Comprehensive Chemical Kinetics, Volume 36: Photokinetics : Theoretical Fundamentals and Applications

COMPREHENSIVE CHEMICAL KINETICS

COMPREHENSIVE Section 1.

THE PRACTICE AND THEORY OF KINETICS (3 volumes)

Section 2.

HOMOGENEOUS DECOMPOSITION AND ISOMERISATION REACTIONS (2 volumes)

Section 3.

INORGANIC REACTIONS (2 volumes)

Section 4.

ORGANIC REACTIONS (5 volumes)

Section 5.

POLYMERISATION REACTIONS (3 volumes)

Section 6.

OXIDATION AND COMBUSTION REACTIONS (2 volumes)

Section 7.

SELECTED ELEMENTARY REACTIONS (1 volume)

Section 8.

HETEROGENEOUS REACTIONS (4 volumes)

Section 9.

KINETICS AND CHEMICAL TECHNOLOGY (1 volume)

Section 10.

MODERN METHODS, THEORY AND DATA

CHEMICAL KINETICS EDITED BY

R.G. COMPTON M.A., D. Phil. (Oxon.) Oxford University The Physical and Theoretical Chemistry Laboratory Oxford, England

G. HANCOCK Oxford University The Physical and Theoretical Chemistry Laboratory Oxford, England

VOLUME 36

PHOTOKINETICS THEORETICAL FUNDAMENTALS AND APPLICATIONS H. MAUSER+ G. GAUGLITZ Eberhard-Karls- Universitdt Tubingen Institutfiir Physikalische und Theoretische Chemie Tubingen, Germany

1998 ELSEVIER AMSTERDAM - LAUSANNE - NEW YORK - OXFORD - SHANNON - SINGAPORE - TOKYO

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211,1000 AE Amsterdam, The Netherlands

ISBN: 0-444-82536-3 (Vol. 36) ©1998 Elsevier Science B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521,1000 AM Amsterdam, The Netherlands. Spedal regulations for readers in the U.S.A. -This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the copyright owner, Elsevier Science B.V., unless otherwise specified. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. © The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Transferred to digital printing 2005

Volumes in the Series Section 1.

Volume 1 Volume 2 Volume 3

The Practice of Kinetics The Theory of Kinetics The Formation and Decay of Excited Species Section 2.

Volume 4 Volume 5

OXIDATION AND COMBUSTION REACTIONS (2 volumes)

Liquid-phase Oxidation Gas-phase Combustion Section 7.

Volume 18

POLYMERISATION REACTIONS (3 volumes)

Degradation of Polymers Free-radical Polymerisation Non-radical Polymerisation Section 6.

Volume 16 Volume 17

ORGANIC REACTIONS (5 volumes)

Proton Transfer Addition and Elimination Reactions of Aliphatic Compounds Ester Formation and Hydrolysis and Related Reactions Electrophilic Substitution at a Saturated Carbon Atom Reactions of Aromatic Compounds Section 5.

Volume 14 Volume 14A Volume 15

INORGANIC REACTIONS (2 volumes)

Reactions of Non-metallic Inorganic Compounds Reactions of Metallic Salts and Complexes, and Organometallic Compounds Section 4.

Volume 8 Volume 9 Volume 10 Volume 12 Volume 13

HOMOGENEOUS DECOMPOSITION AND ISOMERISATION REACTIONS (2 volumes)

Decomposition of Inorganic and Organometallic Compounds Decomposition and Isomerisation of Organic Compounds Section 3.

Volume 6 Volume 7

THE PRACTICE AND THEORY OF KINETICS (3 volumes)

SELECTED ELEMENTARY REACTIONS (1 volume)

Selected Elementary Reactions

Section 8. Volume 19 Volume 20 Volume 21 Volume 22

Simple Processes at the Gas-Solid Interface Complex Catalytic Processes Reactions of Solids with Gases Reactions in the Solid State Section 9.

Volume 23

KINETICS AND CHEMICAL TECHNOLOGY (1 volume)

Kinetics and Chemical Technology Section 10.

Volume 24 Volume 25 Volume 26 Volume 27 Volume 28 Volume 29 Volume 30 Volume 31 Volume 32 Volume 33 Volume 34 Volume 35 Volume 36

HETEROGENEOUS REACTIONS (4 volumes)

MODERN METHODS, THEORY, AND DATA

Modem Methods in Kinetics Diffusion-limited Reactions Electrode Kinetics: Principles and Methodology Electrode Kinetics: Reactions Reactions at the Liquid-Solid Interface New Techniques for the Study of Electrodes and their Reactions Electron Tunneling in Chemistry. Chemical Reactions over Large Distances Mechanism and Kinetics of Addition Polymerizations Kinetic Models of Catalytic Reactions Catastrophe Theory Modem Aspects of Diffusion-Controlled Reactions Low-temperature Combustion and Autoignition Photokinetics. Theoretical Fundamentals and Applications

COMPREHENSIVE CHEMICAL KINETICS

ADVISORY BOARD

Professor C.H. BAMFORD Professor S.W. BENSON Professor LORD DAINTON Professor G. GEE Professor G.S. HAMMOND Professor K.J. LAIDLER Professor SIR HARRY MELVILLE Professor S. OKAMURA Professor Z.G. SZABO Professor O. WICHTERLE

Contributors to Volume 36

All chapters in this volume have been written by

H. MAUSERt G. GAUGLHZ

Eberhard-Karls-Universitat Tubingen Institut fur Physikalische und Theoretische Chemie Auf der Morgenstelle 8 D-72076 Tubingen Germany

Preface

Interest in photochemical reactions has increased tremendously in recent years. Whereas books concerning thermal kinetics can be found in large numbers, only short articles or research reviews have been written on photokinetics. As a consequence, even in respected journals, photokinetic aspects are considered which have already been discussed in detail years ago. One of the few general studies on photokinetics was written by Mauser some years ago in German. Because of the language its content did not become known in English speaking countries, as we realised each time we attended an international meeting. Quite a few times the topics treated in this book were of such interest that, for example, Americans fought with the German language in order to use the formalism for their own research. In addition, in recent years, the combination of spectroscopic and chromatographic methods opened up totally new methods for the measurement and evaluation of photochemical reactions. As a consequence, the topics of the book mentioned above, as well as some experimental set-ups, were considered to be worth a new edition published in English. Since theoretical considerations can be demonstrated best by examples, a variety of applications have been added. Furthermore, in an Appendix many equations are derived in detail and a photokinetic analysis is demonstrated step by step, including some references to an evaluation program as well as some macros which can be embedded in spreadsheet programs such as EXCEL. These supply the necessary calculus. Nowadays detailed kinetic examination of photochemical reactions is unavoidable in the fields of information storage material, in chemical actinometry, photo-polymerisations, photobiology, and laser-induced chemistry. In earlier times, overall turnover yields were thought to provide sufficient information about a reaction to allow an economical appraisal. A tremendous amount of experimental work had to be done to change irradiation conditions or to find derivatives of molecules in order to increase or decrease yields. Mechanistic considerations remained somewhat empirical. Whereas formal kinetics were introduced years ago in thermal reactions of biological material, photobiology only recently detected its advantages.

X

Preface

These methods offer the chance to determine the partial photochemical quantum yields of each step in a multi-step reaction procedure. By way of this quantity, the influence of changed reaction conditions or substitution of the compound in question can be determined quantitatively. By considering the principles and the methods mentioned, it will be possible to reduce the number of experiments to optimise photo-degradation processes or the photostability of products. Both applications are of immense interest in the optimal system for reversible storage of information or to increase photostability of laser dyes. The same is valid for all photochemical processes run in industry or even in laboratories in research groups. Apart from the principles of photophysics and photochemical primary steps of reactions on which a number of books have been published in the past, the principles of photokinetics will be necessary for advanced students to gain a detailed understanding of quantitative considerations of photochemical reactions. For the same reason, researchers as well as professionals in industry, university, or governmental institutions will refer to a book of this type. It provides a concise treatment of photokinetics, its principles, its methods, and also gives a wide range of possible applications as well as a formal description of a large number of possible mechanisms of reactions. This will enable the readers either to find an example for their own application or to set up their own formalism. H, Mauser Reutlingen 1995 G. Gauglitz Tubingen 1996

After the unexpected demise of one of the authors. Prof. Dr. Heinz Mauser, on October 4, 1995, before completion of the final version of the manuscript, the task to finish the book for publication fell upon me. A major difficulty in this process was checking the large number of equations for consistency and accuracy, however. I have to thank many of my graduate students who helped me to check the manuscript for typing errors and inconsistency. I want to thank especially Mrs. Dipl.-phys Albrecht Klotz, DipLphys. Matthias Rothmund, and Dipl.-inf. Fred Rapp. The latter was very much involved in programming a new workstation version of the photokinetic evaluation program.

Preface

XI

This program can be used as a stand-alone version in seminars for chemistry majors, and is available on the Internet at URL www: http.V/barolo.ipc.uni-tuebingen.de/photokinetics (at the moment) Whereas the figures can easily be graphed in spreadsheet programs such as EXCEL, photokinetic evaluation requires some programming using Runge-Kutta algorithms or multi-exponential curve-fitting techniques. Each mechanism requires a different program, this being a major obstacle in developing a common version. The stand-alone version which we are able to provide on request contains some program lines that have to be modified depending on the mechanism used. I am especially indebted to my colleague and personal friend, Prof. Dr. Jtirgen Polster, TU Munich, Weihenstepan, who proof-read part of the manuscript and provided valuable advice as to optimising the didactic aspects of this book and with respect to inconsistency. Finally, I am grateful to the Series Editor, Dr. Hancock as well as the publishing department at Elsevier, where Dr. Dempster (luckily!) forced us to finish the manuscript as fast as possible, and to Lorna O'Brien who was responsible for the production. Last but not least I have to express my thanks to my wife and my children who took the burden of tolerating my spending time to finish this book. Giinter Gauglitz Tubingen, July 1998

Contents

Preface

IX

1

Introduction 1.1 General considerations 1.2 Stoichiometry of chemical reactions 1.3 Photophysics 1.3.1 Absorption of radiation 1.3.2 Primary photophysical processes 1.4 Photochemistry and photokinetics 1.4.1 Chemical primary processes 1.4.2 Definition of the quantum yield 1.4.2.1 Apparent integral quantum yield 1.4.2.2 True integral quantum yield 1.4.2.3 Apparent differential quantum yield 1.4.3 Determination of the amount of light absorbed 1.4.4 The photokinetic factor 1.4.5 Disposition of chapters

1 1 5 7 7 8 13 13 15 15 16 16 16 19 22

2

General approach in formal kinetics 2.1 Fundamentals 2.1.1 Stoichiometry 2.1.1.1 Formalism of the reaction scheme 2.1.1.2 Partial reactions 2.1.2 Quantum yields independent on reaction time 2.1.2.1 True differential quantum yield 2.1.2.2 Partial photochemical quantum yields 2.1.3 Differential equations 2.1.3.1 Thermal rate laws 2.1.3.2 Bodenstein hypothesis 2.1.3.3 Simple photokinetic rate laws 2.1.4 The principle of linear independent reactions 2.1.4.1 Uniform reactions 2.1.4.2 Thermal reactions 2.1.4.3 Photochemical reactions 2.1.4.4 Complex non-uniform photoreactions

25 25 25 27 31 35 36 36 37 37 40 40 44 45 45 51 52

XIV

Contents

2.2

2.3

2.4

2.5

2.1.5 Comparison of the formalism for thermal and photochemical steps of a reaction 2.1.6 Use of symbols 2.1.7 General rate laws for simple thermal reactions 2.1.7.1 Zero order reactions 2.1.7.2 First order reactions 2.1.7.3 Second order reactions 2.1.7.4 Reactions of//th order 2.1.8 Determination of concentrations at the end of the reaction General approach to linear systems of reactions 2.2.1 Differential equations as a general treatment in kinetics 2.2.1.1 Formalism 2.2.1.2 Jacobi matrix 2.2.1.3 Examples for Jacobi matrices (thermal reactions) 2.2.1.4 Photokinetic Jacobi matrix 2.2.2 General solution of the differential equations 2.2.2.1 Determination of coefficients of the Jacobi matrix 2.2.2.2 Solution at two equal eigenvalues 2.2.2.3 Solution for complex eigenvalues 2.2.3 Numerical calculation of functions of concenti-ation Information about reactions supplied by graphs 2.3.1 Essential properties of linear systems 2.3.1.1 General considerations 2.3.1.2 First view: extrema and points of inflection 2.3.2 Number of exponential functions 2.3.3 X- and K-diagrams 2.3.3.1 Initial and final values 2.3.3.2 The initial slope 2.3.3.3 The final slope 2.3.3.4 The extrema 2.3.3.5 Influence of dilution 2.3.3.6 The areas 2.3.3.7 Points of inflection 2.3.3.8 Regions of existence in a K-diagram 2.3.4 Transformation of linear systems Applications to selected systems 2.4.1 Uniform reactions 2.4.1.1 Parallel reaction 2.4.1.2 Back and forward reactions 2.4.2 Two linear independent reactions 2.4.2.1 Independent parallel reactions 2.4.2.2 Consecutive reactions 2.4.2.3 More than two linear independent steps of reaction Methods of approximation

53 55 55 56 57 58 61 63 65 65 65 72 74 74 75 78 81 83 85 87 87 87 87 88 91 92 92 94 96 96 96 98 98 99 102 103 103 105 106 106 107 116 118

Contents 2.5.1 Negligible concentration of an intermediate 2.5.2 The Bodenstein hypothesis 2.5.2.1 Example of a radical chain reaction 2.5.2.2 The Michaelis-Menten reaction 2.5.3 Method of quasi-stationary variables 2.5.4 A criterion 2.5.5 The approximation of the equilibrium 2.5.6 reduction of the number of linear independent steps of a reaction Special considerations for simple and complex photochemical reactions 3.1 Relationship between quantum yield and mechanism 3.1.1 Simple uniform photoreactions 3.1.1.1 Quenched photoisomerisations 3.1.1.2 Photoadditions 3.1.1.3 Photoreduction 3.1.1.4 Sensitised photoreactions 3.1.1.5 Radical photoreactions 3.1.1.6 Parallel photoreactions 3.1.2 Simple non-uniform photoreactions 3.1.2.1 Parallel photoreaction including other reactants 3.1.2.2 Physically sensitised photoreactions 3.1.3 Complex uniform photochemical reactions 3.1.4 Complex non-uniform photoreactions 3.1.4.1 Parallel photoreactions 3.1.4.2 Consecutive photoreaction 3.2 Explicit differential equations in photokinetics 3.2.1 Rates of simple uniform photoreactions 3.2.1.1 The quantum yield does not depend on the amount of light absorbed 3.2.1.2 The quantum yield depends on the amount of light absorbed 3.2.2 Rates of non-simple and/or non-uniform photoreactions 3.2.3 The relation between the concentrations for complicated photoreactions. 3.3 The integration of the differential equation 3.3.1 The absorbance at the wavelength of irradiation stays constant 3.3.2 The relationship between absorbance and concentration 3.3.3 Summary 3.3.4 The integrals of simple uniform photoreactions 3.3.5 The integrals of uniform complicated reactions 3.3.6 The integrals for simple non-uniform reactions 3.3.7 The integrals for non-uniform complicated reactions 3.3.8 The integration of uniform reaction with superimposed thermal reactions 3.3.9 Non-uniform reactions with superimposed thermal reactions 3.3.10 Reactions with a difference in the dependence on intensity 3.3.11 An alternative: the transformation of time

XV 118 122 125 128 130 132 136 141 145 145 145 146 149 152 154 158 160 163 164 175 177 178 178 180 181 181 182 184 185 187 191 192 197 199 204 207 208 210 210 212 214 214

XVI

Contents 3.4 Photoreactions in viscous media 3.4.1 Simple example for photoreactions in viscous media 3.4.2 General solution 3.4.2.1 The quantum yield does not depend on the amount of light absorbed 3.4.2.2 The quantum yield is reciprocally proportional to the square root of the amount of light absorbed 3.4.3 Some further examples 3.4.3.1 The integrals can be solved in a closed form 3.4.3.2 The integrals cannot be solved in a closed form 3.4.4 Complicated photoreactions 3.5 Survey on approaches for selected examples

4

Experimental techniques in photokinetic analysis 4.1 Introduction 4.1.1 Requirements of methods of measurement 4.1.2 Classification of the uv/vis spectroscopic methods 4.2 Absorption measurement 4.2.1 Principle considerations 4.2.2 Apparatus for absorbance measurement 4.2.2.1 Sequential spectrometers 4.2.2.2 Multiplex spectrometers 4.2.2.3 Combined irradiation and measurement devices 4.2.3 Measurement of photoreactions with superimposed thermal reactions 4.2.4 Special devices 4.2.5 Some considerations with respect to photometry 4.2.5.1 Limitations by Lambert-Beer*s law 4.2.5.2 Influence of stray light 4.2.5.3 Multiple reflection 4.2.5.4 Further problems 4.2.6 Absorbance measurements for photokinetic analysis 4.2.6.1 Reaction spectra, absorbance time diagrams 4.2.6.2 The determination of concentrations from absorbance measurements 4.2.6.3 Multicomponent analysis 4.2.7 Derivative spectroscopy 4.3 Fluorescence measurements 4.3.1 Correlation between fluorescence and structure 4.3.2 Fluorescence spectrometer 4.4 Reflectance measurements 4.4.1 Principles of reflectometry 4.4.1.1 Diffuse reflectance 4.4.1.2 Regular reflectance 4.4.2 Set-ups in reflectance measurements

218 218 222 222 223 224 224 230 232 237 241 241 241 244 244 244 245 245 247 249 255 257 258 258 260 261 262 263 263 267 267 273 275 275 277 281 281 281 283 285

Contents

XVII

4.5 Chromatographic methods 286 4.5.1 General considerations and comparison between different chromatographic methods 286 4.5.2 Thin-layer chromatography 287 4.5.3 High performance liquid chromatography 289 4.6 Special methods 291 4.7 Process control in kinetics 292 4.8 Determination of intensity of irradiation 293 4.8.1 Physical methods 294 4.8.2 Chemical actinometers 296 Applications of kinetic analysis to photoreactions 5.1 Evaluation of measurements of concentrations 5.1.1 Determination of the number of linear independent steps of reactions 5.1.1.1 Matrix rank analysis 5.1.1.2 Graphical methods 5.1.2 Evaluation of concentration-time measurements 5.1.2.1 Evaluation of integrated time laws 5.1.2.2 Evaluation of differential equations 5.2 Evaluation of measurements of concentrations 5.2.1 Principles 5.2.2 Matrix rank analysis 5.2.3 Evaluation of time functions 5.2.4 Quantum yields 5.2.5 Formal integration 5.2.6 Missing discrimination between two reactions 5.3 Evaluation of linear dependent measured values 5.3.1 Principle consideration 5.3.2 The fundamental equation of spectroscopy 5.3.2.1 Determination of concentrations 5.3.2.2 Determination of absorption coefficients 5.3.2.3 Other concentration measures 5.3.2.4 Linear relationship between the concentrations 5.3.2.5 Absorbance and degree of advancement 5.3.3 Matrix rank analysis 5.3.3.1 Application of graphical methods using absorbances 5.3.3.2 Absorbance diagrams 5.4 Evaluation of absorbance-time measurements 5.4.1 Evaluation by integrated time laws 5.4.2 Difference equations 5.4.3 Difference equations of higher order 5.4.4 Formal integration 5.4.4.1 Simple uniform photoreactions 5.4.5 Discrimination of two mechanisms

299 299 299 300 304 315 315 323 327 327 327 329 331 332 334 336 336 337 337 337 338 338 342 344 345 363 371 371 371 373 374 374 392

XVIII

5.5

5.6

5.7

5.8

Contents 5.4.5.1 All the concentrations at one reaction time are known 5.4.5.2 Only one component absorbs within the wavelength range 5.4.5.3 Further additional information 5.4.6 Approximations in practice 5.4.6.1 the determination of concentrations from absorbance measurements 5.4.6.2 Application to actinometry 5.4.7 Evaluation of photoreactions superimposed by thermal reactions 5.4.7.1 Photochromic systems of dihydroindolizines 5.4.7.2 Photoisomerisation of (Z,E,E)-4,4'-distyrylazobenzcne 5.4.8 Further approaches Fluorescence 5.5.1 Correlation between fluorescence intensity and concentration 5.5.1.1 Time-dependence of fluorescence signals during reaction 5.5.1.2 Intensity diagrams for mechanistic information 5.5.2 Photokinetic equations for different reactions 5.5.2.1 Uniform reactions 5.5.2.2 Consecutive reactions 5.5.3 Errors due to E' > 0.02 5.5.4 Applications 5.5.4.1 Anthraquinones 5.5.4.2 Photostability of laser dyes Combination of methods 5.6.1 Combination of absorbance and fluorescence measurements 5.6.1.1 Applications to photokinetics of laser dyes 5.6.1.2 Dependence on variation of viscosity of the solvent 5.6.2 Combination of absorbance and chromatography 5.6.2.1 Chromatographic reaction spectra 5.6.2.2 Photochemical quantum yields using chromatographic data 5.6.3 Application of chemometrics 5.6.4 Combination of absorbance with NMR 5.6.5 Combination of absorbance with reflectance 5.6.5.1 Photoresist material 5.6.5.2 Photopolymerisation Photoreactions in viscous material 5.7.1 Microcrystalline systems 5.7.2 Chromophores in polymers 5.7.2.1 Azobenzene in siloxanes 5.7.2.2 Dihydroindolizines in siloxanes 5.7.3 Photokinetics in viscous media 5.7.3.1 Azobenzene 5.7.3.2 Photokinetics in liquid crystals Concluding remarks

393 395 396 397 397 399 404 407 410 413 416 416 417 418 418 418 424 428 431 431 431 438 438 440 441 442 445 447 450 456 457 460 461 462 462 463 463 465 465 465 470 471

Contents

XIX

6

Appendix 6.1 Reaction scheme 6.2 Simple photochemical reactions 6.2.1 Photoreaction via the triplet state 6.2.2 Complex uniform photoreactions 6.2.3 Complex non-uniform photoreactions 6.3 Simple thermal reactions 6.3.1 Reactions including steps of back-reactions 6.3.2 Parallel reactions 6.4 Examples for the determination of concentrations 6.4.1 Second order reactions 6.4.2 At the end of the reaction 6.5 Consecutive reactions with superimposed back-reactions 6.6 Relationship between quantum yield and photomechanism 6.6.1 Details for simple uniform photoreactions 6.6.1.1 Photoaddition 6.6.1.2 Sensitised photoreactions 6.6.2 Details for simple non-uniform photoreactions 6.6.2.1 Physically sensitised parallel photoreaction 6.6.2.2 Physically sensitised-consecutive photoreaction 6.7 Example of a photokinetic evaluation 6.7.1 Measurements 6.7.1.1 Absorbance 6.7.1.2 Chromatography 6.7.2 Graphics 6.7.2.1 Absorbance-time curves 6.7.2.2 Absorbance diagrams 6.7.3 Evaluation of absorbance data 6.7.3.1 Set-up of differential equations 6.7.3.2 Determination of kinetic constants 6.7.4 Evaluation of concentration data 6.7.4.1 Absorbance triangle 6.7.4.2 Data from HPLC 6.7.4.3 Concentration-time curves 6.7.4.4 Photokinetic equations 6.7.4.5 Photochemical quantum yields 6.7.5 Computer programs

473 473 477 477 478 481 482 482 486 488 488 491 495 510 510 510 511 515 515 515 516 518 518 519 520 520 520 522 522 523 524 524 525 525 527 527 528

7

References

533

Subject index

545

Chapter 1

Introduction

1.1 GENERAL CONSIDERATIONS The study of photochemical kinetics is a very important approach for elucidating reaction mechanisms and determining the quantitative parameters characteristic for the progress of the photochemical reaction. Since the excitation of electronically excited states can be achieved more selectively than by thermal pathways, irradiation allows reactions to be guided into specific reaction channels. Solvent interaction, matrices, and variation in chemical derivatisation will influence this reaction coordinate. The intention of classical photochemistry was to determine a turnover at an empirical level. However, detailed kinetic examination supplies data beyond such procedures. The determination of the dependence of rates of reaction on the experimental conditions mentioned induced developments in the fields of solar energy storage, information recording, photobiology, understanding of environmental problems, and optimisation of industrial chemical production. Understanding of mechanistic studies requires fundamental knowledge on • creation and properties of radiation and its interaction with matter, • structure and properties of excited states, • photophysical reaction channels, and • subsequent degradation and relaxation processes. All these aspects are frequently treated in textbooks on physical chemistry or photochemistry. Modem equipment allows the monitoring of reaction down to the time domain of femtoseconds. Thus, application of such time resolved methods allows the determination of intermediates and sometimes even the characterisation of transition states. By these means, in many cases the mechanism of elementary reactions can be determined as in the case of examinations of thermal reactions. However, in thermal kinetics the description of reaction progress by rate laws, rate constants, and thermodynamic data usually avoids the quantitative specification of the different elementary steps of the overall reaction. These

2

Introduction

Ch. 1

examinations give rise to rate equations according to the order of the reaction which are used to argue about mechanisms and support the information obtained by flash spectroscopy. This formal approach requires experimental data, measuring the concentrations of the components resolved in time during the reaction procedure at a high level of quality. Modem analytical methods can supply such data. In principle UV/vis spectroscopy is favoured as it is fast, photometrically very exact, and relatively inexpensive. Rate constants are determined by variation of experimental conditions and chemical composition of the reaction mixture. Data are measured by application of a variety of modem analytical methods. Modem numerical approaches of curve fitting and/or solution of differential equations are applied. Results and consequences influence chemical reaction engineering as well as production costs. Many books cover these formal thermal kinetics in detail, but most are restricted to simple mechanisms. In contrast, analogous treatments of photochemical reactions are restricted to publications of special reactions and examinations. Therefore this book aims to supply an overall treatment of formal photokinetics beyond the scope of normal books on kinetics. In principle the approach in photokinetics can be compared to the handling of thermal reactions: • a model of the mechanism of the reaction is stated, • the differential equations are set up, • an analytical numerical solution is tried, • the kinetics are measured, • changes in concentration of the reactants are calculated, • mathematical tools of parametrisation, iteration, or direct determination of coefficients (rate constants) are applied, and • the steps of the thermal reaction procedure together with rate constants are obtained for further interpretation. Unfortunately a general transfer of this well established procedure in the quantification of thermal reactions is not possible directly to photokinetics. In photokinetics even the simplest pseudo first order reaction cannot be described by a differential equation in a closed form. The amount of light absorbed during the reaction has to be taken into account in addition to a quantity comparable to the rate constant. This amount of light absorbed changes during the reaction because of a change in concentration of the component starting the step of the photoreaction. Thus, the proportionality constant in the rate law, set up in analogy to thermal kinetics, includes a co-

Ch. 1

General considerations

3

efficient dependent on irradiation time. The problem can be demonstrated for the simple reaction

which can be described in the case of a thermal reaction as a = -ka,

(1.1)

where a is the actual concentration of reactant A (measured with time), k is the rate constant, and a is the rate (change of A per time). The comparable rate law for the photochemical reaction is given by a = ^(p/^= ^(pIe\aF{t)

(1.2)

where

4

Introduction

Ch. 1

• Photoreactions are treated as pseudo first order kinetics. As demonstrated in Section 2.1.3.3, besides the photochemical step, a variety of photophysical steps such as radiationless transitions are included in the mechanism. Excited states are intermediates. The considerations typical in thermal kinetics as, for example, Bodenstein's hypothesis, can also be applied to these steps. Thus an overall treatment of the total mechanism becomes possible. • If constant intensity of irradiation with time is assumed, only the photokinetic factor F has to be included in the time variable. It depends on the progress of the reaction normally, as stated above depending on the absorption of the sample. Then the product of irradiation intensity and photochemical quantum yield forms a constant equivalent to the rate constant of thermal reactions. The dependent variable is a product of the factor F and the irradiation time t combined as a variable 0 . This introduction of a transformation in the time axis allows formal kinetics to be applied to thermal and photochemical reactions as well. It even allows the handling of solutions which cannot be homogenised, or solid samples in which the concentration varies locally because of decreasing irradiation intensity in the direction of irradiation by the turnover of the reactants. • As in thermal kinetics, the changes in concentration are usually monitored in photokinetics by spectroscopic measurements. Under these conditions, a distinction between different mechanisms is not possible for many reactions. Just the number of spectroscopically linear independent steps of reaction can be determined (see Chapter 4). • An essential approach is the so-called partial (differential) photochemical quantum yield, which is time independent and refers to a specific step of the photochemical reaction. It can be compared to the rate constant of a reaction step in thermal reactions. The change in concentration of reactants is at the centre of interest in photokinetics as well as the determination of these partial photochemical quantum yields. The time laws cannot be integrated in a closed form. Therefore to avoid the problems with solving these differential equations, the integrals are numerically calculated - a procedure named formal integration. This method also turns out to be advantageous in thermal and photochemical examinations. Since the reader will be more familiar with the treatment of thermal reactions, symbols and kinetic fundamentals are first explained using thermal

Ch. 1

Stoichiometry of chemical reactions

5

formalism with reference to the equivalent photokinetic formula in the second chapter. In this way, the elements of the Jacobi matrices for various relative simple mechanisms are derived. It will become obvious that these elements are, in the case of thermal reactions, the rate constants or combinations thereof. For the photochemical reactions, the products of quantum yields and amount of light absorbed form the equivalent matrix elements. However, only under simple conditions are quantum yields independent of concentrations. Therefore in a further chapter the peculiarities of photochemical mechanisms and rate laws are discussed in detail for complex mechanisms. These derivations use matrix equations and approximations as, for example, the Bodenstein hypothesis, as well as other relationships typical of formal kinetics. All these approaches are found in Chapters 2 and 3. The intention is to introduce this formalism to the reader by help of examples and by rather simple mechanisms. These discussions are based on the principles of the stoichiometry of a reaction as well as on photochemical and photophysical facts which are reviewed briefly next. 1.2 STOICHIOMETRY OF CHEMICAL REACTIONS As discussed in detail in textbooks on kinetics [1,2] chemical equations supply information on the number of steps of a reaction, the number of moles of the reactants taking part in the reaction, and the progress of chemical reactions. It implies that in a closed system the changes in the number of moles A^ and of the different reactants A, are not independent of each other. Furthermore, the turnover depends on the stoichiometric coefficients v, of the reactants A, (A,B,...,D) and on the so-called degree of advancement A. These stoichiometric coefficients correspond to the number of moles of reactants formed (positive v, by definition) or disappearing (negative V;) during one turnover (A = 1). The chemical equation 2A-»B + C symbolises that the chemical reaction contains only one step, in which 2A moles of A react to form A moles B and A moles C. By defining

6

Introduction

Ch. 1

as the change in the number of moles of the ith reactant during the progress of the reaction, one obtains a common variable, the degree of advancement A as 2

A

J

B

J

c

Expressions for the changes in concentrations of each reactant can be substituted by this variable A, whereby Ni and Nio correspond to the number of moles at times t and ^ = 0. The stoichiometric coefficients of the reactants are explicitly given to express the degree of advancement. In a more generalised form, the above chemical equation can be rewritten for a single reaction between the n reactants i4,, i42,...,A;, as v,^,+-—^v,^,+---+v„^„.

(1.3)

This expression can be abbreviated by the sum £v,4=0

0 = 1,2,...,«).

(1.4)

1=1

The following equation defines the degree of advancement, known from thermodynamics, in a more generalised form: —AAAi =—AAT, =...= —AA^,. =—AN=X. V,

V2

V,

(1.5)

v„

By differentiation of this equation with respect to time, one obtains —AA^i =—AA^2 =•••=—AA^, = — / / = A. V/

V2

V,

(1.6)

V„

This A is called the total rate of the above generalised reaction. By use of eqs. (1.5) and (1.6) the equations AA^, = «, A

and

N^ = v, A

(/ = 1,2,.. .,n)

Ch. 1

Photophysics

7

can be derived. However, the turnover of reactions in homogeneous phases can be better described by the volume concentration according to

a,--f>

(1.7)

where V represents the volume of the system. Therefore instead of eq. (1.5) the relations 1 A 1 A 1A 1A —Aai =—Atij =•••=—Aa, =•••=—A<3„ = x V, V2 V, V,

,. O. (l-o)

Aa, =a,-a,o=v,jc

(1.9)

or

are used. In this equation a/o represents the concentration of the reactant >4, at time / = 0. In analogy to X the variable x is named the {kinetic) degree of advancement or the advancement variable. Differentiation of eq. (1.7) with respect to time yields: • ^ A^i ^'~ V

N^V ^ N, F^ " F

a,V F *

In general, an attempt is made to achieve an isochoric condition ( F = 0 ) by temperature control in liquid solution. Such isochoric behaviour is taken for granted in the subsequent discussions. Therefore the change in concentration of each reactant can be expressed by ti,=v,jc,

(1.10)

where x = X/V can be called a specific rate or in short the rate of the reaction. 1.3 PHOTOPHYSICS 1.3.1 A bsorption of

radiation

The properties of electromagnetic radiation are discussed elsewhere [ 3 5]. Absorption of such radiation is essential to start the photoreaction. A c -

8

Introduction

Ch. 1

cording to the mechanism A + /iv->A*

(1.11)

an excited molecule A* is formed by absorption of a photon of appropriate energy. The difference between the energy of the molecule in the ground state A and the formed excited state A* is given by the energy of the photon e = hv where h is Planck's constant. The wavelength A, the frequency v and the velocity of light c depend on each other according to vA = c . The photon energy becomes e:=hv = — = hcv . X V = 1/A is called the wavenumber which is given in cm-* (old textbooks use Kaiser), v and v are directly proportional to the energy of the photon. Taking a mole of photons (sometimes photochemists call this an Einstein) the energy of the photon is given in kJ/mol as Ni^s[^] = ^ — ^ = 2.86xl0-^v[cm-^]. A[nm] A^L is Avogadro's or the Loschmidt number. A comparison between the energies in different units is given in Table 1.1. 7.5.2 Primary photophysical processes This absorption in combination with the excitation of a molecule A from the electronic and vibrational ground state at thermal equilibrium A to an energy level of high vibrational and electronic excitation symbolised by A* is given in Fig. 1.1 [6, 7]. This figure represents a schematic energy level

Ch. 1

Photophysics

TABLE 1.1 Energies of electromagnetic radiation relevant to photochemistry in different units Wavelength [nm]

200 250 300 350 400 450 500 550 600 650 700

Energy [1/nm]

[kcal/mol]

[kJ/mol]

[eV]

50000 40000 33333 28571 25000 22222 20000 18182 16667 15385 14286

143.0 114.4 95.3 81.7 71.5 63.6 57.2 52.0 47.7 44.0 40.9

598.7 479.0 399.1 342.1 299.4 266.1 239.5 217.7 199.6 184.2 171.1

6.21 4.96 4.14 3.55 3.10 2.76 2.48 2.26 2.07 1.91 1.77

diagram, called a Jablonski-term diagram. Electronic energy levels are drawn in bold face, vibrational levels in normal type lines. The rotational levels are omitted to reduce the complexity of the diagram. After absorption of photons a variety of processes can take place. These processes are either isoenergetic (no change in energy) or combined with an energy transfer to other molecules or between different types of energy levels of the molecule itself: 1. It can change its state of rotation. The levels of the rotational energies are closely spaced. The difference amounts to approximately 0.4 kJ. That means a molecule can be excited in a higher rotational state by absorption of radiation in the far infrared. The excitation of rotational states by use of conventional intensities of light cannot induce a photoreaction. 2. The distances between nuclei can be changed by vibrations. The levels of vibrational energy have a larger spacing than those of rotational energy. It amounts to 4-40 kJ. A molecule can be excited to a higher vibrational level by absorption of photons in the near infrared and infrared. A photoreaction by uptake of vibrational energy is unusual except for an isomerisation at low temperatures. An example is the isomerisation of nitric acid at 20 K. 3. The state of electronic energy can be changed by absorption of photons in the visible and ultraviolet. In combination with this energy uptake the

10

Introduction

Ch. 1

rotational and vibrational states are also changed. The necessary energies amount to 80-1000 kJ. After excitation to higher states an isomeric molecule is formed, which has different physical properties (dipole moment, pAT value, etc.) to those of the molecule in the ground state. At room temperature in general a molecule happens to be in the lowest electronic and vibrational, and in a relative low rotational state. In the following, organic molecules are considered [8]. In general their lowest electronic state is a singlet, symbolised by SQ. The time necessary for absorption of radiation (eq. (1.11)) is approximately 10"^^ s. During this time the molecule is raised to a higher electronically excited state Si, S2, etc. (states higher than Si are not included in Fig. 1.1). This process is frequently accompanied by an excitation to a higher vibrational and/or rotational energy state. That

,^

A*

>

r

B Fig. 1.1. Electronic and vibrational energy levels (schematic, rotational are omitted for simplicity), So singlet ground state, Si excited singlet state (higher electronically excited states are omitted in this figure), Ti triplet electronically excited energy state. Straight lines symbolise radiative processes (absorption (Ab) as well as emission: F, fluorescence; P, phosphorescence). Wavy lines give the radiationless transitions, ic, internal conversion; isc, intersystem crossing; sd, radiationless deactivation; te, thermal equilibration.

Ch. 1

Photophysics

11

means the molecule marked as A* in the following and in the figure is 'hot' in comparison to its environment. For this reason it equilibrates its 'temperature' in a short time in the condensed phase. It deactivates without radiation to the lowest vibrational state of the actual electronic state (marked in Fig. 1.1 as te). By isoelectronic internal conversion (ic) it can pass over to a very high vibrational state of the next lower electronic energy state. These deactivations normally take place until the molecule has reached the vibrational ground state of the first excited electronic state Si. This overall deactivation process is called thermal relaxation (sd), which can be divided into the isoenergetic deactivation 'internal conversion' and the non-isoenergetic 'thermal equilibration'. Its time scale is 10"'^ to 10"^^ s. The process can be symbolised by A*->A'.

(1.12)

All these different deactivation pathways are given in Fig. 1.1. The molecule A' is in thermal equilibrium with its environment. Average lifetime lasts from 10"^ to 10"^ s. The excited state can lose its energy by fluorescence (F) according to A ' - > A + /iv'.

(1.13)

As a result the molecule reaches any one of the vibrational levels of the electronic ground state SQ. In a next step it adapts its temperature to that of the environment. Another deactivation pathway from the electronic excited to the ground state is a radiationless one (radiationless deactivation): A'-^A.

(1.14)

Besides the excited singlet states some excited tripled states T,, T2, etc. exist, which cannot be reached directly by absorption of radiation starting from the ground state SQ. This direct transition is forbidden [3]. In the case of radiationless transition this limitation is not valid. For this reason the molecule A' can radiationlessly deactivate to the first triplet state A'' (T,) (see Jablonski's diagram in Fig. 1.1) A'-->A".

(1.15)

12

Introduction

Ch. 1

This process is called intersystem crossing (isc). In general values between 10^ and 10^0 s~^ are discussed as rate constants for a monomolecular reaction as in eq. (1.15). The state A!' is meta-stable. For this reason molecules in this state have a longer lifetime than those in the state A'. In fluids the average lifetime is between 10"^ and 10^ s. It depends on the nature of the molecule as well as on the purity of the solvent. Molecules A" can also emit, and thereby return to the ground state. This process is also forbidden as for the absorptive transition [3]. Therefore its probability is very small. It is called ^- or low-temperature phosphorescence (Pin Fig. 1.1): M'-^A

+ hV.

(1.16)

Another deactivation process is radiationless: A"-^A.

(1.17)

If the energy difference between A!' and A' is less than 40 kJ/mol, the molecule A" can uptake thermal energy and cross over backward to the state A': A"->A'.

(1.18)

If in the next step fluorescence happens, it is called high-temperature or aphosphorescence (delayed fluorescence). All the processes discussed are included in the term diagram given in Fig. 1.1. If appropriate acceptor molecules B are present, the molecules A' and A" can transfer energy according to A' + B ^ B ' + A

(1.19)

A"+B->B" + A

(1.20)

and

respectively. The processes of eqs. (1.19) and (1.20) can be caused by collision. They have to happen within the lifetimes of A' and A", respectively. The maximal bimolecular rate constant k^ turns out to be diffusion controlled and can be approximated according to [3]

Ch. 1

Photochemistry and photokinetics

_ oJ\T *n,=-5— .

13

.^ ^.^ (1.21)

Tis the absolute temperature in Kelvin, R amounts to 8.31 J moh* K"' and rj is the viscosity in kg m"' s~' (Pa s). At room temperature and normal viscosity of the solvent, the rate constant becomes /:^« 10^^ P mol~^ s^K Energy transfer can be caused even by induced resonance without any impact up to distances of 5-10 nm, a process of quantum mechanical origin [8, 9]. Under these conditions the 'bimolecular rate constant' is independent of the viscosity of the solvent and approaches values up to 10^^ P mol"^ s"^ Strictly speaking this process cannot be treated as a normal bimolecular reaction. 1.4 PHOTOCHEMISTRY AND PHOTOKINETICS L4.1 Chemical primary processes Molecules which have been excited by absorption of light or energy transfer can also chemically react, in addition to the physical processes discussed in eqs. (1.12) to (1.20). This chemical reaction can even start during the thermal relaxation, i.e. during the process from A* to A'. That means the cross-over to the term system of the product B takes place starting at a higher vibrational level. Another possibility is the start of the chemical reaction from the states A' or A". In all these examples the term diagram given by reactant A is left and a reaction takes place to a new compound B characterised by the left side of the term diagram (B, see Fig. 1.1). Generalising these states by the symbol A*, the following chemical primary processes can be discussed: (1) the dissociation into radicals according to A"-^B"+C*,

(1.22)

(2) the dissociation into other instable molecules, e.g. carbenes according to R~CHN^ -^N2 +RCH, •• or into imines according to

(1.23)

14

Introduction

RN = N = N' -> N + RN".

Ch. 1

(1.24)

(3) Another pathway is the dissociation into ions according to A'-^B^+C",

(1.25)

(4) the dissociation into a positive ion and an electron according to A"-^A^+e".

(1.26)

(5) the formation of stable molecules A ' - > B + C,

(1.27)

(6) the photoisomerisation A'-^B,

(1.28)

(7) or a photoaddition or photodimerisation according to A"+B(A)-^C.

(1.29)

This bimolecular process requires the impact of the exited molecule during its lifetime with another molecule B(A). Assuming the lifetime of the molecule A' is 10'* s, a bimolecular reaction can only take place if the concentration of the partners of the reaction is lOr^ mol 1~* or larger. If the lifetime is remarkably longer, this bimolecular reaction becomes more probable. For a lifetime of KHs it can only react according to eq. (1.29), if the concentration of the partner amounts to KH mol 1"^ or more. In both cases it is assumed that the rate constant is diffusion controlled (/:„,« 10^^ 1 mol"^ s"^), if the viscosity of the solvent has a normal value. Another possibility for the reaction of the exited molecule is the abstraction of hydrogen to form radicals according to A ' + R H - > A H + R*.

(1.30)

It had been mentioned that the primary reactions (1.13) to (1.30) can also

Ch. 1

Photochemistry and photokinetics

15

start from the molecules B' or B", formed according to eqs. (1.19) and (1.20). If these reactions are followed by another chemical reaction, the photoreaction of B is called a (physically) sensitised reaction. On the other hand, if A^ shows a photoreaction, then the processes (1.19) and (1.20) are competitive reactions. Therefore the molecule B desensitises A^ and quenches the photoreaction starting at A. The process (1.20) is photochemically very important. It allows the transfer of a precise amount of energy to the molecule B. 1.4,2 Definition of the quantum yield In thermal kinetics the rate is proportional to concentration in the most simplest mechanism according to eq. (1.1). The proportionality constant is the rate constant k. In photokinetics, the equivalent proportionality constant is, according to eq. (1.2), the so-called photochemical quantum yield. In the literature, some different definitions of quantum yields are discussed but not always clearly distinguished. Therefore the problems with three different definitions are discussed here. Two others related to the partial reactions and independent of the time of the reaction are given in Section 2.1.2. 1.4,2.1 Apparent integral quantum yield The apparent integral quantum yield YQ is defined for any reaction partner Cby

rc=±^=±fi^.

0.3.)

ANc is the number of moles of the reaction partner C reacting during the time t. Nhv is the number of moles of light quanta (Einstein) absorbed in total by the system in the same time t. c{i) and CQ are the concentrations of C per unit volume at time t and before the irradiation starts (r = 0), respectively. Ij is the amount of light absorbed by the system (not only the interesting reactant) per second and unit of volume, measured in Einstein 1"^ s~^ In general, yields are defined as positive values. For this reason the plus sign is used if C is formed during the reaction and the minus sign is used if the concentration of C decreases. YC can be easily measured but has a complex dependency on the time of the reaction.

16

Introduction

Ch. 1

1.4.2.2 True integral quantum yield The true integral quantum yield is defined by

^^vA

JlUdt

y^ can be distinguished from the apparent integral quantum yield, because its turnover is related to the amount of light absorbed by reactant A. It is assumed that A starts the photochemical reaction. For this reason N/^^ and /^ are the numbers of moles of light quanta absorbed by reactant A per volume and time unit. The latter quantity is discussed in detail in Section 1.4.3. It depends on the experimental conditions. 1.4.2.3 Apparent differential quantum yield The apparent differential quantum yield

?>c=±-f

(1.33)

is calculated for infinitesimal small turnover, c is the change in concentration of C with time. The total amount of light absorbed (by all the reactants) per volume and time unit is taken. All the quantum yields defined so far cannot give a correct view of the photokinetics.
Photochemistry and photokinetics

Ch. 1

17

geometry of the light source and of the reactor is exactly defined. In principle the following statements are assumed: The photoreaction takes place in a cell with planar and parallel windows. The irradiated area is F^ [cm^]. The exciting radiation is monochromatic. Its incidence is perpendicular to the front window. The intensity of the incident radiation /Q [mol s"^ cm-^] is constant with respect to time and over the total front area. 5. The sample is not opaque; it can absorb light, but does not scatter. 6. The exit window does not reflect light. All these statements are taken into consideration in Fig. 1.2. A small volume element parallel to the entrance window is considered according to the given assumptions. This volume element has a front area of F^ and a pathlength Az. It is assumed to be at a distance z [cm] from this entrance window. Its volume is given by F = FAAz[cm^].

cell window l^{d)=

1^(0)- l o ' " ' ' "

optical pathway Fig. 1.2. Schematic drawing to understand the derivation of eq. (1.36). For calculation of the intensity of irradiation, Lambert-Beer's law is used for a wavelength A. Decadic units are taken as used in practise.

18

Introduction

Ch. 1

I{z) [mol s"^ cm~2] is the intensity of irradiation at a distance z from the entrance window per irradiated area. Therefore the amount of light absorbed by reactant A, within the volume element V is (': wavelength of irradiation)

In this relationship K\ is named the Napierian (natural) molar absorption coefficient [1 mol"' cm"*] of reactant A, at wavelength of irradiation. In principle the derivation of the kinetic equations becomes easier if exponential functions and Napierian logarithms are involved. For this reason the Napierian units are used first. Division of the relationship above by the volume of the chosen volume element results in

uMiVy^hiz)

= <«,/(z),

the amount of light absorbed in mole photons per volume and second. The use of the units mole per litre results in /,(z) = 1000^;a,/(z),

(1.34)

where the factor 1000 corrects for the different units (litres, cm^). The change in the number of photons absorbed in the volume element depends on the path length of the radiation in this volume dz. Therefore the following differential equation is valid: dI{z) =

-I{z)J,K\a,dz,

whereby the sum over all those reactants which absorb at the considered wavelength has to be taken. The integration has to take into account the fact that the concentrations a, are functions of the irradiation time as well as of the location z of the molecules irradiated. For this reason the intensity of irradiation has to be integrated starting at 7(0) = /Q at z = 0 /U) = /o e x p ( - j ; i « a , ) d z ) .

(1.35)

Ch. 1

Photochemistry and photokinetics

19

This is a generalised form of Bouguer-Lambert-Beer's law. Inserting eq. (1.35) in eq. (1.34) gives the moles of light quanta which are absorbed by reactant A, at a distance z from the entrance window per second and per litre: /,(z) = IOOO
(1-36)

In this equation the factor 1000 takes care of the fact that the units of concentration (moles/litre) and irradiance /Q [mol s-^ cm^^j ^re combined. Therefore the amount of light absorbed in some volume at a distance z has the dimension [mol 1-^ s"*]. In the course of this formal deduction of the amount of light absorbed, the Napierian molar absorption coefficient K is used. Later in the experimental section and during the practical examples using absorbance measurements, e symbolises the decadic (molar) absorption coefficients obtained in calibration experiments using UV/vis spectrometers. 1.4.4 The photokinetic factor A very simple derivation of the amount of light absorbed is possible using Lambert-Beer's law in solution given by /(rf) = /(0)xlO~''^''^'

(1.37)

using the decadic absorption coefficient and derived in Fig. 1.2 at the wavelength of irradiation k\ which is symbolised by a superscript (') in the following. Using the decadic instead of the Napierian absorption coefficient is symbolised by writing e instead of K. This is advantageous in practical applications, since normal instrumentation supplies data in decadic units. Because monochromatic radiation is a prerequisite to apply this law, only this type of irradiation can be used in photokinetics. The part of radiation lost within the reaction vessel can be determined as the difference between incident I{0) and outgoing I{d) radiation per square area /abs(^) = AO) - I{d) = /(0)[l - 10-^^''^' ] ,

20

Introduction

Ch. 1

where

c/X^:^/=^'

(1-38)

1=1

represents the total absorbance ('extinction') of the solution at the wavelength of irradiation. It has been assumed that only one component starts the photoreaction even though more absorb at this wavelength. Therefore just the absorption by this reactant is essential and it has to be set in a ratio to the overall absorption of the solution:

Obviously this quantity exhibits units in moles per area and time as irradiance does. Taking into account that absorption of radiation is related to a volume element, the relationship /abs(rf,A,) has to be divided by the pathlength of this volume element d. Thus, finally, by use of eq. (1.38) the following relationship is given for the amount of light absorbed per irradiated volume of the reaction vessel and by reactant A and per second of irradiation:

/abs(A.) = ;^#^/(0)[l-10-''2:^:'"], /=!

Substituting the sum in the exponent by eq. (1.38), the amount of irradiation absorbed by that reactant starting the photoreaction (reactant A is assumed) can be written as

/, =
(1.39)

This quantity appears in most of the photokinetic equations. Part of it was named the photokinetic factor. This factor can be defined using different absorption coefficients.

Ch. 1

Photochemistry and photokinetics

21

Derivation of photochemical equations can become easier using the Napierian molar absorption coefficient. This is discussed in detail in Section 5.1.2.2. The result is the photokinetic factor given by ^>'-'f,-^-'.

(1.40)

the index n symbolising the use of Napierian units. The differential equations have to be integrated. As will be seen in Section 3.3 therefore the reciprocal of this factor is frequently used as F(E'„)^-

^f-=Tr. l-exp(~£n)

(1.41)

As can be seen in Chapter 3, this factor makes the solution of the differential equation impossible in a closed form. Therefore either high or low absorptions at the wavelength of irradiation are chosen to allow approximations as discussed in Section 3.3.3. As seen in Chapter 5 giving examples of application of the photokinetic equations, absorption spectroscopy in the UV/vis is the general method to monitor the reactions. Since in this spectral range the molar absorption coefficient is used in decadic units, in practice this photokinetic factor is used as F(0 =

1-10"^'^'^ 7

(1.42)

to demonstrate the dependence on time as used in eq. (1.2). This form of the photokinetic factor is taken in all cases where absorbance measurements are taken to monitor the progress of the photochemical reaction. Since this photokinetic factor is essential in photokinetics it will be further derived more exactly and discussed in Sections 3.3.3 and 5.4.6. The different forms of this photokinetic factor are summarised in Table 1.2. Looking at eq. (1.36) a factor of 1000 has been introduced to correct for the values combining litres per mole with the irradiated area measured in cm^. To avoid too many factors in the equations, the following definition is often applied:

22

Introduction

Ch. 1

TABLE 1.2 Different forms of photokinetic factor (sec Sections 3.3, 3.3.3, 5.1.2.2, 5.4.6, 5.4.6.2)

Napierian

Photokinetic factor

Approximations

Fi - ' - ^ " >

F(r„) = UIE'„

^^E',^

+|.£'„^

+ | . r „ « •.•

^(^n) • -;l-exp(-r„) ^ K T T - /"(^n) = K [l + exp(£i) + exp(2£'„) +• •]

^^^"^=l.exp(-£;)

F(E;) « F(£;o)-H^^^b^""?^^"'^ (^;-^;oo) ^noo-^nO

Decadic

F(t) =

£•(0 , —

F(/)» = [£'(/)] *, F(/) * • = 2.303 F=F(i)+JIZIz£OL(£:'--r(i)) ^^ £:'(2)-£'(l)^

/ = lOOO/o

mol cm 1'^ s"^.

^^^

(1.43)

As demonstrated in Section 5.5.1.1, in some cases of fluorescence measurements, another definition is useful: / = 2.303/o.

All these and other definitions and approximations are discussed in details at the appropriate moments in the text. 7.4.5 Disposition of chapters Because of this common approach to deal with thermal and photochemical reactions, in the following chapters a generalised treatment of thermal reactions is given first, to be transferred to complex photochemical reactions later. The improvement in evaluation by formal photokinetics is demon-

Ch. 1

Photochemistry and photokinetics

23

strated. Finally, the impact using practical examples to obtain reliable quantum yields will become obvious. As in thermal kinetics, these examinations require measurements of high quality at as many times during the reaction as possible. Process control and automated equipment are state of the art nowadays to obtain data at more than 30-50 reaction times even in the case of such simple reactions as given by eq. (1.2). Otherwise different rate equations and mechanisms cannot be differentiated and trends in dependency of the photochemical quantum yields on reaction conditions will not be detected. This is the reason why - at least at the present time - data of flash photolysis result in too large standard deviations. Therefore here we focus on equipment measuring in the second or slightly sub-second time domain and on adequate examples of practical photoreactions. However, the formalism depends not on the time domain but only on the quality of data supplied. Many reaction schemes are discussed and rate equations derived taking into account a variety of possible photophysical processes. To avoid the tiring repetition of reaction schemes, mechanisms, differential equations, and evaluation approaches which are sometimes very similar, the results are summarised in an appendix where the appropriate formalism can be selected. Some of the theoretical derivations and definition are explained in Examples. They are intended to help to set up relationships and to follow the derivation of equations. UV/vis-spectroscopy is the most common method used to obtain data during photoreactions. The advantage of its high quality of photometry is reduced by its non-selectivity with respect to component specific information. Therefore methods of multivariate data analysis in multicomponent analysis are reviewed and demonstrated in practice for some reaction examples. Furthermore the advantages of using combined methods, such as chromatography and spectroscopy, are discussed. Their application allows high quality photometry to be combined with the chance of detecting the reactants individually. Application of HPLC process controlled repetitive experiments has proved successful as have first attempts to use NMR techniques. Measurement and evaluation of photokinetics assume that the reaction solution is spatially homogeneous to be able to take into account the problems with the amount of light absorbed. Homogeneous medium and stirring are prerequisites. However, in photochemistry with lasers or in viscous media the dependence of concentration on the volume element irradiated and observed has to be taken into account. Strategies to consider these problems

24

Introduction

Ch. 1

are also given. Such conditions are of special interest in modem photochemistry working in the solid phase or using liquid crystals. The procedure of an evaluation of a typical photoreaction is given in the Appendix. Furthermore some macros are explained which allow kinetic evaluation; these can be added to spreadsheet programs.

Chapter 2

General Approach in Formal Kinetics

2.1 FUNDAMENTALS 2.7.7 Stoichiometry In Section 1.2 the influence of the stoichiometric coefficients were discussed, the degree of advancement as a generalised variable of the reaction was introduced, and by eqs. (1.9) and (1.10) a relationship between the changes of concentration and this degree of advancement was deduced. To handle complex mechanisms a generalised treatment is advisable to avoid a specific derivation for each reaction. Methods of linear algebra can be an elegant tool to solve problems in formal kinetics. This can be demonstrated even for simple reaction mechanisms. In the following, bold type is used to symbolise vectors. By defining a column vector

(A,\ A=

and a stoichiometric vector v°*

:(Vi V 2 V 3 . . . ) ,

eq. (1.4) can be represented by v^*.A = 0

(2.1)

This expression represents the law of conservation of mass, if the elements A, of the vector A defined above correlate to the molecular weights of

General Approach in Formal Kinetics

26

Ch.2

the various reactants. The transposed vector v*** contains the stoichiometric coefficients of all the reactants involved in a chemical reaction according to eq. (1.3). How these vectors are chosen is a question of definition to some extent. Thus the symbol v**^ (transposed vector) is chosen to define the row vector of the stoichiometric coefficients. This is in contrast to definitions in the past [10], however, it turns out to be more concise for the following overall treatment using a schematic representation of the reaction. As demonstrated in eq. (1.9) for any reactant this relationship can be written in a more generalised form in matrix notation as (2.2)

Aa" = v°

Aa** is the column vector containing the changes of concentrations of all the reactants as elements. The stoichiometric vector v*" corresponds to its transpose v**^ defined above. Eq. (2.2) is explicitly written as follows:

fAa^ Ab ! Ac

M VB

(x) 1

Vc

[M) UDJ This example makes it obvious that the transposed vector of the stoichiometric coefficients is relevant. The symbol " is used if this vector of the reactants is not reduced with respect to linear dependencies given by the law of conservation of mass. In the case of a single step reaction considered here, the differentiation results a

= V

jc

and only a single degree of advancement exists.

(2.3)

Fundamentals

Ch. 2

27

2,LL1 Formalism of the reaction scheme The chemical reaction can be formally written in the following way: A,

On the top, the different reactants are written into the first row. The second row (for the reactants) is formed by the stoichiometric coefficients of these reactants (results v**^). The first element is the degree of advancement of this reaction X\, The last element is determined by the derivative of this degree of advancement with respect to time used in eq. (2.3). A simple example is the reaction A + B~>C + D according to eq. (1.3): B - 1 ~1

D 1

1

Degrees of advancement are formal quantities. However, their use allows us to obtain equations with mass balance by combination of eq. (1.9) and (2.2). In this case, the assumed mechanism is correct and only one single step of a reaction takes place; the progress of the reaction can be described by monitoring the concentration of just one properly chosen reactant (e.g. taking A,). One obtains accordingly Aa, = VjX or

jc = v, Aa,.

Thus, the use of the degree of advancement simplifies the equation and the number of components to be monitored during the reaction to one. This advantage becomes more obvious in multi-step reactions, since in general a chemical reaction proceeds by more than one step. Furthermore some of the reactants take part in more than one reaction step. Each can be represented by a linear equation as given above. These form a system of r different linear equations of type eq. (1.9). Instead of a single degree of advancement, each step is represented by one. Instead of eq. (2.2) the result is

28

General Approach In Formal Kinetics

Ch. 2

a matrix equation, which contains a column vector x*. The stoichiometric vector expands to a matrix v°*. The symbol is used if the vector of degrees of advancement is not reduced with respect to linear dependencies between the different steps of the reaction. These dependencies are handled in detail in Section 2.1.4. In the following, matrices are represented by bold capital letters. The stoichiometric matrix has to be considered with respect to both dependencies. The reaction scheme introduced above expands by an additional line for each step of the reaction (see later in Example 2.2). This approach can be summarised by the following matrix notation: Aa"=v"x*

or

x* =v'*-^ Aa'.

(2.4)

Using systems of linear equations, special care has to be taken that none of these equations depends linearly on each other. This can happen either by the different components of the reactions itself or by a dependency between different steps of the reaction. For this reason the symbols (" ) and (*) have been chosen to point out both these types of possible linear dependencies. First, redundancy in the reactants is treated. As mentioned above, in the special example of a single step reaction, not all the reactants have to be observed during the reaction to describe its progress (reactant A was selected). An index ' is chosen to mark variables, vectors, or matrices which contain a non-reduced set of reactants to describe the progress of the reaction. In the case of a multi-step mechanism, the matrix V* thus includes only the stoichiometric coefficients of the reactants selected and taking part in all the different steps of the reaction. Their changes in concentration are given by the column vector Aa as x*=v*"*.Aa

(2.5)

instead of eq. (2.4). This approach avoids measuring concentrations of components which depend linearly on each other. At the conditions mentioned of a single reaction with interesting reactant A, matrix v (or more general v*)

Ch. 2

Fundamentals

29

contains only one element Vj, the column vector Aa just Aa\. At the present stage of the discussion, considering just one reaction step, the symbol (*) can be omitted. In a generalised treatment, a combination of both the equations above results in A r = v° v ' Aa.

(2.6)

It has to be mentioned that matrix v ' and vector Aa are regular ones. For convenience, the matrix

p = v''v"

(2.7)

is defined, which is singular, since the matrix v° is singular. Thus the final equations, including mass balance, are given by

A r = p Aa.

(2.8)

This procedure is explained in the first example given next. The purpose of this example is to demonstrate the advantage of a reduced set of concentrations and to exercise the application of eqs. (2.2) and (2.8) in the case of a simple mechanism before multi-step reactions are handled. Example 2.1: A simple reaction scheme For the reaction A + B —> C + D, eq. (2.2) has been given explicitly according to the reaction scheme B -1

D

-1

by

Aa =

Ab Ac = \-x

-1 =

1

U

30

General Approach in Formal Kinetics

Ch. 2

using the vector of stoichiometric coefficients related to the transpose "'*-:(-l V'

-1

1 1)

derived for the four reactants according to eq. (1.3). In this example, the vector x contains only one degree of advancement, that of the above assumed single step. Since the observation of just one component is sufficient to describe the progress of the reaction, the vector v" can be reduced. Selection of the reactant A results using eq. (1.9) Aa = (-l)x. By rearrangement one finds ;c = (-l)Aa. This can be inserted in the equation given above resulting in

Ar

(Aa^ f-V\ (-l)(Aa) ri\ (Aa) 1 -1 \Ab = pAa = = Ac = v°v-'Aa = 1 -1 [Ad, .1> .-1;

This can be achieved either by multiplication and following the steps discussed above or, on the other hand, using eq. (2.6) with the vectors Aa', v% and calculating v"* = - 1 . Furthermore Aa reduces to Aa. According to the definition of P, the new vector on the right-hand side in the above equation is obtained as

rn p=

1 -1

Thus a relationship between the changes of all the concentrations and the chosen Aa is given. Therefore using the above equation three linear

Ch. 2

Fundamentals

31

independent equations with mass balance are obtained besides the trivial correlation Aa = Aa: Afe = Aa,

Ac = -Aa

and

Arf = -Aa

2J.L2 Partial reactions In case where the number of reactions between the reactants exceeds one, eq. (1.3) has to be expanded to a system of r partial reactions between n reactants A,. For each of these, eq. (2.1) has to be valid. Therefore the selected kth equation of the reaction is given by ,^kA = 0

(* = l,2,...,r),

including another index for this partial reaction. The transposed stoichiometric matrix v** contains r rows (number of reactions k) and n columns (number of reactants i). In case the molecular weights are included within the column vector, even for complex systems the law of conservation of mass is satisfied according to eq. (2.1). The same is valid for eqs. (2.2) to (2.8), if r degrees of advancement x^ are included in the column vector x* (one for each reaction). Then v becomes a regular quadratic rxr matrix, which can be derived from the matrix v° and represents a matrix which cannot be reduced any more with respect to the reactant. The next example demonstrates the procedure to derive the equations with the mass balance mentioned above by use of eq. (2.8). Example 2.2: Derivation of change in concentration A system of two reactions is given:

X, X2

A -1 0

B 1 -1

C D E 1 0 0 ir, 0 1 1 X2

Thus the transposed matrix of the stoichiometric coefficients v°^ has to be formed for two reactions between the 5 compounds (the stoichiometric coefficients of the reactants form a row for each step):

32

General Approach in Formal Kinetics

Ch.2

- 1 1 1 0 0 V = 0 - 1 0 1 1 This demonstrates clearly that in general matrices v**^ and v% respectively, are singular. In consequence the changes in concentrations are according to eq. (2.4)

(Aa^ (-1 0^ 1 -1 A6 Ac = 1 0 0 1 \Ad ^Ae, . 0 ij In the chosen example the concentrations of two components a and b suffice to describe the progress of the reaction. The vector Aa" can be reduced and the equation above is written as Aa^

-1

0

Ab,

1

-1

V-'^2,

By inversion of the reduced stoichiometric matrix the relationship can be rearranged to an equation according to eq. (2.5): -1

0

Aa Ah

The latter equation can be inserted in the upper first one. The result is a matrix equation according to eq. (2.6):

(Aa" r-1 0^ - 1 0 YAa 1 -1 -1 -if^Ab A& Ac = 1 0 Arf 0 1 \Ae^ . 0 ij

Fundamentals

Ch. 2

33

Obviously the matrix is the inverse of the reduced stoichiometric matrix and the vector Aa is the reduced vector of changes in concentrations of relevant reactants. The more general approach uses eq. (2.6), the reduced Aa as well as v~'. Multiplication of r-1 I v°v-'=p = 1 0 0

0^ r-\ - 1 [-1 0 1

0 -1

(I 0

0^ 1

-1 -1

0 -1

l-l -ij

finally results in the following relationship according to eq. (2.8) fAa^ Ac Ad Ae

(\ 0 -1 -1 -1

OVAa> 1 0 -1 -I)

This is the same result as obtained by the use of eq. (2.5), however, applying matrix calculus with less expenditure. The only disadvantage is the matrix inversion, but modem software provides an easy way to do this (see software MAPLE used in Appendix 6.1). Assuming that the concentrations of the intermediate and of the final products equal zero at the beginning of the reaction (feo = co =rfo= ^Q = 0), the following three linear independent equations with massbalance result: Ac = c = - Aa = ao - a. Ad = d = - Aa — Ab = GQ - a- b, Ae = ^ = - Aa - Afo = aQ - a - fo. The example given above explains the use of eqs. (2.2)-(2.8) as well as the advantage of using reduced reaction schemes. The given procedure re-

General Approach in Formal Kinetics

34

Ch.2

quires one prerequisite. The rank of the stoichiometric matrix has to be equal to r, which is equivalent to the number of the reactions taking place in the system (for further details see Section 2.1.4). This means the matrices and vectors used have to be regular. The next example expands the procedure to three steps of the reaction and to demonstrate once more the approach to deduce the changes in concentration of the reactants by use of eq. (2.8). In addition, the possibility of formulating these changes dependent on the degrees of advancement is given. These allow another approach to obtain the changes in concentration. Example 2.3: Complex reaction scheme A further example with three steps of reaction is discussed in detail step by step in Appendix 6.1 jcp

2A-

B

X2: B 4 - C - » D + E .

x^: D + A - ^ F + G One obtains the reaction scheme

Xl Xy ^3

A -2 0 -1

B 1 -1 0

C 0 -1 0

D 0 1 -1

E F G 0 0 0 1 0 0 0 1 1

-^i X2

-^3

The calculation of the changes in the concentration is given in detail in Appendix 6.1. In principle these values can easily be obtained by a summation of the products of stoichiometric coefficients for the reactant under consideration with the degree of advancement in the column in the above reaction scheme according to eq. (2.4) A a = - 2 J C , - JC3,

At/ = JC2 - X'^y

Afe = jCj - ^ 2 , Ae^

X2,

A c = -jCj

Af = Ag = x^

Ch. 2

Fundamentals

35

Advice: The changes in concentration are simply obtained from the reaction scheme for each reactant by a summation in the respective column of the products of stoichiometric coefficient times the degree of advancement of each step of the reaction. Using the reduced matrix of the stoichiometric coefficients one finds according to eq. (2.8), demonstrated in Appendix 6.1: Ac = -JC2 = 1/3 ^a + 2/3 Afo -1/3 Arf Ae = ^2 = -1/3 ^a - 2/3 ^b + 1/3AJ = -Ac, A/ = ^g = -y3Aa

- 2/3 Afe - 2/3Arf

Thereby the calculation of the relationship between the concentration changes can be formally obtained by use of the reaction scheme and simple matrix algebra. 2.7.2 Quantum yields independent on reaction time In Section 1.4.2, in analogy to the thermal rate constants, photochemical quantum yields have been defined. It was mentioned that the amount of light absorbed varies according to Section 1.4.3 during the photoreaction. Accordingly the definitions given in Section 1.4.2 exhibit a dependence on time. It is assumed that a simple photoreaction stoichiometrically takes place according to the condition given in eq. (1.4). If the reactant Aj = A starts a photoreaction, then the degree of advancement is given according to eq. (1.8) and one finds the rate law as given in eq. (1.2) to

a,=±^f/,.

(2.9)

/^ has been defined by eq. (1.39) in Section 1.4.4. Thus for any reactant by use of eq. (1.10) a correlation with the degree of advancement is found and the quantum yield can be defined in a new way

\vA

vj

I.

36

General Approach in Formal Kinetics

Ch. 2

In eq. (2.10) the amounts of the stoichiometric coefficients have to be inserted, since the quantum yields are defined always as positive numbers. It makes more sense to relate the quantum yield to the degree of advancement X, Therefore by definition
(2.11)

This results the general relationships

K=KK

and d, = v,
(2.12)

In principle the superscript refers to the reactant which starts the photoreaction absorbing the radiation. The subscript refers to the reactant under consideration. In the case of the use of the degree of advancement, its subscript is not necessary. Two other true quantum yields can be defined. 2.7.2.1 True differential quantum yield The true differential quantum yield
(2.13)

is very important for the correct mathematical treatment of a simple photoreaction. Those types of photoreactions are those in which just one reactant starts the photoreaction by absorption of light and all possible thermal reactions take place so fast, that the Bodenstein hypothesis (see Section 2.1.3.2) can be applied to the unstable intermediates. All the other partners of the reaction may absorb light, but are not considered to start any photoreaction. Photoreactions, which do not fulfil one of the conditions - only one reactant starts the photoreactions, all others can absorb radiation, - Bodenstein hypothesis can be applied to instable intermediates are complex photoreactions. 2.7.2.2 Partial photochemical quantum yields As for thermal reactions complex photoreactions can be the result of more than one step of the reaction. By analogy one can correlate a partial photochemical quantum yield to each of the partial steps of the reaction

Ch. 2

Fundamentals Vk^^kllAr

i^h2,,„,n^
* = 1,2,...,5^ < 5 .

37 (2.14)

/^. is the amount of light absorbed by this reactant which starts the photoreaction (see also Section 1.4.2). Under these conditions n^ products, the rate is expressed by the derivative of the degree of advancement:

(2) Therefore the rate is proportional to the concentration of starting material A. kk is defined as the rate constant of this partial step k. In analogy Xk = kj^ ab

or

ij^ ~ ^k ^^'

is obtained for a bimolecular reaction, given by either of the mechanisms A + B -> products,

2A —> products.

(3) The rate equation can be generalised as a system of r elementary reactions, which can be represented by

General Approach in Formal Kinetics

38

Ch.2

i-'H

Xt=k,YlaH

Vy,<0

(k = l,2

r).

(2.15)

i=I

Therefore the rate of the *th partial reaction is determined by the product of the concentrations of the different disappearing reactants, the amount of the stoichiometric coefficients being the exponents. The concentrations of products, formed in the different k partial reactions, are not included in this equation. In consequence the problem is to find a method which allows the derivation of the differential equations by a systematic procedure. In the following, such a convenient and systematic method is discussed [10-12], which was used in part in Section 2.1.1.1: (a) A representation similar to matrices is constructed. (b) In the first row the n initial products, intermediates and end-products are inserted. (c) In the first column the r degrees of advancement x^ of the partial reaction steps are arranged. (d) Then the stoichiometric coefficients for the different steps are put into this matrix representation accordingly. (e) Finally the vector of the rates i , calculated according to eq. (2.15), is listed in the last column of this scheme. This procedure allows any differential equation to be set up schematically. The systematic approach helps to avoid errors with stoichiometric coefficients, signs in front of the rate constants, and will be useful to find linear dependencies between the different partial reactions later on. The application of this scheme is demonstrated in the following two examples. Example 2.4: Multi reactant reaction scheme According to (a)-(c) the first row and first column are filled with the information from the two reactions: A + B->2C + D C + E->F

A Xx - 1 0 X'y

B -1 0

C 2

D 1

-1

0

E

F

0 -1

0 kiob 1 k^ce

Xk

Using the chemical reactions, the stoichiometric coefficients are put into the rows below the reactants according to (d). Finally concentra-

Ch. 2

Fundamentals

tions of components with negative coefficients are written as a product together with the rate constant (index: step of reaction) into the last row according to (e). The scheme is used to find the correlation between the change in concentration and the degree of advancement by use of eqs. (1.9), (2.2) or (2.4) and by stoichiometric summation of the ith column: Aa^a-QQ^

-jc,, a = GQ — jc,

Ab = b-bQ=

-JC,, fo = ^Q - JC|

Ac = c - 0 = 2JC, -X2,

C = 2JC,

- JCj

and accordingly (do = 0,/o = 0) rf = X,, ^ = ^o--^2'

/ = ^2-

Taking the concentration c as an example, in the first step of the reaction the stoichiometric coefficient amounts to +2 (therefore + 2JC, ) and in the 2nd step to -1 (therefore -X2) In analogy the rates are extracted by stoichiometric summation of the contents of last column to a = - i , = -k^aby b = -k^ab, c = Ik^ab - ^2^^* d = k^ab, e = -k2ce,

f = k2ce

The stoichiometric coefficient of the considered reactant A, is multiplied by the element of the last column, if this reactant takes part in the partial reaction. In particular the relationship for c explains the procedure (v,c= 2, V2c=-1). Example 2.5: Catalytic reaction

A ^ B +C A + B->2B + C

X,

Xo

A -1

B C Xk 1 1 fc,a -1 -1 1 k^ab +2

39

40

General Approach in Formal Kinetics

Ch. 2

The difference in the formal approach to these two reactions in comparison to the former examples is caused by the catalytic action of component B. One mole of this component is consumed in the second step first; later on two moles are formed. Therefore the two stoichiometric coefficients have to be included in the scheme for jCj to get the correct results for equations according to eqs. (1.9) or (2.2). On the other hand, in the stoichiometric matrix only the sum of these two factors ( - 1 + 2 = +!) is found and used for the formation of the rate equation according to (e). Thus a = —k^a - k2ab,

b = k^a + k2ab,

c = k^a + k2ab.

2.1.3.2 Bodenstein hypothesis The Bodenstein hypothesis [13] can also be called a hypothesis of the stationary or quasi-stationary state. It makes the following assumptions: (1) The concentrations z\ of the intermediates Z, are negligible within the limits of measurement in comparison to the concentrations of starting material and products, (2) changes of concentrations with time of the intermediates are negligible with respect to the changes in the concentrations of initial products and end-products with time. That means both the equations z,=0,

z,=0

(2.16)

are valid for all the intermediates which fulfil this hypothesis. 2.1.3.3 Simple photokinetic rate laws In contrast to thermal reactions, any photochemical reaction is accompanied by a number of photophysical processes which all have to be taken into account in the reaction scheme. Most of these processes are thermal reactions. The mechanism of the photoreaction and - as we will see in the examples - the photochemical quantum yield depends on these photophysical steps. One of the most simple photoreactions is a photo-isomerisation A—^il->B which can be assumed to consist of the following partial steps

Fundamentals

Ch. 2

A + hv

A* x\

A*

-^

A'

A'

->

A + hv'

A'

-» A

X\

A'

-^ B

x's

X'2 X',

41

A -1 0 1 1 0

A* 1 -1 0 0 0

A 0 1 -1 -1 -1

B 0 0 0 0 1

^\ JA

M* k^d k^d k^d

As in the case of thermal reactions, the reaction scheme introduced in Section 2.1.1.1 can be used to set up the differential equations. However, the degrees of advancement are primed, since the number of steps can be reduced as will be demonstrated by use of the Bodenstein hypothesis. In the last column of this scheme, the number of moles of light quanta are written for a photochemical step, which are absorbed by the reactant starting this photochemical step. According to this assumption and the different photophysical relaxation processes discussed in Section 1.3 the primary exited molecule A* completely deactivates into the lowest level of vibrational energy of the first exited singlet state. Three further steps are possible: (1) deactivation by fluorescence, (2) radiationless deactivation and (3) isomerisation to reactant B. Reactant A reacts by step 1 and will be re-formed by reactions 3 and 4. In consequence, the change of A in dependence on time is (summation of all the products stoichiometric coefficient in the first row times the element of the last row: reactions 1,3,4) a = -If^ +(^3

•¥k^)a\

(2.17)

During this process all the molecules of reactant A will react, since they absorb light. The amount of light absorbed is given according to the definition: If^ in mole 1"^ s-^ The rate laws for the photophysically exited molecules can be given accordingly a* = /^ - *2« *

a' = jk2a*-(Jfc3-h)k4+/:5)a' as well as the change in concentration of B

(2.18)

42

General Approach in Formal Kinetics

Ch. 2

The fluorescence quanta emitted (fluorescence intensity F) according to the radiative process (3) amount per second to

since each molecule A* reacting according to the third step emits a quantum of light. The lifetimes of the exited molecules A* and A' are extremely short in comparison to the time the reaction needs. Therefore both concentrations as well as their changes with time are very small as long as normal light sources are used for irradiation. For this reason, the Bodenstein hypothesis can be assumed, and one can set

In consequence, the stationary concentrations according to eq. (2.18) amount to ^*=_A_^

^ ^ _

Ky

Z K'i

I

KA

_ • Kc

A K-i

v

KA

' rCe

Therefore the change in concentration of compound A is (a' inserted in eq. (2.17)): a = -/A+(jfe2+*3)

= Ko "» f^A i" iCr

/A. K-y "T

KA

> Kr

andof reactantB: b=

'^ ACa "T*

KA

/A. • fCc

A comparison of the above equation with eq. (2.12) shows that the quantum yield for this simple photoreaction is given by

Ch. 2

Fundamentals

43

for the proposed mechanism. In this case - the quantum yield (p^ is smaller than 1, ~ it does not depend on the concentration of the reactants, and - it does not depend on 7^. It has been assumed that all primarily exited molecules reach the state A'. Therefore the quantum yield cannot depend on the wavelength of the exciting light. The approaches to use the reaction scheme introduced in thermal kinetics can be very helpful and allow the transformation of the mechanism into a differential equation. All the elementary processes have been included in the scheme. For this reason, constants of these elementary processes can be explicitly determined from the quantum yields by these means. However, many of these steps cancel out by using the Bodenstein hypothesis (see Section 2.1.3.2) and only effective steps of the reaction are found. Nevertheless, the total scheme explains dependencies of quantum yields on concentration as discussed in Chapter 3 and in the following example. Conclusion: Most of the photophysical steps are thermal reactions, for which the Bodenstein hypothesis is valid. The photochemical quantum yield depends on these elementary steps. The dependence of the photochemical quantum yield on the types of photophysical processes becomes obvious comparing the simple photoreaction

for two cases: the start of the photoreaction from the singlet and triplet state, respectively. The results for the latter are summarised in brief in Example 2.6 which is discussed in detail in Appendix 6.2.1 Example 2.6: Simple photoisomerisation Instead of the photoreaction of an isomerisation from the singlet state for

44

General Approach in Formal Kinetics

Ch. 2

a pathway via the triplet state is considered. According to the procedure explained in Section 2.1.1.1 the reaction scheme is set up. All the excited molecules are supposed to form A' first. Therefore the both steps A + Av

>A* and A *

>A'

can be combined to give the single step A-hAv

^A'.

Application of the Bodenstein hypothesis to the intermediates A' and A" results in y> =

K^K {k2-^k^+k^){ks+k^)

and the quantum yield depends on other photophysical processes than in the case of a pathway via the singlet state. 2.1.4 The principle of linear independent reactions Some multi-step reactions have been discussed above in the examples. In all cases the different partial steps were elementary reactions. None of the rate equations of these steps had any dependence on one another. It is obvious that this number of linearly independent steps of reaction (partial reactions) plays an essential part in the process to set up the rate equations. This number has to be known for the calculation of the yields as well as for a kinetic analysis of the reaction system. Linear dependencies exist between different rows of the stoichiometric matrix v*, if its rank is smaller than the number r of all the partial reactions possible. Under such circumstances the advancement of the reaction can be described by a reduced number 5" < r of degrees of advancement or concentrations {s defines the rank of the stoichiometric matrix). In the following, two examples are given to demonstrate the problems, which arise by this restriction. They are chosen to give information about various aspects of linear dependence.

Fundamentals

Ch. 2

45

2.1.4.1 Uniform reactions The number of linearly independent steps of a reaction is given by the number s of the independent degrees of advancement and therefore according to the statement above to the rank s of the reduced stoichiometric matrix. The following definitions are used, especially in the case of photokinetics: - A reaction of just one linear independent step (5 = 1) is call uniform. - A thermal reaction with just a single elementary step is call simple uniform', in the case of a photochemical reaction elementary reactions for which the intermediates can be omitted because of the Bodenstein hypothesis are taken also as a simple one. Examples are both the photoreactions given in Section 2.1.3.3. - When the reactions steps depend on each other linearly and by these means 5^ = 1 is obtained, such reactions are called complex uniform. Examples for such complex uniform reactions are given in the following sections. 2.1.4.2 Thermal reactions The simplest case of a complex uniform thermal reaction is the reversible reaction according to the following reaction scheme: A A^B,

4>

^1

Xy

B

-1 I 1 -1

• •

k^a k^b

The two rows of the reactants are proportional to each other (stoichiometric coefficients can be converted into each other by multiplication by the factor (-1)). jCj* and x*2 are linearly dependent. Therefore the rank of the matrix v"* reduces to 1. This is valid for all the reactants A, in the first two linear dependent steps 1 and 2: Vu = - V 2 / .

