Electrode Kinetics for Chemists, Chemical Engineers and Materials Scientists

IEL-1C. 114 C1IDE.

for Chemists, Chemical Engineers, and 71WWWIEV, MaterilsScn

Offering a thorough explanation of electrode kinetics, this textbook emphasizes physical phenomena rather than mathematical formalism. The underlying principles of the different experimental techniques are stressed over their • technical details. Assuming an elementary knowledge of thermodynarbicS and chemical kinetics, and minimal mathematical skills, coverage explores the arguments of two primary schools of thought: electrode kinetics and interfacial electrothemistry viewed as a branch of physical chemistry, and from the perspective . of analytical chemistry. . This book is recommended for scientists and engineers in chemistry, chemical engineering, materials science, corrosion, battery development, and electroplating. Graduate students in electrochemistry, electrochemical engineering, and materials science will also find it a challenging and stimulating introduction to electrode kinetics.

VCHO ISBN 1-5608 1-62 6-0 VCI-1 Publishers, Inc. ISBN 3-527-89626-0 Verlagsgesellschaft

VCH

Cover Design: Anna Lee

411111•1011INKIN

I F Z F.

G I L

F • "

Electrode Kinetics for Chemists, Chemical Engineers, and Materials Scientists

Eliezer Gileadi

Professor of Chemistry School of Chemistry Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University Ramat-Aviv 69978 Israel

Dedicated to my parents, who valued the book above all treasures of the human race.

Library of Congress Cataloging-in-Publication Data CIP pending.

harin-Meitner-Institut Berlin Grnbl-i

ZentrabbIlothek

1993 VCH Publishers, Inc. This work is subject to copyright. All rights reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Registered names, trademarks, etc., used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printed in the United States of America ISBN 1-56081-561-2 VCH Publishers, Inc. ISBN 1-56081-626-0 VCH Publishers, Inc. (paper cover) ISBN 3-527-89561-2 Verlagsgesellschaft ISBN 3-527-89626-0 Verlagsgesellschaft (paper cover) Printing History: 10 9 8 7 6 5 4 3 2 1 Published jointly by VCH Publishers, Inc. 220 East 23rd Street New York, New York 10010

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PREFACE The purpose of this book is to explain electrode kinetics, not to derive it. Consequently, the main emphasis is on the physical phenomena rather than the mathematical formalism. It is written from the point of view of an experimentalist, the emphasis being on the underlying principles of the different experimental techniques, not necessarily on their technical details. It is impossible to write an advanced text in any area of 'physical chemistry without resort to some mathematical derivations, but these have been kept to a minimum consistent with clarity, and used mostly when several steps in the derivation involve approximations, or some other physical assumption, which may not be obvious to the reader. Thus, the theories of the diffuse-double-layer capacitance and of electrocapillary thermodynamics are derived in some detail, while the discussion of the diffusion equation is limited to the translation of the conditions of the experiment to the corresponding initial and boundary conditions and the presentation of the final results, while the sometimes tedious mathematical methods of solving the equations are left out. The mathematical skills needed to comprehend this book are minimal, and it should be easily followed by anybody with an undergraduate degree in science or engineering. An elementary knowledge of thermodynamics and of chemical kinetics is assumed, however. This book is intended both for self study and as a graduate textbook. Each of the two parts can serve as the basis of a onesemester course. In Part one I have included the very minimum needed for developing a basic understanding of interfacial electrochemistry. In Part two some of the same subjects are dealt with in further detail and new subjects are introduced, to provide a broader appreciation of this area of science. It is recommended for scientists and engineers in chemistry, chemical engineering, materials science, corrosion, battery vi i

PREt'AL-12.

viii

development and electroplating. It can also be useful as a textbook for graduate students in electrochemistry, electrochemical engineering or

ACKNOWLEDGMENTS It is a great pleasure to acknowledge the contribution of several

materials science. Special efforts were taken to make the figures informative. Some

colleagues who have helped with advice and criticism to improve the

are based on data reported in the literature. Others, which are derived from equations given in the text, are all simulated data, employing

Kirowa-Eisner of Tel-Aviv University for her critical comments and

reasonable parameters, which one could encounter in an actual experiment. So-called schematic diagrams have been avoided as much as possible No book encompassing such a broad field of science can be comprehensive and yet manageable in size. A choice has to be made. This depends to some extent on the personal preferences of the author. Here an effort was made to find a good balance between the two major schools of thought: that viewing electrode kinetics and interfacial electrochemistry as a branch of physical chemistry and that approaching it from the point of view of analytical chemistry. Thinking of it philosophically, what matters is how one teaches, not what one teaches. A good teacher is one who leaves his students excited about the subject, challenged, stimulated and intrigued by it. Understanding part of the field in depth is to be preferred over learning most of it superficially. It is this goal which I have tried to achieve when writing the book and it is by these criteria that it should be judged by the reader.

quality of this book. First and foremost thanks are due to Professor E. suggestions and her help in the preparation of the numerical data for many of the figures. Her help in the preparation of this book has been invaluable. My colleagues in the Department of Materials Science at the University of Virginia, Professor G. E. Stoner, Dr. S. R. Taylor, and Dr. R. G. Kelly, read parts of the manuscript and helped in streamlining it and eliminating errors. Professor E. Peled of Tel-Aviv University made valuable suggestions on the chapter discussing batteries, and Dr. J. Penciner, also of Tel-Aviv University, read the manuscript with great care and improved it linguistically. The dedicated efforts of Mrs. D. Markovski in preparing the figures is greatly appreciated. I also thank Professor Stoner, Director of the Center of Electrochemical Science and Engineering of the University of Virginia and the Center for Innovative Research of the State of Virginia on the one hand, and Professor M. Costa, Head of the Laboratoire d'Electrochimie Interfaciale and the French Centre National de la Recherche Scientifique (CNRS) for financial support during the time that parts of this book were written. Finally, thanks to my wife Dalia, because she was always there.

E. Gileadi Tel-Aviv January 1993

ix

CONTENTS PART ONE 1

A. INTRODUCTION 1. GENERAL CONSIDERATIONS

1

1 1 The Current-Potential Relationship 1.2 The Resistance of the Interphase Can Be Infinite

2

1.3 The Transition from Electronic to Ionic Conduction.

3

1.4 Mass Transport Limitation

4

1.5 The Capacitance at the Metal-Solution Interphase

6

2. POLARIZABLE AND NONPOLARIZABLE INTERPHASES 2.1

8

Phenomenology

2.2 The Equivalent Circuit Representation

10

2.3 The Electrochemical Timer

12

B. THE POTENTIALS OF PHASES

15

3. THE DRIVING FORCE 3.1

General Considerations

15

3.2 Definition of the Electrochemical Potential 3.3

Separability of Chemical and Electrical Terms

16 16

4. TWO CASES OF SPECIAL INTEREST 4.1

Equilibrium of a Species Between Two Phases in Contact

4.2 Two Identical Phases Not at Equilibrium

19 21

5. COMPONENTS OF THE MEASURED POTENTIAL 5.1

A Cell with Two Different Electrodes

5.2 The Metal-Solution Potential Difference at Redox Electrodes

xi

22 24

CONTENTS 5.3

A Cell with the Same Redox Reaction on Different Electrodes

L uiv

9.4 The Nernst Diffusion Layer Thickness 26

6. THE MEANING OF THE NORMAL HYDROGEN ELECTRODE (NIIE) SCALE

10. METHODS OF MEASUREMENT 10.1

Potential Control versus Current Control

10.2 The Need to Measure Fast Transients 6.1

Thermodynamic Approach

27

6.2 Which Potential Difference Is Defined as Zero?

29

6.3 The Modified Normal Hydrogen Electrode (MNHE) Scale

30

C. FUNDAMENTAL MEASUREMENTS IN ELECTROCHEMISTRY

33

10.3

Polarography and the Dropping Mercury Electrode

10.5 Further Aspects of the RDE and Similar Configurations

91

E. SINGLE-STEP ELECTRODE REACTIONS

Definition and Physical Meaning of Overpotential

11.1

7.2

Use of a Nonpolarizable Counter Electrode

11.2 Types of Overpotential

33

7.3 The Three-Electrode Measurement

34

7.4 Residual iR

35

REFERENCE ELECTRODES

8.3

37 38

Calculation of the Uncompensated Solution iR s Potential 39

8.4 Positioning the Reference Electrode

44

8.5

47

The Experimental Tafel Equation

106

108 109

12.3 The Equation for a Single-Step Electrode Reaction

111

12.4 Limiting Cases of the General Equation

116

13. THE SYMMETRY FACTOR IN ELECTRODE KINETICS The definition of

p

13.2 The Numerical Value of 13 13.3

Is the Symmetry Factor Potential Dependent?

F. MULTI STEP ELECTRODE REACTIONS D. ELECTRODE KINETICS: SOME BASIC CONCEPTS

101

12.2 The Absolute Rate Theory

13.1

Drop for a Few Simple Geometries Primary and Secondary Current Distribution

101

12. FUNDAMENTAL EQUATIONS OF ELECTRODE KINETICS 12.1

8. CELL GEOMETRY AND THE CHOICE OF

8.2 The Use of Auxiliary Reference Electrodes for the Study of Fast Transients

72 82

The Cell Voltage Is the Sum of Several Potential Difference 33

Types of Reference Electrodes

64

10.4 The Rotating Disc Electrode (RDE)

7.1

8.1

61

11. THE OVERPOTENTIAL

7. MEASUREMENT OF CURRENT AND POTENTIAL

Potential Drop in a Three-Electrode Cell

57

120 123 124 127

51 14. MECHANISTIC CRITERIA

9. RELATING ELECTRODE KINETICS TO CHEMICAL KINETICS 9.1

The Relation of Current Density to Reaction Rate

51

9.2 The Relation of Potential to Energy of Activation

53

9.3

55

Mass Transport versus Charge-Transfer Limitation

14.1

The Transfer Coefficient a and Its Relation to

127

14.2

Steady State and Quasi-Equilibrium

131

14.3

Calculation of the Tafel Slope

134

14.4 Reaction Orders in Electrode Kinetics

140

CONTENTS

xiv

CONTENTS

xv

14.5 The Effect of pH on Reaction Rates

144

17.3 Derivation of the Electrocapillary Equation

230

14.6 Isotope Effects Depend on the Mechanism

148

14.7 The Stoichiometric Number

149

17.4 The Electrocapillary Equation for a Reversible Interphase

238

14.8 The Enthalpy of Activation

150

14.9 Some Experimental Considerations

154

18. METHODS OF MEASUREMENT AND SOME RESULTS 18.1 The Electrocapillary Electrometer

241

18.2 The Drop-Time Method

248

161

18.3 Integration of the Double-Layer Capacitance

249

15.2 The Hydrogen-Evolution Reaction on Platinum

164

18.4 Some Experimental Results

252

15.3 Hydrogen Storage and Hydrogen Embrittlement

169

15.4 Possible Paths for the Oxygen Evolution Reaction

172

15.5 The Role and Stability of Adsorbed Intermediates

177

I. INTERMEDIATES IN ELECTRODE REACTIONS

15.6 Catalytic Activity: the Relative Importance of i o and b

179

19. ADSORPTION ISOTHERMS FOR INTERMEDIATES FORMED BY CHARGE TRANSFER

15.7 Adsorption Energy and Catalytic Activity

182

15. SOME SPECIFIC EXAMPLES 15.1 The Hydrogen Evolution Reaction on Mercury

G. THE IONIC DOUBLE-LAYER CAPACITANCE C dl

185

16. THEORIES OF DOUBLE-LAYER STRUCTURE

PART TWO

19.1 The Langmuir Isotherm and Its Limitations

261

19.2 The Frumkin and Temkin Isotherms

266

19.3 Introducing the Temkin Isotherm into the Equations of Electrode Kinetics

271

19.4 Calculating the Tafel Slopes and Reaction Orders Under Temkin Conditions

273 276 280

16.1 Phenomenology

185

16.2 The Parallel-Plate Model of Helmholtz

188

16.3 The Diffuse-Double-Layer Theory of Gouy-Chapman

190

16.4 The Stern Model

195

19.5 Some Special Aspects of the Use of the Temkin Isotherm in Electrode Kinetics

16.5 The Role of the Solvent in the Interphase

200

19.6 Underpotential Deposition

16.6 Diffuse-Double-Layer Correction in Electrode Kinetics

205

16.7 Application of Diffuse-Double-Layer Theory in Plating

211

16.8 Modern Instrumentation for the Measurement of C dl

213

H. ELECTROCAPILLARITY

225

17. THERMODYNAMICS 17.1 Adsorption and Surface Excess

225

17.2 The Gibbs Adsorption Isotherm

228

261

20. THE ADSORPTION PSEUDOCAPACITANCE C 20.1 Formal Definition of C and Its Physical Significance (1) 20.2 The Equivalent Circuit Representation 20.3 Calculation of C as a function of 0 and E (i) 20.4 The Case of a Negative Value of the Parameter f

291 293 296 303

)t, .-.1TS

X

J. ELECTROSORPTION

307

21. PHENOMENOLOGY 21.1 What Is Electrosorption?

,JNTEN

L. EXPERIMENTAL TECHNIQUES: 2

403

25. LINEAR POTENTIAL SWEEP AND CYCLIC VOLTAMMETRY 307 309

21.2 Electrosorption of Neutral Organic Molecules 21.3 The Potential of Zero Charge and its Importance in Electrosorption

318

21.4 Methods of Measurement of Coverage on Solid Electrodes

322

22. ISOTIIERMS

25.1 Three Types of Linear Potential Sweep 25.2 Double Layer Charging Currents 25.3 The Form of the Current-Potential Relationship

403 405 409

25.4 Solution of the Diffusion Equation

410

25.5 Uses and Limitations of the Linear Potential Sweep Method 25.6 Cyclic Voltammetry for Monolayer Adsorption

414 420

22.1 General Comments

329

22.2 The Parallel-Plate Model of Frumkin

332

22.3 The Water-Replacement Model of Bockris, Devanathan and Muller

26.1 Introduction

428

335

26.2 Graphical Representation

431

22.4 The Combined Adsorption Isotherm of Gileadi

340

26.3 The Effect of Diffusion Limitation

436

22.5 Application of the Gileadi Combined Adsorption Isotherm to Electrode Kinetics

344

26.4 Some Experimental Results

440

K. EXPERIMENTAL TECHNIQUES: 1

349

23. FAST TRANSIENTS

26. ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY (EIS)

27. MICROELECTRODES 27.1 The Unique Features of Microelectrodes 27.2 Enhancement of Diffusion at a Microelectrode

443 445

23.1 The Need for Fast Transients

349

27.3 Reduction of Solution Resistance

447

23.2 Small-Amplitude Transients

354

27.4 Single Microelectrodes versus Ensembles

448

23.3 The Sluggish Response of the Electrochemical Interphase

356

27.5 Shapes of Microelectrodes and Ensembles

453

23.4 How to Overcome the Slow Response of the Interphase

357

M. APPLICATIONS

23.5 Analysis of the Information Content of Fast Transients

366

28. BATTERIES AND FUEL CELLS

24. LARGE-AMPLITUDE TRANSIENTS 24.1 Open-Circuit-Decay Transients 24.2 The Diffusion Equation and Its Boundary Conditions 24.3 Single-Pulse Techniques 24.4 Reverse-Pulse Techniques

374 376 390 397

28.1 General Considerations 28.2 The Maximum Energy Density of Batteries 28.3 Types of Batteries 28.4 Design Requirements and Characteristics of Batteries 28.5 Primary Batteries

455

455 456 458 460 462

CONTENTS

xviii 28.6 Secondary Batteries

470

28.7 Fuel Cells

476

28.8 Porous Gas-Diffusion Electrodes

484

28.9 The Polarity of Batteries

488

Scope and Economics of Corrosion

29.2 Fundamental Electrochemistry of Corrosion 29.3

Micro Polarization Measurements

PART ONE A. INTRODUCTION 1. GENERAL CONSIDERATIONS

29. CORROSION 29.1

490 492 499

29.4 Potential/pH Diagrams

502

29.5 Passivation and Its Breakdown

513

29.6 Localized Corrosion

519

29.7 Corrosion Protection

526

1.1 The Current-Potential Relationship From a phenomenological point of view, the study of electrode kinetics involves the determination of the dependence of current on potential. It is therefore appropriate that we start this book with a general qualitative description of such a relationship, as shown in Fig. 1A.

30. ELECTROPLATING General Observations

14

538

30.2 Macro Throwing Power

540

Micro Throwing Power

552

30.3

30.4 Plating from Nonaqueous Solutions

558

BIBLIOGRAPHY

565

LIST OF ACRONYMS

579

LIST OF SYMBOLS

581

SUBJECT INDEX

589

Curren tdens ity/ mA crn '

30.1

1

A. INTRODUCTION

I ac 12

10

a 8

6

Infinite resistance

Activation control

Mixed control

4

Mass— transport control

2

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

18

Potential /V

Fig. IA 11E plot for the electrolysis of a dilute (0.01 M) solution of K1 in H SO , employing two Pt electrodes. The minimum potential 2 4 for current flow is 0.59 volt. Line a is the purely activation controlled current 1 , line b is the actual current which will ac

be measured, having a mass transport limited value 1c

L.L.LL 11(4._,IJE iuNeries

In the simplest case E is the potential applied between two

3

A. 1N4

namely, 2H 0 2

electrodes in solution and / is the current flowing in the circuit. Curve a in Fig. 1A represents the dependence of current on potential

2H

2

+

0

(1A)

2

This is "up the free energy ladder."

Evidently this reaction

when the process is controlled by the kinetics of the reaction alone.

proceeds spontaneously in the opposite direction (the burning of

Curve b takes into account the effects of mass transport. These

hydrogen); and hence electrical energy must be supplied to make the

concepts are explained in detail in the section that follows. In actual

reaction happen, according to the well-known relationship:

measurement the potential E is always measured versus a fixed reference AG = — nFE rev

electrode and instead of the current one refers to the current density on the electrode being studied, but at this point we need not concern ourselves with these refinements.

(2A)

The negative sign in this equation shows that when the free energy decreases, the potential is positive — the cell acts as a source of

It is immediately obvious from Fig. IA, that Ohm's law does not

electrical energy, and vice versa. The free energy for water electroly-

apply, not even as a rough approximation. This observation is not as

sis at room temperature is 237.16 kJ/mol, leading to a minimum potential

trivial as it may seem when we recall that in the study of conductivity

of 1.229 V required for the reaction to occur.

of electrolytic solutions, Ohm's law is strictly obeyed over a very

Replacing the platinum electrodes with copper, and adding some

large range of potentials and frequencies. The difference is that Fig. IA pertains to measurements conducted under dc conditions, whereas

CuSO4 , changes the situation radically. Passing a current between the electrodes causes no net chemical change (copper is dissolved off one

ionic conductivity is measured, as a rule, with an alternating current

electrode and deposited on the other). In this case current is observed

or potential. The implication is that the impedance of the metal-

as soon as a potential, small as it may be, is applied to the

solution interphase is partially capacitive — a subject to be dealt with

electrodes.

in considerable detail shortly. 1.3 The Transition from Electronic to Ionic Conduction 1.2 The Resistance of the Interphase Can Be Infinite

If one were to describe the essence of electrode kinetics in one

Looking at Fig. 1 A carefully, one observes that up to a certain

short phrase, it would be: the transition from electronic to ionic

potential the current is zero. This is not a matter of limited sensi-

conduction, and the phenomena associated with and controlling this

tivity of the measuring instrument. The current is exactly zero

process. Conduction in the solution is ionic, whereas in the electrodes

(disregarding minor impurity effects), corresponding effectively to an

and the connecting wires it is electronic. The transition from one mode

infinite resistance of the interface. The reason for this observation

of conduction to the other requires charge transfer across the inter-

is thermodynamic. Curves such as those shown in Fig. IA are obtained

faces. This is a kinetic process. Its rate is controlled by the

when electrolysis causes a net chemical reaction. Passing a current

catalytic properties of the surface, the chemisorption of species, the

between two platinum electrodes in a pure solution of sulfuric acid is a

concentration and the nature of the reacting species and all other

good example. The reaction taking place is the electrolysis of water,

parameters that control the rate of heterogeneous chemical reactions.

4

ELECTRODE KINETICS

In addition, the potential plays an important role.

This is not

surprising, since charge transfer is involved, which may be accelerated

and that controlled by mass transfer iL' we can write for the observed current density i the simple relationship:

by applying a potential difference of the right polarity across the

1/i = 1/i ac + 1/i L

interphase. The current would continue to rise exponentially with potential, along line a in Fig. 1A, were it not for mass transport limitation, represented by the horizontal part of line

b.

controlled" or "activation controlled". The detailed dependence of current on potential in this region is discussed later.

1.4 Mass Transport Limitation The rate of charge transfer can be increased very much by increasing the potential, but charge can be transferred over a very short distance (of the order of 0.5 nm) only. Another process is required to bring the reacting species close enough to the surface, and to remove the species formed at the surface into the bulk of the solution. This process is called mass transport. Mass transport and charge transfer are two consecutive processes. It is therefore always the slower of the two that determines the overall rate observed experimentally. When the potential applied is low, charge transfer is slow and one can ignore the mass transport limitation. The bottleneck is in transferring the charge across the interphase to the electroactive species, not in getting the species to the surface. At high potentials, charge transfer becomes the faster process and ceases to influence the overall rate. Increasing the potential further will increase the rate of charge transfer, but this will have no effect on the overall rate, which is now limited by mass transport. The result is a current that is independent of potential, which is referred to as the

limiting current /L. If we denote the current density controlled by charge transfer i ac

(3A)

Clearly, the smaller of the two currents is dominant. The masstransport-limited current density can be written in the form

In the initial rising

part of the curve the reaction is said to be "charge-transfer

5

A. INTRODUCTION

i = nFDC°/8

(4A)

D is the diffusion coefficient (cm 2/s), C° is the concentration (mol/cm 3 ), and 8 in which nF is the charge transferred per mole (C/mol),

is the Nernst diffusion layer thickness (cm). Calculated in these units, the current density is obtained in amperes per square centimeter (A/cm2). Now, the essence of mass transport is the quantity 8. In certain favorable cases it has been calculated theoretically, in others it can only be determined experimentally. Sometimes it is a function of time, while under different circumstances it is essentially constant during an experiment. Stirring the solution and flowing it at, past, or through the electrode all decrease the effective value of 8, hence increase

i .

Moving the electrode (rotation, vibration, ultrasonics) has a similar effect. In quiescent solutions 8 increases linearly with t l/2, hence iL canbeirsdytkgmauenshorti. In typical electrochemical measurements, the Nernst diffusion layer thickness attains values in the range of 8 = 10 3-10-1 cm. Since in aqueous solutions at room temperature the diffusion coefficient is on the order of 10 5 cm2/s, this yields limiting current densities in the range of 0.01-1.0 A/cm 2 , when the concentration of the electroactive species in solution is 1.0 M. The two most important things to notice in Eq. 4A are (a) that the limiting • current density is independent of potential, and (b) that it depends linearly on the bulk concentration. A less obvious, but equally

6

ELEC a. )DE KINETICS

A.

INTRODUCTION

important, consequence of this equation is that i is independent of the kinetics of the reaction (i.e., of the nature of the surface and its

a plane separating the two phases. This is, however, an ill-defined

catalytic activity). These characteristics make it an ideal tool for

quantity, since it is not possible to define exactly the boundary of a

probing the concentration of species in solution. This is why most

phase, even a metal, on the subatomic level. In other words, one does

electroanalytical methods depend in one way or another on measurement of the mass-transport-limited current density.

not know exactly "where the metal ends". Is it the plane going through

1.5 The Capacitance at the Metal-Solution Interphase When a metal is dipped in solution, a discontinuity is formed. This affects both phases to some degree, so that their properties near the contact are somewhat different from their bulk properties. We define the interphase as the region in which the properties change from those found in the bulk of one phase (the metal) to those in the bulk of the other phase (the solution). This is shown schematically in Fig. 2A, where some property X (which could be charge density or potential, for example) is plotted as a function of distance.

One may be tempted to discuss the system in terms of the interface:

the center of the outermost layer of atoms, is it one atomic radius farther out, or is it even farther, where the charge density wave of the free electrons has decayed to essentially zero? Fortunately, we do not need to know the position of this plane for most purposes when we discuss the properties of the interphase, as defined earlier. One such property is the capacitance, which is observed whenever a metal-solution interphase is formed. This capacitance, called the double layer capacitance, Cdt is a result of the charge separation in the interphase. Since the interphase does not extend more than about 10 ,

nm in a direction perpendicular to the surface (and in concentrated solutions it is limited to 1.0 nm or less), the observed capacitance depends on the structure of this very thin region, called the double If the surface is rough, the double layer will follow its layer. curvature down to atomic dimensions, and the capacitance measured under suitably chosen conditions is proportional to the real surface area of the electrode. The double-layer capacitance is rather large, on the order of 10-30 tif/cm2 . This presents a serious limitation on our ability to

x metal

study fast electrode reactions. Thus, a 10 1..tF capacitor coupled with a 1.0 SI resistor yield a time constant of T e = 10 µs. It is possible to solution

Distance

Fig. 2A Schematic representation of an interphase, showing the gradual change of a property X from its value in the bulk of one phase to that in the bulk of the other phase.

take measurements at shorter times using special techniques, but even so, the lower limit at present seems to be about 0.05 six orders of magnitude slower than that currently achievable in the gas phase. The double-layer capacitance depends on the potential, the composition of the solution, the solvent and the metal. It has been the subject of numerous investigations, some of which are discussed later.

ELECTRODE KINETICS

8

A. INTRODUCTION

2. POLARIZABLE AND NONPOLARIZABLE INTERPHASES

; Nonpolarizable

2.1 Phenomenology When a small current or potential is applied, the response is in many cases linear. The effective resistance can, however, vary over a wide range. When this resistance is high, we refer to it as a polarizable interface, since a small current generates a high potential across it (i.e., it polarizes the interphase to a large extent). When the effective resistance is low, the interphase is said to be

Fig. 3A i/E plots for polarizable & nonpolarizable interphases. It is hard to pass a current across a polarizable

rr

•

Polarizable

E

interphase, while it is hard to change the potential of a nonpolarizable interphase.

nonpolarizable. In this case a significant current can be passed with only minimal change of the potential across the interphase. The current-potential relationship for the two cases is shown schematically in Fig. 3A. A nonpolarizable electrode is, in effect, a reversible electrode. The potential is determined by the electrochemical reaction taking place and the composition of the solution, through the Nernst equation. For a

Polarizable interphases behave differently. Their potential is not fixed by the composition in solution, and it can be changed at will over a wide range (until a potential is reached at which the interphase is no longer polarizable). For such system the potential may be viewed as an additional degree of freedom in the thermodynamic sense, as used in the Gibbs phase rule. To be sure, a so-called nonpolarizable interphase can

copper electrode in a solution containing CuSO 4 this is

be polarized, if we force a potential across it. This, however, causes (5A)

a current to flow, which alters the concentration of species on the

in which E° = 0.340 V, versus NHE is the standard reversible potential for the Cu 2+/Cu couple, on the Normal Hydrogen Electrode scale, and

solution side of the interphase, in agreement with the Nernst equation.

E = E° + (2.3RT/nF)log(a cu.)

acu++ is the activity of cupric ions in solution, which can often be represented approximately by the corresponding concentration. A good reference electrode is always a reversible (i.e. non polarizable) electrode. The converse is not necessarily true. Not every reversible electrode is suitable as a reference electrode. For example, the correct thermodynamic reversible potential of a metalmetal-ion electrode may be hard to reproduce, due to impurities in the metal or complexing agents in the solution, even when the interphase is highly non-polarizable.

To clarify this point, let us use the example given above, of Cu dipped in a solution containing Cu 2+ ions at a concentration of 0.1 M. The potential, according to Eq. 5A is 0.31 V versus NHE. Now if we polarize this electrode negatively to, say, 0.28 V, copper will be deposited and the concentration of Cu 2+ ions near the interphase will decrease. Steady state is attained when the concentration on the solution side of the interphase has reached roughly 0.01 M, which corresponds to the potential of 0.28 V, NHE. If the solution originally used had been 0.01 M with respect to Cu 2, + this potential would have been observed at open circuit, with no current flowing. In the present example the bulk concentration is 0.1 M and the surface concentration is maintained at

10

KINETICS

A. IN I KC)

0.01 M only as long as a potential is forced across the interphase and a

Now, the equivalent circuit shown in Fig. 4A represents a gross

current is flowing. The important point to remember is that the

oversimplification, and interphases rarely behave exactly like it. It

potential responds reversibly (i.e. according to the Nernst equation) to the concentration of the electroactive species at the surface, not in

does, nevertheless, help us gain some insight concerning the properties of the interphase.

the bulk of the solution.

The combination of the double layer capacitance Cdi and the faradaic resistance R represents the interphase. How do we know that

2.2 The Equivalent Circuit Representation

C

dl

and R Fmust be put in a parallel rather than in a series combina-

We have already seen that the metal-solution interphase has some

tion? Simply because we can observe a steady direct current flowing

capacitance associated with it, as well as a (non ohmic) resistance.

when the potential is high enough (above the minimum prescribed by

Also, the solution has a finite resistance that must be taken into

thermodynamics). Also when the resistance is effectively infinite under dc conditions, we can still have an ac signal going through.

account. Thus, a cell with two electrodes can be represented by the

The equivalent circuit just described also makes it clear why

equivalent circuit shown in Fig. 4A. Usually one considers only the part of the circuit inside the

conductivity measurements are routinely done by applying an ac signal.

dashed line, since the experiment is set up in such a way that only one

If the appropriate frequency is chosen, the capacitive impedances associated with C

of the electrodes is studied at a time.

resistance R

can be made negligible compared to the ohmic dl which is thus effectively shorted, leaving the solution

F resistance R s as the only measured quantity. As pointed out earlier, the equivalent circuit shown in Fig. 4A is meant to represent the simplest situation only. It does not take into C

C dl

1--

I I—

—

---I— AAMW---

the occurrence of reaction intermediates absorbed at the surface. Some of these factors are discussed later. Even in the simplest cases, in

MANARF

account factors such as mass transport, heterogeneity of the surface and

which this circuit does represent the response of the interphase to an RF

electrical perturbation reasonably well, one should bear in mind that both C and RFdepend on potential and, in fact, RFdepends on potendl tial exponentially over a wide range of potentials, as will be discussed

Fig. 4A Equivalent circuit for a two-electrode cell. A Single interphase is usually represented by the elements inside the dashed rectangle. CdI , R F and R s represent the double-layer capacitance, the faradaic resistance and the solution resistance, respectively.

later. The difference between polarizable and nonpolarizable interphases can be easily understood in terms of this equivalent circuit. A high value of R Fis associated with a polarizable interphase, whereas a low value of R represents a nonpolarizable interphase. F

ELECTRODE KINETICS

12

13

A. INTRODUCTION

The anodic process is oxygen evolution, whereas the cathodic

2.3 The Electrochemical Timer

process changes from thallium deposition to include cadmium deposition

The electrochemical timer is a device that can be set to switch a circuit on or off at a given time. It was of great practical importance

and eventually also hydrogen evolution, as the potential is gradually increased.

until the development of microelectronic digital devices, since it could

Imagine now a cell having two gold electrodes and a solution of

be set to operate for periods of minutes to months, with an accuracy of better than 1%. We describe it here to show how an understanding of the

AgNO in HNO3 . One of the electrodes is coated with an exactly known 3 amount of silver, and this electrode serves as the anode. The cell is

fundamental electrochemical processes taking place can lead to the

connected to a galvanostat (which supplies a constant current) and the

development of a simple and very useful device.

potential across it is measured. This setup constitutes an electro-

To understand the operation of the "electrochemical timer" we must

chemical timer. At first the reactions are the same at both electrodes,

review the current-potential relationship shown in Fig. 1,A. What

but going in opposite directions, namely, silver dissolution at the

happens when there is more than one electroactive species in solution? This modification is shown in Fig. 5A for a solution containing T1 + and

anode and its deposition at the cathode. The cell potential is the sum of three main potentials: the reversible potential corresponding to the

Cd2+ ions, employing a mercury cathode.

reaction taking place, the polarization at both electrodes, and the potential drop across the solution resistance,

■

0

iR s . In the cell just described the reversible potential is zero, since no net chemical change takes place. The polarization is small, since the kinetics of silver

/V

TI 4/TI(Hg)

–20

dissolution and deposition is fast, yielding low values of R F(the interphase is nonpolarizable). The iR potential drop depends on

s

geometry but is irrelevant for our purpose, as will be shown in a –40

Cd 4-4/Cd(Hg)

moment. Now, after a certain length of time, the charge passed will be great enough to dissolve all the silver from the anode. The next

–60

possible anodic reaction that can take place is oxygen evolution.

–80 –18

I –1.6

The

cell reaction will now be

0/H 2

–1.4

A,

–0.8

–0.6

–0.4

–0 2

E/V vs NHE

Fig. 5A ilE plot for a system containing two reducible ions. The curves were calculated for a dropping mercury electrode, with the Tr ion concentration twice that of the Cd 2+ ion, in order to obtain a nearly equal current wave.

anodic

2H 0 —4 0 + 4H + + 4e 2 2 N4

E° = 1.23 V

(6A)

cathodic

4Ag+ + 4em —› 4Ag

E° = 0.79 V

(7A)

AE° = 0.44 V

(8A)

2H 2 0

+

4Ag+ --> 0 + 4Ag + 4H+ 2

The potential must increase suddenly by at least 0.44 V, because of the change in the reaction taking place in the cell.

An additional

14

1.......:11tODE KINETICS

increase is observed in practice, since the oxygen evolution reaction on

B. 'FHB PO

..

.,,^

B. THE POTENTIALS OF PHASES

gold is much slower than silver dissolution and a substantial polarization results. The iR

s

potential drop across the solution remains

unchanged, since the current has not changed. Thus, a sudden change of cell potential of about 1.0 V will occur, enough to activate an electronic switch. How sudden is the potential jump? What happens when most of the gold surface serving as the anode is bare, and only a small fraction is still covered with silver? The answer to the first question depends on the current density and the double-layer capacitance. The rate of change of potential is given by

3. THE DRIVING FORCE 3.1 General Considerations Knowledge of the driving force is of utmost importance for the understanding of any system. It determines the direction in which the system can move spontaneously, as well as its position of equilibrium, at which the driving force is zero along all coordinates. In mechanics, the driving force is the gradient of the potential energy, U. That is why a marble rolls to the bottom of a bowl and water falls over a dam, allowing us to produce hydroelectric power.

i = Cdi(dE/dt)

(9A)

In chemistry we are used to thinking of the gradient of the chemical potential, t, as the driving force.

For 20 1.1F/cm 2 and 50 11A/cm2 , Eq. 9A yields dE/dt = 2.5 V/s, corresponding to a switching time of about 0.4 second in the present case.

driving force = — grad 1.1.

(1B)

The second question is more interesting, from the point of view of

How can this be? Surely the laws governing physical and chemical

fundamental electrochemistry. What happens to an electrode on which two

phenomena must be the same, and indeed they are. The answer is that in

reactions can occur simultaneously? In the present case both kinetics

mechanics, as well as in chemistry, one can consider the gradient of

and thermodynamics favor silver dissolution over oxygen evolution, hence

chemical potential to be the real driving force. The chemical poten-

this will be the main reaction, until the surface is covered with well

tial, being the free energy per mole for a pure substance, differs from

below 1% of silver. As the area of the electrode still covered with

the energy by an entropy term.

silver decreases, the effective current density for its dissolution increases, along with the corresponding polarization, while the total current is forced to remain constant. At a certain point the polarization will be sufficiently high for the next reaction (oxygen evolution) to occur. The potential does not change much until most of the silver has been removed from the surface, but then changes rapidly while the last small amount (say, 1% or less) is dissolved and the completely bared gold electrode is exposed to the solution.

A = U — TS

(2B)

Now, when a body slides down a slope, the entropy remains constant. Thus, the change in energy equals the change in free energy or chemical potential. Summing it up, the gradient of chemical potential (or free energy) is the driving force in chemistry as well as in mechanics. In the latter the entropy does not change, therefore the gradient of free energy equals the gradient of potential energy.

16

ELEL. i RODE KINETICS

17

B. THE POTENTIALS OF PHASES

the phase 4). But can such separation be made?

3.2 Definition of the Electrochemical Potential Let us turn our attention now to processes involving charged species, in particular charge-transfer processes. We recall that the

One recalls that the potential

4)xyz at some point in space is defined as the energy required to bring a unit positive test charge from infinity to this point. This is fine as long as we move the charge in

chemical potential relates to the activity of the species:

free space or inside one homogeneous phase. But what happens when we (3B)

try to determine the potential inside a phase, with respect to a point

The activity is related to the concentration via the activity coeffi-

at infinity in free space, or the difference in potential between points

cient 7., which itself is a function of concentration:

in two different phases? As the "test charge" crosses the boundary of a

+ RT ln(ai)

=

(4B)

ai = y•Ci Now it would seem clear that

phase, it interacts with the molecules in the phase, and it is impossible to distinguish between the so-called "chemical" and

as given in Eq. 3B, does not account

for the effect of the electrical field or its gradient, unless we include that implicitly in the activity coefficient.

We must conclude from the preceding considerations that, while the electrochemical potential Ft is a measurable quantity, its components

It has been found more expedient to define a new thermodynamic function, the electrochemical potential Ft i , which includes a specific term to account for the effect of potential on a charged species: (5B)

z i Ft0

"electrical" interactions in this region.

in which 4) is the inner potential of a phase. When charged species are involved, the driving force is the gradient of electrochemical potential

and 4) cannot be separately measured. The potential 4) in Eq. 5B is called the inner potential of a phase. For the same reason that it cannot be measured, one also cannot measure the value of \4), the difference of inner potentials between different phases. The foregoing statement may seem odd, since we are accustomed to measuring potential differences say, the potential difference (i.e., the

along some coordinate:

ai

driving force = — grad j

voltage) between two terminals of a battery. To do this we connect the (6B)

two terminals of a suitable voltmeter with copper wires to the terminals

By now it should be clear that this does not constitute a new law. If

of the battery. We are therefore measuring, in effect, the potential

the charge on the particle is zero, the chemical and the electrochemical

difference between two identical phases, which, as we shall show, is possible.

potentials are equal, and so are their gradients.

Now consider an attempt to measure the potential difference across 3.3 Separability of the Chemical and the Electrical Terms

the metal-solution interphase (1)"1 —

MAN.% Assume for simplicity

It was noted above that Eq. 5B is an attempt to separate chemical

that the metal used is copper, connected with a copper wire to one of

interactions, represented by 1.t, and electrical interactions, represented

the terminals of a voltmeter. This terminal will then be at the

by the product of charge (per mole of species) zF and the potential of

potential (1) rs". Now, to determine the potential of the solution phase 4s,

18

11_1C REIVt.l 1,a

tit"

we would have to use a copper wire connected to the other terminal of

If the two terms on the right-hand side of Eq. 5B cannot be

the meter, and dip it in the solution. This, however, would create a

separately measured, what is the point of using this equation? It turns

new metal-solution interphase, and the meter would show the sum of two

out that in some special cases of great practical importance, this

metal-solution potential differences. It is important to realize that this is not a technical limitation,

equation does lead to results that can be tested by experiment. The

which may be overcome as instrumentation is improved. In any "thought

usefulness of the electrochemical potential as defined by Eq. 5B is in

single interphase necessarily creates at least one more interphase. To end this section on a positive note, it should be pointed out

distinguishing between "short-range" interactions, represented by j.t, and "long-range" interactions, represented by zF4. The energy of interaction for the former typically decays with ( 6 , while that for the latter decays with r - I . This behavior is shown schematically in Fig. 1B.

that while 04) cannot be measured, changes in AO (caused, e.g., by passing a current) are readily measurable. Indeed when ilE measurements

In Fig. 1B an initial value of 200 kJ/mol is assumed for a chemical bond and the electrostatic energy is taken as 20 kJ/mol at the same

are recorded in electrochemical research, it is commonly the change of

distance. Starting off with a chemical interaction energy 10 times

potential difference 6(4) at one of the electrodes, rather than the

higher than the electrostatic interaction, the two are equal when the

change in cell potential, which is measured.

distance has been increased by 58%, and the electrostatic energy becomes

experiment" one may devise, an attempt to measure the value of 64 at a

10 times larger than the chemical energy when the distance has increased r/r, 1.0

1.2

2.5

2.0

1.5

3

by a factor of 2.5. Thus, electrostatic interactions between charged species predominate everywhere, except very close to the boundary between two phases.

3.0 10 3

10 2 7

zq F

on 1

4. TWO CASES OF SPECIAL INTEREST 10

0

RT

4.1 Equilibrium of a Species Between Two Phases in Contact

c. 10 0

70

0.1

0.2

0.3

0.4

-t

0.5

log (r/r o )

Fig. I B Variation of bonding energy with distance for short range, covalent interactions and long range, electrostatic interactions, marked by u. and zF4, respectively. The average thermal energy at 25° C, RT, is given for comparison.

Consider Eq. 5B for the case of a species at equilibrium in two different phases — for instance, an electron in a copper wire and a nickel wire welded together. Since equilibrium is presumed, we can write — Cu

— Ni

ti e = ge

(7B)

Combining with Eq. 5B we have Cu

—

Ni F(I) = e Ni—FA N' Cu

(8B)

20

ELECTRODE KINETICS

cuNi p.e/F = cuAN i 4)

(9B)

21

B. THE POTENTIALS OF PHASES

Remembering that the activity of a species in a pure phase a m +, is m always defined as unity, we can write Eq. 13B in simplified form as follows:

The physical significance of Eq. 9B is that at the contact between two dissimilar metals, a certain potential difference will develop, generated by the difference in chemical potential of the electrons in the two metals. It might at first seem odd to have a potential drop inside a metal, unless a current is flowing. In the present case, however, one might consider 64 as the potential difference needed to oppose the flow of electrons in the direction of decreasing chemical

the Nernst equation in its usual form (i.e., in terms of the measurable cell potential).

potential. Consider another example, that of a metal ion in solution at

4.2 Two Identical Phases Not at Equilibrium

equilibrium with the same ion in the crystal lattice. In this case m+

+ Pl)

m =

m+

+ F(l) s

(10B) (11B)

which is a form of the Nernst equation, written for a single interphase. Combining two such equations corresponding to two half-cells leads to

which the electrochemical potential of a species (an electron this time) is considered in two identical phases that are not at equilibrium. In this case we have m

which is similar to Eq. 9B, except for the sign. We recall that AO in Eqs. 9B and 11B is not measurable, since any attempt to measure this quantity leads to the creation of at least one more interphase, with its own 04. Equation 11B does, however, lead to some very interesting conclusions, as we shall presently see. To do this, let us substitute the expression for chemical potential from Eq. 3B into Eq. 11B o,M

(15B)

We turn our attention now to another special case of Eq. 5B, in

This leads to m Asm.mIF = — mg(I)

L) = AO° + (RT/F)In(a ms )

'=

m' e — 1-71)

and

=- Fo

m

and since

we obtain

_ m 'Am t-e/F = rvi• Am The quantity m 'Amn (1) in Eq. 18B is the actual potential measured between,

s = — F(mAs ) + RT In am+ — p m+s — RT In a m+ m As(1) = (mAs(1)) ° — (RT/F)In(a m + 4 54)

(12B) (13B)

that potential is measured as a rule with a device having a very high input resistance which, in effect, prevents equilibrium between the electrons in its two terminals. The important physical understanding we can gain from Eq. 18B is that the potential difference we measure is nothing but the difference

in which we have defined (W( ° as follows: ( mAs [tm+ )°/F ( m.As0) °

say, two copper wires attached to the terminals of a battery. We note

(14B)

in the electrochemical potentials of the electrons in the two terminals

EL.z (RODE KINETICS

of the measuring instrument (divided by the Faraday F, for consistency of units). It is important to understand clearly the difference between the two special cases of Eq. 5B. First we discussed equilibrium between

B. THE POTENTIALS OF PHASES

in which we have ignored the potential drop across the membrane (dashed line in Fig 2B) which is usually negligible. The measured potential E is the sum of four potential differences across interphases, none of

dissimilar phases. Then we discussed nonequilibrium between identical

which can be individually determined. The potential measured is related to the free energy of the reac-

phases. The former led to:

tion taking place through the appropriate Nernst equation:

A

= 0, hence m 'Am "il/F e = m' Am "4)

m' " ' Am" = 0, and hence A m i.te/F = m Am 4)

(22B)

As such, it cannot depend on the connecting wires, yet it would appear

where m' Am "4) is not measurable. The latter led to: rat

) 2 /a E = E° + (2.3RT/2F)log(a Ag+ 2n++

(19B)

from Eq. 21B that the potential differences at the Ag/Cu' and Cu"/Zn (20B)

interphases do contribute to the measured potential. In this context we might ask ourselves how Eq. 21B would be modified if we used another

where m 'Am 4) is the potential measured between the two (identical)

metal wire, say nickel, to connect the electrodes to the terminals of

terminals of a suitable voltmeter.

the voltmeter. The two metal-solution potential differences would not be affected, but instead of

5. COMPONENTS OF THE MEASURED POTENTIAL 5.1 A Cell with Two Different Electrodes Consider a cell consisting of an Ag/Ag 4 and a Zn/Zn2+ half-cell

Fig. 2B Schematic represen-

measured with a voltmeter. Furthermore, assume that the wires connec-

tation of a Ag/Zn cell. The dashed line in the

ting the electrodes to the voltmeter, as well as the voltmeter termi-

middle represents a memb-

nals, are made of copper. This setup is shown schematically in Fig. 2B.

rane which prevents mixing

We already know that the potential measured by the voltmeter is

of the two solutions but

equal to the difference in the electrochemical potential of electrons in

allows free movement of the ions from one compartment

with a suitable membrane separator. Assume that the potential is

the two copper wires, marked as Cu' and Cu" (cf. Eqs. 18B and 20B). What are all the potential differences that make up this measured quantity? Going around the cell counter clockwise, we have E cu" Acu

=

Cu ' ,eg o AgAS 0 S AZn et, ZnACu" 0

(21 B)

to the other.

