4-50), written for the special case of an ideal solution, may be applied to Eqs. (4-89) through (4-92): i
i
i
id
i
i
i
i
i
i
When this equation and Eq. (4-74) are combined with Eq. (4-89), Γi(T) is eliminated, and the resulting expression reduces to fˆiid = xi fi (4-97) This equation, known as the Lewis/Randall rule, applies to each species in an ideal solution at all conditions of T, P, and composition. It shows that the fugacity of each species in an ideal solution is proportional to its mole fraction; the proportionality constant is the fugacity of pure species i in the same physical state as the solution and at the
i
i
G iid = Γi (T ) + RT ln xi fi
i
A simple equation for the fugacity of a species in an ideal solution follows from Eq. (4-89). Written for the special case of species i in an ideal solution, Eq. (4-77) becomes µ id G id = Γ (T) + RT ln fˆid i
By difference
fˆi G Giid = RT ln ᎏ i − xi fi The left-hand side is the partial excess Gibbs energy GiE; the dimensionless ratio fˆi/xifi appearing on the right is called the activity coefficient of species i in solution, and is given the symbol γi. Thus, by definition, fˆi γi ᎏ (4-102) xi fi and
G iE = RT ln γi
(4-103)
Comparison with Eq. (4-78) shows that Eq. (4-103) relates γi to G iE exactly as Eq. (4-78) relates φˆ i to GiR. For an ideal solution,G iE =0, and therefore γi = 1.
SOLUTION THERMODYNAMICS VE ∂(GE/RT) ᎏ = ᎏᎏ RT ∂P
An alternative form of Eq. (4-101) follows by introduction of the activity coefficient through Eq. (4-103):
E
E
E
nH nG nV d ᎏ = ᎏ dP − ᎏ2 dT + ln γi dni RT RT RT i
(4-117)
T,x
HE ∂(GE/RT) ᎏ = −T ᎏᎏ RT ∂T
(4-104)
4-11
(4-118)
P,x
and from Eq. (4-104) SUMMARY OF FUNDAMENTAL PROPERTY RELATIONS For convenience, the three other fundamental property relations, Eqs. (4-16), (4-80), and (4-82), expressing the Gibbs energy and related properties as functions of T, P, and the ni, are collected here: d(nG) = nV dP − nS dT + µi dni
(4-16)
(4-80)
i
G nG nV nH i d ᎏ = ᎏ dP − ᎏ2 dT + ᎏ dni RT RT RT RT i
R
R
R
nG nV nH d ᎏ = ᎏ dP − ᎏ2 dT + ln φˆ i dni RT RT RT i
(4-82)
These equations and Eq. (4-104) may also be written for the special case of 1 mole of solution by setting n = 1 and ni = xi. The xi are then subject to the constraint that i xi = 1. If written for 1 mole of a constant-composition solution, they become: dG = V dP − S dT
(4-105)
(4-106)
(4-107)
(4-108)
G V H d ᎏ = ᎏ dP − ᎏ2 dT RT RT RT GR VR HR d ᎏ = ᎏ dP − ᎏ2 dT RT RT RT GE VE HE d ᎏ = ᎏ dP − ᎏ2 dT RT RT RT
These equations are, of course, valid as a special case for a pure species; in this event they are written with subscript i affixed to the appropriate symbols. The partial-property analogs of these equations are: dG Vi dP − Si dT i = dµi = µi G V H i i i d ᎏ = d ᎏ = ᎏ dP − ᎏ2 dT RT RT RT RT
G Vi H d ᎏ = d ln φˆ i = ᎏ dP − ᎏ2 dT RT RT RT R i
R
R i
H G ViE iE iE d ᎏ = d ln γi = ᎏ dP − ᎏ2 dT RT RT RT
(4-110)
∂ ln γi
ᎏ ∂T
H iE = − ᎏ2 (4-121) P,x RT Finally, an especially useful form of the Gibbs/Duhem equation follows from Eq. (4-116):
x d ln γ = 0 i
(constant T,P)
i
Since ln γi is a partial property with respect to GE/RT, the following form of the summability equation is valid: GE (4-123) ᎏ = xi ln γi RT i The analogy between equations derived from the fundamental residual- and excess-property relations is apparent. Whereas the fundamental residual-property relation derives its usefulness from its direct relation to equations of state, the excess-property formulation is useful because VE, HE, and γi are all experimentally accessible. Activity coefficients are found from vapor/liquid equilibrium data, and VE and HE values come from mixing experiments. PROPERTY CHANGES OF MIXING If M represents a molar thermodynamic property of a homogeneous fluid solution, then by definition, ∆M M − xi Mi
(4-124)
i
(4-111) (4-112)
where by definition
(4-113)
G V H i ᎏ dP − ᎏ2 dT = xi d ᎏ RT RT RT i
(4-114)
VR HR ᎏ dP − ᎏ2 dT = xi d ln φˆ i RT RT i
(4-115)
∆M = xi ∆ M i
(4-125)
Mi − Mi ∆ M i
(4-126)
i
All three quantities are for the same T, P, and physical state. Eq. (4-126) defines a partial molar property change of mixing, and Eq. (4-125) is the summability relation for these properties. Each of Eqs. (4-93) through (4-96) is an expression for an ideal solution property, and each may be combined with the defining equation for an excess property (Eq. [4-99]), yielding GE = G − xiGi − RT xi ln xi
(4-127)
V = V − xiVi
(4-128)
i
i
E
HE VE ᎏ dP − ᎏ2 dT = xi d ln γi RT RT i
(4-122)
i
where ∆M is the property change of mixing, and Mi is the molar property of pure species i at the T and P of the solution and in the same physical state (gas or liquid). The summability relation, Eq. (4-50), may be combined with Eq. (4-124) to give
i
(4-109)
Finally, a Gibbs/Duhem equation is associated with each fundamental property relation: V dP − S dT = xi dµi
∂(nGE/RT) ln γi = ᎏᎏ (4-119) T,P,nj ∂ni The last relation demonstrates that ln γi is a partial property with respect to GE/RT. The partial-property analogs of Eqs. (4-117) and (4-118) follow from Eq. (4-112): V ∂ ln γi iE (4-120) ᎏ =ᎏ ∂P T,x RT
(4-116)
This depository of equations stores an enormous amount of information. The equations themselves are so general that their direct application is seldom appropriate. However, by inspection one can write a vast array of relations valid for particular applications. For example, Eqs. (4-83) and (4-84) come directly from Eq. (4-107); Eqs. (4-86) and (4-87), from (4-111). Similarly, from Eq. (4-108),
i
SE = S − xiSi + R xi ln xi
(4-129)
HE = H − xiHi
(4-130)
i
i
i
In view of Eq. (4-124), these may be written G E = ∆G − RT xi ln xi i
(4-131)
4-12
THERMODYNAMICS V E = ∆V
(4-132)
S = ∆S + R xi ln xi E
H E = ∆H
i
(4-133) (4-134)
where ∆G, ∆V, ∆S, and ∆H are the Gibbs energy change of mixing, the volume change of mixing, the entropy change of mixing, and the enthalpy change of mixing. For an ideal solution, each excess property is zero, and for this special case ∆Gid = RT xi ln xi ∆Vid = 0
i
(4-135) (4-136)
∆S = −R xi ln xi id
(4-137)
i
(4-138) ∆Hid = 0 Property changes of mixing and excess properties are easily calculated one from the other. The most commonly encountered property changes of mixing are the volume change of mixing ∆V and the enthalpy change of mixing ∆H, commonly called the heat of mixing. These properties are directly measurable and are identical to the corresponding excess properties. Pertinent examples are given in Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., Sec. 11.4, McGraw-Hill, New York, 1996). BEHAVIOR OF BINARY LIQUID SOLUTIONS Property changes of mixing and excess properties find greatest application in the description of liquid mixtures at low reduced tempera-
tures, that is, at temperatures well below the critical temperature of each constituent species. The properties of interest to the chemical engineer are V E ( ∆V), H E ( ∆H), S E, ∆S, G E, and ∆G. The activity coefficient is also of special importance because of its application in phase-equilibrium calculations. The behavior of binary liquid solutions is clearly displayed by plots of M E, ∆M, and ln γi vs. x1 at constant T and P. The volume change of mixing (or excess volume) is the most easily measured of these quantities and is normally small. However, as illustrated by Fig. 4-1, it is subject to individualistic behavior, being sensitive to the effects of molecular size and shape and to differences in the nature and magnitude of intermolecular forces. The heat of mixing (excess enthalpy) and the excess Gibbs energy are also experimentally accessible, the heat of mixing by direct measurement and GE (or ln γi) indirectly as a product of the reduction of vapor/liquid equilibrium data. Knowledge of HE and GE allows calculation of SE by Eq. (4-13) written for excess properties, HE − GE SE = ᎏ T
(4-139)
with ∆S then given by Eq. (4-133). Figure 4-2 displays plots of ∆H, ∆S, and ∆G as functions of composition for 6 binary solutions at 50°C. The corresponding excess properties are shown in Fig. 4-3; the activity coefficients, derived from Eq. (4-119), appear in Fig. 4-4. The properties shown here are insensitive to pressure, and for practical purposes represent solution properties at 50°C (122°F) and low pressure (P ≈ 1 bar [14.5 psi]).
FIG. 4-1 Excess volumes at 25°C for liquid mixtures of cyclohexane(1) with some other C6 hydrocarbons.
SOLUTION THERMODYNAMICS
(a)
(b)
(c)
(d)
(e)
(f)
Property changes of mixing at 50°C for 6 binary liquid systems: (a) chloroform(1)/n-heptane(2); (b) acetone(1)/ methanol(2); (c) acetone(1)/chloroform(2); (d) ethanol(1)/n-heptane(2); (e) ethanol(1)/chloroform(2); ( f) ethanol(1)/water(2). FIG. 4-2
(a)
(b)
(c)
(d)
(e)
(f)
Excess properties at 50°C for 6 binary liquid systems: (a) chloroform(1)/n-heptane(2); (b) acetone(1)/methanol(2); (c) acetone(1)/chloroform(2); (d) ethanol(1)/n-heptane(2); (e) ethanol(1)/chloroform(2); ( f) ethanol(1)/water(2). FIG. 4-3
4-13
4-14
THERMODYNAMICS
(a)
(b)
(c)
(d)
(e)
(f)
Activity coefficients at 50°C for 6 binary liquid systems: (a) chloroform(1)/n-heptane(2); (b) acetone(1)/ methanol(2); (c) acetone(1)/chloroform(2); (d) ethanol(1)/n-heptane(2); (e) ethanol(1)/chloroform(2); ( f) ethanol(1)/ water(2).
FIG. 4-4
EVALUATION OF PROPERTIES RESIDUAL-PROPERTY FORMULATIONS The most satisfactory calculational procedure for thermodynamic properties of gases and vapors requires PVT data and ideal gas heat capacities. The primary equations are based on the concept of the ideal gas state and the definitions of residual enthalpy and residual entropy: H = Hig + HR
and
S = Sig + SR
The enthalpy and entropy are simple sums of the ideal gas and residual properties, which are evaluated separately. For the ideal gas state at constant composition, dH = C dT ig P
ig
dT dP dSig = CPig ᎏ − R ᎏ T P Integration from an initial ideal gas reference state at conditions T0 and P0 to the ideal gas state at T and P gives: Hig = H0ig +
C
ig P
dT
C
ig P
dT P ᎏ − R ln ᎏ T P0
T
T0
Sig = S0ig +
T
T0
H = H0ig +
C T
T0
ig P
dT + HR
(4-140)
T
T0
ig P
CPig = A + BT + CT 2 + DT −2
(4-142)
where A, B, C, and D are constants characteristic of the particular gas, ig and either C or D is 0. Evaluation of the integrals ∫ CP dT and ig ig ∫ (CP /T)dT is accomplished by substitution for CP , followed by formal integration. For temperature limits of T0 and T the results are conveniently expressed as follows: T B C D τ−1 ig CP dT = AT0(τ − 1) + ᎏ T02(τ2 − 1) + ᎏ T03 (τ3 − 1) + ᎏ ᎏ T0 2 3 T0 τ (4-143) ig T CP D τ+1 2 and ᎏ dT = A ln τ + BT0 + CT0 + ᎏ ᎏ (τ − 1) T0 T τ 2T02 2 (4-144) T where τᎏ T0
Substitution into the equations for H and S yields
C
dT P (4-141) ᎏ − R ln ᎏ + SR T P0 The reference state at T0 and P0 is arbitrarily selected, and the values assigned to H0ig and S0ig are also arbitrary. In practice, only changes in H and S are of interest, and the reference-state values ultimately cancel in their calculation. ig The ideal-gas–state heat capacity CP is a function of T but not of P. ig For a mixture, the heat capacity is simply the molar average i xi CP . ig Empirical equations giving the temperature dependence of CP are available for many pure gases, often taking the form S = S0ig +
EVALUATION OF PROPERTIES Equations (4-140) and (4-141) may sometimes be advantageously expressed in alternative form through use of mean heat capacities: H = H0ig + 〈CPig〉H(T − T0) + HR (4-145) T P ig R ig S = S0 + 〈CP 〉S ln ᎏ − R ln ᎏ + S (4-146) T0 P0 where 〈CPig〉H and 〈CPig〉S are mean heat capacities specific respectively to enthalpy and entropy calculations. They are given by the following equations: B C D 〈CPig〉H = A + ᎏ T0 (τ + 1) + ᎏ T02(τ 2 + τ + 1) + ᎏ2 (4-147) 2 3 τ T0
D 〈CPig〉S = A + BT0 + CT02 + ᎏ τ 2 T02
τ+1 τ−1 ᎏ ᎏ 2 ln τ
(4-148)
LIQUID/VAPOR PHASE TRANSITION When a differential amount of a pure liquid in equilibrium with its vapor in a piston-and-cylinder arrangement evaporates at constant temperature T and vapor pressure Pisat, Eq. (4-16) applied to the process reduces to d(niGi) = 0, whence ni dGi + Gi dni = 0
B ln P sat = A − ᎏ (4-152) T+C A principal advantage of this equation is that values of the constants A, B, and C are readily available for a large number of species. The accurate representation of vapor-pressure data over a wide temperature range requires an equation of greater complexity. The Wagner equation, one of the best, expresses the reduced vapor pressure as a function of reduced temperature: Aτ + Bτ 1.5 + Cτ 3 + Dτ 6 ln Prsat = ᎏᎏᎏ (4-153) 1−τ where here τ 1 − Tr and A, B, C, and D are constants. Values of the constants either for this equation or the Antoine equation are given for many species by Reid, Prausnitz, and Poling (The Properties of Gases and Liquids, 4th ed., App. A, McGraw-Hill, New York, 1987). LIQUID-PHASE PROPERTIES Given saturated-liquid enthalpies and entropies, the calculation of these properties for pure compressed liquids is accomplished by integration at constant temperature of Eqs. (4-34) and (4-35):
(4-149)
where Gil and Giv are the molar Gibbs energies of the individual phases. If the temperature of a two-phase system is changed and if the two phases continue to coexist in equilibrium, then the vapor pressure must also change in accord with its temperature dependence. Since Eq. (4-149) holds throughout this change, dGil = dGiv Substituting the expressions for dGil and dGiv given by Eq. (4-16) yields Vil dPisat − Sil dT = Viv dPisat − Siv dT which upon rearrangement becomes dPisat Siv − Sil ∆Silv =ᎏ ᎏ=ᎏ v l dT Vi − Vi ∆Vilv The entropy change ∆Silv and the volume change ∆Vilv are the changes which occur when a unit amount of a pure chemical species is transferred from phase l to phase v at constant temperature and pressure. Integration of Eq. (4-18) for this change yields the latent heat of phase transition: ∆ Hilv = T∆Silv Thus, ∆Si = ∆Hi /T, and substitution in the preceding equation gives lv
lv
dPisat ∆ Hilv (4-150) ᎏ=ᎏ dT T∆Vilv Known as the Clapeyron equation, this is an exact thermodynamic relation, providing a vital connection between the properties of the liquid and vapor phases. Its use presupposes knowledge of a suitable vapor pressure vs. temperature relation. Empirical in nature, such relations are approximated by the equation B ln P sat = A − ᎏ (4-151) T where A and B are constants for a given species. This equation gives a rough approximation of the vapor-pressure relation for its entire temperature range. Moreover, it is an excellent basis for interpolation between values that are reasonably spaced. The Antoine equation, which is more satisfactory for general use, has the form
−
P
Hi = Hisat +
Since the system is closed, dni = 0 and, therefore, dGi = 0; this requires the molar (or specific) Gibbs energy of the vapor to be identical with that of the liquid: Gil = Giv
4-15
sat
Vi(1 − βiT)dP
(4-154)
βiVi dP
(4-155)
Pi
Si = Sisat
P sat
Pi
where the volume expansivity of species i at temperature T is 1 ∂Vi βi ᎏ ᎏ (4-156) Vi ∂T P Since βi and Vi are weak functions of pressure for liquids, they are usually assumed constant at the values for the saturated liquid at temperature T.
PROPERTIES FROM PVT CORRELATIONS The empirical representation of the PVT surface for pure materials is treated later in this section. We first present general equations for evaluation of reduced properties from such representations. Equation (4-83), applied to a pure material, may be written GR VR d ᎏ = ᎏ dP (constant T) RT RT Integration from zero pressure to arbitrary pressure P gives P GR VR (constant T) ᎏ = ᎏ dP 0 RT RT where at the lower limit GR/RT is set equal to zero on the basis that the zero-pressure state is an ideal gas state. The residual volume is related directly to the compressibility factor: ZRT RT RT V R V − V ig = ᎏ − ᎏ = (Z − 1) ᎏ P P P
VR Z − 1 ᎏ=ᎏ RT P
whence
(4-157)
P GR dP (constant T ) (4-158) ᎏ = (Z − 1) ᎏ 0 RT P Differentiation of Eq. (4-158) with respect to temperature in accord with Eq. (4-84), gives P HR ∂Z dP (constant T ) (4-159) ᎏ = −T ᎏ ᎏ 0 RT ∂T P P Equation (4-13) written for residual properties becomes SR HR GR (4-160) ᎏ=ᎏ−ᎏ R RT RT
Therefore
4-16
THERMODYNAMICS
In view of Eq. (4-75), Eqs. (4-158) and (4-160) may be expressed alternatively as P dP ln φ = (Z − 1) ᎏ (constant T ) (4-161) 0 P
Again, in alternative form, HR (HR)0 (HR)1 ᎏ=ᎏ+ωᎏ RTc RTc RTc
SR HR and (4-162) ᎏ = ᎏ − ln φ R RT Values of Z and of (∂Z/∂T)P come from experimental PVT data, and the integrals in Eqs. (4-158), (4-159), and (4-161) may be evaluated by numerical or graphical methods. Alternatively, the integrals are expressed analytically when Z is given by an equation of state. Residual properties are therefore evaluated from PVT data or from an appropriate equation of state. Pitzer’s Corresponding-States Correlation A three-parameter corresponding-states correlation of the type developed by Pitzer, K.S. (Thermodynamics, 3d ed., App. 3, McGraw-Hill, New York, 1995) is described in Sec. 2. It has as its basis an equation for the compressibility factor:
where
Z = Z0 + ωZ1 0
(4-163)
1
where Z and Z are each functions of reduced temperature Tr and reduced pressure Pr. The acentric factor ω is defined by Eq. (2-23). The Tr and Pr dependencies of functions Z0 and Z1 are shown by Figs. 2-1 and 2-2. Generalized correlations are developed here for the residual enthalpy, residual entropy, and the fugacity coefficient. Equations (4-161) and (4-159) are put into generalized form by substitution of the relationships P = Pc Pr
T = Tc Tr
dP = Pc dPr
dT = Tc dTr
The resulting equations are: ln φ =
Pr
0
dP (Z − 1) ᎏr Pr
HR ᎏ = −T r2 RTc
and
∂Z ᎏ ∂T
(4-164)
Pr
0
r
Pr
dPr ᎏ Pr
(4-165)
(HR)0 ᎏ = −T r2 RTc
∂Z ᎏ ∂T ∂Z ᎏ ∂T Pr
0
0 r
Pr
(4-168)
dPr ᎏ Pr
Pr 1 (HR)1 dPr ᎏ = −T r2 ᎏ 0 RTc r Pr Pr The residual entropy is given by Eq. (4-162), here written SR 1 HR (4-169) ᎏ = ᎏ ᎏ − ln φ R Tr RTc Pitzer’s original correlations for Z and the derived quantities were determined graphically and presented in tabular form. Since then, analytical refinements to the tables have been developed, with extended range and accuracy. The most popular Pitzer-type correlation is that of Lee and Kesler (AIChE J., 21, pp. 510–527 [1975]). These tables cover both the liquid and gas phases, and span the ranges 0.3 ≤ Tr ≤ 4.0 and 0.01 ≤ Pr ≤ 10.0. Shown by Figs. 4-5 and 4-6 are isobars of −(HR)0/RTc and −(HR)1/RTc with Tr as independent variable drawn from these tables. Figures 4-7 and 4-8 are the corresponding plots for −ln φ0 and −ln φ1. Figures 4-9 and 4-10 are isotherms of φ0 and φ1 with Pr as independent variable. Although the Pitzer correlations are based on data for pure materials, they may also be used for the calculation of mixture properties. A set of recipes is required relating the parameters Tc, Pc, and ω for a mixture to the pure-species values and to composition. One such set is given by Eqs. (2-80) through (2-82) in Sec. 2, which define pseudoparameters, so called because the defined values of Tc, Pc, and ω have no physical significance for the mixture. Alternative Property Formulations Direct application of Eqs. (4-159) and (4-161) can be made only to equations of state that are solvable for volume, that is, that are volume explicit. Most equations of state are in fact pressure explicit, and alternative equations are required.
The terms on the right-hand sides of these equations depend only on the upper limit Pr of the integrals and on the reduced temperature at which they are evaluated. Thus, values of ln φ and HR/RTc may be determined once and for all at any reduced temperature and pressure from generalized compressibility factor data. Substitution for Z in Eq. (4-164) by Eq. (4-163) yields ln φ =
Pr
0
dP (Z0 − 1) ᎏr + ω Pr
Pr
0
dP Z1 ᎏr Pr
This equation may be written in alternative form as ln φ = ln φ0 + ω ln φ1 where
ln φ0
ln φ1
Pr
0 Pr
0
(4-166)
dP (Z0 − 1) ᎏr Pr dP Z1 ᎏr Pr
Since Eq. (4-166) may also be written φ = (φ0)(φ1)ω
(4-167)
correlations may be presented for φ0 and φ1 as well as for their logarithms. Differentiation of Eq. (4-163) yields ∂Z ∂Z = ᎏ ᎏ ∂T ∂T 0
r
Pr
r
Pr
∂Z1 +ω ᎏ ∂Tr
Pr
Substitution for (∂Z/∂Tr)Pr in Eq. (4-165) gives: HR ᎏ = −T r2 RTc
∂Z ᎏ ∂T Pr
0
0 r
dPr ᎏ − ωT r2 Pr Pr
∂Z ᎏ ∂T Pr
0
1 r
Pr
dPr ᎏ Pr
Correlation of −(HR)0/RTc, drawn from the tables of Lee and Kesler (AIChE J., 21, pp. 510–527 [1975]).
FIG. 4-5
EVALUATION OF PROPERTIES
4-17
Correlation of −(HR)1/RTc, drawn from the tables of Lee and Kesler (AIChE J., 21, pp. 510–527 [1975]).
FIG. 4-6
Equation (4-158) is converted through application of the general relation PV = ZRT. Differentiation at constant T gives P dV + V dP = RT dZ which is readily transformed to dP dZ dV ᎏ=ᎏ−ᎏ P Z V
(constant T )
(constant T )
FIG. 4-8 Correlation of [−ln φ1] vs. Tr, drawn from the tables of Lee and Kesler (AIChE J., 21, pp. 510–527 [1975]).
Substitution into Eq. (4-158) leads to V GR dV (4-170) ᎏ = Z − 1 − ln Z − (Z − 1) ᎏ ∞ RT V The molar volume may be eliminated in favor of the molar density, ρ = V−1, to give ρ GR dρ (4-171) ᎏ = Z − 1 − ln Z + (Z − 1) ᎏ 0 RT ρ For a pure material, Eq. (4-75) shows that GR/RT = ln φ, in which case Eqs. (4-170) and (4-171) directly yield values of ln φ: V dV ln φ = Z − 1 − ln Z − (Z − 1) ᎏ (4-172) ∞ V
ρ
dρ (Z − 1) ᎏ (4-173) ρ where subscript i is omitted for simplicity. The corresponding equations for HR are most readily found from Eq. (4-107) applied to a pure material. In view of Eqs. (4-75) and (4-157), this equation may be written HR dP d ln φ = (Z − 1) ᎏ − ᎏ2 dT RT P Division by dT and restriction to constant V gives, upon rearrangement, HR Z − 1 ∂P ∂ ln φ ᎏ2 = ᎏ ᎏ − ᎏ RT P ∂T V ∂T V Differentiation of P = ZRT/V provides the first derivative on the right and differentiation of Eq. (4-172) provides the second. Substitution then leads to V HR ∂Z dV (4-174) ᎏ=Z−1+T ᎏ ᎏ ∞ RT ∂T V V ln φ = Z − 1 − ln Z +
0
FIG. 4-7 Correlation of [−ln φ0] vs. Tr, drawn from the tables of Lee and Kesler (AIChE J., 21, pp. 510–527 [1975]).
4-18
THERMODYNAMICS
FIG. 4-10 Correlation of φ1 vs. Pr, drawn from the tables of Lee and Kesler (AIChE J., 21, pp. 510–527 [1975]).
Direct application of these results is possible only to equations of state explicit in volume. For pressure-explicit equations of state, alternative recipes are required. The basis is Eq. (4-82), which in view of Eq. (4-157) may be written nGR n(Z − 1) nHR d ᎏ = ᎏ dP − ᎏ2 dT + ln φˆ i dni RT P RT i Division by dni and restriction to constant T, nV, and nj ( j ≠ i) leads to ∂(nGR/RT) n(Z − 1) ∂P ln φˆ i = ᎏᎏ −ᎏ ᎏ T,nV,nj ∂ni P ∂ni T,nV,nj But P = (nZ)RT/nV, and therefore ∂P P ∂(nZ) =ᎏ ᎏ ᎏ ∂ni T,nV,nj nZ ∂ni T,nV,nj Combination of the last two equations gives ∂(nGR/RT) Z − 1 ∂(nZ) ln φˆ i = ᎏᎏ − ᎏ ᎏ (4-178) T,nV,nj ∂ni Z ∂ni T,nV,nj Alternatively, ∂(nGR/RT) Z − 1 ∂(nZ) ln φˆ i = ᎏᎏ − ᎏ ᎏ (4-179) T,ρ/n,nj ∂ni Z ∂ni T,ρ/n,nj These equations may either be applied to the results of integrations of Eqs. (4-170) and (4-171) or directly to Eqs. (4-170) and (4-171) as written for a mixture. In the latter case the following analogs of Eq. (4-176) are obtained: V ∂(nZ) dV ln φˆ i = − − 1 ᎏ − ln Z (4-180) ᎏ ∞ ∂ni T,nV,nj V
FIG. 4-9 Correlation of φ0 vs. Pr, drawn from the tables of Lee and Kesler (AIChE J., 21, pp. 510–527 [1975]).
ρ
H ∂Z dρ (4-175) ᎏ=Z−1−T ᎏ ᎏ 0 RT ∂T ρ ρ As before, the residual entropy is found by Eq. (4-162). In applications to equilibrium calculations, the fugacity coefficients of species in a mixture φˆ i are required. Given an expression for GR/RT as determined from Eq. (4-158) for a constant-composition mixture, the corresponding recipe for ln φˆ i is found through the partialproperty relation ∂(nGR/RT) ln φˆ i = ᎏᎏ (4-85) T,P,nj ∂ni There are two ways to proceed: operate on the result of the integration of Eq. (4-158) in accord with Eq. (4-85) or apply Eq. (4-85) directly to Eq. (4-158), obtaining P dP ln φˆ i = (Z (4-176) i − 1) ᎏ 0 P where Zi is the partial compressibility factor, defined as ∂(nZ) Z (4-177) i ᎏ ∂ni T,P,nj R
Alternatively,
∂(nZ) ln φˆ = − ᎏ ∂n ρ
i
0
i
T,ρ/n,nj
dρ − 1 ᎏ − ln Z ρ
(4-181)
EVALUATION OF PROPERTIES Virial Equations of State The virial equation in density is an infinite-series representation of the compressibility factor Z in powers of molar density ρ (or reciprocal molar volume V−1) about the real-gas state at zero density (zero pressure): Z = 1 + Bρ + Cρ2 + Dρ3 + ⋅ ⋅ ⋅
(4-182)
The density-series virial coefficients B, C, D, . . . , depend on temperature and composition only. The composition dependencies are given by the exact recipes B = yiyjBij
(4-183)
C = yiyjykCijk
(4-184)
i
i
j
j
k
where yi, yj, and yk are mole fractions for a gas mixture, with indices i, j, and k identifying species. The coefficient Bij characterizes a bimolecular interaction between molecules i and j, and therefore Bij = Bji. Two kinds of second virial coefficient arise: Bii and Bjj, wherein the subscripts are the same (i = j); and Bij, wherein they are different (i ≠ j). The first is a virial coefficient for a pure species; the second is a mixture property, called a cross coefficient. Similarly for the third virial coefficients: Ciii, Cjjj, and Ckkk are for the pure species; and Ciij = Ciji = Cjii, and so on, are cross coefficients. Although the virial equation itself is easily rationalized on empirical grounds, the “mixing rules” of Eqs. (4-183) and (4-184) follow rigorously from the methods of statistical mechanics. The temperature derivatives of B and C are given exactly by dB dBij (4-185) ᎏ = yiyj ᎏ dT dT i j dC dCijk (4-186) ᎏ = yiyjyk ᎏ dT dT i j k An alternative form of the virial equation expresses Z as an expansion in powers of pressure about the real-gas state at zero pressure (zero density): Z = 1 + B′P + C′P 2 + D′P 3 + ⋅ ⋅ ⋅ (4-187) Equation (4-187) is the virial equation in pressure, and B′, C′, D′, . . . , are the pressure-series virial coefficients. Like the density-series coefficients, they depend on temperature and composition only. Moreover, the two sets of coefficients are related: B B′ = ᎏ (4-188) RT
P (4-196) ln φˆ i = 2 ykBki − B ᎏ RT k Equation (4-191) is explicit in pressure, and Eqs. (4-173), (4-175), and (4-181) are therefore applicable. Direct substitution of Eq. (4-191) into Eq. (4-173) yields 3 ln φ = 2Bρ + ᎏ Cρ2 − ln Z (4-197) 2
Whence
(4-189)
∂Z
dC =ᎏρ+ᎏρ ᎏ ∂T dT dT
Moreover,
dB
2
ρ
and so on Application of an infinite series to practical calculations is, of course, impossible, and truncations of the virial equations are in fact employed. The degree of truncation is conditioned not only by the temperature and pressure but also by the availability of correlations or data for the virial coefficients. Values can usually be found for B (see Sec. 2), and often for C (see, e.g., De Santis and Grande, AIChE J., 25, pp. 931–938 [1979]), but rarely for higher-order coefficients. Application of the virial equations is therefore usually restricted to two- or three-term truncations. For pressures up to several bars, the two-term expansion in pressure, with B′ given by Eq. (4-188), is usually preferred: BP Z=1+ᎏ (4-190) RT For supercritical temperatures, it is satisfactory to ever-higher pressures as the temperature increases. For pressures above the range where Eq. (4-190) is useful, but below the critical pressure, the virial expansion in density truncated to three terms is usually suitable:
(4-191)
Equations for derived properties may be developed from each of these expressions. Consider first Eq. (4-190), which is explicit in volume. Equations (4-159), (4-161), and (4-176) are therefore applicable. Direct substitution for Z in Eq. (4-161) gives BP ln φ = ᎏ (4-192) RT Differentiation of Eq. (4-190) yields ∂Z dB B P ᎏ = ᎏ−ᎏ ᎏ ∂T P dT T RT Whence, by Eq. (4-159) HR P B dB (4-193) ᎏ=ᎏ ᎏ−ᎏ RT R T dT and by Eq. (4-162), SR P dB (4-194) ᎏ=−ᎏᎏ R R dT Multiplication of Eq. (4-190) by n gives P nZ = n + (nB) ᎏ RT Differentiation in accord with Eq. (4-177) yields ∂(nB) P Z i = 1 + ᎏ ᎏ ∂ni T,nj RT Whence, by Eq. (4-176), ∂(nB) P ln φˆ i = ᎏ ᎏ ∂ni T,nj RT Equation (4-183) can be written 1 nB = ᎏ nknlBkl n k l from which, by differentiation, ∂(nB) = 2 ykBki − B (4-195) ᎏ ∂ni T,nj k
and so on
C − B2 C′ = ᎏ (RT)2
Z = 1 + Bρ + Cρ2
4-19
HR dB T dC (4-198) ᎏ = B − T ᎏ ρ + C − ᎏ ᎏ ρ2 RT dT 2 dT The residual entropy is given by Eq. (4-162). Application of Eq. (4-181) provides an expression for ln φˆ i. First, from Eq. (4-191), whence
∂(nZ) ᎏ ∂n
∂(nB) ∂(nC) =1+ B+ ᎏ ρ + 2C + ᎏ ∂ni T,nj ∂ni Substitution into Eq. (4-181) gives, on integration, i
T,ρ/n,nj
ρ
2
T,nj
∂(nB) 1 ∂(nC) ln φˆ i = B + ᎏ ρ + ᎏ 2C + ᎏ ρ2 − ln Z ∂ni T,nj 2 ∂ni T,nj The mole-number derivative of nB is given by Eq. (4-195); the corresponding derivative of nC, similarly found from Eq. (4-184), is ∂(nC) = 3 ykylCkli − 2C (4-199) ᎏ ∂ni T,nj k l
4-20
THERMODYNAMICS
Finally, 3 ln φˆ i = 2ρ ykBki + ᎏ ρ2 ykylCkli − ln Z (4-200) 2 k k l In a process calculation, T and P, rather than T and ρ (or T and V), are usually the favored independent variables. Application of Eqs. (4-197), (4-198), and (4-200) therefore requires prior solution of Eq. (4-191) for Z or ρ. Since Z = P/ρRT, Eq. (4-191) may be written in two equivalent forms: BP CP 2 Z3 − Z2 − ᎏ Z − ᎏ2 = 0 (4-201) RT (RT)
B 1 P (4-202) ρ3 + ᎏ ρ2 + ᎏ ρ − ᎏ = 0 C C CRT In the event that three real roots obtain for these equations, only the largest Z (smallest ρ) appropriate for the vapor phase has physical significance, because the virial equations are suitable only for vapors and gases. Generalized Correlation for the Second Virial Coefficient Perhaps the most useful of all Pitzer-type correlations is the one for the second virial coeffieient. The basic equation (see Eq. [2-68]) is BPc (4-203) ᎏ = B0 + ωB1 RTc where for a pure material B0 and B1 are functions of reduced temperture only. Substitution for B by this expression in Eq. (4-190) yields P Z = 1 + (B0 + ωB1) ᎏr (4-204) Tr By differentiation, ∂Z dB0/dTr B0 dB1/dT B1 + ωPr ᎏr − ᎏ ᎏ = Pr ᎏ − ᎏ 2 ∂Tr Pr Tr Tr Tr Tr2 Substitution of these equations into Eqs. (4-164) and (4-165) and integration gives P ln φ = (B0 + ωB1) ᎏr (4-205) Tr or
HR dB0 dB1 (4-206) ᎏ = Pr B0 − Tr ᎏ + ω B1 − Tr ᎏ RTc dTr dTr The residual entropy follows from Eq. (4-162): SR dB0 dB1 (4-207) ᎏ = −Pr ᎏ + ω ᎏ R dTr dTr In these equations, B0 and B1 and their derivatives are well represented by 0.422 B0 = 0.083 − ᎏ (4-208) Tr1.6 and
0.172 B1 = 0.139 − ᎏ Tr4.2 dB0 0.675 ᎏ=ᎏ dTr Tr2.6
(4-209) (4-210)
dB1 0.722 (4-211) ᎏ=ᎏ dTr Tr5.2 Though limited to pressures where the two-term virial equation in pressure has approximate validity, this correlation is applicable to most chemical-processing conditions. As with all generalized correlations, it is least accurate for polar and associating molecules. Although developed for pure materials, this correlation can be extended to gas or vapor mixtures. Basic to this extension is the mixing rule for second virial coefficients and its temperature derivative: B = yiyjBij
(4-183)
dB dBij ᎏ = yiyj ᎏ dT dT i j
(4-185)
i
j
Values for the cross coefficients and their derivatives in these equations are provided by writing Eq. (4-203) in extended form: RTcij 0 Bij = ᎏ (B + ωij B1) (4-212) Pcij where B0, B1, dB0/dTr, and dB1/dTr are the same functions of Tr as given by Eqs. (4-208) through (4-211). Differentiation produces dBij RTcij dB0 dB1 ᎏ = ᎏ ᎏ + ωij ᎏ dT Pcij dT dT
0
1
dBij R dB dB (4-213) ᎏ = ᎏ ᎏ + ωij ᎏ dT Pcij dTrij dTrij where Trij = T/Tcij. The following are combining rules for calculation of ωij, Tcij, and Pcij as given by Prausnitz, Lichtenthaler, and de Azevedo (Molecular Thermodynamics of Fluid-Phase Equilibria, 2d ed., pp. 132 and 162, Prentice-Hall, Englewood Cliffs, N.J., 1986): ωi + ωj ωij = ᎏ (4-214) 2 Tcij = (Tci Tcj)1/2(1 − kij) (4-215) Zcij RTcij Pcij = ᎏ (4-216) Vcij Zci + Zcj (4-217) with Zcij = ᎏ 2 1/3 1/3 3 Vci + Vc j (4-218) Vcij = ᎏᎏ 2 In Eq. (4-215), kij is an empirical interaction parameter specific to an i-j molecular pair. When i = j and for chemically similar species, kij = 0. Otherwise, it is a small (usually) positive number evaluated from minimal PVT data or in the absence of data set equal to zero. When i = j, all equations reduce to the appropriate values for a pure species. When i ≠ j, these equations define a set of interaction parameters having no physical significance. For a mixture, values of Bij and dBij /dT from Eqs. (4-212) and (4-213) are substituted into Eqs. (4183) and (4-185) to provide values of the mixture second virial coefficient B and its temperature derivative. Values of HR and SR for the mixture are then given by Eqs. (4-193) and (4-194), and values of ln φˆ i for the component fugacity coefficients are given by Eq. (4-196). Cubic Equations of State The simplest expressions that can (in principle) represent both the vapor- and liquid-phase volumetric behavior of pure fluids are equations cubic in molar volume. All such expressions are encompassed by the generic equation or
RT a(V − η) P = ᎏ − ᎏᎏᎏ (4-219) V − b (V − b)(V 2 + δV + ε) where parameters b, θ, δ, ε, and η can each depend on temperature and composition. Special cases are obtained by specification of values or expressions for the various parameters. The modern development of cubic equations of state started in 1949 with publication of the Redlich/Kwong equation (Redlich and Kwong, Chem. Rev., 44, pp. 233–244 [1949]): RT a(T) P=ᎏ−ᎏ (4-220) V − b V(V + b) a a(T) = ᎏ T1/2 and a and b are functions of composition only. This equation, like other cubic equations of state, has three volume roots, of which two may be complex. Physically meaningful values of V are always real, positive, and greater than the constant b. When T > Tc, solution for V at any positive value of P yields only one real positive root. When T = Tc, this is also true, except at the critical pressure, where there are three roots, all equal to Vc. For T < Tc, only one real positive root exists at high pressures, but for a range of lower pressures there are three real positive roots. Here, the middle root is of no significance; the smallest root is a liquid or liquidlike volume, and the largest root is a vapor or vaporlike volume. The volumes of saturated liquid and satuwhere
EVALUATION OF PROPERTIES rated vapor are given by the smallest and largest roots when P is the saturation vapor pressure Psat. The application of cubic equations of state to mixtures requires expression of the equation-of-state parameters as functions of composition. No exact theory like that for the virial coefficients prescribes this composition dependence, and empirical mixing rules provide approximate relationships. The mixing rules that have found general favor for the Redlich/Kwong equation are: with aij = aji, and
a = yi yj aij
(4-221)
b = yi bi
(4-222)
i
j
i
The aij are of two types: pure-species parameters (like subscripts) and interaction parameters (unlike subscripts). The bi are parameters for the pure species. Parameter evaluation may be accomplished with the equations 0.42748R2 Tcij2.5 aij = ᎏᎏ (4-223) Pcij 0.08664RTci (4-224) bi = ᎏᎏ Pci where Eqs. (4-215) through (4-218) provide for the calculation of the Tcij and Pcij. Multiplication of the Redlich/Kwong equation (Eq. [4-220]) by V/RT leads to its expression in alternative form: a 1 h Z=ᎏ−ᎏ (4-225) ᎏ 1 − h bRT1.5 1 + h
a h h Z−1=ᎏ−ᎏ ᎏ 1 − h bRT1.5 1 + h
Whence
(4-226)
bP h=ᎏ (4-227) ZRT Equations (4-170) and (4-174) in combination with Eq. (4-226) give GR a ln (1 + h) (4-228) ᎏ = Z − 1 − ln (1 − h)Z − ᎏ RT bRT1.5 where
R
H 3a ln (1 + h) (4-229) ᎏ=Z−1− ᎏ RT 2 bRT 1.5 Once a and b are determined by Eqs. (4-221) through (4-224), then for given T and P values of Z, GR/RT, and HR/RT are found by Eqs. (4-225), (4-228), and (4-229) and SR/R by Eq. (4-160). The procedure requires initial solution of Eqs. (4-225) and (4-227) for Z and h. The original Redlich/Kwong equation is rarely satisfactory for vapor/liquid equilibrium calculations, and equations have been developed specific to this purpose. The two most popular are the Soave/ Redlich/Kwong (SRK) equation, a modification of the Redlich/Kwong equation (Soave, Chem. Eng. Sci., 27, pp. 1197–1203 [1972]), and the Peng/Robinson (PR) equation (Peng and Robinson, Ind. Eng. Chem. Fundam., 15, pp. 59–64 [1976]). Both equations are designed specifically to yield reasonable vapor pressures for pure fluids. Thus, there is no assurance that molar volumes calculated by these equations are more accurate than values given by the original Redlich/Kwong equation. Written for pure species i the SRK and PR equations are special cases of the following: RT ai(T ) P = ᎏ − ᎏᎏ (4-230) Vi − bi (Vi + εbi)(Vi + σbi) and
where
Ωaα(Tri; ωi)R2 Tci2 ai(T) = ᎏᎏ Pci
Ωb RTci bi = ᎏ Pci and ε σ, Ωa, and Ωb are equation-specific constants. For the Soave/ Redlich/Kwong equation: α(Tri; ωi) = [1 + (0.480 + 1.574ωi − 0.176ω2i ) (1 − Tri1/2)]2
4-21
For the Peng/Robinson equation: α(Tri; ωi) = [1 + (0.37464 + 1.54226ωi − 0.26992ω i2) (1 − Tri1/2)]2 Written for a mixture, Eq. (4-230) becomes RT a(T) P = ᎏ − ᎏᎏ (4-231) V − b (V + εb)(V + σb) where a and b are mixture values, related to the ai and bi by mixing rules. Equation (4-170) applied to Eq. (4-231) leads to ai bi bi (V − b)Z a/bRT V + σb ln φˆ i = ᎏ (Z − 1) − ln ᎏ + ᎏ 1 + ᎏ − ᎏ ln ᎏ b a b V ε−σ V + εb (4-232)
where ai and bi are partial parameters for species i, defined by ∂(na) (4-233) ai = ᎏ ∂ni T,nj
∂(nb) b = ᎏ ∂n
and
(4-234)
i
i
T,nj
These are general equations that do not depend on the particular mixing rules adopted for the composition dependence of a and b. The mixing rules given by Eqs. (4-221) and (4-222) can certainly be employed with these equations. However, for purposes of vapor/liquid equilibrium calculations, a special pair of mixing rules is far more appropriate, and will be introduced when these calculations are treated. Solution of Eq. (4-232) for fugacity coefficient φˆ i at given T and P requires prior solution of Eq. (4-231) for V, from which is found Z = PV/RT. Benedict/Webb/Rubin Equation of State The BWR equation of state with Z as the dependent variable is written A0 C0 a Z = 1 + B0 − ᎏ −ᎏ ρ + b − ᎏ ρ2 RT RT 3 RT
c aα + ᎏ ρ5 + ᎏ3 ρ2 (1 + γρ2) exp (−γρ2) (4-235) RT RT All eight parameters depend on composition; moreover, parameters C0, b, and γ are for some applications treated as functions of T. By Eq. (4-171), the residual Gibbs energy is GR C0 3 a A0 6aα ᎏ = 2 B0 − ᎏ − ᎏ3 ρ + ᎏ b − ᎏ ρ2 + ᎏ ρ5 RT RT RT 2 RT 5RT
c + ᎏ3 (2γ 2ρ4 + γρ2 − 2) exp (−γ ρ2) + 2 − ln Z (4-236) 2γRT With allowance for T dependence of C0, b, and γ, Eq. (4-175) yields HR 1 dC0 2A0 4C0 ᎏ = B0 − ᎏ − ᎏ3 + ᎏ2 ᎏ ρ RT RT RT RT dT
1 db 3a 6aα − ᎏ T ᎏ − 2b + ᎏ ρ2 + ᎏ ρ5 2 dT RT 5RT
c + ᎏ3 (2γ 2ρ4 − γρ2 − 6) exp (−γρ2) + 6 2γRT
c dγ −ᎏ (4-237) ᎏ (γ 2ρ4 + 2γρ2 + 2) exp (−γρ2) − 2 2γ 2RT 2 dT The residual entropy is given by Eq. (4-160). Computation of ln φˆ i is done via Eq. (4-181). The result is A0 + A 0 C0 + C 0i ln φˆ i = B0 + B0i − ᎏi − ᎏ ρ RT RT 3
4aα + aα 2a + ai 2 1 i + αai ρ5 bi − ᎏ ρ + ᎏᎏ + ᎏ 2b + 5RT RT 2
c + ᎏ3 2γRT
2 γi
1 + ᎏγ γ ρ + ᎏγ − ᎏc ρ
ci γi −2 1+ᎏ −ᎏ c γ
γi
2 4
ci
2
γi
exp (−γρ ) + 2 1 + ᎏc − ᎏγ − ln Z 2
ci
(4-238)
4-22
THERMODYNAMICS
Here the quantities with overbars are partial parameters for species i, defined for arbitrary parameter π by ∂(nπ) πi ᎏ (4-239) ∂ni T,nj Application of these equations requires specific mixing rules. For example, if
π=
y π
r
1/r k k
(4-240)
The symmetrical nature of these relations is evident. The infinitedilution values of the activity coefficients are ln γ 1∞ = ln γ 2∞ = B. If D = ⋅ ⋅ ⋅ = 0, then GE ᎏ = B + C(x1 − x2) = B + C(2x1 − 1) x1 x2 RT and in this case GE/x1 x2 RT is linear in x1. The substitutions, B + C = A21 and B − C = A12 transform this expression into the Margules equation: GE/x1 x2 RT = A21 x1 + A12 x2
k
where r is a small integer, the recipe for πi is πi 1/r π (4-241) i = π r ᎏ − (r − 1) π Specifically, if r = 3 for π c; then ci 1/3 −2 ci = c 3 ᎏ c where ci is the parameter for pure i and c is the parameter for the mixture, given by
c=
y c
3
1/3 k k
k
Equation-of-state examples are given in Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., Secs. 3.4–3.7 and 6.2–6.6, McGraw-Hill, New York, 1996). EXPRESSIONS FOR THE EXCESS GIBBS ENERGY In principle, equation-of-state procedures can be used for the calculation of liquid-phase as well as gas-phase properties, and much has been accomplished in the development of PVT equations of state suitable for both phases. However, a widely used alternative for the liquid phase is application of excess properties. The excess property of primary importance for engineering calculations is the excess Gibbs energy GE, because its canonical variables are T, P, and composition, the variables usually specified or sought in a design calculation. Knowing GE as a function of T, P, and composition, one can in principle compute from it all other excess properties (see, for example, Eqs. [4-117] through [4-119]). As noted with respect to Fig. 4-1, the excess volume for liquid mixtures is usually small; the pressure dependence of GE may then be safely ignored. Thus, the engineering efforts at describing GE center on representing its composition and temperature dependence. For binary systems at constant T, GE is a function of just x1, and the quantity most conveniently represented by an equation is GE/x1x2RT. The simplest procedure is to express this quantity as a power series in x1: GE ᎏ = a + bx1 + cx 12 + ⋅ ⋅ ⋅ (constant T) x1 x2 RT An equivalent power series with certain advantages is known as the Redlich/Kister expansion (Redlich, Kister, and Turnquist, Chem. Eng. Progr. Symp. Ser. No. 2, 48, pp. 49–61 [1952]): GE ᎏ = B + C(x1 − x2) + D(x1 − x2)2 + ⋅ ⋅ ⋅ x1x2 RT In application, different truncations of this series are appropriate. For each particular expression representing GE/x1x2 RT, specific expressions for ln γ1 and ln γ2 result from application of Eq. (4-119). When all parameters are zero, GE/RT = 0, and the solution is ideal. If C = D = ⋅ ⋅ ⋅ = 0, then GE ᎏ=B x1 x2 RT where B is a constant for a given temperature. The corresponding equations for ln γ1 and ln γ2 are ln γ1 = Bx22
(4-242)
ln γ2 = Bx 21
(4-243)
(4-244)
Application of Eq. (4-119) yields ln γ1 = x 22 [A12 + 2(A21 − A12)x1]
(4-245)
ln γ2 = x 21 [A21 + 2(A12 − A21)x2]
(4-246)
An alternative equation is obtained when the reciprocal quantity x1x2RT/GE is expressed as a linear function of x1: x1x2 ᎏ = B′ + C′ (x1 − x2) = B′ + C′ (2x1 − 1) GE/RT This may also be written: x1x2 = B′ (x1 + x2) + C′ (x1 − x2) = (B′ + C′)x1 + (B′ − C′)x2 ᎏ GE/RT The substitutions B′ + C′ = 1/A′21 and B′ − C′ = 1/A′12 produce x1x2 x1 x2 A′12 x1 + A′21 x2 = ᎏ + ᎏ = ᎏᎏ ᎏ GE/RT A′21 A′12 A′12 A′21 or
GE A′12 A′21 ᎏ = ᎏᎏ x1x2 RT A′12 x1 + A′21 x2
(4-247)
The activity coefficients implied by this equation are given by A′12 x1 −2 (4-248) ln γ1 = A′12 1 + ᎏ A′21 x2
A′21 x2 ln γ2 = A′21 1 + ᎏ A′12 x1
−2
(4-249)
These are known as the van Laar equations. When x1 = 0, ln γ 1∞ = A′12; when x2 = 0, ln γ 2∞ = A′21. The Redlich/Kister expansion, the Margules equations, and the van Laar equations are all special cases of a very general treatment based on rational functions, that is, on equations for GE given by ratios of polynomials (Van Ness and Abbott, Classical Thermodynamics of Nonelectrolyte Solutions: With Applications to Phase Equilibria, Sec. 5-7, McGraw-Hill, New York, 1982). Although providing great flexibility in the fitting of VLE data for binary systems, they are without theoretical foundation, with no rational basis for their extension to multicomponent systems. Nor do they incorporate an explicit temperature dependence for the parameters. Modern theoretical developments in the molecular thermodynamics of liquid-solution behavior are often based on the concept of local compositon, presumed to account for the short-range order and nonrandom molecular orientations that result from differences in molecular size and intermolecular forces. Introduced with the publication of a model of GE behavior known as the Wilson equation ( J. Am. Chem. Soc., 86, pp. 127–130 [1964]), it prompted the development of alternative local-composition models, most notably the NRTL (Non-Random-Two-Liquid) equation of Renon and Prausnitz (AIChE J., 14, pp. 135–144 [1968]) and the UNIQUAC (UNIversal QUAsi-Chemical) equation of Abrams and Prausnitz (AIChE J., 21, pp. 116–128 [1975]). A further significant development, based on the UNIQUAC equation, is the UNIFAC method (UNIQUAC Functional-group Activity Coefficients). Proposed by Fredenslund, Jones, and Prausnitz (AIChE J., 21, pp. 1086–1099 [1975]) and given detailed treatment by Fredenslund, Gmehling, and Rasmussen (Vapor-Liquid Equilibrium Using UNIFAC, Elsevier, Amsterdam, 1977), it provides for the calculation of activity coefficients from contributions of the various groups making up the molecules of a solution.
EVALUATION OF PROPERTIES The Wilson equation, like the Margules and van Laar equations, contains just two parameters for a binary system (Λ12 and Λ21), and is written: GE (4-250) ᎏ = −x1 ln (x1 + x2Λ12) − x2 ln (x2 + x1Λ21) RT Λ12 Λ21 ln γ1 = −ln (x1 + x2Λ12) + x2 ᎏᎏ − ᎏᎏ x1 + x2Λ12 x2 + x1Λ21
Λ21 Λ12 ln γ2 = −ln (x2 + x1Λ21) − x1 ᎏᎏ − ᎏᎏ x1 + x2Λ12 x2 + x1Λ21
whence
(4-252)
Both Λ12 and Λ21 must be positive numbers. The NRTL equation contains three parameters for a binary system and is written: GE G21τ21 G12τ12 (4-253) ᎏ = ᎏᎏ + ᎏᎏ x1x2 RT x1 + x2 G21 x2 + x1 G12 G21 ln γ1 = x 22 τ21 ᎏᎏ x1 + x2 G21
G τ + ᎏᎏ (x + x G )
(4-254)
G τ + ᎏᎏ (x + x G )
(4-255)
G12 ln γ2 = x 21 τ12 ᎏᎏ x2 + x1 G12 Here
G12 = exp (−ατ12)
12 12
2
1
12
2
21 21
1
2
21
2
b12 b21 τ21 = ᎏ τ12 = ᎏ RT RT where α, b12, and b21, parameters specific to a particular pair of species, are independent of composition and temperature. The infinite-dilution values of the activity coefficients are given by the equations:
and
ln γ 1∞ = τ21 + τ12 exp (−ατ12) ln γ 2∞ = τ12 + τ 21 exp (−ατ 21)
j
Function g contains pure-species parameters only, whereas function g R incorporates two binary parameters for each pair of molecules. For a multicomponent system, Φ θ gC = xi ln ᎏi + 5 qi xi ln ᎏi (4-260) xi Φi i i gR = − qi xi ln i
where
θ τ
(4-261)
j ji
j
xi ri Φi ᎏ xj rj
(4-262)
xi qi θi ᎏ xj qj
(4-263)
j
and
Subscript i identifies species, and j is a dummy index; all summations are over all species. Note that τji ≠ τij; however, when i = j, then τii = τjj = 1. In these equations ri (a relative molecular volume) and qi (a relative molecular surface area) are pure-species parameters. The influence of temperature on g enters through the interaction parameters τji of Eq. (4-261), which are temperature dependent: −(uji − uii) τji = exp ᎏᎏ (4-264) RT Parameters for the UNIQUAC equation are therefore values of (uji − uii). An expression for ln γi is found by application of Eq. (4-119) to the UNIQUAC equation for g (Eqs. [4-259] through [4-261]). The result is given by the following equations: ln γ i = ln γ iC + ln γ iR
The local-composition models have limited flexibility in the fitting of data, but they are adequate for most engineering purposes. Moreover, they are implicitly generalizable to multicomponent systems without the introduction of any parameters beyond those required to describe the constituent binary systems. For example, the Wilson equation for multicomponent systems is written: GE (4-256) ᎏ = − xi ln xj Λij RT i j and
(4-259)
j
G21 = exp (−ατ21)
xkΛki ln γi = 1 − ln xj Λij − ᎏ xj Λkj j k
g = gC + g R C
ln γ 2∞ = −ln Λ21 + 1 − Λ12
2
binary (in contrast to multicomponent) systems. This makes parameter determination for the local-composition models a task of manageable proportions. The UNIQUAC equation treats g GE/RT as comprised of two additive parts, a combinatorial term g C, accounting for molecular size and shape differences, and a residual term g R (not a residual property), accounting for molecular interactions:
(4-251)
ln γ 1∞ = −ln Λ12 + 1 − Λ21
2
4-23
(4-257)
where Λij = 1 for i = j, and so on. All indices in these equations refer to the same species, and all summations are over all species. For each ij pair there are two parameters, because Λij ≠ Λji. For example, in a ternary system the three possible ij pairs are associated with the parameters Λ12, Λ21; Λ13, Λ31; and Λ23, Λ32. The temperature dependence of the parameters is given by: −a V Λij = ᎏj exp ᎏij (i ≠ j) (4-258) Vi RT where Vj and Vi are the molar volumes at temperature T of pure liquids j and i, and aij is a constant independent of composition and temperature. Thus the Wilson equation, like all other local-composition models, has built into it an approximate temperature dependence for the parameters. Moreover, all parameters are found from data for
(4-265)
J J ln γ i = 1 − Ji + ln Ji − 5qi 1 − ᎏi + ln ᎏi Li Li C
τij ln γ iR = qi 1 − ln si − θj ᎏ sj j
where in addition to Eqs. (4-263) and (4-264) ri Ji = ᎏ rj xj
(4-266) (4-267)
(4-268)
j
qi Li = ᎏ qj xj
(4-269)
si = θl τ li
(4-270)
j
l
Again subscript i identifies species, and j and l are dummy indicies. Values for the parameters ri, qi, and (uij − ujj) are given by Gmehling, Onken, and Arlt (Vapor-Liquid Equilibrium Data Collection, Chemistry Data Series, vol. I, parts 1–8, DECHEMA, Frankfurt/Main, 1974–1990). The Wilson parameters Λij, NRTL parameters Gij, and UNIQUAC parameters τij all inherit a Boltzmann-type T dependence from the origins of the expressions for GE, but it is only approximate. Computations of properties sensitive to this dependence (e.g., heats of mixing and liquid/liquid solubility) are in general only qualitatively correct.
4-24
THERMODYNAMICS
EQUILIBRIUM CRITERIA The equations developed in preceding sections are for PVT systems in states of internal equilibrium. The criteria for internal thermal and mechanical equilibrium are well known, and need not be discussed in detail. They simply require uniformity of temperature and pressure throughout the system. The criteria for phase and chemical-reaction equilibria are less obvious. Consider a closed PVT system, either homogeneous or heterogeneous, of uniform T and P, which is in thermal and mechanical equilibrium with its surroundings, but which is not initially at internal equilibrium with respect to mass transfer or with respect to chemical reaction. Changes occurring in the system are then irreversible, and must necessarily bring the system closer to an equilibrium state. The first and second laws written for the entire system are dU t = dQ + dW dQ dSt ≥ ᎏ T
dW = −P dV t dU t + P dV t − T dSt ≤ 0
The inequality applies to all incremental changes toward the equilibrium state, whereas the equality holds at the equilibrium state where any change is reversible. Various constraints may be put on this expression to produce alternative criteria for the directions of irreversible processes and for the condition of equilibrium. For example, it follows immediately that ≤0
Alternatively, other pairs of properties may be held constant. The most useful result comes from fixing T and P, in which case d(Ut + PVt − TSt)T,P ≤ 0 t ≤0 dGT,P
or
i
dn″i = −dn′i
(µ′ − µ″) dn′ = 0
Therefore
i
i
i
i
Since the dn′i are independent and arbitrary, it follows that µ′i = µ″i This is the criterion of two-phase equilibrium. It is readily generalized to multiple phases by successive application to pairs of phases. The general result is
i
Since mechanical equilibrium is assumed,
dU
i
Since the system is closed and without chemical reaction, material balances require that
(4-272)
Substitution for each µi by Eq. (4-77) produces the equivalent result (4-273) fˆ ′ = fˆ ″ = fˆ ′′′ = ⋅ ⋅ ⋅
dU t − dW − T dSt ≤ 0
t St,V t
t dG T,P d(nG)T,P = µ′i dn′i + µ″i dn″i = 0
µ′i = µ″i = µ′′′ i = ⋅ ⋅ ⋅
Combination gives
Whence
If the two-phase system is at equilibrium, then application of Eq. (4-271) yields
This expression shows that all irreversible processes occurring at constant T and P proceed in a direction such that the total Gibbs energy of the system decreases. Thus the equilibrium state of a closed system is the state with the minimum total Gibbs energy attainable at the given T and P. At the equilibrium state, differential variations may occur in the system at constant T and P without producing a change in Gt. This is the meaning of the equilibrium criterion t =0 dGT,P
(4-271)
This equation may be applied to a closed, nonreactive, two-phase system. Each phase taken separately is an open system, capable of exchanging mass with the other, and Eq. (4-16) may be written for each phase: d(nG)′ = −(nS)′ dT + (nV)′ dP + µ′i dn′i i
d(nG)″ = −(nS)″ dT + (nV)″ dP + µ″i dn″i i
where the primes and double primes denote the two phases and the presumption is that T and P are uniform throughout the two phases. The change in the Gibbs energy of the two-phase system is the sum of these equations. When each total-system property is expressed by an equation of the form nM = (nM)′ + (nM)″
i
i
These are the criteria of phase equilibrium applied in the solution of practical problems. For the case of equilibrium with respect to chemical reaction within a single-phase closed system, combination of Eqs. (4-16) and (4-271) leads immediately to
µ dn = 0 i
i
(4-274)
i
For a system in which both phase and chemical-reaction equilibrium prevail, the criteria of Eqs. (4-272) and (4-274) are superimposed. THE PHASE RULE The intensive state of a PVT system is established when its temperature and pressure and the compositions of all phases are fixed. However, for equilibrium states these variables are not all independent, and fixing a limited number of them automatically establishes the others. This number of independent variables is given by the phase rule, and is called the number of degrees of freedom of the system. It is the number of variables which may be arbitrarily specified and which must be so specified in order to fix the intensive state of a system at equilibrium. This number is the difference between the number of variables needed to characterize the system and the number of equations that may be written connecting these variables. For a system containing N chemical species distributed at equilibrium among π phases, the phase-rule variables are temperature and pressure, presumed uniform throughout the system, and N − 1 mole fractions in each phase. The number of these variables is 2 + (N − 1)π. The masses of the phases are not phase-rule variables, because they have nothing to do with the intensive state of the system. The equations that may be written connecting the phase-rule variables are: 1. Equation (4-272) for each species, giving (π − 1)N phaseequilibrium equations. 2. Equation (4-274) for each independent chemical reaction, giving r equations. The total number of independent equations is therefore (π − 1)N + r. In their fundamental forms these equations relate chemical potentials, which are functions of temperature, pressure, and composition, the phase-rule variables. Since the degrees of freedom of the system F is the difference between the number of variables and the number of equations,
this sum is given by
F = 2 + (N − 1)π − (π − 1)N − r
d(nG) = (nV) dP − (nS) dT + µ′i dn′i + µ″i dn″i i
i
or
F=2−π+N−r
(4-275)
EQUILIBRIUM The number of independent chemical reactions r can be determined as follows: 1. Write formation reactions from the elements for each chemical compound present in the system. 2. Combine these reaction equations so as to eliminate from the set all elements not present as elements in the system. A systematic procedure is to select one equation and combine it with each of the other equations of the set so as to eliminate a particular element. This usually reduces the set by one equation for each element eliminated, though two or more elements may be simultaneously eliminated. The resulting set of r equations is a complete set of independent reactions. More than one such set is often possible, but all sets number r and are equivalent. Example 1: Application of the Phase Rule a. For a system of two miscible nonreacting species in vapor/liquid equilibrium, F=2−π+N−r=2−2+2−0=2 The two degrees of freedom for this system may be satisfied by setting T and P, or T and y1, or P and x1, or x1 and y1, and so on, at fixed values. Thus, for equilibrium at a particular T and P, this state (if possible at all) exists only at one liquid and one vapor composition. Once the two degrees of freedom are used up, no further specification is possible that would restrict the phase-rule variables. For example, one cannot in addition require that the system form an azeotrope (assuming this possible), for this requires x1 = y1, an equation not taken into account in the derivation of the phase rule. Thus, the requirement that the system form an azeotrope imposes a special constraint and reduces the number of degrees of freedom to one. b. For a gaseous system consisting of CO, CO2, H2, H2O, and CH4 in chemical-reaction equilibrium, F=2−π+N−r=2−1+5−2=4 The value of r = 2 is found from the formation reactions: C + a O2 → CO C + O2 → CO2 H2 + a O2 → H2O C + 2H2 → CH4 Systematic elimination of C and O2 from this set of chemical equations reduces the set to two. Three possible pairs of equations may result, depending on how the combination of equations is effected. Any pair of the following three equations represents a complete set of independent reactions, and all pairs are equivalent. CH4 + H2O → CO + 3H2 CO + H2O → CO2 + H2 CH4 + 2H2O → CO2 + 4H2 The result, F = 4, means that one is free to specify, for example, T, P, and two mole fractions in an equilibrium mixture of these five chemical species, provided nothing else is arbitrarily set. Thus, it cannot simultaneously be required that the system be prepared from specified amounts of particular constituent species.
Since the phase rule treats only the intensive state of a system, it applies to both closed and open systems. Duhem’s theorem, on the other hand, is a rule relating to closed systems only: For any closed system formed initially from given masses of prescribed chemical species, the equilibrium state is completely determined by any two properties of the system, provided only that the two properties are independently variable at the equilibrium state. The meaning of completely determined is that both the intensive and extensive states of the system are fixed; not only are T, P, and the phase compositions established, but so also are the masses of the phases. VAPOR/LIQUID EQUILIBRIUM Vapor/liquid equilibrium (VLE) relationships (as well as other interphase equilibrium relationships) are needed in the solution of many engineering problems. The required data can be found by experiment, but such measurements are seldom easy, even for binary systems, and they become rapidly more difficult as the number of constituent species increases. This is the incentive for application of thermodynamics to the calculation of phase-equilibrium relationships.
4-25
The general VLE problem involves a multicomponent system of N constituent species for which the independent variables are T, P, N − 1 liquid-phase mole fractions, and N − 1 vapor-phase mole fractions. (Note that i xi = 1 and i yi = 1, where xi and yi represent liquid and vapor mole fractions respectively.) Thus there are 2N independent variables, and application of the phase rule shows that exactly N of these variables must be fixed to establish the intensive state of the system. This means that once N variables have been specified, the remaining N variables can be determined by simultaneous solution of the N equilibrium relations: fˆ l = fˆ v (i = 1, 2, . . . , N) (4-276) i
i
where superscripts l and v denote the liquid and vapor phases, respectively. In practice, either T or P and either the liquid-phase or vapor-phase composition are specified, thus fixing 1 + (N − 1) = N independent variables. The remaining N variables are then subject to calculation, provided that sufficient information is available to allow determination of all necessary thermodynamic properties. Gamma/Phi Approach For many VLE systems of interest the pressure is low enough that a relatively simple equation of state, such as the two-term virial equation, is satisfactory for the vapor phase. Liquid-phase behavior, on the other hand, may be conveniently described by an equation for the excess Gibbs energy, from which activity coefficients are derived. The fugacity of species i in the liquid phase is then given by Eq. (4-102), written fˆ l = γ x f i
i i i
while the vapor-phase fugacity is given by Eq. (4-79), written fˆiv = φˆ iv yi P Equation (4-276) is now expressed as (i = 1, 2, . . . , N ) γi xi fi = φˆ i yiP
(4-277)
The identifying superscripts l and v are omitted here with the understanding that γi and fi are liquid-phase properties, whereas φˆ i is a vapor-phase property. Applications of Eq. (4-277) represent what is known as the gamma/phi approach to VLE calculations. Evaluation of φˆ i is usually by Eq. (4-196), based on the two-term virial equation of state, but other equations, such as Eq. (4-200), are also applicable. The activity coefficient γi is evaluated by Eq. (4-119), which relates ln γi to GE/RT as a partial property. Thus, what is required for the liquid phase is a relation between GE/RT and composition. Equations in common use for this purpose have already been described. The fugacity fi of pure compressed liquid i must be evaluated at the T and P of the equilibrium mixture. This is done in two steps. First, one calculates the fugacity coefficient of saturated vapor φiv = φisat by an integrated form of Eq. (4-161), written for pure species i and evaluated at temperature T and the corresponding vapor pressure P = Pisat. Equation (4-276) written for pure species i becomes f iv = f il = f isat
(4-278)
sat
where fi indicates the value both for saturated liquid and for saturated vapor. The corresponding fugacity coefficient is fisat φisat = ᎏ (4-279) Pisat This fugacity coefficient applies equally to saturated vapor and to saturated liquid at given temperature T. Equation (4-278) can therefore equally well be written φ iv = φ il
(4-280)
The second step is the evaluation of the change in fugacity of the liquid with a change in pressure to a value above or below Pisat. For this isothermal change of state from saturated liquid at Pisat to liquid at pressure P, Eq. (4-105) is integrated to give Gi − Gisat =
P
sat Pi
Vi dP
Equation (4-74) is then written twice: for Gi and for Gisat. Subtraction provides another expression for Gi − Gisat:
4-26
THERMODYNAMICS
fi Gi − Gisat = RT ln ᎏ fisat Equating the two expressions for Gi − Gisat yields P 1 fi ln ᎏ = ᎏ sat Vi dP sat RT Pi fi Since Vi, the liquid-phase molar volume, is a very weak function of P at temperatures well below Tc, an excellent approximation is often obtained when evaluation of the integral is based on the assumption that Vi is constant at the value for saturated liquid, Vil: fi Vil(P − Pisat) ln ᎏ = ᎏᎏ sat fi RT Substituting fisat = φ isatPisat (Eq. [4-279]), and solving for fi gives Vil(P − Pisat) fi = φ isat Pisat exp ᎏᎏ (4-281) RT The exponential is known as the Poynting factor. Equation (4-277) may now be written
yiPΦi = xiγi Pisat
(i = 1, 2, . . . , N)
(4-282)
φˆ i −Vi (P − Pi ) Φi = ᎏ exp ᎏᎏ (4-283) φisat RT If evaluation of φisat and φˆ i is by Eqs. (4-192) and (4-196), this reduces to PB i − PisatBii − Vil(P − Pisat) Φi = exp ᎏᎏᎏ (4-284) RT where Bi is given by Eq. (4-195): ∂(nB) Bi ᎏ = 2 ykBki − B (4-285) ∂ni T,nj k with B evaluated by Eq. (4-183). The N equations represented by Eq. (4-282) in conjunction with Eq. (4-284) may be used to solve for N unspecified phase-equilibrium variables. For a multicomponent system the calculation is formidable, but well suited to computer solution. The types of problems encountered for nonelectrolyte systems at low to moderate pressures (well below the critical pressure) are discussed by Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., McGraw-Hill, New York, 1996). When Eq. (4-282) is applied to VLE for which the vapor phase is an ideal gas and the liquid phase is an ideal solution, it reduces to a very simple expression. For ideal gases, fugacity coefficients φˆ i and φisat are unity, and the right-hand side of Eq. (4-283) reduces to the Poynting factor. For the systems of interest here this factor is always very close to unity, and for practical purposes Φi = 1. For ideal solutions, the activity coefficients γi are also unity. Equation (4-282) therefore reduces to
where
l
sat
yiP = xiPisat
(i = 1, 2, . . . , N)
an equation which expresses Raoult’s law. It is the simplest possible equation for VLE, and as such fails to provide a realistic representation of real behavior for most systems. Nevertheless, it is useful as a standard of comparison. When an appropriate correlating equation for G E is not available, reliable estimates of activity coefficients may often be obtained from a group-contribution correlation. The Analytical Solution of Groups (ASOG) method (Kojima and Tochigi, Prediction of Vapor-Liquid Equilibrium by the ASOG Method, Elsevier, Amsterdam, 1979) and the UNIFAC method are both well developed. Additional references of interest include Hansen et al. (Ind. Eng. Chem. Res., 30, pp. 2352– 2355 [1991]), Gmehling and Schiller (Ibid., 32, pp. 178–193 [1993]); Larsen et al. (Ibid., 26, pp. 2274–2286 [1987]); and Tochigi et al. ( J. Chem. Eng. Japan, 23, pp. 453–463 [1990]). Data Reduction Correlations for GE and the activity coefficients are based on VLE data taken at low to moderate pressures. The ASOG and UNIFAC group-contribution methods depend for validity on parameters evaluated from a large base of such data. The process
of finding a suitable analytic relation for g (GE/RT) as a function of its independent variables T and x1, thus producing a correlation of VLE data, is known as data reduction. Although g is in principle also a function of P, the dependence is so weak as to be universally and properly neglected. Given here is a brief description of the treatment of data taken for binary systems under isothermal conditions. A more comprehensive development is given by Van Ness ( J. Chem. Thermodyn., 27, pp. 113–134 [1995]; Pure & Appl. Chem., 67, pp. 859–872 [1995]). Presumed in all that follows is the existence of an equation inherently capable of representing correct values of GE for the liquid phase as a function of x1: g GE/RT = (x1; α, β, . . .)
(4-286)
where α, β, and so on, represent adjustable parameters. The measured variables of binary VLE are x1, y1, T, and P. Experimental values of the activity coefficient of species i in the liquid are related to these variables by Eq. (4-282), written: y*i P* γ*i = ᎏ Φi (i = 1, 2) (4-287) xiPisat where Φi is given by Eq. (4-283), and the asterisks denote experimental values. A simple summability relation analogous to Eq. (4-123) defines an experimental value of g*: g* x1 ln γ 1* + x2 ln γ 2*
(4-288)
Moreover, Eq. (4-122), the Gibbs/Duhem equation, may be written for experimental values in a binary system as d ln γ1* d ln γ 2* x1 ᎏ + x2 ᎏ = 0 (4-289) dx1 dx1 Because experimental measurements are subject to systematic error, sets of values of ln γ 1* and ln γ 2* determined by experiment may not satisfy, that is, may not be consistent with, the Gibbs/Duhem equation. Thus, Eq. (4-289) applied to sets of experimental values becomes a test of the thermodynamic consistency of the data, rather than a valid general relationship. Values of g given by the correlating equation, Eq. (4-286), are called derived values, and associated derived values of the activity coefficients are given by specialization of Eqs. (4-58): dg γ1 = exp g + x2 ᎏ (4-290) dx1
dg (4-291) γ2 = exp g − x1 ᎏ dx1 These two equations may be combined to yield dg γ1 (4-292) ᎏ = ln ᎏ dx1 γ2 This equation applies to derived property values. The corresponding experimental values are given by differentiation of Eq. (4-288): dg* d ln γ 1* d ln γ 2* ᎏ = x1 ᎏ + ln γ 1* + x2 ᎏ − ln γ 2* dx1 dx1 dx1 dg* γ 1* d ln γ 1* d ln γ 2* ᎏ = ln ᎏ + x1 ᎏ + x2 ᎏ dx1 γ 2* dx1 dx1 Subtraction of Eq. (4-293) from Eq. (4-292) gives
or
(4-293)
γ1 dg dg* γ 1* d ln γ 1* d ln γ 2* ᎏ − ᎏ = ln ᎏ − ln ᎏ − x1 ᎏ + x2 ᎏ dx1 dx1 γ2 γ 2* dx1 dx1 The differences between like terms represent residuals between derived and experimental values. Defining these residuals as γ γ γ 1* δ g g − g* and δ ln ᎏ1 ln ᎏ1 − ln ᎏ γ2 γ2 γ 2* puts this equation into the form dδg γ1 d ln γ 1* d ln γ 2* ᎏ = δ ln ᎏ − x1 ᎏ + x2 ᎏ dx1 γ2 dx1 dx1
EQUILIBRIUM If a data set is reduced so as to make the δg residuals scatter about zero, then the derivative on the left is effectively zero, and the preceding equation becomes γ d ln γ1* d ln γ 2* δ ln ᎏ1 = x1 ᎏ + x2 ᎏ (4-294) γ2 dx1 dx1 The right-hand side of this equation is exactly the quantity that Eq. (4-289), the Gibbs/Duhem equation, requires to be zero for consistent data. The residual on the left is therefore a direct measure of deviations from the Gibbs/Duhem equation. The extent to which values of this residual fail to scatter about zero measures the departure of the data from consistency with respect to this equation. The data-reduction procedure just described provides parameters in the correlating equation for g that make the δg residuals scatter about zero. This is usually accomplished by finding the parameters that minimize the sum of squares of the residuals. Once these parameters are found, they can be used for the calculation of derived values of both the pressure P and the vapor composition y1. Equation (4-282) is solved for yiP and written for species 1 and for species 2. Adding the two equations gives x1γ1P1sat x2γ2P2sat P=ᎏ +ᎏ (4-295) Φ1 Φ2 whence by Eq. (4-282), x1γ1P1sat y1 = ᎏ (4-296) Φ1P These equations allow calculation of the primary residuals: δP P − P*
and
δy1 y1 − y1*
If the experimental values P* and y1* are closely reproduced by the correlating equation for g, then these residuals, evaluated at the experimental values of x1, scatter about zero. This is the result obtained when the data are thermodynamically consistent. When they are not, these residuals do not scatter about zero, and the correlation for g does not properly reproduce the experimental values P* and y1*. Such a correlation is, in fact, unnecessarily divergent. An alternative is to process just the P-x1 data; this is possible because the P-x1-y1 data set includes more information than necessary. Assuming that the correlating equation is appropriate to the data, one merely searches for values of the parameters α, β, and so on, that yield pressures by Eq. (4-295) that are as close as possible to the measured values. The usual procedure is to minimize the sum of squares of the residuals δP. Known as Barker’s method (Austral. J. Chem., 6, pp. 207–210 [1953]), it provides the best possible fit of the experimental pressures. When the experimental data do not satisfy the Gibbs/Duhem equation, it cannot precisely represent the experimental y1 values; however, it provides a better fit than does the procedure that minimizes the sum of the squares of the δg residuals. Worth noting is the fact that Barker’s method does not require experimental y1* values. Thus the correlating parameters α, β, and so on, can be evaluated from a P-x1 data subset. Common practice now is, in fact, to measure just such data. They are, of course, not subject to a test for consistency by the Gibbs/Duhem equation. The world’s store of VLE data has been compiled by Gmehling et al. (VaporLiquid Equilibrium Data Collection, Chemistry Data Series, vol. I, parts 1–8, DECHEMA, Frankfurt am Main, 1979–1990). Solute/Solvent Systems The gamma/phi approach to VLE calculations presumes knowledge of the vapor pressure of each species at the temperature of interest. For certain binary systems species 1, designated the solute, is either unstable at the system temperature or is supercritical (T > Tc ). Its vapor pressure cannot be measured, and its fugacity as a pure liquid at the system temperature f1 cannot be calculated by Eq. (4-281). Equations (4-282) and (4-283) are applicable to species 2, designated the solvent, but not to the solute, for which an alternative approach is required. Figure 4-11 shows a typical plot of the liquidphase fugacity of the solute fˆ1 vs. its mole fraction x1 at constant temperature. Since the curve representing fˆ1 does not extend all the way to x1 = 1, the location of f1, the liquid-phase fugacity of pure species 1, is not established. The tangent line at the origin, representing Henry’s
FIG. 4-11
4-27
Plot of solute fugacity fˆ1 vs. solute mole fraction.
law, provides alternative information. The slope of the tangent line is Henry’s constant, defined as fˆ1 k1 lim ᎏ (4-297) x1→0 x 1 This is the definition of k1 for temperature T and for a pressure equal to the vapor pressure of the pure solvent P2sat. The activity coefficient of the solute at infinite dilution is fˆ1 1 fˆ1 lim γ1 = lim ᎏ = ᎏ lim ᎏ x1→0 x1→0 x f x1→0 x f 1 1 1 1 In view of Eq. (4-297), this becomes γ1∞ = k1/f1, or k1 f1 = ᎏ (4-298) γ ∞1 where γ 1∞ represents the infinite-dilution value of the activity coefficient of the solute. Since both k1 and γ 1∞ are evaluated at P2sat, this pressure also applies to f1. However, the effect of P on a liquid-phase fugacity, given by a Poynting factor, is very small, and for practical purposes may usually be neglected. The activity coefficient of the solute, given by fˆ1 y1P φˆ 1 γ1 ᎏ =ᎏ x1 f1 x1 f1 then becomes y1P φˆ 1γ1∞ γ1 = ᎏ x1k1 For the solute, this equation takes the place of Eqs. (4-282) and (4-283). Solution for y1 gives x1(γ1/γ1∞)k1 y1 = ᎏᎏ (4-299) φˆ 1P For the solvent, species 2, the analog of Eq. (4-296) is x2γ 2 P2sat y2 = ᎏ (4-300) Φ2P Since y1 + y2 = 1, x1(γ1/γ1∞)k1 x2γ2 P2sat P = ᎏᎏ +ᎏ (4-301) φˆ 1 Φ2 Note that the same correlation that provides for the evaluation of γ1 also allows evaluation of γ ∞1 . There remains the problem of finding Henry’s constant from the available VLE data. For equilibrium fˆ1 fˆ1l = fˆ1v = y1Pφˆ 1 Division by x1 gives fˆ1 y1 ᎏ = Pφˆ 1 ᎏ x1 x1
4-28
THERMODYNAMICS
Henry’s constant is defined as the limit as x1 → 0 of the ratio on the left; therefore y k1 = P2sat φˆ 1∞ lim ᎏ1 x1→0 x 1 The limiting value of y1/x1 can be found by plotting y1/x1 vs. x1 and extrapolating to zero. K-Values A measure of how a given chemical species distributes itself between liquid and vapor phases is the equilibrium ratio: y Ki ᎏi (4-302) xi
with the theoretically based Wong/Sandler mixing rules for parameters a and b these equations provide the means for accurate correlation and prediction of VLE data. The first of the Wong/Sandler mixing rules relates the difference in mixture quantities b and a/RT to the corresponding differences (identified by subscripts) for the pure species: a b − ᎏ = xp xq E pq (4-305) RT p q where
1 ap aq E pq ᎏ bp − ᎏ + bq − ᎏ (1 − k pq) 2 RT RT
(4-306)
Usually called simply a K-value, it adds nothing to thermodynamic knowledge of VLE. However, its use may make for computational convenience, allowing formal elimination of one set of mole fractions {yi} or {xi} in favor of the other. Moreover, it characterizes lightness of a constituent species. For a light species, tending to concentrate in the vapor phase, K > 1; for a heavy species, tending to concentrate in the liquid phase, K < 1. Empirical correlations for K-values found in the older literature have little relation to thermodynamics. Their proper evaluation comes directly from Eq. (4-277): γifi y Ki ᎏi = ᎏ (4-303) xi φˆ i P
Binary interaction parameters kpq are determined for each pq pair (p ≠ q) from experimental data. Note that kpq = kqp and kpp = kqq = 0. Since the quantity on the left-hand side of Eq. (4-305) represents the second virial coefficient as predicted by Eq. (4-231), the basis for Eq. (4-305) lies in Eq. (4-183), which expresses the quadratic dependence of the mixture second virial coefficient on mole fraction. The second Wong/Sandler mixing rule relates ratios of a/RT to b: a (4-307) ᎏ=1−D bRT
When Raoult’s law applies, this becomes Ki = Pisat/P. In general, Kvalues are functions of T, P, liquid composition, and vapor composition, making their direct and accurate correlation impossible. Those correlations that do exist are approximate and severely limited in application. The DePriester correlation, for example, gives K-values for light hydrocarbons (Chem. Eng. Prog. Symp. Ser. No. 7, 49, pp. 1–43 [1953]). Equation-of-State Approach Although the gamma/phi approach to VLE is in principle generally applicable to systems comprised of subcritical species, in practice it has found use primarily where pressures are no more than a few bars. Moreover, it is most satisfactory for correlation of constant-temperature data. A temperature dependence for the parameters in expressions for GE is included only for the localcomposition equations, and it is at best only approximate. A generally applicable alternative to the gamma/phi approach results when both the liquid and vapor phases are described by the same equation of state. The defining equation for the fugacity coefficient, Eq. (4-79), may be applied to each phase:
The quantity GE/RT is given by an appropriate correlation for the excess Gibbs energy of the liquid phase, and is evaluated at the mixture composition, regardless of whether the mixture is liquid or vapor. The constant c is specific to the equation of state. The theoretical basis for these equations can be found in the literature (Wong and Sandler, op. cit.; Ind. Eng. Chem. Res., 31, pp. 2033–2039 [1992]; Eubank, et al., Ind. Eng. Chem. Res., 34, pp. 314–323 [1995]). Elimination of a from Eq. (4-305) by Eq. (4-307) provides an expression for b: 1 b = ᎏ xp xq Epq (4-309) D p q
Liquid: Vapor:
fˆil = φˆ il xiP fˆiv = φˆ iv yiP (i = 1, 2, . . . , N)
(4-304)
This introduces the compositions xi and yi into the equilibrium equations, but neither is explicit, because the φˆ i are functions, not only of T and P, but of composition. Thus Eq. (4-304) represents N complex relationships connecting T, P, the xi, and the yi, suitable for computer solution. Given an appropriate equation of state, one or another of Eqs. (4-178) through (4-181) provides for expression of the φˆ i as functions of T, P, and composition. Because of inadequacies in empirical mixing rules, such as those given by Eqs. (4-221) and (4-222), the equation-of-state approach was long limited to systems exhibiting modest and well-behaved deviations from ideal solution behavior in the liquid phase; for example, to systems containing hydrocarbons and cryogenic fluids. However, the introduction by Wong and Sandler (AIChE J., 38, pp. 671–680 [1992]) of a new class of mixing rules for cubic equations of state has greatly expanded their useful application to VLE. The Soave/Redlich/Kwong (SRK) and the Peng/Robinson (PR) equations of state, both expressed by Eqs. (4-230) and (4-231), were developed specifically for VLE calculations. The fugacity coefficients implicit in these equations are given by Eq. (4-232). When combined
GE ap D 1 + ᎏ − xp ᎏ cRT bp RT p
(4-308)
Mixture parameter a then follows from Eq. (4-307): a = bRT(1 − D)
(4-310)
Equations (4-233) and (4-234) may now be applied for the evaluation of partial parameters ai and bi: 1 ln γ ai bi = ᎏ 2 xj Eij − b 1 + ᎏi − ᎏ D c biRT j
and
Equation (4-276) now becomes xi φˆ il = yi φˆ iv
where
ai ln γi bi ai = bRT ᎏ − ᎏ + a ᎏ − 1 bi RT c b
(4-311) (4-312)
For pure species i, Eq. (4-232) reduces to (Vi − bi)Zi ai /biRT Vi + σbi ln φi = Zi − 1 − ln ᎏᎏ + ᎏ ln ᎏ Vi ε−σ Vi + εbi
(4-313)
This equation may be applied separately to the liquid phase and to the vapor phase to yield the pure-species values φil and φiv. For vapor/ liquid equilibrium (Eq. [4-280]), these two quantities are equal. Given parameters ai and bi, the pressure P in Eq. (4-230) that makes these two values equal is Pisat, the equilibrium vapor pressure of pure species i as predicted by the equation of state. The correlations for α(Tri; ωi) that follow Eq. (4-230) are designed to provide values of ai that yield pure-species vapor pressures which, on average, are in reasonable agreement with experiment. However, reliable correlations for Pisat as a function of temperature are available for many pure species. Thus when Pisat is known for a particular temperature, ai should be evaluated so that the equation of state correctly predicts this known value. The procedure is to write Eq. (4-313) for each of the phases, combining the two equations in accord with Eq. (4-280), written ln φil = ln φiv
EQUILIBRIUM The resulting expression may be solved for ai: Vi − bi bi RT(ε − σ) ln ᎏ ᎏ + Ziv − Zil Viv − bi ai = (Vil + σbi)(Viv + εbi) ln ᎏᎏᎏ (Vil + εbi)(Viv + σbi)
l
(4-314)
where ln γi∞ comes from the GE correlation and ln φi is given by Eq. (4-313) written for the liquid phase. Equation (4-315) supplies a value for ln φˆ i∞ which, when used with Eq. (4-232), ultimately (see following) leads to values for kpq. For a binary system comprised of species p and q, Eqs. (4-232), (4-312), and (4-315) may be written for species p at infinite dilution. The three resulting equations are then combined to yield b p∞ ln γ p∞ + ln φp − Mp ᎏ = ᎏᎏ bq Zq − 1
(4-316)
where (Vq − bq)Zq 1 ap ln γ p∞ Vq + σbq Mp −ln ᎏᎏ +ᎏ ᎏ −ᎏ ln ᎏ (4-317) Vq ε − σ bp RT c Vq + εbq By Eq. (4-311) written for species p at infinite dilution in a pq binary, 2Epq ln γ p∞ ap ᎏᎏ − 1 − ᎏᎏ + ᎏᎏ ∞ b b c RT q p bp ᎏ = ᎏᎏᎏ (4-318) aq 1 − ᎏᎏ bq bqRT
Equations (4-316) and (4-318) are set equal, Epq is eliminated by Eq. (4-306), and kpq is replaced by kp, its infinite-dilution value at xp → 0. Solution for kp then yields ln γ + ln φ − M ln γ a ᎏ + b 1 + ᎏᎏ − ᎏᎏ b − ᎏRaᎏT ᎏ Z −1 c b RT q
q
∞ p
p
p
∞ p
q
p
kp = 1 − ᎏᎏᎏᎏᎏᎏ ap aq bp − ᎏᎏ + bq − ᎏᎏ RT RT (4-319) q
p
where ln φp comes from Eq. (4-313). All values in Eq. (4-319) are for the liquid phase at P = Pqsat. The analogous equation for kq, the infinitedilution value of k pq at xq → 0 is written ln γ + ln φ − M ln γ a ᎏ + b 1 + ᎏᎏ − ᎏᎏ b − ᎏRaᎏT ᎏ c Z −1 b RT p
p
∞
q
q
q
∞
p
q
q
kq = 1 − ᎏᎏᎏᎏᎏᎏ ap aq bp − ᎏᎏ + bq − ᎏᎏ RT RT (4-320) p
A second advantage is that the procedure, applied for infinite dilution of each species, yields two values of kpq from which a composition-dependent function can be generated, a simple linear relation proving fully satisfactory: kpq = kp xq + kq xp
where Ziv = Pisat Viv/RT and Zil = Pisat Vil/RT. Values of Viv and Vil come from solution of Eq. (4-230) for each phase with P = Pisat at temperature T. Since a value of ai is required for these calculations, an iterative procedure is implemented with an initial value for ai from the appropriate correlation for α(Tri; ωi). The binary interaction parameters kpq are evaluated from liquidphase GE correlations for binary systems. The most satisfactory procedure is to apply at infinite dilution the relation between a liquid-phase activity coefficient and its underlying fugacity coefficients, γ i∞ = φˆ i∞/φi. Rearrangement of the logarithmic form yields ln φˆ i∞ = ln γ i∞ + ln φi (4-315)
q
where Mq is given by an equation analogous to Eq. (4-317) but with subscripts reversed. All values in Eq. (4-320) are for the liquid phase at P = Ppsat. One advantage of this procedure is that kp and kq are found directly from the pure-species parameters ap, aq, bp, and bq. In addition, the required values of ln γp∞ and ln γq∞ can be found from experimental data for the pq binary system, independent of the correlating expression used for GE.