For this reason the relationship between concentrations and the degrees of advancement is reduced from (stoichiometric summation within one column) Aa, = v,,x

+ V2,JC2

46

General Approach in Formal Kinetics

Ch. 2

to Aa,. =v,,.(x,*-X2). Only the difference, taken from the degrees of advancement jc^* and jCj, arises in all the equations. The x* themselves can neither be measured nor their dependencies on time calculated independently of each other. For this reason it makes sense to define the following new, linear independent degrees of advancement (without the index *) •

*

Xt = X\ """ X2 •

In consequence the reaction scheme reduces to (the new element in the last column is obtained using both the equations above) B -1

k^a-k2b

As a result the new time dependence of the degree of advancement jc, becomes i, = x\ - x*2- k^a(t) - k2b{t) according to the definition. Advice: In practice one cancels the linear dependent line(s) in the scheme and reconsiders the expressions in the last column. In the following, three other examples are discussed to demonstrate this procedure of reduction of the reaction scheme. Details are given in Appendix 6.3.1. The first two cases are mechanisms with back-reactions. The last one treats the case of a parallel reaction.

Ch. 2

Fundamentals

Example 2.7: Consecutive reaction: linear dependency For the reaction A ^ B ~» C, jc,* and x^ are linearly dependent. Therefore the rank of the matrix v*** reduces to 2.

For this reason the relationship between concentrations and the degrees of advancement is reduced to

A new linear independent degree of advancement is defined X\ — X\

Xy )

X'^ — ^-1

and the reaction scheme is reduced to

X\ X2

A -1 0

B 1 -1

C 0 1

Xk kiO - ^ 2 ^ • kjb

As a result, the new time dependence of the degree of advancement jc, becomes ij = i* - i* = /cj . a{t) - k2b{t) according to the definition. Further details can be found in Appendix 6.3.1. Advice: In practice one cancels the linear dependent line(s) in the scheme and reconsiders the expressions in the last column. Example 2.8: Reaction cycle Even reactions with more steps and fully reversible forming a reaction cycle can be treated

47

General Approach in Formal Kinetics

48

Ch.2

A ;^ B ^r^ C 'Fi^ A.

The backward reactions cause a reduced reaction scheme. In addition, the third step of the reaction can be cancelled and one obtains two linear independent degrees of advancement Xi = Xi ~- x^

ano X2 ^^ Xy -- x-y

and the following representation

X\

A -1

B 1

X'j

0

-1

C Xk 0 kiO- /jj^ - k^c + k^a 1 k^b - k^c — k^c + k^a

is found. The reduced stoichiometric matrix is symbolised by v. In consequence eq. (2.8) with the definition of matrix p can be further used as (2.7)

p = v--v-\

only taking into account that stoichiometric matrices are reduced with respect to the number of linear independent steps of a reaction. Thus, with Ar=v'*.x*

and

Aa'=p.Aa

one finds (see Appendix 6.3.1) the following relationships: Aa = -jc,,

Afc = jc, - JC2,

Ac = jCj

and Ac = - Aa - Afe.

Conclusion: The stoichiometry fits, if the number of degrees of advancement is reduced to the rank of the stoichiometric matrix.

Fundamentals

Ch. 2

49

Such a simplification can be used in all mechanisms in which a backreaction takes place. However, further reduction might be necessary in kinetic analysis. This becomes obvious looking at a further example, which is discussed in detail in Appendix 6.3.2 for the following reaction scheme: Example 2.9: Parallel reaction The rank of the matrix V is equal to the number of the degrees of advancement in the following mechanism. Therefore one would expect that the conditions of equal number of r and rank of the matrix stated at the beginning of Section 2.1.4 is sufficient for the optimal reduced matrix.

A + B-^C A + B->D

JC, 2

A -1 -1

B -1 -1

C 1 0

D Xk 0 kfOb 1 k2ab

However, if the concentrations a and b are chosen to be independent, eq. (2.8) can be used to obtain the non-trivial balance-equations: Ab = Aa and

Ac = - Aa - Ad.

Both the degrees of advancement show a linear dependence for kinetic reasons. A comparison of the rates in the last column of the scheme yields

Integration results in

and Aa, = v,.x; + V2,X2 = (V,, + jV2,)x,

General Approach in Formal Kinetics

50

Ch.2

It turns out that this reaction system can be described by a single degree of advancement according to the scheme

A

c

B

D

Xk

-(!+;»:) -(\ + x) 1 X k^ab

Xl

Advice: The simplified scheme is obtained by multiplication of the second row by X ^^d addition of the result to the first row. In consequence the last row is cancelled according to the method explained above. This last column contains the rates and is not influenced by this procedure: Considering a as the independent concentration, the changes in the different concentrations are given by Afc = Aa,

Ac = —

Jti -^k^

Aa,

AJ = —

-Aa.

jfc, +Jt2

A and B have the same stoichiometric coefficients in the reduced scheme. For C and D the ratios have to be taken according to Ac ^a

k^-¥k2

and

Ad — =• Aa k^ -{- ^2

Even though some arbitrariness exists in the definition of linear independent degrees of advancement, the measured value is not influenced at all. For example, the above given degree of advancement x* can be substituted by

if all other stoichiometric coefficients are divided by (1 +x) in consequence. Therefore the brutto equations are given by

Fundamentals

Ch. 2

(^).

: ^ L ± ^ ( A + B)->C + ^ D ,

(;c'):

A+B — ^ C

51

+ -^^—D. fC|1 "v ^ ^#C' ^2

fCj "T" fC^

There exist some other possibilities to reduce the number of linear independent reactions, degrees of advancement, or concentrations at conditions specifically obtained in kinetics. Further information is given in Section 2.3. Conclusion: If an extreme difference between the rate constants of different steps exists a simplification of the mechanism is possible.

2.1.4.3 Photochemical reactions The reduction of the reaction scheme because of linear dependencies between partial steps works for photoreactions in the same way as for thermal reactions. In addition it has to be considered that intermediates are negligible and allow the application of the Bodenstein hypothesis. Therefore the number of steps reduces to one per photoreactive step. Between these steps, linear relationships can happen. Photoreactions and true thermal reactions (not the photophysical steps) can exhibit linear dependencies. The simplest case is a photoisomerisation superimposed on a thermal back-reaction. Example 2.10: Photoisomerisation with superimposed thermal backreaction The combined photochemical and thermal reaction mechanism according to A ->B B-

->A


1 -1

Kb

* -1 Xy

. •

B 1

is the most simple type possible. A forms reactant B by absorption of

General Approach in Formal Kinetics

52

Ch.2

light and this converts back to A in a slow thermal reaction. The reaction scheme explained in detail in Appendix 6.2.2 results in - a = Z7 =

^3^5^A

{k2-^k,){k,+k,)

-Kb^ip^l^-k^b

(2.20)

Example 2.11: Reversible photoisomerisation In the case of a reversible photoreaction one can define A to be the trans and B the cis form, and the following reaction scheme applies:

B-

hv

^B

hv

^A

A * -1 * 1

B 1 -1

^^IA

•

As derived in Appendix 6.2.2, the partial quantum yields are found as k^kg


(/:2+^3)(^7+^8)'

(P2 =

(2.21) (^5+*6)(*7+*8)

The law of conservation of mass is permanently valid during the reaction procedure a + 6 = ^Q, Therefore both the reactions discussed are uniform. 2.1.4,4 Complex non-uniform photoreactions The number of such reactions is large. The differential equations for the changes of the concentrations with time can be given in general. In the following example a consecutive reaction which contains two photoisomerisation steps (forward, backward) is followed by a consecutive thermal step. This reaction can be chosen to demonstrate the consequence of the definition of partial photochemical quantum yields (see Section 2.1.2.2), the linear dependencies, and the influence of the thermal reaction: hv

A^B—^C. hv

The reaction contains three steps, two of them are linearly dependent (the photochemically induced ones). According to eq. (2.14) one finds:

Ch. 2

Fundamentals

number of partial steps number of photochemical induced steps number of linear independent steps number of photochemical linear independent steps

53

n n^ s 5^

= = = =

3 2 2 1

Using eqs. (2.15) and (2.14) the following equations are obtained from the reaction scheme: .• A/ 1 ~~ T\ A'

•* B/ 2 ~~ r 2 B '

•• L- 1% 3 — K-ily *

In consequence one finds the rate laws a = -x't +xl= -ftU I*

••

•«

.4r

+ ^2 ^B Ar

Br

f i

fo = Xj - ^ 2 "• -^3 =^ ^ 1 ' A "" ^ 2 ' B "" ^ 3 ^

2.1.5 Comparison of the formalism for thermal and photochemical steps of a reaction It is obvious that these relationships can be deduced from the stoichiometric scheme explained in Sections 2.1.1.1 and 2.1.3.3. A comparison with the consecutive reaction given in Example 2.7 makes it obvious how similar the discussions of thermal and photochemical reactions are. For this reason in Table 2.1 some of the kinetic parameters discussed in the previous sections are compared. In the following the use of the term quantum yield implies that it is either a true differential or a partial one; however, another special definition is given. Derivations are carried out for thermal reactions and dependencies demonstrated and explained. This is just a matter of simplicity, since in the application of the formalism the photochemical quantum yields in contrast to the rate constants contain potential dependencies on the mechanism. This is demonstrated in Chapter 3 for complex reactions.

54

X/i

ft« tA^ 73

8

^ 43 Q

'fcS

x: a.

4>

•s ? §

c: o "3 c o

•i3

u ed

« ^ > 'zi »=»

s

o o 2j

M

O.

^ (/)

13 u

§

u •c

^

+

o -c 04

13

^
S

0 *2

^ V 13

4 0

1—*

1

0

_ I 7

»-H

CJ

-

^

7 « 0

<

>Q

^

0

7

-

1

U

0

II

• n

0

Qu

&

0 0

1

+ <^ I II

I PQcs

'H 11

L

I II

•H

i^

-0

n

"^ ^ •*< 1 + '^ ^ Q

/—V

II II

.>?

l^

I II

-3

-iT II • ^ • fS 1 m 1 'H II I II 1 « «-« • ro ^ ^ ts 1 •H 'H II •>1< •K,^ •H II II II <s II 'H • Q •^

"^1

CQ

7

K1

«s»

<

'H

^ ^1

7

0

•>?

-Q

«—< 1

1-^

0

H\

11

General Approach in Formal Kinetics

0 1—«

CQn

U

-

•X

PQ

7

1

<

-Q

0

^ ^

CO ( J 0

7 • r< • r*^ H H

0

K

*-H

1

1-H

-

^ v

^ <

*

PQ CQ

1

<

c

^

c

•s

0 jd 0 0

>

i

13 cd 0)

(4-4

C^

iX 13

^

1 U 0!i

G 0

1> Ui

,C 0

cd

j<

1

<1

'^ CQ !

-s € 2 ^ S 3

Ui

ed

s

£ •S 4> C

.O

O

S

U-i

O.

•^ c c4 S HJ

% U1 H

Ch.2

Ch. 2

Fundamentals

55

Conclusion: (a) Photochemical steps of reaction include many thermal degradation processes, which do not need to be considered in the rate law, if the Bodenstein hypothesis is valid, (b) The mechanism has to be reduced as far as possible to avoid linear dependencies, (c) Thermal and photochemical reactions can be treated in principle by the same formalism. Rate constants times concentrations have to be substituted by the product of the partial photochemical quantum yield times the amount of light absorbed, which contains the concentration of the reactant which starts the photoreaction. 2.1.6 Use of symbols Different symbols have been used for the degree of advancement and the concentrations depending on the conditions. Furthermore vectors have been symbolised by bold letters with arrows, matrices by bold letters, variables as italics. Stoichiometric coefficients and the matrix p are written in Greek symbols. In addition, the concentrations of reactants to be determined and the number of degrees of advancement can be reduced because of the law of conservation of mass and linear dependencies between different steps of the reaction, respectively. Therefore some indices have been introduced to characterise the different variables. The different symbols are summarised in Table 2.2. 2.1.7 General rate laws for simple thermal reactions The differential equation of the overall thermal reaction can be expressed for an elementary reaction by i: = ika^^fc^«c^^...

(2.22)

Under these conditions the parameters r = 1 and /i, = |v,| have to be inserted into eq. (2.15). In general eq. (2.22) allows more complicated reaction schemes to be described, and //, need not be an integer (see Section 3.1.7). The exponents //, show the order with respect to the different reactants. //^ is the order of the reaction with respect to A, fi^ with respect to B and so on. A summation over all these orders results in the total order of the reaction

56

General Approach in Formal Kinetics

Ch. 2

TABLE 2.2 List of symbols used in the following chapters, the derivatives with respect to time of the degree of advancement (i:,) have to be handled in the same way Symbol

Meaning

a°, (Aa*)

Of all reactants taking part in the reaction scheme, no reduction with respect to conservation of mass (changes in the concentrations) Vector of concentrations of selected reactants taking part in the reaction scheme, reduction with respect to conservation of mass considered (changes in the concentrations) Vector of degrees of advancement of all the partial steps taking part in the reaction scheme, no reduction with respect to linear dependencies between some of these steps Degree of advancement of one partial step taking part in the reaction scheme, no reduction with respect to linear dependencies between some of these steps Degree of advancement of one photophysical step taking part in the reaction scheme, no reduction by approximation according to Bodenstein considered Vector of degrees of advancement of those partial steps taking part in the reaction scheme, which show no linear dependencies Degree of advancement of one partial step after reduction to avoid linear dependencies Stoichiometric coefficient of the kxh step of the reaction scheme for the ith reactant Vector of stoichiometric coefficients of all reactants in one reaction step Vector of stoichiometric coefficients of those reactants in one reaction step remaining after reduction according to the law of conservation of mass Matrix of stoichiometric coefficients of all reactants in all reaction steps, no reduction Matrix of stoichiometric coefficients of all reactants in those reaction steps, which do not linearly depend on each other Matrix of stoichiometric coefficients of reactants in all reaction steps, reduction with respect to conservation of mass Matrix of stoichiometric coefficients of reactants after reduction with respect to mass conservation and linear dependence between steps of reaction

a , (Aa) X* X* xj X xi Vi^i v° V o*

v° V* V

order of reaction = /
Ch. 2

Fundamentals

57

The dimension of the rate constant k is [mol 1-^ s"']. Integration of this equation, starting at ;c = 0 and t = 0, resuhs in X = kt. Using of a = AQ - jc one obtains a = aQ — kt

The time in which the starting material has decreased to half of its original concentration is called the half-life ti/2. In the case of a reaction of zero order, therefore _

"0

For convenience new dimensionless coordinates are defined using Greek symbols for concentration, degree of advancement, and time normalised by division by the initial concentration UQ (called general coordinates): a = a/ao,

^ = X/UQ,

r = kt/ao

one can express these variables in terms of ^ = r,

a = 1 - r,

ri/2 = 1/2.

No elementary reaction can proceed with a rate law of zero order, since mechanistically speaking - at least one type of molecule must be involved in any chemical elementary reaction. Furthermore, after a finite time (that means at r = 1) a has to become zero. In consequence an overall reaction can only be called zero order during a certain time range. During that time the degree of advancement has to be characterised by x (by a linear combination of jc*), since the reactions considered need not necessarily be elementary reactions. 2.1.7,2 First order reactions The elementary reaction A —> B yields according to eq. (2.22) the differential equation -d = x = l? = ka.

General Approach in Formal Kinetics

58

Ch.2

In this case the rate constant k has the dimension [s~^]. By integration within the limits r = 0, a = ao,x = 0 and r, respectively, one obtains a = aQe'kt

jc = 6 = ^0(1-e *0»

(2.23)

First order reactions show no dependence of the half-life on the initial concentration. According to eq. (2.23), the half-life yields In 2

0.693

h/2 ~"

The slope at the beginning of the reaction is CIQ = -ka. The definition of the general coordinates a =

a

?=

P-

r -kt

yields the following relationship: ^ = /8 = 1 - exp(-r),

a = exp(-r).

Furthermore the initial slope of a with respect to r becomes da

= -1

at time approaching zero. 2,1.7.3 Second order reactions Two possible cases have to be distinguished: (1) Initially only a single compound is present: B 2A-»B

-2

ka'

(2) On the other hand a reaction with two different starting components,

Fundamentals

Ch. 2

59

A I B IC -1

A+ B^C

-1

1 kab'

takes place. The differential equation of the rate has to be solved by a separation of the variables and by formation of partial fractions. The derivations and calculations for both cases are given in detail in Appendix 6.4.1. The different parameters, the obtained rates, concentrations, and their values in generalised coordinates are given in Table 2.3. The rate constant has the unit 1 moh' s-^ The limits of integration are given by jc = 0, a = aQ, r = Oand t In the case of two different initial components, the dependence of a on r for different values of u = &o/«o is represented in Fig. 2.1. It can be seen that the turnover drastically increases with increasing u, the ratio of the initial concentrations. [^0 ~^] becomes constant in a first approximation, if b^ »aQ, The differential equation can be simplified under these conditions to i = - a = k{bQ - x){aQ - jc) = k^a^ - x). As can be seen, the reaction formally takes place according to first order kinetics. Eq. (2.25) cannot be used when GQ equals b^. This specific case simplifies the differential equation to (see Appendix 6.4.1) JC = c = —a = —fo = ka^. Therefore an analogous solution to eq. (2.24) (only one initial reactant) is found: GQ

kt

c = A: = 1 + GQkt'

(2.24)

a = -1 + a^kt

whereby the general coordinates are defined by

a=

a

^ b ^ =

Gn

C

y = —, ^0 Gn

r,

X

§ = —. ^0 Gn

X=

G^t

General Approach in Formal Kinetics

60

Ch.2

TABLE 2.3 Rates, half-lives, concentrations, and general coordinates of second order reactions, for details see Appendix 6.4.1 2A-^B

A+B

Rate 2 Concentration -time relationship

J

=^ k{aQ - x){bQ - x)

l_ _ ^kt

a I.

In

GQ

(^o--^K

V^^

b ^ X=

1 + la^kt (2.24) 1 + la^kt

. = — ^

^-floexp(-(^o-^)^0 ^-aoexp(-(^0-^)^0 ^o(^o - Q Q ) ^= ^0 - «o exp(-(^o - «o)^0

Half-life Concentration given in general coordinates

ha = larsk _

^1/2 =

^ QQ * x

ln(2-^)

«0 M=-

«0

dx

^0 flo

1 1 + T'

= -1

They depend on each other according to

l + r'

{ao
a=

«0

^-P-~' 21 + T

l=/3 =

-—!

(2.25)

a=/3 =

1 1 + T'

^1/2 =

-^ln(2 M-1

^.^^^^n-exp(-(i/-l)T)] (I - exp(-(M - 1)T) ^^((i~l)exp(-(ti~l)T) M - exp(-(M - 1)T) uiu-V) )5 = u - exp(-(M - 1)T)

) U

Fundamentals

Ch. 2

0.0 ' ' 0.0

61

'

Fig. 2.1. Correlation between a and r for the reaction A + B —> C, u = bofaQ is the used parameter. With increasing u the turnover is accelerated.

2.1.7.4 Reactions offith order The rate of an overall reaction (a single initial component) results in the following differential equation:

whereby |v^| is the amount of the stoichiometric coefficient of the component A. Under these conditions the rate law is given by

with a half-life of ^ "'

ao'-^2^-'-l) (/i-l)|vjA: •

General coordinates are defined as above to a = —,

with

T = (/i -

\)\^f\aQ^-^kt

62

General Approach in Formal Kinetics H

1

1

H-H

1

1

1

1

1

1

1

Ch.2 1

1 ^-

identical initial slopes

I

Fig. 2.2. The concentrations of the initial component a are graphed as a function of time r for the reactions of different order//. The rate constants k are chosen such, that the initial slopes are all the same. The numbers at the different curves relate to the order of the reaction//.

a = (1 + r)'^^*"^^

and

r„2 = (1/2)'"^ - 1 .

In Figs. 2.2 and Fig. 2.3 two examples of reactions of//th order are presented. The above rate law is applied in the form <—I—I—I—I—I—I—H

identical half-lives

H

1

1

1

1

1

1

\

Fig. 2.3. Concentrations of the initial component a are graphed as a function of time r for the reactions of different order//. The rate constants k are chosen such that the half-lives TJ/2 for all the reactions are the same.

Ch. 2

Fundamentals

63

For a better comparison x is chosen such, that - in Fig. 2.2 the initial slopes of a versus r are equivalent, - in Fig. 2.3 the half-lives r|/2 of all reactions for different order ju are the same. In both cases % is used as a parameter. The rate constant k can be eliminated, if eq. (2.25) is used for three equidistant times t,t + A, and t + 2A. Under these conditions a correlation exists between the three different concentrations according to

„(»A)--J'""'"''^°<'^^'^''"'''

(2.26)

Eq. (2.26) demonstrates that necessarily the concentration of compound A at time r + A is equivalent to a suitable mean value of concentrations taken for the times / and / + 2A. The harmonic mean has to be taken in the case of 2nd-order reactions, the geometric one for Ist-order reactions. 2.7.5 Determination of concentrations at the end of the reaction At the end of the reaction the changes with time of all the concentrations become zero: a,^=0

(/=l,2,...,n).

(2.27)

Since the concentrations themselves cannot become negative physically, their values at the end of the reaction are determined by a,^>0

(i = l,2,...,n).

By use of the differential equations, eqs (2.3) and (2.15), the final concentrations can be calculated as well as the relationship between the concentrations and the degrees of advancement (eq. (2.2)). The detailed procedure is given in Appendix 6.4.2. The results are summarised in the following in brief. The numbers of examples correlate to those used in the appendix, where the different mechanisms are presented and the equations derived in detail for those reactions given in Table 2.4.

General Approach in Formal Kinetics

64

Ch.2

Example 2.12: Simple reaction For the reaction

X

A~>B + C

A -1

B C Xk 1 1 ka

one finds at the end of the reaction by combination of eqs. (2.15) and (2.27) and the procedure explained above -"
A:^

= i ^ = c„ =

UQ

are obtained. Further results are given in Table 2.4 and in Appendix 6.4.2.

TABLE 2.4 Rates and final concentrations at the end of the reaction for some mechanisms discussed in detail in Appendix 6.4.2 Example

Reactions

Concentration

Final concentration

2.13

A + B->C

a^OQ-x, b^bo-x, c=x a = -kiab, b = -k\ab, c = kxab

- Q ^ = - i ^ =c«, ^ ka^b^ = 0 c,. = J:«» = Min(aQ,£?^))

2.14

A^B B~>C

a = -A:,a, i = Jt,a - kib, c^kib

« - = 0 and

2.15

A-^B B + C->D

a = -kiQ,

2.16

A + B-^C C -^ A + B C + D->E

b-kxa-

kibc, c = -d = -kibc

X\oo = flfo

and Ar2oo = Min(a^,co).

a = ^ = -it,a^ + V . ^ ^ f^^^ij ^ f^^c _ ^^^^^

Coo = Xx^ - Xioo - X-^^ = 0

-d = e = k'^cd.

floo =ao~^loo+JC2oo = 0

^o«=:v3oo = Min(ao,^o)

Ch. 2

General Approach to Linear Systems of Reactions

65

In particular Example 2.16 depends on the starting concentration d^ being higher or smaller than a^ and b^ which influences e^ and x^. In this table just one condition is summarised. Details are given in Appendix 6.4.2. Assuming that the initial concentration of D is smaller than that of A as well as of B, an equilibrium is obtained between the remaining amounts of reactants A, B, and C according to

c^ =

a^b^

(2.28)

This equation is the typical form of the law of conservation of mass. According to thermodynamics, eq. (2.28) is only valid in ideal or ideal diluted systems. The consequence is, that any fundamental assumptions in kinetics (especially eq. (2.15)) will only be valid in ideal systems. 2.2 GENERAL APPROACH TO LINEAR SYSTEMS OF REACTIONS In Sections 2.1.1.1, 2.1.3.1, and 2.1.3.3 simple relationships for sequences of elementary steps of reactions have been introduced and discussed. A generalised system has been given by eq. (2.15) and has been demonstrated using Examples 2.4-2.10. This approach can be used to consider linear reactions in general. It is of principal interest, since it can also be used in the treatment of quasi linear photoreactions [10,14-16]. 2.2.1 Differential equations as a general treatment in kinetics 2.2.1.1 Formalism In linear systems of reactions, the differentiations of the degrees of advancement with respect to time t depend linearly on the concentrations. According to the principles mentioned in Sections 2.1.3.1 and 2.1.4.2, the last column of the stoichiometric matrix representation (see Example 2.7) becomes for the ^th of the r different partial reactions (eq. (2.15)) K =*it^)k

(^ = l,2,...,r).

This linear system of equations can be written as a matrix equation:

66

General Approach in Formal Kinetics

i*=K;.a'.

Ch. 2

(2.29)

As defined in Table 2.2, in this equation x* symbolises the vector of all r degrees of advancement differentiated with respect to time, a is the vector of the concentrations of all the reactants. KQ represents the matrix of all the rate constants. This matrix will have n columns and r rows. It can be directly derived from the scheme discussed in Section 2.1.1.1. This differential equation is valid for all concentrations. By combination of eqs. (2.3) and (2.29) one obtains the differential equation v°*i* = d" = v'*Koa' = K'c^\

(2.30)

It has been mentioned that linear dependencies between partial reactions have to be avoided (see Section 2.1.4) and the conservation of mass between the concentrations has to be used. Under these conditions according to eq. (2.6) and the definition of P (defined by eq. (2.7) the following relationship is obtained: r=ao+PAa. The reduced matrix v has to be used, otherwise v"* cannot be formed, since a quadratic matrix is necessary for inversion! Therefore the change in concentration with time is given according to eq. (2.30) by ^°=KJ.(ao+pAa) = ^o + K^PAa

(2.31)

and the following symbols are defined as io=K^ao,

KJ,=vKo,

Aa = a - a o .

Furthermore the reaction scheme has to be written in the reduced form. Then the reduced stoichiometric coefficients, degrees of advancement, and matrix of rates have to be taken i = Koa

(2.32)

Because of matrix algebra, in the following equation, according to eq. (2.5),

Ch. 2

General Approach to Linear Systems of Reactions

67

the reduced stoichiometric matrix v for the reduced reaction scheme of the system of linear independent steps of reaction has to be chosen: Aa = vx. Accordingly eq. (2.30) has to be written as v i = ^ = vKoa = vKo(ao+pAa).

(2.33)

By defining a matrix in analogy to K^ as Kc=vKoP, the above equation can be written as ^= ^o + K^Aa.

(2.34)

with the definition ao=vKoao. This differential equation represents a set of linear independent concentrations. The matrix K^ becomes a regular sxs matrix. Eq. (2.33) allows a direct transformation to the differential equations for the linear independent degrees of advancement. Multiplying by the inverse of the reduced matrix of the stoichiometric coefficients v~\ one obtains v"**vx = v"*a = V^ao + v'^K^vx, i = v-^^o+v-^KcVx,

(2.35)

x = Xo +K-X. The matrix of the rate constants is defined by K = v-^KcV.

(2.36)

Both the matrices K and K^ have to be regular matrices with sxs rows and columns. In the example of a consecutive reaction, all the equations above

General Approach in Formal Kinetics

68

Ch. 2

are derived explicitly to demonstrate the principle of this approach. Some of these equations are summarised in Table 2.5 for different mechanisms. In future discussions such reduced vectors of concentrations and matrices in differential equations are assumed. Therefore from now on the symbols ^ and * are only used, if either linear dependencies exist between the partial reactions and/or the number of concentrations of reactants is not reduced to the necessary ones. Thus in general the following equation^ will be valid: XQ + K - X a = a^ + KQ ' Aa, Example 2.17: Matrix of rate constants for consecutive reactions For the reaction A ^ B •-> C, the reaction scheme can be written as or as a reduced case of Table 2.1: A * -1 ^1 •

X2

*

B 1 -1 -1

1 0

C 0 k^a 0 k^b 1 kjb

Xl

A -1

X2

0

B 1 -1

C ^k 0 kiO - k2b 1 k^b

containing two linear independent steps of the reaction (.; = 2). It cannot be reduced any more. The stoichiometric matrices are therefore according to the defmitions given in Section 2.1.1.1 M v°* = 1

1 -1

0^ -1

0

0

1

and

M v° = 1 0

0^ -1 1

The matrix of the thermal rate constants are given by (kx

K=

0

0 Qi] k2

[O k.

and

K„ =

,0

L

Ch. 2

General Approach to Linear Systems of Reactions

69

Next, the equations are given in matrix notation / ' • * \

X * = K ; a° = K^3J

fki 0 O'jj 0 Jfc^ 0 b \^ 0 kj 0 j

<-l v°*i* = a" = v°*K:a'= 1 .0

1 0^ 'a^ -1 -1 • * = b \X2 0 l-*3> <^)

r'A

ij

0

OYa\ b

OYa^

-it. -*2-*3

0

b

*3

0

y-j

with

f-k.

0

Kc = k,

0

and the following rate equations are obtained:

1

1 -1

0 V;.•^ -1

.0

0

1 yXjj

(a") b =

r-1

;c.

oYa^ -k2 - Jkj

0

k,

0 0

\^J

a = -jc,* +xl= -kfa{t) + k2b{t), b = xl - x*2- xl=^ k\a{t) - (^2 + ^3 )b{t), c = x\= k^b{t). In the following, the matrices are reduced because of the linear dependencies, therefore -1 v=

0

v-l =

-1 -1

0 -1

General Approach in Formal Kinetics

70

and

r-1 p = v°-v-' = 1

0^ -1

.0

U

:: -.]= -01

1 -ij

one finds

M

Aa 1 Ab -l)

a" = ao + pAa = \b = 0 + 0 ,-1 .0) c = —Aa - Ab and with

K'C=V-K; =

finally i- = K^(a'+PAa) = ^;+K^PAa -K

n

0\ 0

(-k,aA

f-k, ACi

0 /

"^l^O '

A:,ao

0 -1

0 '

^I

K^

V ^

/C^

3

o^

Aa

0 -1 0" y V"

1 -1

Ab

"^2

Aa

—k.2 — Kj

Afe

^3

y

1 - 1V

Aa Ab

oYi w

'v^

0^

Ch.2

General Approach to Linear Systems of Reactions

Ch. 2

71

or a = -k^a^ - k^Aa 4- kj^, b = k^aQ-hk^JSa — [k2 + k-^ )Ab, c = k-^tsh. The matrix of the rate constants reduces to fife, Ko =

-*2 0

0

Jtj

0

Thus

(d\ x = Kor =

0

^^)

itj

i , = k^aif) - k2b(t) or

0

X2 = k^b{t)

V-)

Therefore vx = a = vK„a° =vKo(ao+pAa) -ifej

0

itj

0

/t,

-yfej

0

0

itj

0

rn\

-1

0

A,

-/fej

0

1

-1,

0

*3

0

-/ -J /"-^IOQ'

\^ ^JOQ ^

0

^ 0

ikj

Afe -Aa-Afc

-A:,Aa + fcjA^ \k^^-{k2

0

Aa

+^3)Afe

^1 0

0^ Aa 1

v-1

-h

^

A^

General Approach in Formal Kinetics

72

Ch. 2

with -1 Kc=vKoP = 1 -1

0 Yk,

1

-U 0

0 ^K -h -1 . 0 k. -k

0^

(\ ^\

0

1

V .-1 -ij

'-it, k^

-(^2+^3);

the rate matrix is obtained K=

-1 -1

0 (-k, - 1 {k,

-1

-1 -1

0 r k^+k^ -l{-{ki+k^+k,)

-{k,+k,)ll

0 -1^

-k^ \J-{k^+k:,) k^+k,) [ ^3

k/ -kj

In consequence, using eq. (2.35), the following relationships are found:

^=^—(:;]<:; -.ra0

^-{ki+k^} V

k V^, - J f c . 37

^3-^1 ""^3-^2

The use of all the equations given in this section have been demonstrated in detail to make the principle procedure to set up all the linear differential equations obvious [16]. In the following, other mechanisms can only be handled briefly. Some help on the derivation is given in the Appendix 6.5. Results are summarised Table 2.5 in Section 2.2.1.3. 2.2.7.2 Jacobi matrix Both the matrices K and K^. have to be regular. They are called Jacobi matrices. In general the Jacobi matrix for two linear independent steps of a reaction (s = 2) has the following form:

General Approach to Linear Systems of Reactions

Ch. 2

K=

dx2 dxt

73

(2.37) 0X2

dx2 J

As mentioned above, the reduced degrees of advancement are used (without linear dependencies). K^ can be obtained accordingly. For tjiis reason, one can understand eqs. (2.31) and (2.35) as a development of x and a into a Taylor series to x and Aa at time t = 0, respectively. The differential equation (eq. (2.35)) can even be simplified. At time r -> 00 the change in the degree of advancement has to become = 0.

For this reason (2.38) results. Subtracting eq. (2.38) from eq. (2.35) yields i = K-(x-x^)

(2.39)

The relationship (2.39) is useful, since this equation allows the calculation of the values of the degrees of advancement at the end of the reaction using the initial slopes. This result can be compared to the measurement. Since the Jacobi matrix has to be regular, using eq. (2.38) (2.40)

x^=~K-^ i .

is immediately obtained. In analogy to eq. (2.35), eq. (2.31) can be simplified. Some equations become similar to eqs. (2.38)-(2.40) 0 = ao + KcAa^, Aa^=-Kc ao-

a = Kc(a-a^)

(2.41) (2.42)

74

General Approach in Formal Kinetics

Ch. 2

Eqs. (2.40) and (2.41) are systems of s linear homogeneous differential equations with constant coefficients for all the s unknown jc^^ and Aa, respectively. These systems of equations are equivalent to a single differential equation of 5^th order for a single unknown [17]. Taking eq. (2.41), this differential equation for the /th concentration is given by is)

(5-1)

(j-2)

a, --a^M, -^-a.M^

("-l)^(a, -a,^)M, = 0

(^.43)

In this differential equation the coefficients M^ {i = 1,2,...,5") represent the sums of the main minors [18] of the Jacobi matrix K^ (for comparison see eq. (2.48)). 2,2. L3 Examples for Jacobi Matrices (thermal reactions) The results for some more mechanisms of different consecutive reactions are derived in Appendix 6.5. The results are sunmiarised in Table 2.5. The results for the mechanism A ^ B ^ C ^ A are not included because of the extensive relationships; they can be found in the Appendix as a further example (see Example 2.18). 2.2,1.4 Photokinetic Jacobi matrix All the equations used above can be taken in principle to describe photochemical reactions also. However, the relationships become more complicated, since the amount of light absorbed has to be considered. Furthermore the quantum yield can depend on the concentration which makes the formulae more complex. Therefore in a simple example of a consecutive photoreaction, the correlation is demonstrated. Details are discussed in Chapter 3. According to the comments in Sections 1.4.2, 1.4.3, and 2.1.4.3 the consecutive photochemical reaction A—iil^B—!^C yields

Using eq. (1.39), relationships are obtained in the case of Napierian units for the absorption coefficients K\ and the absorbance E'^ at the wavelength of irradiation k'.

General Approach to Linear Systems of Reactions

Ch. 2

dt

75

t^ K'b

dt

= i:^ = lOOOipf - ^ / o [ l - exp(-£;)] = lOOO^p^V^fe/oF;

Both quantum yields do not depend on the intensity of irradiation. Integration is done with respect to the local volume element. According to eqs. (2.9) and (1.40) the rate law is given by a = -1000
(2.44)

c = 1000
It is obvious that thermal and photochemical laws can be treated equivalently (see Table 2.5 for comparison). However, the derivation of the photokinetic equations causes more expenditure, therefore in this chapter thermal equations are chosen for convenience. 2.2.2 General solution of the differential equations In eq. (2.39) a general equation has been given which allows the calculation of eigenvalues of the Jacobi matrix (that means reaction constants or their combinations) taking the degree of advancement. In the case of consecutive reactions, special solutions have been given. However, this system of differential equations has a general solution [17a], (2.45)

X = x^ + p exp(rt) whereby the matrices are defined as |'exp(r,0'|

^Pu'-Pu^ exp(rt) = \Ps\'"PssJ

|^exp(r,Oj

o

"cc«

o c •a j2

c/l

x;

a

S

I

•a,

76

«r>

t I < u3

t i

BQ "a.

< i2

a\

t •^
U

t rVi CQ 'St < X

o

1

o

V

M

.U

3

n

<

H C/5

o 1

^

O

-sT

1

15 7

o

7

7

General Approach in Formal Kinetics

N

n

o

O

^

Ch.2

General Approach to Linear Systems of Reactions

Ch. 2

77

It is assumed, that all the s eigenvalues r^ (A: = 1,2,...,5) are real and different. These eigenvalues are the solutions of the characteristic equation of the matrix K: ••

II-'-

A:,2

K2I

*22 ""'*

*,.

...

*„

=0

(2.46)

*« - n

This determinant (eq. (2.46)) is called the characteristic polynomial of the Jacobi matrix. Its rank takes on the degree s. The determinant is explicitly given by s-2

r' - M^r'-' -h M^r'-'-.^X-iy

(2.47)

M^ =0

The different coefficients M^ (A: = 1,2,...,5) represent the sum of the main minors of the Jacobi matrix K (see eq. (2.43)). Thereby

M,=J^kjj,

M,=X j<j

\l<-n JJ 1 •'-'

jfc,. JJ

(2.48)

Af,=|K|.

JJ

By use of Vieta's [17] theorem, the determinant (eq. (2.47)) can be rearranged to ( r - r , ) ( r - r 2 > - ( r - r J = r'-5,r'-*+52r'"'

(-1)^5^=0

(2.49)

taking the solutions for r^ to r^. The coefficients are given by

'Si=X'*y'

•^2=X00'—'•^.^ = 'i''2...'i.

(2.50)

Comparison of the coefficients of eqs. (2.47) and (2.49) results in S^^M^

(it = 1,2,..,,5).

(2.51)

The derived relationships between these coefficients are of great impor-

78

General Approach in Formal Kinetics

Ch. 2

tance in the spectroscopic analysis of linear reactions. One realises that the coefficients of the determinant of a Jacobi matrix using the degree of advancement or spectroscopic data (signals) can be the same. In the literature, generally the determination of the coefficients p^, is described using the eigenvectors of the Jacobi matrix [17]. In the following, another approach is discussed, which is more convenient and more useful for a numerical calculation of the time dependencies of the degrees of advancement as well as the concentrations. These calculated concentrations are used to construct graphs of the concentration of one reactant (a.) versus that of another {a), each taken at a large number of reaction times. This representation is called a concentration diagram (abbreviated K-diagram) [19]. This method of evaluation is explained later in detail. 2.2.2.1 Determination of coefficients of the Jacobi matrix To obtain the interesting coefficients p^., the general solution of the differential equation, eq. (2.45), is differentiated for each degree of advancement s-l times. Thus for the ki\\ degree of advancement at time ^ = 0, the following system of linear equations is obtained [16]:

^*=nP*i+"- + 'iP*5 ^*=nViti+-+'iVib,

(2.52)

This system includes the original equation at the differentiation of zero order and contains the eigenvalues r^, which can be easily calculated by use of eq. (2.47). Higher derivatives of the degrees of advancement Xj^ are possible. They can be achieved by use of the fundamental equation, eq. (2.35), in a convenient way. This is shown in Section 2.2.2.2. Next, the derivatives on the left-hand side of eq. (2.52) are summarised as a column vector (s-l)

Ch. 2

General Approach to Linear Systems of Reactions

79

Then the above system of differential equations is a matrix equation (s-l)

X, = V p , .

(2.53)

In this equation, the matrix V represents the so-called Vandermond's matrix [17] with the determinant 1 |V| =

(2.54)

=("-irn(o"0').s-y

.s-i

Pkis the column vector with the elements p^., which represent the coefficients to be determined. This column-vector is given by (s-l)

(2.55) Up to now degrees of advancement have been used. If the equation is set up for concentrations, one finds for the elements p^.^ the following relationship: a = a^+pexp(rt).

(2.56)

The explicit expressions for the constants p^^ are listed in Table 2.6. All the given derivations can be applied to the concentrations and the degrees of advancement. If the latter is used, one has to consider that by definition the degree of advancement of any step of the ^h reaction Xf^ becomes zero at time r = 0. For the number of independent steps of reaction s one finds a general solution given by .X(*-i)

,.(j-2)

, ,.

IIV.J

P,r = ( - ! ) " * [«/o-5,* fl,o + - + (-ir'5*-,(«,o -«,~)J-f^.

(2-57)

80

General Approach in Formal Kinetics

Ch. 2

TABLE 2.6 Solutions for the elements of the p-matrix for reactions with 1-3 linear independent steps of reaction

Pii =

^ _ gto-n(^/o-^/oo) Pi2

^'•*^

(n-2)(n-3) ('•2-n)('2-''3) (^3-n)(r3'-'"2)

The elements in the brackets in eq. (2.57) have the form of the differential equation, eq. (2.43), however, the rank is reduced to 5 - 1. Furthermore the relationships of eq. (2.51) are used to calculate the values of 5f

(/ = 1,2,...,5-1)

by use of the s eigenvalues according to eq. (2.50). To get the correct results the eigenvalues with the index k' have to be omitted. As discussed in eq. (2.54), W) is the Vandermond determinant given by

X(

)•••(

)

(2.58)

X('3-'2)('^-n) X(r2--r,).

Taking all brackets, of which the eigenvalues r^/ are part, equal to one, the different values of V^. are obtained from the Vandermond determinant V.

Ch. 2

General Approach to Linear Systems of Reactions

81

Example 2.22 A simple consecutive reaction, as given in Example 2.19, has the eigenvalues r, = —/:,

and

rj = —^2

looking at the elements on the diagonal of the K matrix. Therefore, according to Table 2.6, for the concentrations of the three reactants, one finds the following relationships according to eq. (2.56): a = aQ exp(-A:,0»

c = -^[k2(\ 1

b=

ka ^ ^ [exp(~/:,0 -

txpi-kjt)].

- exp(-A:,0) - *, d - expC-^^r))].

I

The derivations are given in detail in Appendix 6.5. 2,2.2.2 Solution at two equal eigenvalues If equal eigenvalues result in a Vandermond determinant, which equals zero, eq. (2.53) can no longer be solved with respect to p^.^. With h = 'i' = '* the kX\\ and k'ih element in eqs. (2.45) and (2.56) respectively are substituted by .. • + p,.^ exp(rO -h p.^,t exp(rr) + • • •.

(2.59)

Differentiation by an order of j - 1 changes the matrix Vonly in the column k and k'. These two can be represented by

1 r

0 ... 1 ...

r^

2r ...

r'-'

{s-\y-'^

(2.60)

82

General Approach in Formal Kinetics

Ch. 2

Example 2.23 For a reaction with two linear independent steps (s = 2), eq. (2.53) is replaced by

r^/o-^/oo'i n 0 V

^«o J

r

iV'r

The rearrangement yields ^'^="'^""-'

(2.61)

For this reason the simple consecutive reaction discussed in Example 2.22 results in a = ^Q exp(-/:0»

^ = ^^o^ exp(-/:0»

c: = ^Q (1 - (1 + kt) cxp(-kt))

if the relationship r = -/: is used. General solutions are possible if the two eigenvalues are formally combined by a variable A. This general solution is inserted for the intermediate B in Example 2.22: k^=k

and

A:2 = /: + A.

This approach allows A to be approximated gradually to zero. Using the equation derived for the intermediate B in Example 2.22 one obtains ka b = —^(exp(-/:0 - cxp{-kt) exp(-Af)), A which can be rearranged to kn

b = ^!^[exp(-Jkr)(l - exp(~AO)]. A By development of the second exponential function into a series, either

General Approach to Linear Systems of Reactions

Ch. 2

83

\A

A

exp(-Jt0| A / - - ^ ^ + J)

or h = ka^ exp(~A:r)

rA^

are found. The more A approaches zero, the better this equation equals the relationship for concentration h in Example 2.22. 2.2.2.3 Solution for complex eigenvalues Thermal reactions, which take place in closed systems, do not show complex eigenvalues. In contrast, in the case of quasilinear photoreactions, such solutions cannot be excluded in principle. Complex eigenvalues always appear in pairs. In the following, a general approach to solve this problem is given for two linear independent steps of reaction {s = 2). The two eigenvalues are r,, r2 = M ± ii^. Under these conditions, the solution with respect to concentrations is given by [17] «, = «,oo + P\ exp(MO sin(i;r -f P2)-

(2.62)

Taking the first derivative at time r = 0, the values p, and p^ are determined by the following system of linear equations: «/o-«,oo=PiSin(p2), a^o = ^ 1 sin(p2) + i;p, cos(p2). Division of the second equation by the first one yields *i^

= M + i; cot(p2).

84

General Approach in Formal Kinetics

Ch.2

Therefore the values of p are given by

p2 = arctg

sin(/>2)

(2.63) l^io/(«/o-^ioo)-wj

Example 2.24 In the case of the mechanism A — ! — » B — ^ C — ^ A with the Jacobi matrix •""fCj

rC-i

K= the solution

can turn out to be negative as in the case of ACi — Ky — K-i — 1 •

Then the eigenvalues become complex: 3 2

V3 2

and

3

are found. Under these conditions the time functions amount to a ( . ) = ^ ( U 2 e x p ( - | ) sm

2

2j

Ch. 2

General Approach to Linear Systems of Reactions

m=^\ l-h2exp[

sin

rV3

85

K^

.(.)=t 2.2.3 Numerical calculation of functions of concentration Analytical expressions for the time dependence of the degrees of advancement or the concentrations can be preferably obtained by the methods discussed in the last paragraphs. Nowadays personal computers may be used for numerical and graphical calculations. In the following, a strategy for a program is discussed, which allows one to calculate or to graph all the kinetically relevant relations for any linear reaction system [20]. The required input are the Jacobi matrix, the first derivatives of the degrees of advancement at time r = 0, and the stoichiometric matrix, if concentrations are to be calculated. It is assumed that all eigenvalues are real and different. The possible number of linear independent partial reactions is only limited by the storage capacity of the computer and by its numerical exactness. The following procedures of the program can be used: (1) input the number of linear independent degrees of advancements; (2) arrangement of the necessary storage for vectors and matrices; (3) input of the Jacobi matrix and the slope at r = 0; (4) inversion of the Jacobi matrix; (5) calculation of the final values x^ according to eq. (2.40); (6) calculation of higher derivatives at r = 0. In the last procedure, eq. (2.35) is differentiated with respect to time. Therefore the second derivative is obtained i = Ki. Substituting x*in this equation by use of eq. (2.35), one obtains i = Kio + KKx. According to the definition XQ becomes 0 at time r = 0; therefore

86

General Approach in Formal Kinetics

Ch. 2

XQ = K X Q .

This method can repeatedly applied. One finds for the kXh differentiation at time r = 0, (k)

XQ—K

XQ.

The next steps are: (7) control, if the Jacobi matrix is a triangular one; the answer *yes' allows the transfer of the diagonal elements to (9), *no' causes the call of procedure (8); (8) calculation of the eigenvalues. The elements of the Jacobi-matrix are named Z:^.. The trace T of the Jacobi matrix K is equal to the sum of all eigenvalues. It amounts to

For physical reasons all eigenvalues have to be negative. Otherwise x would increase with time to infinity according to eq. (2.45). Therefore eigenvalues have to range between zero and T. Eq. (2.46) can be rewritten in the form 3; = | K - r E | = 0. The program calculates >; for 5^ + 1 equidistant supporting points from r = 0 to r = r. A parabola of the order of the number of linear independent reaction steps s is fitted to the supporting points. The values for the s intersections of *zero' represent the required eigenvalues. They can be obtained by use of common numerical procedures such as the Newton method, nest of intervals, and so on. The program checks whether all s intersections of zero are found. The procedure continues with the following steps: (9) calculation of the Vandermond matrix; (10) numerical inversion of the Vandermond matrix; (11) determination of the elements of all p,^ vectors according to eq. (2.55). All quantities that are used to calculate the time functions or to graph diagrams, are stored after this procedure.

Ch. 2

Information about Reactions Supplied by Graphs

87

2.3 INFORMATION ABOUT REACTIONS SUPPLIED BY GRAPHS

2.3.1 Essential properties of linear systems 2.3.1.1 General considerations In this section the properties of linear systems relevant to kinetic analysis are discussed. Using this approach information can be extracted from the characteristics of diverse graphical representations. By looking for extrema and points of inflection in the concentration-time diagrams one can draw conclusions with respect to the mechanism - a prerequisite to determine the number of exponential functions of the overall rate law. 2.3.1.2 First view: extrema and points of inflection If all eigenvalues are different, the following equations are valid in the concentration-time diagrams: for extrema cii^Yi''kPik^^Viht) = 0,

(2.64)

and for points of inflection
(2.65)

If at least one of the s parameters p^j^ differs from zero, neither extrema nor points of inflection are found in concentration-time diagrams. For example, assuming the values p,, and p^2 are different from zero, eqs. (2.64) and (2.65) can be solved with respect to t. As soon as the concentrations either reach a minimum or a maximum (index m) one obtains ^m=

In

If they reach a point of inflection (index w), one obtains r^=

ln|'

r^P. riPn,

General Approaches in Formal Kinetics

88

Ch.2

In consequence, one finds, for any component A, for which exactly two of the parameters p^y^ differ from zero as the distance between the extrema and the point of inflection within a concentration-time diagram in general.

1 'w

^m

(2.66)

-In

r, - r ,

V'*2,

For the special case of two linear independent steps of a reaction {s = 2), the following two equations are valid [16]:

t^=-

1

-In t'2('i(a,o.-a.o) + «,o)J

K=-

-In r, - r ,

['•2^('i(«/~-«,o) + «,o)j

These two equations demonstrate that at time f = 0 an extreme is found for a,Q = 0 . Then a point of inflection is found according to eq. (2.66) at the time r„=-

1

-in^.

To - r , The special condition a,^ = a,o = 0 yields

C =

1

, /I

and

t ^2t

,

(2.67)

2.3,2 Number of exponential functions The method given in Section 2.2.2 for the calculation of concentrationtime laws turns out to be successful, if the eigenvalues are different and real. The number e, gives the number of exponential functions, which are necessary to determine the concentration the compound Ai in dependence on time

Ch. 2

Information about Reactions Supplied by Graphs

89

ttiit). This number e^ is obtained by calculation and is useful in any kinetic analysis. Sometimes the calculation can be tedious. There is another way to determine this number of exponential functions. It uses simple rules. They are derived from the fact that differential equations which describe the time dependence of concentrations have to be always linear, their coefficients being constant. If these differential equations are not homogeneous, a formal interference function can be introduced. It is represented by a sum of exponential functions. Then, the solution of the differential equations is a linear combination of exponential functions. They consist of powers of the eigenvalues of the homogeneous system and of powers of the interference functions. In practice, first the differential equation or the total system of differential equations have to be set up. Next the number of eigenvalues of the homogeneous differential equations have to be determined. Finally the sum of exponential functions is given by this value and the number of interference functions. The principle of this calculation is explained in the following. It is assumed that the compound A, reacts according to the mechanism i

> A^'^i — '

' >

^—> i4,'^l i

>

Ai <—> A-.i - ^ ^ ^

(2.68) - ^ A>i <—>.

This scheme has the following meaning: reactant A^ is formed by just one or even many compounds A,_j,A,._j It can react to form A,^^,,A..^,,... Backward reactions are allowed in principle. However, they are not allowed to form Ai or to start from A,. Under these assumptions the differential equation (with respect to a,) is given by a,. + {k, +/:,.+... )a,.(0 = A:,._,a,_, (r) + ^,^^,a..,, ( 0 -

(2.69)

In the equation above, the aj{t) represent the interference functions (j = 1 - 1 , /' - 1 , . . . ) . They exhibit the dependence on time for those reactants which form A,. It is assumed that they are known. Therefore the result is the number of exponential functions

90

General Approaches in Formal Kinetics

Ch. 2

The exponential functions in the angle brackets represent the number of different exponential functions of those reactants produced immediately in the reaction sequence before the reactant Ai. This means that only the sum over all reactants formed has to be determined. In consequence the additional new exponential function takes the exponent

Therefore (1) The exponential function for any initial product which is not formed by any back reaction amounts to one, since the differential equation, eq. (2.69), is a homogeneous one under these conditions:

(2) Any final product A^ yields the exponential function

since under these circumstances the differential quotient a^ is the only element on the left-hand side of eq. (2.69). The solution of this equation is the integral of the right-hand side of the equation. (3) Therefore a simple non-branched consecutive reaction A,

>A^

>Aj^

>

>A^

yields ^. = /

(/ = 1,2,3,...,/!-!),

e^-n-\

(4) The following is always valid according to Section 2.2.2: \<ei<s. If any reactant A-^ is the starting material taking part in one or more back reactions, eq. (2.69) has to be replaced by a system of differential equations.

Ch. 2

Information about Reactions Supplied by Graphs

91

The following equation can be taken as an example, if the above model of the reaction system, eq. (2.68), is extended to a back reaction from Af to y4/_i: /

X

(2.71)

a, +^/-i«/-, +(/:, +/:Ja, = V,a,...,(0. k- symbolises the rate constant for the back reaction from A^ to Ai^i. One finds instead of eq. (2.70) ^/=(^i-2»^r-i--) + 2,

(2.72)

since the homogeneous system adds two further eigenvalues. ^, stays the same, because €^,2 ^^^ ^^ be inserted in eq. (2.72) instead of e,_,. In the case of a back reaction from A,^., to i4,, eq. (2.71) has to be modified to a,. + k.a^ - k^^^a,^^ = /:,.,«,_, (0 + k^._^a.._^ (0,

One finds instead of eq. (2.70) or (2.72) the following relationship ^ . = ( ^ i - P ^ r - P - - ) + 2.

(2.73)

A comparison of the results of eqs. (2.70), (2.72), and (2.73) makes obvious the fact that the exponential function e, is only enlarged, if the back reaction or an uninterrupted sequence of back reactions take place to the considered reactant At. 2.3,3 X' and K-diagrams Further information can be obtained, if instead of concentration-time diagrams, other types of graphical representations are used: ~ An X'diagram represents a plot of two degrees of advancement versus each other for a given system of reactions. - The graph of two concentrations versus each other is called a concentration-diagram [19], which is abbreviated in the following by the term Kdiagram.

92

General Approaches in Formal Kinetics

Ch. 2

In many cases both these representations characterise a system in a better way than concentration-time diagrams. A special advantage of these diagrams is their limited length in their axes. For example, it represents the total advancement of the reaction in contrast to the time axis of concentrationtime diagrams which in principle extends to infinity. Concentration diagrams as well as X-diagrams are given by two of the equations, eqs. (2.45) and (2.56), respectively. Thus the K-diagram of ^,(0 versus aj{t) is given by

t^

(2.74)

2.3.3.1 Initial and final values Initial and final points of a K-diagram are determined by the initial and final concentrations of reactants A, and A,. If the concentrations of two totally degrading intermediates are graphed versus each other, the results are closed curves starting and ending in the origin of the diagram. Under these conditions no additional information is obtained in comparison to the related a/(0 diagram. 2.3.3.2 The initial slope One can calculate the initial slope in a K-diagram by use of the system of differential equations according to da^ I _ a,0 daj

ajQ

In some cases, a division of zero by zero can happen. Under these circumstances the formalism of Bemoulli-L'Hospital has to be applied, which is discussed in detail in the next example. The following reaction mechanism is assumed: k!

' kUt

'^^ ku,

.

Thus the two differential equations can be derived:

(2.75)

Information about Reactions Supplied by Graphs

Ch. 2

93

The rate constants of the back reaction are symbolised by kj (y = 1,...,rj. For 7 = 1 (that means A, is the starting material), one obtains

and one finds La 10 da.

= -1

^1^10

as well as ^ ^ | = / - =0

(r = 3.4.5...).

Whereas the condition i = 2 gives the same result as above. 0 yda^

=0

(r = 3,4,5...).

ktU 1"10

the graph of the concentration of reactant A^ versus the third reaction product A3 (/ = 3) happens to cause a zero division: dttj.

da^

I

(r = 4,5,...).

According to Bernoulli and L'Hospital both the numerator and denominator are differentiated with respect to t. As soon as the time t approaches zero the numerator becomes zero. Then the denominator is given by ^30 •" ^ 1 ^ 2 ^ 1 0 -

94

General Approaches in Formal Kinetics

Ch. 2

Thus

K 1 =0

(r = 4,5....).

Accordingly, the initial slopes of a K-diagram can be calculated for any combination of reactants. 2. i. 3.3 The final slope In general the slope of a K-diagram is given by —- = =

d^j

—-.

(2.76)

2^^*P;*exp(r^0

Eq. (2.76) requires a summation from 1 to s. The final slope in eq. (2.76) becomes indefinite at the limit t->oo. Assuming ri is the smallest eigenvalue (by amount), one can divide the nominator as well as the denominator by exp(-r,0: da, ^ r,Pi, 4-^r^p,^exp[(r^ ~ri>]

In this case the sums have to be taken from 2 to s. Then all the exponents are negative. Consequently at the limit r —> «> all the exponential functions become zero; therefore the final slope is given by da, daj I-

=-^.

(2.77)

Pji

The slope depends on the value p,,. The following cases can be considered: (a) Pfi = 0, pji ^ 0: the final slope is zero. (b) p .J 9t 0 and py, ^ 0: the final slope is finite. (c) Pj\ = 0 and in consequence p,, ?& 0: the final slope becomes infinite. Examples for the values p,., and py, respectively are given in Table 2.6 and were used to calculate the concentration's dependence on time in Appendix 6.4.1.

Ch. 2

Information about Reactions Supplied by Graphs

95

Example 2.25 The determination of these final slopes is demonstrated using the mechanism of a multi-step consecutive reaction A—!-^B—^C-^-^D. The final slopes are given in Table 2.7 for a variety of cases. In the first column the concentrations are listed which are graphed versus each other to form the K-diagram [aXaj)]. The next three columns represent the three cases for which different rate constants are assumed to result in the smallest eigenvalue. The data used in Table 2.6 supply the values p,, and p^, which are taken to calculate the ratios of the different rate constants according to eq. (2.77). Thus the final slopes can be determined. The full calculation procedure is demonstrated in Appendix 6.4.2.

TABLE 2.7 Slopes at final values for 3 different rate constants taking the smallest value Smallest eigenvalue ks

*2-*l

c{d)

*1*2

{ki-k2){kf d(a)

-*3)

hh (*3-*l)(*I- -*2)

c(b)

h *3-*l

k2ki k^k^

d{b)

die)

^_^

k^

k^^

-1

96

General Approaches in Formal Kinetics

Ch. 2

2.3.3.4 The extrema According to eq. (2.74) different cases can be distinguished in a Kdiagram aXo,): - for a, = 0, the result is a horizontal tangent and - for fly = 0, a vertical tangent. For the compound Aj, which reacts according to the mechanism, eq. (2.75), the maximum concentration amounts to I

_ ^1-1 ^ M

•^^i-H^f-H

The maximum of the k-diagram is given by a: a:1-1

_

^M

jfc,-hit;

if a,^., is not the starting material of the backward reaction. This relationship can be easily visualised in a K-diagram ei,(a,_,): in the maximum of the Kdiagram the ratio of ordinate to abscissa equals the right-hand side of the relationship above. 2.3.3.5 Influence of dilution The vector of derivatives comparable to eq. (2.53) (s)

5,=Vp,, given by a, to the order s, is proportional to the initial concentrations. Since V contains no concentrations, the p, must also be proportional to the initial concentrations. After division of eq. (2.74) by the initial concentration GQ on both sides of the equation, values are found which are independent of concentration. For this reason a graph of aja^ versus ajja^ results in a reduced concentration diagram, which becomes independent of dilution. 2.3.3.6 The areas In the K-diagram some equations can be set up for the areas below the concentration functions. In the case of a simple consecutive reaction

Ch. 2

Information about Reactions Supplied by Graphs

97

A, — ^ ^2 — ^ A3 — ^

one obtains for the areas below the curves a^Ca,)

This equation can be verified by complete induction, since obviously the integral is given by

Jo 2

'

it,+Jk, 2

Since eq. (2.78) has been verified, one obtains for / + 1

Integration of this equation gives ~*i j^«i da^^i +*i£^i+i da, =*, J^«, da, -*,>i J^^w da,. Taking into account that the first integral on the left-hand side of the equation vanishes at the lower and upper integration limits and that the first integral on the right-hand side is given by eq. (2.78), a substitution of 1 by 1 + 1 gives eq. (2.78) back again after simple rearrangement. If the ith reactant is the final product, then the rate constant k^ becomes 0. In consequence the following equation is obtained:

Jo

Since n

i=2

da,=

2

3 * *'

i—1

2 '

98

General Approaches in Formal Kinetics

Ch. 2

is valid, for the simple consecutive reaction discussed above one obtains

In the case of the more complicated reactions the areas can be calculated accordingly. 2.3.3.7 Points of inflection In the a^{aj) diagram a point of inflection is found, if

These points of inflection are typical for the K-diagram. They are not found in concentration-time diagrams. It is difficult to define general rules. However, one is able to simulate Kdiagrams using a desk-top computer for assumed mechanisms and to analyse the curves for points of inflection. The principle approach was discussed in Section 2.3.2. Points of inflection are allowed in K- or X-diagrams only, if more than two linear independent steps of reaction (s > 2) take place. 2.3.3.8 Regions of existence in a K-diagram The K-diagram is usually limited to a triangular area below the straight hne «/=«o-^y

(2.79)

At this line the sum of the concentrations of A, and Aj equals the sum of the concentrations a^ at the beginning of the reaction. For this reason, in general, the diagram stays within this triangle given by the coordinates 0, 0; 0, aQ-, and UQ, 0. Exceptions to this theorem can be expected if two reactants are produced within one step of a reaction. The sum of the concentrations of the reactants can no longer stay constant, if the mechanism contains the following partial step:

Ch. 2

Information about Reactions Supplied by Graphs

>A,

>B + C

>

99

(2.80)

For this reason the above-mentioned straight line (eq. (2.79)) can no longer describe the above-mentioned limitation. The partial step, eq. (2.80), cannot be reversible since the system loses its linearity. 2,3.4 Transformation of linear systems K and K' are assumed to be regular Jacobi matrices of two different linear systems of equal rank s. Furthermore, all eigenvalues are assumed to be different. Under these conditions, similarity transformations S and S' are possible: D = SKS"'

and D' = S'K'S'"^

D and D' are diagonal matrices. If all eigenvalues of K and K' are equal, that means r^ = r/,

for A: = 1,..., 5,

then D = D' can be obtained. In this case

or by use of T = S-'S'

and

T'^S'-^S,

one finds K = TK'T-'

or

KT = TK'.

(2.81)

The transformation T is linear, if the primed system can be transformed into the non-primed system. In this case.

General Approaches in Formal Kinetics

100

x = Tx'

Ch.2

(2.82)

and x^=Txl,

x = Tx',

Xo=Txl..

(2.83)

If this transformation is applied to the fundamental eq. (2.35), then T i ' = i = T i i + TK'x' = *« + TK'T-*x = io + Kx.

(2.84)

The matrices K and K' are similar matrices; they have the same eigenvalues and principal minors [18]. Example 2,26 A system of two independent reactions A'

>B',

C

>D'

results in the simple rate law y, = a^(l - exp(-A:,0), x^ = Co(l ~ tx^{-k^t)). (2.85) One looks for the transformation T which transforms the system into the reaction ->B-

->C.

This reaction is represented by [14]

or

MO ^20

^11

^12 if "0 I "'22A^0/

•^21

•'22

•'21

(T^1

^12

T;1

^12

^21

^22, V^2^0>

V^21

^22

0

Information about Reactions Supplied by Graphs

Ch. 2

101

and

0 YT;, "^ijKTlX

T.AJT,, T,^\(-k{ ^22 7

V21

'22

0

0 -k'^

These equations are useful for the determination of the four elements r^, and for the correlation of the eigenvalues. They depend on each other because of eq. (2.38). The last equation can be written explicitly (a)

'^\^\\ —K^u

*2^12 ""^2•'22 —""^2^22

(^/

Equation (a) means that either Jt, =ik;

or

7;i=0.

In the first case 7,2 becomes 0 because of equation (b), since k^ ^^kj. Equation (d) gives A:2 = Acj. According to equation (c) one finds T -

'^ T =

k'

Eq. (2.85) results in ao = r,,a^

and

T^.k^^ + 722*2^0 = 0.

By this means the elements of T are found to be 0 x=

ki Ky

Kt

GQ

Ky

a^(l-exp(-A:,'0)^ c^(l-exp(-A:^0)

ao K\

CQ

j

The result fits into the formalism found in Section 2.2.2 and discussed in Example 2.22:

102

General Approaches in Fonnal Kinetics

Ch. 2

jc, =ao(l-exp(-A:,0) ^2 = T ^ V ( * 2 ( ^ " ^*P(-*i^)) - *i(l - exp(-*20))' Ky "~ /Cf If r,, becomes 0, then according to equation (b) ^12 ~

^22 •"

7

^22

77

and one finds the transformation T ^

0

^

T= ^1

^0

^2

^0

The same result as above is obtained. These transformations can be used to determine the rate equations of complex systems by use of the known differential equations of simple systems as has been demonstrated above. However, this procedure does not have any advantages in comparison to the other methods described previously. In the case of two linear independent partial reactions, one can set up the following theorems for linear transformations: (1) Straight lines are transformed into straight lines, therefore (a) tangents become tangents and (b) asymptotes become asymptotes. (2) Parallel straight lines are transformed into parallel straight lines. (3) The ratio of distances on straight lines stays constant. (4) The ratio of areas stays constant. (5) Conic sections are transformed into other conic sections. 2.4 APPLICATIONS TO SELECTED SYSTEMS In this section some of the essentials stated in the previous sections are reviewed and summarised. The equations derived are demonstrated for some

Applications to Selected Systems

Ch. 2

103

simple selected systems. As mentioned, the result does not depend in principle on whether thermal or photochemical reactions are considered and the quantum yield does not depend on the concentration. 2,4.1 Uniform reactions For 5 = 1 the differential equation, eq. (2.35), degenerates to the simple equation

Therefore eq. (2.38) is reduced to

whereby *

,

=

•

a,iO a- —alO

The simple reaction A —>B has been discussed in Section 2.1.7.2. Therefore only two further reactions have to be compared. 2.4.1.1 Parallel reaction The reaction A * -1 • -1 X2

>B >C

. •

B C 1 0 k^a 0 1 k2a

stays uniform, since ^2=X^\y

where% = -^.

In consequence one finds for the second variable by integration xl = x^x.

General Approaches in Formal Kinetics

104

Ch.2

Thus the reaction scheme can be reduced by analogy to Example 2.9 in Section 2.1.4.2: B

- a + l)

X k^a

For this reason the differential equation is given by X = k^{a^ - (x +1)^) = k^a^ - (/:, + k^)x (2.oo) and ^ = -(pc +1)^1^ = -(^1 + ^2 )^»

c = k^a. According to eq. (2.86) the Jacobi matrix is represented by K = /: = -(/:, + ^2) • By use of eq. (2.55) the following relationship is obtained:

^

k^^k^

The integral for the degree of advancement is ^Q/V,

A: = -

-(l~exp(-(/:,+A:2)0).

A:, + ^2

One obtains either by direct integration or by use of above equation, a = ^0 exp(-(/:i 4- k^ )0, /Cj ~r /C2

Applications to Selected Systems

Ch. 2

105

2.4.1.2 Back and forward reactions In Section 2.1.4.2 an equilibrium reaction has been mentioned, A ^ B with the reaction scheme

-1

B 1 k^a — k2b

consistent with the assumed mechanism. Therefore the degree of advancement depends on time according to X = k^a — k2b = k^(aQ — JC) —/c2-^ = ^i^o ~ ( ^ i

+^2)'^-

For this reason, the Jacobi matrix K is reduced to one element, k = -{k^ -••^i)-

The degree of advancement amounts in the photostationary steady state (t -^ oo) to

AC|

"T" K-J

By integration one obtains x = b = - ^ ^ ( 1 - exp(-(fc, + k^ )0) /Ci " r AC2

and a=

—{k2 + k^ exp(-(A:, + k2)t)).

For this discussion it was assumed that at time r = 0 the concentration of the second reactant A2 becomes zero (&o = 0)-

106

General Approaches in Formal Kinetics

Ch. 2

2A.2 Two linear independent reactions 2A,2,1 Independent parallel reactions Two parallel steps demonstrate the simplest example A->B,

C^D

for two linear independent steps of a reaction. Their rate laws are A:J = Z? = ao(l - exp(-/:,r)),

a^a^ exp(-/:,0,

;c2 = t/ = CQ (1 - exp(~^2^))'

C^CQ exp(-/:2^)-

By use of the time equations, the X- and K-diagrams of the system are given in parametric form. By an elimination of time one obtains fln

Jt^ln^

M

=_ /:, In

Cn

X'y

or in general coordinates ^ • = - ^

S2 ~

*2

»

^2=l-(l-^,)^

(2.87)

In these equations, the symbols I represents the degree of advancement reduced with respect to the initial concentration. In consequence in the ^diagrams (reduced X-diagrams) the initial |o and the final |„ slopes are given according to eq. (2.87) as ^, ^.

0, X>1 =X

and

d^i

4,

«,

A:
Applications to Selected Systems

Ch. 2

107

Fig. 2.4. ^-diagram for the parallel reaction A -• B, C -• D. The parameter is^ = k2lk\.

In Section 2,4.1 further reactions were mentioned. Their combination results in further systems with two independent parallel reactions. These types of reaction characteristically yield Jacobi matrices, which are diagonal. According to eq. (2.87) a ^ -diagram is obtained, which is shown in Fig. 2.4. 2.4.2.2 Consecutive reactions The simple consecutive reaction A —> B -^ C yields the Jacobi matrix K=

-A:. 2

0

(2.88) 2

and the rate equations

a = k^a, c = ^2^.

108

General Approaches in Formal Kinetics

Ch. 2

By integration one obtains (see Example 2.19 in Section 2.2.1.3) Xt =ao(l-exp(-A:,0) JC2 = c = -r^{k2 K<j

(1 - exp(-*,0) - *, (l - cxp(-k2t)))

ACi

(2.89)

a = OQ exp(-/:,r) _

b=

QQ^I /Cj

(exp(-/:,r) - exp(-"^2^))

'Cj

Both concentrations can be graphed in a reduced form depending on the ratio of the rate constants of the individual steps. The functions are given in Figs. 2.5 and 2.6. Eq. (2.89) gives X- and K-diagrams in parametric form. The elimination of t using

^,=-^, an

^2=^ and

x-¥^1

results in the relationship ^ 2 = 1 . -

1.0

(i-i.r-(i-^.)

(2.90)

i-x -i—I—I—I—I—I—I—I—I—I—I—I—I—I

A

I

> B

Fig. 2.5.fi= b/ciQ. graphed as a function ofx according to eq. (2.89).

Ch. 2

Applications to Selected Systems

109

Fig. 2.6. y = c/flQ graphed as a function o{% according to eq. (2.89).

The eq. (2.90) above represents the solutions for the case xi^\. The case ;f = 1 yields dependencies discussed in Example 2.17 in Section 2.2.1.1. The corresponding ^-diagram is given in Fig. 2.7 using

Fig. 2.7. ^-diagram according to eq. (2.90). x >s the chosen parameter.

110

General Approaches in Formal Kinetics

Ch.2

^2=e,+(l-f.)ln(l-e.). The slopes in the ^-diagram yield %

d^. = 0 and d^x

x-i

d^.

X>1 X = iX<1

dir

Another example is a mechanism containing two linear independent steps, the latter including a back reaction A -» B ?* C

JC, ^2

A B C Xk -1 1 0 k^a 0 -1 1 k^b — kjC

The Jacobi matrix remains a triangular matrix. The only difference from eq. (2.88) is given by a change in the rate constant k22 which amounts to ^22 =~(^2 +^3)- Th^ concentration-time curve is given in Fig. 2.8. The eigenvalues are the elements of the diagonal. By use of the final values calculated in Example 2.20 in Section 2.2.1.3 one finds the following time functions: ^, =ao(l-exp(-A:,0),

k^a 2^0 Ky

' ^3

J

f^y

exp(-/:,0

^3 """ 1

^1^2^0

(*2 +^3X^2 +*3 - * l )

exp(-(A:2+^3)0,

a = aQexp(-/:jO,

b = ar V ^ 2 "^'^33

k —k ' ^ (exp(-A:iO - exp(-(A:2 + k-i)t)) #v^ "1 fCa """ /Ci

(2.91)

Applications to Selected Systems

Ch. 2

111

Fig. 2.8. Graph of/8 = bluQ as a function oir-k\t for the reaction A -• B ?* C according to eq. (2.91) kilkx = 1, the parameter is;i^ = k-^lki.

Concentration can only reach a maximum according to eq. (2.67) if fe, >k2^ The final slope is similar to that of a simple consecutive reaction: -,

/:2+*3>0

K'y "T" K-i "~ K\

d^2

/C2 "1 K."^ -~ K.\

^,

The other case is a back reaction superimposed on the first partial reaction. A -1 0

X, X2

B 1 -1

C Xk 0 kja-k2b 1 kjb

Then the Jacobi matrix is no longer a triangular one. In contrast one finds _ / - ( * ! +^2)

K=

^2

-A:,

General Approaches in Formal Kinetics

112

Ch.2

Under these conditions the eigenvalues are complicated functions of the rate constants. They are given by

K =

1 —

and - ( / : , + ^2 + ^3) + V(^1 + ^2 •+• *3 )^ " 4/:,Jtj r-, =

2

Under these conditions the number of exponential functions e, equals two. The concentrations depend on time according to

rj~r2 ,

b=

^0^1

r\-h

(exp(r,0-exp(r20).

(2.92)

r2-r, c = af 1 V

—exp(r,0 + —^—exp(r20 r^ - T i '2-1

In any case the final slope for this type of reaction in an X-diagram amounts to dxn dx,

_

^1

/:,+r,

and stays finite under all conditions. This diagram is demonstrated in Fig. 2.9. The last reaction discussed in this section is a more complex one:

In a first step the reaction schema can be set up as

Ch. 2

Applications to Selected Systems

113

Fig. 2.9. ^-diagram of the reaction A ^ B -^ C. k2lk\ (first symbol) and k-^lk^ are the parameters used.

h •

^3

* ^4

A -1

B 1

0 -1

-1 0

C D 0 0 1 0 0 1

Xk

kja — k2b kjb

k^a

Since the following relationship exists Z> = —^

and

a = -^,

by use of jc, the second step of the reaction given (by x^) depends linearly on jCj as well as on x*2 according to

/C j fC-i

General Approaches in Formal Kinetics

114

Ch.2

By integration •

^4

^2 4

X'y — "~"~" Xt

•

"T" —"""^ X2

is formed. The concentration-time diagram is given in Fig. 2.10. Therefore this scheme of reaction can be reduced to A

B

C

Xl

1

0

X'y

-1

1

13

D

Xk

A

^2

4

k^a — ^2^

k^b

13

In consequence the Jacobi matrix is

K = ~~V^1 "^ ^ 2 "^ ^ 4 /

^ 2 ""J

(2.93)

Fig. 2.10. Concentration-time diagram of the reaction D «— A *» B —» C, with *i = 1, *2 = 2, fcs = 2,A:4 = 4.

Ch. 2

Applications to Selected Systems

115

The trace of the matrix is

whereas the determinant amounts to

Therefore the eigenvalues are

S + V5^-4D r-, = •

The final values of concentration a, the degrees of advancement jc, and concentration d yield a^ =fo^= 0, _ ^

_ ^1^3

"^loo " " "^2«o ~~ ^oo ~"

a

=

D

—^-^fln.

The related />-values can be determined by use of Table 2.6. The concentration-time diagrams are given in Fig. 2.10. They have been calculated with the program presented in Section 2.2.3. This mechanism allows the equivalence between thermal and photokinetic rate laws to be demonstrated. If one can assume that the quantum yields do not depend on the concentration of the reactants, the Jacobi matrix given by eq. (2.93) can be written in a similar form according to Section 2.2.1.4 as K = 1000/(0) "i^A^l +^B^2 + ^ A ^ 4 j ^B^2

£B^3

Fit)

^B^3

(2.94)

General Approaches in Formal Kinetics

116

Ch.2

Nevertheless it becomes obvious that these equations become more complex. Therefore the details are handled in total in the next chapter. In Section 2.5 approximations are introduced which simplify some of the equations given and allow reduction in the expenditure of setting up photokinetic equations. 2,4.2 J More than two linear independent steps of reaction Just one reaction is considered as an example in the following: A -^ B -> C -^ D. Its Jacobi matrix is given by

1 ^^2

0 -k.

0 0

[0

*3

-*3

f~^' K=

(2.95)

The elements of the main diagonal are the eigenvalues. Using Table 2.6 one can find ^i=ao(l-exp(-/:,0),

X.

_k2{l- exp(-A:,0) - *i (l - expC-Jtj/)) =

«o» ^2 ""^1

At)

«o»

whereby / ( 0 = ^2^3(^2 -*3)exp(~*,0-*i*3(*,

-k^)txp{-k2t)

+^1^2 (^1 - ^2) expC-Z^jr). The dependence on time of the concentrations is shown in Fig. 2.11 calculated by eq. (2.95). Some of the resulting concentration diagrams are presented in Fig. 2.12.

Applications to Selected Systems

Ch. 2 r

ai

f-

1

H— 1

.

1

1——1

—>- B —

• ^

1

1I

1 i^

1

117

1

1

1

1

^ u

1

A

1 1

0.5 -

B

c"

^'^^^ 0.0 0.0

1

—1

h— 1

1

h-•

1

— 1

1 — h—

¥— 1

1.0

1

1

.

2.0

1

'

3.0

Fig. 2.11. Dependence of the concentration on time for the consecutive reaction A —> B C -^ D. The parameters ki = 1, A:2 = 2, /:3 = 3 are assumed.

1.0

_i

A

• B

1

1

1

{-

>- C

. 1.0

Fig. 2.12. Concentration diagrams of the consecutive reaction A -> B -> C -> D. The symbols at the curves mean: 1. b(a)y 2. c(a), 3. d(a)y 4. c(b). The assumptions ki=^ 1, A:2 = 2/3, k^ = 1/3 are valid.

118

General Approach in Formal Kinetics

Ch. 2

2.5 METHODS OF APPROXIMATION Linear reaction systems allow the rate laws to be presented in a closed form even if the reaction procedure is complex. But non-linear systems cause extreme difficulties in the integration of even simple equations. Therefore quite a few methods are described in the literature to approximate the solution of the differential equation. Nowadays such iterations are no longer necessary, since the relationship between concentrations can be calculated in an easy way for given parameters. Nevertheless in kinetic analysis two questions are essential: (1) Up to what amount can one simplify the rate laws and the relationships between the concentrations, if the rate constants of the partial reactions differ? (2) At what conditions can one distinguish between a simple and a complex mechanism? 2.5.7 Negligible concentration of an intermediate For the consecutive reaction A -> B -> C, the concentration of the intermediate B is given by the difference between the degrees of advancement of the two steps

The maximum value of its concentration is reached at the time at which the change in concentration b becomes zero: b = k^a — k2b-0. Therefore the maximum concentration amounts to JIC2-

tn ~^\^i

Since a^ <«o is valid, the relationship tn h-

-•^2

=

k2

Ch. 2

Methods of Approximation

119

is found. In Section 2.4.2.1 general coordinates were defined. Their values modify the above equation to

In the following the exactness of the measurement is assumed to be approximately 1%. If the rate of the second step is only 1/100 of the rate of the first step, one finds

k^

100

Therefore one obtains for the generalised coordinates y = ?i=?2=l-exp(-T). Under these conditions the concentration of the intermediate becomes negligible and the consecutive reaction degenerates to a simple one-step process A->C. The reproducibility of the measurement is assumed to be limited to 1%. Therefore a discrimination between the consecutive and the simple reaction mechanism is only possible, if kjk2 is larger than 0.01. The ^-diagram has to be considered for the time functions cc = a(t) and y = y(r) accordingly. It degenerates for small values of kjk2 to a diagonal between the corners ^ , = ^ 2 = 0 and ^j = ^2 = 1* Then the following relationship between the concentrations is valid: a-{-c=^aQ.

Depending on the reproducibility, the two linear independent steps of reaction of the consecutive mechanism (existing in reality) appear as a uniform reaction. In the following, conditions are given for some examples of consecutive reactions, which can be treated in the same way. The conditions are chosen such that their apparent mechanisms cannot be distinguished from uniform overall reactions at the experimental limitation mentioned above (see mechanisms A-D):

120

General Approach in Formal Kinetics

Ch. 2

(A) 2A >B >C, (B) A + B >C >D, (C)A >B, 2B >D, (D)2A >B, 2B >C. The different equations for these four different consecutive reactions are summarised in Table 2.8 using the same nomenclature as above. All these reactions are examples for overall reactions proceeding stoichiometrically. Their rates can be expressed by equations derived in Section 2.1.7. In some examples the stoichiometry makes it obvious that the mechanism is not given by elementary reactions. The overall mechanism summarises many elementary reactions. The method explained has the advantage that it is free of any additional hypothesis. Its disadvantage is, that it cannot be applied in general. This fact will be discussed for the reaction A ^ B —> C. Reducing the number of linear independent steps of the reaction by X = X* - xl

and

X2 = ^ 3 ,

one obtains x^-X2^b
Therefore the difference between the two linear independent degrees of advancement amounts to

Using the reduced degrees of advancement

«0

^0

Ch. 2

0)

I g .2

>

1

< «5

eg «

0^

1 §

4>

>

JUj)

il

I

11

o II ^ I

II

II

o

« I

^ j

"^1

VI I

^

t

w

II I

H

w U

t t

rj

I

r X 4J

VI

II

<3^

I

o 4?

ri II CM

S^ ^ '

It H 4U7


t + < < r5

c

Methods of Approximation

o II

I

II

U

u t + t <

V

II

II

U

t <

II

II

M

+ tO

^

o

r

CQ

u

A\^

CQ

t ^

U

t

<

121

122

General Approach in Formal Kinetics

Ch. 2

the difference becomes

Assuming the condition ^l_<10-^ the difference between the two reduced degrees of advancement becomes zero within the limits of measurement ^,-^2=0.