24

ELECTRODE KINETICS

25

B. THE POTENTIALS OF PHASES

Zn1Cu" 0

Cu'AAgo

solution potential difference in the above example does depend on the we would now have four terms, namely ZnANi4

NiA AN)

Cu' ANi(i)

nature of the metal, and so does the observed potential E, since the two electrodes participate in the cell reaction.

NiACu"0

Consider now a different type of cell, in which the electrodes Acu Yet we know from thermodynamics that the measured potential E = cu"

serve only as the source or sink of electrons, and in some cases as

must remain unaltered.

catalysts for the charge-transfer reaction taking place across the

There are several ways to explain this apparent discrepancy. One

interphase. As a specific example we shall consider two half-cells

way to look at it is to say that the electrochemical potentials of the

connected through a membrane or salt bridge. One half-cell consists of

electrons in the silver and the zinc electrodes are determined by the

a platinum electrode dipping into a mixture of Fe 2÷/Fe3+ in H2SO4 and

thermodynamics of the equilibrium with their respective ions in solu-

the other is a Pt electrode in a mixture of Ce 3+/Ce4+ also in H SO . 2 4

tion. Since the electrons are at equilibrium between the silver or zinc

The two half-cell reactions are:

electrode and any metal wire connecting them to the terminal of the Ce

voltmeter, we can always write — Ag

i Cu ' = "e

Fe

Zn — II Cu"

and

"e

4+

2+

(23B)

=

(26B)

e —› Ce3+ rvi 3+ Fe + e m

+

(27B)

leading to the overall reaction

Cu Act' "(I) = –Cu' A Cu" / F must also be the Hence the measured potential Ce4

same, irrespective of which metal is used to connect the electrodes to The same conclusion can be reached on the basis of a formal algebraic arguments, considering that NiAAgo = ( 430Cu'

ii)Ni)

Cu' rAg = Cu ' AAg4, = (1) — ip

A

(1) +

+

(28B)

Fe3+

any of the last three equations, but it is implicit in the half-cell

( 4:0Ni

metal, and we might expect the metal-solution potential difference to

(t) Ag)

include a term for the chemical potential of electrons in platinum. (24B)

show this, consider Eq. 27B at equilibrium.

and similarly Ni

Ce3+

reactions, since the electrons are either taken from or returned to the —

a

Fe2+

Now, the metal used as the electrode does not appear explicitly in

the terminals of the voltmeter.

Cu'ANi4)

+

NiCu"

(I) = znAcu" (I)

S "Fe+2

(25B) la

s Fe+2

+ 2F4

=

= 1.ts

We can write

— 11 Pt "e

S "Fe+3

3F.1) S

To

(29B) ift

Fe+3

Rrt

(30B)

5.2 The Metal-Solution Potential Difference at Redox Electrodes S

We have just seen that the observed cell potential does not depend on the leads used to connect the cell to the voltmeter. The metal-

[i I S

PI

=

"Fe+2

IS

"Fe+3

Pt

/F

ji

(31B)

26

PIECCRODE K1NEi1ICS

Similar equations can be derived for the equilibrium shown in Eq. 26B. In both cases sAPt(1) depends, among other things, on kt Pte , namely, on the nature of the metal used. The overall cell reaction represented by Eq. 28B is not in any way related to the electrodes being used. The measured cell potential must

THb

;LS

process. The difference in potential between these interphases is: P

'

I ACu — A u /A C‘'

(1/0[61c:'µe`) — ( µ cu"

"

gae u)]

=

ptAAuile/F

(33B)

therefore be independent of the electrodes used, since it is strictly a function of the free energy of the chemical reaction.

Thus, the differences in AO values at the two metal-solution interphases

We summarize the findings in this section as follows: In any redox

metal-metal interphases, leading to an overall measured potential of

are exactly compensated for by the differences in AO values at the two

reaction, the metal-solution potential difference is a function of the metal used, whereas the total cell potential is independent of it.

zero, as expected from simple thermodynamic considerations.

5.3 A Cell With the Same Redox Reaction on Different Electrodes

6. THE MEANING OF TIIE NORMAL HYDROGEN

To resolve the apparent inconsistency of the summary statement of Section 5.2, let us consider yet another example. Assume that a gold electrode and a platinum electrode are dipped into the same solution of Fe2iFe3VH SO and both electrodes behave reversibly with respect to 2 4 this redox couple. We have already established that the metal-solution potential difference at the two interphases is different, since each includes the term i_tm , as seen in Eq. 31B. Yet the measured potential of this cell must be zero, since both electrodes are assumed to behave reversibly in the same solution. We could tentatively state that since electrons are neither produced nor consumed in the overall cell reaction, their chemical potential in the different phases cannot affect the measured potentials. Looking at this example in more detail, we note that the difference between the two metal-solution potential differences is: SAPto SAAuo

PIA Aull e /F

(32B)

ELECTRODE (NHE) SCALE 6.1 Thermodynamic Approach Since it is impossible to determine experimentally the metalsolution potential difference at a single interphase, it has been necessary to measure all such values against a commonly accepted reference. The reversible hydrogen electrode, operating under standard conditions (pH = 0; Po1 2) = 1 atm) has been chosen as the reference, and its potential has been assigned an arbitrary value of zero. The potential quoted for any redox couple on the normal hydrogen electrode (NHE) scale is then the actual potential measured in a cell made up of the desired redox couple in one half-cell (under standard conditions of concentration and pressure) and the NHE in the other half-cell. Giving the E° values of CuiCu 2+ and Fe/Fe2+ as + 0.340 V and — 0.440 V, NHE, respectively, implies that these values will be measured versus an NHE. But what about sign convention? It has been internationally agreed that all potentials listed in the literature will refer to the reduction

To measure the potential, we connect the two electrodes to the copper terminals of the voltmeter, creating two additional interphases in the

process. For the preceding two examples we can therefore write:

28

ELECTRODE KINETICS

Cu2+ + 2e

Cu

29

B. TEE POTENTIALS OF PHASES

kinetic information to determine what will happen at a rate that may be

E° = 0.340 V

(34B)

E° = 0.000 V

(35B)

For the corrosion scientist it will be easy to remember that any

E° = 0.340 V

(36B)

metal for which E ° is negative is liable to corrode in acid, while those having a positive value of E ° will not. This rule of thumb should not

of practical interest.

H2

2H+ + 2e

-->

leading to Cu2+ + H 2 ---> 2H+ + Cu

be taken as being exact, since in situations of practical interest the

and similarly Fe2+ + H

2

--->

2H+ + Fe

E° = – 0.440 V

(37B)

system is rarely, if ever, under standard conditions. Pipelines rarely carry 1.0 M acid, and metal structures are not, as a rule, in contact

A positive value of E° indicates that the reaction proceeds sponta-

with a one-molar solution of their ions. For any specific system of

neously in the direction shown, since for E ° > 0 one has AG° < 0. We

known composition and pH, the reversible potential can readily be

conclude from Eqs. 36B and 37B that copper ions in solution can be

calculated from the Nernst equation, and the thermodynamic stability

reduced by molecular hydrogen while Fe 2+ ions cannot. Looking at the same reactions proceeding in the opposite direction, we note that

with respect to corrosion can be determined.

6.2 Which Potential Difference Is Defined as Zero?

metallic copper cannot be dissolved in acid at pH = 0, while iron can. We have seen that there is no ambiguity in the thermodynamic

If we were to combine copper with iron we would find:

definition of the NHE scale. A detailed analysis, in terms of the Cu Fe

2+

+

2e —> Cu m 2+ Fe + 2e m

Cu2+ + Fe

Cu + Fe2+

E° = + 0.340 V

(38B)

E° = + 0.440 V

(39B)

various 04 values at different interphases in the cell, is less straightforward. At first we might be tempted to assign a zero value to the metal-solution potential difference at the normal hydrogen electrode

E° = + 0.780 V

(40B)

Pt AS 4(NHE) = 0

(41B)

showing that this reaction proceeds spontaneously in the direction of

This is clearly not an acceptable choice, since we have seen that the

dissolution of metallic iron and precipitation of copper. This is, in

metal-solution potential difference at a redox electrode depends on the

fact, used as one of the industrial processes for copper recovery from

metal employed. Thus, if the condition given by Eq. 41B would define

ores. Note that Eq. 37B is written in the direction of reduction of

the zero for the NHE scale, we could set up a different normal hydrogen

Fe2+ ions, while Eq. 39B is written in the direction of oxidation of

electrode employing, say, iridium instead of platinum. Since

iron, hence the reversal of sign. These observations are based on thermodynamic considerations alone.

IrLA S A_

(NHE) #

Pt A S

t1(NHE)

(42B)

Thermodynamics can provide only the negative answers; it allows us to

we could not have them both defined as zero, yet the potential measured

calculate and determine which reactions will not happen. We need

against any reference electrode would be the same, as long as the two metals behaved reversibly with respect to hydrogen.

30

ELECTRODE KINETICS

31

B. THE POTENTIALS OF PHASES

that this value is not zero in reality.

Since we have no way of

determining it experimentally, and since every measurement we make yields the difference between the potentials of two half-cells and is independent of the value chosen for the NHE, a value of zero can be used. Considering tables of "standard electrode potentials", sometimes referred to as the "electromotive series", we find the NHE roughly in the middle, with potentials ranging from roughly — 3.0 V to + 3.0 V. For the purpose of calculations, it is somewhat more convenient to change this scale so that all standard potentials will be positive. This is Fig. 3B Illustration of the definition of the "zero" on the NHE scale. The potential in the copper wire on the left, O cu ' is the same in

readily done by redefining the NHE as having a potential of 3.000 V rather than 0.000 V. Table 1B shows the resulting scale, which we shall

both cells shown, irrespective of the metal used to form the

call the modified normal hydrogen electrode (MNHE) scale, along with the NHE scale, for some redox couples. Any additional E ° value on the

reversible hydrogen electrode. This potential is defined as

MNHE scale (not shown in Table 1B), can readily be obtained by adding

being zero. The quantity that is defined as zero is the sum of all potential

3.00 V to the value given in standard tables. The practical implications of the values of E ° on the MNHE scale are straightforward. A higher value always indicates a stronger

differences between the solution and the terminals of the voltmeter

oxidizing power. Thus, for example, chlorine gas can be used to produce

connected to the hydrogen electrode half-cell. In the example shown in

liquid bromine in a solution containing bromide ions, since

Fig. 3B, this is the sum of two potential differences: SAPt4)(NHE)

P

tAeu'

= Sgr4(NHE) -F IrACII

=4

° = 4.35 V and E Br

E°

C12/Cl-

S— (I)Cu a 0

(43B)

Defined in this way, the sum of the potentials will be the same, irrespective of the metal used to build the normal hydrogen electrode, which is consistent with thermodynamics. 6.3 The Modified Normal Hydrogen Electrode (MNIIE) Scale

We have emphasized that the value of zero chosen for the NHE is quite arbitrary. Considering Eq. 43B we have every reason to believe

2

/Br-

= 4.087 V

(44B)

Similarly, zinc will be deposited and magnesium will be dissolved in a solution containing both metals and their ions, since E°

= 2.237 V and E°M g/Mg + + = 0.625 V

Zn IZn++

(45B)

In other words, Zn 2+ will oxidize Mg to Mg2+ and will itself be reduced in the process to metallic zinc.

32

ELECTRODE KINETICS

Table IB Abbreviated EMF Series on the Normal Hydrogen Electrode Scale and on the Modified Normal Hydrogen Electrode Scale

33

C. FUNDAMENTAL MEASUREMENTS IN ELEC IKOCHEMISTRY

C. FUNDAMENTAL MEASUREMENTS IN ELECTROCHEMISTRY 7. MEASUREMENT OF CURRENT AND POTENTIAL

Standard potential

Redox reaction

(NNE)

(MNHE)

7.1 The Cell Voltage Is the Sum of Several Potential Differences The measured potential is a sum of several potential differences. When a current is made to flow through the cell, these potential

C 1 2 + 2e

-i 2 C 1

1.350

4.350

+ 2e

-9 2B r

1.087

4.350

potential reflects the sum of all these changes. If we consider one of

Ag+ +

e

Ag

0.799

3.799

the examples shown in Fig. 3B we can express the change of cell poten-

F e 3+ +

e

->

Fe 2+

0.770

3.770

C u 2+ + 2e

->

Cu

0.340

3.340

2H + +

e

-->

11

0.000

3.000

The last two terms on the right-hand side of this equation may be

P b 2+ + 2e

->

Pb

- 0.126

2.874

considered to be negligible, since metal-metal interphases behave as

S n 2+ + 2e

->

Sn

- 0.136

2.864

N i 2+ + 2e

-

Ni

- 0.230

2.770

well-defined geometries. In most cases it is measured and often can be

C d 2+ + 2e

-p

Cd

- 0.403

2.593

compensated for electronically. One is still left with 8E, representing

F e 2+ + 2e

->

Fe

- 0.440

2.560

G a 3+ + 3e

-9

Ga

- 0.530

2.470

M43, at one of the interphases (at the so-called working or test elect-

Z n 2+ + 2e

->

Zn

- 0.763

2.237

rode) have been devised and are discussed later.

Mn 2+ + 2e

-->

Mn

- 1.029

1.971

7.2 Use of a Nonpolarizable Counter Electrode

A 1 3+ + 3e

-->

A1

- 1.663

1.337

If we combine the working electrode with a highly nonpolarizable

Mg 2+ + 2e

-9

Mg

- 2.375

0.625

counter electrode, the change of potential 8(A(19 at the counter

Na+ +

->

Na

- 2.711

0.289

Br

2

e

2

differences are affected to different degrees, and the change in cell

tial resulting from an applied current i as follows: 8E = Eo) - E(i=o) =

6Agziso iR5 8sApto 8ptAcuo 8cue

g4) (1C)

ideally nonpolarizable interphases. The voltage drop across the solution resistance,

iR s, can be calculated in certain simple and

the sum of the changes of potential across two metal-solution interphases. Several methods to overcome this problem and to relate SE to

electrode will be negligible compared to that at the working electrode, and practically all the change in potential observed will occur at the working electrode

34

ELECTRODE KINETICS

SE = 8wAs4) scAso 8wAs

(2C)

as seen in Fig. 3A. This can be achieved either by using a highly

C. FUNDAMENTAL mEASlii(LNIL

ELECTROCHEMISTRY

Current source

Voltmeter

W.E.

C.E.

R.E.

reversible counter electrode or by making the counter electrode much larger than the working electrode. Since the same current must flow through both electrodes, the current density at the counter can be made

, • . • • • • ..

. ••.•.

much smaller. The polarization (i.e., change of potential) resulting from an applied current is determined by the current density, not the total current; hence, it can be made very small for the counter electrode compared with that for the working electrode (satisfying Eq. 2C), even if the two electrodes are chemically identical and have the same inherent polarizability.

Fig. IC Schematic representation of a three-electrode circuit, showing the working electrode, (W.E.), reference electrode, (R.E.) and

7.3 The Three-Electrode Measurement

counter electrode (C.E.).

A better method of measuring changes in the metal-solution potential difference at the working electrode (which we shall refer to from

potential of the working electrode. As stated earlier, although

now on as "changes in the potential of the working electrode") is to use a three-electrode system, like the one in Fig. 1C.

cannot be measured, its variation S(0(1)) can readily be determined. It should be noted that during measurement in a three-electrode

A variable current source is used to pass a current through the

cell, the potential of the counter electrode may change substantially.

working and counter electrodes. Changes in the potential of the working electrode are measured versus a reference electrode, which carries

This, however, does not in any way influence the measured potential of

practically no current. In this way the polarizing current flows

The three-electrode arrangement can be used equally well if the

through one circuit (which includes the working and the counter

potential between the working and reference electrodes is controlled and

electrodes) while the resulting change in potential is measured in a

the current flowing through the working and counter electrodes is

different circuit (consisting of the working and reference electrodes),

measured. Details of this mode of measurement are discussed later.

through which the current is essentially zero. Since no current flows through the reference electrode, its potential can be considered to be constant, irrespective of the current passed through the working (and counter) electrodes. Thus, the measured change in potential (between working and reference electrodes) is truly equal to the change of

the working electrode with respect to the reference.

7.4 Residual iR s Potential Drop in a Three-Electrode Cell Regarded superficially, it might appear that making a currentpotential measurement in a three-electrode cell eliminates the need to consider any correction for the iR s potential drop in the solution, since there is practically no current flowing through the circuit used

36

ELECTRODE KINETICS

to determine the potential. Unfortunately, this is not quite true. The reference electrode (or the tip of the Luggin capillary leading to it)

C. FUNDAMENTAL MEASUREMENTS IN ELECTROCHEMISTRY

37

8. CELL GEOMETRY AND THE CHOICE OF REFERENCE ELECTRODES

is situated somewhere between the working and counter electrodes. As a result, the potential it measures includes some part of the potential

8.1 Types of Reference Electrodes

drop in solution between these electrodes. This is shown schematically

A good reference electrode consists of a reversible electrode with

in Fig. 2C, in which a parallel-plate cell geometry is chosen for

a readily reproducible and stable potential. A typical commercial

simplicity.

reference electrode (such as a calomel electrode) is a complete system,

Placing the reference electrode near the working electrode can

with electrode and electrolyte enclosed in a small compartment and

decrease, but not totally eliminate, the residual potential drop due to

connected to the rest of the cell through a porous plug. The latter is

solution resistance. The methods used to determine this potential drop

designed to allow passage of ions, yet keep the flow of solution to a

are discussed later. It is important to understand that the

iR s

minimum. In the case of calomel, a mercury electrode in contact with

potential drop cannot be determined by measuring the resistance between

mercurous chloride (Hg 2C12), commonly known as calomel, is placed in a

the terminals of the working and reference electrodes, since such a

saturated solution of KCI. Leaving some solid KCl in contact with the

measurement includes resistive elements through which there is no flow

saturated solution ensures that its composition will be constant,

of current during determination of the i/E relationship.

leading to a stable reference potential. Compatibility with the various components in solution is, however, important. Thus, a calomel refe-

W.E.

R.E.

C.E.

rence electrode should not be used in a solution of HCIO or of AgNO 4 3 because KC1O4 or AgCI, respectively, could precipitate in the porous plug and isolate the reference electrode from the solution in the main cell compartment. Also, since chloride ions are strongly adsorbed on electrode surfaces and can hinder the formation of passive films in corrosion studies, a calomel reference electrode should not be used, unless the test solution itself contains chloride ions, such as in seawater corrosion studies. Another class of reference electrodes, often called

ween the working and the reference electrodes. The total poten-

indicator electrodes are reference electrodes in direct contact with the solution. The most common among these is the reversible hydrogen electrode, formed

tial drop in the cell, between the working and the counter elec-

by bubbling hydrogen over a large-area platinized Pt electrode in the

trodes is also shown.

test solution. This electrode is reversible with respect to the

Fig. 2C Schematic representation of the residual iR s potential drop bet-

hydronium ion H 3 0-1 , serving, in effect, as a p11 indicator electrode.

Lbt.:1KOLi3 KINETICS

C.

I" UNDAMI...4

1AL

IS

LI

MULL LL-"I.)

Similarly one could use a silver wire coated with AgC1 in a chloridecontaining test solution as a reversible Ag/AgCl/C1 — electrode, which

resistance of the reference electrode. The problem is aggravated in the study of transients. The inherent reason for this is that there is a

responds to the concentration of Cl ions in solution following the

tradeoff between response time and input impedance in all measuring

Nernst equation. The advantage of indicator electrodes is that they

instruments. Whereas for steady-state measurements an input impedance

always measure the reversible potential with respect to the ion being studied, regardless of its concentration in solution.

of 10 12 SI is commonplace, fast oscilloscope and transient recorders may have an input impedance as low as 10 5-106 O.

The disadvantage of using indicator-type reference electrodes is that they must be prepared for each experiment and often end up being

This problem can be alleviated by the use of an indicator-type reference electrode. When this is not possible (because of chemical

less stable and less reliable than commercial reference electrodes.

incompatibility), an "auxiliary reference electrode" can be used. This

Also, being in intimate contact with all ingredients in the test

usually consists of a platinum wire placed near the working electrode.

solution, they are more prone to contamination, either by impurities or

While the potential of such an electrode is not stable or well defined,

by components of the test solution, such as additives for plating and

it can be measured just before application of the transient, and it can

corrosion inhibitors.

be safely assumed to be constant during the transient. The transient is then performed with the platinum wire (which has a very small internal

8.2 Use of an Auxiliary Reference Electrodes

resistance) acting momentarily as the reference electrode.

for the Study of Fast Transients One of the advantages of making measurements in a three-electrode configuration is that the resistance of the reference electrode should

8.3 Calculation of the Uncompensated Solution iR s PotenialDrpfFwSmeGotris

not affect the measured potential, since the current passing through

As a rule, the iR s potential drop is measured and a suitable

that circuit is extremely low. This allows us to use reference

correction is made, either directly during measurement or in the

electrodes well separated from the main electrolyte compartment, thus

analysis of the data. When the geometry of the cell is simple, it is

minimizing the danger of mutual contamination. Typical values of the

possible to calculate this quantity. Such calculations are important

resistance of the reference electrode assembly may be 10 4-106 O.

because they can yield clear criteria for the design of cells and for

Combined with an input resistance of 10 12 S2 for the voltmeter, this

positioning the reference electrode with respect to the working

would introduce an error of 1 [IV or less in the measured potential. As

electrode, as will be seen below.

is often the case, practical considerations are more intricate, and it is found that decreasing the resistance of the reference electrode is advantageous. For one thing, the electrode and its connecting wires may act as an antenna and pick up stray currents. The resulting noise in the measurement of potential will then increase with increasing

(a) Planar Configuration The planar configuration was shown in Fig. 2C. The iR s potential drop is given by

40

ELECTRODE KINETICS

iRs= i•d/x

(3C)

C. FUNDAMENTAL MEASUREMENTS IN ELECTROCHEMISTRY

41

which brings us back to the equation for planar configuration. This should not be surprising, since any probe looking at the surface from a

in which i is the current density, x is the specific conductivity of the solution, and d is the effective distance between the working electrode and the reference electrode (or the tip of the Luggin capillary connec-

distance, which is short compared to the radius of curvature, responds to it as though it were flat. For large distances from the electrode surface, when di,. » 1, one has i•R = (ir/x)ln(d/r) s

ting to it). The way to decrease iR s is to increase the conductivity

(7C)

(e.g., by adding an inert supporting electrolyte, when appropriate), or

There is not much to be gained by decreasing the distance d, (unless it

to position the reference electrode closer to the surface of the working

can be reduced to well below the radius of the electrode), since

electrode. The latter approach can result in a local distortion of the current density, which could introduce a larger error than iR s, which we are trying to decrease, as shown later (Fig. 6C).

iR s changes logarithmically with the distance. On the other hand, we note that iR decreases in this case with decreasing radius of the working s electrode. There is, therefore, a clear advantage in using a very fine

Changing the electrode area in this configuration does not affect

iR (or the total potential between working and counter electrode) as s long as the current density remains constant.

wire in this type of measurement. (c) Spherical symmetry Next we consider the case of an electrode in the shape of a drop

(b) Cylindrical Configuration

located at the center of a spherical counter electrode. The potential The cylindrical configuration was shown in Fig. 1C. A thin-wire

drop across the solution resistance is expressed in this case by,

working electrode of radius r is positioned at the center of a cylindrical counter electrode. The equation relating

iR

s

to the distance d iR — (")( r ) s r+d

and the radius of the working electrode r, is iR

s

= (i.r/K)141 + d/r)

(4C)

(8C)

For very short distances (d/r « 1) Eq. 8C reverts to the equation for planar configuration, as for the cylindrical case. For large

It is interesting to consider two extreme cases of this equation. For

distances, however, the same equation yields

very short distances of the reference electrode from the working

iR

electrode, where d/r « 1, we can write, to within a good approximation,

s

i•r/K

(9C)

What this equation implies is that for spherical symmetry, most of 141 + d/r) = d/r

(5C)

iR = i•d/x s

(6C)

and hence,

the iR

s potential drop occurs in the vicinity of the working electrode (within, say, 5 radii). Beyond that it approaches a constant value, independent of distance.

40

ELECTRODE KINETICS

iR

s

(3C)

= i.d/x

C. FUNDAMENTAL MEASUREMENTS IN ELECTROCHEMISTRY

41

which brings us back to the equation for planar configuration. This should not be surprising, since any probe looking at the surface from a

in which i is the current density, ic is the specific conductivity of the solution, and d is the effective distance between the working electrode and the reference electrode (or the tip of the Luggin capillary connec-

distance, which is short compared to the radius of curvature, responds to it as though it were flat. For large distances from the electrode surface, when d/r » 1, one has i•R = (ir/x)In(d/r) s

ting to it). The way to decrease iR s is to increase the conductivity

(7C)

(e.g., by adding an inert supporting electrolyte, when appropriate), or

There is not much to be gained by decreasing the distance d, (unless it

to position the reference electrode closer to the surface of the working

can be reduced to well below the radius of the electrode), since

electrode. The latter approach can result in a local distortion of the current density, which could introduce a larger error than iR s, which we

changes logarithmically with the distance. On the other hand, we note

are trying to decrease, as shown later (Fig. 6C). Changing the electrode area in this configuration does not affect

iR (or the total potential between working and counter electrode) as s long as the current density remains constant.

iR

s

that iR

decreases in this case with decreasing radius of the working s electrode. There is, therefore, a clear advantage in using a very fine wire in this type of measurement.

(c) Spherical symmetry Next we consider the case of an electrode in the shape of a drop located at the center of a spherical counter electrode. The potential

(b) Cylindrical Configuration The cylindrical configuration was shown in Fig. 1C. A thin-wire

drop across the solution resistance is expressed in this case by,

working electrode of radius r is positioned at the center of a cylindrical counter electrode. The equation relating

iR

s

to the distance d iR

and the radius of the working electrode r, is iR

s

= (i•r/K)141 + d/r)

(4C)

s

(

K

cl )( r ) i\r +

(8C)

For very short distances (d/r « 1) Eq. 8C reverts to the equation for planar configuration, as for the cylindrical case. For large

It is interesting to consider two extreme cases of this equation. For

distances, however, the same equation yields

very short distances of the reference electrode from the working electrode, where d/r « 1, we can write, to within a good approximation,

iR = i.r/x s

What this equation implies is that for spherical symmetry, most of

141 + d/r) = d/r

(5C)

the iR

iR = i•d/x s

(6C)

independent of distance.

and hence,

(9C)

s potential drop occurs in the vicinity of the working electrode (within, say, 5 radii). Beyond that it approaches a constant value,

42

'MODE KINETICS

Hi

ECM, .

C. FUNDAME, ,

a

E 12 N

E

o

k'

0

c

12

0 -

8

't = .-

o

4 4

o

E

_c

O

41 0

cE 8 .2 4.1

6

0

0 4

173

E 0

0.2

0.4

0.6

0.8

10

Distance from the surface d/cm

0 z

0

0.2

0.4

0.6

0.8

1.0

Distance from the surface /cm

Fig. 3C Uncompensated solution resistance, in units of L2•cm 2 , and the corresponding potential drop, for a current density of

b 160

surface. Calculated for a solution having a specific conductivity of lc = 0.01 Slcm, and an electrode of radius 0.05 cm. The variation of potential with distance is shown in Fig. 3C for the three configurations just discussed. It should be clear that the spherical configuration is the best in reducing the error that results

So lution res ista nce/ Q

0.4 mAlcm 2 , as a function of the distance from the electrode 120

1. 80

40

from a residual iR

s potential drop, and the planar is the worst. In spite of this, the cylindrical configuration is often used in research, because it is only a little worse than the spherical but much better than the planar configuration, and is easier to set up experimentally than the spherical configuration. The foregoing discussion may require some clarification. Thus, Eq. 9C for the spherical configuration implies that the resistance

decreases with decreasing radius of the electrode, yet one would think

0.4

0.6

0.8

Distance from the surface /cm

Fig. 4C A comparison between (a) the normalized and (b) the total resistance at a spherical electrode, as a function of distance from the electrode surface, shown for different radii.

44

ELEL. DIODE KINETICS

45

C. FUNDAMENTAL MEASUREMENTS IN ELELIItOCHEMISTRY

that a smaller electrode should have a larger resistance. The apparent discrepancy is resolved by noting that R s in Eq. 9C, is given in units of ohm-square centimeters (0.cm 2), namely it is the uncompensated

R.E.

solution resistance, normalized to the area of the electrode. Multip2 lied by the current density, in the usual units (A/cm ), this yields the Porous plug

uncompensated solution potential drop for a given current density. The meaning of Eq. 9C is that for a fixed current density, the iR s potential drop is proportional to the radius. The total solution

Luggin

resistance is inversely proportional to the radius. The normalized and the total resistances are shown for three

capillary W.E.

different radii of a spherical electrode in Fig 4C. Although the normalized resistance decreases with the radius, the total resistance increases, as expected. In electrode kinetics we are interested in

Fig. 5C Schematic representation of a Luggin capillary. W.E. — working electrode, R.E. — reference electrode.

obtaining the potential as a function of the current density; thus it is the normalized resistance shown in Fig. 4C(a) that counts. The same holds true for the cylindrical configurations. In both cases the error introduced by the potential drop due to the uncompensated solution resistance can be reduced by reducing the radius of the electrode.

current to the reference electrode is essentially zero, the potential anywhere inside the capillary (and up to the reference electrode compartment) is the same as the potential at its outer rim, a distance d away from the working electrode. But how close should the tip of the capillary be to the surface of

8.4 Positioning the Reference Electrode In the design of an electrochemical cell, the position of the reference electrode has to fulfill two contradicting requirements. On the one hand, it must be far from the working electrode and well separated from the solution in the main compartment to reduce, as far as possible, the possibility of mutual contamination. On the other hand, it should be as close as possible to the working electrode, to reduce the residual iR

s potential drop. This set of requirements is partially solved by the use of a Luggin capillary, shown in Fig. 5C. The reference electrode compartment is separated from the rest of the solution by a porous plug and the tip of the capillary. Since the

the electrode? Should it be in the middle of the electrode or near the edge? The former position measures a more typical value of the potential, not influenced by edge effects, but the body of the Luggin capillary may disturb the flow of current to the working electrode. The latter position is influenced by edge effects but interferes less with the current flow. Many configurations have been suggested in the literature, and they all share the following drawbacks (albeit to varying degrees), which follow from the basic laws of electrostatics. Bringing the Luggin capillary close to the surface causes a nonuniformity of the current density in that area (usually it is a decrease in local current density, caused by the existence of a nonconducting body,

the glass capillary, in the path of the flow of current).

.: MODE KINETICS

C. FUNDAMLN

MEN . : S

1120CliLivi...> 02Y

60

in industrial cells, a Luggin capillary will cause distortion in the

55

current flow and potential profiles, as shown in Fig. 6C.

50

Since modern instruments allow accurate measurement and compensa-

45

Fig. 6C Primary current distribution and potential profiles for a parallel-plate configuration, with the Luggin capillary placed close to the working electrode. K = 50 mS cm 1 . Top: equipotential

35 30

distance of about five times the radius of the Luggin capillary is

25

usually enough), to minimize the inhomogeneities of the current density

20

10

1 5 10 15 20 25 30 35 40 45 50 55 60

lines. Reprinted with

55

:1

lines. Bottom: current

Weinberg and Gileadi, J. Electrochem. Soc. 135,

distribution. Then one can correct for the resulting higher value of

15

60

permission from Landau,

potential drop, it is better in most cases to s move the Luggin capillary farther away from the working electrode, (a

tion for the residual iR

40

potential is not sensitive to the exact position of the Luggin capillary, leading to better reproducibility in measurements.

50 95

8.5 Primary and Secondary Current Distribution

40

Uniformity of the current density over the whole area of the

35 J_

396. Copyright 1988, the Electrochemical Society.

iR s by proper use of the instruments. In the case of cylindrical and spherical electrodes this approach has the added advantage that the iR s potential drop changes little with d (at dlr. 5). Thus, the measured

1

electrode is important for the interpretation of current-potential data.

30 25

We recall that the measured quantities are the total current and the

20 15

potential at a certain point in solution, where the reference electrode

10

(or the tip of the Luggin capillary) is located. From this we can calculate the average current density, but not its local value. As for the

5 0 35 40 45 50 55 60

potential, we have already noted that unless the cell is properly

If the probe (i.e., the Luggin capillary) is small, this anomaly may not

designed, the potential measured may be grossly in error if the refe-

affect the total current to a significant extent. However, the poten-

rence electrode is located at a point where the local current density

tial is measured at a point where the deviation of the current density

deviates significantly from its average value, as shown in Fig. 6C.

is a maximum. For parallel-plate electrodes, which are commonly used *

In the first approximation we consider primary current distribu-

tion, which refers to the case in which the local current density is determined only by the voltage applied to the cell (i.e., the potential A tenfold decrease in current density on 1% of the surface causes

only a 0.9% change in total current but may decrease the measured potential by 0.1 V or more.

between the working and the counter electrode) and the ohmic resistance of the solution. This corresponds to R F = 0 in the equivalent circuit

48

ELECTRODE KINETICS

representation. Primary current distribution is a function of cell geometry only. Where the electrodes are closer, the current density is higher, and vice versa. An interesting case to be discussed in the context of primary current distribution is the edge effect calculated for the point of con-

C. FUNDAMENTAL MEASUREMENTS IN ELECTROCHEMISTRY

polished to be coplanar with it, such as in a rotating disc configuration ((p = 27c), gives rise to a highly nonuniform current distribution near the edges. Now, the assumption of R F= 0 is equivalent to assuming that the electrochemical reaction rate approaches infinity. When a finite reaction rate is assumed, R is finite and one is in the realm of

tact between an electrode and the insulator in which it is set. The equation describing the current density as a function of distance r from

secondary current distribution.

the point of contact is

about primary current distribution if RiR

Kr (it/2- (p)

(10C)

49

In practice it is customary to talk s < 0.1 and about secondary

distribution if R

> 10. s Maintaining a uniform current distribution is of great importance

and the three limiting cases of interest are shown in Fig., 7C. It is

in the electrochemical industry. In plating it determines the uni-

easy to see that this equation yields a constant current density, inde-

formity of the deposit; in electroorganic synthesis it affects the

pendent of the distance from the edge (i.e., a uniform current distribu-

uniformity of the products. A nonuniform current distribution can lead

tion) only for y = ith.. For larger angles, the current density at the

to the formation of undesired side products and to waste of energy in

point of contact between the electrode and the insulator should go to infinity, and for smaller angles it should be zero. This type of beha-

all areas of the electrolytic industry. It will be shown below that the faradaic resistance R Fdecreases with increasing current density. Thus,

vior is indeed observed experimentally and is well known in the plating

a particular cell configuration, which may correspond to secondary

industry. Thus, a geometry such as shown in Fig. 2C yields a uniform

current distribution at low current densities (in the range typically

current distribution, while an electrode set in a plastic holder and

used in the research laboratory), may move gradually into the realm of primary current distribution as the current density is increased. This is an important factor in determining the optimum current density in an industrial plating process: for example, when it may be necessary to

j

0 0

► co

i=const.

/i

seek a compromise between high plating rate and better uniformity of 0

0

//

3

3

C

metal

meld

metal

Fig. 7C The angle cp between metal and insulator determines the uniformity of current distribution near the edge of an electrode.

plating on a complex-shaped part.

D. ELECIROLA., if&

1

D. ELECTRODE KINETICS: SOME BASIC CONCEPTS 9. RELATING ELECTRODE KINETICS TO CHEMICAL KINETICS 9.1 The Relation of Current Density to Reaction Rate The current density is proportional to the rate of the heterogeneous reaction taking place across the interphase. The relationship, i = nFv

(1D) follows directly from dimensional analysis, since its right-hand side yields (equiv/mol)(C/equiv)(mol/s•cm 2) = C/s•cm2 = A/cm2 Substituting the appropriate numbers into Eq. 1D shows that the electrochemical reaction rate can be measured with very high sensitivity, without causing significant changes in the concentration of reactants or products in solution. This follows from the high sensitivity in measurement of current, hence also of charge. Thus, a current of 1 IAA, which can readily be measured accurately, corresponds to an extremely low reaction rate of about 10 11 equiv/s. Hence, one can ordinarily measure the rate of an electrode reaction for long minutes without causing a significant change in the concentrations of reactants or products. As a result, most electrochemical reactions can be studied under what may be called quasi-zero-order conditions, since the change in concentration can be maintained negligible during measurement of the current. Consider the following example, showing the high sensitivity that can be achieved by measurement of the current. The charge required to form a monolayer of adsorbed hydrogen atoms on platinum in the reaction

52

ELECTRODE KINETICS

H 0+ 3

+

Pt + e

Pt—H + H 0 2

D. ELEC I RODE KINETICS: SOME BASIC CONCEPTS

53

9.2 The Relation of Potential to Energy of Activation

(2D)

In kinetics the rate constant can be written in the form, can be estimated as follows. The area taken up by a single platinum atom on the surface is of the order of 10 A 2. Hence there are about

k = k oexp(—AG°4t/RT)

10 15 platinum atoms per square centimeter of the metal surface. Since

(3D)

where AG°4 is the standard free energy of activation and

k is a constant. Now we have already shown that for a charge-transfer process

one electron is discharged per platinum atom in Eq. 2D, the total charge required to form a monolayer of hydrogen atoms is

it is advantageous to separate the free energy into so-called "chemical" and "electrical" terms.

(1.6x10 19)x10 15 = 1.6x10 4 C/cm2 = 0.16 mC/cm 2 This is an order-of-magnitude type of calculation. The correct number,

tti + zini)

obtained experimentally for hydrogen atoms adsorbed on platinum is

In much the same way we can define the electrochemical free energy of a

0.22 mC/cm 2 . Thus, it is necessary to pass a current of, say, 10 RA/cm 2

reaction as follows: for2secndtmaolyerfhdgnatmsoeurfc.A

far as the electrical measurement is concerned, one could easily measure

AG = AG -T- zFAO

(5B)

(4D)

For the standard electrochemical free energy of activation we may write,

a very small fraction of a monolayer. Now, 0.22 mC equals about 2x10 9 equiv,whcntasofydrgem,untso2gfade weight. Thus we conclude that the measurement of the current and time makes it possible to determine quantities of adsorbed material in the range of 0.1-1 ng/cm 2, which is much better than the sensitivity achieved by the quartz crystal microbalance, itself by far the most sensitive method of determining very small amounts of materials adsorbed on a surface. The same high sensitivity that makes measurements so convenient causes great difficulty in the electrolytic industry. Thus, it takes

9.65x104 C or 26.8 A•h to generate one equivalent of a product formed by

AG)°1 = AG°4t f3FAO

Although Eqs. 4D and 5D look similar, the transition from one to the other is by no means trivial and is the subject of detailed discussion later. Here we shall limit ourselves to a brief discussion of two points. First, the charge on the particle z, which appears in Eq. 4D has been dropped from Eq. 5D, since it is tacitly assumed that electrode reactions occur by the transfer of one electron at a time. Thus, for any rate-determining step, the value of z is always taken as unity. Second, the parameter

p,

called the symmetry factor, has been intro-

duced. By definition it can take values from zero to unity,

electrolysis. A cylinder of compressed H2 contains about a pound of the gas. It would take a water electrolyzer running at 10 3 A about 12 hours to produce this small amount of hydrogen!

(5D)

0<

p<

1

(6D)

and it can be viewed as representing the fraction of the total change in free energy of the reaction that is applied to its free energy of activation:

54

ELECIRODE KINETICS

= (86,:6°4786,40/(8,6086,4) = (86,G °4/8AG1

D. ELECTKODE KINETICS: SOM. U. . -..INCEPTS

(7D) exp

The important consequence of Eq. 5D for our present discussion is that the energy of activation is a linear function of potential. The negative sign in Eqs. 4D and 5D is applicable to anodic reactions, in which the rate is enhanced by increasing the potential in the positive direction, and the positive sign is applicable to cathodic reactions. In either case, the electrochemical free energy of activation decreases when the potential is changed "in the right direction", namely positive for an anodic process and negative for a cathodic process. We can thus

0.5x9. 65x10 4 8 .13x298

1

= 4.5x108

The effect of temperature on the reaction rate depends on the enthalpy of activation. Taking a reasonable range of 40-80 kJ/mol for this quantity, we find that the rate of a reaction at 100 °C is larger than at 0°C by a factor of 1.3x10 2-1.6x104 . Thus, the effect on the reaction rate of raising the temperature by 100 °C is many orders of magnitude less than the effect of changing the potential by 1.0 V in the right direction.

relate the anodic current density to the exponent of the potential: 9.3 Mass Transport Versus Charge-Transfer Limitation i = nFv = nFC°Icoexp(—AG°41/RT)exp((3A0F/RT)

(8D)

We started this book with a schematic presentation (Fig. 1A) of the

Now consider how we can alter the rate of a chemical or an electro-

current-potential relationship in an electrolytic cell from the region

chemical reaction. The best means of accelerating a chemical reaction

where no current is flowing, in spite of the applied potential, to the

is by increasing the temperature. This approach does have its limita-

region where the current rises exponentially with potential, following

tions, however, because of possible decomposition of reactants or

an equation such as Eq. 8D and through the limiting current region,

products, the range of stability of reaction vessels, the cost of energy

where the current has a constant value, determined only by the rate of

and so on. Increasing the concentration of reactants is another means;

mass transport to the electrode surface or away from it.

and employing a suitable catalyst (which decreases the value of AO') is a very effective method, most often used in industrial processes.

Mass transport limitation is more often encountered in electrode kinetics than in any other field of chemical kinetics because the

For an electrochemical reaction, there is an additional means (very

activation-controlled charge-transfer rate can be accelerated (by

powerful, as we shall see in a moment) of controlling the rate of the

applying a suitable potential) to the point that it is much faster than

reaction by adjusting the potential. From Eq. 8D we can calculate the

the consecutive step of mass transport, and therefore no longer controls

magnitude of this effect. To do this, we assign a numerical value to

the observed current. From the laboratory research point of view, mass

the symmetry factor

p.

For this type of calculation a value of 0.5 can

transport is an added complication to be either avoided or corrected for

be chosen, although we shall see later that this may not always be the case, and in any event, one should not take the symmetry factor to be p = 0.500 (i.e., exactly one half). For a change of potential of MO = — 1.0 V, Eq. 8D yields a change of current density by a factor of:

quantitatively, in order to obtain the true kinetic parameters for the charge-transfer process. From the engineering point of view, mass transport is usually the main factor determining the space-time yield of an electrochemical reactor, that is, the amount of product that can be generated in a given

56

ELECTRODE KINETICS

57

The task of the applied

electrode, or pumping the liquid at, past, or through the electrode.

First to find better catalysts (or rather

Most efficient convective mass transport is achieved when the flow at

reactor (or a given plant) per unit time. scientist is then twofold.

D. ELECTRODE KINETICS: SOME BASIC CONCEPTS

electrocatalysts, which are electrodes acting as catalysts), which

the electrode surface is turbulent; yet - turbulent flow tends to be

accelerate the rate of charge transfer, and then to design cells with

sensitive to minor changes in cell configuration, electrode area, the

high rates of mass transport, to take full advantage of the improvement

nature of its surface and so on, often making it difficult to obtain

in catalytic activity.

reproducible results.

Mass transport to the interphase can occur through three independent mechanisms: migration, convection and diffusion.

Mass transport by diffusion can be regarded as the last resort.

The driving force

When movement of the electroactive species is not promoted by the input

for migration is the electric field in solution. In setting up an

of external energy, either electrical (migration) or mechanical (convec-

experiment, one usually tries to eliminate this effect by adding a high

tion), diffusion takes over. The driving force in this case is the

concentration of supporting electrolyte (compared to the concentration

gradient in chemical potential caused by the gradient in concentration.

This helps reduce the electric field by

It is a relatively slow process, with diffusion coefficients ranging

In addition, most of the

from about 5x10 -6 to 8x10 5 cm2/s for ions and small molecules in dilute

of the electroactive ions).

increasing the conductivity in solution.

electricity is carried by the inert ions of the supporting electrolyte,

aqueous solutions at room temperature.

so that the contribution of the electrical field to the rate of mass transport is reduced to a negligible level. In the following discussion

9.4 The Nernst Diffusion Layer Thickness

it is assumed that mass transport by migration has been essentially

We alluded to the Nernst diffusion layer thickness 8, in the first

eliminated, unless we refer to it specifically. It should be borne in

chapter. It relates to the mass-transport limited current density

mind, however, that this is not always the case. Specifically, when

through the equation

films are formed on the surface of electrodes (such as anodic oxide

iL= nfiDC6/8

(4A)

films on aluminum and some other metals or the so-called solid electro-

lyte interphase, which is a salt film formed spontaneously on Li anodes in nonaqueous batteries), mass transport through the film may depend

For a diffusion-controlled process, 8 is proportional to the square root of time

exclusively on migration. Also in many industrial processes, a high concentration of the electroactive ions is used to allow high rates of

5 = (rapt) 1/2

(9D)

production (e.g., C1 2 from NaCI, Cu plated from an acid CuSO 4 bath), and migration is viewed favorably as an additional mode of mass transport. Convective mass transport is caused by the movement of the solution as a whole. The driving force in this case is external energy, usually in the form of mechanical energy of stirring the solution, rotating the

and the limiting current decreases gradually with time. The concept of the diffusion layer thickness can also be applied, however, when the main mode of mass transport is convection. In such cases one assumes that there is a thin layer of liquid at the surface

rckda., L.