4-29
(4-321)
The two values kp and kq are usually not very different, and kpq is not strongly composition dependent. Nevertheless, the quadratic dependence of b − (a/RT) on composition indicated by Eq. (4-305) is not exactly preserved. Since this quantity is not a true second virial coefficient, only a value predicted by a cubic equation of state, a strict quadratic dependence is not required. Moreover, the compositiondependent kpq leads to better results than does use of a constant value. The equation-specific constants for the SRK and PR equations are given by the following table:
ε σ Ωa Ωb c
SRK equation
PR equation
0 1 0.42748 0.08664 0.69315
−0.414214 2.414214 0.457235 0.077796 0.62323
Outlined below are the steps required for of a VLE calculation of vapor-phase composition and pressure, given the liquid-phase composition and temperature. A choice must be made of an equation of state. Only the Soave/Redlich/Kwong and Peng/Robinson equations, as represented by Eqs. (4-230) and (4-231), are considered here. These two equations usually give comparable results. A choice must also be made of a two-parameter correlating expression to represent the liquid-phase composition dependence of GE for each pq binary. The Wilson, NRTL (with α fixed), and UNIQUAC equations are of general applicability; for binary systems, the Margules and van Laar equations may also be used. The equation selected depends on evidence of its suitability to the particular system treated. Reasonable estimates of the parameters in the equation must also be known at the temperature of interest. These parameters are directly related to infinite-dilution values of the activity coefficients for each pq binary. Input information includes the known values of T and {xi}, as well as the equation-of-state and GE-expression parameters. Estimates are also needed of P and {yi}, the quantities to be evaluated, and these require some preliminary calculations: 1. For the chosen equation of state (with appropriate values of ε, σ, and c), find values of bi and preliminary values of ai for each species from the information following Eq. (4-230). 2. If the vapor pressure Pisat for species i at temperature T is known, determine a new value for ai by Eqs. (4-314) and (4-230). 3. Evaluate kp and kq by Eqs. (4-319) and (4-320) for each pq binary. 4. Although pressure P is to be determined, an estimate is required to permit any VLE calculations at all. A reasonable initial value is the sum of the pure-species vapor pressures, each weighted by its known liquid-phase mole fraction. 5. The vapor-phase composition is also to be determined, and it, too, is required to initiate calculations. Assuming both the liquid and vapor phases to be ideal solutions, Eqs. (4-98) and (4-304) combine to give φil yi = xi ᎏ φiv Evaluation of the pure-species values φil and φiv by Eq. (4-313) then provides values for yi. Since these are not constrained to sum to unity, they should be normalized to yield an initial vapor-phase composition. Given estimates for P and {yi} an iterative procedure can be initiated: 1. At the known liquid-phase composition, evaluate D by Eq. (4-308), b and a by Eqs. (4-309) and (4-310), and {b i} and {ai} by Eqs. (4-311) and (4-312).
4-30
THERMODYNAMICS
2. Evaluate {φˆ il }. The mixture volume V is determined from the equation of state, Eq. (4-231), applied to the liquid phase at the given composition, T, and P. 3. Repeat the two preceding items for the vapor-phase composition, thus evaluating {φˆ vi }. 4. Eq. (4-304) is now written φˆ l yi = xi ᎏvi φˆ i The values of yi so calculated are normalized by division by i yi. 5. Recalculate the φˆ iv, and continue this iterative procedure until it converges to a fixed value for i yi. This sum is appropriate to the pressure P for which the calculations have been made. Unless the sum is unity, the pressure is adjusted and the iteration process is repeated. Systematic adjustment of pressure P continues until i yi = 1. The pressure and vapor compositions so found are the equilibrium values for the given temperature and liquid-phase composition as predicted by the equation of state. A vast store of liquid-phase excess-property data for binary systems at temperatures near 30°C and somewhat higher is available in the literature. Effective use of these data to extend G E correlations to higher temperatures is critical to the procedure considered here. The key relations are Eq. (4-118),
GE HE d ᎏ = − ᎏ2 dT RT RT
(constant P,x)
and the excess-property analog of Eq. (4-31), dH E = CPE dT
Integration of the first of these equations from T0 to T gives
T
GE GE HE (4-322) ᎏ= ᎏ − ᎏ dT RT RT T0 T0 RT 2 Similarly, the second equation may be integrated from T1 to T:
T
E
CP dT
(4-323)
T1
In addition, we may write ∂CPE dC PE = ᎏ ∂T Integration from T2 to T yields
CPE = CPE2 +
dT
P,x
∂C ᎏ ∂T T
E P
dT
P,x
T2
Combining this equation with Eqs. (4-322) and (4-323) leads to H T T ᎏ − 1 ᎏ − ᎏ RT T T
GE GE ᎏ= ᎏ RT RT
E
T0
1
T1
0
∂C 1 I ᎏ ᎏ RT ∂T
CPE2 T T T −ᎏ ln ᎏ − ᎏ − 1 ᎏ1 − I R T0 T0 T
T
where
T0
2
T
T
T1
T2
E P
(4-324)
dT dT dT
P,x
This general equation makes use of excess Gibbs-energy data at temperature T0, excess enthalpy (heat-of-mixing) data at T1, and excess heat-capacity data at T2. Evaluation of the integral I requires informaE tion with respect to the temperature dependence of CP . Because of the relative paucity of excess heat-capacity data, the most reasonable assumption is that this quantity is constant, independent of T. In this event, the integral is zero, and the closer T0 and T1 are to T, the less the influence of this assumption. When no information is available with E respect to CP , and excess enthalpy data are available at only a single temperature, the excess heat capacity must be assumed zero. In this case only the first two terms on the right-hand side of Eq. (4-324) are retained, and it more rapidly becomes imprecise as T increases. Our primary interest in Eq. (4-324) is its application to binary systems at infinite dilution of one of the constituent species. For this pur-
E P
C T T T − ᎏ ln ᎏ − ᎏ − 1 ᎏ1 x1x2R T0 T0 T As shown by Smith, Van Ness and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., Chap. 11, McGraw-Hill, New York, 1996), GE ln γi∞ ᎏ x1x2 RT xi = 0 The preceding equation may therefore be written HE T T1 ln γi∞ = (ln γi∞)T0 − ᎏ ᎏ−1 ᎏ x1x2 RT T1,xi = 0 T0 T
E P
C T T T − ᎏ ln ᎏ − ᎏ − 1 ᎏ1 (4-325) x1x2 R xi = 0 T0 T0 T The methanol(1)/acetone(2) system serves as a specific example in conjunction with the Peng/Robinson equation of state. At a base temperature T0 of 323.15 K (50°C), both VLE data (Van Ness and Abbott, Int. DATA Ser., Ser. A, Sel. Data Mixtures, 1978, p. 67 [1978]) and excess enthalpy data (Morris, et al., J. Chem. Eng. Data, 20, pp. 403– 405 [1975]) are available. From the former, (ln γ 1∞)T0 = 0.6281
(constant P,x)
HE = H1E +
E
pose, we divide Eq. (4-324) by the product x1x2. For CP independent of T (and thus with I = 0), it then becomes GE GE HE T T1 ᎏ= ᎏ − ᎏ ᎏ−1 ᎏ x1x2RT x1x2RT T0 x1x2RT T1 T0 T
and
(ln γ 2∞)T0 = 0.6557
and from the latter HE HE = 1.3636 and = 1.0362 ᎏ ᎏ x1x2 RT T0,x 1 = 0 x1x2 RT T0 ,x 2 = 0 The Margules equations (Eqs. [4-244], [4-245], and [4-246]) are well suited to this system, and the parameters for this equation are given as
A12 = ln γ 1∞
and
A21 = ln γ 2∞
This information allows prediction of VLE at 323.15 K and at the higher temperatures, 372.8, 397.7, and 422.6 K, for which measured VLE values are given by Wilsak, et al. (Fluid Phase Equilibria, 28, pp. 13–37 [1986]). Values of ln γi∞ and hence of the Margules parameters at the higher temperatures are given by Eq. (4-325) with CPE = 0. The pure-species vapor pressures in all cases are the measured values reported with the data sets. Results of these calculations are displayed in Table 4-1, where the parentheses enclose values from the gamma/ phi approach as reported in the papers cited. The results at 323.15 K (581.67 R) show both the suitability of the Margules equation for correlation of data for this system and the capability of the equation-of-state method to reproduce the data. Results for the three higher temperatures indicate the quality of predictions based only on vapor-pressure data for the pure species and on mixture data at 323.15 K (581.67 R). Extrapolations based on the same data to still higher temperatures can be expected to become progressively less accurate. When Eq. (4-325) can no longer be expected to produce reasonable values, better results are obtained for higher temperatures by assuming that the parameters, A12, A21, k1, and k2, do not change further at still-higher temperatures. This is also the course to be followed for extrapolation to supercritical temperatures. Only the Wilson, NRTL, and UNIQUAC equations are suited to the treatment of multicomponent systems. For such systems, the parameters are determined for pairs of species exactly as for binary systems. Examples treating the calculation of VLE are given in Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., Chap. 12, McGraw-Hill, New York, 1996). LIQUID/LIQUID AND VAPOR/LIQUID/LIQUID EQUILIBRIA Equation (4-273) is the basis for both liquid/liquid equilibria (LLE) and vapor/liquid/liquid equilibria (VLLE). Thus, for LLE with superscripts α and β denoting the two phases, Eq. (4-273) is written
EQUILIBRIUM TABLE 4-1
VLE Results for Methanol(1)/Acetone(2)
T, K
ln γ1∞
ln γ 2∞
k1
k2
RMS δP, kPa
RMS % δP
323.15
0.6281 (0.6281) 0.4465 (0.4607) 0.3725 (0.3764) 0.3072 (0.3079)
0.6557 (0.6557) 0.5177 (0.5271) 0.4615 (0.4640) 0.4119 (0.3966)
0.1395
0.0955
0.12
0.1432
0.1056
0.1454
0.1118
0.1480
0.1192
0.08 (0.06) 0.85 (0.83) 2.46 (1.39) 7.51 (2.38)
372.8 397.7 422.6
fˆiα = fˆiβ
(i = 1, 2, . . . , N)
(4-326)
Eliminating fugacities in favor of activity coefficients gives x αi γ iα = xiβ γ iβ
(i = 1, 2, . . . , N)
(4-327)
For most LLE applications, the effect of pressure on the γ i can be ignored, and thus Eq. (4-327) constitutes a set of N equations relating equilibrium compositions to each other and to temperature. For a given temperature, solution of these equations requires a single expression for the composition dependence of GE suitable for both liquid phases. Not all expressions for GE suffice, even in principle, because some cannot represent liquid/liquid phase splitting. The UNIQUAC equation is suitable, and therefore prediction is possible by the UNIFAC method. A special table of parameters for LLE calculations is given by Magnussen, et al. (Ind. Eng. Chem. Process Des. Dev., 20, pp. 331–339 [1981]). A comprehensive treatment of LLE is given by Sorensen, et al. (Fluid Phase Equilibria, 2, pp. 297–309 [1979]; 3, pp. 47–82 [1979]; 4, pp. 151–163 [1980]). Data for LLE are collected in a three-part set compiled by Sorensen and Arlt (Liquid-Liquid Equilibrium Data Collection, Chemistry Data Series, vol. V, parts 1–3, DECHEMA, Frankfurt am Main, 1979–1980). For vapor/liquid/liquid equilibria, Eq. (4-273) gives fˆiα = fˆiβ = fˆiv (i = 1, 2, . . . , N) (4-328) where α and β designate the two liquid phases. With activity coefficients applied to the liquid phases and fugacity coefficients to the vapor phase, the 2N equilibrium equations for subcritical VLLE are xiα γiα fiα = yi φˆ iP (all i) (4-329) xiβ γiβ fiβ = yi φˆ iP
E
As for LLE, an expression for G capable of representing liquid/liquid phase splitting is required; as for VLE, a vapor-phase equation of state for computing the φˆ i is also needed.
0.32 0.55
∆nN, j ∆n1, j ∆n2, j (4-331) ᎏ=ᎏ=⋅⋅⋅=ᎏ ν1, j ν2, j νN, j Since all of these terms are equal, they can be equated to the change in a single quantity εj, called the reaction coordinate for reaction j, thereby giving i = 1, 2, . . . , N ∆ni, j = νi, j ∆εj (4-332) j = I, II, . . . , r
Since the total change in mole number ∆ni is just the sum of the changes ∆ni,j resulting from the various reactions, ∆ni = ∆ni, j = νi, j ∆εj j
sign (νi,j ) =
(4-333)
If the initial number of moles of species i is ni0 and if the convention is adopted that εj = 0 for each reaction in this initial state, then ni = ni0 + νi, j εj
(i = 1, 2, . . . , N)
(4-334)
j
Equation (4-334) is the basic expression of material balance for a closed system in which r chemical reactions occur. It shows for a reacting system that at most r mole number–related quantities εj are capable of independent variation. Note the absence of implied restrictions with respect to chemical-reaction equilibria; the reactioncoordinate formalism is merely an accounting scheme, valid for tracking the progress of each reaction to any arbitrary level of conversion. The reaction coordinate has units of moles. A change in εj of 1 mole signifies a mole of reaction, meaning that reaction j has proceeded to such an extent that the change in mole number of each reactant and product is equal to its stoichiometric number. CHEMICAL-REACTION EQUILIBRIA The general criterion of chemical-reaction equilibria is given by Eq. (4-274). For a system in which just a single reaction occurs, Eq. (4-334) becomes dni = νi dε
whence
Substitution for dni in Eq. (4-274) leads to
νµ = 0 i i
where the |νi| are stoichiometric coefficients and the Ai stand for chemical formulas. The νi themselves are called stoichiometric numbers, and associated with them is a sign convention such that the value is positive for a product and negative for a reactant. More generally, for a system containing N chemical species, any or all of which can participate in r chemical reactions, the reactions can be represented by the equations:
where
(i = 1, 2, . . . , N)
j
|ν1|A1 + |ν2|A2 + ⋅ ⋅ ⋅ → |ν3|A3 + |ν4|A4 + ⋅ ⋅ ⋅
( j = I, II, . . . , r)
0.004 (0.006) 0.014 (0.013) 0.009 (0.006)
ni = ni0 + νiε
Consider a phase in which a chemical reaction occurs according to the equation
0 = νi, j Ai
RMS δy1
0.22
CHEMICAL-REACTION STOICHIOMETRY
i
4-31
(4-330)
− for a reactant species
+ for a product species
If species i does not participate in reaction j, then νi,j = 0. The stoichiometric numbers provide relations among the changes in mole numbers of chemical species which occur as the result of chemical reaction. Thus, for reaction j:
(4-335)
i
Generalization of this result to multiple reactions produces
ν
i, j
µi = 0
( j = I, II, . . . , r)
(4-336)
i
Standard Property Changes of Reaction A standard property change for the reaction aA + bB → lL + mM is defined as the property change that occurs when a moles of A and b moles of B in their standard states at temperature T react to form l moles of L and m moles of M in their standard states also at temperature T. A standard state of species i is its real or hypothetical state as a pure species at temperature T and at a standard-state pressure P°. The standard property change of reaction j is given the symbol ∆Mj°, and its general mathematical definition is ∆Mj° νi, j Mi° i
(4-337)
4-32
THERMODYNAMICS
For species present as gases in the actual reactive system, the standard state is the pure ideal gas at pressure P°. For liquids and solids, it is usually the state of pure real liquid or solid at P°. The standard-state pressure P° is fixed at 100 kPa. Note that the standard states may represent different physical states for different species; any or all of the species may be gases, liquids, or solids. The most commonly used standard property changes of reaction are ∆Gj° νi, jGi° = νi,j µi°
(4-338)
∆Hj° νi, j Hi°
(4-339)
∆C°Pi νi, j C°Pi
(4-340)
i
i
i
i
The standard Gibbs-energy change of reaction ∆Gj° is used in the calculation of equilibrium compositions. The standard heat of reaction ∆Hj° is used in the calculation of the heat effects of chemical reaction, and the standard heat-capacity change of reaction is used for extrapolating ∆Hj° and ∆Gj° with T. Numerical values for ∆Hj° and ∆Gj° are computed from tabulated formation data, and ∆C°Pi is determined from empirical expressions for the T dependence of the C°Pi (see, e.g., Eq. [4-142]). Equilibrium Constants For practical application, Eq. (4-336) must be reformulated. The initial step is elimination of the µi in favor of fugacities. Equation (4-74) for species i in its standard state is subtracted from Eq. (4-77) for species i in the equilibrium mixture, giving µi = Gi° + RT ln âi
(4-341)
where, by definition, âi fˆi /fi° and is called an activity. Substitution of this equation into Eq. (4-341) yields, upon rearrangement,
[ν
(Gi° + RT ln âi)] = 0
i, j
i
or
(ν
Gi°) + RT ln âiνi,j = 0
i, j
i
i
− (νi,jGi°)
ln âiνi, j = ᎏᎏ RT i The right-hand side of this equation is a function of temperature only for given reactions and given standard states. Convenience suggests setting it equal to ln Kj; whence i
or
(y φˆ ) i i
νi, j
i
νj
P ᎏ P°
= Kj
(all j)
(4-344)
where νj i νi,j and P° is the standard-state pressure of 100 kPa, expressed in the same units used for P. The yi may be eliminated in favor of equilibrium values of the reaction coordinates εj. Then, for fixed temperature Eqs. (4-344) relate the εj to P. In principle, specification of the pressure allows solution for the εj. However, the problem may be complicated by the dependence of the φˆ i on composition, that is, on the εj. If the equilibrium mixture is assumed an ideal solution, then each φˆ i becomes φi, the fugacity coefficient of pure species i at the mixture T and P. This quantity does not depend on composition and may be determined from experimental data, from a generalized correlation, or from an equation of state. An important special case of Eq. (4-344) is obtained for gas-phase reactions when the phase can be assumed an ideal gas. In this event φˆ i = 1, and P νj = Kj (all j) (4-345) i (yi)νi, j ᎏ P° In the general case the evaluation of the φˆ i requires an iterative process. An initial step is to set the φˆ i equal to unity and to solve the problem by Eq. (4-345). This provides a set of yi values, allowing evaluation of the φˆ i by, for example, Eq. (4-196), (4-200), or (4-231). Equation (4-344) can then be solved for a new set of yi values, and the process continues to convergence. For liquid-phase reactions, Eq. (4-342) is modified by introduction of the activity coefficient, γi = fˆi /xi fi, where xi is the liquid-phase mole fraction. The activity is then f fˆi âi ᎏ = γi xi ᎏi fi° fi° Both fi and fi° represent fugacity of pure liquid i at temperature T, but at pressures P and P°, respectively. Except in the critical region, pressure has little effect on the properties of liquids, and the ratio fi/fi° is often taken as unity. When this is not acceptable, this ratio is evaluated by the equation f 1 P Vi(P − P°) ln ᎏi = ᎏ Vi dP ᎏᎏ fi° RT P° RT When the ratio fi /fi° is taken as unity, âi = γi xi, and Eq. (4-342) becomes
(γ x ) i i
νi,j
= Kj
(all j)
(4-346)
i
â
i
νi, j
= Kj
(all j)
(4-342)
i
−∆Gj° (4-343) Kj exp ᎏ RT Quantity Kj is the chemical-reaction equilibrium constant for reaction j, and ∆Gj° is the corresponding standard Gibbs-energy change of reaction (see Eq. [4-338]). Although called a “constant,” Kj is a function of T, but only of T. The activities in Eq. (4-342) provide the connection between the equilibrium states of interest and the standard states of the constituent species, for which data are presumed available. The standard states are always at the equilibrium temperature. Although the standard state need not be the same for all species, for a particular species it must be the state represented by both G°i and the fi° upon which the activity âi is based. The application of Eq. (4-342) requires explicit introduction of composition variables. For gas-phase reactions this is accomplished through the fugacity coefficient: âi fˆi/fi° = yi φˆ iP/fi° where
and Eq. (4-342) becomes
However, the standard state for gases is the ideal gas state at the standard-state pressure, for which fi° = P°. Therefore yi φˆ iP âi = ᎏ P°
Here the difficulty is to determine the γi , which depend on the xi . This problem has not been solved for the general case. Two courses are open: the first is experiment; the second, assumption of solution ideality. In the latter case, γi = 1, and Eq. (4-346) reduces to
(x ) i
νi,j
= Kj
(all j)
(4-347)
i
the law of mass action. The significant feature of Eqs. (4-345) and (4-347), the simplest expressions for gas- and liquid-phase reaction equilibrium, is that the temperature-, pressure-, and compositiondependent terms are distinct and separate. Example 2: Single-Reaction Equilibrium Consider the equilibrium state at 1,000 K and atmospheric pressure for the reaction CO + H2O → CO2 + H2 Let the feed stream contain 3 mol CO, 1 mol H2O, and 2 mol CO2 for every mole of H2 present. This initial constitution forms the basis for calculation, and for this single reaction, Eq. (4-334) becomes ni = ni0 + νiε. Whence nCO = 3 − ε nH2O = 1 − ε nCO2 = 2 + ε nH2 = 1 + ε
n =7 i
i
Each mole fraction is therefore given by yi = ni/7.
EQUILIBRIUM At 1,000 K, ∆G° = −2680 J per mole of reaction; whence by Eq. (4-343) 2680 K = exp ᎏᎏ = 1.38 (8.314)(1000) For the given conditions, the assumption of ideal gases is appropriate; Eq. (4-345) written for a single reaction (subscript j omitted) with ν = 0 becomes
ᎏ2 7+ᎏε ᎏ1 7+ᎏε = K = 1.38 y = ᎏᎏ ᎏ3 7−ᎏε ᎏ1 7−ᎏε i
or
νi i
limited to gas-phase reactions for which the problem is to find the equilibrium composition for given T and P and for a given initial feed. 1. Formulate the constraining material-balance equations, based on conservation of the total number of atoms of each element in a system comprised of w elements. Let subscript k identify a particular atom, and define Ak as the total number of atomic masses of the kth element in the feed. Further, let aik be the number of atoms of the kth element present in each molecule of chemical species i. The material balance for element k is then
na
i ik
na
or
ε = 0.258
i ik
2.
nCO = 2.74 mol nH2O = 0.74 mol nCO2 = 2.26 mol nH2 = 1.26 mol
yCO = 0.391 yH2O = 0.106 yCO2 = 0.323 yH2 = 0.180
i ni = 7.00 mol
i yi = 1.000
(4-353)
− Ak = 0
(k = 1, 2, . . . , w)
− Ak = 0
(k = 1, 2, . . . , w)
λ n a k
i ik
k
∆CP° ᎏ dT R
(4-351)
where for simplicity subscript j has been suppressed. A convenient integrated form of Eq. (4-349) is −∆G° ∆H0° − ∆G0° ∆H° 1 ln K = ᎏ = ᎏᎏ −ᎏ+ᎏ RT RT RT T
i ik
Summed over k, these equations give
Integration of Eq. (4-350) from reference temperature T0 (usually 298.15 K) to temperature T gives T0
n a i
For an endothermic reaction ∆Hj° is positive; for an exothermic reaction it is negative. The temperature dependence of ∆ Hj° is given by d∆Hj° (4-350) ᎏ = ∆C°Pj dT
T
Multiply each element balance by λk, a Lagrange multiplier: λk
The effect of temperature on the equilibrium constant follows from Eq. (4-106): d(∆Gj°/RT) −∆Hj° (4-348) ᎏᎏ = ᎏ dT RT 2 The total derivative is appropriate here because property changes of reaction are functions of temperature only. In combination with Eq. (4-343) this gives d ln Kj ∆ Hj° (4-349) ᎏ=ᎏ dT RT 2
(k = 1, 2, . . . , w)
i
Thus, for the equilibrium mixture,
∆H° = ∆H0° + R
= Ak
i
(2 + ε)(1 + ε) ᎏᎏ = 1.38 (3 − ε)(1 − ε)
whence
4-33
T
T0
CP° ᎏ dT R
(4-352)
where ∆H°/RT is given by Eq. (4-351). In the more extensive compilations of data, values of ∆G° and ∆H° for formation reactions are given for a wide range of temperatures, rather than just at the reference temperature of 298.15 K. (See in particular TRC Thermodynamic Tables—Hydrocarbons and TRC Thermodynamic Tables—Non-hydrocarbons, serial publications of the Thermodynamics Research Center, Texas A & M University System, College Station, Tex.; “The NBS Tables of Chemical Thermodynamic Properties,” J. Physical and Chemical Reference Data, 11, supp. 2 [1982]. Where data are lacking, methods of estimation are available; these are reviewed by Reid, Prausnitz, and Poling, The Properties of Gases and Liquids, 4th ed., Chap. 6, McGraw-Hill, New York, 1987. For an estimation procedure based on molecular structure, see Constantinou and Gani, Fluid Phase Equilibria, 103, pp. 11–22 [1995]. (See also Sec. 2.) Complex Chemical-Reaction Equilibria When the composition of an equilibrium mixture is determined by a number of simultaneous reactions, calculations based on equilibrium constants become complex and tedious. A more direct procedure (and one suitable for general computer solution) is based on minimization of the total Gibbs energy Gt in accord with Eq. (4-271). The treatment here is
3.
i
− Ak = 0
Form a function F by addition of this sum to Gt: F = Gt + λk k
n a
i ik
i
− Ak
t
Function F is identical with G , because the summation term is zero. However, the partial derivatives of F and Gt with respect to ni are different, because function F incorporates the constraints of the material balances. 4. The minimum value of both F and Gt is found when the partial derivatives of F with respect to ni are set equal to zero: ∂F ∂Gt = ᎏ + λkaik = 0 ᎏ ∂ni T,P,nj ∂ni T,P,nj k The first term on the right is the definition of the chemical potential; whence
µi + λk aik = 0
(i = 1, 2, . . . , N)
(4-354)
k
However, the chemical potential is given by Eq. (4-341); for gas-phase reactions and standard states as the pure ideal gases at P°, this equation becomes fˆi µi = G°i + RT ln ᎏ P° If Gi° is arbitrarily set equal to zero for all elements in their standard states, then for compounds G°i = ∆Gf°, the standard Gibbs-energy i change of formation for species i. In addition, the fugacity is eliminated in favor of the fugacity coefficient by Eq. (4-79), fˆi = yiφˆ iP. With these substitutions, the equation for µi becomes yi φˆ iP µi = ∆Gf°i + RT ln ᎏ P° Combination with Eq. (4-354) gives yi φˆ iP ∆Gf°i + RT ln ᎏ + λkaik = 0 P° k
(i = 1, 2, . . . , N)
(4-355)
If species i is an element, ∆Gf°i is zero. There are N equilibrium equations (Eqs. [4-355]), one for each chemical species, and there are w material-balance equations (Eqs. [4-353]), one for each element—a total of N + w equations. The unknowns in these equations are the ni (note that yi = ni / i ni), of which there are N, and the λk, of which there are w—a total of N + w unknowns. Thus, the number of equations is sufficient for the determination of all unknowns. Equation (4-355) is derived on the presumption that the φˆ i are known. If the phase is an ideal gas, then each φˆ i is unity. If the phase is an ideal solution, each φˆ i becomes φi, and can at least be estimated. For real gases, each φˆ i is a function of the yi , the quantities being calculated. Thus an iterative procedure is indicated, initiated with each φˆ i
4-34
THERMODYNAMICS
set equal to unity. Solution of the equations then provides a preliminary set of yi. For low pressures or high temperatures this result is usually adequate. Where it is not satisfactory, an equation of state with the preliminary yi gives a new and more nearly correct set of φˆ i for use in Eq. (4-355). Then a new set of yi is determined. The process is repeated to convergence. All calculations are well suited to computer solution. In this procedure, the question of what chemical reactions are involved never enters directly into any of the equations. However, the choice of a set of species is entirely equivalent to the choice of a set of independent reactions among the species. In any event, a set of species or an equivalent set of independent reactions must always be assumed, and different assumptions produce different results. Example 3: Minimization of Gibbs Energy Calculate the equilibrium compositions at 1,000 K and 1 bar of a gas-phase system containing the species CH4, H2O, CO, CO2, and H2. In the initial unreacted state there are present 2 mol of CH4 and 3 mol of H2O. Values of ∆Gf° at 1,000 K are ∆G°f CH = 19,720 J/mol
The five equations for the five species then become: CH4:
19,720 nCH λC 4λH ᎏ + ln ᎏ4 + ᎏ + ᎏ = 0 i ni RT RT RT
H2O:
−192,420 nH2O 2λH λO +ᎏ+ᎏ=0 ᎏᎏ + ln ᎏ i ni RT RT RT
CO:
−200,240 nCO λC λO ᎏᎏ + ln ᎏ + ᎏ + ᎏ = 0 i ni RT RT RT
CO2:
−395,790 nCO λC 2λO ᎏᎏ + ln ᎏ2 + ᎏ + ᎏ = 0 i ni RT RT RT
H2:
nH2 2λH ln ᎏ +ᎏ=0 i ni RT
The three material-balance equations (Eq. [4-353]) are:
∆G°f H O = −192,420 J/mol
C:
nCH4 + nCO + nCO2 = 2
∆G°f CO = −200,240 J/mol
H:
4nCH4 + 2nH2O + 2nH2 = 14
∆G°f CO = −395,790 J/mol
O:
nH2O + nCO + 2nCO2 = 3
4
2
2
The required values of Ak are determined from the initial numbers of moles, and the values of aik come directly from the chemical formulas of the species. These are shown in the accompanying table.
Simultaneous computer solution of these eight equations, with RT = 8,314 J/mol and
n = n i
Carbon
Oxygen
Hydrogen
Ak = no. of atomic masses of k in the system AC = 2 Species i CH4 H2O CO CO2 H2
AO = 3
AH = 14
aik = no. of atoms of k per molecule of i aCH4,C = 1 aH2O,C = 0 aCO,C = 1 aCO2,C = 1 aH2,C = 0
aCH4,O = 0 aH2O,O = 1 aCO,O = 1 aCO2,O = 2 aH2,O = 0
CH4
+ nH2O + nCO + nCO2 + nH2
i
Element k
aCH4,H = 4 aH2O,H = 2 aCO,H = 0 aCO2,H = 0 aH2,H = 2
At 1 bar and 1,000 K the assumption of ideal gases is justified, and the φˆ i are all unity. Since P = 1 bar, Eq. (4-355) is written: ∆Gf° ni λk ᎏi + ln ᎏ + ᎏ aik = 0 RT Σi ni RT k
produces the following results (yi = ni/i ni): yCH4 = 0.0196
λC ᎏ = 0.7635 RT
yH2O = 0.0980 yCO = 0.1743
λO ᎏ = 25.068 RT
yCO2 = 0.0371 yH2 = 0.6711
λH ᎏ = 0.1994 RT
y = 1.000 i
i
The values of λk/RT are of no significance, but are included to make the results complete.
THERMODYNAMIC ANALYSIS OF PROCESSES Real irreversible processes can be subjected to thermodynamic analysis. The goal is to calculate the efficiency of energy use or production and to show how energy loss is apportioned among the steps of a process. The treatment here is limited to steady-state, steady-flow processes, because of their predominance in chemical technology. CALCULATION OF IDEAL WORK In any steady-state, steady-flow process requiring work, a minimum amount must be expended to bring about a specific change of state in the flowing fluid. In a process producing work, a maximum amount is attainable for a specific change of state in the flowing fluid. In either case, the limiting value obtains when the specific change of state is accomplished completely reversibly. The implications of this requirement are: 1. The process is internally reversible within the control volume. 2. Heat transfer external to the control volume is reversible. The second item means that heat exchange between system and surroundings must occur at the temperature of the surroundings, presumed to constitute a heat reservoir at a constant and uniform temperature
Tσ. This may require Carnot engines or heat pumps internal to the system that provide for the reversible transfer of heat from the temperature of the flowing fluid to that of the surroundings. Since Carnot engines and heat pumps are cyclic, they undergo no net change of state. The entropy change of the surroundings, found by integration of Eq. (4-3), is ∆Sσ = Qσ/Tσ; whence Qσ = Tσ ∆Sσ (4-356) Since heat transfer with respect to the surroundings and with respect to the system are equal but of opposite sign, Qσ = −Q. Moreover, the second law requires for a reversible process that the entropy changes of system and surroundings be equal but of opposite sign: ∆Sσ = −∆St. Equation (4-356) can therefore be written Q = Tσ∆St. In terms of rates this becomes ˙ = Tσ ∆(Sm) Q ˙ fs (4-357) ˙ where Q = rate of heat transfer with respect to the system m ˙ = mass rate of flow of fluid In addition, ∆ denotes the difference between exit and entrance streams, and fs indicates that the term applies to all flowing streams.
THERMODYNAMIC ANALYSIS OF PROCESSES The energy balance for a steady-state steady-flow process resulting from the first law of thermodynamics is 1 ˙ +W ˙s ∆ H + ᎏ u2 + zg m ˙ =Q (4-358) fs 2 where H = specific enthalpy of flowing fluid u = velocity of flowing fluid z = elevation of flowing fluid above datum level g = local acceleration of gravity Ws = shaft work ˙ in Eq. (4-358) by Eq. (4-357) gives Eliminating Q
1 ˙ s(rev) ∆ H + ᎏ u2 + zg m ˙ = Tσ ∆(Sm) ˙ fs + W fs 2 ˙ s(rev) indicates that the shaft work is for a completely where W ˙ ideal. Thus reversible process. This work is called the ideal work W 1 2 ˙ ideal = ∆ H + ᎏ W u + zg m ˙ − Tσ ∆(Sm) ˙ fs (4-359) fs 2 In most applications to chemical processes, the kinetic- and potential-energy terms are negligible compared with the others; in this event Eq. (4-359) is written ˙ ideal = ∆(Hm) W ˙ fs − Tσ ∆(Sm) ˙ fs (4-360)
For the special case of a single stream flowing through the system, Eq. (4-360) becomes ˙ ideal = m(∆H W ˙ − Tσ ∆S) (4-361) Division by m ˙ puts this equation on a unit-mass basis Wideal = ∆H − Tσ ∆S (4-362) A completely reversible processes is hypothetical, devised solely to find the ideal work associated with a given change of state. Its only connection with an actual process is that it brings about the same change of state as the actual process, allowing comparison of the actual work of a process with the work of the hypothetical reversible process. Equations (4-359) through (4-362) give the work of a completely reversible process associated with given property changes in the flowing streams. When the same property changes occur in an actual ˙ s (or Ws) is given by an energy balance, and process, the actual work W comparison can be made of the actual work with the ideal work. When ˙ ideal (or Wideal) is positive, it is the minimum work required to bring W about a given change in the properties of the flowing streams, and is ˙ s. In this case a thermodynamic efficiency ηt is defined smaller than W as the ratio of the ideal work to the actual work: ˙ ideal W ηt (work required) = ᎏ (4-363) ˙s W ˙ ideal| is the maximum work ˙ ideal (or Wideal) is negative, | W When W obtainable from a given change in the properties of the flowing ˙ s|. In this case, the thermodynamic effistreams, and is larger than |W ciency is defined as the ratio of the actual work to the ideal work: ˙s W ηt(work produced) = ᎏ (4-364) ˙ ideal W LOST WORK Work that is wasted as the result of irreversibilities in a process is ˙ lost, and is defined as the difference between the called lost work W actual work of a process and the ideal work for the process. Thus, by definition, Wlost Ws − Wideal (4-365) In terms of rates this is written ˙ lost W ˙ s −W ˙ ideal W (4-366) The actual work rate comes from Eq. (4-358) 1 2 ˙ s=∆ H+ᎏ ˙ W u + zg m ˙ −Q fs 2 Subtracting the ideal work rate as given by Eq. (4-359) yields ˙ ˙ lost = Tσ∆(Sm) ˙ fs − Q (4-367) W
4-35
For the special case of a single stream flowing through the control volume, ˙ lost = mT ˙ W ˙ σ∆S − Q (4-368) Division of this equation by m ˙ gives Wlost = Tσ∆S − Q
(4-369)
where the basis is now a unit amount of fluid flowing through the control volume. The total rate of entropy increase (in both system and surroundings) as a result of a process is ˙ Q S˙ total = ∆(Sm) ˙ fs − ᎏ (4-370) Tσ For a single stream, division by m ˙ provides an equation based on a unit amount of fluid flowing through the control volume: Q Stotal = ∆S − ᎏ (4-371) Tσ Multiplication of Eq. (4-370) by Tσ gives ˙ ˙ fs − Q TσS˙ total = Tσ∆(Sm) Since the right-hand sides of this equation and of Eq. (4-367) are identical, it follows that ˙ lost = TσS˙ total W (4-372) For flow of a single stream on the basis of a unit amount of fluid, this becomes Wlost = TσStotal
(4-373)
Since the second law of thermodynamics requires that S˙ total ≥ 0 and Stotal ≥ 0 it follows that ˙ lost ≥ 0 W
and
Wlost ≥ 0
When a process is completely reversible, the equality holds, and the lost work is zero. For irreversible processes the inequality holds, and the lost work, that is, the energy that becomes unavailable for work, is positive. The engineering significance of this result is clear: The greater the irreversibility of a process, the greater the rate of entropy production and the greater the amount of energy that becomes unavailable for work. Thus, every irreversibility carries with it a price. ANALYSIS OF STEADY-STATE, STEADY-FLOW PROCESSES Many processes consist of a number of steps, and lost-work calculations are then made for each step separately. Writing Eq. (4-372) for each step of the process and summing gives
W˙ lost = Tσ S˙ total Dividing Eq. (4-372) by this result yields ˙ lost S˙ total W =ᎏ ᎏ ˙ lost S˙ total W Thus, an analysis of the lost work, made by calculation of the fraction that each individual lost-work term represents of the total lost work, is the same as an analysis of the rate of entropy generation, made by expressing each individual entropy-generation term as a fraction of the sum of all entropy-generation terms. An alternative to the lost-work or entropy-generation analysis is a work analysis. This is based on Eq. (4-366), written ˙ lost = W ˙ s −W ˙ ideal W (4-374) For a work-requiring process, all of these work quantities are positive ˙ s >W ˙ ideal. The preceding equation is then expresed as and W ˙ ideal + W ˙ lost ˙ s =W (4-375) W A work analysis gives each of the individual work terms on the right as ˙ s. a fraction of W
4-36
THERMODYNAMICS TABLE 4-2
States and Values of Properties for the Process of Fig. 4-12*
Point
P, bar
T, K
Composition
State
H, J/mol
S, J/(mol ⋅ K)
1 2 3 4 5 6 7
55.22 1.01 1.01 55.22 1.01 1.01 1.01
300 295 295 147.2 79.4 90 300
Air Pure O2 91.48% N2 Air 91.48% N2 pure O2 Air
Superheated Superheated Superheated Superheated Saturated vapor Saturated vapor Superheated
12,046 13,460 12,074 5,850 5,773 7,485 12,407
82.98 118.48 114.34 52.08 75.82 83.69 117.35
*Properties on the basis of Miller and Sullivan, U.S. Bur. Mines Tech. Pap. 424 (1928).
˙ s and W ˙ ideal are negative, and For a work-producing process, W ˙ ideal | > |W ˙ s|. Equation (4-374) in this case is best written: |W ˙ s| + ˙ lost ˙ ideal| = |W W (4-376) |W
A work analysis here expresses each of the individual work terms on ˙ ideal|. A work analysis cannot be carried out the right as a fraction of |W ˙ ideal is negative, indiin the case where a process is so inefficient that W ˙ s is positive, indicating that the process should produce work, but W cating that the process in fact requires work. A lost-work or entropy-generation analysis is always possible. Example 4: Lost-Work Analysis Make a work analysis of a simple Linde system for the separation of air into gaseous oxygen and nitrogen, as depicted in Fig. 4-12. Table 4-2 lists a set of operating conditions for the numbered points of the diagram. Heat leaks into the column of 147 J/mol of entering air and into the exchanger of 70 J/mol of entering air have been assumed. Take Tσ = 300 K. The basis for analysis is 1 mol of entering air, assumed to contain 79 mol % N2 and 21 mol % O2. By a material balance on the nitrogen, 0.79 = 0.9148 x; whence x = 0.8636 mol of nitrogen product 1 − x = 0.1364 mol of oxygen product Calculation of Ideal Work If changes in kinetic and potential energies are neglected, Eq. (4-360) is applicable. From the tabulated data, ∆(Hm) ˙ fs = (13,460)(0.1364) + (12,074)(0.8636) − (12,407)(1) = −144 J ∆(Sm) ˙ fs = (118.48)(0.1364) + (114.34)(0.8636) − (117.35)(1) = −2.4453 J/K
Thus, by Eq. (4-360), ˙ ideal = −144 − (300)(−2.4453) = 589.6 J W Calculation of Actual Work of Compression For simplicity, the work of compression is calculated by the equation for an ideal gas in a three-stage reciprocating machine with complete intercooling and with isentropic compression in each stage. The work so calculated is assumed to represent 80 percent of the actual work. The following equation may be found in any number of textbooks on thermodynamics: nγRT1 P2 (γ − 1)/nγ ˙ s = ᎏᎏ W −1 ᎏ (0.8)(γ − 1) P1
where
n = number of stages, here taken as 3 γ = ratio of heat capacities, here taken as 1.4 T1 = initial absolute temperature, 300 K P2 /P1 = overall pressure ratio, 54.5 R = universal gas constant, 8.314 J/(mol⋅K)
The efficiency factor of 0.8 is already included in the equation. Substitution of the remaining values gives (3)(1.4)(8.314)(300) ˙ s = ᎏᎏᎏ W (54.5)0.4/(3)(1.4) − 1 = 15,171 J (0.8)(0.4) The heat transferred to the surroundings during compression as a result of intercooling and aftercooling to 300 K is found from the first law:
Diagram of simple Linde system for air separation.
FIG. 4-12
˙ s = (12,046 − 12,407) − 15,171 = −15,532 J Q˙ = m(∆H) ˙ −W Calculation of Lost Work Equation (4-367) may be applied to each of the major units of the process. For the compressor/cooler, ˙ lost = (300)[(82.98)(1) − (117.35)(1)] − (−15,532) W = 5,221.0 J For the exchanger, ˙ lost = (300)[(118.48)(0.1364) + (114.34)(0.8636) + (52.08)(1) W − (75.82)(0.8636) − (83.69)(0.1364) − (82.98)(1)] − 70 = 2,063.4 J Finally, for the rectifier, ˙ lost = (300)[(75.82)(0.8636) + (83.69)(0.1364) − (52.08)(1)] − 147 W = 7,297.0 J Work Analysis Since the process requires work, Eq. (4-375) is appropriate for a work analysis. The various terms of this equation appear as entries in the following table, and are on the basis of 1 mol of entering air. ˙s % of W ˙ ideal W ˙ lost: W ˙ lost: W ˙ lost: W ˙s W
Compressor/cooler Exchanger Rectifier
589.6 J 5,221.0 J 2,063.4 J 7,297.0 J
3.9 34.4 13.6 48.1
15,171.0 J
100.0
The thermodynamic efficiency of this process as given by Eq. (4-363) is only 3.9 percent. Significant inefficiencies reside with each of the primary units of the process.
Section 5
Heat and Mass Transfer*
James G. Knudsen, Ph.D., Professor Emeritus of Chemical Engineering, Oregon State University; Member, American Institute of Chemical Engineers, American Chemical Society; Registered Professional Engineer (Oregon). (Conduction and Convection; Condensation, Boiling; Section Coeditor) Hoyt C. Hottel, S.M., Professor Emeritus of Chemical Engineering, Massachusetts Institute of Technology; Member, National Academy of Sciences, American Academy of Arts and Sciences, American Institute of Chemical Engineers, American Chemical Society, Combustion Institute. (Radiation) Adel F. Sarofim, Sc.D., Lammot du Pont Professor of Chemical Engineering and Assistant Director, Fuels Research Laboratory, Massachusetts Institute of Technology; Member, American Institute of Chemical Engineers, American Chemical Society, Combustion Institute. (Radiation) Phillip C. Wankat, Ph.D., Professor of Chemical Engineering, Purdue University; Member, American Institute of Chemical Engineers, American Chemical Society, International Adsorption Society. (Mass Transfer Section Coeditor) Kent S. Knaebel, Ph.D., President, Adsorption Research, Inc.; Member, American Institute of Chemical Engineers, American Chemical Society, International Adsorption Society. Professional Engineer (Ohio). (Mass Transfer Section Coeditor)
HEAT TRANSFER Modes of Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-8
HEAT TRANSFER BY CONVECTION
HEAT TRANSFER BY CONDUCTION Fourier’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three-Dimensional Conduction Equation . . . . . . . . . . . . . . . . . . . . . . . Thermal Conductivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steady-State Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-Dimensional Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conduction through Several Bodies in Series . . . . . . . . . . . . . . . . . . . Conduction through Several Bodies in Parallel. . . . . . . . . . . . . . . . . . Several Bodies in Series with Heat Generation . . . . . . . . . . . . . . . . . Example 1. Steady-State Conduction with Heat Generation . . . . . . . Two-Dimensional Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unsteady-State Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-Dimensional Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Dimensional Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conduction with Change of Phase. . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-8 5-8 5-9 5-9 5-9 5-9 5-10 5-10 5-10 5-10 5-10 5-10 5-11 5-11
Coefficient of Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Individual Coefficient of Heat Transfer. . . . . . . . . . . . . . . . . . . . . . . . Overall Coefficient of Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . Representation of Heat-Transfer Film Coefficients . . . . . . . . . . . . . . Natural Convection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nusselt Equation for Various Geometries . . . . . . . . . . . . . . . . . . . . . . Simplified Dimensional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . Simultaneous Loss by Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enclosed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forced Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analogy between Momentum and Heat Transfer . . . . . . . . . . . . . . . . Laminar Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transition Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2. Calculation of j Factors in an Annulus . . . . . . . . . . . . . . . Jackets and Coils of Agitated Vessels . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-12 5-12 5-12 5-12 5-12 5-12 5-12 5-12 5-12 5-12 5-14 5-14 5-15 5-16 5-16 5-17 5-19
* The contribution to the section on Interphase Mass Transfer of Mr. William M. Edwards (editor of Sec. 14), who was an author for the sixth edition, is acknowledged. 5-1
5-2
HEAT AND MASS TRANSFER
Nonnewtonian Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-19 5-19
HEAT TRANSFER WITH CHANGE OF PHASE Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Condensation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Condensation Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boiling (Vaporization) of Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boiling Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boiling Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-20 5-20 5-20 5-22 5-22 5-22
HEAT TRANSFER BY RADIATION General References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature for Radiative Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . Nature of Thermal Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Blackbody Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radiative Exchange between Surfaces and Solids . . . . . . . . . . . . . . . . . Emittance and Absorptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Black-Surface Enclosures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3: Calculation of View Factor . . . . . . . . . . . . . . . . . . . . . . . . Example 4: Calculation of Exchange Area. . . . . . . . . . . . . . . . . . . . . . Non-Black Surface Enclosures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 5: Radiation in a Furnace Chamber . . . . . . . . . . . . . . . . . . . Emissivities of Combustion Products . . . . . . . . . . . . . . . . . . . . . . . . . . . Gaseous Combustion Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 6: Calculation of Gas Emissivity and Absorptivity . . . . . . . . Flames and Particle Clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radiative Exchange between Gases or Suspended Matter and a Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 7: Radiation in Gases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Gas-Zone/Two-Surface-Zone Systems . . . . . . . . . . . . . . . . . . . The Effect of Nongrayness of Gas on Total-Exchange Area . . . . . . . Example 8: Effective Gas Emissivity . . . . . . . . . . . . . . . . . . . . . . . . . . Treatment of Refractory Walls Partially Enclosing a Radiating Gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combustion Chamber Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 9: Radiation in a Furnace . . . . . . . . . . . . . . . . . . . . . . . . . . . MASS TRANSFER General References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fick’s First Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mutual Diffusivity, Mass Diffusivity, Interdiffusion Coefficient . . . .
5-23 5-23 5-24 5-24 5-25 5-25 5-27 5-29 5-29 5-29 5-31 5-32 5-33 5-34 5-35 5-36 5-36 5-37 5-37 5-38 5-39 5-40 5-41
5-42 5-42 5-42 5-46
Self Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tracer Diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass-Transfer Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem Solving Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuity and Flux Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Material Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flux Expressions: Simple Integrated Forms of Fick’s First Law . . . . Stefan-Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diffusivity Estimation—Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Binary Mixtures—Low Pressure—Nonpolar Components . . . . . . . . Binary Mixtures—Low Pressure—Polar Components . . . . . . . . . . . . Self-Diffusivity—High Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supercritical Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-Pressure/Multicomponent Mixtures . . . . . . . . . . . . . . . . . . . . . . Diffusivity Estimation—Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stokes-Einstein and Free-Volume Theories . . . . . . . . . . . . . . . . . . . . Dilute Binary Nonelectrolytes: General Mixtures . . . . . . . . . . . . . . . Binary Mixtures of Gases in Low-Viscosity, Nonelectrolyte Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dilute Binary Mixtures of a Nonelectrolyte in Water . . . . . . . . . . . . . Dilute Binary Hydrocarbon Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . Dilute Binary Mixtures of Nonelectrolytes with Water as the Solute. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dilute Dispersions of Macromolecules in Nonelectrolytes . . . . . . . . Concentrated, Binary Mixtures of Nonelectrolytes . . . . . . . . . . . . . . Binary Electrolyte Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multicomponent Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diffusion of Fluids in Porous Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interphase Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass-Transfer Principles: Dilute Systems . . . . . . . . . . . . . . . . . . . . . . Mass-Transfer Principles: Concentrated Systems . . . . . . . . . . . . . . . . HTU (Height Equivalent to One Transfer Unit) . . . . . . . . . . . . . . . . NTU (Number of Transfer Units) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions of Mass-Transfer Coefficients ˆkG and ˆkL . . . . . . . . . . . . . Simplified Mass-Transfer Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass-Transfer Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of Total Pressure on ˆkG and ˆkL . . . . . . . . . . . . . . . . . . . . . . . . . Effects of Temperature on ˆkG and ˆkL . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of System Physical Properties on ˆkG and ˆkL . . . . . . . . . . . . . . Effects of High Solute Concentrations on ˆkG and ˆkL . . . . . . . . . . . . . Influence of Chemical Reactions on ˆkG and ˆkL . . . . . . . . . . . . . . . . . . Effective Interfacial Mass-Transfer Area a . . . . . . . . . . . . . . . . . . . . . Volumetric Mass-Transfer Coefficients Kˆ Ga and Kˆ La . . . . . . . . . . . . . Chilton-Colburn Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-46 5-46 5-46 5-46 5-46 5-46 5-47 5-47 5-48 5-48 5-49 5-49 5-49 5-50 5-50 5-50 5-50 5-51 5-52 5-52 5-52 5-52 5-52 5-53 5-54 5-54 5-54 5-54 5-56 5-57 5-57 5-57 5-57 5-58 5-61 5-64 5-66 5-69 5-69 5-74 5-78 5-79
HEAT AND MASS TRANSFER
5-3
Nomenclature and Units Specialized heat transfer nomenclature used for radiative heat transfer is defined in the subsection “Heat Transmission by Radiation.” Nomenclature for mass transfer is defined in the subsection “Mass Transfer.” Symbol a ax a′ A
b b′ bf B c1, c2, etc. c, cp C Cr dm D Dc De Dj Dotl Dp Ds Dt D1, D2 EH EM f F Fa Fg Fc Ft Fs Fw FD FT g, gL gc G Gmax Gmf h ham, hlm hb hcg hc + hr
Definition Proportionality coefficient Cross-sectional area of a fin Proportionality factor Area of heat transfer surface; Ai for inside; Ao for outside; Am for mean; Aavg for average; A1, A2, and A3 for points 1, 2, and 3 respectively; AB for bare surface of finned tube; Af for finned portion of tube; Auf for external area of unfinned portion of finned tube; Aof for external area of finned tube before fins are attached, equals Ao; Aoe for effective area of finned surface; AT for total external area of finned tube; Ad for surface area of dirt (scale) deposit Proportionality coefficient Proportionality factor Height of fin Material constant = 5D−0.5 Constants of integration Specific heat at constant pressure; cs for specific heat of solid; cg for specific heat of gas Thermal conductance, equals kA/x, hA, or UA; C1, C2, C3, Cn, thermal conductance of sections 1, 2, 3, and n respectively of a composite body Correlating constant; proportionality coefficient Depth of divided solids bed Diameter; Do for outside; Di for inside; Dr for root diameter of finned tube Diameter of a coil or helix Equivalent diameter of a cross section, usually 4 times free area divided by wetted perimeter; Dw for equivalent diameter of window Diameter of a jacketed cylindrical vessel Outside diameter of tube bundle Diameter of packing in a packed tube Inside diameter of heat-exchanger shell Solids-processing vessel diameter Diameter at points 1 and 2 respectively; inner and outer diameter of annulus respectively Eddy conductivity of heat Eddy viscosity Fanning friction factor; f1 for inner wall and f2 for outer wall of annulus; fk for ideal tube bank; skin friction drag coefficient Entrance factors Dry solids feed rate Gas volumetric flow rate Fraction of total tubes in cross-flow; Fbp for fraction of cross-flow area available for bypass flow Factor, ratio of temperature difference across tube-side film to overall mean temperature difference Factor, ratio of temperature difference across shell-side film to overall mean temperature difference Factor, ratio of temperature difference across retaining wall to overall mean temperature difference between bulk fluids Factor, ratio of temperature difference across combined dirt or scale films to overall mean temperature difference between bulk fluids Temperature-difference correction factor Acceleration due to gravity Conversion factor Mass velocity, equals V or W/S; Gv for vapor mass velocity Mass velocity through minimum free area between rows of tubes normal to the fluid stream Minimum fluidizing mass velocity Local individual coefficient of heat transfer, equals dq/(dA)(∆T) Film coefficient based on arithmetic-mean temperature difference and logarithmic-mean temperature difference respectively Film coefficient delivered at base of fin Effective combined coefficient for simultaneous gas-vapor cooling and vapor condensation Combined coefficient for conduction, convection, and radiation between surface and surroundings
SI units
U.S. customary units
Dimensionless m2
Dimensionless ft2
m2
ft2
m
ft
J/(kg⋅K)
Btu/(lb⋅°F)
J/(s⋅K)
Btu/(h⋅°F)
Dimensionless m m
Dimensionless ft ft
m m
ft ft
m m m m m m
ft ft ft ft ft ft
J/(s⋅m⋅K) Pa⋅s Dimensionless
Btu/(h⋅ft⋅°F) lb/(ft⋅h) Dimensionless
kg/(s⋅m2) m3 (s⋅m2 of bed area)
lb/(h⋅ft2) ft3/(h⋅ft2 of bed area)
Dimensionless
Dimensionless
Dimensionless
Dimensionless
Dimensionless
Dimensionless
Dimensionless
Dimensionless
981 m/s2 1.0 (kg⋅m)/(N⋅s2) kg/(m2⋅s) kg/(m2⋅s)
(4.18)(108) ft/h2 (4.17)(108)(lb⋅ft)/(lbf⋅h2) lb/(h⋅ft2) lb/(h⋅ft2)
kg/(m2⋅s) J/(m2⋅s⋅K) J/(m2⋅s⋅K)
lb/(h⋅ft2) Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F)
J/(m2⋅s⋅K) J/(m2⋅s⋅K)
Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F)
J/(m2⋅s⋅K)
Btu/(h⋅ft2⋅°F)
5-4
HEAT AND MASS TRANSFER
Nomenclature and Units (Continued ) Symbol hdo, hdi hf hfi hfo hF, hs hi, ho hk h1 h′ hlm hr hT hw j J Jb, Jc, Jl, Jr k kv kavg, km kf kw K′ lc L Lo Lu LF LH Lp m M M n nt nr n′ nb Nr N
NB Nd NGr NNu NPe NPr NRe NSt Nss p pf p, p′
Definition Film coefficient for dirt or scale on outside or inside respectively of a surface Film coefficient for finned-tube exchangers based on total external surface Effective outside film coefficient of a finned tube based on inside area Film coefficient for air film of an air-cooled finned-tube exchanger based on external bare surface Effective film coefficient for dirt or scale on heat-transfer surface Film coefficient for heat transfer for inside and outside surface respectively Film coefficient for ideal tube bank; hs for shell side of baffled exchanger; hsv for coefficient at liquid-vapor interface Condensing coefficient on top tube; hN coefficient for N tubes in a vertical row Film coefficient for enclosed spaces Film coefficient based on log-mean temperature difference Heat-transfer coefficient for radiation Coefficient of total heat transfer by conduction, convection, and radiation between the surroundings and the surface of a body subject to unsteady-state heat transfer Equivalent coefficient of retaining wall, equals k/x Ordinate, Colburn j factor, equals f/2; jH for heat transfer; jH1 for inner wall of annulus; jH2 for outer wall of annulus; jk for heat transfer for ideal tube bank Mechanical equivalent of heat Correction factors for baffle bypassing, baffle configuration, baffle leakage, and adverse temperature gradient respectively Thermal conductivity; k1, k2, k3, thermal conductivities of bodies 1, 2, and 3 Thermal conductivity of vapor; k1 for liquid thermal conductivity; ks for thermal conductivity of solid Mean thermal conductivity Thermal conductivity of fluid at film temperature Thermal conductivity of retaining-wall material Property of non-Newtonian fluid Baffle cut; ls for baffle spacing Length of heat-transfer surface Flow rate Undisturbed length of path of fluid flow Thickness of dirt or scale deposit Depth of fluidized bed Diameter of agitator blade Ratio, term, or exponent as defined where used Molecular weight Weight of fluid Position ratio or number Number of tubes in parallel in a heat exchanger Number of rows in a vertical plane Flow-behavior index for nonnewtonian fluids Number of baffle-type coils Speed of agitator Number of tubes in a vertical row; or number of tubes in a bundle; Nb for number of baffles; NT for total number of tubes in exchanger; Nc for number of tubes in one cross-flow section; Ncw for number of cross-flow rows in each window Biot number, hT ∆x/k Proportionality coefficient, dimensionless group Grashof number, L3ρ2gβ ∆t/µ2 Nusselt number, hD/k or hL/k Peclet number, DGc/k Prandtl number, cµ/k Reynolds number, DG/µ Stanton number, NNu/NRe NPr Number of sealing strips Pressure Perimeter of a fin Center-to-center spacing of tubes in tube bundle (tube pitch); pn for tube pitch normal to flow; pp for tube pitch parallel to flow
SI units
U.S. customary units
J/(m2⋅s⋅K)
Btu/(h⋅ft2⋅°F)
J/(m2⋅s⋅K)
Btu/(h⋅ft2⋅°F)
J/(m2⋅s⋅K) J/(m2⋅s⋅K)
Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F)
J/(m2⋅s⋅K) J/(m2⋅s⋅K)
Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F)
J/(m2⋅s⋅K)
Btu/(h⋅ft2⋅°F)
J/(m2⋅s⋅K)
Btu/(h⋅ft2⋅°F)
J/(m2⋅s⋅K) J/(m2⋅s⋅K) J/(m2⋅s⋅K) J/(m2⋅s⋅K)
Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F)
J/(m2⋅s⋅K) Dimensionless
Btu/(h⋅ft2⋅°F) Dimensionless
1.0(N⋅m)/J
778(ft⋅lbf)/Btu
J/(m⋅s⋅K)
(Btu⋅ft)/(h⋅ft2⋅°F)
J/(m⋅s⋅K)
(Btu⋅ft)/(h⋅ft2⋅°F)
J/(m⋅s⋅K) J/(m⋅s⋅K) J/(m⋅s⋅K)
(Btu⋅ft)/(h⋅ft2⋅°F) (Btu⋅ft)/(h⋅ft2⋅°F) (Btu⋅ft)/(h⋅ft2⋅°F)
m m kg/s m m m m
ft ft lb/h ft ft ft ft
kg/mol kg Dimensionless
lb/mol lb Dimensionless
rad/s
r/h
Dimensionless
Dimensionless
kPa m m
lbf/ft2 abs ft ft
HEAT AND MASS TRANSFER
5-5
Nomenclature and Units (Continued ) Symbol ∆p P P′ ∆Pbk, ∆Pwk q q′ (q/A)max Q Q r
rj R
Rj S
Sr s t t1, t2, tn t′ t′1, t′2 t″1, t″2 tb tH, tL ts tsv tw t∞ TH, TL T ∆T, ∆t
∆tam, ∆tlm ∆tom ∆TH, ∆tH ∆TL, ∆tL ∆Tm, ∆tm u u° U U1, U2 Uco, Ucv, Uct, Ura Um v Vr
Definition Pressure of the vapor in a bubble minus saturation pressure of a flat liquid surface Absolute pressure; Pc for critical pressure Spacing between adjacent baffles on shell side of a heat exchanger (baffle pitch) Pressure drop for ideal-tube-bank cross-flow and ideal window respectively; ∆Ps for shell side of baffled exchanger Rate of heat flow, equals Q/θ Rate of heat generation Maximum heat flux in nucleate boiling Quantity of heat; rate of heat transfer Quantity of heat; QT for total quantity Radius; cylindrical and spherical coordinate; distance from midplane to a point in a body; r1 for inner wall of annulus; r2 for outer wall of annulus; ri for inside radius of tube; rm for distance from midplane or center of a body to the exterior surface of the body Inside radius Thermal resistance, equals x/kA, 1/UA, 1/hA; R1, R2, R3, Rn for thermal resistance of sections 1, 2, 3, and n of a composite body; RT for sum of individual resistances of several resistances in series or parallel; Rdi and Rdo for dirt or scale resistance on inner and outer surface respectively Ratio of total outside surface of finned tube to area of tube having same root diameter Cross-sectional area; Sm for minimum cross-sectional area between rows of tubes, flow normal to tubes; Stb for tube-to-baffle leakage area for one baffle; Ssb for shell-to-baffle area for one baffle; Sw for area for flow through window; Swg for gross window area; Swt for window area occupied by tubes Slope of rotary shell Specific gravity of fluid referred to liquid water Bulk temperature; temperature at a given point in a body at time θ Temperature at points 1, 2, and n in a system through which heat is being transferred Temperature of surroundings Inlet and outlet temperature respectively of hotter fluid Inlet and outlet temperature respectively of colder fluid Initial uniform bulk temperature of a body; bulk temperature of a flowing fluid High and low temperature respectively on tube side of a heat exchanger Surface temperature Saturated-vapor temperature Wall temperature Temperature of undisturbed flowing stream High and low temperature respectively on shell side of a heat exchanger Absolute temperature; Tb for bulk temperature; Tw for wall temperature; Tv for vapor temperature; Tc for coolant temperature; Te for temperature of emitter; Tr for temperature of receiver Temperature difference; ∆t1, ∆t2, and ∆t3 temperature difference across bodies 1, 2, and 3 or at points 1, 2, and 3; ∆To, ∆to for overall temperature difference; ∆tb for temperature difference between surface and boiling liquid Arithmetic- and logarithmic-mean temperature difference respectively Mean effective overall temperature difference Greater terminal temperature difference Lesser terminal temperature difference Mean temperature difference Velocity in x direction Friction velocity Overall coefficient of heat transfer; Uo for outside surface basis; U′ for overall coefficient between liquid-vapor interface and coolant Overall coefficient of heat transfer at points 1 and 2 respectively Overall coefficients for divided solids processing by conduction, convection, contact, and radiation mechanism respectively Mean overall coefficient of heat transfer Velocity in y direction Volume of rotating shell
SI units
U.S. customary units
kPa
lbf/ft2 abs
kPa m
lbf/ft2 ft
kPa
lbf/ft2
W, J/s J/(s⋅m3) J/(s⋅m2) J/s J m
Btu/h Btu/(h⋅ft3) Btu/(h⋅ft2) Btu/h Btu ft
Dimensionless (s⋅K)/J
Dimensionless (h⋅°F)/Btu
m2
ft2
K K
°F °F
K K K K
°F °F °F °F
K
°F
K K K K K
°F °F °F °F °F
K
°R
K
°F, °R
K K K K K m/s m/s J/(s⋅m2⋅K)
°F °F °F °F °F ft/h ft/h Btu/(h⋅ft2⋅°F)
J/(s⋅m2⋅K) J/(s⋅m2⋅K)
Btu/(h⋅ft2⋅°F) Btu/(h⋅ft2⋅°F)
J/(s⋅m2⋅K) m/s m3
Btu/(h⋅ft2⋅°F) ft/h ft3
5-6
HEAT AND MASS TRANSFER
Nomenclature and Units (Concluded ) Symbol V V′, Vs VF Vg, Vl V′max w w W Wr W1, Wo xq x X y y+ Y z zp ZH
Definition
SI units
Velocity Velocity Face velocity of a fluid approaching a bank of finned tubes Specific volume of gas, liquid Maximum velocity through minimum free area between rows of tubes normal to the fluid stream Velocity in z direction Flow rate Total mass rate of flow; mass rate of vapor generated; WF for total rate of vapor condensation in one tube Weight rate of flow Total mass rate of flow on tube side and shell side respectively of a heat exchanger Vapor quality, xi for inlet quality, xo for outlet quality Coordinate direction; length of conduction path; xs for thickness of scale; x1, x2, and x3 at positions 1, 2, and 3 in a body through which heat is being transferred Factor Coordinate direction Wall distance Factor Coordinate direction Distance (perimeter) traveled by fluid across fin Ratio of sensible heat removed from vapor to total heat transferred
U.S. customary units
m/s m/s m/s m3/kg m/s
ft/h ft/s ft/h ft3/lb ft/h
m/s kg/s kg/s
ft/h lb/h lb/h
kg/(s⋅tube) kg/s
lb/(h⋅tube) lb/h
kg/s m
lb/h ft
Dimensionless m Dimensionless Dimensionless m m Dimensionless
Dimensionless ft Dimensionless Dimensionless ft ft Dimensionless
m2/s
ft2/h
K−1 ° kg/(s2 − n′⋅m) kg/(s⋅m)
°F−1 ° lb/(ft⋅s2 − n′) lb/(h⋅ft)
m m m2/s
ft ft ft2/h
s
h
J/kg m Pa⋅s
Btu/lb ft lb/(h⋅ft)
m2/s kg/m3
ft2/h lb/ft3
N/m
lbf/ft
N/m2
lbf/ft2
rad Dimensionless
rad Dimensionless
Greek symbols α β β′ γ Γ δs δ δsb ε εv η θ θb λ λm µ
ν ρ σ τ φ φp Φ ω Ω
Thermal diffusivity, equals k/ρc; αe for effective thermal diffusivity of powdered solids Volumetric coefficient of thermal expansion Contact angle of a bubble Fluid consistency Mass rate of flow of a falling film from a tube or surface per unit perimeter, equals w/πD for vertical tube, w/2L for horizontal tube Correction factor, ratio of nonnewtonian to newtonian shear rates Cell width Diametral shell-to-baffle clearance Eddy diffusivity; εM for eddy diffusivity of momentum; εH for eddy diffusivity of heat Fraction of voids in porous bed Fluidization efficiency Time Baffle cut Latent heat (enthalpy) of vaporization (condensation) Radius of maximum velocity Viscosity; µw for viscosity at wall temperature; µb for viscosity at bulk temperature; µf for viscosity at film temperature; µG, µg, and µv for viscosity of gas or vapor; µL, µl for viscosity of liquid; µw for viscosity at wall; µl for viscosity of fluid at inner wall of annulus Kinematic viscosity Density; ρL, ρl for density of liquid; ρG, ρv for density of gas or vapor; ρs for density of solid Surface tension between a liquid and its vapor Term indicating summation of variables Shear stress τw for shear stress at the wall Velocity-potential function Particle sphericity Viscous-dissipation function Angle of repose of powdered solid Fin efficiency
GENERAL REFERENCES: Becker, Heat Transfer, Plenum, New York, 1986. Bejan, Convection Heat Transfer, Wiley, New York, 1984. Bird, Stewart, and Lightfoot, Transport Phenomena, Wiley, New York, 1960. Carslaw and Jaeger, Conduction of Heat in Solids, Clarendon Press, Oxford, 1959. Chapman, Heat Transfer, 2d ed., Macmillan, New York, 1967. Drew and Hoopes, Advances in Chemical Engineering, Academic, New York, vol. 1, 1956; vol. 2, 1958; vol. 5, 1964; vol. 6, 1966; vol. 7, 1968. Dusinberre, Heat Transfer Calculations by Finite Differences, International Textbook, Scranton, Pa., 1961. Eckert and Drake, Heat and Mass Transfer, 2d ed., McGraw-Hill, New York, 1959. Gebhart, Heat Transfer, McGraw-Hill, New York, 1961. Irvine and Hartnett, Advances in Heat Transfer, Academic, New York, vol. 1, 1964; vol. 2, 1965; vol. 3, 1966. Grigull and Sandner, Heat Conduction, Hemisphere Publishing, 1984. Jakob, Heat Transfer, Wiley, New York, vol. 1, 1949; vol. 2, 1957. Jakob and Hawkins, Elements of Heat Transfer, 3d ed., Wiley, New York, 1957. Kakac, Bergles, and Mayinger, Heat Exchangers: Thermal Hydraulic Fundamentals and Design, Hemisphere Publishing, Washington, 1981. Kakac and Yener, Convective Heat Transfer, Hemisphere Publishing, Washington, 1980. Kay, An Introduction to Fluid Mechanics and Heat Transfer, 2d ed., Cambridge University Press, Cambridge, England, 1963. Kays, Convective Heat and Mass Transfer, McGraw-Hill, New York, 1966. Kays and London, Compact Heat Exchangers, 3d ed., McGraw-Hill, New York, 1984. Kern, Process Heat Transfer, McGrawHill, New York, 1950. Knudsen and Katz, Fluid Dynamics and Heat Transfer, McGraw-Hill, New York, 1958. Kraus, Analysis and Evaluation of Extended Surface Thermal Systems, Hemisphere Publishing, Washington, 1982. Kutatladze, A Concise Encyclopedia of Heat Transfer, 1st English ed., Pergamon, New York, 1966. Lykov, Heat and Mass Transfer in Capillary Porous Bodies, translated from Russian, Pergamon, New York, 1966. McAdams, Heat Transmission, 3d ed., McGraw-Hill, New York, 1954. Mickley, Sherwood, and Reed, Applied Mathematics in Chemical Engineering, 2d ed., McGraw-Hill, New York, 1957. Rohsenow and Choi, Heat, Mass, and Momentum Transfer, Prentice-Hall, Englewood Cliffs, N.J., 1961. Schlünder (ed.), Heat Exchanger Design Handbook, Hemisphere Publishing, Washington, 1983 (Book 2; Chapter 2.4 [Conduction], Chapter 2.5 [Convection], Chapter 2.6 [Condensation], Chapter 2.7 [Boiling]). Skelland, Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967. Taborek and Bell, Process Heat Exchanger Design, Hemisphere Publishing, Washington, 1984. Taborek, Hewitt, and Afghan, Heat Exchangers: Theory and Practice, Hemisphere Publishing, Washington, 1983. TSederberg, Thermal Conductivity of Liquids and Gases, M.I.T., Cambridge, Mass., 1965. Welty, Wicks, and Wilson, Fundamentals of Momentum, Heat and Mass Transfer, 3d ed., Wiley, New York, 1984. Zenz and Othmer, Fluidization and Fluid Particle Systems, Reinhold, New York, 1960.
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Lee and Thodos, Ind. Eng. Chem. Fundam., 22, 17–26 (1983). Lee and Thodos, Ind. Eng. Chem. Res., 27, 992–997 (1988). Lees and Sarram, J. Chem. Eng. Data, 16, 1, 41 (1971). Leffler and Cullinan, Ind. Eng. Chem. Fundam., 9, 84, 88 (1970). Lugg, Anal. Chem., 40, 1072 (1968). Marrero and Mason, AIChE J., 19, 498 (1973). Mathur and Thodos, AIChE J., 11, 613 (1965). Matthews and Akgerman, AIChE J., 33, 881 (1987). Matthews, Rodden and Akgerman, J. Chem. Eng. Data, 32, 317 (1987). Olander, AIChE J., 7, 175 (1961). Passut and Danner, Chem. Eng. Prog. Symp. Ser., 140, 30 (1974). Perkins and Geankoplis, Chem. Eng. Sci., 24, 1035–1042 (1969). Pinto and Graham, AIChE J., 32, 291 (1986). Pinto and Graham, AIChE J., 33, 436 (1987). Quale, Chem. Rev., 53, 439 (1953). Rathbun and Babb, Ind. Eng. Chem. Proc. Des. Dev., 5, 273 (1966). Reddy and Doraiswamy, Ind. Eng. Chem. Fundam., 6, 77 (1967). Riazi and Whitson, Ind. Eng. Chem. Res., 32, 3081 (1993). Robinson, Edmister, and Dullien, Ind. Eng. Chem. Fundam., 5, 75 (1966). Rollins and Knaebel, AIChE J., 37, 470 (1991). Siddiqi, Krahn, and Lucas, J. Chem. Eng. Data, 32, 48 (1987). Siddiqi and Lucas, Can. J. Chem. Eng., 64, 839 (1986). Smith and Taylor, Ind. Eng. Chem. Fundam., 22, 97 (1983). Sridhar and Potter, AIChE J., 23, 4, 590 (1977). Stiel and Thodos, AIChE J., 7, 234 (1961). Sun and Chen, Ind. Eng. Chem. Res., 26, 815 (1987). Tanford, Phys. Chem. of Macromolecule, Wiley, New York, NY (1961). Taylor and Webb, Comput. Chem. Eng., 5, 61 (1981). Tyn and Calus, J. Chem. Eng. Data, 20, 310 (1975). Umesi and Danner, Ind. Eng. Chem. Process Des. Dev., 20, 662 (1981). Van Geet and Adamson, J. Phys. Chem., 68, 2, 238 (1964). Vignes, Ind. Eng. Chem. Fundam., 5, 184 (1966). Wilke, Chem. Eng. Prog., 46, 2, 95 (1950). Wilke and Chang, AIChE J., 1, 164 (1955). Wilke and Lee, Ind. Eng. Chem., 47, 1253 (1955).
REFERENCES FOR DIFFUSIVITIES IN POROUS SOLIDS, TABLE 5-20 66. Ruthven, Principles of Adsorption & Adsorption Processes, Wiley, 1984. 67. Satterfield, Mass Transfer in Heterogeneous Catalysis, MIT Press, 1970. 68. Suzuki, Adsorption Engineering, Kodansha—Elsevier, 1990. 69. Yang, Gas Separation by Adsorption Processes, Butterworths, 1987.