(2.96)

For r = k^t the rate equations can be written in general coordinates by

4l- = a-^y8 = (l-|,)-:^(^,-^,). If the equation above is used without consideration one obtains ^

= (1-^,)

or § , = | , = r = l-exp(-T).

This result is only valid, if k^ is much larger than k2. The result is valid even if the use of eq. (2.16) can be justified by one of the conditions like k^ « k2 or /:, « k^. A more general solution is offered by the Bodenstein hypothesis, which was introduced in Section 2.1.3.2, and this is treated in more detail next. 2,5.2 The Bodenstein hypothesis The Bodenstein hypothesis [13] can be stated as a hypothesis of a station-

Ch. 2

Methods of Approximation

123

ary state or the quasi-stationary state. In Section 2.1.3.2 two assumptions were stated as (1) negligible concentrations of intermediates (z, = 0) (2) negligible changes in concentrations of intermediates with time (z,. = 0 ) . Using these assumptions given in eq. (2.16) and applying this hypothesis to the reaction discussed above, one finds in case of A ^ B ~> C for the change in the concentration of the "intermediate" B & = Jk,a-(Jk2+Jt3)fo = 0. The differential equation has taken the form of a normal algebraic equation. It can be used to obtain the actual concentration of B by

Thus the time-dependent concentration of the initial product can be calculated as a = -k^a + k2b = -k^a + —^--^—a =

^-^—a = -;fa,

which results in a = aoexp(~xO. Accordingly the change in the concentration of C can be calculated as c-k^b = xci. Since the first assumption according to Bodenstein is valid (the concentration of intermediates is negligible compared to those initial and final products), one can assume

124

General Approach in Formal Kinetics

Ch. 2

Therefore the concentration of C can be calculated from

which can be integrated to c = ao(l--exp(-;^;0). This solution obtained by the application of the Bodenstein hypothesis can be used in a more generalised form given in Section 2.5.1. It is true that the concentration of B is very small. But it has to be multiplied by the rate constants ^2» which can reach very high values. The validity of the Bodenstein hypothesis has been discussed in great detail (see e.g. [2]). However, it has proven its applicability in some cases. Besides, its correctness has been proven in systems which can be mathematically treated in a closed form. TTiis point can be demonstrated in the example given above. The exact solution is derived by eq. (2.92). Using the eigenvalues r, and rj in the form -(/:,+A:2+*3) i±

1-

4k,k. 1R.3

W

{k,+k^+k,) j

^\/2 ~ •

one obtains ^ f C i fCa

«1

(/:, +^2 +^3)

with the assumption that /:, « ^2 ov (and) ^, « k-^ which has been discussed above. Under these conditions the square-root can be expanded in a series. If only the first term is taken, then

{k.+k^'^kA 2 - (^

r, = •

ib'/Cl fCa

(k^+k^+k,)

) ^^ Kt "T Ky *t* K-i

Methods of Approximation

Ch. 2

125

and

(A:, +ik2 -^kA 1-1 +

2Lk^ (/:, +*2 +^3)

ro = -

KtK,A,3

/._. /C|

I

fC'i "t" fC'i

fC-j

r" fC'

In consequence fCifC'

^0^1

a = -AC| ~r AC2

1^3

(*2+*3)exp(-;i;01 fCo

-/:, exp(-(/:, +^2 +^3)0

V^2 + * 3

Since the amount of the first eigenvalue is larger than the second one, the second exponential function decreases faster than the first one. For this reason a = aexp(-xO is valid even at a shorter time. The approximation to the exact solution becomes better the larger the sum fcj + ^3 is in comparison to fc,. The differential equation makes it obvious that the Bodenstein hypothesis becomes valid just after a certain period of induction. In the model reaction examined above one obtains under the conditions r = 0, a — a^, b = c=^0 the following time-dependencies: d = -kyaQ,

b = k^aQ,

(7 = 0 .

These equations violate the second assumption of the Bodenstein hypothesis (changes in the concentration of intermediates are negligible compared with those of the initial or final products). A further disadvantage of the Bodenstein hypothesis is that one cannot predict how large the difference between the two rate constants has to be to restrict the error to a limited value by using the approximation. For further details see Section 2.5.4. 2.5.2.7 Example of a radical chain reaction The reaction

General Approach in Formal Kinetics

126

A

-

->2B

^1

B + C-

•^D + E

X>j

E + A-

•^D + B

X-i

2B

-^A

XA

-

A -1 0 -1 1

B 2 -1 1 -2

Ch.2 C 0 -1 0 0

D E 0 0 1 1 1 -1 0 0

Xk

k^a *2^C

kjoe

k^b^

is a chain reaction. B and E react in a second and third partial step of reaction with the starting material A and C, respectively. According to the mechanism they are regenerated by this means. The overall reaction is given by A + C-

->2D.

If B and E are unstable short-lived intermediates and the Bodenstein hypothesis can be applied, the following relationships are obtained, since b = 0: JsLX\

Xy

i" X'l ~~' ^XA

~~ ^ »

and since e = 0

Using the second Bodenstein assumption (fc = e = 0), the changes in the degree of advancement amount to i , = x^

and

This means k^a^k^b^y

or rearranged

jCj = ^ 3 .

Ch. 2

Methods of Approximation

127

In consequence the changes in the concentrations of A and B become a = c=^ -k2bc = -kj / - ^ c. Introducing the common degree of advancement ;c = ATJ = JC3 for the overall reaction, then dx

^k,l^dt.

The use of the substitution y-a^-x allows this differential equation to be solved. Introducing the reduced concentrations a

c

, \k.a(.

«0

«0

V *4

the reduced concentration of starting material A depends on time for M>1:

a=(M~l)tg^ arctg / \^ Vw~l exp( Vl -ux) 4-

forM
a=(M-l)exp( Vl - M r)

forM = l:

2

Vw-lr , )

1-vr^ l+ VT^

l^VT^ l-hVr^

v2 »

a=-

In Fig. 2.13 the relationship between a and r is graphed, taking M as the parameter. It turns out that the time dependence of a for «> 1 can be repre-

General Approach in Formal Kinetics

128

H

\—I

1—I

Ch. 2

1—I—I—h

Fig. 2.13. a given as a function of r for the chain reaction, u = CQ/QQ is the parameter.

sented by a trigonometric function. Besides, the reaction is finished after a finite time. This fact gives a hint that the Bodenstein hypothesis can no longer be used for w > 1 at the end of the reaction. 2.5.2.2 The Michaelis-Menten reaction The reaction

A + B-

->C

C

->A + B

C

-»B-fD

A -1 1 ^2 * 0 *3

B -1 I 1

C 1 -1 -1

D 0 k^ab 0 k^c 1 kjc

is an enzymatic model mechanism [21]. The enzyme B reversibly forms a complex C together with the substrate A. The complex C produces the final product D giving the enzyme B back again. According to Briggs and Haldane [22] it is assumed that the total concentration of enzyme b^ is smaller than the concentration UQ of the substrate at the beginning of the reaction. A short period of induction is found. The reason is not any assumption regarding the values of the rate constants but the consequence of the ratio taken for the concentrations. After this period of induction one finds

Ch. 2

Methods of Approximation

b«a,d

and

129

c«a,d.

According to the Bodenstein hypothesis

c«d,d can be assumed. Furthermore c can be taken as zero as c = k^ab — ^2^"~ ^3^ = 0. The law of conservation of mass requires b-hc = bQ. In consequence one obtains

k^a-¥k2 +^3

^ +a

and a constant defined as

According to the hypothesis one can neglect c in the equation including the mass balance a + c +t/ = aQ. Therefore • . k^bQa d = -a = k^c= ^ ^ K-\-a is valid. Integration within the limits a = aQ and r = 0 results in a-a^ + ATln—^-k^b^t.

General Approach in Formal Kinetics

130

Ch.2

Fig. 2.14. a as a function of ur or different values of r = K/OQ .

Taking reduced concentrations a a=-

K r=an

,-^

x^k-a

the final relationship a - l + rlncr = Mr is obtained. This dependence of a and ur is shown in Fig. 2.14 for the parameter r. 2,5.3 Method of quasi-stationary variables The second assumption {z = 0) of the Bodenstein hypothesis as a consequence of the first (z = 0) is reasonable. Just arguing from a mathematical point of view assumption 1 is neither a necessary nor sufficient condition for the validity of assumption 2. For this reason in the case of the method of quasi-stationary variables one avoids assumption 1 and takes for the changing concentration

i,=0

Ch. 2

Methods of Approximation

131

for the quasi-stationary variable Z,. The consequence of this assumption is explained in the following example. For the reaction A +B

>C

>D + B

it is assumed that k^ »k2 and &Q «aQ. Under these conditions the total amount of B has been transformed into C after a short period of induction. As long as the solution contains free A, one finds

In consequence C = 0,

d = ^2^ =" ^2^0

and a-^-c-^-d^aQ,

Therefore one obtains

For this reason the following time dependencies for the change in the concentration of the starting material A and the product D result: d = k2bQt and a = ^Q ~feo"" ^2^0^ • As long as the condition for the quasi stationary state is fulfilled, the reaction follows an order of zero. At time

^2^0

all starting material A is consumed. Then C forms B and D by a first order reaction. At this "post steady state" the following rate laws are valid:

132

General Approach in Formal Kinetics

Ch. 2

fe = 6o(l-exp(-/:2(r--0))'

d=

aQ-boexp{''k2{t-t%

The example proves the validity of the method of quasi-stationary variables under certain conditions. But this fact does not imply the answer to the question on the relative values of rate constants k^ and k2 and concentrations UQ to fco respectively, to be able to describe the progress of the reaction by the approximation correctly. 2.5.4 A criterion A criterion should be mentioned, which has been used successfully in the numerical integration of complex differential equations [23]. Without going into details of new mathematical considerations [24,25] with respect to these problems, this criterion can help to discuss the conditions of assumption 2 in eqs. (2.16) in a quantitative way. At first an intermediate Z is assumed and its concentration is assumed to stay stationary:

a,=y;.(a;,4 .

1,1 = ( ! , . . . , n - l ) .

The above conditions give the complete set of equations. The approximation z = 0 is used. Applying the last algebraic equation, z gives the stationary values Zj. Thus the remaining differential equations can be solved. The approximations become reasonable if intermediate Z reaches the stationary value Zs as long as the other concentrations a^ do not change. Taking these other concentrations a^ as a constant then z can be expanded into a Taylor series: dz r \ 1 d^z / \2 z = Z o + — ( ^ ~ ^ O ) + T T - T ( ^ - ^ O ) +•••• dz 2 oz Taking the first approximation step, ZQ = ^s» ^^d

Ch. 2

Methods of Approximation

133

A z = z - Zjj

the Taylor series becomes z =—Az, dz

since

z, = 0.

Az is the deviation of the concentration of the intermediate from the stationary value. For this reason Az is an expression for the quality of the approximation, dz/dz can be obtained from the differential equation for z. As a first guess, z will be taken equal to Zj, which can be obtained by the differentiation of z with respect to time. The relative error is given by Az = Az/z,

(2.97)

and amounts to Az =

Zs{dz/dz)

If other intermediates are assumed to be also quasi-stationary, they have to be treated in an equivalent way. For the hypothesis z, = 0

and Z2 = 0

one finds z, =T^Az, +T^Az2, dZi

Z2 = T ^ A z , +T^Az2.

dZ2

dZi

dZ2

Both equations are to be solved with respect to Az, and Az2. The relative errors can be obtained by use of eq. (2.97). Example 2.27 The hypothesis 6 = 0, c = 0 is going to be applied to the reaction A

>B

>C

>D,

134

General Approach in Formal Kinetics

The rate equations are given by a = —k^Cy

b = k^a - ^2^'

Therefore

^2

^3

^3

in consequence k^a

= -

k}a

The differential1 equations give

db

"

dc _ -k. db'

dc

7

dc

For this reason the set of equations is given by — • — = A:2 Afe,

^— = ^2 A& - ^3 Ac. ^^2

In consequence one finds k^

k^

k^

/Co

fCy

fC^/C-i

The relative error amounts to

*'•

Ch. 2

Ch. 2

I i cd

a

1 i 2

I §

1 U

J ^

- 3

H osi

\^

^

"-rh?

I

<3

^-1 .

t t

-^1

CO

>K

I II

I

^

ri

Methods of Approximation

«3

+

+

-iirn?

+

1CO

t + t <

a| o

^

B3

ffl t <

•§

135

^h

136

General Approach in Formal Kinetics

Ch. 2

In Ref. [27] the equivalent result was obtained by explicit calculation. The results for some consecutive reactions are summarised in brief in the Table 2.9. _ It turns out that At always decreases during the second reaction and becomes infinity AZ?

>oo

at the end of the third reaction. This fact exactly represents the form of the ^diagrams of such systems. If the given criterion is applied to the chain reaction of Section 2.5.2.1, some very complex expressions are obtained. They make it obvious that the Bodenstein hypothesis fails at the end of the reaction for CQ >flto• ^^^ ^^is reason the chosen approximate solution of the system can no longer describe the progress of the reaction for large turnovers. For this reason the reaction does not come to an end in a finite time. 2.5,5 The approximation of the equilibrium If an equilibrium precedes or succeeds any reaction or intersects a sequence of reaction steps, the relationships can be simplified, since the rate constants of forward and backward reactions are larger than those of the other partial reaction steps. This fact is demonstrated for the reaction A ^ B —>C. In the case where A and B are always in equilibrium, a^Kb

with

K^^

is valid. Furthermore the mass balance is given by a +fo+ c = aQ+fcQ and the single kinetic equation

Ch. 2

Methods of Approximation

137

can describe the reaction. By combination of these equations one finds

or for the concentration of the component B

Introducing this result in the kinetic equation, one finds .-

^3

K-hl

-(«o+^o~c)»

which will yield after integration \A

r

c = K"+-^o) l - e x p — JJ

The equation with mass balance results a4-fe + (: = 0

or b = -

_

^3

-b.

The second equation gives, after integration,

and by use of the first equation a = Kb = aQ exp

(K + l)

In contrast to a treatment according to the Bodenstein hypothesis the changes b of concentration with time are finite. That means the factor

138

General Approach in Formal Kinetics

Ch. 2

in the exponent has to be substituted by the factor 1 3 Kt

r K-j

In the case where an equilibrium reaction intersects a consecutive reaction as in the system A —» B ^ C -^ D, one finds b = Kc,

b = Kc

by the equilibrium approximation discussed. The respective kinetic equations read

The equation with mass balance yields a + b + c-\-d = aQ or ti + (Ar + l)c +rf= aoi therefore a + (/i:-fl)c + d = 0. The kinetic equation for the first step is a = aQexp(-^,0

The differential equation of the Product C results in

Methods of Approximation

Ch. 2

139

Thus, by integration of this linear differential equation beginning at c = 0, r = 0, one finds

c=-

^1^0

ik4-it,(if + l)

exp(~^,0-exp -

k.

(^+1)))

and b=

Kkjap exp(-A:,0-exp k^-k^{K + l) I

(^ + 1) ))

as well as d = a, l - e x p ( - ^ , 0 -

K{K + \) exp(-A:,0-expl- ^ ^ k^-ktiK + i)

f )))

The examples demonstrate that the number of linear independent partial steps of a reaction is reduced by one, if the equilibrium approximation can be applied to isomeric equilibria. This statement is not valid for more complex equilibria. Treating the reaction 2A ^ B -» C according to the equilibrium approximation, one finds b^Ka^,

b = 2Kaa,

c = k^b

and

Therefore a + 2& + 2c = a(l + 4Ka) + 2k^b = 0 or a{l + 4Ka) = 2k^Ka\

140

General Approach in Formal Kinetics

Integration beginning at a = AQ, / = 0 yields - - + — + 4Kln— = -2k3Kt. a

OQ

OQ

Using reduced concentrations

the following relationship is obtained:

J_(l-l).2.na=. 2X The concentration of C is given by

or

The final concentration of C (c^) is reached, if a becomes 0. It is

c^=Y + %aoDefining c

the following relationship results:

Ch. 2

Methods of Approximation

Ch. 2

141

Fig. 2.15. y= cjc^ as function of r for the reaction 2A ^ B —> C (equilibrium approximation). X- KQQ is used as a parameter.

y=-

1

-(l-a)-f-

1 + 2;^^

'

2X ii-a^y \ + 2X

It is given in Fig. 2.15 as a function of r. 2,5.6 Reduction of the number of linear independent steps of a reaction If either the Bodenstein hypothesis can be applied to a reaction system or one of the steps is an isomeric equilibrium, the number of linear independent steps of reaction is reduced. The concentration of an unstable intermediate is according to the Bodenstein hypothesis with z = 0:

^=x^^^*=^k=\

This equation can be solved with respect to ^c^^, if Vj^ ^ 0:

(2.98)

142

General Approach in Formal Kinetics

Ch. 2

For any component A, the general relationship is valid: Aa, = ^uXi+-- • + Vri^r k-l

Aai

=

^^'1^1

X^:.

iXf

<:'=*+!

with

K-i = n'i-V«

(2.99)

The following rules for the reduction of the stoichiometric scheme are valid. In principle the kth line is cancelled and the stoichiometric coefficients v^.,. are replaced by vJJ,^ according to eq. (2.99). According to eq. (2.99) the following facts can be stated: (1) v^,^ becomes zero for all k\

(2) v;,=v,,forv^.=0. (3) All v^,. stay unchanged if v^,^ = 0. (4) When v^ and v^.^ have the same sign, the multiplied columns are to be subtracted from each other. If both stoichiometric coefficients have different sign, the columns have to be added. (5) The i:^. stay unchanged. Example 2.28 B and E are intermediates in the radical chain reaction treated in Section 2.5.2.1. The Bodenstein hypothesis can be applied. By use of e=0 one obtains

Substitution of x^ by X2 does not change the first and fourth row of the scheme. Since

Ch.2

Methods of Approximation

143

'IE '3E

is valid, the scheme is reduced to

^1

X2

JC4

A -1

B 2

-1 1

0 -2

C 0 -1 0

D E Xk 0 0 ^,a 2 0 fcj^C 0 0 *4&^

Since 6 = 0, one obtains X, - X4 = 0 .

If the row which contains x^ is cancelled, and one uses eq. (2.98), the first row disappears and the second stays unchanged. This result correlates to the overall reaction proceeding stoichiometrically. The examples in this chapter demonstrate that thermal and pseudo-linear photochemical reactions can be treated equivalently. The approach is the same in both cases: - extract as much information as possible by use of graphical representations, - set up the rate equations using matrix notation, and - try to simplify the relationships taking approximations into consideration that are valid because of kinetic data. As demonstrated the Jacobi matrices are more complex in the case of photokinetics, since instead of the rate constant the quantum yield together with the photokinetic factor and the absorption coefficient have to be taken. Furthermore all the photochemical steps are accompanied by photophysical steps. Tliese problems are dealt with in the next chapter.

Chapter 3

Special Considerations for Simple and Complex Photochemical Reactions

3.1 RELATIONSHIP BETWEEN QUANTUM YIELD AND MECHANISM Based on the fundamental considerations in Sections 1.3 and 1.4, the principle of quantum yield was introduced in Section 2.1.2 to allow a treatment of photochemical reactions in a way comparable to thermal reactions. The difference from thermal reactions has been demonstrated by taking account of the photophysical steps in Sections 2.1.3.3 and 2.1.4.3. These have to be considered in detail to find out whether the partial photochemical quantum yield depends on the intensity of the irradiation source or even on concentrations. Furthermore the definitions derived in Chapter 2.1 and summarised in Table 2.2 are used. In particular the definitions for the degrees of advancement for partial steps in general x*, for photophysical steps in special jc/, and for linearly independent steps jc, of the reaction procedure have to be remembered. In the following in addition to the examples in the sections mentioned above and the derivations in Appendix 6.2 many of the possible mechanisms are reviewed and classified in brief. To avoid repetition, in some cases details of the derivations are found in Appendix 6.6. However, since the formulae derived depend on the chosen numbering of the photophysical steps and the reaction scheme, in most cases both have to be set up to make the relationships obtained obvious. 3.LI Simple uniform photoreactions In Section 2.1.3.3 the details for two simple uniform photoreactions are derived [10]. These are summarised in Table 3.1.

146

Simple and Complex Photochemical Reactions

Ch. 3

TABLE 3.1 Mechanisms and photochemical quantum yields of the simple photoisomerisation in dependence on the photophysical pathway Mechanism

Rate law and quantum yield

A

^ _, _^>^/

^

^pA ^

^ = ^^A j

^

^A ^ ^4 '^6 "^ (^2+^3+^4)-(^5+^6)

> B via singlet state

A — 5 ^ B via triplet state

^

^5

3,1.1.1 Quenched photoisomerisations The two examples of a photoisomerisation given here are the simplest cases of photoreactions. As demonstrated above and in Section 2.1.3.3 the photophysical pathway can proceed via the singlet or the triplet state. During these isomerisation reactions the intermediates A' or A" are of great interest. Besides the given mechanism the stationary concentration of the excited photophysical intermediate is influenced in contact with a partner B. Energy transfer to an acceptor molecule B opens a new reaction channel, but does not form a new reaction product itself. Looking at the photoisomerisation, this energy transfer will reduce the concentration of the excited state. By this step the photoreaction is "desensitised" or quenched. This quenching can either happen in the singlet state A' or at the triplet state A". Both mechanisms are discussed in the following examples.

Example 3.1: Quenching of the photoisomerisation via the singlet state Using the mechanism A—!^^C the following photophysical steps and the reaction scheme can be derived:

Relationship Between Quantum Yield and Mechanism

Ch. 3

A + hv A'

>A'

<

>A

A' + B

X2

>B' + A

B'

>B

A'

>C

^3

x'.

4

A -1 1

A' 1 -1

1 0 0

-1 0 -1

B 0 0 -1 1 0

147

B' C X', 0 0 /A 0 0 k2a' 1 0 k^a'b' - 1 0 k^b' 0 1 k^a'

These steps result in the following rates and quantum yields: - a = c = y) lyv.


k2 + kjb + k^

(3.1)

This mechanism can be compared with a reaction procedure, in which the quencher is not present. A definition of two different quantum yields, one in the presence of quenching molecules B ((p^) and one in the absence of the quenching molecules (^0 )» gives the ratio
^2 +^5

1

^2 + k^b + ^5 ^2 "^^5

This equation defines the efficiency of the quencher. The value by2 is defined as

equivalent to that concentration of quencher B which reduces the photochemical quantum yield to one half. This B concentration can be calculated to

by2-io-^mo\r\ For this calculation some assumptions have been made:

Simple and Complex Photochemical Reactions

148

-

Ch.3

the bimolecular rate constant k^ is diffusion controlled, its value is taken as 10^^ 1 mol"^ s~S and A' is a singlet state, its average lifetime in general is assumed to be /„ =1/^2 = 1 0 - 8 .

Thus the rate constant of the bimolecular step 3 is kj ~ 10'" mol 1-' S-' and the deactivation rate constants of the singlet states are taken as 8„-l Jt,-it, ~10»s

Example 3.2: Quenching of the photoisomerisation via the triplet state A more complex mechanism is obtained, if the photoisomerisation takes place via a triplet state A" which can be quenched. According to the reaction steps and the reaction scheme

A + /IV

»A'

<

A'

>A

^2

A'

>A"

X',

A"

>A

X',

A" + B

>B" + A

B"

>B

A"

>C

^5

< X'-,

A -1 1 0 1 1 0 0

A' 1 -1 -1 0 0 0 0

A" 0 0 1 -1 -1 0 -1

B B" 0 0 0 0 0 0 0 0 -1 1 1 -1 0 0

C xi 0 IA 0 kja' 0 kja' 0 k^a" 0 ksa"b 0 Kb" 1 kjo"

the following expressions are obtained: -a = c = (p / A .

y,A =

3 7

{k2 + k^){k^ + k^b-¥ k^)

(3.2)

with the ratio between the quantum yield of the normal isomerisation and the quenched one

Ch. 3

Relationship Between Quantum Yield and Mechanism


k^ + k^b + k^ '

^^^

149

ifcj

The concentration by2 can be calculated to be 1(H mol 1-^ In the case of the participation of the triplet excited state its lifetime is considered to be 10"^ s. Therefore the deactivation constants become ^4 ^ kj -iC^s"*. As above the diffusion controlled bimolecular step has a rate constant of ^5 ~ 10*^1 mol"* s'"^ The defined concentration by2 can become so small that any impurity can inhibit the planned reaction. This fact explains some non-conformity in the photochemical literature. Some other simple uniform mechanisms are the basis of photochemical reactions. These are photoadditions, the photoreduction and some sensitised photoreactions which are discussed in the following. 3.1.1.2 Photoadditions As in the case of photoisomerisations, the reaction can proceed either via the singlet or the triplet intermediate. The component B is part of the reaction in contrast to the sensitisation mechanism discussed above. In addition both pathways via the singlet or triplet state can be quenched. The photophysical steps are summarised in Table 3.2 and the reaction scheme is derived in Appendix 6.6.1.1. In the following examples this information is used to derive the time laws and the quantum yields for 4 types of photoaddition reactions according to either the normal addition reaction A +B

^ )C,

or the case in which a quencher is involved A + C —^^^ D and quencher B.

Example 3.3: Photoaddition via the singlet state The photoaddition via the singlet state results by application of the Bodenstein hypothesis to the photophysical intermediates in the rate law

150

Simple and Complex Photochemical Reactions

Ch. 3

TABLE 3.2 Reaction steps for the different photoaddition mechanisms Via singlet

Via triplet

without quenching 1: A + /IV >A'

without quenching 1: A + /iv >A'

2: A '

2: A'

>A

3: A '

•A"

4: A"

>A

>A

3: A ' + B

>C

1: A + hy 2: A '

>A' >A

3: A' + B 4: B'

>B' + A >B

5: A' + C

>D

5: A" + B

>C

1: A + /iv

•A'

2: A '

>A

3: A '

^A"

4: A "

^A

5: A" + B 6: B " 7: A" + C

.

.

.

k,b

>B" + A >B •D

,

and for the photochemical quantum yield .A _

k^b ^2 + Ic^b

In this photoaddition the quantum yield depends on the concentration of the reactant B. Therefore this quantum yield can be used to estimate the minimal concentration which yields a reasonable turnover at optimal conditions. The dependence of the quantum yield on the concentration of compound B turns out to be

Ch. 3

Relationship Between Quantum Yield and Mechanism A
151

b =•

if the above equation is rearranged. In consequence the concentration of B requires an order of magnitude of 10-2 niol 1-^ to yield a reasonable turnover using the assumptions for the rate constants as above. Accordingly the quantum yield is determined to be A

b

fo + lO" Therefore any type of excited state of A having a shorter lifetime as 10-^ s cannot follow the pathway of photoaddition starting at this state. Example 3.4: Photoaddition via triplet state In the case when the photoaddition proceeds via the triplet state, the rate law differs from that in the above example -.a = -.fe = c = —^^

^ ^ / A .

(3.3)

The photoaddition favours the pathway via A" in competition to A' because of the longer lifetime of the triplet state. An appropriate consideration similar to that in Example 3.3 demonstrates that an average lifetime for triplet A" in the order of magnitude of 1(H s can yield a reasonable turnover, if the concentration of B is in the order of magnitude of 10-^mol l-^ In the following examples the results given above are compared to mechanisms in which either the singlet intermediate A' or the triplet intermediate A" are quenched by an additional species reducing the life times of these excited states and reducing their concentration. In both cases quenching by a molecule B is assumed. Example 3.5: Photoaddition with quenching of the singlet intermediate Assuming the quenching of the excited molecule in the singlet state, one finds the relationships

152

Simple and Complex Photochemical Reactions

Ch. 3

k^c ^2 + k^b + k^c The ratio of non-quenched to quenched quantum yield is obtained as
k2+

k^c

_

1

L

_ ^2 "*• ^ 5 ^

^2 "^ ^5^

Example 3.6: Photoaddition with quenching of the triplet intermediate Taking into account the quenching of A" the following rates, quantum yields, and the concentration of quencher, which reduces the quantum yield to one half, are obtained:

-a = - c = ^ = ^^/;„

^^=7

1^^ k^+k^b-^k.c" ^ ,

^P^

M^'

r

(3.4)

^'^'~Ts

^4 + k-jC

In both the photoaddition reactions the quantum yield depends on the concentrations of the second reactant C as well as of the quencher B. 3J ,1,3 Photoreduction The next example uses a mechanism for photoreductions, in which radicals are formed as intermediates. A typical example is the reaction of benzophenone with isopropanol. Example 3.7: Photoreduction with radical intermediates The photoreduction 2A + B

>C + D

is chosen as an example. It forms radical intermediates according to

Relationship Between Quantum Yield and Mechanism

Ch. 3

A + hv A'

>A' >A

Xy

A' + B 2E

x{

>E + F

3

<

>C

F+A

X5

>E + D

A -1 1 0 0 -1

A' B 1 0 -1 0 -1 -1 0 0 0 0

c 0 0 0 1

0

D E 0 0 0 0 0 1 0 -2 1 1

153

F 1 ^i 0 /A 0 k2a' 1 k^a'b 0 k^e^ - 1 [kjaf

In this specific case the symbols represent: A, benzophenone; B, isopropanol; C, benzpinacol; D, acetone; E, ketylradical; F, alkoxylradical. If the Bodenstein hypothesis can be applied to the radical intermediates one obtains Cl ^ X* ~~ X2 "" X'j -— U,

C ^ X-j ~" ^X/^ "T X^ —- U,

J —- X-i

Xc

therefore Xt

"~ Xy

'~~ X-y -~

XA

~~" Xe

~~ X •

Since the unstable intermediates appear in negligible concentrations, the following relationship is valid

Inserting the values listed in the last column of the above scheme into the variables jc^^. the stationary concentrations of the intermediates are given by 2

k^a^b

a =-k2 + k^b

K4

Consequently ./__._ f Xi

"~" A — KA C

2 _^

k-^b

—- —————— y

^

k^a'b k^a

154

Simple and Complex Photochemical Reactions

Ch. 3

and the quantum yield is determined to be ^

=

^^^—

(3.5)

and one finds a = -2JC = -2(p^I^,

c = i = (p^I^.

In the case of disproportion of the radical F instead of the impact step 5 (in the scheme) above using a 6th step 2F -> B + D with k^, the quantum yield becomes

^

2k^^k^h

If the radicals deactivate according to step 5 as well as step 6, the relationship becomes more complex A_ 1__M__L M V>^= ^ +-

I

^

2/:3fc/A

k^ik^^-k^b)

k^a

2k^j

In this case the quantum yield depends on the concentrations a and b as well as on the amount of light absorbed by A. 3,1,1.4 Sensitised photoreactions In the following, four examples of different types of sensitised photoreactions are discussed first taking into account energy transfer. In all these cases lifetimes and rate constants of the intermediate steps, respectively, determine the concentrations necessary to obtain a noticeable turnover. The different parameters for these necessary concentrations are listed for these mechanisms in Table 3.3. Details are discussed in Appendix 6.6.1.2 for these four examples. A further example of a sensitised photoreaction is the case of an aromatic ketone (A), which acts as a chemical sensitiser in the photooxidation by molecular oxygen (C) of a hydrocarbon (B) to form a peroxide R-OOH (D)

Ch. 3

o -a

I

o x: ex

O

x: o

H CO

A

o

T CO o

O

B

m -a<

4-»

o "o o

2

r<

•^

s. ^

II •u II •-Si

I

00

:^

X

-.

Q

6

o

o

i^ «-^

1-H

II

-f + ^ •^ ^ ^ < ^^

II

.^

ir ^

II

Tf j ^

< II •o II •-c> I o" II •a

u

>n

-S 6 B o

1

I

^

O

1

^ 1 ^ »

-Ci

-^

+

X

+

•Ad

X

oo

•^ ^

+

U r•JX

5 <:

o %

'2 .••

g2

t

B N

"V

o

"o

CO

T

-Ci

f^

1

O

6 o

-n

^ • ^

*^

II

<5 ^

ltd

< -- "^

<

U

II •o 1

•-Ci

II

II

<

N

"O

T

'o

B *s ^

11

+ ^ < ^^

II

X

+

s<

*—• u

tt ' o

u +

• ^1

II •o II

^ ^ ^ <

^"^

NO

Relationship Between Quantum Yield and Mechanism

B

I I u PQ

156

Simple and Complex Photochemical Reactions

156

Ch. 3

and radical intermediates. E is a ketyl-radical, F an alkyl-radical R*, and G a peroxide-radical R - 0 0 \ Example 3.12: Chemically sensitised photoreaction The mechanism B + C-

hv

4D

contains the following steps of the reaction and results in the reaction scheme

^A'

A + AvA' A' + B

>A >E + F

F+C

>G

G+E

>A + D

A x[ - 1 A 1 X',

A A

0 0 1

A'

B

C

1 -1 -1 0 0

0 0 -1 0 0

0 0 0 -1 0

E

F

G

0 0 0 1 0 0 1 -1

0 0 1 -1 0

0 0

D 0 0

0 1 -1

K /A

^a' k^a'b kjc *5«^

Applying the Bodenstein hypothesis to A' and to the radical intermediates E, F, and G, one obtains Xt — Xy "^ X-r ^ U,

X-y *"" X^ - - U,

X-i ~~ Xc -— Uj

X^ ~ X^ ^ U,

or •^1 " X2 ^ X-^ = X^ =

X^,

Inserting the values for i^ the following relationship can be found:

Therefore

(i = JC5 = k^eg = k^a'b.

Ch. 3

Relationship Between Quantum Yield and Mechanism

157

Since a =-

/Co "t" K-iO

the result is A (p^ =

-^b^-c^d^ip^I^,

kob ^

(3.9)

^2 + kjb

In this case the quantum yield depends on the concentration of B and is independent of the concentration of C. Photoisomerisations can also be sensitised by radicals. A typical example is the photoisomerisation from cis- (B) to fmn^-stilbene (C) by iodine (A), forming as intermediates an atomic iodine and the radical J I P h — C — C — Ph. I I H H

Mechanism and details are given in the following example. Example 3.13: Sensitised photoisomerisation The photoisomerisation sensitised by radicals according to B

^ )C

gives the scheme

>2D

A + hv D + B-

-»E

E

>B + D

E

>C + D

2D-

->A

xl X2

x; x'. x's

A -1

B 0

0 0 0 1

-1 1 0 0

D 0 2 0 -1 0 1 1 1 0 -2

c

E 0 1 -1 -1 0

x'.

rU k^bd k^e k^e k,d'

Simple and Complex Photochemical Reactions

158

Ch. 3

The first line of the scheme notes if = r/^^. This term takes into account that the radicals D which are produced in the photochemical primary process stay in a solvent cage and can recombine to form A. For this reason the fraction of quanta causing photoreaction becomes r < 1. Applying the Bodenstein hypothesis to all intermediates the following relationships can be derived: e= therefore (3.10) In consequence the quantum yield is reciprocally proportional to the square root of the amount of light absorbed /A. 3A. 1.5 Radical photoreactions The products can be formed either by the reaction of a radical (produced within the photo-process) with the initial product or in a radical chain reaction. These two possible mechanisms are discussed in the following examples. Example S-M: Interaction with intermediate radicals The reaction 2A-

kv

•>B + C

can be represented by A A + /IV

>D + E

D+A

>B + E

X2

E+A

>C + D

3

D+E

>A

E

i*

-1

B C D 0 0 1 1 0 -1

1 1

riA kjod

-1 1

0 0

1

-1 -1

k^ae k^de

x{ - 1

<

0

1 -1

Relationship Between Quantum Yield and Mechanism

Ch. 3

159

where B is formed by interaction of A with radical D produced by photolysis, and C with radical E. If one applies the Bodenstein hypothesis to D and E, the following equations are obtained: Xt

""* X2

• X'y ""* XA

"~ V/,

Xi

"T X2

~" X-y ~- XA

^~ v/

or X | — JC4,

X-J

""• X"!

Thus

\M4 and in consequence ci=x^=b

=c=— =a / ^ ^

(3.11)

Example 3-15: Radical chain reaction The radical chain-reaction A + B-

Av

->2C

is given by

A + hv

>2D

<

D+B

>C + E

^2

E +A

>C + D

^3

2D

>A

^;

A -1 0

B 0 -1

C D 0 2 1 -1

E 0 1

k2bd

-1 1

0

1

0

0

-1 0

k^ae k^d^

1 -2

K h

A represents chlorine, B hydrogen, C hydrochloric acid, D chlorine at-

160

Simple and Complex Photochemical Reactions

Ch. 3

oms and E hydrogen atoms. When the Bodenstein hypothesis is applicable to all the intermediates the result is 2ji;, - ^2 + -^3 - 2^4 = 0,

^2-^3=0

Thus

and

with the quantum yield of (3.12) All the gross reactions given as examples in this section take place stoichiometrically. Under these conditions the turnover can be expressed by a quantum yield for all the reactants which refers to the total reaction. 3,LL6 Parallelphotoreactions For the parallel photoreactions A—^^B,

A—!^^C

different mechanisms can be assumed: - both the reactions can proceed via the singlet excited A', - both the reactions can proceed via the triplet excited A'', and finally - the formation of the two products B and C can proceed via different excited states, one via the singlet, the other via the triplet. All these reactions do not proceed stoichiometrically. However, all of them are uniform, that means the partial steps linearly depend on each other.

Relationship Between Quantum Yield and Mechanism

Ch. 3

161

The relationship eq. (2.12) cannot be used under these conditions. Each of both the partial steps of the photochemical reactions require their own quantum yield. Example 3.16: Parallel photoreaction via the singlet state The reactions steps and the reaction scheme are given by: A x[ - 1 1 X2 x; 0 0 <

^A'

A + hv A'

>A

A'

>B

A'

>C

A' B 1 0 -1 0 -1 1 -1 0

C K 0 h 0 k2a' 0 k^a' 1 k^a'

Taking into account the linear dependencies of photophysical processes, the overall mechanism can be represented by the gross reactions * B C 1 0
A * -1

A—!2^B

*

A—5:L^C

Xy

-1

The quantum yields are defined by

^N-^^ Kj

' K-i "T KA

92 =

(3.13) Ky

V K'i

I

KA

using the Bodenstein hypothesis and the changes in concentration as b=

/^,

Jl Ky "T" AC-j "r K^

c=

'^

/A.

K2 ' ACj + K^

Therefore one can write b =
c^ip^If,

and a = -(v>f+^2 )'A-

162

Simple and Complex Photochemical Reactions

Ch. 3

Furthermore one finds dc _
= - ^ _= -^4^

or

c-'^b.

The isomerisation to B and C via the triplet state leads to similar consequences: A >A'

A + hv

x[

A'

>A

A'

>A'

^3

A"

>A

x:

A"

>B

xl

A"

>C

X2

A

-1 1 0 1 0 0

A' A" B c i ; 1 0 0 0 /A - 1 0 0 0 k^a' -1 1 0 0 k^a' 0 - 1 0 0 k^a" 0 - 1 1 0 k^a" 0 - 1 0 1 Ka"

As above one obtains two different quantum yields in two linear dependent steps of a reaction which form a uniform reaction with the following dependencies of concentrations on time: b=

'h'^5

^^6

/A=^./A. I

_ .«A

^A=^2^A.

(3.14)

and result in the relationship b = -^c. k. The third example is the formation of B via singlet A' and C via triplet A". This reaction is uniform also:

Relationship Between Quantum Yield and Mechanism

Ch.3

»A'

A + hvA'

>A

A'

>A"

A"-

-^A

A"-

^B

A"-

-^C

A -1 < 1 Xy x; 0 1 X', x's 0 0 <

A' 1 -1 -1 0 -1 0

A" 0 0 1 -1 0 -1

B 0 0 0 0 1

_2J

163

C 1 xi 0 0 k2a' 0 k,a' 0 k^a" 0 k^a' 1 k,a"

u

One finds 'A^'PiU' c=

{k^+k^){k2+k^+ki)

/A=^2/A'

(3.15)

and kjk^ c = k,(k^+k^) b.

These examples of different simple photoreactions demonstrate that the rate laws will be similar and simple in many cases. However, the photochemical quantum yields depend on the photophysical processes which govern the reaction procedure. Therefore they differ and depend on the lifetimes of the excited states and the deactivation mechanism. 3,1.2 Simple non-uniform photoreactions In this type of reaction a single reactant which is excited by absorption of light is able to start more than one gross reaction. The requirement is that the Bodenstein hypothesis can be applied to all intermediates. Whereas in the previous section in the case of the treated parallel photoreactions the gross

Simple and Complex Photochemical Reactions

164

Ch. 3

reactions are linearly dependent, in this section examples are given in which the mechanism does not allow a reduction of the reaction scheme to one single linear independent step of reaction to be uniform. As demonstrated this section handles parallel reactions, also. However, the compound starting the photoreaction either reacts with different partners within the parallel steps or its excited states are quenched according to different mechanisms. 3.L2J Parallel photoreaction including other reactants In the following examples parallel photoreactions starting from the same initial product but including different reactants are treated. Example 3-17: One of the parallel photoreactions includes a photoaddition The photoreaction hv

^B,

A + C-

/iv

^D

assumes the following reaction steps and results in the reaction scheme A x; - 1 4 1 x; 0 0 X'.

^A'

A + hv A'

>A

A'

>B -^D

A' + C

A' 1 -1 -1 -1

B c 0 0 0 0 1 0 0 -1

D K 0 /A 0 ifeja' 0 k,a' 1 k^a'c

which can be reduced to the two gross reactions

X,

A -1

B 1

C 0

X2

-1

0

-1

D Xk 0
Taking the Bodenstein hypothesis, one obtains a =•Kij

"• fC-y ~r

KAC

Ch. 3

Relationship Between Quantum Yield and Mechanism

165

and

i,=^=

VA

^^A^^^ (3.16) ^4^'A

I

A »

^2 = - C = ^ = T — T - T ^ = ^ 2 ' A . Ky "T ACT "T" AC^C

Therefore the change in the concentration of reactant A

depends on two reactions with two partial photochemical quantum yields. One can find a relationship between the concentrations b and d as well as between the degrees of advancement of the two linear independent reactions jc, and X2 respectively: dX'j

KAC

KA

/

\

—^ = —^ = ~^(co -•^2)Using reduced degrees of advancement according to Section 2.4.2 the following definitions are taken:

GQ

a^

^3

QQ

and the relationship between these degrees of advancement is obtained as e2=«(l-exp(~xe,)).

(3.17)

This equation is represented by Fig. 3.1 which has to be seen in comparison to that in Section 2.4.2.1 as Fig. 2.4. Similar relationships are obtained, if B and D are formed via the triplet A". However, different equations arise if B is formed from the singlet A' and D from the triplet A'' according to

Simple and Complex Photochemical Reactions

166

Ch.3

Fig. 3.1. ^-diagram according to eq. (3.17) for u = CfJoQ = 0.9. % is the chosen parameter.

A + /IV

>A'

xi

A

A'

A"

B

c

D

1

0

0

0

0

u

x[

-1

A'

>A

A

1

-1

0

0

0

0

kjo'

A'

>A"

X',

0

-1

1

0

0

0

kja'

A'

>B

A A

0

-1

0

1

0

0

k^a'

1

0

-1

0

0

0

ksa"

x'e

0

0

-1

0

-1

1

k,a"c

A" A" + C

>A >D

This mechanism results in

Ch. 3

Relationship Between Quantum Yield and Mechanism

-C = d =

^3^6 ^A

167

Aj

^

(3.18)

^ = -K+^2)/A

and the following differential equation after reduction to two linear independent steps

dx,

k^ {k^ 4- k^c)

k^ (*5 + k^ (co - ^2))'

which describes the relationship between b and d as well as ;c, and Xj respectively. Integration starting at the limit X, = ^2 = 0

and use of the definitions &

t

4

AC3

5

^6^0

results in l,=X,?2-Z.%2ln[l-^]

(3.19)

which is represented in Fig. 3.2. In the next example both steps of the photochemical reaction include a partner, both being photoaddition reactions. In the case where in both of the reactions the same initial product starts the photoreaction and reacts with the same partner via the same type of excited state, the reactions become uniform, i.e. just one linear independent step is observed. This mechanism can be compared with a thermal reaction discussed in Section 2.4.1.1.

Simple and Complex Photochemical Reactions

168 ^ ^

I

1

1

1

1

I

I

1

Ch. 3 h-

•1 via singlet:

A

^ B hv

via triplet: A + C

•

D

0.5 4

Fig. 3.2. ^-diagram for parallel photoreactions according to eq. (3.19) for^i = 1, w = 1.5. X2 is the chosen parameter.

Example 3-18: Two parallel photoaddition reactions using the same partner via different excited states The photoreaction

A + B-

->C

A -1

A + B-

->D

* -1

B -1

C D 1 0

-1

0

1

.* Xk
is uniform, if C and D are formed either via A' or A". But, if the following scheme is valid, in which the excited state for the reaction with B is different for the two steps

Relationship Between Quantum Yield and Mechanism

Ch. 3

^A'

A + hv A'

xi

>A

X'j

x;

A -1 1

A'

>A'

X',

A"

>A

A

0 0 1

<

0

»C

A' + B

A" + B

•^D

A' 1 -1 -1 -1

A" 0 0 0 1

0 0

-1 -1

B |C 0 0 0 0 0 1 0 0 0 0 -1 0

D 0 0 0 0 0

169

1 X', /A

^2«'

k^a'h k^a'

k^a" 1 k^a"b

then a = k2 + k^b + ^4

^/.^ M ' ^5 + k^b

is obtained. Under these conditions the mechanism is no longer uniform. In the reaction scheme above the * have to be omitted, both the reactions being linearly independent forming two such steps. Therefore the rates are Kbl 3^M = ^f/A» ^2 + k'^b + k^

c =-

d=

(A:2 •¥k^b^k^)(k^

(3.20)

^k^b)

In consequence the following relationship is valid:

—

dxi

-^^{ks-^Kb). *4*6

This equation can be rearranged by the following definitions:

5,=^

t

- -^2

S2 - T ^ »

3 5 AC4AC6

Simple and Complex Photochemical Reactions

170

Ch. 3

to

d^2

^Xi+Xii^-^i'^i)'

By integration of this differential equation starting at the limit ^j = ^2 = 0 one finds (3.21) X2

\ t the end of the reaction (&„ < ag) one flnds

5,~=l-^2-.

^2~= I

Xi

H 1-\

Xl

Xi+;f2 + i.

I—I—I— 0.1

hv via singlet: A + B 0.5

•

B

•

D

hv via triplet:

A +B

1

Fig. 3.3. ^-diagram by use of eq. (3.21). ^2 is chosen as the parameter. The calculation is done for;^i = 1. That means that;if i can be neglected with respect to X2 because of its size.

Ch. 3

Relationship Between Quantum Yield and Meclianism

171

At a < &() one can define u = aQ/ b^, therefore ^ 1 - = " - ? 2~»

S2~ ~

_i_

(

«;t2

In 1 - -

X2

Xl+X2+^

The final slope at f —> oo amounts to d^^ d^2 This typical ^-diagram is given for different values of X2 ^^ X\-^ Fig. 3.3. Example 3.19: Two parallel photoaddition partners via different excited states In the next mechanism

A + B-

kv

A + D-

hv

^C

^I

^E

"^2

A -1 -1

B -1 0

C D 1 0 0 -1

'"

reactions using different

E Xk 0
B and D form an addition reaction with the excited states A' or A", respectively, in two linear independent steps of the reaction. In this case the correlation between the degrees of advancement is equivalent to that in the case of analogous thermal reactions treated in Section 2.4.2.1 as independent parallel reactions. If the reaction takes place according to this mechanism, the relationship will differ from that in the example given above.

^A'

A + Ziv

JC,'

A'

>A

Xj

A'

>A

Xj

A' + BA" A'' + D

->C

JC4

x'f

>A >E

4

A -1 1 0

A' 1 -1 -1

A" 0 0 1

B 0 0 0

c

0 1 0

-1 0 0

0 -1 -1

-I 0 0

1 0 0 0 0 -1

0 0 0

D 0 0 0

E xi 0 JK 0 ^2^' 0 k^a' 0 k^a'b 0 k^a" 1 k^a"d

172

Simple and Complex Photochemical Reactions

Ch.3

The following rate laws are obtained: -b = c = •

——

k2+lc^+

k^b

=
k^hdlK =
(3.22)

-d = e = -

The differential equation becomes dx^ ^ k^b{ks + kf,d) _k^+ k^dp -k^x^ ^ ^ _^ \ dx^

kji^d

*3*6K--^2)

using the definitions Si —"r"»

_ ^4^S

?2

da

Xi =

kjk^

X2

_Kd
Integration starting at ^, = ^j = ^ results in | , = l - ( l - | , f exp(-;f2l2) with the final values ?i~=^2- = l

at ao>(*o+'^o)-

Furthermore using ^~i=-?—r-^2~ bo

Ofl

the initial slope is

d^2

=

X\+X2-

at

ao<(^o+^o)

(3.23)

Ch. 3

Relationship Between Quantum Yield and Mechanism

173

and the final slope at ^Q > AQ 4- d^ 0,

Xi>^

d^2 can be derived. If ;|j, < 1, a point of inflection is found at

_ S2w ~ *

(4x'^-X^) :

•

This relationship is represented in Fig. 3.4 for X\ = 0.1 and for different values of Xi ^ 0.1.

1.0 I

1 —•— — 4 —

1

^, 4

—•— —•— ^

1

4

f ^ 1 %\

^

T

via singlet: hv

j/X 4

ly 0.5 4

y/^ 4

\y/^

A +B

•

C

•

E

via triplet: hv A +D

y 4

0.5^ 0.1

0.0 MTt^ 0.0

—4— — 4 —

1

1

0.5

— h - —•— - 4 —

—•—1

1.0

Fig. 3.4. ^-diagram according to eq. (3.23). ;f i = 0.1. ;if2 is the chosen parameter, f j is the ordinate. The points of inflection are marked.

Simple and Complex Photochemical Reactions

174

Ch. 3

TABLE 3.4 Comparison of simple non-uniform photoreactions Reduced degree of advancement

Mechanism A + /IV

>A'

A'

>A

A'

>B

A'-hC

>D

A^M^

,A'

A'-

->A"

A'-

->B

A"

>A

A" + C

>D

A + hv

^A'

A'

>A

A' + B-

->C

A'-

>»A"

A"

-^A

A" + B-

•^D

A + ;rv

>A'

A'

->A

A'

-»A"

A' + B A" A" + D

>C

Slopes

fi =«0-exp(-xf2))

^,^x^S2'X^X2l4l''^]

fl =

J \X\^ w>l loo Ui+0-");i^2. «
Xi

x(l-exp(-X2?2))-^2

?i=l-(l-?2)^'exp(-;t2?2)

^ = ;ci+;^2 ^^2 and for OQ > ( ^ + ^o)• 0. ; f i > i ^2

>A >E

These mechanisms given in this section are again compared in Table 3.4 with respect to mechanism, resulting in degrees of advancement and final slopes.

Relationship Between Quantum Yield and Mechanism

Ch. 3

175

3.L2.2 Physically sensitised photoreactions Both the photo-parallel reaction and the consecutive photoreaction, being physically sensitised, form two linear independent steps of the reaction. Their mechanisms and the reduced reaction scheme are listed in Table 3.5. The overall reaction schemes are derived in Appendix 6.6.2. Example 3-20: Physically sensitised parallel photoreaction The parallel photoreaction according to the details given in Table 3.5

TABLE 3.5 Mechanisms and reaction schemes of parallel and consecutive photoreactions, physically sensitised Mechanism

Reduced reaction scheme and dependencies of the degrees of advancement

B~^^C ^1

D ~ ^ E

^2

B -1 0

D 1 0 0 -1

c

E 0 1

^k


Au

^2=1-0-1.)'

Reaction steps

A + /IV

A'

•A'

>A

A' + B

>B' + A

B'

>B

B'

>C

A' + D

B--^C A

^1

^D

^2

B -1 0

c 1 -1

D Xk 0
•D' + A

D'

>D

D'

>E

A + /IV

A'

>A'

>A

A' + B

>B' + A

B'

>B

B'

>C

A' + C C

>C' + A >C

a—>D

176

Simple and Complex Photochemical Reactions

Ch. 3

and derived in Appendix 6.6.2.1 gives the following dependencies of the concentrations of the excited intermediates: /l'U

^A

—

'

,

_ k^a'd

h'-

U -

^2 + k^b + k^d

k^ +k^'

Kn

i" /Co

Therefore ^ _ ^ _

^3^5^^A

-(D^F

{k^ + k^b + k(^d){k^ ^M""^' ^ ^d = e = .

^6^8^^

- ^ = ¥>2U

(3.24)

is valid. The following differential equations give the relationship between jc, and.X j : dX2 _
.

6 8\

4

3 S\

7 • '^J

Defining

bo one finds ?2=1-(1-

-^.r

(3.25)

The corresponding ^-diagram is given in Fig. 2.4 in Section 2.4.2.1. Example S-ll: Physically sensitised consecutive photoreaction The consecutive photoreaction according to the details given in Table 3.5 and the reaction scheme in Appendix 6.6.2.2 gives the concentrations of the excited intermediates as

Ch. 3

Relationship Between Quantum Yield and Mechanism

177

and the changes in concentrations consecutive -fo = i , = k^a'b - k^b' =

^ ^ ^

r = ^f/A (3.26)

j = i2=V = 7

^^^^^^^

7 = V>2^'A

are obtained. Next the differential equation for the relationship between jci and X2 is given by

Defining

one finds

This differential equation is equivalent to that used for thermal reactions. The solution is represented by eq. (2.90) and is shown in Fig. 2.7 in Section 2.4.2.2. 5.7.5 Complex uniform photochemical

reactions

In Section 2.1.4.3 two examples are used to demonstrate the principles of photochemical reactions. Both are summarised in Table 3.6. The quantum yields of the two linear dependent steps for the photoisomerisation are given in eq. (2.21).

178

Simple and Complex Photochemical Reactions

Ch. 3

TABLE 3.6 Survey on the relationships between quantum yield and mechanisms for complex uniform photoreactions

p A ^^ photochemical step superimposed by a thermal backward reaction Example 2.10

^A

A - ^ B

^ = -^^/A+^2/B

. , . . , , ,

(^2+^3X^4+^5) ^3^5 {ki + k-^ )(k^ + ^5)

via tnplet state B—^A E-^"^PJ^2.11

s, _ Ic^k^
^

_ k^k^ ^^ - (^5 +^6)(^7 +^8)

5. lA Complex non-uniform photoreactions Besides the reversible photoisomerisation with a consecutive thermal step according to Section 2.1.4.4 and summarised in Table 2.1 a large number of other reactions exist. Just a few mechanisms are discussed in the following. The differential equations for the changes of the concentrations with time can be given in all cases. 3,1,4.1 Parallel photoreactions

Example 3-22: Parallel photoreaction starting at different components In the case of a parallel photoreaction, in which the two steps are independent of each other A—.^B

A

B

c

D

Xk

•^1

-1

0


Xl

0

1 0 0 -1

1

^ih

the two gross reactions are represented by the following elementary steps for the overall mechanism:

Relationship Between Quantum Yield and Mechanism

Ch. 3

A'

>A

x'l

A -1 1

A'

>B

X',

0

A' 1 -1 -1

<

0 0 0

0 0 0

A + hv

>A'

C + hv

x;

>C'

C

>C

C

>D

^5 ^6

B

0 0 0

0 0 1

c0 c0 0

0

0 -1 1

1 1 -1

0

-1

D 0 0 0 0 0 1

179

1h ^' k^a' kjo'

Ic ksC'

KC

and result in a =•iCj

L

=•

c

I /C'l

^5+^6

and the following changes in concentration with time - a =fc= JC| =

^-^ ^2

- c = rf = j c , - ^ - ^ ^

(3.27)

^5 "*" ^6

"^^3

Example 3-23: Mixed parallel photoreaction In the following photoreaction both the steps are run parallel, but in one step the initial reactant reacts with an additional compound given by the scheme

A + BB-

hv

hv

>D

^C

A x^ -1 0 X'j

B -1 -1

C D 1 0 0 1

^k

vtU 9\h

in two linear independent steps, represented by the overall mechanism:

Simple and Complex Photochemical Reactions

180

A + Ziv

A -I 1

>A' ^;

A'

>A

A'

^A"

A"

>A

4 <

A" + B

>C

4

B + hf

>B'

<

B'

>B

B'

>C

0 1

X-i

0 0 0 0

X', Xg

A' A" 1 0 -1 0 -1 1 0

-1

0 0 0 0

-1 0 0 0

B 0 0

C B' 0 0 0 0

0 0 -1 -1 1 0

0 0 0 0 1 0 0 1 0 -1 0 -1

Ch. 3 D 0 0 0 0 0 0 0 1

1 ^'^ /A

k^a' kjo'

k,a" ksa"b /B

kjb'

hb'

The use of a

a" =

=•

ikj + * 3

- ^ k^ + k^b

y='

k-j -^ kg

results the quantum yields A_ ^'

k,k,b (k^-^k^Xk.+k.b)'

(p^ -

*«

(3.28)

k-j + kg

and the time dependencies —
2 / B •

3.1.4,2 Consecutive photoreaction Example 3-24: Normal consecutive photoreaction The consecutive mechanism gives two linear independent steps according to

A—5L-»B

Xl

B-JJL^C

X2

A -1 0

B -1 -1

C D Xk 1 0 ^f^A. 0 1 VZ^B

Ch. 3

Explicit DifTerential Equations in Photokinetics

181

also. The overall mechanism consists of

^A'

A + hv A'

>A

A'

>B ^B'

B + Av B'

>B

B'

>C

A x; - 1 1 Xj 0 ^3 0 x's 0 0

4

<J

A' B 1 0 -1 0 -1 1 0 -1 0 1 0 0

B' 1 K 0 0 /A 0 0 ATJO' 0 0 k^a' 1 0 /B -1 0 k^b' -1 1 Kb'

c

and uses a' = — ^

b'' =— _ _ _ B kj+^e

to give the following quantum yields A _

^3

B _

y>2 =

^6

(3.29)

^ 2 "^ ^3

and the differential equations

3.2 EXPLICIT DIFFERENTIAL EQUATIONS IN PHOTOKINETICS 5.2.7 /?ar^5 of simple uniform photoreactions Based on the derivation in Section 1.4.3 of the amount of light absorbed and the equations derived therein for a simple photokinetic formalism, rates and related photochemical quantum yields are discussed in further detail in the following sections.

182

Simple and Complex Photochemical Reactions

Ch. 3

3.2, L1 The quantum yield does not depend on the amount of light absorbed The simple uniform photoreaction can be described according to eq. (2.12) by

whereby the index / represents any partner of the reaction. It is assumed that the component A starts the photoreaction. The quantum yield ip^ is supposed not to depend on /^^ but it can depend on the concentration. Since /^ (z) depends on the volume element Z in which the reaction takes place, the change in concentration with time a, is also a function of Z. Using eqs. (1.35) and (1.36), at this volume element z for the local rate, the following relationship is obtained as a partial differential (': wavelength of irradiation): ^

= 1000v,y>^/o/c;a, exp(-JX(*:;a, Vz)

(3.30)

accounting for both the variables t and z. As described in Section 1.4.3 the formalism is derived for Napierian absorption coefficients and using the factor of 1000 to take into account the exponential function and the different units (litres, cm^) used. In general, the change in concentration of component / in the cell a,(^z) is not the point of interest but a^it) which represents the mean value of A,, averaged within the cell as a function of the time of irradiation. A mean value is given according to definition by a,(0 = yjj«/(^z)rf^,

(3.31)

whereby / represents the path length of the cell. By partial differentiation with respect to t one obtains da- 1 t^dadar f l = i.[fri.rf2 = ffffL dt /Jo dt dt

(3.32) ^ '

Accordingly the change in the average concentration with time is equal to the mean rate. By use eqs. (3.30) and (3.32) one obtains

Explicit Differential Equations in Photokinetics

Ch. 3

lOOOv.ATl/n f'l G; = -

r ^ J o [ ^ ' « exp[^-j;X«a.V.J^eJ

183

(3.33)

For simplification, the change with respect to time of the local mean value is symbolised by a,. This mean value given in eq. (3.33) can only be calculated, if certain assumptions on the photochemical reactor are taken. - It is assumed that the irradiated solution is stirred so intensively that the different reactants are homogeneously distributed within the cell at each moment of the reaction. - Furthermore according to the Bodenstein hypothesis it is assumed that the short living intermediates do not absorb light and the gross reactions are formed as described in the previous sections. - Specifically it is assumed that the reactant A starts the photoreaction, therefore just the amount of light absorbed by this reactant is considered as well as its concentration in the integral according to eq. (1.36). Under these conditions the integral in the exponent can be solved. Since the different a, no longer depend on the element z, one finds

1

1=1

Therefore eq. (3.33) can be written as

iooov;^;/n f

a: =-

/

Jo


i=l

(

i^n

\dz

and further integration gives . ^ 1000v,K:>^/Afl i=/i

ll,
\A

1 - exp -X('^^^i)^ V 1-1 ))

(3.34)

1=1

This equation could be directly obtained if the mean value for /^ is used in eq. (2.12), and averaged for the total cell. However, this approximation is

184

Simple and Complex Photochemical Reactions

Ch. 3

only valid if the quantum yield does not depend on the intensity of light absorbed. This is not a general case. In contrast, the derivation given above is applicable in general. As stated above

1=1

represents the absorbance of the solution in the cell in Napierian units (natural logarithm) at the wavelength of irradiation. In formal kinetics the symbols A, B,... are used for the reactants. The symbol E is used to represent absorbance in this book. Napierian units are used in this chapter (see Section 1.4.3). In Chapters 4 and 5 decadic units are used for convenience and the decadic molar absorption coefficient is indicated by E as used in Section 1.4.4. Inserting the relationship for the absorbance in eq. (3.34), one obtains, for the photokinetic rate law, in the case when the photochemical quantum yield does not depend on the amount of light absorbed

a, =

'Jy

^ (l~exp(~£;))

(3.35)

Derivations assume that reactant A absorbs the radiation and starts the photoreaction, Napierian units are used. Intense stirring of the solution is required. The Bodenstein hypothesis is valid for all the photophysical intermediates. 3.2, L 2 The quantum yield depends on the amount of light absorbed If the assumption of independence of the quantum yield on the amount of light absorbed is not fulfilled, the equations derived above change. In Section 3.1.1.5 radical photoreactions are discussed. According to eq. (3.11) or eq. (3.12) the photochemical quantum yields depend reciprocally on the amount of light absorbed 7^. That means eq. (2.12) has be transformed to ^/=^/^A/^.

(3.36)

Ch. 3

Explicit Differential Equations in Photokinetics

185

TABLE 3.7 Comparison for different mechanism Quantum yield independent on intensity

Quantum yield depends on intensity

at = V ; ^ ^ / A

Gi = v j ^ ^

cii =

' r

(2.12) ^ (l-exp(-£;))

a, =

^

^^.

(3.36) ^ ^ (l - e x p ( - E ^ / 2 ) )

(3.35)

(3.38)

/"^ is that part of the quantum yield which is independent of intensity but dependent on concentration. Assuming radical intermediates, for which the Bodenstein hypothesis can be applied, and assuming that these radicals degrade within the volume element V = FAz in spite of intense stirring, eqs. (1.36) and (3.36) can be combined to ^

= vJ^^\0O0K\I„a ^J-\fJ^(
(3.37)

Using the same assumptions as before, integration results in a,^^'''^'^r^C-exp(-^;/2».

,3.38,

When the quantum yield depends on the amount of light absorbed in a more complicated way it can happen that the differential equation cannot be solved in a closed form for some mechanisms. An example for the complex relationship is demonstrated in Section 3.1.1.3 in Example 3.7. This mechanism can be solved in closed form. However, the differential equation is rather complicated and lacks application in practice for this reason. Both mechanisms are compared in Table 3.7. 3,2.2 Rates of non-simple and/or non-uniform photoreactions Explicit differential equations can be derived without problems as long as the quantum yields do not depend on the amount of light absorbed at all or

186

Simple and Complex Photochemical Reactions

Ch. 3

just in a simple manner. According to eq. (2.12) one obtains in principle

The amount of light absorbed /^ is substituted according to eq. (1.36) and the local average according to eq. (3.32) is formed. The result is an explicit expression for the change of the degree of advancement jc^ of the kth partial reaction step with time. This relationship can be used to set up the differential equations for the concentrations ^, of the reactants A; according to eq. (2.5). This procedure is discussed in Section 2.2.1 and its subsections. The results for photoreactions are compared to the thermal Jacobi matrices in Section 2.2.1.4. There the results for the two linear independent steps of a consecutive photoreaction

are given as derived in Section 3.1.4.2. Taking eq. (3.29) one finds ^i=^r^A and ^2= ^2Reusing eq. (1.36) one finds for the change of the degree of advancement with time for the two linear independent steps

dX2

= lOOOip^IoK'^b exp

-}y,{Kiai)dz

By treatment of this equation according to the considerations given in Section 3.2.1.1 one obtains equations in analogy to eq. (3.35) ^

= i,=1000^r^/o[l-exp(-£;)].

^

= i3=1000^»4?^o[l-exp(-£;)].

at

r.„

Ch. 3

Explicit DifTerential Equations in Photokinetics

187

Thus, using eq. (2.5) and following the procedure in Section 2.2.1 one finds the three differential equations for the concentrations of the reactants in the chosen consecutive photoreaction

b = \m){ip^K\a

- y%fe)/o^^^''^.^"^^^^

(3.39)

3,2.3 The relation between the concentrations for complicated photoreactions If time is eliminated from the explicit differential equations, the resulting differential equations give the relationships between the concentrations and the degrees of advancement, respectively. In many applications simple solutions are obtained, if the quantum yields of the partial steps of reaction do not depend on the intensity of irradiation and no slow dark reactions are involved. A typical example is the consecutive photoreaction treated above and discussed next. Example 3.25: Consecutive photoreaction The ratio of the changes in the degree of advancement is given by

dx^

(ptfc'^a

(ptiCA^o-^i

using the knowledge that the photochemical quantum yields do not depend on the concentrations. One defines ^B '

^2

'

A

A / '

jf-

^

1

f-

188

Simple and Complex Photochemical Reactions

Ch. 3

By integration the same result as for the analogous dark reaction (eq. (2.89)) is obtained for it 561:

5,=5,.(lziilMLiM

,3.40,

which is identical with eq. (2.90) The corresponding ^-diagram is the same as presented for a thermal consecutive reaction in Fig. 2.7 in Section 2.4.2.2. Example 3.26: Parallel photoreaction Based on Example 3.22 for a parallel photoreaction starting at different components with two linear independent steps A—^^B,

C—^Ji^D

one finds using eqs. (3.27) and (3.35)

^,=-!5«2<^(,-exp(-£,',).

Elimination of time results in

By defining reduced degrees of advancement si

»

hi -

and the relationship ^

A /

»

Ch. 3

Explicit Differential Equations in Photokinetics

189

one obtains ?2=l-tl-§i)'

(3.41)

In this case the relationship between the degrees of advancement depends on the wavelength of irradiation, since the expression X contains the ratio of absorption coefficients The relationship is equivalent to that obtained for the respective dark reactions in Section 2.4.2.1 and given by eq. (2.87). The dependency for the two the degrees of advancement is given in Fig. 2.4. Example 3.27: Mixed parallel photoreaction In addition to thermal reactions derived in Section 2.4.2.1 another mechanism with two linear steps of reaction is given in Section 3.1.4.1 and Example 3.23 following a mixed parallel mechanism according to A+B — ^ C ,

B—i^D.

It results using eqs. (3.28) and (3.35) in a ratio between the degrees of advancement as dx^ ^ [k^ 4- kr^)k^K'^{k^ 4- k^b)b ^ {k^^ -f kr^)k^K'j, (k^ i k^)^b^-. dx^

k^{k'j + k^)K^p^k^ab

^3(^7 + ^8)'^A

after the elimination of the reacting time. Defining

bo

^5^0

bQ

^0

V^7 +^8)^3'^ A

^0 ~ ^1

x^-x^

Simple and Complex Photochemical Reactions

190

<

1

1

1

1

1

h—H

Ch.3

H

5i 4

Fig. 3.5. ^-diagram for the reaction A + B—^^^C, B — ^ ^ D with w = 1, X\ =0.25. The parameter is ;f 2 •

one obtains the ratio

In Fig. 3.5 the reduced x-diagram is graphed for this reaction mechanism. Integration of this linear differential equation starting at the limits §1 = ^2 = 0» results in

«,=fc-») + ^ ( « - f , ) - ^ ^ ^ ^ f ; ^ ( « - l , ) " The initial slope at time f = 0 amounts to d^2

XiXi

^.

u

(3.42)

Ch. 3

The Integration of the Differential Equation

191

TABLE 3.8 Comparison of some complicated photoreactions Mechanism

Equations

Consecutive photoreaction A_!!L-,B-!!^C

h=^i-

(\ ^PA^ _ f j _ f,) ', ; "

(3.40)

with «2^^gl"g2 Parallel photoreactions

f 9 = 1 - (l - ^i )^ ^1 - -7-1

Mixed parallel photoreactions A^B-.i^C,B-^!^D

(3-41)

^2 - ^ ~ '

X - —A

^ , ;ti / t \ 1 X2U1-O + "/ t x;ti ^2=(X2-e.)-f^.^gO-;]^ ^^,, (^^^1) (3.42) ^0

^

^

At the end of the reaction the reduced degrees of advancement will be given by ?2oo = 1 - 5 , 0 0 ,

5,00 ^ « .

3.3 THE INTEGRATION OF THE DIFFERENTIAL EQUATION

In Section 3.2.1 the differential equations for simple uniform photochemical reactions are derived and given by eqs. (3.35) and (3.38). Their integrals can be solved assuming the reactant A starts the photoreaction. - If the quantum yield does not depend on the intensity of irradiation, direct integration of eq. (3.35) is possible between the limits of concentration a,(Oand a,o =af(r = 0) using Napierian units and the symbol ' to mark the wavelength of irradiation:

192

Simple and Complex Photochemical Reactions E:

-

da, = v,.1000/c;^/o. exp(-£;))

Ch. 3

(3.43)

When the quantum yield is reciprocally proportional to the square root of the amount of light absorbed by reactant A the integration results in: r — ^n d^i ^ ^ ^ JmoiPJI Ja,o2/^V^(l-.exp(-£;/2)) 'V AC

(3.44)

However evaluation of the integrals (3.43) and (3.44) in a closed form is not possible, if the absorbance E'^ at the wavelength of irradiation does not change during the photoreaction. Thus, just in a few exceptional cases a solution is possible. Some examples are discussed in the next section. 3.3,1 The absorbance at the wavelength of irradiation stays constant The integrals (3.43) and (3.44) can be simplified, if in a sensitised photoreaction just the sensitiser A absorbs the incident light. In this case the photokinetic factor (for its derivation and its meaning see Section 1.4.4), given by eq. (1.41) is constant E{E:)^-—^^T7T

(1-41)

l-exp(-£„) can be put in front of the integral. During a sensitised reaction the concentration of A stays constant a = aQ. The absorbance is assumed to amount to (only reactant A absorbs)

according to eq. (1.38), but using Napierian units. Taking into account these facts, eq. (3.43) can be simplified to

Ch. 3

The Integration of the Differential Equation

193

p da, ^ v,1000/c;,ao/o ^^ v,1000/o(l - e x p ( - £ ; ) ) Ktp'^

F{E'„)

.345.

d

'

and eq. (3.44) to prfa,

LT=

v^yiOOO/c^ao/o ^ 2v,yJOOOTp'O - e x p ( - £ ; / 2)) ^

F{E:I2)

'=

4E^

'

<'•"'>

In the following examples some mechanisms are discussed which work according to the stated assumptions. Example 3.28: Radical sensitised photoisomerisation As derived in Example 3.13 the mechanism of a photoreaction sensitised by radicals results in a quantum yield dependent on concentration and on intensity of irradiation. One finds according to eqs. (3.10) and (3.36) for the intensity independent factor f^

Using eq. (3.46) and the defined factor x ont obtains f^ db ^

2xVTo007o"(l - exp(~£; / 2)) ^

or after integration

Inf A 1 = 2):Vl0O07;(l-exp(-£:/2))^

UoJ

4Kd

-

A photoisomerisation sensitised by radicals acts as a first order thermal reaction

194

Simple and Complex Photochemical Reactions

Ch. 3

Example 3.29: Chemically sensitised photoreaction The chemically sensitised photoreaction according to Example 3.12, using eq. (3.9) for the reciprocal of the photochemical quantum yield, gives

Therefore the relationships

[' (xL ^ Xf, = iooo/o(i-exp(-g;))^ k[b ) d or

,>|„A,(,-,„),J°««'oO-exp(-^^», feo

p,3,

d

can be derived according to eq. (3.45). Taking reduced coordinates ^^A,

;f = i : ,

^_1000/o(l-exp(^£;))^

the final relationship can be given as r = -xlni8-()8-l)

(3.49)

The dependence on concentration derived above corresponds to the relationship of the Michaelis-Menten reaction, which was given in Section 2.5.1.2. Example 3.30: Physically sensitisedphotoaddition A physically sensitised photoaddition (see Example 3.10) gives according to eq. (3.7) 1 _ ^2 "*• *3^ K "^ *5^

(p^

k^b

k^c

Ch. 3

The Integration of the Differential Equation

195

By definition of X2 =

K

A = Co - ^0

one finds using c =fe+ A that ^ _ 2 I Xi , X2 I X\X2

b


be

_2 , Xi I X2 , X\X x{xi(i b b+ A

L_l

A U

(3.50)

b+Aj

Insertion of eq. (3.45) gives

(3.51) or using the definitions

V = Al

^ ,

»

V = A2

^0

M -=± 0 .

^2 t

^0

1000/0(1-expC-E^))

one finds

^ - 1 + Xi + X\X2 ini8 + M-1

A

/

u

= -r.

(3.52)

196

Simple and Complex Photochemical Reactions

Ch. 3

If ^0 =Co eqs. (3.51) and (3.52) can no longer be used. A corresponding calculation gives f\

r = ^-\

+

{Xi+X2)^^^-XtX:

\ri

''-''^

According to the definition the absorbance E|^ stays constant, if the sample is irradiated at a wavelength at which the reaction spectrum shows an isosbestic point. Under these conditions ai^a^. Therefore one obtains in the case of a quantum yield not dependent on irradiation

instead of eq. (3.45). If the quantum yield depends on the irradiation instead of eq. (3.46) the following relationship is found f

da:

VJIOOOKUQ

Finally the most simple photoreaction irradiated at an isosbestic point is discussed. Thus the absorbance at the wavelength of irradiation stays constant during the reaction. Example 3.31: Normalphotoisomerisation In case of a photo-isomerisation A—^!1^B the absorbance is given by E ; = d{K'f,a + K'^b) = dK'^a^ -f d[K'j^ -ic'^)a. Taking an irradiation wavelength at which K\ equals /Cg (isosbestic point), one finds the absorbance

Ch. 3

The Integration of the Differential Equation

197

to stay constant. In eq. (2.19) the quantum yield has been determined to be independent of concentration. Therefore using eq. (3.54) one finds j^^ ^

IOOO/C^VQ^

^

1000y>Vo(l~exp(-£;))^

^^^^^

This photochemical reaction also formally behaves as a dark reaction of first order. Furthermore, reactions using a triplet mechanism as in Example 2.6 in Section 2.1.3.3 or equilibrium reactions treated in Example 2.10 or Example 2.11 in Section 2.1.4.3 using eqs. (2.20) or(2.21) can be considered accordingly. 3.3.2 The relationship between absorbance and concentration In general absorbance varies during a photochemical reaction. Therefore within the rate laws either the concentrations have to be substituted by the absorbances or the absorbance at the wavelength of irradiation by the concentrations to be able to calculate the integrals, eqs. (3.43) and (3.44), respectively. In principle the relationship between absorbance and concentration is given by the Bouguer-Lambert-Beer law as derived in Section 1.4.3 by Fig. 1.2. For uniform reactions, absorbance at the wavelength of irradiation is defined in Section 1.4.4 by eq. (1.38) to

£„'=rfX<«/

(3-57)

using Napierian units. Taking the relationship between concentration and the degree of advancement according to eq. (1.9), one finds

Defining

198

Simple and Complex Photochemical Reactions

Ch. 3

as the absorbance at the beginning of the irradiation and Q = dJ^^i<

(3.58)

as the stoichiometric sum of the Napierian absorption coefficients of all absorbing species in the solution multiplied by the path length d, one can derive (3.59)

E:=E:^^QX.

The solutions of this equation with respect to the degree of advancement are F' — /*"

x=

"

"» Q

F'

and x = =^. Q

(3.60)

At the end of the reaction the degree of advancement x^ and the absorption E^^ will amount to

-"

Q

•

Therefore we have (3.61)

X-X^={E:-E'^)Q.

A combination of eqs. (1.9) and (3.61) allows rearrangement with respect to concentrations a,. = a,o +

v.(g;-g;:o) Q

«i~=«,0+

^

A combination of these two equations results in

Ch. 3

The Integration of the Differential Equation

199

(3.62) or

E:-\E„^

M^:-EL)

a,=-^ Q

(3.63)

Q

In this last equation E^^ has been defined according to noo

fioa

i.J.i Summary In general integrals like eqs. (3.43) or (3.44) cannot be integrated in a closed form, and since they contain functions such as F(£^) defined by eq. (1.41) or FiE^/2) according to Section 3.3.1 [28]. Mathematicians prefer to define new functions in such cases. One could consider generalising the dependencies between absorbance and time functions as [29]

DM'EL)-=j

E'„dE'„

(3.64)

(l-exp(-£„'))(^;-£;„)"

Such functions are useful in the presentation of absorbance-time dependencies. The exponent m will take the values 1, 2, 1/2 and 3/2 in many cases. The disadvantage of these functions is that they contain the parameter E'„„. For this reason their use in practise would require extensive tabulations. Another possibility is to develop eq. (1.41) as a power series according to Mac Laurin [10,30-32] F(E:)=i+-E' 2

+ -^£„" +—E:* 2! 4!

/6

6!

(3.65)

Simple and Complex Photochemical Reactions

200

Ch. 3

The factors B^ (the ith derivative of F{E'J at E'„=0) are called Bernoulli constants. They are defined as B2 =1/6,54 =-1/30,56 =1/42 ^^^' [31,33]. This series converges for E'^<2jt, In decadic units of absorbance ( f ) at the wavelength of irradiation this value amounts to approximately 2.7. F(E'^) becomes 1 using only the first element of the power series. The result is a very simple approximation. It is used preferably in the literature for the case of low absorption. The advantage of the application of this approximation using the power series is that the concentration-time functions result in the same form as for irradiation at an isosbestic wavelength. Un-

F(E'n)

0

1

6

E;

7

Fig. 3.6. Function F(E^) plotted vs. E^ for approximations using different elements of the power series according to eq. (3.65). ( ) is the asymptote to the photokinetic function given by F(E') = E^ The different curves are given by the steps of approximation: Bo (l)F(£;) = l,(2)F(£;) = l + ^ £ ; , ( 3 ) F ( £ ; ) = l + ~ £ ; + ~ ^ £ ; 2!

(4)F(£i) = l + l £ ; + ^ £ : 2 ^ - ^ £ : 2!

(5) F(£:;) = 1 + ~ F ; +^^'^

4!

_._J^ir/4 ^^6_ C./6

4! "

6!

Ch. 3

The Integration of the Differential Equation 5.0% I

I

I

1

J

I

201 I

2.5% 4

0.0 % -4

•2.5%-I

5.0%

Fig. 3.7. Relative error for an approximation of F(E^) according to eq. (3.65) in percent deviation. The numbers give the elements up to which the series is used and are the same as in Fig. 3.6.

fortunately this approximation is very poor. The error becomes larger than 5% even at an absorbance of £^ = 0.1 (that means in decadic units 0.04!!). The function eq. (1.41) can be approximated up to an absorbance of £^ = 4 (in decadic units £ ' = 1.47) with an error less than 1% if additional elements of the power series up to the 6th power are considered. In Fig. 3.6 F(E'^) is given for the different approximations according to eq. (3.65). In Fig. 3.7 the relative errors of these approximations are demonstrated. In the case of strong absorption the exponential function in the nominator of eq. (1.41) vanishes. For this approximation at "total absorption" one finds

nE'j^E:

(3.66)

This approximation gives better results than the Mac Laurin series, since eq. (3.66) represents the asymptote to the curve, eq. (1.41). The error stays below 1% above an absorbance E'„ = 4.7 (2.04 in decadic units). In the range of natural absorbance E'^=4 to J?^ = 4.7 an error less than 2% is obtained.

202

Simple and Complex Photochemical Reactions

Ch. 3

0%

-10 %4

- 15% Fig. 3.8. Relative error at an approximation of F(E'„) by eq. (3.67) in percent deviation. The numbers at the curves refer to the numbers of the elements considered in the series.