.

which is stationary, while the rest of the solution is stirred. Concentration profiles near the interphase, following the application of a potential step, are shown in Fig. 1D.

a 1.0

.4

a

D. 1131.SCIROui

ICS: SOME BASIC Coiv...:

- IS

In Fig. 1D(a) the concentration profile is presented in dimensionless form, as the relative concentration, C/C° as a function of the dimensionless distance, x/(4Dt) I/2 . This is an elegant way of representing the results calculated from a function that depends on several parameters. Thus, this curve is independent of the initial concentration in solution, of the diffusion coefficient of the reacting species, and of time. A great deal of information is condensed into a single

0.8 0

curve, which can then be used to calculate values of the concentration

0.6 C.) 0.4 0.2

0

0.5

1.0

1.5

2.0

2.5

3.0

Dimensionless distance x/(4Dt)'

as a function of distance and time for any specific system. But how far is one unit of dimensionless distance in real terms? That depends on the diffusion coefficient and on time. After 1 ms it is about 1.6 gm from the electrode surface, but after 1 s it corresponds to 50 gm. Figure 1D(b) shows concentration profiles calculated for the same system, but plotted versus the distance in centimeters Each curve corresponds to a different time after application of the potential

b

pulse, and the evolution of the concentration profile with time is 1.0

presented. All the information contained in the curves in Fig. 1D(b)

0.8

exists, of course, in the single curve shown in Fig. 1D(a). This is

0 0 0.6

both the advantage and the disadvantage of presenting the data in dimensionless form! The current flowing across the interphase is given by

0.4 0.2

i = — nFD(dC/dx) x=o 0

0.01

0.02

0 03

Distance, cm Fig. ID Calculated concentration profiles in the Nernst diffusion layer following a potential step. D =

8x10 -6cm 21s. (a) dimensionless coordinates (b) distance in cm, with the time marked on each curve.

(10D)

It is determined by the gradient of concentration at the electrode surface, which decreases with time, as seen in Fig. 1D(b) In a stirred solution a steady state is reached when the concentration inside the diffusion layer varies linearly with distance. Under such conditions we can write Eq. 10D in the form i = nFD C

° — C(s) S

(11D)

El FCTRODE KINETICS

60

It is evident that this current can reach its maximum value iLwhen C(s) = 0, namely

D. ELECTRODE KINETICS: SOME BASIC CONCEPTS

61

of an anodized aluminum electrode, to he deposited there electrochemically.

i = nFDC°/S L

Combining the last two equations, we have i/iL = 1 — C(s)/C°

10. METHODS OF MEASUREMENT 10.1 Potential Control versus Current Control An experiment in electrode kinetics usually consists of determining the current-potential relationship under a given set of fixed conditions

or C s) = C° (1 — i/i L) (

Since the reactant is consumed at the interface, its surface concentration C(s) is always smaller than the bulk concentration C. The ratio between the two depends on the ratio i/iL. For proper kinetic measurements, without the influence of mass transport, this ratio must be maintained below a chosen level, say 0.01 or 0.05, depending on the accuracy desired. This can be done by making measurements at very low current densities, by increasing i L , or both. Alternatively, the

(temperature, concentrations). The measurement may then be repeated under a set of gradually changing conditions, to obtain the i/E plots as a function of temperature or concentration. In many cases the experiment can be performed either by controlling the current externally and measuring the resulting changes in potential at the working electrode (i.e., between the working electrode and a suitable reference) or by controlling the potential and measuring the resulting current. The former is referred to as a galvanostatic measurement and the latter is called a potentiostatic measurement. These

from its bulk value, as long as this deviation is taken into account

commonly used terms are misnomers to some extent, since they include cases in which either i or E is changed abruptly or modulated conti-

quantitatively, by solving the appropriate equations for mass transport. For a diffusion-controlled process, 8 is a rough estimate of the

nuously during the course of measurement, and the suffix "static" does not really apply.

distance over which molecules can diffuse in a given time. For = 0.1 cm and D. 8x10 6 we find from Eq. 9D a value of

In Fig. 2D we show schematically the current-potential relationships for a system undergoing corrosion and passivation. If the

t = 398 seconds. It may perhaps take 10 times longer for the concentra-

experiment is conducted potentiostatically, the curve A—B—C—D—E is

tion to become uniform over a distance of 0.1 cm, while it may take only a tenth of that time for molecules to start to "show up" at a distance

observed, in which both the active dissolution region (A—B) and the passive region (C—D) are seen. If measurement is conducted galvano-

of 0.1 cm from a source. This kind of simple calculation allows us to

statically, all the information concerning the passive region is lost,

estimate how long it would take, for example, for chloride ions from a

and the potential follows the dashed line, from B to E. Clearly, a

calomel reference electrode to penetrate the test solution through a

system exhibiting the type of behavior shown in Fig. 2D should be

side arm of given length, or how fast ions will diffuse into the pores

studied potentiostatically. On the other hand, if one is studying only

concentration at the surface can be allowed to deviate significantly

62

ELECTRODE KINETICS

D. ELECTRODE KINETICS: SOME BASIC CONCEPTS

experimental parameters (concentration, temperature, the nature of the catalyst, and in electrochemistry also the potential) are fixed exterI .0

—

w

Eq. 8D). Just as in chemical kinetics we might repeat the experiment

Cr)

Fig. 2D Schematic RE plots for a system undergoing corrosion and passivation. Active dissolution region: A—B.

_J 'ct

I

The potential of an electrode with respect to a reference represents its oxidizing or reducing power. In a mixture of, say, silver and

z

copper ions, one could set the potential at a value that ensures that

LJ.J I— 0.0 0

only silver ions can be reduced, producing a pure deposit of this metal,

n_

even if we leave the system operating for an indefinite length of time.

A

Transpassive region

-0.5

changing one of the concentrations or the temperature, in electrochemistry we have the additional option of changing the potential.

0. 5

Passive region: C—D. where pitting occurs: D—E.

nally, and the current, namely, the reaction rate, is measured (cf.

B

10 1 10 2 10 3 10 4 Current Density/p.A.cm -2

Similarly it should be possible, in principle at least, to determine the course of an electroorganic synthesis to produce one desired product or the other, by prudent choice of the potential. A galvanostatic measurement represents a different situation, unparalleled in chemical kinetics. Here the rate of the reaction (i.e., the current density) is controlled externally, and the potential is allowed to assume a value appropriate to the rate. The choice between galvanostatic and potentiostatic measurements depends on circumstances. From the instrumentation point of view,

the active region (A—B), or if passivity does not occur, galvanostatic

galvanostats are much simpler than potentiostats. This is not only a

measurements can yield equally accurate results and in certain cases may be experimentally advantageous, as we shall see.

matter of cost, but also a matter of performance. Thus, where it is desired to measure very low currents (e.g., on single microelectrodes),

Even though galvanostatic and potentiostatic measurements may yield

a battery with a variable resistor may be all that is needed to set up a

the same results in many cases, it is important to understand the inherent difference between them and to determine the advantages and disadvantages of each.

low-noise galvanostat. At the other extreme, when large currents must be passed — for instance, in an industrial pilot plant for electro-

Controlling the potential in measurements of electrode kinetics is in many ways the natural way" to conduct a measurement. It is similar to any other measurement in chemical kinetics in that all the

synthesis — power supplies delivering controllable currents in the range of hundreds of amperes are readily available, whereas potentiostats of comparable output are either nonexistent or extremely expensive.

ELECTRODE KINETICS

64

A potentiostat is inherently a more complex apparatus. It operates with a feedback loop in which the potential between the working and the

D. ELECTRODE KINETICS: SOME BASIC CONCEPTS

i = nFC°1( exp(—AG`IRT)exp(flA)F/RT) 8C

0

65

(8D)

difference is amplified, and a current is passed between the working and

To maintain the ratio i /i small (say, < 0.05) over a wide range of ac L potentials, we would like to find ways of increasing i L without changing

counter electrodes, of such magnitude and sign as to make that diffe-

iac, or at least in such a way that i

rence close to zero. Potentiostats did not become commercially available until the late

This can be done by improving the efficiency of stirring, or by taking

1950s. Most earlier work was conducted either galvanostatically or

measurements in unstirred solutions at short times. As stated earlier,

potentiostatically, but with a two-electrode cell, in which one

in quiescent solutions, in which diffusion is the only mode of mass

electrode served as both counter and reference electrode. Because of

transport, the diffusion layer thickness is given by an equation of the form:

reference electrodes is compared to an externally set value, the

their complexity, potentiostats tend to have slower response times than

will grow faster than i . The 8C best way to achieve this is to decrease the diffusion layer thickness 8.

8 = (mDt) 1/2

galvanostats. It should be pointed out, though, that some of the limitations of potentiostats alluded to above are a matter of the past.

(9D)

With present day (1993) electronic components, it is possible to build

Taking measurements at short times increases the value of i L, (which is equal in this case to id ), hence allowing us to study i over a wider

home-made potentiostats, or to purchase commercial units, that make use

range of potentials.

of all the inherent advantages of potentiostatic measurements with

Equation 9D is strictly applicable only in unstirred solutions, but

little instrumental limitation, or none.

for short transients the situation is better. As long as the value of

10.2 The Need to Measure Fast Transients

8, as calculated from Eq. 9D for a purely diffusion-controlled process,

So far we have concentrated our attention on activation-controlled processes. Since the rate of such processes increases exponentially with potential, it is usually possible to drive them fast enough, to ensure that mass transport becomes the limiting factor. For a measurement to be truly activation controlled, the current must be small compared to the mass-transport-controlled limiting current. The latter is given, as we have seen by

is small compared to the diffusion layer thickness set up by stirring, the latter will have no effect on i . L

This may he further clarified by using a numerical example. In a cell stirred by bubbling an inert gas, the effective diffusion-layer thickness is typically about 100 gm. For D = 8x10-6 cm2/s, Eq. 9D yields a diffusion-layer thickness of 10 tm in about 40 ms. Thus, for measurements taken at t < 40 ms, stirring the solution by bubbling a gas has little or no effect on the observed current density.

(4A)

Having established the importance of transient measurements in

in which 8 is the Nernst diffusion layer thickness, and the activation-

electrode kinetics, it is of interest to discuss the effect of the residual iR potential drop in solution on the analysis of such trans sients.

i = nFCD°/8

controlled current density is given by an equation of the form

ELECTRODE KINETICS

D. ELECTRODE KINETICS: SOME BASIC CONCEPTS

Consider first a galvanostatic measurement. When the current density i is applied, the potential measured as a function of time is

this happens, a reaction starts to take place and the interphase is

always the true value, 84, plus a constant given by iRs . The shape of the transient is not affected, and as long as R s is known, a simple

After reaching a peak at short time, the current decays, as shown by

correction can be applied, as shown in Fig. 3D, where line 1 represents

consistent with the equivalent circuit assumed for the iaterphase, shown

the measured potential and line 2 is the change in electrode potential

in Fig. 5D.

depleted from the reactant as a result of mass transport limitation. curve 1 in Fig. 4D, where the shape of the transient is qualitatively

8(i )4:1, corrected for the residual solution resistance. The distance between the curves is constant, since the applied current is constant. Now consider the course of events when a potential step is applied.

300

The current first rises very quickly with time, while the main process taking place is the charging of the double-layer capacitance. Even as

I

1

I

I

250

E 200 1-cr) z I50

20

15

H 100 z

iR s

N to ow

cr

10

20

30

40

50

60

t/ iu sec

Fig. 3D Variation of E with time following a current step, (galvanostatic transient). The residual ohmic potential drop iR is constant throughout the transient. Line 1 — measured; s line 2 — corrected for iR . s

2

50

0

0

I

1

0

2

1

8 10 6 4 TIME /ms

Fig. 4D Variation of current with time following a potential step. (1) without iR s (2) with significant iR s . Lines are calculated for a fast charge-transfer reaction, taking into account partial limitation by diffusion. e = 50 m114, D = 6x10 6cm21s. From Gileadi, Kirowa-Eisner and Penciner, "Interfacial Electrochemistry — An Experimental Approach", Addison Wesley, 1975, with permission.

ELECTRODE KINETICS

68

D. ELECTRODE KINETICS: SOME BASIC CONCEPTS

69

delivers a step function between the working and the reference electrodes (solid line in Fig. 6D), the potential actually applied to the interphase (dashed line) varies much more slowly with time. Since the potential step is not really a sharp step, the current grows more C dl

slowly, and the shape of the curve, particularly its initial parts, is badly distorted.

A

R B

C

A similar situation arises in the application of the linear

•

potential sweep method, to be discussed. In this case a potential that varies linearly with time is applied by the potentiostat. A typical

RF

Fig. 5D The equivalent circuit for an interphase, showing the difference and the potential actually between the measured potential EAC applied across the interphase which drives the chargeEBC transfer process. It is important to' realize that, although the potential that is applied in a potentiostatic measurement (or being measured in a galvanostatic measurement) is between the points A and C in Fig. 5D, the

100 4

80

iR s

E 0 a)

E AC

E BC

60 40 20

potential driving the charge-transfer process is that between the points B and C. The difference between them is EAB = iR

s

0 (14D)

This is an easy correction to make in galvanostatic measurements or in

I

I

4

8

1 12

16

TIME/ms

steady-state potentiostatic measurements, but the situation is much more complex when the controlled potential is changed with time. In Fig. 6D we show schematically the change of the measured potential

and the potential across the interphase when a EAC EBC potential step is applied, corresponding to the current transient shown by curve 2 in Fig. 4D. It is clear that, whereas the potentiostat

Fig. 6D The change of potential across the double layer

EBC following the application of a potential step function, EAC at the output terminals of the potentiostat. Note that the residual solution potential drop, iR s = EAR is a function of time, since the current is not constant in this experiment.

iv

tancwouu KINETICS: SOME BASIC CONCEPTS

ELECTRODE KINETICS

plot obtained in linear-sweep measurement is based Analysis of the a assumption that the potential applied to the interphase varies an the with time, which can be grossly in error if the iR s potential 'linearly

current time response is shown in Fig. 7D and the corresponding variations of the potential E applied externally and the potential

EBB

AC

imposed on the interphase are shown in Fig. 8D. Whiletponsadvrcteofhang potential with time,

v = dE/dt = constant

(15D)

the actual rate of change of potential imposed on the interphase (dashed line in Fig. 8D) varies during the transient. This can cause a major

11

drop becomes significant anywhere along the transient. it may be concluded that potentiostatic transients can be Thus, correctly only if the value of iR s is negligible throughout the

analyzed or if it is dynamically compensated for in real time, during transient the course of the transient.

shift in the position of the peak current density along the potential axis and modify the numerical value of the peak current i as seen in Fig. 7D.

0.60

2 E

0 I

j 0.45

(7-3 z 0 LLJ

ujr_ 0.30

E >-

O

a_ cc 0.15

z —I Lu cc

(-) -2

0.00 -0.4 -0.2

0

0.2 OVERPOTENTIAL/volt

0.4

Fig. 7D Variation of current density with time during linear potential sweep measurement. Quinhydrone (5 mM) in 1 mM sulfuric acid. v = 75 mVls. (a) with dynamic iR s compensation (b) without iR s

potential across the interphase, EBC . compensati.FrGld,Kowa-EisnerPc,

"Interfacial Electrochemistry - An Experimental Approach", Addison Wesley, 1975, with permission.

Fig. 8D Variation of potential with time during linear potential sweep. Dashed line: the Solid line: the applied potential, EAc

ELECTRODE KINETICS

72

10.3 Polarography and the Dropping Mercury Electrode (DME)

D. ELECTRODE KINETICS: SOME BAS IC CONCEPTS

73

and structureless surface, which is renewed periodically, at intervals

One of the most important tools in electrochemical research, the

of a few seconds. Its second asset is that it does not corrode in most

dropping mercury electrode, warrants some special attention. Originally

aqueous solutions; moreover it is highly polarizable in the cathodic

the device was formed by connecting a fine capillary to a mercury

direction, leaving a relatively large window of potential in which

reservoir and dipping the end of the capillary into the test solution.

various reactions can be studied, without interference by decomposition of the solvent (i.e., by the hydrogen evolution reaction).

In the early technique a mercury pool served as both the counter and reference electrodes in a two-electrode-cell arrangement. Now the dropping mercury electrode is used in the usual three-electrode configu-

For a reversible process, the potential depends on the current according to

ration. The flow rate is determined by the pressure (which can be controlled by the height of the mercury in the reservoir over the end of

E = E 112 + (2.3RT/nOlog[(i d/i) — 1]

(17D)

the capillary or by some other means of applying pressure) and the dimensions of the capillary, particularly its diameter. A drop falls when the gravitational force acting on it just exceeds the force of adhesion due to its surface tension. When the two forces are equal, one can write approximately 2nry = M•g

in which E , is the polarographic half-wave potential, to be discussed IR * later. The diffusion-limited current i obtained with the dropping

d

mercury electrode at 25 °C is given by the Ilkovic equation:

(16D) d

where y is the surface tension, r is the radius of the capillary, M is the mass of the drop and g is the acceleration due to gravity. The

=

708m

2/3 I/6 I/2 o t D C

(18D)

size of the drop also depends on potential. The flow rate is nearly

in which m is the flow rate of Hg (mg/s), t is the drop time (s), D is the diffusion coefficient (cm 2/s) and C° is the concentration (mM). The

independent of surface tension, although there is some small residual

current calculated from Eq. 18D with these units is given in micro-

effect, which is most noticeable when the pressure is low.

amperes. Under typical polarographic conditions, the value of i d,

surface tension is a function of potential, as we shall see; hence the

The dropping mercury electrode has been studied extensively. Its

calculated from the Ilkovic equation, is of the order of 3 1.1A for a drop

greatest advantage is that it represents a highly reproducible, clean

*

* For laminar flow in a tube, the flow rate is proportional to the fourth power of the radius and is inversely proportional to the length.

Note that i and i in Eqs. 17D and I8D represent total currents in microamperes, whereas in the rest of this book they represent current deniities, usually in units of amperes per square centimeter. We are making the exception here to conform to the notation commonly used in the vast literature on polarography.

ELEC1RODE KINETICS

time of 1 second in a 1 mM solution. It should be remembered that this is a total current, not a current density. The area of the drop, just before it falls, is of the order of 0.01-0.04 cm 2 , yielding current densities in the range of 0.08-0.30 mA/cm 2 . A typical polarogram,

ELECTRODF

);

It may be appropriate to ask here why the potential at a reversible electrode should change at all with current density. This does not

electroanalytical tool, to measure the concentration of many different species in solution. This is the simplest form of polarography, called The field has been extended to increase sensitivity dc polarography. to decrease the time of measurement. A host of related techniques, and as normal-pulse polarography, square-wave polarography, reversedsuch pulse polarography, and ac polarography, have been introduced. We note that under conditions of diffusion control, the current 1/6 depends on t . Thus, a small change in drop time, resulting from a

occur because "no system is really ideally polarizable", and one is

change in surface tension with potential, does not produce a large

observing a small polarization. Indeed the relationship shown in

difference in the diffusion current. If the error caused by this effect is considered troublesome, it is possible to knock the drops off at

calculated for a solution containing two reducible ions, was shown in Fig. 5A.

Eq. 17D holds strictly only when the interphase is ideally nonpolarizable. Each value of the potential given by Eq. 17D represents the reversible potential for the concentration of the species at the surface, C(s). These concentrations deviate, however, from the corresponding bulk concentrations C ° as a result of mass-transport limitations, according to Eq. 13D. The quantity E 112 is called the half-wave potential, which is observed at a point corresponding to i/i = 0.50. For a reversible d polarographic wave, which can simply be defined as one for which the current-potential relationship follows Eq. 17D, the half-wave potential is nearly equal to the standard reversible potential E ° (not the reversible potential in the solution!): E 112 = E°

+ (2.3RT/nOlog[(yo.hre)(Dred/Do.)1/1

(19D)

Hence the measured value of E can serve to identify the species being 112 reduced. In a typical polarographic measurement, the potential is changed slowly with time (1-3 mV/s) and the current at the end of the life of a drop is plotted as a function of potential. The Ilkovic equation yields the diffusion-limited current, which has been used extensively as an

fixed intervals, yielding drops of exactly equal size, irrespective of the surface tension. This mode of operation becomes of particular importance for kinetic studies conducted at the foot of the polarographic wave, since the activation-controlled current is proportional to 2/3 (the volume the surface area, which is itself proportional to t increases linearly with time). We stated earlier that for a diffusion-controlled process the 1/2 and hence the diffusiondiffusion layer thickness grows with t -112 . This might seem to be at limited current density declines with t 1/6 t . It odds with the Ilkovic equation, in which i d increases with total current at should, however, be realized that i d in Eq. 18D is the d by the area of the drop and not the current density. If we divide i 213 , we find that the current density is the drop, which grows with t -1/2 , as expected for a diffusion cont1/6 2h proportional to t /t = t rolled process. Although measurements of the diffusion-limited current in dc polarography and variations of this technique provide a variety of means to measure the concentration of reducible species in solution, it is also possible to use the dropping mercury electrode to obtain kinetic

ELECTRODE KINETICS

76

information. Measurement at the foot of the polarographic wave, where the current is activation controlled, can yield the most accurate

i/E

relationship achievable with any electrode, because of the very high reproducibility of the surface. Equations have also been developed that allow quantitative correction for mass transport limitation up to higher

77

D. ELECTRODE KINETICS: SOME BASIC CONCEPTS

i/E relationship does not follow Eq. 17D. The problem was treated by Koutecky, who was able to express the activation-controlled current iac in terms of a dimensionless parameter x, which is linearly related to the heterogeneous rate constant of the reaction When the reaction being studied is relatively slow, the

currents along the polarographic wave, extending the region over which X = (12/7) 1/2 0/0 1t2k

very accurate measurements of the activation-controlled current can be made.

h

Values of x were evaluated numerically as a function of //id .

(20D) From

these, the activation-controlled current is obtained as a function of the measured current, hence also as a function of potential. In fact, the Tafel slope can be obtained directly from a plot of log x versus E.

1

An example of the treatment of polarographic data following the theory developed by Koutecky is shown in Fig. 9D, for the case of reduction of

0

hydroxylamine. These results were obtained by analysis of the polarographic data from values of ifid ranging from 0.001 (at the —1 0

so-called foot of the wave) up to Wi d = 0.9, where the activation

b/ mV

controlled current is nine times larger than the diffusion-controlled

1. 89.2+0.2 —2

current (cf. Table 1D in Section 10.4).

2. 102.8+0.1

The current-potential relationship for irreversible polarographic

3. 112.9±0.2 —3 —0 5

waves is more complex than that for the reversible situation. Instead * of Eq. 17D one can use the empiric expression E=E

1/2

+ (0.916b)log[(id/i) — 1]

(21D)

Fig.9D Semi-logarithmic plots of the dimensionless rate constant x = (1217) 05(tID)05k h versus E for the reduction of hydroxylamine on a dropping mercury electrode. NH2 OH 2mM, phosphate buffer

*

Strictly speaking, this is correct only for a stationary drop, for

0.2 M, pH = 6.8, KCl 0.6 M. Drop-time 1 s. Reprinted with

which the parameter x is replaced by the parameter x = kh(tID) ° ' 5 , and

permission from Kirowa-Eisner, Schwarz and Gileadi, Electrochim.

there is an analytical expression for the variation of x with time. For

Acta 34, 1103. Copyright (1989). Pergamon Press.

the expanding drop, which is used in polarography, only numerical solutions are available.

ELECTRODE KINETICS

in which the numerical parameter of 0.916 is chosen to best fit the calculated values of log x vs log[(i d/i) — it This equation can be used in the range of 0.05 5. i/id .5 0.95. The biggest difference between the equations for reversible and irreversible waves is hidden in the expression for the half-wave potential. For the so-called "totally irreversible" or "linear Tafel" region one has, instead of Eq. 19D E ia = E° + b.log[1.32k s,h(t/D) 1/21

(22D)

in which k

s,h is the value of the heterogeneous rate constant at the standard potential E° . The important points to note here are that the

79

D, ELECTRODE KINETICS: SOME BASIC CONCEPTS

polarogram using a mechanical knocker, to ensure uniform size of the 10 2 cm/s drops at the time of measurement. Data calculated for k

s,h

coincide with the reversible curve and are not shown. For values of 10 15 cm/s the system is in the totally irreversible region, with the i/E relationship following Eq. 22D, while the curves for

k sh

ks,h= 10 23 and 10 3 cm/s represent the quasi-reversible situation.

In Fig. 10D(b) the curves are calculated for k s,h = 10 2 cm/s but for different measurement times. We note that a reaction that would appear quite reversible if measurements were taken at 1 s gives rise to a totally irreversible wave if the current is obtained at 1 ms. Thus, reversibility depends not only on the intrinsic kinetic parameters but

half-wave potential is a function of the heterogeneous rate constant and

also on the time of measurement. Which is better? Should one make an

of the time of measurement: it is no longer determined by the standard potential alone.

effort to measure under reversible conditions or when the reaction

This leads us to a discussion of the definition of reversibility or of a polarographically reversible reaction, terms that are often encountered in the literature. It would be tempting to state that a reaction is reversible if the i/E relationship follows Eq. 17D, the

ment. Evidently, there is no kinetic information in an ideally rever-

equation for a reversible polarographic wave. This is correct, but it does not contain enough information. For one thing, reversibility depends on the time scale considered. In Eq. 22D the time t is not the drop time but the time at which the current is actually measured. This could be on the order of seconds in classical dc polarography, where the current is measured just before the drop falls off; but it could also be

behaves irreversibly? The answer depends on the purpose of the measuresible polarographic plot. On the other hand, if the experiment is performed for analytical purposes, then reversible conditions are better, since in this case E 1/2 has a clear thermodynamic significance

(cf. Eq. 17D), and the overlap between successive wave (which occurs if there is more than one electroactive species in solution) is less. One should not be surprised to find that the distinction between so-called reversible and irreversible processes is not sharp and may depend on experimental conditions. The physical meaning of reversibility is that the concentrations of reactants and products

at the

in the range of milliseconds in pulse polarography. Second, one would like to know the order of magnitude of the rate constants for which a polarographic wave is reversible. A series of polarographic waves, calculated for different values of the standard heterogeneous rate constant k sh is shown in Fig. 10D(a), ,

for a measurement time of 1.0 second, which corresponds to a typical dc

*

One should not be misled to think that polarography for analytical

purposes is always conducted at long titnes. Sensitivity can be enhanced by using pulse polarography, with measuring times of the order of milliseconds. As long as interference with other reactions does not occur, this will be the better way to do the experiment.

ELECTRODE KINETICS

80

81

D. ELECTRODE KINETICS: SOME BASIC CONCEPTS

interphase are equal at all times to their equilibrium value, corresponding to the applied potential. For the system to be maintained under reversible conditions, the rate of change of concentrations must be high enough that, at the time of measurement, it will have reached its equilibrium value at the interphase. This naturally depends not only on the heterogeneous rate constant but also on the experimental conditions,

a 1.0 0.8 s- 0.6

such as the time at which measurement is made, the rate of change of 0.4

potential with time, and the rate of diffusion. The condition of reversibility can be quantified in terms of the

0.2

shift of the half-wave potential from its reversible value. This is 0

—300

—200

0

—100

100

shown in Fig. 11D, which is plotted for a measurement time of 1 second.

Potential, (E-E °)/mV

b Dimensionless rate constant (t/D) 1/2 k s,h

1.0

10 -1

10 °

0.8 -0

0.6 0.4 0.2 0 —200

—100

0

100

Potential, (E-E °)/mV 10 -3

Fig. IOD Polarographic waves for reversible, quasi-reversible and totally irreversible processes, as related to different values of the standard heterogeneous rate constant ks,h . (a) curves calculated for different rate constant and a measuring time of 1 s (b) curves calculated for k = 10 2 cmls and different s,h measuring tunes.

1 0 -2

k s ,h /cm sec -1

Fig. 11D The half-wave potential as a function of the standard heterogeneous rate constant for a measurement time of one second. The dimensionless rate constant (tID) 112k

s,h is also shown.

6.3

D. ELECTRODE KINETICS:

ELECTRODE KINETICS

Reversibility is reached

asymptotically, but for k o,

10-2 m/s the

a

system can be considered to be reversible. To generalize this curve, we have shown the values of the rate constant also in dimensionless form as the parameter (t1D)

112 K

s,h

b

. A system behaves reversibly if (t1D) 112 k

3. This corresponds to k = 3x10- 3 cm/s and 1.0 cm/s at 10 seconds and 0.1 ms, respectively.

10.4 The Rotating Disc Electrode (RDE) There are many ways of increasing the rate of mass transport by stirring. As a rule, moving the electrode in solution turns out to be more efficient than moving the solution by bubbling a gas, using a magnetic stirrer, and so on. One of the best methods of obtaining efficient mass transport in

a

highly reproducible manner is by the use of the rotating disc electrode. The RDE consists of a cylindrical metal rod embedded in a larger cylindrical plastic (e.g., Teflon) holder. The electrode is cut and

RDE

RRDE

polished flush with its holder, so that only the bottom end of the metal cylinder is exposed to the solution.

Fig.

The configuration of a rotating disc electrode is shown schematically in Fig. 12D(a), compared to a rotating ring disc electrode, which

12D The rotating disk (a) and the rotating ring-disk (b) electrode. The electrodes and insulated material are marked by filled and clear areas, respectively.

is discussed in Section 10.5(d). The most important feature of the RDE is that it acts as a "uniformly accessible surface," which in simple

characteristic velocity, v, by a characteristic is the product of a This v. The latter is defined length I, divided by the kinematic viscosity, viscosity divided by the density and has the dimensions of square

language means that the rate of mass transport to the surface is uniform. This is by no means self evident, considering that the linear velocity of points on the surface increases with their distance from the

as the centimeter per second (the same as the dimensions of the diffusion

center of rotation. The other important property of the rotating disc is that flow of the solution around it is laminar up to rather high rotation rates.

coefficient) (23D)

In hydrodynamics the transition from laminar to turbulent flow is characterized by a dimensionless number called the Reynolds number, Re.

i

s

ELECTRODE KINETICS

84

85

D. ELECTRODE KINETICS: SOME BASIC CONCEPTS

The characteristic velocity and length must be defined separately

One should be careful in applying the foregoing considerations to

for different geometries, and in each case there is a critical value of

electrodes of widely different dimensions. For example, if one were to

the Reynolds number at which laminar flow changes into turbulent flow.

employ the RDE in an industrial cell using, say, an electrode area of 1000 cm 2 , the critical Reynolds number would be reached at the rim of

For a tube, v is the average linear velocity of the fluid in the tube, 1 = r and the corresponding critical Reynolds number is about 2x10 3 . In

the electrode at a rotation rate of only 30 rpm, and turbulence may have

dilute aqueous solutions v, is close to 10 -2 cm2/s. Thus, flow in a major pipeline is always turbulent (for r = 25 cm, Re = 2.5x103 at

occurred at even lower rotation rates, as pointed out earlier. Such

v = 1 cm/s) whereas in small capillaries it is almost always laminar

long as one is aware that mass transport may become turbulent beyond a

(for r = 0.1 cm, Re = 1x103 at v = 100 cm/s). In the case of the rotating disc, the characteristic velocity is

certain radius and that it will not necessarily be uniform on all parts

the linear velocity at the outer edge of the disc, given by NT' = wr, were

At the other extreme, if one wishes to use rotating microelect-

w is the angular velocity, expressed in radians per second. The

rodes, the minimum rotation rate may be dictated by the requirement that

characteristic length is r and the critical Reynolds number is about

the diffusion layer thickness be small compared to the radius of the

1x105 . The condition for laminar flow is then:

electrode. This becomes a limitation, however, only for r

Re = (w.r)r/v = to•r 2/v < 1x105

electrodes may still be of practical value in an industrial process, as

of the electrode.

0.025 cm. Actually, the limitations imposed on the minimum radius of an RDE by the

(24D)

For a typical radius of, say, 0.3 cm, we find w < 1x10 4 rad/s, which corresponds to a rotation rate of about 1x10 5 rpm. It should be noted that critical Reynolds numbers represent, as a rule, the upper

assumptions made in solving the hydrodynamic equations may be somewhat more severe, and under most circumstances one would be ill-advised to use an RDE having a radius of less than about 0.2 cm. Since flow is laminar, it is possible to calculate rigorously the

limits for laminar flow at ideally smooth surfaces. If the surface is

rate of mass transport. The corresponding equation for the limiting

rough, turbulence may set in at a lower Reynolds number. For this reason most RDEs are set to operate at a maximum speed of lx10 4 rpm (to

current density, developed by Levich, is

-,-- 1x103 rad/s, well within the range of laminar flow). The lower limit for the rotation rate is determined by the requirement that the limiting current density resulting from rotation be large compared to that which would exist in a stagnant solution due to natural convection. In

i = 0.62nFD

2/3 - I/6 1/2 o v w C

(25D)

The angular velocity in this equation is given in radians per second. We can rewrite Eq. 25D in the more convenient form:

practice this corresponds to a lower limit of about 400 rpm, which can be extended to 100 rpm under carefully controlled experimental conditions.

1L= 0.200D2/3 v-1/6 (rpm) 1/2C°

(26D)

where the angular velocity is given in revolutions per minute. Comparing these to the equation for mass-transport-limited current density

lib

ELECTRODE KINETICS

i = nFDC°/S

(4A)

we find for the diffusion layer thickness 8 at the RDE the expression

S = 1.61D 1/3v1/66)-1/2

(27D)

with w in radians per second or

S

5.00 D thv i/o(rpm)-t/2

(28D)

Fig. 13D. So far we have discussed the properties of the limiting current density at the RDE. What about activation or mixed control? For a

purely activation-controlled process, the current should be independent of rotation rate, or should at least become independent of it beyond a certain value. If one has mixed control, the activation and masstransport-controlled current densities combine to yield the total current density as the sum of reciprocals, namely

Since i Lis proportional to (0 1/2

I

100-200 i_tm obtained by simple stirring. For a diffusion-controlled process S = 5 gm after about 8 ms, showing the advantage of even enhancing the rate of mass transport. On the other hand, measurement with an RDE is a steady-state measurement, which is often advantageous. Looking at Eq. 25D we note that i L is a linear function of concentration. Hence the rotating disc electrode can be used as a tool in determine the diffusion coefficients of different electroactive species in solution.

Fig. 13D Motion of the liquid at the surface of an RDE. (a) tangential

Now, one may wonder why the movement of the RDE in a fixed plane

motion in the plane of

has an effect on mass transport in a direction perpendicular to the

the electrode. (b) per-

plane. This comes about because the disc drags the solution nearest to

pendicular motion towards

it and imparts to it momentum in the tangential direction. As a result,

the electrode.

solution is pushed out of the surface sideways (in a plane parallel to it) and is replaced by solution moving in from the bulk, in a direction perpendicular to the surface. The rotating surface acts, in effect, as a pump, pulling the liquid up toward it, as shown schematically in

, this can be rewritten as:

1/i =1/i ac + 11130) 1

moderately fast transients (say, in the range of 0.1-1.0 ms) in

electroanalytical measurements. It has also been used extensively to

(3A)

1/i = 1/i ac + 1/i t,

for w expressed in rpm. For typical values of D = 105 cm2/s and v = 0.01 cm2/s, the diffusion layer thickness turns out to be 5-50 gm for angular rotation rates of 10 2-104 rpm. This range should be compared to values of

8i

D. ELECTRODE KINETICS: SOME BASIC CONCEPTS

2

(29D)

88

ELECTRODE KINETICS

15 -10

Equation 29D is very useful for the study of electrode kinetics. 112 should yield a straight It is clear that a plot of 1/i versus 11(0 Repeating the experiment at line having an intercept of 1/i ac . potential, different potentials, one can obtain the dependence of i 1/2 = as shown in Fig. 14D. It should be noted that extrapolation to 1/0) 0 is equivalent to measuring at a point where the rotation rate, hence the limiting current density, approach infinity. At this point the current would evidently be controlled only by the rate of charge transfer.

10

3

.10 3

E N N,

N .—

The constant B, as defined here (Eqs. 3A , 25D and 29D), appears to

89

D. ELECTRODE KINETICS: SOME BASIC CONCEPTS

■-•

5.10

3

be independent of potential. This is indeed true in most cases, but a detailed analysis shows that B is a function of potential when the system is studied close to equilibrium. It is reassuring to note, nevertheless, that the value of i

ac

obtained by extrapolating 1/i versus

co-" in the foregoing manner is correct, irrespective of the dependence of B on potential. An alternative way of correcting for mass transport limitation is to replace the concentration term

C in the rate equation by the

concentration at the surface, C(s), given by Eq. 13D. The measured current density

i is smaller than the activation-controlled current

0 0

0.02

0.04

0.06

0.08

0 10

N -1 /rpm - 1h Fig. 14D Analysis of the results obtained on an RDE, in the region of mixed control. The activation controlled current density is obtained from the intercept of the lines at infinite rotation rate (N-112

=

0).

density inc, because the concentration at the surface is lower than in the bulk. The ratio between the two is given by C(s)/C = 1 —i/i L

It should be noted here that Eq. 30D can be used to correct for (13D)

mass transport only when steady-state measurements are concerned, such as those obtained with the RDE. It is not applicable for polarography

which leads to

ix i i = 11C

1 — 1/1 L

1L— 1

(30D)

or for any other method in which the current varies with time. The reason is rather subtle: when such methods are used, the activation

This equation can be used to correct for mass transport and thus obtain

controlled current and the diffusion controlled current depend diffe-

as a function of potential from the experimental steady-state

rently on time. As a result, the dependence of measured current on time

current-potential relationship, provided the limiting current is known

varies with potential in the region of mixed control, and a simple

with sufficient accuracy.

correction for diffusion limitation, following Eq. 30D is not valid.

i

ac

ELECTRODE KINETICS

D. ELECTRODE KINETICS: SOME BASIC CONCEPTS

One consequence of the foregoing analysis, which may at first seem

It should also be noted that i

ac

91

can deviate significantly from the

somewhat surprising, is that the activation-controlled current density sac , determined by proper correction for mass transport, can actually be

measured current, even when the latter is quite small compared to the

much larger than the mass-transport-limited current density i L. This follows directly from Eq. 30D. If we substitute m = ail, in this

transport limitation should be applied.

equation, we can write

make use of the basic idea of the RDE but improve upon it or extend it,

is c

m/(1 - m)

=

and

iaci = 1/(1 - m)

(31 D)

As might be expected, several methods have been developed that for the purpose of studying electrode kinetics. Some of these are discussed later. Some caution should be exercised, however, in the

Some values of the two ratios given in Eq. 31D are shown as a function of m in Table 1D. We can readily understand why i can be larger than aC

iL' by recalling that the very reason for mass transport limitation is that the rate of charge transfer (i.e., the activation-controlled rate) has become much larger than the rate of mass transport. In other words, the measured current density i approaches the limiting current density iLexactly when i » i . ac

limiting current density, and an appropriate correction for mass-

L

The percent error given in Table 1D refers to the error expected in the calculated value of i , for a random error of 0.2 % in both i and i . It is clear that, although Eq. 30D can be very useful in extending the range of values of i can be determined experimentally, an attempt to calculate i ac from current densities too close to i may lead L to large experimental errors.

Table ID Comparison of i acto the Measured and the Limiting Current Densities for Different Values of the Ratio of ili

analysis of such data, because of nonuniformity of current distribution. Thus, although the RDE represents a uniformly accessible surface, implying that the limiting current density is uniform, the same cannot be said for the activation-controlled current density. The configuration of a metal electrode and its insulating holder in the same plane leads inherently to large edge effects (cf. Eq. 10C and Fig. 7C), if primary current distribution is important. The resulting error can be decreased, but not entirely eliminated, by placing a ring around the electrode, separated from it electrically and held at the same potential. The best way of avoiding the problem of nonuniform current distribution on the electrode surface is to operate under conditions of secondary current distribution. This can be achieved by operating at lower current densities (which increases the value of R , as we shall see) or by increasing the conductivity in solution, to decrease R s .

10.5 Further Aspects of the RDE and Similar Configurations m = i/iL

0.01

0.05

0.10

0.50

0.80

0.90

i

0.01

0.05

0.11

1.00

4.00

9.00

1.01

1.05

1.11

2.00

5.00

0.2

0.2

0.2

0.5

1.4

0.95

0.98 The usefulness of the RDE stimulated the development of several

ac

L

ac

% error

19.00 49.00

other rotating configurations that are worth mentioning. Some of these

10.00 20.00 50.00

have evolved in response to specific experimental needs, whereas others

2.8

5.7

14.0

serve as possible extensions of the technique.

92

ELECTRODE KINETICS

(a) Irregular configurations In attempting to build an RDE out of highly refractory material,

D. ELECTRODE KINETICS: SOME BASIC CONCEPTS

93

A better and easier solution to this problem is to use a rotating cone electrode. It turns out that both the rotating ring and the

which cannot be readily machined, it is important to know how a devia-

rotating cone electrodes are similar to the RDE in that they act as

tion of the shape of the electrode from a perfect disc may affect the

uniformly accessible surfaces, producing a uniform rate of mass trans-

result. It turns out that the equations for the RDE hold, irrespective

port to all parts of the electrode surface. For a cone having an opening angle of 0 the current density is given by

of the shape of the electrode embedded in the insulating holder, except that the numerical constant appearing in Eq. 25D is different. In fact, the equation can be used unchanged and the correction can be included in what might be called an effective area of the electrode, which can be determined by calibration with a solution of known properties.

I (cone) = i (disc) cos0 —

(33D)

with the limiting current density for the disc i (disc) calculated from the equation for the RDE. When a bubble forms at the rotating cone, it is driven to its

(b) Rotating Ring and Rotating Cone Electrodes

center, namely to the tip of the cone. However, it is in a rather

Often bubbles are generated at the electrode surface (e.g., when the reaction being studied is the hydrogen evolution reaction) or reach

precarious state there, and if it is not dislodged spontaneously by some minor instability in rotation, it can be removed easily.

it from the bulk of the solution, from the gas used to deaerate it, or from degassing of the solvent. A bubble trapped at the surface of the

(c) Rotating cylinder electrode

RDE is driven toward its center by centrifugal force. This phenomenon

Using a rotating cylinder electrode is a good way to achieve high

can cause major errors in measurement because (a) part of the surface of

rates of mass transport. This is very different, however, from the RDE

the electrode is blocked and (b) the hydrodynamic flow in the vicinity

in that the flow is turbulent rather than laminar. As a result, it is

of the electrode can be greatly distorted.

not possible to derive theoretical equations that relate the rate of

An attempt to solve the first problem entails the use of a rotating

mass transport to the various parameters in the reaction, and one must

ring electrode, instead of the rotating disc. Both theory and experiment show that the rotating ring electrode follows the equations for the

on dimensions and on the specific configuration of the cell, hence are

RDE, with an additional factor related to the dimensions of the ring.

less reproducible. A typical equation for mass transport to a rotating

The current at a ring of inner radius r 2 and outer radius r 3 is:

inner cylinder of radius r is:

i = 0.62nFit(r3

3

r

3)2/3 D2hv- i/6(01/2c0 21

(32D)

If a small gas bubble is formed at the center of the rotating disc, it does not isolate the ring electrode from the solution, although it may still interfere with hydrodynamic flow at the electrode surface.

resort to empirical correlations. These tend to be critically dependent

iL = 0.062nFCra4 D0.65v-0.34 (00.7

(34D)

It should be noted specifically that i L is proportional to (0(17 and not .5 to to° , as in the case of the RDE. Laminar flow around a rotating cylinder does not enhance the rate of mass transport to it, because the

LLEC IR O LIE KITE fICS

O. •LECTRODE, KINETICS: SUMS BASIC CONCEPTS

liquid moves in circles parallel to the surface, with no components of

best way to understand this is to view both the ring and the disc as

movement vertical to it. On the other hand, turbulence sets in at very

working electrodes, with a common counter electrode and a common

low rotation rates, of the order of 10-20 rpm for electrodes of sizes typically used in the laboratory.

reference electrode. The important point is that the four-electrode

(d) The rotating ring disc electrode (RRDE) -

potentiostat controls the potential of the disc and the ring with respect to the reference electrode independently and measures the current going through each of them separately. We can, for example,

The rotating ring-disc electrode is perhaps the most useful

hold the potential of the ring at a fixed value, and vary the potential

extension of the idea of the rotating disc. We have already seen (cf.

of the disc to analyze the products formed on it by studying the ring

Eq. 32D) that the ring current has the same form as the disc current,

current as a function of the disc potential, or vice versa.

namely that it is also proportional to the bulk concentration of the electroactive species C ° and to the square root of the rotation rate. A numerical comparison of Eqs. 32D and 25D shows that the current density

The RRDE is calibrated by determining its collection efficiency N. This parameter is defined as the fraction of a species formed at the

at the ring is higher than at the disc, indicating that some increase in

disc that arrives at the ring and reacts there. To do this, consider a simple oxidation reaction, say, Fe 2+ to Fe3+ . The disc potential is set

analytical sensitivity may be attained. This, however, is not the

at a positive value, where the oxidation process occurs at some rate i,

reason for using the rotating ring-disc electrode. The great advantage

while the ring potential is set at a very negative value, at which the

of the RRDE is that it can be used to analyze short-lived intermediates in a steady-state measurement.

oxidized species is reduced at its limiting current density. Assuming that there is initially no Fe 3+ in solution, the ring current results

The RRDE is constructed in such a way that both electrodes are in

only from Fe3+ ions produced at the disc and transported to the ring by convection (there is always some diffusion, but its rate is negligible

the same plane, and in close proximity to each other. The dimensions of the electrode are defined by three radii: the radius of the disc (r 1 ), the inner radius of the ring (r 2 ) and its outer radius (r 3), as shown in

compared to the rate of convection in an RRDE experiment). The ratio of

Fig. 12D(b). Since both electrodes are in the same flow regime

collection efficiency N.

ring to disc current observed under these conditions is defined as the

(evidently the convective flow of the liquid cannot distinguish between the disc electrode, the insulator, and the ring electrode), the equations for the limiting current at both the disc and the ring electrode can be solved simultaneously. To operate an RRDE, one needs a four-electrode potentiostat, often referred to also as a "double" potentiostat. Now, we started this book with a simple two-electrode cell, then went on to the three-electrode configuration. How does the fourth electrode fit into this scheme? The

N

1R /i D = f(r I , r 2 , r3)

(35D)

The important thing to notice in this equation is that the collection efficiency N is not a function of the rotation rate; rather it depends exclusively on the dimensions of the electrodes. The function f(r ,r ,r ) is rather complicated, but its values have been tabulated, I 2 3

and the collection efficiency can be obtained from the dimensions of the

electrode. In a practical experiment (aimed at using the RRDE, not at

96

ELECTRODE KINETICS

97

D. ELECTRODE KINETICS: SOME BASIC CONCEPTS

verifying the validity of the equations derived for it), it is always

which moves the liquid toward the surface, in a direction perpendicular

preferable to evaluate N experimentally, since it depends strongly on

to it, and (c) the tangential velocity ve , which is a rotational

the width of the gap (r 2 — r 1 ), which cannot be measured very accu-

movement imparted to the liquid adjacent to the disc, by the angular

rately. Note that i R and ir) in Eq. 35D represent the total measured

momentum transferred from the solid surface.

currents at the ring and the disc, not the current densities. Typical

In Fig. 15D we show two of these velocities for an electrode having

values of the collection efficiency for commercial RRDEs are in the

a radius of 0.2 cm, which is rotated at 2.5x10 3 rpm. We chose to

range of 0.1-0.3. Decreasing the insulating gap between the disc and

present the velocities as a function of distance from the electrode

the ring, (r2 — r 1) increases N.