REFERENCES FOR DIFFUSIVITIES 1. Akita, Ind. Eng. Chem. Fundam., 10, 89 (1981). 2. Asfour and Dullien, Chem. Eng. Sci., 41, 1891 (1986). 3. Blanc, J. Phys., 7, 825 (1908). 4. Brokaw, Ind. Eng. Chem. Process Des. and Dev., 8, 2, 240 (1969). 5. Caldwell and Babb, J. Phys. Chem., 60, 51 (1956). 6. Catchpole and King, Ind. Eng. Chem. Res., 33, 1828 (1994). 7. Chen and Chen, Chem. Eng. Sci., 40, 1735 (1985). 8. Chung, Ajlan, Lee and Starling, Ind. Eng. Chem. Res., 27, 671 (1988). 9. Condon and Craven, Aust. J. Chem., 25, 695 (1972). 10. Cullinan, AIChE J., 31, 1740–1741 (1985). 11. Cullinan, Can. J. Chem. Eng., 45, 377–381 (1967). 12. Cussler, AIChE J., 26, 1 (1980). 13. Darken, Trans. Am. Inst. Mining Met. Eng., 175, 184 (1948). 14. Debenedetti and Reid, AIChE J., 32, 2034 (1986); see errata: AIChE J., 33, 496 (1987). 15. Elliott, R. W. and H. Watts, Can. J. Chem., 50, 31 (1972). 16. Erkey and Akgerman, AIChE J., 35, 443 (1989). 17. Ertl, Ghai, and Dullien, AIChE J., 20, 1, 1 (1974). 18. Fairbanks and Wilke, Ind. Eng. Chem., 42, 471 (1950). 19. Fuller, Schettler and Giddings, Ind. Eng. Chem., 58, 18 (1966). 20. Ghai, Ertl, and Dullien, AIChE J., 19, 5, 881 (1973). 21. Gordon, J. Chem. Phys., 5, 522 (1937). 22. Graham and Dranoff, Ind. Eng. Chem. Fundam., 21, 360–365 (1982). 23. Graham and Dranoff, Ind. Eng. Chem. Fundam., 21, 365–369 (1982). 24. Gurkan, AIChE J., 33, 175–176 (1987). 25. Hayduk and Laudie, AIChE J., 20, 3, 611 (1974). 26. Hayduk and Minhas, Can. J. Chem. Eng., 60, 195 (1982). 27. Hildebrand, Science, 174, 490 (1971). 28. Hiss and Cussler, AIChE J., 19, 4, 698 (1973). 29. Jossi, Stiel, and Thodos, AIChE J., 8, 59 (1962). 30. Krishnamurthy and Taylor, Chem. Eng. J., 25, 47 (1982).
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Perez and Sandall, AIChE J., 20, 770 (1974). Petrovic and Thodos, Ind. Eng. Chem. Fundam., 7, 274 (1968). Pinczewski and Sideman, Chem. Engr. Sci., 29, 1969 (1974). Prandtl, Phys. Zeit., 29, 487 (1928). Prasad and Sirkar, AIChE J., 34, 177 (1988). Rahman and Streat, Chem. Engr. Sci., 36, 293 (1981). Ranz and Marshall, Chem. Engr. Prog., 48, 141, 173 (1952). Reiss, Ind. Eng. Chem. Process Des. Develop, 6, 486 (1967). Riet, Ind. Eng. Chem. Process Des. Dev., 18, 357 (1979). Rowe, Chem. Engr. Sci., 30, 7 (1975). Rowe, Claxton, and Lewis, Trans. Inst. Chem. Engr. London, 43, 14 (1965). Ruckenstein and Rajagopolan, Chem. Engr. Commun., 4, 15 (1980). Ruthven, Principles of Adsorption & Adsorption Processes, Wiley, 1984. Satterfield, AIChE J., 21, 209 (1975). Schluter and Deckwer, Chem. Engr. Sci., 47, 2357 (1992). Schmitz, Steiff, and Weinspach, Chem. Engng. Technol., 10, 204 (1987). Sherwood, Brian, Fisher, and Dresner, Ind. Eng. Chem. Fundam., 4, 113 (1965). Sherwood, Pigford, and Wilke, Mass Transfer, McGraw-Hill, 1975. Shulman, Ullrich, Proulx, and Zimmerman, AIChE J., 1, 253 (1955). Shulman and Margolis, AIChE J., 3, 157 (1957). Siegel, Sparrow, and Hallman, Appl. Sci. Res. Sec. A., 7, 386 (1958). Sissom and Pitts, Elements of Transport Phenomena, McGraw-Hill, 1972. Skelland, Diffusional Mass Transfer, Wiley (1974). Skelland and Cornish, AIChE J., 9, 73 (1963). Skelland and Moeti, Ind. Eng. Chem. Res., 29, 2258 (1990). Skelland and Tedder, “Extraction—Organic Chemicals Processing” in Rousseau, Handbook of Separation Process Technology, Wiley, 1987, pp. 405–466. Skelland and Wellek, AIChE J., 10, 491, 789 (1964). Slater, “Rate Coefficients in Liquid-Liquid Extraction Systems” in Godfrey and Slater, Liquid-Liquid Extraction Equipment, Wiley, 1994, pp. 45–94. Steinberger and Treybal, AIChE J., 6, 227 (1960). Steiner, L., Chem. Eng. Sci., 41, 1979 (1986). Taylor and Krishna, Multicomponent Mass Transfer, Wiley, 1993. Tournie, Laguerie, and Couderc, Chem. Engr. Sci., 34, 1247 (1979). Treybal, Mass Transfer Operations, 3d ed., McGraw-Hill, 1980. Von Karman, Trans. ASME, 61, 705 (1939). Wakao and Funazkri, Chem. Engr. Sci., 33, 1375 (1978). Wankat, Equilibrium-Staged Separations, Prentice Hall, 1988. Wankat, Rate-Controlled Separations, Chapman-Hall, 1990. Wilson and Geankoplis, Ind. Eng. Chem. Fundam., 5, 9 (1966). Yagi and Yoshida, Ind. Eng. Chem. Process Des. Dev., 14, 488 (1975). Yang, Gas Separation by Adsorption Processes, Butterworths, 1987. Yoshida, Ramaswami, and Hougen, AIChE J., 8, 5 (1962).
HEAT TRANSFER MODES OF HEAT TRANSFER There are three fundamental types of heat transfer: conduction, convection, and radiation. All three types may occur at the same time, and it is advisable to consider the heat transfer by each type in any particular case. Conduction is the transfer of heat from one part of a body to another part of the same body, or from one body to another in physical contact with it, without appreciable displacement of the particles of the body.
Convection is the transfer of heat from one point to another within a fluid, gas, or liquid by the mixing of one portion of the fluid with another. In natural convection, the motion of the fluid is entirely the result of differences in density resulting from temperature differences; in forced convection, the motion is produced by mechanical means. When the forced velocity is relatively low, it should be realized that “free-convection” factors, such as density and temperature difference, may have an important influence. Radiation is the transfer of heat from one body to another, not in contact with it, by means of wave motion through space.
HEAT TRANSFER BY CONDUCTION FOURIER’S LAW Fourier’s law is the fundamental differential equation for heat transfer by conduction: dQ/dθ = −kA(dt/dx)
direction of the flow of heat, i.e., the temperature gradient. The factor k is called the thermal conductivity; it is a characteristic property of the material through which the heat is flowing and varies with temperature.
(5-1)
where dQ/dθ (quantity per unit time) is the rate of flow of heat, A is the area at right angles to the direction in which the heat flows, and −dt/dx is the rate of change of temperature with the distance in the
THREE-DIMENSIONAL CONDUCTION EQUATION Equation (5-1) is used as a basis for derivation of the unsteady-state three-dimensional energy equation for solids or static fluids:
HEAT TRANSFER BY CONDUCTION ∂t ∂t ∂ ∂t ∂ ∂t ∂ cρ ᎏ = ᎏ k ᎏ + ᎏ k ᎏ + ᎏ k ᎏ + q′ ∂θ ∂x ∂x ∂y ∂y ∂z ∂z
(5-2)
where x, y, z are distances in the rectangular coordinate system and q′ is the rate of heat generation (by chemical reaction, nuclear reaction, or electric current) in the solid per unit of volume. Solution of Eq. (5-2) with appropriate boundary and initial conditions will give the temperature as a function of time and location in the material. Equation (5-2) may be transformed into spherical or cylindrical coordinates to conform more closely to the physical shape of the system. THERMAL CONDUCTIVITY Thermal conductivity varies with temperature but not always in the same direction. The thermal conductivities for many materials, as a function of temperature, are given in Sec. 2. Additional and more comprehensive information may often be obtained from suppliers of the materials. Impurities, especially in metals, can give rise to variations in thermal conductivity of from 50 to 75 percent. In using thermal conductivities, engineers should remember that conduction is not the sole method of transferring heat and that, particularly with liquids and gases, radiation and convection may be much more important. The thermal conductivity at a given temperature is a function of the apparent, or bulk, density. Thus, at 0°C (32°F), k for asbestos wool is 0.09 J/(m⭈s⭈K) [0.052 Btu/(hr⭈ft⭈°F)] when the bulk density is 400 kg/m3 (24.9 lb/ft3) and is 0.19 (0.111) for a density of 700 (43.6). In determining the apparent thermal conductivities of granular solids, such as granulated cork or charcoal grains, Griffiths (Spec. Rep. 5, Food Investigation Board, H. M. Stationery Office, 1921) found that air circulates within the mass of granular solid. Under a certain set of conditions, the apparent thermal conductivity of a charcoal was 9 percent greater when the test section was vertical than when it was horizontal. When the apparent conductivity of a mixture of cellular or porous nonhomogeneous solid is determined, the observed temperature coefficient may be much larger than for the homogeneous solid alone, because heat is transferred not only by the mechanism of conduction but also by convection in the gas pockets and by radiation from surface to surface of the individual particles. If internal radiation is an important factor, a plot of the apparent conductivity as ordinate versus temperature should show a curve concave upward, since radiation increases with the fourth power of the absolute temperature. Griffiths noted that cork, slag, wool, charcoal, and wood fibers, when of good quality and dry, have thermal conductivities about 2.2 times that of still air, whereas a highly cellular form of rubber, 112 kg/m3 (7 lb/ft3), had a thermal conductivity only 1.6 times that of still air. In measuring the apparent thermal conductivity of diathermanous substances such as quartz (especially when exposed to radiation emitted at high temperatures), it should be remembered that a part of the heat is transmitted by radiation. Bridgman [Proc. Am. Acad. Arts Sci., 59, 141 (1923)] showed that the thermal conductivity of liquids is increased by only a few percent under a pressure of 100,330 kPa (1000 atm). The thermal conductivity of some liquids varies with temperature through a maximum. It is often necessary for the engineer to estimate thermal conductivities; methods are indicated in Sec. 2. Equation (5-2) considers the thermal conductivity to be variable. If k is expressed as a function of temperature, Eq. (5-2) is nonlinear and difficult to solve analytically except for certain special cases. Usually in complicated systems numerical solution by means of computer is possible. A complete review of heat conduction has been given by Davis and Akers [Chem. Eng., 67(4), 187, (5), 151 (1960)] and by Davis [Chem. Eng., 67(6), 213, (7), 135 (8), 137 (1960)]. STEADY-STATE CONDUCTION For steady flow of heat, the term dQ/dθ in Eq. (5-1) is constant and may be replaced by Q/θ or q. Likewise, in Eq. (5-2) the term ∂ t/∂θ is zero. Hence, for constant thermal conductivity, Eq. (5-2) may be expressed as ∇ 2 t = (q′/k)
(5-3)
5-9
One-Dimensional Conduction Many heat-conduction problems may be formulated into a one-dimensional or pseudo-onedimensional form in which only one space variable is involved. Forms of the conduction equation for rectangular, cylindrical, and spherical coordinates are, respectively, ∂2 t q′ (5-4a) ᎏ2 = − ᎏ ∂x k
1 d dt q′ ᎏᎏ rᎏ =−ᎏ r dr dr k
(5-4b)
1 d dt q′ (5-4c) ᎏ2 ᎏ r 2 ᎏ = − ᎏ r dr dr k These are second-order differential equations which upon integration become, respectively, t = −(q′x2/2k) + c1x + c2
(5-5a)
t = −(q′r /4k) + c1 ln r + c2
(5-5b)
t = −(q′r 2/6k) − (c1/r) + c2
(5-5c)
2
Constants of integration c1 and c2 are determined by the boundary conditions, i.e., temperatures and temperature gradients at known locations in the system. For the case of a solid surface exposed to surroundings at a different temperature and for a finite surface coefficient, the boundary condition is expressed as hT (ts − t′ ) = −k(dt/dx)surf
(5-6)
Inspection of Eqs. (5-5a), (5-5b), and (5-5c) indicates the form of temperature profile for various conditions and geometries and also reveals the effect of the heat-generation term q′ upon the temperature distributions. In the absence of heat generation, one-dimensional steady-state conduction may be expressed by integrating Eq. (5-1): x2 t2 dx q (5-7) ᎏ = − k dt x1 t1 A Area A must be known as a function of x. If k is constant, Eq. (5-7) is expressed in the integrated form
q = kAavg (t1 − t2)/(x2 − x1)
(5-8)
x2
1 dx (5-9) Aavg = ᎏ ᎏ x2 − x1 x1 A Examples of values of Aavg for various functions of x are shown in the following table. where
Area proportional to
Aavg
Constant x
A1 = A2 A2 − A1 ln (A2/A1) 2 A1 A
x2
Usually, thermal conductivity k is not constant but is a function of temperature. In most cases, over the ranges of values used the relation is linear. Integration of Eq. (5-7), with k linear in t, gives x2 dx q (5-10) ᎏ = kavg(t1 − t2) x1 A where kavg is the arithmetic-average thermal conductivity between temperatures t1 and t2. This average probably gives results which are correct within the precision of the data in the majority of cases, though a special integration can be made whenever k is known to be greatly different from linear in temperature. Conduction through Several Bodies in Series Figure 5-1 illustrates diagrammatically the temperature gradients accompanying the steady conduction of heat in series through three solids.
5-10
HEAT AND MASS TRANSFER For fuel, at x = 0, t = 570°C (1050°F), dt/dx = 0 (this follows if the temperature is finite at the midplane). For fuel and cladding, at x = x1, tf = tc, kf (dt/dx) = kc (dt/dx) For cladding, at x = x2, tc − 400 = −(kc /42,600)(dt/dx) For the fuel, the first integration of Eq. (10-4a) gives dtf /dx = −(q′/kf )x + c1 which gives c1 = 0 when the boundary condition is applied. Thus the second integration gives tf = −(q′/2kf )x2 + c2 from which c2 is determined to be 570 (1050) upon application of the boundary condition. Thus the temperature profile in the fuel is tf = −(q′/2kf )x2 + 570
FIG. 5-1
Temperature gradients for steady heat conduction in series through three solids.
The temperature profile in the cladding is obtained by integrating Eq. (10-4a) twice with q′ = 0. Hence
Since the heat flow through each of the three walls must be the same,
There are now three unknowns, c1, c2, and q′, and three boundary conditions by which they can be determined. At x = x1,
(dtc /dx) = c1
q = (k1 A1 ∆t1/x1) = (k2 A2 ∆ t2 /x2) = (k3 A3 ∆ t3 /x3)
(5-11)
then
∆t1 = qR1
∆ t2 = qR2
(5-12) ∆ t3 = qR3
(5-13)
Adding the individual temperature drops, noting that q is uniform, q(R1 + R2 + R3) = ∆t1 + ∆ t2 + ∆ t3 = ∆ t
(5-14)
q = ∆ t/RT = (t1 − t4)/RT
(5-15)
or
where RT is the overall resistance and is the sum of the individual resistances in series, then RT = R1 + R2 + ⋅ ⋅ ⋅ + Rn
(5-16)
When a wall is constructed of several layers of solids, the joints at adjacent layers may not perfectly exclude air spaces, and these additional resistances should not be overlooked. Conduction through Several Bodies in Parallel For n resistances in parallel, the rates of heat flow are additive: q = ∆t/R1 + ∆ t/R2 + ⋅ ⋅ ⋅ + ∆ t/Rn
1 1 1 q= ᎏ+ᎏ+⋅⋅⋅+ᎏ R1 R2 Rn
∆t
q = (C1 + C2 + ⋅ ⋅ ⋅ + Cn)∆ t = C ∆ t
tc = c1 x + c2
q′x12 /2kf + 570 = c1x1 + c2 − kf q′x1/kf = kc c1 At x = x2,
Since, by definition, individual thermal resistance R = x/ kA
and
(5-17a) (5-17b)
c1 x 2 + c2 − 200 = −(kc /42,600)c1 From which q′ = (2.53)(109) J/(m3⋅s)[(2.38)(108)Btu/(h⋅ft3)] c1 = −(1.92)(105) c2 = 724
Two-Dimensional Conduction If the temperature of a material is a function of two space variables, the two-dimensional conduction equation is (assuming constant k) ∂ 2t/∂x2 + ∂ 2 t/∂y2 = −q′/k
(5-18)
When q′ is zero, Eq. (5-18) reduces to the familiar Laplace equation. The analytical solution of Eq. (10-18) as well as of Laplace’s equation is possible for only a few boundary conditions and geometric shapes. Carslaw and Jaeger (Conduction of Heat in Solids, Clarendon Press, Oxford, 1959) have presented a large number of analytical solutions of differential equations applicable to heat-conduction problems. Generally, graphical or numerical finite-difference methods are most frequently used. Other numerical and relaxation methods may be found in the general references in the “Introduction.” The methods may also be extended to three-dimensional problems. UNSTEADY-STATE CONDUCTION
(5-17c)
where R1 to Rn are the individual resistances and C1 to Cn are the individual conductances; C = kA/x. Several Bodies in Series with Heat Generation The simple Fourier type of equation indicated by Eq. (5-15) may not be used when heat generation occurs in one of the bodies in the series. In this case, Eq. (5-5a), (5-5b), or (5-5c) must be solved with appropriate boundary conditions. Example 1: Steady-State Conduction with Heat Generation A plate-type nuclear fuel element, consisting of a uranium-zirconium alloy (3.2)(10−3) m (0.125 in) thick clad on each side with a (6.4)(10−4)-m- (0.025-in-) thick layer of zirconium, is cooled by water under pressure at 200°C (400°F), the heat-transfer coefficient being 42,600 J/(m2⋅s⋅K) [7500 Btu/(h⋅ft2⋅°F)]. If the temperature at the center of the fuel must not exceed 570°C (1050°F), determine the maximum rate of heat generation in the fuel. The zirconium and zirconium alloy have a thermal conductivity of 21 J/(m⋅s⋅K) [12 Btu/(h⋅ft2)(°F/ft)]. Solution. Equation (5-4a) may be integrated for each material. The heat generation is zero in the cladding, and its value for the fuel may be determined from the integrated equations. Let x = 0 at the midplane of the fuel. Then x1 = (1.6)(10−3) m (0.0625 in) at the cladding-fuel interface and x2 = (2.2)(10−3) m (0.0875 in) at the cladding-water interface. Let the subscripts c, f refer to cladding and fuel respectively. The boundary conditions are:
When temperatures of materials are a function of both time and space variables, more complicated equations result. Equation (5-2) is the three-dimensional unsteady-state conduction equation. It involves the rate of change of temperature with respect to time ∂ t/ ∂θ. Solutions to most practical problems must be obtained through the use of digital computers. Numerous articles have been published on a wide variety of transient conduction problems involving various geometrical shapes and boundary conditions. One-Dimensional Conduction The one-dimensional transient conduction equations are (for constant physical properties) ∂ t/∂θ = α(∂2t/∂x2) + q′/cρ ∂t α ∂ ∂t q′ ᎏ=ᎏᎏ rᎏ +ᎏ ∂θ r dr ∂r cρ
(rectangular coordinates)
(5-19a)
(cylindrical coordinates)
(5-19b)
q′ ∂t α ∂ ∂t (spherical coordinates) (5-19c) ᎏ = ᎏ2 ᎏ r2 ᎏ + ᎏ ∂θ r ∂r ∂r cρ These equations have been solved analytically for solid slabs, cylinders, and spheres. The solutions are in the form of infinite series, and usually the results are plotted as curves involving four ratios [Gurney and Lurie, Ind. Eng. Chem., 15, 1170 (1923)] defined as follows with q′ = 0:
HEAT TRANSFER BY CONDUCTION Y = (t′ − t)/(t′ − tb)
X = kθ/ρcrm2
5-11
Since each ratio is dimensionless, any consistent units may be employed in any ratio. The significance of the symbols is as follows: t′ = temperature of the surroundings; tb = initial uniform temperature of the body; t = temperature at a given point in the body at the time θ measured from the start of the heating or cooling operations; k = uniform thermal conductivity of the body; ρ = uniform density of the body; c = specific heat of the body; hT = coefficient of total heat transfer between the surroundings and the surface of the body expressed as heat transferred per unit time per unit area of the surface per unit difference in temperature between surroundings and surface; r = distance, in the direction of heat conduction, from the midpoint or midplane of the body to the point under consideration; rm = radius of a sphere or cylinder, one-half of the thickness of a slab heated from both faces, the total thickness of a slab heated from one face and insulated perfectly at the other; and x = distance, in the direction of heat conduction, from the surface of a semi-infinite body (such as the surface of the earth) to the point under consideration. In making the integrations which lead to the curves shown, the following factors were assumed constant: c, hT, k, r, rm, t′, x, and ρ. The working curves are shown in Figs. 5-2 to 5-5 for cylinders of infinite length, spheres, slabs of infinite faces, and semi-infinite solids respectively, with Y plotted as ordinates on a logarithmic scale versus X as abscissas to an arithmetic scale, for various values of the ratios m and n. To facilitate calculations involving instantaneous rates of cooling or heating of the semi-infinite body, Fig. 5-5 shows also a curve of dY/dX versus X. Similar plots to a larger scale are given in McAdams,
Brown and Marco, Schack, and Stoever (see “Introduction: General References”). For a solid of infinite thickness (Fig. 5-5) and with m = 0, 2 z (5-21) Y = ᎏ exp (−z2) dz π 0 where z = 1/ 2X and the “error integral” may be evaluated from standard mathematical tables. Various numerical and graphical methods are used for unsteadystate conduction problems, in particular the Schmidt graphical method (Foppls Festschrift, Springer-Verlag, Berlin, 1924). These methods are very useful because any form of initial temperature distribution may be used. Two-Dimensional Conduction The governing differential equation for two-dimensional transient conduction is q′ ∂t ∂2t ∂2t (5-22) ᎏ = α ᎏ2 + ᎏ2 + ᎏ ∂θ ∂x ∂y cρ McAdams (Heat Transmission, 3d ed., McGraw-Hill, New York, 1954) gives various forms of transient difference equations and methods of solving transient conduction problems. The availability of computers and a wide variety of computer programs permits virtually routine solution of complicated conduction problems. Conduction with Change of Phase A special type of transient problem (the Stefan problem) involves conduction of heat in a material when freezing or melting occurs. The liquid-solid interface moves with time, and in addition to conduction, latent heat is either generated or absorbed at the interface. Various problems of this type are discussed by Bankoff [in Drew et al. (eds.), Advances in Chemical Engineering, vol. 5, Academic, New York, 1964].
Heating and cooling of a solid cylinder having an infinite ratio of length to diameter.
FIG. 5-4 Heating and cooling of a solid slab having a large face area relative to the area of the edges.
m = k/hTrm
n = r/rm
FIG. 5-2
(5-20a,b) (5-20c,d)
Heating and cooling of a solid of infinite thickness, neglecting edge effects. (This may be used as an approximation in the zone near the surface of a body of finite thickness.)
FIG. 5-5 FIG. 5-3
Heating and cooling of a solid sphere.
5-12
HEAT AND MASS TRANSFER
HEAT TRANSFER BY CONVECTION COEFFICIENT OF HEAT TRANSFER In many cases of heat transfer involving either a liquid or a gas, convection is an important factor. In the majority of heat-transfer cases met in industrial practice, heat is being transferred from one fluid through a solid wall to another fluid. Assume a hot fluid at a temperature t1 flowing past one side of a metal wall and a cold fluid at t7 flowing past the other side to which a scale of thickness xs adheres. In such a case, the conditions obtaining at a given section are illustrated diagrammatically in Fig. 5-6. For turbulent flow of a fluid past a solid, it has long been known that, in the immediate neighborhood of the surface, there exists a relatively quiet zone of fluid, commonly called the film. As one approaches the wall from the body of the flowing fluid, the flow tends to become less turbulent and develops into laminar flow immediately adjacent to the wall. The film consists of that portion of the flow which is essentially in laminar motion (the laminar sublayer) and through which heat is transferred by molecular conduction. The resistance of the laminar layer to heat flow will vary according to its thickness and can range from 95 percent of the total resistance for some fluids to about 1 percent for other fluids (liquid metals). The turbulent core and the buffer layer between the laminar sublayer and turbulent core each offer a resistance to heat transfer which is a function of the turbulence and the thermal properties of the flowing fluid. The relative temperature difference across each of the layers is dependent upon their resistance to heat flow. The Energy Equation A complete energy balance on a flowing fluid through which heat is being transferred results in the energy equation (assuming constant physical properties): ∂t ∂t ∂t ∂t cρ ᎏ + u ᎏ + v ᎏ + w ᎏ ∂θ ∂x ∂y ∂z
∂2 t ∂ 2 t ∂ 2 t = k ᎏ2 + ᎏ2 + ᎏ2 + q′ + Φ (5-23) ∂x ∂y ∂z where Φ is the term accounting for energy dissipation due to fluid viscosity and is significant in high-speed gas flow and in the flow of highly viscous liquids. Except for the time term, the left-hand terms of Eq. (523) are the so-called convective terms involving the energy carried by the fluid by virtue of its velocity. Therefore, the solution of the equation is dependent upon the solution of the momentum equations of flow. Solutions of Eq. (5-23) exist only for several simple flow cases and geometries and mainly for laminar flow. For turbulent flow the difficulties of expressing the fluid velocity as a function of space and time coordinates and of obtaining reliable values of the effective thermal conductivity of the flowing fluid have prevented solution of the equation unless simplifying assumptions and approximations are made.
Individual Coefficient of Heat Transfer Because of the complicated structure of a turbulent flowing stream and the impracticability of measuring thicknesses of the several layers and their temperatures, the local rate of heat transfer between fluid and solid is defined by the equations dq = hi d Ai (t1 − t3) = ho d Ao (t5 − t7)
(5-24)
where hi and ho are the local heat-transfer coefficients inside and outside the wall, respectively, and temperatures are defined by Fig. 5-6. The definition of the heat-transfer coefficient is arbitrary, depending on whether bulk-fluid temperature, centerline temperature, or some other reference temperature is used for t1 or t7. Equation (5-24) is an expression of Newton’s law of cooling and incorporates all the complexities involved in the solution of Eq. (5-23). The temperature gradients in both the fluid and the adjacent solid at the fluid-solid interface may also be related to the heat-transfer coefficient: dt dt dq = h i dAi (t1 − t3) = −k ᎏ = −k ᎏ (5-25) dx fluid dx solid Equation (5-25) holds for the liquid only if laminar flow exists immediately adjacent to the solid surface. The integration of Eq. (5-24) will give out out dq dq Ai = or Ao = (5-26) ᎏ ᎏ in in hi ∆ ti ho ∆ to which may be evaluated only if the quantities under the integral can be expressed in terms of a single variable. If q is a linear function of ∆t and h is constant, then Eq. (5-26) gives hA(∆tin − ∆ tout) q = ᎏᎏ (5-27) ln (∆ tin/∆ tout) where the ∆t factor is the logarithmic-mean temperature difference between the wall and the fluid. Frequently experimental data report average heat-transfer coefficients based upon an arbitrarily defined temperature difference, the two most common being hlm A(∆tin − ∆ tout) q = ᎏᎏ (5-28a) ln(∆ tin/∆ tout) ham A(∆tin + ∆ tout) q = ᎏᎏ (5-28b) 2 where hlm and ham are average heat-transfer coefficients based upon the logarithmic-mean temperature difference and the arithmeticaverage temperature difference, respectively. Overall Coefficient of Heat Transfer In testing commercial heat-transfer equipment, it is not convenient to measure tube temperatures (t3 or t4 in Fig. 5-6), and hence the overall performance is expressed as an overall coefficient of heat transfer U based on a convenient area dA, which may be dAi, dAo, or an average of dAi and dAo; whence, by definition,
dq = U dA (t1 − t7)
(5-29)
U is called the “overall coefficient of heat transfer,” or merely the “overall coefficient.” The rate of conduction through the tube wall and scale deposit is given by kdAavg(t3 − t4) dq = ᎏᎏ = hd dAd(t4 − t5) (5-30) x Upon eliminating t3, t4, t5 from Eqs. (5-24), (5-29), and (5-30), the complete expression for the steady rate of heat flow from one fluid through the wall and scale to a second fluid, as illustrated in Fig. 5-6, is t1 − t7 dq = ᎏᎏᎏᎏ = U dA (t1 − t7) (5-31)* 1 x 1 1 ᎏᎏ + ᎏᎏ + ᎏᎏ + ᎏᎏ hi dAi k dAavg hd dAd ho dAo Temperature gradients for a steady flow of heat by conduction and convection from a warmer to a colder fluid separated by a solid wall.
FIG. 5-6
* Normally, dirt and scale resistance must be considered on both sides of the tube wall. The area dA is any convenient reference area.
HEAT TRANSFER BY CONVECTION
tally and are given in Table 5-1. Fluid properties are evaluated at tf = (ts + t′)/2. For vertical plates and cylinders and 1 < NPr < 40, Kato, Nishiwaki, and Hirata [Int. J. Heat Mass Transfer, 11, 1117 (1968)] recommend the relations 0.36 0.175 NNu = 0.138NGr (NPr − 0.55) (5-33a)
Representation of Heat-Transfer Film Coefficients There are two general methods of expressing film coefficients: (1) dimensionless relations and (2) dimensional equations. The dimensionless relations are usually indicated in either of two forms, each yielding identical results. The preferred form is that suggested by Colburn [Trans. Am. Inst. Chem. Eng., 29, 174–210 (1933)]. It relates, primarily, three dimensionless groups: the Stanton number h/cG, the Prandtl number cµ/k, and the Reynolds number DG/µ. For more accurate correlation of data (at Reynolds number <10,000), two additional dimensionless groups are used: ratio of length to diameter L/D and ratio of viscosity at wall (or surface) temperature to viscosity at bulk temperature. Colburn showed that the product of the Stanton number and the two-thirds power of the Prandtl number (and, in addition, power functions of L/D and µw /µ for Reynolds number <10,000) is approximately equal to half of the Fanning friction factor f/2. This product is called the Colburn j factor. Since the Colburn type of equation relates heat transfer and fluid friction, it has greater utility than other expressions for the heat-transfer coefficient. The classical (and perhaps more familiar) form of dimensionless expressions relates, primarily, the Nusselt number hD/k, the Prandtl number cµ/k, and the Reynolds number DG/µ. The L/D and viscosityratio modifications (for Reynolds number <10,000) also apply. The dimensional equations are usually expansions of the dimensionless expressions in which the terms are in more convenient units and in which all numerical factors are grouped together into a single numerical constant. In some instances, the combined physical properties are represented as a linear function of temperature, and the dimensional equation resolves into an equation containing only one or two variables.
for NGr > 109, and 0.25 0.25 NNu = 0.683NGr NPr [NPr /(0.861 + NPr)]0.25
h = b(∆t)mL3m − 1 (5-34) Values of b in SI and U.S. customary units are given in Table 5-1 for air, water, and organic liquids. Simultaneous Loss by Radiation The heat transferred by radiation is often of significant magnitude in the loss of heat from surfaces to the surroundings because of the diathermanous nature of atmospheric gases (air). It is convenient to represent radiant-heat transfer, for this case, as a radiation film coefficient which is added to the film coefficient for convection, giving the combined coefficient for convection and radiation (hc + hr). In Fig. 5-7 values of the film coefficient for radiation hr are plotted against the two surface temperatures for emissivity = 1.0. Table 5-2 shows values of (hc + hr) from single horizontal oxidized pipe surfaces. Enclosed Spaces The rate of heat transfer across an enclosed space is calculated from a special coefficient h′ based upon the temperature difference between the two surfaces, where h′ = (q/A)/ (ts1 − ts2). The value of h′L/k may be predicted from Eq. (5-32) by using the values of a and m given in Table 5-3. For vertical enclosed cells 10 in high and up to 2-in gap width, Landis and Yanowitz (Proc. Third Int. Heat Transfer Conf., Chicago, 1966, vol. II, p. 139) give
Natural convection occurs when a solid surface is in contact with a fluid of different temperature from the surface. Density differences provide the body force required to move the fluid. Theoretical analyses of natural convection require the simultaneous solution of the coupled equations of motion and energy. Details of theoretical studies are available in several general references (Brown and Marco, Introduction to Heat Transfer, 3d ed., McGraw-Hill, New York, 1958; and Jakob, Heat Transfer, Wiley, New York, vol. 1, 1949; vol. 2, 1957) but have generally been applied successfully to the simple case of a vertical plate. Solution of the motion and energy equations gives temperature and velocity fields from which heat-transfer coefficients may be derived. The general type of equation obtained is the so-called Nusselt equation: L3ρ2gβ ∆t cµ m hL (5-32a) ᎏ = a ᎏᎏ ᎏ k µ2 k
q
0.84
(NGr NPr)0.28
3
7
hδ δ ᎏ = 0.2 + 0.145 ᎏ NGr k D1
0.25
δ exp −0.02 ᎏ D1
b, air at Configuration
Y = NGrNPr
a
m
Vertical surfaces L = vertical dimension < 3 ft
<104 104 < Y < 109 >109 <10−5 −5 10 < Y < 10−3 10−3 < Y < 1 1 < Y < 104 104 < Y < 109 >109 105 < Y < 2 × 107(FU) 2 × 107 < Y < 3 × 1010(FU) 3 × 105 < Y < 3 × 1010(FD)
1.36 0.59 0.13 0.49 0.71 1.09 1.09 0.53 0.13 0.54 0.14 0.27
1⁄ 5
Horizontal flat surface
(5-36)
for 0.55 < δ/D1 < 2.65, where NGr is based upon gap width δ and D1 is the core diameter of the annulus.
Values of a, m, and b for Eqs. (5-32) and (5-34)
Horizontal cylinder L = diameter < 8 in
(5-35)
for 2 × 10 < NGr NPr (δ/L) < 10 , where q/A is the uniform heat flux and ∆ t is the temperature difference at L/2. Equation (5-35) is applicable for air, water, and silicone oils. For horizontal annuli Grugal and Hauf (Proc. Third Int. Heat Transfer Conf., Chicago, 1966, vol. II, p 182) report 3
(5-32b) NNu = a(NGr NPr) Nusselt Equation for Various Geometries Natural-convection coefficients for various bodies may be predicted from Eq. (5-32). The various numerical values of a and m have been determined experimen-
NOTE:
δ
= 0.123(δ/L) ᎏA ᎏ k ∆t
m
TABLE 5-1
(5-33b)
for NGr < 109. Simplified Dimensional Equations Equation (5-32) is a dimensionless equation, and any consistent set of units may be used. Simplified dimensional equations have been derived for air, water, and organic liquids by rearranging Eq. (5-32) into the following form by collecting the fluid properties into a single factor:
NATURAL CONVECTION
5-13
d s 0 1⁄25 1⁄ 10 1⁄ 5 d s d s d
b, water at
b, organic liquid at
21°C
70°F
21°C
70°F
21°C
70°F
1.37 1.24
0.28 0.18
127
26
59
12
1.32 1.24 1.86
0.27 0.18 0.38
0.88
0.18
FU = facing upward; FD = facing downward. b in SI units is given in °C column; b in U.S. customary units, in °F column.
5-14
HEAT AND MASS TRANSFER siderable effort has been directed toward deriving some simple relationship between momentum and heat transfer. The methodology has been to use easily observed velocity profiles to obtain a measure of the diffusivity of momentum in the flowing stream. The analogy between heat and momentum is invoked by assuming that diffusion of heat and diffusion of momentum occur by essentially the same mechanism so that a relatively simple relationship exists between the diffusion coefficients. Thus, the diffusivity of momentum is used to predict temperature profiles and thence by Eq. (10-25) to predict the heat-transfer coefficient. The analogy has been reasonably successful for simple geometries and for fluids of very low Prandtl number (liquid metals). For highPrandtl-number fluids the empirical analogy of Colburn [Trans. Am. Inst. Chem. Eng., 29, 174 (1933)] has been very successful. A j factor for momentum transfer is defined as j = f/2, where f is the friction factor for the flow. The j factor for heat transfer is assumed to be equal to the j factor for momentum transfer j = h/cG(cµ/k)2/3
(5-37)
More involved analyses for circular tubes reduce the equations of motion and energy to the form (ν + εM)du τgc (5-38a) ᎏ = − ᎏᎏ dy ρ (α + εH)dt q/A (5-38b) ᎏ = − ᎏᎏ dy cρ where εH is the eddy diffusivity of heat and εM is the eddy diffusivity of momentum. The units of diffusivity are L2/θ. The eddy viscosity is EM = ρεM, and the eddy conductivity of heat is EH = εHcρ. Values of εM are determined via Eq. (5-38a) from experimental velocitydistribution data. By assuming εH/εM = constant (usually unity), Eq. (10-38b) is solved to give the temperature distribution from which the heat-transfer coefficient may be determined. The major difficulties in solving Eq. (5-38b) are in accurately defining the thickness of the various flow layers (laminar sublayer and buffer layer) and in obtaining a suitable relationship for prediction of the eddy diffusivities. For assistance in predicting eddy diffusivities, see Reichardt (NACA Tech. Memo 1408, 1957) and Strunk and Chao [Am. Inst. Chem. Eng. J., 10, 269 (1964)]. Internal and External Flow Two main types of flow are considered in this subsection: internal or conduit flow, in which the fluid completely fills a closed stationary duct, and external or immersed flow, in which the fluid flows past a stationary immersed solid. With internal flow, the heat-transfer coefficient is theoretically infinite at the location where heat transfer begins. The local heat-transfer coefficient rapidly decreases and becomes constant, so that after a certain length the average coefficient in the conduit is independent of the length. The local coefficient may follow an irregular pattern, however, if obstructions or turbulence promoters are present in the duct. For immersed flow, the local coefficient is again infinite at the point where heating begins, after which it decreases and may show various irregularities depending upon the configuration of the body. Usually in this instance the local coefficient never becomes constant as flow proceeds downstream over the body. When heat transfer occurs during immersed flow, the rate is dependent upon the configuration of the body, the position of the body, the proximity of other bodies, and the flow rate and turbulence of the
FIG. 5-7 Radiation coefficients of heat transfer hr. To convert British thermal units per hour-square foot-degrees Fahrenheit to joules per square meter– second–kelvins, multiply by 5.6783; °C = (°F − 32)/1.8.
FORCED CONVECTION Forced-convection heat transfer is the most frequently employed mode of heat transfer in the process industries. Hot and cold fluids, separated by a solid boundary, are pumped through the heat-transfer equipment, the rate of heat transfer being a function of the physical properties of the fluids, the flow rates, and the geometry of the system. Flow is generally turbulent, and the flow duct varies in complexity from circular tubes to baffled and extended-surface heat exchangers. Theoretical analyses of forced-convection heat transfer have been limited to relatively simple geometries and laminar flow. Analyses of turbulent-flow heat transfer have been based upon some mechanistic model and have not generally yielded relationships which were suitable for design purposes. Usually for complicated geometries only empirical relationships are available, and frequently these are based upon limited data and special operating conditions. Heattransfer coefficients are strongly influenced by the mechanics of flow occurring during forced-convection heat transfer. Intensity of turbulence, entrance conditions, and wall conditions are some of the factors which must be considered in detail as greater accuracy in prediction of coefficients is required. Analogy between Momentum and Heat Transfer The interrelationship of momentum transfer and heat transfer is obvious from examining the equations of motion and energy. For constant fluid properties, the equations of motion must be solved before the energy equation is solved. If fluid properties are not constant, the equations are coupled, and their solutions must proceed simultaneously. ConTABLE 5-2
Nominal pipe diameter, in 1 3 5 10
Values of (hc + hr)*
Btu/(h⋅ft2⋅°F from pipe to room) For horizontal bare standard steel pipe of various sizes in a room at 80°F Temperature difference, °F
30
50
100
150
200
250
300
350
400
450
500
550
600
650
700
2.16 1.97
2.26 2.05 1.95 1.87
2.50 2.25 2.15 2.07
2.73 2.47 2.36 2.29
3.00 2.73 2.61 2.54
3.29 3.00 2.90 2.82
3.60 3.31 3.20 3.12
3.95 3.69 3.54 3.47
4.34 4.03 3.90 3.84
4.73 4.43
5.16 4.85
5.60 5.26
6.05 5.71
6.51 6.19
6.98 6.66
1.80
*Bailey and Lyell [Engineering, 147, 60 (1939)] give values for (hc + hr) up to ∆ts of 1000°F. °C = (°F − 32)/1.8; 5.6783 Btu/(h⋅ft2⋅°F) = J/(m2⋅s⋅/K).
HEAT TRANSFER BY CONVECTION TABLE 5-3
Values of a and m for Eq. (5-32)
Configuration Vertical spaces Horizontal spaces
NGr NPr(δ/L)3
a
m
2 × 104 to 2 × 105 2 × 105 to 107 104 to 3 × 105 3 × 105 to 107
0.20 (δ/L)−5/36 0.071 (δ/L)1/9 0.21 (δ/L)−1/4 0.075
d s d s
stream. The heat-transfer coefficient varies over the immersed body, since both the thermal and the momentum boundary layers vary in thickness. Relatively simple relationships are available for simple configurations immersed in an infinite flowing fluid. For complicated configurations and assemblages of bodies such as are found on the shell side of a heat exchanger, little is known about the local heattransfer coefficient; empirical relationships giving average coefficients are all that are usually available. Research that has been conducted on local coefficients in complicated geometries has not been extensive enough to extrapolate into useful design relationships. Laminar Flow Normally, laminar flow occurs in closed ducts when NRe < 2100 (based on equivalent diameter De = 4 × free area ÷ perimeter). Laminar-flow heat transfer has been subjected to extensive theoretical study. The energy equation has been solved for a variety of boundary conditions and geometrical configurations. However, true laminar-flow heat transfer very rarely occurs. Natural-convection effects are almost always present, so that the assumption that molecular conduction alone occurs is not valid. Therefore, empirically derived equations are most reliable. Data are most frequently correlated by the Nusselt number (NNu)lm or (NNU)am, the Graetz number NGz = (NRe NPr D/L), and the Grashof (natural-convection effects) number NGr. Some correlations consider only the variation of viscosity with temperature, while others also consider density variation. Theoretical analyses indicate that for very long tubes (NNu)lm approaches a limiting value. Limiting Nusselt numbers for various closed ducts are shown in Table 5-4. Circular Tubes For horizontal tubes and constant wall temperature, several relationships are available, depending on the Graetz number. For 0.1 < NGz < 104, Hausen’s [Allg. Waermetech., 9, 75 (1959)], the following equation is recommended. 0.8 µb 0.14 0.19NGz (NNu)lm = 3.66 + ᎏᎏ (5-39) ᎏ 0.467 1 + 0.117NGz µw
For NGz > 100, the Sieder-Tate relationship [Ind. Eng. Chem., 28, 1429 (1936)] is satisfactory for small diameters and ∆t’s: 1/3 (NNu)am = 1.86NGz (µb /µw)0.14
(5-40)
A more general expression covering all diameters and ∆ t’s is 1/3 obtained by including an additional factor 0.87(1 + 0.015NGr ) on the right side of Eq. (5-40). The diameter should be used in evaluating NGr. An equation published by Oliver [Chem. Eng. Sci., 17, 335 (1962)] is also recommended.
TABLE 5-4 Values of Limiting Nusselt Number in Laminar Flow in Closed Ducts Limiting Nusselt number NGr < 4.0 Configuration Circular tube Concentric annulus Equilateral triangle Rectangles Aspect ratio: 1.0 (square) 0.713 0.500 0.333 0.25 0 (parallel planes)
For laminar flow in vertical tubes a series of charts developed by Pigford [Chem. Eng. Prog. Symp. Ser. 17, 51, 79 (1955)] may be used to predict values of ham. Annuli Approximate heat-transfer coefficients for laminar flow in annuli may be predicted by the equation of Chen, Hawkins, and Solberg [Trans. Am. Soc. Mech. Eng., 68, 99 (1946)]:
Constant wall temperature
Constant heat flux
3.66
4.36 Eq. (10-42) 3.00
2.89
3.63 3.78 4.11 4.77 5.35 8.24
3.39 7.60
µb
ᎏ ᎏ D µ
0.45 0.5 De (NNu)am = 1.02N Re N Pr ᎏ L
δ = cell width, L = cell length.
5-15
0.4
D2
0.8
1
0.14 0.05 NGr
(5-41)
1
Limiting Nusselt numbers for slug-flow annuli may be predicted (for constant heat flux) from Trefethen (General Discussions on Heat Transfer, London, ASME, New York, 1951, p. 436): 8(m − 1)(m2 − 1)2 (NNu)lm = ᎏᎏᎏ 4m4 ln m − 3m4 + 4m2 − 1
(5-42)
where m = D2/D1. The Nusselt and Reynolds numbers are based on the equivalent diameter, D2 − D1. Limiting Nusselt numbers for laminar flow in annuli have been calculated by Dwyer [Nucl. Sci. Eng., 17, 336 (1963)]. In addition, theoretical analyses of laminar-flow heat transfer in concentric and eccentric annuli have been published by Reynolds, Lundberg, and McCuen [Int. J. Heat Mass Transfer, 6, 483, 495 (1963)]. Lee [Int. J. Heat Mass Transfer, 11, 509 (1968)] presented an analysis of turbulent heat transfer in entrance regions of concentric annuli. Fully developed local Nusselt numbers were generally attained within a region of 30 equivalent diameters for 0.1 < NPr < 30, 104 < NRe < 2 × 105, 1.01 < D2 /D1 < 5.0. Parallel Plates and Rectangular Ducts The limiting Nusselt number for parallel plates and flat rectangular ducts is given in Table 5-4. Norris and Streid [Trans. Am. Soc. Mech. Eng., 62, 525 (1940)] report for constant wall temperature 1/3 (NNu)lm = 1.85NGz
(5-43)
for NGz > 70. Both Nusselt number and Graetz numbers are based on equivalent diameter. For large temperature differences it is advisable to apply the correction factor (µb/µw)0.14 to the right side of Eq. (5-43). For rectangular ducts Kays and Clark (Stanford Univ., Dept. Mech. Eng. Tech. Rep. 14, Aug. 6, 1953) published relationships for heating and cooling of air in rectangular ducts of various aspect ratios. For most noncircular ducts Eqs. (5-39) and (5-40) may be used if the equivalent diameter (= 4 × free area/wetted perimeter) is used as the characteristic length. See also Kays and London, Compact Heat Exchangers, 3d ed., McGraw-Hill, New York, 1984. Immersed Bodies When flow occurs over immersed bodies such that the boundary layer is completely laminar over the whole body, laminar flow is said to exist even though the flow in the mainstream is turbulent. The following relationships are applicable to single bodies immersed in an infinite fluid and are not valid for assemblages of bodies. In general, the average heat-transfer coefficient on immersed bodies is predicted by NNu = Cr(NRe)m(NPr)1/3
(5-44)
Values of Cr and m for various configurations are listed in Table 5-5. The characteristic length is used in both the Nusselt and the Reynolds numbers, and the properties are evaluated at the film temperature = (tw + t∞)/2. The velocity in the Reynolds number is the undisturbed free-stream velocity. Heat transfer from immersed bodies is discussed in detail by Eckert and Drake, Jakob, and Knudsen and Katz (see “Introduction: General References”), where equations for local coefficients and the effects of unheated starting length are presented. Equation (5-44) may also be expressed as m−1 NSt NPr2/3 = Cr NRe = f/2
(5-45)
where f is the skin-friction drag coefficient (not the form drag coefficient). Falling Films When a liquid is distributed uniformly around the periphery at the top of a vertical tube (either inside or outside) and allowed to fall down the tube wall by the influence of gravity, the fluid
5-16
HEAT AND MASS TRANSFER
TABLE 5-5
Laminar-Flow Heat Transfer over Immersed Bodies [Eq. (5-44)] Configuration
Characteristic length
Flat plate parallel to flow Circular cylinder axes perpendicular to flow
Plate length Cylinder diameter
Non-circular cylinder, axis Perpendicular to flow, characteristic Length perpendicular to flow
Square, short diameter Square, long diameter Hexagon, short diameter Hexagon, long diameter
Sphere*
Diameter
NRe
NPr
Cr
m
103 to 3 × 105 1–4 4 – 40 40 – 4000 4 × 103 – 4 × 104 4 × 104 – 2.5 × 105 5 × 103 – 105 5 × 103 – 105 5 × 103 – 105 5 × 103 – 2 × 104 2 × 104 – 105 1 – 7 × 104
>0.6
0.648 0.989 0.911 0.683 0.193 0.0266 0.104 0.250 0.155 0.162 0.0391 0.6
0.50 0.330 0.385 0.466 0.618 0.805 0.675 0.588 0.638 0.638 0.782 0.50
>0.6
>0.6 0.6 – 400
*Replace NNu by NNu − 2.0 in Eq. (5-44).
does not fill the tube but rather flows as a thin layer. Similarly, when a liquid is applied uniformly to the outside and top of a horizontal tube, it flows in layer form around the periphery and falls off the bottom. In both these cases the mechanism is called gravity flow of liquid layers or falling films. For the turbulent flow of water in layer form down the walls of vertical tubes the dimensional equation of McAdams, Drew, and Bays [Trans. Am. Soc. Mech. Eng., 62, 627 (1940)] is recommended: hlm = bΓ1/3
(5-46)
where b = 9150 (SI) or 120 (U.S. customary) and is based on values of Γ = WF/πD ranging from 0.25 to 6.2 kg/(m⭈s) [600 to 15,000 lb/(h⭈ft)] of wetted perimeter. This type of water flow is used in vertical vaporin-shell ammonia condensers, acid coolers, cycle water coolers, and other process-fluid coolers. The following dimensional equations may be used for any liquid flowing in layer form down vertical surfaces: For
k3ρ2g 4Γ ᎏ > 2100 hlm = 0.01 ᎏ µ µ2
For
4Γ k2ρ4/3cg2/3 ᎏ < 2100 ham = 0.50 ᎏᎏ µ Lµ1/3
4Γ ᎏk ᎏ µ 1/3
cµ
1/3
µ
1/3
(5-47a)
ᎏ ᎏ µ µ 1/3
1/4
4Γ
1/9
(5-47b)
w
Equation (5-47b) is based on the work of Bays and McAdams [Ind. Eng. Chem., 29, 1240 (1937)]. The significance of the term L is not clear. When L = 0, the coefficient is definitely not infinite. When L is large and the fluid temperature has not yet closely approached the wall temperature, it does not appear that the coefficient should necessarily decrease. Within the finite limits of 0.12 to 1.8 m (0.4 to 6 ft), this equation should give results of the proper order of magnitude. For falling films applied to the outside of horizontal tubes, the Reynolds number rarely exceeds 2100. Equations may be used for falling films on the outside of the tubes by substituting πD/2 for L. For water flowing over a horizontal tube, data for several sizes of pipe are roughly correlated by the dimensional equation of McAdams, Drew, and Bays [Trans. Am. Soc. Mech. Eng., 62, 627 (1940)]. ham = b (Γ/D0)1/3
4, 91 (1934)] fits both the laminar extreme and the fully turbulent extreme quite well. D 2/3 µb 0.14 2/3 (NNu)am = 0.116(NRe − 125)NPr1/3 1 + ᎏ (5-49) ᎏ L µw between 2100 and 10,000. It is customary to represent the probable magnitude of coefficients in this region by hand-drawn curves (Fig. 5-8). Equation (5-40) is plotted as a series of curves ( j factor versus Reynolds number with L/D as parameters) terminating at Reynolds number = 2100. Continuous curves for various values of L/D are then hand-drawn from these terminal points to coincide tangentially with the curve for forced-convection, fully turbulent flow [Eq. (5-50c)].
Turbulent Flow Circular Tubes Numerous relationships have been proposed for predicting turbulent flow in tubes. For high-Prandtl-number fluids, relationships derived from the equations of motion and energy through the momentum-heat-transfer analogy are more complicated and no more accurate than many of the empirical relationships that have been developed. For NRe > 10,000, 0.7 < NPr < 170, for properties based on the bulk temperature and for heating, the Dittus-Boelter equation [Boelter, Cherry, Johnson and Martinelli, Heat Transfer Notes, McGrawHill, New York (1965)] may be used: 0.8 0.4 NNu = 0.0243 NRe NPr (µ b /µw)0.14
(5-50a)
For cooling, the relationship is 0.14 0.3 NNu = 0.0265 N 0.8 Re N Pr (µb /µw)
(5.50b)
The Colburn correlation is 2/3 −0.2 (µw /µb)0.14 = 0.023NRe jH = NSt NPr
(5-50c)
In Eq. (5-50c), the viscosity-ratio factor may be neglected if properties are evaluated at the film temperature (tb + tw)/2.
(5-48)
where b = 3360 (SI) or 65.6 (U.S. customary) and Γ ranges from 0.94 to 4 kg/m⋅s) [100 to 1000 lb/(h⋅ft)]. Falling films are also used for evaporation in which the film is both entirely or partially evaporated (juice concentration). This principle is also used in crystallization (freezing). The advantage of high coefficient in falling-film exchangers is partially offset by the difficulties involved in distribution of the film, maintaining complete wettability of the tube, and pumping costs required to lift the liquid to the top of the exchanger. Transition Region Turbulent-flow equations for predicting heat transfer coefficients are usually valid only at Reynolds numbers greater than 10,000. The transition region lies in the range 2000 < NRe < 10,000. No simple equation exists for accomplishing a smooth mathematical transition from laminar flow to turbulent flow. Of the relationships proposed, Hausen’s equation [Z. Ver. Dtsch. Ing. Beih. Verfahrenstech., No.
Graphical representation of the Colburn j factor for the heating and cooling of fluids inside tubes. The curves for NRe below 2100 are based on Eq. (5-40). L is the length of each pass in feet. The curve for NRe above 10,000 is represented by Eq. (5-50c).
FIG. 5-8
HEAT TRANSFER BY CONVECTION For the transition and turbulent regions, including diameter to length effects, Gnielinski [Int. Chem. Eng., 16, 359 (1976)] recommends a modification of an equation suggested by Petukhov and Popov [High Temp., 1, 69 (1963)]. This equation applies in the ranges 0 < D/L < 1, 0.6 < NPr < 2000, 2300 < NRe < 106. ( f/2)(NRe − 1000) NPr D 1+ ᎏ NNu = ᎏᎏᎏ 2/3 1 + 12.7( f/2)0.5(NPr − 1) L
µb
ᎏ µ 2/3
0.14
(5-51a)
w
The Fanning friction f is determined by an equation recommended by Filonenko [Teploenergetika, 1, 40 (1954)] f = 0.25 (1.82 log10 NRe − 1.64)−2
(5-51b)
Any other appropriate friction factor equation for smooth tubes may be used. Approximate predictions for rough pipes may be obtained from Eq. (5-50c) if the right-hand term is replaced by f / 2 for the rough pipe. For air, Nunner (Z. Ver. Dtsch. Ing. Forsch., 1956, p. 455) obtains frough (NNu)rough (5-52) ᎏᎏ = ᎏ (NNu)smooth fsmooth Dippery and Sabersky [Int. J. Heat Mass Transfer, 6, 329 (1963)] present a complete discussion of the influence of roughness on heat transfer in tubes. Dimensional Equations for Various Conditions For gases at ordinary pressures and temperatures based on cµ/k = 0.78 and µ = (1.76)(10−5) Pa⋅s [0.0426 lb/(ft⋅h)] h = bcρ0.8(V 0.8/D0.2)
(5-53)
where b = (3.04)(10−3) (SI) or (1.44)(10−2) (U.S. customary). For air at atmospheric pressure h = b(V 0.8/D0.2)
(5-54)
where b = 3.52 (SI) or (4.35)(10−4) (U.S. customary). For water [based on a temperature range of 5 to 104°C (40 to 220°F)] h = 1057 (1.352 + 0.02t) (V 0.8/D0.2)
(5-55a)
in SI units with t = °C, or h = 0.13(1 + 0.011t)(V 0.8/D0.2)
(5-55b)
in U.S. customary units with t = °F. For organic liquids, based on c = 2.092 J/kg⋅K)[0.5 Btu/(lb⋅°F)], k = 0.14 J/(m⋅s⋅K) [0.08 Btu/(h⋅ft⋅°F)], µb = (1)(10−3) Pa⋅s (1.0 cP), and ρ = 810 kg/m3 (50 lb/ft3), h = b(V 0.8/D0.2)
(5-56)
where b = 423 (SI) or (5.22)(10 ) (U.S. customary). Within reasonable limits, coefficients for organic liquids are about one-third of the values obtained for water. Entrance effects are usually not significant industrially if L/D > 60. Below this limit Nusselt recommended the conservative equation for 10 < L/D < 400 and properties evaluated at bulk temperature −2
0.8 NNu = 0.036NRe NPr1/3(L/D)−0.054
(5-57)
It is common to correlate entrance effects by the equation hm /h = 1 + F(D/L)
(5-58)
where h is predicted by Eq. (5-50a) or (5-50b), and hm is the mean coefficient for the pipe in question. Values of F are reported by Boelter, Young, and Iverson [NACA Tech. Note 1451, 1948] and tabulated by Kays and Knudsen and Katz (see “Introduction: General References”). Selected values of F are as follows: Fully developed velocity profile 1.4 Abrupt contraction entrance 6 90° right-angle bend 7 180° round bend 6
5-17
For large temperature differences different equations are necessary and usually are specifically applicable to either gases or liquids. Gambill (Chem. Eng., Aug. 28, 1967, p. 147) provides a detailed review of high-flux heat transfers to gases. He recommends 0.8 0.4 0.021NRe NPr NNu = ᎏᎏ (5-59) (Tw /Tb)0.29 + 0.0019 (L/D) for 10 < L/D < 240, 110 < Tb < 1560 K (200 < Tb < 2800°R), 1.1 < (Tw /Tb) < 8.0, and properties evaluated at Tb. For liquids, Eq. (5-50c) is generally satisfactory. Annuli For diameter ratios D1/D2 > 0.2, Monrad and Pelton’s equation [Trans. Am. Inst. Chem. Eng., 38, 593 (1942)] is recommended for either or both the inner and outer tube: 0.8 1/3 NNu = 0.020NRe NPr (D2/D1)0.53 (5-60a) Equation (5-51a) may also be used for smooth annuli as follows: (NNu)ann D1 (5-60b) ᎏ=φ ᎏ (NNu)tube D2 The hydraulic diameter D2 − D1 is used in NNu, NRe, and D/L is used for the annulus. The function on the right of Eq. (5-60b) is given by Petukhov and Roizen [High Temp., 2, 65 (1964)] as follows: Inner tube heated 0.86 (D1 /D2)−0.16 Outer tube heated 1 − 0.14 (D1 /D2)0.6 If both tubes are heated, the function is the sum of the above two functions divided by 1 + D1/D2 [Stephan, Chem. Ing. Tech., 34, 207 (1962)]. The Colburn form of relationship may be employed for the individual walls of the annulus by using the individual friction factor for each wall [see Knudsen, Am. Inst. Chem. Eng. J., 8, 566 (1962)]: jH1 = (NSt)1NPr2/3 = f1 /2 (5-61a)
jH2 = (NSt)2 NPr2/3 = f2 /2 (5-61b) Rothfus, Monrad, Sikchi, and Heideger [Ind. Eng. Chem., 47, 913 (1955)] report that the friction factor f2 for the outer wall bears the same relation to the Reynolds number for the outer portion of the annular stream 2(r22 − λm)Vρ /r2 µ as the friction factor for circular tubes does to the Reynolds number for circular tubes, where r2 is the radius of the outer tube and λm is the position of maximum velocity in the annulus, estimated from r22 − r12 λm = ᎏ (5-62)* ln (r2/r1)2 To calculate the friction factor f1 for the inner tube use the relation f2 r2(λm − r12) f1 = ᎏᎏ (5-63) r1(r22 − λm) There have been several analyses of turbulent heat transfer in annuli: for example, Deissler and Taylor (NACA Tech. Note 3451, 1955), Kays and Leung [Int. J. Heat Mass Transfer, 6, 537 (1963)], Lee [Int. J. Heat Transfer, 11, 509 (1968)], Sparrow, Hallman and Siegel [Appl. Sci. Res., 7A, 37 (1958)], and Johnson and Sparrow [Am. Soc. Mech. Eng. J. Heat Transfer, 88, 502 (1966)]. The reader is referred to these for details of the analyses. For annuli containing externally finned tubes the heat-transfer coefficients are a function of the fin configurations. Knudsen and Katz (Fluid Dynamics and Heat Transfer, McGraw-Hill, New York, 1958) present relationships for transverse finned tubes, spined tubes, and longitudinal finned tubes in annuli. Noncircular Ducts Equations (5-50a) and (5-50b) may be employed for noncircular ducts by using the equivalent diameter De = 4 × free area per wetted perimeter. Kays and London (Compact Heat Exchangers, 3rd ed., McGraw-Hill, New York, 1984) give charts for various noncircular ducts encountered in compact heat exchangers. Vibrations and pulsations generally tend to increase heat-transfer coefficients. Example 2: Calculation of j Factors in an Annulus Calculate the heat-transfer j factors for both walls of an annulus for the following condi-
* Equation (5-62) predicts the point of maximum velocity for laminar flow in annuli and is only an approximate equation for turbulent flow. Brighton and Jones [Am. Soc. Mech. Eng. Basic Eng., 86, 835 (1964)] and Macagno and McDougall [Am. Inst. Chem. Eng. J., 12, 437 (1966)] give more accurate equations for predicting the point of maximum velocity for turbulent flow.
5-18
HEAT AND MASS TRANSFER
tions: D1 = 0.0254 m (1.0 in); D2 = 0.0635 m (2.5 in); water at 15.6°C (60°F); µ/ρ = (1.124)(10−6) m2/s [(1.21)(10−5) ft2/s]; velocity = 1.22 m/s (4 ft/s). 0.06352 − 0.02542 λm = ᎏᎏᎏ2 = (4.621)(10−4) m2 (0.716 in2) 4 ln (0.0635/0.0254) 2(r 22 − λm)Vρ 2[0.03182 − (4.621)(10−4)(1.22)] Re2 = ᎏᎏ = ᎏᎏᎏᎏ = (3.74)(104) r2µ (0.0318)(1.124)(10−6) From Eq. (5-51b), f2 = 0.0055. Hence 2/3 jH2 = (NSt)2 N Pr = 0.00275 From Eq. (5-63), (0.0055)(0.0318)[(4.621)(10−4) − 0.01272] = 0.00754 f1 = ᎏᎏᎏᎏᎏ (0.0127)[0.03182 − (4.621)(10−4)] 2/3 from which jH1 = (NSt)1N Pr = 0.00377. These results indicate that for this system the heat-transfer coefficient on the inner tube is about 40 percent greater than on the outer tube.
Coils For flow inside helical coils, Reynolds number above 10,000, multiply the value of the film coefficient obtained from the applicable equation for straight tubes by the term (1 + 3.5 Di /Dc). For flow inside helical coils, Reynolds number less than 10,000, substitute the term (Dc /Di)1/2 for (L/Di) where the latter appears in the applicable equation for straight tubes (frequently as part of the Graetz number). For flat spiral (pancake) coils, in which the ratio Dc /Di varies for each turn, a different value of coefficient will be obtained for each turn; a weighted average based on length per turn is used. For flow outside helical coils use the equation for flow normal to a bank of tubes, in-line flow. Finned Tubes (Extended Surface) When the film coefficient on the outside of a metal tube is much lower than that on the inside, as when steam condensing in a pipe is being used to heat air, externally finned (or extended) heating surfaces are of value in increasing substantially the rate of heat transfer per unit length of tube. The data on extended heating surfaces, for the case of air flowing outside and at right angles to the axes of a bank of finned pipes, can be represented approximately by the dimensional equation derived from 0.6 VF0.6 p′ hf = b ᎏ (5-64) ᎏ 0.4 D0 p′ − D0 −3 where b = 5.29 (SI) or (5.39)(10 ) (U.S. customary); hf is the film coefficient of heat transfer on the air side; VF is the face velocity of the air; p′ is the center-to-center spacing, m, of the tubes in a row; and D0 is the outside diameter, m, of the bare tube (diameter at the root of the fins). In atmospheric air-cooled finned tube exchangers, the air-film coefficient from Eq. (5-64) is sometimes converted to a value based on outside bare surface as follows: Af + Auf A hfo = hf ᎏ = hf ᎏT (5-65) Aof Ao in which hfo is the air-film coefficient based on external bare surface; hf is the air-film coefficient based on total external surface; AT is total external surface, and Ao is external bare surface of the unfinned tube; Af is the area of the fins; Auf is the external area of the unfinned portion of the tube; and Aof is area of tube before fins are attached. Fin efficiency is defined as the ratio of the mean temperature difference from surface to fluid divided by the temperature difference from fin to fluid at the base or root of the fin. Graphs of fin efficiency for extended surfaces of various types are given by Gardner [Trans. Am. Soc. Mech. Eng., 67, 621 (1945)]. Heat-transfer coefficients for finned tubes of various types are given in a series of papers [Trans. Am. Soc. Mech. Eng., 67, 601 (1945)]. For flow of air normal to fins in the form of short strips or pins, Norris and Spofford [Trans. Am. Soc. Mech. Eng., 64, 489 (1942)] correlate their results for air by the dimensionless equation of Pohlhausen: hm cpµ 2/3 zpGmax −0.5 = 1.0 ᎏ (5-66) ᎏ ᎏ cpGmax k µ for values of zpGmax/µ ranging from 2700 to 10,000.
For the general case, the treatment suggested by Kern (Process Heat Transfer, McGraw-Hill, New York, 1950, p. 512) is recommended. Because of the wide variations in fin-tube construction, it is convenient to convert all film coefficients to values based on the inside bare surface of the tube. Thus to convert the film coefficient based on outside area (finned side) to a value based on inside area Kern gives the following relationship: hfi = (ΩAf + Ao)(hf /Ai)
(5-67)
in which hfi is the effective outside film coefficient based on the inside area, hf is the outside film coefficient calculated from the applicable equation for bare tubes, Af is the surface area of the fins, Ao is the surface area on the outside of the tube which is not finned, Ai is the inside area of the tube, and Ω is the fin efficiency defined as Ω = (tanh mbf)/mbf
(5-68)
m = (hf pf / kax)1/2 m−1 (ft−1)
(5-69)
in which and bf = height of fin. The other symbols are defined as follows: pf is the perimeter of the fin, ax is the cross-sectional area of the fin, and k is the thermal conductivity of the material from which the fin is made. Fin efficiencies and fin dimensions are available from manufacturers. Ratios of finned to inside surface are usually available so that the terms A f, Ao, and Ai may be obtained from these ratios rather than from the total surface areas of the heat exchangers. Banks of Tubes For heating and cooling of fluids flowing normal to a bank of circular tubes at least 10 rows deep the following equations are applicable: Colburn type:
h cµ ᎏ ᎏ cGmax k
2/3
a = ᎏᎏ =j (DoGmax/µ)0.4
(5-70)
cµ ᎏ k
(5-71)
Nusselt type:
hD DoGmax ᎏ=a ᎏ k µ
0.6
1/3
The dimensionless constant a in these equations varies depending upon conditions. Conditions, Reynolds number > 3000
Value of a
Flow normal to apex of diamond, staggered arrangement No leakage Normal leakage in baffled exchanger Flow normal to flat side of diamond, not staggered (in-line) arrangement No leakage Normal leakage in baffled exchanger
0.330 0.198 0.260 0.156
For Reynolds number less than 3000, Eq. (5-70) would give conservative results, but greater accuracy (if desired) may be obtained by using the following equation.
h cµ ᎏ ᎏ cGmax k
2/3
a = ᎏᎏ =j (DoGmax /µ)m
(5-72)
in which the constant a and exponent m are as follows: Reynolds number
m
Tube pitch
100–300
0.492
Staggered In-line
1–100
0.590
Staggered In-line
Leakage
a
None Normal None Normal None Normal None Normal
0.695 0.416 0.548 0.329 1.086 0.650 0.855 0.513
HEAT TRANSFER BY CONVECTION The following dimensional equations (5-73 to 5-77) are based on flow normal to a bank of staggered tubes without leakage. Multiply the values obtained for h by 0.6 for normal leakage and, in addition, by 0.79 for in-line (not staggered) tube arrangement. c1/3k2/3ρ0.6 V 0.6 max h = b ᎏᎏ (5-73) µ0.267 D00.4 where b = 0.33 (SI) or 0.261 (U.S. customary). For gases at ordinary pressures and temperatures, based on cµ/k = 0.78; µ = (1.76)(10−5) Pa⭈s [0.0426 lb/(ft⭈h)], 0.6 Gmax h = bc ᎏ (5-74) 0.4 D0 where b = (4.82)(10−3) (SI) or 0.109 (U.S. customary). For air at atmospheric pressure 0.6 Vmax h=bᎏ (5-75) D00.4 where b = 5.33 (SI) or (5.44)(10−3) (U.S. customary). For water based on a temperature range 7 to 104°C (40 to 220°F) 0.6 Vmax h = 986(1.21 + 0.0121t) ᎏ (5-76a) D00.4 in SI units and t in °C. 0.6 Vmax h = 1.01(1 + 0.0067t) ᎏ (5-76b) D00.4 in U.S. customary units and t in °F. For organic liquids, based on c = 2.22 J/(kg⭈K) [0.53 Btu/(lb⭈°F)], k = 0.14 J/(m⭈s⭈K) [0.08 Btu/ (h⭈ft⭈°F)], µb = (1)(10−3) Pa⭈s (1.0 cP), ρ = 810 kg/m3 (50 lb/ft3), V 0.6 max h=bᎏ (5-77) D00.4 where b = 400 (SI) or 0.408 (U.S. customary).
5-19
LIQUID METALS Liquid metals constitute a class of heat-transfer media having Prandtl numbers generally below 0.01. Heat-transfer coefficients for liquid metals cannot be predicted by the usual design equations applicable to gases, water, and more viscous fluids with Prandtl numbers greater than 0.6. Relationships for predicting heat-transfer coefficients for liquid metals have been derived from solution of Eqs. (5-38a) and (5-38b). By the momentum-transfer-heat-transfer analogy, the eddy conductivity of heat is kNPr(EM /µ) ≈ k for small NPr. Thus in the solution of Eqs. (5-38a) and (5-38b) the knowledge of the thickness of various layers of flow is not critical. In fact, assumption of slug flow and constant conductivity (=k) across the duct gives reasonable values of heat-transfer coefficients for liquid metals. For constant heat flux: NNu = 5 + 0.025(NRe NPr)0.8
(5-81)
For constant wall temperature: NNu = 7 + 0.025(NRe NPr)0.8
(5-82)
For 0.003 < NPr < 0.05 and constant heat flux, Sleicher and Rouse [Int. J. Heat Mass Transfer, 18, 677 (1975)] obtained the correlation 0.85 NNu = 6.3 + 0.0167 NRe NPr0.93
(5-83)
For parallel plates and annuli with D2 /D1 < 1.4 and uniform heat flux, Seban [Trans. Am. Soc. Mech. Eng., 72, 789 (1950)] obtained the equation NNu = 5.8 + 0.020(NRe NPr)0.8
(5-84)
JACKETS AND COILS OF AGITATED VESSELS
For annuli only, application of a factor of 0.70(D2 /D1)0.53 is recommended for Eqs. (5-81) and (5-82). For more accurate semiempirical relationships for tubes, annuli, and rod bundles, refer to Dwyer [Am. Inst. Chem. Eng. J., 9, 261 (1963)]. Hsu [Int. J. Heat Mass Transfer, 7, 431 (1964)] and Kalish and Dwyer [Int. J. Heat Mass Transfer, 10, 1533 (1967)] discuss heat transfer to liquid metals flowing across banks of tubes. Hsu recommends the equations
See Sec. 18.
NNu = 0.81NRe NPr (φ/D)1/2
(for uniform heat flux)
NONNEWTONIAN FLUIDS
NNu = 0.096NRe NPr (φ/D)
(for cosine surface temperature)
1/2
A wide variety of nonnewtonian fluids are encountered industrially. They may exhibit Bingham-plastic, pseudoplastic, or dilatant behavior and may or may not be thixotropic. For design of equipment to handle or process nonnewtonian fluids, the properties must usually be measured experimentally, since no generalized relationships exist to predict the properties or behavior of the fluids. Details of handling nonnewtonian fluids are described completely by Skelland (NonNewtonian Flow and Heat Transfer, Wiley, New York, 1967). The generalized shear-stress rate-of-strain relationship for nonnewtonian fluids is given as d ln (D ∆P/4L) n′ = ᎏᎏ (5-78) d ln (8V/D) as determined from a plot of shear stress versus velocity gradient. For circular tubes, NGz > 100, n′ > 0.1, and laminar flow 1/3 (NNu)lm = 1.75 δ1/3 s NGz
(5-79)
where δs = (3n′ + 1)/4n′. When natural-convection effects are considered, Metzer and Gluck [Chem. Eng. Sci., 12, 185 (1960)] obtained the following for horizontal tubes: NPr NGr D 0.4 1/3 γb 0.14 (NNu)lm = 1.75 δ 1/3 NGz + 12.6 ᎏ (5-80) ᎏ s L γw where properties are evaluated at the wall temperature, i.e., γ = gc K′8n′ − 1 and τw = K′(8V/D)n′. Metzner and Friend [Ind. Eng. Chem., 51, 879 (1959)] present relationships for turbulent heat transfer with nonnewtonian fluids. Relationships for heat transfer by natural convection and through laminar boundary layers are available in Skelland’s book (op. cit.).
(5-85) (5-86)
where the heat-transfer coefficient is based on the average circumferential temperature around the tubes, the Reynolds number is based on the superficial velocity through the tube bank, D is the tube outside diameter, and φ is a velocity potential function having the following values:
D/p′
φ/D square pitch
φ/D equilateral triangular pitch
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
2.00 2.02 2.07 2.16 2.30 2.52 2.84 3.34 4.23
2.00 2.02 2.06 2.15 2.27 2.45 2.71 3.11 3.80
Equations (5-85) and (5-86) are useful in calculating tube-surface temperatures. Further information on liquid-metal heat transfer in tube banks is given by Hsu for spheres and elliptical rod bundles [Int. J. Heat Mass Transfer, 8, 303 (1965)] and by Kalish and Dwyer for oblique flow across tube banks [Int. J. Heat Mass Transfer, 10, 1533 (1967)]. For additional details of heat transfer with liquid metals for various systems see Dwyer (1968 ed., Na and Nak supplement to Liquid Metals Handbook) and Stein (“Liquid Metal Heat Transfer,” in Advances in Heat Transfer, vol. 3, Academic, New York, 1966).
5-20
HEAT AND MASS TRANSFER
HEAT TRANSFER WITH CHANGE OF PHASE In any operation in which a material undergoes a change of phase, provision must be made for the addition or removal of heat to provide for the latent heat of the change of phase plus any other sensible heating or cooling that occurs in the process. Heat may be transferred by any one or a combination of the three modes—conduction, convection, and radiation. The process involving change of phase involves mass transfer simultaneous with heat transfer. CONDENSATION Condensation Mechanisms Condensation occurs when a saturated vapor comes in contact with a surface whose temperature is below the saturation temperature. Normally a film of condensate is formed on the surface, and the thickness of this film, per unit of breadth, increases with increase in extent of the surface. This is called film-type condensation. Another type of condensation, called dropwise, occurs when the wall is not uniformly wetted by the condensate, with the result that the condensate appears in many small droplets at various points on the surface. There is a growth of individual droplets, a coalescence of adjacent droplets, and finally a formation of a rivulet. Adhesional force is overcome by gravitational force, and the rivulet flows quickly to the bottom of the surface, capturing and absorbing all droplets in its path and leaving dry surface in its wake. Film-type condensation is more common and more dependable. Dropwise condensation normally needs to be promoted by introducing an impurity into the vapor stream. Substantially higher (6 to 18 times) coefficients are obtained for dropwise condensation of steam, but design methods are not available. Therefore, the development of equations for condensation will be for the film type only. The physical properties of the liquid, rather than those of the vapor, are used for determining the film coefficient for condensation. Nusselt [Z. Ver. Dtsch. Ing., 60, 541, 569 (1916)] derived theoretical relationships for predicting the film coefficient of heat transfer for condensation of a pure saturated vapor. A number of simplifying assumptions were used in the derivation. The Reynolds number of the condensate film (falling film) is 4Γ/µ, where Γ is the weight rate of flow (loading rate) of condensate per unit perimeter kg/(s⭈m) [lb/(h⭈ft)]. The thickness of the condensate film for Reynolds number less than 2100 is (3µΓ/ρ2g)1/3. Condensation Coefficients Vertical Tubes For the following cases Reynolds number < 2100 and is calculated by using Γ = WF /πD. The Nusselt equation for the heat-transfer coefficient for condensate films may be written in the following ways (using liquid physical properties and where L is the cooled length and ∆t is tsv − ts): Colburn type: h cµ 5.35 ᎏ ᎏ=ᎏ cG k 4Γ/µ Γ WF2 ρ2g where G = ᎏᎏ = ᎏ (3µΓ/ρ2g)1/3 29.6D2µ
(5-87)
1/3
kg/(s⭈m2) [lb/(h⭈ft2)]
1/4
L3ρ2g = 0.925 ᎏ µΓ
1/3
(5-88)
Dimensional: h = b(k3ρ2D/µbWF)1/3
(5-89)
where b = 127 (SI) or 756 (U.S. customary). For steam at atmospheric pressure, k = 0.682 J/(m⭈s⭈K) [0.394 Btu/(h⭈ft⭈°F)], ρ = 960 kg/m3 (60 lb/ft3), µb = (0.28)(10−3) Pa⭈s (0.28 cP), h = b(D/WF)1/3
h = b(D/WF)1/3
(5-91)
where b = 457 (SI) or 1080 (U.S. customary). Horizontal Tubes For the following cases Reynolds number < 2100 and is calculated by using Γ = WF /2L. Colburn type: h cµ 4.4 (5-92) ᎏᎏ=ᎏ cG k 4Γ/µ Γ WF2 ρ2g 1/3 kg/(s⭈m2) [lb/(h⭈ft2)] G = ᎏᎏ = ᎏ (3µΓ/ρ2g)1/3 12L2µ Nusselt type: D3ρ2g 1/3 hD D3ρ2gλ 1/4 = 0.76 ᎏ (5-93)* ᎏ = 0.73 ᎏ k kµ ∆t µΓ Dimensional:
h = b(k3ρ2L/µbWF)1/3
(5-90)
(5-94)
where b = 205.4 (SI) or 534 (U.S. customary). For steam at atmospheric pressure h = b(L/WF)1/3
(5-95)
where b = 2080 (SI) or 4920 (U.S. customary). For organic vapors at normal boiling point h = b(L/WF)1/3
(5-96)
where b = 324 (SI) or 766 (U.S. customary). Figure 5-9 is a nomograph for determining coefficients of heat transfer for condensation of pure vapors. Banks of Horizontal Tubes (NRe < 2100) In the idealized case of N tubes in a vertical row where the total condensate flows smoothly from one tube to the one beneath it, without splashing, and still in laminar flow on the tube, the mean condensing coefficient hN for the entire row of N tubes is related to the condensing coefficient for the top tube h1 by hN = h1N−1/4
(5-97)
Dukler Theory The preceding expressions for condensation are based on the classical Nusselt theory. It is generally known and conceded that the film coefficients for steam and organic vapors calculated by the Nusselt theory are conservatively low. Dukler [Chem. Eng. Prog., 55, 62 (1959)] developed equations for velocity and temperature distribution in thin films on vertical walls based on expressions of Deissler (NACA Tech. Notes 2129, 1950; 2138, 1952; 3145, 1959) for the eddy viscosity and thermal conductivity near the solid boundary. According to the Dukler theory, three fixed factors must be known to establish the value of the average film coefficient: the terminal Reynolds number, the Prandtl number of the condensed phase, and a dimensionless group Nd defined as follows: 2/3 2 0.553 0.78 0.16 Nd = (0.250µ1.173 ρG ) L µG /(g D ρL
Nusselt type: hL L3ρ2gλ ᎏ = 0.943 ᎏ k kµ ∆t
where b = 2954 (SI) or 6978 (U.S. customary). For organic vapors at normal boiling point, k = 0.138 J/(m⭈s⭈K) [0.08 Btu/(h⭈ft⭈°F)], ρ = 720 kg/m3 (45 lb/ft3), µb = (0.35)(10−3) Pa⭈s (0.35 cP),
(5-98)
Graphical relationships of these variables are available in Document 6058, ADI Auxiliary Publications Project, Library of Congress, Washington. If rigorous values for condensing-film coefficients are desired, especially if the value of Nd in Eq. (5-98) exceeds (1)(10−5), it is suggested that these graphs be used. For the case in which interfacial shear is zero, Fig. 5-10 may be used. It is interesting to note that, according to the Dukler development, there is no definite transition Reynolds number; deviation from Nusselt theory is less at low Reynolds numbers; and when the Prandtl number of a fluid is less * If the vapor density is significant, replace ρ2 with ρl(ρl − ρv).
HEAT TRANSFER WITH CHANGE OF PHASE
5-21
Chart for determining film coefficient hm for film-type condensation of pure vapor, based on Eqs. 5-88 and 5-93. For 4 ρ 2k3/µ is in U.S. customary units; to convert feet to vertical tubes multiply hm by 1.2. If 4Γ/µf exceeds 2100, use Fig. 5-10. λ meters, multiply by 0.3048; to convert inches to centimeters, multiply by 2.54; and to convert British thermal units per hour– square foot–degrees Fahrenheit to watts per square meter–kelvins, multiply by 5.6780.
FIG. 5-9
than 0.4 (at Reynolds number above 1000), the predicted values for film coefficient are lower than those predicted by the Nusselt theory. The Dukler theory is applicable for condensate films on horizontal tubes and also for falling films, in general, i.e., those not associated with condensation or vaporization processes.