Even the consideration of powers up to the 6th in approximation, eq. (3.65), yields an error of at least 2.7%. Many approaches have been tried to improve this approximation at total absorption. To achieve this task eq. (1.41) has been extended into exponential functions [34] E: HK) = — 7-= £::(l + exp(-£;) + exp(~2£;)+.-.) ^ '^^ l - e x p ( - £ ; ) '^ ^ "^ Fv „y y

(3. 67)

As one can see in Fig. 3.8 the improvement is poor. The integration causes exponential integrals, error functions, and other higher order functions [10,28], which is a disadvantage of this approximation. Taylor series will yield better results than the Mac Laurin series, eq. (3.65). Beginning at E'^Q or E'^^ one finds [28]

(3.68)

The Integration of the Differential Equation

Ch. 3

203

TABLE 3.9 Maximal error for a Tayl or scries ofE^o cut off at the quadratic term 0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

F' j

1 0.1

0.3 0.0

0.8 0.2 0.0

1.5 0.6 0.2 0.0

2.5 1.4 0.7 0.3 0.0

3.9 2.5 1.6 0.7 0.2 0.0

5.7 3.9 2.7 1.5 0.6 0.2 0.0

7.7 5.7 4.2 2.6 1.3 0.5 0.1 0.0

10.1 12.7 7.7 10.0 8.0 5.9 4.0 5.6 2.3 3.4 1.1 1.8 0.4 0.8 0.1 ! 0.3 0.0 0.1

0^ I

^nO

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

4.5

1

In this case reasonable results are obtained even taking only the square element into consideration. This is demonstrated by Table 3.9. The function F(£^) shows a weak curvature with an initial slope of 0.5 and an asymptote, eq. (3.66). For this reason it makes sense to choose a linear interpolation between the starting value E'^Q and the final value £^^. In a first approximation [28,35] one finds

Ffe)-F(£;o) F(£;)==F(£„'o) +

{E:-E'^)

(3.69)

One can define the degree of advancement x in such a way that it becomes at the end of the reaction • ^ 0 0 = ^ 0 .

Under these conditions using eq. (3.60)

is obtained. Defining the reduced degree of advancement

Simple and Complex Photochemical Reactions

204

Ch.3

TABLE 3.10 Relative error table for a linear interpolation between supporting points according to eq. (3.69) 0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0.5

1.6 0.3

3.2 1.2 0.3

5.0 2.4 0.9 0.2

6.8 3.6 1.7 0.6 0.1

8.7 4.9 2.6 1.2 0.4 0.1

10.4 6.2 3.5 1.8 0.8 0.3 0.1

11.9 7.3 4.3 2.4 1.2 0.5 0.2 0.0

13.3 8.4 5.1 3.0 1.6 0.8 0.4 0.1 0.0

14.6 9.3 5.8 3.5 2.0 1.1 0.5 0.2 0.1 0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 _ |

one finds E:=E:,+Q'^

(3.70)

Using this assumption the first approximation yields within the total range of absorbances the simple relationship F(E;)

= FiE^o) -h [ F ( £ ; J - F(£;o)]^

(3.71)

The maximum relative errors caused by this approximation are given in Table 3.10 for the extremely non-optimal range of natural absorbances between £;=5and£;=0. 3.3,4 The integrals of simple uniform photoreactions All the approximations discussed in the previous section might be out of the range of interest, since nowadays numerically powerful electronic calculators can be used. Therefore it is more convenient to solve numerically the integrals, eqs. (3.43) and (3.44). It is useful to present the result in general coordinates. In this case, of all photoreactions which show no dependence of the quantum yields on the intensity of the exciting irradiation, eqs. (1.41) and (3.70) define E^ and F(E'^), Thus in general coordinates uniform

Ch. 3

The Integration of the Differential Equation

205

photoreactions with quantum yields are not dependent on the irradiation result according to eq. (3.34)

^-'^rhiv

= t/r,

(3.72)

with the modified time axis r = 1000x^;V.

(3.73)

The relationship (3.72) can be integrated using numerical methods, e.g., Simpson's [31,33]. On the left-hand side of eq. (3.72) the reciprocal quantum yield is used. Thereby the symbol x includes some reaction constants derived from the reaction scheme and depending on the photophysical mechanism used. The following examples demonstrate this. Example 3.32: Photoaddition in general coordinates In Section 3.1.1.2 in Example 3.4 the equations for a photoaddition via the triplet state are given. Using eq. (3.3) one derives /=-(£;)(l + ^ ] : f 3 | =

rf^

(3.74)

using the following definitions: __

^4

^3

_

0

for the coefficients in eqs. (3.73) and (3.74).

Another example of such a mechanism is the photophysically sensitised photoreaction, which is derived in Section 3.1.1.4. In the following example, the mechanism via the triplet is chosen (see Table 3.3). However, according to Example 3.9 one can set up an equation in general coordinates which differs from eq. (3.74). One finds

206

Simple and Complex Photochemical Reactions

F{E'Jl + -^]d^

= dr

Ch. 3

(3.75)

with dv = lOOOxK\^Iodt ^0

and the definitions t_

"

•^

fco'

y -

"

*3*7

(*2+^3)(*6+*7)'

.. _ *4

ksbo

If the quantum yield does not depend on the intensity of the irradiation, many of the photoreactions discussed in Section 3.1.1 can be evaluated using a single program. To achieve this eq. (3.72) is written in a more generalised form as

^(^<'*:^I

l j _ Y l + ^ l j ^ = ^r.

(3.76)

t-()i-(

In this case X\ and/or Xi ^^^ be taken as zero. In consequence many applications can be realised. If the quantum yield is reciprocally proportional to the square root of the intensity of the amount of light absorbed, one defines instead of eq. (3.72) according to eq. (3.44)


''^

' =A

(3.77)

whereby

As discussed above x can include constants, which depend on the spe-

Ch. 3

The Integration of the Differential Equation

207

cific reaction. Accordingly one finds for the photochemical reaction in Section 3.1.1.5 using eq. (3.11) ^ ^ ] — ^

= X^|lOOOK'^I,dt

(3.78)

and for the radical chain-reaction according to Example 3.15 and eq. (3.12)

Furthermore the radically sensitised photoisomerisation Section 3.1.1.4 in Example 3-13 gives, using eq. (3.10), F{^^^-^

= X^|iOOO^^^dt.

(3.80)

3,3.5 The integrals of uniform complicated reactions Photoreactions are called complicated if more than one component starts the photo-reaction by absorption of light and/or if slow dark reactions have to be taken into consideration, to which the Bodenstein hypothesis cannot be applied. These complicated reactions can be treated straightforwardly, if - two or more components start the photoreaction, - the partial photochemical quantum yields depend in the same way on the intensity of irradiation, and - no dark reactions take place. This fact can be demonstrated for Example 2.11 in Section 2.1.4.3 as well as in Section 3.1.3. According to Table 3.6

JC =

F{K)

_ iooo/o(^,V;ao -(^;^/c; +
208

Simple and Complex Photochemical Reactions

Ch. 3

is valid. Using this equation, the final value of the degree of advancement is found to be

Therefore

x=-

HK)

or F{K)^

= dt

(3.81)

using the definition r = 1000/o(^f/c;+^^/c^V

and ? = — .

3,3.6 The integrals for simple non-uniform reactions In the case of non-uniform reactions, the relationship between the absorbances at the wavelength of irradiation and the degrees of advancement is given by E'^=E:,+^Q,X,.

(3.82)

In this case all the .$• linear independent degrees of advancement have to be summed up. The relationship between the different values of jc^ can be calculated for simple photoreactions under these conditions. A typical example is a parallel photoreaction as demonstrated in Section 3.1.2.

Ch. 3

The Integration of the DifTerential Equation

209

Example 3'33: Integration of a parallel photoreaction A photoreaction is given in Example 3.17 in which both partial steps of the reaction are induced by absorption of the reactant A: A—^*^B,

A+ C—^D.

The following relationships are necessary to be able to integrate the rate equation £ ; = iP;o+G,?,+e2?2»

(3.83)

whereby the relationship between the ^-values is given according to eq. (3.17) ^2=u{l-cxp(-X^,))

(3.84)

whereby x = ^4^0/^3 • Integration results according to eq. (3.16)i F{E:)

f >

{l + x^{u-^2))-dr

(3.85)

whereby the symbols

. = .ooo,
„=J^, , - j ^

are defined. The physically sensitised consecutive photoreaction can be treated in the same way, as discussed in Example 3.21. Instead of eq. (3.84) the following relationship

,,=j,.(lziilMlzM i-x is valid with the definition

,3.86,

210

Simple and Complex Photochemical Reactions

Ch. 3

One finds the integral

HKi ,.,,_J_.,3^]k,

(3.87)

=^r

using kc

bo

'

y

ky

z^

y KA "T"

Km

:s

k^ y

zz

k^Or\

K^

3.3,7 The integrals for non-uniform complicated reactions The consecutive photoreaction is discussed in Example 3.25 in Section 3.2.3. Instead of eq. (3.40) derived there, eq. (3.83) can be used. In this case the parameter % has a different meaning. As demonstrated above it becomes

A "•

A

/

»

and one has to integrate

F{K)

• = dr

(3.88)

1-^.

using r = 1000/c;V.

3.3.8 The integration of uniform reaction with superimposed thermal reactions In Example 2.10 in Section 2.1.4.3 a photoisotnerisation is presented, which is superimposed by a thermal backward reaction. According to eqs. (2.20) and (3.35) the rate laws are given by

Ch. 3

The Integration of the Differential Equation

211

IOOOK^/O

The steady state value of f = x/a^ can be determined by

whereby ik 1000/c;/o ^^ cannot be calculated in a direct manner because of the factor F{E'„) in the nominator, but it can be obtained by a simple iteration method. In general coordinates the differential equation is given by

|^x(l-g)_^

(3.90)

It has to be integrated numerically. A typical method is the procedure according to Runge and Kutta. In a similar way the reaction hv

A^2B V

can be treated. It is X —

-.

"7

KX

.

The steady state concentration is given by

212

Simple and Complex Photochemical Reactions

Ch. 3

Jl + 4aF(E'„)-l ^~

2aF{E'„)

^'-^'^

whereby a = ^ - ^ lOOO/c^/o is defined. Once more this differential equation must be integrated numerically. 3,3.9 Non-uniform reactions with superimposed thermal reactions For the consecutive reaction containing a photochemical and a thermal step of reaction A—5L->B—^C

the differential equations 1000/c>f/o(ao --Xt) F{E'„)

(3.92)

and X'y ~~ rCI At "" ^0

)

are valid. The relationship between the degrees of advancement can no longer be calculated in such a system in a direct way, since the function F(E'^) is part of the differential equation. Using general coordinates the system of both differential equations

'

nK)

(3.93)

Ch. 3

The Integration of the Differential Equation

213

with the independent variables ^, and ^2 "^ust be numerically integrated, whereby eqs. (2.7) and (1.41) have to be taken into consideration. Once more the method according to Runge and Kutta is a useful procedure. The mechanism A—^B—5^C can be treated in a similar way. In general coordinates the differential equations amount to

,

.X(g,-g.)

(3.94)

with the definition ;f = 1000/c^^2%In this case the first equation can be solved directly. But this advantage has no consequence. For the reaction

one finds

nE„)

with

(3.95)

214

Simple and Complex Photochemical Reactions

Ch. 3

3,3.10 Reactions with a difference in the dependence on intensity In the assumed photochemical consecutive mechanism A—!iL->B—~^C the first partial reaction is assumed to be a simple photoisomerisation with constant quantum yield. The second partial reaction is taken to be a radically sensitised reaction as described in Example 3.9 in Section 3.1.1.4. One assumes that the E-radicals are produced by a photolysis of a reactant D according to D—^E. In this case the differential equations .

1000/c;y)f/o(l-g,) ^ "^

^2

(3.96)

F{E'J2)

can be integrated according to Runge-Kutta as discussed in the previous section. 3.3.11 An alternative: the transformation of time Up to now the function F(£^) was considered to be primarily a function of concentration. For this reason it was integrated together with the concentrations themselves. If no slow dark reactions have to be considered, and if all partial photochemical reactions depend in the same way on the intensity of irradiation, the fundamental differential equation, eq. (3.72), can be rearranged as ^

^ ^ ] = lOOOxK\Io-^ ^^'F{E:)

l-^[
(3.97)

Ch. 3

The Integration of the Differential Equation

215

and by d@ = - ^

(3.98)

a transformed time-axis 0 can be defined [36]. The left-hand side of eq. (3.97) can be integrated in a closed form in many cases. One finds under these conditions

®(^)=r;

^

^^

(3.99)

In this case the principal task is the calculation of t according to r = jV(£;)J0.

(3.100)

In consequence one calculates the relevant A 0 as well as F(E'„) according to eq. (3.99) for a given A§, beginning at § = 0. By this means one obtains the different elements of the integral in eq. (3.100). This method has only a few advantages in the calculation of the turnover in comparison to the method given above. However, eq. (3.99) allows a qualitative impression of the time dependence to be obtained. Furthermore the approximation Ar=F(^)A0, gives a first idea, whereby E^ is an absorbance between E^n and E^g^ for example. The advantage of this transformation appears in the case of the general approaches. It allows quasi-linear systems of photoreactions to be treated in the same way as systems of linear thermal reactions. One understands quasilinear photoreactions as reactions in which the quantum yield is constant and no dark reactions are superimposed. In consequence all the relationships derived in Section 2.2 can be applied to these types of reactions.

216

^

1

JO

S u

I i

o

'^ § H CO

en

o

*2 ^ T3 ed 2 O OH

II 3

II

55

I I

I II

II X II

1

I

a ON

o <s '•§ en € S g

ii

en

n I

+ I

II V*

II

II

1

I

II

II

xUc>

>» o w

I ^&

-a

ii

X

II

II

if

CO

•§1

^ E

a'g

CA

-I -a

^^

+

, ^

Simple and Complex Photochemical Reactions

o 'a

J3

B CO *5

1

II

-* -ft 11

oo

^

11

Ch. 3

I

24> Tt »-H tS en O

o

1

H i2

r4

Ch. 3

\^\

CO

m

II

1-^

o

1 o

II

'^

o

oo

O

o

S 2

pSi

X I

I

^

f^

-^^

00

II

II

3

>

si >>

lit

I

II

I

I

<^J^

+

I +

y^

II

M

II

I

t3

II

ON

cd

a o -a

CO

.9 -S ' C

I

I

Ox fO

II • JHJ\

o a*

u

U 1^9 9

u

O

^ OQ >j

8 1* '>
I S 1 •a <

.S3 73 fc ou, -S

<

The Integration of the Differential Equation

I

MP

+ a. ^

J^"^

+ o

U

+ < PQ

2 ^^ ^ >< CL, U

u

217

218

Simple and Complex Photochemical Reactions

Ch. 3

3.4 PHOTOREACTIONS IN VISCOUS MEDIA

In all the derivations above two experimental principles have been assumed: - a simple geometry of light source and reactor, - an intense stirring of the solutions during irradiation. This second assumption is neglected in the following. However, it is assumed that the concentration at a certain volume element of the reactor can change only by the photoreaction and not by diffusion or convection. In the next section this principle is discussed using a simple example [10,12,37], 3.4,1 Simple example for photoreactions in viscous media In Section 3.2 the relationship for the local rate of a simple photoreaction ^

= \(mv,ip\K\ txv(-\Y^{ic\a,)dz)

(3.30)

is derived at position z of the cell assuming a non-dependency of the photochemical quantum yield on the intensity of irradiation. Excluding diffusion and convection, eq. (3.30) determines a^ as a function of position and time at the given boundary conditions. First the simple photoisomerisation A—^B is considered, in which the photochemical quantum yield (p^ =(p is constant, and the Napierian absorption coefficients are defined by ic\^0 and /Cg = 0. Taking a, to be equal to a, eq. (3.30) can be rearranged to

dt

= -1000V?A:;/O^ expf-JV^a dz\

or dlna dt

= -1000^/c'^/o expf-JV^a dz\

(3.101)

Ch. 3

Photoreactions in Viscous Media

219

The partial differentiation with respect to z gives d^

— = lOOOipK'^I^a cxipi-^^K^a Jzl. dtdz

A combination of the last two equations allows the elimination of the exponential function d^lna dtdz

,

d\i\a dt

, da ^ dt

An integration of this partial differential equation with respect to t yields _ _ = -/CAa-/(z), or —^-K^a -af{z), dz dz whereby J{z) is the integration constant. This differential equation turns out to be a Bernoulli differential equation. It can be transformed to the differential equation dz using a = \/y. This equation has the solution y = exp(j;/(z) dz)[H(t) + K\ | > x p ( - / ^ / ( Z ) dz) dz]. whereby the integration constant H{t) is a function of time. Defining exp(^-JJ/(z)rfzj = G(z), one finds the general solution «(Z'0 =

^ ,

•

Hit) + K'^[G{z)dz

(3.102)

220

Simple and Complex Photochemical Reactions

Ch. 3

The intention is to determine the functions G{z) and H(t) in such a way that eq. (3.102) fulfils the boundary conditions, which are given by a(z.O) = aQ.

This means that the component A is homogeneously distributed within the cell at the beginning of the irradiation. On the other hand, at the limit z = 0 one finds using eq. (3.101) that In—= -1000^/c;V an or

since at the limit z = 0 the exponential function in eq. (3.101) becomes 1. These conditions are fulfilled, if H{t) = exp(1000^/CA/oO and G(z) = exp(ao<^)are valid. Then one obtains ^cpKK^^) exp(1000^/c'^/oO + cxpia^K^z) - 1 or by defining a = —,

Tj-a^K^z,

the new relationship

r = 1000^/c^V»

^3^^3^

Photoreactions in Viscous Media

Ch. 3

221

Fig. 3.9. Turnover a of the photoisomerisation (^^ T^ 0, KT^ = 0) as a parameter for the variation with time Tand position ?].

exp(;7)

a(i/,T) =

exp(T) + expitj) -1

(3.104)

using non-dimensional coordinates. The dependency of the turnover on the volume element in the cell is presented for a simple photoisomerisation in Fig. 3.9. Taking the local mean value introduced in Section 3.2.1 —

1 fZ

(3.31)

a •'0

one finds using eq. (3.103) a{t) = —

ln[l + exp(-1000^/c';^/oO(exp(A:;ao^) ~ Ol-

(3.105)

Kj^d

The same result will be obtained for the change in concentration for the mechanism considered in a stirred solution, if the component A is the only

222

Simple and Complex Photochemical Reactions

Ch. 3

absorbing one. This means that in this special case the turnover does not depend on stirring and the photoreaction in viscous medium can be evaluated by the same formalism. 3.4.2 General solution 3.4.2.1 The quantum yield does not depend on the amount of light absorbed If eq. (3.30) is written in the form

dt

= lOOO/c^^/oa expr-£(£;o + Qx) dz\

(3.106)

one finds using the definition dx _

dx

_

d^

_

the relationship ^ = exp[-£(£;o+Gao?)rfz], if £^ as well as Q are related to a path length of 1 cm. Furthermore the two definitions ^ = x/a and dr = lOOOK\lQ dt are used. In the case where the above equation is differentiated with respect to z

then the exponential function can be eliminated. One obtains d^u

/„. _ u\du = -{£:;o+e«ol)—. dTdz ' " " 'dt since all the values of z amount to f(z.O) = 0.

(3-107)

Ch. 3

Photoreactions in Viscous Media

223

At time r = 0 the integration of eq. (3.106) gives

within the limits r = 0 to r for a constant position z. On the left-hand side of eq. (3.108) the lower limit becomes 0 because of the boundary conditions. The second integration at constant r within the limits z = 0 to z results in the fundamental equation

At the position z = 0 the time r can be directly calculated as a function of the turnover, since the exponential function becomes 1 at this position: f^(0,r)

cl^

^=1 A . .. J|(o.o)^A(l-§)

(3.110) ^

Either z can be calculated as a function of time for constant turnover using eqs. (3.109) and (3.110) or the turnover can be determined for constant time according to eq. (3.109). 3.4.2.2 The quantum yield is reciprocally proportional to the square root of the amount of light absorbed Using eq. (3.37) instead of eq. (3.30) one obtains ^ = /^7lOOOv^/o(ao - x ) exp[-l£(E;;o -f Gao§)rfzj.

(3.111)

The derivation proceeds as above and results in the general solution , =.f""

y

„

,

.

(3.112)

224

Simple and Complex Photochemical Reactions

Ch. 3

At the position z = 0 ?(0.r)

d^

J?(o.o)7\/lis obtained. 3 A3 Some further examples 3.4.3,1 The integrals can be solved in a closed form Example 3-34: Photoisomerisation by irradiation at the isosbestic point In the case of a simple photoisomerisation, an irradiation at the wavelength of the isosbestic point results in

for the inner integral of eq. (3.109), if the quantum yield is constant, since Q becomes 0:

£.'o(i-aMi-« Further integration gives 77 = £:;o2 = - l n l n ( l - ? ( ^ ) ) + lnln(l-^(0)). The second integral contains the time function, since at the position z = 0 one obtains ln(l~^) = - r . Therefore

Photoreactions in Viscous Media

Ch. 3

225

-«3

Fig. 3.10. Turnover a of the photoisomerisation as a parameter for the variation with time T and position 77. Irradiation takes placed at the isosbestic wavelength.

or a = 1 - ^ = exp[-r exp(-7;)]

(3.113)

is the solution for the turnover, which is shown in Fig. 3.10 as a parameter in the functional dependency between r and 77. The mean turnover in the cell is given by __

1 rz

a =-

/ •'0

exp[-r exp(-^)] dz =

1 rVo

n^ Jo

exp[~r exp(-7;)] drj,

according to eqs. (3.113) and (3.31), whereby by definition ri =

K\aQd.

Therefore the relative concentration in the newly defined time axis r is given by

226

Simple and Complex Photochemical Reactions

Ch. 3

a(r) = -[Ei(-i^) - Ei(-rexp(-77o))], whereby Ei stands for the special case of an exponential integral equation [31,33]. This equation has to be seen in comparison to the solution for a stirred solution (index R) according to eq. (3.56) /

a^ =exp

(l-exp(-77o))

In Fig. 3.11 a comparison is given between the photoisomerisation in viscous media and in a stirred solution. Furthermore another graph demonstrates the difference between a viscous medium and a stirred solution. In Fig. 3.12 the reaction time of a stirred solution f^ is calculated by iterative procedures for an equivalent of the turnover in viscous medium as a function of its irradiation time.

1.0 1

1

1

a 10

0.5-

^ 0.0 - 1

1

1

1

— 1

1

Fig. 3.11. The mean turnover of a photoisomerisation is shown as a function of the irradiation time irradiating at the isosbestic wavelength. (—) viscous medium, ( ) stirred solution. The parameter is ri = K^OQZ .

Photoreactions in Viscous Media

Ch. 3

I

5.0

2.5

227

I

-f

0.0 Fig. 3.12. The time of irradiation in a stirred solution (ordinate) TR has been calculated by iteration for the same turnover in viscous medium as a function of the irradiation time. The parameter is rj .

Example 3.35: Chemically sensitised photoaddition According to eq. (3.9) the following equation is valid for the chemically sensitised photoaddition discussed in Example 3.12.

^ = ^ + l = - i L + i. ^^

fo

"

1-^

If the sensitiser A is the only absorbing component, Q = 0 is valid and eq. (3.30) can be written as

dt

• = I000
by defining M = —^

228

Simple and Complex Photochemical Reactions

Ch. 3

The calculation of the inner integral gives

i'G

+ 1 d? = - X l n ( l - ^ ) + ^.

Further integration gives ;; = £^z =-ln(-;f ln(l - ^) + ^) + Inr,

(3.114)

if one takes into account that at the position z = 0 the following equation is valid: ~Xln(l-^) + ^ = 10{)0/c;/oMr = r. The turnover a of this chemically sensitised photoaddition is given in Fig. 3.13.

10

T

5 4

Fig. 3.13. The turnover of a chemically sensitised photoaddition as a function of position and time. The sensitiser is the only absorbing component.

Ch. 3

Photoreactions in Viscous Media

229

Example 3-36: Radical chain reaction In Section 3.1.1.5 the radical chain reaction is discussed in Example 3.15. The following is valid:

Using a = aQ-^

and

§= —

one finds ^ = (1 -e)'"2xVlOOO/c;ao/o e x p [ - | j J ( E : o + e ' a o ^ ) ^ z ] Using the new time axis r = 2xVlOOO/c;ao/o t one finds at the limit z = 0 r=

'

-1.

If the starting material A is the only absorbing material then the integrand of the exponent becomes

Under these conditions the inner integral of eq. (3.112) becomes 2E'"nO X

(Vr:?-!).

Therefore one finds the solution, which is shown in Fig. 3.14

Simple and Complex Photochemical Reactions

230

Ch.3

1 0 ^ Fig. 3.14. The turnover a of a radical chain-reaction is shown as a parameter in a function of position and time. The starting material is the only absorbing one.

z£;;o=77 = r-f 1-f Inr--

In

-1

If Q equals zero, a corresponding calculation results in 7; = l n r - l n

VW

~1.

(3.115)

In this equation the rj and r are defined as above. The relationship, eq. (3.115), is explained by Fig. 3.15. 3.4,3.2 The integrals cannot be solved in a closed form The integrals given in the previous section can be solved in a closed form only in very few cases. Under such conditions the integrals in eq. (3.109)

Ch. 3

Photoreactions in Viscous Media "^

1

fi

3

1

0

231

^

Fig. 3.15. The turnover aof a radical chain-reaction is shown as a parameter in a function of position and time. Irradiation takes place at the isosbestic wavelength.

have to be calculated numerically for example using Simpson's method. This is demonstrated in the following examples. The simple photoisomerisation A—!^B is assumed to show constant quantum yield
-E'„o+Qao$

(l-€

1-1

(3.116)

d?

232

Simple and Complex Photochemical Reactions

Ch. 3

The inner integral J can be solved in a closed form using z = constant as explained in Section 3.4.1. It is given by

= -[£;:. in(i-^)+fc-£;o)4 E'^^ is the product of concentration and absorption coefficient at the end of the reaction. Using

for J = ^ (for r kept constant), one finds the solution

^

%o.r)(l-.y)[ln(l-3;) + ( l - r ) j ] *

According to eq. (3.110) one finds at 77 = 0 ln(l-^) = - r .

(3.118)

In Fig. 3.16 the relationship between t} and r is given for some values of r at constant degree of advancement § = 0.1. r is calculated according to eq. (3.118) for some assumed values of ^(0,r). 77 is determined by numerical integration using eq. (3.117). In Fig. 3.17 a different presentation is chosen. One assumes at turnover ^ = 0.95 at ;; = 0 and the corresponding time (r = 3.00) is kept constant. Assuming different values of r the turnover ^ is given as a function of 77. These values can be obtained by numerical integration according to eq. (3.117) for a constant lower limit and a variable upper limit. The values for r-~>oo (f*^^ = 0 ) cannot be obtained by numerical integration. Under these conditions the exceptional case happens where the integral can be solved in a closed form. 3AA Complicated photoreactions Complicated photochemical reactions can be treated in the same way as

Ch. 3

Photoreactions in Viscous Media

233

Fig. 3.16. T as a function of tj for constant turnover ^ = 0.1. r is the parameter.

simple reactions if the photochemical quantum yield of the partial steps of reactions depend on the intensity of irradiation in the same way. This can be demonstrated using the following example of a consecutive

Fig. 3.17. Degree of advancement^ as a function of ;; for constant time T = 3. r is the parameter.

234

Simple and Complex Photochemical Reactions

Ch. 3

photochemical reaction: A~J2L^B—^C. It is assumed that the partial photochemical quantum yields tp^ and ip^ are constant. Then ^

= l{)OO^f
rfzj

^

= 10()Oy)2V3/ofeexp(^-|j£; Jzj

(3.119)

and (3.120)

are obtained using E'^ =£^o "^Qx^x -^Qz^i- ^ ^ division of eq. (3.120) by (3.119) gives

•

\

dx2

'

~

A

/

At


"^ At

a

'

CQ- JC,

Using the general coordinates ^,=^

and

?2=i^

one finds the equation

as the solution of this differential equation. This result correlates with the relationships between the reduced degrees of advancement in a stirred solution. But the equation above is only valid for a single volume element. Eq. (3.119) can be treated in the same way as eq. (3.106). One finds an analogous differential equation instead of eq. (3.115)

Ch. 3

Photoreactions in Viscous Media

= {E:o+Qiao^y+Q2ao$2)— oToz

235

(3121)

dt

with

If this equation is integrated in the same way as eq. (3.107), one finds a result which corresponds to eq. (3.109) \

r!zi

(3 122)

The inner integral has to be solved within the limits ?i(z,0) to ?,(z,r). By these means it depends on two degrees of advancement. Thus ^2 can be calculated using eq. (2.90). One finds

= -{^:.ln(l-^.) + ( G . - * - G . ) a o ^ l . - . ^ [ l - ( l - | , r ] ] The following abbreviations are used: 1~P2 '•.=»-^.=

l_;^'

^ _ I-P2 -

(1_;^);^

and

Therefore, in a similar way as in Section 3.4.3.2 according to eq. (3.122) one obtains

Simple and Complex Photochemical Reactions

236

Ch.3

(3.123) In Fig. 3.18 the curves for a = a/aQ, fi = b/aQ = §i ~ ?2 ^"^ V = ^/^o = ^2 are shown as functions of t} at time r = 3.00 for ^, (0,3) = 0.95. The ratio of the partial photochemical quantum yields;^' = ^2 ^ ^Pi" is assumed to be 0.5. For this reason tj has been calculated as a function of ^, according to eq. (3.115). The respective relative concentrations can be obtained using eq. (2.90). In Fig. 3.19 r is given as a function of ;; for constant turnover fj =0.1. The ratio pjp2 is chosen as a parameter, ;c' = 0.5 is used. The depth of penetration becomes larger with increasing ratio of p, to pjThe method presented here can be applied to many but not all complicated photoreactions. The derivations discussed above can be used in principle if the photochemical quantum yields of all partial steps of reactions do not depend on the amount of light absorbed, or if they depend on this amount of light absorbed in the same way. An example for the latter condi-

1.0

o.s 4 .

0.0

Fig. 3.18. Dependence of a,/5 and y and on rj. x' = 0.5, rj = 0.2 and r2 = 1.0. The time is taken to be constant.

Survey on Approaches for Selected Examples

Ch. 3

237

Fig. 3.19. r shown as a function of rf for a turnover f | = 0.1. x' = 0.5, ri/r2 is the parameter.

tion is a dependence on the square root of the amount of light absorbed according to
{k=\,2,...,5).

If the differential equations for the degrees of advancement have to be solved in a numerical manner, the computational expenditure can be increased drastically. If the quantum yields of the partial steps of a reaction depend on the amount of light absorbed in different ways, for example if slow dark reactions have to be considered, in these cases the method will fail. Then, the partial differential equations have to be solved numerically. 3.5 SURVEY ON APPROACHES FOR SELECTED EXAMPLES

In this and the previous chapter different approaches have been demonstrated to solve the differential equations for a variety of reaction mechanisms. Besides the set-up of differential rate equations for concentrations, the degree of advancement is introduced to reduce complex systems. Furthermore reduced degrees of advancement are demonstrated to result in gen-

238

PQ

c o

•c

I

o o

•a

t

c«

g

O

-II- 2O CO

<

PQ

H U

o

«o

^

CO

1b

en

2 c4

ON CO

.-4

X I

y

11

MP

II

en

6

«

II

I

1§ ^ 1^ *"

VO

y=

to

I li

f •i

^

c X

li

V|i
1 ^a' .
o

II

!.«

+

^

CO

_ !S

II

+ i^^'huf io.

1*4

> "C

br

II

» o

^^i^

? ^ --

1 ^w "TT

s

^

I

i

I

o X

+ -<

II

= -15

n.

I.

**

Is?

o

CO

*^

en

VO

Simple and Complex Photochemical Reactions

en

-I. -^^

4>

II 9.

ts

15 CO

t

Ch. 3

Ch. 3

U

s

I

Ul

I

o

II

I

II

N5

I

^hy^ -I II N5

t*4

9^

5'

+

o ^ H I a II •iij)

ttj

en

I, I

X

PPrJ

I Vf

9-

<^

II

§ ^< i 11 5 is;

I"

^.

II

M

I

I

+

I

-3

11

I

CI.

I

I

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«"«"

Survey on Approaches for Selected Examples

CO

§

1

4

3'I H U

239

240

Simple and Complex Photochemical Reactions

Ch. 3

eral coordinates. These consider the photokinetic factor and modify the time axis to find a general solution for photoreactions with quantum yields which do not depend on the intensity of the irradiation source. Most integrals cannot be solved in a closed form because of the time dependence of the photokinetic factor (except at specific reaction conditions). Therefore the integrals have to be solved numerically. When the solution cannot be homogenised during the photoreaction by stirring (viscous media) dependencies on time and position have to be taken into account. For two typical photoreactions these approaches are compared and summarised in Tables 3.12 and 3.13. In Chapter 5 experimental results are discussed for many of the mechanisms given in this and the previous chapter. These examples require information about the methods of photokinetic measurements. Therefore experimental techniques are discussed in the next chapter.

Chapter 4

Experimental Techniques in Photokinetic Analysis

4.1 INTRODUCTION 4J,1 Requirements of methods of measurement In the previous chapters methods were given to derive rate laws and to find correlations between concentration and degree of advancement. According to an assumed mechanism a system of linear independent differential equations can be set up. By these means the Jacobi matrix is obtained. Its elements are the rate constants or photochemical quantum yields as well as combinations thereof in the case of complex mechanisms. The prerequisite to set up these differential equations is some knowledge about the mechanism of the reaction, about the reactants, and the molecularity of the elementary steps. This requires information about the reaction obtained by time-resolved measurements during the progress of the reaction. General chemical knowledge and insight into reaction kinetics combined with data obtained about reactants and their time-dependent behaviour allows a mechanism to be proposed. Accordingly a system of differential equations is set up. Next the measured data are used to control the assumptions. Thus, the general approach consists of three steps which can be summarised as: (a) The first step has to be a check on whether the data fit to the assumed number of linear independent steps of the reaction scheme. As discussed in the previous chapters, linear dependencies between reactants or partial steps of a reaction have to be excluded. This can be done either by a numerical or graphical principle component analysis which supplies this number of linear independent steps [38,39]. This principle is discussed in more detail in Section 5.3.3 and for the applications demonstrated in Chapter 5.

242

Experimental Techniques in Photokinetic Analysis

Ch. 4

(b) In the next step, the measured data are used to control whether measured data and the proposed set of differential equations fit. Two approaches are discussed in the next chapter. The first is to solve the differential equations by means of a "formal" (numerical) integration and to generate simulated sets of data which have to fit as well as possible to the measured ones. The other approach is to fit the differential equations directly to the set of data and to deduce the parameters by these means. This requires the differential equations to be solved in a closed form, a task which is very difficult in photokinetics as discussed in the previous chapter. It can only be achieved by certain approximations transferring the photokinetic equations to pseudo-first order equations as in thermal kinetics. (c) Finally the rates and yields of the different partial steps must be determined. Thus, it is obvious that high quality of measured data is essential. It would be preferable to be able to monitor each reactant separately during the progress of the reaction. This determination of the different concentrations with necessary exactness and time resolution is one of the main problems in kinetic analysis. In brief the following requirements can be stated: (1) In liquid or solid phase photoreactions the solvent should not absorb at the wavelength of irradiation. (2) The reaction mixture has to be homogenised rapidly by stirring to reduce the complexity of the evaluation. (3) The detection method chosen is expected to selectively detect the single components of the reaction mixture. (4) The detection should be quantitative and the signal intensity should vary linearly with concentration. (5) The detection method should be fast enough to obtain all the data necessary during the progress of the reaction. (6) Automation of the monitoring of the reaction progress should be easily obtainable. In principle spectroscopic methods have turned out to be preferable in kinetic analysis. Nowadays a large variety of spectroscopic methods exist which allow the progress of the photochemical reaction to be monitored. However, none of them perfectly satisfies under all conditions the requirements stated above. Even though it is difficult to state a general procedure, some generalised ideas are discussed next to find arguments for the selection of certain analytical tools.

Ch. 4

Introduction

243

The low energy methods in spectroscopy, such as microwave, NMR, ESR, or IR spectroscopy, result in patterns which are very specific to different chemicals [40]. The same is valid for mass spectrometry. In principle these methods allow the concentrations of the reactants to be measured separately. However, the disadvantage of these methods is their rather complex set-up, some problems with exact correlation between signal and concentration as well as problems with requirement (1). Normally irradiation takes place in the UV or visible range. Water or alcoholic solvents are therefore favourable. However, these absorb in the infrared. Thus, IR spectroscopy exhibits disadvantages even though it is preferable because of its selectivity and some progress has been achieved in recent years using Fourier transform methods. With respect to photometry neither microwave, NMR, ESR nor mass spectroscopy really fulfil the requirements in photokinetics. Nevertheless, in Chapter 5 an example of the successful application of NMR spectroscopy is given. On the other hand, spectroscopy in the ultraviolet or visible is a very fast, simple, and photometrically exact method. Because of the broad bands in this wavelength region of electronic spectroscopy, however, the signals obtained give a summation over the concentrations of all the reactants in the solution. Therefore requirement (3) cannot be fulfilled except in special cases. On the other hand, on-line analysis can be easily achieved. The lack in selectivity is overcome by two approaches: - multicomponent analysis or - combination of UVA^is with separation techniques. Computers allow multicomponent analysis to separate the concentrations of the different compounds in the solution - but, the spectra of all the reactants have to be known. This latter requirement causes quite a few problems in kinetic analysis, since in general, intermediates or final products are not known a priori. This method is discussed with its possibilities and limitations in Section 4.2.6.3. On the other hand, combination of separation techniques such as thin layer chromatography and high pressure liquid chromatography with spectroscopic detection results in time-resolved signals for the different reaction products. In general chromatography uses UVA^is spectroscopy as the detection technique. However, the application of separation techniques hinders in-line monitoring. Under these conditions a quasicontinuous measurement technique has to be developed to fulfil the requirements mentioned above as demonstrated in Section 4.5.

244

Experimental Techniques in Photokinetic Analysis

Ch. 4

4 J. 2 Classification of the UV/Vis spectroscopic methods Taking into account all the considerations above, UVA^is spectroscopy [41,42] is frequently considered to be preferable for the examination of photochemical reactions. The method is simple, can be automated easily, and supplies data with high quality of photometry. It permits exact measurements, even during fast kinetics. Years ago most of the measurements were taken in absorption. Nowadays fluorescence as well as reflection spectroscopy have generated great interest. Since fluorescence measurements are very sensitive and allow detection of very low concentrations of compounds, their use can enlarge the applicability of absorption spectroscopy in many cases. Furthermore photokinetics in thin films or in solids are considered interesting. For this reason reflection spectroscopy has become the method of choice in many applications [43,44]. Thus, any successful kinetic analysis requires an optimal choice of the spectroscopic detection method, whereby special methods (e.g. pre-separation by chromatography) or modem approaches (e.g. interferometry) must also be considered. Therefore in the following sections different approaches in spectroscopy are presented, some combined irradiation and measurement devices introduced, and different set-ups classified with respect to their principle of operation (sequential, multiplex, single and double beam equipment). In addition some special devices are presented which allow an automated examination even of complex photochemical reactions (in part superimposed by thermal reactions) at a highly sophisticated level using various combinations of modem equipment and supplying data for multicomponent analysis.

4.2 ABSORPTION MEASUREMENT 4,2,1 Principle considerations Absorption spectroscopy in the UVA^is is a method which has been used for a very long time [41,43]. In principle each spectrometer applicable to photokinetics consists of four components: - the light source, - the dispersion element (monochromator).

Ch. 4

Absorption Measurement

245

-

the sample compartment (containing a stirrer for homogenising the sample), and - the detector. In addition modem equipment has interfaces to personal computers, work stations, or data stations. Either hydrogen or deuterium light sources are used in the UV, and tungsten halogen sources in the visible [45]. Both wavelength ranges can be covered by xenon lamps, but below 250 nm deuterium lamps are preferable. At wavelengths higher than 500 nm deuterium lamps show very poor intensity. Nowadays most of the spectrometers use gratings as dispersion elements [46]. Prisms are only built in expensive equipment with double monochromators, in which the first monochromator is used to reduce problems with overlapping higher orders and to increase wavelength resolution. The monochromator disperses the spectrum like in a rainbow across the exit slit. This slit selects one wavelength out of this dispersed spectrum in the case of sequential spectroscopy. Oxygen containing air restricts the possible region of measurement down to 195 nm unless the light source, all optical pathways, and the sample as well as the monochromator can be degassed by passing pure nitrogen. The sample compartment of the spectrometer should be spacious enough to allow the cell holder to be kept at a constant temperature. All of the differential equations mentioned in the previous chapters require at any time an irradiation of a homogeneous solution by parallel radiation. For this reason the solutions have to be stirred perfectly and continuously. The sample compartment has to be equipped with a stirrer and temperature control. The solvent should not be too viscous. 4.2.2 Apparatus for absorbance measurement 4.2.2.1 Sequential spectrometers Sequential spectroscopy selects one wavelength after the other by moving the spectrum dispersed by the grating across this exit slit. Either single or double beam spectrometers can be used in UVA^is absorption spectroscopy. In old single beam instrumentation, the wavelength had to be selected, sample and reference cells had to be measured one after another, and the next wavelength had to be chosen. In this equipment the wavelength drive had to be advanced manually. Modem single beam spectrometers automatically scan the total wavelength region, normally first for the reference cell. After-

Experimental Techniques in Photokinetic Analysis

246

light source switch-over mirror

Ch.4

quartz halogen lamp filter wheel

Deuterium lamp

entrance

aperture

4) n D <>

condensing lenses /

— ^

camnli* sample

Photomultiplier

exit slit chopper

concave holographic grating 1500 lines/mm

Fig. 4.1. Optical geometry of a single beam instrument, the Uvikon 710 by Kontron Instruments, Eching. wards the wavelength is reversed and the scan for the sample cell is started [47]. Even though fast scans are able to record wavelength ranges up to 1200 nm in less than 1 min., this type of equipment needs more time to scan one spectrum than double beam instrumentation. On the other hand, a less expensive optical geometry is an advantage (for this reason opaque samples can be measured more easily), and the measurement beam is not modulated by the beam splitter. In addition, polarisation of the measurement light can be avoided. In Fig. 4.1 the schematic diagram of a typical single beam instrument which is commercially available is given. In double beam instruments the radiation of the light source is divided into a reference and a sample beam either by a beam splitter or a rotating mirror. This equipment therefore continuously takes reference and sample signals just one directly after the other during the wavelength scan. The scan of the total wavelength range takes less time than with single beam instruments. However, optical and mechanical equipment is more expensive. Fig. 4.2 shows a typical instrument which has been used to examine photoreactions in thin films and for photokinetics in the student's lab.

Ch. 4

Absorption Measurement

247

halogen lamp

deuterium lamp

monochromator Fig. 4.2. Schematic diagram of the optical pathway in a double beam instrument Lambda 2, Perkin Elmer, Uberlingen.

Such double beam instruments can be controlled more easily in processes by relays or microprocessors. No mechanics for automatic exchange of sample and reference cells have to be included. The energetic efficiency of the light paths is lower. A double monochromator supplies higher quality photometry. The spectral resolution can be increased and the amount of stray light is drastically decreased. The slit in-between the two monochromator parts is essential. A high performemce instrument is shown in Fig. 4.3. Such spectrometers are rather expensive but are very useful in the examination of complex photoreactions as well as in the measurement of problematic samples such as turbid solutions, viscous samples, or thin films. 4.2,2,2 Multiplex spectrometers Whereas in sequential spectroscopy one wavelength after the other is selected by moving the spectrum across the exit slit of the monochromator, multiplex spectroscopy measures the whole spectrum at a time. In these instruments the white light of the measurement source is resolved with respect to wavelength either by a numerical approach as for example Fourier transform spectroscopy (preferable in the IR) or by imaging the dispersed spec-

248

Experimental Techniques in Photokinetic Analysis

Ch.4

halogen lamp monochromator 2

deuterium lamp

reference

monoUV/Vis chromator I Fig. 4.3. Double beam set-up with a double monochromator for very precise photometric measurements and the possibility of taking high resolution spectra (Lambda 9, Perkin Elmer, Uberlingen). This instrument covers the wavelength range from the UV until the near infrared (NIR).

trum to a diode array type of detector (laterally resolving element) [42]. In the latter case the exit slit does not exist (then the element of dispersion is called a polychromator). The different wavelengths fall onto different areas of a plane of observation. This plane consists of an array of photo diodes (128-1024 columns, 1 to many rows each). Each column of diodes will "see" a different wavelength of the dispersed spectrum. Fig. 4.4 shows an application to a HP 8452A spectrometer [48]. The total spectrum can be taken in less than 1 s. This set-up is therefore very useful in the measurement of fast processes. In general, in sequential UVA^is spectroscopy the sample is placed between the monochromator and the detector to avoid extreme exposure to high energy UV-radiation. The intensity reduced by the sample falls onto the detector. However, in the case of a diode array spectrometer a sample position behind the polychromator would result in a "wavelength gradient" (laterally resolved wavelengths) in the sample across the plane of observation, since at the exit of the polychromator a correlation exists between wavelength and space. For this reason, in multiplex spectroscopy the sample is placed directly in front of the polychromator. Under these conditions

Absorption Measurement

Ch.4

249

irradiation source water filter shutter — filter fibre optics

diode array

shutter deuterium lamp

©

tlo

cell holder temperature controlled monochromator

stirrer

sample compartment

Fig. 4.4. Modified commercial photo diode array spectrometer HP 8452A. SL, slit; L, lens; S, stirred and temperature controlled sample cell [49,50].

white light falls onto the sample and the radiation intensity is fairly high [42,43]. However, multiplex spectroscopy allows the observation of all wavelengths at a time. Therefore the overall exposure to white light can be reduced to as short as the time to measure at a single wavelength in sequential spectrometry. 4.2.2.3 Combined irradiation and measurement devices The simplest way to follow a photoreaction is to irradiate the sample for some time and to place it thereafter into the spectrometer to take a spectrum. This procedure is very time consuming, cannot be automated, and causes problems with reproducibility. For this reason in photokinetics a direct irradiation into the sample compartment of the spectrometer is preferable as has been demonstrated in the case of a diode array spectrometer (see Fig. 4.4).

Experimental Techniques in Photokinetic Analysis

250

Ch.4

The irradiation can take place either by incident light from the side or via fibre optics. Most of the instruments commercially available do not directly fit photokinetic requirements. For this reason a combined irradiation and measurement device was set up some years ago. Its schematic block diagram is given in Fig. 4.5. It combines a commercial double beam UVA^is spectrometer and a control unit. Its microprocessor clocks reaction and irradiation times, controls the wavelength drive of the spectrometer, acquires the digitised signals, controls the shutters in irradiation and measurement beams, transforms the digital information to an analogous output onto an XY-recorder, and stores the acquired data on tape [51]. The central processing unit (CPU) interfaces to the printer via a serial RS 232 C (V24) and to the tape via an 8 bit parallel IEEE 488 (lEC). The microprocessor which controls the spectrometer is programmed in assembler (approximately 32 KByte). This instrumentation allows direct irradiation of the sample compartment and the observation of the photoreaction without removing the sample cell for measurement. The instrumentation allows different types of measurement cycles:

stabilised power supply

ST75 clocks

digital voltmeter X (wavcl.) DMR 10

n

D R

PS2

command bus

y (data)

absolute reaction hv tape unit lEC control CPU

terminal

printer V 24

XY-recorder

Fig. 4.5. Block-diagram of a combined in^adiation and measurement device. DMR 10 (C. Zeiss, Oberkochen) process controlled by microprocessor TM 990 (Texas Instruments, Freising). PSl, PS2, alternatively active shutters; F, interference filter; B, beam splitter to P (photo diode) and sample cell (S), which is stirred; R, reference cell; ST 75, high-pressure mercury arc.

Ch. 4

-

Absorption Measurement

251

A measurement during thermal or photochemical reactions at selected wavelengths whereby the irradiation is stopped during the total time of all the wavelengths within the measurement cycle (synchronous measurement). - The irradiation is only cut off during the measurement at the selected wavelength itself. The irradiation is continued during the time the spectrometer needs to move to the next wavelength (asynchronous measurement). - The total spectrum (not at selected wavelengths as above) is digitised during the recording of thermal or photochemical reactions. In this case all wavelengths are taken at the same irradiation time, since the shutter in the irradiation pathway is closed during the recording. Different types of program cycles can be chosen to allow a relatively variable control of different types of processes. Even though optical pathways of irradiation and measurement are crossed, the intense light of the irradiation source can be scattered onto the photomultiplier. Therefore this set-up avoids the measurement of transmittance during the time interval when the irradiation beam falls onto the cell. This is achieved by two shutters, PSl and PS2, which are alternatively opened and closed, respectively (see Fig. 4.5). By these means during measurement the cell is not irradiated and during the irradiation the photomultiplier is protected against scattered photons. The choice of the second type of measurement cycle avoids long periods without irradiation. But a disadvantage must be overcome. Each wavelength is measured at a different irradiation time. In consequence the different wavelengths have to be synchronised with respect to a common irradiation time for all the wavelengths. Therefore the transmittance signals are taken for each wavelength and plotted versus the irradiation time (absorbancetime curves; see Chapter 3 and below). The absorbances at the curves for different wavelengths are interpolated between the measured absorbances. Thus the interpolated absorbances show the same time of irradiation for all wavelengths. This principle is demonstrated in Fig. 4.6. In fast photochemical reactions the measured data will have some time lag which is determined by the speed of the wavelength drive. Then, since synchronisation has to relay to just a few supporting points, interpolation becomes erroneous. Nevertheless this program must be chosen when dealing with thermal reactions or with the examination of photoreactions with superimposed thermal reactions.

Experimental Techniques in Photoltinetic Analysis

252

Ch.4

8i •e E

s i

' t

«

» 11 1

1 11

1

1 1

1

•

'I

' l l 1—1—1—1—1—1—1

1 1 1

r-r-l—<

I

» 11 1 1 11 1 1 1

I I I

1 ' 1—1—1—11—1—r

irradiation time Fig. 4.6. Schematic representation of absorbance/transmittance versus time curves. The symbols give the data points for three wavelengths, measured repetitively one after another. Synchronisation is obtained by taking interpolated values at equal times (perpendicular lines).

The sample compartment contains a specially constructed cell holder. It can be temperature controlled as well as stirred and supplies positions for photo diodes which allow continuous monitoring of the intensity of the irradiation source. This specific cell holder is demonstrated in Fig. 4.7 in more detail than in Fig. 4.5. Stirring of the sample at a constant temperature is essential in the case of highly intense irradiation. A thermoelement in the cell measures an increase in temperature from 20*^0 to more than SS'^C by long irradiation. The cell compartment permits the use of a flow cell, which can be used to circulate an externally irradiated solution, for example by a laser. This extended setup (see Section 4.6) has been used in the examination of the photostability of laser dyes by combined transmittance and laser-power measurements [52J. As mentioned above a specific control cycle is chosen for the recording of spectra or selected wavelengths. No wavelength scan takes place during the integration time of the signal. The microprocessor stops the stepper motor either at the selected wavelength or, in the case of digitising the spectrum, at any other 0.2 nm (or other wavelength distances) as long as the transmit-

Ch. 4

Absorption Measurement

253

Fig. 4.7. Cell holder with temperature control, stirred sample cell (S), and flow cell arrangement. Q, high pressure mercury arc ST75; IQ and / j , measurement beam with attenuated intensity, PSl, PS2, shutters; F, interference filter; B, beam splitter in the irradiation pathway for the control of the stability of the arc by two photo diodes Dl, D2.

tance signal is integrated and taken. Afterwards the wavelength scan continues. This principle is demonstrated in Fig. 4.8. The central processing unit of the microprocessor (CPU) selects the time until the next measurement cycle has to be started. In kinetic experiments the change in signal lessens during the course of the reaction. For this reason after each cycle of measurement the time until the next one is increased by a certain percentage. Beginning at the long wavelength end, the stepper motor advances the monochromator to the first wavelength marked ( i ) in the schematic spectrum in Fig. 4.8. The shutter PSl is closed and PS2 is opened. The transmittance value is taken. Then PS2 is closed and PSl is reopened. The wavelength scan advances to the next wavelength.

Experimental Techniques in Photokinetic Analysis

254

Ch.4

E

200nm

500nm restart

Z

start wait

stop

M PS 1 closed measurement PS I open continue stepping data transfer times, irradiation, transmisson, intensity

count steps

count steps next wavelength

end

return to H start wl

Fig. 4.8. Scheme of the control process during the wavelength scan and the measurement of photokinetic data. CPU, central processing unit; wl, wavelength; PSl, PS2, photo-shutters.

Meanwhile, the data are transferred to the memory of the CPU. It is possible to record the reaction and irradiation times at each data point as well as the intensity of the light source besides the transmittance signal. As soon as the last wavelength has been measured, the wavelength scan rewinds to the long wavelength side of the spectrum [51]. The scan stops and the next cycle of measurement is awaited. If full spectra are taken, the signals are converted to an analogous format for a XY-recorder. This type of recording allows exact overlay of a large number of spectra. Otherwise after a few spectra, isosbestic points begin to smear if the spectrometer strip chart recorder is used. The combination of an irradiation and measurement device needs some modification of the sample compartment of the spectrometer. In some cases problems with the geometry can arise. For this reason another approach was tried using the advantage of fibre optics. A bundle of quartz fibres (approx. 1 cm in diameter) allows process-controlled irradiation of the cell in a conventional double beam spectrometer (see Fig. 4.9) without any modification of the sample compartment. Only the cell holder has to be modified to allow the output of the fibre to be fixed [53].

Ch.4

Absorption Measurement

255

Fig. 4.9. Fibre optics conducting the photons of a high-pressure mercury arc into the cell compartment. The cell holder is temperature controlled and stirred (M). The circular area of thefibreoptic (L) is transformed to a rectangular cross section (T) fitting to the dimensions of the cell. In front of the light source a shutter PS and a lens are positioned. The interference filter F selects the wavelength. The irradiation device isfixedon top of the spectrometer by an optical bench (OB).

4.2.3 Measurement of photoreactions with superimposed thermal reactions In the case of photochemical reactions superimposed by fast thermal reactions during the time of measurement the irradiation is cut off and the irradiation time stopped. But the thermal reaction proceeds simultaneously and the overall reaction time will exceed the irradiation time. This principle of two time scales will increase problems with kinetic analysis. For this reason one tries to avoid interruption of the irradiation for such photochemical reactions. A set-up is therefore required in which absorbance is measured without interruption of the irradiation. Then the reaction time for irradiation and that for the overall reaction can be taken equally. This fact is a prerequisite in photokinetic analysis. These isochronic conditions are achieved by the setup given in Fig. 4.10. The sample is placed between two monochromators. Therefore the stray light of the irradiation source is reduced drastically. For this reason it is not necessary to interrupt the irradiation during the measurement of absorbance.

Experimental Techniques in Photokinetic Analysis

256

dataacquisition

electronics

100% slit motor control

power supply

m MM

Ch.4

S

D

L - 4 LI

D

L2

R

SA

IT

M

XY-recorder

W

HS2

h

SI

Fig. 4.10. Two-monochromator arrangement to reduce scattering of the irradiation source in the case of the measurement of absorbance without interruption of the irradiation. LI, L2: Deuterium and tungsten lamp, respectively; S,R, sample and reference cell; M,MM, single and double monochromator; SI, fixed slit width of the first monochromator; SA, S2, automatic slit stepper motor to adjust the reference to 100%; L, lens; W, wavelength stepper motor; P, photomultiplier; F,filter;Q, high pressure mercury arc.

Both the monochromators are adjusted to the same wavelength, the reference is taken, and the absorbance of the sample is measured. Afterwards the next wavelength is selected. In this type of measurement the irradiation times differ for the various wavelengths. Therefore the data points have to be synchronised as explained in Fig. 4.6. The above-mentioned modified HP photo diode array spectrometer (see Section 4.2.2.2) allows the interruption of the irradiation to be abandoned, since the measured stray light turns out to be negligible. In addition a shutter closes the pathway of measurement at those times when no signal has to be recorded. The sample is not illuminated by the spectral light of measurement but rather in the fraction of a second during which the spectrum is taken. Negligible interference of stray light makes the interruption of irradiation unnecessary. The synchronous recording of all wavelengths renders synchronisation superfluous.

Absorption Measurement

Ch. 4

257

4.2.4 Special devices In principle the combination of fibre optics and photodiode array spectrometers shows great advantages in photokinetics. Fibre optic technology allows a wide variation of such set-ups. In the 1960s one tried to achieve a versatile spectrometer by combining modules such as light source, monochromator, cell compartment, and detector on an optical bench. This arrangement of the different components allowed the measurement of absorbance, fluorescence, and reflectance. The same can be achieved nowadays by the use of a so-called simultaneous spectrometer module, schematically presented in Fig. 4.11. A ceramic block contains a grating and the diode array. A fibre optic guides the measurement light to the sample, the second arm of the bifurcated fibre optic observes the sample and transfers the information to the entrance slit of the spectrometer.

xenon light source

photodiode array Computer

(1),(2) (3)

(1) transmittance (2) fluorescence (3) reflectance

sample Fig. 4.11. Schematic diagram of a modular modem photodiode spectrometer containing a light source, bifurcated fibre optics, and the optical components in a ceramic block. With a different arrangement of the fibres a transmittance, fluorescence, and reflectance device can be built up.

Experimental Techniques in Photokinetic Analysis

258

Ch.4

fibre shutter

sample compartment

mirror with hole

Fig. 4.12. Modular diode array spectrometer with fibre optic for rapid spectral measurements of combined photochemical and thermal reactions.

This modular apparatus allows even very special applications. As a further example a combined measurement and irradiation device can be set up where the sample is placed in a Dewar and kept at low temperature. Fast shutters, a beam splitter, and a repositioning device for sample and reference allow the uninterrupted examination of photochemical reactions under specific conditions (see Fig. 4.12) [54]. The application of photodiode arrays in photokinetics results in many advantages, but they require some optical understanding to obtain photometricly correct data. For this reason their principles are explained in a later chapter in more detail. 4,2,5 Some considerations with respect to photometry 4,2,5,1 Limitations by Lambert-Beer's law According to Bouguer-Lambert-Beer's law (see Section 3.3.2, eq. (3.57)) the absorbance in decadic units at any wavelength is measured as a sum of the partial absorbances of all the absorbing components:

Ch. 4

Absorption Measurement

259

i'=n

Ex = d^e,,c,

(4.1)

1=1

In Chapter 3 Napierian units were used for absorption coefficients and absorbance. The reason was a simpler handling of the equations. Since instrumentation reads decadic units, they are used in most equations in this chapter. The decadic molar spectral absorption coefficient € of the compounds (i) varies with wavelength A. For this reason Bouguer-Lambert-Beer's law is restricted to monochromatic radiation. This restriction and the interactions between molecules in concentrated solutions cause two problems in applying the law to quantitative evaluation: (a) If the bandwidth of the measurement light is broader than 1/10 of the natural bandwidth of the sample, the measured absorption appears too low [41]. An "averaged absorption coefficient" is used which results in a convolution function in combination with the slit function of the spectrometer (intensity non-constant over the wavelength range). This gives absorbance values which are too low. If (in an extreme case) the wavelength range of the measurement light is broader than the natural bandwidth of the sample to be measured, the absorbance obtained and reality will not agree at all. As Fig. 4.13a demonstrates, the light source offers photons in a wide wavelength range (s). If the slit of the monochromator is too broad, photons at many wavelengths are not affected by the sample at all. Nevertheless, the same value for the absorption coefficient is presumed to be taken for all the wavelengths. The result is a totally incorrect absorbance value. In Fig. 4.13b the opposite case is given. The bandwidth of the measurement light is smaller than the natural bandwidth. The smaller the bandwidth of the monochromator the better the absorbance value obtained will be, even at shoulders and at steep slopes of the band. The "error" by a too broad band measurement light is called a "physical deviation" of Lambert-Beer's law [41]. (b) Chemical deviations happen if the compounds in the solution dissociate, vary their spectrum by pH, or form charge-transfer complexes. This means the real absorption coefficient will not be equal to the one used for the calculation of the concentrations. In addition to these deviations, which influence the linear relationship between concentration and the signal, any inadequate equipment can cause

260

Experimental Techniques in Photokinetic Analysis

(a)

Ch.4

(b)

Fig. 4.13. Gaussian shape of an absorbance band (p) and triangular slit function (s). (a) The light of measurement is very broad-band, therefore the absorption of the sample only covers a small part of the offered spectrum of light; (b) the offered light is absorbed proportional to the concentration of the sample for this specific ratio of slit function and natural band width. Therefore the somewhat polychromatic light is affected by various different absorption coefficients.

errors in measurement. Either stray light, bad photometry, or poor spectral resolution will impair the quality of the data [55]. 4.2.5,2 Influence of stray light Missing reproducibility of the wavelength scale causes extreme changes in the absorbance signal in the slopes of absorbance bands, even at small spectral bandwidths. Stray light, caused either by the solution itself or by the optical components of the spectrometer, also influences the absorbance signal. Even small percentages of such stray light produce drastic errors in spectral regions where the intensity of the light source is poor, the sensitivity of the detector is decreased, and the optical transmittance of the optical components is small. For this reason the measurement of absorbances higher than 1.5 at the low- or high-end region of the wavelength scale of any spectrometer are due to errors (see Fig. 4.14). Bit&y light ii uiUally ^piieified at 220 fimi In the Mse of low eost instrumiHfftit@R 11 mmm m /« M 6 . I # , tnmti of n nifii tmnimtitanet cif tht sample of 1% (absorbance E»2) the spectrometer reads 1.2% (i?== 1.922). That means the absolute error amounts to 4%. (For real £ = 3, /. =0.2%,

Ch. 4

Absorption Measurement

261

Fig. 4.14. Relative error in absorbance measurement for increasing absorbance values. Parameter is an increasing stray light; (A) ±0.075%; (B) ±0.05%, (C) ±0.025%, (D) ±0.01%, (E) 0.0025%, (F) ±0.001%.

measured E^ =2.523, A £ / £ = 18.9%.) This problem becomes worse if one measures the change in the main absorption band which varies drastically during a photochemical reaction. Under these conditions the error in measurement will vary during the photoreaction. The systematic residuals of curve-fitting techniques to the exponential absorbance-time curves will favour the inclusion of an additional step of the reaction to improve the fits. Additional errors can arise if the sample measured is not homogeneous (streak formation in micro cells). Furthermore interference to the beam of measurement by either the walls, the bottom of the cells, or the interface between solution and air must be avoided. 4,2.5.3 Multiple reflection In absorption spectroscopy the sample is usually measured with respect to a reference, normally a cell with pure solution. By these means multiple reflections at the cell walls can be eliminated [42,56,57]. In Fig. 4.15 some of these multiple reflections are given for the light paths in the cell. They can amount to some percent depending on the absorption in the cell. In the UVA^is range the refractive indices at the air/quartz or quartz/solution inter-

262

Experimental Techniques in Photokinetic Analysis

Ch.4

Fig. 4.15. Multiple reflections in a cell. /?], R2 are the reflectivities at the quartz/solution and quartz/air interfaces, respectively. The refractive indices of the media are marked by Hi, ^2, /I3. Only the essential light paths are drawn (/2, /4,..., /g). The transmittance coefficient tj of the solution influences the intensity of the multiple reflected partial beams superimposing to form I^.

faces cause reflections of approximately between 2 and 4% depending on the refractive index of the solvent. The following recursion formula, eq. (4.2), calculates the intensity /^ after the cell to be calculated relative to the intensity IQ falling onto the cell. Multiple reflections cause a too high value for measured L: / , = ( ! - /?,)(! - /?2)r,/o[l H- Rlr^ + R^r^ + /?2^f+-],

(4.2)

where the symbols are defined according to Fig. 4.15. 4.2,5A Further problems Another problem is measurement in turbid solutions. This problem arises in biological systems, e.g. in lipid membranes or in the examination of photo cross-linking of proteins. The photochemistry in micelles is also frequently examined. In all these cases the chromophore which is supposed to show the photoreaction can interact with the lipids. This interaction can either cause

Ch. 4

Absorption Measurement

263

scattering, a change in the spectrum (different maximum) or a change in the absorption coefficients by chemical interactions between chromophore and lipids. Micelles can sediment with time causing further problems with the correct measurement in dependence on time. In all these cases a linear relationship between the concentration and the signal is lost unless the coefficient of scattering follows an exponential relationship equivalent to Lambert-Beer's law. Even in this case the interaction between the chromophore and the support material (micelle) can depend on the concentration or the wavelength. Both facts require intense examination of the dependencies of the system before quantitative evaluation. Further details on problems in photometry and instrumentation are given in the following literature [40,42,43,58,59]. 4.2,6 Absorbance measurements for photokinetic analysis 4.2.6.1 Reaction spectra, absorbance time diagrams Even though applications are given in the next chapter, in this section the principle approach of the first steps of photokinetic analysis are discussed. By these means the necessity for spectral data becomes obvious. During the progress of a photoreaction, numerous spectra are recorded to observe spectral changes caused by the formation and degradation of the reactants. Spectral recording is advantageous compared to measurements at just a few wavelengths. Progress in a reaction can be better visualised. Normally, at the beginning of the photoreaction large spectral changes take place. For this reason automatic recording has to take spectra in such a way that in the interesting wavelength regions the absorbance values vary noticeably but not too drastically. Approximately 30-50 spectra should be recorded during each step of a reaction to allow a high quality evaluation. All the recorded spectra result in an overlay of spectra representing the progress of the photoreaction. This sequence of spectra is called a reaction spectrum. An example is given in Fig. 4.16 for the photoisomerisation of trans- to c/5-stilbene with consecutive rearrangement to phenanthrene. Since the dihydrophenanthrene is very unstable at the chosen conditions and time domain, it is not observable as an intermediate as in flash photolysis experiments [60,61]. In liquids and solids, absorption spectra do not permit the characterisation of the individual reactants. Therefore a reaction spectrum merely informs the photochemist about the complexity of a reaction at first glance. In addition, wavelengths can be selected which best represent the

264

Experimental Techniques in Photokinetic Analysis

Ch.4

<

320 340 360 Wavelength [nm] Fig. 4-16. Reaction spectrum of the photoreaction of /ranj-stilbene to phenanthrene via cisstilbene. A 10"^ molar solution is irradiated at 313 nm in n-hexane. The wavelength maximum of rran5-stilbene at 292 nm decreases and at 250 nm the new maximum of phenanthrene shows up during the photoreaction.

progress of the reaction and contain information about the different steps of the reaction. These wavelengths have to be selected from all wavelength regions to avoid loss of information. The wavelengths selected are taken for graphical and numerical evaluation of the reaction. If a reaction spectrum exhibits isosbestic points, the reaction is assumed to be a single step process. During complex reactions, time intervals can be found during which quasi-isosbestic points are recorded. In both cases the assumption of a simple reaction can be erroneous because of similar absorption spectra of reactants or lack of spectral resolution of the spectrometer. Therefore the simplicity of any reaction has to be proven by other methods. In any case an irradiation wavelength close to an isosbestic point or exactly at the wavelength of the isosbestic point reduces the expenditure in evaluation as discussed in Section 3.3.1 (Example 3.31).

Absorption Measurement

Ch.4 8

265

0.8 A

320 nm — • — 3 1 3 nm — • — 3 0 6 nm —A—292 nm —T—273 nm — • — 2 5 0 nm

o

< 0.6

0.4

0.2

0.0

100

200

300

Irradiation time [s] Fig. 4.17. Absorbance-time diagram demonstrating the change in absorbance at some selected characteristic wavelengths of the stilbene photoreaction. The wavelength 313 nm corresponds to the isomerisation of /ran5-stilbene; at 250 nm the formation of phenanthrene can be observed.

The first step in a graphical evaluation is to construct absorbance-time diagrams. For this purpose the absorbance values either measured at selected wavelengths or taken out of the reaction spectrum are plotted versus the irradiation time. Such an absorbance-time diagram is given in Fig. 4.17. If necessary it can be used to synchronise the different wavelengths to the same irradiation time (see Fig. 4.6). The information from such absorbancetime diagrams is used to construct so-called absorbance diagrams. In various combinations the absorbance values at two wavelengths are taken for one identical time of irradiation and entered in a diagram in which the absorbance at one wavelength is graphed versus the absorbance at another (absorbance diagram, E-diagram). The points for all the measured times form a straight line or a curve. The form of the curve obtained determines the number of linearly independent steps of the reaction. This graphical rank analysis is a prerequisite for any further kinetic evaluation (see Sections 2.1.4.4 and 5.3.3). A straight line in such a diagram is caused by a reaction which is at least spectroscopically uniform. Whereas an isosbestic

Experimental Techniques in Photokinetic Analysis

266

Ch.4

point in a reaction spectrum does not prove uniformity (especially if the isosbestic point is just in the long wavelength range of the reactions spectrum), the information taken from the absorbance diagrams is more certain. To be on the safe side, wavelength combinations for such diagrams have to be chosen out of the total spectral range. If curved absorbance diagrams are obtained, the dependencies of the absorbances at different wavelengths are linearised in absorbance diagrams of higher order [62] by mathematical manipulation. In Fig. 4.18 an example of a curved absorbance diagram is given representing the photoreaction of stilbene at two selected wavelengths. These visualise the two different steps of a linear independent reaction very specially. At either of these two wavelengths one of the steps is barely observable. The determination of the number of linear independent steps of a reaction does not solve the problem that many mechanisms result in the same number of these steps. This problem has been discussed extensively in Section 3.2.3. As an example the following two mechanisms cannot be distinguished by these means:

A^B-^C,

B^A->C

ON

CO

u u c

<

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Absorbance at 250 nm Fig. 4.18. Absorbance diagram of the photochemical reaction of rrans-stilbene.

Ch. 4

Absorption Measurement

267

In consequence without further knowledge such types of consecutive and parallel reactions cannot be distinguished. To evaluate the rate laws derived in Chapters 2 and 3, the concentrations of the reactants have to be known. For this reason the next step is to get the concentrations dependent on time out of the reaction spectrum or the absorbance-time curves. For further details see Sections 5.1 and 5.3.3. 42,62 The determination of concentrations from ahsorhance measurements An absorbance of just one reactant in the solution to be examined is a trivial case. Then eq. (4.1) can be directly used to transform the concentration in the rate equation to a relationship containing the absorbances. In general under practical conditions it is very exceptional to find wavelengths at which just one reactant absorbs. For this reason methods have to be developed to extract the different concentrations out of the overall absorption spectrum. This is very difficult since absorption spectra in the UVA^is are very broad band and most of the spectra of the reactants overlap. If the reactants and their absorptions are known, a method called multicomponent analysis can be used. 42,6.3 Multicomponent analysis Using eq. (4.1) and defining the path length of the cell as d and taking decadic units for the absorption coefficient, the absorbance can be given in general in decadic units by E = {e^a^ -¥6202 +---+£,a, +---+e„a„)rf

(4.3)

Measuring various wavelengths A (A = 1,2,3,..., /i) one obtains the equation system for i reactants (/ = 1,2,...,n)

Eh={^hi^\

+^ft2«2

'^""^^hn^n)d

268

Experimental Techniques in Photolcinetic Analysis

Ch. 4

which can be written as a matrix equation E=e a d .

(4.4)

If the examined system is allowed to be in different states / (/ = 1,2,.. .,m), eq. (4.4) can be expanded to E=e a J

(4.5)

The elements of the matrices E, e and a are represented by Ej^i^e^., and a^i, whereby the states can symbolise different reaction times, pH values, etc. Taking the photoreaction of rmn^-stilbene as an example, absorbances for three selected wavelengths at a chosen time of irradiation are given for an optical path-length of J = 1 cm by the following summation over the partial absorptions of three reactants: ^ 3 2 0 ~ ^320,A^ "^ ^320,B^ "*" ^320,C^ ^ 2 9 2 ~ ^292,A^ "^ ^ 2 9 2 , 5 ^ "^ ^292,C^ ^ 2 5 0 ~ ^250,A^

(4.0)

"*• ^250.fl^ "^ ^250,C^

In reality the linear equations contain a value C;^ which takes into account that the measured absorbance includes some error caused by noise or by errors in the determination of the absorption coefficients. These coefficients are obtained by calibration. The pure reactant is weighted many times and its absorption coefficient is determined by a calibration curve as a result of many measurements at different concentrations. The absorption coefficient has to be evaluated statistically and contains a variance in dependence on the number of measurements of the blank, the different concentrations, and the exactness of the calibration itself [39,42]. As demonstrated above this system of linear equations can be simplified using vectors and matrices. The absorbances measured at the three wavelengths give the vector E. The concentrations of the three components A, B, and C produce the vector of concentrations a. Finally the nine molar decadic absorption coefficients of the three components at the three wavelengths of measurement form a matrix e in eq. (4.5), whereby

Absorption Measurement

Ch.4

269

I ^313 \

E=

and

'292

a= b

V • J represent the vectors just for a single reaction time. The matrix e has to be determined using the pure substances by calibration methods discussed above. According to these equations the concentration of each compound can be determined by use of a measured spectrum E of the solution. Since neither E nor 8 can be determined without any error, it has to be included in the matrix equation as

E=earf + e

(4.7)

Neglecting errors e at first, the matrix equation must be solved with respect to a. The procedure is multiplication of the equation from the left by the transposed e -matrix, which is obtained by an interchange of rows and columns of the original matrix. This approach is called linear regression [39,63,64]. The result of the first step of the solution of the system of linear equations is the so-called normal matrix N. A prerequisite is h>n. For /i = n the matrix becomes quadratic. e a=E

-^

e*ea = e'E

N-a = e'-E

(4.8) (4.9)

The determination of a is required. For this reason in the above equation the normal matrix has to vanish. Multiplication of a matrix by its inverse results in a unity matrix with 1 along the main diagonal. Therefore in the next step the normal matrix is inverted and its inverse is multiplied from the left by eq. (4.9). The result is N-a = e'-E

•-»

N"'•N.a = N"^e'-E

->

l - a = N-^-e'-E

-^

a = N-'-8'-E

(4.10)

270

Experimental Techniques in Pliotokinetic Analysis

Ch. 4

Consequently a measurement of the absorbances at some wavelengths is necessary. Knowledge of the molar decadic absoq)tion coefficients at these wavelengths enables the vector of the concentrations of the reactants to be obtained at the chosen irradiation time. However, the inversion of the matrix causes some numerical problems. Therefore in the literature quite a few algorithms are described which try to successfully invert matrices even in the case of badly conditioned matrices of the absorption coefficients [39,41]. The calculation of the a -vector according to the procedure given above should result in values which are mainly influenced by numerical problems with respect to the inversion process. However, neither the measured absorbances in the course of kinetic analysis nor the absorption coefficients obtained by calibration procedures are faultless. On the contrary these errors have to be taken into consideration in eq. (4.4) by an additional vector e which takes into account noise in measurement and errors in calibration. It is a residual vector. In consequence, by use of the measured absorbances, in the following the concentration vector obtained is used to recalculate absorbances. The difference between calculated and measured absorbances correlates to the residual vector e=E-8a.

(4.11)

a illustrates that the calculated concentrations just result in approximate values. Optimal concentrations are obtained as soon as the variance is minimised. For that the residual vector is multiplied by its transpose: |e|^=(E-e-Sy(E-e.a).

(4.12)

This procedure is called multi-linear regression and uses least squares approximations. The calculation can be improved with respect to numerical problems by applying pivoting in the course of matrix inversion. Weighting of the wavelengths used dependending on the relative error takes into account that the standard deviation in the calibration matrix varies with wavelength. The variance becomes constant [39]. The result depends on the condition of the calibration matrix given, e.g. by Hadamard's condition value or by the specifity of the calibration matrix obtained by the determinant of the quadratic calibration matrix [65-67].

Ch. 4

Absorption Measurement

271

Another possibility to select optimal wavelengths is the variancecovariance matrix [68]. The fundamental problem of this approach is that the algorithm tries to minimise the residual vector. If an additional unknown concentration is hidden in the spectrum measured, interactions between the reactants exist which were not taken into account in the calibration. Therefore the algorithm has to result in an error. This means the solution I has to be erroneous. In this case the minimisation of the least squares sums, the variance, has to give an erroneous result in principle. This means that for some of the concentrations, a wrong amount is added to compensate for the additional compound or the interaction. To avoid these problems an inverse approach is sometimes used [69]. During the calibration procedure no pure reactants are used but various mixtures of them [39]. For this reason the calibration returns a matrix of absorbances for the different mixtures at many wavelengths. In this case the parameter vector does not represent the concentrations but its portion of the mixtures. According to the least squares approximation the variance is also minimised. However, in this case the variance represents the difference between the estimated and the real portion. Since mixtures are used for calibration, in principle - a priori non-linearities are not excluded, - unknown components are automatically included, and - the inversion of the covariance matrix (e' e) gives a relatively bad solution because of possible linear dependencies between the mixtures. In consequence the condition number can turn out to be high. In general two approaches are used in the literature to find optimal parameters [39,42,70]: - The principle component regression (PCR), which uses an eigenvectoreigenvalue decomposition or a single value decomposition or a singular value decomposition of the matrix E [38,71,72]. Both require a lot of calculation time. - In contrast, partial least squares (PLS) [73] reduces the calculation time by a different type of reduction of the E-matrix by use of the Lanczos bidiagonalisation. Both approaches adjust a different method of calculation to determine the estimated parameter value. In both cases a systematic error (bias) is superimposed to reduce the high variances, caused by the inversion of badly conditioned matrices. In Fig. 4.19 the different algorithms are compared schematically [38].

Experimental Techniques in Photokinetic Analysis

272

fMLR as BLUE

6VCR

Multivariate Linear Regression MLR

Pricipal Components Regression PCR

Ch.4 S^LS

Partial Least-Squares Regression PLS

D

n

Matrix inversion

Singular Value decomposition

Lanczosbidiagonalisation

a

0

D

Classic approach c = Xb

^

n Inverse approach c = Xb

a Multivariate data (e.g. spectroscopic data), Bouguer-Lambcrt-Bcer

Fig. 4.19. Schematic comparison of methods of multivariate data analysis using the classical and inverse approach. BLUE, best linear unbiased estimator; PRESS, predictive residual error sum of squares.

In principle both the classical and the inverse approach use a multivariate data set. But in the classical approach the variance is minimised, whereas in the inverse approach one tries to find an equilibrium between bias and variance. Therefore the bias is reduced and by the procedure of predictive receivable error sum of squares either via a singular value decomposition or the bidiagonalisation method estimated values, either according to principle component regression or partial least squares, are found. The multilinear regression on the other hand will find the best linear unbiased estimation as an approach to a "true" concentration. Besides applications in absorption spectroscopy, fluorescence spectra can also be evaluated [74]. Another approach to multicomponent analysis is the use of Kalman filters [75,76], applying neural networks [77,78], or even fuzzy logic [79]. The quality of these approaches depends on the number of reaction components, the condition of the calibration matrix, the errors in determination, the absorption coefficients, and the measurement of the absorbances during kinetics. In any case the use of any of these algorithms can cause problems depending on the chosen algorithm and the specific behaviour of the reaction

Ch. 4

Absorption Measurement

273

spectrum. Therefore in practice such muhicomponent analyses are restricted to well behaving matrices with absorption coefficients given by significant changes in absorbance at the wavelengths of measurement during the overall reaction. In general the more reactants are to be considered the greater the problems will be. Applications of various types of muhicomponent analysis in kinetics are published for principle component analysis combined with target transformation [80] and using neural networks [81]. It is obvious that during kinetics one or the other reactant will turn up in relatively small concentrations. This fact is an additional problem compared to muhicomponent analysis in stationary systems as in the normal analysis of sample mixtures in an analytical laboratory. The dynamic process - first hinders the experimentalist from taking as much time as necessary to increase the integration time for the measured signals and - second causes the relative change in concentrations, therefore one or the other selected wavelength will become better or worse for observation. For this reason one takes as many wavelengths as possible out of the reaction spectrum, uses this over-determined system of equations to calculate the best condition of the matrix, and tries for an optimisation of the covariance matrices. The use of pattern recognition principles [82] is intended to classify and discover the main features of the experiment. In this cases elements of factor analysis [38], [72] must also be used. It becomes obvious that the more complex a reaction is the more difficult evaluation will be. In any case one finds that the less the spectra of the various reactants differ, the fewer the number of steps of reactions and included components can be. The above-mentioned approaches take for granted that the spectra of all the reactants are known. This is not at all the case in kinetic analysis, although usually at least the intermediates are unknown. As is demonstrated in Chapter 5, even reactions can be evaluated under these circumstances. But usually the constants obtained from the over-determined system of differential equations are only proportional to the interesting kinetic constants or a combination of them. For this reason absorption spectroscopy is likely to be combined with other methods. An example of combination with fluorescence and with chromatographic principles is given in Chapter 5. 4.2 J Derivative spectroscopy In analytical chemistry derivative spectroscopy is frequently used if the

Experimental Techniques in Photokinetic Analysis

274

Ch.4

absorption bands show poor characteristics. A typical application is the examination of biological material in turbid solution. Differentiation of the signal visually enhances small but sharp peaks compared to broad bands with little curvature. In the literature the different principles of differentiation are discussed as optical, electronic, and mathematical methods [42,83,84]. In contrast to integration each differentiation increases the noise of the signal. For this reason opinions in the literature [85], that the amount of information increases with the order of differentiation, are infrequent. In general mathematical differentiation is superior to the differentiation of the analogue signal, because in the latter case the original data are lost and not available for later further manipulation. Furthermore differentiation happens with respect to time instead of wavelength in the case of used analogue signals. Time constants and any fluctuations of the electronic parts of the spectrometer influence the result. However, these aspects are discussed controversially in the literature [85,86].