Decreasing the width of the ring,

(r — r ) leads to a lower collection efficiency but improves the 3 2 accuracy of measurement of the lifetime of unstable intermediates formed at the disc. If an unstable intermediate or product is formed at the disc, only

250

Vz /cmsec -1 1.5 1.0 0.5

a fraction will reach the ring, and the ratio iR/iD will be smaller than

N. The extent by which the collection efficiency is decreased is a function of the rate of rotation. The dependence of N on w can be used to evaluate the lifetime (or rate of decomposition) of the unstable intermediate. As in any kinetic measurement of this type, one attempts to design the system for the fastest possible transition time from disc to ring, to allow detection of short-lived intermediates. The gap in commercial RRDEs is of the order of 0.01 cm, but electrodes with substantially narrower gaps have been built. We may be tempted to use modern techniques of microelectronics to construct an RRDE with a very small gap, say, 0.1 [tm. A closer examination of the hydrodynamics

200 Fig. 15D Radial and perpendicular velocity components of

E

the liquid at a RDE, v and v z, respectively, as a function of the distance from the surface. w = 2500 rpm, v = 0.01 cm21s and r = 0.2 cm.

(1)

o

100

w

0 50

involved reveals that this may not work and, in fact, there is little or no advantage in reducing the gap below about 5 jim. To understand this, we must consider the flow velocities of the liquid near the electrode surface. Convective flow near a rotating disc electrode is described by three velocity components: (a) the

radial velocity vr, which moves the liquid from the center of rotation outward, in a plane parallel to the surface, (b) the perpendicular velocity v

0 10 20 30 40 5C Vr /cmsec-1

I.:LEM-RODE KINETICS

99

D. ELECTRODE KINETICS: SOME BASIC CONCEPTS

surface in real numbers, rather than in the dimensionless form most often used in the literature, for easier visualization. Conversion to other rotation rates and electrode dimensions is easy when we remember that v is proportional to (r0.)) I/2 and v is proportional to 0.) 1/2 . The

E 40

radial velocity starts from zero at the electrode surface, rises quite quickly to a maximum of just under 10 cm/s, at a distance of about 40 p.m, and decays back toward zero slowly. The perpendicular velocity

M

30

0

20

component is also zero at the surface and rises to a constant value of about 1.4 cm/s at a distance of 250 p.m. This distance, which is proportional to co -° • 5 , can be defined as the hydrodynamic layer thickness. It should be noted that the Nernst diffusion layer thickness at

v.

the same angular velocity is only 10 p.m, about 25 times thinner than the 0

hydrodynamic layer, and this ratio is independent of the rotation rate.

physical meaning of the Nernst diffusion layer becomes clear in this

10

15

20

Vr /cmsec -I

An enlarged view of the two velocity components near the electrode surface is shown in Fig. 16D, for the same numerical parameters. The

5

Fig. 16D An expanded view of the curves shown in Fig. 15D, for short distances from the surface of the RDE. Note that at a

the Nernst diffusion layer is very small, not exceeding 2% of its value

distance corresponding to the thickness of the Nernst diffusion layer 8 D the perpendicular velocity component is close to

far away from the surface. This is the justification for the assumption

zero.

form of presentation. Thus, the perpendicular velocity component inside

that inside this diffusion layer the solution is practically stagnant, even though the solution as a whole is well stirred by rotating the electrode.

typically 0.01 cm, is only 3.3 ms, but diffusion across the Nernst

Now consider the path of a chemical species formed at the rim of

distance is traversed twice, the total time for transition of the ms, deterspecies from the disc to the ring is ti = 2x32 + 3.3 = 67.3

the disc electrode as it travels toward the inner edge of the ring electrode. There are two transport processes involved: convective flow at a velocity v and diffusion. The species must diffuse a certain distance z into the solution, be transported across the gap at a

diffusion layer, through a distance of 10 lam, takes 32 ms. Since this

mined mostly by the diffusion time. Consider another pathway, much closer to the electrode surface, say at a distance of 1 pm. Here the radial velocity is only about 0.3 cm/s,

velocity vr , and diffuse back to the surface. The radial velocity at a

so it will take the species 33 ms to cross the gap. On the other hand,

distance of 10 pm from the surface (i.e., just outside the Nernst

diffusion over a distance of 1 pm takes only 0.3 ms, leading to a total 2x0.3 + 33 = 33.6 ms. Very close to the surface, transition time of ti =

diffusion layer) is 3 cm/s. The time taken to cross the gap, which is

ELECTRODE KINETICS

100

the total transition time is thus controlled by the rate of convection

F. SINGLE-STEP ELECTRODE REACTIONS

101

E. SINGLE-STEP ELECTRODE REACTIONS

across the gap. Evidently, there must be an optimum distance from the surface somewhere in between, where the transition time is a minimum.

11. THE OVERPOTENTIAL tl

In the present example this turns out to be close to 3 gm, for which

11.1 Definition and Physical Meaning of Overpotential

= 16.9 ms. A similar calculation can be made for other angular velocities and for different dimensions of the RRDE. When the gap between the disc and the ring is made very small

ti

The reversible potential for a given electrode reaction is related to the difference in free energy between reactants and products

(e.g., < 1 p.m), direct diffusion from the disc to the ring may become

E = - AG/nF rev

the fastest mode of transition. In this case, however, the system acts as two very closely spaced stationary electrodes, and the fact that both are rotated has little or no effect on the transition time. The rotating ring-disc electrode can also be very useful in other studies, where the species formed at the disc are stable. For example, the corrosion of a metal is always associated with a cathodic reaction, which in deaerated solutions is hydrogen evolution and in the presence of air is the reduction of molecular oxygen. The rate of corrosion of a metal (used as the disc) can therefore be followed indirectly by measuring the amount of molecular hydrogen reaching the ring electrode or by determining the decline in the amount of molecular oxygen reaching the ring. The RRDE can also be used to study complex reactions in which two processes occur in parallel. For example, electrodeposition of Zn is accompanied by hydrogen evolution. The partial current for hydrogen evolution at the disc can then be determined by measuring the amount of molecular hydrogen reaching the ring. Similarly, when a Ni/Ni0OH electrode (used in a Ni/Cd battery) is charged, oxygen evolution is known to start before the electrode has been fully charged. The partial current for charging can be determined by monitoring the amount of oxygen reaching the ring electrode electrochemically and deducting the corresponding current, corrected for the collection efficiency, from the total observed disc current.

(2A)

Its dependence on concentrations in solution is expressed by the Nernst equation, which has the general form E

rev

= E° + (2.3RT/nF)logQ

(1E)

where Q represents the ratio of concentrations of products to reactants, and the activity coefficients have been taken as unity, for simplicity of presentation. When at open circuit, a highly nonpolarizable electrode assumes its reversible potential, whereas a highly polarizable electrode may deviate from it significantly. In either case, the overpotential is defined as the difference between the actual potential measured (or applied) and the reversible potential rl = E - E

rev

(2E)

This definition of overpotential is phenomenological and is always valid, irrespective of the reasons for the deviation of the potential from its reversible value. The overpotential is always defined with respect to a specific reaction, for which the reversible potential is known. When more than one reaction can occur simultaneously on the same electrode, there is a different overpotential with respect to each reaction, for any value of the measured potential. This situation is encountered most commonly during the corrosion of metals. When iron corrodes, for example, in a neutral solution, the overpotential may be

ELECTRODE KINETICS

+ 0.4 V with respect to metal dissolution and — 0.8 V with respect to oxygen reduction. In most studies of electrode kinetics, however, the

B. SINGLE-STEP ELECTRODE REACTIONS

the molecules or by changing the temperature suddenly. The rate at which the reaction will proceed toward its new equilibrium is given by

experiment is set up in such a manner that only one reaction can take place in the range of potential studied, and the overpotential is unambiguously defined.

103

v = v eq (A/RT)

(5E)

where v is the exchange rate, namely the rate at which the reaction eq

The overpotential is a measure of the distance of a reaction from its equilibrium state, on the free-energy scale. Multiplied by the charge consumed per mole of reactants nF, it is equal to the affinity A,

proceeds back and forth at equilibrium. For a simple charge-transfer process we can similarly write

of a reaction, which is a measure of the driving force that exists to make the reaction occur

exchange where we have taken n = 1. In this equation io is called the current density and it represents, much as v eq, the current density in both anodic and cathodic direction when the system is at equilibrium. Equations 5E and 6E are special cases of a general rule according

A = nFri

(3E)

Now, we have already seen that the electrochemical free energy of activation is linearly related to the applied potential, giving us a powerful tool with which we can control the rate of electrode reactions over many orders of magnitude. At the other extreme we can also use the

i = ioeriF/RT)

(6E)

to which, whenever a system is perturbed to a small extent, the response

It will be remembered that equilibrium does not imply total

is proportional to the magnitude of the perturbation. But how small is "small" in the present context? This must be defined in some unitless form, comparing affinity to thermal energy or rate to exchange rate. For Eq. 5E the perturbation is small if A/RT << 1. Likewise for the case of charge transfer we should have r1F/RT << 1. Clearly, a small << 1. This latter perturbation also leads to v/v eg << 1 and i/i.

freezing of the reaction. It is characterized by a dynamic situation in

criterion is, however, mainly relevant to electrode kinetics, since it

which the reaction can proceed in both directions at a high rate.

is there and only there that one can control the reaction rate (in a galvanostatic measurement) and observe the resulting affinity (i.e.,

potential to probe the reaction under conditions close to equilibrium, by applying small values of the overpotential in both directions around zero and measuring the resulting current density.

Equilibrium is reached when the forward and backward rates are equal and the net rate, which one would observe, say, as a change of concentration in a chemical reaction or a flow of current in the external circuit in an electrochemical reaction, is zero. Consider the following simple chemical reaction at equilibrium, with reactants and products all in the gas phase I2 + H E 2

21-11

(4E)

By definition, the affinity A = 0 and the net reaction rate v = 0. We can disturb the equilibrium slightly by adding a small amount of one of

overpotential). To obtain a better feel for the quantities involved, we note that an overpotential of 1.0 mV corresponds to an affinity of = 1.0x10 -3x9.65x104 = 96.5 J/mol

(7E)

This should be compared to the thermal energy RT which, at room temperature, amounts to 2.42 kJ/mol or about 25 mV (multiplied by F). For an overpotential of 5 mV we then have

104

ELECTRODE KINETICS

TIF/RT = 0.2

be expected between n = —5 and +5 mV. The value of the exchange current density, i, can readily be obtained from the slope of such plots, using Eq. 6E. It may be of interest to compare the effect of the overpotential on the work performed in, say, charging a battery with the work needed to compress a gas isothermally at a finite rate. We recall that the work required to compress a gas is:

2

to

Fig. 1E Loss of energy as

a

s. a result of irreversibility in processes occurring at a finite rate.

Volume

(a) expansion and compression of gas. (h) charging and discharging a •E

a)

battery. Dashed lines (9E)

For an ideal gas being compressed reversibly and isothermally from a volume of V

105

(8E)

at room temperature. We conclude that a linear i./t1 relationship is to

W = f PdV

E. SINGLE-STEP ELECTRODE REACTIONS

0

a_

correspond to the reversible paths.

Charge

V I , this reduces to W = nRT1n(V I/V2)

(10E)

The same is true when a battery is cycled. We could charge and

If the gas is then allowed to expand back to its initial volume under

discharge it reversibly with a vanishingly small current, maintaining

the same conditions, the same amount of work will be regained. For the

the potential at its reversible value throughout the process. Such a

process to be reversible, it must occur very slowly, in fact,

infinitely

cycle does not require the input of net energy, but it takes "forever"

slowly for true reversibility. But infinitely slow processes last an

to occur. Now batteries are built to operate at a finite rate, in fact,

infinitely long time! If we wish to have the process occur at a finite

often at a high rate. The energy (i.e. electrical work) needed to

rate (as in a real engine), we must sacrifice efficiency. More work

charge a battery is given by

(than shown in Eq. 10E) is needed to compress the gas, and less is regained during its expansion. The difference represents the irrever-

W = f EdQ

(11E)

and since

sibility of the process, or the price we pay, in terms of energy, to

E=E

secure a desired rate of operation.

rev

+ >1

(12E)

the total energy lost as a result of irreversibility, namely due to the overpotential, in one charge-discharge cycle, is

*

It should he remembered that Eq. 6E is applicable only for a

energy loss =

f [n(charge) — ri(discharge)) dQ

(13E)

single-step electrode reaction. The corresponding equation for multi step electrode reactions is Eq. 51F in Section 14.7.

[note that n(discharge) is negative].

The loss of energy due to

1

VU

ELECTRODE KINETICS

irreversibility for the two preceding cases is shown schematically in Fig. 1E.

B. SINGLE-STEP ELECTRODE REACTIONS

utis 4 by SmAs4), namely: nao

MA

N) MAS orev smAso

10

(16E)

11.2 Types of Overpotential In Section 11.1 we viewed the overpotential as the stimulus (or perturbation), which causes the reaction to proceed at a certain rate. We could equally well have inverted the roles and looked at the current density as the stimulus and the resulting overpotential as the response. In this case the value of the overpotential is an indication of the sluggishness of the reaction. To be more precise, the ratio between the response 11 and the perturbation i is a measure of the effective resistance of the system to proceed at the desired rate. In the example given earlier, applicable to small perturbations, the ratio (i/i) is constant, but in general it is a function of potential, as implied by Fig. 1A. It will be noted that has the dimensions of an electrical resistance, and indeed it represents the total reaction resistance R F. For the simple case discussed here we can write RF=

(14E)

but, in general, it would seem preferable to define the faradaic resistance in differential form as follows: R bearing in mind that

RF

F

= (8ll/80 CT,P

(15E)

is, as a rule, a function of

In proper kinetic studies one needs to isolate this type of overpotential. A mechanistic study is then based on determining the dependence of on current density over a wide range of experimental ac

conditions. The second type of overpotential often mentioned in the literature which results from the residual iR s is the resistance overpotential r1R, potential drop in the solution. In laboratory research experiments this

type of overpotential can be largely eliminated by proper positioning of the Luggin capillary, combined with automatic correction for most of the remaining resistance, in real time. It does, however, have a major effect in practical electrochemistry, with respect to its influence on the performance of batteries and on the energy consumption of electrolytic processes. About one-third of the power consumption in chlorine production and two-thirds in aluminum production are due to the ohmic resistance in the cell, namely to the resistance overpotential. The immense magnitude of this effect may be appreciated when it is realized that the amount of electricity lost to heat in the chlorine and aluminum industries is about 2-3% of the total electric power production in the United States costing about $1.5 billion a year at current prices ($0.05/kW•h). Being an electrochemist, one should perhaps not feel too badly about the staggering loss of energy and money due to the ineffi-

The overpotential observed experimentally can result from several unrelated physical phenomena, all hindering the reaction in one way or

ciency of these electrochemical processes, when compared to the ineffic-

another. Here we shall discuss the three main causes of overpotential. Activation overpotential ac is the kinetically significant

utility power stations operate at an average efficiency of about 40%, which means that 60% of the thermal energy of the fuel is wasted. This

quantity that acts on the electrochemical energy of activation, lowering it in one direction and increasing it in the opposite direction; TI ac is

amounts to a direct loss of about $25 billion a year in the United States alone, not to mention the cost involved in removing this heat

the potential acting across the interphase and changing the value of

efficiently, which is essential for the operation of the plant.

iency of electricity production. Consider the following: electric

ELECTRODE KINETICS

108

The third important type of overpotential is caused by mass-

109

E. SINGLE-STEP ELECTRODE REACTIONS

magnitude by applying a relatively small potential to the electrode. With present-day understanding of electrochemistry, one wonders whether

transport limitation. It is called concentration overpotential Ti c because, when the reaction is influenced by the rate of mass transport,

Tafel was lucky in choosing this particular system for his studies, or

the concentration at the surface is different from its bulk value (cf.

whether he had some deep insight, not reflected in his original paper:

Eq. 13D). Since the reaction rate depends only on the concentration at

the conditions under which the Tafel equation can be observed, such as a

the surface, it is necessary to apply a higher overpotential to maintain the same rate when the concentration at the surface is lower than in the

high overpotential, no mass transport limitation, secondary current

bulk. In conclusion, one may be tempted to write the overpotential as the

distribution RJR

s » 1 and no interference by film formation, all happen to exist in the particular experimental system chosen by him. For many

years the Tafel equation was viewed as an empirical equation. A theoretical interpretation was proposed soon after the absolute rate

sum of three terms: n = lac + 11R +

(17E)

This is conceptually correct, but it must be remembered that Ti n and ri c R ) are interrelated, and an attempt to obtain 11 (andtolesrx1 by calculating the three contributions separately and adding them up may lead to gross errors.

theory for chemical kinetics was developed by Eyring, Polanyi and Horiuchi, in the early 1930s. Since it is the most important fundamental equation of electrode kinetics, we shall derive it first for a single-step process and then extend the treatment for multiple consecutive steps. Before we do that, however, we shall review very briefly the derivation of the equations of absolute rate theory.

12. FUNDAMENTAL EQUATIONS OF ELECTRODE KINETICS

12.2 The Absolute Rate Theory Consider the simple isotope-exchange reaction in which a hydrogen

12.1 The Empirical Tafel Equation

atom reacts with a deuterium molecule D The Tafel equation was first written in 1905 in the form a – b.log i

2

to form a molecule of HD,

leaving a free deuterium atom: (18E)

D–D + H

D–H + D

(19E)

One could imagine the reaction occurring by first splitting the as an empirical equation relating the observed overpotential r1 to the

deuterium molecule into two atoms and then combining one of the atoms

current density during hydrogen evolution on mercury cathodes. It shows

with the hydrogen atom. This is an unlikely course of events, since a

a linear relationship between the overpotential and log i with a slope

great deal of energy is required to break this bond. A more likely

b, which came to be known as the Tafel slope. The numerical value of b observed by Tafel was about 0.1 V per decade of current density. This

route is to form an intermediate, such as an unstable D — D — H

is consistent with our previous discussion in which we showed that the

molecule dissociate with equal probability to the original reactants or

rate of an electrochemical reaction can be increased by many orders of

to the products. The standard free energy needed to form the unstable

molecule, in which both bonds are of about equal strength, and let this

110

rk Dli IU.,L i ICS

E. SINGLE-STEP ELECI1(ODE REAc loN;

111

intermediate, which is called the activated complex, is the standard

Equation 24E gives the chemical rate constant of the forward reaction as

free energy of activation for the reaction.

a function of the standard free energy of activation. To use this

We shall now write the reaction in more general form as A

X# --> B

(20E)

in which A and B are the reactant and product molecules, respectively, and X# represents the activated complex. Two main assumptions are made in the framework of the absolute rate theory: (a) the reaction rate is assumed to be proportional to the concentration of the activated complex and (b) the activated complex is assumed to be at equilibrium with the

equation for electrode kinetics, it is necessary to relate the standard free energy of activation to the potential drop i4 across the interphase. Before we do that, it would be interesting to consider the question of activity coefficients in the framework of the foregoing theory of chemical kinetics. It is well known that all equations for equilibrium should be written in terms of activities rather than concentrations. Thus, we recognize that Eq. 22E should be replaced by:

reactant. With these assumptions one can write for the reaction rate, v v=kC f A and for the equilibrium

= (k T/11)C E

C#/C A = K# = exp(—AG"/RT)

(21 E)

(22E)

In these equations kf , kB and h are the forward rate constant, Boltzmann's constant and Plank's constant, respectively, and ,AG' °# is the standard free energy of activation. The latter equals, by definition,

Cgy# /CAyA = K# = exp(-6.e/RT)

(25E)

in which yit and 7A represent the activity coefficients of the corresponding species. The rate of the reaction is still proportional to the concentration of the activated complex, not its activity, since it is assumed that once formed, the activated complex will decompose in the first vibration toward the products. if now we use Eq. 25E instead of Eq. 22E to substitute the concentration of X 41 in Eq. 21E , we obtain

the standard free energy of the reaction in which the activated complex

is formed from the reactant. The term k Tlh in Eq. 21E follows from an added assumption, namely that the critical bond in the activated

v = kfCA= aC(yo/yA )exp(—AG"/RT)

(26E)

It is usually unnecessary to correct for activity coefficients in

complex, which must be severed to form the product, is very weak and

chemical kinetics. However, when such a correction is deemed necessary,

behaves "classically" (in that the energy levels are so close to each

it is well to remember that it cannot be made by simply replacing the

other that there appears to he a continuum). We need not concern

concentrations of the reactants by the respective activities in the rate

ourselves with this in the context of electrode kinetics and this term,

equation, and the activity coefficient of the activated complex must

which has the units of frequency, is replaced by w in all further equations. Substituting C if from Eq. 22E into Eq. 21E we have

also be considered.

12.3 The Equation for a Single-Step Electrode Reaction v=k

f

C

A

= (D . 0

exp(—AG ° "/RT) A

(23E)

or kf= co-exp(—AG (4/EZT)

(24E)

We discussed the relationship between free energy and potential in Section 9.2. For the electrochemical free energy of a reaction, we have

112

ELEC I RODE KINETICS

written Eq. 4D and for the electrochemical free energy of activation,

E SINGLE-STEP ELECT RODE REACTIONS

113

Equation 30E can also be written in the form

Eq. 5D. Taking z = 1 and using the standard free energies, we can write i = Fk C •exp((3FTVRT) h A

these equations with only minor modification as follows:

l

AG° = AG ° — FAO

(4D)

AG° = AG° — 13FAO

(5D)

(31E)

in which khis the heterogeneous electrochemical rate constant at the reversible potential. Equation 31E is an expression for the current density (which, we recall, is the rate of the reaction in electrical

in Eq. 5D is discussed in Section 12.4. Equations

units) in terms of the concentration and the overpotential, both of

4D and 5D are applicable to anodic (oxidation) reactions, for which

which can be measured and controlled experimentally. It should be noted

changing the potential in the positive direction shifts the equilibrium

that in deriving Eq. 31E we have tacitly assumed that the concentration

toward the products and increases the reaction rate. For cathodic

at the surface is independent of the current density, namely, we have

(reduction) reactions, a positive sign should be used in both equations.

neglected mass transport limitations. Strictly speaking we should

When Eq. 5D is substituted into the rate equation we have

therefore replace ri by Ti in these equations. For the sake of simpli-

The symmetry factor

i = F•v = F•(o.0 •exp(—AG°/RT)expO3FA(I)/RT) A

(27E)

city we shall leave it as it is, and refer specifically to mass transport limitation in a separate section.

There are two ways in which we wish to modify this equation.

Equation 31E represents the rate of an anodic oxidation reaction

First, we simplify it by replacing w.exp(—AG °/RT) by a "chemical" rate

for which the overpotential is, by definition, positive. For a cathodic

constant k° , which is the value of the heterogeneous rate constant at

reduction we write a similar equation with a negative sign in the

= 0,

exponent, and we take the overpotential, by definition, to be negative.

to yield i = FIc ° C .exp(PFAO/RT) h A

(28E)

In either case the current density increases exponentially with increasing absolute value of the overpotential.

Second, we would like to get rid of the awkward term AO, which, as we

The complete expression for the current density is obtained as the

recall, cannot be measured. Now, we have gone into some length to show

difference between the anodic and the cathodic current densities. This

(cf. Section 7) that although AO cannot be measured, changes in it can

is the current measured in an external circuit. Now, if we have used

be readily determined. Bearing this in mind, we can write the over-

Eq. 5D to represent the potential dependence of the electrochemical

potential as follows:

energy of activation for the anodic direction, what would we use for the tl = '44) — 4re v =

where A(I)

rev

(29E)

opposite sign, since the same potential that enhances the rate of the

is the value of AO at the reversible potential (Ti = 0).

Substituting AO from this equation into Eq. 28E, we obtain: i = Fk'h CA° • exp((3AOre v F/RT)exp(fliF/RT)

cathodic reaction? Evidently the term including AO must have the reaction in one direction must retard it in the opposite direction. What about the symmetry factor [3? We noted earlier (cf. Eq. 6D) that it

(30E)

must have a value between zero and unity. Also, we have seen in Eq. 7D

ELECTRODE KINETICS

that 13 represents the ratio of the effect of potential on the electrochemical free energy of activation to its effect on the electrochemical free energy of the overall reaction. If this fraction is 0 in the anodic direction, it must be (1 — (3) in the cathodic direction. Thus we have, for the anodic and cathodic reactions, respectively: AG`A = AG'" — f3FA(1) A 60#

AG0# + (1 —

13)FA(1)

(5D) (32E)

Following the derivation that led to Eq. 31E we now obtain: i=

l — i = a c

->
Consider now what happens at equilibrium. The overpotential is, by definition, zero and so is the net current density. This leads to
h CB

i

0

at a high rate. The exchange current density can be orders of magnitude larger than the current measured, which may be a small difference between large anodic and cathodic currents. An alternative way to derive an expression for i o is to start with Eq. 30E and arrive at an expression such as Eq. 33E, in which k: and AO are used instead of k . Setting i = 0 and tI = 0 one has rev ‹-> F/RT] nFlehCA . ex p ((34 ,e , F/RT) = nFlehC B.exp [41-0)4rev

A(I) re v 6,(1,0

(34E)

(35E)

The physical meaning of 1 , which is called the exchange current density, should be clear from its definition in Eq. 34E. It represents the rate at which the electrochemical reaction proceeds back and forth at equilibrium when the net reaction rate, observed as a current flowing through the external circuit, is zero. It is similar to the exchange rate discussed earlier in connection with E9. 5E. We also note that i is the heterogeneous electrochemical rate constant at rl = 0, multiplied by the appropriate concentration. One cannot overemphasize that, while the measured current is zero, the reaction is not "frozen" and can occur

(RT/F)1n(C B/CA)

(37E)

Substituting this value of AO rev into the expression for i° , we obtain an expression of the form i o = Fk s,h C A(1-13) CP in which

— exp[—(1—(3)Fi3/RT])

(36E)

Now, we can write a Nernst-type equation for the absolute metal-solution potential difference at the reversible potential, namely

We can substitute this in Eq. 33E to obtain i 0 exp (‘ PFIVRT

.3

SINGLE-STEP ELECTRODE REACTIONS

k s,h

-> °F/RT) = le-exp [—(1 —(3) AO° F/RT] = le-exp(0,64 h

(38E)

(39E)

Equation 38E shows that the exchange current density is a function of the concentration of both reactants and products. This relationship is also implicit in Eq. 34E, considering that the rate constants in this equation depend exponentially on AO rev' which itself is a function of the ratio of concentrations of reactants and products. Some variations may be found in the sign convention used in different texts on electrode kinetics. In this book we shall consistently define anodic currents and anodic overpotentials as positive and the corresponding cathodic quantities as negative. The foregoing equations are consistent with this notation. Thus, if ri > 0, the anodic

112

ELEC. i RODE KINETICS

written Eq. 4D and for the electrochemical free energy of activation,

P.. SINGLE-STEP ELEC. DCODE REACTIONS

113

Equation 30E can also be written in the form

Eq. 5D. Taking z = 1 and using the standard free energies, we can write i = Fk C •exp((3FIVRT) h A

these equations with only minor modification as follows: AG° = AG° — Ft

AG' - [3E4

(4D) (5D)

in which k

h

(31E)

is the heterogeneous electrochemical rate constant at the

reversible potential. Equation 31E is an expression for the current density (which, we recall, is the rate of the reaction in electrical

The symmetry factor 13 in Eq. 5D is discussed in Section 12.4. Equations 4D and 5D are applicable to anodic (oxidation) reactions, for which changing the potential in the positive direction shifts the equilibrium toward the products and increases the reaction rate. For cathodic

units) in terms of the concentration and the overpotential, both of which can be measured and controlled experimentally. It should be noted that in deriving Eq. 31E we have tacitly assumed that the concentration at the surface is independent of the current density, namely, we have

(reduction) reactions, a positive sign should be used in both equations.

neglected mass transport limitations. Strictly speaking we should

When Eq. 5D is substituted into the rate equation we have

therefore replace t1 by ri ac in these equations. For the sake of simplicity we shall leave it as it is, and refer specifically to mass tran-

i = Fv = F.(0.0 .exp(—AG°4/RT)exp((3EAtIVRT) A

(27E)

sport limitation in a separate section.

There are two ways in which we wish to modify this equation.

Equation 31E represents the rate of an anodic oxidation reaction

First, we simplify it by replacing •exp(—AG °4/RT) by a "chemical" rate

for which the overpotential is, by definition, positive. For a cathodic

constant

kh,

which is the value of the heterogeneous rate constant at

= 0, to yield

reduction we write a similar equation with a negative sign in the exponent, and we take the overpotential, by definition, to be negative.

i = Fk° C •exp((3FAVRT) h A

(28E)

In either case the current density increases exponentially with increasing absolute value of the overpotential.

Second, we would like to get rid of the awkward term A(1), which, as we

The complete expression for the current density is obtained as the

recall, cannot be measured. Now, we have gone into some length to show

difference between the anodic and the cathodic current densities. This

(cf. Section 7) that although A4) cannot be measured, changes in it can

is the current measured in an external circuit. Now, if we have used

be readily determined. Bearing this in mind, we can write the over-

Eq. 5D to represent the potential dependence of the electrochemical

potential as follows:

energy of activation for the anodic direction, what would we use for the

rl =

— AOre, =

(29E)

cathodic reaction? Evidently the term including 44 must have the opposite sign, since the same potential that enhances the rate of the

where 64

rev

is the value of All) at the reversible potential (ri = 0).

Substituting AO from this equation into Eq. 28E, we obtain: i = Fkh C° .exp(134

TeV

F/RT)exp([34F/RT)

reaction in one direction must retard it in the opposite direction. What about the symmetry factor [3? We noted earlier (cf. Eq. 6D) that it

(30E)

must have a value between zero and unity. Also, we have seen in Eq. 7D

4

that

LLECIRODE KINETICS

p

a SINGLE-STEP ELECTRODE REACTIONS -

i.5 1

represents the ratio of the effect of potential on the electro-

at a high rate. The exchange current density can be orders of magnitude

chemical free energy of activation to its effect on the electrochemical

larger than the current measured, which may be a small difference

free energy of the overall reaction. If this fraction is in the

between large anodic and cathodic currents.

anodic direction, it must be (1 — f3) in the cathodic direction. Thus we have, for the anodic and cathodic reactions, respectively: AG °' = AG °11 — (3FAci) Ad°# = 6.G°4 + (1 — (3) FAQ

An alternative way to derive an expression for i° is to start with Eq. 30E and arrive at an expression such as Eq. 33E, in which k: and Ail)re, are used instead of k h . Setting i = 0 and Ti = 0 one has

(5D) (32E)


(36E)

h

Now, we can write a Nernst-type equation for the absolute metal-solution

Following the derivation that led to Eq. 31E we now obtain:

potential difference at the reversible potential, namely i = i — i = Fk C .exp(f3FIVIZT) — Fk C .expE(1—(3)(FIVRT)] (33E) a h A h B Consider now what happens at equilibrium. The overpotential is, by definition, zero and so is the net current density. This leads to -> Fk

<= Fk h CA h CB

i

AOre : AO° + (RT/F)1n(C B/CA)

(37E)

Substituting this value of 64 rev into the expression for io, we obtain an expression of the form

(34E)

= Fks,h CA(1-13) C 13

(38E)

We can substitute this in Eq. 33E to obtain in which i = i [exp(fiFtl/RT) — exp[—(1-13)Fil/RT]) 0

(35E)

k

sh

= le.exp(PAO° F/RT) = le.exp[—(1-10A0 ° F/RT]

(39E)

Equation 38E shows that the exchange current density is a function of The physical meaning of i , which is called the exchange current density, should be clear from its definition in Eq. 34E. It represents the rate at which the electrochemical reaction proceeds back and forth at equilibrium when the net reaction rate, observed as a current flowing through the external circuit, is zero. It is similar to the exchange rate discussed earlier in connection with Eq. 5E. We also note that i is the heterogeneous electrochemical rate constant at 11 = 0, multiplied by the appropriate concentration. One cannot overemphasize that, while the measured current is zero, the reaction is not "frozen" and can occur

the concentration of both reactants and products. This relationship is also implicit in Eq. 34E, considering that the rate constants in this equation depend exponentially on AO re v , which itself is a function of the ratio of concentrations of reactants and products. Some variations may be found in the sign convention used in different texts on electrode kinetics. In this book we shall consistently define anodic currents and anodic overpotentials as positive and the corresponding cathodic quantities as negative. The foregoing equations are consistent with this notation. Thus, if 11 > 0, the anodic

116

ELECTRODE KINETICS

E. SINGLE-STEP ELECTRODE REACTIONS

117

partial current is larger than the cathodic partial current, the overall observed current will be positive, and vice versa. Equation 35E is the general equation for a single-step chargetransfer process occurring under pure activation control. We shall now proceed to discuss some special cases of this equation, one of which leads to the Tafel equation.

12.4 Limiting Cases of the General Equation In many simple electrode reactions, the symmetry factor 13 is found to be close to one-half, and this value is usually assumed in the kinetic analysis of complex reactions. If this is substituted into Eq. 35E, we obtain: i = 2i -sinh(nF/2RT) 0

(40E)

Current-potential curves calculated from Eq. 35E are shown in Fig. 2E, for three values of 13.

The plots are linear near the equilibrium

Fig. 2E Current-potential plots for a single step charge-transfer process, for three values of f3, at low overpotentials. Note that

potential, and the current density increases exponentially at large

linearity is maintained longer for

overpotentials.

values of the symmetry factor.

13 = 0.50 than for other

Consider the case of low overpotential, often referred to as the micropolarization region or the

linear current-potential region. The

What is the width of the linear region represented by Eq. 6E.

exponential terms in Eq. 35E can he linearized, using the relationship:

There is no unique answer to this question, since it depends on the

exp x = 1 + x, which is valid for x « 1. This yields a linear relation-

level of accuracy desired. We can readily calculate the range of

ship between the current density and the overpotential, namely:

overpotential for which the deviation from linearity of the Uri plot

i = io(TIF/RT)

(6E)

Note that we have derived here Eq. 6E, which was written intuitively in Section 11.1. It is also interesting that the symmetry factor 13 disappeared from Eq. 35E in the process of linearization. Thus, the rate of reaction near equilibrium does not depend on the detailed shape of the energy barrier for activation (which determines the value of (3, as we shall see). It does, however, depend on the magnitude of the energy of activation, which manifests itself in the value of i . 0

does not exceed, say, 5%. This turns out to be about ± 28 mV for 13 = 0.5. For other values of 13 the curve is no longer symmetrical and the linear region is shorter, as can he seen in Fig. 2E, Thus, while 13 does not appear in the equation for the low overpotential linear approximation, its numerical value does influence the region over which this linear approximation applies. It is interesting to note that in Section 11.1 we estimated the linear region to be approximately ± 5 mV, whereas here we find it to be ± 28 mV for 13 = 0.5. This discrepancy arises

118

S INGLE-SI

ELEC.: I k ti or. r N ET

because in Section 11.1 we treated the matter in a tentative and qualitative manner, and here the calculation is quantitative. Also it turns out that the relatively long linear region applies only (rather fortuitously) to the case of 13 = 0.5 and is shorter for other values of of the symmetry factor. We turn out attention now to the case of high overpotential. One of the two exponential terms in Eq. 35E becomes negligible with respect to the other. For a large anodic overpotential one has i = io-exp((3FTVRT) (41E)

t ODE I:ACTIONS

9

2. The extrapolated anodic and cathodic lines intersect at 11 = 0, irrespective of the value of 0. This property could serve to determine the reversible potential, but it rarely does, since in most cases it is difficult to determine the current-potential relationship both at high anodic and high cathodic overpotentials. More often, the reversible potential for the reaction studied is known, and the experimental line is extrapolated to this value to obtain io. 3. The Tafel plot is presented in terms of the current density, while the quantity determined experimentally is the total current. An uncertainty in the real surface area of the electrode (which often exists, as

Written in logarithmic form, this equation becomes log i = log io + 13TIF/2.3RT = log io +

(42E)

where the Tafel slope b is defined as b = 2.3RT/I3F

(43E)

Equation 42E can be written in the form = b.log(i/i .)

(44E)

which is the same as the Tafel equation (Eq. 18E) introduced earlier, except for a change of sign, because here we refer to an anodic reaction. In Fig. 2E we showed the i/n relationship in both anodic and cathodic directions on a linear scale. In Fig. 3E the same data are shown on a semi-logarithmic plot (Tafel plot) as T1 versus log i.

A

close look at this figure reveals several interesting points: 1.

The Tafel plot is linear only at high values of the overpotential.

*

Mathematically this happens because the first nonlinear term in the Taylor expansion series of exp(x) is x 2/2 while for the hyperbolic function sinh(x) it is .0/6.

plots calculated from Eq. 35E for three Fig. 3E Semi-logarithmic values of 13. Note that the lines are symmetrical around 11 = 0 only for p = 0.5, but the lines extrapolated from high overpotential all meet at T1 = 0 and at i = i0 .

120

ELECTRODE KINETICS

121

E. SINGLE-STEP ELECTRODE REACTIONS

we shall see) causes a comparable uncertainty in the calculated value of

concern ourselves with it further. For the reaction to occur, a unit

i , but does not affect the Tafel slope. Similarly, if we do not know the reversible potential, we cannot determine io, but the Tafel slope can be obtained. 4. While i represents the value of the current density extrapolated 0 to ti = 0, the parameter a in the Tafel equation is the overpotential extrapolated to a high current density of i = 1.0 A/cm 2 . It is deter-

charge must cross the interphase, a process that requires electrical

k , the concentration of the reactants and the rate of change of current with potential, given by

metal, or if the electron jumps across to neutralize the ion on the

the Tafel slope b.

between. The electrochemical free energy is a state function and

mined by the electrochemical rate constant

energy amounting to 17(0 m — os). Any change in this potential difference will modify the electrochemical free energy of the reaction by the amount FS4. It does not matter, for the purpose of this argument, whether the reaction occurs by way of the positive ion crossing the interphase (from (O s to Om) to be neutralized by the electron in the solution side of the interphase, or the particles meet somewhere in changes in it depend only on the initial and final states, not on the

13. THE SYMMETRY FACTOR IN ELECTRODE KINETICS

route the system has taken to go from one to the other. We do not know the exact nature of the activated complex in such a

13.1 The Definition of (3

process and fortunately we do not need to know, for the present purpose.

The symmetry factor has already been defined (Eq. 7D) in terms of

All we have to assume is that it is an intermediate state, somewhere

the ratio between the effect of potential on the electrochemical free

between the initial and the final states which, by definition, it is.

energy of activation and its effect on the electrochemical free energy

As such we can say that the activated complex occurs somewhere in the

of the reaction:

interphase, where the potential has an intermediate value 0 # between O m

13 = (SAW/84)/06,MS*

(7D)

s , as shown schematically in Fig. 4E. Since the formation of the a activated complex is treated as a chemical reaction, we can write, for

It would seem appropriate to justify this definition here, and to

an anodic reaction:

compare it to some other ways in which this quantity has been defined in 6,G°4 AG°4/ — F44

the literature. To do this, consider a simple reaction of the type Ag+ + e Ag (45E)

and for the overall reaction we have, as before AG° = AG° — F4

(4D)

(S6, W - /84)/(S6,6 °/84) = SAS"/SAS

(47E)

To be specific, we must note the positions of the various species before and after the reaction has taken place. The Ag + ion is on the solution

Hence we obtain

side of the interphase, at a potential O s . The electron is in the metal, at a potential (1) m . The resulting silver atom will become part of the electrode and will be at the same potential O M, but since it is a neutral species, it is not affected by the potential, and we need not

(46E)

Comparing Eqs. 47E and 7D we see that the symmetry factor

p

is

given by

13 = 8A0V8,600

(48E:

122

LLEC: KO

n:04

L. S 111GLE-S 1E1-• Lir-L.:MODE REM:IIONS

Now, what is the physics behind these equations? We note that the potential difference A(1) across the interphase affects the free energy of

energy of activation, the rest being "lost" as far as our endeavor to accelerate the reaction rate is concerned. Why is it then that we call

the reaction. A fraction of this potential affects the free energy of

it the symmetry factor? It could be argued that a value of 13 = 0.5

formation of the activated complex (i.e., the free energy of activation When we conduct an

corresponds to a symmetrical situation, in which the activated complex is formed exactly halfway between the reactant and the product. Also,

experiment, we can control the changes in AO, but we have no control

if we look again at Figs. 2E and 3E, we note that the value of 13

of the reaction). This fraction is defined as

0.

over the changes in 64 #, which amounts to a fraction

13 of the former.

determines the degree of symmetry of the plots around the reversible

Therefore, it might be most appropriate to refer to the parameter 13

potential. The important thing to remember is that 13 is determined by

as an efficiency factor. We try to accelerate the rate of an electro-

the shape of the free-energy barrier, which the system must cross along the reaction coordinate, as it is transformed from reactants to

chemical reaction by applying an amount of electrical energy 8,64F, but only a fraction 0 of this electrical energy is used to reduce the free

products. 13.2 The Numerical Value of 13 The determination of the numerical value of the symmetry factor

0

is a thorny problem in electrode kinetics. We might start with the conclusion: namely, that it is common practice to use the value of 13 0.5 in the study of electrode reactions. It is hard to come up with a satisfactory theory showing why this should be so, but there seems to be 0

some good experimental evidence that it is, at least in a large number

a) if

MASI)

c

S

0

of experimental systems. The most reliable data are from studies of hydrogen evolution on mercury cathodes in acid solutions. This reaction has been studied most extensively over the years. The use of a renewable surface (a dropping mercury electrode, in which a new surface is formed every few seconds),

e m xo

g+

Distance

Fig. 4E Charge transfer across the Helmholtz double layer. The reactant is at the potential 4S , the product is at a potential (I)M and the activated complex is at an intermediate position where the potential is

e, which is partway between 4) s and 4)M .

our ability to purify the electrode by distillation, the long range of overpotentials over which the Tafel equation is applicable and the relatively simple mechanism of the reaction in this system all combine to give high credence to the conclusion that 13 = 0.5. This value has been used in almost all mechanistic studies in electrode kinetics and has led to consistent interpretations of the experimental behavior. It

124

ELECTRODE KINETICS

125

a SINGLE-STEP ELECTRODE REACTIONS

is reasonable to adopt this practice and to use the value of 13 = 0.5, in spite of the lack of solid theoretical evidence to support it. 13.3 Is the Symmetry Factor Potential Dependent? The other difficulty concerning the symmetry factor is its dependence on potential, if any. Here again there is highly reliable and extensive experimental evidence showing that the symmetry factor is

0 C 0 0

independent of potential. But what about theory? If we can adopt a simple model such as shown in Fig. 4E, it follows that 13 should indeed be independent of potential. This is shown schematically in Fig. 5E for two cases, one in which the potential difference across the iliterphase is linear and one in which it decays exponentially. In the linear case it would seem obvious that the change of (1) # is proportional to the change in (1)"1 , so that 13 = SA(1) #/6A(1) turns out to be independent of C

potential. For the exponential case, we could write: = O mexp(—r/d)

Distance

0

(49E)

a_

where r is the distance from the metal surface and d is a characteristic length. The potential at the plane of the activated complex, a distance

r# from the metal surface is accordingly: 4i)4/ = (1) mexp(—rlt/d)

Distance

(50E)

metal-solution potential difference A(I) is the sum of its value at equilibrium A(1),,v , and the value S(AO)added when an over-

Defining the potential in the solution as zero we have: etAl)m = [(I) mexp(—r#/d)]/41 m = exp(—r#/d)

Fig. SE Model showing how 13 can be independent of potential. The

(51E)

potential is applied. The corresponding values for the

This ratio depends on the distance of the activated complex from the

activated complex are marked as A(I) # , A(I) rev and

metal surface and on the shape of the dependence of potential on

respectively.

05(A4 # ),

distance in the interphase; it does not depend on the value of (I) m. Thus, 13 can be independent of potential, even if the variation of the potential inside the interphase in nonlinear.

One cannot rule out the possibility that the change in applied potential SAO will cause a significant change in the shape of the energy barrier, thereby effectively changing the position of the activated

ILO ELECTRODE KINETICS

127

F. MULTI STEP ELECTRODE REACTIONS

complex in the interphase, giving rise to a potential-dependent symmetry factor. Several theories proposing such behavior have indeed been put forward. As we have done regarding the numerical value of 13, however, we rely on solid experimental evidence and treat 13 as a constant parameter, independent of potential.