Vapor Shear Controlling For vertical in-tube condensation with vapor and liquid flowing cocurrently downward, if gravity controls, Figs. 5-9 and 5-10 may be used. If vapor shear controls, the Carpenter-Colburn correlation (General Discussion on Heat Transfer, London, 1951, ASME, New York, p. 20) is applicable:
5-22
HEAT AND MASS TRANSFER
Dukler plot showing average condensing-film coefficient as a function of physical properties of the condensate film and the terminal Reynolds number. (Dotted line indicates Nusselt theory for Reynolds number < 2100.) [Reproduced by permission from Chem. Eng. Prog., 55, 64 (1959).] FIG. 5-10
1/2 hµl / kl ρl1/2 = 0.065(NPr)1/2 l Fvc
(5-99a)
Fvc = fG2vm /2ρv 2 1/2 Gvi2 + GviGvo + Gvo Gvm = ᎏᎏᎏ 3 and f is the Fanning friction factor evaluated at
where
(5-99b)
(5-99c)
(NRe)vm = DiGvm /µv
(5-99d)
and the subscripts vi and vo refer to the vapor inlet and outlet, respectively. An alternative formulation, directly in terms of the friction factor, is h = 0.065 (cρkf/2µρv)1/2Gvm
(5-99e)
expressed in consistent units. Another correlation for vapor-shear-controlled condensation is the Boyko-Kruzhilin correlation [Int. J. Heat Mass Transfer, 10, 361 (1967)], which gives the mean condensing coefficient for a stream between inlet quality xi and outlet quality xo: hDi DiGT 0.8 (ρ /ρ (ρ /ρ m)i + m)o (5-100a) ᎏ = 0.024 ᎏ (NPr)l0.43 ᎏᎏᎏ kl µl 2 where GT = total mass velocity in consistent units ρ ρl − ρv (5-100b) ᎏ = 1 + ᎏ xi ρm i ρv
ρ ρ −ρ =1+ᎏx ᎏ ρ ρ l
and
v
(5-100c)
o
m
o
v
For horizontal in-tube condensation at low flow rates Kern’s modification (Process Heat Transfer, McGraw-Hill, New York, 1950) of the Nusselt equation is valid: k 3l ρl (ρl − ρv)gλ 1/4 Lk 3l ρl(ρl − ρv)g 1/3 hm = 0.761 ᎏᎏ = 0.815 ᎏᎏ (5-101) WF µl π µl Di ∆t where WF is the total vapor condensed in one tube and ∆ t is tsv − ts . A more rigorous correlation has been proposed by Chaddock [Refrig. Eng., 65(4), 36 (1957)]. Use consistent units. At high condensing loads, with vapor shear dominating, tube orientation has no effect, and Eq. (5-100a) may also be used for horizontal tubes. Condensation of pure vapors under laminar conditions in the presence of noncondensable gases, interfacial resistance, superheating, variable properties, and diffusion has been analyzed by Minkowycz and Sparrow [Int. J. Heat Mass Transfer, 9, 1125 (1966)].
BOILING (VAPORIZATION) OF LIQUIDS Boiling Mechanisms Vaporization of liquids may result from various mechanisms of heat transfer, singly or combinations thereof.
For example, vaporization may occur as a result of heat absorbed, by radiation and convection, at the surface of a pool of liquid; or as a result of heat absorbed by natural convection from a hot wall beneath the disengaging surface, in which case the vaporization takes place when the superheated liquid reaches the pool surface. Vaporization also occurs from falling films (the reverse of condensation) or from the flashing of liquids superheated by forced convection under pressure. Pool boiling refers to the type of boiling experienced when the heating surface is surrounded by a relatively large body of fluid which is not flowing at any appreciable velocity and is agitated only by the motion of the bubbles and by natural-convection currents. Two types of pool boiling are possible: subcooled pool boiling, in which the bulk fluid temperature is below the saturation temperature, resulting in collapse of the bubbles before they reach the surface, and saturated pool boiling, with bulk temperature equal to saturation temperature, resulting in net vapor generation. The general shape of the curve relating the heat-transfer coefficient to ∆tb, the temperature driving force (difference between the wall temperature and the bulk fluid temperature) is one of the few parametric relations that are reasonably well understood. The familiar boiling curve was originally demonstrated experimentally by Nukiyama [ J. Soc. Mech. Eng. ( Japan), 37, 367 (1934)]. This curve points out one of the great dilemmas for boiling-equipment designers. They are faced with at least six heat-transfer regimes in pool boiling: natural convection (+), incipient nucleate boiling (+), nucleate boiling (+), transition to film boiling (−), stable film boiling (+), and film boiling with increasing radiation (+). The signs indicate the sign of the derivative d(q/A)/d ∆ tb. In the transition to film boiling, heat-transfer rate decreases with driving force. The regimes of greatest commercial interest are the nucleate-boiling and stable-film-boiling regimes. Heat transfer by nucleate boiling is an important mechanism in the vaporization of liquids. It occurs in the vaporization of liquids in kettle-type and natural-circulation reboilers commonly used in the process industries. High rates of heat transfer per unit of area (heat flux) are obtained as a result of bubble formation at the liquid-solid interface rather than from mechanical devices external to the heat exchanger. There are available several expressions from which reasonable values of the film coefficients may be obtained. The boiling curve, particularly in the nucleate-boiling region, is significantly affected by the temperature driving force, the total system pressure, the nature of the boiling surface, the geometry of the system, and the properties of the boiling material. In the nucleate-boiling regime, heat flux is approximately proportional to the cube of the temperature driving force. Designers in addition must know the minimum ∆t (the point at which nucleate boiling begins), the critical ∆t (the ∆t above which transition boiling begins), and the maximum heat flux (the heat flux corresponding to the critical ∆t). For designers who do not have experimental data available, the following equations may be used. Boiling Coefficients For the nucleate-boiling coefficient the Mostinski equation [Teplenergetika, 4, 66 (1963)] may be used: q 0.7 P 0.17 P 1.2 P 10 h = bPc0.69 ᎏ 1.8 ᎏ + 4 ᎏ + 10 ᎏ (5-102) A Pc Pc Pc where b = (3.75)(10−5)(SI) or (2.13)(10−4) (U.S. customary), Pc is the critical pressure and P the system pressure, q/A is the heat flux, and h is the nucleate-boiling coefficient. The McNelly equation [J. Imp. Coll. Chem. Eng. Soc., 7(18), (1953)] may also be used: 0.33 qc 0.69 Pkl 0.31 ρl h = 0.225 ᎏl (5-103) ᎏ ᎏ−1 Aλ σ ρv where cl is the liquid heat capacity, λ is the latent heat, P is the system pressure, kl is the thermal conductivity of the liquid, and σ is the surface tension. An equation of the Nusselt type has been suggested by Rohsenow [Trans. Am. Soc. Mech. Eng., 74, 969 (1952)].
hD/k = Cr(DG/µ)2/3(cµ/k)−0.7 in which the variables assume the following form: 1/2 1/2 hβ′ gcσ β′ gcσ W 2/3 cµ = Cr ᎏ ᎏᎏ ᎏ ᎏ ᎏ ᎏ k g(ρL − ρv) µ g(ρL − ρv) A k
(5-104a) −0.7
(5-104b)
The coefficient Cr is not truly constant but varies from 0.006 to 0.015.* It is possible that the nature of the surface is partly responsible for the variation in the constant. The only factor in Eq. (5-104b) not readily available is the value of the contact angle β′. Another Nusselt-type equation has been proposed by Forster and Zuber:† 0.62 1/3 NNu = 0.0015NRe NPr (5-105) which takes the following form:
α W 2σ cρLπ ᎏᎏ ᎏ kρv A ∆p
ρL
ᎏ ∆pg α ρ cρ ∆T π cµ = 0.0015 ᎏ ᎏᎏ ᎏ µ λρ k 1/2
1/4
c
L
2 0.62
L
1/ 2
(5-106)
v
where α = k/ρc (all liquid properties) ∆p = pressure of the vapor in a bubble minus saturation pressure of a flat liquid surface Equations (5-104b) and (5-106) have been arranged in dimensional form by Westwater. The numerical constant may be adjusted to suit any particular set of data if one desires to use a certain criterion. However, surface conditions vary so greatly that deviations may be as large as ⫾25 percent from results obtained. The maximum heat flux may be predicted by the KutateladseZuber [Trans. Am. Soc. Mech. Eng., 80, 711 (1958)] relationship, using consistent units: (ρl − ρv)σg 1/4 q = 0.18gc1/4ρv λ ᎏᎏ (5-107) ᎏ A max ρ2v Alternatively, Mostinski presented an equation which approximately represents the Cichelli-Bonilla [Trans. Am. Inst. Chem. Eng., 41, 755 (1945)] correlation:
HEAT TRANSFER BY RADIATION
5-23
1 − ᎏPP
(5-108)
(q/A)max P ᎏ=b ᎏ Pc Pc
0.35
0.9
c
where b = 0.368(SI) or 5.58 (U.S. customary); Pc is the critical pressure, Pa absolute; P is the system pressure; and (q/A)max is the maximum heat flux. The lower limit of applicability of the nucleate-boiling equations is from 0.1 to 0.2 of the maximum limit and depends upon the magnitude of natural-convection heat transfer for the liquid. The best method of determining the lower limit is to plot two curves: one of h versus ∆ t for natural convection, the other of h versus ∆ t for nucleate boiling. The intersection of these two curves may be considered the lower limit of applicability of the equations. These equations apply to single tubes or to flat surfaces in a large pool. In tube bundles the equations are only approximate, and designers must rely upon experiment. Palen and Small [Hydrocarbon Process., 43(11), 199 (1964)] have shown the effect of tube-bundle size on maximum heat flux. p q gσ(ρl − ρv) 1/4 = b ᎏ ρv λ ᎏᎏ (5-109) ᎏ Do NT A max ρ2v
where b = 0.43 (SI) or 61.6 (U.S. customary), p is the tube pitch, Do is the tube outside diameter, and NT is the number of tubes (twice the number of complete tubes for U-tube bundles). For film boiling, Bromley’s [Chem. Eng. Prog., 46, 221 (1950)] correlation may be used: kv3(ρl − ρv)ρv g h = b ᎏᎏ µv Do ∆ tb
1/4
(5-110)
where b = 4.306 (SI) or 0.620 (U.S. customary). Katz, Myers, and Balekjian [Pet. Refiner, 34(2), 113 (1955)] report boiling heat-transfer coefficients on finned tubes.
HEAT TRANSFER BY RADIATION GENERAL REFERENCES: Much of the pertinent literature on radiative heat transfer has been surveyed in the following texts: Goody, Atmospheric Radiation, Clarendon Press, Oxford, 1964. Sparrow and Cess, Radiation Heat Transfer, Brooks/Cole Publishing Company, Belmont, Calif., 1966. Hottel and Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967. Love, Radiative Heat Transfer, Merrill, Columbus, 1968. Siegel and Howell, Thermal Radiation Heat Transfer, NASA SP-164, GPO, Washington, 1968. Edwards, Radiation Heat Transfer Notes, Hemisphere Publishing Corp., 1981; Howell and Siegel, Thermal Radiative Heat Transfer, McGraw-Hill, 3d ed., 1992; Brewster, Thermal Radiative Transfer and Properties, Wiley, 1992; Modest, Radiative Heat Transfer, McGraw-Hill, 1993. Additional sources are the Journal of Applied Optics and the Journal of the Optical Society of America, particularly for surface properties; the Journal of Quantitative Spectroscopy and Radiative Transfer for gas properties; the Journal of Heat Transfer and the International Journal of Heat and Mass Transfer for broad coverage; and the Journal of the Institute of Energy for applications to industrial furnaces.
Thermal radiation—electromagnetic energy in transport—is emitted within matter excited by temperature; it is absorbed in other matter at distances from the source which depend on the mean free path of the photons emitted. The ratio of the mean free path involved in an energy-transport process to a characteristic dimension of the system of interest determines the mathematical structure of the formulation. In molecular conduction this ratio is minute (unless the system or the density of matter is minute, which is the case of free molecular flow), and a differential equation of energy diffusion is involved. In gas radiation the ratio is generally large enough to give rise to an integral equation, with an unknown function inside the integral. Solids gener-
ally have small enough photon mean free paths (high enough absorption coefficients) for the radiation escaping through the surface to have originated close to the surface; radiative loss is then identifiable with its surface temperature, but an integral equation is still involved if all the surfaces of an enclosure filled with a diathermanous medium like air are not specified as to temperature or are not black. Radiation differs from conduction and convection not only in mathematical structure but in its much higher sensitivity to temperature. It is of dominating importance in furnaces because of their temperature, and in cryogenic insulation because of the vacuum existing between particles. The temperature at which it accounts for roughly half of the total heat loss from a surface in air depends on such factors as surface emissivity and the convection coefficient. For pipes in free convection, this is room temperature; for fine wires of low emissivity it is above red heat. Gases at combustion-chamber temperatures lose more than 90 percent of their energy by radiation from the carbon dioxide, water vapor, and particulate matter. NOMENCLATURE FOR RADIATIVE TRANSFER Terms that are defined at specific places in the text are excluded. a = effective energy fraction of blackbody spectrum in which a nongray gas absorbs. A = area. c = number concentration of particles in a cloud. c1, c2 = first and second Planck-law constants.
* Reported by Westwater in Drew and Hoopes, Advances in Chemical Engineering, vol. I, Academic, New York, 1956, p. 15. † Forster, J. Appl. Phys., 25, 1067 (1954); Forster and Zuber, J. Appl. Phys., 25, 474 (1954); Forster and Zuber, Conference on Nuclear Engineering, University of California, Los Angeles, 1955; excellent treatise on boiling of liquids by Westwater in Drew and Hoopes, Advances in Chemical Engineering, vol. I, Academic, New York, 1956.
5-24
HEAT AND MASS TRANSFER
C = axis-to-axis distance of separation of tubes. Cb = mean specific heat of combustion products from base temperature To to leaving-gas temperature TE. C = cold-surface fraction of a furnace enclosure. CW = correction factor for pressure broadening of radiation from water vapor. d = particle diameter. D = tube diameter; characteristic dimension; dimensionless firing density. D′ = reduced firing density. E = hemispherical emissive power of a blackbody. f = fraction of blackbody radiation lying below λ. fv = volume fraction of space occupied by particles. F = direct view factor; Fij, fraction of isotropic radiation from Ai intercepted directly by Aj. F = total view factor from black source to black sink, with allowance for refractory surfaces (subscripts identify source and sink). Ᏺij = total view factor, radiation from i to j both directly and indirectly, expressed as fraction of blackbody radiation from Ai. gs = direct-exchange area between gas volume and surface. G S = total-exchange area between gas and surface; subscript R indicates allowance for radiatively adiabatic surfaces. h = coefficient of convective heat transfer. H˙ = enthalpy of fuel plus air entering combustion chamber. I = intensity, radiant-energy-flux density per unit solid angle of divergence. ij = shorthand for sisj. k = absorption or emission coefficient; or thermal conductivity. K = constant defined in connection with Eq. (5-147). L = mean beam length; L0, at vanishingly small optical thickness; Lm, average value. L = wall-loss group. Lc = dimensionless convective loss. Lo = dimensionless wall-opening loss. Lr = dimensionless refractory-wall loss. ˙ = mass flow rate. m n = refractive index. p = partial pressure, atm.; subscript c, CO2; subscript w, water vapor. P = total pressure atm. q = heat-flux density, energy per time-area. ˙ = heat flux, energy per time. Q r = separating distance; or electrical resistivity; or refractory (radiatively adiabatic) surface. ss AF, direct-exchange area (subscripts identify surface zones). SS AᏲ, total-exchange area. T = absolute temperature. Subscript 1 (or G), radiating surface (or gas) temperature; subscript E, exit-gas; subscript o, base temperature; subscript F, pseudoadiabatic flame temperature based on C p averaged from To to TE. U = overall coefficient of heat transfer, gas convection to refractory wall to ambient air. W = total leaving-flux density (also radiosity). α = absorptivity or absorptance; α12, absorptance of surface 1 for radiation from surface 2. ∆ = difference between radiating temperature and leaving-gas temperature divided by the pseudoadiabatic flame temperature TF. ε = emissivity or emittance. η = thermal efficiency. Subscript G, gas-side; subscript 1, sinkside. θ = polar angle. λ = wavelength. µm = micrometer (m−6). ρ = reflectance; ρs, specular reflectance. σ = Stefan-Boltzmann constant. τ = ratio of temperature to TF. Subscript G, gas; subscript 1 sink; subscript o, base. τ = transmittance. Ω = solid angle. ω = albedo of a surface.
NATURE OF THERMAL RADIATION Consider a pencil of radiation, defined as all the rays passing through each of two small widely separated areas dA1 and dA2. The rays at dA1 will have a solid angle of divergence dΩ1 equal to the apparent area of dA2 viewed from dA1, divided by the square of the separating distance. Let the normal to dA1 make the angle θ1 with the pencil. The flux density q (energy per time-area) normal to the beam and per unit solid ˙1 angle of its divergence is called the intensity I, and the flux dQ (energy per time) through the area dA1 (of apparent area dA1 cos θ1 normal to the beam) is therefore given by ˙ 1 = dA1(cos θ1)q = I dA1(cos θ1) dΩ1 dQ (5-111) The intensity I along a pencil, in the absence of absorption or scatter, is constant (unless the beam passes into a medium of different refractive index n; then I1/n21 = I2 /n22). The emissive power* of a surface is the flux density (energy per time-surface area) due to emission from it throughout a hemisphere. If the intensity I of emission from a surface is independent of the angle of emission, Eq. (5-111) may be integrated to show that the surface emissive power is πI, though the emission is throughout 2π sr. Blackbody Radiation Engineering calculations of thermal radiation from surfaces are best keyed to the radiation characteristics of the blackbody, or ideal radiator. The characteristic properties of a blackbody are that it absorbs all the radiation incident on its surface and that the quality and intensity of the radiation it emits are completely determined by its temperature. The total radiative flux throughout a hemisphere from a black surface of area A and absolute temperature T is given by the Stefan-Boltzmann law: ˙ = AσT 4 Q or q = σT 4 (5-112) The Stefan-Boltzmann constant σ has the value (0.1713)(10−8) Btu/ (ft2 ⭈h⭈°R4); (1.00)(10−8) CHU/(ft2 ⭈h⭈K4); (4.88)(10−8) kcal/(m2 ⭈h⭈K4); (1.356)(10−12) cal (cm2 ⭈s⭈K4); (5.67)(10−12) W/(cm2 ⭈K4); (5.67)(10−8) W/(m2 ⭈K4); or in terms of Planck constants, c1(π/c2)4/15. From the definition of emissive power, σT 4 is the total emissive power of a blackbody, called E; the intensity IB of blackbody emission is E/π or σT 4/π. The spectral distribution of energy flux from a black body is expressed by Planck’s law: Eλ dλ = (2πhc2 n2 λ−5)/(ehc/kλT − 1) dλ
(n2c1λ−5)/(ec / λT − 1) dλ 2
(5-113) (5-114)
where Eλ dλ is the hemispherical flux density lying in the wavelength range λ to λ + dλ; h is Planck’s constant, (6.6256)(10−27) erg⭈s; c is the velocity of light in vacuo, (2.9979)(1010) cm/s; k is the Boltzmann constant, (1.3805)(10−16) erg/K; λ is the wavelength measured in vacuo; and n is the refractive index of the emitter (λ = nλm, where λm is the wavelength measured in the medium; Eλ d λ = Eλm d λm, where Eλ and Eλm are both measured in the medium; engineers commonly use Eλ). Equation (10-191) may be written Eλ c1(λT)−5 =ᎏ (5-115) ᎏ 2 5 nT ec /λT − 1 The first and second Planck-law constants c1 and c2 are respectively (3.740)(10−16) (J⭈m2)/s and (1.4388)(10−2) m⭈K. The term Eλ/n2T 5, clearly a function only of the product λT, is given in Fig. 5-11 which may be visualized as the monochromatic emissive power versus wavelength measured in vacuo of a black surface at 1 K discharging in vacuo. The wavelength of maximum intensity is seen to be inversely proportional to the absolute temperature. The relation is known as Wien’s displacement law: λmaxT = (2.898)(10−3) m⭈K. This can be misleading, however, since the wavelength of maximum intensity depends on whether intensity is defined in terms of wavelength interval or frequency interval. More useful displacement laws refer to the value of λT corresponding to maximum energy per unit fractional change in wavelength or frequency [(3.67)(10−3) m⭈K] or to the value of λT corresponding to half of the energy [(4.11)(10−3) m⭈K]. Figure 2
* Variously called, in the literature, emittance, total hemispherical intensity, or radiant flux density.
HEAT TRANSFER BY RADIATION
5-25
Distribution of energy in the spectrum of a blackbody. To convert microns to micrometers, multiply by unity. To convert ergs per square centimeter–second–micron–K5 to watts per square meter, per meter, per K5, multiply by 10−3. FIG. 5-11
5-11 carries, at the top, a scale giving the fraction f of the total energy in the spectrum that lies below λT. A generalization useful for identifying the spectral range of greatest interest in evaluations of radiative transfer is that roughly half of the energy from a black surface lies within the twofold range of λT geometrically centered on 3.67 × 10−3, i.e., from λT = (3.67/2)(10−3) to (3.67 × 2)(10−3) m⭈K. One limiting form of the Planck equation, approached as λT → 0, is the Wien equation [Eqs. (5-113) and (5-114)] with the 1 missing in the denominator. The error is less than 1 percent when λT < (3)(10−3) m⭈K or when T < 4800 K if an optical pyrometer with red screen (λ = 0.65µm) is used. RADIATIVE EXCHANGE BETWEEN SURFACES OF SOLIDS Emittance and Absorptance The ratio of the total radiating power of a real surface to that of a black surface at the same temperature is called the emittance of the surface (for a perfectly plane surface, the emissivity), designated by ε. Subscripts λ, θ, and n may be assigned to differentiate monochromatic, directional, and surfacenormal values respectively from the total hemispherical value. If radi-
ation is incident on a surface, the fraction absorbed is called the absorptance (absorptivity), a term to which two subscripts may be appended, the first to identify the temperature of the surface and the second to identify the spectral energy distribution of the surface. According to Kirchhoff’s law, the emissivity and absorptivity of a surface in surroundings at its own temperature are the same for both monochromatic and total radiation. When the temperatures of the surface and its surroundings differ, the total emissivity and absorptivity of the surface often are found to be different, but, because absorptivity is substantially independent of irradiation density, the monochromatic emissivity and absorptivity of surfaces are for all practical purposes the same. The difference between total emissivity and absorptivity depends on the variation, with wavelength, of ελ and on the difference between the emitter temperature and the effective source temperature. Consider radiative exchange between a body of area A1 and temperature T1 and black surroundings at T2. The net interchange is given by ˙ 1 = 2 = A1 Q
∞
[ελEλ(T1) − αλEλ(T2)] dλ
0
= A1(ε1σT 41 − α12σT 42 )
(5-116)
5-26 where
HEAT AND MASS TRANSFER
ε df = ε df
ε1 =
1
λ
λT 1
(5-117)
λ
λT 2
(5-118)
0
and
α12
1
0
The value of ε1 (or α12, the absorptivity of surface A1 for blackbody radiation at T2) is the area under a curve of ελ versus f, the latter read as a function of λT1 (or λT2) from the top ordinate of Fig. 5-11. For a gray surface, ε1 = α12 = ελ. A selective surface is one whose ελ changes dramatically with wavelength. If this change is unidirectional, ε1 and α12 are, according to Eqs. (5-116) to (5-118), markedly different when the absolute-temperature ratio is far from 1; e.g., when T1 = 294 K (530°R; ambient temperature), and T2 = 6000 K (10,800°R; effective solar temperature), ε1 = 0.9 and α12 = 0.1 to 0.2 for a white paint, but ε1 can be as low as 0.12 and α12 above 0.9 for a thin layer of copper oxide on bright aluminum. The effect of radiation-source temperature on the low-temperature absorptivity of a number of additional materials is presented in Fig. 5-12. It will be noted that polished aluminum (curve 15) and anodized (surface-oxidized) aluminum (curve 13), representative of metals and nonmetals respectively, respond oppositely to a change in the temperature of the radiation source. The absorptance of surfaces for solar
FIG. 5-12 Variation of absorptivity with temperature of radiation source. (1) Slate composition roofing. (2) Linoleum, red brown. (3) Asbestos slate. (4) Soft rubber, gray. (5) Concrete. (6) Porcelain. (7) Vitreous enamel, white. (8) Red brick. (9) Cork. (10) White dutch tile. (11) White chamotte. (12) MgO, evaporated. (13) Anodized aluminum. (14) Aluminum paint. (15) Polished aluminum. (16) Graphite. The two dashed lines bound the limits of data on gray paving brick, asbestos paper, wood, various cloths, plaster of paris, lithopone, and paper. To convert degrees Rankine to kelvins, multiply by (5.556)(10−1).
radiation may be read from the right of Fig. 10-45, if solar radiation is assumed to consist of blackbody radiation from a source at 5800 K (10,440°R). Although values of emittance and absorptance depend in very complex ways on the real and imaginary components of the refractive index and on the geometrical structure of the surface layer, the generalizations that follow are possible. Polished Metals 1. ε λ in the infrared is governed by free-electron contributions, is quite low, and is a function of the resistivity-wavelength quotient r/ λ (Fig. 5-13). For λ > 8µm, ε λ,n is approximately 0.0365 r/ λ , where r is in ohm-meters and λ in micrometers (the Drude or HagenRubens relation). At shorter wavelengths, bound-electron contributions become significant and ελ increases, sometimes exhibiting maxima; values of 0.4 to 0.8 are common in the visible spectrum (0.4 to 0.7 µm). ελ is approximately proportional to the square root of the absolute temperature (ελ ∝ r, and r ∝ T) in the far infrared (λ > 8µm), is temperature-insensitive in the near infrared (0.7 to 1.5 µm), and decreases slightly as temperature increases in the visible. 2. Total emittance is substantially proportional to absolute temperature; at moderate temperature, εn = 0.058T rT, where T is in kelvin. 3. The total absorptance of a metal at T1 for radiation from a black or gray source at T2 is equal to the emissivity evaluated at the geometric mean of T1 and T2. Figure 5-13 gives values of ελ, ελ,n, and their ratio as a function of r/λ (dashed lines); and total emissivities ε, εn and their ratio as a function of rT (solid lines). Although the figure is based on free-electron contributions to emissivity in the far infrared, the relations for total emissivity are remarkably good even at high temperatures. Unless extraordinary pains are taken to prevent oxidation, however, a metallic surface may exhibit several times the emittance or absorptance of a polished specimen. The emittance of iron and steel, for example, varies widely with degree of oxidation and roughness; clean metallic surfaces have an emittance of from 0.05 to 0.45 at ambient temperatures to 0.4 to 0.7 at high temperatures; oxidized and/or rough surfaces range from 0.6 to 0.95 at low temperatures to 0.9 to 0.95 at high temperatures. Refractory Materials Grain size and concentration of trace impurities are important. 1. Most refractory materials have an ελ of 0.8 to 1.0 at wavelengths beyond 2 to 4 µm; ελ decreases rapidly toward shorter wavelengths for materials that are white in the visible but retains its high value for black materials such as FeO and Cr2O3. Small concentrations of FeO
FIG. 5-13 Hemispherical and normal emissivities of metals and their ratio. Dashed lines: monochromatic (spectral) values versus r/λ. Solid lines: total values versus rT. To convert ohm-centimeter-kelvins to ohm-meter-kelvins, multiply by 10−2.
HEAT TRANSFER BY RADIATION and Cr2O3 or other colored oxides can cause marked increases in the emittance of materials that normally are white. The sensitivity of the emittance of refractory oxides to small additions of absorbing materials is demonstrated by the results of calculations, shown in Fig. 5-14, of the emittance of a semi-infinite absorbing-scattering medium as a function of its albedo: the ratio of the scatter coefficient to the sum of scatter and absorption coefficients. The results, pertinent to the radiative properties of fibrous materials, paints, oxide coatings, and refractories, show that when absorption accounts for only 0.5 percent (10 percent) of the total attenuation within the medium, the emittance is greater than 0.15 (0.5). ελ for refractory materials varies little with temperature, with the exception of some white oxides which at high temperatures become good emitters in the visible spectrum as a consequence of the induced electronic transitions. 2. Refractory materials generally have a total emittance which is high (0.7 to 1.0) at ambient temperatures and decreases with increase in temperature; a change from 1000 to 1570°C (1850 to 2850°F) may cause a decrease in ε of one-fourth to one-third. 3. The emittance and absorptance increase with increase in grain size over a grain-size range of 1 to 200 µm. 4. The ratio ε/εn of hemispherical to normal emissivity of polished surfaces varies with refractive index n from 1 at n = 1.0 to 0.93 at n = 1.5 (common glass) and back to 0.96 at n = 3. 5. The ratio ε/εn for a surface composed of particulate matter which scatters isotropically varies with ε from 1 when ε = 1 to 0.8 when ε = 0.07 (see Fig. 5-14). 6. The total absorptance shows a decrease with increase in temperature of the radiation source similar to the decrease in emittance with increase in the specimen temperature. Figure 5-12 shows a regular variation of α12 with T2. When T2 is not very different from T1, α12 may be expressed as ε1(T2 /T1)m. It may be shown that Eq. (5-116) is then approximated by m ˙ 1,net = σA1εav 1 + ᎏ Q (T 14 − T 42) (5-119) 4 where εav is evaluated at the arithmetic mean of T1 and T2. For metals m is about 0.5; for nonmetals it is small and negative. Table 5-6, based on a critical evaluation of early data, is illustrative of the emittance of materials encountered in engineering practice; it shows the wide variation possible in the emissivity of a particular material due to variations in surface roughness and thermal pretreatment. (With few exceptions the values refer to emission normal to the surface; see above for conversion to hemispherical values.) More recent data support the range of emittance values given in Table 5-6 and their dependence on surface conditions. Extensive compilations of data are provided by Schmidt and Furthmann (Mitt. KaiserWilhelm-Inst. Eisenforsch., 109, 225), covering data to 1928; by Gubareff, Jansen, and Torborg [Thermal Radiation Properties Sur-
5-27
vey, Honeywell Research Center, Minneapolis, 1960), covering data to 1940; and by Goldsmith, Waterman, and Hirschhorn [Thermophysical Properties of Matter, Purdue University (Touloukian, Ed.), Plenum, 1970]. For opaque materials, the reflectance ρ is the complement of the absorptance. The directional distribution of the reflected radiation depends on the material, its degree of roughness or grain size, and, if a metal, its state of oxidation. Polished surfaces of homogeneous materials reflect specularly. In contrast, the intensity of the radiation reflected from a perfectly diffuse, or Lambert, surface is independent of direction. The directional distribution of reflectance of many oxidized metals, refractory materials, and natural products approximates that of a perfectly diffuse reflector. A better model, adequate for many calculational purposes, is achieved by assuming that the total reflectance ρ is the sum of diffuse and specular components ρD and ρ S . Black-Surface Enclosures View Factor and Direct-Exchange Area When several surfaces are present, the need arises for evaluating a geometrical factor F, called the direct view factor. In the following discussion, restriction is to black surfaces, the intensity from which is independent of angle of emission. Define F12 as the fraction of the radiation leaving surface A1 in all directions which is intercepted by surface A2. Since the net interchange between A1 and A2 must be zero when their temperatures are alike, it follows that A1F12 = A2F21. This product, having the dimensions of area, is called the direct-exchange area and is designated for brevity by 12( 2 1). It is sometimes designated s1s2. Clearly, 11 + 12 + 13 + ⭈⭈⭈ = A1; and when A1 cannot “see” itself, 11 = 0. From Eq. (5-111) and the definition of F: Q1 − 2 dA1(cos θ1) dΩ1 A1F12 s1s2 ᎏ = ᎏᎏ A1 A2 E1 π
dA1(cos θ1) dA2 (cos θ2) (5-120) ᎏᎏᎏ πr 2 where A(cos θ) is the projection of A normal to r, the line connecting dA1 and dA2. Values of s1s2 (or of F12) may be obtained by integrating either Eq. (5-120) or an equivalent contour integral (see Hottel and Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967, chap. 2). Such values are given for opposed parallel disks or rectangles in Fig. 5-15. For rectangles of dimensions L1 and L2, F F 1F2, where F1 and F2 are for squares of sides L1 and L2. The view factor for rectangles in perpendicular planes and having a common edge length x and ratios Y and Z of widths to common length is given by =
A1
1 FYZ = ᎏ πY
1 Y2 ᎏ ln (1 + Y 2) ᎏ2 4 1+Y
A2
Z 1+Z ᎏᎏ ᎏ Y + Z 1 + Y + Z Y2
Z2
2
2
2
1 − Y 2 − Z2
2
2
2
1 1 1 2+ + Y tan−1 ᎏ + Z tan−1 ᎏ − Y Z2 tan−1 ᎏᎏ Y2 + Z2 Y Z
(5-121) The direct-exchange area is given by x2 sysz = ᎏ (5-122) π When the maximum dimensions of each of two plane surfaces is small relative to their center-to-center separating distance r, Eq. (5-120) gives A1(cos θ1)A2(cos θ2) 12 = ᎏᎏᎏ (5-123) πr2 and when, in addition, the normals to A1 and A2 are in a common plane, 12 = A1A2n1n2/πr2
Hemispherical emittance εh and the ratio of hemispherical to normal emittance εh/εn for a semi-infinite absorbing-scattering medium.
FIG. 5-14
(5-124)
where n1 is the normal-to-A1 component of the distance to A2. Equation (5-124) is, for example, in error only by +7 percent for the case of opposed squares separated by 3 times their side dimension. The view factors are given for finite coaxial coextensive cylinders in Fig. 5-16,
5-28
HEAT AND MASS TRANSFER
TABLE 5-6
Normal Total Emissivity of Various Surfaces A. Metals and Their Oxides Surface
Aluminum Highly polished plate, 98.3% pure Polished plate Rough plate Oxidized at 1110°F Aluminum-surfaced roofing Calorized surfaces, heated at 1110°F. Copper Steel Brass Highly polished: 73.2% Cu, 26.7% Zn 62.4% Cu, 36.8% Zn, 0.4% Pb, 0.3% Al 82.9% Cu, 17.0% Zn Hard rolled, polished: But direction of polishing visible But somewhat attacked But traces of stearin from polish left on Polished Rolled plate, natural surface Rubbed with coarse emery Dull plate Oxidized by heating at 1110°F Chromium; see Nickel Alloys for Ni-Cr steels Copper Carefully polished electrolytic copper Commercial, emeried, polished, but pits remaining Commercial, scraped shiny but not mirrorlike Polished Plate, heated long time, covered with thick oxide layer Plate heated at 1110°F Cuprous oxide Molten copper Gold Pure, highly polished Iron and steel Metallic surfaces (or very thin oxide layer): Electrolytic iron, highly polished Polished iron Iron freshly emeried Cast iron, polished Wrought iron, highly polished Cast iron, newly turned Polished steel casting Ground sheet steel Smooth sheet iron Cast iron, turned on lathe Oxidized surfaces: Iron plate, pickled, then rusted red Completely rusted Rolled sheet steel Oxidized iron Cast iron, oxidized at 1100°F Steel, oxidized at 1100°F Smooth oxidized electrolytic iron Iron oxide Rough ingot iron
t, °F*
Emissivity*
440–1070 73 78 390–1110 100
0.039–0.057 0.040 0.055 0.11–0.19 0.216
390–1110 390–1110
0.18–0.19 0.52–0.57
476–674 494–710 530
0.028–0.031 0.033–0.037 0.030
70 73 75 100–600 72 72 120–660 390–1110 100–1000
0.038 0.043 0.053 0.096 0.06 0.20 0.22 0.61–0.59 0.08–0.26
176
0.018
66
0.030
72 242
0.072 0.023
77 390–1110 1470–2010 1970–2330
0.78 0.57 0.66–0.54 0.16–0.13
440–1160
0.018–0.035
350–440 800–1880 68 392 100–480 72 1420–1900 1720–2010 1650–1900 1620–1810
0.052–0.064 0.144–0.377 0.242 0.21 0.28 0.435 0.52–0.56 0.55–0.61 0.55–0.60 0.60–0.70
68 67 70 212 390–1110 390–1110 260–980 930–2190 1700–2040
0.612 0.685 0.657 0.736 0.64–0.78 0.79 0.78–0.82 0.85–0.89 0.87–0.95
Surface Sheet steel, strong rough oxide layer Dense shiny oxide layer Cast plate: Smooth Rough Cast iron, rough, strongly oxidized Wrought iron, dull oxidized Steel plate, rough High temperature alloy steels (see Nickel Alloys). Molten metal Cast iron Mild steel Lead Pure (99.96%), unoxidized Gray oxidized Oxidized at 390°F. Mercury Molybdenum filament Monel metal, oxidized at 1110°F Nickel Electroplated on polished iron, then polished Technically pure (98.9% Ni, + Mn), polished Electroplated on pickled iron, not polished Wire Plate, oxidized by heating at 1110°F Nickel oxide Nickel alloys Chromnickel Nickelin (18–32 Ni; 55–68 Cu; 20 Zn), gray oxidized KA-2S alloy steel (8% Ni; 18% Cr), light silvery, rough, brown, after heating After 42 hr. heating at 980°F. NCT-3 alloy (20% Ni; 25% Cr.), brown, splotched, oxidized from service NCT-6 alloy (60% Ni; 12% Cr), smooth, black, firm adhesive oxide coat from service Platinum Pure, polished plate Strip Filament Wire Silver Polished, pure Polished Steel, see Iron. Tantalum filament Tin—bright tinned iron sheet Tungsten Filament, aged Filament Zinc Commercial, 99.1% pure, polished Oxidized by heating at 750°F. Galvanized sheet iron, fairly bright Galvanized sheet iron, gray oxidized
t, °F* 75 75 73 73 100–480 70–680 100–700
2370–2550 2910–3270 260–440 75 390 32–212 1340–4700 390–1110 74
Emissivity* 0.80 0.82 0.80 0.82 0.95 0.94 0.94–0.97
0.29 0.28 0.057–0.075 0.281 0.63 0.09–0.12 0.096–0.292 0.41–0.46 0.045
440–710
0.07–0.087
68 368–1844 390–1110 1200–2290
0.11 0.096–0.186 0.37–0.48 0.59–0.86
125–1894
0.64–0.76
70
0.262
420–914 420–980
0.44–0.36 0.62–0.73
420–980
0.90–0.97
520–1045
0.89–0.82
440–1160 1700–2960 80–2240 440–2510
0.054–0.104 0.12–0.17 0.036–0.192 0.073–0.182
440–1160 100–700
0.0198–0.0324 0.0221–0.0312
2420–5430 76
0.194–0.31 0.043 and 0.064
80–6000 6000
0.032–0.35 0.39
440–620 750 82 75
0.045–0.053 0.11 0.228 0.276
260–1160
0.81–0.79
1900–2560 206–520 209–362
0.526 0.952 0.959–0.947
B. Refractories, Building Materials, Paints, and Miscellaneous Asbestos Board Paper Brick Red, rough, but no gross irregularities Silica, unglazed, rough Silica, glazed, rough Grog brick, glazed See Refractory Materials below.
74 100–700 70 1832 2012 2012
0.96 0.93–0.945 0.93 0.80 0.85 0.75
Carbon T-carbon (Gebr. Siemens) 0.9% ash (this started with emissivity at 260°F. of 0.72, but on heating changed to values given) Carbon filament Candle soot Lampblack-waterglass coating
HEAT TRANSFER BY RADIATION TABLE 5-6
5-29
Normal Total Emissivity of Various Surfaces (Concluded ) B. Refractories, Building Materials, Paints, and Miscellaneous Surface
Same Thin layer on iron plate Thick coat Lampblack, 0.003 in. or thicker Enamel, white fused, on iron Glass, smooth Gypsum, 0.02 in. thick on smooth or blackened plate Marble, light gray, polished Oak, planed Oil layers on polished nickel (lube oil) Polished surface, alone +0.001-in. oil +0.002-in. oil +0.005-in. oil Infinitely thick oil layer Oil layers on aluminum foil (linseed oil) Al foil +1 coat oil +2 coats oil Paints, lacquers, varnishes Snowhite enamel varnish or rough iron plate Black shiny lacquer, sprayed on iron Black shiny shellac on tinned iron sheet Black matte shellac Black lacquer Flat black lacquer White lacquer
t, °F*
Emissivity*
260–440 69 68 100–700 66 72
0.957–0.952 0.927 0.967 0.945 0.897 0.937
70 72 70 68
0.903 0.931 0.895 0.045 0.27 0.46 0.72 0.82
212 212 212
0.087† 0.561 0.574
Surface Oil paints, sixteen different, all colors Aluminum paints and lacquers 10% Al, 22% lacquer body, on rough or smooth surface 26% Al, 27% lacquer body, on rough or smooth surface Other Al paints, varying age and Al content Al lacquer, varnish binder, on rough plate Al paint, after heating to 620°F. Paper, thin Pasted on tinned iron plate On rough iron plate On black lacquered plate Plaster, rough lime Porcelain, glazed Quartz, rough, fused Refractory materials, 40 different poor radiators
t, °F*
Emissivity*
212
0.92–0.96
212
0.52
212
0.3
212 70 300–600
0.27–0.67 0.39 0.35
66 66 66 50–190 72 70 1110–1830
0.924 0.929 0.944 0.91 0.924 0.932
good radiators 73 76 70 170–295 100–200 100–200 100–200
0.906 0.875 0.821 0.91 0.80–0.95 0.96–0.98 0.80–0.95
Roofing paper Rubber Hard, glossy plate Soft, gray, rough (reclaimed) Serpentine, polished Water
69 74 76 74 32–212
0.65 – 0.75 0.70 0.80 – 0.85 0.85 – 0.90 0.91
} }{
0.945 0.859 0.900 0.95–0.963
*When two temperatures and two emissivities are given, they correspond, first to first and second to second, and linear interpolation is permissible. °C = (°F − 32)/1.8. †Although this value is probably high, it is given for comparison with the data by the same investigator to show the effect of oil layers. See Aluminum, Part A of this table.
and for an infinite plane parallel to a system of rows of parallel tubes as curves 1 and 3 of Fig. 5-17. The exchange area between any two area elements of a sphere is independent of their relative shape and position and is simply the product of the areas divided by the area of the whole sphere; i.e., any spot on a sphere has equal views of all other spots. For surfaces in two-dimensional systems (with third dimension infinite), A1F12 per unit length in the third dimension may be obtained simply by evaluating, in a cross-sectional view, the sum of lengths of crossed strings from the ends of A1 to the ends of A2 less the sum of uncrossed strings from and to the same points, all divided by 2. The strings must be so drawn that all the flux from one surface to the other must cross each of a pair of crossed strings and neither of a pair of uncrossed ones. If one surface can see the other around both sides of an obstruction, two more pairs of strings are involved.
Example 3: Calculation of View Factor Evaluate the view factor between two parallel circular tubes long enough compared with their diameter D or their axis-to-axis separating distance C to make the problem twodimensional. With reference to Fig. 5-18, the crossed-strings method yields, per unit of axial length, 1/2 2(EFGH − HJ) D C 2 C A1F12 = ᎏᎏ = D sin−1 ᎏ + ᎏ − 1 − ᎏ 2 C D D
Results for a large number of other cases are given by Hottel and Sarofim (op. cit., chap. 2) and Hamilton and Morgan (NACA-TN2836, December 1952). A comprehensive bibliography is provided by Siegel and Howell (Thermal Radiation Heat Transfer, McGraw-Hill, 1992). The view factor F may often be evaluated from that for simpler configurations by the application of three principles: that of reciprocity, AiFij = AjFji; that of conservation, Fij = 1; and that due to Yamauti [Res. Electrotech. Lab. (Tokyo), 148, 1924; 194, 1927; 250, 1929], showing that the exchange areas AF between two pairs of surfaces are equal when there is a one-to-one correspondence for all sets of symmetrically placed pairs of elements in the two surface combinations. Example 4: Calculation of Exchange Area The exchange area between the two squares 1 and 4 of Fig. 5-19 is to be evaluated. The following exchange areas may be obtained from Eq. 5-121. F for common-side rectangles: 13 = 0.24, 24 = 2 × 0.29 = 0.5S, (1 + 2)(3 + 4) = 3 × 0.32 = 0.96. Expression of (1 + 2)(3 + 4) in terms of its components yields (1 + 2)(3 3+ 4) = 13 + 14 + 23 + 24. And by the Yamauti principle 14 = 23, since for every pair of elements in 1 and 4 there is a corresponding pair in 2 and 3. Therefore, (1 1+ 2)(3 3+ 4) − 13 − 24 14 = ᎏᎏᎏ = 0.07 2 Figure 5-16 may be used in the same way.
FIG. 5-15
Radiation between parallel planes, directly opposed.
Non-Black-Surface Enclosures In the following discussion we are concerned with enclosures containing gray sources and sinks, radiatively adiabatic surfaces, and no absorbing gas. The calculation of interchange between a source and a sink under conditions involving successive multiple reflections from other source-sink surfaces in the
5-30
HEAT AND MASS TRANSFER
(a) FIG. 5-16
(b)
View factors for a system of two concentric coaxial c to inner cylinder. (b) Inner surface of outer cylinder to itself.
FIG. 5-18
Direct exchange between parallel circular tubes.
FIG. 5-19
Illustration of the Yamauti principle.
Distribution of radiation to rows of tubes irradiated from one side. Dashed lines: direct view factor F from plane to tubes. Solid lines: total view factor F for black tubes backed by a refractory surface. FIG. 5-17
enclosure, as well as reradiation from refractory surfaces which are in radiative equilibrium, can become complicated. Zone Method Let a zone of a furnace enclosure be an area small enough to make all elements of itself have substantially equivalent “views” of the rest of the enclosure. (In a furnace containing a symmetry plane, parts of a single zone would lie on either side of the plane.) Zones are of two classes: source-sink surfaces, designated by numerical subscripts and having areas A1, A2, . . . , and emissivities ε1, ε2, . . . ; and surfaces at which the net radiant-heat flux is zero (fulfilled by the average refractory wall in which difference between internal convection and external loss is minute compared with incident radiation), designated by letter subscripts starting with r, and having areas Ar, As,. . . . It may be shown (see, for example, Hottel and Sarofim, op. cit., chap. 3) that the net radiation interchange between source-sink zones i and j is given by ˙ i = j = AiᏲijσ T i4 − AjᏲjiσ T 4j Q (5-125)
Ᏺij is called the total view factor from i to j, and the term AiᏲij, sometimes designated S iS j, is called the total interchange area shared by areas Ai and Aj and depends on the shape of the enclosure and the emissivity and absorptivity of the source and sink zones. Restriction here is to gray source-sink zones, for which AiᏲij = AjᏲji; the more general case is treated elsewhere (Hottel and Sarofim, op. cit., chap. 5). Evaluation of the AᏲ’s that characterize an enclosure involves solution of a system of radiation balances on the surfaces. If the assumption is made that all the zones of the enclosure are gray and emit and reflect diffusely,* then the direct-exchange area ij, as evaluated for the black-surface pair Ai and Aj, applies to emission and reflections between them. If at a surface the total leaving-flux density, emitted plus reflected, is denoted by W (and called by some the radiosity and by others the exitance), radiation balances take the form: * So-called Lambert surfaces, which emit or reflect with an intensity independent of angle; approximately satisfied by most nonmetallic, tarnished, oxidized, or rough surfaces.
HEAT TRANSFER BY RADIATION For source-sink j, AjεjEj + ρj (ij)Wi = AjWj
(5-126)
i
For adiabatic surface r,
(ir)W = A W i
r
(5-127)
r
i
where ρ is reflectance and the summation is over all surfaces in the enclosure. In matrix notation, Eq. (10-196) becomes, with source or sink zones represented by 1, 2, 3 . . . and adiabatic zones by r, s, t . . . ,
A 11 − ᎏᎏ1 ρ1
12
1r
1s
12
A 22 − ᎏᎏ2 ρ2
2r
2s
1r 1s
2r 2s
rr − Ar rs
rs ss − As
W2
=
Refractory temperature is obtained from Wr = Er = σTr4. The more general use of Eq. (5-128) is to obtain the set of total interchange areas AᏲ⋅ which constitute a complete description of the effect of shape, size, and emissivity on radiative flux, independent of the presence or absence of other transfer mechanisms. It may be shown that Aiεi Ajεj D′ij AiᏲij AjᏲji SiS j = ᎏ ᎏ − ᎏ − δijεj ρi ρj D
(5-130)
where D is the determinant of the square coefficient matrix in Eq. 5-128) and D′ij is the cofactor of its ith row and jth column, or −1)i + j times the minor of D formed by crossing out the ith row and ith column, and δij is the Kronecker delta, 1 when i = j, otherwise 0. As an example, consider radiation between two surfaces A1 and A2, which together form a complete enclosure. Equation (5-130) takes the form
12 A 11 − ᎏᎏ1 ρ1 12
12 A 22 − ᎏᎏ2 ρ2
(5-131)
Only one direct-view factor F12 or direct-exchange area 12 is needed because F11 equals 1 − F12 and F22 equals 1 − F21 equals 1 − F12 A1 /A2. Then 11 equals A1 − 12 and 22 equals A2 − 21. With these substitutions, Eq. (5-131) becomes A1 A1Ᏺ12 = ᎏᎏᎏ 1 1 A1 1 ᎏᎏ + ᎏᎏ − 1 + ᎏᎏ ᎏᎏ − 1 F12 ε1 A2 ε2
Ᏺ12 = ε1
(5-135)
F12( A2 F21) represents the total interchange where the expression A1 area for the limiting case of a black source and black sink (the refractory emissivity is of no moment). The factor F is known exactly for a few geometrically simple cases and may be approximated for others. If A1 and A2 are equal parallel disks, squares, or rectangles, connected by nonconducting but reradiating refractory walls, then F is given by Fig. 5-15, curves 5 to 8. If A1 represents an infinite plane and A2 is one or two rows of infinite parallel tubes in a parallel plane and if the only other surface is a refractory surface behind the tubes, F12 is given by curve 5 or 6 of Fig. 5-17. If an enclosure may be divided into several radiant-heat sources or sinks A1, A2, etc., and the rest of the enclosure (reradiating refractory surface) may be lumped together as Ar at a uniform temperature Tr, then the total interchange area for zone pairs in the black system is given by (1r)(r2) A1F (5-137) 12( A2F 21) = 12 + ᎏ Ar − rr For the two-source-sink-zone system to which Eq. (5-136) applies, Eq. (5-137) simplifies to
(5-128)
0 0
This represents a system of simultaneous equations equal in number to the number of rows of the square matrix. Each equation consists, on the left, of the sum of the products of the members of a row of the square matrix and the corresponding members of the W-column matrix and, on the right, of the member of that row in the third matrix. With this set of equations solved for Wi, the net flux at any surface Ai is given by Q˙ i,net = (Aiεi /ρi)(Ei − Wi) (5-129)
A1ε1 A2ε2 A1Ᏺ12 = ᎏ ᎏ ρ1 ρ2
Aε − ᎏ1ᎏ1 E1 ρ1 A2ε2 − ᎏᎏE2 ρ2
Wr Ws
2. Sphere of area A1 concentric with surrounding sphere of area A2. F12 = 1. Then A1 A1Ᏺ12 = ᎏᎏ (5-134) 1 A1 1 ᎏᎏ + ᎏᎏ ᎏᎏ ε1 A2 ε2 − 1 3. Body of surface A1 having no negative curvature, surrounded by very much larger surface A2. F12 = 1 and A1/A2 → 0. Then Many furnace problems are adequately handled by dividing the enclosure into but two source-sink zones A1 and A2 and any number of no-flux zones Ar, As,. . . . For this case Eq. (5-130) yields 1 1 1 1 1 1 1 ᎏ ᎏ = ᎏ ᎏ − 1 + ᎏ ᎏ − 1 + ᎏ (5-136) A1Ᏺ12 A2Ᏺ21 A1 ε1 A2 ε2 A1F 12
W1
5-31
(5-132)
Special cases include 1. Parallel plates, large compared to clearance. Substitution of F12 = 1 and A1 = A2 gives A1 (5-133) A1Ᏺ12 = ᎏᎏ 1 1 ᎏᎏ + ᎏᎏ − 1 ε1 ε2
A 1F 12 = 12 + 1/(1/1 r + 1/2 r)
(5-138)
and if A1 and A2 each can see none of itself, there is further simplification to 1 A1F 12 = 12 + ᎏᎏᎏ 1/(A1 − 12) + 1/(A2 − 12) A1A2 − (1 2)2 = ᎏᎏ (5-139) A1 + A2 − 2(1 2) which necessitates the evaluation of but one geometrical factor F. Equation (5-136) covers many of the problems of radiant-heat interchange between source and sink in a furnace enclosure. The error due to single zoning of source and sink is small even if the views of the enclosure from different parts of each zone are quite different, provided the emissivity is fairly high; the error in F is zero if it is obtainable from Fig. 5-15 or 5-17, small if Eq. (5-137) is used and the variation in temperature over the refractory is small. An approach to any desired accuracy can be made by use of Eqs. (5-126) and (5-130) with division of the surfaces into more zones. From the definitions of F, F, and Ᏺ it is to be noted that F11 + F12 + F13 + ⋅⋅⋅ + F1r + F1s + ⋅⋅⋅ = 1 F F12 + F13 + ⋅⋅⋅ = 1 11 + Ᏺ11 + Ᏺ12 + Ᏺ13 + ⋅⋅⋅ = ε1 Example 5: Radiation in a Furnace Chamber A furnace chamber of rectangular parallelepipedal form is heated by the combustion of gas inside vertical radiant tubes lining the sidewalls. The tubes are of 0.127-m (5-in) outside diameter on 0.305-m (12-in) centers. The stock forms a continuous plane on the hearth. Roof and end walls are refractory. Dimensions are shown in Fig. 5-20. The radiant tubes and stock are gray bodies having emissivities of 0.8 and 0.9 respectively. What is the net rate of heat transmission to the stock by radiation when the mean temperature of the tube surface is 816°C (1500°F) and that of the stock is 649°C (1200°F)?
5-32
FIG. 5-20
HEAT AND MASS TRANSFER
Furnace-chamber cross section. To convert feet to meters, multiply
by 0.3048.
This problem must be broken up into two parts, first considering the walls with their refractory-backed tubes. To imaginary planes A2 of area 1.83 by 3.05 m (6 by 10 ft) and located parallel to and inside the rows of radiant tubes, the tubes emit radiation σ T 14A1Ᏺ12, which equals σT 14A2 Ᏺ21. To find Ᏺ21, use Fig. 5-17, curve 5, from which F 21 = 0.81. Then from Eq. (10-200) 1 ᎏᎏᎏᎏ Ᏺ21 = 1 1 = 0.702 12 1 ᎏᎏ − 1 + ᎏᎏ ᎏᎏ − 1 + ᎏᎏ 1 5π 0.8 0.81 This amounts to saying that the system of refractory-backed tubes is equal in radiating power to a continuous plane A2 replacing the tubes and refractory back of them, having a temperature equal to that of the tubes and an equivalent or effective emissivity of 0.702. The new simplified furnace now consists of an enclosure formed by two 1.83by 3.05-m (6- by 10-ft) radiating sidewalls (area A2, emissivity 0.702), a 1.52- by 3.05-m (5- by 10-ft) receiving plane on the floor A3, and refractory surfaces Ar to complete the enclosure (ends, roof, and floor side strips). The desired heat transfer is Q˙ 2 3 = σ (T 14 − T 34)A2 Ᏺ23
To evaluate Ᏺ23, start with the direct interchange factor F23. F23 = F from A2 to (A3 + a strip of Ar alongside A3, which has a common edge with A2) minus F from A2 to the strip only. These two F’s may be evaluated from Fig. 5-20 and Eq. (5-121). For the first F, Y = 1.83/3.05, Z = 1.98/3.05, and F = 0.239; for the second F, Y = 1.83/3.05, Z = 0.46/3.05, and F = 0.100. Then F23 = 0.239 − 0.10 = 0.139. Now F may be evaluated. From Eq. 5-137 et seq., 1 1 A2F23 = 23 ÷ ᎏᎏ F 23 = F23 + ᎏᎏᎏ 1/2r + 1/3r (1/F2r) + (A2/A3)(1/F3r) Since A2 “sees” Ar, A3, and some of itself (the plane opposite), F2r = 1 − F22 − F23. F22, the direct interchange factor between parallel 1.83- by 3.05-m (6- by 10-ft) rectangles separated by 2.44 m (8 ft), may be taken as the geometric mean of the factors for 1.83-m (6-ft) squares separated by 2.44 m (8 ft) and for 3.05-m (10-ft) squares separated by 2.44 m (8 ft). These come from Fig. 5-15, curve 2, according to which F22 = 0 .13 ×0 .255 = 0.182. Then F2r = 1 − 0.182 − 0.139 = 0.679. The other required direct factor is F3r = 1 − F32 = 1 − F23 A2/A3 = 1 − (0.139)(11.14)/4.65 = 0.666. Then 1 F 23 = 0.139 + ᎏᎏᎏᎏ = 0.336 (1/0.679) + (11.14/4.65)(1/0.666) Having F23, we may now evaluate the factor Ᏺ23: 1 Ᏺ23 = ᎏᎏᎏᎏᎏᎏ = 0.273 (1/0.336) + [(1/0.702) − 1] + (11.14/4.65)[(1/0.9) − 1] ˙ net = σ(T 14 − T 43)A2 Ᏺ23 = 5.67(10.894 − 9.224)(11.15)(0.273) Q = 118,000 J/s (402,000 Btu/h) A result of interest is obtained by dividing the term A2 Ᏺ23(11.15 × 0.273, or 3.04) by the actual area A1 of the radiating tubes (0.127π)(18.3)(2) = 14.6 m2 (157 ft2). Thus 3.04/14.6 = 0.208, which means that the net radiation from a tube to the stock is 20.8 percent as much as if the tube were black and completely surrounded by black stock.
Integral Formulation The zone method has the purpose of dodging the solution of an integral equation. If in Eq. (5-126) the zone on which the radiation balance is formulated is decreased to a differential element, that equation becomes dAi dAf (cos θi)(cos θj)Wi dA jεj Ej + ρj ᎏᎏᎏ = dA jWj (5-140) r2
which is an integral equation with the unknown function W inside the integral. Integration is over the entire surface area. Exact solutions have been carried out for only a few simple cases. One of these is the evaluation of emittance of an isothermal spherical cavity, for which dAi dAf (cos θi)(cos θj)/r2 in the integral of Eq. (5-140) becomes dAi dAj /4πR2, where R is the sphere radius. For this special case W is, from Eq. (10-202), constant over the inner surface of the cavity and given by εE W = ᎏᎏ (5-141) 1 − ρ(1 − A1/4πR2) where A1 is the curved area of a hole in the sphere’s surface. The ratio W/E is the effective emittance of the hole as sensed by a narrow-angle receiver viewing the cavity interior. If the material of construction of the cavity is a diffuse emitter and reflector and has an emissivity of 0.5 and the cavity is to appear at least 98 percent black, the curved area A1 of the hole must be smaller than 2 percent of the total surface area of the sphere. Enclosures of Surfaces That Are Not Diffuse Reflectors If no restriction that the surfaces be diffuse emitters and reflectors is imposed, Eq. (5-140) becomes much more complex. The W’s are replaced by πI’s and εj, Ii, and Ij all become functions of the angle of the leaving beam, and ρj goes inside the integral and becomes a function of angles of incidence and reflection. Seldom are such details of reflectance known. When they are and a solution is needed, the Monte Carlo method of tracing the history of a large number of beams emitted from random positions and in random initial directions is probably the best method of obtaining a solution. Another approach is possible, however, because of the tendency of most surfaces to fit a simpler reflection model. The total reflectance ρ( 1 − ε) can be represented by the sum of a diffuse component ρD and a specular component ρS. For applications see Hottel and Sarofim (op. cit., chap. 5). The method yields the following relation for exchange between concentric spheres or infinite cylinders: A1Ᏺ12 S1S 2 1 = ᎏᎏᎏᎏᎏᎏ (5-142) 1/A1ε1 + (1/A2)(1/ε2 − 1) + [ρs2 /(1 − ρs2)](1/A1 − 1/A2) When there is no specular reflectance, the third term in the denominator drops out, in agreement with Eqs. (5-134) and (5-135). When the reflectance is exclusively specular, the denominator becomes 1/A1ε1 + ρs2 /A1(1 − ρs2), easily derivable from first principles. EMISSIVITIES OF COMBUSTION PRODUCTS The radiation from a flame is due to radiation from burning soot particles of microscopic and submicroscopic dimensions, from suspended larger particles of coal, coke, or ash, and from the water vapor and carbon dioxide in the hot gaseous combustion products. The contribution of radiation emitted by the combustion process itself, so-called chemiluminescence, is relatively negligible. Common to these problems is the effect of the shape of the emitting volume on the radiative flux; this is considered first. Mean Beam Lengths Evaluation of radiation from a nonisothermal volume is beyond the scope of this section (see Hottel and Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967, chap. 11). Consider an isothermal gas confined within the volume bounded by the solid angle dΩ with vertex at dA and making the angle θ with the normal to dA. The ratio of the emission to dA from the gas to that from a blackbody at the gas temperature and filling the field of view dΩ is called the gas emissivity ε. Clearly, ε depends on the path length L through the volume to dA. A hemispherical volume radiating to a spot on the center of its base represents the only case in which L is independent of direction. Flux at that spot relative to hemispherical blackbody flux is thus an alternative way to visualize emissivity. The flux density to a small area of interest on the envelope of an emitter volume of any shape can be matched by that at the base of a hemispherical volume of some radius L, which is called the mean beam length. It is found that although the ratio of L to a characteristic dimension D of the shape varies with opacity, the variation is small enough for most engineering purposes to permit use of a constant ratio LM/D,
HEAT TRANSFER BY RADIATION where LM is the average mean beam length. LM can be defined to apply either to a spot on the envelope or to any finite portion of its area. An important limiting case is that of opacity approaching zero (pD → 0, where p = partial pressure of the emitter constituent). For this case, L (called L0) equals 4V/A (V = gas volume; A = bounding area) when interest is in radiation to the entire envelope. For the range of pD encountered in practice, the optimum value of L (now LM) lies between 0.8 to 0.95 times L0. For shapes not reported in Table 5-7, a factor of 0.88 (or LM = 0.88L0 = 3.5V/A) is recommended. Instead of using the average-mean-beam-length concept to approximate A1[ε(Lm)] (the flux per unit black emissive power from a gas volume partially bounded by a surface of area A1), one may calculate the flux rigorously by integration, over the gas volume and over A1, of the expression 4k dv τ(r) dA cos θ/πr 2. Here k is the emission coefficient of the gas, and τ(r) is the transmittance through the distance r between dv and dA. The result has the dimensions of area and, by analogy to ss, is called gs1, the direct-exchange area between the gas zone and the surface zone (Hottel and Sarofim, op. cit., chap. 7). The use of A1[ε(Lm)] instead of gs1 is adequate when the problem is such that all the gas can be treated as a single zone in contact with A1, and having a mean radiating temperature, but the gs concept is clearly useful if allowance is to be made for temperature variations within the gas. Gaseous Combustion Products Radiation from water vapor and carbon dioxide occurs in spectral bands in the infrared. In magnitude it overshadows convection at furnace temperatures. Carbon Dioxide The contribution εc to the emissivity of a gas containing CO2 depends on gas temperature TG, on the CO2 partial pressure-beam length product pc L and, to a much lesser extent, on the total pressure P. Constants for use in evaluating εc at a total pressure of 101.3 kPa (1 atm) are given in Table 5-8 (more on this later). The gas absorptivity αc equals the emissivity when the absorbing gas and the emitter are at the same temperature. When the emitter surface temperature is T1, αc is (TG /T1)0.65 times εc, evaluated using Table 5-8 at T1 instead of TG and at pc LT1 /TG instead of pc L. Line broadening, due to TABLE 5-7
increases either in total pressure or in partial pressure of CO2, makes a correction necessary. However, at a total pressure of 101.3 kPa (1 atm) the correction factor may be ignored, since it decreases with increase in temperature and is never more than 4 percent at temperatures above 1111 K(2000°R). Estimations of the correction in systems up to 1013.3 kPa (10 atm) are given by Hottel and Sarofim (op. cit., p. 228), and by Edwards [J. Opt. Soc. Am., 50, 617 (1960)] who in addition presents data on CO2-band emission for use in calculations involving spectrally selective surfaces. The principal emission bands of CO2 are at about 2.64 to 2.84, 4.13 to 4.5, and 13 to 17 µm. Water Vapor The contribution εw to the emissivity of a gas containing H2O depends on TG and pw L and on total pressure P and partial pressure pw. Table 5-8 gives constants for use in evaluating εw. Allowance for departure from the special pressure conditions is made by multiplying εw by a correction factor Cw read from Fig. 5-21 as a function of (pw + P) and pw L. The absorptivity αw of water vapor for blackbody radiation is εw evaluated from Table 5-8 but at T1 instead of TG and at pw LT1 /TG instead of pw L. Multiply by (TG /T1)0.45. The correction factor Cw still applies. Spectral data for water vapor, tabulated for 371 wavelength intervals from 1 to 40 µm, are also available [Ferriso, Ludwig, and Thompson, J. Quant. Spectros. Radiat. Transfer, 6, 241–273 (1966)]. The principal emission is in bands at about 2.55 to 2.84, 5.6 to 7.6, and 12 to 25 µm. Carbon Dioxide–Water-Vapor Mixtures When these gases are present together, the total radiation due to both is somewhat less than the sum of the separately calculated effects, because each gas is somewhat opaque to radiation from the other in the wavelength regions 2.7 and 15 µm. Allowance for spectral overlap, the effect of pressure, and the effect of soot luminosity would make computation tedious. Table 5-8 gives constants for use in direct calculation, for H2O/CO2 mixtures, of the product εGT . The product term is used because it varies much less with T than does εG alone. Constants are given for mixtures, in nonradiating gases, of water vapor alone, CO2 alone, and four pw /pc mixtures.
Mean Beam Lengths for Volume Radiation Shape
Sphere Infinite cylinder Semi-infinite cylinder, radiating to: Center of base Entire base Right-circle cylinder, ht. = diam. radiating to: Center of base Whole surface Right-circle cylinder, ht. = 0.5 diam. radiating to: End Side Total surface Right-circle cylinder, ht. = 2 × diam. radiating to: End Side Total surface Infinite cylinder, half-circle cross section radiating to middle of flats Rectangular parallelepipeds: 1:1:1 (cube) 1:1:4, radiating to: 1 × 4 face 1 × 1 face Whole surface 1:2:6, radiating to: 2 × 6 face 1 × 6 face 1 × 2 face Whole surface Infinite parallel planes Space outside bank of parallel tubes on equilateral triangular centers Tube diam. = clearance Tube diam. = a clearance Tube centers on squares, diam. = clearance
5-33
Characteristic dimension, D
L 0/D
L M /D
Diameter Diameter
0.67 1.0
0.63 0.94
Diameter Diameter
1.0 0.81
0.90 0.65
Diameter Diameter
0.76 0.67
0.71 0.60
Diameter Diameter Diameter
0.47 0.52 0.50
0.43 0.46 0.45
Diameter Diameter Diameter
0.73 0.82 0.80
0.60 0.76 0.73
Radius
1.26
Edge
0.67
0.60
Shortest edge Shortest edge Shortest edge
0.90 0.86 0.89
0.82 0.71 0.81
Shortest edge Shortest edge Shortest edge Shortest edge Clearance
1.18 1.24 1.18 1.2 2.00
1.76
Clearance 3.4 Clearance 4.45 Clearance 4.1
2.8 3.8 3.5
0.82 0.85 0.85
5-34
HEAT AND MASS TRANSFER
TABLE 5-8
Emissivity eG of H2O:CO2 Mixtures Limited range for furnaces, valid over 25-fold range of pw + cL, 0.046–1.15 m atm (0.15–3.75 ft. atm)
pw /pc pw ᎏ pw + pc
0
a
1
2
3
∞
0
s(0.3–0.42)
a(0.42–0.5)
w(0.6–0.7)
e(0.7–0.8)
1
corresponding to (CH6)x, covering future high H2 fuels
H2O only
CO2 only
corresponding to (CH)x, covering coal, heavy oils, pitch
corresponding to (CH2)x, covering distillate oils, paraffins, olefines
corresponding to CH4, covering natural gas and refinery gas
Constants b and n of Eq., εGT = b(pL − 0.015)n, pL = m atm, T = K T, K
b
n
b
n
b
n
b
n
b
n
b
n
1000 1500 2000
188 252 267
0.209 0.256 0.316
384 448 451
0.33 0.38 0.45
416 495 509
0.34 0.40 0.48
444 540 572
0.34 0.42 0.51
455 548 594
0.35 0.42 0.52
416 548 632
0.400 0.523 0.640
T, °R
b
n
b
n
b
n
b
n
b
n
b
n
1800 2700 3600
264 335 330
0.209 0.256 0.316
467 514 476
0.33 0.38 0.45
501 555 519
0.34 0.40 0.48
534 591 563
0.34 0.42 0.51
541 600 577
0.35 0.42 0.52
466 530 532
0.400 0.523 0.640
Constants b and n of Eq., εGT = b(pL − 0.05)n, pL = ft. atm, T = °R
Full range, valid over 2000-fold range of pw + cL, 0.005–10.0 m atm (0.016–32.0 ft. atm) Constants of Eq., log10 εGTG = a0 + a1 log pL + a2 log2 pL + a3 log3 pL pL = m atm, T = K pw ᎏ pc
pw ᎏ pw + pc
0
0
a
s
1
a
2
w
3
e
∞
1
pL = ft. atm, T = °R
T, K
a0
a1
a2
a3
T, °R
a0
a1
a2
a3
1000 1500 2000 1000 1500 2000 1000 1500 2000 1000 1500 2000 1000 1500 2000 1000 1500 2000
2.2661 2.3954 2.4104 2.5754 2.6451 2.6504 2.6090 2.6862 2.7029 2.6367 2.7178 2.7482 2.6432 2.7257 2.7592 2.5995 2.7083 2.7709
0.1742 0.2203 0.2602 0.2792 0.3418 0.4279 0.2799 0.3450 0.4440 0.2723 0.3386 0.4464 0.2715 0.3355 0.4372 0.3015 0.3969 0.5099
−0.0390 −0.0433 −0.0651 −0.0648 −0.0685 −0.0674 −0.0745 −0.0816 −0.0859 −0.0804 −0.0990 −0.1086 −0.0816 −0.0981 −0.1122 −0.0961 −0.1309 −0.1646
0.0040 0.00562 −0.00155 0.0017 −0.0043 −0.0120 −0.0006 −0.0039 −0.0135 0.0030 −0.0030 −0.0139 0.0052 0.0045 −0.0065 0.0119 0.00123 −0.0165
1800 2700 3600 1800 2700 3600 1800 2700 3600 1800 2700 3600 1800 2700 3600 1800 2700 3600
2.4206 2.5248 2.5143 2.6691 2.7074 2.6686 2.7001 2.7423 2.7081 2.7296 2.7724 2.7461 2.7359 2.7811 2.7599 2.6720 2.7238 2.7215
0.2176 0.2695 0.3621 0.3474 0.4091 0.4879 0.3563 0.4561 0.5210 0.3577 0.4384 0.5474 0.3599 0.4403 0.5478 0.4102 0.5330 0.6666
−0.0452 −0.0521 −0.0627 −0.0674 −0.0618 −0.0489 −0.0736 −0.0756 −0.0650 −0.0850 −0.0944 −0.0871 −0.0896 −0.1051 −0.1021 −0.1145 −0.1328 −0.1391
0.0040 0.00562 −0.00155 0.0017 −0.0043 −0.0120 −0.0006 −0.0039 −0.0135 0.0030 −0.0030 −0.0139 0.0052 0.0045 −0.0065 0.0119 0.00123 −0.0165
NOTE: pw /(pw + pc) of s, a, w, and e may be used to cover the ranges 0.2–0.4, 0.4–0.6, 0.6–0.7, and 0.7–0.8, respectively, with a maximum error in εG of 5 percent at pL = 6.5 m atm, less at lower pLs. Linear interpolation reduces the error generally to less than 1 percent. Linear interpolation or extrapolation on T introduces an error generally below 2 percent, less than the accuracy of the original data.
Four suffice, since a change halfway from one mixture ratio to the adjacent one changes the emissivity by a maximum of but 5 percent; linear interpolation may be used if considered necessary. The constants are given for three temperatures, adequate for linear interpolation since εGT changes a maximum of one-sixth due to a change from one temperature base halfway to the adjacent one. The interpolation relation, with TH and TL representing the higher and lower base temperatures bracketing T, and with the brackets in the term [A(x)] indicating that the parentheses refer not to a multiplier but to an argument, is [εGT H(pL)](TG − TL) + [εGT L(pL)](TH − TG) εGT G = ᎏᎏᎏᎏᎏ (5-143) 500 Extrapolation to a temperature that is above the highest or below the lowest of the three base temperatures in Table 5-8 uses the same formulation, but one of its terms becomes negative. The gas absorptivity may also be obtained from the constants for emissivities. The product αG1T1 (gas absorptivity for black surface radiation), x (surface temperature), is εGT 1 evaluated at T1 instead of TG and at pLT1/TG instead of pL, then multiplied by (TG /T1)0.5, or pLT1 TG 0.5 αG1T1 = εGT (5-144) 1 ᎏ ᎏ TG T1
The exponent 0.5 is an adequate average of the exponents for the pure components. The interpolation relation for absorptivity is TG 0.5 T1 − TL pLTH αG1T1 = εGT ᎏ H ᎏ ᎏ TG TH 500
pLTL + εGT L ᎏ TG
T T −T ᎏ ᎏ T 500 G
0.5
H
1
(5-145)
L
The base temperature pair TH and TL can be different for evaluating εG and αG1 if TG and T1 are far enough apart. Example 6: Calculation of Gas Emissivity and Absorptivity This example will use only SI units, except that pressure will be in atm, not kPa. Flue gas containing 6 percent CO2 and 11 percent H2O vapor, wet basis, flows through a bank of tubes of 0.1016 in (4-in) outside diameter on equilateral 0.2032 m (8-in) triangular centers. In a section in which the gas and tube surface temperatures are 691°C (964 K) and 413°C (686 K), what are the emissivity and absorptivity of the gas? From Table 5-8, Lm = (2.8)(0.01016) = 0.2845 m (only SI units will be used in this example). p = pw + pc = 0.17 atm; pL = 0.0484 m atm, barely large enough to justify the short method, the top part of Table 5-8. pw /pc = 11/6, near enough to 2 to use col. 5. Since both TG and T1 are below the lowest T in the top part of the table, use the nearest pair, TH = 1500 K and TL = 1000 K. At TH, b = 540, n = 0.42. εGT H = 540(0.0484 − 0.015)0.42 = 129.5. At TL,
HEAT TRANSFER BY RADIATION
5-35
cles are small relative to the wavelength of the radiation of interest [diameters (2)(10−8) to (1.4)(10−7) m (200 to 1400 Å)], the monochromatic emissivity ελ depends on the total particle volume per unit volume of space fv regardless of particle size. It is given by ελ = 1 − e−Kf v L/λ
(5-146)
where L is the path length. Use of the perfect gas law and a material balance allows the restatement of the above to ελ = 1 − e−KPSL/λT
Correction factor for converting emissivity of water vapor to values of Pw and PT other than 0 to 1 atm respectively. To convert atmosphere-feet to kilopascal-meters, multiply by 30.89; to convert atmospheres to kilo-pascals, multiply by (1.0133)(102).
FIG. 5-21
(5-147)
where P is the total pressure (atm) and S is the mole fraction of soot in the gas. S depends on the fractional conversion fc of the fuel carbon to soot and is the mole fraction, wet basis, of carbon in gaseous form (CO2, CO, CH4, etc.) times fc /(1 − fc) or, with negligible error, times fc, which is a very small number. Evaluation of K is complex, and its numerical value depends somewhat on the age of the soot, the temperature at which it is formed, and its hydrogen content. It is recommended that K = 0.526 [K/atm]. The total emissivity of soot εs is obtained by integration over the wavelength spectrum, giving KPSL 15 εs = 1 − ᎏ ψ(3) 1 + ᎏ , (5-148) 4 c2 where ψ(3)(x) is the pentagamma function of x. It may be shown that an excellent approximation to Eq. (5-148) is
εs = 1 − [1 + 34.9SPL]−4 b = 444, n = 0.34, εGT L = 444(0.0484 − 0.015)0.34 = 139.8. From interpolation Eq. (5-143) (here extrapolation), εGT G = [129.5 (964 − 1000) + 139.5 (1500 − 964)]/500 = 140.2. For αG1T1 with T1 = TH = 1500, pLTH /TG = 0.0753. From Eq. (5-144), αGHT αGLT H = 540 (0.0753 − 0.015)0.42(964/1500)0.5 = 133.1. Similarly, L = 444 (0.0502 − 0.015)0.34 (964/1000)0.5 = 139.7. From Eq. (5-145) 133.1(686 − 1000) + 139.7(1500 − 686) αG1T1 = ᎏᎏᎏᎏᎏ = 143.8 500 Then εG = 140.2/964 = 0.145 and αG1 = 143.8/686 = 0.210. If the longer method (the bottom part of Table 5-8) were used, εG = 0.141 and αG1 = 0.206.
Other Gases Because of their practical importance, the emissivities of CO2 and H2O have been studied much more extensively than those of other gases, and the values summarized in the preceding paragraphs are based on extensive measurement of both total and integrated spectral values. Correction for pressure has reduced the disagreement among experimenters. A summary of the less adequate information on other gases appears in Table 5-9. Flames and Particle Clouds Luminous Flames Luminosity conventionally refers to soot radiation; it is important when combustion occurs under such conditions that the hydrocarbons in the flame are subject to heat in the absence of sufficient air well mixed on a molecular scale. Because soot parti-
TABLE 5-9
(5-149)
where PL is in atm m. The error is less the lower εs, and is only 0.5 percent at εs = 0.5 and 0.8 percent at 0.67. There is at present no method of predicting soot concentration of a luminous flame analytically; reliance must be placed on experimental measurement on flames similar to that of interest. Visual observation is misleading; a flame so bright as to hide the wall behind it may be far from a “black” radiator. The chemical kinetics and fluid mechanics of soot burnout have not progressed far enough to evaluate the soot fraction fc for relatively complex systems. Additionally, the soot in a combustion chamber is highly localized, and a mean value is needed for calculation of the radiative heat transfer performance of the chamber. On the basis of limited experience with fitting data to a model, the following procedure is recommended when total combustion chamber performance is being estimated: (1) When pitch, or a highly aromatic fuel, is burned, one percent of the fuel carbon appears as soot. This produces values of εs of 0.4–0.5 and εG + s of 0.6–0.7. These values are lower than some measurements on pitch flames, but the measurements are usually taken through the flame at points of high luminosity. (2) When No. 2 fuel oil is burned, 33 percent of the fuel carbon appears as soot (but that number varies greatly with burner design). (3) When natural gas is burned, any soot contribution to emissivity may be ignored. Admittedly, the numbers given should be functions of burner design and excess air, and they should be considered tentative, subject to change when good data show they are off target. The Inter-
Total Emissivities of Some Gases 1000°R
1600°R
2200°R
2800°R
Temperature Px L, (atm)(ft)
0.01
0.1
1.0
0.01
0.1
1.0
0.01
0.1
1.0
0.01
0.1
1.0
NH3a SO2b CH4c COd NOe HCl f
0.047 0.020 0.020 0.011 0.0046 0.00022
0.20 0.13 0.060 0.031 0.018 0.00079
0.61 0.28 0.15 0.061 0.060 0.0020
0.020 0.013 0.023 0.022 0.0046 0.00036
0.120 0.090 0.072 0.057 0.021 0.0013
0.44 0.32 0.194 0.10 0.070 0.0033
0.0057 0.0085 0.022 0.022 0.0019 0.00037
0.051 0.051 0.070 0.050 0.010 0.0014
0.25 0.27 0.185 0.080 0.040 0.0036
(0.001) 0.0058 0.019 (0.012) 0.00078 0.00029
(0.015) 0.043 0.059 (0.035) 0.004 0.0010
(0.14) 0.20 0.17 (0.050) 0.025 0.0027
NOTE: Figures in this table are taken from plots in Hottel and Sarofim, Radiative Transfer, McGraw-Hill, New York, 1967, chap. 6. Values in parentheses are extrapolated. To convert degrees Rankine to kelvins, multiply by (5.556)(10−1). To convert atmosphere-feet to kilopascal-meters, multiply by 30.89. a Total-radiation measurements of Port (Sc.D. thesis in chemical engineering, MIT, 1940) at 1-atm total pressure, L = 1.68 ft, T to 2000°R. b Calculations of Guerrieri (S.M. thesis in chemical engineering, MIT, 1932) from room-temperature absorption measurements of Coblentz (Investigations of Infrared Spectra, Carnegie Institution, Washington, 1905) with poor allowance for temperature. c Band measurements of Lee and Happel [Ind. Eng. Chem. Fundam., 3, 167 (1964)] at T up to 2050°R plus calculations to extrapolate temperature to 3800°R. d Total-radiation measurements of Ullrich (Sc.D. thesis in chemical engineering, MIT, 1953) at 1-atm total pressure, L = 1.68 ft, T to 2200°R. e Calculations of Malkmus and Thompson [J. Quant. Spectros. Radiat. Transfer, 2, 16 (1962)], to T = 5400°R and PL = 30 atm ⋅ ft. f Calculations of Malkmus and Thompson [J. Quant. Spectros. Radiat. Transfer, 2, 16 (1962)], to T = 5400°R and PL = 300 atm ⋅ ft.