I 0.04

-0.12 200

220

240

260

280

300

320

340

360

Wavelength [nm] Fig. 4.20. Derivation reaction spectrum of the photoreaction rra/w-stilbene to phenanthrene via cw-stilbene in n-hexane by irradiation at 313 nm.

Ch. 4

Fluorescence Measurements

275

Even derivative spectra do not produce an increase in information compared to zero order absorption spectra. Differentiation can visualise the wavelength regions in which spectral changes are exceptional. For this reason derivative spectroscopy has been applied to kinetic analysis [87]. Fig. 4.20 demonstrates an example of a derivative reaction spectrum given for the above mentioned photoreaction of stilbene. The second derivatives are shown. The advantage in selection of characteristic wavelengths improves the kinetic evaluation as discussed in Chapter 5. However, in each application the optimum between increased noise and extracted information has to be found.

4.3 FLUORESCENCE MEASUREMENTS 4.3.1 Correlation between fluorescence and structure As mentioned in Section 1.3.2 the chemical constitution of an organic molecule determines whether it fluoresces or not [88,89,90]. In the vapour phase and in solvents, intra-molecular quenching effects determine whether emission will take place in addition to radiationless transitions. Furthermore in solvents the interaction with non-excited molecules (mostly solvent molecules) can cause an intermolecular deactivation process, which also reduces the probability of fluorescence. In principle, high fluorescence yield can be expected, if - The absorption coefficient of the long wavelength absorption band is high. This correlates with a high oscillator strength (given by the integration of the absorption coefficients over wave number) and a high rate of fluorescence decay. - The transition at the lowest energy is at wavelengths as long as possible to reduce the probability of photo-dissociation. - The emission starts from a relatively weak bonded orbital. - The molecule lacks such functional groups which increase radiationless transition. These requirements can be summarised in the following rules: - Fluorescence overcomes other processes the better, the smaller the excitation energy is in comparison to the energy of bonding, the more delocalised the jr-orbital is, and if n ~> jr transitions are at shorter wavelength than JT —> jr transitions.

276

-

Experimental Techniques in Photokinetic analysis

Ch. 4

Furthermore low temperatures and solvents at high viscosity will increase the yield of fluorescence. - On the other hand, radiationless transitions will reduce fluorescence, if the molecule has substituents which are more weakly bonded than hydrogen; e.g. substituents like -I, -NOj, -NH3*. They cause predissociation, a process which also reduces fluorescence. One can distinguish a phenomenological and a physical theory. According to the first one gets a more qualitative impression of the correlation between fluorescence yield and chemical constitution. The term fluorophore is introduced in combination with auxochromic groups, whereby any possibility of exhibition of torsion or hindered rotation will reduce fluorescence drastically. In some cases, such as anthracene or perylene, substitution hinders fluorescence. But in erythrosin containing four iodine substituents one finds very good fluorescence. Any additional bonding between two rings via a second bridge, especially including heteroatoms, results in fluorescence. According to the physical theory, fluorescence is treated as a concurrent process to photochemistry, intersystem crossing to the triplet state, and radiationless transition. Therefore, any intramolecular processes will reduce the emission. Interaction and energy transfer either in the molecular skeleton itself or to other molecules can be considered reasons for reduced fluorescence. Some typical examples are given by molecules in which electron excitation energy is transformed into vibrational energy. The reason can be either energy levels of the atomic vibration at close distance or weak bonded substituents of large mass. Furthermore fluorescence yield is reduced, if vibrational equilibrium distances differ greatly for the ground and excited electronic states, since any electronic excitation is combined with highly excited vibrational states. In principle the potential hypersurfaces as well as coupling between the different states have to be considered. In general high energy of excitation in combination with high bond strength will increase the fluorescence. Furthermore good resonance can stabilise the system of bondings. This means normally any substitution reduces the bond strength. On the other hand, good selection of auxochromes can even increase the stability of the jtsystem. The following effects can cause a reduction of the fluorescence yield by intermolecular deactivation: - The solvent will drastically affect the fluorescence intensity of the solute. This can be either caused by its polarity which influences the ground

Ch. 4

-

-

Fluorescence Measurements

277

and the excited states differently, whereby a drastic wavelength shift happens. Solvents containing heavy atoms increase the intersystem crossing rates drastically. The pK values are different for the ground and the excited state, sometimes varying many orders of magnitude. Therefore any change in the pH will effect fluorescence yields. Another reduction of fluorescence can be caused by hydrogen bridging. Since the concurrent processes such as radiationless deactivation are dependent on temperature, a variation of the temperature will also influence the fluorescence. Paramagnetic O2, metal ions, or high concentration will reduce the fluorescence intensity either by electron energy charge transfer or by impact deactivation.

4.3.2 Fluorescence spectrometer In gases and liquids, molecules are randomly distributed. For this reason those organic molecules which exhibit fluorescence emit the absorbed electronic energy into the space with radiant distribution. Therefore an observation in the optical pathway of the beam of excitation adds both the fluorescence intensity and the amount of light penetrating the sample. In principle the wavelengths of excitation and fluorescence differ but there are many cases where the fluorescence spectrum overlaps with the excitation wavelength. For this reason the arrangement of fluorescence measurement is somewhat changed in comparison to absorption measurements. The sample is usually observed at a W angle with respect to the optical excitation light beam. The principle set-up is given in Fig. 4.21. Good instrumentation uses a monochromator in the emission pathway, since normally some amount of scattering is superimposed on the fluorescence. In contrast to absorption spectroscopy the use of a reference is rather difficult. It requires a sample for which absorption and fluorescence spectra both exhibit the same spectral distribution as the probe. This is quite impossible. For this reason in some instrumentation one tries to eliminate fluctuations in the intensity of the light source. The excitation beam is split and a small amount of the intensity of excitation is focused on a photodiode. Nevertheless neither the changes in the optics of the instrumentation nor the variance of the transmission curve with wavelength can be overcome. However there is a way to calibrate the spectral distribution of a fluorimeter in-

278

Experimental Techniques in Photokinetic analysis

?

fluorescent light

F or Ml

Selection of excitation wavelength \

Ch.4

^o» ••• ^ n ForM2

1

Selection of emission wavelength 7^

v±yPM Fig. 4.21. General set-up of a fluorimeter system containing filters or monochromators in both the excitation and emission pathway. Q, light source emitting polychromatic light in the wavelength range A], A^,..., A„; F, filters; or Ml, M2, monochromators; P, photodiode for control of excitation intensity fluctuations; A^, A^, emission and observation wavelength; fluorescence light in the wavelength range A^,..., AQ; P M , photomultiplier.

strumentation using standard fluorophores with known fluorescence quantum yield [91]. This task is very tiresome and expensive. Fortunately in reaction kinetics the absolute value of fluorescence intensity does not have to be known. Since fluorescence intensity is proportional to the concentration of the fluorophore in dilute solutions, changes in time dependence of fluorescence can be correlated with a change in concentration. If even one of the components fluoresces, photokinetic evaluation becomes very simple. Nevertheless the measurement of fluorescence intensity to obtain concentration changes in reaction kinetics has been considered rather inexact. In absorption measurements small spatial fluctuations of the light sources do not cause extreme problems, since the diameter of the measurement beam is small in comparison to the width of the cell and the overall dimension of the active surface of the photomultiplier. On the other hand, small fluctuations of the excitation light beam drastically influence the geometry in the case of fluorescence measurements. Therefore small changes in overall intensity of fluorescence caused by the fluctuation of the plasma light sources become large with respect to the different volume elements in the cell. These volume elements are monitored by the entrance slit of the observation monochro-

Ch.4

Fluorescence Measurements

279

mator. The small slit causes an extreme dependence on the radiant geometry. For this reason very dilute solutions are chosen to reduce an intensity distribution in the cell [92]. But good stabilisation of the light source, parallel measurement of the intensity of the excitation light source, and exact geometry of the optical pathway allow quantitative evaluation of fluorescence signals. In principle the wavelength of irradiation and excitation are the same. The excitation light source is used to cause the photoreaction. By these means monochromatic irradiation is used automatically. A typical cell compartment is given in Fig. 4.22. The sample is stirred and temperature controlled. Three positions are mounted tilted at 45'' relative to the optical axis of the excitation light beam. One position is for the cell with the solvent, one for the fluorescence standard, and the third for the sample itself. These three positions absorbance measurement thermostat

photomuiti plier monochromator

amplifier 1

photodiode

YT recorder ADC unit

TM 990 labdatastation

St 46 stable power supply

Fig. 4.22. Combined absorption,fluorescencemeasurement, and irradiation compartment, /Q is controlled by a split-beam and a photodiode. The 45° tilted bar contains three positions: (1) for afluorescencestandard (S), (2) for the sample (R), (3) for a reference blank (v). The sample can be stirred. The cell holder is temperature controlled. The irradiation source is either a ST 46 or ST 75 (Quarzlampengesellschaft Hanau).

Experimental Techniques in Photokinetic analysis

280

Ch.4

St 41

M i

1

I E i

1

VI

1

S 1

I)MR 2 1

Fig. 4.23. Combined absorption andfluorescencespectrometer device in a schematic representation. An automatic dual beam DMR 21-spectrometer and a single beam PMQIIphotometer are used. The cell holder ZFM4 combines both instruments.

allow the control of the relative intensity of the light source for different photoreactions by use of the fluorescence standard. Therefore comparable results for different reactions are possible [93]. This specific 45"* arrangement also allows transmission measurements. Another arrangement for combined measurement of absorption and fluorescence information is given in Fig. 4.23, using a DMR 21-spectrometer, a PMQII-photometer and a fluorescence cell holder (ZFM4) [94]. The principle fluorescence arrangement given in Fig. 4.21 allows the measurement of fluorescence and excitation spectra. In the case of fluorescence measurements, the wavelength of excitation is kept constant. Only the emission wavelength is varied to take a fluorescence spectrum. In the case of an excitation spectrum, the wavelength of observation is set to the maximum of the fluorescence spectrum and kept constant. During the measurement of the excitation spectrum, the wavelength of excitation is varied. This proce-

Ch. 4

Reflectance Measurements

281

dure allows spectra to be obtained which are caused by the variation of the absorption coefficient with wavelength. Accordingly the measured spectra show a resemblance to absorption spectra. This method can be used in multicomponent solutions in which just one of the unknowns fluoresces. In this case the principle appearance of its absorption spectra can be obtained.

4.4 REFLECTANCE MEASUREMENTS When the samples are transparent, reactions at the interfaces have to be monitored by reflection spectroscopy. In Fig. 4.11 a device is shown which, using fibre optics, allows the direct measurement of reflectance at the interface with a commercially available photodiode array. Some conventional recording spectrometers are also equipped with fibre optics. Such devices are used to measure colours at the surfaces of metals or clothes directly [95]. In principle in Section 4.2.5 it was discussed that reflectance at an interface usually disturbs the absorption measurement. In the present applications, however, this reflectance allows reactions at the surfaces to be monitored. Since nowadays solid samples are exhibited to photo-induced processes, such measurements are a convenient application. In principle two types of reflectance take place [42,96]: (1) The regular or specular reflectance at a mirror surface for which the geometric laws are valid which results in the same angle of exit as for the incidence. Examples of this type of reflectance are metals, glasses or crystals. (2) Diffuse reflectance at interfaces which are structured and are microscopically inhomogeneous such that a large number of micro-reflecting surfaces distribute the reflected light statistically into space. There are quite a few cases where these two types of reflectance appear in combination. 4,4,1 Principles of reflectometry 4,4.1.1 Diffuse reflectance The quantitative description of diffuse reflectance measurements is known from thin-layer chromatography. Taking a powdery sample with such a large thickness (rf = ©o) that no reflection at the support material can be observed, the reflectivity R^ of the sample is connected to the stray coefficients as well as with the absorption coefficient k . Kubelka and Munk [97]

Experimental Techniques in Photolcinetic analysis

282

Ch.4

have developed a correlation between these absorption and stray coefficients and the dependence of the reflectivity R^ according to

s

(4.13)

2R^

The theory requires the following prerequisites [96]: The Lambert cosine-law has to be valid (isotropic distribution of stray light), that means the regular reflectance can be neglected. - The scattering particles of the layer are distributed statistically. Their dimensions are smaller than the thickness of the layer examined. - This layer is irradiated by diffuse monochromatic light or by parallel light at 60** incidence. If the layers are too thin, the stray coefficients become so small that no isotropic distribution of stray light is formed. The equation given above allows a quantitative determination of the concentrations of the particles in such a layer. It can be used to follow a photoreaction in powdery medium. This type of layer is schematically represented in Fig. 4.24. The light path for the incident and diffuse reflected radiation are included in the figure. Problems with quantification are well known from the application of thin-

k

R

•I t'- '1 r i T Fig. 4.24. Schematic representation of the propagation of incident and reflected radiation in a diffuse scattering layer. I, light path in the incident direction; J, light path in the opposite direction; X, local element for calculation; I^, incident intensity; R, intensity of reflected radiation; R,, reflected radiation at the support layer; T, radiation penetrating the layer.

Ch. 4

Reflectance Measurements

283

layer chromatography. A principle problem of the direct observation of photoreactions on a silica or alumina thin-layer as used in thin-layer chromatography is that both types of substrate activate organic molecules, especially during irradiation. Thus, such a type of kinetic analysis will vary the kinetic mechanism. However, this fact has to be taken into account in any thin-layer chromatographic analysis of solutions done to separate the reaction solution. 4A. 1.2 Regular reflectance Diffuse reflectance spectrometers offer a tool to examine on-line photoprocesses on diffuse reflecting substances or to control the photostability of coloured materials as pigment or car lacquer. Either the set-up using fibre optics (see Fig. 4.11) or modified thin-layer equipment (Fig. 4.25) can be used.

photomultipiier helping mirror slit filter

uv-achromate

X y carrier with sample 3

filter

mirror quartz lens Fig. 4.25. Schematic pathway of radiation in a typical thin-layer chromatography scanner. This set-up allows the measurement in transmission, reflection and fluorescence. The optical arrangement and the light source are varied according to the problem.

284

Experimental Techniques in Photokinetic analysis

Ch. 4

At mirror surfaces, incident radiation can be reflected regularly. This reflectivity (the ratio of the reflected to the incident radiation power) depends on the refractive indices of the two layers forming the interface [96,98]. According to the Fresnel relationships, the reflectivity is given by R - ^ i ^

(4.14)

The equation above gives only the simplest case where the irradiation is incident perpendicularly and the medium is not absorbing. However, in photokinetics normally conditions where the sample shows absorption are of interest. For this reason another equation has to be used, given by /? = U

iZ-

L_

(4.15)

where K is the index of absorption [43,98]. This type of measurement requires a lot of expenditure to evaluate the quantitative changes in concentration. In principle, a transmission and a reflection measurement have to be combined [99]. However, in some cases the reflection spectra can be used to obtain the concentrations by measurement of regular reflection, if the absorption coefficients of the components are known. This type of evaluation has been applied to reaction dynamics and thin films [100]. It is rather difficult. Nevertheless, it offers the only chance to examine photoprocesses in photoresists on circuit boards, which are not transparent [101]. In contrast to this rather difficult determination of changes in the concentrations of reactant, in a few applications reflection spectroscopy is simply used for the measurement of absorbance by an indirect method using a mirror behind the cell. The light beam of measurement passes twice through the cell. It is either coUimated by lenses, or fibre optics are used. A typical application of such a type is reflectometry in chemical sensors [102]. Another prospective application of reflectance measurements in the case of strongly absorbing solutions takes advantage of the principle of attenuated total reflection (ATR) [98].

Ch.4

Reflectance Measurements

285

4,4.2 Set-ups in Reflectance Measurements In Fig. 4.25 a typical arrangement of the measurement of diffuse reflectance is given. Such a set-up is commercially available and any thin-layer spectrometer can be used which normally scans the thin-layer plates. The coUimated optics of the set-up demonstrated can be substituted by fibre optics. This type of equipment is shown in Fig. 4.11 using a bifurcated fibre and a diode array. Under these conditions white light has to be used to produce a reflection spectrum monitored by the diode array. However, white light causes undefined photochemical conditions. For this reason a third fibre arm is used, which guides monochromatic radiation to the sample. If the white radiation is blocked by a shutter, it may not disturb the monochromatic photoirradiation. Since modem diode arrays take spectra within a few milliseconds the time of measurement stays short compared to the total photochemical process time. This type of equipment can be used to monitor the photoprocess during irradiation of photoresists [101]. The spectroscopic observation of rather concentrated solutions is of broad interest in photometry. Because of the necessary transmission of solutions, the concentrations in normal direct spectroscopy measurements in photokinetics are restricted to approximately 10"^ molar solutions. In many experiments - especially in the case of preparative photochemistry (where often

Fig. 4.26. ATR-rod (attenuated total reflection) monitoring changes in the sample via multiple total reflection and evanescent fields in the reaction solution. Fibers are hooked to a highly optically polished glass, in which the radiation is guided in contact with the solution in the surroundings.

286

Experimental Techniques in Photokinetic analysis

Ch. 4

the optical turnover is intended to be controlled) - this concentration is too low. Either cells with extremely small optical thicknesses have to be used, which cause problems with leakage and flow-through effects, or the principle of attenuated total reflectance measurements has to be applied. If irradiation passes the interface from an optically denser medium to one of smaller refractive index, at angles larger than a certain limiting one (with respect to the perpendicular optical axis) the radiation will no longer leave the medium. It is totally reflected at the interface. Electrodynamics explain the fact that the reflected radiation couples to an evanescent field, which exponentially decreases into the medium of lower optical density. Thus the reflected beam gathers information about the optical properties of this medium across the interface. Such a device is demonstrated in Fig. 4.26 for a highly absorbent solution. In a fibre the radiation is guided by multiple total reflection. This analytical "finger" can measure changes in the solution [98,103]. 4.5 CHROMATOGRAPHIC METHODS 4.5.1 General considerations and comparison between different chromatographic methods In complex photoreactions many products can be produced simultaneously during the photoprocess. For this reason separation techniques are desirable in principle. However, these techniques must allow an automatic observation during the reaction. The first applications of gas chromatography [104] demonstrated some problems. Liquid chromatography offers better results, since photoreactions are usually examined in liquid phases. Two principles are presented in the literature in detail. Both high performance liquid chromatography (HPLC) and thin-layer chromatography (TLC) are frequently used in analytical applications [105,106]. However, a few essential points with respect to their applicability to photokinetics have to be discussed: - first, a possible automation has to be compared; - second, the correlation between the signal (peak area or peak high and concentration has to be examined; - furthermore, any separation technique will interact with the reaction solution. Therefore, neither TLC nor HPLC techniques can be used in such a way

Ch.4

Chromatographic Methods

287

that after detection the volume element containing the reactants is fed back into a photoprocess cycle. Recently, spectra identification by use of spectra libraries [107] and the use of multicomponent analysis using chromatographic data have increased applicability of chromatography in combination with UVA'^is spectrometry. 4,5.2 Thin-layer chromatography Commercially available thin-layer chromatography equipment is shown in Fig. 4.25. It allows the determination of absorbent and/or fluorescent products after a small part of the photosolution has been extracted and applied onto a thin-layer plate which is developed by a suitable eluent. This is an interactive process comparable with early techniques in kinetics where small portions of the reaction solution were frozen and conventionally analysed afterwards off-line. Since the determination of peak areas, peak detection, and baseline correction are well established methods, the determination of concentration is a smaller problem and results in values with small standard deviations. Nevertheless, TLC is not suitable for automation in its application to photokinetics since the photoactivation of the sample by the chromatographic material is not negligible [108]. The light paths for reflection and fluorescence measurements are given in principle in Fig. 4.27. The result of an examination using a TLC device with spectral detection using a photodiode array is given in Fig. 4.27. The advantage is the combined observation of absorption and fluorescence spectra using a xenon light fluorescence F

reflection

filter

Fig. 4.27. Lightpath for observing a thin-layer plate in reflection orfluorescence.In the case offluorescencemeasurements, the directly reflected intensity is cut off by a filter.

Experimental Techniques in Photokinetic analysis

288

Ch.4

Os

corr

I.O

•

absorption

^••^*mm,

-

luK.

0.8

SOOs fluorescence

500 sjM

0.6 0.4 0.2

\ I y os 1

,

,

,

1

1

1

1

1 —

700 wavelength [nm] Fig. 4.28. DEMC placed as a spot on a silica TLC plate irradiated by a xenon source to obtain spectra. The photodiode array allows the combined measurement of absorbance (reduced reflectance) and fluorescence (increased reflectance). Icon is the reflectivity corrected with respect to a spot without sample. Many spectra are taken during 500 s of irradiation. 350

400

500

600

source for the measurement. The diode array supplies both types of spectra taken from a spot containing a laser dye. The intense polychromatic measurement light causes a photoreaction catalysed by a silica sorbent material. Laser dyes are assumed to be relatively photostable. However, as Fig. 4.28 shows, the spectra change within a few seconds. Under these conditions 7,7'-diethylamino-4-methylcoumarine (DEMC) undergoes photodegradation. The different photodegradation products demonstrate a change in absorption and fluorescence maxima. In addition, their fluorescence quantum yields differ. This specific technique allows the determination of the different reactants and metabolites for different reaction conditions. However, in many cases the rates of photoreaction processes are not comparable to those in solution due to the photocatalytic behaviour of the sorbent material. Detectivity and limit of determination can be optimised by pixel bunching of the diode array. It measures concentrations as low as in the case of monochromatic detection using commercial equipment [108].

Chromatographic Methods

Ch.4

289

4,5.3 High performance liquid chromatography HPLC causes the same problem with a time delay between analysis and taking a specimen out of the photoreaction solution. However, in HPLC automation is easier to realise [109]. In Fig. 4.29 a combined irradiation and HPLC apparatus is given as a block diagram. It is controlled by a microprocessor and contains a cycle for the photoreaction including a mercury arc, a photoshutter with an interference filter, and a flow pump at normal pressure. The reaction solution circulates through a flow cell with 1 cm optical thickness, in which it is irradiated. The irradiation time is controlled by the microprocessor via the shutter. The volume of the circulating solution amounts to approximately 8 ml of solution, of which 3 ml are irradiated at a time in the cell. At chosen reaction times lOjulof the reaction solution are extracted using a manifold valve (see Fig. 4.29). This volume is injected into a high pressure

lamp

photoshutter

0

gradient ^ former

reactor

Uh

solvent

C

column

—

UV/Vis detector

inject injection valve

iwaste H

LXu X-t- recorder

selenoid valve

air-

I/O interface

TM 990 CPU

A/D converter

dialog terminal floppy controller plotter

floppy-disk

graphic interface

Mjigraphic monitor

Fig. 4.29. Block diagram of a combined HPLC apparatus and an irradiation set-up, which is controlled by a microprocessor.

290

Experimental Techniques in Photolcinetic analysis

Ch.4

A

50 inj ection Fig. 4.30. Dilution by injection of 20/41 of reaction solution and re-injection of the same volume of eluent into the photocycle. The straight line is reproducible.

flow cycle of a HPLC apparatus. In addition it can make use of a gradient former. Under these conditions, the lOjul volume taken out of the photocycle cannot be substituted, since the eluent would vary the solvent properties, most probably influencing the photoreaction. However, when the HPLC analysis uses just a defined eluent which is the same as the solvent of the photoreaction, the re-injection into the photocycle just causes a dilution, but keeps the volume irradiated constant. This dilution can be accounted for. As Fig. 4.30 demonstrates, a graph of the integrated peak area of a compound measured under such conditions (mercury arc stays switched off) linearly depends on the number of injections [110]. The relationship turns out to be linear. Since the calibration is reproducible, the dilution can take account of each calculation of concentration (see Section 5.2.1). Therefore the concentration of all the components is possible. The concentration-time curves ^ e directly obtained. The evaluation becomes simpler, in principle. However, the noise is larger than in pure photometric determinations using UVA^is absorption spectroscopy. Some experiments prove that the reproducibility of such a processcontrolled device fulfils the requirements for a photokinetic evaluation.

Ch. 4

Special Methods

291

4.6 SPECIAL METHODS

Many further analytical methods exist for monitoring a photochemical reaction. Some even offer some selectivity with respect to the reactants in the solution. However, in most cases on-line combinations of instrumental analysis and photochemical examination cause problems. Thus, infrared spectroscopy would be a preferable method with respect to the selective determination of reactants, especially after development of Fourier instrumentation, which allows quantification with similar standard deviations as UVA^is-photometry. However, solvents used in IR spectroscopy absorb at longer wavelengths in the UV, absorbing the irradiation of the photolamp and causing unwanted concurrent reactions (e.g., CI' from CHCI3 or CCI4). On the other hand, solvents suitable for UV and the photoreaction will absorb interesting regions of the IR spectrum. Furthermore, IR measurements in the liquid phase cause some problems with the cells depending on the solvent. In the examination of optically active compounds, optical rotational dispersion (ORD) and circular dichroism (CD) have been applied with success [111]. Analysis took place in a conventional way extracting samples out of the reaction solution with off-line analysis in CD equipment. NMR equipment has also been used in kinetic examinations. The recent development of pulse Fourier spectroscopy allows the application of NMR even to solids and in flow systems. Thus, *^C-NMR spectroscopy can offer the advantage of monitoring chemical pathways. Even quantitative results can be achieved to determine rates of reaction [112]. Another possibility of the application of NMR spectroscopy is via chemically induced dynamic nuclei polarisation (CIDNP), if a radical pair is formed during the photoreaction. The intermediate formation of a radical pair or a biradical in a magnetic field causes intermediate emission and enhanced absorption signals, respectively, which show up in the nuclear magnetic spectra of diamagnetic reactants or products. A nuclear spin selection takes place for the electronic transition. A direct relationship is found between the type of NMR signal, and the sign of the hyperfine coupling constant [113]. In *^C-NMR apparatus, a UV fibre optic is focused on the sample tube inside the magnetic field [114]. By this means information on the structure of photochemical primary products and their dynamics can be obtained.

292

Experimental Techniques in Photolcinetic analysis

Ch. 4

4.7 PROCESS CONTROL IN KINETICS

Quite a few different set-ups have been demonstrated, in which an irradiation apparatus was combined with different methods of spectroscopy. In all these combinations, process control either by microprocessors or by personal computers, was achieved. Modem instruments contain a microprocessor for process control of the spectrometer and data transfer to a dedicated personal computer. Quite a few years ago data stations were integrated in the spectrometers. However, their flexibility is poor and their software is restricted to the instrumentation making adaptations to user specified problems very difficult. In general nowadays most instruments contain a certain "intelligence" without a display. Process control and data acquisition is done by a personal computer interfaced to the instrument via an analog to digital converter board (ADU), which contains some input-output (lO) ports with TTL levels for control handshake. This board is plugged into the computer bus. The control menu is usually handled under Windows to make a convenient user interface available. Very few of the software packages allow subroutines to be added by the users to handle flow pumps, photoshutters, valves or other switches. For this reason, special applications need self-written programs in the personal computer software (if possible) or an additional microprocessor (programmed in assembler) for the special task. In Fig. 4.31 an example is given controlling a dye laser by one microprocessor and a UVA^is-spectrometer by another. These two microprocessors are interconnected via data and control lines. Their programs are synchronised. The first microprocessor controls the spectrometer, the second an optical multichannel analyser, which rapidly takes emission spectra during the time the laser pumps the dye solution. Both have to be synchronised with the spectrometer which takes the absorbance spectra. The laser has to be triggered by the optical multichannel analyser. Thus, the circulating dye solution flashed by the laser can be examined by a vidicon (during the flash) with respect to its laser activity. Using a flow cell with 0.1 mm optical path length because of high absorbance of the dye laser solution, in the spectrometer the photodegradation can be monitored at the same time. Fig. 4.7 gives a view of the cell holder. In the case of diode arrays used in applications given in Figs. 4.4, 4.11, 4.12, and 4.28, the overall process control and data handling was programmed in C running on personal computers. In this case the control of

Ch.4

Determination of Intensity of Irradiation

293

I r S 3 . SL—polychromator h T A I m-rrr-—uJlli control

Fig. 4.31. Block diagram of a dye laser using an irradiation source and UVA^is spectrometer for time-resolved examination of photo-degradations of the dye solution at laser conditions. Two synchronised microprocessors are used for control. The assembler program amounts to 32 KB of code.

additional parameters (pump, shutters) was handled by macros, which are user definable [115]. 4.8 DETERMINATION OF INTENSITY OF IRRADIATION In Chapters 2 and 3 partial photochemical quantum yields have been introduced as parameters which give information on photochemical reactions. According to eqs. (2.12) and (2.14) they depend on the change in concentration of the reactant which undergoes the photoreaction and on the light absorbed by this reactant. For this reason an essential of any photochemical examination is the determination of this amount of light absorbed. Two principles are known: - either time-resolving methods can be used controlling the intensity of the irradiation source by photodiodes, or

294

Experimental Techniques in Photokinetic analysis

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-

time integration methods have to be applied which determine the intensity falling onto the surface of the photoreaction cell. These two principles can be correlated to either physical or chemical methods. Both offer advantages and disadvantages which are discussed below [116].

4.8,1 Physical methods Totally absorbing material is heated by incident irradiation (bolometer). A relationship between intensity and temperature effects exists. Since temperature measurement are very sensitive, this effect is used to determine intensities. Semiconductor detectors either use the internal or external photoeffect [117,118]. In a photodiode, an incident photon causes a photocurrent by charge separation. It can be amplified and depends linearly on the number of incident photons. In photocathodes or photomultipliers the incident photons force electrons to leave the material. This external photoeffect can be calibrated and amplified by acceleration and multiplication of the electrons in multi-electrode arrangements (dynodes). These devices have very short response times and can be successfully used to control the stability of a light source. Therefore such devices are frequently included in commercially available set-ups. An example is given in Fig. 4.32 combining an irradiation source with a measurement set-up. This is commercially available [119] and allows simple control of a photoreaction. However, due to geometry and inhomogeneity of the sensitive layer of the photodiode, non-homogeneous irradiation can cause errors. These thermoelements, photoelements, and other physical measurement devices can cause some problems for the photochemist: - Sensitivity of the physical system depends on wavelength. - Sensitivity drastically decreases in the ultraviolet (UV). - Correction factors as well as the compensation plots in operation manuals only supply limited precision. - Calibration of such devices has to be repeated frequently since the material of the detector head exhibits photodegradation. - This photodegradation is extreme, if the device is exposed to highly intense sources which can damage the detector. This problem has to be taken into account when measuring laser sources.

Ch. 4

Determination of Intensity of Irradiation .14

295

13

Fig. 4.32. Irradiation set-up in which the photoreaction is controlled by spectroscopy. The intensity of the radiation source is monitored by a physical device measuring photocurrent (supplied by AMKO. Tomesch) [119].

-

The linear dynamic range is limited; it can be enlarged by using a preamplifier. - Any absolute determination of the radiant flux density or radiant flux needs frequent and troublesome re-calibration due to changing sensitivity of the detector surface. - Detector areas show spatial distribution of sensitivity. This amounts to 40% and more. This fact will cause problems especially if the light is focused only onto part of the target. - Multiple reflections in the solution or in thin layers influence the effective intensity in the sample. Physical methods cannot take account of this. These facts makes it obvious that physical methods are perfect to control the intensity of an irradiation source as a function of time. That means, they

296

Experimental Techniques in Photokinetic analysis

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can be conveniently used for a relative determination of radiant flux permanently. 4,8.2 Chemical actinometers Since the absolute intensity of an irradiation source is difficult to determine by physical methods, photochemists prefer the use of a simple photochemical reaction for the measurement of intensity. Any photoreaction for which the mechanism is known to be simple can be used as a chemical actinometer. It requires that the photochemical quantum yield is determined and independent of intensity or other effects. For many years chemists have looked for such types of reactions and have proposed a large variety of actinometric systems. The first were based on a colorimetric principle. A product is formed by the photoreaction which results in a coloured complex by addition of a chelating agent. The absorbance of this complex is linearly dependent on the intensity of irradiation [7,120]. Another possibility is the formation of a gas which can be measured volumetrically. A further possibility is the use of photochromic systems in which the thermal backward reaction is suppressed or very slow. These reactions can be monitored by applications of UVA^is-spectroscopy quantifying the change in absorbance at the chromophore's maximum. The advantages of such chemical actinometers are the following: - Absolute intensity of the irradiation source can be determined, since the number of photons incident onto the sample are measured. - Most of the chemicals necessary are very cheap, thus offering to replace the "detector" in case of damage. - There exist a lot of substances, which are utilisable many times. They exhibit photochromic behaviour and photoisomerise. - Chemical actinometry is especially suitable for photochemists, since the actinometric solution can be substituted by the sample of interest without changing the geometry and experimental conditions. This is of special value in the case of liquid solutions. By these means the problems with multiple reflection in solution or at thin layers as well as with inhomogeneous distribution of the irradiation or special geometries of the photoreactor can be overcome. Two typical reactions which are used in chemical actinometry, are the photoreactions of iron oxalate (Parker's solution) [7,121] and of potassium Reinecke's salt [122]. In both cases, photochemistry and analysis are sepa-

Ch. 4

Determination of Intensity of Irradiation

297

rated. In the literature some errors are reported which can be made [123] if optimal conditions are not achieved. Whereas Reinecke's salt is hardly used any more, Parker's solution has become the classical actinometer for photochemists. Iron(III) ions which are produced by irradiation have to be chelated by 1,10-phenanthroline in an acetate buffer after the photoreaction. The coloured complex must be photometrically determined at a wavelength of 510 nm after a complete conversion has taken place. The time delay has to be at least 1 h, overnight is preferable. The actinometric solution is restricted to wavelength areas below 450 nm. Some skill of the experimenter is required otherwise the errors amount to more than 10% [123,124]. Another example is azobenzene which is even commercially available [125]; its photoreaction has been examined in detail [126,127]. The same is valid for other systems such as heterocoerdianthrone [128] and mesodiphenylhelianthrene [129]. These are also commercially available [125] as are the aberchrome 540 and similar compounds [130] applying a simple evaluation using totally absorbent solutions. The methods demonstrated in this chapter are used in the applications discussed in the next chapter to obtain the necessary data for the following photokinetic evaluations.

Chapter 5

Applications of Kinetic Analysis to Photoreactions

Different approaches are given in Chapters 2 and 3 for the calculation of the relationship between the concentrations of the reactants and the reaction time, assuming mechanism and kinetic constants are known. Such rate equations can be formed for any complex mechanism, at least in principle, since there is a well defined route from mechanism to concentration and time laws. The aim of any kinetic analysis is to determine the mechanism using concentration-time measurements. Therefore a variety of different methods have been described in Chapter 4 to obtain these data. A kinetic analysis has turned out to be successful if the coefficient scheme according to Example 2.1 and explained in Section 2.1.1.1 together with the numerical values of the rate constants k,^ are determined for the reaction under examination. In general many mechanisms can result in the same functional relationship between concentration and time. For this reason the mechanism determined and in consequence the rate constants obtained are not all unambiguous. Furthermore accuracy and reproducibility of the measurements are limited. They vary with the method used for the determination of the concentrations with time. Thus the limitation in the measured signals and the variety of rate laws increase the many mechanisms fitting to the signals measured. For this reason the only positive result of any kinetic analysis can be in principle: The values measured do not contradict an assumed mechanism

5.1 EVALUATION OF MEASUREMENTS OF CONCENTRATIONS 5. LI Determination of the number of linear independent steps of reactions Any kinetic analysis starts with a hypothesis. One assumes a certain mechanism, determines the concentration and time laws, and checks whether

300

Applications of Kinetic Analysis to Photoreactions

Ch. 5

the measured values fulfil the calculated functional relationship. If the result is positive, the kinetic constants can be determined. In the case of a negative result, the hypothesis has to be discarded. First, the number of linear independent reactions has to be known before the examination of proposed rate laws can be achieved. The result depends on the existing information and is limited by the quality of the measurement. In the present section, it is assumed that the concentrations of all the reactants as functions of time are known. 5.7.7.7 Matrix rank analysis According to Sections 2.1.1, 2.1.4, and 2.5.6 the relationship between the concentrations and the linear independent degrees of advancement can be given by (see Section 2.2.1.1) Aa = vx.

(5.1)

Thus the vector of the differences of concentrations is a product of a matrix for the general stoichiometric coefficients and the vector of the linear independent degrees of advancements. This relationship is valid for each time ti. For various times given as f/ (/ = 0,l,2,...,m) these measured differences in concentration can be arranged in a matrix

'Aa„ Aa =

Aa2,

Aa,2 ...


...

• •• Aa,„^

(5.2)

•• Aa^J

In this case it is essential that all the concentrations are measured at the same reaction time, that means simultaneously. If this prerequisite cannot be experimentally achieved the measured values have to be synchronised with respect to equivalent "wavelength"-independent reaction times (see Fig. 4.6 in Section 4.2.2.3). Usually the concentration at the time t^ is taken as a reference: Aan^an-a^Q. However, this procedure is not necessary in the case of the matrix rank

Evaluation of Measurements of Concentrations

Ch. 5

301

analysis discussed in the following. Any reaction time can be used as a reference. If the differences in concentration have been determined a sufficient number of times one will find, in general, that the number of times m has to exceed the number of reactants n which exceeds the number of linear independent concentrations or steps s, respectively: m>n

and

m>s

Under these conditions the rank of the matrix in eq. (5.2) equals s according to definition. That means s equals the number of linear independent concentrations and the number of linear independent partial steps of reaction according to eq. (5.1), respectively. The numerical rank analysis is explained by an example [15]. Example 5.1: Example of a matrix rank analysis The following mechanism is assumed:

Only two of the four reactions are linear independent according to Section 2.1.4. The reaction can even be described by a single degree of advancement, if for example B is a short lived intermediate, which can be treated by use of the Bodenstein hypothesis. The experimentally determined concentration-difference matrix contains the following elements (each column correlates to a different time of reaction): 1 2 3 A 7 -0.21 -0.43 -0.86 0 Afe 0 0 0.14 0.28 Ac 0.06

4 5 -1.21 -1.61 0 0 0.41 0.53

Ad

0.80

0.15

0.29

0.56

1.07

Under these conditions, the number of measured signals is too small for a kinetic analysis. However, it is sufficient for the demonstration of

302

Applications of Kinetic Analysis to Photoreactions

Ch. 5

the principle. One realises that the reactant B occurs in analytically negligible quantities. This means the Bodenstein hypothesis has to be used, if the assumed mechanism really takes place. Under these conditions the rank of the matrix has to be 1. It will be determined as follows: (1) Rows and columns are rearranged that the absolutely largest element, the so-called Pivot element (-1.61), is placed in the left upper comer (Ab is omitted): 5 1 2 3 A 7 -1.61 -0.21 -0.43 -0.86 0.14 0.28 Ac 0.53 0.06 0.15 0.56 Ad 1.07 0.29

4 -1.21 0.41 0.80

(2) The new first row obtained is divided by this Pivot element and new values of the elements in the first row are given as 1 0.130 0.267

0.534

0.752

(3) The /th element (/ = 5,1,...,4) of the first row thus obtained is multiplied by the first element of the second row (0.53). Next this product is subtracted from the Zth element of the second row:

0.53-1 0.53-* 0.06-0.130 0.53

1 0.130 0.267 0.543 0 -0.009 0.002 0.003

-^

0.752 0.012

T

(4) The third and any other row are treated accordingly. 1

0.130

0 0

-0.009 0.010

0.267

0.534

0.752

0.002 0.003 0.012 0.004 0.012 0.004

(5) The sub-matrix which is printed in italics (a) containing the 2nd and 3rd row from column 2 to 5 is treated next according the preceding paragraphs 1 to 4.

(a)

Ch. 5

Evaluation of Measurements of Concentrations

303

One obtains: 1 0 0

0.130 0.267 -0.782 1 1

0.010

0.534 0.752 0.133 0.265

(b)

0.004 0.072

and by another repetition of the procedure: 1 0 0

0.130 0.267 -0.782 1 1 0

0.534 0.752 0.133 0.265 0.900 0.366

(c)

This treatment has to be redone until either the values 1 are found on the diagonal / = 1 (see (c)) or all elements starting at a certain row become 0. Then the rank of the matrix equals the number of rows which cannot be neglected (all elements: zeros). In the example given, the rank of the used measured values is 3. Nevertheless the mechanism cannot be discarded. This fact becomes obvious if

Fig. 5.1. Concentration-difference diagram for the measured values used in the Example 5.1: {\)Ad, (2) Ac plotted versus Aa.

304

Applications of Kinetic Analysis to PhotoreacUons

Ch. 5

one plots the concentration differences Aa, versus each other in a diagram. Fig. 5.1 makes it obvious that the differences in concentrations depend linearly on each other within a reproducibility of the measured signals of ±0.01. For this reason the reaction is uniform, and the rank of the matrix of the signals becomes 1. The errors in measurement are the reason that in (a) the elements of the second and third row do not become 0. Therefore, after each 4th step of calculation one has to check whether the remaining elements of the succeeding rows can be taken to 0 in the limit of error of measurement or not. Computer programs have been developed for the rank analysis and the error estimation. 5.7.7.2 Graphical methods The formalism of a numerical rank analysis can be substituted by graphical methods. Even though this graphical approach is formally equivalent to the numerical rank analysis, diagrams frequently are more descriptive and result in more information. Concentration diagrams [10] For a uniform reaction {s = 1) the concentrations of each reactant A^ form a linear function of the concentration of each reactant A, according to eq. (1.8): (5.3) This relationship can be graphically checked. One plots the concentration of one component versus the concentrations of all the other components in pairs in a rectangular coordinate system. If linear concentration diagrams (K-diagrams) result, the reaction is uniform. Additionally one obtains informations on the stoichiometric coefficients, the equilibrium constants, and constants of competitive reactions (if the reaction contains competitive steps). Example 5-2: K-diagrams for a single reaction step For the reaction 2A + B

>C

Evaluation of Measurements of Concentrations

Ch.5

305

Fig. 5.2. K-diagrams for the reactions 2A + B -> C. The initial concentrations are a^ = 2, ^^ = 1.5, c„ = 0 (in relative units).

the relationships

fo = &Q - c are valid according to eq. (5.3). Figs. 5.2a,b demonstrate the resulting K-diagrams. Since B exists at the start of the reaction in excess, at the end of the reaction the vessel also contains B. Example 5-3: K-diagramfor consecutive reaction For the reaction A^F^B-^C,

C-^D

one finds

if the equilibrium approximation is applied. Or

Applications of Kinetic Analysis to Photoreactions

306

Ch.5

a^Kb. Since the relationship exists,

In consequence the concentrations of C and D depend according to

and one finds

The use of the K-diagrams (see Fig. 5.3) for the dependence of a{b) allows the determination of K (a is the ordinate, b is the abscissa). Furthermore, the competitive constant ;f can be found for the K-graph d versus c. The latter relationship can be used for control. In this figure the concentrations are added in relative units at the axes.

Fig. 5.3. K-diagram for the reaction A ^ B ~ > C , B->Cat equilibrium approximation: a^ 0.6,^„=10,/i:=:0.6,;t = 0.2.

Ch. 5

Evaluation of Measurements of Concentrations

307

Even if 5 > 1, linear K-diagrams may result for the concentrations of some components. This can happen if the columns of these components depend on each other in the scheme in Section 2.1.1.1. Thus for the reaction A-^B+C,

B->D

with a + c = ^0 b+d =c a linear diagram a{c) will be obtained, whereas the diagrams b{c) and b{d) are curved. Non-linear K-diagrams are discussed in Section 2.3.3 in detail. The relationships given in this section for reactions of first order are valid without limitation also for quasi-linear photoreactions. Additional information in the case of photoreactions can result in an examination of the dependence of K-diagrams on intensity and wavelength of the irradiation source. The dependence on intensity K-diagrams do not depend on the intensity of the irradiation source, if the quantum yields of both the partial reactions depend in the same way on the amount of light absorbed. However, if there is a difference in the dependences of the quantum yields of the two partial reactions on the light intensity, or if linear independent photochemical and thermal reactions compete with each other, then the K-diagrams will depend on /Q. The dependence on wavelength Since /Q varies with the wavelength of the irradiation source, the Kdiagrams depend on the wavelength in those cases in which they depend on the intensity. Furthermore the K-diagram changes if more than one component absorbs the light and starts the photoreaction by this means. Example 5.4: Simple consecutive reaction For the reaction A—^!^B—5^C

308

Applications of Kinetic Analysis to Photoreactions

Ch. 5

which follows the mechanism treated in Section 3.2.3 in Example 3.25, the same initial slopes are obtained as for the analogous thermal reactions:

da 0

= 0, d,a

^

=0. to

The partial quantum yields do not depend on irradiation intensity according to eq. (3.29). For this reason the same situation is valid for Kdiagrams. The form of the K-diagrams is determined by eq. (3.40). The related constantx contains the ratio K'Q I ic\, which varies with wavelength. For this reason the K-diagrams have to depend on the wavelength of the irradiation source. Concentration difference quotient diagrams K-diagrams lose linearity, if 5 > 1. If j = 2 (just 2 linear independent steps of the reaction form the mechanism), the concentration of each component A, is a linear function of two suitable chosen components A., and A,-. For this reason a^^ p-\' a^a^ + ajfl/-,

(5.4)

whereby p, a, and a2 are constants which are determined by the mechanism for each combination of components. For example one obtains for the reaction

the relationship b = bQ-c d = dQ-aQ+a-{-c

if a and c are chosen to be independent concentrations. The diagram c{b) is linear, if all the concentrations fulfil eq. (5.3). Equations for d and e are examples for eq. (5.3). They result at r = 0 in

Ch. 5

Evaluation of Measurements of Concentrations

309

«,o = P + ^i^,o+<^2«roIf this value is subtracted from eq. (5.3), one obtains Aa,. =a,Aaf. + Aaja,". By dividing both sides by Aa, one finds Aa.. Aa... -7-^ = ^ 1 + ^ 2 ^ -

(5.5)

For this reason a graph of Aa,/Aa,, versus Aa,./Aa,. results in general in a straight line for s = 2. The slope determines a j , the intercept is a,. This restriction is necessary, since concentration difference quotient diagrams (in brief called KDQ-diagrams) can degenerate. If the differences in concentration of two components are proportional to the difference in concentration of a third component, the KDQ-diagram degenerates to just one point. This especially happens when the three components only take part in one partial step of the reaction. Example 5.5: Two bimolecular parallel reactions A further example is the reaction mechanism A + B - > D + E,

A + C - > D + E.

The differences in concentrations are Aa = - A^,

Ae = A
Accordingly the following ratios are obtained: Aa/Arf = - 1 ,

Ae/Ad^l

and the KDQ-diagram degenerates to the point (1,-1). One can directly read out of the scheme of the stoichiometric coefficients those combinations of components for which the KDQ-diagram degenerates:

310

-

-

Applications of Kinetic Analysis to Photoreactions

Ch. 5

If the columns of three components are proportional to each other, then the KDQ-diagram degenerates to a single point. The KDQ-diagram degenerates either to a zero point line or to a line parallel to one of the two coordinate axes, if those two components, for which the columns are proportional to each other, are combined with a third one which is not proportional to the other two. If in the previous example Aa, Ab and Ad and are combined, then one finds always Ad/Aa = -1 Therefore one obtains a parallel to the abscissa for any ratio Ab/Aa. Furthermore Ad _

Aa

Afc^^'Afe is always valid. The result is a zero point straight line. If a reaction yields non-linear K-diagrams, then KDQ-diagrams allow the cases 5 = 2 and s>2 to be distinguished. If KDQ-diagrams are either linear or degenerate for all combinations of components, then the number of partial reactions amounts to ^ = 2. If one finds at least one KDQ-diagram non-linear and non-degenerate, then the number of independent partial reactions is s>2. In practise it is essential that there is no need to relate the differences in concentration to time / = 0. By taking indices / and /' for any two times one obtains in analogy to eq. (5.5)

hi - bi.

bi - bi,

as a generalised equation. This relationship proves that KDQ-diagrams result in straight lines independent of the choice of reference time. The constants o^ and a j used above can be calculated for a given mechanism. If they can be determined experimentally, a criterion for the control of the hypothesis is possible.

Ch. 5

Evaluation of Measurements of Concentrations

311

Example 5.6: Initial slopes of a simple consecutive photoreaction In Section 3.1.4.2 a consecutive photoreaction had been treated in Example 3.24. Using the law of conservation of mass a +fo+ c = aQ. eq. (5.5) can be rearranged to

(a-do)

(«~«o)*

Slope and intercept in this diagram become - 1 . The range of the straight line, which is formed by the measured values, can be determined as follows. By reducing the time interval for the first measurement after the start of the reaction to an extremely small value and by application of the rule of Bemoulli-L'Hospital one obtains using eq. eq. (3.39)

,.

Urn

b

iy (^r^^-^^'^B^) —

r = -— =

,

= — 1,

since b becomes 0 at r = 0. Accordingly one finds

ci->flo(a-ao)

a

[-(p^K^a)

Therefore the reaction starts at the point (0,-1). Obviously this point cannot be calculated from the measured values directly. However, it can be determined using the initial slopes of the K-diagrams b{a) and c{a). At the end of the reaction a = 0,

6 = 0,

c-a^

is valid. Therefore, for r ~> oo one finds the relationships

a — an

a— an

312

Applications of Kinetic Analysis to Photoreactions

Ch. 5

and

For this reason the end point of the reaction is given by (-1,0). If the differences are calculated with reference to the end of the reaction one finds according to eq. (5.6)

a

a

At the beginning of the reaction one finds ^ =0 a

and

SZ^=^-l a

and for the end of the reaction ,,

b

b

(^r^^-^i'^B^)

lim—= —=

;

T

r

.

Applying the rule of Bemoulli-L'Hospital for the right-hand side of this relationship one finds for a -4 0 db

^

db

— = -l-hx— da da

/

using the abbreviation

One solution of eq. (5.7) is db/da —> oo,

^v

(«-»0)

,_, _.

(5.7)

Ch. 5

Evaluation of Measurements of Concentrations

313

a second solution is db

1

da

x-1

Accordingly one finds dc/da —> —oo and ^ ^ da

X_ x~l

One can summarise: The first solution is valid for % < 1, as a closer examination proves. The second solution is given for X - ^ • P^^ this condition the measured values are situated between the points (1,0) and [-x/iX"" ^)*y(X " Ijj- On the other hand a condition of ;f < 1 results in values on a straight line in the fourth quadrant. -

KDQ(s)'diagram If all the KDQ-diagrams are either linear or degenerate, only two linear independent partial reactions take place. This graphical method can be expanded to more complex mechanisms as discussed in the following example. Example 5.7: Multi-step consecutive reaction The reaction A~>B->C->D consists of three linear independent partial reactions. For this reason the KDQ-diagrams are curved. Applying mass balance results in a + b-\-c + d — aQ. Therefore, one finds

314

Applications of Kinetic Analysis to Photoreactions

Ch. 5

Aa + A& + Ac + Arf = 0 for taking the difference to any chosen time interval. This equation can be divided by Ad for example and the following relationship is formed: Aa

Afc

Ac _

which can be given in a different notation DQia^d) + DQ{b,d) -h DQic^d) = - 1 . The above equations demonstrate that the three differential quotients are situated in a plane in three-dimensional space. Taking any time as a reference (index *) the following equation is obtained:

DQ{a,dl+DQ{b,dl+DQ{c,d),=-'L Defining the differences ADQ{a,d) = DQ{a,d) - DQ{a,d),, one finds ADQ{a,d) + ADQ{b,d) + ADQ{c,d) = 0. As done above using the quotients, one finds KDQ(3)-diagrams ADQja^d) ^ ADQjb.d) ^ ^ ADQ{c,d) ADQicd) which are obviously linear or degenerate, if 5 = 3. This method can be expanded to more complex reactions in principle. However, the errors in measurement will limit this type of evaluation.

Ch. 5

Evaluation of Measurements of Concentrations

315

5.1.2 Evaluation of concentration-time measurements 5.1.2.1 Evaluation of integrated time laws As demonstrated in previous shapters the time laws for photoreactions can only be integrated in a closed form for a few special mechanisms. To avoid this problem one can make use of a time transformation as demonstrated in Section 3.3.11. Thus for many applications the concentrations can be calculated as an analytical function of the transformed time 0 . Example 5.8: Physically sensitised photoisomerisation and use of time transformation The physically sensitised photoisomerisation which had been discussed in Example 3.9 in Section 3.1.1.4 yields a differential equation according to eqs. (3.6) and (3.35) 1000/c;ao/oX- *'^

X is defined by eq. (3.75). Rearrangement yields \ ^ + l\db = -\OOOxK\ao - ^ d t = - KdS. By integration one obtains

an expression which can be evaluated by linear regression methods. If for example ln(fc/feo)/(& ~ ^o) ^^ plotted versus 0/(fc - fco) ^^^ obtains a straight line for this reaction with the slope -Kk^/k^ and the intercept -k^/k^. Quasi-linear photoreaction According to Section 3.2.2 one obtains for the kth partial step in the case of a simple geometry

316

Applications of Kinetic Analysis to Photoreactions

Ch. 5

dx, _ 1000/o/c;a,.y)i

if the quantum yields are independent of the intensity of the irradiation source. The use of the time transformation

de=

'lOOO/o'

IHK) J

dt

(5.9)

and the abbreviation

kk=«,vi results in

d&

k^Oi,

A: = 1 , 2 , .

.,k

(5.10)

Eq. (5.10) represents a system of linear differential equations with constant coefficients, if the quantum yield does not depend on the concentrations. Any photoreaction, which can be rewritten by the transformation eq. (5.9) to such a system, is called a quasi-linear photoreaction in the following. All the relationships derived in Section 2.2 can be used, if the real reaction time is substituted by the treuisformed time 0 . Difference equations [15] All the linear photoreactions can be evaluated using difference equations. The general solution of such a differential equation is according to eq. (2.56) a = a^+pexp(r0)

(5.11)

Introducing a difference in pseudo-time 0 by choice but constant and symbolised by A one finds Aa s a(0 + A) - a(0) = p • (exp(rA) - 1 ) exp(r0)

(5.12)

Ch. 5

Evaluation of Measurements of Concentrations

317

The matrix e x p ( r A ) - l is a diagonal matrix with the elements exp(r^A)-l. Since the chosen concentrations are linear independent the matrix p is a regular one. Therefore using eq. (5.11) exp(r0) = p"*(a-a^), and Aa = p ( e x p ( r A ) - l ) p ~ * ( a - a ^ ) , or defining K^=p(exp(rA)-l)p-* one obtains Aa = K^(a-a^)

(5.13)

This system of difference equations is set-up just in analogy to the system of differential equations. It can be transformed to those if both sides of eq. (5.13) are divided by A and if 0 approaches 0. Recursion equations [15] The system of difference equations, eq. (5.13), can be substituted by a system of recursion equations. Eq. (5.13) can be explicitly written as 5(0 + A) - 5(0) = p • exp(rA) • p"* • 5(0) - 5(0) - p • exp(rA) • p"* • 5^ + 5^, or using K'=pexp(rA)p-^ and the definition fi = 5^ -K^a^ as

318

Applications of Kinetic Analysis to Photoreactions

a(0 + A) = K'a(0) + u

(5.14)

Eq. (5.14) is a system of recursion equations. Example 5.9: Simple consecutive photoreaction For the consecutive photoreaction A—^B—^C , one obtains as

V

According to eq. (2.89) one defines (

ao

0

P =

By inversion one obtains 0 and exp(-/:,A)-l K^ = /:, (exp(-/:i A) - exp(-/:2^))

0 exp(-/:2'^) " 1

^2 " ^ 1

and exp(-/:,A) K' = *i(expH,A)-exp(-*2A)) ^2

^1

Ch. 5

0 ^^/j^^N

Ch. 5

Evaluation of Measurements of Concentrations

319

With a^ =fo^= 0 one yields u = 0. The system of difference equations is given by Aa(0) = (exp(-ik, A) - l)a(0), AM0) = k, e x p K A ) - - e x p H , A ) ^ ^ ^ ^ ^ exp(^it,A)M0).

The system of recursion equations is given accordingly by Aa(0) = (exp(-it, A) - l)a(0), A^(0) = k, exp(-A:,A)-expH,A)^^^^^exp(-*,A)Me). ^2

*1

The evaluation according to Sections 3.2.2 and 5.1.2.1 needs in general the measured values of many components A,. If the ^,(0) values are just known for a few components, the methods given can fail. The a{@) values can be evaluated by themselves for the given example of a consecutive photoreaction. However, the b{Q) and c(0) values need another set of measurements each. The next paragraph demonstrates how a single set of concentration data has to be examined. Difference equations of higher order [15] In analogy to eq. (5.11) differences of higher order can be defined. Thus the difference of second order is given by A%sAa,.(0 + A)-Aa,(0). In general one defines for any kx\i difference ^^a, = A^'-'^a,(0 + A) - A^*-*^a,(0). Using this definition one finds according to eqs. (5.11) and (5.12) for the ^th difference of the component A,

320

Applications of Kinetic Analysis to Pliotoreactions

Ch.5

A*a, = (exp(r,A) - l)*p/i exp(r,0) + (exp(r2A)-l) yo,.2 exp(r20)

(5.15)

+ (exp(r,A) - l)*p„ exp(r,0) The use of the abbreviation 'i'^=exp(r^A)-l and eq. (5.15) allows us to rewrite the system of higher differences as a matrix equation A''a,=V'^p,exp(f0), which can be written as

r1

A^a,

I

A'«,. „AJ

A^a,

-AJ

Pn exp(r,0) P,-2exp(r20)

(5.16)

/>,.,exp(r,0)

The zero difference is defined according to A°a,- =a, -a^. Eq. (5.16) represents a system of s + 1 equations for the s unknowns P;jt exp(r40). For physical reasons the system has to give a defmite solution. Therefore one finds according to Kronecker and Capelli

1

1 = 0.

Wa,

-AJ

-AJ

Ch. 5

Evaluation of Measurements of Concentrations

321

Multiplication yields A^a, - 5 f A^-'a, +... + (-ir5,^(a, - a , „ ) = 0

(5.17)

whereby the 5f represent the elementary symmetric functions of the r^ [10,33] S

st = I. j
cA

A A

A

For 5" = 2 one finally finds A\

= {[exp(r, A) -1] -f [exp(r,A) - iJJAa, -{[exp(r,A) -1] + [exp(r,A) - l])(a, - «,^).

Recursion equation of higher order [15] Using eq. (5.11) and the definition r;=exp(r^A) one can construct a system of higher recursion equations a, equivalent to eq. (5.16) a, ( e + kA) - a,^ = V"*Pi exp(r0). Using the same argumentation as in the previous section the recursion equations are given by a,.(0 + jA)-5;a,.(e + (j~l)A) + --.-h(-l)'5>,(0) + a = O

322

Applications of Kinetic Analysis to Photoreactions

Ch. 5

with

(5.18)

=-«.-n(i-'•*)• In analogy to above the S[ represent the elementary symmetric functions r/. Therefore one finds for 5 = 2 a,(0 + 2A) - (exp(r,A) + exp(r,A))a,(0 + A) + exp(r, A)exp(r2A)a.(0) + a = 0 with o' = -^/oofl - Gxp(r, A) - exp(r2A) + exp(riA) exp(r2A)j = -a/oo[l - exp(r, A)][exp(r2A)] By evaluation of concentration-time measurements with difference or recursion equations of higher order, one finds the final concentrations, the number and the amount of the eigenvalues which determine the a,(©) function of the different components. In this case a knowledge of the total reaction procedure is not necessary. Known eigenvalues allow the calculation of the coefficients p^^ using eq. (5.11) and linear regression. If the system is limited to two eigenvalues, graphical methods can be used. For this reason one can evaluate experimental values, which can be represented by linear combinations of experimental functions, using eqs. (5.11) and (5.17) in two steps by linear regression. But, if all values of r^ and p,^ are known, one can determine the derivatives at time 0 = 0 using eq. (5.11). Thus all coefficients can be obtained which determine the progress of the concentration of the ith component. Example 5.10: Application of recursion equations to consecutive reactions For a certain component Z sufficient exact values of the function z(0) have been measured. However, the values at the beginning and the end

Ch. 5

Evaluation of Measurements of Concentrations

323

of the reaction are missing. In this case the measurement can be evaluated by a recursion equation of second order. One finds two eigenvalues - they cannot be correlated further - and z^ equals 0. The two exponential functions determined are used to calculate the coefficients p^, and p^2 according to eq. (5.11) by linear regression. At time 0 = 0 one finds ^^ - ^^ = 0; and thus z^ becomes 0. Furthermore one finds a finite value for the first derivative at the beginning of the reaction. Thus Z is characterised as a first intermediate. Nevertheless quite a few mechanisms are imaginable which can contain such an intermediate. Even in the case it is known that the simple unambiguously consecutive reaction follows the mechanism

the two eigenvalues cannot be correlated to the two partial steps of the reaction. 5.1.2,2 Evaluation of differential equations Determination of quantum yields The true differential quantum yields can no longer be localised in the integrated equations of photokinetics. The different rate constants are distributed to different terms in the equation. On the other hand one can correlate quantum yields in the differential equations. If the quantum yield does not depend on the intensity of the irradiation source on finds according to eq. (3.35) lOOOv^Wocp

The true differential quantum yield has been symbolised as

324

Applications of Kinetic Analysis to Photoreactions

Ch. 5

-

the absorbance E^ at the wavelength of the irradiation source is known. Section 2.1.2 demonstrated that the quantum yields depend on the concentration of the reactants in a characteristic way (in general the true differential and the partial quantum yields respectively). This correlation can be used for kinetic analysis. In practice in general the wavelengths of measurement and irradiation are not taken the same. Therefore A' is taken as the wavelength of irradiation. The applications are treated accordingly in the further sections of this chapter. Example 5.11: Photochemical quantum yields of a photoaddition reaction In Example 3.4 the photo-addition had been treated according to eq. (3.3). One finds for the quantum yield y
(5.21)

with X, = 1 + ^2/^3

and

X2=*4/*5-

The coefficients X\ ^"d Xi ^^^ ^^ determined according to eq. (5.21). Assuming that the reaction follows the postulated mechanism, the reciprocal quantum yield can be plotted versus the reciprocal concentration of B. Then a straight line with intercept X\ and slope Xi will be obtained. Formal integration The measured values contain errors. Thus the differential quotients determined will contain some uncertainty. This turns out to be an essential disadvantage in the evaluation of the differential equation. Integration of empirical functions smoothes these errors to a large extent. This method is explained according to a simple example. For the reaction A -» v^B, b^kb^-

kb

or

db = kb^ dt - kb dt

are valid. Formal (numerical) integration of these equations within the limits ti and ti> yields

Ch. 5

Evaluation of Measurements of Concentrations

K^r)-

^(^/) = *^<-(^/' - ^ / ) - f' l^dt.

325

(5.22)

If the concentrations b(tQXb{t^),...yb{tf),,.,,b(t^) are measured at different times ti (/ = 0,1,2,...,m), the integral can be calculated by graphical and numerical means within different limits. Thus eq. (5.22) yields a linear relationship between the quantities, which are directly determinable from measured data. Variation of /' and / respectively yields a system of linear equations for the determination of the parameters kb^ and k Af mis large enough the system of equations is over-determined. It can be solved, if the measured values fulfil the relationship given by the differential equation. If only two constants are necessary, the relationship can be evaluated by graphical means. If one notes eq. (5.22) in the form

one obtains a straight line for a graph of Y versus X, if the relationship is correct. The graph can be used to take the slope and intercept to determine k and b^. In general one is able to transform any differential equation by formal integration to

^ ( 0 = ^,^,(0 + - + ^/;(0 + - + ^,^,(0.

(5.23)

The quantities Jj[t) are integrals over pure measured signals. The Zj are the parameters to be determined. Taking the eq. (5.23), a linear regression will yield Zy. This procedure has the following advantage in comparison to the evaluation of difference equations: - the differential equations have not to be solvable in a closed form, and - no equidistant signals have to be determined. Quasi-linear photoreactions can be easily evaluated by formal integration. If all the concentrations have been determined, one can use eq. (2.30). If the the transformed time 0 is used instead of the real time t for differentiation one finds

Applications of Kinetic Analysis to Photoreactions

326

Ch.5

taking the definitions of 0 according to eq. (5.9) and F(E^) according to eq. (1.41)

For simplification the definition

r(E)^-Q^ ^

lOOO/o

was used. The integration of this equation yields j;>'(£)rfa'=K;,j;rrf/

(5.24)

This vector equation contains n normal equations. Example 5.12: Formal integration of a simple consecutive reaction In Example 2.19 in Section 22 A A the consecutive photoreaction

was discussed. One finds according to the derivations in Section 2.2.1.1 as given in Table 2.5 and discussed in detail in Appendix 6.5 -it, Kc = k^ 0

0 -k2 k^

0 0 0

Therefore the following formal integrals can be calculated: ^F\E)da = -k^^adt

J F\E)db = k^^adt - k^ jbdt JF'{E)dc = k2Jbdt

Ch. 5

Evaluation of Measurements of Concentrations

327

These equations can be evaluated by linear regression for suitably chosen lower and upper limits of integration. The kinetic constants of the system are obtained directly. Literature with respect to formal integration is found in the following citations [60,127,131-141].

5.2 EVALUATION OF MEASUREMENTS OF CONCENTRATIONS 5.2.7 Principles In this section it is not the concentration itself that is measured as a function of time but rather physical properties Af, are observed which are given by M,=m,a,

(/ = 1,2,...,«)

(5.25)

The signal obtained in high performance liquid chromatography (HPLC) represents such a measured value. M^ is the signal of the reactant A,, m, is a proportionality constant; its value is not known in general. Thus one can also give the n scalar equations (eq. (5.25)) in the form of matrix equations: M = ina,

(5.26)

where M is the vector of the measured values, a is the vector of the concentrations, and m the regular diagonal matrix with the diagonal elements m,. Thus one obtains using eq. (5.26) a = m"'M.

(5.27)

The reciprocal matrix m"* gives another diagonal matrix with the diagonal elements l/m^. 5,2,2 Matrix rank analysis In Section 5.1.1.1 it was demonstrated that a matrix Aa defined according to eq. (5.2) has rank s. The same is valid for the matrix AM with the elements

328

Applications of Kinetic Analysis to Photoreactions

Ch. 5

since obviously AM = i n A a .

(5.28)

The rank of Aa is not influenced by multiplication with the regular matrix m. Therefore the measured values can be directly used to determine the number of linear independent degrees of advancement without knowledge of the proportionality constant. In the same way the Af,. values can be taken for a graphical rank analysis. Instead of the K-diagrams related M-diagrams are obtained. Example 5.13: Transformation of signals to concentrations for photoaddition For the reaction 2A + B

a

>C

one obtains A/A=A/AO-2MBO^ + 2M3^^ mg mg

M^=A/^o-2Mc-^ m,c MB=A^BO-A^C

(5.29)

—

m^

if the concentrations according to eq. (5.25) are substituted by the M^values for the balanced masses in Example 5.2 in Section 5.1.1.2. In this case linear relationships between measured values are obtained. Besides the stoichiometric coefficients, the proportionality coefficients are included in the slope and the intercept. Eq. (5.29) can be used for the determination of m^, if the m values of the initial components are known.

Ch. 5

Evaluation of Measurements of Concentrations

329

Example 5.14: Combined parallel and consecutive mechanism The balanced masses in the Af-system for Example 5.3 in Section 5.1.1.2 are given by

me M^ = X^^Mc

(5.30)

K -hi

m^

Evidently the structure of the balanced equations is maintained. However, the constants x ^^^ ^ cannot be taken directly out of the graphics. The M-diagrams have the same form as the K-diagrams. By variation of the scaling of the two axes both the diagrams can be transformed into each other. In particular, the dependencies on intensity and wavelength correspond to those in K-diagrams. Similarly MDQ-diagrams refer to KDQ-diagrams. 5.2,3 Evaluation of time functions Since it had been assumed that the measured values are proportional to concentration, the evaluation by integrated time laws causes no essential difficulties. In Section 5.1.2.1 the physically sensitised photoisomerisation was treated as an example. Instead of the relationship given there one finds -^mgln —S- -\-Mj,'-M^^ = Km^e *5 K'^mJ by graphing ln[(AfB / M^Q) I (Afg - Mgo)] versus ©/Af^ - M^Q • One obtains a straight line with a slope -Kk^/k^ and the intercept -k^K I k^m^, and k^ I k^ cannot be determined separately without knowledge of the proportionality constants. However, since B represents the initial compound, m^ is obtainable in general.

330

Applications of Kinetic Analysis to Photoreactions

Ch. 5

In the same way the kinetic analysis of quasi-linear photoreactions by difference equations is possible using M values (signals of reactants at various times t). By multiplication of eq. (5.13) from the left by m and taking into account eq. (5.27) one finds m Aa = AM = in K"^ m"* ( M - M ^ ) or using K^smK'^iii""*

(5.31)

AM = K ^ . ( M - M ^ ) .

(5.32)

If the M values are evaluated according to eq. (5.32) one can determine the elements of K^. According to eq. (5.31) the matrices K^ and K^ are similar, that means they have the same eigenvalues. Since m is a diagonal matrix, for the above transformation the general relationship is valid Z = m K m"^ with the elements Zi^ z,,=*,,^.

(5.33)

The use of eq. (5.33) has the following consequences: (1) If kif^ = 0, then Zn, = 0 , also. (2) Z and K have the same diagonal elements. (3) In all cases kij^k,^^ = Z/j^Zj^,. This means that the consecutive photoreaction examined in Section 5.1.2.1 (Example 5.9) can be analysed in the M system using difference equations. The kinetic constants /:, and /TJ can be determined and correlated with the two steps of the reaction. Furthermore recursion equations can be used. By multiplication of eq. (5.14) by m and taking into account eq. (5.27) one finds

Ch. 5

Evaluation of Measurements of Concentrations

331

ma(04-A) = M(0 + A) = in-K'in"'M(0) + in-u. With

and

one obtains M ( 0 + A) = K""^ . M(0) + u ^ .

(5.34)

Accordingly difference and recursion equation of higher order can be transferred into the Af-system also. Multiplying eq. (5.17) by m^ one finds A^M, - 5 f A^-*M, +...4-(-l)^5,^(M, - Af,^) = 0,

(5.35)

since the higher order differences are given by

Eq. (5.35) correlates perfectly with eq. (5.17). The 5f values have the same meaning as in eq. (5.17). In eq. (5.18) a formula for recursion equations of higher order was given. It can be transformed into the M-system accordingly. One has just to substitute all the Sj by M, in this relationship. 5.2.4 Quantum yields Taking eq. (5.20) and eq. (5.24) one obtains

''-—m^If the components A and A, are the same, the compound starting the photoreaction is monitored itself by the measurement of the change in signal.

332

Applications of Kinetic Analysis to Photoreactions

Ch. 5

Under these conditions tp can be directly obtained. If eq. (5.21) is valid for the quantum yield tp as is the case in Example 5.11 in Section 5.1.2.2 one obtains in the M-system - =Z,+;f2^ ip

(5.37)

MB

A graph of 1/^ versus M^ allows control of the functional relationship and determination of X\* In contrary Xi is only obtainable if m^ is known. Since component B represents the starting material, m^ will be measurable in general without problems. 5,2.5 Formal integration These M, values can also be evaluated in general by formal integration. This fact is going to be proven in the case of the photoaddition treated above. Eq. (5.36) can be rearranged to the form m^ F ( £ J

MB

if the quantum yield is given by eq. (5.37). A, has been taken as B. If M^, MB, and F{El^) have been measured as a function of time, one can integrate within the limits of the variables t and f. In consequence the equation

MB(0 M^{t)--M^{t')

^

y, Ximj,

iooo
Mj,{t) •• Mj,{t')

can be evaluated by linear regression. This is another example where quasi-linear photoreactions can be conveniently evaluated by formal integration. If the values of all concentrations have been measured, one can use eq. (2.30). By multiplication of

on both sides by m, using the definition of 0 according to eq. (5.9) and for F(£';J)according to eq. (3.67) and the definition used in eq. (5.24), one obtains

Ch. 5

Evaluation of Measurements of Concentrations

333

F'(£)JM = mK'cmMc/r. Under these conditions it is necessary to differentiate with respect to the transformed time scale 0 . For simplification FXE) is defined as F(E'^)/ 1000/Q. By formal integration one obtains

JlFXE)dM = Kj^,Mdt

(5.39)

K^ = m • K • m

(5.40)

with

The vector equation, eq. (5.39), consists of n equations. Example 5.15: Consecutive photoreaction with equilibrium in the 2nd step For the consecutive photoreaction hv

hv

A—>B^C hv

one finds using eq. (2.30) the matrix of the reaction constants ^-it,

0

0^

Kj

—k'^ J

K^ = 0

Taking into account the factors m, one obtains

0 K^ =

0 ntf.

0

334

Applications of Kinetic Analysis to Pliotoreactions

Ch. 5

and the following linear integral equations

JF'{E)dM^=-kiJM^dt, \F\E)dMr, = - ^ ^ [M.dt-k^

[Mr,dt-¥^^^ [MrdU ntQ

jr{E)dMc =^^^jbdt'-k,JMcdt. This system of equations can be evaluated by linear regression. The rateconstants A:,, ^2 ^"d ^3 can be determined directly. If the proportionality constant of the initial component m^ is known, in addition the constants m^ and niQ can be determined. 5,2,6 Missing discrimination between two reactions If measured values of a certain type are used for kinetic analysis, the general question arises, up to what amount a discrimination between different reaction pathways is possible for these measured values used in evaluation. In general no answer can be given to this question. However, for the quasilinear photoreactions examined above a criterion can be stated. The properties of the transformation according to eq. (5.40) allow the conclusion that in the A/-system two reactions R and R' cannot be distinguished, if a diagonal matrix T exists which allows the two Jacobi matrices to be transformes into each other according to the following equation: (5.41)

K „ = T K ; T - '

To do this the following conditions are necessary according to eq. (5.33): (1) The diagonal element in the ^-matrices must be equal. (2) For both the reactions the zero-elements must be placed at the same positions in the matrices. Example 5.16: Comparison of parallel and consecutive mechanisms Two different mechanisms are assumed. The first /?':

A - ^ B + D,

B->C

Ch. S

Evaluation of Measurements of Concentrations

335

is given by the following matrix equation:

^d'\ f-k; b' K

0 —"fCn

0

c'

^2

0

\d'j

0 oY-'^ a 0 0 0 0 0 0

b' c'

whereas the second reaction

R: A—^B—^C,

A-^D

can be described by

(-{k\+h) b c ydj

0

0 oVa

-kj^

0

0

b

0

ik2

0

0

c

jfc,

0

0

0 d)

The necessary conditions stated above kU ^12 = 0

and

*,<, 12 = 0

^3, = 0

and ^3, = 0

^42 = 0

and

^42 = 0

are fulfilled. If a transformation T exists, one finds k{ = A:, -f A:3 and

A:2 = A:2.

The diagonal elements t^^ have to follow the conditions according to eq. (5.33)

336

Applications of Kinetic Analysis to Photoreactions

Ch.5

The condition can be met if the element t, is set to 1, obtaining 0 0

0 0 T= 0

0

0

0

it, +it3

0 0

^

0

^1 "^ *3 y

Both reactions can be transformed into each other by the matrix T. Therefore

is valid also. For the reaction R' one finds c^ = a^ and d^=aQ. Applying the transformation one finds and

k,-hk^

5.3 EVALUATION OF LINEAR DEPENDENT MEASURED VALUES 5J.1 Principle consideration In this section methods are demonstrated to evaluate measured values which depend linearly on the concentrations of the reactants. However, the measured value is not necessarily proportional just to the concentration of one single component. It may happen that many (in general all) reactants influence the value of the measured signal, proportional to each concentration. Absorbance is such a typical measured value which can conveniently be measured in the visible and the ultraviolet wavelength range. The use of spectroscopy to evaluate photoreactions makes sense since in any case the absorbance at the wavelength of irradiation source is necessary to be able to evaluate such photoreactions. In the following, this type of evaluation of spectroscopic measurements is discussed. Other measured values which linearly depend on the concentrations may be evaluated in analogy.

Ch. 5

Evaluation of Linear Dependent Measured Values

337

5.3,2 The fundamental equation of spectroscopy The fundamental equations according to Lambert-Beer's law have been given in Sections 1.4.3 and 3.3.2. The general problem in using spectroscopy data in UV and Vis is the poor selectivity in this wavelength range. In principle, a multicomponent analysis has to be done to extract concentrations of the different components. 5.3.2.1 Determination of concentrations As derived in Section 4.2.6.2 the use of normal equations like eq. (4.10) allows concentrations to be determined. Some numerical problems and advice are given there. 5.3.2.2 Determination of absorption coefficients As described in Section 4.2.6.3 the absorption coefficients can be determined, if concentrations and absorbance at a distinct wavelength A are known at a sufficient number of states m. From now on the decadic absorption coefficients are used, since they are experimentally obtained. Transposition of the matrices in eq. (4.4) for some reaction times results

that means for a distinct wavelength A E;^=a'i;^d.

(5.42)

The index A makes it obvious that the column vectors have to be formed at one wavelength for m different states (e.g. times). If m>n, the absorption coefficients 6;^,. can be determined using eq. (5.42). If m = n, the matrix a* becomes regular and one finds

338

Applications of Kinetic Analysis to Photoreactions

Ch. 5

However, if m>n the system is overdetermined. Therefore one finds accordingly using eq. (5.42) after multiplication by a a system of normal equations aE;^ = aa^t/E;^ s N8;^,

(5.43)

and therefore £,=—~^. a

(5.44)

The matrix N describes the coefficient matrix of the normal equations. For further information see Section 4.2.6. 53.2.3 Other concentration measures The volume concentration in mol/1 is the typical concentration measure in spectroscopy as well as in kinetics. However, the transition to other measures may happen without problems. For example one finds for the relationship between a, and g,. (g/1): g,. = Mfi.^ or g,. = M;^ai.

(5.45)

In this equation M^ is the molecular weight of the compound A^. If one defines the diagonal matrix M with the elements A/,., one can rewrite eq. (5.45) in the form a = M"^g

(5.46)

The vector g is defined in analogy to vector a. Thus, using eqs. (5.42) amd (5.46) one obtains for a single reaction time t^dzW^i^dz'i.

(5.47)

5.3.2.4 Linear relationship between the concentrations Frequently linear correlations exist between the concentrations as has been stated in Section 2.1.1. The index ° symbolises that the law of conser-

Ch. 5

Evaluation of Linear Dependent Measured Values

339

vation of mass has not been applied yet and the vector a° contains linear dependencies between the concentrations. Examples can be found in kinetics, analytics, etc. If linear independent equations of the form

^pl«l+^p2^2+"- + /pn«n=*jP

between the n concentrations exist, then s-n- p linear independent and p linear dependent concentrations result. The above equation can be written as a matrix equation L a = b.

(5.48)

If one combines the columns of the matrix L (which belong to the vector of the linear independent concentration a ' ) to the matrix L I and combines the columns, which belong to the vector of the linear dependent concentrations a°, to the matrix L2, then one can rewrite eq. (5.48) in the following form: Ll.a'+L2.a°=b

(5.49)

Under these conditions L I is a /? x 5 matrix with the elements

/„

/,.

and L2 is a quadratic regular matrix pxp matrix with the elements

The concentration vector a' consists of the elements a^ to a^. The vector

340

Applications of Kinetic Analysis to Photoreactions

Ch. 5

a** consists of the elements a^^^ to a^. Since the matrix L2 is regular, the inverse exists. Therefore one can rewrite eq. (5.49) in the form a" = L2""'. b - L2~* • LI • a'

(5.50)

This equation determines the linear dependent concentrations a" as a function of the linear independent ones a'. The matrix L2"* • LI has the dimension pxs. Next a combination matrix of the dimension nxs

is defined. The element I is the .s: x .y identity matrix. Furthermore the vector

is defined. Under these conditions the total vector a can be written as a function of the vector of the linear independent concentrations a': a = b"-H.a'

(5.53)

At first this vector b' consists of s zero elements, p elements follow, which are created by multiplication of the inverse matrix with the vector b . Combining eq. (5.53) with the fundamental equation (5.42) one finds E = d E h' "d e H r =E' -F r

(5.54)

Example 5.17: Transformation from concentration to absorbance One gram of binary mixture may consist of pj grams of compound 1 and /?2 grams of compound 2 /72=1-Pi.

(5.55)

If G grams of the binary mixture are dissolved in V litres of a solvent one finds

Ch. 5

Evaluation of Linear Dependent Measured Values

g,=M

and

341

g,=^.