. MULTI STEP ELECTRODE REACTIONS 14. MECHANISTIC CRITERIA 14.1 The Transfer Coefficient a and Its Relation to 13 So far we have discussed the current-potential relationship for a simple electrode reaction, which occurs in one step and in which a single electron is transferred. We might add here, in parentheses, that there can be only one electron transferred in each elementary step, so that all single-step reactions involve the transfer of only one electron. Real systems are, however, more complex. Most electrode reactions occur in several steps, and the transition from the reactants to the products requires the transfer of several electrons. Even an apparently simple reaction, such as silver plating, cannot be described by the simple equation Ag+ + eM

(IF)

Ag

It might therefore be since the silver ion in solution is solvated. closer to reality to describe this electrodeposition process by the following series of equations [Ag(H 2 0)n]

[Ag(H20) n _ 1 ] + H2O

[Ag(H 20) ,, 2] + 2H20

Ag+ + eM

Ag

(2F)

in which the intermediate species are stabilized by adsorption on the surface, and partial neutralization of the ion may, in effect, take place as the solvent molecules are removed, one at a time, from the solvated ion. Even this scheme may not represent the end of this apparently simple process, since a metal atom is not stable everywhere

128

PI ECTRODE KINETICS

on the surface. Having been deposited, it usually moves on the surface

F, MULTI STEP ELECTRODE REACTIONS

a = — (2.3 RT/F) (

129

(6F)

to a stable position, where it is incorporated into the crystal lattice. In contrast, when one considers a complex reaction, such as the

coefficient a . Equation 5F relates it directly to the measured quantity (ar/a log i), while Eq. 6F can be regarded as a formal defini-

oxidation of methanol in a fuel cell, the overall reaction is: CH OH + H 2 O -> CO 2 + 6H+ + 6e rvt 3

Either Eq. 5F or 6F can be considered to be a definition of the transfer

(3F)

which, as we have pointed out, must proceed in at least six steps and in

tion, indicating that the transfer coefficient a c , is simply the reciprocal Tafel slope in dimensionless form. Although the definition of a and the difference between it and 13

reality requires more steps to go from reactants to products. We shall not discuss here the mechanism of either of these reac-

should be quite clear from the preceding explanation, there has been a

tions; they were mentioned only to show the complexity of typical

fair amount of confusion in the literature concerning these two para-

electrode reactions. It is evident then that the equations of electrode

meters. It is therefore appropriate to state again the definitions of

kinetics derived in Chapter E must be generalized to describe multi step

these quantities and to highlight their different physical meanings. The symmetry factor 13 is a fundamental parameter in electrode

electrode processes, which involve the transfer of several electrons. We recall that the Tafel equation was originally observed for the

kinetics. It must always be discussed with respect to a specific single

hydrogen evolution reaction on mercury. We should therefore have an

step in a reaction sequence, and its value (which, by definition, must

equation similar to Eq. 41E to describe the current-potential relation-

be between zero and unity) is related to the shape of the free-energy

ship. For a single cathodic step we wrote:

barrier and to the position of the activated complex along the reaction

i = io.exp(—(3iFiRT)

(41E)

coordinate. When the potential across the interface is changed (by application of an external current or potential), both the electro-

But the symmetry factor [3 has been defined strictly for a single step

chemical free energy of activation and the electrochemical free energy

and is related to the shape of the free-energy barrier and to the

of the reaction are altered. The symmetry factor [3 represents the ratio

position of the activated complex along the reaction coordinate. To

between the changes in these quantities, as given by Eq. 7D , namely:

describe a multi step process, 13 must be replaced by an experimental parameter, which we call the cathodic transfer coefficient a . Instead of Eq. 41E we then write:

(4F)

P,T,C

— 2.3(RT/a F)

This can be turned around and written as follows:

from the current-potential relationship. Just that and nothing more! It is equal to the inverse of the Tafel slope b expressed in units of

It is now evident that the Tafel slope will take the form: (81-ila log i)

( 7D)

The transfer coefficient a is an experimental parameter obtained

i = io.exp(—acTiFiRT)

b

13 = ( SAG°V.564)/(86,08,#) = (86,G°V8AG °)

(2.3RT/F), that is, to the inverse Tafel slope in dimensionless form.

(5F)

It is shown later that the relationship between a and 13 depends on the mechanism of the reaction. The transfer coefficient is therefore one of

the parameters that allow us to evaluate the mechanism of electrode

130

Ft FCTRODE KINETICS

14.2 Steady State and Quasi-Equilibrium

reactions or to distinguish between different plausible mechanisms. It cannot be overemphasized that one can measure only the transfer coefficient a, not the symmetry factor 0. The latter can be inferred from the former by making a suitable set of assumptions. For example, for the hydrogen evolution reaction on mercury, it is commonly assumed that a = 0. One often tends to refer in this case to the measurement of the symmetry factor; even in this simple case, however, only the transfer coefficient can be measured, and the symmetry factor must be calculated from it (even if in this case "calculation" simply means making them equal), on the basis of certain assumptions. There is an additional difference of great importance between 13 and a, which is often overlooked. The way in which f3 is defined requires that the sum of the symmetry factors in the anodic and cathodic directions be unity; if it is 0 for the cathodic reaction, it must be 1 — 13 for the anodic reaction, and vice versa. The same is not true with respect to the transfer coefficient. To begin with, a is a parameter obtained from experiment. One therefore must find its value, not decide what it should be. When we wish to write a rate equation, such as Eq. 35E, for a multi-step reaction, the correct form will be

i = i o [exp(a atiFiRT) — exp(—acriFiRT)]

(7F)

* in which a + a does not necessarily add up to unity. Indeed, one can readily write mechanisms for which the transfer coefficient is greater than unity, as we shall see below. *

In general (a. +

131

F. MULTI S !El" ELECTROull I
= n/v in which n is the number of electrons transferred in the overall reaction and v is the stoichiometric number, to be discussed later.

Consider a simple anodic reaction, such as the oxidation of C1. to Cl , which can occur in the following two steps: 2 Cl ads + e M C11- Clads

C1

2

(8F) (9F)

eM

The concentration of the adsorbed intermediate can best be expressed in terms of the partial surface coverage 0, which is defined as the surface concentration F (mol/cm2 ) divided by the maximum surface concentration Fmax (expressed, of course, in the same units): 0

(10F)

max

The net rate of formation of the adsorbed intermediate can be written as (11F)

dO/dt = k i (E).(1-0)Cc1 — k i (E)0 — k2(E)OCci_+ k 2(a)•(1-0)Ca 2

in which ki (E) are the potential-dependent electrochemical rate constants. This can be solved readily under steady-state conditions by setting dO/dt = 0. It is, however, easier to treat multi step reaction sequences in the framework of the quasi-equilibrium assumption, as shown below. When a reaction occurs in several consecutive steps, the rate of all steps must be equal at steady state (otherwise the system would not be at steady state). This rate is determined by the slowest step in the sequence, which we refer to as the rate-determining step (rds). In the preceding example, if k i (E), the specific rate constant for step 8F, is much smaller than k 2 (E), the specific rate constant for step 9F, the rate of the second step will effectively be limited by the supply of adsorbed intermediates Cl ads , namely, by the rate of the first step. This is somewhat like fluid flowing through a maze of pipes and valves,

132

ELECTRODE KINETICS

the constant rate of flow being determined by the narrowest constriction along the way. For a better description, however, consider a potential applied to several resistors in series. The current is determined by the overall resistance, which is simply the sum of the resistances in series (R = R + R

1

2

+

R3

....

R ).

If one of these is much larger

133

F. MULTI STEP ELECTRODE REACTIONS

to observe a single rate-determining step, at least over a certain potential range. It is therefore both sensible and common to treat the mechanisms of electrode reactions on the assumption that there is a well-defined rate-determining step in each potential region studied, with possible transitions from one rate determining step to another as the potential is changed.

than all others, it will be dominant, and one has R = R i . To use this simple equivalent circuit, it must be realized that the rate constants of the various steps can be represented by the inverse of the corresponding resistances: the higher the resistance, the lower the rate constant, and vice versa. Thus, the overall effective rate constant

Now we come to the concept of quasi-equilibrium. If there is a distinct rate-determining step in a reaction sequence, then

all other

steps before and after it must be effectively at equilibrium.

This

comes about because the overall rate is, by definition, very slow compared to the rate at which each of the other steps could proceed by

is given by

itself, and equilibrium in these steps is therefore barely disturbed. 1/k = 1/k 1 + 1/k2 + 1/k3

+ 1/kn

(12F)

It is evident then that the overall rate constant is determined by the

lowest rate constant. We note, in passing, that Eq. 12F is similar to Eq. 3A, which correlates the overall current to the activation and masstransport controlled currents.

To see this better, consider the specific example given earlier for chlorine evolution. Assume, for the sake of argument, that the values of the exchange current density i for steps 8F and 9F are 250 and 1.0 2 mA/cm , respectively. Assume now that we apply a current density of 0.5 2 mA/cm . We can calculate the overpotential corresponding to each step in the sequence, using Eq. 6E, namely

1/i = 1/i + ac

L

(3A)

This should not be surprising, since the activation and mass-transport

i = io 6F/RT) We find for step 8F:

processes always occur in series and should combine to determine the overall rate in the same way as several-activation controlled steps in

(6E)

= (RT/F)(0.5/250) = 0.05 mV and for step 9F:

series.

rl = (RT/F)(0.5/1.0) = 12.83 mV

How do we know that there is a distinct rate-determining step, one for which ki is much smaller than all other rate constants? The fact is

The total overpotential is the sum of these values, namely rl = 12.88 mV,

that we do not know a priori, and indeed one could envisage situations

of which only 0.05 mV (0.4%) is associated with the first step. Hence

in which several rate constants in a sequence could be comparable in magnitude. But in electrode kinetics, one recalls, the rate constant is a function of potential, and in a reaction sequence all the rate constants can depend differently on potential. Thus, in electrochemistry, more than in other fields of chemical kinetics, one is likely

we can consider this step to be effectively at equilibrium. Thus we proceed to calculate the kinetic parameters for the reaction sequence, assuming that all steps other than the rate-determining step are at equilibrium.

134

ELECTRODE KINETICS

Why call it the quasi-equilibrium assumption? This point has nothing to do with the mathematical treatment that follows; it is used only to pacify our conscience.

Strictly speaking, the preceding

assumption is self-contradictory.

Equilibrium is defined as the state in which no net reaction takes place. There is an exchange reaction (the rate of which is represented in electrode kinetics by i) but the rates of the forward and backward reactions are equal. How can we say that a step in a reaction sequence is at equilibrium while it is proceeding at a finite rate in one direction? We use the term quasiequilibrium in recognition that this is only an approximation, which serves our purpose well, even as we remain fully aware of its logical limitations.

F. MULTI STEP ELECTRODE REACt IONS

either negligible or has been corrected for in a quantitative manner. It should be noted that we have used the symmetry factor, (3, not the transfer coefficient a. This is because we are referring to a specific step in the reaction sequence, not the whole reaction. Coming back to Eq. 14F, we have good reason to assume that the partial coverage 0 is very small, since the intermediate is formed in the rate-determining step and is removed by the following step, which is much faster. Taking 0 « 1, and hence (1 — 0) = 1, we can rewrite Eq. 14F to within a very good approximation as: i 1 = FklCCI- exp OFF/12T)

i = nFk Ca: exp (a.EF/RT) = io•exp(aariF/RT)

proceed to calculate the Tafel slopes and some other kinetic parameters for a few very simple hypothetical cases, to show how such calculations are made. In Section 15 we shall discuss the kinetics of several reactions that either have been important in the development of the theory of electrode kinetics or are of Current practical importance. Consider again the chlorine evolution reaction and let us assume first that step 8F is rate determining k

CI

rds Clads

eM

(13F)

FkICCI - (1 — 0)exp((3EF/RT)

(16F)

It is important to understand the logic of the transition from Eq. 15F to 16F. In the first we used the partial current i 1 for a step in the reaction sequence; in the second we used the total current i. In the present case i = 2i 1 , since for every electron transferred in the first step, there is another one transferred in the second step (n = 2). Also we have changed from 13 to a , since Eq. 15F refers to an elementary step a

in the reaction sequence, while Eq. 16F refers to the overall reaction. In this particular case we find that aa = 13, but that is besides the * point. It should also be noted here that the exchange current density is related to the rate constant, to the concentration of the reactant, and to the metal-solution potential difference at the reversible

The rate of this step is given by i1=

(15F)

For the overall reaction we can then write:

14.3 Calculation of the Tafel Slope Equipped with the assumption of quasi-equilibrium, we can now

135

(14F)

This equation is applicable at high overpotentials, where the reverse reaction can be ignored. Also, it is assumed in writing this equation (and all following kinetic equations) that mass transport limitation is

*

It may seem odd that we designate as for the anodic process and do not use the same notation for 13. This follows from the definition of 0, according to which Pc = I — 13. . The same relationship cannot be used for a, as we have pointed out earlier.

136

ELECTRODE KINETICS

potential (since E = E + 117, and the metal-solution potential reV difference at the reversible potential 64rev differs from Erev only by a constant). i = nFic C .exp(a E FAT) o 1 C1a rev

We shall presently show that Eq. 19F is equivalent to the Nernst equation, but first let us work out the kinetic equations. For the rate-determining step (Eq. 9F) we write i 2 = Fk 2 C Cl- 0-exp((3EF/RT)

(17F)

We can readily calculate the Tafel slope for this case, if we assign a numerical value to the symmetry factor P. This, as we have said before, is commonly taken to be 0.5. The Tafel slope can then be obtained either from Eq. 5F, namely

137

F MULTI STEP ELECTRODE REACTIONS

(20F)

in which we have to substitute the potential-dependent value of 0 from Eq. 19F. To simplify things, we consider two extreme cases: 1. For 0 « 1, we can write (1 — 0) = 1. Then, by combining Eqs. 19F and 20F we have i = n Fk2K Cc21: exp [(1 + (3)EF/RT]

(21F)

b a = 2.312Ticta F = 2.3RT/PF = 2(2.3RT/F) = 118 mV (at 25 °C) from which it follows that, under these assumptions, or directly from Eq. 16F, by calculating the value of (aE/alog )

P,T,C.

We can now proceed to obtain the kinetic equation for the same reaction in a somewhat more complicated case, when the second step (Eq. 9F) is assumed to be the rate determining step, and the first step (Eq. 8F) is at quasi-equilibrium. For quasi-equilibrium in step 8F we can write k Ca_(1 — 0)exp(PEF/RT) = k 1 0.exp[—(1 — 13)EF/RTj( 1

18F)

0

— KI C ct-•exp(EF/RT)

+1i= 1.5

and b. = 2.3RT/(1 + 0)1? = 40 mV 2. When the surface coverage approaches unity and can no longer change significantly with potential, we can substitute 0 = 1 in Eq. 20F, to obtain (22F) .exp((3EF/RT) C 2 Cland b = 2(2.3RT/F) = 118 mV. This result seems to

i = nFk

from which it follows that 1 _ 0

aa = 1

which leads to a = a be identical to that derived by assuming that the first step in the (19F)

in which K

= k /k 1 1 -1 is the equilibrium constant. The absence of the symmetry factor R from this equation is not an error, neither is it a

coincidence. The symmetry factor is strictly related to the shape of the free-energy barrier and the position of the activated complex along the reaction coordinate. Equation 19F describes an equilibrium, which is independent of the preceding considerations and is related only to the difference in free energy between the initial and the final states.

reaction sequence is rate determining. There is one important difference, however. Equation 22F was derived for the case of high surface coverage (0 = 1), while Eq. 15F applies for limitingly low coverage, where (1 — 0) can replaced by unity. To distinguish between these possibilities, we must make independent measurements of the surface coverage. The two limiting cases just discussed are approximately applicable for 0 5_ 0.1 and 0

0.9, respectively. In the intermediate region one

138

ELECTRODE KINETICS

b9

F. MULTI STEP ELECTRODE REACTIONS

log

can readily solve the equation by substituting 0 from Eq. 19F into

[1 — 0

= logK1 + logCa + EF/2.3RT

(24F)

Eq. 20F, but the result (besides being cumbersome) leads to a transfer coefficient that decreases gradually with increasing overpotential, from

ae = 1 + 3 to a s

=

13. This also means that the Tafel slope changes with potential. In other words, the Tafel plot (which is the plot of log i

which is transformed to E = — (2.3RT/F)logK 1+ (2.3RT/F)log [[

1

°

11 fi - 0LI L

(25F)

versus E is not linear in this intermediate range of coverage. It

One can consider 0 as being equivalent to the concentration of the

should be noted that the potential dependence of a shown here is due to the variation of the partial coverage 0 with potential, not because 13

products (adsorbed intermediates) in this reaction and (1 — 0) can be

itself is potential dependent. In our calculations of the transfer coefficients and Tafel slopes

sites on the surface), making the second term in Eq. 25F a typical ratio of concentrations of products to reactants. Also, we know from thermo-

for very low and very high coverage, we omitted the intermediate range

dynamics that the equilibrium constant is related to the standard free

of 0.1 5_ 0 0.9. Have we missed the most interesting region? What is

energy of the reaction, and the latter is related to the standard

the probability that, for the reaction being studied, the range of

potential:

potential accessible for experiment will happen to correspond to values

considered equivalent to the concentration of one of the reactants (free

(2.3RT/F)logK = — AGIF = Eo°

(26F)

of 0 that are either below 0.1 or above 0.9? We can use Eq. 19F to calculate the change in potential required to increase 0 from 0.1 to 0.9. This turns out to be quite small: E(0 = 0.9) — E(0 = 0.1) = (2.3RT/F)log(81) = 113 mV

E = Eo° + (2.3RT/F)log

C1 J

( i

0

11

(27F)

(23F)

Generally speaking, the potential range accessible for measurement is of the order of 2-3 V, depending on the solvent, the solute and the electrodes used. Thus, on the basis of a purely statistical argument, it can be said that there is more than a 95% chance that the coverage will be either below 0.1 or above 0.9, where one of the two limiting cases discussed above is applicable. We have stated that Eq. 19F is equivalent to the Nernst equation. This can be proved in the following simple way. We write this equation in logarithmic form as:

Substituting in Eq. 25F we thus have

where E0° is the standard potential for a reaction involving the formalion of an adsorbed species by charge transfer such, as Eq. 8F. We note that since the standard state for the adsorbed species is chosen as 0 = 0.5, the term 0/(1 — 0) will be unity. Equation 27F is the Nernst equation, written for a single electron transfer. There is nothing unusual in this result, of course. We treated Eq. 8F as an electrochemical equilibrium and obtained the correct equation for an electrochemical reaction at equilibrium. The point is that the Nernst equation is commonly derived from thermodynamic considerations. It is reassuring to be able to start from kinetic equations and reach the same result, in the limit where equilibrium is assumed!

140

ELECTRODE KINETICS

F. MULTI STEP ELEC I RODE REACTIONS

14.4 Reaction Orders in Electrode Kinetics

141

a

The reaction order is defined in chemical kinetics by the partial derivative (28F)

p = (alog via logC) Tp

which measures the dependence of the reaction rate v on the concentra-

N

tion of one species in solution, keeping the concentrations of all others (as well as the temperature and pressure) constant. The experiment implied by this equation can be performed only approximately, since the rate of the reaction is measured by observing the change of concentration of reactants or products or the change in some physical property caused by it, and the condition of constant concentrations, required by the partial derivative (Eq. 28F), is hard to maintain. In electrochemistry the reaction order is defined in a similar

40 60 80 an arbitrary reference

100

b

manner, but, in addition to keeping the temperature and the pressure constant, one should maintain a constant potential. As a result, there are two reaction orders in electrochemistry, one taken at constant

potential and the other taken at constant overpotential: p1 =

(a log ValogC

(29F) J ul

and p

2

= (alog lialOgC ) i n,Tx,C

(30F)

It is important to distinguish between these two parameters, since the reversible potential changes as we change the concentration of the reactant, and the overpotential can thus change while the applied potential remains constant.

Fig.IF Calculation of the reaction order from the Tafel plots. (a) Tafel

In Eq. 29F the potential E, which we keep constant, is the measured

lines for a series of concentrations of the electroactive

potential, with respect to a fixed reference electrode. It does not

species. (b) Reaction-order plots derived from (a) for different values of the potential.

matter which reference electrode is used, as long as it is the same throughout the experiment. Actually, we would like to obtain the

ELLt.' I RODE KINETICS

reaction order while keeping the metal-solution potential difference

KulJE REAL ...

MULTI STEP

reversible potential on concentration is well known.

Ao

To show the difference between these two reaction order parameters,

constant. This would seem to be impossible, since Ail) cannot be measured,

we return to the equations derived in Section 14.3 for the chlorine

as explained in detail at the beginning of the book. But the absolute

evolution reaction.

metal-solution potential difference differs from the measured potential

Consider the case in which the first chargeEquation 16F in loga-

transfer step (Eq. 13F) is rate determining.

only a constant! Thus, by keeping the potential E constant, we can be

rithmic form, is:

assured that the metal-solution potential difference AO is also maintained onstant, even though its value is not known.

log i = log(nFk 1 ) + logCa_ + ccaEF/2.3RT

(31F)

p 1 = (alog i/alogCci_) E = 1

(32F)

We recall that the current is a very sensitive measure of the rate hence

of an electrochemical reaction. It is therefore quite easy to determine the current-potential relationship without causing a significant change

To calculate p 2 , the reaction order at constant overpotential, we in Eq. 31F and express the reversible potential substitute E = Erev + in terms of the Nernst equation, namely:

in the concentration of either reactants or products. Thus, measurements in electrode kinetics are conducted effectively under quasi-zeroorder kinetic conditions. It would be wrong to infer from this that electrode reactions are independent of concentration. To determine the

— (2.3RT/2F) log [Cci Erev = E°

concentration dependence (i.e., the reaction order), one must obtain a

/

(33F)

Ca21

series of ilE or 01 plots and derive from them plots of log i versus logCi at different potentials, as shown in Fig. IF. The slopes in

Assuming that the solution is saturated with respect to C1 2, we can take

Fig. 1F(b) yield the parameter p since p 1 = (alog i/alogCi) E is 1 measured at constant potential E. Here, and in all further equations,

form:

this concentration to be constant and rewrite Eq. 31F in the following

log i = log(nFk 1 ) +

oc.Erev F/2.3RT + aanF/2.3RT

we shall assume that T, P, and the concentration of all other species in solution C

j#1 are kept constant, to permit us to write the equations in a more concise form.

= log(nFk) + 1ogCci + a.E°F/2.3RT — aalogCci _ + aaTIF/2.3RT (34F) Differentiating with respect to logC a_ while keeping

Which of the two reaction order parameters should one prefer? When measurement is made with a constant reference electrode (e.g., calomel),

p2 = (alog i/alogCci )T1 = 1 — as

the reaction order at constant potential is obtained. If, however, an indicator-type reference electrode is employed (e.g., a reversible

(35F)

We note that the reaction order at constant overpotential is a little more complex and depends also on the transfer coefficient oc a .

hydrogen electrode, often used in the study of the hydrogen evolution reaction), the overpotential can be kept constant and the parameter p 2 istheondrclyba.Ieiths,oparmcnbe calculated from the same experimental data, since the dependence of the

constant yields:

One may be tempted to obtain both p 1 and p2 from the experimental data, in order to calculate the transfer coefficient from the difference. This can be done, in principle, but the measurement of reaction orders

144

F1 FCTRODE KINETICS

is inherently much less accurate than the measurement of the Tafel slope, hence this approach is not expedient, unless the transfer

of the reactant (H 0+ or 01-I) over many orders of magnitude, without 3 running into solubility limitations on the high concentration end and mass transport limitations on the low concentration end (as long as the

coefficient cannot be measured directly. One often encounters in the literature two additional types of reaction orders, which are actually derived from the two we have just discussed. The first relates to the variation of the exchange current

density with concentration, namely

145

F. MULTI STEP ELECTRODE REACTIONS

(a log ilalogC). It is easy to see io, one has actually

that this parameter is equal to p 2, since by using

specified a constant overpotential of tl = 0. This parameter is useful

solution is buffered). The second reason is that in aqueous solutions the solvent itself can be the reactant or the product in the reaction being studied. A few words of caution may be appropriate in regard to the use of this concept, particularly for readers who did not major in chemistry. The pH of an aqueous solution is formally defined as follows:

for the study of fast electrode reactions, where measurements can be

pH

7 -

taken only at low overpotentials and the Tafel region is not accessible experimentally. The second reaction order parameter occasionally used is (aE/alogC) i . Applying the rule of partial derivatives, we can write:

(38F)

log[a(H3o I )]

in which a(ii3o+ ) is the activity of the hydronium ion.

*

In dilute

solutions, the activity can be approximated by the concentration, but this is far from being true in concentrated solutions. Thus, for

Eialogq(a logCia log i) E(alog itaE) c = — 1

(36F)

example, in a 7.0 M solution of KOH the pH is about 16, far from the value of 14.85 calculated from Eq. 38F, with the use of concentration

Hence (aEialogC) i= — bp i

(37F)

instead of activity. Also, if one uses a mixed solvent (e.g., methanol and water), the concept of pH becomes somewhat more complicated, and

There is nothing wrong with using any of these reaction order parameters

must be employed with great care. In such cases, even if the pH can be

as long as we remember that only two of them (any two, of course) are

measured by a conventional method (e.g., with a glass electrode), the

independent, and the other two can always be derived from these.

result may represent quite different concentrations of H 30+ ions in different solvent mixtures.

14.5 The Effect of pH on Reaction Rates

When the temperature is changed, the equilibrium constant for the

The influence of pH on reaction rates may be looked upon as just

dissociation of water is also changed. As a result, pH 7.0 only

another concentration effect, which can be dealt with in terms of the

represents the point of neutrality at 25 °C. At higher temperatures the

reaction orders just discussed. It merits special attention, however,

point of neutrality moves to lower pH values. Similarly, in mixed

for two reasons: first, because it allows us to change the concentration

solvents the point of neutrality is not necessarily at pH 7.0.

*

This is the case for fast electrode reactions, where measurements are confined to the linear Uri region.

*

We shall not discuss here the difficult problem of defining the

activity of a single ionic species.

146

Li:

ERODE KINETICS

Let us now return to the effect of pH on electrode kinetics, using concentrations instead of activities. Consider the hydrogen evolution reaction, and assume that it proceeds in the following two steps, with the second step being rate determining. HI- e 2H

ads

H rd s ----3

H

ads

2

(39F) (40F)

In these equations we have replaced H 30+ with lit for simplicity. This is not meant to imply that a bare proton can exist in solution. In fact, there is ample evidence to show that li t is energetically unstable and probably exists as the hydronium ion 11 30+ , or in a more solvated form, such as H 0+. 94

F. MULTI STEP ELECTRODE

1—0

= K I C H+ exp(—EF/RT)

into Eq. 42F to obtain i = nFK2.k 2 C 2 .exp(-2EFIRT)

p 1 = (alog iialogC Ht) E = — (alog iiapH) E = 2

(41F)

necessary because we are now dealing with a cathodic reaction. For the rate-determining step we write: =

Fk

2

02

(44F)

Also, since we can write the Nernst equation in this case as: RT I log [130-12)/C2Hil E° — 2.3 I- 2F = E° — r 2.3RT } pH _ r 2.3 RT logPoi 2 I- 2 F

E E re

(45F)

we have, from Eq. 43F, (taking Pot 2) = 1) log i = log(nFK2i .k2) + 2.1og(CH+)

evolution, except for the change in the sign of the exponent, which is

2

(43F)

from which we obtain directly

— 2(E° F/2.3RT) — 2-log(C H+) — 2(fiF/2.3RT)

(46F)

= log(nFK2i k2) — 2(E° F/2.3RT) — 2(fiF/2.3RT)

This is identical to Eq. 19F, derived for the first step in chlorine

1

141

If we assume low coverage (0 « 1), we can substitute 0 from Eq. 41F

For (quasi) equilibrium in the first step we can write: 0

REAt.:iluNS

We note that in this particular case, the reaction order at constant overpotential p 2 becomes independent of pH. When the pH is lowered, the effect of increasing concentration of the hydronium ion is exactly compensated by the influence of the variation of the reversible poten-

(42F)

This is an interesting case, which may need some clarification. Equation 40F, as written, is not an electrochemical step in the sense that it does not involve charge transfer. Nevertheless we are justified in expressing the rate of this step in terms of a current density, since every time step 40F occurs, two electrons must have been transferred in the preceding step.

tial on the reaction rate. did not appear It is interesting to note that the symmetry factor in any of these equations. This is because the rate-determining step assumed here does not involve charge transfer. The current depends indirectly on potential, through the potential dependence of the fractional coverage 0. The transfer coefficient is ac = 2, as can be seen in Eq. 43F, corresponding to a Tafel slope of b = — 30 mV at room temperature.

148

ELECTRODE KINETICS

Note that the transfer coefficient obtained here is not in any way related to the symmetry factor. It follows from the quasi-equilibrium assumption and should therefore be a true constant, independent of potential and temperature, as long as the assumptions leading to Eq. 43F

149

F. MULTI STEP ELECTRODE REACTIONS

such as platinum, palladium and nickel in alkaline solutions. Thus, it has become accepted that the separation factor depends on mechanism. The theory of the isotopic separation factor and its dependence on mechanism is rather involved and is of no interest to us here; particularly since the argument is still open, the different views have not

are valid.

been unified, and the subject is not ripe for introduction in textbooks. It is generally accepted, though, that the separation factor depends on

14.6 Isotope Effects Depend on the Mechanism Isotope effects have been of great interest in electrochemistry, both from the fundamental and the applied points of view. Most work was naturally conducted on the hydrogen evolution reaction, since the isotope effect is expected (and indeed found) to be the highest there,

mechanism, equal values of the separation factor implying identical mechanism and vice versa. Thus, the measurement of the separation factor can serve as an important tool in mechanistic studies, even if the theory behind it is not entirely clear. One could use the method of electrolysis to separate other iso-

because the ratio of masses of the different isotopes is the highest. When a mixture of ordinary and heavy water (14 20 and D20) is electrolyzed, the isotope ratio H/D is higher in the gas phase than in the liquid. In other words, the gas phase is enriched with the lighter isotope, while the solution phase is enriched with the heavier isotope. Electrolysis served as the basis for preparation of heavy water during

topes, for instance, to produce 18 0, but the separation factors are low,

and the method is not expedient, unless the product has a very high value and other competing methods of separation (e.g., gas diffusion) cannot be used for one reason or another.

14.7 The Stoichiometric Number

World War II, but has since been superceded by other, more efficient, The stoichiometric number v is defined as the number of times the

methods. The separation factor S H/D is defined as the ratio of concentra-

rate determining step must occur, for the overall reaction to occur once. This is best explained by showing a few simple examples. For the

tions of the isotopes in the gas phase over that in the solution

reaction sequence shown earlier for hydrogen evolution S„ , — tv D

[H]/[D] gas [H]/[D]

(47F)

Fl+ + e

H

Iiq

Already in the earliest studies it was realized that the separation

211

rds ads

ads

u

.

2

(39F) (40F)

factor depends on the system being considered, mainly on the metal used as the cathode and on pH. From experimental evidence, it also became apparent that the separation factors measured fall in groups, the lowest being observed for mercury and some of the soft metals, such as tin, lead and bismuth in acid solutions; the highest, for catalytic metals

the stoichiometric number v is 1. If we rewrite this so that the first step is assumed to be rate-determining, namely

1.JU

ELECTRODE KINETICS

H+ + e

i = iSTI F/ RT)

H M

(48F)

ads

2H ---> H

F. MULTI STEP ELECTRODE RE.ALI

Clearly n

Ili

(6E)

v, so that taking n = 1 in Eq. 6E, requires that v also be

(49F)

unity. To obtain the stoichiometric number experimentally, it is necessary

the stoichiometric number is 2, since step 48F must take place twice, for the overall reaction to occur once.

to measure i 0 by two independent methods: from extrapolation of the linear Tafel region to Ti = 0 and from micropolarization measurements. Equation 51F can then be used to calculate v, since the value of n is

ads

2

If now we leave step 48F, but replace the following step, such that the reaction sequence is H + + e _LciL) H Al

H+ + e

n4

+H

(48F)

ads

ads

11

2

determined independently. Unless one considers a very complex system, both n and v are integers, and it is easy to distinguish experimentally among a small number of possible values of v.

(50F)

14.8 The Enthalpy of Activation

the stoichiometric number is again unity.

Comparing the last two examples shows an interesting feature of the stoichiometric number. Whereas most kinetic parameters depend on what happens

up to and including the rate-determining step, the stoichiometric number can also depend on the step following the rate-determining step, yielding further

To discuss the enthalpy of activation in electrode kinetics, we make use of the fundamental rate equation i = nFC w.exp(—AG"/IIT)exp(cc a EF/RT)

(52F)

information on the reaction sequence, which cannot be obtained by other methods.

which is equivalent to Eq. 27E discussed earlier, except that it is

How can the stoichiometric number be determined experimentally? It

activation into its enthalpic and entropic parts and rewrite Eq. 52F in

is introduced into the rate equations in the transition from a single step to a multistep process. The derivation is somewhat cumbersome, but the final result, applicable to the micropolarization region (i.e., to low overpotentials, where the iirt relationship is linear) has the simple form

written for a multistep anodic reaction. We split the free energy of logarithmic form as: log i = log(nFC (,)) — AH °# /2.3RT + AS"/2.3R + oc a EF/2.3RT (53F) from which it follows that a plot of log i versus 1/T should be a straight line having a slope of

= io (n/v)(11FIRT)

(51 F)

in which n is the number of electrons transferred in each act of the overall reaction. This is the general form of Eq. 6E, which was written for a one-electron, single-step electrode reaction

[alog iia(1/1 = — Ode— a aEF)/2.3R = — AI-1 °"/2.3R JJ E

(54F)

This is similar to the usual treatment in chemical kinetics, except that the enthalpy of activation is found to be a function of potential, as shown in Fig. 2F.

152

ELECTRODE KINETICS

dependence of log i on 1/T. The effect is small, however, and can be

T/ K 1000

153

F. MULTI STEP ELECTRODE REACTIONS

500 400

300

neglected in the narrow temperature range accessible to experiment in

250

most solutions. A further source of uncertainty in the calculation of the enthalpy

E/mV ..................... . .........

7 E

10 -4

90

of activation, which is unique to electrochemistry, relates to the

70

temperature dependence of the potential of the reference electrode.

Are

we determine log i versus 1/T at a constant

"ct

Thus, to obtain

N 10 -5

metal-solution potential difference A(I). Now, at any one temperature 64) is constant as long as the potential with respect to a given reference electrode is constant. When the temperature is changed, this is no longer true, since the metal-solution potential difference at the

to -7 0

1

2

3

4

reference electrode has changed by an unknown amount.

1000/T

Fig. 2F Arrhenius plots, showing the variation of the apparent electrochemical enthalpy of activation (calculated from the slope of the straight lines) as a function of potential. All lines extrapolate to the same value at infinite temperature, showing

There are two ways to approach this problem: measurement can be conducted isothermally (i.e., keeping the working and the reference electrodes always at the same temperature) or nonisothermally (i.e., keeping the temperature of the reference electrode constant while that of the working electrode is scanned). In this case the reference and

that the entropy of activation is independent of potential.

working electrode compartments are connected through a salt bridge, and Often, reference is made in the literature to the

energy of

activation, instead of the enthalpy of activation. It follows from elementary thermodynamics that the former applies if the reaction is conducted at constant volume, whereas the latter is applicable to conditions of constant pressure. In aqueous solutions the difference between the two is negligible, and when making measurements at ambient

there must be a temperature gradient somewhere along this bridge, causing a small thermal junction potential. There is some merit to each of these methods, and both have been used. In isothermal mode, we estimate the change in the value of

with T for the reference electrode. In nonisothermal mode, we estimate the additional potential drop generated in the salt bridge, as a result of the thermal gradients. Neither can be measured directly. Subse-

pressure the terms can be used interchangeably. . The other point to remember is that the frequency term w, in Eqs. 27E and 52F, is generally considered to he a function of tempe-

quently, there is always some uncertainty in the value of the enthalpy of activation of electrode reactions. We note (cf. Eq. 54F) that the enthalpy of activation decreases

rature (0 = k TA' 13

(55F)

with potential (or rather with increasing overpotential) and is proportional to the transfer coefficient oc : a

where k

As.

is the Boltzman constant. This should lead to a nonlinear

,a4

ELL,: i RODE KINETICS

F. MULTI STEP ELECTRODE REACTIONS

155

the solution and the range of potential studied. There could also be a — AH" = — (AH °4— aEF) .

(56F)

The transfer coefficient a could be obtained from Eq. 56F, but since a direct measurement of as (from the current-potential relationship) is much more accurate, it is better to introduce into Eq. 56F the value of

a obtained that way, to test the accuracy of the measured values of the a electrochemical enthalpy of activation, An". 14.9 Some Experimental Considerations In all the discussions so far it has been tacitly assumed that we know the reaction taking place and that only one reaction is occurring in the potential range of interest. Unfortunately, this is not always the case. In the electroplating industry, for example, one must specify the so-called faradaic efficiency, which is the fraction of the current utilized for metal deposition (the rest is usually taken up by hydrogen evolution). It must be realized that the measured current in itself does not yield any information on the fraction of the current supporting each reaction. If two or more reactions occur simultaneously, the current-potential relationship may become rather complicated, since the different partial currents may depend on potential in different ways. In the case of metal deposition, one could obtain the partial current density by simultaneously measuring the weight of metal deposited or the volume of hydrogen evolved (or both, for double checking). However, the high sensitivity and ease of determination of the rate of electrode kinetics, offered by the simple measurement of the current density, are lost in this case. In other cases, it may not be obvious how far the reaction can proceed under a given set of conditions. Oxygen, for instance, can be reduced either to hydrogen peroxide (n = 2) or to water, (n = 4) depending on the nature of the electrode, the composition and purity of

region in which both reactions occur to varying degrees, depending on potential, in which case analysis of the ilE relationship would become rather useless. This brings us to the need for extreme conditions of purity in electrochemical measurements. Why is high purity necessary and how clean should the system be to obtain reliable results? While it may be

a good idea to conduct any experiment in a clean system, it must be remembered that purity has a price, and for extreme purity the price may be quite high, in terms of both the cost of electrode materials and other chemicals and the time and effort required for each experiment. Impurities can be divided into two groups: those that are electroactive in the range of potential of interest, and those that may interfere with measurement by adsorbing on the surface and poisoning it (a more general and perhaps less ominous phrase, would be "changing its catalytic activity"). For the former group, the allowed level of impurity is relatively easy to assess. It should be several orders of magnitude less than the concentration of the material being studied, and the mass-transportlimited current density at which the impurity can react should be smaller than the smallest current we wish to observe. Let us clarify this argument by an example — the need to remove oxygen from solution during the study of the hydrogen evolution reaction on mercury. The concentration of oxygen in a dilute aqueous solution at equilibrium with air is about 0.25 mM. For a dropping mercury electrode which is renewed once a second, this should yield a limiting current of about 3 i.tA, calculated for the 4-electron reduction of oxygen to water, which is the reaction taking place at high overpotentials with respect to oxygen reduction, in the range in which hydrogen evolution is studied.

156

ELECTRODE KINETICS

For the hydrogen evolution reaction on mercury, one typically tries to obtain the ilE relationship from 0.01 to 100 p.A. It is necessary, therefore, to reduce the concentration of oxygen by two to three orders of magnitude, so that the limiting current for this reaction will be lower than the lowest current we wish to measure. This can be done by bubbling purified nitrogen or argon though the solution for some minutes before the experiment is started. The partial currents for oxygen reduction and for hydrogen evolution are shown in Fig. 3F for two

F. MULTI STEP ELECTRODE REACTIONS

157

The numbers just given should be taken as order-of-magnitude type estimates only, since as in any other measurement, it is the fluctuation in the background current, not its absolute value, that determines the lowest current that can he reliably measured. The importance of reducing the concentration of the electroactive impurity should, however, be clear. Now, consider the effect of an impurity that is adsorbed on the surface and alters its catalytic properties. We recall that it takes a very small amount of material to form a monolayer (about 1.5x10 15

concentrations of oxygen.

2 = 2.5x10-9 mol/cm 2). Moreover, very often a fraction of a molecus/ monolayer is enough to change the properties of the surface significantly. In a typical experiment there will be 10-100 cm 3 of solution per square centimeter of electrode area. If the concentration of impurity 10 4

is 1.0 1.1.M, there will he (1-10)x10 8 mol of impurity in solution for each square centimeter of electrode area, enough to form 4-40 mono-

2 10

layers. This is too high, even if we accept that only a fraction of these impurity molecules reach the surface and adhere to it during measurement. Thus, a higher level of purity is required if the solution

J C_)

is to be in contact with the electrode for a long time. A relatively 10-2

simple way to ease the requirement for very high purity is to decrease the volume of the solution per unit surface area. An order of magnitude

-13

-1.1

-0.9

-0.7

-0.5

-0 3

Overpotential/ V

decrease (to 1 cm 3 of solution per unit surface area) can readily be achieved. Moreover, thin-layer cells have been built with a gap of 10 ttm, corresponding to a volume of 1x10 3 cm3 per unit surface area.

Fig. 3F The current calculated for hydrogen evolution at a DME, assuming -12 i = 3.3x10 A (1x10 -1° Alcm 2 ,) in the presence of two 0 concentrations of dissolved oxygen. Dotted line shows the

With an impurity level of 1.0 j.tM, this corresponds to a mere 1.0x10 12 mol per unit surface area, or about 0.04% of a monolayer. Thus, by decreasing the volume of the solution per unit surface area of

current which would be observed in the total absence of oxygen.

the electrode, it is possible to maintain a pure surface during the

The concentration of 0.25 mM is the saturation value for oxygen

experiment. Employing a thin-layer cell, one could relax the require-

in a dilute aqueous solution at equilibrium with air at room temperature.

ment of purity by one or two orders of magnitude and still be able to generate reliable data.

158

ELECTRODE KINETICS

At the other extreme, we find the case of studies on single microelectrodes, which will be discussed further in Section 27. For a small (but not the smallest built so far) microelectrode of, say, 10 lim2 , even if studied in a thin-layer cell, there will be about 104 cm3 of solution per unit surface area. The level of impurity required to ensure that this small surface area could not be contaminated simply cannot be achieved. The other aspect of allowed impurity level relates to the rate of its adsorption, as compared to the duration of the experiment, or rather to the time interval between successive renewals of the surface. We shall explain this in relation to studies on the dropping mercury electrode, which is typically renewed once a second, although the argument can be applied also to solid electrodes, under certain favorable conditions, as we shall see.

F. MULTI S'; • .

JuNS

159

since the rate of diffusion is the maximum rate for a given concentration, and the assumption that each species reaching the surface is adsorbed may not be warranted. Thus, the degree of coverage could be less than the value calculated in this manner, but it could not exceed it. Considering now the dropping mercury electrode, the limiting current is on the order of 3.0 ttA/mM, for a drop time of 1.0 s, corresponding to a surface area of A = 0.02 cm2 and to n = 1. The flux of the impurity reaching the surface, in units of mole per second per square centimeter, is equal to the diffusion-limited current divided by the charge per mole, nF. Assuming, as before, an impurity concentration of 1.0 1AM, this leads to a flux of _

FA

(3X10-6 A/MM)(1X10 3 MM) _ 1 .6X 1 0- 12 MOVS'CM2 (0.02 cm2)(9.65x104 C/mol)

The rate of adsorption of an impurity may be kinetically controlled, in which case it depends on the specific system being studied. This cannot be discussed in general terms, but we can calculate the maximum rate of adsorption (i.e., that controlled by mass transport) as

Since a monolayer amounts to about 2.5x10 9 mol/cm2 , nearly 1.6x103 s would be necessary to cover the electrode surface at this rate. Since the drop is renewed every second, the maximum coverage by an impurity

a function of the concentration of impurity. This rate is given by the

that exits in solution at a concentration of 1.0 p.M cannot exceed

flux of the impurity molecules reaching the surface. The logic behind this argument is very simple. If every molecule of impurity reaching the

= 1/1600. The foregoing calculation shows that one can relax the requirements

surface is instantaneously adsorbed, the concentration of these mole-

for purification substantially. Thus, allowing a tenfold increase in the

cules in the solution nearest to the surface (i.e., at x = 0) will be

impurity level (to 10 1AM) would still limit the maximum coverage by

zero. The rate of adsorption will then depend on the rate of supply of molecules to the surface, namely it will be totally mass transport

impurity during the lifetime of a drop to 0 0.006. These conditions can be realized in the simplest way on a dropping

controlled. The situation is similar to the case in which an electro-

mercury electrode, but they are not totally limited to liquid metals.

active material is oxidized or reduced at the limiting current. In the case of adsorption there is no charge transfer taking place, but the adsorbed molecules are removed from the system just as if they had

The surface of a solid electrode can be cleaned periodically by a suitable series of pulses of potential, by mechanical abrasion in situ,

reacted at the surface. Clearly, this is a "worst-case calculation"

surface before each measurement, not leaving enough time for the

or by sudden heating with a laser pulse, to create effectively a "new"

160

ELECTRODE KINETICS

impurity in solution to be adsorbed on it. On the other hand, when a mercury pool or a solid electrode is used, and the surface is not reproduced periodically, the electrode may be in contact with the solution for 10 3 to 104 seconds, allowing plenty of time for the impurity to diffuse to the surface and be adsorbed on it. The requirements for solution purity are much more stringent in this case, of course. So far we have only discussed the maximum extent and rate of adsorption. This represents the worst case, and one cannot go wrong if the preceding requirements are satisfied. In general, however, adsorption depends on potential. This can best be discussed in the context of electrosorption of organic materials, in Chapter J. We shall anticipate that discussion here by saying that adsorption depends mostly on the charge density on the metal. It is obviously expected that impurities that carry a negative charge will be adsorbed when the excess charge density on the electrode is positive, and vice versa. It is also true, although not obvious from first principles, that neutral species tend to be adsorbed mostly in the region in which the excess charge density is the lowest. The so-called potential of zero charge E, where the excess charge density on the metal is zero, can be measured for most metals. It can be said qualitatively that a region of about 0.5 V on either side

F. MULTI STEP ELECTRODE REACTIONS

161

If the impurity is not electroactive, there will purified electrolyte. be plenty of time for it to adsorb on the surface; thus it can be removed from solution with the electrode used for preelectrolysis, at the end of this purification procedure. One can estimate the time required for preelectrolysis, but it is probably best to develop the procedure by trial and error, choosing a time beyond which further preelectrolysis does not help. Care should be exercised in applying this purification technique. Carelessly done, it can introduce more impurities than it removes. For example, if platinum is used as the anode during preelectrolysis of a chloride-containing solution, minute amounts of this metal may be dissolved and then redeposited at the cathode during measurement, changing its surface properties significantly. We conclude this discussion by stating that maintaining high purity conditions is essential in measurements of electrode kinetics. Experimental results obtained without due control of the impurity level cannot be trusted. Yet the level of impurity required in each case depends on the system being studied and on the method of measurement. One must prudently combine the use of high purity solvents and chemicals with suitable measuring techniques, to reach the desired level of purity

during measurement at the lowest cost in materials and effort.

of this potential is the most susceptible to interference by impurity adsorption of neutral species.. When very high solution purity is required, the last stage of purification should be preelectrolysis, with the use of a large-surfacearea electrode, of the same metal as that selected for the experiment. The idea behind this procedure is simple: if there is an electroactive impurity in solution, let it be consumed during preelectrolysis (which is typically conducted overnight or even for several days), so that none will be left to interfere with the reaction to be studied in the

15. SOME SPECIFIC EXAMPLES

15.1 The Hydrogen Evolution Reaction on Mercury Mercury electrodes have been studied more than any other type of electrode, because of their ease of purification and the high degree of reproducibility attainable when they are employed. All aspects of hydrogen evolution on mercury have probably been studied at one time or another. On the basis of all experimental evidence it is commonly accepted that in this case, the first charge-transfer step is rate

162

ELECTRODE KINETICS

163

IONS F. MULTI STEP ELECTRODE REACTIONS

determining, and it is followed by fast ion-atom recombination H+ + e

1

H M

(57F)

ads

0

1-1+ + H

ads

+ e

M

f H

(58F)

2

—1

The coverage by adsorbed hydrogen atoms must be very low, since none was ever detected, even at the highest overpotentials measured. This also

On —2

rules out atom-atom recombination as the fast second step, since the

—3

b/mV

rate of the reaction

105.3 ± 0.1 H

ads

+

—4 H ----) --. —

H

120.7 ± 0.2

(59F)

2

129.6 ± 0.2

—5

—18

is proportional to 0 2 , while the rate of step 58F is proportional to the first power of 0. The Tafel slope for this mechanism is 2.3RT/PF, and this is one of namely, that the

Fig. 4F Tafel plots for the h.e.r. at a DME in 3 mM HC! and 0.8 M KC1. x is the dimensionless rate constant given by x = (12t17D) 112kh .

experimentally measured transfer coefficient is equal to the symmetry

It is proportional to the activation controlled current density

factor. A plot of log i versus E for the hydrogen evolution reaction

i corrected for mass transport. From Schwarz, Ph.D. Dissertaac tion, Tel-Aviv University, 1993.

the few cases offering good evidence that

p = a,

(h.e.r.), obtained on a dropping mercury electrode in a dilute acid solution is shown in Fig. 4F. The accuracy shown here is not common in electrode kinetics measurements, even when a DME is employed. On solid

The exchange current density for this system depends on the

electrodes, one must accept an even lower level of accuracy and reprodu-

composition of the solution, but generally it is in the range of

cibility. The best values of the symmetry factor obtained in this kind

10 12 — 10 10

A/cm2.