5-36
HEAT AND MASS TRANSFER
where a/v is the projected area of the particles per unit volume of space. If the particles have no negative curvature (a particle can see none of itself) and are randomly oriented, a is a′/4, where a′ is the actual surface area; and if the particles are uniform, a/v = cA = cA′/4 where A and A′ are the projected and total areas of each particle and c is the number concentration of particles. For spherical particles, this gives εM = 1 − e−(π/4)cd L = 1 − e−1.5fvL/d (5-153)
Evaluation of αG1 is unnecessary when T1 is less than one-half TG; αG1 may then be assumed equal to εG. If the bounding surface is gray rather than black, multiplication of Eq. (5-154) by surface emissivity ε1 allows properly for reduction of the primary beams, gas-to-surface or surface-to-gas, but secondary reflections are ignored. The correction then lies between ε1 and 1, and for most industrially important surfaces with ε1 > 0.8 a value of (1 + ε1)/ 2 is adequate. Rigorous allowance for this and other factors is presented later, e.g., Eq. (5-163). If the bounding walls are mostly sink-type surfaces of area A1 and temperature T1, but in small part refractory surfaces of area Ar in radiative equilibrium at unknown temperature Tr, an energy balance on Ar is in principle necessary to determine Tr and the effect on energy flux. However, the total heat transfer to the sink may be visualized as corresponding to its having an effective area equal to its own plus a fraction x of that of the refractory, with the only temperatures involved being those of the gas and the heat sink. The fraction x varies from zero when the ratio of refractory to heat-sink surface is very high to unity when the ratio is very low and the value of εG is low. If Ar is small compared with A1, a value for x of 0.7 may be used in the approximate method. Long Exchanger This case, in which axial radiative flux is ignored, includes most radiatively modified heat exchangers of interest to chemical engineers. When the gas temperature transverse to the flow direction is reasonably uniform and the chamber is long compared with its mean hydraulic radius, the opposed upstream and downstream fluxes through the flow cross section will substantially cancel (hot combustion products through tubes or across tube banks, tunnel kilns, billet-reheating furnaces, Example 7). Under these conditions, the radiative contribution to local flux density q may be formulated in terms of local temperatures and beam lengths or exchange areas evaluated for a two-dimensional system infinite in the flow direction. The local flux density at the sink A1 is then
As an example, consider heavy fuel oil (CH1.5, specific gravity, 0.95) atomized to a surface mean particle diameter of d, burned with 20 percent excess air to produce coke-residue particles having the original drop diameter and suspended in combustion products at 1204°C (2200°F). The flame emissivity due to the particles along a path of L m will be, with d in micrometers, εM = 1 − e−24.3L/d (5-154)
where h is the local convective heat-transfer coefficient and qr(TG, T1) the radiation contribution calculated from TG, T1, εG, and ε1 by using the approximate treatment in the preceding subsection or the more rigorous treatment in the following subsection. If m ˙ Cp is the hourly heat capacity of the gas stream, the temperature of which changes by dTG over the sink-area increment dA1, then
national Flame Foundation has recorded data on many luminous flames from gas, oil, and coal (see J. Inst. Energy, formerly J. Inst. Fuel, 1956 to present). Combined Soot, H2O, and CO2 Radiation The spectral overlap of H2O and CO2 radiation has been taken into account by the constants for obtaining εG. Additional overlap occurs when soot emissivity εs is added. If the emission bands of water vapor and CO2 were randomly placed in the spectrum and soot radiation were gray, the combined emissivity would be εG plus εs minus an overlap correction εGεs. But monochromatic soot emissivity is higher the shorter the wavelength, and in a highly sooted flame at 1500 K half the soot emission lies below 2.5 µm where H2O and CO2 emission is negligible. Then the correction εGεs must be reduced, and the following is recommended: εG + s = εG + εs − MεGεs (5-150) where M depends mostly on TG and to a much less extent on optical density SPL. Values that have been calculated from this simple model can be represented with acceptable error by T M = 1.07 + 18 SPL − 0.27 ᎏ (5-151) 1000 Clouds of Large Black Particles The emissivity εM of a cloud of particles with a perimeter large compared with wavelength λ is εM = 1 − e−(a/v)L (5-152)
2
With 200-µm particles and an L of 3.05 m (10 ft), the particle contribution to emissivity will be 0.31. Soot luminosity will increase this; particle burnout will decrease it. Clouds of Nonblack Particles The correction for nonblackness of the particles is complicated by multiple scatter of the radiation reflected by each particle. The emissivity εM of a cloud of gray particles of individual surface emissivity ε1 can be estimated by the use of Eq. (5-151), with its exponent multiplied by ε1, if the optical thickness (a/v)L does not exceed about 2. Modified Eq. (5-151) would predict an approach of εM to 1 as L → ∞, an impossibility in a scattering system; the asymptotic value of εM can be read from Fig. 5-14 as εh, with albedo ω given by particle-surface reflectance 1 − ε1. Particles with a perimeter lying between 0.5 and 5 times the wavelength of interest can be handled with difficulty by use of the Mie equations (see Hottel and Sarofim, op. cit., chaps. 12 and 13). Summation of Separate Contributions to Gas or Flame Emissivity Flame emissivity εG + s due to joint emission from gas and soot has already been treated. If massive-particle emissivity εM, such as from fly ash, coal char, or carbonaceous cenospheres from heavy fuel oil, are present, it is recommended that the total emissivity be approximated by εG + s + εM − (εG + s)(εM) RADIATIVE EXCHANGE BETWEEN GASES OR SUSPENDED MATTER AND A BOUNDARY ˙ between an Local Radiative Exchange The interchange rate Q isothermal gas mass at TG and its isothermal black bounding surface of area A1 is given by ˙ = A1σ(TG4 εG − T 41 αG1) Q (5-155)
q(TG, T1) = qr(TG, T1) + h(TG − T1)
˙ Cp dTG [q(TG, T1)]dA1 = −m from which
(5-156)
(5-157)
TG,inlet
Cp dTG (5-158) ᎏ q(TG, T1) The area under a curve of Cp /q versus TG or 1/q versus the specific enthalpy i may be used to solve for the area A1 required to obtain a given outlet temperature or to obtain the outlet temperature given A1. Three points generally suffice to determine the area under the curve within 10 percent. Instead of using graphical integration, which can handle any complexity of variation of flux density q with TG and T1 along an interchanger flow path, one may evaluate a mean flux density based on mean gas and sink temperatures, based in turn on terminal temperatures. It has been found empirically that fair results are obtained by the use of a mean surface temperature equal to the arithmetic mean of the terminal surface temperatures and by the use of a mean gas temperature equal to the mean surface temperature plus the logarithmic mean of the temperature difference, gas to surface, at the two ends of the exchanger. When radiation dominates the transfer process, however, graphical integration is safer. ˙ A1 = m
TG,outlet
Example 7: Radiation in Gases Flue gas containing 6 percent carbon dioxide and 11 percent water vapor by volume (wet basis) flows through the convection bank of an oil tube still consisting of rows of 0.102-m (4-in) tubes on 0.203-m (8-in) centers, nine 7.62-m (25-ft) tubes in a row, the rows staggered to put the tubes on equilateral triangular centers. The flue gas enters at 871°C (1144 K, 1600°F) and leaves at 538°C (811 K, 1000°F). The oil flows in a countercurrent direction to the gas and rises from 316 to 427°C (600 to 800°F). Tube surface emissivity is 0.8. What is the average heat-input rate, due to gas radiation alone, per square meter of external tube area?
HEAT TRANSFER BY RADIATION With each row of tubes there is associated (0.203)(3/2) = 0.176 m (0.577 ft) of wall height, of area [(0.203)(9)(2) + (7.62)(2)]0.176 − (9)(2)(π)(0.0508)2 = 3.18 m2 (34.2 ft2). One row of tubes has an area of (π)(0.102)(7.62)(9) = 22.0 m2 (236 ft2). If the recommended factor of 0.7 on the refractory area is used, the effective area of the tubes is [22.0 + (0.7)(3.18)]/22.0 = 1.10 m2/m2 of actual area. The exact evaluation of the outside tube temperature from the known oil temperature would involve a knowledge of the oil-film coefficient, tube-wall resistance, and rate of heat flow into the tube, the evaluation usually involving trial and error. However, for the present purpose the temperature drop through the tube wall and oil film will be assumed to be 41.7°C (75°F), making the tube surface temperatures 357°C (675°F) and 468°C (875°F); the average is 412°C (775°F). The radiating gas temperature is (871 − 468) − (538 − 357) tg = 412 + ᎏᎏᎏᎏ 2.3 log [(871 − 468)/(538 − 357)] = 412 + 278 = 690°C (1274°F) These temperatures, partial pressures, and dimensions were used in Example 6 to determine gas emissivity and absorptivity, εG = 0.145; αG1 = 0.210. The approximate effective emissivity of the boundary is (0.8 + 1)/2 = 0.9. Then from Eq. (5-155), modified to allow for sink emissivity and for the presence of a small amount of refractory boundary, ˙ 1 = q = (0.9)(1.10)σ(TG4 εG − T41αG) Q/A = (0.9)(1.10)(5.67)[(9.63)4(0.145) − (6.85)4(0.210)] = 4405 J/(m2 tube area⋅s) [1396, Btu/(ft2 tube area⋅h)] This is equivalent to a convection coefficient of 4405/278, or 15.85 W/(m2)(K) which is of the order of magnitude expected of the convection coefficient itself. Radiation rapidly becomes dominant as the system temperature rises.
Total-Exchange Areas SS GS and The arguments leading to the development of the interchange factor AiᏲij ( S iS j) between surfaces apply to the case of absorption within the gas volume if in the evaluation of the direct-exchange areas allowance is made for attenuation of the radiant beam through the gas. This necessitates nothing more than redefinition, in Eqs. (5-126) to (5-130), of every term ij( Si S j AiFij) to represent, per unit black emissive power, flux from Ai through an absorbing gas to Aj. This may be visualized as multiplication of AiFij by the mean gas transmittance Tij(= 1 − εG for a gray gas). In a system containing an isothermal gas and source-sink boundaries of areas A1 . . . An, the total emission from A1 per unit of its black emissive power is ε1A1, of which S1S 1 + S 1S 2 + ⋅⋅⋅ + S 1S n is absorbed in the various source-sink surfaces by multiple reflections. The difference has been absorbed in the gas and is called the gas-surface totalexchange area GS 1 G S1 = A1ε1 − S 1S i
(5-159)
i
Note that though S 1S i is never used in calculating radiative exchange, its value is necessary for use of Eq. (5-159) to calculate GS . If the gas volume is not isothermal and is zoned, an additional magnitude, the gas-to-gas total-exchange area GiG j, arises (see Hottel and Sarofim. Radiative Transfer, McGraw-Hill, New York, 1967, chap. 11). Space does not permit derivations of special cases; only the single-gas-zone system is treated here. Single-Gas-Zone/Two-Surface-Zone Systems An enclosure consisting of but one isothermal gas zone and two gray surface zones can, properly specified, model so many industrially important radiation problems as to merit detailed presentation. One can evaluate the total radiation flux between any two of the three zones, including multiple reflection at all surfaces. ˙ G⇔1 = Q GS 1σ(TG4 − T 41) ˙ 1⇔2 = Q S1S 2σ(T14 − T 42 ) The total-exchange area takes a relatively simple closed form, even when important allowance is made for gas radiation not being gray and when a reduction of the number of system parameters is introduced by assuming that one of the surface zones, if refractory, is radiatively adiabatic. Before allowance is made for these factors, the case of a gray gas enclosed by two source-sink surface zones will be presented. Modification of Eq. 5-130, as discussed in the first paragraph of this subsection, combined with the assumption that a single mean beam length applies to all transfers; that is, that there is but one gas transmittance τ(= 1 − εG), gives
5-37
A1ε1ε2F12 S (5-160) 1S 2 = ᎏᎏᎏᎏᎏ 1/τ + τρ1ρ2(1 − F12 /C2) − ρ1(1 − F12) − ρ2(1 − F21) A1ε12(F11 + ρ2τ(F12 /C2 − 1) (5-161) S 1S 1 = ᎏᎏᎏᎏᎏ 1/τ + τρ1ρ2(1 − F12 /C2) − ρ1(1 − F12) − ρ2(1 − F21) A1ε1εG(1/τ + ρ2(F12 /C2 − 1)) (5-162) G S 1 = ᎏᎏᎏᎏᎏ 1/τ + τρ1ρ2(1 − F12 /C2) − ρ1(1 − F12) − ρ2(1 − F21) These three expressions suffice to formulate total-exchange areas for gas-enclosing arrangements which include, for example, the four cases illustrated in Table 5-10. An additional surface arrangement of importance is a single-zone surface enclosing gas. With the gas assumed gray, the simplest derivation of G S 1 is to note that the emission from surface A1 per unit of its blackbody emissive power is A1ε1, of which the fractions εG and (1 − εG)ε1 are absorbed by the gas and the surface, respectively, and the surface-reflected residue always repeats this distribution. Therefore, εG A1 G GS (5-163) S single surface 1 = A1ε1 ᎏᎏ = ᎏᎏ zone surround1 1 εG + (1 − εG)ε1 ing gray gas ᎏᎏ + ᎏᎏ − 1 εG ε1 Alternatively, G S1 could be obtained from Case 1 of Table 5-10 by letting plane area A1 approach 0, leaving A2 as the sole surface zone. Departure of gas from grayness has so marked an effect on radiative transfer that the subject will be presented prior to discussion of the systems covered by Table 5-10. The Effect of Nongrayness of Gas on Total-Exchange Area A radiating gas departs from grayness in two ways: (1) Its transmittance τ through successive path lengths Lm due to surface reflection, instead of being constant, keeps increasing because at the wavelengths of high absorption the incremental absorption decreases with increasing pathlength; (2) Gas emissivity εG and absorptivity αG1 are not the same unless T1 equals TG. The total emissivity of a real gas, the spectral emissivity, and absorptivity ελ that varies in any way with λ can be expressed as the a-weighted mean of a suitable number of gray-gas emissivity or absorptivity terms εG,i or αG,i, representing the gray-gas emissivity or absorptivity in the energy fractions ai of the blackbody spectrum. Then n
n
0
0
εG = aiεG,i = ai(1 − e−k i pL)
(5-164)
For simplicity, n should be as low as is consistent with small error. The retention of but two terms is feasible when one considers that if αG1 is so fitted that the first absorption and the second following surface reflection are correct, then further attenuation of the beam by successive surface reflections makes the errors in those absorptions decrease in importance. Let the gas be modeled as the sum of one gray gas plus a clear gas, with the gray gas occupying the energy fraction a of the blackbody spectrum and the clear gas the fraction (1 − a). Then [εG(pL)] = a(1 − e−kpL) + (1 − a)0 [εG(2pL)] = a(1 − e−2kpL) + (1 − a)0
(5-165)
Solution of these gives εG(pL) a = ᎏᎏ εG(2pL) 2−ᎏ ᎏ εG(pL) εG(pL) kpL = −ln 1 − ᎏ a
(5-166)
Note that these values are specific to the subject problem in which the mean beam length is Lm, with εGs evaluated from basic data, such as Table 5-8. (1 − e−kpL) in Eq. (5-165) represents the emissivity of a gray gas, which will be called εG,i. For later use, note that, εG(pL) εG,i = ᎏ (5-167) a To allow for the difference between emissivity and absorptivity and combine them into a single emissivity-absorptivity term called effec-
5-38
HEAT AND MASS TRANSFER
TABLE 5-10
Total-Exchange Areas for Four Arrangements of Two-Zone-Surface Enclosures of a Gray Gas
A plane surface A1 and a surface A2 completing the enclosure
Infinite parallel planes
Concentric spherical or infinite cylindrical surface zones, A1 inside
Two-surface-zone spherical surface, each zone one or more parts; or speckled enclosure, any shape
F12 = 1
F12 = F21 = 1
F12 = 1; F21 = A1/A2
F12 = F22 = C2; F21 = F11 = C1
S 1S 2 ε1ε2 ᎏ=ᎏ A1 D1
S 1S 2 ε1ε2 ᎏ=ᎏ A1 D2
S 1S 2 ε1ε2 ᎏ=ᎏ A1 D3
S 1S 2 ε1ε2C2 ᎏ=ᎏ A1 D4
G S 1 ε1εG(1/τ + ρ2A1/A2) ᎏ = ᎏᎏ A1 D1
G S 1 ε1εG(1/τ + ρ2) ᎏ = ᎏᎏ A1 D2
G S 1 ε1εG(1/τ + ρ2A1/A2) ᎏ = ᎏᎏ A1 D3
G S 1 ε1εG/τ ᎏ=ᎏ A1 D4
G S 2 ε2εG(1/τ + ρ1A1/A2) ᎏ = ᎏᎏ A2 D1
S 1S 1 ε 12 ρ2τ ᎏ=ᎏ A1 D2
G S 2 ε2εG(1/τ + ρ1A1/A2) ᎏ = ᎏᎏ A2 D3
S 1S 1 ε 21 C1 ᎏ=ᎏ A1 D4
S 1S 1 ε 12 τρ2A1/A2 ᎏ = ᎏᎏ A1 D1
D2 1/τ − τρ1ρ2
S 1S 1 ε 12 ρ2τA1/A2 ᎏ = ᎏᎏ A1 D3
D4 1/τ − ρ1C1 − ρ2C2
1 A1 D1 ᎏ − ρ2 1 − ᎏ (1 − τρ1) τ A2
1 A1 D3 ᎏ − ρ2 1 − ᎏ (1 − τρ1) τ A2
tive emissivity εG,e, one must first evaluate absorptivity αG1 using Eq. (5-161). Formulation of the net direct exchange can then be used to define εG,e: σ(εGTG4 − αG1T 41) σεG,e(T G4 − T 41) εG,e − αG1(T41/ TG4 ) (5-168) or εG,e = ᎏᎏ 1 − (T1 / TG)4 The emissivity and absorptivity of use in converting gray-gas totalexchange areas to real-gas values are εG,e and ae, the latter obtained by using Eq. (5-166), except that εG,e (pL) replaces εG(pL); the same for εG,e (2pL). This means that, for the conversion, four terms will have to be formulated: εG(pL), εG(2pL), αG1(pL), and αG1(2pL). The gray emissivity term εG,i of Eq. (5-167) now becomes εG,e /ae. Conversion of gray-gas total exchange areas GS SS and to their nongray form depends on the fact that the relation between radiative transfer and blackbody emissive power σT 4 is linear and proportional. The gray-gas-equivalent emissivity εG,e /ae is applicable only to the energy fraction ae of σT 4. In consequence, to convert GS SS or to its nongray form, wherever εG or τ appears in G S it must be replaced by εG,e /ae or (1 − εG,e /ae), respectively; the overall result is then multiplied by ae. The converted S S is in two parts: The gray-gas contribution involves, as above, replacement of εG by εG,e /ae and τ by (1 − εG,e /ae), and the result multiplied by ae; for the clear-gas contribution, εG is replaced by 0 and τ by 1, and the result is multiplied by (1 − ae) and added to the gray-gas contribution. The simplest application of this simple gray-plus-clear model of gas radiation is the case of a single gas zone surrounded by a single surface zone, Eq. (5-163) for a gray gas. The gray-plus-clear model gives ae GS 1 (5-169) ᎏ = ᎏᎏ ae 1 A1 ᎏᎏ + ᎏᎏ − 1 εG,e ε1
at 1000 K. The surrounding gas is at 1500 K and is well stirred. Find the effective gas emissivity εG,e, the weighting factor ae, and the surface radiative flux density. Solution: Combustion is 1CH4 + 2 × 1.2 O2 + 2 × 1.2 × 79/21 N2 + 2 × 1.2 × 100/21 × 0.0088 H2O, going to 1 CO2 + [2 + 2 × 1.2 × (100/21) × 0.0088] H2O + 0.4 O2 + 9.03 N2 = 12.53 moles per mole of CH4; pc + pw = (1 + 2.1)/12.53 = 0.2474 atm. The mean beam length, Lm = 0.88 × 4V/AT = 0.88 × 4(10 × 3 × 5)/ [2 × (10 × 3 + 10 × 5 + 3 × 5)] = 2.779 m. pLm = 0.2474 × 2.779 = 0. 6875 m atm. From emissivity Table 5-8, b(1500) = 540; n(1500) = 0.42; b(1000) = 444; n(1000) = 0.34. εG(pL) = 540(0.6875 − 0.015)0.42/1500 = 0.3047; αG1(pL) = 444(0.6875 × 1000/1500 − 0.015)0.34 (1500/1000)0.5/1000 = 0.4124. Then εG,e(pL) = [0.3047 − 0.4124(1000/1500)4]/[1 − (1000/1500)4] = 0.2782. Repeat all 3 computations for pL = 2 × 0.6875 to give εG(2pL) = 0.4096, αG1 (2pL) = 0.5250, εG,e(2pL) = 0.3812. Then ae = 0.2782/(2 − 0.3812/0.2782) = 0.4418 and the emissivity substitute = 0.2782/0.4418 = 0.6297. For a single enveloping surface zone, the total-exchange area comes from Eq. (5-169); G S 1/A1 = ae/(ae/εG,e + 1/ε1 − 1) = ˙ 0.4418/(0.4418/0.2782 + 1/0.8 − 1) = 0.2404. The flux density is Q/A = q = 4 4 (G S 1/A)σ(TG − T 1) = 0.2404 × 56.7 × [(1500/1000)4 − (1000/1000)4] = 55.37 2 kW/m (17,550 Btu/sq ft hr). (Note that allowing for average humidity in air adds 5 percent to H2O and approximately 2 percent to gas emissivity.)
Example 8: Effective Gas Emissivity Methane is burned to completion with 20 percent excess air (air half-saturated with water vapor at 298 K (60°F), 0.0088 mols H2O/mol dry air) in a furnace chamber of floor dimensions 3 × 10 m and height 5 m. The whole surface is a gray-energy sink of emissivity 0.8
Total-exchange areas for the basic one-gas two-surface model [Eqs. (5-160) to (5-162)], used to evaluate the cases in Table 5-10, take the following form when converted by the above described procedure to their nongray form: ae A1F12ε1ε2 S 1S 2 = ᎏᎏᎏᎏᎏ 1/(1 − εG,e /ae) + (1 − εG,e /ae)ρ1ρ2(1 − F12C2) − ρ1(1 − F12) − ρ2(1 − F21) (1 − ae)A1F12ε1ε2 + ᎏᎏᎏᎏᎏ 1 + ρ1ρ2(1 − F12C2) − ρ1(1 − F12) − ρ2(1 − F21) A1ε1εG,e(1/(1 − εG,e /ae) + ρ2(F12 /C2 − 1) G S 1 = ᎏᎏᎏᎏᎏ 1/(1 − εG,e /ae) + (1 − εG,e /ae)ρ1ρ2(1 − F12C2) − ρ1(1 − F12) − ρ2(1 − F21)
(5-170) (5-171)
Modification of Table 5-10 to make the total-exchange areas conform to the gray-plus-clear gas model is straightforward, following the instructions presented above. The results are given in Table 5-11.
HEAT TRANSFER BY RADIATION TABLE 5-11
5-39
Conversion of Some Total-Exchange Areas to Their Gray-Plus-Clear Values
1. Plane slab A1 and surface A2, completing an enclosure of gas (F12 = 1) 1S S 2 aε1ε2 (1 − a)ε1ε2 ᎏ=ᎏ + ᎏᎏ A1 D1 1 − ρ2(1 − ε1A1/A2) ρ1εG
− ρ 1 − ᎏ ε + ᎏ D = 1ᎏ − ε /a A a 1
A1
1
2
1
G
2
S G 1 ε1εG(1/(1 − εG /a) + ρ2A1 /A2) ᎏ = ᎏᎏᎏ A1 D1 S G 2 ε2εG(1/(1 − εG /a) + ρ1A1/A2) ᎏ = ᎏᎏᎏ A2 D1 2. Infinite parallel planes, gas between (F12 = F21 = 1) S S aε1ε2 (1 − a)ε1ε2 1 2 ᎏ=ᎏ + ᎏᎏ A1 D2 1 − ρ1ρ2 εG
− 1 − ᎏ ρ ρ D = ᎏ 1 − ε /a a 1
2
1 2
G
S G 1 ε1εG(1/(1 − εG /a) + ρ2) ᎏ = ᎏᎏᎏ A1 D2 3. Concentric spherical or infinite cylindrical surface zones, A1 inside (F12 = 1; F21 = A1/A2) 1S S 2 aε1ε2 (1 − a)ε1ε2 ᎏ = ᎏ + ᎏᎏ A1 D3 1 − ρ2(1 − ε1A1/A2) 1 ρ1εG A D3 = ᎏᎏ − ρ2 1 − ᎏ1 ε1 + ᎏ (1 − εG /a) A2 a
S G 1 ε1εG(1/(1 − εG /a) + ρ2A1/A2) ᎏ = ᎏᎏᎏ A1 D3 S G 2 ε2εG(1/(1 − εG /a) + ρ1A1/A2) ᎏ = ᎏᎏᎏ A2 D3 4. Spherical enclosure of two surface zones or “speckled” A1:A2 enclosure (F12 = F22 = C2; F21 = F11 = C1) S S aε1ε2C2 (1 − a)ε1ε2C2 1 2 ᎏ = ᎏ + ᎏᎏ A1 D4 1 − ρ1C1 − ρ2C2 −ρC −ρC D = ᎏᎏ (1 − ε /a) 1
4
1
1
2
2
G
C = Aᎏ +A A1
1
1
2
GS 1 ε1εG /(1 − εG /a) ᎏ = ᎏᎏ A1 D4
Treatment of Refractory Walls Partially Enclosing a Radiating Gas Another modification of the results in Table 5-10 becomes important when one of the surface zones is radiatively adiabatic; the need to find its temperature can be eliminated. If surface A2, now called Ar, is radiatively adiabatic, its net radiative exchange with A1 must equal its net exchange with the gas.
or
4 4 G SS S r(TG4 − T 4r ) = r(T 1 r − T 1) TG4 − T 4r T 4r − T 41 T G4 − T 41 ᎏ = ᎏ = ᎏᎏ 1/G 1/S 1/G S r rS 1 S r + 1/S rS 1
(5-172)
GS The net flux from gas G is GS 1σ(TG4 − T14) + rσ(TG4 − Tr4) which, with replacement of the last term using Eq. (5-172), gives the single term
1 σ(TG4 − T14) G S1 + ᎏᎏ 1 1 ᎏᎏ + ᎏᎏ G Sr S rS 1
The bracketed term is called (G S 1)R, the total exchange area from G to A1 with assistance from a refractory surface. In summary,
1 ˙ G ⇔ 1 = (G S 1 + ᎏᎏ σ(TG4 − T14) Q S 1)Rσ(TG4 − T 14) = G 1 1 ᎏᎏ + ᎏᎏ G Sr S rS 1
(5-173)
Table 5-10 supplies the forms for the three terms needed to formulate (G S 1)R, with Ar substituted for A2. If, in addition, allowance is to be made for the gas not being gray, εG,e and ae are evaluated using values of the emissivity and absorptivity calculated using Table 5-8, and the procedure described in the previous subsection is followed with εG,e /ae replacing εG together with the addition of a clear-gas contribution, when S S is at issue. It is tempting to say that a surface A2 (or Ar) could be made radiatively adiabatic simply by assigning its reflectance ρ a value of 1, making the terms in the brackets of Eq. (5-173) much easier to evaluate and the result much simpler. This is valid only if the gas is gray. If it is not, Ar is a net absorber of radiation
5-40
HEAT AND MASS TRANSFER
in the spectral energy fraction a (or ae) and a net emitter in the cleargas fraction (1 − a). Conversion of GS to (G S 1)R will be carried out for two of the four cases of Table 5-10. Case 1 is an idealization of a metal-heating slab furnace or glass furnace, with its plane sink A1 combining with refractory surface Ar to complete the enclosure. With insertion into Eq. (5173) of G SrS S 1, G S r, and 1 after converting each to its gray plus clear form, one obtains
(G C1 1 S 1)R εG ᎏ = ᎏ ε1 ρ2 ᎏ + ᎏ A1 D1 Cr 1 − ᎏεGᎏ a
+
εr εr + ρrε1(C1/Cr) 1 ᎏᎏᎏ + ρ1 + (Cr /C1)[1 − (εG/a)] ε1 C1 ᎏ a εr + ρrε1 ᎏ + (1 − a)D1 εG Cr
(5-174) where D1 = 1/(1 − εG/a) − ρr{1 − (C1/Cr)[1 − ρ1(1 − εG/a)]}. Conversion of (G S ) 1 R to applicability to a gray gas comes by making a equal 1, producing the enormous simplification to ᎏ A (GS1)R 1
gray gas
1 = ᎏᎏᎏ 1 + C1(1/εG − 2) ρ1 ᎏᎏ + ᎏᎏ 1 − C1εG ε1
(5-175)
Note that the emissivity and reflectance of the refractory are without effect on (G S 1)R if the gas is gray. The second conversion of G S ) S to (G 1 R will be Case 4 of Table 5-10, the two-surface-zone enclosure with computation simplified by assuming that the direct-view factor from any spot to a surface equals the fraction of the whole enclosure that the surface occupies (the speckled-furnace model). This case can be considered an idealization of many processing furnaces such as distilling and cracking coil furnaces, with parts of the enclosure tube-covered and part left refractory. (But the refractory under the tubes is not to be classified as part of the refractory zone.) Again, one starts with substitution into Eq. (5-173) of the terms G SrS S 1, G S r, and 1 from Table 5-10, Case 4, with all terms first converted to their gray-plus-clear form. To indicate the procedure, one of the components, SrS 1, wil be formulated. Sr S1 Crε1εr Crε1εr ᎏ = a ᎏ + (1 − a) ᎏᎏ A1 D4 1 − ρ1C1 − ρrCr Crε1εr εG(1 − a)(a − εG) = ᎏ 1 + ᎏᎏ D4 1 − ρ1C1 − ρrCr With D4 = 1/(1 − εG/a) − ρ1C1 − ρrCr, the result of the full substitution simplifies to
1 (G S 1)R ᎏ = ᎏᎏᎏᎏ A1 1 1 1 1/a − 1 C1 ᎏᎏ − ᎏᎏ + ᎏᎏ + ᎏ ᎏ εG a ε1 ε1 + εr(Cr /C1)
For a gray gas (a = 1), the above becomes 1 (G S 1)R ᎏ = ᎏᎏ 1 1 A1 C1 ᎏᎏ − 1 + ᎏᎏ εG ε1
(5-176)
(5-177)
Eq. (5-176) has wide applicability. COMBUSTION CHAMBER HEAT TRANSFER Treatment of radiative transfer in combustion chambers is available at varying levels of complexity, including allowance for temperature variation in both gas and refractory walls (Hottel and Sarofim,
Radiative Transfer, McGraw-Hill, New York, 1967, chap. 14). A less rigorous treatment suffices, however, for handling many problems. There are two limiting cases: the long chamber with gas temperature varying only in the direction of gas flow (already treated) and the compact chamber containing a gas or a flame to which can be assigned an effective or average radiating temperature. The latter will be considered. Stirred-Chamber Model; Refractory Wall Loss Negligible What furnace engineers most need is a closed-form solution of the problem, theoretically sound in structure and therefore containing a minimum number of parameters and no empirical constants and, preferably, physically visualizable. They can then (1) correlate data on existing furnaces, (2) develop a performance equation for standard design, or (3) estimate performance of a new furnace type on which no data are available. An equation representing an energy balance on a combustion chamber of two surface zones, a heat sink A1 at temperature T1, and a refractory surface Ar assumed radiatively adiabatic at Tr, is most simply solved if the total enthalpy input H is expressed as mC ˙ p(TF − To); m ˙ is the mass rate of fuel plus air; and TF is a pseudoadiabatic flame temperature based on a mean specific heat from base temperature To up to the gas exit temperature TE rather than up to TF. The heat transfer rate Q˙ out of the gas is then H˙ − mC ˙ p(TE − To) or mC ˙ p(TF − TE). The energy balance, with ambient temperature taken as conventional base To, is ˙ − mC (Q˙ =)H ˙ p(TE − To) = (G S 1)R σ(TG4 − T 14 ) + h1A1(TG − T1) + AoFoσ(TG4 − To4) + UAr(TG − To)
(5-178)
S 1)RσTF4 To make the relation dimensionless, divide through by (G and let all temperatures, expressed as ratios to TF, be called τs. For clarity, the terms are tabulated: mC ˙ p/(G S 1)RσTF3 = D, dimensionless firing density l.h.s. term = D(1 − τE) 1st r.h.s. term = τ G4 − τ 14 (h1A1/(G GS 1)R σTF3) = Lc, convection number (dimensionless) AoFo /(G S 1)R = Lo, wall-openings loss number, (dimensionless) UAr /(G S 1)RσTF3 = Lr, refractory-wall loss number (dimensionless) The equation then becomes D(1 − τE) = τG4 − τ14 + Lc(τG − τ1) + Lo(τG4 − τo4) + Lr(τG − τo)
(5-179)
This equation has two unknowns (τG and τE), and an empirical relation between them is needed. Many have been tried, and one of the best is to assume that the excess of TG over TE expressed as a ratio to TF (zero for a perfectly stirred chamber) is a constant ∆ [ (TG − TE)/ TF]. Although ∆ should vary with burner type, the effects of firing rate and percent excess air are small. In the absence of performance data on the kind of furnace under study, assume ∆ = 300/TF, °R or 170/TF, K. The left side of Eq. (5-178) then becomes D(1 − τG + ∆), and with coefficients of τG and τ G4 collected, the equation becomes D + Lc + Lr τ14 + Lcτ1 + Loτ 4o + Lrτo + D(1 + ∆) τG4 + ᎏᎏ τG − ᎏᎏᎏᎏ = 0 1 + Lo 1 + Lo (5-180)
Though this is a quartic equation, it is capable of explicit solution because of the absence of second and third degree terms. Trial-anderror enters, however, because (G S 1)R and C p are mild functions of TG and related TE, respectively, and a preliminary guess of TG is necessary. An ambiguity can exist in interpretation of terms. If part of the enclosure surface consists of screen tubes over the chamber-gas exit to a convection section, radiative transfer to those tubes is included in the chamber energy balance, but convection is not, because it has no effect on chamber gas temperature. With Eq. (5-180) solved, the gas-side efficiency ηG is (1 − τG + ∆)/ (1 − τo). The sink-side efficiency η1 is less by the amount (Lo(τG4 − τL4) + Lr(τG − τo))/D(1 − τo) and is also given by [(G GS 1)Rσ(TG4 − T 41) + h1A1(TG − T1)]/H˙. It must be remembered that the efficiency η1
HEAT TRANSFER BY RADIATION includes the losses through the wall from the backside of any wall–mounted heat sinks. Though the results must be considered approximations, depending as they do on the empirical ∆, the equation may be used to find the effect of firing rate, excess air, and air preheat on efficiency. With some performance data available, the small effect of various factors on ∆ may be found. The first term on the right side of Eq. (5-179) is so nearly dominant for most furnaces that consideration of the main features of chamber performance is clarified by ignoring the loss terms Lo and Lr or by assuming that they and Lc have a constant mean value. The relation of a modified chamber efficiency ηG(1 − τo) to a modified firing density D/(1 − τo) and to the normalized sink temperature τ = T1 /TF is shown in Fig. 5-23, which is based on Eq. (5-178), with the radiative and convective transfer terms (G S 1)Rσ(TG4 − T 41) + h1A1(TG − T1) replaced by a combined radiation/conduction term (G GS 1)R,cσ(TG4 − T 41), where 3 (G GS GS ; TG1 is adequately approximated by the 1)R,c = (G 1)R + h1A1/4σTG1 arithmetic mean of TG and T1. Example 9: Radiation in a Furnace Consider a furnace 3 m × 10 m × 5 m fired with methane and 20 percent excess air, at a methane firing rate of 2500 kg/hr. Two rows of 5-inch (0.127 m) tubes (outer diameter) are mounted on equilateral triangular centers, with center-to-center distance twice the tube diameter, on 60 percent of the interior surface of the chamber. The radiative properties of the gases for an enclosure of these dimensions, containing the same combustion products, have been estimated in Example 8 for a gas temperature of 1500 K and a sink temperature of 1000 K: 12.53 moles of combustion products are generated per mole of fuel, with a mean molar heat capacity between a base temperature of 298 K and the exit gas exit temperature TE, adequately represented for this example by M C p = 7.01 + 0.875 (T/1000) over a TE range of 800 to 1600 K. The lower heating value of CH4 is 191,760 cal/g mole. The air is preheated to 600°C, and has a mean MC p of 7.31 cal/g mole. The alloy tube emissivity ε1 is 0.7 and may be assumed gray; the mean tube surface temperature is 700°C. The convection coefficient, gas-to-tube plane and to refractory surface is 0.0170 kW/m2 °C; hc + r on the outside surface is 0.0114 kW/m2 °C. The 0.343-m-thick refractory walls and roof have a k of 0.00050 kW/m °C and an assumed εr of 0.6; the walls are pierced by four 0.10-m × 0.23-m peepholes. The gas exit area, 1 m × 10 m, is tube-screen-covered. What is the sink-side efficiency η1, the gas exit temperature TE, and the mean flux density through the tube surface? Solution: Temporary basis—1 mole entering CH4. Since no molal change occurs when CH4 burns completely with half-saturated entering air 20 percent in excess of stoichiometric, the total number of moles produced equal 11.53 moles. Entering enthalpy = 191,760 + 11.53 × 7.31(600 − 25) = 240,220 Kcal/kg mol CH4. H˙ [of Eq. (5-178)] = 240,220 × (2500/16.04) = 37.44E6 Kcal/hr × 4.186/3600 = 43.54E3 Kw. m ˙ = [16.04 + 2 × 1.2(100/21)(29 + 0.0088 × 18.016)](2500/16.04) = 54,440 kg/hr. Trial and error solution necessitates several sets of computations of TG to check assumed TE; only the last of these will be given. The first, to save time by using results attained elsewhere, assumes TG = 1500 K; the resulting TG is 369 K higher. The second set assumes TG = 2017 K; the resulting TG is 80 K lower. Linear interpolation indicates the third set should assume TG = 1934 K. That set is presented: TE = 1934 − 170 = 1764. M Cp = 7.01 + 0.875 × (1764/1000) = 8.554 cal/(gmol)(K). TF = 240,220/(12.53 × 8.554) +
5-41
298 = 2539 K. Cp = (8.554/16.04)(4.186/3600) = 0.6201E-3 kw-hr/(kg)(K). For εG, pLm = 0.247 × 2.779 = 0.6875 m atm. Use of Table 5-8, with TH = 2000 K (b = 572, n = 0.51) and TL = 1500 (b = 540, n = 0.42) gives εG(pL) = [572(0.6875 − 0.015)0.51(1934 − 1500) + 540(0.6725)0.42(2000 − 1934)]/(500 × 1934) = 0.2409. For αG1, with T1 = 1000, Table 5-8 gives b = 444, n = 0.34. αG1(pL) = 444(0.6875 × 1000/1934 − 0.015)0.34(1934/1000)0.5/1000 = 0.4281. εG,e(pL) = [0.2409 − 0.4281(1000/1934)4]/(1 − 1/1.9344) = 0.2265. 2pLm = 2 × 0.6875 = 1.375 m atm. εG(2pL) = [572(1.375 − 0.015)0.51 × 434 + 540(1.36)0.42 × 66]/(500 × 1934) = 0.3422. αG1(2pL) = 444 × (1.375 × 1000/1934 − 0.015)0.34(1.934)0.5/1000 = 0.5459. εG,e(2pL) = [0.3422 − 0.5459(1/1.934)4/(1 − 1/1.9344) = 0.3265. ae = 0.2265/(2 − 0.3265/0.2265) = 0.4056. Of all these, only εG,e(pL) and ae will be used from here on. From Eq. (5-176), (G S1)R / A1 = 1/[0.6(1/0.2265 − 1/0.4056) + 1/0.87 + (1/0.4056 − 1)/(0.87 + 0.6 (0.4/0.6))] = 0.2879. A1 = (3 × 10 + 3 × 5 + 2 10 × 5) × 2 = 190 × 0.6 = 114 m . Ar (with floor area omitted for loss) = 190 × 0.4 − 10 × 3 = 46 m2. (G S 1)R = 0.2879 × 114 = 32.82 m2. (G S 1)RσTF3 = 32.82 × (56.7E − 12) × 25393 = 30.46 kw/K. D = 54,440 × (0.6201E − 3)/30.46 = 1.1083. ∆ = 170/2539 = 0.06696. Lc = 0.017 × 114/30.46 = 0.0636. Lo = 4 × 0.1 × 0.23 × 0.335/32.82 = 9.4E − 4. Lr = 0.0012 × 46/30.46 = 0.001812. In Eq. (5-180), the coefficient of τG equals (1.1083 + 0.0636 + 0.0018)/1.00094 = 1.1726. The constant in the equation equals [(100/2539)4 + 0.0636(1/2.539) + 0.00094 × 0.11744 + 0.00181(298/2539) + 1.1083(1 + 170/2539)]/1.00094 = 1.2307. The equation to solve is: τG4 + 1.1726τG − 1.2307 = 0. Solution gives τG = 0.7620; TG = 0.762 × 2539 = 1935 K. Te = 1765 K, only 1 K above value assumed for obtaining C p and εG. ηG = (1 − τG)/(1 − τo) = (1 − 1765/2539)/(1 − 298/2539) = 0.3454. Sink-side efficiency η1 0.3454 − [0.00094(0.7624 − 0.117444)] + 0.001812(0.762 − 0.1174)/1.1083(1 − 0.1174) = 0.344, not including convection to screen tubes ˙ η1)/A1 = 43,540 × 0.3439/114 = 131.3 kw/m2. covering gas exit. qplane of sink = (H qtube surf = 131.3 (2D/2πD) = 41.8 kw/m2 × 3412 × 0.30482 = 13,300 Btu/(ft2)(hr). TE = 1765 K = 1492°C = 2717°F.
In Fig. 5-22, the shaded areas indicate the operating regimes of a wide range of furnace types. Note the significant properties of the function presented. (1) As firing rate D′ goes down, the efficiency rises and approaches 1 − τ1 in the limit. (This conclusion is modified if wall losses are significant.) (2) Changes in sink temperature have little effect if τ1 < 0.3. (3) As the furnace walls approach complete coverage by a black sink [Cε1 → 1 in Eqs. (5-176) and (5-177)] and as convection becomes unimportant, the effect of flame emissivity on D becomes one of inverse proportionality; thus at very high firing rates at which efficiency approaches inverse proportionality to D, the efficiency of heat transfer varies directly as εG (gas-turbine chambers), but at low firing rates εG has relatively little effect. (4) When Cε1 << 1 because of a nonblack sink or much refractory surface, the effect of changing flame emissivity is to produce a much less than proportional effect on heat flux. The factor ∆, the allowance for imperfect stirring, must be estimated. Values in the range of 93 to 149°C (200 to 300°F) have been found to produce data correlation for a series of tests on marine boilers. Equation (5-179) and Fig. 5-22 serve as a framework for correlating the performance of furnaces with flow patterns—plug flow, parabolic
FIG. 5-22 Thermal performance of well-stirred furnace chambers; reduced efficiency as a function of reduced firing density D and reduced sink temperature τ1. (a) Radiant section, oil tube stills, cracking coils. (b) Domestic boiler combustion chambers. (c) Open-hearth furnaces. (d) Soaking pits.
5-42
HEAT AND MASS TRANSFER
FIG. 5-23
Effect of wall-loss factor L on combustion-chamber performance; L′ = 0.02, and τ = 0.5 and 0.7.
profile, and recirculatory flow—which differ from the well-stirred model [Hottel and Sarofim, Int. J. Mass Heat Transfer, 8, 1153 (1965)]. As expected, plug-flow furnaces show somewhat higher efficiency, mild-recirculation types somewhat lower efficiency, and strong-recirculation furnaces a performance closely similar to that of the well-stirred model.
If data on several furnaces of a single class are available, a similar treatment can lead to a partially empirical equation based on simplified rules for obtaining (G S1)R,c or an effective ∆. Because Eq. (5-178) has a structure which covers a wide range of furnace types and has a sound theoretical basis, it provides safer structures of empirical design equations than many such equations available in the engineering literature.
MASS TRANSFER GENERAL REFERENCES: Bird, Stewart, and Lightfoot, Transport Phenomena, Wiley, New York, 1960. Cussler, Diffusion: Mass Transfer in Fluid Systems, Cambridge University Press, Cambridge, 1984. Danner and Daubert, Manual for Predicting Chemical Process Design Data, AIChE, New York, 1983. Daubert and Danner, Physical and Thermodynamic Properties of Pure Chemicals, Taylor and Francis, Bristol, PA, 1989–1995. Fahien, Fundamentals of Transport Phenomena, McGraw-Hill, New York, 1983. Foust, Wenzel, Clump, Maus, and Andersen, Principles of Unit Operations, 2d ed., Wiley, New York, 1980. Gammon, Marsh, and Dewan, Transport Properties and Related Thermodynamic Data of Binary Mixtures, AIChE, New York, Part 1, 1993; Part 2, 1994. Geankoplis, Transport Processes and Unit Operations, 3d ed., Prentice Hall, Englewood Cliffs, NJ, 1993. Hines and Maddox, Mass Transfer: Fundamentals and Applications, Prentice Hall, Englewood Cliffs, NJ, 1985. Kirwan, “Mass Transfer Principles,” Chap. 2 in Rousseau, R. W. (ed.), Handbook of Separation Process Technology, Wiley, New York, 1987. McCabe, Smith, and Harriott, Unit Operations of Chemical Engineering, 5th ed., McGraw-Hill, New York, 1993. Reid, Prausnitz, and Poling, The Properties of Gases and Liquids, 4th ed., McGrawHill, New York, 1987. Schwartzberg and Chao, Food Technol., 36(2), 73 (1982). Sherwood, Pigford, and Wilke, Mass Transfer, McGraw-Hill, New York, 1975. Skelland, Diffusional Mass Transfer, Wiley, New York, 1974. Taylor and Krishna, Multicomponent Mass Transfer, Wiley, New York, 1993. Treybal, Mass-Transfer Operations, 3d ed., McGraw-Hill, New York, 1980.
INTRODUCTION This part of Sec. 5 provides a concise guide to solving problems in situations commonly encountered by chemical engineers. It deals with
diffusivity and mass-transfer coefficient estimation and common flux equations, although material balances are also presented in typical coordinate systems to permit a wide range of problems to be formulated and solved. Mass-transfer calculations involve transport properties, such as diffusivities, and other empirical factors that have been found to relate mass-transfer rates to measured “driving forces” in myriad geometries and conditions. The context of the problem dictates whether the fundamental or more applied coefficient should be used. One key distinction is that, whenever there is flow parallel to an interface through which mass transfer occurs, the relevant coefficient is an empirical combination of properties and conditions. Conversely, when diffusion occurs in stagnant media or in creeping flow without transverse velocity gradients, ordinary diffusivities may be suitable for solving the problem. In either case, it is strongly suggested to employ data, whenever available, instead of relying on correlations. Units employed in diffusivity correlations commonly followed the cgs system. Similarly, correlations for mass transfer correlations used the cgs or English system. In both cases, only the most recent correlations employ SI units. Since most correlations involve other properties and physical parameters, often with mixed units, they are repeated here as originally stated. Common conversion factors are listed in Table 1-4. Fick’s First Law This law relates flux of a component to its composition gradient, employing a constant of proportionality called a
MASS TRANSFER
5-43
Nomenclature and Units—Mass Transfer Symbols a
Definition
Acs A′ ap c ci cP d db dc d drop dimp dpore DA′A DAB D°AB Deff DK DL DS E ES f F g gc G Ga GM h′ hT H H
Effective interfacial mass transfer area per unit volume Cross-sectional area of vessel Constant (see Table 5-28-I) See a Concentration = P/RT for an ideal gas Concentration of component i = xi c at gas-liquid interface Specific heat Characteristic length Bubble diameter Column diameter Sauter mean diameter Impeller diameter Pore diameter Self-diffusivity (= DA at xA = 1) Mutual diffusivity Mutual diffusivity at infinite dilution of A in B Effective diffusivity within a porous solid = εpD/τ Knudson diffusivity for gases in small pores Liquid phase diffusion coefficient Surface diffusivity Energy dissipation rate/mass Activation energy for surface diffusion Friction factor for fluid flow Faraday’s constant Acceleration due to gravity Conversion factor Gas-phase mass flux Dry air flux Molar gas-phase mass flux Heat transfer coefficient Total height of tower packing Compartment height Henry’s law constant
H′
Henry’s law constant
HG HOG HL HOL HTU jD jH jM m JA
J JSi k k k k′
Height of one transfer unit based on gas-phase resistance Height of one overall gas-phase mass-transfer unit Height of one transfer unit based on liquid-phase resistance Height of one overall liquid-phase mass-transfer unit Height of one transfer unit (general) Chilton-Colburn factor for mass transfer, Eq. (5-291) Chilton-Colburn factor for heat transfer See jD Mass flux of A by diffusion with respect to the mean mass velocity Molar flux of A by diffusion with respect to mean molar velocity Molar flux of A with respect to mean volume velocity Molar flux by surface diffusion Boltzmann’s constant Film mass transfer coefficient Thermal conductivity Mass-transfer coefficient for dilute systems
kG
Gas-phase mass-transfer coefficient for dilute systems
k′G
Gas-phase mass-transfer coefficient for dilute systems
kGa kˆ Ga
Volumetric gas-phase mass-transfer Overall volumetric gas-phase mass-transfer coefficient for concentrated systems Liquid phase mass transfer coefficient for pure absorption (no reaction) Liquid-phase mass-transfer coefficient for dilute systems
J
M A
V A
ˆL k° kL k′L kˆ L kLa K K KG
Liquid-phase mass-transfer coefficient for dilute systems Liquid-phase mass-transfer coefficient for concentrated systems Volumetric liquid-phase mass-transfer coefficient for dilute systems Overall mass transfer coefficient α/R = specific conductance Overall gas-phase mass-transfer coefficient for dilute systems
SI units
U.S. customary units
m2/m3
ft2/ft3
m2 or cm2
ft2
mol/m3 or mol/l or gequiv/l mol/m3 or mol/l or gequiv/l kJ/(kg⋅K) m or cm m m or cm m m m or cm m2/s or cm2/s m2/s or cm2/s m2/s or cm2/s m2/s m2/s or cm2/s m2/s m2/s or cm2/s
lbmol/ft3 lbmol/ft3 Btu/(lb⋅°F) ft ft ft ft ft ft ft2/h ft2/h ft2/h ft2/h ft2/h ft2/h ft2/h
J/mol or cal/mol Dimensionless 96,487 Coulomb/gequiv m/s2 1.0 kg/(s⋅m2) kg/(s⋅m2) kmol/(s⋅m2) W/(m2⋅K) = J/(s⋅m2⋅K) m m kPa/(mole-fraction solute in liquid phase) kPa/[kmol/(m3 solute in liquid phase)]
Dimensionless
m m m m m Dimensionless Dimensionless
ft/h2 4.17 × 108 lb ft/[lbf⋅h2] lb/(h⋅ft2) lb/(h⋅ft2) (lbmol)/(h⋅ft2) Btu/(h⋅ft2⋅°F) ft ft (lbf/in2)/(mole-fraction solute in liquid phase) (lbf/in2)/[(lbmol)/(ft3 solute in liquid phase)] or atm/[(lbmole)/ (ft3 solute in liquid phase)] ft ft ft ft ft Dimensionless Dimensionless
kmol/(m2⋅s) or mol/(cm2⋅s)
lbmol/(ft2⋅h)
kmol/(m2⋅s) or mol/(cm2⋅s)
lbmol/(ft2⋅h)
kmol/(m2⋅s) kmol/(m2⋅s) or gmol/(cm2⋅s) 8.9308 × 10−10 gequiv ohm/s m/s or cm/s (J⋅m)/(s⋅m2⋅K) kmol/[(s⋅m2)(kmol/m3)] or m/s kmol/[(s⋅m2)(kPa solute partial pressure)] kmol/[(s⋅m2)(mole fraction in gas)] kmol/[(s⋅m3)(mole fraction)] kmol/(s⋅m3)
lbmol/(ft2 ⋅h) lbmol/(ft2⋅h) ft/hr Btu/(h⋅ft⋅°F) lbmol/[(h⋅ft2)(lbmol/ft3)] or ft/hr lbmol/[(h⋅ft2)lbf/in2 solute partial pressure)] lbmol/[(h⋅ft2)(mole fraction in gas)] (lbmol)/[(h⋅ft3)(mole fraction)] lbmol/(h⋅ft3)
kmol/(s⋅m2)
lbmol/(h⋅ft2)
kmol/[(s⋅m2)(mole-fraction solution in liquid)] kmol/[(s⋅m2)(kmol/m3)] or m/s kmol/(s⋅m2)
(lbmol)/[(h⋅ft2)(mole-fraction solute in liquid)] (lbmol)/[(h⋅ft2)(lbmol/ft3)] or ft/h lbmol/(h⋅ft2)
kmol/[(s⋅m3)(mole fraction)]
(lbmol)/[(h⋅ft3)(mole fraction)]
m/s or cm/s ohm/cm kmol/[(s⋅m2)(mole fraction)]
ft/h (lbmol)/[(h⋅ft2)(mole fraction)]
5-44
HEAT AND MASS TRANSFER
Nomenclature and Units—Mass Transfer (Continued ) Symbols Kˆ G
Definition
KGa
Overall gas-phase mass-transfer coefficient for concentrated systems Overall volumetric gas-phase mass-transfer dilute systems
K′Ga
Overall volumetric gas-phase mass-transfer dilute systems
(Ka)H KL Kˆ L
Overall enthalpy mass-transfer coefficient Overall liquid-phase mass-transfer coefficient Liquid-phase mass-transfer coefficient for concentrated systems Overall volumetric liquid-phase mass-transfer coefficient for dilute systems Overall volumetric liquid-phase mass-transfer coefficient for concentrated systems Liquid-phase mass flux Molar liquid-phase mass flux Slope of equilibrium curve = dy/dx (mole-fraction solute in gas)/(mole-fraction solute in liquid) Molality of solute Molecular weight of species i Mass in a control volume V Valences of cationic and anionic species See Table 5-28-I Mass flux of A with respect to fixed coordinates Impeller speed Number deck levels Interphase mass-transfer rate of solute A per interfacial area with respect to fixed coordinates Number of components Froude Number (dimp N2/g)
KLa Kˆ La L LM m m Mi M |n+ n −| n′ nA N N′ NA Nc NFr
NOG NOL NTU NKn NPr NRe NSc NSh NSt NW p pB,M pi pT P P Pc Per Q rA R
ρ∞ gx3 Grashof number ᎏ2 ᎏ −1 (µ/ρ) ρo Number of overall gas-phase mass-transfer units Number of overall liquid-phase mass-transfer units Number of transfer units (general) Knudson number = l/dpore Prandtl number (cpµ/k) Reynolds number (Gd/µG) Schmidt number (µG/ρGDAB) or (µL/ρLDL) Sherwood number (ˆkGRTd/DABpT) Stanton number (ˆkG/GM) or (ˆkL/LM) Weber number (ρcN2d3imp/σ) Solute partial pressure in bulk gas Log mean partial pressure difference of stagnant gas B Solute partial pressure at gas-liquid interface Total system pressure Pressure Power Critical pressure Perimeter/area Volumetric flow rate Radius of dilute spherical solute Gas constant
R Ri s S t tf T Tb Tc Tr u, v uo u∞ uL vs vT vTS V V Vb Vi vi
Solution electrical resistance Radius of gyration of the component i molecule Fractional surface-renewal rate Tower cross-sectional area = πd2/4 Contact time Formation time of drop Temperature Normal boiling point Critical temperature Reduced temperature = T/Tc Fluid velocity Blowing or suction velocity Velocity away from object Superficial liquid velocity in vertical direction Slip velocity Terminal velocity Stokes law terminal velocity Packed volume in tower Control volume Volume at normal boiling point Molar volume of i at its normal boiling point Partial molar volume of i
NGr
SI units
U.S. customary units
kmol/(s⋅m2)
lbmol/(h⋅ft2)
kmol/[(s⋅m3)(mole-fraction solute in gas)] kmol/[(s⋅m3)(kPa solute partial pressure)] kmol/[(s⋅m2)(mole fraction)] kmol/[(s⋅m2)(mole fraction)] kmol/(s⋅m2)
(lbmol)/[(h⋅ft3(mole-fraction solute in gas)] (lbmol)/[(h⋅ft3)(lbf/in2 solute partial pressure)] lb/[(h⋅ft3)(lb water/lb dry air)] (lbmol)/[(h⋅ft2)(mole fraction)] (lbmol)/(h⋅ft2)
kmol/[(s⋅m3)(mole-fraction solute in liquid)] kmol/(s⋅m3)
(lbmol)/[(h⋅ft3)(mole-fraction solute in liquid)] (lbmol)/(h⋅ft3)
kg/(s⋅m2) kmol/(s⋅m2) Dimensionless
lb/(h⋅ft2) (lbmol)/(h⋅ft2) Dimensionless
mol/1000 g solvent kg/kmol or g/mol kg or g Dimensionless Dimensionless kg/(s⋅m2) Revolution/s Dimensionless kmol/(s⋅m2)
lb/lbmol lb Dimensionless Dimensionless lb/(h⋅ft2) Revolution/min Dimensionless (lbmol)/(h⋅ft2)
Dimensionless Dimensionless
Dimensionless Dimensionless
Dimensionless
Dimensionless
Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless kPa Dimensionless kPa kPa Pa Watts Pa m−1 m3/s Å 8.314 J/mol K = 8.314 Pa m3/(mol K) = 82.057 atm cm3/mol K ohm Å s−1 m2 s s K K K Dimensionless m/s or cm/s m/s m/s m/s m/s m/s m/s m3 m3 or cm3 m3/kmol or cm3/mol m3/kmol or cm3/mol m3/kmol or cm3/mol
Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless lbf/in2 Dimensionless lbf/in2 lbf/in2 lbf/in2 or atm lbf/in2 or atm ft−1 ft3/h 10.73 ft3 psia/lbmol⋅h
h−1 ft2 h h °R °R °R Dimensionless ft/h ft/h ft/h ft/h ft/h ft/h ft/h ft3 ft3 ft3/lbmol ft3/lbmol ft3/lbmol
MASS TRANSFER
5-45
Nomenclature and Units—Mass Transfer (Concluded ) Symbols Vmli Vtower w x x xA xo xBM o BM
x
xi y yBM o yBM yi yio z
Definition
SI units
Molar volume of the liquid-phase component i at the melting point Tower volume per area Width of film Length along plate Mole-fraction solute in bulk-liquid phase Mole fraction of component A Mole-fraction solute in bulk liquid in equilibrium with bulk-gas solute concentration y Logarithmic-mean solvent concentration between bulk liquid and interface values Logarithmic-mean inert-solvent concentration between bulk-liquid value and value in equilibrium with bulk gas Mole-fraction solute in liquid at gas-liquid interface Mole-fraction solute in bulk-gas phase Logarithmic-mean inert-gas concentration (5-262) Logarithmic-mean inert-gas concentration Mole fraction solute in gas at interface Mole-fraction solute in gas at interface in equilibrium with the liquid-phase interfacial solute concentration xi Direction of unidimensional diffusion
U.S. customary units
m3/kmol or cm3/mol
ft3/lbmol
m3/m2 m m (kmol solute)/(kmol liquid) kmole A/kmole fluid (kmol solute)/(kmol liquid)
ft3/ft2 ft ft (lbmol solute)/(lb mol liquid) lbmole A/lb mole fluid (lbmol solute)/(lbmol liquid)
(kmol solvent)/(kmol liquid)
(lbmol solvent)/(lbmol liquid)
(kmol solvent)/(kmol liquid)
(lbmol solvent)/(lbmol liquid)
(kmol solute)/(kmol liquid) (kmol solute)/(kmol gas) (kmol inert gas)/(kmol gas) (kmol inert gas)/(kmol gas) (kmole solute)/(kmol gas) (kmol solute)/(kmol gas)
(lbmol solute)/(lbmol liquid) (lbmol solute)/(lbmol gas) (lbmol inert gas)/(lbmol gas) (lbmol inert gas)/(lbmol gas) (lbmol solute)/(lbmol gas) (lbmol solute)/(lbmol gas)
m
ft
Dimensionless cm−1 Dimensionless m Dimensionless
Dimensionless
m3/m3 Dimensionless Dimensionless Dimensionless Dimensionless cm2/(gequiv⋅ohm) cm2/ohm gequiv cm2/gequiv ohm Debeyes cP or Pa s kg/(s⋅m) kg/(s⋅m) m2/s kg/m3 or g/cm3 kg/m3 or g/cm3 kg/m3 kg/m3 kmol/m3 kg/m3 or g/cm3 Dimensionless
ft3/ft3 Dimensionless Dimensionless Dimensionless Dimensionless
Dimensionless Dimensionless dyn/cm Å dyn/cm Å Dimensionless Dimensionless Radians/s Dimensionless
Dimensionless Dimensionless lbf/ft
Greek symbols α α β δ ε ε εA εAB γi γ⫾ λ+ λ− Λ Λo µi µi µG µL ν ρ ρc ρc ρG ρ L ρp ρr ψi ψ Ψ σ σi σi σAB τ ω ω Ω
1 + NB/NA Conductance cell constant (measured) 5/6 MA1/2 P1/3 c /T c Effective thickness of stagnant-film layer Fraction of discontinuous phase in continuous phase for two-phase flow Void fraction available for gas flow or fractional gas holdup Characteristic Lennard-Jones energy (εAεB)1/2 Activity coefficient of solute i Mean ionic activity coefficient of solute Infinite dilution conductance of cation and anion 1000 K/C = λ+ + λ− = Λo + f(C) Infinite dilution conductance Dipole moment of i Viscosity of pure i Gas-phase viscosity Liquid-phase viscosity Kinematic viscosity = ρ/µ Density of A Critical density of A Density continuous phase Gas-phase density Average molar density of liquid phase Particle density Reduced density = ρ/ρc Parachor of component i = Viσ1/4 Parameter, Table 5-28-G Shape factor, Table 5-27-A Interfacial tension Characteristic length Surface tension of component i Binary pair characteristic length = (σA + σB)/2 Intraparticle tortuosity Pitzer’s acentric factor = −[1.0 + log10(P*/Pc)] Rotational velocity Diffusion collision integral = f(kT/εAB) Subscript
A B G m L super
Solute component in liquid or gas phase Inert-gas or inert-solvent component Gas phase Mean value Liquid phase Superficial velocity
*
At equilibrium
Superscript
Dimensionless ft Dimensionless
lb/(h⋅ft) lb/(h⋅ft) lb/(h⋅ft) ft2/h lb/ft3 lb/ft3 lb/ft3 lb/ft3 (lbmol)/ft3 lb/ft3 Dimensionless
Dimensionless Dimensionless
5-46
HEAT AND MASS TRANSFER
diffusivity. It can be written in several forms, depending on the units and frame of reference. Three that are related but not identical are dcA dxA dwA (5-181) V JA = −DAB ᎏ ≈ M JA = −cDAB ᎏ ∝ m JA = −ρDAB ᎏ dz dz dz The first equality (on the left-hand side) corresponds to the molar flux with respect to the volume average velocity, while the equality in the center represents the molar flux with respect to the molar average velocity and the one on the right is the mass flux with respect to the mass average velocity. These must be used with consistent flux expressions for fixed coordinates and for NC components, such as: NC
NC
NC
J + wA ni
m A
i=1
NA = V JA + cA NiV i = M JA + xA Ni = ᎏᎏ MA i=1 i=1
(5-182)
In each case, the term containing the summation accounts for conveyance, which is the amount of component A carried by the net flow in the direction of diffusion. Its impact on the total flux can be as much as 10 percent. In most cases it is much less, and it is frequently ignored. Some people refer to this as the “convective” term, but that conflicts with the other sense of convection which is promoted by flow perpendicular to the direction of flux. Mutual Diffusivity, Mass Diffusivity, Interdiffusion Coefficient Diffusivity is denoted by DAB and is defined by Fick’s first law as the ratio of the flux to the concentration gradient, as in Eq. (5-181). It is analogous to the thermal diffusivity in Fourier’s law and to the kinematic viscosity in Newton’s law. These analogies are flawed because both heat and momentum are conveniently defined with respect to fixed coordinates, irrespective of the direction of transfer or its magnitude, while mass diffusivity most commonly requires information about bulk motion of the medium in which diffusion occurs. For liquids, it is common to refer to the limit of infinite dilution of A in B using the symbol, D°AB. When the flux expressions are consistent, as in Eq. (5-182), the diffusivities in Eq. (5-181) are identical. As a result, experimental diffusivities are often measured under constant volume conditions but may be used for applications involving open systems. It turns out that the two versions are very nearly equivalent for gas-phase systems because there is negligible volume change on mixing. That is not usually true for liquids, however. Self Diffusivity Self-diffusivity is denoted by DA′A and is the measure of mobility of a species in itself; for instance, using a small concentration of molecules tagged with a radioactive isotope so they can be detected. Tagged and untagged molecules presumably do not have significantly different properties. Hence, the solution is ideal, and there are practically no gradients to “force” or “drive” diffusion. This kind of diffusion is presumed to be purely statistical in nature. In the special case that A and B are similar in molecular weight, polarity, and so on, the self-diffusion coefficients of pure A and B will be approximately equal to the mutual diffusivity, DAB. Second, when A and B are the less mobile and more mobile components, respectively, their self-diffusion coefficients can be used as rough lower and upper bounds of the mutual diffusion coefficient. That is, DA′A ≤ DAB ≤ DB′B. Third, it is a common means for evaluating diffusion for gases at high pressure. Self-diffusion in liquids has been studied by many [Easteal AIChE J. 30, 641 (1984), Ertl and Dullien, AIChE J. 19, 1215 (1973), and Vadovic and Colver, AIChE J. 18, 1264 (1972)]. Tracer Diffusivity Tracer diffusivity, denoted by DA′B is related to both mutual and self-diffusivity. It is evaluated in the presence of a second component B, again using a tagged isotope of the first component. In the dilute range, tagging A merely provides a convenient method for indirect composition analysis. As concentration varies, tracer diffusivities approach mutual diffusivities at the dilute limit, and they approach self-diffusivities at the pure component limit. That is, at the limit of dilute A in B, DA′ B → D°AB and DB ′A → DB ′ B; likewise at the limit of dilute B in A, DB ′A → D°BA and DA ′B → DA′A. Neither the tracer diffusivity nor the self-diffusivity has much practical value except as a means to understand ordinary diffusion and as
order-of-magnitude estimates of mutual diffusivities. Darken’s equation [Eq. (5-220)] was derived for tracer diffusivities but is often used to relate mutual diffusivities at moderate concentrations as opposed to infinite dilution. Mass-Transfer Coefficient Denoted by kc, kx, Kx, and so on, the mass-transfer coefficient is the ratio of the flux to a concentration (or composition) difference. These coefficients generally represent rates of transfer that are much greater than those that occur by diffusion alone, as a result of convection or turbulence at the interface where mass transfer occurs. There exist several principles that relate that coefficient to the diffusivity and other fluid properties and to the intensity of motion and geometry. Examples that are outlined later are the film theory, the surface renewal theory, and the penetration theory, all of which pertain to idealized cases. For many situations of practical interest like investigating the flow inside tubes and over flat surfaces as well as measuring external flow through banks of tubes, in fixed beds of particles, and the like, correlations have been developed that follow the same forms as the above theories. Examples of these are provided in the subsequent section on mass-transfer coefficient correlations. Problem Solving Methods Most, if not all, problems or applications that involve mass transfer can be approached by a systematic course of action. In the simplest cases, the unknown quantities are obvious. In more complex (e.g., multicomponent, multiphase, multidimensional, nonisothermal, and/or transient) systems, it is more subtle to resolve the known and unknown quantities. For example, in multicomponent systems, one must know the fluxes of the components before predicting their effective diffusivities and vice versa. More will be said about that dilemma later. Once the known and unknown quantities are resolved, however, a combination of conservation equations, definitions, empirical relations, and properties are applied to arrive at an answer. Figure 5-24 is a flowchart that illustrates the primary types of information and their relationships, and it applies to many mass-transfer problems. CONTINUITY AND FLUX EXPRESSIONS Material Balances Whenever mass-transfer applications involve equipment of specific dimensions, flux equations alone are inadequate to assess results. A material balance or continuity equation must also be used. When the geometry is simple, macroscopic balances suffice. The following equation is an overall mass balance for such a unit having Nm bulk-flow ports and Nn ports or interfaces through which diffusive flux can occur: Nm Nn dM (5-183) ᎏ = mi + ni Acsi dt i = 1 i=1 where M represents the mass in the unit volume V at any time t; mi is the mass flow rate through the ith port; and ni is the mass flux through the ith port, which has a cross-sectional area of Acsi. The corresponding balance equation for individual components includes a reaction term: Nm Nn dMj (5-184) ᎏ = mij + nij Acsi + rj V dt i=1 i=1 For the jth component, mij = miwij is the component mass flow rate in stream i; wij is the mass fraction of component j in stream i; and rj is the net reaction rate (mass generation minus consumption) per unit volume V that contains mass M. If it is inconvenient to measure mass flow rates, the product of density and volumetric flow rate is used instead. In addition, most situations that involve mass transfer require material balances, but the pertinent area is ambiguous. Examples are packed columns for absorption, distillation, or extraction. In such cases, flow rates through the discrete ports (nozzles) must be related to the mass-transfer rate in the packing. As a result, the mass-transfer rate is determined via flux equations, and the overall material balance incorporates the stream flow rates mi and integrated fluxes. In such instances, it is common to begin with the most general, differential material balance equations. Then, by eliminating terms that are negligible, the simplest applicable set of equations remains to be solved.
MASS TRANSFER
5-47
Flowchart illustrating problem solving approach using mass-transfer rate expressions in the context of mass conservation.
FIG. 5-24
Table 5-12 provides material balances for Cartesian, cylindrical, and spherical coordinates. The generic form applies over a unit crosssectional area and constant volume: ∂ρ j (5-185a) ᎏ = −∇ ⋅ nj + rj ∂t where nj = ρvj. Applying Fick’s law and expressing composition as concentration gives ∂cj (5-185b) ᎏ = −v ⋅ ∇cj + Dj∇2cj + rj ∂t Flux Expressions: Simple Integrated Forms of Fick’s First Law Simplified flux equations that arise from Eqs. (5-181) and (5-182) can be used for unidimensional, steady-state problems with binary mixtures. The boundary conditions represent the compositions xAL and xAR at the left-hand and right-hand sides of a hypothetical layer having thickness ∆z. The principal restriction of the following equations is that the concentration and diffusivity are assumed to be constant. As written, the flux is positive from left to right, as depicted in Fig. 5-25. 1. Equimolar counterdiffusion (NA = −NB) dxA DAB NA = M JA = −DAB c ᎏ = ᎏ c (xAL − xAR) (5-189) dz ∆z 2. Unimolar diffusion (NA ≠ 0, NB = 0) 1 − xA DAB NA = M JA + xANA = ᎏ c ln ᎏR (5-190) ∆z 1 − xAL 3. Steady state diffusion (NA ≠ −NB ≠ 0) NA ᎏᎏ − xAR NA + NB NA DAB ᎏᎏ (5-191) NA = M JA + xA(NA + NB) = ᎏ ᎏ c ln NA NA + NB ∆z ᎏᎏ − xAL NA + NB TABLE 5-12
Nc
Ni = −cDim ∇xi + xi Nj
In the late 1800s, the development of the kinetic theory of gases led to a method for calculating multicomponent gas diffusion (e.g., the flux of each species in a mixture). The methods were developed simultaneously by Stefan and Maxwell. The problem is to determine the diffusion coefficient Dim. The Stefan-Maxwell equations are simpler in principle since they employ binary diffusivities: Nc
1 ∇xi = ᎏ (xiNj − xj Ni) j = 1 cDij
(5-186)
∂nzj 1 ∂rnrj 1 ∂nθj ∂ρj ᎏ = − ᎏ ᎏ + ᎏ ᎏ + ᎏ + rj ∂t ∂z r ∂r r ∂θ
Cylindrical
(5-187)
2 ∂nθj sin θ ∂nφj ∂ρj 1 ∂r nrj 1 1 ᎏ = − ᎏ2 ᎏ + ᎏ ᎏᎏ + ᎏ ᎏ + rj ∂t ∂r r sin θ ∂θ r sin θ ∂φ r
(5-193)
If Eqs. (5-192) and (5-193) are combined, the multicomponent diffusion coefficient may be assessed in terms of binary diffusion coefficients [see Eq. (5-204)]. For gases, the values Dij of this equation are approximately equal to the binary diffusivities for the ij pairs. The Stefan-Maxwell diffusion coefficients may be negative, and the method may be applied to liquids, even for electrolyte diffusion [Kraaijeveld, Wesselingh, and Kuiken, Ind. Eng. Chem. Res., 33, 750 (1994)]. Approximate solutions have been developed by linearization [Toor, H.L., AIChE J., 10, 448 and 460 (1964); Stewart and Prober, Ind. Eng. Chem. Fundam., 3, 224 (1964)]. Those differ in details but yield about the same accuracy. More recently, efficient algorithms for solving the equations exactly have been developed (see Taylor and Krishna, Krishnamurthy and Taylor, and Taylor and Webb).
Equation ∂nyj ∂nzj ∂nxj ∂ρj ᎏ = − ᎏ + ᎏ + ᎏ + rj ∂t ∂x ∂y ∂z
Cartesian
(5-192)
j=1
Continuity Equation in Various Coordinate Systems
Coordinate System
Spherical
The unfortunate aspect of the last relationship is that one must know a priori the ratio of the fluxes to determine the magnitudes. It is not possible to solve simultaneously the pair of equations that apply for components A and B because the equations are not independent. Stefan-Maxwell Equations Following Eq. (5-182), a simple and intuitively appealing flux equation for applications involving Nc components is
(5-188)
5-48
FIG. 5-25
HEAT AND MASS TRANSFER Before using diffusivities from either data or correlations, it is a good idea to check their reasonableness with respect to values that have been commonly observed in similar situations. Table 5-13 is a compilation of several rules of thumb. These values are not authoritative; they simply represent guidelines based on experience. Diffusivity correlations for gases are outlined in Table 5-14. Specific parameters for individual equations are defined in the specific text regarding each equation. References are given after Table 5-19. The errors reported for Eq. (5-194) through (5-197) were compiled by Reid et al., who compared the predictions with 68 experimental values of DAB. Errors cited for Eqs. (5-198) to (5-202) were reported by the authors. Binary Mixtures—Low Pressure—Nonpolar Components Many evaluations of correlations are available (Elliott and Watts; Lugg; Marrero and Mason). The differences in accuracy of the correlations are minor, and thus the major concern is ease of calculation. The Fuller-Schettler-Giddings equation is usually the simplest correlation to use and is recommended by Reid et al. Chapman-Enskog (Bird et al.) and Wilke-Lee The inherent assumptions of these equations are quite restrictive (i.e., low density,
Hypothetical film and boundary conditions.
DIFFUSIVITY ESTIMATION—GASES Whenever measured values of diffusivities are available, they should be used. Typically, measurement errors are less than those associated with predictions by empirical or even semitheoretical equations. A few general sources of data are Sec. 2 of this handbook, Schwartzberg and Chao; Reid et al.; Gammon et al.; and Daubert and Danner. Many other more restricted sources are listed under specific topics later in this subsection. TABLE 5-13
Rules of Thumb for Diffusivities (See Cussler, Reid et al., Schwartzberg and Chao) Di magnitude
Di range
m2/s
cm2/s
m2/s
Gas at atmospheric pressure
10−5
0.1
10−4–10−6
Liquid Liquid occluded in solid matrix Polymers and glasses
10−9 10−10 10−12
10−5 10−6 10−8
10−8–10−10 10−8–10−12 10−10–10−14
10−4–10−6 10−4–10−8 10−6–10−10
Solid
10−14
10−10
10−10–10−34
10−6–10−30
Continuous phase
TABLE 5-14
cm2/s 1–10−2
Comments Accurate theories exist, generally within ⫾10%; DiP constant; Di ∝ T1.66 to 2.0 Approximate correlations exist, generally within ⫾25% Hard cell walls: Deff /Di = 0.1 to 0.2. Soft cell walls: Deff /Di = 0.3 to 0.9 Approximate theories exist for dilute and concentrated limits; strong composition dependence Approximate theories exist; strong temperature dependence
Correlations of Diffusivities for Gases
Authors*
Equation
Error
1. Binary Mixtures—Low Pressure—Nonpolar Chapman-Enskog
1/2 0.001858T 3/2 MAB DAB = ᎏᎏ Pσ 2AB ΩD
(5-194)
7.3%
Wilke-Lee [65]
1/2 1/2 (0.0027 − 0.0005 MAB ) T 3/2MAB DAB = ᎏᎏᎏᎏ 2 PσAB ΩD
(5-195)
7.0%
Fuller-Schettler-Giddings [19]
1/2 0.001T1.75MAB DAB = ᎏᎏᎏ 2 P [( v)A1/3 + ( v)1/3 B ]
(5-196)
5.4%
(5-197)
9.0%
2. Binary Mixtures—Low Pressure—Polar Brokaw [4]
1/2 0.001858T 3/2MAB DAB = ᎏᎏ 2 Pσ ABΩD
3. Self-Diffusivity—High Pressure Mathur-Thodos [37]
10.7 × 10−5T DAA = ᎏᎏr {ρr ≤ 1.5} βρr
(5-198)
5%
Lee-Thodos [31]
0.77 × 10−5Tr DAA = ᎏᎏ {ρr ≤ 1} ρrδ
(5-199)
0.5%
Lee-Thodos [32]
(0.007094G + 0.001916)2.5Tr DAA = ᎏᎏᎏ , [ρr > 1, G < 1] δ
(5-200)
17%
4. Supercritical Mixtures Sun and Chen [56]
1.23 × 10−10T DAB = ᎏᎏ µ0.799VC0.49 A
(5-201)
5%
Catchpole and King [6]
(ρ−0.667 − 0.4510) (1 + MA/MB) R DAB = 5.152 DcTr ᎏᎏᎏᎏ (1 + (VcB /VcA)0.333)2
(5-202)
10%
*References are listed on pages 5-7 and 5-8.
MASS TRANSFER TABLE 5-15
5-49
Estimates for εi and σi (K, Å, atm, cm3, mol)
Critical point
ε/k = 0.75 Tc
1/3 σ = 0.841 V1/3 c or 2.44 (Tc/Pc)
Critical point
ε/k = 65.3 Tc zc3.6
1.866 V1/3 c σ = ᎏᎏ z1.2 c
Normal boiling point Melting point
ε/k = 1.15 Tb ε/k = 1.92 Tm
σ = 1.18 V1/3 b σ = 1.222 V1/3 m
Acentric factor
ε/k = (0.7915 + 0.1693 ω) Tc
T σ = (2.3551 − 0.087 ω) ᎏc Pc
1/3
NOTE: These values may not agree closely, so usage of a consistent basis is suggested (e.g., data at the normal boiling point).
spherical atoms), and the intrinsic potential function is empirical. Despite that, they provide good estimates of DAB for many polyatomic gases and gas mixtures, up to about 1000 K and a maximum of 70 atm. The latter constraint is because observations for many gases indicate that DABP is constant up to 70 atm. The characteristic length is σAB = (σA + σB)/2 in A˚. In order to estimate ΩD for Eqs. (5-194) or (5-195), two empirical equations are available. The first is: ΩD = (44.54T *−4.909 + 1.911T *−1.575)0.10 (5-203a)
the gas pairs [e.g., (CH3)2O & CH3Cl]. Despite that, Reid, op. cit., found the average error was 9.0 percent for combinations of mixtures (including several polar-nonpolar gas pairs), temperatures and pressures. In this equation, ΩD is calculated as described previously, and other terms are:
where T* = kT/εAB and εAB = (εA εB)1/2. Estimates for σi and εi are given in Table 5-15. This expression shows that ΩD is proportional to temperature roughly to the −0.49 power at low temperatures and to the −0.16 power at high temperature. Thus, gas diffusivities are proportional to temperatures to the 2.0 power and 1.66 power, respectively, at low and high temperatures. The second is: G A C E ΩD = ᎏ + ᎏᎏ + ᎏᎏ + ᎏᎏ (5-203b) T*B exp (DT*) exp (FT*) exp (HT*)
Self-Diffusivity—High Pressure The criterion of high pressure is vague at best. For most “permanent” gases, such as the major constituents of air, it would mean P > 70 atm. For less volatile components, the criterion would be lower. At present, accurate prediction of mutual diffusion coefficients for dense gas mixtures is not possible. One major reason for this is the scarcity of data. Most high-pressure diffusion experiments have measured the self-diffusion coefficient. The general observation is that the product DP is near constant at low pressure, is not constant at high pressure, but rather decreases as pressure increases. In addition, although there are usually negligible composition effects on diffusivity of gases at low pressures, the effects are not negligible at high pressures. Mathur-Thodos showed that for reduced densities less than unity, the product DAAρ is approximately constant at a given temperature. Thus, by knowing the value of the product at low pressure, it is possible to estimate its value at a higher pressure. They found at higher pressures the density increases, but the product DAAρ decreases rapidly. In their correlation, β = MA1/2PC1/3/T C5/6. Lee-Thodos presented a generalized treatment of self-diffusivity for gases (and liquids). These correlations have been tested for more than 500 data points each. The average deviation of the first is 0.51 percent, and that of the second is 17.2 percent. δ = MA1/2/ 2 5/6 0.1 P1/2 c V c , s/cm , and where G = (X* − X)/(X* − 1), X = ρr /T r , and X* = ρr /T r0.1 evaluated at the solid melting point. Lee and Thodos expanded their earlier treatment of self-diffusivity to cover 58 substances and 975 data points, with an average absolute deviation of 5.26 percent. Their correlation is too involved to repeat here, but those interested should refer to the original paper. Supercritical Mixtures Debenedetti-Reid showed that conventional correlations based on the Stokes-Einstein relation (for liquid phase) tend to overpredict diffusivities in the supercritical state. Nevertheless, they observed that the Stokes-Einstein group DABµ/T was constant. Thus, although no general correlation applies, only one data point is necessary to examine variations of fluid viscosity and/or temperature effects. They explored certain combinations of aromatic solids in SF6 and CO2. Sun-Chen examined tracer diffusion data of aromatic solutes in alcohols up to the supercritical range and found their data correlated with average deviations of 5 percent and a maximum deviation of 17 percent for their rather limited set of data. Catchpole-King examined binary diffusion data of near-critical fluids in the reduced density range of 1 to 2.5 and found that their data correlated with average deviations of 10 percent and a maximum deviation of 60 percent. They observed two classes of behavior. For the first, no correction factor was required (R = 1). That class was comprised of alcohols as solvents with aromatic or aliphatic solutes, or carbon dioxide as a solvent with aliphatics except ketones as solutes, or
where A = 1.06036, B = 0.15610, C = 0.1930, D = 0.47635, E = 1.03587, F = 1.52996, G = 1.76474, and H = 3.89411. Fuller-Schettler-Giddings The parameters and constants for this correlation were determined by regression analysis of 340 experimental diffusion coefficient values of 153 binary systems. Values of vi used in this equation are in Table 5-16. Binary Mixtures—Low Pressure—Polar Components The Brokaw correlation was based on the Chapman-Enskog equation, but σAB* and ΩD* were evaluated with a modified Stockmayer potential for polar molecules. Hence, slightly different symbols are used. That potential model reduces to the Lennard-Jones 6-12 potential for interactions between nonpolar molecules. As a result, the method should yield accurate predictions for polar as well as nonpolar gas mixtures. Brokaw presented data for 9 relatively polar pairs along with the prediction. The agreement was good: an average absolute error of 6.4 percent, considering the complexity of some of TABLE 5-16 Atomic Diffusion Volumes for Use in Estimating DAB by the Method of Fuller, Schettler, and Giddings Atomic and Structural Diffusion–Volume Increments, vi (cm3/mol) C H O (N)
16.5 1.98 5.48 5.69
(Cl) (S) Aromatic ring Heterocyclic ring
19.5 17.0 −20.2 −20.2
Diffusion Volumes for Simple Molecules, Σvi (cm3/mol) H2 D2 He N2 O2 Air Ar Kr (Xe) Ne
7.07 6.70 2.88 17.9 16.6 20.1 16.1 22.8 37.9 5.59
CO CO2 N2O NH3 H2O (CCl2F2) (SF5) (Cl2) (Br2) (SO2)
18.9 26.9 35.9 14.9 12.7 114.8 69.7 37.7 67.2 41.1
Parentheses indicate that the value listed is based on only a few data points.