By use of eq. (5.55) one finds the relationship (5.49) G 82= —-gx

=8-81

and in analogy to eq. (5.53)

Therefore using eq. (5.47) at a single wavelength one obtains

or

— = cfe; + 4 ^ ; - f ; ) p , +4^2 -^0/^2

(5.56)

8 In both equations the index ' does not refer to the wavelength of irradiation but to the degree of linear dependence between the different components according to the law of conservation of mass. Example 5.18: Application to an addition reaction For the concentrations of the compounds taking part at the reaction A + B-^C the following equations are valid: a~fo + 0 = aQ -foQ, a + 0 + c = aQ, 0-f & + c = fco. Thus two of this equations are linearly independent. The first equation

Applications of Kinetic Analysis to Photoreactions

342

Ch.5

is equal to the difference of the second and the third. If one cancels the first equation, one finds eq. (5.48) 1 0 1> 0 1 r

y.^0

and therefore

crn: :i:]=e:)Thus, the inverse of the matrix containing the columns of the linear dependent concentrations becomes L2-> =

-1

1

1 or

and eq. (5.50) can be written explicitly

^^-1

lY«o)_f-l

[l

OXbJ [l

lYlMa)Jbo-aoYflMa

OAOJ

t a, J [-\)

Therefore eq. (5.53) can be written as

^a\

{

0 ^

rn 1

\^J

l-ij

Thus one finds using eq. (4.4) at A = 1

5.3,2.5 Absorbance and degree of advancement The change in absorbances during a chemical reaction can be calculated via the balanced equations as has been demonstrated in the proceeding ex-

Ch. 5

Evaluation of Linear Dependent Measured Values

343

ample. More convenient is to treat the absorbances as a function of linear independent degrees of advancement. The correlation between the concentrations and the linear independent degrees of advancement yields according to the treatment in Section 2.1.1.1 and eq. (2.5) a =ao+vx,

(5.57)

where linear dependencies between different steps of the reaction have also been omitted (see considerations in Section 2.2.1.1). By multiplication of both sides from the left by the absorption coefficients matrix e one obtains Je-a =E=

rfea0+rfe-vx

or E = Eo + Qx,

(5.58)

with the definitions Eosrfeao

and

Qsdev.

(5.59)

The vector x consists of exactly s elements. The vector E is represented by h elements, h can be any number. Under these conditions the matrix Q consists of h rows and s columns. The special case s = h is very essential. If the wavelengths A had been chosen such that Q is regular, a reciprocal matrix exists and using eq. (5.58) one obtains (5.60)

X= Q-*(E-EO).

Under these conditions one can determine the linear independent degrees of advancement by use of absorbance measurements if the elements of Q are known. Example 5.19: Transformation consecutive photoreaction For the reaction

concentration

to absorbance for

a

Applications of Kinetic Analysis to Photoreactions

344

Ch.5

one finds the following relationships according to the derivations given in Section 2.2.1.4 and including a reversible isomerisation for the first step, using the equations discussed above, v is given by

(-1 0^ v = 1 -1

u

0

whereby s equals 2. The measurement at two wavelengths results £ =

^lA

« IB

^2A

^ 2B

^IX

For Q =rf• e • V one obtains

Q=

(""^aA "^ ^2B j

( ^2B "^ ^2C j

According to the definition, eq. (5.59), the elements of Q are represented by n

(5.61) They are the stoichiometric sums of the elements C;^. and can be directly taken out of the general scheme explained in Section 2.1.1.1. For this reason one multiplies the stoichiometric coefficients in the *th row with the absorption coefficients of those reactants in the related columns. 5,3,3 Matrix rank analysis If h absorbencies E;^{t) (A = 1,2,..., A) have been measured at m different times tf (/ = 1,2,..., m), the matrix AE of the measured values

^;u^Ex{u)-E.M

Ch. 5

Evaluation of Linear Dependent Measured Values

345

can be rearranged according to eq. (4.4) to the form AE = rf e Aa.

(5.62)

If the rank of e is not smaller than the rank of Aa, then AE and Aa will take the same rank. For this reason one is able to determine the number s of linear independent steps of a reaction by measurement of absorbances without knowledge of the absorption coefficients. 5.5.5.1 Application of graphical methods using absorbances Isosbestic points [12] During thermal- and photoreaction periods a registration of spectra is performed. The obtained reaction spectra (see Section 4.2.6.1) can at one or some wavelength cross in just one point. These points are called isosbestic points. An example is given by the reaction spectrum of azobenzene (see Fig. 5.4). Such a point is found at a certain wavelength A if at all times A£,(r) = 0. o:u^^

E 0.7 0.6 0.5

/////^^

0.4 0.3 0.2 0.1

//y^

z^:;^^^! ^ ^ ^ 236

f//y^

7*"^^^ 268 ^ ^

^ ^ ^ ^ " " " ^ ' " ^ "^^^ 340

320

300

280

260

240

A[nm]

Fig. 5.4. Reaction spectrum of the photoisomerisation of azobenzene in methanol, irradiated at 313 nm.

346

Applications of Kinetic Analysis to Photoreactions

Ch. 5

Therefore according to eq. (5.58) A £ , ( 0 = eAi^i +GA2^2 -^'"^QXS^S = 0

(5.63)

has to be valid. Since according to the definition the linear independent degrees of advancement jc^ cannot be zero during the total reaction eq. (5.63) is only fulfilled if at the wavelength A GAit=^Z^/it^Ai=0

for /: = l , 2 , . . . , ^

(5.64)

is given. In the case 5 = 1, i.e. the reaction is uniform, eq. (5.64) can be fulfilled very easily. One finds an isosbestic point at all wavelengths at which the spectra of the starting compounds cross with the spectra of the final products. If 5> 1 and all reactants absorb then it is rather improbable that eq. (5.64) is fulfilled at any wavelength. For the consecutive reaction

for example

has to be fulfilled at the same time for one wavelength. If just a few reactants absorb in the certain spectral region one can find an isosbestic point even though ^ > 1. An example is the reaction system A+B—^D +E A+C—^D +F under the condition that, in the spectral range considered, only the reactants A and B absorb: A£:A(0 = 4 ^ A D - ^ A A ) ( ^ I + ^ 2 ) -

If the condition s^^ = E;^^ is fulfilled at just one single wavelength, one

Evaluation of Linear Dependent Measured Values

Ch. 5

220

240

260

280

347

300 wavelength [nm]

Fig. 5.5. Reaction spectrum of the photo-induced hydratisation of l-/?-D-arabinofuranosylcytosine, irradiation at 254 nm with a mercury arc (/Q = 2.78 x 10~^ Einstein cm~^ s~^).

finds an isosbestic point at that wavelength. In another spectral region in which further compounds absorb, the spectra will cross without forming an isosbestic point. In Fig. 5.5 an example is given for a reaction spectrum in which the isosbestic point is valid during the total reaction just in part of the spectrum. During the photo-induced addition of water to l-)8-D-arabinofuranosylcytosine and subsequent thermal consecutive reactions an isosbestic point is only valid for wavelengths longer 280 nm [142]. Two others below vanish during the reaction procedure. In general the existence of an isosbestic point or its absence can be used to draw the following conclusions: (1) If spectra cross in a single isosbestic point a possibility exists for reaction to be uniform. (2) The crossing of spectra in more than one isosbestic point increases the probability of the reaction to be uniform. (3) The crossing of spectra in certain wavelength regions without an isosbestic point requires that the reaction is not uniform even though in other spectral ranges isosbestic points show up. (4) The absence of any crossing of spectra hinders the possibility of uniformity of a reaction.

348

Applications of Kinetic Analysis to Photoreactions

Ch. 5

Sometimes one can obtain further information from reaction spectra. For the reaction

according to eqs. (5.58) and (5.61) ^Ej^ (t) = d{e^^ - e^f^ )x, -h d[e^^ - e^^^ )x^ is valid. If at a certain wavelength e^^^ equals e^^, one finds

At this wavelength the concentration of the intermediate is directly proportional to the change in absorbance. One can recognise such a wavelength by the fact that the initial spectrum for r = 0 will cross with the final spectrum at r —> 00, since at this wavelength the condition is given by ^ A ( O ) = ^AA^O = ^ A ( ~ ) = ^^AC^O-

Absorbance difference diagrams [134] As demonstrated above isosbestic points only give limited information about the uniformity of a chemical reaction. More reliable expressions can be made using absorbance difference diagrams (ED-diagrams). For two different wavelengths 1 and 2 according to eq. (5.63) the change in absorbance with time is given by (J.OJ)

AE^it) = 021^1 + Q22X2 + • • • + Qls^s

This equation states the following. A graph in a rectangular coordinate system of AE^ and AE2 results an affine distorted image for {s>2 degenerate) of the relationship between the linear independent degrees of advancement Xf^ in two-dimensional space. Whereas only the ratio of the axis is valid in the case of a graph taking the diagonal matrices, in this type of diagram the

Evaluation of Linear Dependent Measured Values

Ch. 5

349

axes are even rotated and stretched. The most essential properties of affine or linear transformations had been summarised in Section 2.3.4. Linear absorbance difference diagram If the reaction is uniform, the number of linear independent reactions becomes s = \. Therefore using eq. (5.65) one obtains ^E, = ^E^

(5.66)

The ED-diagram results in a straight line passing the zero point with the slope QuIQix (see Fig. 5.6). By these means any reaction can be tested in a simple way for uniformity using diagrams with statistically dependence of the wavelength on g^*' • It has been demonstrated that ED-diagrams always result in zero point straight lines if the rank of Q A[nml ^ 320 y / \ 310

ii

1.0 . 0.8 .

M 300

^

0.6 . ^ ^

0.4 -

290

0.2 • 0^

1»0

AE ^313

-0.2 4

^^^""^^"^^ 250 1

, 200

, , 500 800

1

Ms]

•

Fig. 5.6. Linear absorbance difference diagram for the photoisomerisation of azobenzene, irradiated in methanol at 313 nm.

350

Applications of Kinetic Analysis to Photoreactions

Ch. 5

is smaller than 2. This is always valid for 5^ = 1. For jr = 2 a single wavelength combination can result by chance |Q| = Gi,Q22-612621=0

(5.67)

Thereby a certain analogy to the discussion of the isosbestic point in the reaction spectrum can be found. Condition eq. (5.67) can be easily fulfilled, if in a certain wavelength range just a few compounds absorb. It is sufficient if in this spectral range all other elements QA* except two become zero in one row. This will happen if independent parallel reactions take place whose reactants do not absorb in the spectral range examined. However, the rank of Q can become 1 if the J2A* become equal in two columns. This can happen for the parallel reaction treated in Section 2.1.4.2, if in some spectral regions only compounds A and D absorb. Under these conditions

is valid. Therefore the rank of Q becomes 1. Even though if in a certain spectral region just one compound absorbs, the Qj^j^ in two columns become equal. This fact proves that one has to examine the ED-diagrams at as many wavelength combinations as possible (see Fig. 5.7). Absorbance difference diagrams for s ^2 By the transformation of eq. (5.65) the correlation between the degrees of advancement jCj and Xj are rotated and distorted. Thereby the characteristic properties of the X- and ^-diagrams are lost. Fig. 5.7 gives the standard example for a photoreaction with two linear independent steps (^ = 2) [60]. Fig. 5.8 is an example of an ED-diagram of a consecutive photoreaction which proceeds according to eq. (3.39) and fulfils eq. (3.40) with ;f <1 [131,143]. The different combinations of wavelengths allow two linear areas to be realsied in which one of the partial steps of reaction dominates. However, the typical slopes of Fig. 2.7 of Section 2.4.2.2 given by

Evaluation of Linear Dependent Measured Values

Ch. 5

351

246 nm 256.5 nm 320 nm

Fig. 5.7. Absorbance difference diagram for the photo-consecutive reaction of trans'StWbene in perfluorohydrocarbons.

dxo

=0

and

dXt

^ dx.

are no longer observable by the affine transformation. In general for s = 2 the correlation between the slope of the image and the original is given by dX'^

dAEo2

dE^ dE2

£2 E2

Qii^x^Qii^i 621^1 + 222-^2

n^, I n_ ^ 2 C21+222 dxx

(5.68)

This relationship is much more complicated than the related relationship for values which are proportional to concentration. Even though their slopes are varied also, the special slopes zero and 00 are not influenced.

352

Applications of Kinetic Analysis to Photoreactions

Ch.5

Fig. 5.8. Absorbance difference diagram of the photo-consecutive reaction of 2,3-dimethylantraquinone in alkaline methanol for two intensities of irradiation at 313 nm. ao= 1.05 x 10-^ mol l-^ /o(B) = 1.24 X lO'^, /o(0) = 5.46 x lO-'^Einstein cm'^ s'^

Initial slopes in ED-diagrams In the case of a consecutive reaction the initial slope amounts to dX'^

dxt

= 0.

Thus using eq. (5.68) one obtains dAE, dAE^

Qix

The initial slopes in ED-diagrams only depend on the Q values of the first partial step of the reaction. Therefore it is independent of the initial concentration. Varying the concentration of the starting compounds in different se-

Evaluation of Linear Dependent Measured Values

Ch. 5

353

ries of measurement one will always obtain the same initial slope in the case of a consecutive reaction. In the case of linear independent parallel reactions the initial slope of an X-diagram depends at least on the initial concentration of one of the starting compounds. In the case where one does not find any dependencies of the quantum yields of two partial reactions on the intensity of radiation, one can shorten the equation for the derivative of i^ with respect to time according to Xu

=

F{E)

Thus for two reactions one finds

If a, and aj differ, the initial slope in a X-diagram depends on the ratio of ^2(/^io- The same is evident for an ED-diagram. Thus the initial slope becomes dependent on concentration for any type of quantum yield. On the other hand if flfj = a^ (both photochemical partial reactions start with the same compound) at least one of the quantum yields has to be a function of concentration. Otherwise dx2ldxx becomes constant and x, and X2 would linearly depend on each other which is contradictory to the assumption. Points of inclination in an ED-diagram The second derivative of an ED-diagram results in

dAE^ dAE,
dXf

dAE^ dx.I /

(61262. -611622y'^2 _ 622611-612621 d^Xi dx. dxl 6 2 1 ^ + 6: 621+622 ^^dx^ !^22 dx. dxo (5.69)

A point of inclination in an ED-diagram requires that the left-hand side of

354

Applications of Kinetic Analysis to Photoreactions

Ch. S

eq. (5.69) becomes zero. This happens if either the determinant of eq. (5.67) becomes zero (in this case the ED-diagram is linear) or if

(in this case the X-diagram shows a point of inclination). Thus if the nominator in eq. (5.69) does not vanish by chance, a point of inclination in the original as well as in the image arises. The final slope in ED-diagrams According to eq. (5.68) the final slope in an ED-diagram depends on the initial concentrations if the final slope in the X-diagram does the same. Whereas the initial slope is given by the differential equation it turns out to be difficult to calculate the final slope in many cases. Whereas in the case of thermal reactions general methods for the determination of the final slopes [23,144] exist, if the partial reactions follow first or second order kinetics, such methods are missing in photochemical reactions. In the following some examples are given to demonstrate the possibilities. Example 5.20: Simple consecutive photoreaction For the consecutive photoreaction

according to eqs. (3.40) and (2.90) one finds dx2 =- ^ dxi oe

for x > l

and dx2 dxi

>oo for %<1 oe

The constant x was defined by eq. (3.40). Since this x does not contain any initial concentrations, the final slope in an ED-diagram will

Ch. 5

Evaluation of Linear Dependent Measured Values

355

always be independent of the initial concentrations. For x > 1 the slope depends on Q , in a very complicated matter. In the case of x 5 1 one finds

which means that the final slope in an ED-diagram follows the direction of the x,-axis. In such a reaction normally the direction of the x,axis is known since the initial slope is zero. Therefore one can take the relative values 5-, = x, /a, and 5, = x,/a, out of the ED-diagram . Reduced ED-diagrams If one varies the initial concentration for different series of measurements, but keeps constant the ratio of initial concentrations for many initial compounds, then one finds for the graph AE, I a,, versus AE, / a,, a type of reduced ED-diagram. These ED-diagrams have similar properties to the reduced K-diagrams. For a thermal reaction, reduced ED-diagrams are invariant against dilution if both partial reactions proceed with the same order of reaction. The diagrams vary if the two partial reactions exhibit different orders of reaction. In the case of photoreactions one has to examine whether the differential equations depend explicitly on the initial concentrations for the relationship in the 6 -diagrams. An example explains this in the following. Example 5.21: Reduced X-diagram for a simple consecutive photoreaction The above-mentioned consecutive photoreaction results in

Or using 5, = x, / a,,

5, = x,

1 a,

356

Applications of Kinetic Analysis to Photoreactions

Ch.5

^„.' i 8 ^395

6 -

4 -

^484

2 •

f\

0

"

>^^^ ^

4

2

.

L^ 10-

-2 -j

^313

^

Fig. 5.9. Reduced ED-diagram of the consecutive photoreaction on 2-methyl-anthraquinone in alkaline methanolic solution, NaOH (0.1 mol l'); wavelength of irradiation is 313 nm, the intensity of the light source is /^ = 1.24 x 10"* Einstein s"' cm"\ cone. (•), a^, = 5.81 x 10"* mol/1. (O): a„= 1.16 X 10"^ mol/l.

is given. The consecutive photoreaction of 2-methyl-anthraquinone in alkaline methanolic solution is a typical example. It is demonstrated in Fig. 5.9. If the quantum yields are independent of concentration - as has been presumed - then the reduced ED-diagram has to be invariant against dilution since no initial concentrations show up in the differential equations. This can be seen in Fig. 5.9 for two different concentrations. Dependencies on intensity and wavelength For the dependencies on intensity and wavelength in the ED-diagrams the same rules are valid as for K-diagrams. For comparison see Section 5.1.1.2. In Fig. 5.8 the dependence on intensity has been marked, demonstrating that the ED-diagrams do not differ.

Ch. 5

Evaluation of Linear Dependent Measured Values

357

Combined photo and thermal reactions If the photochemical reactions are superimposed by thermal reactions, in some cases additional information on the nature of the partial reactions can be obtained. This can be demonstrated in the case of the photoisomerisation of rmn^-azoxybenzene (A) to <7rr/ic?-hydroxy-azoxybenzene (C) [145]. In methanolic solution this reaction does not proceed uniformly, cisAzoxybenzene (B) is formed as an intermediate. Using EDQ-diagrams explained in the next section one can demonstrate that just two linear independent partial reactions take place. The reaction can either follow the mechanisms,

hv

or A—5L^B—!2L^C.

In both cases the slow thermal reaction B—^A, is superimposed. If the irradiation is interrupted, after some time the uniform thermal reaction can be monitored separately. Its slope in the ED-diagram is given by

For iht first reaction

X2=
is valid. Therefore dx2 ^

dx,



Applications of Kinetic Analysis to Photoreactions

358

Ch.5

will be obtained. At r = 0 one finds b = 0. Therefore I^ becomes 0, that means the initial slope is dx^ dx^ For the second reaction

X2 = (P2

/B

is valid. Thus dx2 __ .

=0

dxt

are obtained. The initial slope in the ED-diagram becomes dAE^ ^Qx\ . i i B - _ £lA dAE2 21 ^2A -2B

a

A comparison of the initial slopes shows that as long the first reaction takes place the initial slope of the photoreaction is different from the slope of the thermal reaction. However as long as the second reaction takes place both slopes become the same. Fig. 5.10 gives an ED-diagram of this photoisomerisation. The irradiation was interrupted many times. The slopes of the thermal reactions are parallel to each other, and their slopes differ from the initial slope of the photochemical reaction [145]. For this reason the second mechanism is discarded. The ED-diagram accounts for the first mechanism. However, it does not prove it, since the reaction could proceed in a more complicated way.

Ch. 5

Evaluation of Linear Dependent Measured Values

359

Fig. 5.10. ED-diagram of the photoreaction of azoxybenzene in methanol for irradiation at wavelength 313 nm. a^ = 6.0 x 10"^ mol/l, x—^x, photoreaction; # — • , thermal period [145].

Absorbance difference quotient diagrams [128] If just two linear independent reactions take place in a system, the absorbance differences for three different wavelengths (A = 1, 2, 3) are given according to eq. (5.65)

AE2 = 621^1 + 022^2

(5.70)

AE3 = 231^1 + 632^2

The equation system, eq. (5.70), is overdetermined. These equations can contradict each other for physical reasons. If this does not happen according

360

Applications of Kinetic Analysis to Photoreactions

Ch. 5

to a theorem of Kronecker and Capelli [146] the determinant becomes

A£,

G„ G.2

AEj

22.

G22 = 0.

(5.71)

A^s 031 G32 By multiplication eq. (5.71) can be rearranged to iD23A£:, + DjjAEj + AiA^a = 0.

(5.72)

With the definitions ^23 ~ QixQzi "" QixQii

^12 "^QxxQii "QxiQix eq. (5.72) results in a 3-dimensional area spanned by the absorbance differences AE^,AE2 ^^d AE3. To present this relationship graphically in two dimensions, eq. (5.72) can be rearranged to ^E. AE. L A J^ .-- ^^ ^ "^

(5J3)

usmg and a = - A 3 A2 Fig. 5.11 presents an example for such a diagram according to eq. (5.73). It is called an '*absorbance difference quotient diagram" abbreviated in the following as EDQ-diagram. The figure demonstrates the photoreaction of stilbene in perfluoromethylcyclohexane by irradiation at the wavelength 313 nm. In this diagram both the consecutive reaction JlL^^—hl^C hv

Evaluation of Linear Dependent Measured Values

Ch. 5

10

20

30

361

ED^ EDi

Fig. 5.11. EDQ-diagram of the photoreaction of stilbene in perfluoromethylcyclohexane by irradiation at wavelength 313 nm. x—^x, consecutive reaction; O—O, parallel reaction [60].

and the parallel reaction B:

hv hv

±A,

B-

hv

^C

were included (A, rmn^-stilbene; B, cw-stilbene; C, phenanthrene). The measured values fall onto the same straight line for both reactions. This was done on purpose since all the points of the 2-dimensional X-diagram which are conditioned by x^ and jCj, fall onto the same area by the transformation eq. (5.70) given by eq. (5.72). EDQ-diagrams become linear if either just two linear independent reactions take place or if in more than two linear independent partial reactions the rank of the matrix Q reduces to two by chance. For a district discrimina-

362

Applications of Kinetic Analysis to Photoreactions

Ch. 5

tion between the cases s = 2 and 5 < 2 as many combinations as possible have to be chosen in a large wavelength range extended far into the short wavelength region. If a linear EDQ-diagram intersects the coordinate system at the zero point or is parallel to one of the axes it is called degenerate and does not allow any information on the number of linear independent partial reactions. Under these conditions one of the three D^^ becomes zero. In consequence one of the three possible ED-diagrams degenerates to be linear. For example A3 = QuQn "" Qi\Q\i becomes zero by chance or Q^^ = 612 = ^ can become zero by chance. That means the compounds taking part in the second partial reaction do not absorb at wavelengths 1 and 2. However, under these conditions one finds

A£,

e„'

In consequence the EDQ-diagram degenerates - even rather independently of the fact whether AEj is parallel to the axis AEj / A£^i. As in the case of KDQ-diagrams for the absorbance differences any time t can be taken as a reference. In analogy to eq. (5.6) for any two reaction times / and /' one obtains

E^L-EM^ = p + a^^' ^"' El, - E^,. ^u " ^\r

(5.74)

According to this equation the points always have to fall onto the same straight line for the case ^ = 2, independently of which way the differences were taken. If the differences in the time r = 0 have been taken the origin of the coordinates is given by

.E, <^"*^"t AEi

dX',

dx^ and the final point at f —> oo yields the coordinates

(5.75)

Evaluation of Linear Dependent Measured Values

Ch. 5

363

(5.76) On the other hand, if the differences are referred to the final point at f —> oo, one can use the definition A„E =

E^{t)-E^,

and one obtains with the coordinates of the first points at / = 0 (5.77) A^£,

eii-«i-+Ci2-«2-

and for the coordinates at the final point at t

A.^, A„5,

a,

-Qu^ 0

(5.78)

0

The form of the straight lines is just determined by the few ^A* values. The form does not change if the parameters of the reaction (initial concentration, wavelengths, intensity of irradiation) are changed. The region in which the measured value arises will only change if the quantities and OX,

^ dxx

depend on the parameters of the reaction. EDQ(s)-diagrams of higher order can also be constructed, as demonstrated for concentration diagrams and Mdiagrams. 53,3,2 Absorbance diagrams [131,133] If at two different wavelengths the absorbancies themselves instead of the absorbance differences are plotted versus each other one obtains an absor-

Applications of Kinetic Analysis to Photoreactions

364

Ch.5

bance or "extinction" diagram (E-diagram). The difference from the EDdiagrams is the intercept of the zero point. However, these new types of diagrams show all the characteristic features of the ED-diagram. Nevertheless one can obtain additional information using E-diagrams. These are demonstrated in the following. Just one compound absorbs If just one compound A,, {e^. ^ 0) absorbs for a defined concentration one obtains in the E-diagram one point with the coordinates E^^de^ai

^2=^^2i«i-

If the concentration of A. is varied either by addition of further A,., by dilution of the solution, or by vaporisation of the solvent etc., all the points in the E-diagram will be placed on a zero point straight line: E, -=

^1/

If the compound A^ reacts and forms products which do not absorb at the wavelengths 1 and 2, the E-diagram will give a straight line from the point of initial absorbance in the direction of the zero point. This will be reached if A^. is totally converted. By this means one can recognise in an E-diagram all wavelength regions in which just one compound absorbs. Uniform reactions If a reaction is uniform for the absorbencies E,:=E,o+QiX

(5.79)

^2 = ^20 + Ql^ is valid. Using the vectors E=

^1^ ^2 J

Eo =

and

E> = ^2ooJ

one can rearrange eq. (5.79) as a vector equation

Q=

Ch. 5

Evaluation of Linear Dependent Measured Values

365

(5.80) This is an equation of one straight line in vector representation. For the final absorbance one finds E

(5.81)

= E O - H Q ;c

This relationship is schematically presented in Fig. 5.12. The point P^ i^ the initial point, the point Pj the final point of the E-diagram. Rearrangement of the relationships eqs. (5.80) and (5.81), as well as taking the ratio of the magnitudes, gives E-EQ JC =

E.-E.

P,P RP. r 2

(5.82)

^ o o .

Using eq. (5.82) for any uniform reaction, the degree of advancement can P.

Fig. 5.12. Schematic absorbance diagram of a uniform reaction.

Applications of Kinetic Analysis to Photoreactions

366

Ch.5

be determined as a function of time, if the absorbances as a function of time were measured and if Eo, E- and x^ are known. Special reactions will result in special properties of E-diagrams. For the reaction

one finds EQ = E/^daQ and

E^ = e^daQ

with

E; = ^u

Thus, if many measurement series with different initial concentrations are presented in a single E-diagram, all initial values of the reaction will be placed on a zero point straight line OA and all the final values on a zero point straight line OB. This is demonstrated in Fig. 5.13. The different measured values can be found in the area spanned by straight lines which are parallel to each other and intersecting those OAand OB. By addition of in-

Fig. 5.13. Schematic E-diagram of the reaction A —> B for two differential initial concentrations (O—O). In the case where some final product is added, the absorbances are situated on the line 05 (•—•). The broken line connects all data at different concentrations reached at equivalent reaction times.

Ch. 5

Evaluation of Linear Dependent Measured Values

367

creasing amounts of the final product B to the reaction solution points on the zero point straight line OB are found. As long as the reaction is not finished, the points are placed on a parallel to OB, If the reaction proceeds at first order, all the points of the different measurement series are situated on a zero point straight line obtained by equivalent reaction times. Two linear independent partial reactions For two linear independent partial reactions the vector E is given by (5.83)

E = Eo+QiX,+Q2^2

The relationship eq. (5.83) is schematically presented in Fig. 5.14. The measured points are placed on that part of a curve which starts at E© and ends at Eoo. The figures supply the relationship E^ ^loo>

X-y

=

(5.84)

^0^2

Fig. 5.14. Schematic E-diagram for two linear independent reactions. The figure visualises eq. (5.83).

Applications of Kinetic Analysis to Photoreactlons

368

Ch.5

A

Q,^

BV 1

^I

•• I #<w

1

EB /

\ ^ ^

p,'

V M C

/

;

/ ^"Z

^*^^^..--""^

Ec C ^ '^ Q 2 ^ 2

//.•••"" ry*''*

»-

0

Fig. 5.15. Schematic E-diagram for the reaction A -^ B —> C, including an absorbance triangle according to a distorted Gibbs phase diagram.

Using these relationships the degrees of advancement jc, and Xj can be determined if the direction of the vectors Q, and Qi as well as the final values x^^ and jCjoo are known. In the case of consecutive reactions these values are relatively easy to obtain. This can be demonstrated for the reaction A -^ B -> C, for which the E-diagram in Fig. 5.15 is presented schematically. On the straight lines OA, OB and OC the absorbances of solutions are contain nothing else than the pure reactants A, B or C. For a certain initial concentration CQ QA = E^ = e^dcQ,

0 5 = Eg = e^daQ,

is valid for the vectors. Thus A5 = E B - E A = ( e B - £ A y a o = Qiao

OC = Ep = Erdan

Ch. 5

Evaluation of Linear Dependent Measured Values

369

and S C = / \ C = E C - E B = ( £ C - ^ B V « O = Q 2 « (0

is obtained. For this reason the degrees of advancement amount in the point P to

'

^

JR A5 ^

W ^

'AP" BC "^ AC ^

Considering the X-diagrams, one obtains

'

AB

AC'

^

yS = ^ , - ^ 3 = ^ = ^

AC

^

AC

BC

AB

AC

AC

BC

= 4J^

(5.85)

AB

BC

with

«0

^0

^0

«0

^0

According to eq. (5.85) the triangle ABC is a distorted Gibbs phase diagram. It can be used to determine the concentrations, if the point A, B and C are known. This procedure is demonstrated in Fig. 5.15. The corners of this triangle can be determined as follows: - A is given by the initial absorbance at r = 0, i.e. in the case of photoreactions before the irradiation starts. In the case of thermal reactions A can be determined by an extrapolation of the absorbance-time functions to the time r = 0 if necessary. - C is the final absorbance at r -> ©o. If the reactant C can be obtained, the straight line OC can be determined by measuring the absorbances of C at different concentrations. The absolute concentrations of C need not be known in this case. The point C can be determined as a crossing point of

370

Applications of Kinetic Analysis to Photoreactions

Ch. 5

the straight line OC with the more or less linear final part of the Ediagram. Furthermore the final point can be determined during the evaluation of the absorbance-time series with the methods presented in the next paragraph. - The point B is placed on a tangent to the curve APC in point A. If the straight line OB can be experimentally determined, point B represents a crossing point of this straight line with the tangent to A. If the photoreaction is quasi-linear and if X = ^2/^1 '^ ^ ^"^^^ ^^e straight line CB will touch the curve APC at point C. Under these conditions B is found as a crossing point of the two tangents at the points A and C (For further information see Fig. 2.7 in Section 2.4.2.2). Using the reaction spectra one can control whether the assumption % < 1 is fulfilled or not. If any wavelength is found, at which the absorbance monotonically decreases, one can determine the time t' at this wavelength after which the absorbance has been reduced to one half. However, during this time t' the concentration has been reduced from A to

2

Initially just the experimentally determined curve APC and the points A and C are known in the E-diagram. A tangent can be constructed at the curve in point A. Drawing a parallel to this straight line passing the point P{t') with the coordinates E2(t') and E^{t') one finds according to eq. (5.85)

AC However, eq. (2.90) requires that for a < 0 . 5 and y(^')< 0.153 the condition ;f
Ch. 5

Evaluation of Absorbance-Time Measurements

371

5.4 EVALUATION OF ABSORBANCE-TIME MEASUREMENTS

5,4,1 Evaluation by integrated time laws The physically sensitised photoisomerisation discussed in Section 5.1.2 and 5.2.3 is used as an example. The integrated time law amounts to ^\n\^

+ b-b^^-m,

(5.86)

whereby 0 is the transformed time. Using x-c-b^-b

and x^^b^,

one obtains E-E^=-Qb, which can be rearranged, inserting eq. (5.86), as

^In ^"^k,

Eo-E^

^ElzA^^KQ. Q

If one graphs P— F I In g _^ /(^o " ^ )

ve>*s"s

0 Eo-E

a linear dependence is also obtained in the absorbance system. It represents a straight line with the slope -Kk^/k^ and the intercept k^/Qk^. If one knows Q, one finds K and k^/k^ as in the case of the evaluation by concentrations. This is the case if E^ and E^ are known. 5.4,2 Difference equations [15] Taking the solution, eq. (2.45), one finds the difference equations for the degrees of advancement in analogy to eq (5.13)

372

Applications of Kinetic Analysis to Photoreactions

A x s x ( 0 + A ) - x ( 0 ) = K^(x~x^),

Ch. 5

(5.87)

whereby K^=p(exp(rA)-l)p-V The matrices p are given by eq. (2.45). If one multiplies the difference equations (eq. (5.14)) from the left by Q and from the right by Q"* one can transform the above equation into the absorbance system. One finds Q.Ax = Q . K ^ . ( x - x ^ ) or AE = Q . K ^ . Q - * ( E - E O ) = Z ^ . ( E - E ^ ) .

(5.88)

All the elements of the K^ matrix are varied by this transformation in such a way that the calculation is only possible if all the elements of the matrix Q are known. Since this procedure is a similarity transformation, the matrices K^ and Z^ have the same eigenvalues. This means that the consecutive photoreaction treated in Section 5.1.2.1 (Example 5.9) results in exp(-/:, A) - 1

and

exp(-A:2^) " 1

as eigenvalues of Z as well as of K^. The absorbance measurements can also be evaluated by recursion equations. Rearrangement of eq. (5.88) in analogy to eq. (5.13) results in correspondence to eq. (5.14) x ( 0 + A) = K' • x(0) + u with K'sp(exp(rA))p->. By defining

(5.89)

Ch. 5

Evaluation of Absorbance-Time Measurements

373

one finds the recursion equation E(0 + A) = Z' • E(0) + u'

(5.90)

with u' = E ^ - Z ' E^. In this equation the elements of K"^ are also varied strongly by transformation. However, the eigenvalues of Z'^and K*^ are the same once more. 5.4.3 Difference equations of higher order [15] If one forms the difference equation of higher order for the degree of advancement one finds by analogy to eq, (5.17)

^'x,-st^'-'x,^'-^{-•\Ys^{x, ^x,jf

(5.91)

whereby the elements 5f have the same meaning as in eq. (5.17). Multiplication of eq. (5.91) by QAJ^ yields the difference equation of higher order for the absorbances A^£,-5,^A^-^£,+... + ( - l ) ' 5 , ^ ( £ , - £ , ^ ) .

(5.92)

Using this equation the absorbance measurements can be evaluated by linear regression. By this means one obtains the s eigenvalues r^ using the quantities 5f. The same methods can be used to transform the recursion equation, eq. (5.18), into another one for the absorbances. In the course of the evaluation of concentrations according to eq. (5.17) or of measured values proportional to concentration according to eq. (5.35), one obtains the number e^ which gives the number of the exponential functions necessary to describe the function a^{t). Furthermore the related eigenvalues are received. However, using these equations just the number / and all the eigenvalues can be determined. Concentrations can no longer be obtained. This is a disadvantage of this calculation.

374

Applications of Kinetic Analysis to Photoreactions

Ch. 5

5A,4 Formal integration 5,4,4,1 Simple uniform photoreactions Photoisomerisation In the case of the photoisomerisation A—!!^B with a quantum yield constant according to eq. (3.35), one finds, using Napierian units for the derivation, . _ 1000/c>^/oa

In this and the following example K\ and i?'describe the absorption coefficients and the absorbances at the wavelength of irradiation as derived in Chapter 3 (Napierian units). For any wavelength of measurement A one finds according to eqs. (3.56) and (3.62)(since a^ = 0)

Qx Thus one obtains £,

1000/c>^/o(£,,-£,)F(£0-'

Qx

Qx

However, it is preferable to use decadic units; then the above equation becomes £, Qx

1000g>Vo(£,,-£,)F(0 Qx

using the different definition for the photokinetic factor according to eq. (1.42). The absorbance-time differential equations result in the form

Ch. 5

Evaluation of Absorbance-Tlme Measurements

E,=z,,F{t)

+ z,^E,F{t)

375

(5.93)

with z,,^\Q00e\ip^hE,^

and

z,, = - 1 0 0 0 E : , ^ ^ / O .

The constants Zx\ and Zx2 can be determined by formal integration for the different wavelengths of measurement. Zxi does not depend on the wavelength of measurement. Furthermore one finds for the final absorbance

If £\ and /Q are known, the quantum yield tp^ can be determined using z^iThe photoreduction of substituted 9,10-anthraquinones in neutral methanol has been examined by application to this method [143]. The quantum yields for the photochemical reactions of some of the derivatives are listed in Table 5.1. In Fig. 5.16 the absorbance difference diagram is given as an example for a substituted anthraquinone. Fig. 5.17 demonstrates the absorbance-time curves. All wavelengths have a linear relationship. In Table 5.1 the result of an evaluation according to eq. (5.93) is given for the same compounds. The 1-chloro-substituted compounds demonstrated a comparable small differential yield ip^ in neutral methanol.

TABLE 5.1 Quantum yields of various anthraquinone derivatives -Anthraquinone

Quantum yield ^ j ^

1-Chloro 2-Chloro 1,5-Dichloro 2-Methyl 2,3-Dimethyl 2-Carbonic acid

0.22 0.72 0.28 0.66 0.53 0.50

Applications of Kinetic Analysis to Photoreactions

376

I I

I

0 10" 30"

I

I

I

r

V

4'

Ch. 5

I

w

8'

/

Fig. 5.16. ED-diagram of the photoreaction of 2-chloro-anthraquinone in neutral methanol,

'380

.^313

20

> 30 \ [min]

Fig. 5.17. Absorbance-time diagram of the photoreaction of 2-chloro-anthraquinone in neutral methanolic solution, irradiated at 313 nm with /o= 1.26 x 10"^ Einstein cm~^ 8"^ initial product concentration GQ = 8.24 x 10"^ mol T^

Ch. 5

Evaluation of Absorbance-Time Measurements

377

Physically sensitised photoisomerisation In the physically sensitised photoisomerisation treated in Section 3.1.1.4 the quantum yield is given according to eq. (3.6) by A

Kb k^ + kj^b

X contains the part of the quantum yield independent of concentration. Thus according to Example 3.8 one obtains, using decadic units, ^^

\m)e\xIoaok,b ^^^^ kj + k^b

Using eq. (3.63) and assuming b^^O one finds

or

E

-hilInEx.Fit)

with

^3

Rearrangement of eq. (5.94) into the form

(5.94)

378

Applications of Kinetic Analysis to Photoreactions ["' dE, = z„ r F{t)dt + z,,f Jti

Jt,

E,Fit)dt - zJ'' Jt,

Ch. 5

E,dE,

Jti

allows the determination of the constants Zxi^o z^^ by formal integration. The following relationships are valid: ~

"" ^Aoo»

ii2. = iooOe;/„aoxG„ ^A3 GA^O =^AOO -^AO-

If the quantities e'^,lQ,aQ,bQ and £;IQ are known, one can determine xFurthermore one finds according to

the ratio of the constants /:2/*3- Th© level of agreement of the result for the evaluation of different series of measurements for different wavelengths of observation gives an idea of the accuracy of the determined constants. Furthermore it proves whether the mechanism has been determined correctly. Photoaddition For the photoaddition A+ B—^C, the quantum yield is given according to eq. (3.3) by

k2 + k^b One finds using eqs. (3.35) and (1.42) that

Ch. 5

Evaluation of Absorbance-Time Measurements

a=

379

I000€'lr,xk.ba ^, , dLM^—/r(^). k2 H- k^b

If GQ < b^ (that means a^ = 0 and 6^ = feg - aj,) one finds according to eq.

(3.63) a = —^ Qx

and

b = —^=^ Qx

with

In consequence one obtains ^A =

1 0 0 0 e ; / o X ^ 3 ( £ , , - £ , ) ( £ , ^ - E,) . ^ • , /^ ;r^^ ^(0

or ^Ji "^ ^ J 9 ^ J "*" ^ J i ^ i ^^^^M-^A2^A-^A3-A^(^)

(5.95)

1 + ZA4^A

With ^ 1000e;/oXfe3£,^£;t, ^"

"

^ ^1000£;/oX^3(£^ + g , , )

KQx^K^x^

k^Q,+k,E,^

^''

'*

l^iQx+k^E,^

k^Q,+k,E,„

the constants Z},^ to Z;^4 can be determined by formal integration. The constants Zx\, Zx2 snd Zxj can be used to calculate the values of EJ^„ and E•^„. x is obtained using the relationship - ^ = 1000e;/oZ. ZA4

380

Applications of Kinetic Analysis to Photoreactions

Ch. 5

The ratio ^2/^3 can be determined by

The differential equation of all these reactions results in the general form 1

M

Ex=-^^ j^p+a

^(0-

(5.96)

This equation describes a large number of uniform photoreactions with quantum yields, which do not depend on the amount of light absorbed. Complex uniform photoreactions In Section 3.1.3 and Example 2.11 the photoisomerisation

is discussed. Its partial quantum yields are constant according to eq. (2.21). Using eq. (3.81) one finds the differential equation

Taking eq. (3.63) one obtains E, = 1000/o(^f £; + ^f«B)(^A- -

=

{zxx+Zx2E,)F{t)

defining z„=1000/o(^fe;+^%)£,..

E,)Fit)

Ch. 5

Evaluation of Absorbance-Time Measurements

381

z,2=-iooo/o(^^;-fy>%). This equation can be evaluated by formal integration. Zx2 has to be independent of the wavelength of measurement. £^^ can be calculated according to

^A2

Using z^r^ one finds

lOOO/o

^^ ^

^^ ^

The final absorbance amounts to ^Aco = ^«AA«o + GA^OO =rfe;iBao" CA^OO. Thus one obtains according to eq. (5.103) (5.98) If «o' ^AA» ^AB» ^A ^"d ^B ^^^ known the ratio of the quantum yields can be determined in this case. Fig. 5.18 represents the graphical evaluation of the absorbance measured for the trans'cis isomerisation of azobenzene in methanol according to eq. (5.97) [127,147]. The differential equation was evaluated directly. The differential quotients have been calculated by regression parabola which were constructed for any seven points of measurement next to each other. In Figs. 5.19 and 5.20 the method of formal integration had been applied. In Fig. 5.19 the integration took place between any two sequential points of measurement. In this case the points vary greatly since the errors have not been equalised. In Fig. 5.20 the integration took place between / = 0, /' = 1,2,..., m and / = m, /' = m, m ~ 1, m - 2,..., 1,0 (this procedure is called forward- and backward-integration, respectively).

382

Applications of Kinetic Analysis to Photoreactions

Ch. 5

Y

1.0

0.5

Fig. 5.18. Evaluation according to the differential equation, eq. (5.97) [127,147]. The axis is given by dE' 1 K= 7-[10~^5"M. dt 1-10"^'

The numerical evaluation according to the method of formal integration at eight wavelengths of observation and variance of evaluation prove that the measured values fulfil eq. (5.97) with good results. The constants determined according to the two integration principles mentioned above agree with each other to a large extent. Zx2 is independent of the wavelength of observation within the error of measurement. A further good argument for the simple differential equation, eq. (5.97), is that the final absorbance of the total concentration and thus the absorbance at the isosbestic are proportional (see Fig. 5.21). Furthermore the relationship, eq. (5.98), is fulfilled. Using the exactly determined absorbance coefficients of the cis and the trans form, the ratio of the quantum yields has been calculated according to eq. (5.98) taking the final absorbances at various wavelengths.

Evaluation of Absorbance-Time Measurements

Ch. 5

383

A£'

1(1-10"') J/ (io''r'i i.o

^***^^
1

.^N

** •••^ /•

1 '

, • ••

c^ ^ ^ ^ ^ . • • » •

0.5 1

x„=v

X :

•

E'

J(i-io-

>j

0.5 1.0 1.5 X Fig. 5.19. Evaluation according to eq. (5.97) by formal integration between two consecutive points of measurement.

5-20

Fig. 5.20. Evaluation according to eq. (5.97) by formal integration between different times (see text). Axes are the same as in Fig. 5.19.

384

Applications of Kinetic Analysis to Photoreactions

0

0.5

Ch.5

1.0 £266

Fig. 5.21. Final absorbances as a function of the absorbance in the isosbestic point [127,147].

Table 5.2 demonstrates that the ratio of the quantum yields (px^/(p\ is independent of the wavelength of observation. For further proof of the significance of the method, other differential equations for alternative mechanisms have been used. Taking into account bimolecular quenching, the differential equation, eq. (5.97), has to be enlarged to the form of eq. (5.96). The six differential equations, eq. (5.96), were evaluated according to different methods with the parameters p = 2, a = 0;

/? = 2, or = l;

/7 = 3, a = l;

p = 3, a = 2;

/7 = 4, a = 2;

p = 4, a = 3.

The result was surprising. The differential equation with an odd number of constants was not fulfilled by the measured values. The "mean error" of the constants amounted to more than 100%. The values of the constants determined according to different methods fluctuated drastically. In some cases even the sign was changed. The result is that the mechanisms used to propose the differential equations have to be discarded. In contrary all the dif-

Ch. 5

Evaluation of Absorbance-Time Measurements

385

TABLE 5.2 Ratio of quantum yields of photoisomerisation [127,147].

A[nm]

^iV^?

A[nm]


250 260 270 280 290 300

0.346 0.346 0.346 0.346 0.346 0.351

310 320 330 340 350

0.348 0.348 0.344 0.349 0.343

ferential equations with an even number of constants fitted very well to the measured values. The same experience was obtained for the evaluation of the photoisomerisation of oxindigo in benzene [133]. However, typical criteria have been found so that the measured values only fit to differential equations with two constants [133]. This means that bimolecular processes cannot be proven at the given conditions of the reaction. If one splits the polynomial of the quotients of eq. (5.96) into factors, one can rewrite eq. (5.97) in the following form:

B.-'X-^

m-

(5.99)

;f is a constant, the Ej^^^ are the zero points of the denominator, and the Ej^^y the zero points of the nominator. At common zero points of the nominator and denominator for the example means Ej^^^ = Ej^^y the differential equations, eqs. (5.96) and (5.99), respectively can be shortened by

In this case the equation with p + 2 constants is just a trivial expansion of the differential equation with p constants. This case eventually happens for the evaluation of the measured values. One can determine whether zero points in the nominator and denominator fall together without solving the polynomial of the nominator and denomi-

386

Applications of Kinetic Analysis to Photoreactions

Ch. 5

nator, if it happens that correlations exist between the constants of the differential equations. For the equation with four constants one finds

^A4

and for the one with six constants I ^A3 "" ^A4^A5 ^Al ~"

^ ^A6

'

^ _ ^A1^A5 "^^A4 ^A2 • 2A6

Furthermore the following relationships between the constants z\x and z^j exist for the correct equation with two constants and that with four constants z\^ ~Zxv

ZX2 ' ZA4

and that with six constants respectively: z'n ~Zx\y

ZX2 '

With these relationships it can easily be determined whether the equations with higher number of constants are just trivial solutions. For further literature, see [144,148]. Complex non-uniform photoreactions For the evaluation of quasi-linear photoreactions the method of formal integration is especially well suitable. If eq. (2.39) is rearranged to $=:K.F(E;)~^(X-X^),

(5.100)

whereby F(E^) is a diagonal matrix with the elements F[E'^), one finds according to eq. (5.58) after multiplication by Q

Ch. 5

Evaluation of Absorbance-Time Measurements

387

i = Q.KF(E;)'^Q-^(E-E^)sZF(E;)"^(E-E^)

(5.101)

EF(E;) = Z E - Z E ^ .

(5.102)

or

Formal integration for s chosen wavelengths (written without the factor 1000/Q, which is included in the general equations in the factor F(t) for decadic units) results in jdE,=f^Z;ucj{E,-E^)Fit)dt

(A = l,2....,^)

(5.103)

k=\

or

(5.104) s systems of overdetermined equations are obtained for suitable limits of integration, which allow the determination of the sxs constants Zx^ ^ind 5x(5 + l) constants z^k^^d Zxo respectively. These equations can be evaluated applying methods of regression. Taking eq. (5.104), final concentrations are not necessary for evaluation. They can be calculated for control using Zxo' If the s variables are combined to the vector Z© one finds ZQ = Z E ^ ,

that means by rearrangement

which can be used for checking the evaluation. The different elements of the matrix K do not resemble those in Z, since this matrix is transformed with the matrix Q. Since the transformation is affine, the matrix Z which can be determined experimentally, contains the

388

Applications of Kinetic Analysis to Photoreactions

Ch. 5

same main minors as the matrix K. For this reason spectroscopic measurements can be used to determine the eigenvalues of the reaction without knowledge of the absorption coefficients. Example 5.22: Application of formal photoreaction

integration to a

consecutive

If the consecutive photoreaction A±=z;B-^i^C proceeds with a constant quantum yield the use of eq. (3.39) results in ^

A

ir2 B /

V /

(5.105)

The concentrations are substituted by the degrees of advancement. One obtains the matrix eq. (5.100) for these quasi-linear consecutive photoreactions. The dependence on time of the two degrees of advancement is given by

Jt2 =1000/O^?£'B(;:, -x,^)F(r)~1000/o^?e^(ac2 -^2-.)^(0. The elements of Z and Zo are represented by

^12 =[(G,2^3 -Gii^2)(^i2 "-'22KB -Gii^r^A'ia} ^21 =[(222^3 -221^2X'il - ' i l K -G2iy>NA'il} ^22 =[(222^3 - 621^2 X''^ - '22>B " Qll^t^A^n]

^20 ~ (""^21^1 CO •" ^22^200 )•

Ch. 5

Evaluation of Absorbance-Time Measurements

389

For abbreviation the elements of the reciprocal matrix of Q are symbolised by A;^ . These constants fulfil the relationship (without the factor lOOO/o)

The irradiation of azobenzene in 66% sulfuric acid follows this reaction mechanism. The reaction scheme of the partial reactions is written according to Section 3.1.4 A-

hv

B-

hv

B-

hv

^B

*

->A

* 2

*

-»c

B+C

^3

*

JC4

>D + E

•

^5

A -1 1 0 0 0

B 1 -1 -1 -1 0

C D E 0 0 0 0 0 0 0 0 0 1 1 1 0 0 -1

4>

F 0 ^f/A 0 ^2^8 0 ^3'B 0 k^bc 1 k^e

E >F (A, rran^-azobenzene; B, cw-azobenzene; C, dihydrobenzo[c]cinnoline; D, benzo[c]cinnoline; E, hydrazobenzene; F, benzidine; all the compounds are protonised). The Bodenstein hypothesis can be applied to the intermediates C and E, since the thermal reactions 4 and 5 are very fast. Accordingly :X3 -;C4 = 0,

e—

x^-x^—0

and c = i j - JC4 = 0,

^ = i:4 - ^5 = 0 .

Thus the scheme can be reduced to

X,

A -1

X2

0

B 1 -2

C D Xk 0 0
390

Applications of Kinetic Analysis to Photoreactions

Ch.5

For the photochemical examination of te reaction in a first series of measurements rmnj-azobenzene was irradiated at 405 nm. In a second series cw-azobenzene was irradiated under the same conditions. Taking the same starting concentrations, both experiments result in the same final spectrum after sufficiently long irradiation time. The absorbancetime diagrams of both the reactions are given in Figs. 5.22 and 5.23. The assumption of two linear independent partial reactions was controlled by an EDQ-diagram for both reactions. These are plotted in combination in Fig. 5.24. The photoreaction of the rran^-azobenzene was evaluated according to the method of formal integration using the following equations: Ex = U o +^11^1 4-z,2^2H0lOOO/o, K = (^20 + ^21^1 + Z22E:,)F{t)\QOQI^.

Standard deviations are small. The constants obtained are used to recalculate absorbance-time curves for both series of measurement by numerical integration. The curves obtained are plotted in Figs. 5.22 and

150 Mrnin] Fig. 5.22. Absorbances plotted versus time of irradiation for the reaction of fran^-azobenzene in 66% sulfuric acid, GQ = 5.51 X 10"^ mol l"^ /Q = 10.36 x 10~^° Einstein cm"^ s~^ The experimental absorbances are shown as dots. The line gives the calculated absorbances using the constants zxi-

Ch. 5

Evaluation of Absorbance-Time Measurements

391

Fig. 5.23. Absorbances versus irradiation time for cw-azobenzene in 66% sulfuric acid. ^0 = 5.51 X 10"^ mol r ' , /Q = 10.36 x 10"'^ Einstein cm"^ s~^ The experimental absorbances are shown as dots. The line represents absorbances calculated using constants zxj determined in the rra/i5-azobenzene irradiation.

^250

405

Fig. 5.24, EDQ-diagram of the photoreaction of azobenzene in 66% sulfuric acid. #, cisazobenzene; O, irradiation of cw-azobenzene.

392

Applications of Kinetic Analysis to Photoreactions

Ch. 5

5.23. They are in good agreement with the measured absorption independence on time. The constants are used to calculate the partial quantum yields which amount to (p^ = 0.27,


v>? = 0.02.

Further details is given in Ref. [141]. 5.4.5 Discrimination of two mechanisms If the absorption coefficients in a quasi-linear photoreaction are not known, the rank of the Jacobi matrix, the eigenvalues of the reactions and the final absorbances can be calculated. Furthermore the initial values of the absorbances and their derivatives with respect to time can be determined. Any additional statements require additional information since for linear systems the following theorem is valid [15]: Two reactions, for which Jacobi matrices result in the same rank, cannot be discriminated by spectroscopic means, if the absorption coefficients of the reaction partners are unlcnown This can be easily proved, since any system can be rearranged into another system of the same rank by linear transformation as has been demonstrated in Section 2.3.4. Since the transformation of the system of degrees of advancement to a system of absorption coefficients is done by affine representation, there is no chance that one can realise which system originally exists without additional information. Various additional information can be used. An optimal fact is to know all the absorption coefficients of the components participating in the reaction. This can be achieved if the reaction is interrupted at least n times and the solution analysed to determine quantitatively the concentrations of all the components. In this case concentrations a and the absorbances E are known at these times. For this reason e can be determined according to eq. (5.44). Qualitative analysis has to prove that only n components take part in the reaction. Using the absorbance coefficients for each reaction time the concentrations can be calculated according to eq. (4.9). One can proceed according to Section 5.3.1 in the following way. Because of the errors it is recommended first to use absorbances according to Section 5.3 for evaluation. Since interruption of the reaction and the determination of the concentrations

Ch. 5

Evaluation of Absorbance-Time Measurements

393

is not trivial, unknown absorption coefficients cause a lot of expenditure in evaluation. A possibility is the combination of spectroscopic and chromatographic measurements, examples are given later in Section 5.6. Further helpful information can be obtained if just a single compound absorbs in a certain spectral range or if the eigenvalues can be varied. Furthermore a certain mechanism can be assumed or discarded by pure chemical knowledge. Example 5.23: Discrimination of different mechanisms for two independent steps Between the three compounds A, B, C two linear independent quasilinear photoreactions are assumed to take place. A list of such reaction is given in Table 5.3. The points of concentration in the absorbance diagram are given in a distorted Gibbs triangle. This can be used to determine the concentrations according to eq. (5.85), if the corners of this diagram are known. Fig. 5.25 represents the situation of the kinetic analysis of such a system. The absorbances are measured from the beginning to the end of the reaction. At first one just knows the curve in the absorbance diagram which starts at point A and ends in point T. The point C can either be represented by T, 2\ 3' or 4\ the point B might be represented as one of the points 1-5. The situation depends on the mechanism as well as on the ratio of the rates of the reaction. If one succeeds in fixing the points B and C in the diagram, the reaction can be evaluated. By chance one can extract from the spectra some of the necessary information. 5.4.5.1 All the concentrations at one reaction time are known If the composition of the solution is known at any time, t as well as the concentration of the starting material and that of the final products, one can TABLE 5.3 Survey on possible reaction with two linear independent steps 1. 2. 3. 4. 5.

AA^ AA< A<-

->B-»C ^B->B< >C >Bf >B, A >C

6. 7. 8. 9. 10

A<- ^ B A - ->B< A< >B<

-»C, ->C, >C, ->C,

AAAA-

Applications of Kinetic Analysis to Photoreactions

394

Ch.5 6JL

A

5" r ^

\

/

^^T*

/

A/y / ~*i>^^<5v-7 tangents.

V-^"*/^ ^^^'^r

^^^^SaBS^

/ /

7

\A

^ \ s / '

/ /

/

^

•

Fig. 5.25. Schematic representation of an evaluation of the reaction shown in Table 5.3. A is the initial point of the absorbance curve. The points B and C are situated within the given ranges [15].

determine the comers in the diagram Fig. 5.25 by graphical means. One connects point A with the final point T. In the case where a^ = 0, this point r is placed on the straight line AB, One looks for the point which intersects the line VA with the ratio a = aifyja^ . Next this point is connected with the equivalent point on the curve of measurement. The side BC of the triangle is found on a parallel to the line mentioned above passing point T, then one of the reactions 1, 3, 6 or 8 of Table 5.3 takes place. If the parallel is not a tangent, the reactions 2, 5 or 7 can also take place. In the case where a^ ^ 0, one lengthens AV up to point V in such a way that the sections T'T to VA have the same ratio as a^ to a^. In this case point V is situated on BC, Next the point V'A is looked for, which intersects this section with the ratio a^a{t)l a^. Once more the parallel to the interconnection line passing point V drawn. The side BC is found on this straight line. Under these conditions one of the reactions 4, 9 or 10 takes place.

Ch. 5

Evaluation of Absorbance-Time Measurements

395

The direction AB is determined in the same way by intersecting AV with the ratio y = c{t) I c^. This point is connected with P. The comer P is situated on the parallel passing A. It forms the intersection point with the parallels passing the points T or 1". By these means the length of the section AB becomes known. If it is intersected with the ratio p = b(t) I a^, one finds the connection line to point B in the direction of AC. The parallel passing A intersects the straight line passing B and V or 1" in point C. By these methods all concentrations are found as functions of time. 5,4.5.2 Only one component absorbs within the wavelength range A wavelength range in which just one compound A absorbs can be realised looking at the absorbance diagrams: all points of measurement are found on the concentration line starting at A. If A completely reacts, the final point of the reaction is situated in the zero point of the diagram. In consequence, for each reaction time, one can read from the diagram the values of a. Knowing this value one can determine the straight line passing points B and C as described in the previous section. If only B absorbs in the wavelength range the absorbance diagram will result in a zero point line. The measured absorbances start in the zero point, will reach a maximum, and can return to the zero point in some cases. Since the point of concentration of B is not reached in this diagram, in general just the relative concentration P^{t)^b{t)l b^ can be taken out of the diagram. b^ is the maximum concentration of i?. If fc^ = 0 , point C coincides with point r , only the reactions 1, 2, 5, 6 and 7 can take place. If for two times P^(t)=^P^t'), the interconnection line passing these two absorbance points gives the direction of the straight line passing A and C. In consequence for ^oo = 0 the connection must also be parallel to the section Al'. The same information is obtained if one can measure any kind of information which increases or decreases monotonously with b and vanishes forfe= 0. If only C absorbs the absorbances in the absorbance diagram are also situated on a zero point line. The measurement starts in the zero point and ends at c^. One can determine from the diagram Vr-cIc^, if the section AV is intersected with the ratio y^ to 1 . The resulting point is connected with the related point P of measurement. Then one has determined the parallel to the section AB. If this parallel is a tangent to the curve in A one of the reactions 1-4 will take place. If the parallel is not a tangent, one of the reactions 5-10 is going to happen.

396

Applications of Kinetic Analysis to Photoreactions

Ch. 5

5.4.5.3 Further additional information Point 1 is the intersection of the tangent at the points A and V of the curve of measurement. It can be determined in any case. Because of the affme transformation it is always valid that the curve of measurement intersects the triangle AVl with the ratio of the two eigenvalues. If it is known that a simple consecutive reaction 1 takes place, the points V and C coincide. In addition if /:, >^2> the point 1 relates to the comer B. However, if k^
A!"

it,

The right-hand side is equivalent to the ratio of the large area to the small one. As had been demonstrated in Section 5.2.6 one can distinguish between these two possibilities at several conditions. If one can treat reactions of higher order as a pseudo-linear dark reaction, the dependence on concentration of trace and determinant offer important information on the mechanism [137,144]. This statement cannot be transferred to photoreactions. Variation of the wavelength of irradiation can influence the "rate constants" for quasi-linear photoreactions, since they depend on the absorption coefficients. Under these conditions the following statements for the final point r of the reaction can be made: (1) If the compounds A and B react completely as in the reactions 1, 2, 5, 6 or 7, the final point stays unchanged. (2) React A but not By completely then this final point moves on the connection line BC. (3) React neither A nor B completely, then the final point is situated within the area ABC. Variation of the wavelength of irradiation changes the initial slope for the reactions 5-10, not for the reactions 1-4. In general the final slope is changed. For the reactions 1, 3, 6 and 8 however time areas exist in which the final slopes stay constant. The Jacobi matrix is a triangular matrix. As long as the second eigenvalue stays smaller than the the first, BC will be a tangent to the final point.

Ch. 5

Evaluation of Absorbance-Time Measurements

397

5.4.6 Approximations in practice 5.4.6.1 The determination of concentrations from absorbance measurements In reaction kinetics the absorbance of just one reactant in the solution to be examined is a trivial case. Then eq. (4.1) can be used directly to transform the concentration in the rate equation to a relationship containing the absorbances. Taking a single step photoreaction according to eq. (3.35) instead of the concentration rate law (using decadic units) ait) = -1000 /o ipt £'A «(O

:!;,,

(5.106)

the following relationship E^{t) = -1000 /o ^ ^ A ^ A ( 0 — — — -

(5.107)

b (/)

can be used. Besides the symbols defined in Sections 1.3 and 1.4.3 the primes at the decadic molar absorption coefficient e\ and the overall absorbance in the solution E* indicate that the absorption units are to be taken at the wavelength of irradiation. As mentioned before, in this chapter all the absorbance units are decadic. For this reason the decadic exponent is used in the photokinetic factor as discussed in eq. (1.42) F(t)= L^ ^

(5.108) E\t)

^

^

in contrast to eq. (3.35). The index A at the absorbance symbol E indicates in eq. (5.107) that absorbance can be measured at a wavelength different to the wavelength of irradiation. In contrast to thermal kinetics the photokinetic equations given above cannot be solved in a closed form, since the photokinetic factor depends on the absorbance of the solution itself and is consequently dependent on time. In the literature approximations either with respect to total absorbance or to very dilute solutions are used. The consequences are discussed in detail in Section 3.3.3. Using the conditions

398

Applications of Kinetic Analysis to Photoreactions

Ch. 5

A—^iL-^B,

where A is the only absorbing component, two approximations are possible neglecting the time dependence of the photokinetic factor: (a) The solution is taken at very high concentrations. Therefore at the wavelength of irradiation one finds total absorption. Under these conditions the photokinetic factor transforms to F(ty

= [E(t)]-\

(5.109)

Measuring at a wavelength at which the absorbance is much lower than that at the wavelength of irradiation, eq. (5.106) can be rearranged, integrated, and reduced to (£* (t) = de' a{t)) a(^t) = J ^ = - 1 0 0 0 / o ^ ^ ; a ( 0 - ^ . E^{t)-E^{0) = -'\OOQlQip^ej^fj .

(5.110)

(b) The absorbance E at the wavelength of the irradiation is very small. According to eq. (3.65) expansion of the photokinetic factor into series using decadic units gives F ( 0 " =2.303.

(5.111)

Using the considerations in Section 3.3.2 and insertion into eqs. (5.106) and (5.107) one obtains: l n A i f ^ = -1000x2.303/o^iV.r.

(5.112)

If B absorbs, similar assumptions can be found. However, if the mechanism is given by A^B,

Ch. 5

Evaluation of Absorbance-Time Measurements

399

these approximations are no longer valid. However, frequently in the literature the assumptions above are taken for the evaluation of the photoreaction just at their beginning. Thereby an error in the determined initial slope is taken into account. It increases with an increasing E;^^ or E^ is less negligible. In general, under practical conditions, it is very exceptional to find wavelengths at which just one reactant absorbs. For this reason methods have to be developed to extract the different concentrations out of the overall absorption spectrum. This is very difficult since absorption spectra in the UVA^is are very broad band and most of the reactants' spectra overlap. If the reactants and their absorptions are known, multicomponent analysis can be used (see Section 4.2.6.3). 5.4.6.2 Application to actinometry Azobenzene The photoisomerisation of azobenzene in methanolic solution is a preferable photoreaction to be used in actinometry, since the mechanism has been well examined. The trans-cis photoreaction is photoreversible. The thermal reaction cis —> trans will not disturb photokinetics at normal conditions because of a half-life of approximately 1 week. Therefore a concentrated solution of fran^-azobenzene in methanol at 6.4 x 10-^ mol l-^ which totally absorbs radiation between 345 and 240 nm, can be used taking eq. (5.107) and the approximation given by eq. (5.109). In a first approximation the change in absorbance with time at a wavelength of observation is proportional to the intensity of radiation. This proportionality includes the photochemical quantum yields of the trans —> cis isomerisation step, the factor 1000 and the absorption coefficients at the observation wavelengths of the trans and cis isomers. This is a relatively crude approximation. However, a graph of the change in absorbance versus time gives a linear slope for the beginning of the reaction under these conditions with a deviation less than 1 % in comparison to the correct evaluation. These factors can be calibrated by other actinometers as for example the Parker solution [121]. The values given in Table 5.4 can be used to determine some of the lines of a mercury arc. The dashed line in the table indicates that for the measurement of the 254 nm line the other band of the azobenzene solution has to be taken. A trans solution must be preirradiated at 313 nm to bring the solution into the photostationary state.

400

Applications of Kinetic Analysis to Photoreactions

Ch. 5

TABLE 5.4 Calibration factors for some mercury lines X [nm]

Factor x 10*^ [Einstein cm"^]

280 313 334 254

4.6 5.2 3.6 2.3

Afterwards irradiation at 254 nm will cause cis —> trans photoreaction. The necessary solutions are commercially available. An approximation using the photokinetic factor given by eq. (3.69) supplies an even better evaluation using azobenzene as an actinometric system. Using a linear interpolation between two points of measurement at time t\ and ^2^ the photokinetic factor according to eq. (5.108) can be approximated within this time domain as F=F(l)-h ^ ^ ^ ^ " " ^ i ^ \ r - r ( l ) ) ,

(5.113)

E\2)-E\\y where the indices 1 and 2 correlate to the two times at which the photokinetic factor F is calculated using the absorbances at the wavelength of irradiation E\ Using the abbreviation /? = 1000(/?,-f/?2) = 1 0 0 0 ( c > f + £ ' B ^ ^ )

(5.114)

and insertion into equations like eq. (5.20), (5.93), or (5.97) results in the intensity measured between times r, and t^

R\t^ -t2) whereby the factor P is given by ^ _ F(l) [£(2) - Ejs)] - F(2) [£(1) - Ejs)] ^^ £(1) - Ejs) £(2)-£(l) E{2)-E(sy

Ch. 5

Evaluation of Absorbance-Time Measurements

401

In the case where the change in absorbance at the wavelength of irradiation is relatively small, this wavelength and that for the observation have to be chosen differently. The method is successfully used in practice using factors P given in literature [116], [124]. Photoreaction of isopropyl-azobenzene If the photoproduct does not absorb at the wavelength of irradiation and that of observation (Cg = 0, £" = Ej), eq. (5.107) can be simplified to EXt) = ~ 1000/o^f £^(1 -10"^ ^'^)

(5.117)

Using a cell path length of 1 cm and integration between the time limits t and r = 0 gives in decadic units ln(10^'^'^ - 1 ) = ln(10^'^^^ - 1 ) - 2.303X 1000/o^f e^r

(5.118)

These assumptions are valid for 2,2',4,4'-tetra-isopropylazobenzene which shows a uniform photoreaction. The thermal backward reaction is negligible at room temperature. Thus a graph of ln(10^('> - 1) versus the reaction time initially exhibits a straight line. This derivative can be evaluated as a simple actinometric system for radiation in the range between 350 and 390 nm [149]. In Fig. 5.26 the reaction spectrum demonstrates the validity of the assumptions above. At high absorbance (3.5 x 10-^ molar solutions; total absorption at 365 nm) evaluation becomes even easier. Taking the approximation of eq. (5.109), for e;i3 = 0 at the beginning of the photoreaction, eq. (5.110) gives a straight line in a graph of absorbance at an observation wavelength nonequal to the irradiation wavelength in the range 350-390 nm versus time. This fact is demonstrated in Fig. 5.27 for 450 nm. Heterocoerdianthrone The photochromic system of heterocoerdianthrone (HCD; dibenzo[q/]perylen-l,16-dione) in toluene exhibits a photooxidation to an endo-peroxide during irradiation at long wavelengths. This reaction has been extensively investigated in the literature [150]. Even though this photoreaction is complex, under certain conditions (aerated solution, total absorption at wavelength of irradiation, photoproducts do not absorb at this wavelength) it can

Applications of Kinetic Analysis to Photoreactions

402

450

Ch.5

500

Fig. 5.26. Reaction spectrum of the photoreaction of 2,2^4,4'-tetra-isopropylazobenzene in n-heptane (5 x 10"^ mol"* 1"^).

' 450

Fig. 5.27. Plot of absorbance at 450 nm versus irradiation time (A' = 365 nm) for the photoreaction of 2,2',4,4'-tetra-isopropylazobenzene in n-heptane (3.5 x 10"^ mol"* l"^).

Evaluation of Absorbance-Time Measurements

Ch. 5

403

be handled by eq. (5.117). Since under these conditions the peroxide does not absorb at any irradiation wavelength between 400 and 580 nm, eq. (5.118) can be used for the evaluation. The error in calculation of the irradiation /Q stays less than 1% if just 20% of the overall reaction time is evaluated. The reaction spectrum of this photoreaction is given in Fig. 5.28. Mesodiphenylhelianthrene Mesodiphenylhelianthrene (MDH) exhibits upon irradiation with wavelengths in the range between 475 and 610 nm a sensitised photooxidation to an endo-peroxide (MDHPO) comparable to the heterocoerdianthrone mentioned above. However, according to a detailed kinetic examination the photochemical quantum yield at high concentration of mesodiphenylhelianthrene depends on the concentration of oxygen alone which can be taken as constant under normal conditions because of its large excess in toluene [128]. Mesodiphenylhelianthrene is soluble to a large enough extent, thus eq. (5.110) can be written as

600 X [nm] Fig. 5.28. Reaction spectnim of the photoreaction heterocoerdianthrone to its endo-peroxide [151,152]. 400

Applications of Kinetic Analysis to Photoreactions

404

Ch.5

1.0

0.8

0.6

0.2

0.0 240

320

400

480

560

640 X[nm]

Fig. 5.29. Reaction spectrum of the photooxidation of mesodiphenylhelianthrene of its endoperoxidc [128,152].

The wavelength 429 nm is chosen for the observation. The reaction spectrum of the well examined sensitised photooxidation is given in Fig. 5.29. The absorbance-time diagram is given in Fig. 5.30. It proves linearity of a graph absorbance versus time for a concentrated solution. Since the photochemical quantum yield is constant within the wavelength range mentioned above the reaction can be used as a quantum counter in the visible range [128]. 5,4 J Evaluation of photoreactions superimposed by thermal reactions The evaluation of kinetic data is more complicated if an additional thermal reaction is superimposed on the photoreaction. In addition to the EDdiagrams given in Section 5.3.3.1 some further examples of kinetic analysis can be given. Typical examples are the reactions of photochromic substances either belonging to the classes of dihydroindolizine derivatives, fulgides, or even azobenzene-derivatives. This thermal step either can be a backward

Ch. 5

Evaluation of Absorbance-Time Measurements

0

30

60

90

120

405

150

180

i/s

Fig. 5.30. Absorbance-time diagram of the photooxidation of MDH in toluene solution.

reaction in competition to the photoisomerisation or a degradation step. Then mechanisms comparable to those given in eq. (3.95) have to be applied. In the first case the photokinetic equation similar to eq. (5.106) has to be corrected by a thermal term: •E'(t)

a(0 = -1000/o^fe'Aa(0^

^^

E\t)

•f k^b

(5.119)

or taking a second photoreaction into account.

a(o = -iooo/o[^N'A«(0+^2«BMo]- E\t)

•¥k2b.

(5.120)

Conservation of mass a(0) = a{t) + b{t)

(5.121)

and application of the Bodenstein hypothesis (see Section 2.1.3.2) leads to an equation for the overall mechanism: a{t) = -RI^FiOait) + R2loF(t)a{0) + k^a^O) - k^ait).

(5.122)

406

Applications of Kinetic Analysis to Photoreactions

Ch. 5

where R is given by eq. (5.114), I^ represents the incident radiation in Einstein cm"2 s"^ and ki is the thermal reaction rate constant. According to eq. (5.25) a linear relationship between concentration and absorbance signal exists assuming the Lambert-Beer law is valid. Similar to eqs. (5.93) and (5.107) the relationship E,(t) = --IoRE,{t)F(t) + loPxE^m + k^E.iO) - k^^E.it)

(5.123)

is found, whereby Px is defined by Px = ^

/?,4-/?2.

(5.124)

K^XAJ

This equation [153] includes the thermal step compared to the pure photochemical approach given by E,{t)^-'I^RE,{t)F{t)

+ I^P,E,(0)F{t).

(5.125)

This is equivalent to eq. (5.97) using the photostationary state {s) and the abbreviations z,. (see eq. (5.131)), if Pj,E^{0) = REj,{s)

(5.126)

is considered to be valid. Since all these equations cannot be solved in a closed form, formal integration is applied. It turns out that dependent on the relative values of the thermal rate constant, the photochemical quantum yields, and the absorption coefficients, the quality of the result depends on the evaluation chosen. Rearrangement and formal integration supplies the following equations [153,154]: \dEx^-z,\Ex{t)F{t)dt'^Z,E^{s)\F{t)dt,

(5.127)

\dE, = - z,JE,it)F{t)dt + Z2JF{t)dt - z,JE,{t)dt + z,E,iO)jdt, (5.128)

Ch. 5

Evaluation of Absorbance-Time Measurements

407

J j £ , = ~ z J £ , ( 0 F ( 0 t / r + zJF(0rfr4-^3J[£,(0)-£,(0]t/^

(5.129)

jdE;^^-x^JE;^{t)dt-\-X2E;,(0)jdt.

(5.130)

In these equations the following coefficients are defined: ^

/? = 1000(/?,+/?2) = 1 0 0 0 ( £ > f 4-6^y?f),

P;i= p€^ / ? , + / ? , V^AA

Z, =(/?,+/?2)/0,

Z2=PxIoE;,(OX

Z,=k2, (5.131)

F is the averaged photokinetic factor within the time interval of measurement. In eq. (5.130) the thermal rate constant is determined independently. 5.4.7.1 Photochromic systems of dihydroindolizines Various derivatives of dihydroindolizines with its reversible photoreaction to betaine and the superimposed thermal reaction have been used to test the algorithms. Program system SIDYS A program system called SIDYS using simulation techniques and parameter identification was applied to the absorption data. It includes random search techniques, Rosenbrock strategy, quasi-Newton methods, and lattice search. It runs on a mainframe computer and provides good agreement between simulated and experimental data. On the other hand, it proves that the data of this type of mechanism are not well conditioned and the quality of the evaluation drastically depends on the parameter set chosen [155]. Evaluation by personal computer Because of small calculation capacity, any evaluation will need more CPU time. Therefore parameter estimation with highly sophisticated algorithms will consume too much time on small computers. The different equations, eqs. (5.127)-(5.130) applying formal integration provide a good tool

Applications of Kinetic Analysis to Photoreactions

408

Ch. 5

TABLE 5.5 Quality of the various differential equations given by eqs. (5.127)-(5.130) ki

Pure photochemical eq. (5.127)

A£:'>0 0 <% >5%

++ + --

A£:' = 0 0 <5% >5%

++ + —

Incl. thermal 3 constant eq. (5.129)

Incl. thermal 4 constant eq. (5.128)

^2 independent determination, eq. (5.130)

to obtain results even for this complex mechanism. In Table 5.5 the benefits of the different equations are summarised for different ratios of z-values. Since the differential equation, eq. (5.127), is more stable, but does not fit a mechanism which includes thermal reactions, it supplies only sufficient

error z (%)

• pure photochem

3-const

Z3/Z1

(%)

Fig. 5.31. Deviation between the calculated parameter Z| and true one for different ratios of thermal to photochemical reaction given by Z3 and z\ using eqs. (5.127) and (5.129).

Evaluation of Absorbance-Time Measurements

Ch. 5

40H

409

A

1

errors [% ]

30H

20 '"•A.

10^3

o OfV

Zl

fid-O-Dti

0.0

0.2

0.4

0.6

z-ifzx

0.8

Fig. 5.32. Errors in all the parameters for both eqs. (5.127) and (5.129) for different ratios of thermal to photochemical reaction given by Z3 and z\.

quality when thermal rate constants are negligible. However, Fig. 5.31 proves that up to a ratio of zjz^ < 0.05 even in the case of a superimposed slow thermal reaction, the "wrong" evaluation by the pure photokinetic equation, eq. (5.127), gives better results for the photochemical parameters. The determination of z^ which refers to the thermal rate constant exhibits large errors (see Fig. 5.32). Therefore in principle an independent determination of this thermal rate constant is advisable [153,155]. Combined determination of reaction constants and absorption coefficients In many cases the absorption coefficients of the photo-product cannot be determined, since B cannot be isolated in a pure form. For these conditions quite a few methods have been proposed in the literature [156-160]. If the conditions required for these approximations do not fit, the thermal reverse reaction is used to shift the photostationary state by variation of temperature. The possibilities of such an approach are demonstrated by the photoreaction of dihydroindolizines [49,161]. Using eq. (5.119) and the approximation at the photostationary state

410

Applications of Kinetic Analysis to Photoreactions

Ch. S

a(s) ko b(s) (plOOOIoC'^Fis) where the Arrhenius equation is used for the dependence on temperature, a linear relationship is found ^-

= i / o - ^ e x p | - ^ l + nP,.

E(s)-EiO) \°Fis)

^^IrJ

(5.132)

P

The parameters are defined as follows: Po=

—r.

^1=-—.

^2=

r^

r - (^ = lcm)

with E^ the activation energy andfc,the frequency factor of the Arrhenius equation. On the other hand, the photochemical quantum yield can also be determined by formal integration of the equation jdE,-k,j[E,(0)-E,(t)]dt

=

ip 1000 /oe;{6,B a(0)JFit)dt -'JE,(t)F(t)dt] where the thermal rate constant is determined independently [49]. As well as the thermodynamic constants for the activation, the photochemical quantum yields, the absorption coefficients, and the thermal rate constants can be obtained. In Fig. 5.33 experimental and simulated data are compared for a reciprocal plot according to eq. (5.132). 5.4,7.2 Photoisomerisation of (Z,E,E)'4,4''distyrylazobenzene 4,4'-Distyrylazobenzenes exhibit a photokinetic equilibrium and a superimposed thermal backward reaction. The absorption coefficient of one of the partners in the equilibrium is principally unknown. Nevertheless, using the dependence of the photostationary state on the intensity of the irradiation, the photochemical quantum yields (p^ and (P2 have been determined, as well as the thermal rate constant k, depending on the wavelength of irradiation and wavelength of observation [50]. Fischer's method [156] to determine absorption coefficients requires in-

Evaluation of Absorbance-Time Measurements

411

1/T[K]

^•^

1/TlKI

Fig. 5.33. Graphical representation of an evaluation according to eq. (5.132) for the superimposed photochemical and thermal reaction of rH-2',3'-dimethoxycabonyl-spiro-[fluorene9,1 '-pyrrolo[ 1,2-6]pyridazin] [161].

dependence of the photochemical quantum yield on the wavelength and can even be used if a thermal backward reaction influences the photostationary state. In this case, the thermal reaction is frozen out by working at low temperatures. Then, it has to be assumed that e^ is independent of temperature. This thermochromism is valid for the Z,E,E-, but not for the Z,Z,£'-isomer of distyrylazobenzene. Even though e^ and e^ could be determined by temperature dependent measurements, Fischer's method can not be used, since first experiments proved a dependence of the photochemical quantum yield
Applications of Kinetic Analysis to Photoreactions

412

Ch.5

all the other values are known (especially s^;€'^), photochemical quantum yields cp^ and <jO^ are independent of the intensity of irradiation in the used intensity range, and besides this isomerisation no other intra- or intermolecular reactions take place: 1

E,(s)-E^(0)

kEy)

gAK+gB
(5.133) k is the rate constant of the thermal backward reaction. The linear dependence is demonstrated in Fig. 5.34, if the reciprocal difference of the absorbances is plottedd versus the intensity of irradiation.

/ E(s)-E(0)

1-10'E '(s)

I

10

[Einstein cm" s' ]

Fig. 5.34. Graphical representation of an evaluation according to eq. (5.133) for the superimposed photochemical and thermal reaction of (Z,£,£)-4,4'-distyrylazobenzene for two temperatures (15°C and 50°C).