Mercury is often referred to in the literature,

of experiment are close to, but not exactly equal to, 0.500. It should

rather loosely, as a "high-overpotential metal."

be noted, however, that the Tafel line is very straight: that is, 13 is

chosen term, since the overpotential clearly depends on the current

strictly independent of potential over 0.6-0.7 V, corresponding to five

density and cannot be said to have a specific value for a particular

to six orders of magnitude of current density.

metal. It would be better to describe the situation by saying that

This is a poorly

The hydrogen evolution reaction on the DME is the best-known case

mercury is "a low-exchange-current-density metal." Of course, if the

in which 13 is experimentally shown to be independent of potential and to

overpotential is compared at a given current density, then it is higher

have a value close to 0.5. This is probably the best experimental

for mercury and similar "high-overpotential metals" than for platinum

evidence in favor of using this value of 13 in the analysis of more

and other "low-overpotential metals."

complex electrode reactions.

164

ELECTRODE KINETICS

Which metals are "similar" to mercury in this respect? It turns out that most of the soft metals in group 5B and 6B of the periodic table (including Pb, Bi, Cd, In and Sn) behave rather similarly to mercury in respect to the hydrogen-evolution reaction. It would be presumptuous to claim that we could have predicted this similarity from theory, but being confronted with the facts, we can reasonably well explain this result on the basis of the catalytic activity of these metals (or rather the lack of it), as shown in Section 15.7. The hydrogen-tritium separation factor on mercury and the other "soft" metals is low (S

6). This low value is believed to be characterisH/T tic of the mechanism just presented.

165

F. MULTI STEP ELECTRODE REACTIONS

The results obtained in acid solutions indicate that there are two distinct mechanisms. At low overpotentials, the atom—atom recombination step (59F) is believed to be rate determining. This should yield a Tafel slope of b 2.3RT/2F = — 30 mV and a reaction order (at constant potential) of p i = 2; in agreement with experiment. As the overpotential is increased, the fractional coverage 0 must also increase (cf. Eq. 41F). This increases the rate of the atom-atom recombination step, but also that of the ion-atom recombination, which occurs in parallel. As 0 approaches unity, the rate of step 40F can no longer increase but the rate of the ion—atom recombination step can grow, since it depends on potential (cf. Eq. 20F). This step then becomes rate

15.2 The Hydrogen Evolution Reaction on Platinum It is much more difficult to study the kinetics of the hydrogen evolution reaction on platinum than on mercury. To begin with, the

L1.1

0.2

exchange current density is found to he many orders of magnitude higher, -4 in the range of io = 10 -10 2 A/cm, depending on the composition of the solution and its purity. It may be recalled that the linear Tafel region (in which the rate of the reverse reaction can be neglected) only

0

> —0. 1

starts when i/i

10 (corresponding roughly to I n/b1 1). Since it is 0 necessary to make measurements over at least two decades of current density to obtain a reliable Tafel slope, the measurement on platinum must be extended to rather high current densities, where interference by the uncompensated solution resistance and by mass transport limitations can be significant. Also it is found that platinum, being a catalytic metal, adsorbs most impurities very well. Moreover, the potential region in which the h.e.r. is studied is close to the potential of zero charge on platinum, where adsorption of neutral species is favored. Thus, a very high level of purification is required in order to obtain reliable data for this reaction on platinum electrodes.

z 0.000

10 1

10 2

10 3

CURRENT DENS1TY/mAcm -2 Fig. 5F Tafel plots observed for the h.e.r. on platinum in acid solutions at low overpotentials, where the slope is close to — 30 niVIdecade. Data based on Adzic, Spasojevic and Despic Electrochim. Acta, 24, 569, (1979).

ELECTRODE KINETICS

k

determining.

*

F. MULTI STEP ELEC1RODE

10i

complete monolayer of adsorbed hydrogen.

The resulting rate equation (at 0 = 1) is:

This interpretation is tentative in the sense that it has not been i = nlIC •exp(-13EF/12T) H-F

(60F)

proved directly. There is, however, indirect evidence that this kind of behavior is possible and even plausible, from the study of the oxidation

This leads to a Tafel slope of b = 2.3RT/i3F = — 0.12 V for 13 = 0.5, and

of iodide in thin-layer cells. There it was shown experimentally that

a reaction order (at constant potential) of unity. The transfer

the first layer of iodide ions adsorbed on the surface of platinum is

coefficient a is equal to the symmetry factor 13 as in the case of

not electroactive and iodine is formed from ions in solution, which

mercury, but we recall that the Tafel slope is calculated here assuming

presumably can be adsorbed on top of the first layer. It was also found

essenti Illy full coverage, whereas that on mercury was obtained assuming

that the first adsorbed layer can be oxidized at higher overpotentials,

a very low value of the coverage. The Tafel plot observed on platinum

as proposed here for the h.e.r.

in acid solutions is shown in Fig. 5F. Platinum is a catalytic metal, on which molecular hydrogen is adsorbed

The hydrogen-tritium separation factor on platinum in acid solu= 10 at low tions is higher than on mercury, reaching values of S.., tirr overpotentials and S titt, = 20 at high values of

0

Is the assumption of high coverage borne out by experiment? spontaneously from the gas phase, forming adsorbed atoms on the surface.

In alkaline solutions, the mechanism is apparently the same as that

The surface coverage can readily be measured electrochemically, and

found in acid solutions at high values of the overpotential, namely the

indeed it is found to be high at high overpotentials. The difficulty

first charge-transfer step is at quasi-equilibrium, with the ion—atom

with the foregoing interpretation is that 0 is found to be close to

recombination step following as the rate-determining step at high

unity already at low overpotentials, making it hard to justify the low

surface coverage. This scheme is also confirmed by the high value of

value of the Tafel slope observed in this region. One way out of this

the isotope separation factor observed in this system.

dilemma is to assume that there are two types of adsorbed hydrogen on

We have shown that the hydrogen evolution reaction on platinum is

platinum: A first monolayer, which is already observed at the rever-

hard to measure in the linear Tafel region because of its high exchange

sible potential, is partially absorbed and becomes, in a sense, a part

current density. The same property makes it convenient to conduct

of the metal surface. This layer can be removed by anodic oxidation,

micropolarization measurements in the vicinity of the reversible

yielding a measure of the coverage, but it does not participate in the

potential. Although one cannot obtain the Tafel slope from these

h.e.r. at low overpotentials. The latter occurs from atoms adsorbed on

measurements directly, it is possible to determine the exchange current

top of the first layer, which are present at a low partial coverage. At

density, and if i can also be estimated from measurements at higher 0 overpotentials, the stoichiometric number, v, can be calculated (cf.

high overpotentials, the first layer of adsorbed hydrogen becomes electroactive, and the observed kinetics is that to be expected on a

*

For parallel steps in a reaction scheme, the faster one is always

predominant; for consecutive steps, the slower one is rate determining.

Eq. 51F). It is "common wisdom" in electrode kinetics that the region of micropolarization, where the i/ri plot is linear, can extend to about

168

FTPCTRODE KINETICS

ri/b S 0.2,

whereas the linear Tafel region starts at about TO 1. This should, of course, be considered as a rough guideline, since the limits depend on the transfer coefficient of the reaction being studied and on the desired accuracy of measurement, as shown in Figs. 2E and 3E. As a result, the intermediate region of 0.2 5 ri/b 5 1 would be left unused for the evaluation of kinetic parameters. For fast reactions, such as the h.e.r. on platinum, this represents a loss of crucial data,

169

F. MULTI STEP ELEC RODE REACTIONS

is most pronounced for palladium, which can adsorb hydrogen to the extent represented by the formula Fal co , but hydrogen is also absorbed in iron, nickel, titanium, aluminum, platinum and many other metals. The hydrogen evolution reaction in these cases should be written in the form: H

H+ + e nt

H ads

abs

(62F) H2

since it may be difficult to extend the measurements to overpotentials much above nib = 1, because of mass transport limitations. Fortunately,

Looking at this equation, it becomes clear that penetration of

modern microcomputers allow us to make use of this intermediate region. To do this, we write the full equation for an activationLcontrolled

hydrogen into the metal acts as a side reaction for the h.e.r. If it is substantial, as in the case of palladium, the ilE plot cannot be

electrode reaction as follows:

analyzed properly unless the relative rates of the two reactions is

= exp(a0F/RT) — exp(—a criF/RT)

determined as a function of potential and time. (61F)

assuming that mass transport has been corrected for. The values of i , 0 a aand a are then calculated by parameter fitting of the data to

15.3 Hydrogen Storage and Hydrogen Embrittlement

Eq. 61F, which is valid over the whole range of overpotentials acces-

evolution reaction. These are the storage of hydrogen in the form of a

sible experimentally, from zero to high values of rl. Moreover, for fast

solid, either as a hydride or as hydrogen dissolved in the metal, and

reactions, it is often possible to make measurements both at positive

hydrogen embrittlement.

Two interesting fields of technology are related to the hydrogen

and at negative overpotentials, extending even farther the range of data from which the kinetic parameters are obtained.

(a) Hydrogen storage

Metals that are known to be highly catalytic for hydrogenation and

Hydrogen is in some respects the ideal fuel. It is clean (leaving

dehydrogenation reactions are similar to platinum with respect to the

only water as its product), it is renewable in unlimited amounts

h.e.r. These include palladium, iridium, rhenium, nickel, and cobalt.

(provided there is a nonpolluting method of producing energy, such as

In the case of palladium the situation is further complicated by the

solar or fusion) and, because of its low equivalent weight, it has the

tendency of atomic hydrogen, which is formed on the surface as an

highest energy density per unit weight, (about three times the energy

intermediate, to diffuse into the metal and dissolve in it. This effect

density of natural gas and other fossil fuels). Much of this advantage is lost, however, as a result of the difficulty of storage. Hydrogen is

*

Note that the ratio nib is always positive, since ri and b have the same sign.

usually stored under high pressure or cryogenically. Both methods are expensive, consume a lot of energy (for compression and cooling, respectively), and increase the weight. There is therefore a clear

170

ELECTRODE KINETICS

incentive to develop better methods to store hydrogen, to permit the gas to be used at room temperature and at moderate pressures. Hydrogen can be stored in palladium and retrieved from it easily, but this is not practical, because of the high price of palladium and its high atomic weight (1 g of hydrogen stored as PdH 06 weighs 177 g). An alloy of iron with titanium can absorb and release hydrogen reversibly. Having the composition of TiFeH L8 at maximum loading, this is equivalent to a weight of only 58.6 gram per gram of hydrogen. (This represents only about 5% of the energy density of fossil fuels, but the advantage, in terms of the effect on the environment, may be worth the loss in energy density). The alloy can be modified by small amounts of other metals, to change the absorption characteristics in a desired fashion. A different alloy developed for the same purpose is made of lanthanum and nickel. The composition at maximum loading is LaNi H , 5 6.7

which corresponds to 65.5 g per gram of hydrogen. The Ti/Fe alloy is more sensitive to impurities (particularly oxygen and water vapor), hence the LaNi alloy seems more promising, in spite of the small 5 disadvantage in specific weight, but neither has so far (1993) been put to large scale commercial use (except in the recently developed nickelhydrogen rechargeable battery). It should be added that the numbers given here for total weight per unit weight of hydrogen do not tell the whole story. Even when hydrogen is stored "chemically" in the manner discussed here, it still must he kept in a suitable container, and heat exchangers must be provided to remove and supply the heat of dissolution during loading and discharging the hydrogen from the hydride,

F. MULTI STEP ELECIRODE REACHON

respectively. Thus, the weight per gram of hydrogen in an actual storage device is significantly higher. This is one of the reasons why such hydrogen-storage devices have not yet been commercialized on a large scale.

(b) Hydrogen embrittlement The best-known case of hydrogen dissolved in a metal in small quantities, causing major changes in its bulk properties, is that of steel. Hydrogen absorbed in steel can cause embrittlement (appropriateAs a rule, the higher the tensile ly called hydrogen embrittlement). strength of the steel, the more susceptible it is to hydrogen embrittlement. It is interesting to note that the amount of hydrogen that can cause severe embrittlement is minute, on the order of 0.01 a/o (atom percent) or less. How can hydrogen get into iron and its alloys? Iron is not particularly catalytic with respect to hydrogenation and dehydrogenation reactions. When placed in contact with molecular hydrogen at room temperature and moderate pressure, nothing much happens. The surface of iron is not catalytic enough to split the H 2 molecules and form Hads on the surface — a necessary precursor for diffusion of hydrogen into the metal. The rate of hydrogen evolution on iron and its alloys is intermediate between that on mercury and on platinum, with i values in the range of 10 -5 -10 7 A/cm? depending on the type of alloy and the solution used. The kinetics of the h.e.r. on iron has been studied extensively, and it has been established that atomic hydrogen is adsorbed on the surface at a substantial level, enough to support the entry of hydrogen atoms into the metal. Naturally, one does not cause hydrogen embrittlement on purpose, by evolving hydrogen on a steel part,

This can also be modified by small amounts of alloying elements, to improve the adsorption-desorption characteristic, or tailor-make it to a specific use.

but hydrogen evolution can occur as a side reaction in several processes. We have already mentioned that it is a side reaction during the electroplating of different metals. It can also occur during corrosion,

172

ELECTRODE KINETICS

k

when the sample is exposed to the environment. Furthermore, in an

OH + 0I-I

attempt to protect the metal against corrosion by applying to it a negative potential (a technology referred to as cathodic protection, cf.

173

F. MULTI STEP ELECTRODE REACTIONS

--3 Oads ads

H2O

+

e

M

(66F)

followed by atom-atom recombination, to yield

Section 29.7), hydrogen is evolved and atomic hydrogen is formed on the

k

2(D

surface as an intermediate. Cleaning of the surface in preparation to

ads

)

0

2

(67F)

electroplating or painting often involves dissolution of a thin layer of

Let us evaluate the kinetic parameters corresponding to this

the surface in acid (a process called pickling which, in effect, could

reaction sequence. For the first charge transfer as the rate-

be viewed as a fast, but controlled, corrosion process). During such

determining step we already know the result, since it is equivalent to

processes hydrogen is evolved, and part of it may penetrate the bulk of

step 57F for hydrogen evolution on mercury. For the second step assumed

the metal, causing severe embrittlement.

to be rate determining, we also know the result, given in Eqs. 20F to

It is clear from Eq. 62F that addition of materials that slow down

22F. The Tafel slope changes from b = (2.3RT/F)/(1 + (3) to (2.3RT/F)/13

(i.e., poison) the h.e.r. enhances the absorption of hydrogen into the

as the partial coverage increases. The reaction order p changes from 2

metal, and vice versa. Thus, kinetic studies of the h.e.r. on iron and its alloys, in the presence of different additives, can be useful in the

to I as 0 increases. The stoichiometric number v decreases from 4 to 2, but unfortunately this change cannot be measured, since the range of

search for ways to decrease hydrogen embrittlement during surface

micropolarization is not accessible experimentally, as a result of the

treatment of the metal.

low exchange current density characteristic of this reaction. Now let us look at the third case, assuming the atom-atom recombi-

15.4 Possible Paths for the Oxygen Evolution Reaction

nation (Eq. 67F) to be rate determining. In this case the two preceding Oxygen evolution is a more complex reaction, involving the transfer

steps are at quasi-equilibrium. The corresponding equations are:

of four electrons. The overall reaction in alkaline solutions is

[ 4(OH)

2H 2 0

+ 0

2

+

4e

M

(63F)

whereas in neutral or acid solutions it is:

21-10

0

2

2

+

0

= K I C OH- exp(EF/RT)

— 0 0H — 0

(68F)

e

4H+ + 4e

M

(64F)

Let us try to write a mechanism (one of many possible) for this

2 °H exp(EF/RT) = KC

(69F)

0 0H For the rate-determining step we have

reaction. The first step would be charge transfer k

0H

ads

+

eM

(65F)

2 exp(2EF/RT) i = nFk30 2 = nFk K 2 C OH— 2 0 OH 3

(70F)

At low values of the total coverage, (1 — 0 0H— 00) = 1.0, which leads to and this can be followed by a further charge-transfer step, such as

174

ELECFRODE KINETIC S

F. MULTI STEP ELEA.-

REALTIONS

1 J.)

the oxide to form. This may sound like a clever experiment, but the i = nFk K2K2C4 .exp(4EF/RT) 3 1 2 OH-

(71F)

For this mechanism, the transfer coefficient tx is 4 and the Tafel slope b is 15 mV. The reaction order at constant potential p i is 4, and the stoichiometric number is unity. One could write many other pathways and rate-determining steps, and calculate the kinetic parameters for each, following the same line of reasoning. It is important to note that in a complex reaction sequence there can be more than one type of adsorbed intermediate on the surface, and some steps may involve the transformation of one kind of adsorbed species to another, by either an electrochemical or a chemical route. Two aspects of the oxygen evolution reaction are common to all electrodes studied so far: (a) the exchange current density is low, of 2 the order of 10 8 A/cm or less, and (b) a reversible oxygen electrode has not yet been built. Both statements are relevant to operation at room temperature. At sufficiently high temperature (say, in molten salts, at about 600 °C or with high temperature solid electrolytes operating around 1,000 °C) the kinetics of the reaction can be sufficiently accelerated to make reversible oxygen electrodes operate as well as reversible hydrogen electrodes operate at room temperature. Oxygen evolution never occurs on the bare metal surface. By the time the reversible oxygen evolution potential is reached, an oxide layer has been formed on all metals. At more anodic potentials, where measurements can actually be conducted, (remember that i is small, and it takes a high overpotential to drive the reaction at a measurable rate), the oxide film may be several molecular layers thick. One may be tempted to circumvent this problem by holding the potential at a relatively negative value, where the oxide is known to be absent, and switching it rapidly to the desired overpotential, to observe the rate of oxygen evolution on the bare surface, not allowing enough time for

results turn out to be impossible to interpret, since the current during the transient is divided between oxygen evolution and oxide formation, and the partial current for each reaction changes with time at constant potential. Oxides on metals can be divided into three groups: those that are highly conducting, such as RuO2 and the oxides formed on platinum and iridium; those that are semiconducting, such as NiO and the oxides formed on tungsten and molybdenum; and those that are insulators, such as the oxides of the valve metals (Al, Ti, Ta, and Nb). Oxygen evolution can occur readily on the conducting oxides, and these are the best catalysts for this reaction. On semiconducting oxides the reaction can still occur, but it may be associated with pitting on the one hand and further buildup of the oxide layer on the other, causing poor reproducibility and making the interpretation rather dubious. On valve metals, oxide formation is the main reaction occurring during anodic polarization and the current either decays with time to zero or reaches a constant value, at which the rate of dissolution of the oxide is equal to the rate of its electrochemical formation. At noble metals (e.g., Pt, Ir, Au) the region of potential in which the oxygen evolution reaction is studied is far removed from the potential of zero charge in the positive direction. Thus, there is little danger of adsorption of neutral or positively charged impurities, but negatively charged impurities will be heavily adsorbed. One may expect, therefore, that the nature of the anion in the electrolyte will influence the data substantially, while the type of cation will have little effect. Also, most organic molecules are rapidly oxidized at these potentials, so that the requirements for solution purification are much less severe than in the case of hydrogen evolution. The preceding statement does not hold true for the oxygen reduction reaction! Using platinum as an example, we note that oxygen evolution

176

ELECTRODE KINETICS

is typically studied in the range of 1.5-2.0 V, versus a reversible hydrogen electrode (RHE) in the same solution, while oxygen reduction is studied in the range of 1.0-0.4 V on the same scale. In most of the latter range, the surface is free of oxide (and very sensitive to impurities) if approached from low potentials, whereas it is mostly covered with oxide (and less sensitive to impurities) when approached from higher potentials. This is due to the high degree of irreversibility of formation and removal of the oxide layer on platinum. If the potential is scanned slowly (say, at a rate of 10 mV/s) in the positive direction, starting from 0.4 V RHE, oxide formation begins at about 0.9 V. If it is scanned in the negative direction, starting, 'say, at 1.5 V RHE, the oxide is not completely removed until a potential of about 0.5 V has been reached. A discussion of oxygen evolution and, even more, of oxygen reduc-

177

F. MULTI STEP ELECTRODE REACTIONS

into potent catalysts for oxygen evolution and reduction, and perhaps for other reactions as well.

15.5 The Role and Stability of Adsorbed Intermediates Intermediates are commonly formed in chemical reactions, as well as

in electrode reactions. The preferred mechanism is that which involves the most stable intermediates, since this is the path of lowest energy of activation. For reactions taking place in the gas phase or in the bulk of the solution, the stability of different species can be calculated, or at least estimated, from existing thermodynamic data. This is not the case for electrode reactions. For the h.e.r. discussed earlier,

a hydrogen atom was assumed to be an intermediate. The standard reversible potential for the formation of this species in solution, that is, for the reaction H+ + e

tion, cannot be complete without mentioning the efforts that have gone

m

H

(72F)

organic catalysts of the phthalocyanine group. These are large planar

is about — 2.1 V, NHE. Thus, this species could not be formed in the potential range over which the hydrogen evolution reaction is studied,

molecules with a metal atom in the center, resembling the structure of

even on mercury electrodes. The real reaction we are looking at,

the porphyrin molecules, which are involved in the breathing processes

however, is not represented by Eq. 72F, but rather by Eq. 57F:

into improving the catalytic activity of metals by the use of metal-

of living organisms. The philosophy behind these experiments is to try

H+ + e

to imitate nature. If molecular oxygen can be reduced efficiently in living organisms at or near ambient temperatures, perhaps the molecule involved in this reaction, or other metal-organic compounds having a similar structure, also will act as good catalyst for oxygen evolution and reduction occurring in vitro, in an electrochemical device.

m

H aas

(57F)

The hydrogen atom formed as an intermediate is stabilized by adsorption. To clarify this point, we should perhaps write Eq. 57F in explicit form, as follows: H+ + e + Pt M

-4

H—Pt

(73F)

Unfortunately, the analogy is rather poor. Nature operates in an

implying that a chemical bond is formed between the hydrogen atom and a

intricate manner, and the porphyrin molecule is only one part of a

metal atom on the surface. The reversible potential for this reaction

complex system, which includes various enzymes and other protein

is evidently different from that for reaction 72F. The difference

molecules. Nevertheless, some progress has been made along these lines,

between them depends on the H—Pt bond energy on the surface. This

and phthalocyanines and similar molecules may eventually be developed

should not be confused with the bond energy of bulk platinum hydride, or

cd_LL fRODE KINE i 10

with the energy of formation of a H—Pt species in the gas phase, since platinum atoms on the surface are energetically different from atoms in the bulk, which, in turn, are different from isolated atoms in the gas phase. It is found that the standard reversible potential for reaction

179

. hi UL F] STEP ELECIRODE R CACI iu,

surface. Many attempts to achieve this goal have been made. The use of electron spin resonance (ESR) techniques to detect radicals on the surface failed, not surprisingly, since a radical adsorbed on the surface and bound to it chemically (e.g., H—Pt) is no longer a radical! More promising are studies by infrared spectroscopy, particularly

73F is E ° = + 0.2 V, RUE. For the hydrogen evolution reaction

using the modern technique of fast Fourier transform infra red (1.1-.11R) 2H+ + 2e

m

H

2

(74F)

spectroscopy. The inherent difficulty is that when water is used as the

the standard reversible potential on the same scale is, of course, zero

solvent, it absorbs heavily in the IR region of the spectrum. This

by definition. We note that the adsorption energy of hydrogen on

problem has been partially solved by constructing cells in which the

platinum is so high that it is easier (i.e., it requires a less cathodic

optical path through the solution is very short, and by the high

potential) to form the adsorbed species H—Pt, than to form 1-I 2 molecules

sensitivity provided by present-day FFTIR instruments. The beauty of IR

in solution. In comparison, the 1I—Hg bond is very weak, and consequent-

measurements is that they provide spectroscopic data, yielding informa-

ly a hydrogen atom is not stabilized on the surface of mercury. The

tion on the type of bonding of the intermediates to the surface, not

reversible potential for reaction 73F — with platinum replaced by

only on its existence there.

mercury - would be close to that for reaction 72F. This is why

With the introduction of scanning tunneling microscopy (STM) to

on 0H mercury cathodes is below the detection level, even at the highest

electrochemistry, additional direct information on adsorbed intermediate

overpotentials studied.

may be gained.

The situation is similar in the case of oxygen evolution, where OH

and O

In the discussion of the hydrogen and oxygen evolution reactions,

ads a& are postulated as adsorbed intermediates, even though these radicals are very unstable in the bulk of the solution. The same

we saw that the current-potential relationship is influenced, and

type of argument has been employed to justify the existence of adsorbed

coverage 0. Thus, one may expect that the kinetic parameters will

intermediates in some complex organic reactions, such as RCOO ads and

depend on the adsorption isotherm, which relates the surface concentra-

Rags in the anodic formation of hydrocarbons from the corresponding

tion to the bulk concentration, and more importantly in electroche-

organic acids (the Kolbe reaction)

mistry, to the potential. In the preceding derivations it was tacitly

2C1-1 C00 3

C 11 + 2CO + 2e 2 6 2 M

(75F)

sometimes determined completely, by the potential dependence of the

assumed that the Langmuir isotherm applies. In Section 19 we discuss the limitations of this assumption and show how the kinetic parameters

and the formation of the so-called reduced CO (which is just a fancy 2

change when different isotherms are applicable.

name for CO or CHO adsorbed on the surface) during the anodic oxidation of alcohols and hydrocarbons.

15.6 Catalytic Activity: The Relative Importance of i and b

This argument would, of course, be more convincing, if one could provide direct evidence for the existence of such intermediates on the

0

When the catalytic activity of different metals for a given reaction is considered, there may be some question regarding how the

180

ELECTRODE KINETICS

comparison should be made. Assume that we are looking for the best catalyst for the oxygen reduction reaction. It would seem that the value of the exchange current density is a good measure of the catalytic activity, since it is an indication of the height of the free-energy barrier for the reaction at the reversible potential. But we recall (cf. Eq. 39E) that the exchange current density depends on the metalsolution potential difference at the reversible potential

mAsOrov , and

this quantity is different for different metals, as we showed in Section 5. Thus, for a fundamental study of the effect of the properties of the metal (e.g., its d-band character, electron density etc.) on its inherent catalytic activity, a comparison of i o values may be

181

P. MULTI STEP ELECTRODE REACTIONS

at this current density, since it is a reasonable design point for batteries, fuel cells, and industrial electrolytic processes (the actual operating current densities are typically between 0.2 and 2.0 A/cm 2 . The so called Tafel parameter a depends on both i and b, and is a good measure of the catalytic activity of a metal, from the engineering point of view. Much is being done to increase the catalytic activity of metals. Certain binary alloys seem to have higher catalytic activity than either of their components separately. Oxides and oxide mixtures have been tried, and electrodes modified by organic or metal-organic molecules hold hope for better catalytic properties. An engineering approach, that has been quite successful (although some may consider it a "brute

somewhat misleading. If one is interested in the practical or engineering aspects of

force" approach) is to increase the roughness of the surface, thus

catalysis, the comparison should be made in the range of current densities in which the device (a fuel cell in the present example) is as important as i in determining the overpotential at current densities 0 of practical interest. In Fig. 6F we compare two hypothetical reactions, 5 to 2 one with i = 10 A/cm and b = 30 mV, and the other with i = 10 2 A/cm and b = 120 mV. At a current density of 1.0 A/cm 2 , the overpotentials are 0.3 and 0.6 V, respectively. The system with the lower values

Overp oten tia l /V

destined to operate. In this case the value of the Tafel slope can be

of i and b has the lower overpotential. From the engineering point of 0 view, a lower overpotential corresponds to higher catalytic activity. In the example shown here, the Tafel slope turns out to be more impor-

-2

0

tant than the exchange current density in determining the overpotential under practical operating conditions.

Fig. 6F The catalytic activity from the engineering point of view, as a

We recall that the Tafel equation can be written as: = a — b•log i = b•log io — b-log i

(18E)

Hence a = b.log i o represents the value of the overpotential at unit current density (i = 1.0 A/cm 2). It is sensible to make the comparison

function of i and b. Note that the overpotential correspon0 ding to the intercept of the lines with the right-hand side axis in this figure (namely the value of 11 at i = 1.0 A/cm 2) is numerically equal the the parameter a of the Tafel equation.

ELECTRODE KINETICS

i64

F. MULTI STEP ELECTRODE REACTIONS

183

presenting a larger surface area per unit geometrical area. From the practical point of view it does not much matter whether the overpotential at the desired current density is decreased by increasing

the

intrinsic catalytic activity of the electrode material or by increasing its surface area, making the real current density smaller. The result is a decrease in overpotential (at a given apparent current density),

E

Pt •Re • Rh:

3

Ir 'I Au

namely an improvement in the operational characteristics of the device. Cu•

It is an approach in which Materials Science, rather than Physical Chemistry, counts most.

a) -C

15.7 Adsorption Energy and Catalytic Activity

_c

When one tries to correlate the electrocatalytic activity of metals with some fundamental property of the system, the result is often a It

volcano- type" plot, as shown in Fig. 7F.

a)

0

Fe

7

4.9 9 cn 0

Ni •Co

•

Sn gi Zne ••A g • Pb Go • • Cd TI / • • / In

Mo

Ti Nb • Ta

Such behavior is easy to understand qualitatively, although a quantitative relationship may be hard to derive. The role of a heterogeneous catalyst is to adsorb the reactant or intermediate and transform

30

50

70

90

M-H BOND STRENGTH/ kcol.mol -I

it to a species that can more readily undergo the desired chemical transformation. If the heat of adsorption is very low, the extent of

Fig. 7F The exchange current density for the hydrogen evolution reaction

adsorption will be very small. Moreover, the molecules adsorbed are

as a function of the metal-hydrogen bond energy. Reprinted with permission from Trasatti, J. Electroanal. Chem. 39, 163. Copyright 1972, Elsevier Sequoia.

bound to the surface weakly and are not affected by it. As the heat of adsorption increases, the fractional surface coverage also increases, and the adsorbed species can be modified and activated by its bond to the surface. But this can he overdone! Beyond a certain heat of

there and release the product to the solution. This leads to a "volcano-

adsorption, the coverage approaches saturation, and the rate of reaction

type" relationship, where the best catalyst is one giving rise to an

can no longer increase with increasing energy of adsorption. Also, if

intermediate value of the energy of adsorption.

the binding energy to the surface is too high, the adsorbed intermediate

A word of caution is appropriate here in relation to making such

or its product may stick to the surface and effectively poison it. In

correlations. The proper comparison would be one in which the energy of

short, the energy of adsorption must be high enough to get the reactant

adsorption was changed, while all other properties of the metal were

to the surface in sufficient amount, yet low enough to allow it to react

kept constant. This cannot be done. When we compare, say, the rate of

184

ELECTRODE KINETICS

hydrogen evolution on gold and platinum, the heat of adsorption of

185

0. THE IONIC DOUBLE-LAYER CAPACITANCE CdI

G. THE IONIC DOUBLE-LAYER CAPACITANCE C

dl

hydrogen for the two metals is different, but so is every other pro-

perty: the electronic work function, the heat of sublimation, the potential of zero charge, the free-electron density, the crystal structure and dimension, and any other property one may think of. Often some of these properties are correlated (e.g., the work function, the

16. THEORIES OF DOUBLE-LAYER STRUCTURE

potential of zero charge, and the heat of adsorption are linearly

impedance of the metal-solution interphase is partially capacitive. In

related). Finding a numerical correlation between any two quantities

simple cases, the equivalent circuit is that shown in Fig. 1G(a). The

does not necessarily prove that one is caused by, or is directly related to, the other.

double-layer capacitance C and the faradaic resistance R dl

16.1 Phenomenology It was pointed out at the very beginning of the book that the

are inherent

properties of the interphase, which we measure experimentally and attempt to interpret theoretically. The solution resistance R s is not a property of the interphase. It can be viewed as an "error term" arising from the fact that the potential in solution is always measured far from the interphase (which does not extend more than about 10 nm from the surface). If the interphase is ideally polarizable, the faradaic resistance approaches infinity, and the equivalent circuit shown in Fig. 1G(a) can be simplified to that shown in Fig. 2G(b). If it is ideally nonpolarizable, the faradaic resistance tends to zero, and the equivalent circuit shown in Fig. 1G(c) results. Real systems never behave ideally, of course; they may approach one extreme behavior or the other, or be anywhere in between. It is also important to remember that both C m and

RFdepend on potential and should be defined in differential form as follows: Cal = q /a E) m

and

R = (aE/ai)

(1G)

where qm is the excess surface charge density on the metal. The doublelayer capacitance can be measured in a number of ways, some of which we shall discuss in Section 16.8.

186

ELECTRODE KINETICS

187

G. THE IONIC DOUBLE' LAYER CAPACITANCE Cdl

60 N

Ideally polarizable --

C di Rs

E 0

WW--1 Rs

W

b

z

C dl

I—

I--

-I

U

Fig. 2G Experimental plots of C vs E. (a) Hg in 1 M

a —MAW— R

40

U

Rs

RF C

Ideally nonpolarizable

Fig. 1G Equivalent circuit for an activation controlled process, showing the three basic circuit elements: the double-layer capacitance, Cdl the faradaic resistance, RFand the residual solution resistance, R . (a) general (b) ideally polarizable (c) ideally s nonpolarizable.

in solution is almost ideally polarizable over a relatively wide range

Data from Hamelin, J. Electroanal. Chem. 138,

of potential, making both experimental measurement and theoretical

395, (1982).

unfortunate choice was made in the early days of electrochemical

-1.0

-1.5

60

(b) Single crystal (210) gold surface with increa-

to being highly reproducible and easy to purify, the interphase it forms

shown. The reader will note that in the case of mercury, the potential scale is reversed, negative to the right and positive to the left. This

-0.5

POTENTIAL/Volt vs SCE

Z. Physik. Chem., N.F. Frankfurt 25, 145, (1960)

NaF, (in moll!) as marked.

shown in Fig. 2G. In the case of mercury, the effect of a strongly adsorbed organic molecule on the capacitance–potential curve is also

0

5x10 -4M. Data from Lorenz, MOcklel and Muller,

The system most widely studied is, of course, mercury. In addition

ducible of solid surfaces. Typical results for mercury and gold are

U

KCl. Concentration of Npropylamine (I) zero, (2) 5x10 -5M, (3) 1x10 -4M, (4)

sing concentrations of

interpretation easier. The best results on solid electrodes have been obtained on single-crystal surfaces of gold, which are the most repro-

20

U

z 40 U

a. U

20

0 -0.5 POTENTIAL/Volt vs SCE

0.5

188

ELECTRODE KINETICS

0. THE IONIC DOUBLE-LAYER CAPACITANCE C

189 JI

research, when the field was dominated by polarography on mercury, and

interphase. Here there can be an excess charge density on the metal,

it is still in use in most papers and textbooks when dealing with the

which we denote q m , and an excess charge density on the solution side of the interphase, denoted q s . The interphase as a whole must be electro-

mercury electrode. We have chosen, somewhat reluctantly, to follow the same convention when studies on mercury are discussed, to avoid further confusion.

neutral. It follows then that at any metal-solution interphase we can write

qm

One of the most important features seen in Fig. 2G(a) is the nearly

qs =

°

(2G)

constant value of the capacitance at the far negative end. This value,

While the Helmholtz model can explain the very existence of a capaci-

of about 16 j_tF/cm 2 , is essentially independent of the electrolyte used.

tance at the interface, it can explain neither its dependence on

This observation played an important role in the development of our

potential nor its actual numerical value at any potential. The

understanding of the structure of the double layer at the metal-solution interphase, as we shall see.

capacitance of a parallel-plate capacitor, per unit surface area, is given by:

The other point of great interest is the effect of the organic material. First we note that the capacitance is much smaller when organic matter is adsorbed. Secondly we observe that this occurs only in a certain range of potential, disappearing both at more negative and at more positive potentials. This result is far from being selfevident, if we consider that the organic species used here is not charged. Moreover, similar behavior is observed for many neutral organic molecules of widely different structure.

These observations

form a further cornerstone of our understanding of the structure of the double layer, and in particular, of the factors controlling the adsorption of neutral molecules.

16.2 The Parallel-Plate Model of Helmholtz The first attempt to explain the capacitive nature of the inter-

C = — (6/4nd)

(3G)

where d is the distance between the plates, which should be equal to the radius of the cations of the electrolyte, according to this model. The capacitance should be independent of potential, contrary to the experimental observation shown in Fig. 2G. Moreover, it should be independent of charge, hence of concentration in solution, while in reality it depends strongly on concentration, as shown in Figs. 2G(b) and 3G. The numerical value obtained from Eq. 3G above, using 6 = 78 for the bulk dielectric constant of water at room temperature and a typical value of 2x10 -8 cm for d, is 0.34 mF/cm2 , more than 20 times the experimentally observed value of 16 pF/cm 2 . If a theory has failed so miserably in describing the experimental results, how could it withstand the test of time, and still merit

phase is credited to Helmholtz, in the middle of the nineteenth century. In his model, the interphase is viewed as a parallel-plate capacitor — a

*

layer of ions on its solution side and a corresponding excess of charge

In the c.g.s. system, the capacitance is expressed in centimeters. For comparison with practical units, we note that I farad = 9x1011 cm.

on the surface of the metal. It should be noted here that electroneutrality must be maintained in the bulk of all phases, but not at the

Note, however, that in Eq. 3G, area.

CH

is the capacitance per unit surface

190

BLL,k, )DE KINETICS

40

U. IIIE IONIC DOUBLE-LAYER CAPACITANCE C

dl

diffuse double layer model is that an excess charge q m, on the metal side of the interphase, must create an equal excess of oppositely

11111111111

charged ions qs , on the solution side of the interphase. These ions are attracted to the surface of the metal electrostatically, but they are

32 .0 916 M

also subject to random thermal motion, which acts to equalize the

24

concentration throughout the solution. The equilibrium between these two opposing tendencies is expressed by the well-known Boltzmann

X0.1 M .. •.

o.o m

equation

16

Ci(x) = C:exp(—z

/12T)

(40)

0.001 M

8

0

I 0.4

I 0.0

I

I -0.4

-0.8

-1.2

-1.6

-2.0

POTENTIAL (Volt vs NCE)

Fig. 3G Double-layer capacitance on Hg in solutions of different concentrations of NaF at 0°C. Grahame and Soderberg, ONR Tech Rep. N ° 14 (1954).

mentioning more than a century later? A hint may be found by observing the extreme negative end of the plots in Fig. 2G(a) and Fig. 3G. Over several tenths of a volt, the double-layer capacitance in this region is almost independent of concentration and of potential, as expected from the simple parallel-plate model. The numerical value is off by more than an order of magnitude, but this can be remedied by a better choice of the values of e and d in Eq. 3G , as we shall see later. 16.3 The Diffuse-Double-Layer Theory of Gouy and Chapman A different approach to interpret the capacitance behavior was taken by Gouy and Chapman in 1913. The fundamental premise of the

where 4) xrepresents the potential at a distance x from the surface of the metal (with respect to the potential in the bulk of the solution, O which is taken as zero), and z• is the charge on the ion. The s' concentration of any ionic species at a distance x from the surface, Ci(x), is determined by the ratio between the electrostatic energy (z F4) ) and the average thermal energy RT. From here on the derivation x is equivalent to the much better known derivation of the Debye-Hiickel limiting law, which was, however, published about 10 years later. The potential (I) x is related to the charge per unit volume, p(x), by the Poisson equation. Since the changes in 4) x and p(x) are considered only in the direction perpendicular to the surface, this equation takes the simple form: d241dx2 = — 41cp(x)/e

(50)

The volume-charge density p(x) is related to the concentration by p(x) = F EziCi(x)

(6G)

When combined, Eqs. 50 and 6G yield d24 xid x2 = — (41c/e)F EzCexp(-z i n¢x /RT)

(7G)

192

ELECTRODE KINETICS

C,. THE IONIC DOUBLE-LAYER CAPACITANCE C

193

dl

which is solved by multiplying both sides by 2(d0 ./dx) and using the relationship 2

d

qm

= 2( do )r d 2 4) 1 \ -dr

dx

2RT E C 1 1/2 . sinh

[ lz I F(1)-1 2RT .

(14G)

(8G)

2

Integrating between x and infinity, and bearing in mind that both (1) ,, and

The differential capacitance is readily derived from this equation, since

(d(1) x/dx) approach zero at infinity, one obtains [ 41 1 2 = —

[

871RT Z-0 [exp(—ziFIVRT) — 11 BART

(9G)

-1 1 / 2 I z I f(i) 0 Cdi = (aq m/S 0) v. = zIF[ 2rERC T Co] cosh{ 2RT j1

(15G)

The surface charge density q m is related to the gradient of potential by the Gauss theorem, namely From Eqs. 140 and 15G we see that, at the potential of zero charge E 4itq m = — e(d4 x/dx) x.0

(10G)

(where qm = 0), the potential O . at the surface equals zero and the differential capacitance has its minimum value, which is proportional to

Combining Eqs. 9G and 10G we write:

the square root of the concentration: qm

f RTE Vc0[exp(—z F4) /RT) — I 0

(11G)

i

Cdi(min) =

l 27ce RT z F[

1 / 2 ( c 0)1/2

(I6G)

where we have substituted O . for 41, since Eq. 100 is solved for x = 0. For a symmetrical electrolyte (e.g., NaF or MgSO 4 ), Eq. 11G can be

This is a beautiful theory, in that it allows us to calculate the excess surface charge density and the double-layer capacitance from

written as follows: r

q

M -

well-known principles of electrostatics (the Poisson equation and the

I/ 2

-

(12G) 1/2

x f[exp( I z I F(!) /RT)] + (exklz I F(1)/RT)] — 21

Gauss theorem) and thermodynamics (the Boltzmann equation). It has, however, one major drawback: it does not predict the correct experimental results! Perhaps it would be more accurate to state that agreement between theory and experiment is found only in dilute solu-

Now, we can use the simple relationship [exp(x) + exp(—x) — 2) = [exp(x/2) — exp(—x/2)] 2 (13G) to simplify Eq. 11G further, yielding:

tions and in a limited range of potentials, near the potential of zero charge, as seen in Fig 4G. Now, we could come up with several reasons to explain why this theory would deviate to some extent from experiment: I. In Eq. 4G we made the simplifying assumption that the only energy involved in bringing an ion from infinity to a distance x from the

194

!RODE KINETICS

a 1mM NaF 40

son between C 2-s

culated

from

c2

cal-

(dashed lines) and

theory and experiment at a potential of, say, 0.5 V on either side of the potential of zero charge (PZC), and in concentrated solutions. As 0 0.3

0

-0.3 -0.6 -0.9 - 1.2 -1 5

Potential vs NCE/V

NaF and (b) 10 mM

b

NaF at 0°C, based on

10mM NaF 40 2-s

16.4 The Stern Model The jigsaw puzzle was put together by Stern in 1926. Agreement

20

C

Ez 0.3

0

-0.3 -0.6 -0.9 -1.2 -15

Potential vs NCE/V

surface is the electrostatic energy zO x F, neglecting ion-ion interactions, which are bound to be important at higher concentrations. 2. When integrating Eq. 7G, it was tacitly assumed that the dielectric is independent of the distance from the electrode. This is not correct, and theoretical estimates indicate that E changes from about 6 to 8 in the first layer of water at the metal surface to its E

situation at the interphase, yields quite good agreement between theory and experiment, as we shall see next.