ΩD* = ΩD + 0.19 δ 2AB/T* σAB* = (σA* σB*)1/2 δAB = (δA δB)1/2 εAB* = (εA*εB*)1/2
T* = kT/εAB* σi* = [1.585 Vbi /(1 + 1.3 δ2i )]1/3 δi = 1.94 × 103 µ2i /VbiTbi εi* /k = 1.18 (1 + 1.3 δ 2i )Tbi
5-50
HEAT AND MASS TRANSFER
ethylene as a solvent with aliphatics except ketones and naphthalene as solutes. For the second class, the correction factor was R = X 0.17. The class was comprised of carbon dioxide with aromatics; ketones and carbon tetrachloride as solutes; and aliphatics (propane, hexane, dimethyl butane), sulfur hexafluoride, and chlorotrifluoromethane as solvents with aromatics as solutes. In addition, sulfur hexafluoride combined with carbon tetrachloride, and chlorotrifluoromethane combined with 2-propanone were included in that class. In all cases, X = (1 + (VCB /VCA)1/3)2/(1 + MA /MB) was in the range of 1 to 10. Low-Pressure/Multicomponent Mixtures These methods are outlined in Table 5-17. Stefan-Maxwell equations were discussed earlier. Smith-Taylor compared various methods for predicting multicomponent diffusion rates and found that Eq. (5-204) was superior among the effective diffusivity approaches, though none is very good. They also found that linearized and exact solutions are roughly equivalent and accurate. Blanc provided a simple limiting case for dilute component i diffusing in a stagnant medium (i.e., N ≈ 0), and the result, Eq. (5-205), is known as Blanc’s law. The restriction basically means that the compositions of all the components, besides component i, are relatively large and uniform. Wilke obtained solutions to the Stefan-Maxwell equations. The first, Eq. (5-206), is simple and reliable under the same conditions as Blanc’s law. This equation applies when component i diffuses through a stagnant mixture. It has been tested and verified for diffusion of toluene in hydrogen + air + argon mixtures and for diffusion of ethyl propionate in hydrogen + air mixtures (Fairbanks and Wilke). When the compositions vary from one boundary to the other, Wilke recommends that the arithmetic average mole fractions be used. Wilke also suggested using the Stefan-Maxwell equation, which applies when the fluxes of two or more components are significant. In this situation, the mole fractions are arithmetic averages of the boundary conditions, and the solution requires iteration because the ratio of fluxes is not known a priori. DIFFUSIVITY ESTIMATION—LIQUIDS Many more correlations are available for diffusion coefficients in the liquid phase than for the gas phase. Most, however, are restricted to binary diffusion at infinite dilution D°AB or to self-diffusivity DA′A. This reflects the much greater complexity of liquids on a molecular level. For example, gas-phase diffusion exhibits negligible composition effects and deviations from thermodynamic ideality. Conversely, liquid-phase diffusion almost always involves volumetric and thermodynamic effects due to composition variations. For concentrations greater than a few mole percent of A and B, corrections are needed to obtain the true diffusivity. Furthermore, there are many conditions that do not fit any of the correlations presented here. Thus, careful consideration is needed to produce a reasonable estimate. Again, if diffusivity data are available at the conditions of interest, then they are strongly preferred over the predictions of any correlations. Stokes-Einstein and Free-Volume Theories The starting point for many correlations is the Stokes-Einstein equation. This equation is derived from continuum fluid mechanics and classical thermodynamics for the motion of large spherical particles in a liquid.
For this case, the need for a molecular theory is cleverly avoided. The Stokes-Einstein equation is (Bird et al.) kT (5-207) DAB = ᎏ 6πrAµB where A refers to the solute and B refers to the solvent. This equation is applicable to very large unhydrated molecules (M > 1000) in lowmolecular-weight solvents or where the molar volume of the solute is greater than 500 cm3/mol (Reddy and Doraiswamy; Wilke and Chang). Despite its intellectual appeal, this equation is seldom used “as is.” Rather, the following principles have been identified: (1) The diffusion coefficient is inversely proportional to the size rA VA1/3 of the solute molecules. Experimental observations, however, generally indicate that the exponent of the solute molar volume is larger than one-third. (2) The term DABµB /T is approximately constant only over a 10-to-15 K interval. Thus, the dependence of liquid diffusivity on properties and conditions does not generally obey the interactions implied by that grouping. For example, Robinson et al. found that: ln DAB ∝ −1/T. (3) Finally, pressure does not affect liquid-phase diffusivity much, since µB and VA are only weakly pressure-dependent. Pressure does have an impact at very high levels. Another advance in the concepts of liquid-phase diffusion was provided by Hildebrand, who adapted a theory of viscosity to selfdiffusivity. He postulated that DA′A = B(V − Vms)/Vms, where DA′A is the self-diffusion coefficient, V is the molar volume, and Vms is the molar volume at which fluidity is zero (i.e., the molar volume of the solid phase at the melting temperature). The difference (V − Vms) can be thought of as the free volume, which increases with temperature; and B is a proportionality constant. Ertl and Dullien [ibid.] found that Hildebrand’s equation could not fit their data with B as a constant. They modified it by applying an empirical exponent n (a constant greater than unity) to the volumetric ratio. The new equation is not generally useful, however, since there is no means for predicting n. The theory does identify the free volume as an important physical variable, since n > 1 for most liquids implies that diffusion is more strongly dependent on free volume than is viscosity. Dilute Binary Nonelectrolytes: General Mixtures These correlations are outlined in Table 5-18. Wilke-Chang This correlation for D°AB is one of the most widely used, and it is an empirical modification of the Stokes-Einstein equation. It is not very accurate, however, for water as the solute. Otherwise, it applies to diffusion of very dilute A in B. The average absolute error for 251 different systems is about 10 percent. φB is an association factor of solvent B that accounts for hydrogen bonding. Component B
φB
Water Methanol Ethanol Propanol Others
2.26 1.9 1.5 1.2 1.0
The value of φB for water was originally stated as 2.6, although when the original data were reanalyzed, the empirical best fit was 2.26.
TABLE 5-17 Relationships for Diffusivities of Multicomponent Gas Mixtures at Low Pressure Authors* Stefan-Maxwell, Smith and Taylor [53]
Blanc [13]
Wilke [63]
Equation
Dim = 1 − xi Dim = Dim =
NC
NC
j=1
xj
NC
*References are listed at the beginning of this subsection.
i
j=1
xiNi
j
ij
(5-204)
i
(5-205)
ij
j=1 j≠i
j
−1
ᎏ D j=1
NC
D N /N / x − ᎏ N /
xj ᎏ Dij
−1
(5-206)
MASS TRANSFER TABLE 5-18
5-51
Correlations for Diffusivities of Dilute, Binary Mixtures of Nonelectrolytes in Liquids
Authors*
Equation
Error
1. General Mixtures Wilke-Chang [64]
7.4 × 10−8 (φBMB)1/2 T D°AB = ᎏᎏᎏ µB VA0.6
(5-208)
20%
Tyn-Calus [59]
8.93 × 10−8 (VA/VB2 )1/6 (ψB/ψA)0.6 T D°AB = ᎏᎏᎏᎏ µB
(5-209)
10%
Umesi-Danner [60]
2.75 × 10−8 (RB/RA2/3) T D°AB = ᎏᎏᎏ µB
(5-210)
16%
Siddiqi-Lucas [52]
9.89 × 10−8 VB0.265 T D°AB = ᎏᎏᎏ V A0.45 µB0.907
(5-211)
13%
ᎏ V
(5-212)
18%
(5-213)
6%
2. Gases in Low Viscosity Liquids Sridhar-Potter [54]
Chen-Chen [7]
Vc D°AB = DBB ᎏB VcA
2/3
VB
mlB
(βVcB)2/3(RTcB)1/2 T (Vr − 1) ᎏ D°AB = 2.018 × 10−9 ᎏᎏ MA1/6 (MBVcA)1/3 TcB
1/2
3. Aqueous Solutions Hayduk-Laudie [25]
13.16 × 10−5 D°AW = ᎏᎏ µ w1.14 VA0.589
(5-214)
18%
Siddiqi-Lucas [52]
D°AW = 2.98 × 10−7 VA−0.5473 µ w−1.026 T
(5-215)
13%
Hayduk-Minhas [26]
A − 0.791) D°AB = 13.3 × 10−8 T1.47 µ(10.2/V VA−0.71 B
(5-216)
5%
Matthews-Akgerman [38]
D°AB = 32.88 MA−0.61VD−1.04 T 0.5 (VB − VD)
(5-217)
5%
Riazi-Whitson [48]
(ρDAB)° µ DAB = 1.07 ᎏ ᎏ ρ µ°
(5-218)
15%
4. Hydrocarbon Mixtures
−0.27 − 0.38 ω + (−0.05 + 0.1 ω)P r
*References are listed on pages 5-7 and 5-8.
Random comparisons of predictions with 2.26 versus 2.6 show no consistent advantage for either value, however. It has been suggested to replace the exponent of 0.6 with 0.7 and to use an association factor of 0.7 for systems containing aromatic hydrocarbons. These modifications, however, are not recommended by Umesi and Danner. Lees and Sarram present a comparison of the association parameters. The average absolute error for 87 different solutes in water is 5.9 percent. Tyn-Calus This correlation requires data in the form of molar volumes and parachors ψi = Viσ1/4 i (a property which, over moderate temperature ranges, is nearly constant), measured at the same temperature (not necessarily the temperature of interest). The parachors for the components may also be evaluated at different temperatures from each other. Quale has compiled values of ψi for many chemicals. Group contribution methods are available for estimation purposes (Reid et al.). The following suggestions were made by Reid et al.: The correlation is constrained to cases in which µB < 30 cP. If the solute is water or if the solute is an organic acid and the solvent is not water or a short-chain alcohol, dimerization of the solute A should be assumed for purposes of estimating its volume and parachor. For example, the appropriate values for water as solute at 25°C are VW = 37.4 cm3/mol and ψW = 105.2 cm3g1/4/s1/2mol. Finally, if the solute is nonpolar, the solvent volume and parachor should be multiplied by 8µB. Umesi-Danner They developed an equation for nonaqueous solvents with nonpolar and polar solutes. In all, 258 points were involved ˚ of the component in the regression. Ri is the radius of gyration in A molecule, which has been tabulated by Passut and Danner for 250 compounds. The average absolute deviation was 16 percent, compared with 26 percent for the Wilke-Chang equation.
Siddiqi-Lucas In an impressive empirical study, these authors examined 1275 organic liquid mixtures. Their equation yielded an average absolute deviation of 13.1 percent, which was less than that for the Wilke-Chang equation (17.8 percent). Note that this correlation does not encompass aqueous solutions; those were examined and a separate correlation was proposed, which is discussed later. Binary Mixtures of Gases in Low-Viscosity, Nonelectrolyte Liquids Sridhar-Potter derived an equation for predicting gas diffusion through liquid by combining existing correlations. Hildebrand had postulated the following dependence of the diffusivity for a gas in a liquid: D°AB = DB′B(VcB /VcA)2/3, where DB′B is the solvent self-diffusion coefficient and Vci is the critical volume of component i, respectively. To correct for minor changes in volumetric expansion, Sridhar and Potter multiplied the resulting equation by VB / VmlB, where VmlB is the molar volume of the liquid B at its melting point and DB′B can be estimated by the equation of Ertl and Dullien (see p. 5-50). Sridhar and Potter compared experimentally measured diffusion coefficients for twenty-seven data points of eleven binary mixtures. Their average absolute error was 13.5 percent, but Chen and Chen analyzed about 50 combinations of conditions and 3 to 4 replicates each and found an average error of 18 percent. This correlation does not apply to hydrogen and helium as solutes. However, it demonstrates the usefulness of self-diffusion as a means to assess mutual diffusivities and the value of observable physical property changes, such as molar expansion, to account for changes in conditions. Chen-Chen Their correlation was based on diffusion measurements of 50 combinations of conditions with 3 to 4 replicates each and exhibited an average error of 6 percent. In this correlation, Vr = VB / [0.9724 (VmlB + 0.04765)] and VmlB = the liquid molar volume at the
5-52
HEAT AND MASS TRANSFER
melting point, as discussed previously. Their association parameter β [which is different from the definition of that symbol in Eq. (5-219)] accounts for hydrogen bonding of the solvent. Values for acetonitrile and methanol are: β = 1.58 and 2.31, respectively. Dilute Binary Mixtures of a Nonelectrolyte in Water The correlations that were suggested previously for general mixtures, unless specified otherwise, may also be applied to diffusion of miscellaneous solutes in water. The following correlations are restricted to the present case, however. Hayduk-Laudie They presented a simple correlation for the infinite dilution diffusion coefficients of nonelectrolytes in water. It has about the same accuracy as the Wilke-Chang equation (about 5.9 percent). There is no explicit temperature dependence, but the 1.14 exponent on µw compensates for the absence of T in the numerator. That exponent was misprinted (as 1.4) in the original article and has been reproduced elsewhere erroneously. Siddiqi-Lucas These authors examined 658 aqueous liquid mixtures in an empirical study. They found an average absolute deviation of 19.7 percent. In contrast, the Wilke-Chang equation gave 35.0 percent and the Hayduk-Laudie correlation gave 30.4 percent. Dilute Binary Hydrocarbon Mixtures Hayduk-Minhas presented an accurate correlation for normal paraffin mixtures that was developed from 58 data points consisting of solutes from C5 to C32 and solvents from C5 to C16. The average error was 3.4 percent for the 58 mixtures. Matthews-Akgerman The free-volume approach of Hildebrand was shown to be valid for binary, dilute liquid paraffin mixtures (as well as self-diffusion), consisting of solutes from C8 to C16 and solvents of C6 and C12. The term they referred to as the “diffusion volume” was simply correlated with the critical volume, as VD = 0.308 Vc. We can infer from Table 5-15 that this is approximately related to the volume at the melting point as VD = 0.945 Vm. Their correlation was valid for diffusion of linear alkanes at temperatures up to 300°C and pressures up to 3.45 MPa. Matthews et al. and Erkey and Akgerman completed similar studies of diffusion of alkanes, restricted to n-hexadecane and n-octane, respectively, as the solvents. Riazi-Whitson They presented a generalized correlation in terms of viscosity and molar density that was applicable to both gases and liquids. The average absolute deviation for gases was only about 8 percent, while for liquids it was 15 percent. Their expression relies on the Chapman-Enskog correlation [Eq. (5-194)] for the low-pressure diffusivity and the Stiel-Thodos correlation for low-pressure viscosity: xAµ°A MA1/2 + xBµ°B MB1/2 µ° = ᎏᎏᎏ xAMA1/2 + xBMB1/2 i where µ°i ξi = 3.4 × 10−4 Tr0.94 for Tr i < 1.5 or µ°i ξi = 1.778 × 10−4 (4.58 2/3 i /Pc i Tr i − 1.67)5/8 for Tr i > 1.5. In these equations, ξi = Tc1/6 M1/2 i , and units are in cP, atm, K, and mol. For dense gases or liquids, the Chung et al. or Jossi-Stiel-Thodos correlation may be used to estimate viscosity. The latter is: (µ − µ°) ξ + 10−4 = (0.1023 + 0.023364 ρr + 0.058533 ρ 2r − 0.040758 ρ3r + 0.093324 ρ4r)4 where
(xA TcA + xBTcB)1/6 ξ = ᎏᎏᎏᎏ (xAMA + xBMB)1/2 (xA PcA + xB PcB)
and
ρr = (xA VcA + xB VcB)ρ.
Dilute Binary Mixtures of Nonelectrolytes with Water as the Solute Olander modified the Wilke-Chang equation to adapt it to the infinite dilution diffusivity of water as the solute. The modification he recommended is simply the division of the right-hand side of the Wilke-Chang equation by 2.3. Unfortunately, neither the WilkeChang equation nor that equation divided by 2.3 fit the data very well. A reasonably valid generalization is that the Wilke-Chang equation is accurate if water is very insoluble in the solvent, such as pure hydrocarbons, halogenated hydrocarbons, and nitro-hydrocarbons. On the other hand, the Wilke-Chang equation divided by 2.3 is accurate for solvents in which water is very soluble, as well as those that have low viscosities. Such solvents include alcohols, ketones, carboxylic acids,
and aldehydes. Neither equation is accurate for higher-viscosity liquids, especially diols. Dilute Dispersions of Macromolecules in Nonelectrolytes The Stokes-Einstein equation has already been presented. It was noted that its validity was restricted to large solutes, such as spherical macromolecules and particles in a continuum solvent. The equation has also been found to predict accurately the diffusion coefficient of spherical latex particles and globular proteins. Corrections to StokesEinstein for molecules approximating spheroids is given by Tanford. Since solute-solute interactions are ignored in this theory, it applies in the dilute range only. Hiss-Cussler Their basis is the diffusion of a small solute in a fairly viscous solvent of relatively large molecules, which is the opposite of the Stokes-Einstein assumptions. The large solvent molecules investigated were not polymers or gels but were of moderate molecular weight so that the macroscopic and microscopic viscosities were the same. The major conclusion is that D°AB µ2/3 = constant at a given temperature and for a solvent viscosity from 5 × 10−3 to 5 Pa s or greater (5 to 5 × 103 cP). This observation is useful if D°AB is known in a given high-viscosity liquid (oils, tars, etc.). Use of the usual relation of D°AB ∝ 1/µ for such an estimate could lead to large errors. Concentrated, Binary Mixtures of Nonelectrolytes Several correlations that predict the composition dependence of DAB are summarized in Table 5-19. Most are based on known values of D°AB and D°BA. In fact, a rule of thumb states that, for many binary systems, D°AB and D°BA bound the DAB vs. xA curve. Cullinan’s equation predicts diffusivities even in lieu of values at infinite dilution, but requires accurate density, viscosity, and activity coefficient data. Since the infinite dilution values D°AB and D°BA are generally unequal, even a thermodynamically ideal solution like γA = γB = 1 will exhibit concentration dependence of the diffusivity. In addition, nonideal solutions require a thermodynamic correction factor to retain the true “driving force” for molecular diffusion, or the gradient of the chemical potential rather than the composition gradient. That correction factor is: ∂ ln γA βA = 1 + ᎏ (5-219) ∂ ln xA Caldwell-Babb Darken observed that solid-state diffusion in metallurgical applications followed a simple relation. His equation related the tracer diffusivities and mole fractions to the mutual diffusivity: DAB = (xA DB + xB DA) βA
(5-220)
Caldwell and Babb used virtually the same equation to evaluate the mutual diffusivity for concentrated mixtures of common liquids. Van Geet and Adamson tested that equation for the n-dodecane (A) and n-octane (B) system and found the average deviation of DAB from experimental values to be −0.68 percent. In addition, that equation was tested for benzene + bromobenzene, n-hexane + n-dodecane, benzene + CCl 4, octane + decane, heptane + cetane, benzene + diphenyl, and benzene + nitromethane with success. For systems that depart significantly from thermodynamic ideality, it breaks down, sometimes by a factor of eight. For example, in the binary systems acetone + CCl 4, acetone + chloroform, and ethanol + CCl 4, it is not accurate. Thus, it can be expected to be fairly accurate for nonpolar hydrocarbons of similar molecular weight but not for polar-polar mixtures. Siddiqi et al. found that this relation was superior to those of Vignes and Leffler and Cullinan for a variety of mixtures. Umesi and Danner found an average absolute deviation of 13.9 percent for 198 data points. Rathbun-Babb suggested that Darken’s equation could be improved by raising the thermodynamic correction factor βA to a power, n, less than unity. They looked at systems exhibiting negative deviations from Raoult’s law and found n = 0.3. Furthermore, for polarnonpolar mixtures, they found n = 0.6. In a separate study, Siddiqi and Lucas followed those suggestions and found an average absolute error of 3.3 percent for nonpolar-nonpolar mixtures, 11.0 percent for polarnonpolar mixtures, and 14.6 percent for polar-polar mixtures. Siddiqi et al. examined a few other mixtures and found that n = 1 was probably best. Thus, this approach is, at best, highly dependent on the type of components being considered.
MASS TRANSFER
5-53
TABLE 5-19 Correlations of Diffusivities for Concentrated, Binary Mixtures of Nonelectrolyte Liquids Authors*
Equation
Caldwell-Babb [5] Rathbun-Babb [46] Vignes [62] Leffler-Cullinan [34]
DAB = (xA DBA ° + xB DAB ° )βA DAB = (xA DBA ° + xB DAB ° )βAn DAB = DAB ° xB DBA ° xA βA DAB µ mix = (DAB ° µ B)xB (DBA ° µA) xA βA
Cussler [12]
K ∂ ln xA DAB = D0 1 + ᎏ ᎏ −1 xA xB ∂ ln aA
Cullinan [10]
kT 2πxA xB βA DAB = ᎏᎏ ᎏᎏ 2πµ mix (V/A)1/3 1 + βA (2πxA xB − 1)
Asfour-Dullien [2]
DAB ° DAB = ᎏ µB
−1/2
(5-221) (5-222) (5-223) (5-224) (5-225)
1/2
ζµ β ᎏ µ xB
DBA °
xA
(5-227)
A
A
DAB = (CB V B D AB ° + CA V A D BA ° )βA
Siddiqi-Lucas [52]
(5-226)
(5-228)
Relative errors for the correlations in this table are very dependent on the components of interest and are cited in the text. *See pages 5-7 and 5-8 for references.
Vignes empirically correlated mixture diffusivity data for 12 binary mixtures. Later Ertl et al. evaluated 122 binary systems, which showed an average absolute deviation of only 7 percent. None of the latter systems, however, was very nonideal. Leffler-Cullinan modified Vignes’ equation using some theoretical arguments to arrive at Eq. (5-224), which the authors compared to Eq. (5-223) for the 12 systems mentioned above. The average absolute maximum deviation was only 6 percent. Umesi and Danner, however, found an average absolute deviation of 11.4 percent for 198 data points. For normal paraffins, it is not very accurate. In general, the accuracies of Eqs. (5-223) and (5-224) are not much different, and, since Vignes’ is simpler to use, it is suggested. The application of either should be limited to nonassociating systems that do not deviate much from ideality (0.95 < βA < 1.05). Cussler studied diffusion in concentrated associating systems and has shown that, in associating systems, it is the size of diffusing clusters rather than diffusing solutes that controls diffusion. Do is a reference diffusion coefficient discussed hereafter; aA is the activity of component A; and K is a constant. By assuming that Do could be predicted by Eq. (5-223) with β = 1, K was found to be equal to 0.5 based on five binary systems and validated with a sixth binary mixture. The limitations of Eq. (5-225) using Do and K defined previously have not been explored, so caution is warranted. Gurkan showed that K should actually be closer to 0.3 (rather than 0.5) and discussed the overall results. Cullinan presented an extension of Cussler’s cluster diffusion theory. His method accurately accounts for composition and temperature dependence of diffusivity. It is novel in that it contains no adjustable constants, and it relates transport properties and solution thermodynamics. This equation has been tested for six very different mixtures by Rollins and Knaebel, and it was found to agree remarkably well with data for most conditions, considering the absence of adjustable parameters. In the dilute region (of either A or B), there are systematic errors probably caused by the breakdown of certain implicit assumptions (that nevertheless appear to be generally valid at higher concentrations). Asfour-Dullien developed a relation for predicting alkane diffusivities at moderate concentrations that employs: Vfm 2/3 MxAMxB ᎏ ζ= ᎏ (5-229) Vf xAVf xB Mm
where Vfxi = V ; the fluid free volume is Vf i = Vi − Vml i for i = A, B, and m, in which Vml i is the molar volume of the liquid at the melting point and xA2 2 xA xB xB2 −1 Vmlm = ᎏ +ᎏ +ᎏ VmlA VmlAB VmlB xi fi
and
1/3 V1/3 mlA + V mlB VmlAB = ᎏᎏ 2
3
and µ is the mixture viscosity; Mm is the mixture mean molecular weight; and βA is defined by Eq. (5-219). The average absolute error of this equation is 1.4 percent, while the Vignes equation and the Leffler-Cullinan equation give 3.3 percent and 6.2 percent, respectively. Siddiqi-Lucas suggested that component volume fractions might be used to correlate the effects of concentration dependence. They found an average absolute deviation of 4.5 percent for nonpolarnonpolar mixtures, 16.5 percent for polar-nonpolar mixtures, and 10.8 percent for polar-polar mixtures. Binary Electrolyte Mixtures When electrolytes are added to a solvent, they dissociate to a certain degree. It would appear that the solution contains at least three components: solvent, anions, and cations. If the solution is to remain neutral in charge at each point (assuming the absence of any applied electric potential field), the anions and cations diffuse effectively as a single component, as for molecular diffusion. The diffusion of the anionic and cationic species in the solvent can thus be treated as a binary mixture. Nernst-Haskell The theory of dilute diffusion of salts is well developed and has been experimentally verified. For dilute solutions of a single salt, the well-known Nernst-Haskell equation (Reid et al.) is applicable: 1 1 1 1 ᎏᎏ + ᎏᎏ ᎏᎏ + ᎏᎏ n− n n RT n+ + − ᎏᎏ = 8.9304 × 10−10 T ᎏᎏ (5-230) D°AB = ᎏ 1 1 F 2 ᎏ1ᎏ + ᎏ1ᎏ ᎏᎏ0 + ᎏᎏ0 λ0+ λ0− λ+ λ−
where D°AB = diffusivity based on molarity rather than normality of dilute salt A in solvent B, cm2/s. The previous definitions can be interpreted in terms of ionicspecies diffusivities and conductivities. The latter are easily measured and depend on temperature and composition. For example, the equivalent conductance Λ is commonly tabulated in chemistry handbooks as the limiting (infinite dilution) conductance Λo and at standard concentrations, typically at 25°C. Λ = 1000 K/C = λ+ + λ− = Λo + f(C), (cm2/ohm gequiv); K = α/R = specific conductance, (ohm cm)−1; C = solution concentration, (gequiv/ᐉ); α = conductance cell constant (measured), (cm−1); R = solution electrical resistance, which is measured (ohm); and f(C) = a complicated function of concentration. The resulting equation of the electrolyte diffusivity is |z+| + |z−| DAB = ᎏᎏᎏ (5-231) (|z−| / D+) + (|z+| / D−) where |z⫾| represents the magnitude of the ionic charge and where the cationic or anionic diffusivities are D⫾ = 8.9304 × 10−10 T λ⫾ / |z⫾| cm2/s. The coefficient is kN0 /F 2 = R/F 2. In practice, the equivalent conductance of the ion pair of interest would be obtained and supplemented with conductances of permutations of those ions and one independent cation and anion. This would allow determination of all the ionic conductances and hence the diffusivity of the electrolyte solution.
5-54
HEAT AND MASS TRANSFER
Gordon Typically, as the concentration of a salt increases from infinite dilution, the diffusion coefficient decreases rapidly from D°AB. As concentration is increased further, however, DAB rises steadily, often becoming greater than D°AB. Gordon proposed the following empirical equation, which is applicable up to concentrations of 2N: ln γ⫾ 1 µB 1+ᎏ (5-232) DAB = D°AB ᎏ ᎏ CBVB µ ln m where D°AB is given by the Nernst-Haskell equation. References that tabulate γ⫾ as a function of m, as well as other equations for DAB, are given by Reid et al. Multicomponent Mixtures No simple, practical estimation methods have been developed for predicting multicomponent liquiddiffusion coefficients. Several theories have been developed, but the necessity for extensive activity data, pure component and mixture volumes, mixture viscosity data, and tracer and binary diffusion coefficients have significantly limited the utility of the theories (see Reid et al.). The generalized Stefan-Maxwell equations using binary diffusion coefficients are not easily applicable to liquids since the coefficients are so dependent on conditions. That is, in liquids, each Dij can be strongly composition dependent in binary mixtures and, moreover, the binary Dij is strongly affected in a multicomponent mixture. Thus, the convenience of writing multicomponent flux equations in terms of binary coefficients is lost. Conversely, they apply to gas mixtures because each Dij is practically independent of composition by itself and in a multicomponent mixture (see Taylor and Krishna for details). One particular case of multicomponent diffusion that has been examined is the dilute diffusion of a solute in a homogeneous mixture (e.g., of A in B + C). Umesi and Danner compared the three equations given below for 49 ternary systems. All three equations were equivalent, giving average absolute deviations of 25 percent. Perkins-Geankoplis
n
Dam µ m0.8 = xj D°A j µ0.8 j
(5-233)
j=1 j≠A
Cullinan This is an extension of Vignes’ equation to multicomponent systems:
applications of interest are outlined in Table 5-20. Applications of these equations are found in Secs. 16, 22, and 23. Diffusion within the largest cavities of a porous medium is assumed to be similar to ordinary or bulk diffusion except that it is hindered by the pore walls (see Eq. 5-236). The tortuosity τ that expresses this hindrance has been estimated from geometric arguments. Unfortunately, measured values are often an order of magnitude greater than those estimates. Thus, the effective diffusivity Deff (and hence τ) is normally determined by comparing a diffusion model to experimental measurements. The normal range of tortuosities for silica gel, alumina, and other porous solids is 2 ≤ τ ≤ 6, but for activated carbon, 5 ≤ τ ≤ 65. In small pores and at low pressures, the mean free path ᐉ of the gas molecule (or atom) is significantly greater than the pore diameter dpore. Its magnitude may be estimated from 3.2 µ RT 1/2 ᐉ = ᎏ ᎏ , in m P 2πM As a result, collisions with the wall occur more frequently than with other molecules. This is referred to as the Knudsen mode of diffusion and is contrasted with ordinary or bulk diffusion, which occurs by intermolecular collisions. At intermediate pressures, both ordinary diffusion and Knudsen diffusion may be important [see Eqs. (5-239) and (5-240)]. For gases and vapors that adsorb on the porous solid, surface diffusion may be important, particularly at high surface coverage [see Eqs. (5-241) and (5-244)]. The mechanism of surface diffusion may be viewed as molecules hopping from one surface site to another. Thus, if adsorption is too strong, surface diffusion is impeded, while if adsorption is too weak, surface diffusion contributes insignificantly to the overall rate. Surface diffusion and bulk diffusion usually occur in parallel [see Eqs. (5-245) and (5-246)]. Although Ds is expected to be less than Deff, the solute flux due to surface diffusion may be larger than that due to bulk diffusion if ∂qi / ∂z >> ∂Ci / ∂z. This can occur when a component is strongly adsorbed and the surface coverage is high. For all that, surface diffusion is not well understood. The references in Table 5-20 should be consulted for further details.
INTERPHASE MASS TRANSFER
DIFFUSION OF FLUIDS IN POROUS SOLIDS
Transfer of material between phases is important in most separation processes in which two phases are involved. When one phase is pure, mass transfer in the pure phase is not involved. For example, when a pure liquid is being evaporated into a gas, only the gas-phase mass transfer need be calculated. Occasionally, mass transfer in one of the two phases may be neglected even though pure components are not involved. This will be the case when the resistance to mass transfer is much larger in one phase than in the other. Understanding the nature and magnitudes of these resistances is one of the keys to performing reliable mass transfer. In this section, mass transfer between gas and liquid phases will be discussed. The principles are easily applied to the other phases. Mass-Transfer Principles: Dilute Systems When material is transferred from one phase to another across an interface that separates the two, the resistance to mass transfer in each phase causes a concentration gradient in each, as shown in Fig. 5-26 for a gas-liquid interface. The concentrations of the diffusing material in the two phases immediately adjacent to the interface generally are unequal, even if expressed in the same units, but usually are assumed to be related to each other by the laws of thermodynamic equilibrium. Thus, it is assumed that the thermodynamic equilibrium is reached at the gas-liquid interface almost immediately when a gas and a liquid are brought into contact. For systems in which the solute concentrations in the gas and liquid phases are dilute, the rate of transfer may be expressed by equations which predict that the rate of mass transfer is proportional to the difference between the bulk concentration and the concentration at the gas-liquid interface. Thus
Diffusion in porous solids is usually the most important factor controlling mass transfer in adsorption, ion exchange, drying, heterogeneous catalysis, leaching, and many other applications. Some of the
where NA = mass-transfer rate, k′G = gas-phase mass-transfer coefficient, k′L = liquid-phase mass-transfer coefficient, p = solute partial pressure in
Dam =
n
(D° ) Aj
xj
(5-234)
j=1 j≠A
Leffler-Cullinan They extended their binary relation to an arbitrary multicomponent mixture, as follows: Dam µm =
n
(D° µ ) Aj
j
xj
(5-235)
j=1 j≠A
where DAj is the dilute binary diffusion coefficient of A in j; DAm is the dilute diffusion of A through m; xj is the mole fraction; µj is the viscosity of component j; and µm is the mixture viscosity. Akita Another case of multicomponent dilute diffusion of significant practical interest is that of gases in aqueous electrolyte solutions. Many gas-absorption processes use electrolyte solutions. Akita presents experimentally tested equations for this case. Graham-Dranoff They studied multicomponent diffusion of electrolytes in ion exchangers. They found that the Stefan-Maxwell interaction coefficients reduce to limiting ion tracer diffusivities of each ion. Pinto-Graham Pinto and Graham studied multicomponent diffusion in electrolyte solutions. They focused on the Stefan-Maxwell equations and corrected for solvation effects. They achieved excellent results for 1-1 electrolytes in water at 25°C up to concentrations of 4M.
NA = k′G(p − pi) = k′L(ci − c)
(5-248)
MASS TRANSFER TABLE 5-20
5-55
Relations for Diffusion in Porous Solids
Mechanism
Equation
Bulk diffusion in pores
ε pD Deff = ᎏ τ
Knudsen diffusion
T DK = 48.5 dpore ᎏ M
Applies to
1/2
in m2/s
References*
(5-236)
Gases or liquids in large pores. NK n = ᐉ/d pore < 0.01
[67]
(5-237)
Dilute (low pressure) gases in small pores. NK n = ᐉ/d pore > 10
Geankoplis, [68, 69]
(5-238)
"
"
"
"
(5-239)
"
"
"
"
ε pDK DKeff = ᎏ τ dC Ni = −DK ᎏi dz Combined bulk and Knudsen diffusion
1 − α xA 1 Deff = ᎏ + ᎏ Deff DKeff
−1
NB α=1+ᎏ NA
1 1 Deff = ᎏ + ᎏ Deff DKeff Surface diffusion
−1
(5-240)
NA = NB
dq JSi = −DSeff ρp ᎏi dz
(5-241)
Adsorbed gases or vapors
ε p DS DSeff = ᎏ τ
(5-242)
"
DSθ = 0 DSθ = ᎏ (1 − θ)
(5-243)
θ = fractional surface coverage ≤ 0.6
(5-244)
"
"
"
"
(5-245)
"
"
"
"
(5-246)
"
"
"
"
(5-247)
"
"
"
"
−E DS = DS′ (q) exp ᎏS RT
Parallel bulk and surface diffusion
Geankoplis, [66, 69]
NA ≠ NB
dp dq J = − Deff ᎏi + DSeff ρp ᎏi dz dz
dp J = −Dapp ᎏi dz
dqi Dapp = Deff + DSeff ρp ᎏ dpi
"
"
[66, 68, 69]
"
[68]
*See pages 5-7 and 5-8 for references.
bulk gas, pi = solute partial pressure at interface, c = solute concentration in bulk liquid, and ci = solute concentration in liquid at interface. The mass-transfer coefficients k′G and k′L by definition are equal to the ratios of the molal mass flux NA to the concentration driving forces (p − pi) and (ci − c) respectively. An alternative expression for the rate of transfer in dilute systems is given by NA = kG(y − yi) = kL(xi − x) (5-249)
where NA = mass-transfer rate, kG = gas-phase mass-transfer coefficient, kL = liquid-phase mass-transfer coefficient, y = mole-fraction solute in bulk-gas phase, yi = mole-fraction solute in gas at interface, x = mole-fraction solute in bulk-liquid phase, and xi = mole-fraction solute in liquid at interface. The mass-transfer coefficients defined by Eqs. (5-248) and (5-249) are related to each other as follows: kG = k′G pT
(5-250)
kL = k′L ρ (5-251) L where pT = total system pressure employed during the experimental determinations of k′G values and ρ L = average molar density of the liquid phase. The coefficient kG is relatively independent of the total system pressure and therefore is more convenient to use than k′G, which is inversely proportional to the total system pressure. The above equations may be used for finding the interfacial concentrations corresponding to any set of values of x and y provided the ratio of the individual coefficients is known. Thus (y − yi)/(xi − x) = kL /kG = k′Lρ L/k′G pT = LMHG /GMHL
FIG. 5-26
Concentration gradients near a gas-liquid interface.
(5-252)
where LM = molar liquid mass velocity, GM = molar gas mass velocity, HL = height of one transfer unit based on liquid-phase resistance, and HG = height of one transfer unit based on gas-phase resistance. The last term in Eq. (5-252) is derived from Eqs. (5-271) and (5-273). Equation (5-252) may be solved graphically if a plot is made of the equilibrium vapor and liquid compositions and a point representing
5-56
HEAT AND MASS TRANSFER
the bulk concentrations x and y is located on this diagram. A construction of this type is shown in Fig. 5-27, which represents a gasabsorption situation. The interfacial mole fractions yi and xi can be determined by solving Eq. (5-252) simultaneously with the equilibrium relation y°i = F(xi) to obtain yi and xi. The rate of transfer may then be calculated from Eq. (5-249). If the equilibrium relation y°i = F(xi) is sufficiently simple, e.g., if a plot of y°i versus xi is a straight line, not necessarily through the origin, the rate of transfer is proportional to the difference between the bulk concentration in one phase and the concentration (in that same phase) which would be in equilibrium with the bulk concentration in the second phase. One such difference is y − y°, and another is x° − x. In this case, there is no need to solve for the interfacial compositions, as may be seen from the following derivation. The rate of mass transfer may be defined by the equation NA = KG(y − y°) = kG(y − yi) = kL(xi − x) = KL(x° − x) (5-253) where KG = overall gas-phase mass-transfer coefficient, KL = overall liquid-phase mass-transfer coefficient, y° = vapor composition in equilibrium with x, and x° = liquid composition in equilibrium with vapor of composition y. This equation can be rearranged to the formula 1 1 y − y° 1 1 yi − y° 1 1 yi − y° ᎏ=ᎏ ᎏ =ᎏ+ᎏ ᎏ =ᎏ+ᎏ ᎏ KG kG y − yi kG kG y − yi kG kL xi − x
(5-254) in view of Eq. (5-252). Comparison of the last term in parentheses with the diagram of Fig. 5-27 shows that it is equal to the slope of the chord connecting the points (x,y°) and (xi,yi). If the equilibrium curve is a straight line, then this term is the slope m. Thus 1/KG = (1/ kG + m/ kL) (5-255) When Henry’s law is valid (pA = HxA or pA = H′CA), the slope m can be computed according to the relationship m = H/ pT = H′ρ (5-256) L/ pT where m is defined in terms of mole-fraction driving forces compatible with Eqs. (5-249) through (5-255), i.e., with the definitions of kL, kG, and KG. If it is desired to calculate the rate of transfer from the overall concentration difference based on bulk-liquid compositions (x° − x), the appropriate overall coefficient KL is related to the individual coefficients by the equation 1/KL = (1/kL + 1/mkG) (5-257) Conversion of these equations to a k′G, k′L basis can be accomplished readily by direct substitution of Eqs. (5-250) and (5-251). Occasionally one will find k′L or K′L values reported in units (SI) of meters per second. The correct units for these values are kmol/
[(s⋅m2)(kmol/m3)], and Eq. (5-251) is the correct equation for converting them to a mole-fraction basis. When k′G and K′G values are reported in units (SI) of kmol/[(s⋅m2) (kPa)], one must be careful in converting them to a mole-fraction basis to multiply by the total pressure actually employed in the original experiments and not by the total pressure of the system to be designed. This conversion is valid for systems in which Dalton’s law of partial pressures (p = ypT) is valid. Comparison of Eqs. (5-255) and (5-257) shows that for systems in which the equilibrium line is straight, the overall mass transfer coefficients are related to each other by the equation KL = mKG
(5-258)
When the equilibrium curve is not straight, there is no strictly logical basis for the use of an overall transfer coefficient, since the value of m will be a function of position in the apparatus, as can be seen from Fig. 5-27. In such cases the rate of transfer must be calculated by solving for the interfacial compositions as described above. Experimentally observed rates of mass transfer often are expressed in terms of overall transfer coefficients even when the equilibrium lines are curved. This procedure is empirical, since the theory indicates that in such cases the rates of transfer may not vary in direct proportion to the overall bulk concentration differences (y − y°) and (x° − x) at all concentration levels even though the rates may be proportional to the concentration difference in each phase taken separately, i.e., (xi − x) and (y − yi). In most types of separation equipment such as packed or spray towers, the interfacial area that is effective for mass transfer cannot be accurately determined. For this reason it is customary to report experimentally observed rates of transfer in terms of transfer coefficients based on a unit volume of the apparatus rather than on a unit of interfacial area. Such volumetric coefficients are designated as KGa, kLa, etc., where a represents the interfacial area per unit volume of the apparatus. Experimentally observed variations in the values of these volumetric coefficients with variations in flow rates, type of packing, etc., may be due as much to changes in the effective value of a as to changes in k. Calculation of the overall coefficients from the individual volumetric coefficients is made by means of the equations 1/KGa = (1/kGa + m/kLa)
(5-259)
1/KLa = (1/kLa + 1/mkGa)
(5-260)
Because of the wide variation in equilibrium, the variation in the values of m from one system to another can have an important effect on the overall coefficient and on the selection of the type of equipment to use. For example, if m is large, the liquid-phase part of the overall resistance might be extremely large where kL might be relatively small. This kind of reasoning must be applied with caution, however, since species with different equilibrium characteristics are separated under different operating conditions. Thus, the effect of changes in m on the overall resistance to mass transfer may partly be counterbalanced by changes in the individual specific resistances as the flow rates are changed. Mass-Transfer Principles: Concentrated Systems When solute concentrations in the gas and/or liquid phases are large, the equations derived above for dilute systems no longer are applicable. The correct equations to use for concentrated systems are as follows: NA = kˆ G(y − yi)/yBM = kˆ L(xi − x)/xBM = Kˆ G(y − y°)/y°BM = Kˆ L(x° − x)/x°BM
(5-261)
where (NB = 0)
Identification of concentrations at a point in a countercurrent absorption tower.
FIG. 5-27
(1 − y) − (1 − yi) yBM = ᎏᎏ ln [(1 − y)/(1 − yi)] (1 − y) − (1 − y°) y°BM = ᎏᎏᎏ ln [(1 − y)/(1 − y°)] (1 − x) − (1 − xi) xBM = ᎏᎏ ln [(1 − x)/(1 − xi)] (1 − x) − (1 − x°) x°BM = ᎏᎏ ln [(1 − x)/(1 − x°)]
(5-262) (5-263) (5-264) (5-265)
MASS TRANSFER and where kˆ G and kˆ L are the gas-phase and liquid-phase mass-transfer coefficients for concentrated systems and Kˆ G and Kˆ L are the overall gas-phase and liquid-phase mass-transfer coefficients for concentrated systems. These coefficients are defined later in Eqs. (5-268) to (5-270). The factors yBM and xBM arise from the fact that, in the diffusion of a solute through a second stationary layer of insoluble fluid, the resistance to diffusion varies in proportion to the concentration of the insoluble stationary fluid, approaching zero as the concentration of the insoluble fluid approaches zero. See Eq. (5-190). The factors y°BM and x°BM cannot be justified on the basis of masstransfer theory since they are based on overall resistances. These factors therefore are included in the equations by analogy with the corresponding film equations. In dilute systems the logarithmic-mean insoluble-gas and nonvolatileliquid concentrations approach unity, and Eq. (5-261) reduces to the dilute-system formula. For equimolar counter diffusion (e.g., binary distillation), these log-mean factors should be omitted. See Eq. (5-189). Substitution of Eqs. (5-262) through (5-265) into Eq. (5-261) results in the following simplified formula: NA = kˆ G ln [(1 − yi)/(1 − y)] = Kˆ G ln [(1 − y°)/(1 − y)] = kˆ L ln [(1 − x)/(1 − xi)] (5-266) = Kˆ L ln [(1 − x)/(1 − x°)] ˆ ˆ ˆ ˆ Note that the units of kG, KG, kL, and KL are all identical to each other, i.e., kmol/(s⋅m2) in SI units. The equation for computing the interfacial gas and liquid compositions in concentrated systems is (y − yi)/(xi − x) = kˆ LyBM / kˆ GxBM = LMHGyBM /GMHL x BM = kL /kG
(5-267) This equation is identical to the one for dilute systems since kˆ G = kGyBM and kˆ L = kLxBM. Note, however, that when kˆ G and kˆ L are given, the equation must be solved by trial and error, since xBM contains xi and yBM contains yi. The overall gas-phase and liquid-phase mass-transfer coefficients for concentrated systems are computed according to the following equations: 1 yBM 1 xBM 1 yi − y° =ᎏ (5-268) ᎏ ᎏ+ᎏᎏ ᎏ Kˆ G y°BM kˆ G y°BM kˆ L xi − x
1 xBM 1 yBM 1 x° − xi (5-269) =ᎏᎏ+ᎏᎏ ᎏ ᎏ Kˆ L x°BM kˆ L x°BM kˆ G y − yi When the equilibrium curve is a straight line, the terms in parentheses can be replaced by the slope m as before. In this case the overall mass-transfer coefficients for concentrated systems are related to each other by the equation Kˆ L = m Kˆ G(x°BM /y°BM) (5-270)
All these equations reduce to their dilute-system equivalents as the inert concentrations approach unity in terms of mole fractions of inert concentrations in the fluids. HTU (Height Equivalent to One Transfer Unit) Frequently the values of the individual coefficients of mass transfer are so strongly dependent on flow rates that the quantity obtained by dividing each coefficient by the flow rate of the phase to which it applies is more nearly constant than the coefficient itself. The quantity obtained by this procedure is called the height equivalent to one transfer unit, since it expresses in terms of a single length dimension the height of apparatus required to accomplish a separation of standard difficulty. The following relations between the transfer coefficients and the values of HTU apply: H = G /k ay = G /kˆ a (5-271) G
M
G
BM
M
G
HOG = GM /KGay°BM = GM/Kˆ Ga H = L /k ax = L /kˆ a
(5-272)
HOL = LM /KLax°BM = LM/Kˆ La
(5-274)
L
M
L
BM
M
L
(5-273)
5-57
The equations that express the addition of individual resistances in terms of HTUs, applicable to either dilute or concentrated systems, are yBM mGM xBM HOG = ᎏ HG + ᎏ (5-275) ᎏ HL y°BM LM y°BM xBM LM yBM (5-276) HOL = ᎏ HL + ᎏ ᎏ HG x°BM mGM x°BM These equations are strictly valid only when m, the slope of the equilibrium curve, is constant, as noted previously. NTU (Number of Transfer Units) The NTU required for a given separation is closely related to the number of theoretical stages or plates required to carry out the same separation in a stagewise or plate-type apparatus. For equimolal counterdiffusion, such as in a binary distillation, the number of overall gas-phase transfer units NOG required for changing the composition of the vapor stream from y1 to y2 is y1 dy (5-277) NOG = ᎏ y2 y − y° When diffusion is in one direction only, as in the absorption of a soluble component from an insoluble gas, y1 y°BM dy (5-278) NOG = ᎏᎏ y2 (1 − y)(y − y°) The total height of packing required is then
hT = HOGNOG
(5-279)
When it is known that HOG varies appreciably within the tower, this term must be placed inside the integral in Eqs. (5-277) and (5-278) for accurate calculations of hT. For example, the packed-tower design equation in terms of the overall gas-phase mass-transfer coefficient for absorption would be expressed as follows: y1 y°BM dy GM hT = (5-280) ᎏ ᎏᎏ y2 KGay°BM (1 − y)(y − y°) where the first term under the integral can be recognized as the HTU term. Convenient solutions of these equations for special cases are discussed later. Definitions of Mass-Transfer Coefficients kˆ G and kˆ L The mass-transfer coefficient is defined as the ratio of the molal mass flux NA to the concentration driving force. This leads to many different ways of defining these coefficients. For example, gas-phase masstransfer rates may be defined as N = k (y − y ) = k′ (p − p ) = kˆ (y − y )/y (5-281)
A
G
i
G
i
G
i
BM
where the units (SI) of kG are kmol/[(s⋅m2)(mole fraction)], the units of k′G are kmol/[(s⋅m2)(kPa)], and the units of kˆ G are kmol/(s⋅m2). These coefficients are related to each other as follows: kG = kGyBM = k′G pT yBM
(5-282)
where pT is the total system pressure (it is assumed here that Dalton’s law of partial pressures is valid). In a similar way, liquid-phase mass-transfer rates may be defined by the relations N = k (x − x) = k′ (c − c) = kˆ (x − x)/x (5-283) A
L
i
L
i
L
i
BM
where the units (SI) of kL are kmol/[(s⋅m2)(mole fraction)], the units of k′L are kmol/[(s⋅m2)(kmol/m3)] or meters per second, and the units of kˆ L are kmol/(s⋅m2). These coefficients are related as follows: (5-284) kˆ L = kLxBM = k′Lρ LxBM is the molar density of the liquid phase in units (SI) of kilowhere ρ L moles per cubic meter. Note that, for dilute solutions where xBM ⬟ 1, kL and kˆ L will have identical numerical values. Similarly, for dilute gases kˆ G ⬟ kG. Simplified Mass-Transfer Theories In certain simple situations, the mass-transfer coefficients can be calculated from first principles. The film, penetration, and surface-renewal theories are attempts to extend these theoretical calculations to more complex sit-
5-58
HEAT AND MASS TRANSFER
uations. Although these theories are often not accurate, they are useful to provide a physical picture for variations in the mass-transfer coefficient. For the special case of steady-state unidirectional diffusion of a component through an inert-gas film in an ideal-gas system, the rate of mass transfer is derived as 1 − yi DABpT (y − yi) DABpT NA = ᎏ (5-285) ᎏ = ᎏ ln ᎏ RT δG yBM RT δG 1−y where DAB = the diffusion coefficient or “diffusivity,” δG = the “effective” thickness of a stagnant-gas layer which would offer a resistance to molecular diffusion equal to the experimentally observed resistance, and R = the gas constant. [Nernst, Z. Phys. Chem., 47, 52 (1904); Whitman, Chem. Mat. Eng., 29, 149 (1923), and Lewis and Whitman, Ind. Eng. Chem., 16, 1215 (1924)]. The film thickness δG depends primarily on the hydrodynamics of the system and hence on the Reynolds number and the Schmidt number. Thus, various correlations have been developed for different geometries in terms of the following dimensionless variables: NSh = kˆ GRTd/DABpT = f(NRe,NSc) (5-286) where NSh is the Sherwood number, NRe (= Gd/µG) is the Reynolds number based on the characteristic length d appropriate to the geometry of the particular system; and NSc (= µG /ρGDAB) is the Schmidt number. According to this analysis one can see that for gas-absorption problems, which often exhibit unidirectional diffusion, the most appropriate driving-force expression is of the form (y − yi)/yBM, and the most appropriate mass-transfer coefficient is therefore kˆ G. This concept is to be found in all the key equations for the design of mass-transfer equipment. The Sherwood-number relation for gas-phase mass-transfer coefficients as represented by the film diffusion model in Eq. (5-286) can be rearranged as follows: NSh = (kˆ G /GM)NReNSc = NStNReNSc = f (NRe,NSc) (5-287) where NSt = kˆ G /GM = k′G pBM /GM is known as the Stanton number. This equation can now be stated in the alternative functional forms NSt = kˆ G /GM = g(NRe,NSc) (5-288) 2 /3 (5-289) jD = NSt ⋅ NSc where j is the Chilton-Colburn “j factor” for mass transfer (discussed later). The important point to note here is that the gas-phase masstransfer coefficient kˆ G depends principally upon the transport properties of the fluid (NSc) and the hydrodynamics of the particular system involved (NRe). It also is important to recognize that specific masstransfer correlations can be derived only in conjunction with the investigator’s particular assumptions concerning the numerical values of the effective interfacial area a of the packing. The stagnant-film model discussed previously assumes a steady state in which the local flux across each element of area is constant; i.e., there is no accumulation of the diffusing species within the film. Higbie [Trans. Am. Inst. Chem. Eng., 31, 365 (1935)] pointed out that industrial contactors often operate with repeated brief contacts between phases in which the contact times are too short for the steady state to be achieved. For example, Higbie advanced the theory that in a packed tower the liquid flows across each packing piece in laminar flow and is remixed at the points of discontinuity between the packing elements. Thus, a fresh liquid surface is formed at the top of each piece, and as it moves downward, it absorbs gas at a decreasing rate until it is mixed at the next discontinuity. This is the basis of penetration theory. If the velocity of the flowing stream is uniform over a very deep region of liquid (total thickness, δT >> Dt), the time-averaged masstransfer coefficient according to penetration theory is given by k′L = 2 D πt (5-290) L/ where k′L = liquid-phase mass-transfer coefficient, DL = liquid-phase diffusion coefficient, and t = contact time. In practice, the contact time t is not known except in special cases in which the hydrodynamics are clearly defined. This is somewhat
similar to the case of the stagnant-film theory in which the unknown quantity is the thickness of the stagnant layer δ (in film theory, the liquid-phase mass-transfer coefficient is given by k′L = DL /δ). The penetration theory predicts that k′L should vary by the square root of the molecular diffusivity, as compared with film theory, which predicts a first-power dependency on D. Various investigators have reported experimental powers of D ranging from 0.5 to 0.75, and the Chilton-Colburn analogy suggests a w power. Penetration theory often is used in analyzing absorption with chemical reaction because it makes no assumption about the depths of penetration of the various reacting species, and it gives a more accurate result when the diffusion coefficients of the reacting species are not equal. When the reaction process is very complex, however, penetration theory is more difficult to use than film theory, and the latter method normally is preferred. Danckwerts [Ind. Eng. Chem., 42, 1460 (1951)] proposed an extension of the penetration theory, called the surface renewal theory, which allows for the eddy motion in the liquid to bring masses of fresh liquid continually from the interior to the surface, where they are exposed to the gas for finite lengths of time before being replaced. In his development, Danckwerts assumed that every element of fluid has an equal chance of being replaced regardless of its age. The Danckwerts model gives
s k′L = D
(5-291)
where s = fractional rate of surface renewal. Note that both the penetration and the surface-renewal theories predict a square-root dependency on D. Also, it should be recognized that values of the surface-renewal rate s generally are not available, which presents the same problems as do δ and t in the film and penetration models. The predictions of correlations based on the film model often are nearly identical to predictions based on the penetration and surfacerenewal models. Thus, in view of its relative simplicity, the film model normally is preferred for purposes of discussion or calculation. It should be noted that none of these theoretical models has proved adequate for making a priori predictions of mass-transfer rates in packed towers, and therefore empirical correlations such as those outlined later in Table 5-28. must be employed. Mass-Transfer Correlations Because of the tremendous importance of mass transfer in chemical engineering, a very large number of studies have determined mass-transfer coefficients both empirically and theoretically. Some of these studies are summarized in Tables 5-21 to 5-28. Each table is for a specific geometry or type of contactor, starting with flat plates, which have the simplest geometry (Table 5-21); then wetted wall columns (Table 5-22); flow in pipes and ducts (Table 5-23); submerged objects (Table 5-24); drops and bubbles (Table 5-25); agitated systems (Table 5-26); packed beds of particles for adsorption, ion exchange, and chemical reaction (Table 5-27); and finishing with packed bed two-phase contactors for distillation, absorption and other unit operations (Table 5-28). Graphical correlations for the Bolles and Fair correlation (Table 5-28-G) are in Figs. 5-28 to 5-30. Although extensive, these tables are not meant to be encyclopedic. For simple geometries, one may be able to determine a theoretical (T) form of the mass-transfer correlation. For very complex geometries, only an empirical (E) form can be found. In systems of intermediate complexity, semiempirical (S) correlations where the form is determined from theory and the coefficients from experiment are often useful. Although the major limitations and constraints in use are usually included in the tables, obviously many details cannot be included in this summary form. Readers are strongly encouraged to check the references before using the correlations in important situations. Note that even authoritative sources occasionally have typographical errors in the fairly complex correlation equations. Thus, it is a good idea to check several sources, including the original paper. The references will often include figures comparing the correlations with data. These figures are very useful since they provide a visual picture of the scatter in the data. Since there are often several correlations that are applicable, how does one choose the correlation to use? First, the engineer must determine which correlations are closest to the current situation. This
MASS TRANSFER TABLE 5-21
5-59
Mass Transfer Correlations for a Single Flat Plate or Disk—Transfer to or from Plate to Fluid Situation
A. Laminar, local, flat plate, forced flow
Laminar, average, flat plate, forced flow
Comments E = Empirical, S = Semiempirical, T = Theoretical
Correlation k′x NSh,x = ᎏ = 0.323(NRe,x)1/2(NSc)1/3 D Coefficient 0.332 is a better fit.
B. Laminar, local, flat plate, blowing or suction and forced flow
C. Laminar, local, flat plate, natural convection vertical plate
[T] Low M.T. rates. Low mass-flux, constant property systems. NSh,x is local k. Use with arithmetic difference in concentration. Coefficient 0.323 is Blasius’ approximate solution.
[100] p. 183 [108] p. 526 [146] p. 79 [150] p. 518
xu ∞ ρ NRe,x = ᎏ , x = length along plate µ
[151] p. 110
Lu ∞ ρ NRe,L = ᎏ , 0.664 (Polhausen) µ
k′m L NSh,avg = ᎏ = 0.646(NRe,L)1/2(NSc)1/3 D k′m is mean mass-transfer coefficient for dilute systems.
j-factors
References*
f jD = jH = ᎏ = 0.664(NRe,L)−1/2 2 k′x NSh,x = ᎏ = (Slope)y = 0 (NRe,x)1/2(NSc)1/3 D
k′x −1/4 1/4 NSh,x = ᎏ = 0.508 N1/2 N Gr Sc (0.952 + NSc) D
is a better fit for NSc > 0.6, NRe,x < 3 × 105. [S] Analogy. Nsc = 1.0, f = drag coefficient. jD is defined in terms of k′m.
[151] p. 271
[T] Blowing is positive. Other conditions as above. uo Re ,x ᎏᎏ N u∞ 0.6 0.5 0.25 0.0 −2.5 ᎏᎏ ᎏᎏᎏ (Slope)y = 0 0.01 0.06 0.17 0.332 1.64
[100] p. 185
[T] Low MT rates. Dilute systems, ∆ρ/ρ << 1. NGr NSc < 108. Use with arithmetic concentration difference. x = length from plate bottom.
[151] p. 120
[150] p. 271
gx 3 ρ∞ NGr = ᎏ2 ᎏ −1 (µ/ρ) ρ0
D. Laminar, stationary disk
Laminar, spinning disk
E. Laminar, inclined, plate
k′d disk 8 NSh = ᎏ = ᎏ D π k′d disk 1/3 NSh = ᎏ = 0.879N 1/2 Re N Sc D
3 2 1/3 x ρ g sinα NSh,avg = 0.783 N1/9 ᎏᎏ Re,film N Sc µ2
[T] Stagnant fluid. Use arithmetic concentration difference.
[146] p. 240
[T] Asymptotic solution for large NSc. NRe < ∼ 104 u = ωddisk/2, ω = rotational speed, rad/s. Rotating disks are often used in electrochemical research.
[117] p. 60
[T] Constant-property liquid film with low masstransfer rates. Use arithmetic concentration difference.
[151] p. 130
[146] p. 240
2/9
4Qρ NRe,film = ᎏ < 2000 µ2 k′m x NSh,avg = ᎏ D
3µQ δ film = ᎏᎏ wρg sinα
F. Turbulent, local flat plate, forced flow
Turbulent, average, flat plate, forced flow
w = width of plate, δ f = film thickness, α = angle of inclination, x = distance from start soluble surface.
1/3
= film thickness
k′x 0.8 NSh,x = ᎏ = 0.0292N Re,x , D k′L 0.8 , average coefficient NSh,avg = ᎏ = 0.0365N Re,L D
Newtonian fluid. Solute does not penetrate past region of linear velocity profile. Differences between theory and experiment.
[146] p. 209
[S] Low mass-flux with constant property system. Use with arithmetic concentration difference. NSc = 1.0, NRe,x > 105
[100] p. 191 [146] p. 201 [151] p. 221
Based on Prandtl’s 1/7-power velocity law,
y u ᎏ= ᎏ u∞ δ G. Laminar and turbulent, flat plate, forced flow
f −0.2 jD = jH = ᎏ = 0.037 N Re,L 2
1/7
Chilton-Colburn analogies, NSc = 1.0, (gases), f = drag coefficient. Corresponds to item 5-21-F and refers to same conditions. 8000 < NRe < 300,000. Can apply analogy, jD = f/2, to entire plate (including laminar portion) if average values are used.
[100] p. 193 [109] p. 112 [146] p. 201 [151] p. 271
5-60
HEAT AND MASS TRANSFER
TABLE 5-21
Mass Transfer Correlations for a Single Flat Plate or Disk—Transfer to or from Plate to Fluid (Concluded ) Situation
H. Laminar and turbulent, flat plate, forced flow
Comments E = Empirical, S = Semiempirical, T = Theoretical
Correlation 0.8 NSh,avg = 0.037N 1/3 Sc (N Re,L − 15,500) to NRe,L = 320,000
k′m L NSh,avg = ᎏ , NSc > 0.5 D Entrance effects are ignored. NRe,Cr is transition laminar to turbulent.
NSh,avg = 0.037N 1/3 Sc
References*
[E] Use arithmetic concentration difference.
0.664 1/2 0.8 0.8 × N Re,L − N Re,Cr + ᎏ N Re,Cr 0.037
[109] p. 112 [146] p. 201
in range 3 × 105 to 3 × 106. I. Turbulent, local flat plate, natural convection, vertical plate Turbulent, average, flat plate, natural convection, vertical plate
k′x 2/5 7/15 NSh,x = ᎏ = 0.0299N Gr N Sc D 2/3 −2/5 × (1 + 0.494N Sc ) 2/5 7/15 2/3 −2/5 NSh,avg = 0.0249N Gr N Sc × (1 + 0.494N Sc )
[S] Low solute concentration and low transfer rates. Use arithmetic concentration difference. gx3 ρ∞ NGr = ᎏ2 ᎏ −1 (µ/ρ) ρ0
[151] p. 225
NGr > 1010 Assumes laminar boundary layer is small fraction of total. k′m L NSh,avg = ᎏ D
J. Turbulent, vertical plate
k′m x x 3ρ 2g 2/9 NSh,avg = ᎏ = 0.327N Re,film N 1/3 Sc ᎏ µ2 D
Q2 δ film = 0.172 ᎏ w 2g K. Turbulent, spinning disk
L. Mass transfer to a flat plate membrane in a stirred vessel
2/9
[E] See 5-21-E for terms.
[151] p. 229
4Qρ NRe,film = ᎏ > 2360 wµ2 Solute remains in laminar sublayer.
1/3
k ′ddisk 1.1 1/3 NSh = ᎏ = 5.6N Re N Sc D
k′dtank b c NSh = ᎏ = aN Re N Sc D a depends on system. a = 0.0443 [73, 165]; b is often 0.65–0.70 [110]. If
[E] Use arithmetic concentration difference. 6 × 105 < NRe < 2 × 106 120 < NSc < 1200 u = ωddisk /2 where ω = rotational speed, radians/s. NRe = ρωd 2/2µ.
[82] [146] p. 241
[E] Use arithmetic concentration difference. ω = stirrer speed, radians/s. Useful for laboratory dialysis, R.O., U.F., and microfiltration systems.
[73] [110] p. 965 [165] p. 738
2 ωd tank ρ NRe = ᎏ µ b = 0.785 [73]. c is often 0.33 but other values have been reported [110].
*See pages 5-7 and 5-8 for references.
involves recognizing the similarity of geometries, which is often challenging, and checking that the range of parameters in the correlation is appropriate. For example, the Bravo, Rocha, and Fair correlation for distillation with structured packings with triangular cross-sectional channels (Table 5-28-H) uses the Johnstone and Pigford correlation for rectification in vertical wetted wall columns (Table 5-22-D). Recognizing that this latter correlation pertains to a rather different application and geometry was a nontrivial step in the process of developing a correlation. If several correlations appear to be applicable, check to see if the correlations have been compared to each other and to the data. When a detailed comparison of correlations is not available, the following heuristics may be useful: 1. Mass-transfer coefficients are derived from models. They must be employed in a similar model. For example, if an arithmetic concentration difference was used to determine k, that k should only be used in a mass-transfer expression with an arithmetic concentration difference. 2. Semiempirical correlations are often preferred to purely empirical or purely theoretical correlations. Purely empirical correlations are dangerous to use for extrapolation. Purely theoretical correlations may predict trends accurately, but they can be several orders of magnitude off in the value of k. 3. Correlations with broader data bases are often preferred.
4. The analogy between heat and mass transfer holds over wider ranges than the analogy between mass and momentum transfer. Good heat transfer data (without radiation) can often be used to predict mass-transfer coefficients. 5. More recent data is often preferred to older data, since end effects are better understood, the new correlation often builds on earlier data and analysis, and better measurement techniques are often available. 6. With complicated geometries, the product of the interfacial area per volume and the mass-transfer coefficient is required. Correlations of kap or of HTU are more accurate than individual correlations of k and ap since the measurements are simpler to determine the product kap or HTU. 7. Finally, if a mass-transfer coefficient looks too good to be true, it probably is incorrect. To determine the mass-transfer rate, one needs the interfacial area in addition to the mass-transfer coefficient. For the simpler geometries, determining the interfacial area is straightforward. For packed beds of particles a, the interfacial area per volume can be estimated as shown in Table 5-27-A. For packed beds in distillation, absorption, and so on in Table 5-28, the interfacial area per volume is included with the mass-transfer coefficient in the correlations for HTU. For agitated liquid-liquid systems, the interfacial area can be estimated
MASS TRANSFER
5-61
TABLE 5-22 Mass Transfer Correlations for Falling Films with a Free Surface in Wetted Wall Columns—Transfer between Gas and Liquid Situation A. Laminar, vertical wetted wall column
Comments E = Empirical, S = Semiempirical, T = Theoretical
Correlation x k′m x NSh,avg = ᎏ ≈ 3.41 ᎏ D δfilm (first term of infinite series)
3µQ δfilm = ᎏ wρg
1/3
[T] Low rates M.T. Use with log mean concentration difference. Parabolic velocity distribution in films. 4Qρ NRe,film = ᎏ < 20 wµ Derived for flat plates, used for tubes if
= film thickness
C. Turbulent, vertical wetted wall column with ripples
[E] Use with log mean concentration difference for correlations in B and C. NRe is for gas. NSc for vapor in gas. 2000 < NRe ≤ 35,000, 0.6 ≤ NSc ≤ 2.5. Use for gases, dt = tube diameter.
0.15
[E] For gas systems with rippling.
4Qρ Fits B for ᎏ = 1000 wµ
4Qρ 30 ≤ ᎏ < 1200 wµ k′m d t 0.8 1/3 NSh,avg = ᎏ = 0.023N Re N Sc D
D. Rectification in vertical wetted wall column with turbulent vapor flow, Johnstone and Pigford correlation
k′G dcol pBM 0.33 NSh,avg = ᎏᎏ = 0.0328(N R′ e) 0.77 N Sc Dv p 3000 < N R′ e < 40,000, 0.5 < NSc < 3 d col vrel ρv N′Re = ᎏ , v rel = gas velocity relative to µv 3 liquid film = ᎏ uavg in film 2
[151] p. 137 [161] p. 50
k′m dt 0.83 0.44 N Sc NSh,avg = ᎏ = 0.023N Re D A coefficient 0.0163 has also been reported using NRe′, where v = v of gas relative to liquid film. k′m d t 0.83 0.44 4Qρ NSh,avg = ᎏ = 0.00814N Re N Sc ᎏ D wµ
[146] p. 78
ρg 1/2 rtube ᎏ > 3.0. σ = surface tension 2σ If NRe,film > 20, surface waves and rates increase. An approximate solution Dapparent can be used. Ripples are suppressed with a wetting agent good to NRe = 1200.
w = film width (circumference in column)
B. Turbulent, vertical wetted wall column
References*
[88] p. 266 [93] [100] p.181 [146] p. 211 [151] p. 265 [159] p. 212 [161] p. 71 [88] p. 266 [106] [146] p. 213
[E] “Rounded” approximation to include ripples. Includes solid-liquid mass-transfer data to find s 0.83 coefficient on NSc. May use N Re . Use for liquids. See also Table 5-23. [E] Use logarithmic mean driving force at two ends of column. Based on four systems with gas-side resistance only. pBM = logarithmic mean partial pressure of nondiffusing species B in binary mixture. p = total pressure Modified form is used for structured packings (See Table 5-28-H).
[105] [146] p. 214
*See pages 5-7 and 5-8 for references.
from the dispersed phase holdup and mean drop size correlations. Godfrey, Obi, and Reeve [Chem. Engr. Prog. 85, 61 (Dec. 1989)] summarize these correlations. For many systems, ddrop/dimp = −0.6 (const)NWe where NWe = ρc N 2d 3imp /σ. Effects of Total Pressure on kˆ G and kˆ L The influence of total system pressure on the rate of mass transfer from a gas to a liquid or to a solid has been shown to be the same as would be predicted from stagnant-film theory as defined in Eq. (5-285), where kˆ G = DABpT /RT δG (5-292) Since the quantity DAB pT is known to be relatively independent of the pressure, it follows that the rate coefficients kˆ G, kGyBM, and k′G pTyBM (= k′G pBM) do not depend on the total pressure of the system, subject to the limitations discussed later. Investigators of tower packings normally report k′Ga values measured at very low inlet-gas concentrations, so that yBM = 1, and at total pressures close to 100 kPa (1 atm). Thus, the correct rate coefficient for use in packed-tower designs involving the use of the driving force (y − yi)/yBM is obtained by multiplying the reported k′Ga values by the value of pT employed in the actual test unit (e.g., 100 kPa) and not the total pressure of the system to be designed.
From another point of view one can correct the reported values of k′Ga in kmol/[(s⋅m3)(kPa)], valid for a pressure of 101.3 kPa (1 atm), to some other pressure by dividing the quoted values of k′Ga by the design pressure and multiplying by 101.3 kPa, i.e., (k′Ga at design pressure pT) = (k′Ga at 1 atm) × 101.3/pT. One way to avoid a lot of confusion on this point is to convert the experimentally measured k′Ga values to values of kˆ Ga straightaway, before beginning the design calculations. A design based on the rate coefficient kˆ Ga and the driving force (y − yi)/yBM will be independent of the total system pressure with the following limitations: caution should be employed in assuming that kˆ Ga is independent of total pressure for systems having significant vapor-phase nonidealities, for systems that operate in the vicinity of the critical point, or for total pressures higher than about 3040 to 4050 kPa (30 to 40 atm). Experimental confirmations of the relative independence of kˆ G with respect to total pressure have been widely reported. Deviations do occur at extreme conditions. For example, Bretsznajder (Prediction of Transport and Other Physical Properties of Fluids, Pergamon Press, Oxford, 1971, p. 343) discusses the effects of pressure on the DABpT product and presents experimental data on the self-diffusion of CO2 which show that the D-p product begins to decrease at a pressure of
5-62
HEAT AND MASS TRANSFER
TABLE 5-23
Mass-Transfer Correlations for Flow in Pipes and Ducts—Transfer is from Wall to Fluid
Situation A. Tubes, laminar, fully developed parabolic velocity profile, developing concentration profile, constant wall concentration
Comments E = Empirical, S = Semiempirical, T = Theoretical
Correlation 0.0668(d t /x)NRe NSc k′d t NSh = ᎏ = 3.66 + ᎏᎏᎏ 1 + 0.04[(d t /x)NRe NSc]2/3 D
[T] Use log mean concentration difference. For x/d t ᎏ < 0.10, NRe < 2100. NRe NSc x = distance from tube entrance. Good agreement with experiment at values
References* [98] [100] p. 176 [108] p. 525 [151] p. 159
π d 104 > ᎏ ᎏt NReNSc > 10 4 x B. Tubes, fully developed concentration profile
k′d NSh = ᎏt = 3.66 D
[T] Subset of 5-23-A for fully developed concentration profile. x/d t ᎏ > 0.1 NRe NSc
C. Tubes, approximate solution
k′d d NSh,x = ᎏt = 1.077 ᎏt D x
E. Laminar, fully developed parabolic velocity profile, constant mass flux at wall
F. Laminar, alternate
(NRe NSc)1/3
[T] For arithmetic concentration difference.
(NRe NSc)1/3
∞
Graetz solution for heat transfer written for M.T.
11 1 ∞ exp [−λ 2j (x /rt)/(NRe NSc)] NSh, x = ᎏ − ᎏ ᎏᎏᎏ 48 2 j = 1 Cj λ 4j
λ 2j 25.68 83.86 174.2 296.5 450.9
[T] Use arithmetic concentration difference. Fits W gas data well, for ᎏ < 50 (fit is fortuitous). Dρx NSh,avg = (k′m d t)/D. a1 = 2.405, a 2 = 5.520, a 3 = 8.654, a4 = 11.792, a 5 = 14.931. Graphical solutions are in references.
H. Vertical tubes, laminar flow, forced and natural convection
cj 7.630 × 10−3 2.058 × 10−3 0.901 × 10−3 0.487 × 10−3 0.297 × 10−3
0.023(dt /L)NRe NSc NSh = 4.36 + ᎏᎏᎏ 1 + 0.0012(dt/L)NRe NSc
[T] Use log mean concentration difference. NRe < 2100
[149] [151] p. 167
k′d NSh,x = ᎏt D vd t ρ NRe = ᎏ µ [T]
k′dt 48 NSh = ᎏ = ᎏ = 4.3636 D 11 (NGr NSc d/L)3/4 1/3 1 ⫾ 0.0742 ᎏᎏ NSh,avg = 1.62N Gz NGz
[91] p. 443 [120] [151] p. 150
−1
k′d Nsh = ᎏt D Use log mean concentration difference. NRe < 2100 G. Laminar, fully developed concentration and velocity profile
[151] p. 166
W ᎏ > 400 ρDx Leveque’s approximation: Concentration BL is thin. Assume velocity profile is linear. High mass velocity. Fits liquid data well.
1/3
−2 a 2j (x/rt) 1 − 4 a −2 j exp ᎏᎏ 1 dt NRe NSc j=1 NSh,avg = ᎏ ᎏ NRe NSc ᎏᎏᎏᎏ ∞ 2 L −2 a 2j (x/rt) −2 1 + 4 a j exp ᎏᎏ NRe NSc j=1
j 1 2 3 4 5
[151] p. 165
1/3
k′d d NSh,avg = ᎏt = 1.615 ᎏt D L
D. Tubes, laminar, uniform plug velocity, developing concentration profile, constant wall concentration
[98]
[98]
[100] p. 176
[T] Use log mean concentration difference. NRe < 2100
[98] [151] p. 167
[T] Approximate solution. Use minus sign if forced and natural convection oppose each other. NRe NSc d NGz = ᎏ L
[140]
1/3
g∆ρd 3 NGr = ᎏ ρν 2 Good agreement with experiment. I. Tubes, laminar, RO systems
J. Tubes and parallel plates, laminar RO
k′m d t ud 2 NSh,avg = ᎏ = 1.632 ᎏt D DL
1/3
Graphical solutions for concentration polarization. Uniform velocity through walls.
Use arithmetic concentration difference. Thin concentration polarization layer, not fully developed. NRe < 2000, L = length tube.
[73] [165] p. 738
[T]
[145] [165] p. 762
MASS TRANSFER TABLE 5-23
5-63
Mass-Transfer Correlations for Flow in Pipes and Ducts—Transfer is from Wall to Fluid (Continued ) Comments E = Empirical, S = Semiempirical, T = Theoretical
References*
Graphical solution
[T] Low transfer rates.
[151] p. 176
k′(2h) NSh = ᎏ = 7.6 D
[T] h = distance between plates. Use log mean concentration difference.
[151] p. 177
Situation K. Parallel plates, laminar, parabolic velocity, developing concentration profile, constant wall concentration L. 5-23-K, fully developed
Correlation
NRe NSc ᎏ < 20 x/(2h) M. Parallel plates, laminar, parabolic velocity, developing concentration profile, constant mass flux at wall N. 5-23-M, fully developed
O. Laminar flow, vertical parallel plates, forced and natural convection
Graphical solution
[T] Low transfer rates.
[151] p. 176
k′(2h) NSh = ᎏ = 8.23 D
[T] Use log mean concentration difference.
[151] p. 177
NRe NSc ᎏ < 20 x/(2h)
(NGr NSc h/L)3/4 NSh,avg = 1.47N 1/3 Gz 1 ⫾ 0.0989 ᎏᎏ NGz
1/3
[T] Approximate solution. Use minus sign if forced and natural convection oppose each other.
uH 2p k′(2Hp) NSh,avg = ᎏ = 2.354 ᎏ D DL
Q. Tubes, turbulent
k′m d t 1/3 NSh,avg = ᎏ = 0.023N 0.83 Re N Sc D
R. Tubes, turbulent
k′m d t 0.44 NSh,avg = ᎏ = 0.023N 0.83 Re N Sc D
S. Tubes, turbulent
k′d t 0.346 NSh = ᎏ = 0.0096 N 0.913 Re N Sc D
T. Tubes, turbulent, smooth tubes, Reynolds analogy
f k′d t NSh = ᎏ = ᎏ NRe NSc D 2 f = Fanning friction faction
1/3
f jD = jH = ᎏ 2 f NSh −0.2 = 0.023N Re If ᎏ = 0.023N −0.2 Re , jD = ᎏ 2 NRe N 1/3 Sc
Thin concentration polarization layer. Short tubes, concentration profile not fully developed. Use arithmetic concentration difference.
[73] [165] p. 738
[E] Use with log mean concentration difference at two ends of tube. 2100 < NRe < 35,000 0.6 < NSc < 3000 From wetted wall column and dissolution data— see Table 5-22-B. Good fit for liquids.
[88] p. 266
[E] Evaporation of liquids. Use with log mean concentration difference. See item above. Better fit for gases. 2000 < NRe < 35,000 0.6 < NSc < 2.5.
[93][100] p. 181 [109] p. 112 [146] p. 211
[E] 430 < NSc < 100,000. Dissolution data. Use for high NSc.
[122] p. 668
[T] Use arithmetic concentration difference. NSc near 1.0 Turbulent core extends to wall. Of limited utility.
[91] p. 438 [100] p. 171 [151] p. 239 [159] p. 250
[T] Use log-mean concentration difference. Relating jD to f/2 approximate. NPr and NSc near 1.0. Low concentration. Results about 20% lower than experiment. 3 × 104 < NRe < 106
[72] p. 400, 647 [80][88] p. 269 [151] p. 264 [159] p. 251
k′d t NSh = ᎏ D jD = jH = f (NRe, geometry and B.C.)
V. Tubes, turbulent, smooth tubes, constant surface concentration, Prandtl analogy *See pages 5-7 and 5-8 for references.
[140]
NRe NSc h NGz = ᎏ L g∆ρh3 NGr = ᎏ ρν 2 Good agreement with experiment.
P. Parallel plates, laminar, RO systems
U. Tubes, turbulent, smooth tubes, Chilton-Colburn analogy
k′d ( f /2)NRe NSc NSh = ᎏt = ᎏᎏ D 1 + 5f2 /(NSc − 1) f −0.25 ᎏ = 0.04 NRe 2
[100] p. 181 [120] [161] p. 72
[E] Good over wide ranges. [72] p. 647 [80] [T] Use arithmetic concentration difference. Improvement over Reynolds analogy. Best for NSc near 1.0.
[100] p. 173 [132] [151] p. 241
5-64
HEAT AND MASS TRANSFER
TABLE 5-23
Mass-Transfer Correlations for Flow in Pipes and Ducts—Transfer is from Wall to Fluid (Concluded )
Situation
Comments E = Empirical, S = Semiempirical, T = Theoretical
Correlation
References*
W. Tubes, turbulent, smooth tubes, Constant surface concentration, Von Karman analogy
( f/2)NReNSc ᎏᎏᎏᎏᎏ NSh = 5 2 (NSc − 1) + ln 1 + ᎏᎏ (NSc − 1) 1 + 5f/ 6 f −0.25 ᎏ = 0.04 N Re 2
[T] Use arithmetic concentration difference. NSh = k′dt /D. Improvement over Prandtl, NSc < 25.
[100] p. 173 [151] p. 243 [159] p. 250 [162]
X. Tubes, turbulent, smooth tubes, constant surface concentration
For 0.5 < NSc < 10:
[S] Use arithmetic concentration difference. Based on partial fluid renewal and an infrequently replenished thin fluid layer for high Nsc. Good fit to available data.
[100] p. 179 [131]
9/10 1/2 NSh,avg = 0.0097 N Re N Sc −1/3 −1/6 × (1.10 + 0.44N Sc − 0.70N Sc )
For 10 < NSc < 1000:
u bulk d t NRe = ᎏ ν
NSh,avg
k′avg d t NSh,avg = ᎏ D
9/10 1/2 −1/3 −1/6 0.0097N Re N Sc (1.10 + 0.44N Sc − 0.70N Sc ) = ᎏᎏᎏᎏᎏ 1/2 −1/3 −1/6 1 + 0.064 N Sc (1.10 + 0.44N Sc − 0.70N Sc )
For NSc > 1000: 9/10 1/3 N Sh,avg = 0.0102 N Re N Sc
Y. Turbulent flow, tubes
NSh NSh −0.12 −2/3 NSt = ᎏ =ᎏ = 0.0149N Re N Sc NPe NRe NSc
Z. Turbulent flow, noncircular ducts
Use correlations with
[E] Smooth pipe data. Data fits within 4% except at NSc > 20,000, where experimental data is underpredicted.
[124]
NSc > 100, 105 > NRe > 2100 Can be suspect for systems with sharp corners.