Ch. 5

Evaluation of Absorbance-Time Measurements

413

5.4,8 Further approaches Many photochromic systems can be handled by the equations given in the previous section. They represent essential examples of photochemical reactions. Their principles and their applications are discussed in detail in the literature. A good review is given in Ref. [162]. Typical examinations deal with the photokinetics of fulgides [163-165] and spiropyrans [166-168]. A different approach to overcome the change in the photokinetic factor during the progress in the photoreaction is reported elsewhere [169]. By variation of the intensity of irradiation in such a way that the change in the photokinetic factor is compensated by this change, in eq. (5.107) the factor ^

l,10-^^(^)

'

EXt)

is kept constant. Recently the kinetic treatment of a photoreversible reaction with additional thermal reaction has been applied to thermo-reversible photochromism with degradation in the case of triphenylimidazolyl dimer reactions [170]. Fit techniques allow the simultaneous determination of quantum yields and absorption coefficients. In the case of a-3,5-dimethoxyphenyl5,8-dimethylfuIgide even photo-equilibria with two superimposed thermal reaction steps have been evaluated by formal integration [171]. Different cyclophanes exhibit special behaviour in Z,E thermal as well as photochemical isomerisations, because of steric or torsion effects in comparison to azobenzene. Thus the S-C cyclophane is thought to be useful as an optical switch. For this reason its photokinetics have been examined in detail [54,172]. The structure of this compound is given by

H5C2OOC

H5C2OOC

By application of absorbance diagrams of higher order and mass spectrometry of the reactants the following mechanism is assumed

Applications of Kinetic Analysis to Photoreactions

414

92

9y A, ~

Ch.5

Qz,z

For the measurements the apparatus given in Fig. 4.12 was used which allows the examination of photochemical with superimposed thermal reactions at room temperature and as low as 240 K. The absorbance difference diagram is given in Fig. 5.35. The absorbance diagram (see Fig. 5.36) demonstrates the dependence of the progress of the reaction between the two photostationary states on the irradiation wavelength and in the case of darkness. Applying formal integration and rate equations derived according to Section 5.4.7 all the photochemical quantum yields and the thermal rate constants are obtained [172]. The diagram presents the overall procedure of the reaction demonstrating the 0.8 ED 332 0.6

0.4

0.2

0.0 0.00

0.02

0.04

*^2 70 Fig. 5.35. ED-diagram of the photoreactions of C-S cyclophane.

0.06

Ch. 5

Evaluation of Absorbance-Time Measurements

415

^330

1.0

0.8

0.6

0.4

0.2 U

0.0 0.00

0.05

0.10

0.15

0.20

0.25 ^380

Fig. 5.36. E-diagram for reactions between the two photostationary states (c,c and t,t) of S-C cyclophane depending on wavelength of irradiation and thermal conditions.

different pathways at different wavelength of irradiation (solid lines) and the thermal steps. The information obtained using this type of diagram is typical. It becomes obvious that a change in wavelength of irradiation just switches the reaction between the two photostationary states. The behaviour of the thermal reactions starting after photochemical reactions are stopped at different times is interesting. Details are given in Ref. [54]. Formal integration was also applied to the determination of the dependence of the quantum yields of the photoisomerisation of previtamin D3 on the wavelength of irradiation [173]. A more simple and approximate approach is given for photochromic glasses [174].

416

Applications of Kinetic Analysis to Photoreactions

Ch. 5

5.5 FLUORESCENCE

5.5.1 Correlation between fluorescence intensity and concentration Time-resolved spectral measurement of fluorescence during the progress of a reaction turns out to be a useful tool in photokinetics, since usually only a few of the reactants fluoresce. Fluorescence measurements have been discussed in Section 4.3. In dilute solutions a linear relationship between the fluorescence signal /^ for a wavelength of observation a and the concentration ai of a compound / can be found. Furthermore, fluorescence depends on the amount of light absorbed by the sample /^bs;, and the fluorescence quantum yield 77,, which determines the ratio between the photons absorbed and the photons emitted. This yield is a property which is determined by the photophysical behaviour of the molecules. Using the equation for the amount of light absorbed /^bs, = - 1 0 0 0 / o g ; a / " " | "

r{E)

(5.134)

one finds a relationship between the fluorescence intensity and the concentration of compound A: / ; = /abs.A^A/aAr(£') = IQmi.T, J,^e\a--^—r{E

)T{E ) . (5.135)

-

This equation takes into account, that all the compounds in the solution can absorb at the wavelength of excitation (marked by the prime) and only the ratio £;,a(l-10-^)/£' is responsible for the amount of light adsorbed by the fluorescent compound; r{E') is a correction for the dependence of the reflectivity at the cell walls on absorbance of the solution (see Fig. 4.15), which is not compensated by a reference as in absorption spectroscopy;

Ch. 5

-

Fluorescence

417

that the measured fluorescence intensity depends on the angle and radiant of observation as well as the volume element at which the entrance slit of the monochromator of observation is focused, therefore the geometric factor r(E) includes all the geometric effects in fluorescence detection; for this reason no absolute determination of the fluorescence intensity is possible by this means.

5.5.1.1 Timedependence of fluorescence signals during reaction Taking into account that the concentrations of reactants vary with time and more than one reactant can fluoresce, eq. (5,107) has to be written in a more generalised form / ; ( 0 = 1000 X 2 3 0 3 / o G - — — — ^ ; ; , / , , e ; a , ( 0 , ^

(^)

(5.136)

1=1

where G = r{E )T{E) corrects for geometry and reflectivity. By the variation of fluorescence intensity over the spectral range of the fluorescence spectra, the so-called spectral fluorescence distribution f^ resembles the variation of fluorescence intensity with observation wavelength. This/ai is similar to £xi used in absorption spectroscopy. The above equations give a correlation between the measured fluorescence intensity and the concentration of one or more reactants. As discussed in Section 1.3.2 this fluorescence intensity depends on the photophysical properties of the reactants and on the measurement set-up. Since in fluorescence spectroscopy no reference measurement is possible, the measured intensity will give a relative value dependent on the geometry, reflectivity, and absorption in the solution. To reduce the dependence on the geometric properties in fluorescence spectroscopy usually very dilute solutions are used. The photokinetic factor F(0 is approximated according to eq. (5.111). Under these conditions neither the change in absorbance at the wavelengths of irradiation E nor changes in the factors riE") and Tip") have to be taken into account. Thus eq. (5.136) can be rewritten as / ; ( 0 = 1000/G£»;,^e;a,(0, 1=1

with

(5.137)

418

Applications of Kinetic Analysis to Photoreactions

/ = 2.303/0,

Ch. 5

(5.138)

where G is the geometric constant, which is assumed not to vary during the progress of the reaction. These fluorescence spectra measured during the progress of the reaction can be used for a muhicomponent analysis accordingly to the principles given for absorption spectroscopy (see Section 4.2.6.3). 5.5.7.2 Intensity diagrams for mechanistic information During the progress of the reaction fluorescence reaction spectra are taken and evaluated in the same way as the absorption spectra. First, fluorescence spectra are recorded at the chosen reaction times, and the intensities at wavelengths with significant changes are plotted versus the reaction time yielding intensity-time diagrams. Second, in accordance with the absorption diagrams, the fluorescence intensities at either of two wavelengths of observation are plotted versus each other. The form of the diagrams depends on the number and sequence of appearance of fluorophores during the progress of the reaction and the relative intensity at the wavelengths chosen. In Fig. 5.37 the intensity diagrams of some mechanisms are given, assuming that the fluorophores fluoresce at both wavelengths chosen [92]. These diagrams allow the determination of the number and the appearance of fluorescent compounds. The variety of the forms of these diagrams is manifold and more characteristic than in absorbance diagrams. As can be seen in Fig. 5.38, even for the same mechanism the fluorescence diagrams are extremely dependent on the wavelength of observations. Different types of diagrams influenced by differences in fluorophore properties as well as diagrams of high order in comparison to absorbance quotient diagrams are discussed in detail elsewhere [92]. 5.5.2 Photokinetic equations for different reactions 5.5.2.1 Uniform reactions Photokinetic equations In the following the fluorescence of the starting material alone as well as very dilute solutions {E' < 0.02) are assumed. For this reason the differential equations (see Section 3.3.3) are reduced to simple ones. The measured fluo-

Ch. 5

Fluorescence

419

Fig. 5.37. Schematic intensity diagram for some different types of reaction. The fluorescent molecules will fluoresce at both wavelengths the intensity diagram is plotted.

rescence intensity can be correlated to the concentration of A''. Using eqs. (5.106), (5.111), (5.137), and (5.138)

/;(o=~//?,/;(o

(5.139)

420

Applications of Kinetic Analysis to Photoreactions

Ch.5

faCffiC^O

Jak

^P

Fig. 5.38. Schematic intensity diagrams for a specific sequence of reaction products. By variation of the wavelength of observation the various spectral fluorescence distributions fa^ will vary. This influences the form of the intensity diagram obtained.

is obtained, whereby ci{t) and a(t) in eq. (5.106) are substituted by the fluorescence intensity /^(O and /^(0» respectively. /?yhas been defined by eq. (5.114). In consequence the geometric factor G as well as fluorescence quantum yield 77A and spectral distribution^^ are eliminated. By integration

Ch. 5

Fluorescence

l n / ; ( 0 = ln/;(0)-//f,r

421

(5.140)

is obtained. The slope of a diagram In 7^(0 versus t results in the partial photochemical quantum yield cpf according to
(^: slope).

(5.141)

In the case where the forward reaction A'' —> B is superimposed by a second backward linear dependent isomerisation step B -> A'', the use of the rate law with Rj and /?2 (see eq. (5.114)) yields / ; ( 0 = -(/?! +/f2)//r(0 + /?2^/a(0).

(5.142)

In the photostationary state (s) the change in fluorescence I^(s) becomes zero and 'a(s) = -^^laiO)

(5.143)

Combination of eqs.(5.142) and (5.143) gives

This equation can be integrated. The slope of the diagram l n [ / ^ ( 0 - /a(^)] versus t gives, using eq. (5.143), the partial photochemical quantum yield of the second step ^2=T7^^S

(5.144)

and by combination of eqs. (5.143) and (5.144) the partial photochemical quantum yield of the first step also:

^' K

I'm

Applications of Kinetic Analysis to Photoreactions

422

Ch.5

This partial photochemical quantum yield of the second step can only be determined if the molar absorption coefficient of the reactant B is known. Another possibility of evaluation is to take the integrated eq. (5.142) in its exponential form and to calculate by curve fitting procedures (p^ and
la it) ~ I^(s) = [/^(O) - /;(^)]exp(-/(/?, + R^)t)

Neglecting the photo-degradation processes in laser dyes the partial photochemical quantum yields of stilbene-1 derivatives have been determined by evaluation procedures as given above [93]. In Section 4.3.2 a microprocessor controlled device is described [176], which allows convenient measurement of the data. Even an optical multi-channel analyser can be used [175]. Examples of intensity diagrams The tautomeric equilibrium of anthrone/anthranol is a good example for a 1 '480

(rel. units) 60

k p

1

1

1

—1

1

1

T

'

r —

1

M+ Anthronc e

-^^^^ <^v

>• 1 -1

Anthranol O/*^

n

Ji

50

-^

40 h

30

-^

20 h
10 h

j

[ "^ .___

1

10

I

1

20

.-L.

L

30

i

1

40

1

J —

50

60

ll^Q (rel. units) Fig. 5.39. Intensity diagram of the photoreaction of anthrone-anthranol in methanol irradiated at 313 nm and synchronised to equal times of measurement [175].

Fluorescence

Ch. 5

423

28 ' 410

24

20

16

12 20

40

60

80

/

Fig. 5.40. Intensity diagram of the photodegradation of CPTC in methanol, irradiated at 365 nm.

linear intensity diagram as proved by Fig. 5.39. Similarly to measurements of absorption according to Section 4.2.2 (see Fig. 4.5) the fluorescence intensities obtained at different wavelengths of observation have to be synchronised to avoid distortion of the diagram. These diagrams helped to understand the photodegradation processes of laser dyes. An example is the "fluorescence spectroscopic" uniform reaction of 7-(4-chloro-3-methylpyrazol-1 -yl)-3-(4-methyl-1,2,4-triazolio)-coumarin sulfate (CPTC), irradiated in methanol at 365 nm. Fig. 5.40 demonstrates this linear relationship. Whereas the intensity diagrams are linear, absorbance diagrams exhibit a complex mechanism. Thus the long excitation wavelength (365 nm) and the fact that not all the degradation products fluorescence cut down the information to the essential interesting photodegradation step, for which quantum yields can be determined. Influence of changes in absorbance Even in the case of a uniform reaction A'' —> B'' sometimes some curvatures instead of straight lines are found in intensity diagrams. This problem arises if the absorbance at the wavelength of the irradiation exceeds 0.02 units. The influence is shown in Fig. 5.41.

Applications of Kinetic Analysis to Photoreactions

424

Ch.5

'a ^ 80

60

40

1

20

1

40

80

120

'^

Fig. 5.41. Calculated intensity diagrams for the reaction A*^ -> B'^ dependent on the change in absorbance at the wavelength of irradiation E". In the following the changes from start to end absorbance are given as: (a) £' = 0.4 -> 1.0, (b) £' 0.4 -> 0.8, (c) f* = 0.4 -> 0.6, (d) E' = constant, (e) £:' = 0.4 -> 0.2, (0 ^ = 0.4 -> 0.0001.

5.5.2.2 Consecutive reactions Photokinetic equations If a consecutive degradation step has to be taken into account hv hv

iB^-

hv

^C,

the rate laws become more complex. Nevertheless using fluorescence-time diagrams the steady state fluorescence intensity Ia(s) can be approximated. In consequence the partial photochemical quantum yields of the first step A^ ^ B^ can be determined, if £^ and fig are known. Usually the intermediate's absorption coefficient is not known. Therefore apart from (pf" (using eq. (5.145)) no further information can be gained. Accordingly the evaluation of the second slope in the intensity time diagrams only allows an approximation of the third partial quantum yield cp^ for known fg. In any case, however, the two slopes have to differ by at least one

425

Fluorescence

OOOOOOOOOOOOOOOOOQ O O O O O

O

o Oo o o o o o o

o o o

o

o

o

o

K O , S - © - ^ C H = C H - Q - ^ S O , K

80,Nft N«0,9-^^—^^CH=CH—^^OCH, OCH,

^^«>oooo o o

2.5

500

0

o

o

o

1000

o

t/s

1500

Fig. 5.42. Diagram of fluorescence intensity versus time for some laser dyes based upon substituted stilbene-1 derivatives [93].

known eg. In any case, however, the two slopes have to differ by at least one order of magnitude to reduce errors. For this reason a bi-exponential fit is preferable. Typically non-straight lines in the logarithmic diagram intensity versus time are given in Fig. 5.42. The consecutive mechanism gives the following rate equations d{t) = I[-R,a{t) + R^b{t)] m

= i[RM0(R2

+ R^Mt)]

c{t) = l[R,b(t)] assuming very dilute solution and using the definitions: /?,=1000£'A^f,

/?3=1000e>3.

426

Applications of Kinetic Analysis to Pliotoreactions

Ch. 5

This set of differential equations can be solved at the boundary conditions a(0) and b(0) = c(0) = 0 [177,178] with Wi2 = O-Sl"/?, + /?2/?3 ± ^ ( ^ , +/?2+/?3f-4/?,/? 1

«(0 = J^^l^

[iW^ - R,)txp(-IW,t) + (R, - W2)cxp(-IW,t)]

Ht) = .^^^[exp(-IW,t)

- cxp(-IW^t)]

c(0 = J ' ^ " ^ [W -W,)-W,

(5.147)

txpi-IW,t) + W^ exp(-/W;0]

With eq. (5.137) / a ' ' ( 0 - ^ ( 0 - ( / ? , ~ W,)exp(-/lV,0 + (W^, -/?,)exp(-/VV20 by measurement of the fluorescence intensity at a wavelength a the change in A'^ concentration with time can be correlated to a bi-exponential function, which contains four parameters pi. If an offset has to be taken into account, a further parameter p^ has to be added ^a ( 0 «= Pi exp(-/720 + P3 exp(-/74r) -h ^5. The defined parameters correlate with the partial photochemical quantum yields according to

and

R2 = W^+W2-R^- R^

(5.148)

Ch. 5

Fluorescence

427

A non-linear regression allows calculation of the parameters p^ to p^ for fluorescence-time curves. These equations correlate with eq. (5.141) in the case of the simple mechanism A''^ B'^. Under such conditions eq. (5.148) reduces to R^^

El^

(5.149)

assuming that in this mechanism p4 and W2 become zero (mono-exponential). Thus eq. (5.149) yields /;(0) = p,+P3+P3

with

/ ; ( 0 ) - / ; ( 5 ) = /7,

and

-^=\y,=^.

The same relationship is obtained, as was derived for eq. (5.145) ^

m/;(0)-/>)

'

/

/;(0)

Intensity diagrams for consecutive reactions For the consecutive reactions intensity diagrams are curved if besides A'" another reactant also fluoresces. Examples are given for laser dyes (see Fig. 5.43) [92,93] and for short wavelength irradiation of anthrone [180]. In the case of CPTC at irradiation by 254 nm even an intensity quotient diagram can be constructed (see Fig. 5.44). These diagrams correlate with those of extinction difference quotient diagrams (see Section 5.3.3.1). They are derived by combination of intensity differences at different reaction times taken at three wavelengths according to the following equations:

762? (0 _= «

Applications of Kinetic Analysis to Photoreactions

428 120

1

Uso [%I

'

' 1• " • •

-

; 550nm«530nm

•

I • •

80 h

,•

• H

'n

• •

" «

5 • • "•

M

• • , •

40 h

•

• •

;

•

A

•

•

•

• •

• '"

.••

4

.•!

«

5

A A A '

:

5 i

'

1

A A A

J

A^75nni A A

•

B •

•••

•

•

: ^

• • :500nm

.MSOnm

•

:

60

1

•

m m

100 L

'

r

Ch.5

A ^

-

J 1

A A A A A

1 1

A

_J

A A

• •

^2^'-" ft* m

1\

*•

.*

»* ft

A'=365nm

20 h

^/

_.

0

1

.

1

20

1

40

L

,

60

1

80

,

H 1

1

1 100

Fig. 5.43. Intensity-diagram of the photoreaction of 7,7'-N-diethylamino-4-methylcoumarin (DEMC), irradiated at 365 nm [181].

5,53 Errors due to E > 0.02 When the assumption F(0** = 2.303 is no longer valid, instead of eq. (5.137) the relationship / ; ( 0 = /^^1000/oa(OF(0

(5.150)

has to be used with (5.151) One can assume that Kf stays constant during the progress of the reaction as a first approximation. Thus the differentiation of this equation yields 1-10"^'

/ ; ( 0 = /^^1000/oa(0—-— b,

-KplOOOIoait)-i^ '^ °

^ ^-^—P. ^ - i (l-exp(-£:'))

(5.152)

Ch. 5

Fluorescence

L

120

429

500

100 80 60

>m

a

,•1*'

• 430nm •475 nm

40 20 0

1

•

^1

Ail!

LA:

1

1

1

.^_^ 450

-20

** \

-40 -60 20

-10

10

10 "*•

20

30

40

50

60

Fig. 5.44. Intensity quotient diagram of the photoreaction of CPTC, irradiated at 254 nm in methanol [182].

with

E

(5.153)

For this derivation the Napierian absorbance units are taken for convenience (the molar absorption coefficient e' is decadic). Using eqs. (4.2), (5.152), (5.153) and a rearrangement of eq. (5.150) a(0 =

/;(0

l-exp(-£0 1000/oJ?^F E'

one finds (5.154) The function S(£') is defined by

430

Applications of Kinetic Analysis to Photoreactions

S(E') = exp(-E') +

E'{s)\

E'

i^^f^-exp(-£-)].

Ch.5

(5.155)

using decadic units, one obtains l - 1 0 " ^•IO\.-r 5(0 = 10-^'+-^^ E' 2.303£'

CS. 156)

Using the mean value theorem within the time limits tx and ^2, eq. (5.154) can be integrated to

ln/;(r2) = ln/;(r,)-/?,/o5(F)(r2-r,)

(5.157)

whereby r,
Em Fig. 5.45. Correction factor S for the slope plotted versus £'(0), the absorbance at wavelength of irradiation at the beginning of the reaction. For calculation the absorbances are used according to eq. (5.156). In the graph decadic absorbances are plotted at the axes.

Ch. 5

Fluorescence

431

In Table 5.6 the correction factors are given for a large variety of starting and stationary state absorbances (in decadic units) by which the slope in diagrams according to eq. (5.141) has to be multiplied. It is evident that the higher the absorbances themselves and the larger the change during the progress of reactions the smaller the factors for the correction of the slope will be. This means that 5(0 deviates more from 1 and the slope determined according to eq. (5.141) becomes too small. As a consequence absorbances higher than 0.02 have to be avoided in fluorescence measurements in photokinetics, since even at the simple mechanism A'' —> B'' the influences are as drastic as demonstrated. 5.5.4 Applications 5.5.4.1 Anthraquinones In photochemical reaction pathways, normally the number of reactants which fluoresce is smaller than those absorbing. For this reason intensity diagrams were introduced in Section 5.5.1.2 to supply additional mechanistic information. The photoreaction of the anthrone to tautomeric anthranol gives a linear intensity diagram because anthranol is the only fluorescent product. However during this photoreaction one realises that the oxygen of the solution will be used to photooxidise the reactant to the non-fluorescent anthraquinone. This reactant is stable as long as oxygen is in the solution. As soon as it is consumed, a blue-green fluorescent anthrahydroquinone is formed. The maximum of the wavelength of fluorescence is shifted to the ultraviolet in this consecutive photoreaction. At least at one certain wavelength of observation a third fluorescent compound can be monitored as is demonstrated by the intensity diagram in Fig. 5.46. In the case of the photoreactions of anthrone and anthraquinone only the intensity diagrams were used to support mechanistic considerations obtained by formal kinetic examinations using absorbance. However, in the process of the examination of the photostability of laser dyes, quantum yields have been determined applying the equations given in Sections 5.5.1 and 5.5.1.1. 5.5.4.2 Photostability of laser dyes The structures of two strongly fluorescent dyes are given as

432

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o

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o

I c o 'a o o

SI

o

d d d d


Q O

CD ^

T f CO CS ^ - H ^ <S CO WO t ^ ON t ^ CO « n t ^ ON ^ CO < n r * r^ 0 0 0 0 0 0 0 0 OO ON O N ON 0 \

Applications of Kinetic Analysis to Photoreactions

8:s 00 CO O

^ r<^ vO r O C> d d

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wo *§

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T^ -s I ts

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Ch.5

Ch. 5

DEMC

Fluorescence

433

Carbo 1

Both are considered to be used as laser dyes. The substituted coumarin 7,7'diethylamino-4-methyl-coumarin (DEMC) had been mentioned before and it has been the focus of intense kinetic examinations. The fluorescence reaction spectrum demonstrates a consecutive photochemical reaction via a second fluorescent intermediate (see Fig. 5.43). Looking at the reaction spectrum, the same information is not so obvious. In contrast, the fluorescence reaction spectrum of carbo 1 (7-(5-butoxy6-methyl-1,2,3-benzotriazol-2-y l)-3-pheny 1-1 -ethy loxycarbonyl-carbostyry 1) exhibits an isostilb (the fluorescence does not vary at a wavelength of obser-

^il^,l%I

A,E Fig. 5.46. Intensity diagram of the photoreaction of anthraquinone in oxygen containing methanol for different wavelengths. Wavelength of irradiation and excitation are the same: 313 nm [180].

434

Applications of Kinetic Analysis to Photoreactions

350

400

450

500

Ch.5

550 Of [nm]

Fig. 5.47. Fluorescence reaction spectrum of carbol in oxygen free 10"' M methanolic solution irradiated at 254 nm by /„=1.37 x 10~* Einstein cm~^ s~*.

vation at least for some reaction time) in oxygen free solution of methanol by irradiation at 254 nm only at the beginning of the reaction (see Fig. 5.47). Carbol was chosen since in general a benztriazole substituent photostabilises compounds. The mechanism given in Section 5.5.2.2 represents a consecutive reaction including a photo-equilibrium. It can be reduced in the case of carbol to the simpler mechanism of A^-J!1L^B^- ^2L^C^

Application of an analogous equation to eq. (5.125): /;(o=D.K.D-7;(o, in which the matrix D contains elements with fluorescence quantum yields rji, the spectral distributions^, and the absorption coefficients e\ of the two

Ch. 5

Fluorescence

435

Fig. 5.48. Fluorescence-time curve for the photoreaction of carbol in oxygen free methanolic solution. The symbols represents the points of measurements for two wavelength of observation. (—) is the theoretically calculated curve to determine the kinetic parameters. Irradiation took place at 254 nm. Intensity /o=1.37 x 10"* Einstein cm'^ s"^.

interesting reactants A'" and B'". This matrix equation can be solved according to the methods using absorption measurements given in Section 5.1.2.2. It allows calculation of the eigenvalues of the K-matrix which contains the kinetic parameters. If irradiation intensity and absorbance coefficients are known, the partial photochemical quantum yields
436

Applications of Kinetic Analysis to Photoreactions

Ch. 5

OH

photodegradation in water at an irradiation of 313 nm results in linear intensity diagrams in contrast to non-linear absorbance diagrams. This means that the reaction is "fluorescence spectroscopically uniform" and the photochemical quantum yield can be determined using the simple equation, eq. (5.139). Working in solutions with an extinction higher than 0.2 at the wavelength of irradiation, the photokinetic factor has to be taken into account according to eq. (5.135). In a first approximation the measured fluorescence intensities can be corrected by this photokinetic factor. Thus eq. (5.150) has to be applied instead of eq. (5.137). Corr(0 = la (tint)]"'

= K,lOOOI,a{t)

(5.158)

can be defined, using the abbreviation Kp given by eq. (5.151). The time dependence of /^(O after correction according to eq. (5.158) is inserted into eq. (5.106) and one obtains [183] /;corr(0 = - / ? , / o / ; ( 0 .

This equation can be integrated numerically to

C ( 0 - C(0)=-/?,/oJ/>If the left-hand side of this equation is plotted versus the integral of the right-hand side, one can determine the photochemical quantum yield tp^ for the degradation step from the slope. This is demonstrated in Fig. 5.49. In Section 5.5.3, a more sophisticated method has been given to correct measured fluorescence intensities influenced by the change in the photokinetic factor during the photoreaction. By these means the photochemical quantum yields of a large variety of laser dyes can be determined using the corrections according to eq. (5.157). Some structures of dyes used are

Ch. 5

Fluorescence

437

KO,S

CPTC

CI

As Fig. 5.50 demonstrates, quantum yields lose their "dependence" on absorbance at the wavelength of irradiation. These results are discussed in detail elsewhere [93,176,180,181]. The application of many of these dyes [183] was of interest in dye lasers. These observations led to the synthesis of new

10' J/Ldt t=o

20 o o

40

••. 60 'Oc

80

" " • ^ o**©« .

/f

W - / f (0)

' korr

100

"""^i

' korr ' J

I

I

Fig. 5.49. Determination of the partial photochemical quantum yield ip'^ of the primary photodegradation step of umbelliferone using the fluorescent intensity corrected by absorbance at the wavelength of irradiation.

Applications of Kinetic Analysis to Photoreactions

438

Ch. 5

-m [10 M

10-

1

9.5 -

9-

8X

7-

6-

""'"''"""""^"'""^^""'^ni^^^""""^"'^^^""^^"^Tr"'^"^"'"""^^"^"^^^"T""^"

0

0.1

X

0.2

Fig. 5.50. Slope determined according to eq. (5.157) instead of eq. (5.140) for the first degradation step of stilbene-3 (distyrylbiphenylsulfonic acid, S3) [180,181] at different concentrations.

bridged indeno-indene derivatives [184], which exhibit less photodegradation, and to embedding in a matrix [182].

5.6 COMBINATION OF METHODS 5.6.7 Combination ofabsorbance and fluorescence measurements The major problem in the evaluation of the partial photochemical quantum yields even of the first step is an absorbing product, its molar absorption coefficient e^ being unknown. According to the treatment given in Section 5.5.2.2 the fluorescence measurements yield the parameters /?i, /?2, and R^ as well as Wx and W2. However, the additional measurement of an absorbance reaction spectrum and the determination of how the absorbances at the interesting wavelengths depend on time at a large number of reaction times allows a multiple linear regression for the additional determination of the absorption coefficients of all three reactants [178].

Ch. 5

Combination of Methods

439

Since absorbances can only be measured in a certain dynamic range, the dilute solutions used in fluorescence measurements cannot be taken. Therefore the assumption is necessary that the photochemical reaction is governed by the same mechanism for 10-^ and lO-^ M solutions. Furthermore the higher concentration forbids an approximation according to eq. (5.111). However, in Section 3.3.11 a principle was introduced substituting the normal time scale by a transformation into a pseudo time @(t). In consequence the three equations of eq. (5.147) have to be substituted by

^^®) = ^^^t

[exp(-/oVy2Q(0) - exp(-/oW,0(r))]

^(®> = uf^L

[(^1 - W,) - W^i exp(-/o1V20(O) + W, exp(-/oW,e(0)]

VV,

-W2

The expression 0 ( 0 is defined according to eq. (3.98) by

e(„ = j i ^ r . Taking this time transformation the measured absorbances depend on the time accordingly: £(6) = e^aiO) + e^biO) + ecc(9), ^(^>= uf^L

[gA(W, -/?.) + gB^.

+ J^^l.

-ScW,]exp(-IoW^eit))

[e^iR, - W,) + e^R, -ecH'2]exp(-/oW,0(O).

This equation can be rearranged to

440

Application of Kinetic Analysis to Photoreactions

^(®^ = ^ A T ; 7 ^ [ ( W ' I •M ~ "'2

Ch. 5

-/?i)exp(-/oVV2e(0) + (/?, - W^2)exp(-/oW,0(O)]

+ ^B ^l^^^l [exp(-/oW^2e(0) - exp(-/o • W, • 0 ( 0 ) ]

+Be i j i T ^ K ^ i - w^2) - w; exp(-/o ^20(0) + w^2 exp(-/o w;e(0)] (5.159) This latter equation is a matrix equation like eq. (4.6). Multi-linear regression allows the determination of the molar absorbance coefficients for any wavelengths which have been measured. The calculated absorption coefficients can be used to allow the determination of the partial quantum yields (Px,(p\ and
I'm

a{oy

Taking c{s) = 0 the concentration of the c/5'-intermediate amounts to b{s)^a{G)-a{s), The parameters of the reaction cause an error of approximately 3% in b{s) by this assumption. At the quasi-stationary state the measured absorbance Ej^{s) allows the calculation of Sj^^ according to _E^{s)-ej^^a{s) b(s)

^AB ~"

Ch. 5

Combination of Metliods j

.

,

.

,

.

,

.

441 ,

.

, , . . . , . , . .

400 350 1

H

1

H

250 4

A

300

200

\

150

1

- V^

-]

100 ...

•

1

1.

. 1

1

t

.

i

.

1

•.

1

1

3

^

0 -3

0

100

200

300

400

500

, , 600

Fig. 5.51. Bi-exponential fit tofluorescencedata using the Marquardt algorithm measured for stilbene-1 in water, irradiated at 365 nm.

The values of I^{s) and E^{s) can be obtained from a bi-exponential fit to 7^(0 and E(d) curves, respectively, according to

Fig. 5.51 demonstrates the result of such a bi-exponential fit with residuals for stilbene-1 in H2O irradiated at 365 nm. Using the equations discussed in the section above the concentrations of the three reactants could be determined in dependence on irradiation time. These are presented in Fig. 5.52. 5.6J. 2 Dependence on variation of viscosity of the solvent By variation of the percentage of water in the water/ethylene glycol mixtures and their temperature, a continuous series of viscosities can be obtained. Therefore the mixture and temperature are varied such that overlapping values of viscosity are found at least for two defined contents of water in the mixture. This type of variation gives a hint on the relative size of polarity and viscosity effects influencing the quantum yields of a photochemical reaction.

Application of Kinetic Analysis to Photoreactions

442

Ch.5

25 1

cis-S-1

Phcnanthrcnc-analoj^ .

trans-S-1 -4-

e[hi Fig. 5.52. Calculated concentrations of trans-S-l (A), cis-S-l (B), and phenanthrenederivative (C) versus the pseudo-time © for an irradiation at 365 nm in water.

It can be proved that the refractive index of the mixtures depends linearly on the percentage of ethylene glycol. In the S-1 photoreaction both the photoisomerisation steps strongly depend on the absolute viscosity, whereas (^3 is very small and stays nearly constant during the photoreaction. Since the determined values of quantum yield fit even for different mixtures of water/ethylene glycol with viscosity adjusted to the same value by variation of temperature (see Fig. 5.53), polarity effects of the solvent will influence the photoreaction less than viscosity effects [178]. A similar examination [185] has been achieved for the photoreaction of stilbene-1 in water/methanol mixtures. The results are demonstrated in Fig. 5.54. 5,6.2 Combination ofabsorbance and chromatography The non-selectivity of UVA^is-spectra with respect to the reactants participating in a reaction causes problems in the photokinetic evaluation. That means the number of partial reaction steps together with the number and sequence of the reactions products has to be determined indirectly. A mechanism has to be assumed and differential equations have to be set up. These have to be verified by a correlation of simulated data according to the pro-

Ch. 5

Combination of Methods 1

08

9

'

1

' — 1 —- I — i

1

0.6

f

1—•—1—•—1—i-pi—1 • J

•

• •

»

443

•

•

0.4

T

0.2 1

0

1

1

5

J

i

1

10

1

15

1

1

20

L—j

L-.J

25

— 1 — 1 — 1 — I — >

1

30

1

I

1

*

1

35

q> 1 0.20 1 — 1 — 1 — r — r ' ' '~] U • 0% Glycol •63.% 1 0.15 LI • 20.4% !/9.4% 1 0.10 A 40.4% k 100% J 0.05 k 0.00 1 I 1 • i_- ^ : i - . 1 * . . _ i . . . ._ 1 v.* L. J 10 15 20 25 30 35 L •

»

0.010 hr 9

1

'

1

[•

'

1

'

0.005 1

^•^^^ 0

0» • 1

o 1

<j

5

•

H

o 1

j

j

4

•

r

1

1 •• 1

1

10

15

1u - - J i

20

I . . - J L. . . J

25

1

30

1

1

35

abs. viscosity [cP] Fig. 5.53. Dependence of the photochemical quantum yields of stilbene-1 on absolute viscosity for an irradiation at 365 nm in water/ethyleneglycol mixtures. Glycol: • , 0%; #, 20.4%; A, 40.4%; O, 63%; D, 79.4%; 0» 100%.

posed rate laws and the experimental data. However, if the absorption coefficients of all the reactants are known, the rate constants and the photochemical quantum yields can be determined with quite good quality as demonstrated in the previous chapters.

444

Application of Kinetic Analysis to Photoreactions

Ch.5


0.10 0.08 0.06 0.04 0.02 0.00 9.

•

[• i h

1

•

1

' "•

""j J

I 1 >

0

20

40

1 « ]

1

60

80

100

0.03

1 ' I

0.02 0.01 0.00

I

f I 1

u.»

•

0

\

1

20

11

1

• i

,„i

...i

1

40 60 80 100 %McOH (%weight)

Fig. 5.54. Dependence of the photochemical quantum yields of stilbene-1 on viscosity for an irradiation at 365 nm in water/methanol mixtures.

Nevertheless, the success is better if methods are combined as discussed in Section 4.5. Whereas spectroscopy results in very good data with low noise, chromatography allows the separation of the reactants during the reaction procedure, although making experimental data less easy to quantify. In Section 4.5.3 an experimental set-up has been demonstrated, which allows the observation of single wavelengths as well as spectra during a photoreaction. These so-called reaction chromatograms can be evaluated, since the measured signals obtained for example in high performance liquid chromatography are linearly proportional to the concentrations of the reactants according to Section 5.2. Chromatographic separation techniques in a continuous flow systems supply a large number of chromatograms during a reaction procedure. Using a single wavelength detector, the signals demonstrate the change in concentration of all the reactants separated [109,110]. The use of a

Combination of Methods

Ch. 5

445

photodiode array detector supplies a series of spectra for eacli chromatogram, which amounts in total to a huge data set. 5.6.2.1 Chromatographic reaction spectra The quality of this combined HPLC apparatus and the irradiation set-up given in Fig. 4.29 can be tested using a well-examined example: the photoreaction of trans- to ci5-stilbene with a consecutive step of photo-ring closure to phenanthrene. In Fig. 5.55 the signal proportional to the concentration of the reactants is plotted in a 3-dimensional plot versus the retention time t for some reaction times. It becomes obvious that the rran^-stilbene vanishes in the first chromatograms forming ci5-stilbene which in a con-

-1

"^-T

6.6 7.2 7.8 8.4 9

tU]

Fig. 5.55. Reaction chromatogram of the photoreaction of stilbene using the microprocessor control set-up given in Fig. 4.29 and correcting for the dilution according to Fig 4.30. The different chromatograms absorbance E versus retention time t after distinct photoreaction times are plotted as an overlay to the back.

Application of Kinetic Analysis to Photoreactions

446

A.'=365nm CH3OH/CH3CN/H2O 10 : 10 : 1

Ch. 5

25 h 55 min 17 h 10 min 11 h 40 min I

Cone. [3 15nm]

tj^lmin] Fig. 5.56. Reaction chromatogram of diphenyloctatetraene as the absorbance signal at 315 nm versus the retention time r^ of the chromatogram for some selected reaction times for an irradiation at 365 nm. As an eluent a mixture of methanol/acetonitrile/water in the ratio 10:10:1 was used.

secutive reaction step forms phenanthrene. If the number of chromatograms taken is large [110,186], an evaluation according to the principles given in Section 5.2.1 becomes possible (see next section). The advantage of this method is the possibility of determining and proving mechanisms by the separation of the reactants. In Fig. 5.56 the even more complex photoreaction of diphenyloctatetraene is discussed [187,188]. The reaction chromatogram allows the elucidation of the mechanism with respect to consecutive and parallel photoreactions, which is interesting in comparison with other diphenyl substituted alkene chains. Cinnamic alcohol and its derivatives are starting materials for the synthesis of coumarin dyes used in food industries and as laser dyes. By irradiation at 254 nm cinnamic acid or the alcohol exhibits an extremely complex reaction spectrum using UVA^is-spectroscopy. A comparison of the reaction chromatograms for different irradiation conditions in various solvents

Ch. 5

Combination of Methods

447

(methanol, cyclohexane, acetonitrile) allows differentiation between possible photochemical reaction pathways. Fig. 5.57 is the example of such a reaction chromatogram [ 189]. In Section 5.6.1.1 quantitative results for the partial photochemical quantum yields of the photoreaction of stilbene-1 were given. A photoisomerisation was assumed to be the mechanism as a first step. This proposal is supported by a reaction chromatogram of the photodegradation reaction of this laser dye. The two parts of the diagram (see Fig. 5.58) prove a photoreversible isomerisation as a first step [190]. This information was used for the evaluation mentioned. 5,6.2,2 Photochemical quantum yields using chromatographic data According to eq. (5.27) a relationship between the concentrations ai of reactants and the signals Af, exists, whereby the proportionality constant 1/m, is to be determined by calibration. According to a. = —A/,.,

(5.160)

using the equations for the conservation of mass, one obtains for t reaction times and n components — M,(0)

= —M,(l)

+ ...

+

/W|

mt

—M„(l) m„

(5.161) — M,(0) m,

=

—A/,(0 m,

+ ••• +

—A/,(0 m„

which resembles a set of inhomogeneous linear equations. It can be solved by matrix operations according to procedures given in Section 4.2.6.3. Using eq. (5.160) the concentrations of the i components at reaction times 7 depend on the calibration factors m, and the signals Af,(/*) according to a,U) = —MjU)^ m..

(5.162)

Application of Kinetic Analysis to Photoreactions

448

Ch.5

^' ilh 50m in 8h 45min 8h 15min 5h 50m in

BEFA

CD I

GH

t^lmm]

Fig. 5.57. Cinnamic alcohol irradiated at 254 nm in methanol: reaction chromatogram with signals observed at 255 nm versus retention time f^.

Stilbene-1 CHjOHrHjO 58 : 42 Cone. [300 nm]

25min. 10 min. 12 min, X.'=365nm

4 min 40 sec. 3 min 20 sec.

"10 sec. /v 0

X,'=365nm

tRim in]

10

Fig. 5.58. Separation of stilbene-1 by HPLC chromatography: bottom left, trans-cis photoreaction by irradiation at 365 nm forming the cis product; top right, the cis-trans photoreaction at 254 nm with removal of the cis product.

Ch. 5

Combination of Methods

449

If the initial concentration a/(0) is known, the values of the signals M,, which depend on the reactants and reaction times 7, can be obtained without knowledge of the calibration factors m/. The only requirement is the experimental ability to separate all the reactants at the chromatographic conditions [110,186]. In a closed HPLC loop, the injected amount of substances should stay constant until detection at the end of the column. That means the peak areas Ai or the signal Af, should be directly proportional to the concentration a, of the reactant i for a constant flow FW, Under these conditions the overall absorbance of the ith reactant £", for a defined injection volume V^ is given for a selected time by FW E,=M,—^.

(5.163)

Using Lambert-Beer's relationship (see eq. (4.5)) one obtains «:=—*

,

(5.164)

taking f, as the calibration factor m, and using the peak area At as the signal. In eq. (5.164) measurement of the path length of the detection cell and the injection volume depend on the set-up, but can be kept constant during the experiment. They are combined to a constant K„, specific for the set-up. Combination with eq. (5.162) results in a,(0 = -^A/,(r),

(5.165)

with A.»/ —

"

IVE

defined as the constant. In analogy to eq. (5.161) one can derive linear equations of the n reactants, in which a/O) is independent of time:

450

Application of Kinetic Analysis to Photoreactlons

Ch. 5

,=1^*

Thus, the set-up of the specific constant KH gives

"

M,(0) '

The advantage of this approach is that Kfj does not necessarily have to be known. It is only required to be constant. In Section 4.5.3 it was mentioned that at each injection, the reaction solution is diluted by the by-pass of the eluent, connecting the reaction and chromatographic cycle. Therefore the concentrations have to be corrected according to A / r ( 0 = Mj(0 1 - - f

L

v\

.

(5.166)

where V represents the reaction volume in the irradiation cycle. Thus, eq. (5.161) is corrected to [110] —M,^°"(0) = —M,'^(l)

+ ... +

—Ml'^il) (5.167)

—M,'°"(0) = —Ml''''{t) ^1

^1

+ ... +

—Mf*"(0 ^n

The reaction cycle and the chromatographic cycle are demonstrated in Fig. 4.29. Using the values taken from the reaction chromatogram of stilbene (see Fig. 5.55) either the peak areas or the peaks heights have been evaluated according to the equations given above and the concentrations determined as a function of time. The result for the three reactants is given in Fig. 5.59 [110]. Experimental data (values for peak areas) and simulated data assuming the reaction constants mentioned are compared. 5,6,3 Application of chemometrics In the previous section a graphical rank analysis has been demonstrated to

Combination of Methods

Ch. 5

451

0.5 Cone. -3 [10 mol/1] { trans-stilbene 0.4

0.3

0.2

0.1

8000 Fig. 5.59. Concentration-time curves obtained by HPLC: experimental data after volume correction (symbols) together with data obtained by parameter estimation according to a Runge-Kutta simulation (curve). In methanol/water (3:1) initially at 4.4 x 10"^ mol/1 solution the quantum yields were determined to ^i^= 0.4, (p^ = 0.2 and ^ ? = 0.03 for irradiation at 313 nm. The detection wavelength was 262.5 nm, at which all three reactants have comparable absorption coefficients.

determine the number of partial reaction steps. Next, the determined rank was used to set up a system of differential equations and using matrix calculus to obtain the reaction constants by formal integration. These were used to check calculated data points with the experimental data set applying Runge-Kutta algorithms. If absorption coefficients of intermediates are unknown the photochemical quantum yields could not be obtained directly. However, a combination of experimental methods supplied more results. This approach can be improved using new methods of multivariate data analysis given briefly in Section 4.2.6.3 and well reviewed in the literature [38,39]. Factor analysis allows the extraction of the rank in a set of data and reduction of the amount of data produced in reaction spectra. This principle was applied to kinetics for the first time [80,191]. During the examinations of eigenvectors and eigenvalues, numerical problems are found using reaction spectra. Addition of a Fourier analysis of the rotated eigenvectors [80] solved the problem. Using a model normalised target transformation

Application of Kinetic Analysis to Photoreactions

452

Ch.5

4> O

o G O 2

3

Xi

1U

s±s

340

240

360

380

wavelength [nm] Fig. 5.60. Spectra of /ra/w-stilbene, cw-stilbene and phenanthrene determined from combined HPLC and UVA^is absorbance measurements applying chemometrics. The spectrum of the intermediate cij-stilbene is of special interest.

o c o o

1.0

U

0.8

U

0.6

L

0.4

U

0.2

0.0 I time/10 s

Fig. 5.61. Concentration versus time curves for the stilbene photoreaction.

Ch. 5

Combination of Methods

453

[191,192] and including the spectra obtained during a reaction chromatogram of HPLC data, the concentrations of the reactants and their absolute pure spectra can be ascertained. This procedure was first applied to the model photoreaction of stilbene [191] to check the algorithms and the considerations. Figs. 5.60 and 5.61 show the results. The photochemical quantum yields determined are in good agreement with those calculated using all the absorption coefficients, taking synthesised pure cw-stilbene and undergoing a tedious procedure to determine its absorption coefficient. Since the application of this combined approach to the photoreaction of pdistyrylbenzene [193]

exhibited relatively high standard deviations, the software and the measurement set-up have been improved [194]. A comparative examination of the photokinetics of para-, meta-, orr/io-substituted diphenylpyrazines [100,194]

has proved the feasibility of the combination of UVA^is data with reaction chromatograms and the application of chemometrics. Without information on the reaction mechanisms and absorption coefficients concentration-time curves can be obtained. Chemometrics even allow the examination of photoreactions with unknown intermediates without combination of different spectroscopic or chromatographic methods under certain conditions. As an example, the photoreaction of the Z-fulgide to a ring compound via the £-derivative [195] is given, using the following scheme of reaction:

Application of Kinetic Analysis to Photoreactions

454

Ch. 5

hl/VB

(£r)-fulgide

(Z)-fulgide

7,7a-dihydrobenzofuran-derivativc (C) (DHBF)

Application of the adaptive Kalman filter [73,75] allows the iterative calculation of the data of a third component; the other two are known. Using the law of conservation of mass, the unknown concentration can be calculated from the known concentrations of the other two. The result is given in Fig. 5.62 for some reaction times. One realises that the third component does not start with a zero slope for the concentration in the origin in contradiction to common behaviour of consecutive reactions. 120 Cone. [^moI/1] 100

80

f

1

' O

r

r Uz a 1

o

0

1 L a j o

•

1

0

—

O

1

0

oJ

DHBF

o"

H

J J

o o

1° 1

O

o»• ° " '

L o

60

•

®

J

o

D

J

O

1 °o°

40 L h

20

'h

r/Tv 1 ? ° \^

0 i

•.

\ , .

, 1000

°?° ° q g •• 2000

•

•

°

t °

3000

•

•

PI

B

•

•

Lfl

ft

4000

• aJ 5000

tls]

Fig. 5.62. Photoreaction of fulgides in n-hexane: concentration versus time curves for the three reactants determined by differential calculation of the third spectrum (DHBF: 7,7adihydrobenzofuran derivative).

Ch. 5

Combination of Methods

455

The reason is a numerical problem at the beginning of the reaction, where the concentration of this third reactant is small. This problem is overcome if a mix spectrum is generated using the spectra of the known reactants and calculating their concentrations. This mix spectrum is subtracted from the experimental spectrum of the reaction mixture. Then, concentration-time curves with the expected slope are obtained [196]. This reaction has been examined in various solvents (n-hexane, toluene, ethylacetate) and at different wavelengths of irradiation. Applying formal integration, Runge-Kutta simulation, and curve fitting techniques, the partial photochemical quantum yields have been determined for the ZE-, the EZ-, EC' and CE-reaction steps (see reaction scheme above). The results have been published [197] and are discussed in detail elsewhere [196]. As discussed in Section 5.1.2.2, in general, the following differential equations can be set up and solved by formal integration: dz Ot

de

dc ot

using the abbreviation / = 1 0 0 0 / Q and the photokinetic factor defined by eq. (5.108). The formation of this system of differential equations is straightforward following Section 5.4.6.1 and Sections 5.1-5.4. The numerical integration is done within an interval / starting at a point of measurement i:

eF(t)dt u -{^E^PEZ + ^EV>ECV\ "' ^f^(0 dt

eFit)dt-e'c
cF(t)dt.

456

Application of Kinetic Analysis to Photoreactions

Ch. 5

TABLE 5.7 Partial quantum yields of the photoreaction of the ^-isomer by irradiation at 365 nm in nhexane and toluene [196], Z-, E-isomers, C, cyclic form. Solvent

^zE

*PEZ

*PEC

H>CE

n-Hexane Toluene

0.12 0.12

0.12 0.13

0.2 0.18

0.11 0.07

as described in Section 5.1.2.2. For convenience the nomenclature of partial photochemical quantum yields is modified, the index indicating initial reactant (absorbing the light) and product rather than the number of the step of the partial reaction. Thus, all the considerations of Section 5.4.4 remain valid. A few of the results are summarised in Table 5.7. 5,6A Combination ofabsorbance with NMR According to Section 4.6 nuclear magnetic resonance spectroscopy can be used to monitor photoreactions and obtain quantitative results on the reaction constants evaluating absorption reaction spectra as additional information [114]. The special method of chemically induced dynamic nuclear polarisation (CIDNP) [198] was applied to examine the photolysis of 1hydroxy-benzotriazole in alkaline or acidic acetonitrile (KOH or KCIO4 present). In Fig. 5.63 a Runge-Kutta simulation is demonstrated using a mechanism according to Example 3.24, the rate, eq. (3.39), and a generalised approach according to Section 5.4.4, which was applied in detail to photoreactions. Since the reaction involves two linear independent steps, two wavelengths of observation have to be used. One has to solve two coupled linear differential equations by formal integration: jdE, = ZoJojP^iOdt +ZuIoJE,(t)dt -hZ2iIoJE2(t)dt jdE^ = ZojIoj^iOdt + z^2loJE,(t)dt +Z22loJE2{t)dt whereby the parameters Zij consist of differences and ratios of the reactions constants Ri and R2 (see eq. (5.131)) as well as of elements and the determi-

Ch. 5

Combination of Methods

457

-§ 281nm

Fig. 5.63. Absorbancc-time diagram of the photo consecutive reaction of l-hydroxybenzotriazole in alkaline acetonitrile with measured (O, A, D) and simulated (—) data [114].

nant of the matrix of the absorption coefficients (for details see [10,17a, 143]). Important relationships can be found

^11^22 ""^21^12 = /?i/?2~ determinant giving the trace and determinant of the Jacobi matrix (treated in Section 2.2.1, Example 2.19). Some of the photokinetic results are given in Table 5.8. 5.6.5 Combination ofabsorbance with reflectance In Section 4.4.1.2 regular reflectance has been announced as a method to follow photoreactions taking place in solid material. Since reflectance at interfaces depends on the refractive indices of matter at both sides of this interface, the signal measured either in transmittance or reflectance depends

458

Application of Kinetic Analysis to Photoreactions

Ch. 5

TABLE 5.8 Photokinetic constants for the photoreaction of 1-hydroxy-benzotriazole, which forms nitrosobenzene, p-nitrophenol, and nitrobenzene A

Medium

^j^

313 313

KOH/acetonitrile KClOVacetonitrile

0.026 0.65


on this refractive index. If thin films are examined, the reflections taking place at both the interfaces of this film can cause interference, if its thickness is in the range of some jum [5]. According to Fig. 5.64 in reflectance the intensity of interfering beams /; and I2 depends on the reflectivity given by eqs. (4.14) and (4.15), respectively, taking into account that the film can absorb (either absorption coefficient ki or absorption index Ki): ^A = A + ^2 + 2 7 v r cos(2jr Y + d)

(5.169)

A

Ij W

Fig. 5.64. Radiation pathways of beams reflected at the two interfaces of a thin film with refractive index /zj and absorption coefficient ki in between medium with HQ < nj. IQ^X) is the incident intensity, I\ and I2 are the reflected interfering beams, the latter varied to I2 by a change in the thickness of the film. In general perpendicular incidence is measured. For visual convenience the graph uses some angle. The support material for the thin film is included, which has a refractive index higher than n\. The transmitted light /|(A) is used for reference measurements.

Combination of Methods

Ch. 5

459

n (d-d) ^

0.8 H

I 0.6-1 '"

0.4 0.2 -|

0-0 "V 400

\ 500

\

r

600 700 wavelength [nm]

Fig. 5.65. Interference spectrum in reflection for modulation of reflected intensity at the interfaces of a thin film with wavelength of measurement. The shift of the extremum by changes of optical thickness in the film is demonstrated.

at the wavelength A, where d represents a phase shift for reflection at the interface to the medium of larger n and A = 2n^d is the geometric path difference, given by the "optical" thickness of the film. Variation of the wavelength of measurement results in a cosine modulation around an average value (/, + /j) by an amplitude B, which is called an interference spectrum given by Fig. 5.65. For convenience only perpendicular irradiation is considered here [199,200]. In the case of absorption of the film, the interfering intensities in Fig. 5.64 and used in eq. (5.169) are given by

/,=/„/? /, = /o(l - R)Ril - /f)exp(^-^A:,rfj with

(5.170)

Application of Kinetic Analysis to Photoreactions

460

Ch.5

The change in the interference spectrum can be used to monitor photochemical processes taking place in solid matter, which influence the complex refractive index n, =nt

(5.171)

• / / : ,

5.6.5.1 Photoresist material A typical example of the application of eq. (5.169) is the photoreactive process taking place in a photoresist. In general this process is monitored by the change in the colour of an added indicator dye. However, this procedure is just a qualitative one. Application of Kramer-Kronigs relationship [5] and using the equations as well as the experimental set-up discussed in Section 4.4.1.2 and in this section, one can calculate the absorption of non- and irradiated photoresists and some intermediate spectra, using the change in interference spectra according to Figs. 5.66 and 5.67 [101,201]. Photoresist material contains a large number of components either responsible for the photoprocess or to influence the substrate fltting best for lamination, stabilising the reaction products and indicating the progress of the photoresist process. Therefore a kinetic treatment can only be used to

1.00 -r

600

650 wavelength [nm]

Fig. 5.66. Interference spectra of a 8 nm thick photoresist material placed on a Si-wafer: (1) before, (2) after irradiation.

Combination of Methods

400

450

500

550

461

600

650 wavelength [nm] Fig. 5.67. Photoresist reaction: calculated absorbances of a resist film taking the interference spectra given in Fig. 5.66.

determine optimal processing times. The complex reactions scheme does not allow the determination of quantum yields, not even of turnover rates. However, model reactions can demonstrate that the interferometric measurement in combination with application of the Kramer-Kronig relationship [5] allows calculation of the absorbance-time curves and the determination of the partial photochemical quantum yield. Thus, the method makes it possibile to monitor photoreactions in a solid matrix. As an application the quantum yield of the photoreaction of a 7,7a-dihydrobenzofuran derivative to the (Z)-fulgide is obtained as ip^E =0.03 [201]. 5.6.5.2 Photopolymerisation Interference spectra can be used to follow the photopolymerisation process [44]. Its photokinetic examinations are troublesome [202,203], since, besides the oligomers, some other components exist as inhibitors in the technical processes. In addition, the absorbances of these polymers do not exhibit characteristic spectra. For this reason, techniques other than UVA^isspectroscopy such as FT-IR-spectroscopy are preferred [204] to monitor the progress of the photopolymerisation. Recently the photopolymerisation of polysiloxanes has gained interest, because photo cross-linking allows the synthesis of indicator-free polymers

462

Application of Kinetic Analysis to Photoreactions

Ch. 5

with defined characteristics [205-207]. In principle these examinations require approaches as given in Section 3.4. Qualitatively the process of photopolymerisation of thin films can be monitored by interference spectroscopy [208,209]. Partial photochemical quantum yields of the photopolymerisation cannot be determined. However, in the next section some results are given for the determination of such quantum yields of reactants undergoing photoreaction in viscous matrices. 5.7 PHOTOREACTIONS IN VISCOUS MATERIAL In Section 3.4 in eqs. (3.101)-(3.123) the relationships for photoreactions taking place in viscous media have been derived and examples of reaction types are given. Furthermore this special problem of "unstirred" solution is treated in the literature [205,206]. In the following these equations are used to determine the parameters of the photoisomerisation of azobenzene in a polysiloxane block of distyrylpyrazine in a microcystalline layer, and to observe the photoreaction of a fulgide liquid crystal adherent to a polymeric backbone. As a first approximation the problem of diffusion is excluded. 5.7.1 Microcrystalline systems In a comparison with the photoreaction of 2,5-distyrylpyrazine (DSP) in solution, which has been evaluated by formalisms given in Section 4.5.3 applying a combination of absorbance and HPLC chromatography measurements, thin layers of DSP (an average of 50 nm thickness) have been examined to study the photoreactions of a variety of m-, p-, o-distyryl compounds. In the case of DSP, the reaction depends on the wavelength of irradiation (436 or 365 nm) forming dimers or oligomers [194,211]. For irradiation at 436 nm, even in the viscous layer absorbance diagrams are linear and the reaction spectrum exhibits an isosbestic point (see Fig. 5.68). Since the layer is extremely thin, Bernoulli's differential equations (see Section 3.4.1) reduce to the normal photokinetic equations (see eq. (5.107)). The boundary conditions of constant /abs,A ^s well as constant a{z,t) within the z-direction of the measurement beam can be used as a first approximation. The only problem is the value of E\d (absorption coefficient of the layer), since d is not really known. The product, however, can be estimated using

Photoreactions in Viscous Material

Ch. 5

200

250

300

350

400

463

450 500 550 wavelength [nm]

Fig. 5.68. 2,5-Distyrylpyrazine as a thin layer coated on quartz substrate: reaction spectrum for the photoreaction by irradiation at 436 nm.

calibration measurements in a conventional spectrometer. The method of formal kinetics can be applied. Fig. 5.69 gives the result of a parameter estimation, which yields a photochemical quantum yield of (p^ with an uncertainty less than ±10%. Because of the close neighbourhood of DSP in the layer (high concentration), the photochemical quantum yield is larger than in solution by one order of magnitude for the formation of dimer [193]. Its absorption peaks are situated at lower wavelength, since the dimerisation reduces the mesomeric system. Thus, at long irradiation wavelengths no further photochemistry occurs [194,211]. 5 J.2 Chromophores in polymers 5J,2 J Azobenzene in siloxanes For the observation of the photoreaction of azobenzene in siloxanes two arrangements are possible. First, the irradiation and measurement beam can pass in a parallel direction [153], the siloxane block with azobenzene embedded. Second, the direction of these can be perpendicular to each other [212]. In the case of the parallel arrangement, the concentration along the axis of irradiation is averaged, which results in summarised information on the photoreaction taking place in all the volume elements. This would be

Application of Kinetic Analysis to Photoreactions

464

Ch.5

AE -0.005 -

-0.01

-0.015

-0.02

-0.015

-0.01

-0.05

integral

Fig. 5.69. Computer evaluation of the photoreaction of 2,5-distyrylpyrazine in a thin layer, irradiated at 436 nm.

advantageous compared to a perpendicular observation, which just monitors a small part of the reaction volume. However, parallel observation cannot obtain information about the change in the photokinetic factor. Its size depends on the volume element considered at a distance from the front face for the irradiation, since according to Lambert-Beer's law the effective irradiation intensity decreases passing the silicon block. Accordingly for both the arrangements the rates have to be written in the form of partial differential equations, in which the concentrations a and b depend on time as well as on space: ^ ^ ^ ^ = -/o/?,a(z,Oexp(|-Jj2.303e;aU,0 + 23Q2>e'^b{xj)]cbX +/o/?2&U,Oexp|-J'[2.303£A«(A:,0 -f 23Q3e'^b{x,t)dx]^

(5.172) with

Ch. 5

Photoreactions in Viscous Material

dt

465

dt

In these equations, the dependence of /g^s on the volume elements is taken into account. Since in the perpendicular arrangement a scanner allows one to determine absorbance as a function of the local element, application of good local resolution supplies the necessary data for the calculation. It requires many measurements at each irradiation time. A prerequisite for the evaluation mentioned is knowledge about the reaction mechanism. Linear absorbance diagrams proved the photoisomerisation taking place as in solutions. However, the siloxane matrix has to be fresh. Different types of siloxanes were tested, some photochemically polymerised, others fabricated by a catalyst induced process. In the latter case the Ptcatalyst must not overcome a concentration limit otherwise it influences the azobenzene photoreaction. Approximate evaluations at low absorption (assuming a irradiation intensity independent of the volume element) do not offer appropriate results because of measurement problems. Therefore a transformation of the "time" scale has been used, discussed in Section 5.7.3. 5.7.2.2 Dihydroindolizines in siloxanes Dihydroindolizines can be embbeded in siloxanes out of an ether solution during the process of polymerisation. Absorbance diagrams allow a qualitative comparison with those obtained in solution. In principle the quantum yield of the photoreaction is reduced for a large number of derivatives. The 1,2-di-cyano-4-methyl-aceto derivative no longer exhibits a backward thermal reaction from the betaine because of steric hindrance. Even though irradiation was chosen co-linear with the measurement direction and thus diffusion is not involved, the combinations of differential equations according to Section 3.4 cause numerical problems at evaluation. Therefore only approximate calculations can be done simplifying the mechanism to estimate reaction constants [153]. 5.7.3 Photokinetics in viscous media 5.7. i. 1 Azobenzene As derived in Chapter 3.4 the photokinetic factor depends on the volume

466

Application of Kinetic Analysis to Photoreactions

Ch. 5

element as the concentrations of the reactants depend on time and space, given as a general solution for a single step uniform photoreaction by eq. (3.102). With the boundary conditions, these equations can be solved calculating the functions H{t) and G{z) stepwise (see Section 3.4.1). In Section 3.4.3.1 possibilities of an evaluation have been given for an isomerisation caused by irradiation into an isosbestic point, thus allowing a relatively simple relationship to be set up which becomes more complex if integrals cannot be solved in a closed form as demonstrated in Section 3.4.3.2 by eqs. (3.116)~(3.123). Using those considerations for eq. (5.97) and using the transformation to a pseudo-time 0 once more, one finds [210] 0(jc,O = /olOOoJ'expj-j'[2.303e;^(;c,0 + 230'ie'j^b{x,t)]dx[dt (5.173) The concentrations dependent on time and location in the polymer are given by partial differential equations and using nomencalture introduced in eq. (5.131):

The solutions are a ( e ) = - ^ ( / J j + /?, exp(-/?e)) biG) = ^^R0.

- exp(--R0))

which are comparable to thermal reactions. As derived in the literature by consecutive differentiation, one obtains with respect to time first [212-214] — = /o lOOOexpf-J'cj + C2 exp(-;?0))d[xl and then with respect to volume element 5^0 30 — - = -(c, +C2exp(-^0))— dtdz dt

(5.175)

Ch. 5

Photoreactions in Viscous Material

467

with c, = - ^ ( 2 . 3 0 3 e ; / ? 2 +2.303£^/?,),

K

One can integrate eq. (5.175) with respect to time t and can determine the integration constant/(z) at initial conditions for time r = 0, where no gradient of concentration is given: boundary conditions:

'— = 0

and

©(z,0) = 0

dz thus:

— = - c , 0 + ^ e x p ( - / ? e ) + f(z). dz R

Integration within the limits 0 and z gives a time-dependent constant g(t), which can be determined at initial condition z = 0:

f Jo

de • = -z + git)

c,0 + ^(l-exp(/?0)) R 'olOOOf

^ 0

c,0 + ^(l-exp(/?e)) R

de

-I c,0 + ^ ( l - e x p ( / ? e ) ) + z = 0 R

This integral cannot be solved explicitly. A Newton-Raphson iteration is used first to determine 6 values, which are used to calculate the concentrations of a{6) and b{6) according to eq. (5.174). In Fig. 5.70 the result of such an examination is given. One finds a systematic deviation, which can be explained by a diffusion process actually happening for cis- and transazobenzene molecules.

Application of Kinetic Analysis to Photoreactions

468 l.I JO

1.0

Xi

0.9

0.8

1

'

1

'

"

1

'

1

1

Ch.5 1

experimental data with diffusion without diffusion

'•{

0.7 0.6

0.5 1

1

11

50

1

1

100

1

'

1

150

200 time [s]

Fig. 5.70. Azobenzene in polysiloxane: experimental data and simulated curves versus time and the residuals. The simulation was calculated with and without diffusion.

As soon as diffusion has to be taken into account, the equations derived above can no longer be used. Diffusion causes a time-dependent gradient and the second Pick's law has to be included in eq. (5.172) by an additional term:

A

dz'

and

A

dz'

Diffusion and photoprocesses can be separated at irradiation intensities large enough and irradiation sections short enough to be able to neglect diffusion during irradiation times. This requires numerical integration over all local elements within two dimensions (diffusion perpendicular to the incident irradiation isotropic). Furthermore all the reaction times have to be considered for the photodifferential equation [210,213,215]. Essential is the optimisation of the step width by which the local elements are taken for calculation. Too small elements cause numerical unsteadiness for the solution of the dif-

Photoreactions in Viscous Material

Ch. 5

469

600 wavelength [nm] Fig. 5.71. Reaction spectrum of the photoreaction of the Z-isomer in the polymer liquid crystal at 100°C. Irradiation at 365 nm [196]. 0.4

0.3 V

V

0.2 H

O.I

D

•

V VV

^ O O O O

0.0

-0.1

500

1000

1500

2000 2500 irradiation time [s]

Fig. 5.72. Concentration-time diagram for the photoreaction of the Z-isomer in the polymer liquid crystal at 100°C; irradiation at 365 nm [196].

470

Application of Kinetic Analysis to Photoreactions

Ch. 5

fusion transport equation. On the other hand, too large steps average the concentrations too much. The improvement in the simulation by taking diffusion processes into account is given in Fig. 5.70 [215,216]. 5.7,3.2 Photokinetics in liquid crystals If the fulgide given in the reaction scheme in Section 5.6.3 is connected to an oligomere backbone and mesomeric groups, liquid crystals are formed which exhibit photoreactivity. Application of photokinetic equations and methods given in Sections 5.6.3, 5.7.2.1 as well as 5.7.3.1 allows an evaluation even above the glass temperature. According to eq. (5.172) the system of differential equations

- s'Ee,iPEcf,F(ED + e'cC,iPcEliF{E^ + ^x T T OX

^

at

= -e',e,
was used. The indices Z, E and C symbolise the Z-isomer, the £-isomer and the ring closure to DHBF. The layer of the liquid crystal was intersected along the x- (radiation) axis in n "sub"-layers with index /. In an iterative process, according to the equations derived in the previous section, concentration-time curves are simulated for various assumptions of diffusion constants D using the experimental data. The reaction spectrum is given in Fig. 5.71. TABLE 5.9 Approximate partial photochemical quantum yields for the photoreaction of theftilgidein the polymer liquid crystal at 1(X)°C; irradiation at 365 nm [196] Quantum yields

AtKxrc

^ZE

^EZ

^EC

0.23

0.29

0.37

Ch. 5

Concluding Remarks

471

Since the determination of absorption coefficients is in only 10-23 /xm thick layers, only relative concentrations can be determined, given in Fig. 5.72. Accordingly the partial photochemical quantum yields can only be approximated. Their values in Table 5.9 demonstrate that the absorption coefficients are determined to be too small. However, their values relatively fit to the data obtained in stirred solutions. These results prove the ability of the formalism presented in this book to treat photokinetic data even under extreme conditions. 5.8 CONCLUDING REMARKS After derivation of the principles of kinetic examinations and especially the fundamentals of photokinetics in this final chapter a large number of examples have been discussed based on the equations derived in Chapters 2 and 3. These examples cover a wide field of types of photochemical reactions that take place in various applications. By use of different types of equipment, it was demonstrated how relevant data can be obtained during the reaction. This knowledge was applied to calculate reaction constants as partial photochemical quantum yields or at least the data for a turn over, if the spectroscopic or other characteristics of the compounds involved in the reaction are not known in detail. All these applications only allow the demonstration of strategies that supply powerful tools for quantification of reaction constants. Nevertheless, the interested reader has to set up the specific equations governing his experiments. The outline can only provide helpful information and demonstrate a common approach by means of the given examples. Additional information is provided in the Appendix. Using the various sections of the book and the examples of the azobenzene and of the transstilbene photoreaction, the process of photokinetic analysis is demonstrated step by step. References to useful software and programs are provided in many literature citations, therefore these are not included in the Appendix. However, it proved helpful to use spreadsheets for simple evaluations. For this reason a Macro for 'formal integration' is included. Its use is demonstrated by means of a simple reaction. The intention is to update the methods of evaluation by extending the list of examples from time to time. These additional examples for photokinetic treatment of reactions as well as potential program listings can be found on the homepage of the author at: http://barolo.ipc.uni-tuebingen.de/photokinetics

Chapter 6

Appendix

6.1 REACTION SCHEME Example 23: Complex reaction scheme A further example in Section 2.1.1.2 with three steps of reaction jc,:

2A-^B

x^\

B + C-^D + E

^3:

D + A->F + G

results in the reaction scheme

x^ Xl 3

A -2

B 1

0 -1

-1 0

C 0 -1

D 0 I

0

-1

E F G 0 0 0 1 0 0 0 1 1

x^ X2 X"!

The transposed matrices of the stoichiometric coefficients v°' is derived from this reaction scheme as f-2 v°' =

0 \^-i

1 0 -1-1 0 0

0 0 0 OA 1 1 0 0 -1 0 1 l y

Using eq. (2.4) one finds

Appendix

474

/ -2

0

- n/^v

1 Ac 0 = lAd = 0 Ae 0 0 A/ ^A^> V 0

-1 -1 1 1 0

0 0 \^3j -1 0 1

(Aa^

lAb

Ar=v'*r

Ch.6

0

^

1

According to the advice given in Section 2.1.1.1 one can easily find the changes in concentrations like Aa = -2A:, -x^, Ad=^X2"'X^,

Afc = jc, - jCj, A^ = ;c2,

Ac = —jCj,

Af^Ag^x^^

Just three concentrations are necessary to observe the progress of the reactions (Aa, Ab, Ad), By insertion, the three degrees of advancement can be calculated as Aa = -2JCI - JC3

jc, =l/3(-Aa + Aft + Arf)

Aft = X, - X2

X2=l/3(-Aa~2Afc + Arf)

Arf = JC2 - JC3

jC3 = l/3(-Aa-2Afe-2Arf)

which results in matrix notation /^v ^ Ai yX^j

The changes in concentration amount to Ac = -^2 = Ae=

1/3 Aa + 2l3Ab-1/3 Ad

;c2 = -1/3 Aa - 2/3 Afc +1/3Arf = -Ac

A/ = Ag

=-y3Aa-2f3Ab-2/3Ad

Ch. 6

Reaction Scheme

475

The more formal approach is given using eqs. (2.4) to (2.8). In principle the matrix (-2 1 V

=

0 -1 -1 1 1 0 0

0 0 0 0 lO

-n 0 0 -1 0 1

1J

can be reduced, since only the reactants A, B, and D have to be monitored:

V

=

f-2

0

-0

1 0

-1 1

0 -1

Thus one finds according to eq. (2.8) with the inverted matrix, which can be easily done using a commercial mathematical tool program package like MAPLE, using the following program lines: > with(linalg): > A:= arrayC [[-2,0,-l],[l,-l,0],[0,l,-l]]); -2 A:= I 0

0 -1 1

> inverse(A);

- ^ Vi

-1 0 -1

Appendix

476

Ch.6

Since no linear dependencies between the steps of the reaction exist, the symbol (*) is unnecessary. The new matrix p can be calculated according to the definition given in Section 2.1.1.

f-K K K ( ^ ° 0 ^ \-'A -% % 0 1 0 % -'A [-% -% %] 'A 0 0 1 -% -% K -y, -% -%\ ij [-'A -% -%\

(-2 0 -1> 1 -1 0 0 - 1 0 1-1 v'v-'=p = 0 0 1 0j 0 0 1

^0 0

This matrix can be inserted into eq. (2.8) and one obtains ( 1 ^b Ac Ad Ae A/

0

0 1

0 ^(Aa^ 0 Ab

A A -A 0

0

yAdj

1

-A -A A -A -A -A V-A -A -A)

finally the same relationships as given above for the different changes in concentration: Ac = -X2 = Ae =

l/3Aa + 2/3 A& - 1/3A^

^2 = -1/3 Aa - 2/3 AZ> + l/3Af/ = -Ac

A/ = Ag = -1/3 Aa - 2/3 A6 - 2/3Arf

Ch. 6

Simple Photochemical Reactions

477

6.2 SIMPLE PHOTOCHEMICAL REACTIONS 6.2.7 Photoreaction via the triplet state Example 2,6: Simple photoisomerisation In Section 2.1.3.3 the photoreaction of an isomerisation from the singlet state for A—!2^B was demonstrated. Here a pathway via the triplet state is considered. It is given by the following scheme A + hv-

xs

A -1 1 1 0 1

<

0

-» A'

A'

- ^ A + Av

A'

2t-^A

A'

*i_ •^A"

x; Xj

A"

- ^ A

A"

•^3

<

A' 1 -1 -1 -1 0 0

A" 0 0 0 1 -1 -1

B 1 xi 0 JA 0 ^JQ' 0 k^d 0 k^d 1 k^a" 1 k^a"

According to the procedure explained in Section 2.1.1.1 rows and columns are filled. All the excited molecules are supposed to form A' first. Therefore both steps A + hv

>A* and

A*

>A'

can be combined to give the single step A + /iv-

^A',

Therefore as in the case of the singlet pathway, one partial reaction can be omitted in the scheme. Application of the Bodenstein hypothesis to the intermediates A' and A" gives

Appendix

478

Ch.6

a = -/A+(/:2+*3+*5K'» b = k^a'\ Using

the following relationships are obtained: k^a'

a =-

*4/A

a =

Thus by insertion fe =

^4^6^A

(/:2+/:3+^4)(/:5+A:5)'

and the quantum yield depends on other photophysical processes than in the case of a pathway via the singlet state: ^4^6

^ =

(/:2+/^3+*4)(*5+*6)

6.2,2 Complex uniform photoreactions Example 2.10 A photochemical with a superimposed backward thermal reaction is the most simple type possible. The following reaction scheme represents the mechanism (see Section 2.1.4.3):

A + Zzv->A

->B

A B * -1 1 ^1 1 -1

•*
Kb

Simple Photochemical Reactions

Ch. 6

479

A forms by absorption of light reactant B, which reacts backward to A in a slow dark reaction. The next scheme gives the elementary reactions A >A'

A + hv

JC,

A'

>A

X2

A'

>K'

^3

A"

>A

^4

A"

>B

B

Xs ^6

>A

-1 1

A' 1 -1

0 1 0 1

-1 0 0 0

A" B 0 0 0 0 1 0 -1 0 1 -I 0 -1

-^* /A Atjfl'

k^a' k,a" k,a"

Kb

As in Section 2.1.3.3 explained by use of the Bodenstein hypothesis a* = a' = 0 can be assumed, therefore one finds /.

a =• k^ + ^5

These equations can be used to calculate a and b as explained in Example 2.6 and introduced in Example 2.4 a = ~/A + k2a' ^-k^d '-\-k^b b--k^d'-k^b and ~a =fe=

^3^5 ^A

-hb^^xlK-Kb

This reaction turns out to be uniform since during the reaction

480

Appendix

Ch.6

a+b-ttn is permanently valid. Example 2.11 In case of a reversible photoisomerisation without a thermal backreaction one can define A to be the trans and B the cis form A Xx - 1 1 Xl

A—!21-^B B—!^A

B 1 -1

Xk

^f/A

^\h

Because of the photochemical back-reaction, a photostationary state is reached. If A " = B" = T is the common intermediate the scheme will be

A + Ziv

>A'

A'

^A

A'

>T

B + hv

>B'

B'

>B

B'

>T

T

>A

T

>B

A x{ - 1 1 x\ 0 X', A 0 A 0 A 0 x'n 1 A 0

A' 1 -1 -1 0 0 0 0 0

T 0 0 1 0 0 1 -1 -1

B

B'

0 0 0 -1 1 0 0 1

0 0 0 1 -1 -1 0 0

A /A

k^a' k^a'

h k^V

Kb' k-jt

kgt

Therefore with the usual assumptions, the concentrations of the intermediates can be caculated

a'=-A. k2 +K3

b'=-

/ p

k,a' + kj}' .^'^y

ks+K

and the change in the concentration of A and B is given by

Simple Photochemical Reactions

Ch. 6

d = ~ / ^ 4- k2a' + A:^^ = (*2+*3)(*7+^8) —

481

(*5+*6)(*7+*8)

s,^^

= -K/A+^2/B

and

In consequence the partial photochemical quantum yields are represented by

^r=-(A:2+*3)(*7+*8)' 3 8

^6^7

^ B = .

"'

(*5+*6)(*7+*8)

because of the law of conservation of mass fl + Z? = AQ .

During the total reaction this photoreaction will be uniform. 623 Complex non-uniform photoreactions A typical example is a photoreaction with a superimposed thermal reaction (see Section 2.1.4.4), in which the reaction scheme is given as A—^B

A -1 * 1 ^2 * 0 •

B—5^A B—^C

B 1 -1 -1

C 0
It is assumed that intermediates have very short lifetimes and the Bodenstein hypothesis is valid. The scheme reduces because of the linear dependency of the first two steps according to Appendix 6.3.1 to A x^ - 1

B 1

0

-1

2

C Xk 0
* Xt ^^^ .^1 ^2 =

X

Appendix

482

Ch.6

Stoichiometric summation of the degree of advancement over the column of the reactant gives

Zi = i , - ^ 2 =•*!* "^2

--^a =^l^^A - ^ 2 ^ B - " * 3 ^

C -- Xy "~ .^3 ^ K-yU»

6.3 SIMPLE THERMAL REACTIONS 6.5.7 Reactions including steps of back-reactions Example 2.7: Consecutive reaction: linear dependency For the consecutive reaction with superimposed back-reaction A;f*B->C the following reaction scheme is valid:

A->B B->A B-^C

• x^* ^3

A

B

-I 1 0

1 -1 -1

C 0 kiO 0 kjb 1 k,b

This scheme represents the three partial reaction steps. However, the first two rows are proportional to each other (stoichiometric coefficients can be transferred into each other by multiplication by a factor (-1)). For this reason JC,* and x^ are linearly dependent. Therefore the rank of the original matrix v* of the stoichiometric coefficients reduces to two columns. This is valid for all the reactants A, in the first two linear dependent steps 1 and 2: V,. = —v^-

For this reason the relationship between concentrations and the degrees

Simple Thermal Reactions

Ch.6

483

of advancement is reduced from (stoichiometric summation within one column)

to

Only the difference, taken from the degrees of advancement jCj* and x^, arises in all the equations. The x^ itself neither can be measured nor their dependencies on time be calculated independently of each other. For this reason it makes sense to define the following new linear independent degrees of advancement •

•

Xt ^^ X% ~~ Xy Xy

• 5^ X'l

In consequence the reaction scheme reduces to (the new element in the last column is obtained using both the equations above)

^1 2

A -1 0

B 1 -1

C Xk 0 kfO - k^b 1 kjb

As a result the new time dependence of the degree of advancement jc, becomes

according to definition. Example 2.8 A rather complex reaction scheme is chosen as the next example, which exhibits a cycle of reversible reactions.

Appendix

484

B->A B-^C C-^B C-»A A-»C

A B * -1 1 1 -1 Xl * 0 -1 X3 * 0 1

X4• X,

* Xe

1 -1

0 0

C 0 0 1 -1 -1 1

Ch.6

. •

Xk

k^a k^b kjb

k^c k^c k^a

Considering the linear dependencies, the reduced matrix representation is given by

A«=>B BoC C<=>A

9

xr9 X2 0

xf

A -1 0 1

•9

B 1 -I

C 0 1

k^a — kjb

0

-1

k^c - k^a

k^b - k^c

To demonstrate the reduction, special symbols are used in the following. According to the explanation given above, the six linear dependent steps are first reduced to three steps: Xy

SX,

• -^2'

X'y

— X"!

' •^4»

X-x — Xs — X^,,

But this scheme can be reduced further, since there is a linear dependence according to

by comparing all three columns. For this reason the mechanism contains just two linear independent partial steps of the reaction. The rank s of the stoichiometric matrix v becomes 2. This reduction in the scheme can be done in this example by cancellation of the third row. Its elements have to be subtracted from the first two elements of the last column:

Simple Thermal Reactions

Ch. 6 X-i

— fCeC

K>f.G> •

The two linear independent degrees of advancement are given by Xt = Xt ""• x-y anQ Xy = Xy ""• jc-j and the following representation

X, Jfj

A B C Xk -1 1 0 k^a — ^2^ ~ ^5^ + k^a 0 -1 1 kjb — k^c - k^c + k^a

is found. The reduced stoichiometric matrix is symbolised by v. In consequence the matrix P can be substituted by P = v°-v-'.

Accordingly the reduced degree of advancement x has to be taken. One finds according to eqs. (2.4) and (2.9) Aa°=v°'x*,

Aa°=v°x

and

Aa° = p A a .

Using the equations and the procedure demonstrated in Example 2.2 the following steps of calculus have to be done: ^Aa^ r-1

Ab yAcj

1

,0 1

Aa = -;c,,

fAa\ Ab = v v vAcy

0 r.. \ -1 \^2J

Ab = Xf-X2,

Aa^ Ab,

Ac = X2

-1 0^ -1 0 VAa 1 -1 -1 -l)[Ab 0 1

485

Appendix

486

(^a\ (I AZ? = 0 {^c) [-1

Ch.6

0^ Aa 1 Ab -1

Ac = -Aa-Ab. This final equation correlates with the three equations above. 6.3,2 Parallel reactions Example 2.9 The rank of the matrix v is equal to the number of the degrees of advancement in the following mechanisms. Therefore one would expect that the conditions of equal number of r and rank of the matrix stated at the beginning of Section 2.1.4 is sufficient for the optimal reduced matrix. .* A B C D * -1 -1 1 0 k^ab * -1 -1 0 1 kjab X2

AH-B->C

A + B->D

One realises that in both the reactions the same dependence on the concentrations exists. If the concentrations a and b are chosen to be independent, eq. (2.6) will become explicitly either

-r\(-^ - XVA-]

rAa\ (-\ Ab Ac

-1

-1

1

0

yAd)

[o

vo 11 XAd)

ij

or

fAa\ (\ Ab Ac

0^i^

1 0 \Ad -1 -1

[Ad] [0

ij

By this means one obtains the non-trivial balance-equations: Ab = Aa

and Ac = -Aa - Ad.