0

constant

in the discussion of the Helmholtz model, we might ask ourselves why a theory that is in almost total disagreement with experiment is discussed at all! It turns out that a clever combination of the Helmholtz and the Gouy-Chapman models, resulting from a clear physical grasp of the

60

Cap ac itance /p. F cm - 2

4819, (1954).

at least in principle, by deriving appropriate correction terms. None of these points could, however, explain the total disagreement between

lines) for (a) / mM

Am. Chem. Soc. 76,

3. Perhaps the biggest error is introduced by using the potential 4) 0 at x = 0, which is equivalent to treating the ions as point charges, for which the distance of closest approach to the surface is taken as zero. All these are valid objections, and the theory could be improved,

s

0 a 0

experiment (solid

data from Grahame, J.

195

-

the

Gouy-Chapman theory

dl

bulk value of 78, over a distance of 1-2 nm.

60

Fig. 4G A Compari-

G. THE IONIC DOUBLE-LAYER CAPACITANCE C

between theory and experiment can be achieved once it is realized that both the Helmholtz and the Gouy-Chapman models are valid and exist simultaneously. Thus, there is a layer of ions on the surface that constitutes the Helmholtz (or the compact) part of the double layer. Outside this layer there is an ionic space charge, which constitutes the Gouy-Chapman (or the diffuse) double layer. For electroneutrality across the interphase, one must still have q m = — q s , but the charge on the solution side is partly in the compact layer and partly in the diffuse layer. With this model in mind, we can think of the potential drop between the metal and the solution as being divided into two segments:

196

ELECTRODE KINETICS

197

0 THE IONIC DOUBLE-LAYER CAPACITANCE C dr

(17G)

agreement between the Gouy-Chapman theory and experiment is found only

The potential (1) 2 is called the potential of the outer Helmholtz plane. *

in dilute solutions and only in the vicinity of the potential of zero

ct.s4 — (13's = ((l) m — (0 + (4) 2 — (1)s)

The diffuse double layer starts at the outer Helmholtz plane, where the potential is (1) 2 . It is this value of the potential, rather than 0 0, that must be used in Eqs. 14G and 15G, to relate the surface charge density

2S

to cosh( I Z ) 2/2RT), it follows from Eq. 19G that it can no longer be observed at high concentration and at potentials far removed from the potential of zero charge.

and the diffuse-double-layer capacitance to potential.

Differentiating Eq. 17G with respect to

charge. Since C2increases linearly with (C)) 1/ and is proportional

A detailed quantitative comparison between theory and experiment

and setting (I)s = 0, one

was conducted by Grahame in the years 1947-1958. This comparison was

has:

done in several steps, which we now list.

aOm aqm

=

[ 3 (0 s4— 0 2 ) aq m

[

+ [ 80 2 aqm

p

(18G)

11

1.

Measure the double-layer capacitance as a function of E.

2.

Integrate the data numerically to obtain the surface charge density

qm as a function of potential, using the relationship

This can also be written in the form 1 C

Here C

M -2

and C

2-S

di

=

1 C

M-2

f C

1 C

(19G)

2-S

dI

dE = q m(E)

(20G)

To perform the integration, it is necessary to know the value of E .

stand for the capacitance of the Helmholtz and the

This is readily obtained, in the case of mercury, from electrocapillary

is the experimentally

measurements, as will be shown. Note that we have used the potential E

measured double-layer capacitance. Equation 19G has the usual form for

measured versus some reference electrode, instead of the potential (1) N4 in

two capacitors connected in series, in agreement with the model postula-

the metal. This can be done because the difference between the two is a

ted by Stern, in which the two parts of the double layer are consecutive

constant, which is eliminated from the final equations as long as all

in space.

potentials are measured versus the same reference electrode.

diffuse double layers, respectively, while

Cdt

The important thing to note in Eq. 19G is that in a series combina-

3.

Use diffuse-double-layer theory (Eq. 14G), with O . replaced by

2

smallest capacitor that will predominantly determine the overall capacitance observed. This can

to calculate (0 2 as a function of q m, and from that obtain 0 2 as a

explain qualitatively the observation shown in Fig. 3G, namely that

4.

tion of capacitors, it is always the

function of the measured potential E. Use double-layer theory again (Eq. 15G, with 0 0 replaced by 0 2 ) to

calculate C

2-s as a function of E.

*

There is also an inner Helmholtz plane, where the potential is (0 1 . This is discussed briefly later.

*

Note that C dl given in Eq. 15G is identical to C 2S in Eq. 19G.

198

ELECTRODE KINETICS

5. With both the measured capacitance Cdt and the diffuse-double-layer capacitance C2_s known as functions of E, the capacitance Cm 2 of the compact double layer can be calculated from Eq. 19G as a function of potential.

G. THE IONIC DOUBLE-LAYER CAPACITANCE C

199 dl

38 N

a

IE 34

_, 301 -

How can we judge whether there is agreement between experiment and theory. For one thing, the capacitance of the compact double layer

w 26

should be independent of concentration. Second, in concentrated

I-

z 22

solution, the diffuse double layer should have very little effect on the observed capacitance except at, or very close to, E .

Thus, repeating 1

these five steps in solutions of different concentrations should yield the same plot of C M-2 versus E, and this should coincide with the

0

-0.4

-0.8

-1.2

-2.0

-1.6

POTENTIAL/Volt vs NCE

capacitance measured in concentrated solutions. We recall that the theory discussed so far was purely electro-

40

static. The charge on the metal side of the interphase attracts the oppositely charged ions, some of which are in the compact part of the double layer; the rest (required to make up for electroneutrality) are in the diffuse part of the double layer. Nothing has been said of

IF

32 <1-1-1 28

specific interactions of a chemical nature between the ions and the metal electrode. Thus, one can expect to find agreement between experiment and theory only if the electrolyte used is not specifically adsorbed. Grahame used NaF for this purpose and found good agreement between theory and experiment, as shown in Fig. 50, for 0.01 and 0.10 M solutions. This electrolyte was chosen because both ions are known to

b

36

H

24 20

< 16 12

0.4

-1.6 -1.2 -0.8 -0.4 POTENTIAL/Volt vs NCE

-2.0

be strongly hydrated; thus there can be little direct chemical interaction between the ion and the surface. ions like Br and I, which are not solvated in aqueous solution, have been shown to be adsorbed specifically. One should not expect to

Fig. 5G Experimental test of the diffuse-double-layer theory. Concentration of NaF: (a) 0.0/M (b) 0.10M. Solid lines: experimental values of C dl . Dashed lines: Cdl calculated from Eq. 19G. Reprinted with permission from Grahame, J. Am. Chem Soc. 76, 4819. Copyright 1954, the American Chemical Society.

200

ELECTRODE KINETICS

0. THE IONIC DOUBLE-LAYER CAPACITANCE C

dl

201

find agreement between experiment and the diffuse-double-layer theory * when the solution contains specifically adsorbed ions.

degree of hydration. Most cations are strongly hydrated and, as a

It is evident now why the Helmholtz and Gouy-Chapman models were

Most anions are not hydrated (or may have on the average only one water

retained. While each alone fails completely when compared with experi-

molecule attached to them). As a result, these ions can be in direct

ment, a simple combination of the two yields good agreement. There is

contact with the surface, allowing specific chemical interactions to play a role. Their adsorption is called specific or contact adsorption.

room for improvement and refinement of the theory, but we shall not deal with that here. The model of Stem brings theory and experiment close

result, their interaction with the surface is mainly electrostatic.

enough for us to believe that it does describe the real situation at the

2. The surface itself is hydrated. It can be viewed as a giant ion, having a large number of charges. The field near the surface (inside

interface. Moreover, the work of Grahame shows that the diffuse-double-

the compact part of the double layer) is comparable to the field near an

layer theory, used in the proper context (i.e., assuming that the two

ion in solution. Thus, hydration of the electrode surface accompanied

capacitors are effectively connected in series), yields consistent

with breakdown of the bulk structure of water is expected to occur, just

results and can be considered to be correct, within the limits of the

as in the inner hydration shells of ions. Since the solvent molecules

approximations used to derive it.

are in direct contact with the surface, short-range chemical interac-

We have failed to discuss so far the numerical value of the capacitance of the compact layer Cm_2 and its dependence on potential

tions with the metal electrode cannot be ignored. (In the nomenclature used for ions, we could say that the solvent is specifically adsorbed.)

(or charge), both of which are in disagreement with the simple parallel-

The chemical part of the adsorption energy of water on mercury is not

plate capacitor model proposed originally by Helmholtz. These issues,

very large, but it is enough to explain the experimentally observed lack

and the important effect of the solvent in the interphase, are discussed in Section 16.5.

of symmetry of the plots of

16.5 The Role of the Solvent in the Interphase The solvent can influence the structure of the interphase in many ways. 1.

versuss E around the potential of zero

charge. It may be much larger on solid electrodes, particularly on catalytic metals such as platinum and nickel. 3. The solvent also acts as a dielectric medium, which determines the field d4/dx and the energy of interaction between charges. Now, the dielectric constant

E

depends on the inherent properties of the mole-

First, different ions in solution are hydrated to different

cules (mainly their permanent dipole moment and polarizability) and on

degrees. The interaction of ions with the surface depends on their

the structure of the solvent as a whole. Water is unique in this sense. It is highly associated in the liquid phase and so has a dielectric

Specific adsorption is characterized by adsorption that can occur in spite of electrostatic interactions, namely a negative ion being adsorbed on a negatively charged surface, and vice versa.

constant of 78 (at 25 ° C), which is much higher than that expected from the properties of the individual molecules. When it is adsorbed on the surface of an electrode, inside the compact double layer, the structure of bulk water is destroyed and the molecules are essentially immobilized

ELECIR ODE KINETICS

G. THE IONIC

1),, i :MAL LAYER Cisi'fie_ Alq

by the high electric field. (This phenomenon is usually referred to as

than at the surface. Our simple calculation is good enough, however, to

electrostriction) Consequently, the appropriate dielectric constant to be used in this region has been estimated to be e = 6-8. Farther out in

show that the parallel-plate model proposed by Helmholtz is not as far

the solution, the bulk structure of water is rapidly regained. The dielectric constant is believed to reach its bulk value within about 1-2 nm of the surface. In the discussion of the parallel-plate model of

from reality as we might have inferred at first. Figure 6G(b) shows the structure of the interphase at positive potentials, in the presence of specifically adsorbed anions. Here the physical meaning of the inner Helmholtz plane (IHP) is illustrated. The

Helmholtz, a capacitance of 340 g/cm 2 was calculated using e = 78 and

b

a

d = 0.2 nm. This value compares poorly with the capacitance of about 16 tF/cm 2 observed experimentally at extreme cathodic potentials (with respect to E). If, however, we use E = 6-8 instead, we find agreement with experiment for d = 0.33-0.42 nm, which is of the right order of magnitude. In other words, with the use of the appropriate value of e inside the compact double layer, the capacitance calculated from the parallel-plate capacitor model can yield values compatible with experiment. What is the "correct" value of the thickness of the parallel plate capacitor in Eq. 3G. This may be seen by reference to Fig. 6G(a), which shows the surface of a negatively charged electrode covered with a layer of water molecules. The distance of closest approach of a cation is the sum of the diameter of a water molecule (0.27 nm) and its own hydrated radius. For the latter we can use the so-called Stokes radii, calculated from electrolytic conductivity data, which are in the range of 0.2-0.3 nm for most ions. Thus, the thickness of the compact double layer (i.e., the distance of the outer Helmholtz plane from the metal) is 0.47-0.57 nm. If we take median values of e Helmholtz parallel-plate model

=

7 and d = 0.52 nm, we find from the

11.9 liF/cm 2, in reasonable agreement with experiment. A correct calculation will be much more Cm 2 =

complicated, taking into account, among other things, the possibility that the value of e at the outer Helmholtz plane may already be higher

OHP

IHP OHP

Fig. 6G Schematic representation of the structure of the double layer. (a) At very negative potentials (with respect to the PZC) (b) At positive potentials, where specific adsorption of anions occurs. The Outer Helmholtz Plane is shown at a distance of (2Rw+ R) from the surface, where R w and R represents the radii of a water molecule and the solvated cation, respectively.

204

ELECTRODE KINETICS

G. THE IONIC DOUBLE-LAYER CAPACITANCE C

205

dl

thickness of this layer is always smaller than that of the outer

demonstrated experimentally for mercury electrodes and should be

Helmholtz plane (OHP), since there is no layer of water molecules

considered to be only a relative measure of specific adsorption of ions

between the ions and the surface. The absence of such a layer can

on other metals.

qualitatively explain why the value of

Cm_2

is higher on the positive

side of the PZC than on the negative side, taking the parallel-plate model one step closer to the experimental observations. Looking at the Cd/E curves in Figs. 2G-4G, it is clear that not all the experimental details have been accounted for. There are further effects resulting from lateral ion-ion, ion-dipole and dipole-dipole interactions. For example, the "hump" observed on the positive side of the PZC can be explained in terms of these interactions, which' we shall not discuss, since they are not essential for a basic understanding of the structure of the double layer. Which ions are specifically adsorbed? It depends, of course, on the metal, but detailed and accurate data are available only for

16.6 Diffuse-Double-Layer Corrections in Electrode Kinetics The diffuse double layer can influence electrode kinetics in two ways. First, we must realize that the initial state for charge transfer, namely the state of the system just before charge transfer has occurred, is an ion at the outer Helmholtz plane, not in the bulk of the solution. Thus, the relevant potential difference influencing the charge-transfer process is not 11) m — (l)s (which we usually write as O m' since we choose (I) = 0) but (1)/vi — 2 . Second, the concentration of the s reacting ion is not its bulk concentration, but rather the concentration at the OHP, which is related to the bulk concentration through Eq. 4G. For a simple cathodic charge-transfer reaction such as

mercury. As a rule, ions that are not hydrated tend to be specifically Fe

adsorbed. This includes most of the anions, but not F. Also, some highly symmetrical anions such as C10 , BF , and PF 6 are not specifi4 4 cally adsorbed on mercury. Most cations are not specifically adsorbed

3+

+ e rvi

Fe

2+

(21G)

we shall have to write, instead of i = FkC°exp(-13(13 mF/RT)

on mercury. Cesium, which was found to be specifically adsorbed to some

(22G)

extent, is an exception. Also, large organic cations of the tetraalkyl ammonium type are found to be specifically adsorbed on mercury. For some ions, specific adsorption may be observed only at high concentrations, and it must always be remembered that such adsorption has been *

At large positive potentials (with respect to E z) the capacitance

observed rises sharply, as seen in Fig. 5G. This is an artifact, due to faradaic processes, which are not related to the double-layer capacitance.

an equation including both diffuse double layer effects discussed above, namely i = FkC°exp(—zF(1) 2/RT)exp [—((F/RT)(4) m— 432)]

(23G)

where C° is the bulk concentration of Fe 3+ and z = 3 is the positive charge on it. This equation can be rearranged to i = FkCexp [--(z — f3)(13 2F/RT)lexp(—(343 1,4F/RT)

(24G)

ELECTRODE KINETICS

We recall that O in Eq. 240 can be replaced by the measured potential m E to within a constant. On the other hand, 4) is a complex function of 2 4)m , since it depends on gm , which itself is a nonlinear function of the

G. THE IONIC DOUBLE-LAYER

function of E.

CAt

, ,

L

20)

dl

The results, shown in Fig. 7G, are often presented in

the form of 4) as a function of the so-called rational potential E 2

=

potential.

E—E

(25G)

In spite of its inherent complexity, it is possible to obtain 4) 2 as a function of potential, by first integrating numerically the relationship between Cdi and E (cf. Eq. 20G) to obtain versus E, and then

How "rational" is the rational potential E? We could have a lengthy discussion of this matter, but it is not worth the time. Nomenclature

using diffuse-double-layer theory (cf. Eq. 14G) to calculate4) 2 as a

definition, an additional word in our vocabulary, useful as long as

should be considered to be no more than it actually is: an arbitrary everybody using it means the same thing, and as long as it is not misleading. The current term is not misleading, but it may sound somewhat presumptuous! At the time it was proposed, it was believed to

—0.3

be the best scale for electrode kinetics, since it relates to some intrinsic property of the metal. It is certainly the best scale of

0.001 M —0.2

potentials to be used in the context of diffuse-double-layer theory, since the excess charge density on the metal, q m , is related to E, not

o.lo om

to the potential measured versus some arbitrary reference electrode (which happens to be convenient from the experimental point of view).

0. 1 0.79SM 0

It is relevant for the discussion of adsorption, which is also related

NaF

to the charge density, as will be shown in Section 21. On the other hand, the PZC varies with the composition of the solution; it is a 0.1

property of the whole interphase, not of the metal alone. Also, a detailed analysis (which is beyond the scope of this book) shows that it

0.2

is better to compare the rate of a reaction occurring on different metals at the same overpotential rather than at the same rational

0.3 0.5

0

—0.5

—1.0

—1.5

—2.0

E/V vs NCE

Fig. 7G Dependence of the potential

of the OHP on potential, at different concentrations of NaF, in the absence of specific adsorption. 02

potential. Coming back to the dependence of 4) 2 on

*

E (or on ti) m), we note two

This calculation is valid only in the absence of specific adsorp-

tion of the ions.

208

ELECTRODE KINETICS

important things: change in

4)2

4)2

and Om always change in the same direction, and the

1.

dl

209

For hydrogen evolution on mercury, we substitute z = + 1 in Eq. 280

and 13 = 0.5 to obtain a = 0.5(d4) 2 /(14) M ) + 0.5. This reaction is studied

is always smaller that the change in O m 1 > (d02/dOm ) > 0

0. THE IONIC DOUBLE-LAYER CAPACITANCE C

(26G)

Additional points to note in Fig. 70 are that the derivative d(1) 2 id4) has M its maximum value at the potential of zero charge, and both 4)2 and d(1) /4 decrease with increasing concentration of the electrolyte. In 2 M other words, diffuse-double-layer effects in kinetics can be minimized

on mercury at high negative rational potentials. From Fig. 7G we find in this region, for a solution containing 1.0 M supporting electrolyte, = 0.08 V. This leads to a = 0.01 + 0.5, namely d(1)2/dO m = 0.02 and 2

an error of 2% in the observed value of a , which probably can be ignored in most cases. The effect of (0 2 on the current itself (hence on

by using a relatively high concentration of an inert electrolyte,

the calculated value of the exchange current density 0i) is quite significant, though, amounting to an almost fivefold increase in the

usually referred to as the supporting electrolyte.

present example.

Now, although the diffuse double layer is very important to our understanding of double-layer structure, we must ask ourselves how important it is for electrode kinetics.

2.

Perhaps the most striking example of the effect of the diffuse

double layer on electrode kinetics was demonstrated by Frumkin in 1955, for the reduction of the persulfate ion (S 2082 ). In this case (and in

Rewriting Eq. 24G in logarithmic form we have, for a cathodic process,

every case in which an anion is reduced or a cation is oxidized), the potential acts oppositely on charge transfer and on the diffuse double

log i = log(Fkco) — (z — (3)(0 2F/2.3RT) — 150mE12.3RT

(27G)

layer. Making the potential more negative increases the rate of charge transfer, but at the same time it decreases the concentration of the

hence, the effect of the diffuse double layer on the transfer coefficient a can he written in the form:

—a=

(2.3RT/F)(alog i/a0 m) = — [(z — (3)(dO 2/d(1) m) + 13] (28G)

It is customary to think that the effect of 4) 2 on electrode kinetics can be ignored, as long as a high concentration of supporting electrolyte (say, 0.5 M or more) is used. This may be correct in many

reacting anion at the interphase.

Substituting z = — 2 in Eq. 28G

yields a transfer coefficient of ac = — 2.5(d42/d0 m) + 0.5

(29G)

which shows that a will change sign if d0 2/dOm exceeds 0.2. The effect in dilute solutions is so pronounced that close to the potential of zero charge, a region of apparent negative resistance

cases, but the validity of this statement depends on the purpose of the

(where the current decreases with increasing cathodic potential) is

measurement, on the system being studied, and on the accuracy desired.

actually observed, as shown in Fig. 8G. At higher concentrations of

We can clarify this point with the aid of a few numerical examples, noting that the potential

4) 2 at the outer Helmholtz plane and its

derivative with respect to (1) 1,4 are obtained from the same data used to generate Fig. 7G.

supporting electrolyte, the region of apparent negative resistance cannot be observed, since d(1) /(14) is smaller, but the effect on the 2 M transfer coefficient is still very significant, so that the Tafel slope cannot be obtained without proper diffuse-double-layer correction.

210

EL-LC1RODE KINETICS

0. THE IONIC DOUBLE-LAYER CAPACITANCE C

21

dl

,

and for the oxidation reaction, an equation equivalent to 26G yields:

a. = - (z + (3)(42/d0s4) + 13 = 3.5(d4)2/d4) M) + 0.5

(32G)

Comparing Eqs. 31G and 32G we note that for a relatively small value of d(I) 2 /d(I) M = 0.02, the apparent transfer coefficients will be a = 0.57 and a = 0.43. If we assume 4)= 0.08 V, as in the first 2

example, we find from Eq. 27G that the exchange current density obtained from the Tafel plot in the anodic region may be too large by a factor of exp[-(z + (3)0 2 FAT] = 5x104 This factor depends on the metal, since a given overpotential for the oxidation of [Fe(CN) 6 ] 4 corresponds to quite different values of the rational potential (and hence to different

0.0

0.0

-0.4

-0.8

-1.2

-1.6

-2.0

on different metals. values of on These effects are quite remarkable, and the diffuse-double-layer correction cannot be neglected for such highly charged ions under any

POTENTIAL/Volt vs NCE

circumstances.

Fig. 8G The reduction of 1 mM S 2028 in solution of different concentrations of K 2SO4 , (a) 1 M, (b) 0.1 M, (c) 0.008 M, (d) 0.0 M. Data from Frumkin and Florianovic, Zhur. Fiz. Khim. 29, 1827 (1955).

16.7 Application of Diffuse-Double-Layer Theory in Plating It is interesting to see how diffuse-double-layer theory can be put to practical use in a field in which progress over the years has been achieved by trial and error, rather than through fundamental research.

As a final example, we should perhaps look at the oxidation and

Consider the electroplating of copper, an established industrial

reduction of the ferri/ferrocyanide couple, which is often used as a

process. It is well known to the expert in the field that fast plating

"test reaction" in electrode kinetics:

of thick layers can be achieved in a so-called acid bath, which consists

3.

s

[Fe(CN)]

4-

s

3

-

{Fe(CN) 6] + em

(30G)

of CuSO4 in H 2 SO4 (with some additives, which need not concern us at this point). If, on the other hand, one wishes to obtain a smooth and uniform deposit on an intricately shaped body, an alkaline cyanide bath

Substituting in Eq. 28G we find, for the reduction reaction,

- a, = - {(z - 1.)042/4 )] + 13 = - 3.5(0/4) + 0.5 (31G)

is better. The alkaline bath consists of copper ions in an excess of KCN (kept at high pH, to prevent the formation of volatile and highly poisonous HCN). In this bath copper exists as the negatively charged complex ion [Cu(CN)Y. Now, we recall that to achieve uniform current

212

ELECTRODE KINETICS

distribution (which is needed to obtain uniform thickness), one must

0. THE IONIC DOUBLE-LAYER CAPACITANCE C d1

213

16.8 Modern Instrumentation for the Measurement of C dl

increase the faradaic resistance (i.e., slow down the reaction), as much

The best method to measure the double-layer capacitance is to use a

as possible. In an acid bath, making the potential more negative will proper and by increasing the concentration of the reactant (Cu 2+) in the

This instrument is sometimes incorporated phase-sensitive voltmeter. into a frequency response analyzer, designed to make electrochemical impedance spectroscopy measurements, but it can also be used independ-

OHP.

In the cyanide bath, the two factors act in opposite directions.

ently. In Part Two we devote a full section to the operation of such

Making the potential more negative causes a decrease in the concentra-

instruments and the analysis of results obtained by them. Here we shall

tion of the negatively charged copper complex ions in the OHP. This

limit the discussion to the measurement of capacitance.

increase the current density by increasing the rate of charge transfer

slows the rate of increase of current density with potential, resulting

When a sinusoidal voltage signal E = E sin(cot) is applied to a

effectively in a higher faradaic resistance. For reasonable values of

cell, the result is a sinusoidal current signal of the same frequency

4) (0.05-0.10 V), this may lead to an increase of R Fby several orders

but displaced somewhat in time, i = i sin(cot + (p), namely, having a

of magnitude.

phase shift cp.

2

If the interphase behaves as a pure capacitor (ideally

Other factors, of course, come into play in an actual plating bath.

polarizable interphase with negligible solution resistance), the phase

For example, plating from an acid bath takes place at around 0.3 V, NHE,

angle will be — rc/2. If it behaves as a pure resistor (ideally nonpola-

whereas in a cyanide bath, copper is deposited at a much more negative

rizable interphase), the phase angle will be zero. Real systems do not

potential. The former occurs at a positive rational potential, while

behave ideally. The actual phase angle will, therefore, be somewhere in

the latter occurs at a negative rational potential.

between. The phase-sensitive voltmeter can measure the absolute value

This affects the

choice of additives and their adsorption characteristics.

Also, the

of the impedance vector I Z I and the phase angle (f) simultaneously. Since

and d4) /d4) may be different in the two cases. The 2 2 M foregoing example is not intended to be a quantitative interpretation of

the impedance of a capacitor alone can be represented by an imaginary

the benefits of cyanide baths, but rather an illustration of how

resistance, which is a real number, the impedance of the interphase as a

considerations of a rather fundamental nature can assist in solving

whole is a complex number.

values of 4)

number, Zc = — j/wC, and the impedance of a resistor is simply its

applied problems. Z(co) = ReZ — j(ImZ)

(33G)

in which j = (-1) 1 /2 , ReZ is the "real" or "in—phase" part of the impedance, and ImZ is the "imaginary" or "out-of-phase" component of the

The faradaic resistance is also increased by formation of the complex, which reduces the concentration of free copper ions in solution, hence reducing the rate of their reduction, but here we choose to discuss only the effect of the diffuse double layer.

impedance (the latter is often referred to as the "quadrature"). For a capacitor and a resistor in series one has Z(w) = R s — jicoC

(34G)

ELEerRODE KINEfCS

0. 'ME

JON it.

LA

1../

'di

It is convenient to display the results in the complex-plane representa. * non. The x-axis on this plot is ReZ, which is the ohmic resistance, and the y-axis is — ImZ, which, in the present case, is the capacitive R s = 1000

Cdr=

10AF

impedance — j/wC. The absolute value of the impedance vector I Z I and the phase angle

Cdi = 10/4F

cp are given by

1---

MAAAM-

a

100

/500

b

E

1500

(37G)

the series and parallel combinations. As the frequency is increased, the capacitive impedance decreases,

1000

6

T ao

tan cp = ImZ/ReZ

This is shown in Fig. 9G, which is a vector representation of I Z I for

80 C

(36G)

and

—

R F =1000 100

I Z I = [(ReZY + (ImZ)1 112

1000

while the resistive impedance is unchanged. In the series combination this makes the circuit behave more and more like a pure resistor, causing a decrease in phase angle, as seen in Fig. 9G(a). In the parallel combination, it makes the circuit behave more and more as a capacitor, causing an increase in phase angle, as shown in Fig. 9G(b). Consider now a more realistic situation, in which both the series

Fig. 9G Vector representation of the impedance of (a) series and (b)

and the parallel resistance must be taken into account. The equivalent

parallel combination of a capacitor and a resistor, showing how the phase angle changes with frequency (expressed in Radis on

circuit and the corresponding complex-plane plot are shown in Fig. 100.

the vectors). The ends of the arrows show the absolute values

applicable to this circuit, as a function of to:

Let us derive the mathematical expression for the impedance

of the vectors, except at the two lowest frequencies in the series combination. *

and for a capacitor and resistor in parallel the impedance is given by: 1/Z (0)) = I /RF — wC/j

(35G)

We shall refer to it as the complex-plane impedance plot, recog-

nizing that the same data can also be represented in the complex-plane capacitance or the complex-plane admittance plots. The terms Cole-Cole plot, Nyquist plot and Argand plot are also found in the literature.

216

ELECTRODE KINETICS

R

1

Z(w) = RRs +

=R + s

1/R F — (1)C dl ij

1+

F

C, THE IONIC DOUBLE-LAYER CAPACITANCE C

(38G) Cdr=10AF

*CdI R F

P s = 10 0

The following simple manipulation allows us to separate the real from the imaginary part of the impedance:

Z(w)

=

R Rs + 1+

1 — jwC d1 FR

F

i TOCdi R F

x 1

—

(39G) j(i)C di R F

R R

s

+ 1 +

(0C

F

COC

dI

R ) F

2

ReZ

jx 1+

d I

R

2 F

(40G)

( WC d RF) 2

-

I I-

Fig. JOG Complex-plane

R F = 100 0

representation of the

GJ R,C di = 1

impedance of an interphase. ReZ and ImZ are the real and imaginary

which leads to the expression

Z( w) =

217

dl

components of the impe-

60 40

E

20

dance, respectively. 20 40 60 80 100;120

I mZ RF

The result is a semicircle having a radius equal to RF/2, with its center on the x axis and displaced from the origin of coordinates by R + RF/2. Each point on the semicircle in Fig. 10G represents a s measurement at a given frequency. At very high frequencies, the fara-

ReZ

daic resistance is effectively shorted out by the double-layer capacitance, leaving the solution resistance in series as the only measured quantity. At very low frequency the opposite occurs, namely, the capacitive impedance becomes very high and one measures the sum of the two resistors in series.

any frequency, by first obtaining ImZ from the plot and then using the appropriate relationship given in Eq. 40G . How will the two methods of calculating the capacitance compare

The double-layer capacitance can be obtained from this plot in

with each other? Is the capacitance obtained by the second method a

different ways. The maximum on the semicircle satisfies the equation

R F Cd1 co ma x = 1 from which C

(41G)

di can readily be evaluated (recall that R F is equal to the diameter of the semicircle). The capacitance also can be calculated at

function of frequency, and if so, at which frequency do we obtain the The answers to these questions depend on the real value of C dl If the equivalent circuit is indeed interphase being studied. equivalent — that is, if it represents correctly the electrical response of the interphase — then the capacitance is independent of frequency. A more complex response of the interphase, which cannot be represented by

218

ELECTRODE KINETICS

the equivalent circuit just described, will show itself as an apparent frequency dependence of Cat . Unless the correct equivalent circuit is known, the double-layer capacitance cannot be rigorously evaluated. Fortunately, it turns out that in most cases studied (excluding porous electrodes and other systems in which the interphase responds like a transmission line), the double-layer capacitance is in a separate parallel branch of the equivalent circuit, with only the solution resistance in series with it. This is shown in Fig. 11G, where we have lumped all other components of the interphase into a "black box," the nature of which need not be specified for our present purpose. For sufficiently high frequencies, such an equivalent circuit behaves just like a resistor and a capacitor in series, for which Cal can readily be determined. In conclusion, when the observed capacitance is independent of frequency, it is enough to measure it at one convenient frequency. This is important, since it allows one to measure Cal as a function of time

0. THE IONIC DOUBLE-LAYER LAVACHANLL t2

219

at constant potential, or while the potential is being changed at a controlled rate. If, on the other hand, the capacitance is found to be a function of frequency, it is best to conduct measurements at the highest frequency offered by available instrumentation, or to measure it over a range of frequencies and extrapolate to infinite frequency. If one studies an (almost) ideally polarizable interphase, such as the mercury electrode in pure acids, there is no need to measure at high frequency. In this case the equivalent circuit is a resistor R s and a capacitor Cat in series. The accuracy of measurement is actually enhanced by making measurements at lower frequencies, since the impedance of the capacitor is higher. The high accuracy and resolution offered by modern instrumentation allows measurement in such cases in very dilute solutions or in poorly conducting nonaqueous media, which could not have been performed until about a decade ago. The frequency response analyzer may be the best instrument for double-layer-capacitance measurements, but it is also the most expensive one. Other methods are available, which are faster and cheaper, but less accurate. For example, the charge on the double layer is related to the potential as: lE = C al (E

C dl

(42G)

hence, the charging current can be written as

Rs

i = dqmidt = C at dEidt)

H Fig. I IG General equivalent circuit with an unspecified element Z in the faradaic resistance branch, showing that the double layer capacitance can be obtained by extrapolation to sufficiently high frequency.

(43G)

This is correct as long as the capacitance is independent of potential. As a rule. the double-layer capacitance is a function of potential, but if one uses a small perturbation, this dependence can be considered to be negligible. When a current step is applied to an interphase, which can be represented by the equivalent circuit shown in Fig. 11G, the change of potential with time is given by

220

E = E [1 —

ELECTRODE KINETIC3

O THE IONIC DOUBLE-LAYER CAPACITANCE C d1

(44G)

E/E. = t/'t

+ iR

where E

221

(45G)

is the steady-state potential and i is a characteristic time constant determined by C dl and by the properties of the impedance represented by the "black box" in Fig. 11G. (For the circuit shown in Fig. 10G, this simplifies to ti = RFCdI). When the time of measurement is short compared to the

Thus, the potential varies linearly with time in this region, and the

characteristic time constant (tit 0.1), the exponent in Eq. 44G can be linearized to yield:

(excluding iR F) will be small, not exceeding about 10 mV and (b) the variation of potential with time will be linear.

capacitance can be calculated from the slope according to Eq. 43G. Usually, a square-wave current signal is employed. The resulting variation of potential with time is shown in Fig. 12G. The parameters of the square wave are chosen so that (a) the resulting potential

dE/dt= 1/Cd1

E

b

a

IR s

time

0 3

0

Potential

time

Fig. I2G Square-wave-current pulses and the resulting transients of potential. Both the residual solution resistance R and the s double-layer capacitance C can be obtained from such di

measurements.

Fig. 13G Current response to a triangular potential sweep, plotted on an X—Y recorder. (a) ideally polarizable interphase (b) real interphase, with finite faradaic current.

I.; .1:CTRODE KINETICS

0. THE IONIC DOUBLE-LAYER CAPACITANCE C

223 dl

It should be obvious at this point that taking a short transient is equivalent to making the measurement at a high frequency. It is not

b

a

easy to compare the frequency of a sine wave to the equivalent frequency ---------

1

of a transient, because of the difference in the shapes of the waveforms. For an order-of-magnitude type of estimate, we can take 1/t i, the reciprocal of the pulse length, as the equivalent frequency. A 0.1 is pulse, which can readily be obtained from a good square-wave ID 0

generator, thus corresponds to a frequency of about 1x10 7 Hz. If we wish to measure the capacitance as a function of potential,

2C dl

it is possible to apply a relatively slow triangular potential waveform, and determine the current as a function of potential. If the interphase is highly polarizable, the result will be that shown in Fig. 13G(a). If there is a significant faradaic current, a plot such as shown in Fig. 13G(b) will be observed. Ideally the box representing the anodic and cathodic charging currents in Fig. 13G(a) should be symmetrical around zero. Unfortunately, this is rarely the case. The best way to calculate the capacitance from such experiments is to conduct measurements over a range of sweep rates, obtaining

Cdl from the plot of

Ai = i — i versus dE/dt, as shown in Fig. 14G. c

Potential

Sweep rate, v

Fig. 14G Current-potential plots obtained at different sweep rates, for a system having a faradaic current in addition to the doublelayer charging current. Using the difference Ai, rather than the individual values of i or i, eliminates most of the error due to the faradaic reaction in the calculation of Cam .

There are many other ways by which the capacitance of the interphase can be determined, with the use of single or repetitive waveforms,

If the series resistance is high and the parallel resistance is low, one

with either the current or the potential being used to perturb the faces an adverse experimental situation. In this case, measurements system. The choice depends on available instrumentation, the nature of should be conducted over a range of frequencies, with the highest the system being studied and often the personal preference and experience of the researcher. For any method to be used prudently, it must be remembered that as a rule, the interphase consists of a resistor in series with a parallel combination of a capacitor and a resistot. If both resistances are high, one should try to conduct measurements at low frequencies. If both are low, the experiment should be conducted at high frequency.

possible accuracy, and the optimum conditions for the experiment should

be carefully chosen. Under such conditions, electrochemical impedance spectroscopy apparatus may be indispensable.

225

H. FIECIROCAPILLARITY

II. ELECTROCAPILLARITY 17. THERMODYNAMICS 17.1 Adsorption and Surface Excess The creation of an interface is a symmetry-breaking process. Even

if the two phases in contact are entirely homogeneous and isotropic, the molecules at the interface experience a different average force in each of the two directions perpendicular to the interface. As a result, the energy of a molecule at the surface is different from its energy in the bulk phase. This effect is not restricted to a single layer of molecules, and it can extend to a distance of several molecular diameters into the solution. Indeed, it is possible to envision a region on either side of the interface in which the energy of the molecules is different from that in the two bulk phase. This region is defined as the interphase, the term we have been using to describe the region between two different phases in contact (cf. Fig. 2A). When only neutral species are involved, the interphase cannot be more than a few molecular layers thick, since the forces involved are "chemical" and decay with a high power of the distance (cf. Fig. 1B). When charged particles (i.e., ions in solution at the metal-electrolyte interphase) are considered, the interphase can extend much further, since the electrostatic Coulomb interactions between charges decrease with the first power of the distance. The best example of such behavior is the diffuse double layer. In the discussion of diffuse-double-layer theory up to now, we have been interested in the variation of the potential 431 2 matheourHlmzpn,withexcsurfagdenityq

E.

If we solve the equations for the varia- ortheainlp

tion of the potential inside the diffuse double layer, we obtain the approximate equation:

226

ELECI RODE KINETICS

4)(x) = 4) 2exp(—Kx) Where

K

2,21

H. ELECIROCAPILLARITY

00

(1H)

(3H)

F = f (C — Cldx

is the so-called reciprocal Debye length (which is better known The surface excess F represents the total amount of the relevant species

in the context of the Debye Hiickel limiting law), given by K

=r

2 2 8rt z F 1 1n ( C °) o r2 )

eRT

(2H)

J

Substituting the numerical values of the constants, we find that for a 0.001 M solution of a symmetrical 1-1 electrolyte, K -1 = 10 nm. Looking again at Eq. 1H, we note that at a distance x = K -1 from the metal, the potential is 4)(K) = 4,/e = 0.374) 2 . How far will the interphase extend into the solution in this case? There is no single answer to this question, since the potential 4)(x) decays exponentially and never

in a cylinder of unit cross section, extending from the interface into the bulk of the solution, less the amount that would have been in the same volume, had there been no interface. Fortunately, we do not have to determine how far exactly the interphase extends. The function (C — C° ) is integrated to "infinity" — far enough into the bulk that its value has become negligible. It should be noted that on the scale of interest for interphases, "infinity" is not very far. In fact, it is less than 1 i.tml For the example just discussed, 4)(1 gm) = 4) 2exp(-100). The definition of the surface excess is shown graphically in Fig. 1H.

reaches zero exactly. Often the thickness of the diffuse double layer is quoted as being equal to K -1 . Taking x = 3K -1 may be more approp-

2.2

riate, since at this distance 4) 2 has decayed to 5% of its original value, but this does not matter, from the conceptual point of view.

1.8

The thickness of the diffuse double layer is seen to be a function of concentration. In 1.0 M solution, K -1 = 0.3 nm, which is of the

0

order of magnitude of the thickness of the compact part of the double

N

0 1.4

layer. This is another manifestation of the observation,alluded to earlier, namely, that the effect of the diffuse double layer is small in

1.0

concentrated solutions. The asymmetry at the surface can affect the energy of different

0.6

species in solution to different degrees. If, for example, we dissolve equal amounts of methanol and butanol in water, the concentration of butanol in the interphase will be higher. This comes about because the transfer of butanol from the bulk to the interphase decreases the total free energy of the system more than does the transfer of methanol. It is necessary, at this point, to define a new quantity, called the surface excess, F, such that

0

4

8

12

16

20

Distance from surface/nm

Fig. 1H Illustration of the concept of the surface excess F, expressed in units of mol per square centimeter. The lines represent the concentrations of the two ions as a function of distance from the metal surface, calculated for 0.01 M KCI, q m = 10 Klcm2.

228

ELECTRODE KINETICS

229

H. ELECTROCAPILLARITY

The surface excess is an integral quantity. This has the advantage

— dy =

(4H)

of relieving us of the need to define the boundary of the interphase. On the other hand, it cannot yield any information on the variation of the concentration inside the interphase. Another point to remember is that the surface excess, as defined here, can have both positive and negative values. This statement is generally correct, but its validity

The function y is the surface tension, given in units of force per length (dyne/cm or N/rn) - It is related to the two-dimensional surface pressure II by the simple equation — dy = do

can most easily be seen in electrochemistry. Thus, a negative charge on

(5H)

the metal (qm < 0) causes a positive surface excess of cations and a

It is also the excess surface free energy per unit area, namely the

negative surface excess of anions, and vice versa, as seen in Fig. 1II.

extra free energy added to a system as a result of formation of the

In the case of adsorption of an intermediate in electrode kinetics,

interphase. When we speak of surface tension, we have a mechanical

we have discussed the extent of adsorption in terms of the fractional

model in mind, as though we had an imaginary membrane on the surface,

surface coverage 0, which was given as:

pulling it together with a force (which can readily be measured in an

0 -= Firmax

(10F)

apparatus of the Langmuir-Blodgett type). The notion of an excess surface free energy is, of course, purely thermodynamic.

Here F denotes the surface concentration of the adsorbed species, and F

We noted early on that the driving force in chemistry is the

is the value of F needed to form a complete monolayer. We recall

decrease of free energy (cf. Eq. 1B). Thus, a system will change

that the intermediates discussed (such as an H atom or an OH radical)

spontaneously in the direction of minimum surface tension. This leads

are unstable in solution and can be formed only on the surface. In this

to two observations:

case the surface concentration is equal to the surface excess, and it is

1.

appropriate to use the same symbol for both. It should be noted,

minimum surface area per unit volume. This is why droplets of a liquid

max

though, that in the general case Eq. 3H does not show a clear value of F , and, in fact, max

r could exceed the amount corresponding to a

A pure phase always tries to assume a shape that creates the

are almost spherical (they would be completely spherical in the absence of gravity), and this is also why the planets and other celestial bodies

monolayer, because the material is distributed over a thickness that can

are spherical.

correspond to several molecular layers.

2.

17.2 The Gibbs Adsorption Isotherm

When a solution is in contact with another phase, the composition

of the interphase is different from that of the bulk in such a manner as to minimize the excess surface free energy, y.

The most fundamental equation governing the properties of interphases is the Gibbs adsorption isotherm:

The second observation represents the essence of the physical meaning of the Gibbs adsorption isotherm. The adsorption of any species in the interphase

(ri > 0) must always cause a decrease in the free

energy of the surface (dy < 0), since it is the reduction in this free

—JO

ELECTRODE KINETICS

R. ELECIROCAPILLARITY

— (a2,y/aE2 )ili= kaqslaql, =

energy that acts as the driving force for adsorption to occur.

C

(8H)

dl

The Gibbs adsorption isotherm is derived in many standard textbooks of physical chemistry and surface chemistry. We shall not repeat this derivation here. Rather, we show how this isotherm is modified when it is applied in electrochemistry.

These equations relate the surface tension or excess surface free energy to two very important electrical characteristics of the interphase: the charge density and the double-layer capacitance

Cdl .

It should be

noted that these are purely thermodynamic relationships, not based on a 17.3 Derivation of the Electrocapillary Equation

model. The only assumption made in the derivation of Eq. 6H (from

The surface tension of an electrode in contact with solution depends on the metal-solution potential difference. The equation describing this dependence is called the electrocapillary equation. It follows from the Gibbs adsorption isotherm, as we shall show in a moment. Before we do that, however, let us write this equation and discuss some of its consequences. A general form of the electrocapillary equation is:

Eq. 4H) is that the interphase is ideally polarizable: that is, charge cannot cross the interphase. It follows from Eq. 7H that at the maximum of the electrocapillary curve, q m = 0. In other words, the potential of zero charge coincides with the potential of the electrocapillary maximum. The double-layer capacitance can be obtained by double differentiation of the surface tension with respect to potential, and the surface tension can be obtained by numerically integrating the C d /E relationship twice. The situation is not entirely symmetrical, however. For double differentiation, all one needs is very accurate data of y

— dy = q mdE + Eridgi

(6H)

This equation is equivalent to Eq. 4H, considering that q m is nothing

versus E. For double integration, one also needs two constants of integration. These are the coordinates of the electrocapillary maximum, namely E and y . For liquid electrodes (and particularly for

but the surface excess of electrons on the metal side of the interphase, and the electrical potential E is the intensive variable determining the

mercury), both can be measured readily with high accuracy. For solid electrodes it is possible, in some favorable cases, to measure Ez , but

electrochemical free energy of electrons in the metal.

max

Several relationships follow immediately from this equation. The

yrmax cannot be measured. Thus, for solid electrodes, one can integrate double-layer capacitance data to obtain q m as a function of potential,

partial derivative of y with respect to potential, at constant composi-

but the second integration, needed to obtain the electrocapillary curve,

tion of the solution, yields the surface charge density:

cannot be performed. The third important relationship that follows from the electro-

— (ay/aE) Il = qm

(7H)

and the second derivative with respect to potential yields the doublelayer capacitance:

capillary equation is: E Lt

"

= =

2.3RT

(ayialog

a)

,.

(9H)

This equation can be used to determine the surface excess (in effect,

232

ELECTRODE KINETICS

233

H. ELECTROCAPILLAR1TY

the extent of adsorption in the interphase) of any one species, from —

variation of the surface tension with the activity of this species in solution, at constant potential. Equation 6H is the electrocapillary equation in general form. To fully appreciate the possible applications of this equation and its limitations, we must derive it for a specific system. We have chosen a relatively simple case, to demonstrate the principles involved. Once these are understood, a more complex case can be derived following the same line of reasoning. Consider a mercury electrode in contact with an aqueous solution of KCI, and an Ag/AgC1 reference electrode. This cell can be represented

—Cu ge

The superscripts in Eqs. 11H, 1211, and the following equations, indicate the phase in which a given species is found. When a species exists in only one phase, the superscript is omitted Equation 11H follows from the fact that electrons in the mercury phase and in the copper wire connected to it are at equilibrium. Also, the chemical potential of mercury is equal to the sum of the electrochemical potentials of its components. Combining Eqs. 1011 and 11H, we find, for the mercury phase: (r

Cu' I Ag I AgC1 I KCI, H 2O I Hg I Cu"

d + Hg+ Hg+

where the solid vertical lines represent separation between different

F dg = e el

Ng+

= F dp. — (F

phases (which are actually in physical contact with each other) and Cu'

Hg+

(d 1.t

Hg

—r

e Hg+ Hg

—

dg}:g ) + r dFing e

(1211)

e

(13H)

q m = F(Flig+ — re)

in explicit form as:

For the aqueous phase we have a similar set of equations, namely:

— dy = (Flig+Cl[tHgt + reaFtel

r

(10H)

rCI-dFtCI- + 1-11204µH20

First we wish to transform this equation to a form that does not involve the electrochemical potentials of single ionic species, which cannot be

['KC'

4_ W

"K+

and q s = F(r.,

r ci- ) = qm

[r K+ Crli K+ = rK+d

A third electrode is not needed in this case, since the mercury-

solution interphase is assumed to be ideally polarizable.