4 cross-sectional area d eq = ᎏᎏᎏ wetted perimeter Parallel plates:
[151] p. 289 [165] p. 738
2hw d eq = 4 ᎏ 2w + 2h *See pages 5-7 and 5-8 for references.
approximately 8100 kPa (80 atm). For reduced temperatures higher than about 1.5, the deviations are relatively modest for pressures up to the critical pressure. However, deviations are large near the critical point (see also p. 5-49). The effect of pressure on the gas-phase viscosity also is negligible for pressures below about 5060 kPa (50 atm). For the liquid-phase mass-transfer coefficient kˆ L, the effects of total system pressure can be ignored for all practical purposes. Thus, when using kˆ G and kˆ L for the design of gas absorbers or strippers, the primary pressure effects to consider will be those which affect the equilibrium curves and the values of m. If the pressure changes affect the hydrodynamics, then kˆ G, kˆ L, and a can all change significantly. Effects of Temperature on kˆ G and kˆ L The Stanton-number relationship for gas-phase mass transfer in packed beds, NSt = kˆ G /GM = g(NRe,NSc) (5-293)
With regard to the liquid-phase mass-transfer coefficient, Whitney and Vivian found that the effect of temperature upon kLa could be explained entirely by variations in the liquid-phase viscosity and diffusion coefficient with temperature. Similarly, the oxygen-desorption data of Sherwood and Holloway [Trans. Am. Inst. Chem. Eng., 36, 39 (1940)] show that the influence of temperature upon HL can be explained by the effects of temperature upon the liquid-phase viscosity and diffusion coefficients. It is important to recognize that the effects of temperature on the liquid-phase diffusion coefficients and viscosities can be very large and therefore must be carefully accounted for when using kˆ L or HL data. For liquids the mass-transfer coefficient kˆ L is correlated in terms of design variables by relations of the form NSt = kˆ L/LM = f(NRe,NSc) (5-294)
indicates that for a given system geometry the rate coefficient kˆ G depends only on the Reynolds number and the Schmidt number. Since the Schmidt number for a gas is independent of temperature, the principal effect of temperature upon kˆ G arises from changes in the gas viscosity with changes in temperature. For normally encountered temperature ranges, these effects will be small owing to the fractional powers involved in Reynolds-number terms (see Tables 5-21 to 5-28). It thus can be concluded that for all practical purposes kˆ G is independent of temperature and pressure in the normal ranges of these variables. For modest changes in temperature the influence of temperature upon the interfacial area a may be neglected. For example, in experiments on the absorption of SO2 in water, Whitney and Vivian [Chem. Eng. Prog., 45, 323 (1949)] found no appreciable effect of temperature upon k′Ga over the range from 10 to 50°C.
A general relation for HL which may be used as the basis for applying temperature corrections is as follows: a HL = bNRe N1/2 Sc
(5-295)
where b is a proportionality constant and the exponent a may range from about 0.2 to 0.5 for different packings and systems. The liquidphase diffusion coefficients may be corrected from a base temperature T1 to another temperature T2 by using the Einstein relation as recommended by Wilke [Chem. Eng. Prog., 45, 218 (1949)]: D2 = D1(T2 /T1)(µ1/µ2)
(5-296)
The Einstein relation can be rearranged to the following equation for relating Schmidt numbers at two temperatures: NSc2 = NSc1(T1 /T2)(ρ1 /ρ2)(µ2 /µ1)2
(5-297)
MASS TRANSFER TABLE 5-24
5-65
Mass Transfer Correlations for Flow Past Submerged Objects
Situation A. Single sphere
B. Single sphere, creeping flow with forced convection
Comments E = Empirical, S = Semiempirical, T = Theoretical
Correlation k′G pBLM RTd s 2r NSh = ᎏᎏ =ᎏ PD r − rs r/rs 2 5 10 50 ∞ (asymptotic limit) NSh 4.0 2.5 2.22 2.04 2.0 k′d NSh = ᎏ = [4.0 + 1.21(NRe NSc)2/3]1/2 D
k′d NSh = ᎏ = a(NRe NSc)1/3 D a = 1.01, 1.0, or 0.991 C. Single spheres, molecular diffusion, and forced convection, low flow rates
1/2 1/3 N Sh = 2.0 + AN Re N Sc A = 0.5 to 0.62
A = 0.60.
[T] Use with log mean concentration difference. r = distance from sphere, rs, ds = radius and diameter of sphere. No convection.
[151] p. 18
[T] Use with log mean concentration difference. Average over sphere. Numerical calculations. (NRe NSc) < 10,000 NRe < 1.0. Constant sphere diameter. Low mass-transfer rates.
[78][109] p. 114 [122] p. 671 [146] p. 214
[T] Fit to above ignoring molecular diffusion.
[117] p. 80
1000 < (NReNSc) < 10,000.
146 p. 215
[E] Use with log mean concentration difference. Average over sphere. Frössling Eq. (A = 0.552), 2 ≤ NRe ≤ 800, 0.6 ≤ Nsc ≤ 2.7. NSh lower than experimental at high NRe. [E] Ranz and Marshall 2 ≤ NRe ≤ 200, 0.6 ≤ Nsc ≤ 2.5. See also Table 5-27-L.
[72] [89] p. 409, 647 [100], p. 194 109 p. 114 151 p. 276 [72] p. 409, 647 [135] [146] p. 217 [151] p. 276 [90][91] p. 446 [146] p. 217 [139][151] p. 276 [102][151] p. 276
[E] Liquids 2 ≤ NRe ≤ 2,000. Graph in Ref. 146, p. 217–218. [E] 100 ≤ NRe ≤ 700; 1,200 ≤ NSc ≤ 1525.
A = 0.95. A = 0.95. A = 0.544.
References*
[E] Use with arithmetic concentration difference. NSc = 1; 50 ≤ NRe ≤ 350.
D. 5-24-C
k′d 0.35 NSh = ᎏs = 2.0 + 0.575N 1/2 Re N Sc D
[E] Use with log mean concentration difference. NSc ≤ 1, NRe < 1.
[94][151] p. 276
E. 5-24-C
k′d 0.53 1/3 NSh = ᎏs = 2.0 + 0.552N Re N Sc D
[E] Use with log mean concentration difference. 1.0 < NRe ≤ 48,000 Gases: 0.6 ≤ NSc ≤ 2.7.
[91] p. 446
[S] Correlates large amount of data and compares to published data. vr = relative velocity between fluid and sphere, m/s. CDr = drag coefficient for single particle fixed in fluid at velocity vr. See 5-27-G for calculation details and other applications.
[125]
[E] Use with arithmetic concentration difference. Liquids, 2000 < NRe < 17,000. High NSc, graph in Ref. 146, p. 217–218.
[91] p. 446 [157] [146] p. 217
k′d 0.6 1/3 NSh = ᎏs = 0.33N Re N Sc D
[E] 1500 ≤ NRe ≤ 12,000.
[151] p. 276
k′d 0.56 1/3 NSh = ᎏs = 0.43N Re N Sc D
[E] 200 ≤ NRe ≤ 4 × 104, “air” ≤ NSc ≤ “water.”
[151] p. 276
k′d 0.514 1/3 NSh = ᎏs = 0.692N Re N Sc D
[E] 500 ≤ NRe ≤ 5000.
[128] [151] p. 276
k′d 1/3 NSh = ᎏs = AN 1/2 Re N Sc , A = 0.82 D
[E] 100 < NRe ≤ 3500, NSc = 1560.
[151] p. 276
A = 0.74
[E] 120 ≤ NRe ≤ 6000, NSc = 2.44.
A = 0.582
[E] 300 ≤ NRe ≤ 7600, NSc = 1200.
[151] p. 276 [152]
jD = 0.600(N Re)−0.487
[E] Use with arithmetic concentration difference.
[151] p. 276
k′d cyl NSh = ᎏ D
50 ≤ NRe ≤ 50,000; gases, 0.6 ≤ NSc ≤ 2.6; liquids; 1000 ≤ NSc ≤ 3000. Data scatter ⫾ 30%.
[91] p. 450
F. Single spheres, forced concentration, any flow rate
E1/3d p4/3ρ k′L d s NSh = ᎏ = 2.0 + 0.59 ᎏ µ D
0.57 1/3 N Sc
Energy dissipation rate per unit mass of fluid (ranges 570 < NSc < 1420): ᎏ ᎏ d s m2
v 3r
CDr E= ᎏ 2
3
p
E d ρ 2 < ᎏᎏ < 63,000 µ
G. Single spheres, forced convection, high flow rates, ignoring molecular diffusion
H. Single cylinders, perpendicular flow
1/3
4/3 p
k′d 0.62 1/3 NSh = ᎏs = 0.347N Re N Sc D
5-66
HEAT AND MASS TRANSFER
TABLE 5-24
Mass Transfer Correlations for Flow Past Submerged Objects (Concluded )
Situation
Comments E = Empirical, S = Semiempirical, T = Theoretical
Correlation Can use jD = jH. Graphical correlation.
I. 5-24-H
J. Rotating cylinder in an infinite liquid, no forced flow
[E] Used with linear concentration difference.
[72] p. 408, 647
k′d s NSh = ᎏ D
[146] p. 236 [151] p. 273
k′ 0.644 −0.30 j′D = ᎏ N Sc = 0.0791N Re v
[E] Used with arithmetic concentration difference.
[85]
Results presented graphically to NRe = 241,000.
112 < NRe ≤ 100,000. 835 < NSc < 11490
[146] p. 238
vdcyl µ ω dcyl NRe = ᎏ where v = ᎏ = peripheral velocity ρ 2
K. Oblate spheroid, forced convection
References*
NSh −0.5 = 0.74 N Re jD = ᎏ NRe N 1/3 Sc
k′ = mass-transfer coefficient, cm/s; ω = rotational speed, radian/s. Useful geometry in electrochemical studies. [E] Used with arithmetic concentration difference. 120 ≤ NRe ≤ 6000; standard deviation 2.1%. Eccentricities between 1:1 (spheres) and 3:1.
total surface area dch vρ NRe = ᎏ , dch = ᎏᎏᎏ µ perimeter normal to flow
[151] p. 284 [152]
e.g., for cube with side length a, dch = 1.27a. k′d ch NSh = ᎏ D L. Other objects, including prisms, cubes, hemispheres, spheres, and cylinders; forced convection
M. Other objects, molecular diffusion limits
jD = 0.692N
Shape is often approximated by drops. [109] p. 115
v d ch ρ , N Re,p = ᎏ µ
−0.486 Re,p
[E] Used with arithmetic concentration difference.
Terms same as in 5-24-J.
k′d ch NSh = ᎏ =A D
500 ≤ N Re, p ≤ 5000. Turbulent. Agrees with cylinder and oblate spheroid results, ⫾15%. Assumes molecular diffusion and natural convection are negligible.
[127, 128] [151] p. 285
[T] Use with arithmetic concentration difference. Hard to reach limits in experiments.
[109] p. 114
[E] Use with logarithmic mean concentration difference.
[133]
Spheres and cubes A = 2, tetrahedrons A = 26 . octahedrons 22 N. Shell side of microporous hollow fiber module for solvent extraction
0.33 NSh = β[d h(1 − ϕ)/L]N 0.6 Re N Sc
Kdh NSh = ᎏ D β = 5.8 for hydrophobic membrane.
d h vρ N Re = ᎏ ,K = overall mass-transfer coefficient µ
β = 6.1 for hydrophilic membrane.
dh = hydraulic diameter 4 × cross-sectional area of flow = ᎏᎏᎏᎏ wetted perimeter ϕ = packing fraction of shell side. L = module length. Based on area of contact according to inside or outside diameter of tubes depending on location of interface between aqueous and organic phases. Can also be applied to gas-liquid systems with liquid on shell side.
See Table 5-27 for flow in packed beds. *See pp. 5-7 and 5-8 for references.
Substitution of this relation into Eq. (5-295) shows that for a given geometry the effect of temperature on HL can be estimated as HL2 = HL1(T1 /T2)1/2(ρ1 /ρ2)1/2(µ2 /µ1)1 − a
(5-298)
In using these relations it should be noted that for equal liquid flow rates HL2 /HL1 = (kˆ La)1/(kˆ La)2 (5-299) ˆ L When ˆ G and k Effects of System Physical Properties on k designing packed towers for nonreacting gas-absorption systems for
which no experimental data are available, it is necessary to make corrections for differences in composition between the existing test data and the system in question. For example, the test data of Fellinger for ammonia-water absorption on various packings are frequently used as a base (see Table 5-28-B). In these tests it is estimated that HG = 0.9HOG, so that one may wish to use these data as the basis for estimating HG or kˆ Ga values for other systems. This may be done by tak0.5 −0.5 ing HG proportional to NSc and kˆ Ga proportional to NSc , based on a value of NSc for NH3-air of 0.66 at 25°C. The coefficient kG varies as the diffusivity DAB to the 0.5 power. It should be noted, however, that
MASS TRANSFER TABLE 5-25
5-67
Mass-Transfer Correlations for Drops and Bubbles
Conditions A. Single liquid drop in immiscible liquid, drop formation, discontinuous (drop) phase coefficient
Correlations ρd kˆ d,f = A ᎏ Md
Dd ᎏ π tf
av
24 A = ᎏ (0.8624) (extension by fresh surface 7 elements) kˆ df = 0.0432 d ρd × ᎏp ᎏ tf Md
−0.334
µd ᎏᎏ d gc d pσ ρ
C. Single liquid drop in immiscible liquid, drop formation, continuous phase coefficient D. 5-25-C
av
0.089
uo ᎏ dp g
G. 5-25-E, continuous phase coefficient, stagnant drops, spherical
H. 5-25-E, oblate spheroid
Dc
av
f
F. 5-25-E
−0.601
kL,c = 0.386 ρc × ᎏ Mc
E. Single liquid drop in immiscible liquid, free rise or fall, discontinuous phase coefficient, stagnant drops
d p2 ᎏ t f Dd
ᎏ πt
ρc kˆ cf = 4.6 ᎏ Mc
Dc ᎏ tf
av
−dp ρd kL,d,m = ᎏ ᎏ 6t M d
0.5
ρcσgc ᎏ ∆ ρgt fµ c
0.407
gt f2 ᎏ dp
0.148
−Dd j 2π 2 t
∞
ln ᎏπ6 ᎏj1 exp ᎏ (d /2)
−d p ρd kˆ L,d,m = ᎏ ᎏ 6t Md
[T,S] Use arithmetic mole fraction difference.
[151] p. 399
Fits some, but not all, data. Low mass transfer rate. Md = mean molecular weight of dispersed phase; tf = formation time of drop. kL,d = mean dispersed liquid phase M.T. coefficient kmole/[s ⋅ m2 (mole fraction)].
A = 1.31 (semiempirical value)
B. 5-25-A
References*
1/2
24 A = ᎏ (penetration theory) 7
Comments E = Empirical, S = Semiempirical, T = Theoretical
2
av
2
j=1
p
2
πD t ln 1 − ᎏ d /2 1/2 1/2 d
av
p
kL,c,m dc ρc NSh = ᎏ = 0.74 ᎏ Dc Mc
N
(NSc)1/3
1/2 Re
av
ρc kL,c,m d3 NSh = ᎏ = 0.74 ᎏ Dc Mc
(N
)1/2(NSc,c)1/3
Re,3
av
[E] Use arithmetic mole fraction difference. Based on 23 data points for 3 systems. Average absolute deviation 26%. Use with surface area of drop after detachment occurs. uo = velocity through nozzle; σ = interfacial tension.
[151] p. 401
[T] Use arithmetic mole fraction difference. Based on rate of bubble growth away from fixed orifice. Approximately three times too high compared to experiments.
[151] p. 402
[E] Average absolute deviation 11% for 20 data points for 3 systems.
[151] p. 402 [154] p. 434
[T] Use with log mean mole fraction differences based on ends of column. t = rise time. No continuous phase resistance. Stagnant drops are likely if drop is very viscous, quite small, or is coated with surface active agent. kL,d,m = mean dispersed liquid M.T. coefficient.
[151] p. 404 [154] p. 435
[S] See 5-25-E. Approximation for fractional extractions less than 50%.
[151] p. 404 [154] p. 435
vs d p ρc [E] NRe = ᎏ , special case Eq. (5-254). µc
[151] p. 407
vs = slip velocity between drop and continuous phase.
[152][154] p. 436
[E] Used with log mean mole fraction. Differences based on ends of extraction column; 100 measured values ⫾2% deviation. Based on area oblate spheroid.
[151] p. 285, 406, 407
[154] p. 434
vsd3ρc NRe,3 = ᎏ µc total drop surface area vs = slip velocity, d3 = ᎏᎏᎏ perimeter normal to flow I. Single liquid drop in immiscible liquid, Free rise or fall, discontinuous phase coefficient, circulating drops
J. 5-25-I
dp 3 kdr,circ = − ᎏ ln ᎏ 6θ 8
∞
B
j=1
2 j
λ j64Ddθ exp − ᎏ d p2
Eigenvalues for Circulating Drop k d d p /Dd
λ1
λ2
3.20 10.7 26.7 107 320 ∞
0.262 0.680 1.082 1.484 1.60 1.656
0.424 4.92 5.90 7.88 8.62 9.08
d p ρd kˆ L,d,circ = − ᎏ ᎏ 6θ Md
λ3
B1
B2
B3
15.7 19.5 21.3 22.2
1.49 1.49 1.49 1.39 1.31 1.29
0.107 0.300 0.495 0.603 0.583 0.596
0.205 0.384 0.391 0.386
R πD θ ln 1 − ᎏᎏ d /2 1/2
av
1/2 1/2 d
p
[T] Use with arithmetic concentration difference.
[86][99][151] p. 405
θ = drop residence time. A more complete listing of eigenvalues is given by Refs. 86 and 99.
[161] p. 523
k′L,d,circ is m/s.
[E] Used with mole fractions for extraction less than 50%, R ≈ 2.25.
[151] p. 405
5-68
HEAT AND MASS TRANSFER
TABLE 5-25
Mass-Transfer Correlations for Drops and Bubbles (Continued )
Conditions K. 5-25-I
ˆk L,d,circ d p NSh = ᎏ Dd ρd = 31.4 ᎏ Mf
2 −0.125 d p v s ρc N Sc,d ᎏ σg c
−0.34
ᎏ d
L. Liquid drop in immiscible liquid, free rise or fall, continuous phase coefficient, circulating single drops
Comments E = Empirical, S = Semiempirical, T = Theoretical
Correlations
4Dd t 2 p
av
−0.37
k′L,c d p NSh,c = ᎏ Dd
d p g1/3 0.484 = 2 + 0.463N Re,drop N 0.339 ᎏ Sc,c D c2/3
1/8 K = N Re,drop
M. 5-25-L, circulating, single drop
µcvs ᎏ σg c
[E] Used with log mean mole fraction difference. dp = diameter of sphere with same volume as drop. 856 ≤ NSc ≤ 79,800, 2.34 ≤ σ ≤ 4.8 dynes/cm.
[154] p. 435 [155]
[E] Used as an arithmetic concentration difference.
[103]
d pv sρc NRe,drop = ᎏ µc
F 0.072
F = 0.281 + 1.615K + 3.73K 2 − 1.874K µc ᎏ µd
References*
Solid sphere form with correction factor F.
1/4
k L,c d p ρc NSh = ᎏ = 0.6 ᎏ Dc Mc
1/6
N
1/2 Re,drop
1/2 N Sc,c
[E] Used as an arithmetic concentration difference. Low σ.
av
[151] p. 407
d pv sρc NRe,drop = ᎏ µc N. 5-25-L, circulating swarm of drops
O. Liquid drops in immiscible liquid, free rise or fall, discontinuous phase coefficient, oscillating drops
P. 5-25-O
Q. Single liquid drop in immiscible liquid, range rigid to fully circulating
ρc k L,c = 0.725 ᎏ Mc
N
−0.43 Re,drop
−0.58 N Sc,c v s (1 − φ d)
av
k L,d,osc d p NSh = ᎏ Dd ρd = 0.32 ᎏ Md
−0.14
t ᎏ d 4Dd 2 p
av
σ 3g c3 ρ2c 0.68 N Re,drop ᎏ gµ 4c∆ρ
0.10
0.00375v k L,d,osc = ᎏᎏs 1 + µ d /µ c
Rigid drops: 104 < NPe,c < 106
[E] Used as an arithmetic concentration difference. Low σ, disperse-phase holdup of drop swarm. φ d = volume fraction dispersed phase.
[151] p. 407 [154] p. 436
[E] Used with a log mean mole fraction difference. Based on ends of extraction column.
[151] p. 406
d pvsρc NRe,drop = ᎏ µc d p = diameter of sphere with volume of drop. Average absolute deviation from data, 10.5%. 411 ≤ NRe ≤ 3114 Low interfacial tension (3.5–5.8 dynes), µc < 1.35 centipoise.
[154] p. 435 [155]
[T] Use with log mean concentration difference. Based on end of extraction column. No continuous phase resistance. kL,d,osc in cm/s, vs = drop velocity relative to continuous phase.
[146] p. 228 [151] p. 405
[E] Allows for slight effect of wake.
[156] p. 58 [158]
[E] Used with log mean mole fraction difference. 23 data points. Average absolute deviation 25%. t f = formation time.
[151] p. 408
[E] Used with log mean mole fraction difference. 20 data points. Average absolute deviation 22%.
[151] p. 409
kcdp 0.5 0.33 NSh,c,rigid = ᎏ = 2.43 + 0.774N Re N Sc Dc 0.33 + 0.0103NReN Sc
Circulating drops: 10 < NRe < 1200, 190 < NSc < 241,000, 103 < NPe,c < 106
2 NSh,c,fully circular = ᎏ N 0.5 Pe,c π 0.5 Drops in intermediate range: NSh,c − NSh,c,rigid ᎏᎏᎏ = 1 − exp [−(4.18 × 10−3)N 0.42 Pe,c] NSh,c,fully circular − NSh,c,rigid R. Coalescing drops in immiscible liquid, discontinuous phase coefficient
S. 5-25-R, continuous phase coefficient
d p ρd kˆ d,coal = 0.173 ᎏ ᎏ t f Md
µ ᎏ ρ D
∆ ρgd p2 × ᎏ σg c
−1.115
d
av
d
d
v ᎏ D 1.302
2 s f
t
0.146
d
ρ kˆ c,coal = 5.959 × 10−4 ᎏ M
ρdu3s
av
d p2 ρc ρd v3s
ᎏᎏ ᎏ g µ µ σg
D × ᎏc tf
0.5
c
0.332
d
c
0.525
MASS TRANSFER TABLE 5-25
5-69
Mass-Transfer Correlations for Drops and Bubbles (Concluded )
Conditions T. Single liquid drops in gas, gas side coefficient
Comments E = Empirical, S = Semiempirical, T = Theoretical
Correlations kˆg Mg d p P 1/2 N 1/3 ᎏ = 2 + AN Re,g Sc,g Dgas ρg A = 0.552 or 0.60.
References*
[E] Used for spray drying (arithmetic partial pressure difference).
[88] p. 489
d pρgvs NRe,g = ᎏ , vs = slip velocity between drop µg
[111] p. 388 [135]
and gas stream. Sometimes written with: MgP ᎏ = RT ρg U. Single water drop in air, liquid side coefficient
DL kL = 2 ᎏ πt
1/2
, short contact times
DL k L = 10 ᎏ , long contact times dp V. Single bubbles of gas in liquid, continuous phase coefficient, very small bubbles
k′c d b NSh = ᎏ = 1.0(NReNSc)1/3 Dc
W. 5-25-V, medium to large bubbles
k′c d b NSh = ᎏ = 1.13(NReNSc)1/2 Dc
X. 5-25-W
k′c d b db NSh = ᎏ = 1.13(NReNSc)1/2 ᎏᎏ Dc 0.45 + 0.2d b
Y. Rising small bubbles of gas in liquid, continuous phase
k′c d b 1/3 NSh = ᎏ = 2 + 0.31(NGr)1/3N Sc , d b < 0.25 cm Dc
[T] Use arithmetic concentration difference. Penetration theory. t = contact time of drop. Gives plot for k G a also. Air-water system.
[111] p. 389
[T] Solid-sphere Eq. (see Table 5-24-B). d b < 0.1 cm, k′c is average over entire surface of bubble.
[122] p. 673 [146] p. 214
[T] Use arithmetic concentration difference. Droplet equation: d b > 0.5 cm.
[146] p. 231
[S] Use arithmetic concentration difference. Modification of above (W), db > 0.5 cm. 500 ≤ NRe ≤ 8000. No effect SAA for dp > 0.6 cm.
[104][146] p. 231
[E] Use with arithmetic concentration difference.
[79][91] p. 451
d |ρG − ρL|g NRa = ᎏᎏ = Raleigh number µ L DL Note that NRa = NGr NSc. Valid for single bubbles or swarms. Independent of agitation as long as bubble size is constant.
[109] p. 119 [161] p. 156
[E] Use with arithmetic concentration difference. Hg = fractional gas holdup, volume gas/total volume. For large bubbles, k′c is independent of bubble size and independent of agitation or liquid velocity. Resistance is entirely in liquid phase for most gas-liquid mass transfer.
[79][91] p. 452 [109] p. 119
3 b
Z. 5-25-Y, large bubbles
k′c d b NSh = ᎏ = 0.42 (NGr)1/3N 1/2 Sc , d b > 0.25 cm Dc 6 Hg Interfacial area ᎏᎏ = a = ᎏ volume db
[114] p. 249
See Table 5-26 for agitated systems. *See pages 5-7 and 5-8 for references.
there is conflicting evidence concerning this exponent (w versus a) as discussed by Yadav and Sharma [Chem. Eng. Sci., 34, 1423 (1979)]. The existing data indicate that kˆ La is proportional to the square root of the solute-diffusion coefficient, and since the interfacial area a does not depend on DL, it follows that kˆ L is proportional to DL0.5. An analysis of the design variables involved indicates that kˆ L should be −0.5 proportional to NSc when the Reynolds number is held constant. It should be noted that the influence of substituting solvents of widely differing viscosities upon the interfacial area a can be very large. One therefore should be cautious about extrapolating kˆ La data to account for viscosity effects between different solvent systems. ˆ G and k ˆ L As disEffects of High Solute Concentrations on k cussed previously, the stagnant-film model indicates that kˆ G should be independent of yBM and kG should be inversely proportional to yBM. The data of Vivian and Behrman [Am. Inst. Chem. Eng. J., 11, 656 (1965)] for the absorption of ammonia from an inert gas strongly suggest that the film model’s predicted trend is correct. This is another indication that the most appropriate rate coefficient to use is kˆ G and the proper driving-force term is of the form (y − yi)/yBM.
The use of the rate coefficient kˆ L and the driving force (xi − x)/xBM is believed to be appropriate. For many practical situations the liquidphase solute concentrations are low, thus making this assumption unimportant. ˆ G and k ˆ L When a chemInfluence of Chemical Reactions on k ical reaction occurs, the transfer rate may be influenced by the chemical reaction as well as by the purely physical processes of diffusion and convection within the two phases. Since this situation is common in gas absorption, gas absorption will be the focus of this discussion. One must consider the impacts of chemical equilibrium and reaction kinetics on the absorption rate in addition to accounting for the effects of gas solubility, diffusivity, and system hydrodynamics. There is no sharp dividing line between pure physical absorption and absorption controlled by the rate of a chemical reaction. Most cases fall in an intermediate range in which the rate of absorption is limited both by the resistance to diffusion and by the finite velocity of the reaction. Even in these intermediate cases the equilibria between the various diffusing species involved in the reaction may affect the rate of absorption.
5-70
HEAT AND MASS TRANSFER
TABLE 5-26
Mass-Transfer Correlations for Particles, Drops, and Bubbles in Agitated Systems Comments E = Empirical, S = Semiempirical, T = Theoretical
References*
k′LT d p 1/3 ᎏ = 2 + 0.6N 1/2 Re,T N Sc D
[S] Use log mean concentration difference. Modified Frossling equation:
[97][146] p. 220
Replace vslip with v T = terminal velocity. Calculate Stokes’ law terminal velocity
v Ts d p ρc NRe,Ts = ᎏ µc
Situation A. Solid particles suspended in agitated vessel containing vertical baffles, continuous phase coefficient
Correlation
d p2|ρ p − ρc|g v Ts = ᎏᎏ 18µ c
(Reynolds number based on Stokes’ law.) v T d p ρc NRe,T = ᎏ µc
and correct: NRe,Ts 1 v T /v Ts 0.9
10 0.65
100 0.37
1,000 0.17
10,000 0.07
100,000 0.023
Approximate: k′L = 2k′LT B. 5-26-A C. Solid, neutrally buoyant particles, continuous phase coefficient
Graphical comparisons experiments and correlations.
k′Ld p d imp 0.36 NSh = ᎏ = 2 + 0.47N 0.62 ᎏ Re,p N Sc D d tank
(terminal velocity Reynolds number.) k′L almost independent of d p. Harriott suggests different correction procedures. Range k′L /k′LT is 1.5 to 8.0.
[87]
[E,S] For spheres. Includes transpiration effects and changing diameters.
[78][146] p. 222
[E] Use log mean concentration difference. Density unimportant if particles are close to neutrally buoyant. [E] E = energy dissipation rate per unit mass fluid
[109] p. 115 [118] p. 132
0.17
Graphical comparisons are in Ref. 109, p. 116.
[161] p. 523
Pgc =ᎏ , P = power Vtank ρc E1/3d p4/3 NRe,p = ᎏ ν Also used for drops. Geometric effect (d imp/d tank) is usually unimportant. Ref. 118 gives a variety of references on correlations. D. 5-26-C, small particles
0.52 NSh = 2 + 0.52N Re,p N 1/3 Sc , NRe,p < 1.0
E. Solid particles with significant density difference
F. Small solid particles, gas bubbles or liquid drops, dp < 2.5 mm
k′L d p d 3p |ρp − ρ c| NSh = ᎏ = 2 + 0.31 ᎏᎏ µ cD D
G. Highly agitated systems; solid particles, drops, and bubbles; continuous phase coefficient
H. Liquid drops in baffled tank with flat six-blade turbine
[E] Use log mean concentration difference. NSh standard deviation 11.1%. vslip calculated by methods given in reference.
[118]
[E] Use log mean concentration difference. g = 9.80665 m/s 2. Second term RHS is free-fall or rise term. For large bubbles, see Table 5-25-Z.
[79][91] p. 451 [114] p. 249
[E] Use arithmetic concentration difference. Use when gravitational forces overcome by agitation. Up to 60% deviation. Correlation prediction is low (Ref. 118). (P/Vtank) = power dissipated by agitator per unit volume liquid.
[79][83] p. 231 [91] p. 452
[E] Use arithmetic concentration difference. Studied for five systems.
[154] p. 437
1/4
1.582
2 NRe = d imp Nρc /µ c , NOh = µ c /(ρc d impσ)1/2
1.025 N 1.929 Re N Oh
φ = volume fraction dispersed phase. N = impeller speed (revolutions/time). For dtank = htank, average absolute deviation 23.8%.
k′c d p 1/3 NSh = ᎏ = 1.237 × 10 −5 N Sc N 2/3 D ρd d p2
ᎏ ᎏ D σ
5/12 d imp × N Fr ᎏ dp
[109] p. 116
1/3
(ND)1/2 k′c a = 2.621 × 10−3 ᎏ d imp d imp × φ 0.304 ᎏ d tank
I. Liquid drops in baffled tank, low volume fraction dispersed phase
0.38 N Sc
(P/Vtank)µ cg c 2/3 k′LN Sc = 0.13 ᎏᎏ ρc2
[E] Terms same as above.
1/2
d pv slip k′L d p NSh = ᎏ = 2 + 0.44 ᎏ D ν
dp
1/2
5/4
φ−1/2
tank
Stainless steel flat six-blade turbine. Tank had four baffles. Correlation recommended for φ ≤ 0.06 [Ref. 156] a = 6φ /dˆ 32, where dˆ 32 is Sauter mean diameter when 33% mass transfer has occurred.
[E] 180 runs, 9 systems, φ = 0.01. kc is timeaveraged. Use arithmetic concentration difference.
2 d impN 2 d imp NSc NRe = ᎏ , NFr = ᎏ µc g
d p = particle or drop diameter; σ = interfacial tension, N/m; φ = volume fraction dispersed phase; a = interfacial volume, 1/m; and kcαD c2/3 implies rigid drops. Negligible drop coalescence. Average absolute deviation—19.71%. Graphical comparison given by Ref. 153.
[153, 156] p. 78
MASS TRANSFER TABLE 5-26
5-71
Mass-Transfer Correlations for Particles, Drops, and Bubbles in Agitated Systems (Concluded )
Situation J. Gas bubble swarms in sparged tank reactors
Comments E = Empirical, S = Semiempirical, T = Theoretical
Correlation ν 1/3 P/VL a qG ν 1/3 b k′L a ᎏ2 =C ᎏ ᎏ ᎏ2 g ρ(νg 4)1/3 VL g Rushton turbines: C = 7.94 × 10−4, a = 0.62, b = 0.23. Intermig impellers: C = 5.89 × 10−4, a = 0.62, b = 0.19.
K. 5-26-J
P k′L a = 2.6 × 10−2 ᎏ VL
L. 5-26-J
P k′L a = 2.0 × 10−3 ᎏ VL
M. 5-26-J, baffled tank with standard blade Rushton impeller
P k′L a = 93.37 ᎏ VL
N. 5-26-M
d 2imp µ eff k′L a ᎏ = 7.57 ᎏ D ρD
References*
[E] Use arithmetic concentration difference. Done for biological system, O2 transfer. htank /Dtank = 2.1; P = power, kW. VL = liquid volume, m3. qG = gassing rate, m3/s. k′L a = s −1. Since a = m2/m3, ν = kinematic viscosity, m2/s. Low viscosity system. Better fit claimed with qG /VL than with uG (see 5-26-K to O).
[143]
u 0.5 G
[E] Use arithmetic concentration difference. Ion free water VL < 2.6, uG = superficial gas velocity in m/s. 500 < P/VL < 10,000. P/VL = watts/m3, VL = liquid volume, m3.
[115, 137]
u 0.2 G
[E] Use arithmetic concentration difference. Water with ions. 0.002 < VL < 4.4, 500 < P/VL < 10,000. Same definitions as 5-26-J.
[115, 117]
[E] Air-water. Same definitions as 5-26-J. 0.005 < uG < 0.025, 3.83 < N < 8.33, 400 < P/VL < 7000 h = Dtank = 0.305 or 0.610 m. VG = gas volume, m3, N = stirrer speed, rpm. Method assumes perfect liquid mixing.
[92, 115]
[E] Use arithmetic concentration difference. CO2 into aqueous carboxyl polymethylene. Same definitions as 5-26-M. µeff = effective viscosity from power law model, Pa⋅s. σ = surface tension liquid, N/m.
[115, 129]
[E] Use arithmetic concentration difference. O2 into aqueous glycerol solutions. O2 into aqueous millet jelly solutions. Same definitions as 5-26-M.
[115, 167]
[E] Use arithmetic concentration difference. Solids are glass beads, d p = 320 µm. ε s = solids holdup m3/m3 liquid, (k′L a)o = mass transfer in absence of solids. Ionic salt solution— noncoalescing.
[71, 144]
[E] Use arithmetic concentration difference. Variety of solids, d p > 150 µm (glass, amberlite, polypropylene). Tap water. Slope very different than item P. Coalescence may have occurred.
[71, 112]
0.4
0.7
0.76
u G0.45
µ ᎏ µ
0.5
d 2impNρL × ᎏ µ eff
G
0.694
eff
u d ᎏ σ 1.11
G
0.447
d imp = impeller diameter, m; D = diffusivity, m2/s O. 5-26-M, bubbles
P. Gas bubble swarm in sparged stirred tank reactor with solids present
Q. 5-26-P
k′Lad 2imp d 2impNρ ᎏ = 0.060 ᎏ µ eff D
d N µ u ᎏ ᎏ g σ 2 imp
2
0.19
eff
k′La ᎏ = 1 − 3.54(εs − 0.03) (k′La)o 300 ≤ P/Vrx < 10,000 W/m3, 0.03 ≤ εs ≤ 0.12 0.34 ≤ uG ≤ 4.2 cm/s, 5 < µ L < 75 Pa⋅s k′La ᎏ = 1 − εs (k′La)o
G
0.6
See also Table 5-25. *See pages 5-7 and 5-8 for references.
The gas-phase rate coefficient kˆ G is not affected by the fact that a chemical reaction is taking place in the liquid phase. If the liquidphase chemical reaction is extremely fast and irreversible, the rate of absorption may be governed completely by the resistance to diffusion in the gas phase. In this case the absorption rate may be estimated by knowing only the gas-phase rate coefficient kˆ G or else the height of one gas-phase transfer unit HG = GM /(kˆ Ga). It should be noted that the highest possible absorption rates will occur under conditions in which the liquid-phase resistance is negligible and the equilibrium back pressure of the gas over the solvent is zero. Such situations would exist, for instance, for NH3 absorption into an acid solution, for SO2 absorption into an alkali solution, for vaporization of water into air, and for H2S absorption from a dilute-gas stream into a strong alkali solution, provided there is a large excess of reagent in solution to consume all the dissolved gas. This is known as the gas-phase mass-transfer limited condition, when both the liquidphase resistance and the back pressure of the gas equal zero. Even when the reaction is sufficiently reversible to allow a small back pres-
sure, the absorption may be gas-phase-controlled, and the values of kˆ G and HG that would apply to a physical-absorption process will govern the rate. The liquid-phase rate coefficient kˆ L is strongly affected by fast chemical reactions and generally increases with increasing reaction rate. Indeed, the condition for zero liquid-phase resistance (m/kˆ L) implies that either the equilibrium back pressure is negligible, or that kˆ L is very large, or both. Frequently, even though reaction consumes the solute as it is dissolving, thereby enhancing both the mass-transfer coefficient and the driving force for absorption, the reaction rate is slow enough that the liquid-phase resistance must be taken into account. This may be due either to an insufficient supply of a second reagent or to an inherently slow chemical reaction. In any event the value of kˆ L in the presence of a chemical reaction normally is larger than the value found when only physical absorption occurs, kˆ L0 . This has led to the presentation of data on the effects of chemical reaction in terms of the “reaction factor” or “enhancement factor” defined as
5-72
HEAT AND MASS TRANSFER
TABLE 5-27
Mass Transfer Correlations for Fixed and Fluidized Beds Transfer is to or from particles.
Situation A. Heat or mass transfer in packed bed for gases and liquids
(shape factor, Ψ)
Comments E = Empirical, S = Semiempirical, T = Theoretical
Correlation −0.51 jD = jH = 0.91ΨN Re , 0.01 < NRe < 50 1/3 Equivalent NSh = 0.91ΨN 0.49 Re N Sc . −0.41 jD = jH = 0.61ΨN Re , 50 < NRe < 1000 1/3 Equivalent NSh = 0.61ΨN 0.59 Re N Sc .
particle 1.00
sphere 0.91
[E] Different constants and shape factors reported in other references. Evaluate terms at film temperature or composition. cylinder 0.81
References* [100] p. 194 [169]
k′d NSh NSh = ᎏs , jD = ᎏ NRe N 1/3 D Sc vsuper ρ NRe = ᎏ , v super = superficial velocity µΨa surface area a = ᎏᎏ = 6(1 − ε)/d p volume For spheres, dp = diameter. ar t. u .A re a For nonspherical: d p = 0.567 P Srf Results are from too-short beds—use with caution.
B. For gases, fixed and fluidized beds, Gupta and Thodos correlation
2.06 jH = jD = ᎏ , 90 ≤ NRe ≤ A εN 0.575 Re
v superd p ρ [E] For spheres. NRe = ᎏ µ
[95, 96]
Equivalent:
A = 2453 [Ref. 151], A = 4000 [Ref. 100]. For NRe > 1900, j H = 1.05j D. Heat transfer result is in absence of radiation.
[100] p. 195 [151]
2.06 0.425 1/3 N Sc NSh = ᎏ N Re ε For other shapes:
k′d NSh = ᎏs D
ε jD ᎏ = 0.79 (cylinder) or 0.71 (cube) (ε j D)sphere Graphical results are available for NRe from 1900 to 10,300. C. For gases, for fixed beds, Petrovic and Thodos correlation
0.357 0.641 1/3 NSh = ᎏ N Re N Sc ε
D. For gases and liquids, fixed and fluidized beds
0.4548 jD = ᎏ , 10 ≤ NRe ≤ 2000 εN 0.4069 Re NSh k′d , NSh = ᎏs jD = ᎏ NReN 1/3 D Sc
E. For gases, fixed beds
F. For liquids, fixed bed, Wilson and Geankoplis correlation
0.499 jD = ᎏ 0.382 εN Re 1.09 jD = ᎏ , 0.0016 < NRe < 55 2/3 εN Re 165 ≤ NSc ≤ 70,600, 0.35 < ε < 0.75 Equivalent: 1.09 1/3 NSh = ᎏ N 1/3 Re N Sc ε 0.25 jD = ᎏ , 55 < NRe < 1500, 165 ≤ NSc ≤ 10,690 εN 0.31 Re Equivalent: 0.25 0.69 1/3 N Sc NSh = ᎏ N Re ε
[E] Packed spheres, deep beds, 3 < NRe < 900 can be extrapolated to NRe < 2000. Corrected for axial dispersion with axial Peclet number = 2.0. Prediction is low at low NRe. NRe defined as in 5-27-A and B.
[130][141] p. 214 [163]
[E] Packed spheres, deep bed. Average deviation ⫾20%, NRe = dpvsuperρ/µ. Can use for fluidized beds. 10 ≤ NRe ≤ 4000.
[85][91] p. 447
[E] Data on sublimination of naphthalene spheres dispersed in inert beads. 0.1 < NRe < 100, NSc = 2.57. Correlation coefficient = 0.978.
[101]
[E] Beds of spheres,
[91] p. 448
d pVsuperρ NRe = ᎏ µ Deep beds. k′d s NSh = ᎏ D
[100] p. 195 [151] p. 287 [166]
MASS TRANSFER TABLE 5-27
5-73
Mass Transfer Correlations for Fixed and Fluidized Beds (Continued )
Situation G. For liquids, fixed beds, Ohashi et al. correlation
Comments E = Empirical, S = Semiempirical, T = Theoretical
Correlation E 1/3d 4/3 k′d p ρ NSh = ᎏs = 2 + 0.51 ᎏ µ D
References*
0.60
N 1/3 Sc
E = Energy dissipation rate per unit mass of fluid
[S] Correlates large amount of published data. Compares number of correlations, v r = relative velocity, m/s. In packed bed, v r = v super /ε.
[125]
v 3r = 50(1 − ε)ε 2 CDo ᎏ , m2/s 3 dp 50(1 − ε)CD = ᎏᎏ ε
ᎏ d v 3super p
General form: α E 1/3 D 4/3 p ρ β NSh = 2 + K ᎏᎏ N Sc µ
applies to single particles, packed beds, two-phase tube flow, suspended bubble columns, and stirred tanks with different definitions of E.
CDo = single particle drag coefficient at v super cal−m culated from CDo = AN Re . i NRe 0 to 5.8 5.8 to 500 >500
A 24 10 0.44
m 1.0 0.5 0
Ranges for packed bed: 0.001 < NRe < 1000 505 < NSc < 70600 E 1/3d p4/3ρ 0.2 < ᎏᎏ < 4600 µ Compares different situations versus general correlation. See also 5-24-F.
H. For liquids, fixed and fluidized beds
1.1068 ε jD = ᎏ , 1.0 < N Re ≤ 10 N 0.72 Re NSh k′d ε jD = ᎏ , NSh = ᎏs NRe N 1/3 D Sc
I. For gases and liquids, fixed and fluidized beds, Dwivedi and Upadhyay correlation
0.765 0.365 +ᎏ ε jD = ᎏ N 0.82 N 0.386 Re Re Gases: 10 ≤ N Re ≤ 15,000. Liquids: 0.01 ≤ N Re ≤ 15,000.
[E] Spheres: d pv superρ NRe = ᎏ µ [E] Deep beds of spheres, NSh jD = ᎏ N Re N 1/3 Sc
[84][91] p. 448
[84][100] p. 196
d pv superρ k′d s NRe = ᎏ , NSh = ᎏ µ D Based on 20 gas studies and 17 liquid studies. Recommended instead of 5-27-D or F. J. For gases and liquids, fixed bed
K. For liquids, fixed and fluidized beds, Rahman and Streat correlation L. For liquids and gases, Ranz and Marshall correlation
−0.415 , 10 ≤ NRe ≤ 2500 jD = 1.17N Re
[E] Spheres:
k′ pBM 2/3 jD = ᎏ ᎏ NSc vav P
d pv superρ NRe = ᎏ µ
Comparison with other results are shown.
Variation in packing that changes ε not allowed for. Extensive data referenced. 0.5 < NSc < 15,000.
0.86 NSh = ᎏ NReN1/3 Sc , 2 ≤ NRe ≤ 25 ε
[E] Can be extrapolated to NRe = 2000. NRe = dpvsuperρ/µ. Done for neutralization of ion exchange resin.
[134]
[E] Based on freely falling, evaporating spheres (see 5-24-C). Has been applied to packed beds. Prediction is low compared to experimental data for packed beds. Limit of 2.0 at low NRe is too high. Not corrected for axial dispersion.
[135][141] p. 214 [163][168] p. 106
k′d 1/2 NSh = ᎏ = 2.0 + 0.6N 1/3 Sc N Re D d pv superρ NRe = ᎏ µ
M. For liquids and gases, Wakao and Funazkri correlation
0.6 NSh = 2.0 + 1.1N 1/3 Sc N Re , 3 < NRe < 10,000
k′film d p NSh = ᎏ t D Graphical comparison with data shown by Refs. 141, p. 215, and 163.
[146] p. 241
ρf vsuperρ [E] NRe = ᎏ µ
[141] p. 214 [163]
Correlate 20 gas studies and 16 liquid studies. Corrected for axial dispersion with:
[165] p. 376 [168] p. 106
εDaxial ᎏ = 10 + 0.5NScNRe D Daxial is axial dispersion coefficient.
5-74
HEAT AND MASS TRANSFER
TABLE 5-27
Mass Transfer Correlations for Fixed and Fluidized Beds (Concluded )
Situation
N. Liquid fluidized beds
Comments E = Empirical, S = Semiempirical, T = Theoretical
Correlation (2ξ/εm)(1 − ε)1/2 2ξ/ε m + ᎏᎏ − 2 tan h (ξ/ε m) [1 − (1 − ε)1/3]2 NSh = ᎏᎏᎏᎏᎏ ξ/ε m ᎏᎏ − tan h (ξ/ε m) 1 − (1 − ε1/2)
where α 1 1/2 − 1 ᎏ N 1/3 ξ= ᎏ Sc N Re (1 − ε)1/3 2
This simplifies to: α2
ε 1 − 2m 1 NSh = ᎏ ᎏ−1 (1 − ε)1/3 (1 − ε)1/3
O. Liquid fluidized beds
ᎏ2 N
0.306 ρ s − ρ NSh = 0.250N 0.023 ᎏ Re N Ga ρ
m = 1 for NRe > 2; m = 0.5 for NRe < 1.0; ε = voidage; α = const. Best fit data is α = 0.7. Comparison of theory and experimental ion exchange results in Ref. 113.
0.410 N Sc
(ε < 0.85)
[E] Correlate amount of data from literature. Compare large number of published correlations.
(ε > 0.85)
k′L d p d p ρvsuper NSh = ᎏ , NRe = ᎏ D µ
0.297 0.404 N Sc
0.300 0.400 N Sc
[113, 123, 138]
Vsuper d pξ NRe = ᎏ µ
2/3 N Sc (NRe < 0.1)
This can be simplified (with slight loss in accuracy at high ε) to 0.323 ρ s − ρ NSh = 0.245N Ga ᎏ ρ
k′L d p NSh = ᎏ D
Re
0.282
−0.057 0.332 ρs − ρ NSh = 0.304N Re N Ga ᎏ ρ
[S] Modification of theory to fit experimental data. For spheres, m = 1, NRe > 2.
References*
[160]
µ d p3 ρ 2g , NSc = ᎏ NGa = ᎏ µ2 ρD 1.6 < NRe < 1320, 2470 < NGa < 4.42 × 106 ρs − ρ 0.27 < ᎏ < 1.114, 305 < NSc < 1595 ρ Predicts very little dependence of NSh on velocity.
P. Liquid film flowing over solid particles with air present, trickle bed reactors, fixed bed
Q. Supercritical fluids in packed bed
kL 1/3 NSh = ᎏ = 1.8N 1/2 Re N Sc , 0.013 < NRe < 12.6 aD
L [E] NRe = ᎏ aµ
two-phases, liquid trickle, no forced flow of gas. 1/2 1/3 NSh = 0.8N Re N Sc , one-phase, liquid only.
L = superficial liquid flow rate, kg/m2s. a = surface area/col. volume, m2/m3. Irregular granules of benzoic acid, 0.29 ≤ dp ≤ 1.45 cm.
2 NSh N Re N 1/3 Sc = 0.1813 ᎏ ᎏᎏ NGr (NSc NGr)1/4
1/4
1/3 3/4 (N 1/2 Re N Sc )
2 1/3 N Re N Sc + 1.2149 ᎏ NGr
R. Supercritical fluids in packed bed
3/4
1/3 (N 1/2 NSh Re N Sc ) = 0.5265 ᎏᎏ ᎏᎏ (NSc NGr)1/4 (NSc NGr)1/4
2 N Re N 1/3 Sc + 2.48 ᎏ NGr
0.6439
− 0.01649
[E] Natural and forced convection, 4 < NRe < 135.
[119]
[E] Natural and forced convection. 0.3 < NRe < 135. Improvement of correlation in Q.
[116]
1/3
1.6808
− 0.8768
[142]
1.553
NOTE: For NRe < 3 convective contributions which are not included may become important. Use with logarithmic concentration difference (integrated form) or with arithmetic concentration difference (differential form). *See pages 5-7 and 5-8 for references.
φ = kˆ L / kˆ L0 ≥ 1 (5-300) 0 ˆ ˆ where kL = mass-transfer coefficient with reaction and kL = masstransfer coefficient for pure physical absorption. It is important to understand that when chemical reactions are involved, this definition of kˆ L is based on the driving force defined as the difference between the concentration of unreacted solute gas at the interface and in the bulk of the liquid. A coefficient based on the total of both unreacted and reacted gas could have values smaller than the physical-absorption mass-transfer coefficient kˆ L0 . When liquid-phase resistance is important, particular care should be taken in employing any given set of experimental data to ensure that the equilibrium data used conform with those employed by the original author in calculating values of kˆ L or HL. Extrapolation to widely different concentration ranges or operating conditions should
be made with caution, since the mass-transfer coefficient kˆ L may vary in an unexpected fashion, owing to changes in the apparent chemicalreaction mechanism. Generalized prediction methods for kˆ L and HL do not apply when chemical reaction occurs in the liquid phase, and therefore one must use actual operating data for the particular system in question. A discussion of the various factors to consider in designing gas absorbers and strippers when chemical reactions are involved is presented by Astarita, Savage, and Bisio, Gas Treating with Chemical Solvents, Wiley (1983) and by Kohl and Ricsenfeld, Gas Purification, 4th ed., Gulf (1985). Effective Interfacial Mass-Transfer Area a In a packed tower of constant cross-sectional area S the differential change in solute flow per unit time is given by −d(GMSy) = NAa dV = NAaS dh
(5-301)
MASS TRANSFER
5-75
TABLE 5-28 Mass Transfer Correlations for Packed Two-Phase Contactors—Absorption, Distillation, Cooling Towers, and Extractors (Packing Is Inert) Situation A. Absorption, counter-current, liquid-phase coefficient HL, Sherwood and Holloway correlation for random packings
Comments E = Empirical, S = Semiempirical, T = Theoretical
Correlations
L n 0.5 HL = a L ᎏ N Sc,L , L = lb/hr ft 2 µL LM HL = ᎏ kˆ L a L M = lbmoles/hr ft 2, kˆ L = lbmoles/hr ft 2, a = ft 2/ft 3, µ L in lb/(hr ft). Ranges for 5-28-B (G and L) Packing
aG
b
c
G
L
aL
References*
[E] From experiments on desorption of sparingly soluble gases from water. Graphs [Ref. 146], p. 606. Equation is dimensional. A typical value of n is 0.3 [Ref. 91] p. 633 has constants in kg, m, and s units for use in 5-28-A and B with kˆ G in kgmole/s m2 and kˆ L in kgmole/s m2 (kgmole/m3). Constants for other packings are given by Refs. 121 p. 187 and 161, p. 239.
[121] p. 187 [122] p. 714 [146] p. 606 [164] p. 660
[E] Based on ammonia-water-air data in Fellinger’s 1941 MIT thesis. Curves: Refs. 121, p. 186 and 146 p. 607. Constants given in 5-28-A. The equation is dimensional.
[91] p. 633 [121] p. 189 [146] p. 607 [164] p. 660
[E] Compared napthalene sublimination to aqueous absorption to obtain kˆ G , a, and kˆ L separately. Raschig rings and Berl saddles. d p = diameter of sphere with same surface area as packing piece. ε Lo = operating void space = ε − φ Li , where ε = void fraction w/o liquid, and φ Li = liquid holdup. Same definition as 5-28-A and B. Onda et al. correlation (5-28-D) is preferred. G = ρGv super,gas
[76][123] p. 174, 186 [151, 158] [161] p. 203
n
Raschig rings 3/8 inch 1 1 2
2.32 7.00 6.41 3.82
0.45 0.39 0.32 0.41
0.47 0.58 0.51 0.45
32.4 0.30 0.811 0.30 1.97 0.36 5.05 0.32
0.74 0.24 0.40 0.45
200–500 200–800 200–600 200–800
500–1500 0.00182 0.46 400–500 0.010 0.22 500–4500 — — 500–4500 0.0125 0.22
Berl saddles 1/2 inch 1/2 1 1.5
200–700 200–800 200–800 200–1000
500–1500 0.0067 0.28 400–4500 — — 400–4500 0.0059 0.28 400–4500 0.0062 0.28
Range for 5-28-A is 400 < L < 15,000 lb/hr ft2 B. Absorption counter-current, gasphase coefficient HG, for random packing
0.5 a G(G) bN Sc,v HG = ᎏᎏ , G = lb/hr ft 2 (L) c
GM HG = ᎏ kˆ G a G M = lbmoles/hr ft 2, kˆ G = lbmoles/hr ft 2.
C. Absorption, counter-current, gasliquid individual coefficients and interfacial area, Shulman data for random packings
2/3 dpG k GN Sc,v ᎏ = 1.195 ᎏᎏ GM µ G(1 − ε Lo)
dp L kˆ L d p ᎏ = 25.1 ᎏ DL µL
−0.36
0.45
N 0.5 Sc,L
Interfacial area a per volume given for Racshig rings and Berl saddles in graphical form by Refs. 78 and 121 p. 178, and in equation form by Ref. 161, p. 205. Liquid holdups are given by Refs. 161 (p. 206), 148, or 121, p. 174. D. Absorption and and distillation, counter-current, gas and liquid individual coefficients and wetted surface area, Onda et al. correlation for random packings
G k′G RT ᎏ=A ᎏ a pDG a pµ G
0.7
−2.0 N 1/3 Sc,G (a pd′p)
A = 5.23 for packing ≥ 1/2 inch (0.012 m) A = 2.0 for packing < 1/2 inch (0.012 m) k′G = lbmoles/hr ft 2 atm [kg mol/s m2 (N/m2)] ρL k L′ ᎏ µ Lg
1/3
L = 0.0051 ᎏ aw µL
2/3
−1/2 N Sc,L (a p d′p)0.4
[E] Gas absorption and desorption from water and organics plus vaporization of pure liquids for Raschig rings, saddles, spheres, and rods. d′p = nominal packing size, a p = dry packing surface area/volume, a w = wetted packing surface area/volume. Equations are dimensionally consistent, so any set of consistent units can be used. σ = surface tension, dynes/cm.
k′L = lbmoles/hr ft 2 (lbmoles/ft 3) [kgmoles/s m2 (kgmoles/m3)] aw ᎏ = 1 − exp ap
σ −1.45 ᎏc σ
ᎏ aµ
L2a p × ᎏ ρ L2 g
0.75
0.1
L
p
L
ᎏ ρ σa −0.05
L
L
p
0.2
Critical surface tensions, σ C = 61 (ceramic), 75 (steel), 33 (polyethylene), 40 (PVC), 56 (carbon) dynes/cm. Graphical comparison with data in Ref. 126.
L 4 < ᎏ < 400 aw µ L G 5 < ᎏ < 1000 ap µG Most data ± 20% of correlation, some ± 50%.
[76][88] p. 399 [111] p. 380 [126][159] p. 355
5-76
HEAT AND MASS TRANSFER
TABLE 5-28 Mass Transfer Correlations for Packed Two-Phase Contactors—Absorption, Distillation, Cooling Towers, and Extractors (Packing Is Inert) (Continued ) Situation E. Distillation and absorption, counter-current, random packings, modification of Onda correlation, Bravo and Fair correlation
Comments E = Empirical, S = Semiempirical, T = Theoretical
Correlations Use Onda’s correlations (5-28-D) for k′G and k′L. Calculate: G L HG = ᎏ , HL = ᎏ k G′ aePMG k′LaeρL HOG = HG + λHL where m λ=ᎏ LM/GM Using
References*
[E] Use’s Bolles & Fair (Ref. 75) data base to determine new effective area ae to use with Onda et al. (Ref. 126) correlation. Same definitions as 5-28-D. P = total pressure, atm; MG = gas, molecular weight; m = local slope of equilibrium curve; LM/GM = slope operating line; Z = height of packing in feet. Equation for ae is dimensional. Fit to data for effective area quite good for distillation. Good for absorption at low values of (Nca,L × NRe,G), but correlation is too high at higher values of (NCa,L × NRe,G).
[76]
[E] Based on oxygen transfer from water to air 77°F. Liquid film resistance controls. (Dwater @ 77°F = 2.4 × 10−5). Equation is dimensional. Data was for thin-walled polyethylene Raschig rings. Correlation also fit data for spheres. Fit ⫾25%. See Reiss for graph.
[136] [142] p. 217
σ0.5 ae = 0.498ap ᎏ (NCa,LNRe,G)0.392 Z0.4 where
6G NRe,G = ᎏ , apµG LµL NCa,L = ᎏ (dimensionless) ρLσgc F. Absorption, co-current downward flow, random packings
Air-oxygen-water results correlated by k′La = 0.12EL0.5. Extended to other systems.
DL k′La = 0.12EL0.5 ᎏ 2.4 × 105
0.5
∆p EL = ᎏ vL ∆L 2-phase −1 k′La = s DL = cm/s EL = ft, lbf/s ft3 vL = superficial liquid velocity, ft/s
∆p ᎏ = pressure loss in two-phase flow = lbf/ft2 ft ∆L k′G a = 2.0 + 0.91EG2/3 for NH3 ∆p Eg = ᎏ vg ∆L 2-phase vg = superficial gas velocity, ft/s.
G. Absorption, stripping, distillation, counter-current, HL, and HG, random packings, Cornell et al. correlation, and Bolles and Fair correlation
For Raschig rings, Berl saddles, and spiral tile: Z 0.15 φCflood 0.5 HL = ᎏ N Sc,L ᎏ 3.28 3.05 Cflood = 1.0 if below 40% flood—otherwise, use Fig. 5-28. φ shown in Fig. 5-29 for different packings and sizes. Range 0.02 < φ < 0.300. Aψ(d′col)mZ0.33N0.5 Sc,G ᎏᎏᎏᎏ HG = µL 0.16 ρwater 1.25 σwater 0.8 n L ᎏᎏ ᎏᎏ ᎏᎏ µwater ρL σL
[E] Ammonia absorption into water from air at 70°F. Gas-film resistance controls. Thin-walled polyethylene Raschig rings and 1-inch Intalox saddles. Fit ⫾25%. See Reiss for fit. Terms defined as above. [E] Z = packed height, m of each section with its own liquid distribution. The original work is reported in English units. Cornell et al. (Ref. 81) review early literature. Improved fit of Cornell’s φ values given by Bolles and Fair (Refs. 74 and 75) and in Fig. 5-29.
A = 0.017 (rings) or 0.029 (saddles) d′col = column diameter in m (if diameter > 0.6 m, use d′col = 0.6) m = 1.24 (rings) or 1.11 (saddles) n = 0.6 (rings) or 0.5 (saddles) ψ is given in Fig. 5-30. Range: 25 < ψ < 190 m.
L = liquid rate, kg/(sm2), µwater = 1.0 Pa ⋅ s, ρwater = 1000 kg/m3, σwater = 72.8 mN/m (72.8 dynes/cm). HG and HL will vary from location to location. Design each section of packing separately.
[136]
[74, 75, 81] [100] p. 428 [111] p. 381 [151] p. 353 [164] p. 651
MASS TRANSFER
5-77
TABLE 5-28 Mass Transfer Correlations for Packed Two-Phase Contactors—Absorption, Distillation, Cooling Towers, and Extractors (Packing Is Inert) (Concluded ) Situation H. Distillation and absorption. Counter-current flow. Structured packings. Gauze-type with triangular flow channels, Bravo, Rocha, and Fair correlation
Correlations
[T] Check of 132 data points showed average deviation 14.6% from theory. Johnstone and Pigford [Ref. 105] correlation (5-22-D) has exponent on NRe rounded to 0.8. Assume gauze packing is completely wet. Thus, aeff = ap to calculate HG and HL. Same approach may be used generally applicable to sheet-metal packings, but they will not be completely wet and need to estimate transfer area. L = liquid flux, kg/s m2. G = vapor flux, kg/s m2. Fit to data shown in Ref. 77.
Equivalent channel:
1 1 deq = Bh ᎏ + ᎏ B + 2S 2S
Comments E = Empirical, S = Semiempirical, T = Theoretical
Use modified correlation for wetted wall column (See 5-22-D) k′vdeq 0.333 NSh,v = ᎏ = 0.0338N0.8 Re,vNSc,v Dv
References* [77] [87] p. 310, 326 [159] p. 356, 362
G L HG = ᎏ , HL = ᎏ k′vapρv k′LapρL
deqρv(Uv,eff + UL,eff) NRe,v = ᎏᎏ µv where effective velocities Uv,super Uv,eff = ᎏ ε sin θ 3Γ ρL2 g UL,eff = ᎏ ᎏ 2ρL 3µLΓ
0.333
L ,Γ=ᎏ Per
Perimeter 4S + 2B Per = ᎏᎏ = ᎏ Area Bh Calculate k′L from penetration model (use time for liquid to flow distance s). k′L = 2(DLUL,eff /πS)1/2. I. High-voidage packings, cooling towers, splash-grid packings
J. Liquid-liquid extraction, packed towers
K. Liquid-liquid extraction in Rotating-disc contactor (RDC)
(Ka)HVtower L −n′ ᎏᎏ = 0.07 + A′N′ ᎏ L Ga A′ and n′ depend on deck type (Ref. 107), 0.060 ≤ A′ ≤ 0.135, 0.46 ≤ n′ ≤ 0.62. General form fits the graphical comparisons (Refs. 146 and 164).
lb water lb/(h)(ft3) ᎏ lb dry air Vtower = tower volume, ft3/ft2. If normal packings are used, use absorption masstransfer correlations or Ref. 88, p. 452. [156] p. 79
[70][156] p. 79
kd,RDC N H ᎏ = 1.0 + 1.825 ᎏ ᎏ kd NCr Dtank
kc, kd are for drops (Table 5-25) N = impeller speed Breakage occurs when N > NCr. Maximum enhancement before breakage was factor of 2.0. H = compartment height, Dtank = tank diameter, σ = interfacial tension, N/m. Done in 0.152 and 0.600 m RDC.
See Table 5-26-F, G, H, and I.
[E]
kc,RDC N ᎏ = 1.0 + 2.44 ᎏ kc NCr
2.5
ᎏ D H
tank
*See pages 5-7 and 5-8 for references.
[107][121] p. 220 [146] p. 286 [164] p. 681
[E]
Use k values for drops (Table 5-25). Enhancement due to packing is at most 20%. Packing decreases drop size and increases interfacial area.
σ NCr = 7.6 × 10−4 ᎏ ddrop µc
L. Liquid-liquid extraction, stirred tanks
[E] General form. Ga = lb dry air/hr ft2. L = lb/h ft2, N′ = number of deck levels. (Ka)H = overall enthalpy transfer coefficient =
5-78
HEAT AND MASS TRANSFER
Liquid-film correction factor (Table 5-28-G) for operation at high percent of flood. [Cornell et al., Chem. Eng. Prog., 56(8), 68 (1960).]
FIG. 5-28
where a = interfacial area effective for mass transfer per unit of packed volume and V = packed volume. Owing to incomplete wetting of the packing surfaces and to the formation of areas of stagnation in the liquid film, the effective area normally is significantly less than the total external area of the packing pieces. The effective interfacial area depends on a number of factors, as discussed in a review by Charpentier [Chem. Eng. J., 11, 161 (1976)]. Among these factors are (1) the shape and size of packing, (2) the packing material (for example, plastic generally gives smaller interfacial areas than either metal or ceramic), (3) the liquid mass velocity, and (4), for small-diameter towers, the column diameter. Whereas the interfacial area generally increases with increasing liquid rate, it apparently is relatively independent of the superficial gas mass velocity below the flooding point. According to Charpentier’s review, it appears valid to assume that the interfacial area is independent of the column height when specified in terms of unit packed volume (i.e., as a). Also, the existing data for chemically reacting gas-liquid systems (mostly aqueous electrolyte solutions) indicate that the interfacial area is independent of the chemical system. However, this situation may not hold true for systems involving large heats of reaction. Rizzuti et al. [Chem. Eng. Sci., 36, 973 (1981)] examined the influence of solvent viscosity upon the effective interfacial area in packed columns and concluded that for the systems studied the effective interfacial area a was proportional to the kinematic viscosity raised to the 0.7 power. Thus, the hydrodynamic behavior of a packed absorber is strongly affected by viscosity effects. Surface-tension effects also are important, as expressed in the work of Onda et al. (see Table 5-28-D). In developing correlations for the mass-transfer coefficients kˆ G and kˆ L, the various authors have assumed different but internally compatible correlations for the effective interfacial area a. It therefore would be inappropriate to mix the correlations of different authors unless it has been demonstrated that there is a valid area of overlap between them. Volumetric Mass-Transfer Coefficients Kˆ Ga and Kˆ La Experimental determinations of the individual mass-transfer coefficients kˆ G and kˆ L and of the effective interfacial area a involve the use of extremely difficult techniques, and therefore such data are not plentiful. More often, column experimental data are reported in terms of overall volumetric coefficients, which normally are defined as follows: and
K′Ga = nA /(hTSpT ∆y°1m)
(5-302)
KLa = nA /(hTS ∆x°1m)
(5-303)
where K′Ga = overall volumetric gas-phase mass-transfer coefficient, KLa = overall volumetric liquid-phase mass-transfer coefficient, nA =
Hl correlation for various packings (Table 5-28-G). To convert meters to feet, multiply by 3.281; to convert pounds per hour-square foot to kilograms per second-square meter, multiply by 0.001356; and to convert millimeters to inches, multiply by 0.0394. [Bolles and Fair, Inst. Chem. Eng. Symp. Ser., no. 56, 3.3/35 (1969).]
FIG. 5-29
overall rate of transfer of solute A, hT = total packed depth in tower, S = tower cross-sectional area, pT = total system pressure employed during the experiment, and ∆x°1m and ∆y°1m are defined as (y − y°)1 − (y − y°)2 ∆y°1m = ᎏᎏᎏ ln [(y − y°)1/(y − y°)2]
(5-304)
MASS TRANSFER
5-79
vary with the total system pressure except when the liquid-phase resistance is negligible (i.e., when either m = 0, or kˆ La is very large, or both). Extrapolation of KGa data for absorption and stripping to conditions other than those for which the original measurements were made can be extremely risky, especially in systems involving chemical reactions in the liquid phase. One therefore would be wise to restrict the use of overall volumetric mass-transfer-coefficient data to conditions not too far removed from those employed in the actual tests. The most reliable data for this purpose would be those obtained from an operating commercial unit of similar design. Experimental values of HOG and HOL for a number of distillation systems of commercial interest are also readily available. Extrapolation of the data or the correlations to conditions that differ significantly from those used for the original experiments is risky. For example, pressure has a major effect on vapor density and thus can affect the hydrodynamics significantly. Changes in flow patterns affect both masstransfer coefficients and interfacial area. Chilton-Colburn Analogy When a fluid moves over either a liquid or a solid surface, the eddy motion that causes mass transfer also causes heat transfer and fluid friction owing to the transfer of thermal energy and momentum respectively. This close similarity among the mechanisms for the transfer of mass, heat, and momentum was brought out in the Reynolds analogy (see Table 5-23-T), which stated that the following dimensionless ratios are equal: kˆ G/GM = h′/cpG = f/2 (5-306) where h′ = heat-transfer coefficient, cp = specific heat, G = mass flux, and f = friction factor. Experimental data for mass transfer into gas streams agree approximately with Eq. (5-306) when the Schmidt number is close to unity and in smooth, straight tubes or along flat plates when the pressure drop is due entirely to skin friction against the surface. It does not, however, agree for cases involving “form” drag as well as skin friction. Also, it does not account for the mass-transfer resistance of the region of fluid near the liquid or solid boundary in which mass transfer occurs principally by molecular (as opposed to turbulent) motion. Colburn [Trans. Am. Inst. Chem. Eng., 29, 174 (1933)] and Chilton and Colburn [Ind. Eng. Chem., 26, 1183 (1934)] showed empirically that the resistance of the laminar sublayer can be expressed by the following modification of the Reynolds analogy: ( kˆ G/GM)NSc2/3 = jM = (h′/cpG)NPr2/3 = jH = f/2 (5-307) Hg correlation for various packings (Table 5-28-G). To convert meters to feet, multiply by 3.281; to convert millimeters to inches, multiply by 0.03937. [Bolles and Fair, Inst. Chem. Eng. Symp. Ser., no. 56, 3.3/35 (1979).] FIG. 5-30
(x° − x)2 − (x° − x)1 ∆x°1m = ᎏᎏᎏ (5-305) ln [(x° − x)2/(x° − x)1] where subscripts 1 and 2 refer to the bottom and top of the tower respectively. Experimental K′Ga and KLa data are available for most absorption and stripping operations of commercial interest (see Sec. 15). The solute concentrations employed in these experiments normally are very low, so that KLa ⬟ Kˆ La and K′GapT ⬟ Kˆ Ga, where pT is the total pressure employed in the actual experimental-test system. Unlike the individual gas-film coefficient kˆ Ga, the overall coefficient Kˆ Ga will and
for turbulent flow through straight tubes (see Table 5-23-U) and across plane surfaces (see Table 5-21-G), and jM = jH ≤ f/2
(5-308)
for turbulent flow around cylinders (see Table 5-24-I), where jM = mass-transfer factor, jH = heat-transfer factor, NPr = cp µ/k = Prandtl number, and k = thermal conductivity; other symbols are as defined earlier. On occasion one will find that heat-transfer-rate data are available for a system in which mass-transfer-rate data are not readily available. The Chilton-Colburn analogy provides a procedure for developing estimates of the mass-transfer rates based on heat-transfer data. Extrapolation of experimental jM or jH data obtained with gases to predict liquid systems (and vice versa) should be approached with caution, however. When pressure-drop or friction-factor data are available, one may be able to place an upper bound on the rates of heat and mass transfer, according to Eq. (5-308).
Section 6
Fluid and Particle Dynamics*
James N. Tilton, Ph.D., P.E., Senior Consultant, Process Engineering, E. I. du Pont de Nemours & Co.; Member, American Institute of Chemical Engineers; Registered Professional Engineer (Delaware)
FLUID DYNAMICS Nature of Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deformation and Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kinematics of Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressible and Incompressible Flow . . . . . . . . . . . . . . . . . . . . . . . Streamlines, Pathlines, and Streaklines . . . . . . . . . . . . . . . . . . . . . . . . One-dimensional Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rate of Deformation Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laminar and Turbulent Flow, Reynolds Number . . . . . . . . . . . . . . . . Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Macroscopic and Microscopic Balances . . . . . . . . . . . . . . . . . . . . . . . Macroscopic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass Balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Momentum Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical Energy Balance, Bernoulli Equation . . . . . . . . . . . . . . . . Microscopic Balance Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass Balance, Continuity Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cauchy Momentum and Navier-Stokes Equations . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Force Exerted on a Reducing Bend. . . . . . . . . . . . . . . . . Example 2: Simplified Ejector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 3: Venturi Flowmeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 4: Plane Poiseuille Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . Incompressible Flow in Pipes and Channels. . . . . . . . . . . . . . . . . . . . . . Mechanical Energy Balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Friction Factor and Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . Laminar and Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Entrance and Exit Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Residence Time Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noncircular Channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonisothermal Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Open Channel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-Newtonian Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Economic Pipe Diameter, Turbulent Flow . . . . . . . . . . . . . . . . . . . . .
6-4 6-4 6-4 6-4 6-5 6-5 6-5 6-5 6-5 6-5 6-5 6-6 6-6 6-6 6-6 6-6 6-6 6-7 6-7 6-7 6-7 6-7 6-8 6-8 6-8 6-8 6-9 6-9 6-9 6-9 6-9 6-10 6-11 6-11 6-11 6-12 6-12 6-12 6-13 6-14
Economic Pipe Diameter, Laminar Flow . . . . . . . . . . . . . . . . . . . . . . Vacuum Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slip Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frictional Losses in Pipeline Elements . . . . . . . . . . . . . . . . . . . . . . . . . . Equivalent Length and Velocity Head Methods . . . . . . . . . . . . . . . . . Contraction and Entrance Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 5: Entrance Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expansion and Exit Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fittings and Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 6: Losses with Fittings and Valves . . . . . . . . . . . . . . . . . . . . Curved Pipes and Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Screens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jet Behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow through Orifices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mach Number and Speed of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . Isothermal Gas Flow in Pipes and Channels. . . . . . . . . . . . . . . . . . . . Adiabatic Frictionless Nozzle Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 7: Flow through Frictionless Nozzle . . . . . . . . . . . . . . . . . . Adiabatic Flow with Friction in a Duct of Constant Cross Section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 8: Compressible Flow with Friction Losses . . . . . . . . . . . . . Convergent/Divergent Nozzles (De Laval Nozzles) . . . . . . . . . . . . . . Multiphase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquids and Gases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gases and Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluid Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perforated-Pipe Distributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 9: Pipe Distributor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slot Distributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turning Vanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perforated Plates and Screens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beds of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Flow Straightening Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluid Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stirred Tank Agitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pipeline Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tube Banks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transition Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-14 6-14 6-15 6-15 6-16 6-16 6-16 6-16 6-17 6-17 6-17 6-18 6-19 6-20 6-21 6-22 6-22 6-22 6-22 6-23 6-23 6-25 6-25 6-26 6-26 6-29 6-32 6-32 6-33 6-33 6-33 6-33 6-34 6-34 6-34 6-34 6-35 6-36 6-36 6-38
* The author acknowledges the contribution of B. C. Sakiadis, editor of this section in the sixth edition of the Handbook. 6-1
6-2
FLUID AND PARTICLE DYNAMICS
Laminar Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beds of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fixed Beds of Granular Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tower Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluidized Beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boundary Layer Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flat Plate, Zero Angle of Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . Cylindrical Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Flat Surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Cylindrical Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vortex Shedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coating Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Falling Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimum Wetting Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laminar Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Surface Traction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flooding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydraulic Transients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Water Hammer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-38 6-38 6-38 6-39 6-40 6-40 6-40 6-40 6-40 6-40 6-41 6-41 6-42 6-42 6-42 6-42 6-43 6-43 6-43 6-44 6-44
Example 10: Response to Instantaneous Valve Closing . . . . . . . . . . . Pulsating Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Closure Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eddy Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computational Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimensionless Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-44 6-44 6-44 6-45 6-45 6-46 6-46 6-47 6-48
PARTICLE DYNAMICS Drag Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Terminal Settling Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spherical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonspherical Rigid Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hindered Settling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-dependent Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Drops in Liquids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liquid Drops in Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wall Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-50 6-50 6-50 6-51 6-52 6-52 6-53 6-53 6-54 6-54
Nomenclature and Units* In this listing, symbols used in this section are defined in a general way and appropriate SI units are given. Specific definitions, as denoted by subscripts, are stated at the place of application in the section. Some specialized symbols used in the section are defined only at the place of application. Some symbols have more than one definition; the appropriate one is identified at the place of application. Symbol
Definition
SI units
U.S. customary units
a A b b c cf C Ca C0 CD d D De Dij
Pressure wave velocity Area Wall thickness Channel width Acoustic velocity Friction coefficient Conductance Capillary number Discharge coefficient Drag coefficient Diameter Diameter Dean number Deformation rate tensor components Elastic modulus Energy dissipation rate Eotvos number Fanning friction factor Vortex shedding frequency Force Cumulative residence time distribution Froude number Acceleration of gravity Mass flux Enthalpy per unit mass Liquid depth Ratio of specific heats Kinetic energy of turbulence Power law coefficient Viscous losses per unit mass Length Mass flow rate Mass Mach number Morton number Molecular weight Power law exponent Blend time number Best number Power number Pumping number Pressure Entrained flow rate Volumetric flow rate Throughput (vacuum flow) Heat input per unit mass Radial coordinate Radius Ideal gas universal constant Volume fraction of phase i Reynolds number Density ratio
m/s m2 m m m/s Dimensionless m3/s Dimensionless Dimensionless Dimensionless m m Dimensionless 1/s
ft/s ft2 in ft ft/s Dimensionless ft3/s Dimensionless Dimensionless Dimensionless ft ft Dimensionless 1/s
Pa J/s Dimensionless Dimensionless 1/s N Dimensionless
lbf/in2 ft ⋅ lbf/s Dimensionless Dimensionless 1/s lbf Dimensionless
Dimensionless m/s2 kg/(m2 ⋅ s) J/kg m Dimensionless J/kg kg/(m ⋅ s2 − n) J/kg m kg/s kg Dimensionless Dimensionless kg/kgmole Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless Pa m3/s m3/s Pa ⋅ m3/s J/kg m m J/(kgmole ⋅ K) Dimensionless Dimensionless Dimensionless
Dimensionless ft/s2 lbm/(ft2 ⋅ s) Btu/lbm ft Dimensionless ft ⋅ lbf/lbm lbm/(ft ⋅ s2 − n) ft ⋅ lbf/lbm ft lbm/s lbm Dimensionless Dimensionless lbm/lbmole Dimensionless Dimensionless Dimensionless Dimensionless Dimensionless lbf/in2 ft3/s ft3/s lbf ⋅ ft3/s Btu/lbm ft ft Btu/(lbmole ⋅ R) Dimensionless Dimensionless Dimensionless
E E˙ v Eo f f F F Fr g G h h k k K lv L m ˙ M M M Mw n Nb ND NP NQ p q Q Q δQ r R R Ri Re s
Symbol
Definition
SI units
U.S. customary units
s S S S St t t T u u U v V V We ˙s W δWs x y z z
Entropy per unit mass Slope Pumping speed Surface area per unit volume Strouhal number Time Force per unit area Absolute temperature Internal energy per unit mass Velocity Velocity Velocity Velocity Volume Weber number Rate of shaft work Shaft work per unit mass Cartesian coordinate Cartesian coordinate Cartesian coordinate Elevation
J/(kg ⋅ K) Dimensionless m3/s l/m Dimensionless s Pa K J/kg m/s m/s m/s m/s m3 Dimensionless J/s J/kg m m m m
Btu/(lbm ⋅ R) Dimensionless ft3/s l/ft Dimensionless s lbf/in2 R Btu/lbm ft/s ft/s ft/s ft/s ft3 Dimensionless Btu/s Btu/lbm ft ft ft ft
α α β β γ˙ Γ
Velocity profile factor Included angle Velocity profile factor Bulk modulus of elasticity Shear rate Mass flow rate per unit width Boundary layer or film thickness Kronecker delta Pipe roughness Void fraction Turbulent dissipation rate Residence time Angle Mean free path Viscosity Kinematic viscosity Density Surface tension Cavitation number Components of total stress tensor Shear stress Time period Components of deviatoric stress tensor Energy dissipation rate per unit volume Angle of inclination Vorticity
Greek symbols
δ δij θ θ λ µ ν ρ σ σ σij τ τ τij Φ φ ω
Dimensionless Radians Dimensionless Pa l/s kg/(m ⋅ s)
Dimensionless Radians Dimensionless lbf/in2 l/s lbm/(ft ⋅ s)
m
ft
Dimensionless m Dimensionless J/(kg ⋅ s) s Radians m Pa ⋅ s m2/s kg/m3 N/m Dimensionless Pa
Dimensionless ft Dimensionless ft ⋅ lbf/(lbm ⋅ s) s Radians ft lbm/(ft ⋅ s) ft2/s lbm/ft3 lbf/ft Dimensionless lbf/in2
Pa s Pa
lbf/in2 s lbf/in2
J/(m3 ⋅ s)
ft ⋅ lbf/(ft3 ⋅ s)
Radians 1/s
Radians 1/s
* Note that with U.S. Customary units, the conversion factor gc may be required to make equations in this section dimensionally consistent; gc = 32.17 (lbm⋅ft)/lbf⋅s2). 6-3
6-4
FLUID AND PARTICLE DYNAMICS
FLUID DYNAMICS GENERAL REFERENCES: Batchelor, An Introduction to Fluid Dynamics, Cambridge University, Cambridge, 1967; Bird, Stewart, and Lightfoot, Transport Phenomena, Wiley, New York, 1960; Brodkey, The Phenomena of Fluid Motions, Addison-Wesley, Reading, Mass., 1967; Denn, Process Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1980; Landau and Lifshitz, Fluid Mechanics, 2d ed., Pergamon, 1987; Govier and Aziz, The Flow of Complex Mixtures in Pipes, Van Nostrand Reinhold, New York, 1972, Krieger, Huntington, N.Y., 1977; Panton, Incompressible Flow, Wiley, New York, 1984; Schlichting, Boundary Layer Theory, 8th ed., McGraw-Hill, New York, 1987; Shames, Mechanics of Fluids, 3d ed., McGraw-Hill, New York, 1992; Streeter, Handbook of Fluid Dynamics, McGraw-Hill, New York, 1971; Streeter and Wylie, Fluid Mechanics, 8th ed., McGraw-Hill, New York, 1985; Vennard and Street, Elementary Fluid Mechanics, 5th ed., Wiley, New York, 1975; Whitaker, Introduction to Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1968, Krieger, Huntington, N.Y., 1981.
NATURE OF FLUIDS Deformation and Stress A fluid is a substance which undergoes continuous deformation when subjected to a shear stress. Figure 6-1 illustrates this concept. A fluid is bounded by two large parallel plates, of area A, separated by a small distance H. The bottom plate is held fixed. Application of a force F to the upper plate causes it to move at a velocity U. The fluid continues to deform as long as the force is applied, unlike a solid, which would undergo only a finite deformation.
A
F V
y
H
Fluids without any solidlike elastic behavior do not undergo any reverse deformation when shear stress is removed, and are called purely viscous fluids. The shear stress depends only on the rate of deformation, and not on the extent of deformation (strain). Those which exhibit both viscous and elastic properties are called viscoelastic fluids. Purely viscous fluids are further classified into time-independent and time-dependent fluids. For time-independent fluids, the shear stress depends only on the instantaneous shear rate. The shear stress for time-dependent fluids depends on the past history of the rate of deformation, as a result of structure or orientation buildup or breakdown during deformation. A rheogram is a plot of shear stress versus shear rate for a fluid in simple shear flow, such as that in Fig. 6-1. Rheograms for several types of time-independent fluids are shown in Fig. 6-2. The Newtonian fluid rheogram is a straight line passing through the origin. The slope of the line is the viscosity. For a Newtonian fluid, the viscosity is independent of shear rate, and may depend only on temperature and perhaps pressure. By far, the Newtonian fluid is the largest class of fluid of engineering importance. Gases and low molecular weight liquids are generally Newtonian. Newton’s law of viscosity is a rearrangement of Eq. (6-1) in which the viscosity is a constant: du τ = µγ˙ = µ (6-2) dy All fluids for which the viscosity varies with shear rate are nonNewtonian fluids. For non-Newtonian fluids the viscosity, defined as the ratio of shear stress to shear rate, is often called the apparent viscosity to emphasize the distinction from Newtonian behavior. Purely viscous, time-independent fluids, for which the apparent viscosity may be expressed as a function of shear rate, are called generalized Newtonian fluids.
x Deformation of a fluid subjected to a shear stress.