Simple Thermal Reactions

Ch. 6

487

Both the degrees of advancement show a linear dependence for kinetical reasons. A comparison of the rates in the last column of the scheme makes it evident that CUi'y

fC'j

dx\

k^

Taking into account that the degrees of advancement are zero at r = 0 (a general rule in kinetics), integration of the above equation gives *

*

where % is defined as

The combination of these equations with eq. (2.4) results in Aa,. = v„.x; + v^,xl = (v^,. 4- xv2i)x, It turns out that this reaction system can be described by a single degree of advancement and the reaction scheme finds the following form:

A Jt,

B

c

D

Xk

-O + X) - 0 + X) 1 X kyob

The reduced scheme contains the same information as any extended one. Considering a as the independent concentration, the changes in the different concentrations are determined by

(^"] (-(i+x)] .Ab -a+x) 1

Ac

[Ad)

K

X

J

+ X))(Aa)

488

Appendix

Ch. 6

and the relationships between the concentrations can be calculated as A6 = Aa,

Ac =

^—Aa,

Arf =

—Aa.

A and B have the same stoichiometric coefficients in the reduced scheme. For C and D the equation above can be used to form the ratios: kx

I "^1 +"^2 1

Aa

*. J

/

Ad _k2 Aa kf

I Kt T Ky

k,

{k,+k,) I

K'2

)

{k^-hk^)

Even though some arbitrariness exists in the defmition of linear independent degrees of advancement, the measured value is not influenced at all. For example, the above given degree of advancement x can be substituted by v = ^(i + ;f), if all other stoichiometric coefficients are divided by (1 +;f) in consequence. The first definition x correlates to the overall equation ^ i ^ ^ ( A + B)-^C + ^ D , ^1

k^

the latter (jc*') to A H - B - > — L c + -^2_j) K^ ~r K2

Kt "1 Ky

6.4 EXAMPLES FOR THE DETERMINATION OF CONCENTRATIONS 6.4.1 Second order reactions Two possible cases have to be distinguished for second order reactions (see Section 2.1.7.3). In the first initially only a single compound is present:

Examples for the Determination of Concentrations

Ch. 6

489

B -2

2>4-»B

ka^

In this case the rate is determined by jc =fo= — = ka^. 2 The dimension of the rate constant k is [1 mol~* S"']. By integration (jc = 0, a = a„, and t = 0) one obtains either 1 1 ^ ,

1

= 2Jtr

a

=»

Gn

X^-laAt

- =

a

^—

an

or by rearrangement (2.24)

a = -1 + la^kt

With bQ=0 one finds by substitution into the above b and integration

,.,^_V^

(1 + la^kty

The half-time is reciprocally proportional to the initial concentration (4/2=^o/2) la^k Frequently the concentrations are related to the initial concentration to obtain relative and comparable values. These are called general coordinates. Transformation to these coordinates yields a = —, an

i8 = — ,

^ = —, an

x^lajiU

Appendix

490

Ch.6

and in consequence a =

21 + r

1 1-fr

1

^

1

1

On the other hand a reaction with two different initial components as

A + B-^C

X

A -1

B -1

C JC 1 kab

takes place. The differential equation of the rate has to be solved by a seperation of the variables and by formation of partial fractions. JC = - a = - ^ = c = kab = k{aQ - x){bQ - jc). Integration at / = 0, jc = 0, and a = a^ yields In-rr— = In—- = -(6o - a^Jki [bQ - x)aQ baQ or using exponential notation: X = C

a^b^jl ~exp(~(fcQ ~ ao)kt)) bo - a^expi-ibQ - aQ)kt)

^ UQ (bQ - UQ ) exp(-(fco - gp )kt)

fc =

(2.25)

feo~aoexp(-(&o~ao)^0'

It is common use to define the half-time with respect to the component with the lower concentration. Thus, if a^ < b^ tm^

1 bo-^o

-In 2^^ ^oy

Ch. 6

Examples for the Determination of Concentrations

491

By transformation to the general coordinates a

^

b

c

^

y = —,

X

M = -^,

? = —,

r = a^/rr

an

one obtains ^ ^ ^ M(l-expKM~l)r)) M - exp(-(M - l)r) ' i8 =

u(u -1) u - exp(-(M - l)r)

^ ^ (M-l)expKM-l)r) w - exp(-(M - l)r)

^1/2 =

1

, x^ 7ln(2

^ ),

da -u. 6,4.2 At the end of the reaction In the following, the equations for the concentrations at the end of the reaction are derived for a variety of mechanisms in addition to the text in Section 2.1.8. Example 2A3: Bimolecular reaction (different reactants) For

A + B"->C

X

A -1

B -1

C Xk 1 kab

the use of the schematic representation gives according to Example 2.12 the following relationships by taking a^ - a^Q = Aa, and combining it with the stoichiometric coefficient and the degree of advancement of the considered reaction (row in the reaction scheme): a = aQ — X,

b = bQ- X,

and at the end of the reaction

c=x ,

492

Appendix

Ch. 6

Thus

Using the relationAa, = v,.;c and a^-^a^.x-^x^ whether we are looking at looking at a or 6 either

in depending on

or Coo = ^ c o =

^0

is obtained. To avoid negative concentrations the following restriction is physically necessary: Coo =JCoo=Min(ao,fco)-

Example 2.14: Consecutive reaction For the mechanism

A- • B B- C

JC, JCj

A -1 0

B 1 -1

C Xk 0 k^a 1 k^b

one finds a = —k^a^

b = k^a - ^j^*

c = k2b.

At the end of the reaction all starting material as well as the intermediate B have disappeared: a^^O

atid

b^^Oi

Examples for the Determination of Concentrations

Ch. 6

493

therefore with c^ = AQ and considering only the initial and end products one has to use the first and the third column of the scheme above

Ac = c^ - Co = do - 0 = X2«., a^ = ^2oo or •^loo — •^2o« ~~ ^oo ~" ^ 0 '

Example 2.15: Complex consecutive reaction For the mechanism

B + C-^D

JC,

A -1

^2

0

B 1 -1

C 0 -1

D ^k 0 A:, a 1 k2bc

the rate law is given by d^-k^a,

b = k^a'-k2bc,

c = -d = '-k2bc.

Therefore

is valid. In consequence one finds at the end of the reaction either boo = ^loo"" -^loo = 0

or c«. = Co - X2^ = 0 ^ 2 o o — <^o •

Thus this example gives ^loo = «o

^^d

X200 = Min(a^, Co ).

Appendix

494

Ch. 6

Example 2.16: Complex consecutive reaction including back'reaction For the mechanism A + B-^C C-^A + B C + D-^E

X2

A -1 1

Xj

0

h

B -1 1 0

C 1 -1

D 0 0

E ^k 0 k^ab 0 he

-1

-1

1 k^cd

the rate law is given by a = b ==-k^ab-h k2C, c — k^ab- k2C - k^cd, -d = e = k^cd. In the case where d^ > MinCa^.^^^, the result is ^oo — -^loo "" - ^ a o o " " -^Soo"~ ^

and either

doo^dQ

or

-aQ>0

is obtained. In consequence the final concentration of compound E is given by ^oo =A:3^=Min(ao,feo). Taking the other assumption d^>Min(a^,b^\ one finds the final concentration of D to be zero:

Ch. 6

Consecutive Reactions with Superimposed Back-Reactions

495

In consequence the final concentration of E is determined by the initial concentration of D: •^300 =

^00 =

" 0 •

Therefore an equilibrium is obtained between the remaining amounts of reactants A, B and C according to c^ =

koodoo-

6.5 CONSECUTIVE REACTIONS WITH SUPERIMPOSED BACK-REACTIONS Using the definitions given in Table 2.2 the following equations derived in Section 2.2.1.1 are taken for the calculations in the next sections: Degrees of advancement for all steps of a reaction In matrix notation

^t = kj^aj. {k = 1,2,..., r)

Differential equations with absorption coefficients Non-reduced concentrations

v°*i* = ^^ = v°*Kna* s Kca*" a° = aj + pAa

And its time dependence

^« = KtCaJ + pAa) = ao + K^pAa

Including the defmitions

i j = K^aJ

^* = Kja"

K'C = VKS

Aa = a ^ = K^a

SQ

Reduced time dependence of degree of advancement Changes of selected concentrations Reduced with respect to mass balance

Aa = vx v i = ^ = v Kni = v Kft(a° + BAa)

And reactions steps

K^ = V K Q P

Changes in concentrations

^ = ^0 + K^^Aa ao = v KoSo

Appendix

496

Ch.6

x = v"^ao+v"'^KcVx x = xo + K-x Jacobi matrix

K = v"*KcV

Example 2.17 In addition to the derivation given in Section 22.1 A aspects of Section 2.2.1.2 are included here for the reaction scheme for the reaction A<->B->C A -1 * 1 X2 *

0

B 1 -I -1

• 4>

C 0 kiO 0 k^b 1 k^b

Using eq. (2.40)

we obtain K=

(-{k^+k^) k,

k

and its inverse K-' =

•^/'' "IT

to find a.

A Xx - 1 0 X2

B 1 -1

C Xk 0 k^a — k^b 1 k^b

Consecutive Reactions with Superimposed Back-Reactions

Ch. 6

497

Example 2.19 For the most simple consecutive reaction A -^ B -> C one finds for the steps given in detail in Example 2.17 the reaction scheme

JC,

A -1

B 1

X2

0

-1

C ^k 0 kfO 1 ^2^

r-1 o\ V

1 0

=

-1 V = 1 0

-1 Ij

X = X* = K„a° =

(-ki a = A:,

n.

fc.

0

V-*2

0

Ky

0 -k^

0

0^ -1 1;

OVa^ 0 b

0^ -1

" 1 V .<^J

with

(-k^

0

0\

0

itj

0,

K^ =

r-it, it,

0 -k^

^ 0

A:2

a = -i:,* = —A:,a(?)

-1

0

1

-1

v-' =

0

(a\ h\

k^ ojtc^

^a^ (-1 1

V=

fki '^o ~ '^o ~

-1

0

-1

-1

OYa'i 0 Oy

v-/

0

0

k2 0

498

Appendix

r-1 o\ P=

1

-1

0

1

-1 -1

^a^ 0

a=

^1

0 -1J=

^1 0

Ch. 6

0^

0

1

-1

-1

0^ 1 tA6

v^y

^"=^o+K;pAa =

'-*.«o' *,ao

^

+

-ife,Aa

^

/:, Aa - ^jA^

I 0 J ^^

k^ti^b

J

a = -^j^o - /:, Aa + k2^b b = ^j^o + /:, Aa - /rjAfc c = ^2 Aft /" ' A

Jt,

V^ay

0

x = Kor =

^a^ ON 6 it^ 0) ^<^> 0

i , = /:,a(0 i j = *2^(0

vx = a = V K„a = v K^ (al + pAa] with

-k^Aa \

Kc=vKoP = *i

0 y

Finally the rate matrix is obtained

r-i oYfe, 0 1,-1 -\ik, -k, and

-1

0

^-)k,

0 ^

-1

-1

V ^2

~*^2y

- -

Consecutive Reactions with Superimposed Back-Reactions

Ch.6

499

to find

0 ;

VM1-M2;

1^200;

\a^

Example 2.20 For the consecutive reaction A->B=?^ C one finds for the reaction scheme A -1 * 0 •^2

B 1 -1

0

1

f-l V

=

K;=

C 0 1 -1

k^b kjC

0 -1 1

0^ 1 -Ij

fkt

0

0^

0 {0

Jkj 0 0 k^

1 0

a =

Kj

0

B 1

X2

0

-1

-1

0^

1

-1

0

1

V

=

Kn =

fe,

r^*\

(L

0

y^ij

0 {0

k^ 0 ky)

X = KQ • a =

f-k.

Xl

A -1

k^a

0 -Jt AC2 AC')

0

-)t.37\^J

^k kfO fCyly ~~ IC-yC

0

0^fa^ 0 b

f-k,

OYa\ ^3

C 0 I

with

Kc =

0 "~fC'>

0

0 ^ AC-i

3 /

Appendix

500

(a'' f-l b = 1

0 -1

.0

1

<^J

V =

p=

oVx;] M,

iA

*.

0 -k.

*3

-V {_XjJ [o

h

-hj

1

1

*

X2

=

0^

r

i^j

(-.*-,)• - ( : ; - . ] -1 1 0

0 -1 -1 -1

0 -1

1

n a

Ch.6

b

0

c

0

1

l-i

-1

o\

0

1

-1

-1

c = - A a - A£> -*,Aa a°=ao+KJ;pAa =

^, Aa - ^jA^ — ^jAa — /^j Afe 0

\^

k2Ab + kyAa + kjAb

j

a = —^jflo ~ kiAa b = kfOo + (*, - kj)Aa -{k^ - k^)Ab c = kjAa + (ATJ + kj)Ab ^^ ^ x = Koa° = K^2j

(k^

0

0

0

*2

-ifcj

^2 = Jkj^CO - ^3^(0

Ch. 6

Consecutive Reactions with Superimposed Back-Reactions

501

^ = vKoa°=vKo(ao+PAa) -k^Aa (L - L )Aa - {k. + k. )Ab

-*i«o

with

Kc = vKoP =

r -k,

0

Finally the rate matrix is obtained

{-\

o y -*i

v-i

0

o^_

-k, *2

0 -(^2+*3)

and

r -V*.

0 1

K '= I

fCi yiCj

~r fC-j y

^

fCj "i rC-j i

to find

x=

and

0

2

1 "" v 2

3/

2

^loo

/r^a, 2"0 V ^ 2 "*• ^3 y

Example 2.21 For the complex reaction with two equilibria A ^F^ B :?^ C according to one obtains the reaction schemes and the matrices B

C

. •

1

0

k^a

0

-1 -1

0 1

0

1

-1

A * -1 •

2 X-y

* JC4

1

^2^

X,

-1

B 1

k^b

X2

0

-1

k^c

A

C ^k 0 A:,a - ^2^ 1 kjb — k^c

Appendix

502

V

r-1 o^

r-1 1 0 0^ = 1 -1 -1 1 0 0 1 -1 0

.2

0

h

0^ 0 0

1,0

0

kj

(k

K

• •

0

T^*

V

and

o

Kft =

=

1 0

^1

~1 1

"1

-^2

-*4J 0

0^ ^a^ 0 0

0

k^j v^v

0 0

-•O

X =Ko-a =

^0

-K

Ch. 6

*2

a =

0 with

0

-*, Kc =

fCj

^^^2 ' '^3)

0

^a^

r-1 1

V'-V

^4

-it.4 /

1 -1 0

0 -1 1

p'1

/ -k. 0> x'2 1 = I ^1 —

-1;

•^3

V 0

0 Ya^ v^2

A:3

'^3)

^4

- A : 4 v^y

Consecutive Reactions witli Superimposed Baclc-Reactions

Ch. 6 V=

-1 1

0 -1

.-I

-1 -1 -1 1

P= 1 0

=(::

-.)

O

0^

0 -1

0 -1

(I 0 + 0

0\ Aa 1 Ab

"0

a = \b =

.0,

1 -1

,-1 -ij

c = -Aa-Afe

f-k,a^ a'=aoH-K^pAa = 0

^

-k^Aa-hk2Ab ^ + (it, - k^ )Aa - (k2 + /:3 + A:4 )Ab k2Ab + ^4 Aa + ^4 Afe

^1^0 —ktAa-^-k-yAb '^1 a = Lan k^a^ + (/:, - k^)Aa - {k^ + ^3 + A:4)Afc

c = k^Aa-{'{k2 •^k^)Ab *,

x = Kor = V-^2,

0

-Jk2

0

Jt3 -A:4

^a^ ft

X,

=kya{t)-k2b{t)

X2 = A:3fc(0 - ^4^(0

a = vKor=vKo(a^+PAa) a b with

-"*i«o

fc, a^ j

'k.Aa-\'k^Ab v(^i - ^4 )A^ - (^2 + ^3 + ^4 )Afe

503

Appendix

504

Ch. 6

~ * i

Kc=vKoP =

Finally the rate matrix is obtained J-l [-1

OY -*, -ij^.(^k^+k2+k,)

*, Y-1 (k^+k,+k,)Xl

0 -1

'-(k,+k^) -(kj+k^) and AC^/C^

1V 3

K-' =

4 -^

^^2 4

1 >^ 3

4 •'

^2 4

k^ +k2 \^

k^ (k-^ + ^4 ) •*• ^2^4

^1 V^3 "*" ^4 ) "^ ^2^4 )

to find x=

^1^0 "I M^i+*2)-^l+*2-^2

0 J y k^x^-(k^-^k^)x2

and f

k^(k^+k^)aQ ^ k^(k^-hk^) + k2k^ kik^ciQ

[ ACj yK^ I K^ ) "T ^^2^4

Example 2.18 The method discussed is applied to the chemical mechanism of a cyclic reaction according t o A ^ B ^ C ^ A given in Example 2.8 with the reaction schemes

Consecutive Reactions with Superimposed Baclc-Reactions

Ch. 6

1

• •

A

B

c

-1

I

Xy

1

-1

0 0

k-^b

*

0

-1

1

kjb

Xt 4>

X"!

k^a

JC4

0

1

-1

k^c

*

1

0

-1

k^c

-1

0

1

k^a

•^5

xl

A

B

C

^i

-1

1

0

k^a — ^2^ - k^c + k^a

X2

0

-1

1

k^b — k^c — k^c + k^a

Xk

an the following matrices:

V

=

K* =

r-i

1

1

u

-1

-1

0

1

ffc.

0

0^

0

Ky

0

0

0 0

0

k.

0

0

k,

^,*^6

0

oj

0

0 1

1 -1^ 0 0

-1-1

and

V

0

l<

ij

KQ

/ r-1 0^ = 1 -1 0 1

-k,

=

-(*4+*5)

0 /'„•^

X =

KQ

\^2J

a =

ki Kg

0

K2

0 0

0

•a =

-ikt+k^)

505

0

0

k,

0

0

k.

V*6

0

k^ -ik2+k^) K3

^a^

v-y

ki

Ya\

k^

lb

—{k^+k^^Kcj

Appendix

506

Ch.6

with -(k^+k,)

k,

K^ =

-(k^+ks))

fa\

f-i

1 1

y'-j

-1 0

0 1

0

-1 1 -1

1 0

0

-1

1

a = -x; +xl+ xl - i ; = -(*, + k^)a{t) + k^m + kjciO b = Xi -xl-xl+xl=

kia(t) -(*2 + kjMO + k^c(,t)

c = i j - ^4 - i j + ig = k^a{t) + k^bit)-{k^ + k^)c{t)

v=

r-1

V

1 -1

0

P = 1 -1 -1 -1 0

^a^ a = b

-1 -1

0

1

=

n 0

-1

J

1 0 + 0 -1 vOy

0 -1

1 -1

0\rAa\ 1 -1

{Ab)

c = -Aa - Ab (-(k,+k,)aA ^° = ao+K^pAa =

-(iki + Jkj + itg )Aa+(jfej - *5 )Ab \ (ki-k^)Aa-(k2+kj+k^)Ab [ik^+k^+k^ )Aa + (ikj +^4+^5 )Ab)

Consecutive Reactions with Superimposed Baclc-Reactions

Ch.6

^ = ^j^o + (A:, - k^ )Aa - (A:2 + ^3 + ^4 )Afe

/^. ^

x = Koa^ =

(ki -\-k^

k2

k^

\-^2j

^a^ V^y

•^2 = ^6^(0 + ^ 3 ^ 0 - (^3 + k^)c{t)

^ = vKor=vKo(a^4-PAa)

with ^-(k^+k^-^k^)

k^-k^

Finally the rate matrix is obtained

(-1

-iX

*,-*4

~(ifc, + ^ 2 + ^ 6 ) ^3 "~ ^6

-(*2+*3+*4)Al *2~"*5

'\

~V^3 + ^4 "^ ^5 ))

and I

k'^ -\r k^ '\- k^

^2 "*'^5

1

itr

J

kr

K-' =

^3 "" ^6

V with

/:r

-ij

507

508

Appendix

Ch. 6

to find

x=

f{k^ 4-fcg)aQ "^ r~(/ci + /:2 "^ ^6 )-^i "^ (^2 "" ^5 )-^2 V

6

/

V ^3

^6 /-^l "" v^3 •*" ^4 •*" ^5 )-^2

and

Jtr [/:t/:3-h/:6(^3-^^2)K jfcr

V-^2oo

Example 2.22 An application of Table 2.6 is the simple consecutive reaction A —> B —> C. According to Section 222A it has the eigenvalues: r, = - / : ,

and

h ~ ""^2

looking at the elements on the diagonal of the K-matrix. Therefore one finds according to Table 2.6 for the concentrations of the three reactants the following relationships: For A: No compound A is present at the end of the reaction, therefore a^ = 0 . Furthermore taking just the first step of the reaction in the initial phase as a boundary condition one can use a^ = —k^a^. Both are introduced into the elements given in Table 2.6 for .y = 2:

^

r,-r2

-/:, + ^2

_^io"-n(^»o~^»oo)_ r2-r,

-/:2 + *,

= ^n

=0

Thus using eq. (2.57) and substituting the calculated elements one finds

Ch. 6

Consecutive Reactions with Superimposed Back-Reactions

509

a = aoexp(-/:,/). For B: Assuming the boundary conditions b^^^Q, b^-Q 60 = -^0 == ^1^0 ^"^ ^^" derive accordingly

"~fC|

fC|

I Ky

_ fc,ao + ^1 (0 - 0) _

I Ky

fcjao

Ky

and

"" fCi

__ k^a^

and obtain for the time-dependence of the intermediate, inserting the coefficients above into eq. (2.57) ka fc = —^(exp(~/:,0~exp(-A:20) • Ky

Kt

For C: Finally considering the concentration of compound C at certain conditions of time one can set CQ = 0, C^ = a^ and CQ = 0 and obtains for the elements p^f^

i\>y

Ky

Kit

"^ rCi

The result is according to eq. (2.57):

c = ao + -. ^^_l e x pexp(-fe,r) ( - ) f c , 0 + - '_° ^ expC-fe^f) Ky Kt Ky ^ K-t

Ky K,y

IKt

c= 1

1

c = 7 - ^ [ * 2 ( l - exp(-^,0) - *, (1 - exp(-*2r))] ^2

^I

Appendix

510

Ch.6

6.6 RELATIONSHIP BETWEEN QUANTUM YIELD AND PHOTOMECHANISM 6.6.1 Details for simple uniform photoreactions 6.6.1.1 Photoaddition Without desensitisation by an additional partner via singlet: A ^A'

A + Av A'

>A -»C

A' + B

A

-1 1

X',

0

<

A' 1 -1 -1

B

C

K

0 0 -1

0

/A

0 k^a' 1 k^a'b

via triplet

>A'

A + AvA'

>A

A'

>A"

A"

>A

A" + B-

•4C

A x[ - 1 x\ 1 A 0 A 1 A 0

A' A" B 0 1 0 -1 0 0 -1 1 0 0 -1 0 0 -1 -1

c

A

0 /A 0 k^a' 0 k^a' 0 k^a" 1 k^a"b

With desensitisation by an additional partner via singlet

A + hv A'

>A' >A

A' + B B' A' + C

»B' + A ^B >D

A' 1

B

B'

x{

A -1

0

0

A

1

-1

X^

1

-1

0 -1

0 1

A A

0 0

0 -1

1

-1

0

0

C 0

D

A

0

IA

0

0

k^a'

0 0 -1

0 0

k^a'b k^b'

1

k^a'c

Ch. 6

Relationship between Quantum Yield and PhotoMechanism

511

via triplet

A + hv

>A'

x;

A 1 A' 1 A"1 B -1 1 0 0 1 -1 0 0 0 -1 1 0

B" 0 0

D

0 0 0

0 0 0 0

k^a' k,a"

0

k^a"b

A'

>A

4

A'

>A"

X-i

A"

>A

<

1

0

-1

0

0 0

4 4

1

0

-1

-1

1

0 0

0

0

0

1

-1

0

0

0

-1

0

0

-1

A" + B B"

>B" + A >B

X',

A" + C

6.6.1.2

>D

Sensitisedphotoreactions

Example 3.8: Physically sensitisedphotoreaction The physically sensitised photoreaction B-

kv

^C

with

^A'

A + /iv A'

>A ->B' + A

A' + B B'

>B

B'

>C

A x[ - 1 A 1 A 1 A 0 A 0

A' B 1 0 -1 0 - 1 -1 0 1 0 0

B' 0 0 1 -1 -1

c

A

0 IK 0 k-jO.' 0 k^a'b 0 k^b' 1 ksb'

yields a = 0,

-^b-c-ip^Ip^,

kjk^b
{k^+k,b){k,+ksy

1 ^*

c

'A /Cj^'

0 Kb" l \ h,a"c

Appendix

512

Ch.6

If Bodenstein's hypothesis is applied to A' and B' and /TJ ^O 10* s"* and k-^ toto 10*0 10*0moh* moh*s~* s~*the thebb concentration concentratio has to amount to 10^ 1 mol"* at k^ least to supply a remarkable turnover. Example 39: Physical sensitisation using the triplet state Using the same mechanism with the exception of an energy transfer from the triplet state of B, one finds the modified scheme

A + hv

>A'

A'

>A

A'

>A"

A"

>A

A'' + B

>B''-f A

B''

>B

B"

>C

A x{ - 1 1 ^2 0 ^3 1 X\ x's 1 0 < 0 X'-,

A' A" B 1 0 0 -1 0 0 -1 1 0 0 -1 0 0 -1 -1 0 0 1 0 0 0

B" 0 0 0 0 1 -1 -1

C 0 0 0 0

K /A

k^a' k^a' k^a" k,a"b

0 0 1

Kb" k^b"

with the rate laws a = 0,

^br:zc^ip^l^,

Jkjfc

(p^ ^ Ky

I K'i KA

t ^s

k-, 6

7

ithin the quantum yield the factor k^b k,-¥k^b

b

ii+* *,

determines the minimal necessary concentration of B to allow energy transfer. Concentration b must reach the order of magnitude of k^/k^. If ^4= 10^ s"* and k^- 10*^ 1 mol"* s~*, the concentration of component B must become approximately 10~^ mol 1"*.

Relationship between Quantum Yield and PliotoMechanism

Ch. 6

513

Example 3.10: Physically sensitised photoreaction with another partner Another possible scheme for the physically sensitised reaction B + C-

hv

^D

takes into account a bimolecular step of reaction and is given by the following partial steps of reaction:

-»A'

A + kv A'

<

>A

X-i

^B' + A

A' + B B'

X<j

x;

>B

B' + C-

4

->D

A -1 1 1 0 0

A' B B' C 1 0 0 0 -1 0 0 0 -1 -1 1 0 0 0 -1 0 0 0 -1 -1

D K 0 /A 0 k^a' 0 k^a'b 0 k^b' 1 k^b'c

One finds
k^kAbc V3/V41 (k2 + k^b)[k^ 4- k^c)

A measurable turnover is obtained, if b-^

and *3

h ^5

Assuming the bimolecular rate constants to be diffusion controlled and having a value of approximately 10^^ 1 mol~^ s-^ and taking the average lifetime of A' and B' to be 10-^ s, then the minimal concentration has to be &-'C~10"^moll-'.

Appendix

514

Ch. 6

Example 3.11: Physically sensitised photoreaction with another partner and via triplet pathway In comparison to the previous example for this mechanism the energy is transferred via the triplet state according to the following scheme

>A'

A + Av

^;

A'

>A

A'

>A'

4 4

A''

>A

<

->B" + A

A" + B B"

<

>B

B" + C

A xf,

A -1

A' A" 1 0

B 0

B" 0

C 0

D 0

xl

1 0 1 1 0 0

-1 -1

0 1

0 0

0 0

0 0

0 0 0 0

-1 -1 0 0

0 -1 1

0 0 0 1 -1 -1

0 0 0 -1

k^a' k^a' k^a"

0 0 ksa"b 0 k,b" 1 k^b"c

0

IK

->D

Thus - c = ~fe = rf = y)^/^

and ,p- =

3 5 7

{k^+k^){k^^•kib){k^+k^c)

are obtained. Turnover becomes measurable, if b~^

and

c--^.

Therefore the minimal concentration has to be fe ~ c ~ 10"* mol r* for bimolecular rate constants of 10'° 1 mol-' s"' (the same as explained in Example 3.1) and for average lifetimes of A" and B" amounting to 10-4 s.

Ch. 6

Relationship between Quantum Yield and PhotoMechanism

515

6.6,2 Details for simple non-uniform photoreactions 6.6.2.1 Physically sensitised parallel photoreaction Example 3.20 B-

hv

D-

^c.

•^1

B -1

C 1

X2

0

0

D 0 -1

•^E

E 0

^^A

1

^2/A

^k

Overall scheme: A x{ - 1 1 ^2 X3 1 0 JC4 JCj 0 JCg 1 x^ 0 4 0

->A'

A + hv A'

>A ^B' + A

A' + BB'

>B

B'

>C

A' + D

>D' + A

D'

>D

D'

>E

A' B 1 0 -1 0 - 1 -1 1 0 0 0 -1 0 0 0 0 0

B' 0 0 1 -1 -1 0 0 0

c

0 0 0 0 1 0 0 -1 0 1 0 0

6.6.2.2 Physically sensitised-consecutive photoreaction Example 3.21

C—^D

B_^C,

Xl

B -1

C 1

D 0

H

0

-1

1
H


D 0 0 0 0

D' E 0 0 0 0 0 0 1 -1 -1

Xk

u

0 ^2^' 0 k^a'b 0 k^b' 0 k,b' 0 k^a'd 0 k^d' 1 hd'

S16

Appendix

Ch.6

The overall scheme: A + hv A'

>A' >A

A' + B

x; X2

»B' + A

^3'

B'

>B

x'.

B'

>C

^5

A' + C

>C' + A

C

>C

C

>D

A -1 1 1 0 0

*;

1 0

•^8

0

^6

A' B 1 0 -1 0 -1 -1 0 1

B' 0 0 1 -1

C 0 0 0 0

0 -1

0 0

-1 0

0 0

0 0

0 0

1 -1 1 0

c0 0 0 0 0

D 0 0 0 0

1

0 0

-1 -1

0 1

xi /A

kjtt'

k^a'b k^b' ksb'

hb'c kyC' kgC

6.7 EXAMPLE OF A PHOTOKINETIC EVALUATION As a typical example of a photokinetic examination the consecutive photoreaction of rmn^-stilbene to phenanthrene is taken. Referring to the different theoretical and experimental sections, some essential equations are reviewed, the relevant figures are redrawn and the procedure of the evaluation is explained stepwise. Computer programs for the evaluation exist. They use measured signals to construct absorbance diagrams of different orders to extract the number of linearly independent reactions and apply Runge-Kutta algorithms as well as exponential curve fitting to determine photochemical quantum yields. Since a general overall program for all possible mechanisms is rather difficult to supply, the given example is supposed to allow the reader to modify the parts where the specific mechanism is treated. Graphical presentation, user interface, and part of the algorithms are independent of the mechanism. The following procedure is proposed. (1) The stirred solution is intermittently irradiated - best in a combined irradiation and measurement device. At least 30 spectra are taken for a simple photoreaction. In the case of more complicated mechanism preferably 25-40 spectra should be recorded per linearly independent step of reaction. Care has to be taken that the total wavelength range is recorded in which during reaction progress spectral changes occur, especially in the UV. (2) Next this reaction spectrum is plotted. Wavelengths are selected at

Ch. 6

Example of a Photokinetic Evaluation

517

which the change in absorbance is typical. The present program on the Internet allows bars to move across the reaction spectrum realising these effects. (3) Absorbance-time curves are plotted. (4) Rank analysis allows the determination of the number of linear independent steps of reaction. Graphical approaches are preferable since deviations of a correct determination of this number become better visible than fuzzy results which make the decision delicate. Absorbance diagrams are constructed at different orders. (5) According to these results and additional information, a mechanism is proposed, which has to be proven next. (6) The rate laws according to the mechanism are set up. (7) Since photokinetic equations cannot be solved in a closed form, equations are preferably rearranged due to allow formal (numerical) integration of the integrals. It is best to choose a matrix representation. The elements of the Jacobi matrix are obtained by (a) Runge-Kutta iteration (b) parametrisation via curve fitting (c) transformation of the time axis to reduce complexity and parametrisation. (8) Only rather simple mechanisms give the quantum yields directly. Furthermore according to Lambert-Beer the absorbances have to be converted to concentrations. Complex Q-matrices are obtained (see Section 5.4). In addition in most cases only products of absorbance coefficient times quantum yield are obtained or sums and differences of these products. Therefore additional information is necessary. (9) It is best to find a way to construct concentration diagrams either by synthesising an intermediate or by using an additional method such as HPLC. (10) For all the results it has to be proven that the mechanism is fulfilled (a) not only with respect to the number of linear independent steps, (b) but also considering backward reactions and superimposed dark reactions. This is rather generalised advice to treat photokinetic data. Some of these points are applied to the stilbene photoreaction as an example. The mechanism of a consecutive photoreaction is obvious. The procedure outlined above is followed in the next paragraphs step by step using a program which can be obtained from the authors on disk by re-

Appendix

518

Ch.6

quest. In this program Runge-Kutta iteration and curve fitting is demonstrated in source code to allow the user to set up his own program lines for other mechanisms. The data measured can also be transfered to EXCEL (Microsoft) or ORIGIN (Microcal). The first is used for the graphics, the second can be used for curve fitting or even formal integration. 6.7.1 Measurements Usually absorbance measurements are taken using the equipment demonstrated in Section 4.2.2.3. Since these are not selective to the different components, in addition other methods are used such as chromatography (see Section 4.5.3). 6.7.1.1 Absorbance First a reaction spectrum is taken according to the statements above.

C3 JO

X)

<

Fig.4-16

Wavelength [nm]

Ch. 6

6JA2

Example of a Photokinetic Evaluation

519

Chromatography

Fig. 5-55

6.6 7.2 7.8 8.4 9

\U]

Loss in concentration is corrected for each injection according to the next graph. Fig. 4-30

50 injection

Appendix

520

Ch.6

6J.2 Graphics 6.7.2.1 Absorbance-time curves Using the reaction spectrum at selected wavelengths absorbance-time curves are plotted first. c

320 nm

Xi

- • — 3 1 3 nin

o JO

<

- • — 3 0 6 nm

0.6

— A — 292 nm — T — 2 7 3 nm —•—250nm

0.4

0.2

0.0 Fig.4-17

200

100

300 Irradiation time [s]

6.7.2.2 Absorbance diagrams Next absorbance diagrams are plotted in various combinations. A typical graph demonstrates curves, which means the reaction contains more than one linear independent step of reaction.

Fig.4-18

1.0

1.2

1.4

1.6

Absorbance at 230 nm

Example of a Photokinetic Evaluation

Ch. 6

521

Furthermore absorbance difference diagrams can be plotted as demonstrated in Fig. 5.7. Whereas the additional information of this type of diagram is small, absorbance difference quotient diagrams can be used to decide whether the number of linear independent reactions amounts to 2 or more as given in Fig. 5.11.

Fig. 5-7

Flg.3-n

Taking the data as used for Fig. 4.18 the following linear relationship is obtained demonstrating that the reaction is represented by two linear independent steps. Therefore any of the mechanisms given in Table 2.5 can be used. On the other hand chemical information requires a consecutive reac-

Appendix

522

Ch. 6

tion pathway. Furthermore one finds starting at cw-stilbene the same reactants. Therefore the first step trans- to cw-stilbene must be an equilibrium. 6.7,3 Evaluation of absorbance data 6.7. i. 1 Set-up of differential equations Accordingly the mechanism A;=^B—!^C can be assumed. As done for a similar reaction in Table 2.1 the schemes of the reaction can be set up as

A -1 * 1 * 0

B 1 -1 -1

. 4>

C 0 'PUK 0 ¥>X 1
Xi

—" Xt

Xy

X'^ -~" Xi

Xi Xy

A -1 0

B 1 -1

C Xk 0
with the differential equations in analogy to Section 3.2.2 as dx. "•1 dt

^

at

•o[l-exp(-£„')]

— =_ i,A =1000

= x, = 1000^«^/o[l-exp(-£;)] t,„

or using decadic absorbance units i , = 1000/o(^^'Aa -
Ch. 6

Example of a Photokinetic Evaluation

523

According to Section 5.4:

+lOOOIo
\dE, = \(mi;f^z^\{E,-E^)F{t)dt

(A = 1.2....,5)

or {(lOOO/o)-' F(0-' dE, = J^z,,\E,dt -\Y,z,,E^dt

= J^z,,\E,dt -

z,,\dt.

where the Z;^J^coefficients are found according to Example 5.22 as

ZM

= [ ( e , 2 ^ ? "-G„^2)('i2

-^22KB

-Gll^f^'A'ia]

^21 =[(022^3 -G21^2)(ni --^2lK -G21^lVAr,,]

^22 = [((322^3 - 021^2 )(n2 ' ^22^8 "

QIX^X^'K^,^]

^20 ~ V"~^21^1oo "• ^22^2oo j

By formal integration one can obtain these Z;^^ elements. To calculate the quantum yields, however, the absorption coefficients of all the reactants at all the wavelengths of measurement as well as at the wavelength of irradia-

524

Appendix

Ch. 6

tion have to be known. Looking at the equations one can understand that the correctness will be rather small because of the many factors in each zelement. For some mechanisms a more simple approach is possible: As explained, the Jacobi matrix and the Z-matrix are similar matrices. Therefore the trace and determinants are equal and one obtains

D = ipx^\(p^e'^ = ^11^22 - ^12^21

For some mechanisms two such equations can be used to calculate the quantum yields, if the absorption coefficients are known. For the transstilbene photoreaction, however, the elements are too complex. 6.7A Evaluation of concentration data 6.7A. I Absorbance triangle The absorbance diagram of the stilbene photoreaction is shown schematically below. Since the first step is an equilibrium, the absorbance line from c/5'-stilbene to phenanthrene does not form a tangent to the measured absorbances. The point B is determined from the absorption coefficients of the c/5"-stilbene at both the wavelengths 250 and 292 nm and the initial concentration aQ. Point C is obtained by the absorbance coefficients of phenanthrene and the initial concentration of rran^-stilbene. The lines ^A, K^ and ^c correspond to the calibration curves for the three reactants absorbance/ concentration. The triangle can be considered as a vector space, given by u, v, , Vj. Using these vectors the concentrations of all the reactants can be obtained for each reaction time according to ii = Eo-E = v,+V2 = eAao-[eAao+(eB~^A)b + ( E c - 8 A ) c ] = (eB-eA)b + (ec-eA)c Using the distances

Example of a Photokinetic Evaluation

Ch. 6 1.0

u

j^__

,

j_ _ -

r

-]-

,

,

r

J

1 - .,„.„j

,-. ,

525 -^,..„.-„ ,

j^.

\

0.9 0.8

e

0.7


0.6 0.5 0.4

f

' ^i\

1

U

.

IM

0.3

<

0.2

1

H

'^«

V' " \*

^

A

^ ^ ^ N . . ^ ^ ^

jl

\

Si

O c« Xi

-J

""A^

a

cs '*-* a o o ced

\

1 ^^ k't^.o J ^

l»

^'''*^^^ J

' h " ^

,Kx:cj

'^j^E(t)

,,,.—-^-r-^"^^"'^'*'^ c 0.1

Ec'Ecao

k VS»o '^' r [ ;

- ' 1 '

0.0

-

' .

^^1—-

— ^ ' T ' ^ " ^ ^ '

\ T

1

,

0.2

0.4

"

'

-_ t.

„ 1

0.6

t

1

1—

0.8

i__

_.

i__ __i

1 _

1.0

Absorbance at 250 nm

AB = | E 5 - E o | = (eB-eA)-ao AC = | E c - E o | = (ec-eA)-ao and knowing the initial concentration |Vl|^|(gB"gA)| b AB |(eB-eA)|ao'

N^ICBc-gA)! C AC |(ec-eA)|ao

the concentration are calculated. 6.7.4.2 Data from HPLC According to Section 4.5.3 concentration data can also be obtained by chromatography. 6.7.4.3 Concentration-time curves The result of these "concentration" determinations are given below for the triangle and HPLC measurements

Appendix

526

Ch.6

cis-stilbene

phenanthrene trans-Stilbene

2000

6000

10000

s

Data from triangle Cone.

0.5

,

,

,

.,

,

1

,

1

|i trans-stilbene

[lO'^moI/1] 0.4

\

1 /'''^'•'•'^^^-^ 0.3 r t f

cis-stilbenc

^

^^^ •"*- '

0.2

L

J

H f\o

r / v>

phenanthrene a

0.1

o

1 B

Fig. 5-59

NOg-

-wlt-*''*''"'^'^ -

2000

4000

r-2

6000

8000

/[si

HPLC-data

Ch. 6

Example of a Photokinetic Evaluation

527

6.7.4.4 Photokinetic equations The differential equations using concentrations are given by a = {-ip^E\a{t)+
A,

B

B , |JlOOO/oF(0 = K•a•1000/oF(0•

6,7.4.5 Photochemical quantum yields Formal integration of these equations is possible using ^da = -lOOO/o^jf e ; jaF(r) dt

jdb = lOOOlQ(p^e\jaF{t) dt - 1000/o(^^ + ^3 )^B J^^(0 dt The different coefficients allow us to calculate the partial photochemical quantum yields knowing the absorption coefficients. Using the absorption coefficients for the three reactants at the wavelength of irradiation (313 nm): f^ = 15900, ^B = 1320 lmol-^cm~* the three quantum yields are obtained as ^f = 0.46,

ip^ = 0.28,

ip^ = 0.21

in perfluoro-compounds as solvents, (p^ is obtained from the Element Zj^ and (jof from a formal integration of the third rate law jdc = 1000/0^3 CBJ^^CO dt and finally using this (pf to get (pf from the element Z22 above. Since at maximal concentration of cw-stilbene &max

528

Appendix

Ch. 6

is valid, a possibility exists to check the obtained quantum yields for plausibility. 6.7,5 Computer programs Modem spreadsheets and data evaluation software can be programmed to solve these kinetic problems. Graphs can be easily obtained as absorbance diagrams of different order using EXCEL. Curve fitting programs such as ORIGIN can be used to evaluate differential equations after transformation of the time axis (see Section 3.3.11). Modem software pools supply numerically stable programs to estimate parameters (e.g. Runge-Kutta approaches) as necessary in formal integration. A demo version of the program is available at the URL in www:http://barolo.ipc.uni-tuebingen.de/photokinetics. Updates of this evaluation can be found at this address as well as future approaches for the use of EXCEL in kinetic evaluations. A simple example is given below. Based on the discussion in Section 5.4.4 and using e.g. eq. (5.93) for a simple uniform reaction, it is possible to determine the reaction constants by means of formal integration. The following EXCEL result has been obtained with the Macro for numerical integration, using data from a thermal reaction first:

rearranged for formal integration:

Y

=Zx\+

Zx2^

l\ I

Ch. 6

i u

PQ

Oil

2 Cuii

<

col

o

Q

ol

<

2 13 Q

c:

X)

<

i

^

*-^

529

'-3v2NOOoooooooor-r*t^r^r-r^vo

0000 •-<mu^w^r^oovocn«or^ ^cn«om cnocnwncso^^OcsmOoooNvO'^r^Ocnt^ r-cS'~"r^soTj-^ONr^Ttcsooot^«om
Example of a Photokinetic Evaluation

^

en m fo 00 en CO en »o m m oo '^ »-^ Tf n

888888888 o o o o o o o o o o o o o o o o o o o o o o o o o

en r^ fnr*<svocn»ncnmrt 00 Ov CS un m mvo r^i-Hvooofncs«ofnr^ » ^ O ON cs en r^vo ..cn'-«--^r^t^fno>oooor-'QONm _ , . ON _ --H i - i T f t ^ o o g s Q O N t ^ T t o « n Q f n v o o o O N O O N O N O O « oONc nC<Jo N vo oo»-o«nio»r>»nvovovovovor^r^r--

S S S ? 8 S CS 0 0 § 8 S CS 0 0 sor^t^ooososOO'^cNfSenen

o o d o o o o o o o o o o o o o o o d o o c > o ^ ' - J

voooeninenvo*-^ t^»o»or*-'-'00oooooNO\ONO\ONOO

?8 SSS?

vo es 00 '-^'-^csmenrf'^u-j

to^enON'^voooen r^»o r^vqenvoinpenrf'^t'-j^ncN ^-H^pw^oo^^vo-^w-jr-.—^ "csvd^o6o6oN'-^»r>dvo^* 0'-Jo6oNen^enr^Ti^'^r^c4d 0\rtQ»o—tr^Ttdt^enO TJ•«o^\Ot^t^OOO^O^O'-< ,-H,-^,—icsesr^enen

CS 0 0

i8S

CM 0 0 ON

o o cs 00 v O O O ^ ^ c S c S e n e n O N '-< '-^ •-< '

?8S

8S

^t^ON'^vO"^r-cS'-'OovovooocsoNvo^oOTt»-Hen^rf»OTj-cs OooTt»-Ht--rnON*r>ON'^ooenr-'—iTfoO'—"ent^Qen«nr*0\'-^en ^^raenenTtrtv->w-)VOvor-r-ooooooONONONpOOOp^'-«

§8S

^^esenenTj-Ti-invovor^r-

s o r«i 0 0

Appendix

530

Ch.6

0,0016 Y 0.0012

0,0008

0,0004

t

Y =-0,00094 X +0,00138 ^

+ J

calculated data

1

1

1

0,2

0.4

0,6

0.8

Since Zxi = kE^^^ and 1^2 - ~^ > ^^^ ^^'^^ constant amounts to 9.4 x 1(H s-' and £;,„= 1.468. In the case of a photoreaction the photokinetic factor has to be included. Various evaluations are possible according to Figs. 5.18-5.20, the simplest one being the use of formal integration like (5.93) f,.l-10-^^'"^ dt + 2,, I E, it„) ^=^J: " e(0

dt

or \"EM„)^—^

AE I •''1

•=

dt

Zi,+Zi

dt

I

E (t )

J/,

E(t„) -

dt

with Zxi=lOOOe\(p''IoE^

and

/ ,^Ai Z;^^ =-1000e>'^/o

Ch. 6

Example of a Photokinetic Evaluation

531

or by analogy with Fig. 5.20

In both cases the Macro can be used to calculate the integrals in a more complex way. 'NumlntVl.O 'Numerical Integration using the trapeziod method 'Author: Felix Heise 'The function requires a field with two rows (or two columns) as input parameter. 'The first row (column) should contain the time-values, the second (e.g.) the absorbance-values within the integral. The algorithm of the Macro requires at least three pairs of data Function Numlnt(ParField) Field = ParField If UBound(Field; 1) = 2 And UBound(Field; 2) > 2 Then For Count = 1 To (UBound(Field; 2) - 1) Numint = Numint + (Field(l; Count + 1) - Field(l; Count)) * (Field(2; Count + 1) + Field(2; Count)) / 2 If Field(l; Count) > Field(l; Count + 1) Then Numint = CVErr(xlErrValue) Exit For End If Next Count End If If UBound(Field; 2) = 2 And UBound(Field; 1) > 2 Then For Count = 1 To (UBound(Field; 1) - 1)

532

Appendix

Ch. 6

Numint = Numint + (Field(Count + 1; 1) - Field(Count; 1)) * (Field(Count + 1; 2) + Field(Count; 2)) / 2 If Field(Count; 1) > Field(Count + 1; 1) Then Numint = CVErr(xlErrValue) Exit For End If Next Count End If If UBound(Field; 1) > 2 And UBound(Field; 2) > 2 Then Numint = CVErr(xlErrValue) End Function This macro was used to calculate the results according to the figure on p. 530 using columns 3 and 4 of the table on p. 529.

Chapter 7

References

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G. P. Misra, D. Lavabre, J. C. Micheau, Mechanistic investigations and spectrokinetic parameter determination during thermoreversible photochromism with degradation, J. Photochem. Photobiol. A: Chem. 80 (1994) 251. [171] S. Gliesing, M. Reichenbacher, H.-D. Ilge, D. FaBler, Welleniangeneffekte auf die Quantenausbeuten der Photoisomerisierung von PrSvitamin D3, Z. Chem. Leipzig 29 (1989)21. [ 172] D, Rottger, PhD thesis, Hohenheim, 1995. [173] H.-D. Ilge, Bestimmung der UVA^IS-Extinktionskoeffizienten und der partiellen Quantenausbeuten eines Vierkomponentensystems, Z. Phys. Chem. Leipzig 262 (1981)385. [174] Th. Flohr, R. Helbig, H. J. Hoffmann, Rate equation for photochromic glasses considering both thermal and optical regeneration, J. Mater. Sci. 22 (1987) 2058. [175] E. Friz, G. Gauglitz, T. Klink, A. Lorch, Measurements of fast changes in absorption and fluorescence spectra using an OMA-system, Comput. Enhanced Spectrosc. 1 (1983)49. [176] G. Gauglitz, R. Goes, Photokinetic evaluation by microprocessor controlled fluorescence data acquisition, Comput. Enhanced Spectrosc. 2 (1984) 159. [177] M. Guther, Software for parameter estimation of consecutive reactions, Tubingen, 1987. [178] E. Uhlmann, Stilben-1: LdsungsmittelabhSngigkeit der partiellen photochemischen Quantenausbeuten, Masters thesis, Tiibingen, 1990. [179] G. Gauglitz, Absorptions- und fluoreszenzspektroskopische Methoden fOr komplizierte Photoreaktionen, PhD-thesis, Tubingen, 1972. [180] G. Gauglitz, Determination of photochemical quantum yields using fluorescence data, demonstrated by laser dyes. Z. Phys. Chem. (Wiesbaden) N. F. 113 (1978) 217. [181] G. Gauglitz, Spektroskopische Untersuchungen zur PhotostabilitSt, dargestellt an Laserfarbstoffen, Habilitationsschrift, Tubingen, 1979. [182] T. Suratwala, Z. Gardlund, K. Davidson, D. R. Uhlmann, J. Watson, S. Bonilla, N. Peyghambarian, Silylated coumarin dyes in sol-gel hosts. 2. Photostability and sol-gel processing, Chem. Mat. 10 (1998) 199-209 [183] R. Raue, Laser dyes, in: Ullmann's Encyclopedia of Industrial Chemistry, Vol. A 15, VCH, Weinheim, 1990, pp. 151ff. [184] K. Franke, Konzeption und Darstellung neuer Laserfarbstoffe auf Indenbasis und ihre photochemische Untersuchung, PhD thesis, TUbingen, 1989. [185] J. Riedt, Die Abhangigkeit der partiellen photochemischen Quantenausbeuten vom Medium dargestellt an einem Laserfarbstoff, Masters thesis, Tubingen, 1985. [186] G. Gauglitz and W. Schmid, HPLC in photokinetics: determination of reaction mechanism and photochemical quantum yields of (E)-l-lphenylpropene, Chromatographia23(1987)395. [187] G. Gauglitz, T. Klink, W. Schmid, Untersuchung der Photochemie von Polyenen mit Hilfe von mikroprozessor-gesteuerter HPLC, Fresenius* Z. Anal. Chem. 318 (1984) 298. [188] W. Schmid, Erweiterung photokinetischer Untersuchungsmethoden durch mikroprozessorgesteuerte HPLC mit spektraler Detektion, PhD thesis, Tubingen, 1987.

542 [189] [190]

[191] [192] [193] [194] [195]

[196] [197] [198] [199] [200] [201] [202] [203]

[204] [205] [206] [207] [208] [209]

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K. HeiB, Kombinierte spektroskpische und chromatographischc Verfahren zur Untersuchung photochemischer Reaktionen, PhD thesis, Tiibingen, 1989. G. Gauglitz, R. Goes, W. Schmid, Photochemical degradation mechnaism of stilbene1 examined by spectroscopic and chromatographic methods, J. Photochem. 34 (1986) 339. S. WeiB, Faktorenanalyse in der Kinetik, PhD thesis, Tiibingen, 1991. J. R. Hurley, R. B. Cattel, The procrustes program: producing direct rotation to test a hypothesized factor structure, Comp. Behav. Sci. 7 (1962) 258. G. Gauglitz, H.-P. Neubauer, Moglichkeiten in der Reaktionskinetik durch prozeBgesteuerte HPLC, GIT Fachz. Lab. 34 (1990) 786. H.-P. Neubauer, PhD thesis, Tubingen, in preparation. Y. Yokoyama, T. Goto, T. Inoue, M. Yokoyama, Y. Kurita, Fulgides as efficient photochromic compounds. Role of the substituent on furylalkylidene moiety of furylfulgides in the photoreaction, Chem. Lett. (1988) 1049. E. Uhlmann, Photokinetik von Aberchrome 540 in isotroper und anisotroper Umgebung, PhD thesis, Ttibingen, 1995. E. Uhlmann, G. Gauglitz, New aspects in the photokinetics of Aberchrome 540, J. Photochem. Photobiol. A: Chem 98 (1996) 45. H. D. Roth, Chemically induced dynamic nuclear polarization as a tool in mechanistic photochemistry, Mol. Photochem. 5 (1973) 91. G. Gauglitz, G. Kraus, A reflectometric sensor for ammonia and hydrocarbons, Fresenius' J. Anal. Chem. 346 (1993) 572. G. Kraus, G. Gauglitz, Application and comparison of algorithms for the evaluation of interferograms, Fresenius' Z. Anal. Chem. 344 (1992) 153. J. Nagel, Interferenzspektroskopische Methode zur kinetischen Untersuchung von Photoresisten und photochromen Systemen, PhD thesis, Tiibingen, 1995. H.-J. Timpe, A. G. Rajendran, Light-induced polymer and polymerization reactions, Eur. Polym.J.27(1991),77. C. Carre, D. J. Lougnot, J. P. Fouassier, Holography as a tool for mechanistic and kinetic studies of photopolymerization reactions: A theoretical and experimental approach, Macromol. 22 (1989) 791. D. B. Yang, Kinetic studies of photopolymerization using real time FT-IR spectroscopy, J. Polymer Sci. A 31 (1993) 199. H. C. Kessler, The kinetics of light absorption in photobleaching media, J. Phys. Chem. 77 (1967), 2736. J. R. Sheats, J. J. Diamond, J. M. Smith, Photochemistry in strongly absorbing media, J. Phys. Chem. 92 (1988), 4922. W. Nahm, WeiBlichtinterferenz an diinnen Schichten, PhD thesis, Tiibingen. St. Kohlhage, Charakterisierung diinner, geordneter und ungeordneter Siloxanfilme, PhD thesis, Tiibingen. V. Hoffmann, D. Frfthlich, U. Hellstem, S. Kohlhage, W. Nahm, K. NuBbaumer, D. Oelkrug, Spectroscopical examination of photopolymerization, structure and permeability of thin polymer films, J. Mol. Struct. 293 (1993) 253-256.

Ch. 7 [210] [211] [212] [213] [214]

[215] [216]

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H. Mauser, Zur formalen Photokinetik: IV. Der Ablauf einfacher Photoreaktionen in zMhen Medien, Z. Naturforschg. 22b (1967) 569. H. Lege, Charakterisierung und Photochemie azaaromatischer Schichten, PhD thesis, Tubingen, 1991. D. Fr5hlich, Photokinetische Untersuchungen an Azobenzol in fester und fltissiger Matrix bei Laserbestrahlung, Masters thesis, Tiibingen, 1986. M. Guther, Software for Photokinetics in Viscous Media, Tiibingen, 1987. G. Gauglitz, Photophysical, photochemical and photokinetic properties of photochromic systems, in: H. Dtirr, H. Bouas-Laurent (eds.), Photochromism, Elsevier, Amsterdam, 1990. D. Fr6hlich, PhD thesis, Tiibingen, in preparation. D. Fr5hlich, G. Gauglitz, Untersuchung des Reaktionsverhaltens von Azobenzol bei Laserbestrahlung in Festkorpern, Proceed. 12. GDCh-Fachgruppentagung Photochemie, Aachen, 1991.

Subject index

Absorbance, 257, 267, 337 - at wavelength of irradiation, 192 -

decadic, 267

- degree of advancement relationship, 343 - interpolated, 251 - Napierian units, 184, 197, 259 absorbance diagram, see E-diagram - higher order, 266 absorbance-difference diagram, see EDdiagram absorbance-concentration relationship, 197, 267, 340, 524 absorbance-time diagram, 251, 261, 265, 374,391,517,520 - synchronisation, 251, 256, 265 absorption, 244 - index of, 284 absorption coefficient, 19, 267, 275, 281, 337, 344, 388, 392, 396,438,471, 517 - decadic, 19, 184, 259, 268, 270, 374, 397, 522 - Napierian, 19, 74, 182, 198, 259 absorption spectroscopy, 244, 261 absorption spectrum, 281 actinometry - 2,2',4,4'-tetra-isopropylazobenzene, 401 - azobenzene, 399 - chemical, 296 ~ dibenzo[aj]perylen-1,16-dione (HCD), 401 - mesodiphenylhelianthrene (MDH), 404 - Parker's solution, 399 - physical methods, 294 advancement variable, 7 amount of light absorbed, 3, 16, 181,192, 237,293,416 approximation, 118

- in practice, 397 - photostationary state, 410 - photokinetic factor, 200 ATR (attenuated total reflection), 285 automation, 289 auxochromic, 276 Bemoulli-L'Hospital, 92, 311 Bodenstein hypothesis, 4, 40, 122, 126, 129, 142, 153, 156, 158, 183, 302,477 bolometer, 294 brutto equation, 50 Calibration, 290 cell compartment - absorption measurement, 249, 253, 255, 280 - fluorescence measurement, 279, 280 chemically induced dynamic nuclear polarisation (CIDNP), 291,456 chemometrics, 453 chromatographic reaction spectrum, 444 chromatography, 286 - high performance liquid (HPLC), 286, 289 - thin-layer, 281, 283, 286, 287 chromophore, 296 circular dichroism (CD), 291 coefficients - stoichiometric, 26,473 - stoichiometric, matrix of reduced, 66, 485 - stoichiometric, matrix representation, 28,475 - stoichiometric, rank of matrix, 44 - stoichiometric, reduction of, 32, 143, 475,483,488

546

Subject index

combination - absorbance and chromatography, 443 - absorbance and fluorescence measurements, 439 - absorbance-NMR, 456 - absorbance-reflectance, 457 combination of thermal and photochemical reactions, 357 concentration - at end of reaction, 63, 67 - final. 67, 141 - linear independent, 67 - reduced, consecutive reaction, 141 - reduced, Michaelis-Menten, 130 - reduced, photoaddition, chemically sensitised, 194 - reduced, photoaddition, physically sensitised, 195 - reduced, radical chain reaction, 127 concentration diagram, see K-diagram concentration measure, 338 concentration measurement, 327 concentration-difference diagram, 303 concentration-difference-quotient diagram. see KDQ-diagrams concentration-time curve, 290,451 - general coordinates, 62 concentration-time diagram, 87,442, 454 - extrema, 87 ~ consecutive reaction, 112, 525 - points of inflection, 87 concentration-time measurements, 322 consecutive photoreaction, 350 conservation of mass, 25 correction factor - fluorescence intensity, 430 curve-fitting techniques, 261,454 Definition of symbols, 495 degree of advancement, 6,7,27,66,73,75, 78, 85,198,203,215, 300,343,483,487 - absorbance relationship, 343 - consecutive photoreaction, 186, 523 - consecutive reaction, 122

-

consecutive, photochemical and thermal, 212 - parallel reaction, 106 - photoisomerisation, reversible, 208 - photoisomerisation, superimposed thermal backward, 211 - radical chain reaction, 127 - reduced, 120.203 - reduced parallel photoadditions, 172 - reduced, consecutive photoreaction. 187.234.240 - reduced, consecutive photoreactions. viscous medium, 234,239 ~ reduced, parallel mixed photoreactions. 189 - reduced, parallel photoadditions, 169, 172 - reduced, parallel photoreactions, different initial products, 188 - reduced, parallel photoreactions, including photoaddition. 165 - reduced, photoisomerisation, 239 - reduced, photoisomerisation physically sensitised, 206 - reduced, photoreactions. survey on mechanisms. 216 - time-dependence, 85 derivative spectra, 275 derivative spectroscopy, 273 detector, 295 determination of intensity of irradiation, - chemical method, 296 - physical method, 294 device - combined irradiation and measurement. 250 difference equations. 316.330. 371 - higher order. 319,331.373 differential equation. 89.118.140. 242. 522 - Bernoulli. 219.462 - complex, consecutive, 134 - complex, numerical integration. 132 - consecutive, fluorescence, 426

Subject index -

consecutive, in part sensitised by radicals, 214 - consecutive, photochemical and thermal, 212 - consecutive, viscous medium, 234 - eigenvalues, 89 - integration of. see integrals - partial, viscous medium, 219 - photoaddiation, general coordinates, 205 - photoaddition, 380 - photoaddition, chemically sensitised, 227 - photoisomerisation, 374 - photoisomerisation, physically sensitised, 315 - photoisomerisation, superimposed thermal backward, 211 - photoisomerisation, viscous medium, 224 - photoreactions, 187 ~ photoreactions, radical chains, 230 - photoreactions, survey on mechanisms, 216,240 - transformation of time, 214 - uniform photoreactions, 181 differentiation, 274 diffusion, 468 diffusion controlled, 12, 14, 149 dilution, 290 - influence of, 96, 450, 519 discrimination of mechanism, 393 E-diagram, 265, 266, 364, 395, 416,457, 517,520 - consecutive photoreaction, 368 ~ final points, 365 -- Gibbs-triangle, 369 - higher order, 265 - uniform reaction, 365 ED-diagram, 348, 352, 375,415, 521 - consecutive photoreaction, 354, 362 - dependence on wavelength and intensity, 356

547

- final slopes, 354 ~ initial slope, 352, 358 - photoisomerisation, 358, 376 - points of inclination, 353 ~ reduced, 355 EDQ(s)-diagram - of higher order, 363 EDQ-diagram, 357, 359, 360, 521 - azobenzene, 391 ~ consecutive photoreaction, 362 eigenvalues, 89, 124, 322, 392, 508 - complex, 83 ~ of differential equation, 89, 435 - equal, 81 Einstein, 8 energy transfer, 146, 512 equation - brutto, 50 - differential, 37, 41, 55, 65, 74 - differential, set-up of, 38 - differential, solution of, 75 - matrix, 440 - normal, 326 - photokinetic-thermal, comparison, 397 equilibrium - approximation of, 137 error ~ approximation, 125, 133 - in absorbance measurement, 261 ESR spectroscopy, 243 evanescent fields, 285 excitation spectra, 280 exponential integral equation, 226 extinction, 20, 259, 267, 337 extrema in concentration-time diagrams, 87 Factor analysis, 273,451 factor of 1000, 182 Pick's law, 468 field - evanescent, 285 flash photolysis, 263

548

Subject index

fluorescence, 11. 244, 257, 275, 277, 287, 416,440 - photokinetic equations, 418 fluorescence excitation spectra, 280 fluorescence quantum yield, 276,278, 288 fluorescence-concentration relationship, 416 fluorescence-time curves, 427 fluorophore, 276,418 formal integration, 4, 324, 332, 374, 381, 382, 386, 451, 455, 517, 523. see integrals - azobenzene, 383 - fluorescence measurements, 436 - photoreaction, consecutive, 326 formal kinetics - photoisomerisation, thermal backward, 407 Fourier analysis, 291,451 Fresnel, 284 fuzzy logic, 272 General coordinates, 57, 60, 61, 119, 369 - concentration-time curves, 62 - consecutive photoreactions, viscous medium, 234 - uniform photoreactions, 204 geometric effects, 417 Gibbs diagram, 369 Hadamard, 270 half-life, 61, 399 - first order reaction, 58 - second order reaction, 59 - zero order reaction, 57 heterocoerdianthrone, 401 high performance liquid chromatography (HPLC), 327 Index of absorption, 284 influence of dilution, 519 integrals, 315 - complex non-uniform photoreactions, 210

-

complex uniform photoreactions, 208 consecutive photoreaction, 210 consecutive photoreaction, physically sensitised, 209 - consecutive photoreaction, viscous medium, 235,240 - fluorescence measurement, 422 - formal, 242 - including photokinetic factor, 199 - numerical, 242 - numerical, not in closed form, 231 - parallel photoreaction including photoaddition, 209 - photoaddition, chemically sensitised, 228 - photoaddition, physically sesitised, 195 - photoisomerisation, 196, 324 - photoisomerisation, reversible, 208 - photoisomerisation, sensitised by radicals, 187, 193,215 -- photoisomerisation, superimposed thermal backward, 211 - photoisomerisation, viscous medium, 232, 239,240 - photoreaction, chemically sensitised, 194 - photoreactions in viscous medium, 221 - photoreactions, radical chain, 192, 207, 229 - photoreactions, simple uniform, 191 - simple non-uniform photoreaction, 208 - simple uniform photoreaction, 192, 204 - transformation of time, 214 integration - foxmdX.see integrals - numerical, see integrals - Runge-Kutta, 211 - using Simpson* method, 205,231 intensity diagrams, 423,427,436 intensity of irradiation, 293 intensity quotient diagram, 429 intensity-time diagrams, 418 interference spectrum, 459

Subject index intermediates, 118, 123, 126, 132, 142, 176,301 - of radical chain reaction, 143 ~ photoaddition, radicals, 153 intermediate radicals, 158 internal conversion, 11 intersystem crossing, 12, 276 IR spectroscopy, 243, 291 irradiance, 17 irradiation, 250, 293 - process-controlled, 254 - requirements, 245 - set-up combined with measurement device, 250 irradiation measurement - chemical method, 296 ~ physical method, 294 isochronic, 255 isoenergetic, 9 isosbestic point, 196, 200, 264, 266, 345, 384, 466 - requirements, 347 Jablonski-term diagram, 9 Jacobi matrix, 5, 72, 75, 84, 85, 99, 145, 241,334,388,392,396,517 - complex photoreaction, 116 - consecutive reaction, 111 - eigenvalues, 75, 78 - eigenvectors, 78 - isomerisation, 105 - main minors, 77 - parallel reaction, 104 - photochemical reactions, 74, 186 - thermal reactions, 74 Kalman filter, 272,454 K-diagram, 78, 91, 304, 329, 517 - areas, 96 - consecutive, 108,305 - dependence on intensity, 307 - dependence on wavelength, 307 - extrema, 87, 96 - final point, 92

549

- final slope, 94, 308, 312 - influence of dilution, 96 - initial point, 92 - initial slope, 92, 308, 311 - isomerisation, 305 - parallel reaction, 106 - points of inflection, 87, 98 - reduced, 96 KDQ(3)-diagram,314 KDQ-diagram, 308, 329 - bimolecular, parallel, 309 - consecutive photoreaction, 311, 362 - consecutive, multi-step, 313 kinetic analysis, 118, 243, 324, 334, 393, 523 Kramer-Kronigs relationship, 460 Kronecker and Capelli theorem, 360 Kubelka-Munk,281 Lambert cosine-law, 282 Lambert-Beer law, 19,197, 258, 337,464 - deviations, 259 - restrictions, 259 Lanczos-bidiagonalisation, 271 laser dye, 288,422 least squares approximation, 270 lifetime - excited state, 14 linear equations - inhomogeneous, 447 linear regression, 322 linear systems ~ transformation, 99 Mac Laurin series, 199 MAPLE - use of, 475 matrix - concentration-differences, 301 - elements for formal integration, 388 - inversion, 269 - normal, 269 - reduced, 68 - regular, 67

550

Subject index

- variance-covariancc, 271 matrix rank analysis, 300, 344 - graphical, 328 M-diagram, 328 MDQ-diagram, 329 measurement - asynchronous, 251 - fluorescence, 275 - synchronous, 251, 256, 265 measurement of intensity of irradiation - chemical method, 296 -- physical method, 294 mechanism, 299, 392, 393, 517, 522 micelles, 262 Michaelis-Menten, 128, 194 microprocessors, 292 microwave spectroscopy, 243 monochromator, 245, 247 - dual arrangement, 281 muUicomponent analysis, 243, 267 multi-linear regression, 270 Napierian units, 259, 374,429 neural networks, 272 NMR spectroscopy, 243, 291 normal equations, 338 numerical integration, 517 Optical rotational dispersion (ORD), 291 order of reaction, 55 oxygen - in air, 245 Parker's solution, 296 partial least squares (PLS), 271 partial reaction, 31, 65, 102 pattern recognition principles, 273 personal computers, 292 phosphorescence, 12 photo diode, 294 photo diode array, 248,256, 257, 285,445 photochemical reaction, 51,280, 291 - 1-hydroxy-benzotriazole, 457 - 2,5-distyrylpyrazine (DSP), 462

-

-

4,4'-distyrylazobenzenes, 411 addition, 149, 168, 324, 328, 341, 378 addition, chemically sensitised, 156, 194 addition, chemically sensitised, viscous medium, 227 addition, physically sensitised, 194, 206,513 addition, radical chain, 159 addition, radical intermediate, 153 addition, via singlet, 150, 510 addition, via triplet, 150, 205, 510 anthrone, 422 azobenzene, 464 Carbol,435 chemically sensitised, 156 cinnamic alcohol, 446 comparison stirred solution, viscous medium, 241 complex, comparison parallel, consecutive, 191 complex, non-uniform, 178 complex, uniform, 177 consecutive, 180, 186, 267, 307, 311, 318, 323, 326, 330, 333, 344, 352, 354, 369,388,517 consecutive and parallel, 301, 329, 334, 361 consecutive, fluorescence, 424 consecutive, in part sensitised by radicals, 214 consecutive, multi step, 313 consecutive, photochemical and thermal, 212 consecutive, physically sensitised, 176, 209,516 consecutive, reduced X-diagram, 355, 369 consecutive, viscous medium, 234 coumarin dyes, 446 coumarin-derivatives, 423 C-S cyclophane, 415 dihydroindolizine, 405,408 diphenyloctatetraene, 446

Subject index diphenylpyrazines, 453 fast, 251 fulgides,405,454,455,471 isomerisation, 14, 239, 374 isomerisation via singlet state, 146 isomerisation via triplet state, 146, 178, 477 isomerisation, oxindigo, 385 isomeriation, physically sensitised, 205, 329, 377 isomerisation, reversible, 208, 380, 480 isomerisation, reversible via triplet, 178 isomerisation, reversible, thermal consecutive, 178 isomerisation, sensitised by radicals, 193, 157,207 isomerisation, superimposed thermal backward, 210,405,478,481 isomerisation, thermal backward, 178 isomerisation, viscous medium, 219, 224,231,238 liquid crystals, 471 mechanisms, distiction of, 334 mechanisms, distinction of, 266, 392, 393 non-uniform, 163 parallel, 160, 163, 178, 267, 334, 353 parallel mixed mechanisms, 179 parallel photoadditions, 171 parallel via singlet and triplet, 162 parallel with different partners, 178 parallel, additional partner, 164 parallel, both photoaddiations, 168 parallel, different initial products, 188 parallel, including photoaddition, 165, 209 parallel, physically sensitised, 175, 515 parallel, via singlet, 161 parallel, via triplet, 162 p-distyryl-benzene, 453 physically sensitised, 154, 194, 511 polymerisation, 462 quasi linear, 215, 316, 330, 332, 386, 392

551

- quenched, 146 - radical, 158, 184 - radical chain, 159, 207 - radical chain, viscous medium, 119 - reduction, 152 - reduction, radical intermediates, 153 - sensitised, 154 - stilbene,275,445,517 - stilbene-1,422,441 - superimposed thermal, 255, 357 - survey on mechanisms, 216 - trans- to cw-stilbene, 275,445, 517 - umbelliferone, 437 - uniform, 145, 181, 197 - viscous media, 218,462,466 photochemical reactor - assumptions, 183 photochromism, 414 photo-degradation, 288 photodimerisation, 14. see photochemical reaction photo-effect - external, 294 photoisomerisation, 14,40, 51 - azobenzene, 349, 381, 389, 399 ~ quenched, 146 - quenched, via singlet, 146 - quenched, via triplet, 146 - reversible, azobenzene, 345 - trans- to cis stilbene, 263, 344, 350, 360 photokinetic factor, 19, 20, 192, 241, 374, 408,417 - approximation, 200, 398,428 - approximation at total absorption, 201 - approximation by interpolation, 203 ~ approximation using error functions, 202 - approximation using Taylor series, 202 - interpolation, 400 photokinetics, 249, 257, 516 - general approach, 3 - requirements, 242 photokinetic-thermal equations, 397 photometry, 243, 247, 290

552

Subject index

photophysical steps - photoisomerisation via singlet state, 146 - -photoisomerisation via triplet state, quenched, 148 photophysics, 7,417 photopolymerisation, 462 photoreaction. see , photochemical reaction photoresist, 284, 285,460 photostability, 283 ~ of laser dyes, 431,440 photostationary state, 410 pivoting, 270 pK values, 277 points - isosbestic, 266 - quasi-isosbestic, 264 points of inflection, 98 polychromator, 248 post steady state, 132 potential hyper surfaces, 276 predissociation, 276 principle component analysis, 241, 273 principle component regression (PCR), 271 process - chemical primary, 13 ~ diffusion controlled, 12 - photophysical, 40, 275 process control, 292 product - final, 90 - initial, 90 proteins, 262 Quantum yield, 15, 35, 293, 296, 331 - apparent differential, 16 - apparent integral, 15 - chemcially sensitised, 157 - consecutive photoreactions, 181, 187, 312,323,353,391 - consecutive photoreactions, viscous medium, 235, 239 - consecutive photoreactions, physically sensitised, 177 - dependent on irradiation, 196

~ -

dependent on amount of light absorbed, 184,223 depending on viscosity of solvent, 441 independent of amount of light absorbed, 182,222 independent of irradiation, 196 laser dyes, 436 liquid crystals, 471 parallel photoadditions, 169,173 parallel photoreaction via singlet, 161, 162 parallel photoreaction via singlet and triplet, 163 parallel photoreactions with mixed mechanisms, 180 parallel photoreactions, both photoadditions, 172 parallel photoreactions, including photoaddition, 165 parallel photoreactions, physically sensitised, 176 partial, 236,461, 527 partial photochemical, 4, 36, 165,421, 481 partial reactions, 438 photoaddition, 168 photoaddition via singlet, 150, 378 photoaddition via singlet, quenched, 151 photoaddition via triplet, 151, 324 photoaddition via triplet, quenched, 152 photoaddition, chemically sensitised, 157, 194 photoaddition, chemically sensitised, viscous medium, 227 photoaddition, radical chain, 160 photoisomerisation, 231,374, 377,385 photoisomerisation via singlet, 43, 238, 380 photoisomerisation via singlet, quenched, 147 photoisomerisation via triplet, 43, 238 photoisomerisation via triplet, quenched, 148

Subject index -

photoisomerisation, physically sensitised, 194, 206 - photoisomerisation, reversible via triplet, 178,^81 - photoisomerisation, sensitised by radicals, 158, 193 - photoisomerisation, thermal backward, 178 ~ photoisomerisation, viscous medium, 218,224,231 - photoreaction, radical, 185 - photoreduction, radical intermediates, 153 - radical chain, viscous medium, 229 - stilbene-1,443 - trans- to ci5-stilbene, 451 - true differential, 36, 323 ~ true integral, 16 - using chromatographic data, 447 quasi-isosbestic points, 264 quasi-stationary variables, 130 Rate - chemically sensitised photochemical reaction, 156 - consecutive phororeactions, 181, 186, 187, 240 - consecutive photoreactions, physically sensitised, 177 - equation, 102 - non-uniform photoreaction, 186 - parallel mixed photoreactions, 189 -- parallel photoreaction via singlet, 161 ~ parallel photoreaction via singlet and triplet, 163 - parallel photoreaction via triplet, 162 - parallel photoreactions with different partners, 179 - parallel photoreactions with mixed mechanisms, 180 - parallel photoreactions, both photoadditions, 169, 172 - parallel photoreactions, different intial products, 188

-

553

parallel photoreactions, including photoaddition, 165 - parallel photoreactions, physically sensitised, 176 - photoaddition, radical chain, 160 - photoaddition, radical intermediates, 153 ~ photoaddition, via singlet, 150 - photoaddition, via singlet, quenched, 152 - photoaddition, via triplet, 151 - photoaddition, via triplet, quenched, 152 - photoisomerisation via singlet, 240, 241 - photoisomerisation via triplet, 241 - photoisomerisation, radical intermediate, 159 - photoisomerisation, reversible via triplet, 178 - photoisomerisation, sensitised by radicals, 158 - photoisomerisation, thermal backward, 178 - thermal reaction, 67 - total, 6 rate constant - bimolecular, 13 - consecutive reaction, 68 - matrix of, 66 rate equations - consecutive, 425, 523 rate laws, 118,517 - concentrations, 397 - general, 55 - number of exponential functions, 89 - photokinetic, 40, 75, 523 - quasi-stationary state, 132 - thermal, 37, 45 rate of reaction, 7, 60 reactants - non-reduced set, 28 - reduction of, 26 reaction - addition, 341

554 -

Subject index

back- and forward, 105 bimolecular, 13, 14 catalytic, 39 complex, thermal, 139 consecutive and parallel, 301 consecutive, 47,68, 84,95,107, 118, 119, 307, 323, 482,492,496,497, 499, 501,508 - consecutive, multi step, 313 - cyclic mechanism, 47,483, 504 - diffusion controlled, 12,14 - first order, 57 - isomerisation, 105 - multi-step, 27 - order, 55 - parallel, 49, 103,486 - partial, 31, 65, 102,313 - photochemical and thermal superimposed, 251, 256, 357 - photochemical consecutive, 74 - photochemical, 3,51, 244. See photochemical reaction - photoisomerisation and superimposed thermal, 51 - photoisomerisation, reversible, 52 - photoisomeristaion, reversible, and consecutive thermal, 52 - radical chain, 126, 143 - reversible and consecutive, thermal, 137 - reversible, thermal and consecutive, 120 - reversible, thermal, 45 - second order, 58, 60,488 - thermal, 3, 251 - uniform, 45, 103, 304 - zero order, 57 reaction chromatogram, 444,445,450 reaction rate, 7 reaction scheme, 27, 38,41,143, 241,473, 483, 522 - reduced, 67, 143 - reduced, complex reaction, 115 reaction spectrum, 263, 264, 345,438, 456, 516,518

- derivative, 275 - fluorescence, 434 reaction steps - linear independent, number of, 4,44, 142,241,266 - spectroscopically linear independent, number of, 4 reaction time, 393 recursion equation, 317, 373 - higher order, 321,331 reduction - of reactants, 32 - of reaction steps, 46 reflectance, 244,257,281 - diffuse, 281 - regular, 283 - specular, 281 reflection, 287 reflection spectroscopy, 281, 284 reflections - multiple, 262 reflectometric interference spectroscopy (RlfS), 458 reflectometry, 282, 284 refractive indices, 261 regression - multi-linear. 270,440 Reinecke's salt, 296 residuals, 261 Runge-Kutta algorithm, 213,435,, 451, 517 Scattering, 263, 277 sensors - chemical, 284 singlet, 10 solutions - turbid. 262 spectrometer - double beam, 246, 254 - fluorescence, 277 - photo diode array, 249,256, 257 - reflectance, 283 - reflection, 285

Subject index - requirements, 245 - simultaneous, 257 ~ single beam, 246 spectroscopic data, 78 spectroscopically linear independent steps of reaction, 4 spectroscopically uniform, 265 spectroscopy, 291 - absorption, 277 - multiplex, 247 - reflection, 281 - sequential, 245, 249 steps of reaction. See reaction steps stoichiometry, 25 stray coefficients, 281 stray light, 260 symbols - definition of, 495 Target transformation, 273 Taylor series, 202 - expansion, 132 temperature control, 252 transformation, 350, 387, 396 transformation of linear systems, 99 transformation of time, 4, 214, 229,238, 315,371,439,466,517 transition - radiationless, 11, 276 - thermal relaxation, 11 triplet, 11,276 turnover, 7 Uniform, 265, 347 - complex, 45 - photoreactions, 204, 380 - simple, 45 uniform reaction, 45, 103, 119, 364, 374 - parallel, 103

555

UV spectroscopy, 243 UVA^is-spectrometer, 292 UV/Vis spectroscopy, 244, 250, 267, 290, 292, 296 Vandermond - determinant, 80, 81 - matrix, 79, 86 vector - reduced for concentrations, 68 - stoichiometric, 26 - transposed, 26 Vieta theorem, 77 Vis spectroscopy, 243 viscosity, 14 - of solvent, 441 viscous medium - general solution of differential equations, 222 visible spectroscopy, 243 X-diagram, 91 - consecutive, 108 - consecutive photoreactions, physically sensitised, 177 - consecutive reaction, 110 - consecutive reaction, slopes, 110 - parallel reaction, 106 - reduced, 106,355,369 ~ reduced, consecutive photoreactions, 188,355 - reduced, consecutive reaction, 113 - reduced, final slopes, 174 - reduced, mixed parallel photoreactions, 190 - reduced, parallel photoreaction, 168, 170, 173 - reduced, parallel photoreactions, physically sensitised, 176