The SI units of viscosity are kg/(m ⋅ s) or Pa ⋅ s (pascal second). The cgs unit for viscosity is the poise; 1 Pa ⋅ s equals 10 poise or 1000 centipoise (cP) or 0.672 lbm/(ft ⋅ s). The terms absolute viscosity and shear viscosity are synonymous with the viscosity as used in Eq. (6-1). Kinematic viscosity ν ⬅ µ/ρ is the ratio of viscosity to density. The SI units of kinematic viscosity are m2/s. The cgs stoke is 1 cm2/s. Rheology In general, fluid flow patterns are more complex than the one shown in Fig. 6-1, as is the relationship between fluid deformation and stress. Rheology is the discipline of fluid mechanics which studies this relationship. One goal of rheology is to obtain constitutive equations by which stresses may be computed from deformation rates. For simplicity, fluids may be classified into rheological types in reference to the simple shear flow of Fig. 6-1. Complete definitions require extension to multidimensional flow. For more information, several good references are available, including Bird, Armstrong, and Hassager, (Dynamics of Polymeric Liquids, vol. 1: Fluid Mechanics, Wiley, New York, 1977); Metzner, (“Flow of Non-Newtonian Fluids” in Streeter, Handbook of Fluid Dynamics, McGraw-Hill, New York, 1971); and Skelland (Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967).
ti c as m tic ha las g p n o Bi pl
τy
Ps eu d
The force is directly proportional to the area of the plate; the shear stress is τ = F/A. Within the fluid, a linear velocity profile u = Uy/H is established; due to the no-slip condition, the fluid bounding the lower plate has zero velocity and the fluid bounding the upper plate moves at the plate velocity U. The velocity gradient γ˙ = du/dy is called the shear rate for this flow. Shear rates are usually reported in units of reciprocal seconds. The flow in Fig. 6-1 is a simple shear flow. Viscosity The ratio of shear stress to shear rate is the viscosity, µ. τ µ= (6-1) γ˙
Shear stress τ
FIG. 6-1
lat Di
t an
an toni Ne w Shear rate |du/dy| FIG. 6-2
Shear diagrams.
Non-Newtonian fluids include those for which a finite stress τy is required before continuous deformation occurs; these are called yield-stress materials. The Bingham plastic fluid is the simplest yield-stress material; its rheogram has a constant slope µ∞, called the infinite shear viscosity. τ = τy + µ∞γ˙ (6-3) Highly concentrated suspensions of fine solid particles frequently exhibit Bingham plastic behavior. Shear-thinning fluids are those for which the slope of the rheogram decreases with increasing shear rate. These fluids have also been called pseudoplastic, but this terminology is outdated and discouraged. Many polymer melts and solutions, as well as some solids suspensions, are shear-thinning. Shear-thinning fluids without yield stresses typically obey a power law model over a range of shear rates. τ = Kγ˙ n (6-4) The apparent viscosity is µ = Kγ˙ n − 1
(6-5)
FLUID DYNAMICS The factor K is the consistency index or power law coefficient, and n is the power law exponent. The exponent n is dimensionless, while K is in units of kg/(m ⋅ s2 − n). For shear-thinning fluids, n < 1. The power law model typically provides a good fit to data over a range of one to two orders of magnitude in shear rate; behavior at very low and very high shear rates is often Newtonian. Shear-thinning power law fluids with yield stresses are sometimes called Herschel-Bulkley fluids. Numerous other rheological model equations for shear-thinning fluids are in common use. Dilatant, or shear-thickening, fluids show increasing viscosity with increasing shear rate. Over a limited range of shear rate, they may be described by the power law model with n > 1. Dilatancy is rare, observed only in certain concentration ranges in some particle suspensions (Govier and Aziz, pp. 33–34). Extensive discussions of dilatant suspensions, together with a listing of dilatant systems, are given by Green and Griskey (Trans. Soc. Rheol, 12[1], 13–25 [1968]); Griskey and Green (AIChE J., 17, 725–728 [1971]); and Bauer and Collins (“Thixotropy and Dilatancy,” in Eirich, Rheology, vol. 4, Academic, New York, 1967). Time-dependent fluids are those for which structural rearrangements occur during deformation at a rate too slow to maintain equilibrium configurations. As a result, shear stress changes with duration of shear. Thixotropic fluids, such as mayonnaise, clay suspensions used as drilling muds, and some paints and inks, show decreasing shear stress with time at constant shear rate. A detailed description of thixotropic behavior and a list of thixotropic systems is found in Bauer and Collins (ibid.). Rheopectic behavior is the opposite of thixotropy. Shear stress increases with time at constant shear rate. Rheopectic behavior has been observed in bentonite sols, vanadium pentoxide sols, and gypsum suspensions in water (Bauer and Collins, ibid.) as well as in some polyester solutions (Steg and Katz, J. Appl. Polym. Sci., 9, 3, 177 [1965]). Viscoelastic fluids exhibit elastic recovery from deformation when stress is removed. Polymeric liquids comprise the largest group of fluids in this class. A property of viscoelastic fluids is the relaxation time, which is a measure of the time required for elastic effects to decay. Viscoelastic effects may be important with sudden changes in rates of deformation, as in flow startup and stop, rapidly oscillating flows, or as a fluid passes through sudden expansions or contractions where accelerations occur. In many fully developed flows where such effects are absent, viscoelastic fluids behave as if they were purely viscous. In viscoelastic flows, normal stresses perpendicular to the direction of shear are different from those in the parallel direction. These give rise to such behaviors as the Weissenberg effect, in which fluid climbs up a shaft rotating in the fluid, and die swell, where a stream of fluid issuing from a tube may expand to two or more times the tube diameter. A parameter indicating whether viscoelastic effects are important is the Deborah number, which is the ratio of the characteristic relaxation time of the fluid to the characteristic time scale of the flow. For small Deborah numbers, the relaxation is fast compared to the characteristic time of the flow, and the fluid behavior is purely viscous. For very large Deborah numbers, the behavior closely resembles that of an elastic solid. Analysis of viscoelastic flows is very difficult. Simple constitutive equations are unable to describe all the material behavior exhibited by viscoelastic fluids even in geometrically simple flows. More complex constitutive equations may be more accurate, but become exceedingly difficult to apply, especially for complex geometries, even with advanced numerical methods. For good discussions of viscoelastic fluid behavior, including various types of constitutive equations, see Bird, Armstrong, and Hassager (Dynamics of Polymeric Liquids, vol. 1: Fluid Mechanics, vol. 2: Kinetic Theory, Wiley, New York, 1977); Middleman (The Flow of High Polymers, Interscience (Wiley) New York, 1968); or Astarita and Marrucci (Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, New York, 1974). Polymer processing is the field which depends most on the flow of non-Newtonian fluids. Several excellent texts are available, including Middleman (Fundamentals of Polymer Processing, McGraw-Hill, New York, 1977) and Tadmor and Gogos (Principles of Polymer Processing, Wiley, New York, 1979).
6-5
There is a wide variety of instruments for measurement of Newtonian viscosity, as well as rheological properties of non-Newtonian fluids. They are described in Van Wazer, Lyons, Kim, and Colwell, (Viscosity and Flow Measurement, Interscience, New York, 1963); Coleman, Markowitz, and Noll (Viscometric Flows of Non-Newtonian Fluids, Springer-Verlag, Berlin, 1966); Dealy and Wissbrun (Melt Rheology and its Role in Plastics Processing, Van Nostrand Reinhold, 1990). Measurement of rheological behavior requires wellcharacterized flows. Such rheometric flows are thoroughly discussed by Astarita and Marrucci (Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, New York, 1974). KINEMATICS OF FLUID FLOW Velocity The term kinematics refers to the quantitative description of fluid motion or deformation. The rate of deformation depends on the distribution of velocity within the fluid. Fluid velocity v is a vector quantity, with three cartesian components vx, vy, and vz. The velocity vector is a function of spatial position and time. A steady flow is one in which the velocity is independent of time, while in unsteady flow v varies with time. Compressible and Incompressible Flow An incompressible flow is one in which the density of the fluid is constant or nearly constant. Liquid flows are normally treated as incompressible, except in the context of hydraulic transients (see following). Compressible fluids, such as gases, may undergo incompressible flow if pressure and/or temperature changes are small enough to render density changes insignificant. Frequently, compressible flows are regarded as flows in which the density varies by more than 5 to 10 percent. Streamlines, Pathlines, and Streaklines These are curves in a flow field which provide insight into the flow pattern. Streamlines are tangent at every point to the local instantaneous velocity vector. A pathline is the path followed by a material element of fluid; it coincides with a streamline if the flow is steady. In unsteady flow the pathlines generally do not coincide with streamlines. Streaklines are curves on which are found all the material particles which passed through a particular point in space at some earlier time. For example, a streakline is revealed by releasing smoke or dye at a point in a flow field. For steady flows, streamlines, pathlines, and streaklines are indistinguishable. In two-dimensional incompressible flows, streamlines are contours of the stream function. One-dimensional Flow Many flows of great practical importance, such as those in pipes and channels, are treated as onedimensional flows. There is a single direction called the flow direction; velocity components perpendicular to this direction are either zero or considered unimportant. Variations of quantities such as velocity, pressure, density, and temperature are considered only in the flow direction. The fundamental conservation equations of fluid mechanics are greatly simplified for one-dimensional flows. A broader category of one-dimensional flow is one where there is only one nonzero velocity component, which depends on only one coordinate direction, and this coordinate direction may or may not be the same as the flow direction. Rate of Deformation Tensor For general three-dimensional flows, where all three velocity components may be important and may vary in all three coordinate directions, the concept of deformation previously introduced must be generalized. The rate of deformation tensor Dij has nine components. In Cartesian coordinates, ∂v ∂v Dij = i + j ∂xj ∂xi
冢
冣
(6-6)
where the subscripts i and j refer to the three coordinate directions. Some authors define the deformation rate tensor as one-half of that given by Eq. (6-6). Vorticity The relative motion between two points in a fluid can be decomposed into three components: rotation, dilatation, and deformation. The rate of deformation tensor has been defined. Dilatation refers to the volumetric expansion or compression of the fluid, and vanishes for incompressible flow. Rotation is described by a tensor ωij = ∂vi / ∂xj − ∂vj / ∂xi. The vector of vorticity given by one-half the
6-6
FLUID AND PARTICLE DYNAMICS
curl of the velocity vector is another measure of rotation. In twodimensional flow in the x-y plane, the vorticity ω is given by ∂v 1 ∂v (6-7) ω = y − x 2 ∂x ∂y Here ω is the magnitude of the vorticity vector, which is directed along the z axis. An irrotational flow is one with zero vorticity. Irrotational flows have been widely studied because of their useful mathematical properties and applicability to flow regions where viscous effects may be neglected. Such flows without viscous effects are called inviscid flows. Laminar and Turbulent Flow, Reynolds Number These terms refer to two distinct types of flow. In laminar flow, there are smooth streamlines and the fluid velocity components vary smoothly with position, and with time if the flow is unsteady. The flow described in reference to Fig. 6-1 is laminar. In turbulent flow, there are no smooth streamlines, and the velocity shows chaotic fluctuations in time and space. Velocities in turbulent flow may be reported as the sum of a time-averaged velocity and a velocity fluctuation from the average. For any given flow geometry, a dimensionless Reynolds number may be defined for a Newtonian fluid as Re = LU ρ /µ where L is a characteristic length. Below a critical value of Re the flow is laminar, while above the critical value a transition to turbulent flow occurs. The geometry-dependent critical Reynolds number is determined experimentally.
冢
冣
CONSERVATION EQUATIONS Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and conservation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential conservation equations, respectively. These are often called macroscopic and microscopic balance equations. Macroscopic Equations An arbitrary control volume of finite size Va is bounded by a surface of area Aa with an outwardly directed unit normal vector n. The control volume is not necessarily fixed in space. Its boundary moves with velocity w. The fluid velocity is v. Figure 6-3 shows the arbitrary control volume. Mass balance Applied to the control volume, the principle of conservation of mass may be written as (Whitaker, Introduction to Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1968, Krieger, Huntington, N.Y., 1981) d ρ dV + ρ(v − w) ⋅ n dA = 0 (6-8) Aa dt Va This equation is also known as the continuity equation.
冕
冕
Area Aa
Volume Va
Simplified forms of Eq. (6-8) apply to special cases frequently found in practice. For a control volume fixed in space with one inlet of area A1 through which an incompressible fluid enters the control volume at an average velocity V1, and one outlet of area A2 through which fluid leaves at an average velocity V2, as shown in Fig. 6-4, the continuity equation becomes V1 A1 = V2 A2 (6-9) The average velocity across a surface is given by V = (1/A)
冕 v dA A
where v is the local velocity component perpendicular to the inlet surface. The volumetric flow rate Q is the product of average velocity and the cross-sectional area, Q = VA. The average mass velocity is G = ρV. For steady flows through fixed control volumes with multiple inlets and/or outlets, conservation of mass requires that the sum of inlet mass flow rates equals the sum of outlet mass flow rates. For incompressible flows through fixed control volumes, the sum of inlet flow rates (mass or volumetric) equals the sum of exit flow rates, whether the flow is steady or unsteady. Momentum Balance Since momentum is a vector quantity, the momentum balance is a vector equation. Where gravity is the only body force acting on the fluid, the linear momentum principle, applied to the arbitrary control volume of Fig. 6-3, results in the following expression (Whitaker, ibid.). d ρv dV + ρv(v − w) ⋅ n dA = ρg dV + tn dA (6-10) Aa Va Aa dt Va Here g is the gravity vector and tn is the force per unit area exerted by the surroundings on the fluid in the control volume. The integrand of the area integral on the left-hand side of Eq. (6-10) is nonzero only on the entrance and exit portions of the control volume boundary. For the special case of steady flow at a mass flow rate m ˙ through a control volume fixed in space with one inlet and one outlet, (Fig. 6-4) with the inlet and outlet velocity vectors perpendicular to planar inlet and outlet surfaces, giving average velocity vectors V1 and V2, the momentum equation becomes
冕
冕
冕
冕
m(β ˙ 2V2 − β1V1) = −p1A1 − p2A2 + F + Mg
n outwardly directed unit normal vector
V2 2
w boundary velocity
V1
v fluid velocity FIG. 6-3
Arbitrary control volume for application of conservation equations.
(6-11)
where M is the total mass of fluid in the control volume. The factor β arises from the averaging of the velocity across the area of the inlet or outlet surface. It is the ratio of the area average of the square of velocity magnitude to the square of the area average velocity magnitude. For a uniform velocity, β = 1. For turbulent flow, β is nearly unity, while for laminar pipe flow with a parabolic velocity profile, β = 4/3. The vectors A1 and A 2 have magnitude equal to the areas of the inlet and outlet surfaces, respectively, and are outwardly directed normal to the surfaces. The vector F is the force exerted on the fluid by the nonflow boundaries of the control volume. It is also assumed that the stress vector tn is normal to the inlet and outlet surfaces, and that its magnitude may be approximated by the pressure p. Equation (6-11) may be generalized to multiple inlets and/or outlets. In such cases, the mass flow rates for all the inlets and outlets are not equal. A distinct flow rate m ˙ i applies to each inlet or outlet i. To generalize the equation, ⴚpA terms for each inlet and outlet, − mβV ˙ terms for each inlet, and mβV ˙ terms for each outlet are included. Balance equations for angular momentum, or moment of momentum, may also be written. They are used less frequently than the lin-
FIG. 6-4
Fixed control volume with one inlet and one outlet.
FLUID DYNAMICS ear momentum equations. See Whitaker (Introduction to Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1968, Krieger, Huntington, N.Y., 1981; or Shames (Mechanics of Fluids, 3d ed., McGraw-Hill, New York, 1992). Total Energy Balance The total energy balance derives from the first law of thermodynamics. Applied to the arbitrary control volume of Fig. 6-3, it leads to an equation for the rate of change of the sum of internal, kinetic, and gravitational potential energy. In this equation, u is the internal energy per unit mass, v is the magnitude of the velocity vector v, z is elevation, g is the gravitational acceleration, and q is the heat flux vector: d dt
冕 ρ冢u + v2 + gz冣 dV + 冕 ρ冢u + v2 + gz冣(v − w) ⋅ n dA = 冕 (v ⋅ t ) dA − 冕 (q ⋅ n) dA (6-12) 2
2
Va
Aa
n
Aa
Aa
The first integral on the right-hand side is the rate of work done on the fluid in the control volume by forces at the boundary. It includes both work done by moving solid boundaries and work done at flow entrances and exits. The work done by moving solid boundaries also includes that by such surfaces as pump impellers; this work is called ˙ S. shaft work; its rate is W A useful simplification of the total energy equation applies to a particular set of assumptions. These are a control volume with fixed solid boundaries, except for those producing shaft work, steady state conditions, and mass flow at a rate m ˙ through a single planar entrance and a single planar exit (Fig. 6-4), to which the velocity vectors are perpendicular. As with Eq. (6-11), it is assumed that the stress vector tn is normal to the entrance and exit surfaces and may be approximated by the pressure p. The equivalent pressure, p + ρgz, is assumed to be uniform across the entrance and exit. The average velocity at the entrance and exit surfaces is denoted by V. Subscripts 1 and 2 denote the entrance and exit, respectively. V 12 V 22 h1 + α1 + gz1 = h2 + α2 + gz2 − δQ − δWS 2 2
(6-13)
Here, h is the enthalpy per unit mass, h = u + p/ρ. The shaft work per ˙ s /m. unit of mass flowing through the control volume is δWS = W ˙ Similarly, δQ is the heat input rate per unit of mass. The factor α is the ratio of the cross-sectional area average of the cube of the velocity to the cube of the average velocity. For a uniform velocity profile, α = 1. In turbulent flow, α is usually assumed to equal unity; in turbulent pipe flow, it is typically about 1.07. For laminar flow in a circular pipe with a parabolic velocity profile, α = 2. Mechanical Energy Balance, Bernoulli Equation A balance equation for the sum of kinetic and potential energy may be obtained from the momentum balance by forming the scalar product with the velocity vector. The resulting equation, called the mechanical energy balance, contains a term accounting for the dissipation of mechanical energy into thermal energy by viscous forces. The mechanical energy equation is also derivable from the total energy equation in a way that reveals the relationship between the dissipation and entropy generation. The macroscopic mechanical energy balance for the arbitrary control volume of Fig. 6-3 may be written, with p = thermodynamic pressure, as d dt
冕 ρ冢v2 + gz冣 dV + 冕 ρ冢v2 + gz冣(v − w) ⋅ n dA = 冕 p ⋅ v dV + 冕 (v ⋅ t ) dA − 冕 Φ dV 2
2
Va
Aa
n
Va
Aa
(6-14)
Va
The last term is the rate of viscous energy dissipation to internal energy, E˙ v = 冕Va Φ dV, also called the rate of viscous losses. These losses are the origin of frictional pressure drop in fluid flow. Whitaker and Bird, Stewart, and Lightfoot provide expressions for the dissipation function Φ for Newtonian fluids in terms of the local velocity gradients. However, when using macroscopic balance equations the local velocity field within the control volume is usually unknown. For such cases additional information, which may come from empirical correlations, is needed.
6-7
For the same special conditions as for Eq. (6-13), the mechanical energy equation is reduced to p 2 dp V12 V 22 (6-15) α1 + gz1 + δWS = α2 + gz2 + + lv p1 2 2 ρ ˙ is the energy dissipation per unit mass. This equation Here lv = E˙ v /m has been called the engineering Bernoulli equation. For an incompressible flow, Eq. (6-15) becomes p2 p1 V12 V22 (6-16) + α1 + gz1 + δWS = + α2 + gz2 + lv ρ ρ 2 2 The Bernoulli equation can be written for incompressible, inviscid flow along a streamline, where no shaft work is done. p1 V12 p2 V22 (6-17) + + gz1 = + + gz2 ρ 2 ρ 2 Unlike the momentum equation (Eq. [6-11]), the Bernoulli equation is not easily generalized to multiple inlets or outlets. Microscopic Balance Equations Partial differential balance equations express the conservation principles at a point in space. Equations for mass, momentum, total energy, and mechanical energy may be found in Whitaker (ibid.), Bird, Stewart, and Lightfoot (Transport Phenomena, Wiley, New York, 1960), and Slattery (Momentum, Heat and Mass Transfer in Continua, 2d ed., Krieger, Huntington, N.Y., 1981), for example. These references also present the equations in other useful coordinate systems besides the cartesian system. The coordinate systems are fixed in inertial reference frames. The two most used equations, for mass and momentum, are presented here. Mass Balance, Continuity Equation The continuity equation, expressing conservation of mass, is written in cartesian coordinates as ∂ρ ∂ρvx ∂ρvy ∂ρvz (6-18) +++=0 ∂t ∂x ∂y ∂z In terms of the substantial derivative, D/Dt, Dρ ∂ρ ∂ρ ∂ρ ∂ρ ∂vx ∂vy ∂vz ⬅ + vx + vy + vz = −ρ + + (6-19) Dt ∂t ∂x ∂y ∂z ∂x ∂y ∂z The substantial derivative, also called the material derivative, is the rate of change in a Lagrangian reference frame, that is, following a material particle. In vector notation the continuity equation may be expressed as Dρ (6-20) = −ρ∇ ⋅ v Dt For incompressible flow, ∂v ∂v ∂v ∇ ⋅ v = x + y + z = 0 (6-21) ∂x ∂y ∂z Stress Tensor The stress tensor is needed to completely describe the stress state for microscopic momentum balances in multidimensional flows. The components of the stress tensor σij give the force in the j direction on a plane perpendicular to the i direction, using a sign convention defining a positive stress as one where the fluid with the greater i coordinate value exerts a force in the positive i direction on the fluid with the lesser i coordinate. Several references in fluid mechanics and continuum mechanics provide discussions, to various levels of detail, of stress in a fluid (Denn; Bird, Stewart, and Lightfoot; Schlichting; Fung [A First Course in Continuum Mechanics, 2d. ed., Prentice-Hall, Englewood Cliffs, N.J., 1977]; Truesdell and Toupin [in Flügge, Handbuch der Physik, vol. 3/1, Springer-Verlag, Berlin, 1960]; Slattery [Momentum, Energy and Mass Transfer in Continua, 2d ed., Krieger, Huntington, N.Y., 1981]). The stress has an isotropic contribution due to fluid pressure and dilatation, and a deviatoric contribution due to viscous deformation effects. The deviatoric contribution for a Newtonian fluid is the threedimensional generalization of Eq. (6-2):
冕
冢
冣
τij = µDij
(6-22)
σij = (−p + λ∇ ⋅ v)δij + τij
(6-23)
The total stress is
6-8
FLUID AND PARTICLE DYNAMICS
The identity tensor δij is zero for i ≠ j and unity for i = j. The coefficient λ is a material property related to the bulk viscosity, κ = λ + 2µ/3. There is considerable uncertainty about the value of κ. Traditionally, Stokes’ hypothesis, κ = 0, has been invoked, but the validity of this hypothesis is doubtful (Slattery, ibid.). For incompressible flow, the value of bulk viscosity is immaterial as Eq. (6-23) reduces to σij = −pδij + τij
(6-24)
Similar generalizations to multidimensional flow are necessary for non-Newtonian constitutive equations. Cauchy Momentum and Navier-Stokes Equations The differential equations for conservation of momentum are called the Cauchy momentum equations. These may be found in general form in most fluid mechanics texts (e.g., Slattery [ibid.]; Denn; Whitaker; and Schlichting). For the important special case of an incompressible Newtonian fluid with constant viscosity, substitution of Eqs. (6-22) and (6-24) lead to the Navier-Stokes equations, whose three Cartesian components are ∂v ∂v ∂v ∂v ρ x + vx x + vy x + vz x ∂t ∂x ∂y ∂z
冢
冣
∂p ∂2vx ∂2vx ∂2vx =−+µ + + + ρgx (6-25) ∂x ∂x2 ∂y2 ∂z2
冢
∂vy ∂vy ∂vy ∂vy ρ + vx + vy + vz ∂t ∂x ∂y ∂z
冢
冣
冣
∂p ∂2vy ∂2vy ∂2vy =−+µ + + + ρgy ∂y ∂x2 ∂y2 ∂z2
冢
∂vz ∂vz ∂vz ∂vz ρ + vx + vy + vz ∂t ∂x ∂y ∂z
冢
冣
(6-26)
冣
∂p ∂2vz ∂2vz ∂2vz =−+µ + + + ρgz (6-27) ∂z ∂x2 ∂y2 ∂z2
冢
冣
Stokes equations, Dirichlet and Neumann, or essential and natural, boundary conditions may be satisfied by different means. Fluid statics, discussed in Sec. 10 of the Handbook in reference to pressure measurement, is the branch of fluid mechanics in which the fluid velocity is either zero or is uniform and constant relative to an inertial reference frame. With velocity gradients equal to zero, the momentum equation reduces to a simple expression for the pressure field, ∇p = ρg. Letting z be directed vertically upward, so that gz = −g where g is the gravitational acceleration (9.806 m2/s), the pressure field is given by dp/dz = −ρg
Example 1: Force Exerted on a Reducing Bend An incompressible fluid flows through a reducing elbow (Fig. 6-5) situated in a horizontal plane. The inlet velocity V1 is given and the pressures p1 and p2 are measured. Selecting the inlet and outlet surfaces 1 and 2 as shown, the continuity equation Eq. (6-9) can be used to find the exit velocity V2 = V1A1/A2. The mass flow rate is obtained by m ˙ = ρV1A1. Assume that the velocity profile is nearly uniform so that β is approximately unity. The force exerted on the fluid by the bend has x and y components; these can be found from Eq. (6-11). The x component gives Fx = m(V ˙ 2x − V1x) + p1A1x + p2 A2x while the y component gives Fy = m(V ˙ 2y − V1y) + p1 A1y + p2 A2y
In vector notation, Dv ∂v (6-28) ρ = + (v ⋅ ∇)v = −∇p + µ∇2v + ρg Dt ∂t The pressure and gravity terms may be combined by replacing the pressure p by the equivalent pressure P = p + ρgz. The left-hand side terms of the Navier-Stokes equations are the inertial terms, while the terms including viscosity µ are the viscous terms. Limiting cases under which the Navier-Stokes equations may be simplified include creeping flows in which the inertial terms are neglected, potential flows (inviscid or irrotational flows) in which the viscous terms are neglected, and boundary layer and lubrication flows in which certain terms are neglected based on scaling arguments. Creeping flows are described by Happel and Brenner (Low Reynolds Number Hydrodynamics, Prentice-Hall, Englewood Cliffs, N.J., 1965); potential flows by Lamb (Hydrodynamics, 6th ed., Dover, New York, 1945) and Milne-Thompson (Theoretical Hydrodynamics, 5th ed., Macmillan, New York, 1968); boundary layer theory by Schlichting (Boundary Layer Theory, 8th ed., McGraw-Hill, New York, 1987), and lubrication theory by Batchelor (An Introduction to Fluid Dynamics, Cambridge University, Cambridge, 1967) and Denn (Process Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1980). Because the Navier-Stokes equations are first-order in pressure and second-order in velocity, their solution requires one pressure boundary condition and two velocity boundary conditions (for each velocity component) to completely specify the solution. The no slip condition, which requires that the fluid velocity equal the velocity of any bounding solid surface, occurs in most problems. Specification of velocity is a type of boundary condition sometimes called a Dirichlet condition. Often boundary conditions involve stresses, and thus velocity gradients, rather than the velocities themselves. Specification of velocity derivatives is a Neumann boundary condition. For example, at the boundary between a viscous liquid and a gas, it is often assumed that the liquid shear stresses are zero. In numerical solution of the Navier-
(6-29)
This equation applies to any incompressible or compressible static fluid. For an incompressible liquid, pressure varies linearly with depth. For compressible gases, p is obtained by integration accounting for the variation of ρ with z. The force exerted on a submerged planar surface of area A is given by F = pc A where pc is the pressure at the geometrical centroid of the surface. The center of pressure, the point of application of the net force, is always lower than the centroid. For details see, for example, Shames, where may also be found discussion of forces on curved surfaces, buoyancy, and stability of floating bodies. Examples Four examples follow, illustrating the application of the conservation equations to obtain useful information about fluid flows.
The velocity components are V1x = V1, V1y = 0, V2x = V2 cos θ, and V2y = V2 sin θ. The area vector components are A1x = −A1, A1y = 0, A 2x = A 2 cos θ, and A 2y = A 2 sin θ. Therefore, the force components may be calculated from Fx = m(V ˙ 2 cos θ − V1) − p1A1 + p2A2 cos θ Fy = mV ˙ 2 sin θ + p2A2 sin θ The force acting on the fluid is F; the equal and opposite force exerted by the fluid on the bend is ⴚF.
Example 2: Simplified Ejector Figure 6-6 shows a very simplified sketch of an ejector, a device that uses a high velocity primary fluid to pump another (secondary) fluid. The continuity and momentum equations may be
V2 θ
V1 F
y x Force at a reducing bend. F is the force exerted by the bend on the fluid. The force exerted by the fluid on the bend is ⴚF. FIG. 6-5
FLUID DYNAMICS
6-9
y H
x
FIG. 6-8
FIG. 6-6
Draft-tube ejector.
applied on the control volume with inlet and outlet surfaces 1 and 2 as indicated in the figure. The cross-sectional area is uniform, A1 = A2 = A. Let the mass flow rates and velocities of the primary and secondary fluids be m ˙ p, m ˙ s, Vp and Vs. Assume for simplicity that the density is uniform. Conservation of mass gives m˙2 = m ˙p +m ˙ s. The exit velocity is V2 = m ˙ 2 /(ρA). The principle momentum exchange in the ejector occurs between the two fluids. Relative to this exchange, the force exerted by the walls of the device are found to be small. Therefore, the force term F is neglected from the momentum equation. Written in the flow direction, assuming uniform velocity profiles, and using the extension of Eq. (6-11) for multiple inlets, it gives the pressure rise developed by the device: (p2 − p1)A = (m˙p + m ˙ s)V2 − m ˙ pVp − m ˙ sVs Application of the momentum equation to ejectors of other types is discussed in Lapple (Fluid and Particle Dynamics, University of Delaware, Newark, 1951) and in Sec. 10 of the Handbook.
Example 3: Venturi Flowmeter An incompressible fluid flows
Plane Poiseuille flow.
ish in the x direction. Since the flow field is infinite in the z direction, all velocity derivatives should be zero in the z direction. Therefore, velocity components are a function of y alone. It is also assumed that there is no flow in the z direction, so vz = 0. The continuity equation Eq. (6-21), with vz = 0 and ∂vx / ∂x = 0, reduces to dvy = 0. dy Since vy = 0 at y = H/2, the continuity equation integrates to vy = 0. This is a direct result of the assumption of fully developed flow. The Navier-Stokes equations are greatly simplified when it is noted that vy = vz = 0 and ∂vx /∂x = ∂vx /∂z = ∂vx /∂t = 0. The three components are written in terms of the equivalent pressure P: ∂P ∂2vx 0=−+µ ∂x ∂y2 ∂P 0=− ∂y ∂P 0=− ∂z
through the venturi flowmeter in Fig. 6-7. An equation is needed to relate the flow rate Q to the pressure drop measured by the manometer. This problem can be solved using the mechanical energy balance. In a well-made venturi, viscous losses are negligible, the pressure drop is entirely the result of acceleration into the throat, and the flow rate predicted neglecting losses is quite accurate. The inlet area is A and the throat area is a. With control surfaces at 1 and 2 as shown in the figure, Eq. (6-17) in the absence of losses and shaft work gives
The latter two equations require that P is a function only of x, and therefore ∂P/∂x = dP/dx. Inspection of the first equation shows one term which is a function only of x and one which is only a function of y. This requires that both terms are constant. The pressure gradient −dP/dx is constant. The x-component equation becomes d 2vx 1 dP = dy2 µ dx
p1 V 12 p2 V 22 +=+ ρ 2 ρ 2 The continuity equation gives V2 = V1A/a, and V1 = Q/A. The pressure drop measured by the manometer is p1 − p2 = (ρm − ρ)g∆z. Substituting these relations into the energy balance and rearranging, the desired expression for the flow rate is found. 2(ρm − ρ)g∆z 1 Q = A ρ[(A/a)2 − 1]
Two integrations of the x-component equation give
冪莦
Example 4: Plane Poiseuille Flow An incompressible Newtonian fluid flows at a steady rate in the x direction between two very large flat plates, as shown in Fig. 6-8. The flow is laminar. The velocity profile is to be found. This example is found in most fluid mechanics textbooks; the solution presented here closely follows Denn. This problem requires use of the microscopic balance equations because the velocity is to be determined as a function of position. The boundary conditions for this flow result from the no-slip condition. All three velocity components must be zero at the plate surfaces, y = H/2 and y = −H/2. Assume that the flow is fully developed, that is, all velocity derivatives van-
1
2
1 dP vx = y2 + C1y + C2 2µ dx where the constants of integration C1 and C2 are evaluated from the boundary conditions vx = 0 at y = H/2. The result is
冢
冣冤 冢 冣 冥
H2 dP 2y 2 vx = − 1 − 8µ dx H This is a parabolic velocity distribution. The average velocity V = H/2 (1/H) 冕 −H/2 vx dy is dP H2 V= − 12µ dx
冢
This flow is one-dimensional, as there is only one nonzero velocity component, vx, which, along with the pressure, varies in only one coordinate direction.
INCOMPRESSIBLE FLOW IN PIPES AND CHANNELS Mechanical Energy Balance The mechanical energy balance, Eq. (6-16), for fully developed incompressible flow in a straight circular pipe of constant diameter D reduces to p2 p1 (6-30) + gz1 = + gz 2 + lv ρ ρ In terms of the equivalent pressure, P ⬅ p + ρgz, P1 − P2 = ρ lv
∆z
FIG. 6-7
Venturi flowmeter.
冣
(6-31)
The pressure drop due to frictional losses lv is proportional to pipe length L for fully developed flow and may be denoted as the (positive) quantity ∆P ⬅ P1 − P2. Friction Factor and Reynolds Number For a Newtonian fluid in a smooth pipe, dimensional analysis relates the frictional pressure drop per unit length ∆P/L to the pipe diameter D, density ρ, and average velocity V through two dimensionless groups, the Fanning friction factor f and the Reynolds number Re.
6-10
FLUID AND PARTICLE DYNAMICS D∆P f⬅ 2ρV 2L
(6-32)
DVρ Re ⬅ (6-33) µ For smooth pipe, the friction factor is a function only of the Reynolds number. In rough pipe, the relative roughness /D also affects the friction factor. Figure 6-9 plots f as a function of Re and /D. Values of for various materials are given in Table 6-1. The Fanning friction factor should not be confused with the Darcy friction factor used by Moody (Trans. ASME, 66, 671 [1944]), which is four times greater. Using the momentum equation, the stress at the wall of the pipe may be expressed in terms of the friction factor: ρV 2 τw = f 2
(6-34)
Laminar and Turbulent Flow Below a critical Reynolds number of about 2,100, the flow is laminar; over the range 2,100 < Re < 5,000 there is a transition to turbulent flow. For laminar flow, the Hagen-Poiseuille equation 16 f = , Re
Re ≤ 2,100
(6-35)
may be derived from the Navier-Stokes equation and is in excellent agreement with experimental data. It may be rewritten in terms of volumetric flow rate, Q = VπD2/4, as π∆PD4 Q = , Re ≤ 2,100 (6-36) 128µL
TABLE 6-1 Values of Surface Roughness for Various Materials* Surface roughness ε, mm
Material Drawn tubing (brass, lead, glass, and the like) Commercial steel or wrought iron Asphalted cast iron Galvanized iron Cast iron Wood stove Concrete Riveted steel
0.00152 0.0457 0.122 0.152 0.259 0.183–0.914 0.305–3.05 0.914–9.14
* From Moody, Trans. Am. Soc. Mech. Eng., 66, 671–684 (1944); Mech. Eng., 69, 1005–1006 (1947). Additional values of ε for various types or conditions of concrete wrought-iron, welded steel, riveted steel, and corrugated-metal pipes are given in Brater and King, Handbook of Hydraulics, 6th ed., McGraw-Hill, New York, 1976, pp. 6-12–6-13. To convert millimeters to feet, multiply by 3.281 × 10−3.
For turbulent flow in smooth tubes, the Blasius equation gives the friction factor accurately for a wide range of Reynolds numbers. 0.079 f= , 4,000 < Re < 105 (6-37) Re0.25 The Colebrook formula (Colebrook, J. Inst. Civ. Eng. [London], 11, 133–156 [1938–39]) gives a good approximation for the f-Re-(/D) data for rough pipes over the entire turbulent flow range: 1.256 1 = −4 log + 3.7D Re兹f苶 兹苶f
冤
冥
Re > 4,000
Fanning Friction Factors. Reynolds number Re = DVρ/µ, where D = pipe diameter, V = velocity, ρ = fluid density, and µ = fluid viscosity. (Based on Moody, Trans. ASME, 66, 671 [1944].)
FIG. 6-9
(6-38)
FLUID DYNAMICS An equation by Churchill (Chem. Eng., 84[24], 91–92 [Nov. 7, 1977]) for both smooth and rough tubes offers the advantage of being explicit in f: 1 0.27 = −4 log + (7/Re)0.9 Re > 4,000 (6-39) D 兹苶f In laminar flow, f is independent of /D. In turbulent flow, the friction factor for rough pipe follows the smooth tube curve for a range of Reynolds numbers (hydraulically smooth flow). For greater Reynolds numbers, f deviates from the smooth pipe curve, eventually becoming independent of Re. This region, often called complete turbulence, is frequently encountered in commercial pipe flows. The Reynolds number above which f becomes essentially independent of Re is (Davies, Turbulence Phenomena, Academic, New York, 1972, p. 37) 20[3.2 − 2.46 ln (/D)] Re = (6-40) (/D) Roughness may also affect the transition from laminar to turbulent flow (Schlichting). Common pipe flow problems include calculation of pressure drop given the flow rate (or velocity) and calculation of flow rate (or velocity) given pressure drop. When flow rate is given, the Reynolds number is first calculated to determine the flow regime, so that the appropriate relations between f and Re (or pressure drop and velocity or flow rate) are used. When pressure drop is given and the velocity is unknown, the Reynolds number and flow regime cannot be immediately determined. It is necessary to assume the flow regime and then verify by checking Re afterward. With experience, the initial guess for the flow regime will usually prove correct. When solving Eq. (6-38) for velocity when pressure drop is given, it is useful to note that the right-hand side is independent of velocity since Re兹苶f = (D3/2/µ)兹苶 ρ苶 ∆苶 P苶 /(苶2苶 L苶). As Fig. 6-9 suggests, the friction factor is uncertain in the transition range 2,100 < Re < 4,000 and a conservative choice should be made for design purposes. Velocity Profiles In laminar flow, the solution of the NavierStokes equation, corresponding to the Hagen-Poiseuille equation, gives the velocity v as a function of radial position r in a circular pipe of radius R in terms of the average velocity V = Q/A. The parabolic profile, with centerline velocity twice the average velocity, is shown in Fig. 6-10. r2 v = 2V 1 − 2 (6-41) R In turbulent flow, the velocity profile is much more blunt, with most of the velocity gradient being in a region near the wall, described by a universal velocity profile. It is characterized by a viscous sublayer, a turbulent core, and a buffer zone in between.
冤
冥
冢
冣
Viscous sublayer u+ = y+
for
y+ < 5
(6-42)
u+ = 5.00 ln y+ − 3.05
for
5 < y+ < 30
(6-43)
for
y+ > 30
(6-44)
Buffer zone Turbulent core u+ = 2.5 ln y+ + 5.5
Here, u+ = v/u* is the dimensionless, time-averaged axial velocity, u* =
r z
R
(
2 v = 2V 1 – r 2 R
v max = 2V FIG. 6-10
velocity V.
Parabolic velocity profile for laminar flow in a pipe, with average
6-11
兹苶 τw/ρ 苶 is the friction velocity and τw = fρV 2/2 is the wall stress. The
friction velocity is of the order of the root mean square velocity fluctuation perpendicular to the wall in the turbulent core. The dimensionless distance from the wall is y+ = yu ρ/µ. The universal velocity * profile is valid in the wall region for any cross-sectional channel shape. For incompressible flow in constant diameter circular pipes, τw = ∆P/4L where ∆P is the pressure drop in length L. In circular pipes, Eq. (6-44) gives a surprisingly good fit to experimental results over the entire cross section of the pipe, even though it is based on assumptions which are valid only near the pipe wall. For rough pipes, the velocity profile in the turbulent core is given by u+ = 2.5 ln y/ + 8.5
for
y+ > 30
(6-45)
when the dimensionless roughness + = u*ρ/µ is greater than 5 to 10; for smaller +, the velocity profile in the turbulent core is unaffected by roughness. For velocity profiles in the transition region, see Patel and Head ( J. Fluid Mech., 38, part 1, 181–201 [1969]) where profiles over the range 1,500 < Re < 10,000 are reported. Entrance and Exit Effects In the entrance region of a pipe, some distance is required for the flow to adjust from upstream conditions to the fully developed flow pattern. This distance depends on the Reynolds number and on the flow conditions upstream. For a uniform velocity profile at the pipe entrance, the computed length in laminar flow required for the centerline velocity to reach 99 percent of its fully developed value is (Dombrowski, Foumeny, Ookawara and Riza, Can. J. Chem. Engr., 71, 472–476 [1993]) Lent /D = 0.370 exp (−0.148Re) + 0.0550Re + 0.260
(6-46)
In turbulent flow, the entrance length is about Lent /D = 40
(6-47)
The frictional losses in the entrance region are larger than those for the same length of fully developed flow. (See the subsection, “Frictional Losses in Pipeline Elements,” following.) At the pipe exit, the velocity profile also undergoes rearrangement, but the exit length is much shorter than the entrance length. At low Re, it is about one pipe radius. At Re > 100, the exit length is essentially 0. Residence Time Distribution For laminar Newtonian pipe flow, the cumulative residence time distribution F(θ) is given by θavg F(θ) = 0 for θ< 2 1 θavg 2 θavg F(θ) = 1 − for θ≥ (6-48) 4 θ 2 where F(θ) is the fraction of material which resides in the pipe for less than time θ and θavg is the average residence time, θ = V/L. The residence time distribution in long transfer lines may be made narrower (more uniform) with the use of flow inverters or static mixing elements. These devices exchange fluid between the wall and central regions. Variations on the concept may be used to provide effective mixing of the fluid. See Godfrey (“Static Mixers,” in Harnby, Edwards, and Nienow, Mixing in the Process Industries, 2d ed., Butterworth Heinemann, Oxford, 1992); Gretta and Smith (Trans. ASME J. Fluids Eng., 115, 255–263 [1993]); Kemblowski and Pustelnik (Chem. Eng. Sci., 43, 473–478 [1988]). A theoretically derived equation for flow in helical pipe coils by Ruthven (Chem. Eng. Sci., 26, 1113–1121 [1971]; 33, 628–629 [1978]) is given by θavg 1 θavg 2.81 F(θ) = 1 − for 0.5 < < 1.63 (6-49) 4 θ θ and was substantially confirmed by Trivedi and Vasudeva (Chem. Eng. Sci., 29, 2291–2295 [1974]) for 0.6 < De < 6 and 0.0036 < D/Dc < 0.097 where De = Re兹苶 D苶 /Dc苶 is the Dean number and Dc is the diameter of curvature of the coil. Measurements by Saxena and Nigam (Chem. Eng. Sci., 34, 425–426 [1979]) indicate that such a distribution will hold for De > 1. The residence time distribution for helical coils is narrower than for straight circular pipes, due to the secondary flow which exchanges fluid between the wall and center regions.
冢 冣
冢 冣冤 冥
6-12
FLUID AND PARTICLE DYNAMICS
In turbulent flow, axial mixing is usually described in terms of turbulent diffusion or dispersion coefficients, from which cumulative residence time distribution functions can be computed. Davies (Turbulence Phenomena, Academic, New York, 1972, p. 93), gives DL = 1.01νRe0.875 for the longitudinal dispersion coefficient. Levenspiel (Chemical Reaction Engineering, 2d ed., Wiley, New York, 1972, pp. 253–278) discusses the relations among various residence time distribution functions, and the relation between dispersion coefficient and residence time distribution. Noncircular Channels Calculation of frictional pressure drop in noncircular channels depends on whether the flow is laminar or turbulent, and on whether the channel is full or open. For turbulent flow in ducts running full, the hydraulic diameter DH should be substituted for D in the friction factor and Reynolds number definitions, Eqs. (6-32) and (6-33). The hydraulic diameter is defined as four times the channel cross-sectional area divided by the wetted perimeter. For example, the hydraulic diameter for a circular pipe is DH = D, for an annulus of inner diameter d and outer diameter D, DH = D − d, for a rectangular duct of sides a, b, DH = ab/[2(a + b)]. The hydraulic radius RH is defined as one-fourth of the hydraulic diameter. With the hydraulic diameter subsititued for D in f and Re, Eqs. (6-37) through (6-40) are good approximations. Note that V appearing in f and Re is the actual average velocity V = Q/A; for noncircular pipes; it is not Q/(πDH2 /4). The pressure drop should be calculated from the friction factor for noncircular pipes. Equations relating Q to ∆P and D for circular pipes may not be used for noncircular pipes with D replaced by DH because V ≠ Q/(πDH2 /4). Turbulent flow in noncircular channels is generally accompanied by secondary flows perpendicular to the axial flow direction (Schlichting). These flows may cause the pressure drop to be slightly greater than that computed using the hydraulic diameter method. For data on pressure drop in annuli, see Brighton and Jones (J. Basic Eng., 86, 835–842 [1964]); Okiishi and Serovy (J. Basic Eng., 89, 823–836 [1967]); and Lawn and Elliot (J. Mech. Eng. Sci., 14, 195–204 [1972]). For rectangular ducts of large aspect ratio, Dean (J. Fluids Eng., 100, 215–233 [1978]) found that the numerator of the exponent in the Blasius equation (6-37) should be increased to 0.0868. Jones (J. Fluids Eng., 98, 173–181 [1976]) presents a method to improve the estimation of friction factors for rectangular ducts using a modification of the hydraulic diameter–based Reynolds number. The hydraulic diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters DE defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36); that is, DE ⬅ (128QµL/π∆P)1/4. Equivalent diameters are not the same as hydraulic diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V ≠ Q/(πDE/4). Equivalent diameter DE is not to be used in the friction factor and Reynolds number; f ≠ 16/Re using the equivalent diameters defined in the following. This situation is, by arbitrary definition, opposite to that for the hydraulic diameter DH used for turbulent flow. Ellipse, semiaxes a and b (Lamb, Hydrodynamics, 6th ed., Dover, New York, 1945, p. 587): 32a3b3 1/4 (6-50) DE = a2 + b2 Rectangle, width a, height b (Owen, Trans. Am. Soc. Civ. Eng., 119, 1157–1175 [1954]): 128ab3 1/4 DE = (6-51) πK a/b = 1 1.5 2 3 4 5 10 ∞ K = 28.45 20.43 17.49 15.19 14.24 13.73 12.81 12
冢
冣
冢
冣
Annulus, inner diameter D1 outer diameter D2 (Lamb, op. cit., p. 587):
D22 − D12 DE = (D22 − D12) D22 + D12 − ln (D2 /D1)
冦
冤
冥冧
1/4
(6-52)
For isosceles triangles and regular polygons, see Sparrow (AIChE J., 8, 599–605 [1962]), Carlson and Irvine (J. Heat Transfer, 83, 441–444 [1961]), Cheng (Proc. Third Int. Heat Transfer Conf., New York, 1, 64–76 [1966]), and Shih (Can. J. Chem. Eng., 45, 285–294 [1967]). The critical Reynolds number for transition from laminar to turbulent flow in noncircular channels varies with channel shape. In rectangular ducts, 1,900 < Rec < 2,800 (Hanks and Ruo, Ind. Eng. Chem. Fundam., 5, 558–561 [1966]). In triangular ducts, 1,600 < Rec < 1,800 (Cope and Hanks, Ind. Eng. Chem. Fundam., 11, 106–117 [1972]; Bandopadhayay and Hinwood, J. Fluid Mech., 59, 775–783 [1973]). Nonisothermal Flow For nonisothermal flow of liquids, the friction factor may be increased if the liquid is being cooled or decreased if the liquid is being heated, because of the effect of temperature on viscosity near the wall. In shell and tube heat-exchanger design, the recommended practice is to first estimate f using the bulk mean liquid temperature over the tube length. Then, in laminar flow, the result is divided by (µa /µw)0.23 in the case of cooling or (µa /µw)0.38 in the case of heating. For turbulent flow, f is divided by (µa /µw)0.11 in the case of cooling or (µa /µw)0.17 in case of heating. Here, µa is the viscosity at the average bulk temperature and µw is the viscosity at the average wall temperature (Seider and Tate, Ind. Eng. Chem., 28, 1429–1435 [1936]). In the case of rough commercial pipes, rather than heat-exchanger tubing, it is common for flow to be in the “complete” turbulence regime where f is independent of Re. In such cases, the friction factor should not be corrected for wall temperature. If the liquid density varies with temperature, the average bulk density should be used to calculate the pressure drop from the friction factor. In addition, a (usually small) correction may be applied for acceleration effects by adding the term G2[(1/ρ2) − (1/ρ1)] from the mechanical energy balance to the pressure drop ∆P = P1 − P2, where G is the mass velocity. This acceleration results from small compressibility effects associated with temperature-dependent density. Christiansen and Gordon (AIChE J., 15, 504–507 [1969]) present equations and charts for frictional loss in laminar nonisothermal flow of Newtonian and non-Newtonian liquids heated or cooled with constant wall temperature. Frictional dissipation of mechanical energy can result in significant heating of fluids, particularly for very viscous liquids in small channels. Under adiabatic conditions, the bulk liquid temperature rise is given by ∆T = ∆P/Cv ρ for incompressible flow through a channel of constant cross-sectional area. For flow of polymers, this amounts to about 4°C per 10 MPa pressure drop, while for hydrocarbon liquids it is about 6°C per 10 MPa. The temperature rise in laminar flow is highly nonuniform, being concentrated near the pipe wall where most of the dissipation occurs. This may result in significant viscosity reduction near the wall, and greatly increased flow or reduced pressure drop, and a flattened velocity profile. Compensation should generally be made for the heat effect when ∆P exceeds 1.4 MPa (203 psi) for adiabatic walls or 3.5 MPa (508 psi) for isothermal walls (Gerard, Steidler, and Appeldoorn, Ind. Eng. Chem. Fundam., 4, 332–339 [1969]). Open Channel Flow For flow in open channels, the data are largely based on experiments with water in turbulent flow, in channels of sufficient roughness that there is no Reynolds number effect. The hydraulic radius approach may be used to estimate a friction factor with which to compute friction losses. Under conditions of uniform flow where liquid depth and cross-sectional area do not vary significantly with position in the flow direction, there is a balance between gravitational forces and wall stress, or equivalently between frictional losses and potential energy change. The mechanical energy balance reduces to lv = g(z1 − z2). In terms of the friction factor and hydraulic diameter or hydraulic radius, 2 f V 2L f V 2L lv = = = g(z1 − z2) (6-53) DH 2RH The hydraulic radius is the cross-sectional area divided by the wetted perimeter, where the wetted perimeter does not include the free sur-
FLUID DYNAMICS face. Letting S = sin θ = channel slope (elevation loss per unit length of channel, θ = angle between channel and horizontal), Eq. (6-53) reduces to 2gSRH V= (6-54) f The most often used friction correlation for open channel flows is due to Manning (Trans. Inst. Civ. Engrs. Ireland, 20, 161 [1891]) and is equivalent to 29n2 f= (6-55) RH1/3 where n is the channel roughness, with dimensions of (length)1/6. Table 6-2 gives roughness values for several channel types. For gradual changes in channel cross section and liquid depth, and for slopes less than 10°, the momentum equation for a rectangular channel of width b and liquid depth h may be written as a differential equation in the flow direction x.
冪莦
dh h db f V 2(b + 2h) (1 − Fr) − Fr = S − dx b dx 2gbh
冢 冣
(6-56)
For a given fixed flow rate Q = Vbh, and channel width profile b(x), Eq. (6-56) may be integrated to determine the liquid depth profile h(x). The dimensionless Froude number is Fr = V 2/gh. When Fr = 1, the flow is critical, when Fr < 1, the flow is subcritical, and when Fr > 1, the flow is supercritical. Surface disturbances move at a wave velocity c = 兹苶 gh; they cannot propagate upstream in supercritical flows. The specific energy Esp is nearly constant. 2
V Esp = h + 2g
(6-57)
This equation is cubic in liquid depth. Below a minimum value of Esp there are no real positive roots; above the minimum value there are two positive real roots. At this minimum value of Esp the flow is critical; that is, Fr = 1, V = 兹苶 gh, and Esp = (3/2)h. Near critical flow conditions, wave motion and sudden depth changes called hydraulic jumps are likely. Chow (Open Channel Hydraulics, McGraw-Hill, New York, 1959), discusses the numerous surface profile shapes which may exist in nonuniform open channel flows. For flow over a sharp-crested weir of width b and height L, from a liquid depth H, the flow rate is given approximately by 2 Q = Cd b兹2苶g苶(H − L)3/2 3
(6-58)
where Cd ≈ 0.6 is a discharge coefficient. Flow through notched weirs is described under flow meters in Sec. 10 of the Handbook.
TABLE 6-2 Eq. (6-55)
Average Values of n for Manning Formula, Surface
n, m1/6
n, ft1/6
Cast-iron pipe, fair condition Riveted steel pipe Vitrified sewer pipe Concrete pipe Wood-stave pipe Planed-plank flume Semicircular metal flumes, smooth Semicircular metal flumes, corrugated Canals and ditches Earth, straight and uniform Winding sluggish canals Dredged earth channels Natural-stream channels Clean, straight bank, full stage Winding, some pools and shoals Same, but with stony sections Sluggish reaches, very deep pools, rather weedy
0.014 0.017 0.013 0.015 0.012 0.012 0.013 0.028
0.011 0.014 0.011 0.012 0.010 0.010 0.011 0.023
0.023 0.025 0.028
0.019 0.021 0.023
0.030 0.040 0.055 0.070
0.025 0.033 0.045 0.057
SOURCE: Brater and King, Handbook of Hydraulics, 6th ed., McGraw-Hill, New York, 1976, p. 7–22. For detailed information, see Chow, Open-Channel Hydraulics, McGraw-Hill, New York, 1959, pp. 110–123.
6-13
Non-Newtonian Flow For isothermal laminar flow of timeindependent non-Newtonian liquids, integration of the Cauchy momentum equations yields the fully developed velocity profile and flow rate–pressure drop relations. For the Bingham plastic fluid described by Eq. (6-3), in a pipe of diameter D and a pressure drop per unit length ∆P/L, the flow rate is given by πD3τw 4τy τ y4 Q= 1−+ 32µ∞ 3τw 3τ w4
冤
冥
(6-59)
where the wall stress is τw = D∆P/(4L). The velocity profile consists of a central nondeforming plug of radius rP = 2τy /(∆P/L) and an annular deforming region. The velocity profile in the annular region is given by 1 ∆P vz = (R2 − r2) − τy(R − r) , rP ≤ r ≤ R (6-60) µ∞ 4L where r is the radial coordinate and R is the pipe radius. The velocity of the central, nondeforming plug is obtained by setting r = rP in Eq. (6-60). When Q is given and Eq. (6-59) is to be solved for τw and the pressure drop, multiple positive roots for the pressure drop may be found. The root corresponding to τw < τy is physically unrealizable, as it corresponds to rp > R and the pressure drop is insufficient to overcome the yield stress. For a power law fluid, Eq. (6-4), with constant properties K and n, the flow rate is given by ∆P 1/n n Q=π (6-61) R(1 + 3n)/n 2KL 1 + 3n and the velocity profile by ∆P 1/n n (6-62) vz = [R(1 + n)/n − r (1 + n)/n] 2KL 1+n Similar relations for other non-Newtonian fluids may be found in Govier and Aziz and in Bird, Armstrong, and Hassager (Dynamics of Polymeric Liquids, vol. 1: Fluid Mechanics, Wiley, New York, 1977). For steady-state laminar flow of any time-independent viscous fluid, at average velocity V in a pipe of diameter D, the RabinowitschMooney relations give a general relationship for the shear rate at the pipe wall. 8V 1 + 3n′ γ˙ w = (6-63) D 4n′ where n′ is the slope of a plot of D∆P/(4L) versus 8V/D on logarithmic coordinates, d ln [D∆P/(4L)] n′ = (6-64) d ln (8V/D) By plotting capillary viscometry data this way, they can be used directly for pressure drop design calculations, or to construct the rheogram for the fluid. For pressure drop calculation, the flow rate and diameter determine the velocity, from which 8V/D is calculated and D∆P/(4L) read from the plot. For a Newtonian fluid, n′ = 1 and the shear rate at the wall is γ˙ = 8V/D. For a power law fluid, n′ = n. To construct a rheogram, n′ is obtained from the slope of the experimental plot at a given value of 8V/D. The shear rate at the wall is given by Eq. (6-63) and the corresponding shear stress at the wall is τw = D∆P/(4L) read from the plot. By varying the value of 8V/D, the shear rate versus shear stress plot can be constructed. The generalized approach of Metzner and Reed (AIChE J., 1, 434 [1955]) for time-independent non-Newtonian fluids defines a modified Reynolds number as Dn′V 2 − n′ρ ReMR ⬅ (6-65) K′8n′ − 1 where K′ satisfies 8V n′ D∆P (6-66) = K′ 4L D With this definition, f = 16/ReMR is automatically satisfied at the value of 8V/D where K′ and n′ are evaluated. Equation (6-66) may be obtained by integration of Eq. (6-64) only when n′ is a constant, as, for
冤
冥
冢
冢
冣 冢
冣 冢
冣
冣
冢
冣
冢 冣
6-14
FLUID AND PARTICLE DYNAMICS
example, the cases of Newtonian and power law fluids. For Newtonian fluids, K′ = µ and n′ = 1; for power law fluids, K′ = K[(1 + 3n)/ (4n)]n and n′ = n. For Bingham plastics, K′ and n′ are variable, given as a function of τw (Metzner, Ind. Eng. Chem., 49, 1429–1432 [1957]). K = τ 1w− n′
冤
µ∞ 1 − 4τy /3τw + (τy /τw)4/3
冥
3n + 1 µeff = K 4n
冢
n′
(6-67)
1 − 4τy /(3τw) + (τy /τw)4/3 n′ = 1 − (τy /τw)4
(6-68)
For laminar flow of power law fluids in channels of noncircular cross section, see Schecter (AIChE J., 7, 445–448 [1961]), Wheeler and Wissler (AIChE J., 11, 207–212 [1965]), Bird, Armstrong, and Hassager (Dynamics of Polymeric Liquids, vol. 1: Fluid Mechanics, Wiley, New York, 1977), and Skelland (Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967). Steady state, fully developed laminar flows of viscoelastic fluids in straight, constant-diameter pipes show no effects of viscoelasticity. The viscous component of the constitutive equation may be used to develop the flow rate–pressure drop relations, which apply downstream of the entrance region after viscoelastic effects have disappeared. A similar situation exists for time-dependent fluids. The transition to turbulent flow begins at ReMR in the range of 2,000 to 2,500 (Metzner and Reed, AIChE J., 1, 434 [1955]). For Bingham plastic materials, K′ and n′ must be evaluated for the τw condition in question in order to determine ReMR and establish whether the flow is laminar. An alternative method for Bingham plastics is by Hanks (Hanks, AIChE J., 9, 306 [1963]; 14, 691 [1968]; Hanks and Pratt, Soc. Petrol. Engrs. J., 7, 342 [1967]; and Govier and Aziz, pp. 213–215). The transition from laminar to turbulent flow is influenced by viscoelastic properties (Metzner and Park, J. Fluid Mech., 20, 291 [1964]) with the critical value of ReMR increased to beyond 10,000 for some materials. For turbulent flow of non-Newtonian fluids, the design chart of Dodge and Metzner (AIChE J., 5, 189 [1959]), Fig. 6-11, is most widely used. For Bingham plastic materials in turbulent flow, it is generally assumed that stresses greatly exceed the yield stress, so that the friction factor–Reynolds number relationship for Newtonian fluids applies, with µ∞ substituted for µ. This is equivalent to setting n′ = 1 and τy /τw = 0 in the Dodge-Metzner method, so that ReMR = DVρ/µ∞. Wilson and Thomas (Can. J. Chem. Eng., 63, 539–546 [1985]) give friction factor equations for turbulent flow of power law fluids and Bingham plastic fluids. Power law fluids: 1−n 1+n 1 1 = + 8.2 + 1.77 ln 1 + n 2 fN 兹苶f 兹苶
冢
where fN is the friction factor for Newtonian fluid evaluated at Re = DVρ/µeff where the effective viscosity is
冣
(6-69)
FIG. 6-11 Fanning friction factor for non-Newtonian flow. (From Dodge and Metzner, Am. Inst. Chem. Eng. J., 5, 189 [1959]).
n−1
冣 冢 D冣 8V
n−1
(6-70)
Bingham fluids: (1 − ξ)2 1 1 = + 1.77 ln + ξ(10 + 0.884ξ) 1+ξ fN 兹苶f 兹苶
冢
冣
(6-71)
where fN is evaluated at Re = DVρ/µ∞ and ξ = τy /τw. Iteration is required to use this equation since τw = fρV 2/2. Drag reduction in turbulent flow can be achieved by adding soluble high molecular weight polymers in extremely low concentration to Newtonian liquids. The reduction in friction is generally believed to be associated with the viscoelastic nature of the solutions effective in the wall region. For a given polymer, there is a minimum molecular weight necessary to initiate drag reduction at a given flow rate, and a critical concentration above which drag reduction will not occur (Kim, Little and Ting, J. Colloid Interface Sci., 47, 530–535 [1974]). Drag reduction is reviewed by Hoyt (J. Basic Eng., 94, 258–285 [1972]); Little, et al. (Ind. Eng. Chem. Fundam., 14, 283–296 [1975]) and Virk (AIChE J., 21, 625–656 [1975]). At maximum possible drag reduction in smooth pipes, 1 50.73 = −19 log (6-72) Re兹苶f 兹苶f
冢
or, approximately,
冣
0.58 f= Re0.58
(6-73)
for 4,000 < Re < 40,000. The actual drag reduction depends on the polymer system. For further details, see Virk (ibid.). Economic Pipe Diameter, Turbulent Flow The economic optimum pipe diameter may be computed so that the last increment of investment reduces the operating cost enough to produce the required minimum return on investment. For long cross-country pipelines, alloy pipes of appreciable length and complexity, or pipelines with control valves, detailed analyses of investment and operating costs should be made. Peters and Timmerhaus (Plant Design and Economics for Chemical Engineers, 4th ed., McGraw-Hill, New York, 1991) provide a detailed method for determining the economic optimum size. For pipelines of the lengths usually encountered in chemical plants and petroleum refineries, simplified selection charts are often adequate. In many cases there is an economic optimum velocity that is nearly independent of diameter, which may be used to estimate the economic diameter from the flow rate. For low-viscosity liquids in schedule 40 steel pipe, economic optimum velocity is typically in the range of 1.8 to 2.4 m/s (5.9 to 7.9 ft/s). For gases with density ranging from 0.2 to 20 kg/m3 (0.013 to 1.25 lbm/ft3), the economic optimum velocity is about 40 m/s to 9 m/s (131 to 30 ft/s). Charts and rough guidelines for economic optimum size do not apply to multiphase flows. Economic Pipe Diameter, Laminar Flow Pipelines for the transport of high-viscosity liquids are seldom designed purely on the basis of economics. More often, the size is dictated by operability considerations such as available pressure drop, shear rate, or residence time distribution. Peters and Timmerhaus (ibid., Chap. 10) provide an economic pipe diameter chart for laminar flow. For non-Newtonian fluids, see Skelland (Non-Newtonian Flow and Heat Transfer, Chap. 7, Wiley, New York, 1967). Vacuum Flow When gas flows under high vacuum conditions or through very small openings, the continuum hypothesis is no longer appropriate if the channel dimension is not very large compared to the mean free path of the gas. When the mean free path is comparable to the channel dimension, flow is dominated by collisions of molecules with the wall, rather than by collisions between molecules. An approximate expression based on Brown, et al. (J. Appl. Phys., 17, 802–813 [1946]) for the mean free path is 8RT 冢 冣 冪莦 πM
2µ λ= p
w
(6-74)
FLUID DYNAMICS The Knudsen number Kn is the ratio of the mean free path to the channel dimension. For pipe flow, Kn = λ/D. Molecular flow is characterized by Kn > 1.0; continuum viscous (laminar or turbulent) flow is characterized by Kn < 0.01. Transition or slip flow applies over the range 0.01 < Kn < 1.0. Vacuum flow is usually described with flow variables different from those used for normal pressures, which often leads to confusion. Pumping speed S is the actual volumetric flow rate of gas through a flow cross section. Throughput Q is the product of pumping speed and absolute pressure. In the SI system, Q has units of Pa ⋅ m3/s. Q = Sp
(6-75)
The mass flow rate w is related to the throughput using the ideal gas law. Mw w=Q (6-76) RT Throughput is therefore proportional to mass flow rate. For a given mass flow rate, throughput is independent of pressure. The relation between throughput and pressure drop ∆ p = p1 − p2 across a flow element is written in terms of the conductance C. Resistance is the reciprocal of conductance. Conductance has dimensions of volume per time. Q = C∆p (6-77) The conductance of a series of flow elements is given by 1 1 1 1 = + + + ⋅⋅⋅ C C1 C2 C3 while for elements in parallel,
(6-78)
C = C1 + C2 + C3 + ⋅⋅⋅
(6-79)
For a vacuum pump of speed Sp withdrawing from a vacuum vessel through a connecting line of conductance C, the pumping speed at the vessel is SpC S= (6-80) Sp + C Molecular Flow Under molecular flow conditions, conductance is independent of pressure. It is proportional to 兹苶 T苶 /Mw苶, with the proportionality constant a function of geometry. For fully developed pipe flow, πD3 RT C= (6-81) 8L Mw For an orifice of area A, RT (6-82) C = 0.40A Mw Conductance equations for several other geometries are given by Ryans and Roper (Process Vacuum System Design and Operation, Chap. 2, McGraw-Hill, New York, 1986). For a circular annulus of outer and inner diameters D1 and D2 and length L, the method of Guthrie and Wakerling (Vacuum Equipment and Techniques, McGraw-Hill, New York, 1949) may be written
冪莦
1 K = 1 + (L/D)
for
0 ≤ L/D ≤ 0.75
6-15 (6-84)
1 + 0.8(L/D) K = 2 for L/D > 0.75 (6-85) 1 + 1.90(L/D) + 0.6(L/D) For L/D > 100, the error in neglecting the end correction by using the fully developed pipe flow equation (6-81) is less than 2 percent. For rectangular channels, see Normand (Ind. Eng. Chem., 40, 783–787 [1948]). Yu and Sparrow ( J. Basic Eng., 70, 405–410 [1970]) give a theoretically derived chart for slot seals with or without a sheet located in or passing through the seal, giving mass flow rate as a function of the ratio of seal plate thickness to gap opening. Slip Flow In the transition region between molecular flow and continuum viscous flow, the conductance for fully developed pipe flow is most easily obtained by the method of Brown, et al. (J. Appl. Phys., 17, 802–813 [1946]), which uses the parameter X=
2µ RT 冪莦π8 冢Dλ 冣 = 冢 M p D 冣 冪莦
(6-86)
m
where pm is the arithmetic mean absolute pressure. A correction factor F, read from Fig. 6-12 as a function of X, is applied to the conductance for viscous flow. πD4pm C=F (6-87) 128µL For slip flow through square channels, see Milligan and Wilkerson (J. Eng. Ind., 95, 370–372 [1973]). For slip flow through annuli, see Maegley and Berman (Phys. Fluids, 15, 780–785 [1972]). The pump-down time θ for evacuating a vessel in the absence of air in-leakage is given approximately by p1 − p0 V θ = t ln (6-88) S0 p2 − p0 where Vt = volume of vessel plus volume of piping between vessel and pump; S0 = system speed as given by Eq. (6-80), assumed independent of pressure; p1 = initial vessel pressure; p2 = final vessel pressure; and p0 = lowest pump intake pressure attainable with the pump in question. See Dushman and Lafferty (Scientific Foundations of Vacuum Technique, 2d ed., Wiley, New York, 1962). The amount of inerts which has to be removed by a pumping system after the pump-down stage depends on the in-leakage of air at the various fittings, connections, and so on. Air leakage is often correlated with system volume and pressure, but this approach introduces uncer-
冢 冣 冢
冣
冪莦
(D1 − D2)2(D1 + D2) C = 0.42K L
RT 冪莦 M
(6-83)
w
where K is a dimensionless constant with values given in Table 6-3. For a short pipe of circular cross section, the conductance as calculated for an orifice from Eq. (6-82) is multiplied by a correction factor K which may be approximated as (Kennard, Kinetic Theory of Gases, McGraw-Hill, New York, 1938, pp. 306–308) TABLE 6-3
Constants for Circular Annuli
D2 /D1
K
D2 /D1
K
0 0.259 0.500
1.00 1.072 1.154
0.707 0.866 0.966
1.254 1.430 1.675
FIG. 6-12 Correction factor for Poiseuille’s equation at low pressures. Curve A: experimental curve for glass capillaries and smooth metal tubes. (From Brown, et al., J. Appl. Phys., 17, 802 [1946].) Curve B: experimental curve for iron pipe (From Riggle, Courtesy of E. I. du Pont de Nemours & Co.)
6-16
FLUID AND PARTICLE DYNAMICS
tainty because the number and size of leaks does not necessily correlate with system volume, and leakage is sensitive to maintenance quality. Ryans and Roper (Process Vacuum System Design and Operation, McGraw-Hill, New York, 1986) present a thorough discussion of air leakage.
Contraction and Entrance Losses For a sudden contraction at a sharp-edged entrance to a pipe or sudden reduction in crosssectional area of a channel, as shown in Fig. 6-13a, the loss coefficient based on the downstream velocity V2 is given for turbulent flow in Crane Co. Tech Paper 410 (1980) approximately by
冢
A K = 0.5 1 − 2 A1
FRICTIONAL LOSSES IN PIPELINE ELEMENTS The viscous or frictional loss term in the mechanical energy balance for most cases is obtained experimentally. For many common fittings found in piping systems, such as expansions, contractions, elbows and valves, data are available to estimate the losses. Substitution into the energy balance then allows calculation of pressure drop. A common error is to assume that pressure drop and frictional losses are equivalent. Equation (6-16) shows that in addition to frictional losses, other factors such as shaft work and velocity or elevation change influence pressure drop. Losses lv for incompressible flow in sections of straight pipe of constant diameter may be calculated as previously described using the Fanning friction factor: ∆P 2 f V 2L lv = = (6-89) ρ D where ∆P = drop in equivalent pressure, P = p + ρgz, with p = pressure, ρ = fluid density, g = acceleration of gravity, and z = elevation. Losses in the fittings of a piping network are frequently termed minor losses or miscellaneous losses. These descriptions are misleading because in process piping fitting losses are often much greater than the losses in straight piping sections. Equivalent Length and Velocity Head Methods Two methods are in common use for estimating fitting loss. One, the equivalent length method, reports the losses in a piping element as the length of straight pipe which would have the same loss. For turbulent flows, the equivalent length is usually reported as a number of diameters of pipe of the same size as the fitting connection; Le /D is given as a fixed quantity, independent of D. This approach tends to be most accurate for a single fitting size and loses accuracy with deviation from this size. For laminar flows, Le /D correlations normally have a size dependence through a Reynolds number term. The other method is the velocity head method. The term V 2/2g has dimensions of length and is commonly called a velocity head. Application of the Bernoulli equation to the problem of frictionless discharge at velocity V through a nozzle at the bottom of a column of liquid of height H shows that H = V 2/2g. Thus H is the liquid head corresponding to the velocity V. Use of the velocity head to scale pressure drops has wide application in fluid mechanics. Examination of the Navier-Stokes equations suggests that when the inertial terms dominate the viscous terms, pressure gradients are expected to be proportional to ρV 2 where V is a characteristic velocity of the flow. In the velocity head method, the losses are reported as a number of velocity heads K. Then, the engineering Bernoulli equation for an incompressible fluid can be written ρV22 ρV12 ρV 2 p1 − p2 = α2 − α1 + ρg(z2 − z1) + K (6-90) 2 2 2 where V is the reference velocity upon which the velocity head loss coefficient K is based. For a section of straight pipe, K = 4 f L/D.
(a) FIG. 6-13
(b)
冣
(6-91)
Example 5: Entrance Loss Water, ρ = 1000 kg/m3, flows from a large vessel through a sharp-edged entrance into a pipe at a velocity in the pipe of 2 m/s. The flow is turbulent. Estimate the pressure drop from the vessel into the pipe. With A2 /A1 ∼ 0, the viscous loss coefficient is K = 0.5 from Eq. (6-91). The mechanical energy balance, Eq. (6-16) with V1 = 0 and z2 − z1 = 0 and assuming uniform flow (α2 = 1) becomes ρV22 ρV22 p1 − p2 = + 0.5 = 4,000 + 2,000 = 6,000 Pa 2 2 Note that the total pressure drop consists of 0.5 velocity heads of frictional loss contribution, and 1 velocity head of velocity change contribution. The frictional contribution is a permanent loss of mechanical energy by viscous dissipation. The acceleration contribution is reversible; if the fluid were subsequently decelerated in a frictionless diffuser, a 4,000 Pa pressure rise would occur.
For a trumpet-shaped rounded entrance, with a radius of rounding greater than about 15 percent of the pipe diameter (Fig. 6-13b), the turbulent flow loss coefficient K is only about 0.1 (Vennard and Street, Elementary Fluid Mechanics, 5th ed., Wiley, New York, 1975, pp. 420–421). Rounding of the inlet prevents formation of the vena contracta, thereby reducing the resistance to flow. For laminar flow the losses in sudden contraction may be estimated for area ratios A2 /A1 < 0.2 by an equivalent additional pipe length Le given by Le /D = 0.3 + 0.04Re
(6-92)
where D is the diameter of the smaller pipe and Re is the Reynolds number in the smaller pipe. For laminar flow in the entrance to rectangular ducts, see Shah (J. Fluids Eng., 100, 177–179 [1978]) and Roscoe (Philos. Mag., 40, 338–351 [1949]). For creeping flow, Re < 1, of power law fluids, the entrance loss is approximately Le/D = 0.3/n (Boger, Gupta, and Tanner, J. Non-Newtonian Fluid Mech., 4, 239–248 [1978]). For viscoelastic fluid flow in circular channels with sudden contraction, a toroidal vortex forms upstream of the contraction plane. Such flows are reviewed by Boger (Ann. Review Fluid Mech., 19, 157–182 [1987]). For creeping flow through conical converging channels, inertial acceleration terms are negligible and the viscous pressure drop ∆p = ρlv may be computed by integration of the differential form of the Hagen-Poiseuille equation Eq. (6-36), provided the angle of convergence is small. The result for a power law fluid is 3n + 1 ∆p = K 4n
冢
8V 1 1 − 冢 冣 冥冧 冣 冢 D 冣 冦 6n tan (α/2) 冤 D n
2
n
D2
2
3n
1
where D1 = inlet diameter D2 = exit diameter V2 = velocity at the exit α = total included angle
(c)
(d)
Contractions and enlargements: (a) sudden contraction, (b) rounded contraction, (c) sudden enlargement, and (d) uniformly diverging duct.
(6-93)
FLUID DYNAMICS Equation (6-93) agrees with experimental data (Kemblowski and Kiljanski, Chem. Eng. J. (Lausanne), 9, 141–151 [1975]) for α < 11°. For Newtonian liquids, Eq. (6-93) simplifies to 8V 1 D 3 ∆p = µ 2 1 − 2 (6-94) D2 6 tan (α/2) D1 For creeping flow through rectangular or two-dimensional converging channels, the differential form of the Hagen-Poiseulle equation with equivalent diameter given by Eq. (6-49) may be used, provided the convergence is gradual. Expansion and Exit Losses For ducts of any cross section, the frictional loss for a sudden enlargement (Fig. 6-13c) with turbulent flow is given by the Borda-Carnot equation: A 2 V12 − V22 V12 lv = = 1 − 1 (6-95) 2 2 A2 where V1 = velocity in the smaller duct V2 = velocity in the larger duct A1 = cross-sectional area of the smaller duct A2 = cross-sectional area of the larger duct
冢 冣冦
冤 冢 冣 冥冧
冢
冣
Equation (6-95) is valid for incompressible flow. For compressible flows, see Benedict, Wyler, Dudek, and Gleed ( J. Eng. Power, 98, 327–334 [1976]). For an infinite expansion, A1/A2 = 0, Eq. (6-95) shows that the exit loss from a pipe is 1 velocity head. This result is easily deduced from the mechanical energy balance Eq. (6-90), noting that p1 = p2. This exit loss is due to the dissipation of the discharged jet; there is no pressure drop at the exit. For creeping Newtonian flow (Re < 1), the frictional loss due to a sudden enlargement should be obtained from the same equation for a sudden contraction (Eq. [6-92]). Note, however, that Boger, Gupta, and Tanner (ibid.) give an exit friction equivalent length of 0.12 diameter, increasing for power law fluids as the exponent decreases. For laminar flows at higher Reynolds numbers, the pressure drop is twice that given by Eq. (6-95). This results from the velocity profile factor α in the mechanical energy balance being 2.0 for the parabolic laminar velocity profile. If the transition from a small to a large duct of any cross-sectional shape is accomplished by a uniformly diverging duct (see Fig. 6-13d) with a straight axis, the total frictional pressure drop can be computed by integrating the differential form of Eq. (6-89), dlv /dx = 2 f V 2/D over the length of the expansion, provided the total angle α between the diverging walls is less than 7°. For angles between 7 and 45°, the loss coefficient may be estimated as 2.6 sin (α/2) times the loss coefficient for a sudden expansion; see Hooper (Chem. Eng., Nov. 7, 1988). Gibson (Hydraulics and Its Applications, 5th ed., Constable, London 1952, p. 93) recommends multiplying the sudden enlargement loss by 0.13 for 5° < α < 7.5° and by 0.0110α1.22 for 7.5° < α < 35°. For angles greater than 35 to 45°, the losses are normally considered equal to those for a sudden expansion, although in some cases the losses may be greater. Expanding flow through standard pipe reducers should be treated as sudden expansions. Trumpet-shaped enlargements for turbulent flow designed for constant decrease in velocity head per unit length were found by Gibson (ibid., p. 95) to give 20 to 60 percent less frictional loss than straight taper pipes of the same length. A special feature of expansion flows occurs when viscoelastic liquids are extruded through a die at a low Reynolds number. The extrudate may expand to a diameter several times greater than the die diameter, whereas for a Newtonian fluid the diameter expands only 10 percent. This phenomenon, called die swell, is most pronounced with short dies (Graessley, Glasscock, and Crawley, Trans. Soc. Rheol., 14, 519–544 [1970]). For velocity distribution measurements near the die exit, see Goulden and MacSporran (J. Non-Newtonian Fluid Mech., 1, 183–198 [1976]) and Whipple and Hill (AIChE J., 24, 664–671 [1978]). At high flow rates, the extrudate becomes distorted, suffering melt fracture at wall shear stresses greater than 105 N/m2. This phenomenon is reviewed by Denn (Ann. Review Fluid Mech., 22, 13–34 [1990]). Ramamurthy (J. Rheol., 30, 337–357 [1986]) has found a dependence of apparent stick-slip behavior in melt fracture to be dependent on the material of construction of the die.
6-17
Fittings and Valves For turbulent flow, the frictional loss for fittings and valves can be expressed by the equivalent length or velocity head methods. As fitting size is varied, K values are relatively more constant than Le /D values, but since fittings generally do not achieve geometric similarity between sizes, K values tend to decrease with increasing fitting size. Table 6-4 gives K values for many types of fittings and valves. Manufacturers of valves, especially control valves, express valve capacity in terms of a flow coefficient Cv, which gives the flow rate through the valve in gal/min of water at 60°F under a pressure drop of 1 lbf/in2. It is related to K by C1d 2 Cv = (6-96) K 兹苶 where C1 is a dimensional constant equal to 29.9 and d is the diameter of the valve connections in inches. For laminar flow, data for the frictional loss of valves and fittings are meager. (Beck and Miller, J. Am. Soc. Nav. Eng., 56, 62–83 [1944]; Beck, ibid., 56, 235–271, 366–388, 389–395 [1944]; De Craene, Heat. Piping Air Cond., 27[10], 90–95 [1955]; Karr and Schutz, J. Am. Soc. Nav. Eng., 52, 239–256 [1940]; and Kittredge and Rowley, Trans. ASME, 79, 1759–1766 [1957]). The data of Kittredge and Rowley indicate that K is constant for Reynolds numbers above 500 to 2,000, but increases rapidly as Re decreases below 500. Typical values for K for laminar flow Reynolds numbers are shown in Table 6-5. Methods to calculate losses for tee and wye junctions for dividing and combining flow are given by Miller (Internal Flow Systems, 2d ed., Chap. 13, BHRA, Cranfield, 1990), including effects of Reynolds number, angle between legs, area ratio, and radius. Junctions with more than three legs are also discussed. The sources of data for the loss coefficient charts are Blaisdell and Manson (U.S. Dept. Agric. Res. Serv. Tech. Bull. 1283 [August 1963]) for combining flow and Gardel (Bull. Tech. Suisses Romande, 85[9], 123–130 [1957]; 85[10], 143–148 [1957]) together with additional unpublished data for dividing flow. Miller (Internal Flow Systems, 2d ed., Chap. 13, BHRA, Cranfield, 1990) gives the most complete information on losses in bends and curved pipes. For turbulent flow in circular cross-section bends of constant area, as shown in Fig. 6-14a, a more accurate estimate of the loss coefficient K than that given in Table 6-4 is K = K*CReCoCf
(6-97)
where K*, given in Fig. 6-14b, is the loss coefficient for a smoothwalled bend at a Reynolds number of 106. The Reynolds number correction factor CRe is given in Fig. 6-14c. For 0.7 < r/D < 1 or for K* < 0.4, use the CRe value for r/D = 1. Otherwise, if r/D < 1, ob