(141-1)

hence +

F

C I -C I-

F

K

KCI

—w ■ —w — dp C I- + rCl- dg C I-

measured. This is done by writing the two relationships:

*

e

We also bear in mind that, by definition:

source of variable potential. The Gibbs adsorption isotherm is written

+ [rK+diTiK+ +

and g Hg e

+

by:

and Cu" are copper wires connected to the two terminals of a suitable

—

— Hg

Hg

11 Hg =

(rK+ rci-)dro- = FK+dilKCI

Substituting all these into Eq. 101-1 yields:

(qs/F)dg ci

(15H)

234

ELECTRODE KINETICS

— dy = F

Hg+ di_tHg

-

(q /F)dru e M +

(qs/F)drci_

K+d

H. aECTROCAPILLARITY

— dy = qmdE + FK+d

rH2odgH20

ilKCI+

r H22 o t H0

(23H)

After this rather lengthy derivation, we should have arrived at the (qs4IF) (dficev —

(16H)

electrocapillary equation in its final form. Equation 23H does indeed look much like Eq. 6H, which we set out to derive. There is, however,

Now we turn to consider the equilibrium pertaining to the Cu' I Ag AgCll KC1, H 2 01 side of the cell. For the two pure phases we can

another subtle point that we must deal with. One of the important uses

= rHg+d 1-t Hg

K+dliKCI

rH 2 0dIII-1 2 0

write: dl-tAg = 0 =

(17H)

dijA A g+ di??

and similarly dli

= 0 = dli

would seem that Eq. 23H is just what we need, since it follows from this relationship that:

- Aga

AgC1

of the electrocapillary equation, we recall, is to determine the surface excess of various species, in the present example that of K + ions. It

dii AgC

Ag+

(18H)

CI-

Since the silver ion is at equilibrium in the two phases (otherwise this electrode would not act as a reversible Ag/AgCl/C1 electrode), it follows from Eqs. 17H and 18H that deg = d egCI

(19H)

— [a IP

'1 E,1.1.H 0

r K+

(24H)

2

Taking a closer look, however, we realize that the partial derivative represents an impossible experiment! There are only two components in the aqueous phase in the system being discussed: water and KC1. We cannot change the concentration of one without simultaneously changing

Also, the electrons in silver are at equilibrium with those in Cu', and

the concentration of the other. The effect may not be important in

the CI ions in AgCI are at equilibrium with those in the solution, hence it follows from Eq. 19H that

dilute aqueous solutions, but it is always there, and it must be taken

dg—cu. =

(20H)

diCl

into account in a rigorous treatment of the problem. To overcome this difficulty, we make use of the Gibbs Duhem equation, which has the following general form:

We can substitute in Eq. 1611

(25H)

Ex,d[ti= 0

(dpi e u" — dkiwci ) = (dlice u — diicec ) = — FdE

(21H)

Applied to the aqueous phase in the present system, it takes the form

to obtain — dy = F

Hg+

dp.

Hg

1-K+ dpi

X

KCI

F

H20 d4

H20

q dE m

(22H)

In the present example, mercury was taken as a pure phase, hence = 0, leading to the simplified equation Hg

KCIKa

+X

H0 2

d pt

H2 0

=0

(26H)

where x is the mole .fraction of the appropriate species. Substituting di.t110 2 from Eq. 26H into Eq. 23H, we have

ELECTRODE KINETICS

236

— dy = q dE + F K+ dp,KCI m

—

F

H20

(X

We now define the relative surface excess,

KC I

/X

H20

)4L

(27H)

KCI

as follows:

ric' 4.

237

H. ELECTROCAPILLARITY

reference electrode that is reversible with respect to the positive ion in solution yields the relative surface excess of the negative ion, and vice versa. Looking at the way the relative surface excess has been defined, it

(28H)

r' and F is going to be negligible in dilute aqueous solutions. The difference can, however, he

and thus obtain the electrocapillary equation for the system discussed

very large in mixed solvents, when the mole fractions of the two

here in final form:

components are comparable.

F'K+

K+

—F

H 0 2

(x

KCI H 0 2

)]

is evident that the difference between

We shall end this section by using the electrocapillary equation to (29H)

— dy = qmdE_ + FK+dpicci

derive some relationships between partial derivatives, which are occasionally used in research in this field. Starting with a simplified

In this equation we added a new subscript, which has not been discussed

form of Eq. 611

so far. Instead of the potential E, used thus far, we have introduced

— dy = qmdE + Fdp

(32H)

E_, to specify that the cell for which this equation is valid has a reference electrode that is reversible with respect to the anion in solution. What would have been the result if a reference electrode

we make use of the fact that dy is a complete differential to write: (aq m /a0 E (aF/aqi

(3311)

reversible with respect to the cation had been used? We shall spare the reader the tedium of working through the equations again, presenting

such that

only the final result, which is similar to Eq. 28H, namely: — dy = qmdE+ +

A different relationship is obtained if we define a new function y'

dp.Ka

y' y + q m E

(34H)

dy' = dy + q mdE + Edq m = — Fdp, + Edqm

(35H)

(30H) from which it follows that

The partial derivatives derived from Eqs. 2911 and 30H

[aY/ 41xcli E = F1'<+

and

(Va llKcjE+=

CI-

(3111)

represent legitimate experiments (unlike that in Eq. 2411), since only the potential needs to be kept constant, and the variation of the concentration of water, caused by varying the concentration of KCI, is taken into account by realizing that we obtain only the relative surface excess

F',

not its absolute value

r.

Thus, the use of an indicator-type

Since y', as defined in Eq. 3411, is also a complete differential, we have from Eq. 35H the relation: (81-/aq m) p. = — (0E/al_t) q

(3611) M

Similarly, we can define another function, y" as:

KiNtiTICS

Y" . Y

r/-1-

(37H)

Eq. 41H? We note that the surface tension is a function of three

(38H)

For the ideally polarizable interphase, For and variables: E , u they are all independent. For the ideally nonpolarizable interphase,

to obtain:

(aqmpr)E =

—

(31,,VaE) F

H. ELECI kOCAPJLLARITY

only two can be controlled independently. We recall that an ideally and finally, by defining a third function 7 * : 7* -a 7 + q m E +

(39H)

nonpolarizable electrode is a reversible electrode. By setting the concentrations (more accurately, the activities) of K + ions in the two phases, we determine the potential. Alternatively, by setting the

we obtain, following the same reasoning, the relation:

potential, we determine the ratio of concentrations of this ion in the

(a War) q = M

(aptpq m) F,

(4011)

two phases. We conclude that the electrocapillary equation for the nonpolarizable interphase must have one less degree of freedom.

The differential relationships just derived represent the equivalent of the Maxwell equations in thermodynamics. Seldom used in electrochemistry, these equations have been employed in relation to the study

To show this, we express the condition of equilibrium at the interphase by setting the electrochemical potential of K + equal in the two phases:

of adsorption, particularly at the mercury-solution interphase.

(42H)

dFt K, =

17.4 The Electrocapillary Equation for a Reversible Interphase

This equation makes all the difference between a nonpolarizable and a

It is interesting to see how the electrocapillary equation is modified if the interphase is ideally nonpolarizable. To do this, let us consider the cell

polarizable interphase. The rest of the derivation follows the lines given in Section 17.3. We write Hg

Cu' I Ag AgCl I KO, H 2O I K, Hg Cu"

modify Eq. 291-1 intuitively, to be applicable to this system, if we

= ail

K+

w K CI

=

d Ftcu"

- Hg

,- Hg

Hg

du

which is similar to the cell discussed earlier, except that it is assumed here that the liquid metal phase is a King amalgam. We can

=.

K+

+—w C 1-

(44H)

d µ K+ +

—w —cu• already shown (cf. Eq. 20H) that du e = du c hence

assume (for the moment) that the Hg/H 2O interphase is ideally polarizable. The corresponding equation is:

—

— dy = q dE + F' w d[tw + F' llg M K+

KO

K+

d µ Kg

(41H)

But the 1-1g/11 20 interphase is not ideally polarizable in this case! flow is the notion of an ideally nonpolarizable interphase introduced into

(4311)

- "KO =

+

dri"=

u

d

-

NKCI

(45H)

4 µe

CV;) =

di.AH K8

FdE

K a into Eq. 4111 yields Substituting this value of di.t w

(46H)

240

ELECTRODE KINETICS

— dy = (gm + FFK' w)dE_ + (FK' Fig + Ftc' w)4111Kg Alternatively, by substituting di

(4711)

from Eq. 45H into Eq. 41H one has:

— dy = (q m — FFK'7)dE_ + (r

;c7g

rICW +)(111 KCI

(48H)

What are the differences between this equation and the correspon-

241

It ELECTROCAPILLARITY

18. METHODS OF MEASUREMENT AND SOME RESULTS 18.1 The Electrocapillary Electrometer The measurement of surface tension is an old trade in science.

There are consequently many methods of determining this quantity. In chemistry, the surface tension is measured as a function of the solvent,

ding equation (Eq. 29H), for the ideally polarizable interphase?

the composition of the solution, and the nature of the two phases. In

1.

The surface tension is still a function of potential, but the

electrochemistry the potential is an added variable. The so-called

electrocapillary maximum no longer coincides with the potential of zero

electrocapillary curve is a plot of surface tension of a liquid metal (usually mercury) electrode versus potential, at a given composition of

charge. Instead, at (dy/dE) = 0 we have: q

m

= F F' Hg = — F F' w = — a K+

K+

(49H)

Also, we can no longer obtain the charge, q m, as a function of potential

species, as discussed earlier. To be more exact, we should be talking about the

from the dependence of y on E. 2.

the solution. This type of measurement is then repeated in solutions of different composition, to obtain the surface excess of the appropriate

Taking the partial derivative of y with respect to the chemical

potential yields the combined surface excess on both sides of the

interfacial

tension, which is the surface tension between two specified phases. In electrochemistry it is customary to use the term surface tension to refer to the interfacial tension at the metal-solution interphase.

interphase

We recall from our elementary science classes the phenomenon of (dAL Kci ) E = (1";(11 4. g

FKW )

= (dyidlifiKOB

(50H)

which cannot be resolved into its components unless one of them is obtained from an independent measurement. Equations 49H and 50H explain why there has been little interest in obtaining the electrocapillary curve for ideally nonpolarizable interphases. On the other hand, this analysis can give us a feel for the

capillary rise, shown in Fig. 2H(a). When a series of glass capillaries is inserted into water, the water will rise in the capillaries to a height that is inversely proportional to the radius of the capillary.

The equation describing this behavior is the Young-Laplace equation, which can be written as: AP = 2y/R

(51H)

type and magnitude of error that may arise when measurements are

This equation gives the pressure difference AP needed to keep two phases

conducted with an electrode that is presumed to be ideally polarizable

in mechanical equilibrium, if the interphase has a radius of curvature

but in fact does allow some faradaic current to flow across the

R. When the interphase is flat, R goes to infinity and AP = 0; that is,

interphase.

the phases are at equilibrium when there is no pressure difference between them. In the present example, the pressure in each capillary is

242

ELLCTR ODE KINETICS

related to the height of the liquid, since 2 2 AP = force/area = rcIZ g•h•p/itI2

43

H. EI-ECI ROCAPILLAR 11 Y

There is no affinity between mercury and glass. Thus mercury must be "forced" to enter the capillary. One would have to apply a pressure =

•h•

(52H)

where p is the density of the liquid and g is the acceleration due to

in the opposite direction to bring the mercury in the capillary to its level in the vessel. Clearly the physics controlling capillary rise and capillary depression is the same. To accommodate the difference between

gravity. The height of the liquid in the capillary is given by:

the two in the framework of the same equations, we note the difference h=

AP P = (2y/R)(1/p-g) g•

(53H)

in the form of the meniscus in Fig. 2H(a) and 2H(b). For water the meniscus is concave, while for mercury it is convex. If we agree to

If we replace water by mercury, there will be a capillary depression instead of a capillary rise, as shown in Fig. 2H(b). The simple physics behind this is that water wets glass whereas mercury does not. In everyday language we could say that there is an affinity between water and glass, causing the liquid to "crawl up" the capillary. To force the water in the capillary back to its level in the vessel, it would be necessary to apply a pressure, that is inversely proportional

define the radius as being negative for the convex interphase and choose the level of the liquid in the outer vessel as zero (leading to negative values of h for capillary depression), we find that the Young-Laplace equation is applicable to wetting as well as nonwetting liquids, and Eq. 53H correctly describes both capillary rise and capillary depression. The reader may object to the use of an artificial concept such as a negative radius, but this is not an uncommon practice in science. For

to the radius of the capillary.

instance, the capacitive impedance is described as the "imaginary" part of the impedance, although it is a very real impedance indeed! a

b

We have obviously taken some shortcuts to keep the foregoing discussion short and, hopefully, clear. For example, we ignored the angle of contact between the liquid and the solid at the edge of the meniscus. This is tantamount to considering water to be an

ideally

wetting liquid (with a contact angle of zero) and mercury to be an ideally nonwetting liquid (having a contact angle of Tr). One may also question the exact definition of the height h, in Fig. 2H. Is it measured to the bottom of the meniscus, to the point at which the meniscus contacts the glass, or somewhere in between? Is the capillary Fig. 2H (a) capillary rise observed for a wetting liquid, such as water in glass and (h) capillary depression observed for a nonwetting liquid, such as mercury in glass.

ideally circular, leading to a spherical meniscus, or should we define two radii of curvature? These are important points when one conducts research in this field, but they are not relevant to the basic physical

244

ELECTRODE RODE KINETICS

H. ELECTROCAPILLAR ITY

245

Measurement of the electrocapillary curve consists of changing the

understanding of the problem. In the classical electrocapillary electrometer the configuration is

potential stepwise and determining the pressure required to return the

inverted. Mercury is placed in a glass tube that ends with a fine

mercury meniscus to the same location in the fine capillary. A plot of

capillary, as shown in Fig. 3H. Since we need pressure to force mercury

this pressure as a function of potential is nothing but the electro-

into a fine capillary, there will be a certain height of mercury column

capillary curve, within a constant, as seen from Eq. 53H. The best way

supported by the capillary in this configuration. This is the exact

to determine the magnitude of this constant is by calibration with a

equivalent of the capillary depression shown in Fig. 2H(b), and the

known system. This requires, of course, one accurate determination of y

height of the column is also given by Eq. 53H. In this equation we note

by an independent method. Very careful experiments were performed by

that h depends on y, and the surface tension depends on potential;

Gouy around the turn of the century, and these results are used even now

hence, the height of the mercury column above the capillary is a

as the primary standard for electrocapillary measurements.

function of potential.

The potential of the mercury electrode is controlled against a suitable reference electrode. It is not essential to use a threeelectrode system in this case, since the mercury-solution interphase is studied in the range of potential where it behaves as an ideally

ressure auge

polarizable interphase. A reference electrode is commonly used never-

Potentiostat

theless, to ensure stability and reproducibility of the measured N2

pressure)

potential. The position of the mercury in the capillary is observed either though a microscope of long focal length or by a video camera connected to a high resolution monitor. We shall not dwell on technical

Hg capillary

details, except to note that building and operating an electrocapillary electrometer at the desired level of accuracy is a delicate matter,

Microscope with long focal length

requiring both skill and experience. Let us make a small detour here to discuss a minor point, which appears to be purely technical but may help us to better understand the physics behind electrocapillary measurements. The question we want to

Optical window

answer is: Should one make an effort to use a perfectly cylindrical capillary (which is not easy to come by), or is a slightly tapered,

Fig. 3H The electrocapillary electrometer. For each potential the pressure of nitrogen is adjusted to bring the mercury inside the capillary back to the same fixed position.

conical capillary satisfactory? The answer cannot be found in Eq. 53H, which relates the height of the mercury column, or the pressure difference, to the radius. Consider, however, the situation in a perfectly

I:La:MODE KIN Errs

H. ELECIROCAPILLARITY

247

cylindrical tube. If we move the meniscus to a different position, it will stay there, since the radius has not been changed. Thus, the

system is perturbed, there is a force acting in the direction opposite

system is at equilibrium with the mercury meniscus anywhere in the cylindrical tube. What we have is a neutral equilibrium. Its mechani-

rium position. Consider now the same tapered capillary, but placed in the inverted position, namely, wider at the bottom. This represents an unstable

cal equivalent is a sphere on a perfectly flat and horizontal surface, as shown in Fig. 4H(a). Now consider a tapered tube, wider at the top. This represents a stable equilibrium. For a given pressure applied, the mercury is stable

to the direction of the perturbing force, bringing it to a new equilib-

equilibrium. The mechanical equivalent is a sphere on the top of a convex surface, as shown in Fig. 4H(c). It is theoretically possible to

in only one position in the capillary. If the pressure is increased

find an equilibrium position for the sphere, but it will be impossible to maintain it in practice. The slightest disturbance will cause it to

momentarily (or the surface tension is decreased, say, by a fluctuation

roll down. Unstable equilibrium is the result of positive feedback.

in the applied potential), the mercury meniscus will move to a lower

Removing the system from its state of equilibrium creates a force in the

point, where the radius is a little smaller, to establish a new equilibrium, in accordance with Eqs. 51H-53H. The mechanical equivalent of

direction of removing it even further from equilibrium. What is the effect of a small increase in pressure on the mercury in

this configuration is a sphere at the bottom of a concave surface, shown

such a capillary? The meniscus will descend slightly, but in its new

in Fig. 4H(b). Stability is attained by negative feedback. If the

position the radius is larger, so that the same pressure will tend to move it farther down, where the radius is even larger, and so on. Thus, while the general equations allow the existence of an equilibrium position in this situation, it will not be possible, in practice, to

a

I

maintain a stable position of the mercury meniscus in the capillary. Fortunately, the normal manufacturing process of glass tubing produces slightly tapered capillaries, and the problem does not arise, unless one connects the capillary in the electrocapillary electrometer in the wrong direction. Let us return now to the main course of our discussion. It is of interest to consider just how accurately we need to determine 7 versus E. This depends on the purpose of the experiment. For example, if all we need to know is the PZC, within say, ± 5 mV, a simple measurement of drop time, described in Section 18.2 may suffice. In most cases,

Fig. 4H Neutral, stable and unstable equilibria (a, b and c, respective-

however, the purpose is to obtain surface excess, charge density, and

ly) in mechanics and in an electrocapillary electrometer.

even double-layer capacitance, as a function of potential. This task,

248

ELECTRODE KINETICS

which involves differentiating the experimental curve numerically,

249

H. ELECTROCAPILLA R ITY

could be used to trigger an electronic clock.

requires the highest attainable accuracy. The range of values of y in

Initially there were great hopes that the drop-time method would

most cases is 250-426 dyne/cm, and the best measurements recorded claim

provide fast and accurate data on the surface tension in general and on

an accuracy of ± 0.1 dyne/cm, namely about 0.02-0.04 %.

its potential dependence in particular. Closer examination, with the use of fast photography, has shown the process of detachment of a drop

18.2 The Drop-Time Method

from the surface to be rather complex. Just before the drop falls off,

Setting up an electrocapillary electrometer is a long and tedious

it becomes elongated. Eventually it breaks off at a small distance from

process. If one can tolerate lower accuracy, enough to determine Ez and

the capillary, not at the capillary itself. Based on these observa-

its change with concentration of a certain component in solution, or to

tions, it is clear that Eq. 55H cannot be exactly correct, and one may

determine qualitatively whether adsorption occurs, the drop-time method

even wonder why the drop time should depend on surface tension at all.

may be used.

The fact is that a dependence such as shown in Eq. 55H is nevertheless

When liquid is dropping from a capillary, the force holding the

obeyed, to a good approximation. The limited accuracy of the drop-time

drops up (2rtRy) is due to the surface tension, while the force making

method is due to this uncertainty, and the results probably can be

them fall (g•M) is due to gravity. The drop will hang on the tip of the

considered to be accurate to within about 2%.

capillary as long as the force due to surface tension is greater than or 18.3 Integration of the Double-Layer Capacitance

equal to the gravitational force. 2nRy g•M = g•m.t

(54H)

The thermodynamic relationships among surface tension, surface excess, surface charge density, and double-layer capacitance are shown

where R is the radius of the capillary, M is the mass of the drop and in

in Fig. 5H, which follows from Eq. 29H. It is clear that the double-

is the rate of flow of the liquid. If the drop falls off just as the

layer capacitance C

gravitational force exceeds the force generated by the surface tension,

electrocapillary curve, and conversely, that the electrocapillary curve

we can express the drop time ti by:

can be obtained by double integration of the dependence of

ti

=

(21tR/g.m).?

dI

can be obtained by double differentiation of the

on Cdl potential. It cannot be overemphasized that Eq. 29H and the relation-

(55H)

Clearly, the drop-time is proportional to the surface tension. It also depends on the parameters of the capillary, which can be calibrated with

ships derived from it are thermodynamic and do not depend on the model assumed for the double layer, except to the extent that it is assumed to be ideally polarizable.

a known system, to obtain the surface tension as a function of poten-

Which is the preferred approach? Should one obtain the surface

tial. There are many obvious ways to measure the drop time. For

charge density, for example, by differentiating the electrocapillary

example, one could have the falling drop interrupt a beam of light; or

curve or by integrating the differential capacitance curve? Theory does

the sudden change in the charging current just after the drop has fallen

hUsCIRODE KINETICS

1

H. ELECTROCAPILLAR1TY

results obtained by numerical differentiation are more sensitive to experimental error than those obtained by numerical integration. Later several authors reported that by conducting very careful electro-

7

capillary measurements, results comparable in accuracy and reproducibility to those obtained from double-layer capacitance measurements can F

qm

(a/a E),,,

(a/8A) E

be had. This argument was current in the 1960s. Subsequent developments in instrumentation have tipped the scale clearly in favor of Grahame's view. Measurements of the capacitance, using a phase-sensitive volt-

(dlya Op.

C dl A914,) E

/q/aE)m

(dcdi /dy.

,

meter (alias lock-in amplifier), can be conducted with very high accuracy, much faster and more conveniently than by the bridge ,method employed by Grahame and his contemporaries. It can be said that the accuracy of such measurements is no longer limited by instrumentation

) e

and depends on the chemistry of the system, mostly its purity. Moreover, although computer-controlled electrocapillary electrometers have

Fig. 5H Thermodynamic relationships between surface tension, the chemical potential and the applied potential. Note that the partial derivatives relating y to m , F. and Cal are purely thermodynamic, and do not depend on a model. not yield the answer, and one must resort to experimental considerations, which may change over the years, as new instrumentation is developed and becomes commercially available. D. C. Grahame, who is revered by electrochemists even today for his

been developed and built in a few laboratories, these have remained rather delicate research instruments, far from being commercially available. There is one important point in favor of making electrocapillary measurements. Consider the relationship between Cal and y given in Eq. 8H.

— (a2y1aE2)

=(aelliaE)[ti = Cat

(8H)

To obtain 'y from Cal , we must integrate twice:

meticulous measurements and detailed and careful analysis of the data, believed that starting from double-layer-capacitance data yields more accurate results. He based his argument on the fact that capacitance can be measured on the dropping mercury electrode, with its periodically renewed surface, whereas electrocapillary measurements are taken on a stationary interphase, which is more prone to contamination. Also,

f

C

dl

dE = q(E)

f qmdE = f — Ez

dy = y(n) — ymax

yma x

(56H)

(57H)

252

ELECTRODE KINETICS

253

H. ELECTROCAPILLARITY

This requires the knowledge of two integration constants: the potential of zero charge E and the value of the surface tension at the PZC. It Thus, the first integration can is relatively easy to measure E .

420

NaF

0.1M o c

electrocapillary measurements. But if one is setting up an electro-

0

capillary electrometer to measure y ax anyway, should one not use it to

2 400

obtain the whole electrocapillary curve? Still, these are long and

H

determined over only a narrow range of potential, in the vicinity of the

C

PZC. Alternatively, it may even be sufficient to determine 'max by the drop-time method, using this value to integrate the capacitance data.

O. 0 I M

I

I

0

and much faster from capacitance measurements, with the surface tension

/

/

0

tedious experiments, and the same data can be obtained more accurately

18.4 Some Experimental Results

\

a,

readily be performed, in many cases even on solid electrodes. The surface tension is more elusive. The best method to obtain y r.. is from

380 0.2

-0.2

-0.6

-1.0 Potentiol/V vs NCE

-I 4

Fig. 6H Electrocapillary curves for two concentrations of NaF. Data from Grahame and Soderberg, Tech. Rep. # 14, ONR, 1954.

The thermodynamic derivations and the experimental procedures for determining the electrocapillary curve were given in Sections 18.1-18.3.

Distinctly different behavior is shown in Fig. 7H for KBr. The PZC

Here we shall describe some of the experimental results obtained for mercury.

is shifted in the cathodic direction as the concentration of the electrolyte increases. The anion is strongly adsorbed, as can be seen

(a) The adsorption of ions

from the depression of the value of 'y with increasing concentration, at

Figure 6H shows electrocapillary curves obtained in NaF solutions The most remarkable feature of these at different concentrations. curves is that both E and 'ymax are practically independent of the concentration of the electrolyte. Also, there is no detectable adsorption of the anion at negative potentials or of the cation at positive

fixed potential, which is observed even on part of the negative branch of the electrocapillary curve. A case of specific adsorption of the cation is shown in Fig. 8H,

potentials. This observation led Grahame and others to believe that

for solutions of T1NO 3 , in a supporting electrolyte consisting of 1.0 M KNO and 0.01 M HNO . The PZC shifts in the anodic direction, showing 3 3 that specific adsorption of the cation is predominant. The values of

neither of these ions is specifically adsorbed at the mercury-

the

electrolyte interphase.

implying high values of the relative surface excess thalium,

surface tension are substantially lowered at all potentials, .

_LD4

ry curves for different concentrations of KBr. Data from Lawrence, Parsons and Payne, J. Electroanal. Chem. 16, 193, (1968).

In ter fac ia lTens ion /dy n cm - 1

Fig. 711 Electrocapilla-

ELECTRODE KINETICS

255

H. ELECTROCAP ILLAR rY

Figure 9H compares the electrocapillary curves obtained at a fixed 420

concentration of six electrolytes; the value of the PZC is that obtained in the presence of NaF, which, as we have noted, is not

400

specifically adsorbed. The shift in the PZC in the negative direction 380

and the extent of change of the surface tension are in the same order,

360

indicating that specific adsorption is increasing in strength in the order OH < NO 3 < Cl < CNS < Br < I. The concept of specific

340

adsorption can be seen very clearly in this figure. For example, adsorption of iodide is observed up to a potential of about — 0.35 V

320 0

-0.4

-0.8

-1.2

-16

Potential/V vs NCE - 420

E 420 U

Fig. 8H Electrocapillary curves for solutions of TWO in 1.0M KNO 3 3 and 0.01M HNO 3 . Data from Frumkin, Trans. Sym. Electrode Kine-

0 o c 400

0.8 H

0

-0.4 -0.8

-1.2

E- E z /V

— 0 380

Fig. 9H Electrocapillary curves taken at 18°C for different electrolytes at a fixed concentration of 0.1 M, showing the relative

0

tics, Yeager, Ed. Wiley, pp. 1-12, (1961).

0.4

strength of specific adsorption. The potential of zero charge

360

-0.1

-0.5

-0.9

Potential/V vs NCE

is that measured for NaF. Reprinted with permission from Grahame, Chem. Rev. 41, 441. Copyright 1947, the American Chemical Society.

256

ELECTRODE KINETICS

257

ELECTROCAPILLAR ITY

versus E in the same solution (which is about — 0.6 V versus E in NaF). The bromide ion is no longer adsorbed when the potential is about — 0.25 V versus Ez in the same solution. Adsorption of an anion at a negative rational potential is a clear indication that "chemical" or some other "nonelectrostatic" interactions are involved. The negative potential needed to drive an anion out of the interphase is a quantitative measure of the strength of the chemical bond holding it there.

different anions as a function of potential. All anions shown (except

A more direct way of comparing the strength of specific adsorption

taken in a solution of NaF. Looking at Fig. 1111 we note that adsorption

F) are specifically adsorbed at the potential of zero charge. (b) Adsorption of neutral molecules The adsorption of neutral molecules is discussed in detail in Chapter J. Here we show electrocapillary curves obtained in the presence of n-butanol and compare them to an electrocapillary curve

is shown in Fig. 10H, which displays the relative surface excess for 430

350

0.0

-0.4

-0.8

-1.2

Potential/V vs RCE

Fig. 10H Relative surface excess of different anions, multiplied by zF, as a function of potential, in O.IN solutions of their respective salts, at 25°C. Reprinted with permission from Grahame and Soderberg, J. Chem. Phys., 22, 449, (1954). Copyright 1954, the American Institute of Physics.

Fig. 1111 Electrocapillary curve in the presence of different concentrations of n-butanol in 0.1 M HCl. Muller, Ph.D. Dissertation, Fig. 37, University of Pennsylvania, 1965.

258

ELECTRODE KINETICS

occurs only in the vicinity of E , and the organic substance seems to be

259

H. ELECTROCAPILLARrry

The corresponding plots of the relative surface excess as a

of the relative surface excess is found, for example, for benzene and

function of potential appear in Fig. 12H. We note that the potential of maximum adsorption E(F ) (which is usually denoted E max , for simplicity) is near E , but does not coincide with it. In fact, E max is at a potential of about — 0.1 V on the rational scale, which corresponds to a surface charge density q m of about — 2 1..1C/cm 2 . The potential of

phenol, although the former has no permanent dipole moment whereas the

maximum adsorption is also found to be the same for quite different

dipole moment of phenol is substantial. Results similar to those shown

molecules, strengthening the view that it is a property of the solvent,

in Fig. 11H were obtained for valeronitrile (C 4H9CN), although its

rather than that of the solute, as will be discussed in Chapter J.

"pushed out of the interphase at both negative and positive rational potentials. This behavior is not a property of the specific compound shown in Fig. 11H, nor can it be related to the structure of the adsorbed molecule or its dipole moment. The same potential dependence

dipole moment is quite different from that of n-butanol. It is logical, then, to conclude that the observed behavior is primarily a property of 0.5 M

the solvent. The adsorption of an organic molecule requires the removal of a certain number of water molecules (depending on the size of the

0 N

molecule being adsorbed), and it is the effect of potential on the

E

energy required to remove these water molecules that determines,

a)

0.IM

or

0 0

appropriate number of water molecules from the interphase is called electrosorption.

4 0.05M o......,o—...0........0

cn

Other interesting features of the curves plotted in Fig. 11H are

a)

the small dependence of y on potential in the presence of the strongly

1.1.1

adsorbed organic molecules (at potentials close to

E ),

and the

/

o

\

a) U 0

relatively sudden convergence of the curves in the presence and in the

o

absence of these molecules at high positive and negative rational

c/

o o

0.0IM o_o__.

double-layer capacitance in the region of adsorption, and to two sharp, so-called adsorption—desorption peaks in the plot of are shown in Fig. 2G(a).

Cdi

versus E, which

° 0--- 0

00 N .....,2,... \ 0

mo o- I

potentials, corresponding to a sudden transition from strong adsorption to practically no adsorption. These effects lead to low values of the

ono

vo

5

indirectly, the potential dependence of adsorption of neutral species. The adsorption of a neutral molecule and the simultaneous removal of an

6

-0.9

-0.5

I

-0.1

Potential / V vs NCE Fig. 1211 The relative surface excess of the electrosorption of tzbutanol, as a function of potential. Data from Blotngren, Bockris and Jesch. J. Phys. Chem. 65, 2004, (1961).

261

L LNTERMEDIATES IN ELECTRODE KINETICS

PART TWO I. INTERMEDIATES IN ELECTRODE KINETICS 19. ADSORPTION ISOTHERMS FOR INTERMEDIATES FORMED BY CHARGE TRANSFER 19.1 The Langmuir Isotherm and its Limitations The Langmuir Isothermis is written in chemistry in the form 0 I—0

— KC

where 0 is the fractional surface coverage, C is the bulk concentration of the species being adsorbed (if adsorption from the gas phase is considered, C is replaced by the partial pressure of the appropriate species), and K is the equilibrium constant for adsorption. The standard free energy of adsorption is related to the equilibrium constant as in every chemical equilibrium, namely: 2.3RTIogK = — AG: ds

(2I)

Before we start our discussion of the application of this and other isotherms to electrochemistry, let us consider the conditions under which it was derived and the resulting limitations in its applicability. I.

The Langmuir isotherm is applicable only to monolayer adsorption.

This seems obvious when the isotherm is discussed in the context of adsorption of intermediates in electrode kinetics, but it should be emphasized here, following a chapter in which we have discussed adsorption in general, in terms of the surface excess F rather than the partial coverage 0. 2. It is assumed that the standard free energy of adsorption is independent of coverage.

This is a severe limitation!

For this

2b2

ELECTRODE KINETICS

263

L INTERMEDIATES IN ELECTRODE KINETICS

condition to apply, we must have (a) a completely homogeneous surface

distance in the bulk of the solution. We have seen that the number -of

and (b) no lateral interaction between the adsorbed species. Solid

metal atoms per unit surface area is about 1.3x10 15 (or 0.22 mC/cm if

surfaces are rarely homogeneous. There are active sites on which the standard free energy of adsorption is high, and there are less active

each atom carries a unit electronic charge). This corresponds to a square array of (3.6x10 7)x(3.6x107) atoms/cm 2 or an average distance of

sites. As the fractional surface coverage increases, the most active

0.28 nm between atoms. The equivalent bulk concentration is

sites are occupied first and the least active ones are occupied last.

(3.6x107)3/6x102° = 78 M, which is evidently much above anything that

(This is much like people entering a movie theater with unmarked seats:

can be observed experimentally. Thus, we find that the molecules are

the best seats are taken first, and the worst remain for late-comers).

very crowded on the surface. A relatively low fractional coverage of,

In this case the standard free energy of adsorption will decrease with increasing 0.

say, 0 = 0.1, is equivalent to a bulk concentration of about 2.5 M, in

The purists among us may object to the last statement. A standard free energy is defined for some standard state, and it cannot, from a formal point of view, be said to depend on any variable. Yet, if we

which interactions between solute molecules are expected to be the rule, rather than the exception. 3. The surface is assumed to have well-defined sites, with each molecule of the adsorbate occupying a single site, and each site being

think of a heterogeneous surface as consisting of a very large number of

able to accommodate only a single adsorbate molecule.

minute homogeneous patches, we would have a different standard free

4.

energy for each little patch, and the experimentally observed standard

surface is assumed. Considering the foregoing limitations, it is perhaps surprising

free energy of adsorption is, in effect, a function of coverage. To

Finally, equilibrium between the species in the bulk and on the

overcome this problem, it is customary to talk about the variation of the apparent standard free energy of adsorption with coverage. This

that the Langmuir isotherm is ever found to apply experimentally. The

distinction does not change the ensuing arguments, but it helps to

between physically adsorbed molecular hydrogen and chemically adsorbed

soothe our scientific conscience. (Remember the case of

atomic hydrogen. For the former one writes

quasi-

fact is that it does. One of its early triumphs was in distinguishing

equilibrium?)

For brevity, we shall drop the word "apparent" therefore in the discussion that follows. What about lateral interactions? These depend on the nature of the adsorbed species and on their average distance apart. Ion-ion interac-

H+M 2

M—H

2

(31)

which should follow Eq. 1I, with C replaced by Po-1 2),, For the adsorption of atomic hydrogen the appropriate equilibrium is

tions are long range, since they decay with the first power of the distance. Dipole dipole interactions decay with r -3 , while chemical interactions decay with r -6 , and are expected to be felt only at higher values of the coverage. Now, it is interesting to estimate the average distance between molecules on the surface and compare it to the average

II

2

+

2M

M—H + M—H

(41)

In this case the rate of adsorption is proportional to (1 — 0) 2 , and the rate of desorption is proportional to 0 2 , since we consider empty and

ELECTRODE KINETICS

The

a Frac tiona l cover ag e/e

occupied sites to he reactants for the respective reactions. corresponding isotherm is: 2 [1

0

= K•P(H ) 2

(51)

Thus, at low coverage, dissociative adsorption leads to a dependence of on the square root of the partial pressure of hydrogen, while adsorption of molecular hydrogen leads to a linear dependence of 0 on the pressure. The adsorption of molecular hydrogen was studied in this manner on catalytic metals, such as platinum, palladium and nickel and found to be dissociative, following Eq. 51. If a larger molecule, taking up n sites on the surface, is being adsorbed, we could follow the same kind of reasoning and write:

0

= KC

Fig. 11 The Langmuir isotherm. Plots of the fractional coverage 0

(61)

This is a better approximation than Eq. II to describe the adsorption of

log C, for three values of the equilibrium constant K = 1, 5 and 25.

larger molecules. Consider the application of the Langmuir isotherm to electrochemistry. For a simple charge transfer process, such as the formation of an adsorbed chlorine atom, we write

0.8 0.6 0.4 0.2

0 0

0.4

0.8

1.2

1.6

2.0

C/ M M

b E/mV vs arbitrary reference 0

60

120

-3

-2

-1

1.0

180

240

300

0.8 0.6 0.4 0.2 0

log C/mM

(71)

The corresponding isotherm, as we have shown before, is

1.0

versus (a) concentration and (b) potential or

(1 — 0)"

Cl + M < M—Cl + e m

265

1. INTERMEDIATES IN ELECA RODE KINETICS

Frac tio na l co verag e/ e

264

as the Langmuir isotherm, as just discussed. Plots of 0 versus C and versus logC (or E) are shown in Fig. II, for three values of the equilibrium constant K.

[ 1 —0

0 = KC•exp(EF/RT)

(17F)

in which C is the concentration of Cl ions. This is the Langmuir isotherm, as applied to electrochemistry. It has the same limitations

Increasing the equilibrium constant causes a change in the shape of the plot of 0 versus C whereas it causes only a parallel shift in the plot of 0 versus logC.. If 0 is plotted versus E, the same plot is obtained as for 0 versus logC. Changing the potential by 2.3RT/F has the same effect on 0 as changing the concentration by an order of magnitude.

ELECTRODE KINETICS

266

It may be recalled (cf. Eq. 16F and Eq. 22F) that we have used the

we replace AG: ds in Eq. 21 by AGo° and substitute from Eq. 81, to obtain

Langmuir isotherm in electrode kinetics only under the special conditions of 0 approaching either zero or unity. This relieves us of the most difficult assumption, namely, that the standard free energy of adsorption is independent of coverage. Thus, even if AG ° changes

267

1. INTERMEDIATES IN ELECTRODE KINETICS

2.3RTlogK = — AG8 = — (AGE + r0)

(10I)

which can also be written as:

significantly with 0, this change can be considered to be negligible

K = exp

(AG°0 + r0)/RT] = Ko.exp(— rO/RT)

(III)

when 0 changes from, say, lx10 -5 to lx10-3 , or from 0.99900 to 0.99999. It is not surprising, then, that the Langmuir isotherm, in spite of its

Substituting this value of K into Eq. 19F we have

many limitations, is often applicable in electrode kinetics, albeit at extreme values of 0. There are, however, cases in which it is not applicable, as discussed in Section 19.2. 19.2 The Frumkin and Temkin Isotherms

- 0 exp(rO/RT) [ 1 -(3

K o C.exp(EF/RT)

This equation is known as the Frumkin isotherm.

The assumption that the standard free energy of adsorption is independent of coverage may be viewed as the "zeroth-order" approximation. The first-order approximation will then be a linear dependence of

(120

It is clear that the

Langmuir isotherm is a special case of the Frumkin isotherm, which can be derived from it by setting r = 0. It can also be seen that, for reasonable values of the parameter r, the exponential term in this equation approaches unity for very small values of 0 and becomes

AG° on 0:

constant when 0 is close to unity. Thus, at extreme values of 0, the AG ° = AG°o + r0

(8I)

where r is the rate of change of free energy of adsorption with coverage. Both AGo° and AG°0 are negative numbers, hence Eq. SI, which can

Frumkin and the Langmuir isotherms lead to the same dependence of coverage on potential, hence to the same rate equations in electrode kinetics. What is a "reasonable" value of the parameter r? In adsorption of

also be written in the following form:

hydrogen on nickel from the gas phase, values as high as 150 kJ/mol have I AG; = I AG °0 I —r0

(91)

been observed. In electrochemistry, both the absolute value of AG ° and its variation with 0 are smaller, for reasons to be discussed later. A

represents a decrease in the absolute value of the standard free energy of adsorption (i.e., a decrease in the corresponding equilibrium constant). Keeping in mind the relationship between standard free energy and the equilibrium constant, it. is very easy to modify the Langmuir isotherm to take into account the variation of AG ° with 0. To do this,

range of 20-60 kJ/mol includes most values reported in the literature and can be considered to be "reasonable." The dependence of 0 on potential is shown in Fig. 21 for three values of the parameter r, and for the case of r = 0, namely for the Langmuir isotherm.

tl

268

-200

0

200

400

600

Potential/mV vs E c

e '

269

INTERMEDIATES IN ELECTRODE KINETICS

Fr ac tiona l c o verag e/e

Frac tiona l coverag e/0

ELEC. I RODE KINETICS

10 -2

10 -3

-200

0

200

400

Potential/mV vs E °0

Fig. 21 The Frumkin isotherm. Dependence of 0 on E for different values of the parameter f = r/RT. (f = 0 corresponds to the Langmuir isotherm).

Fig. 31 The Frumkin isotherm, plotted as log0 versus E, for different Negative values of f correspond to a values of f = r/RT. lateral attraction interaction, (i.e., to an increase of IAGo° 1

A different way of presenting the same data is shown in Fig. 31,

with coverage.) The standard potential for adsorption, Ec°, is defined as the potential where [0/(1 — 0)] = 0 for r = 0 and

where log0 is plotted versus E. The advantage of this rather uncommon

unit concentration in solution.

way of showing the isotherms is that the data for very low values of 0 are also seen. In this form of presentation it is clear that at such

constant for adsorption with increasing coverage. This can account for

values of 0 the isotherm becomes independent of the parameter r.

two-dimensional phase formation, as we shall see.

In Fig. 31 we show positive as well as negative values of the parameter r.

What can be learned from Fig. 21? First we note that the frac-

What is the physical meaning of r < 0, namely of an

tional coverage increases more slowly with potential as the value of r

increase in the absolute value of the standard free energy of adsorption with coverage? We have seen that a positive value of r can be due

is increased. It takes only 0.11 V to change the coverage from 0.1 to 0.9 when f = 0. The same increase in 0 occurs over about 0.52 V when

either to a surface inhomogeneity or to lateral repulsion interactions.

f = 20. Second, we note that the fractional surface coverage is a

A negative value of the same parameter hence must correspond to lateral

linear function of the potential at intermediate values of 0, and the linear region increases with increasing value of the parameter f. This

attraction interactions, which lead to an increase of the equilibrium

behavior can be shown to follow from Eq. 121. At intermediate values of

270

ELECT RODE KINETICS

271

L INTERMEDIATES IN ELECTRODE KINETICS

the coverage, the preexponential term 0/(1 — 0) varies little with 0

than predicted from the foregoing argument, since even for the Langmuir

compared to the variation of the exponential term exp(r0/RT). Taking

isotherm, there is a short region in which 0 is nearly linear with E.

0/(1 — 0) = I, we can write the Frumkin isotherm in approximate form:

This is shown in Fig. 41 for two values of the parameter f.

(131)

exp(rO/RT) = K oC•exp(EF/RT)

The slope

dO/dE is not equal to F/r, as expected from Eq. 141, but the relationship is nonetheless linear.

or in logarithmic form:

Disregarding for a moment the electrochemical aspect of this (141)

0 = (2 .3 RT/r) log (K oC) + (F/r)E

isotherm, we note that 0 is proportional to logC, (as opposed to the Langmuir isotherm, where it is proportional to a linear function of the

This equation predicts the linear dependence of 0 on E, which is shown

concentration.) A similar "logarithmic isotherm" was developed by

Actually, the linear relationship turns out to be better

Temkin. His derivation is much more complex, but in the final analysis

in Fig. 21.

it is based on the same physical assumptions. It has, therefore, become common to refer to Eq. 141 as the Temkin isotherm, although Temkin has never used it in this form. It is this approximate form of the Frumkin

1.0

Frac tion a l c o verag e/ 0

isotherm which is applied to electrode kinetics, as we shall see below. 0.8

19.3 Introducing the Temkin Isotherm into the Equations of Electrode Kinetics

0.6

We noted earlier that the equilibrium constant is an exponential function of coverage, when adsorption follows the Frumkin isotherm:

0.4

K = exp (AG °0 + r0)/RT1 = K oexp(— rO/RT)

0.2

0 —200

(11I)

We can write similar relationships for the rate constant in a reaction 0

200

400

600

involving the formation or removal of an adsorbed intermediate, such as:

Potential/mV vs E

k k

Fig. 41 The Frumkin isotherm. Plots of 0 versus E showing the effect of the preexponential term in 0 on the slope dOidE, at intermediate

+

e

(8F)

M

-I

These will have the form

coverage. Full lines are based on the complete isotherm, k = exp 1

whereas dashed lines are drawn with a slope of (Fir), totally neglecting the preexponential term in Eq. 121.

Clads

and

, -› (AG" g+ (3rO)/RT} = k1

0

exp(— [3r0/RT)

(151)

272

k

ELEC 1 RODE KINETICS

-1

= expf— [AG°# — (1—(3)r0)/RT} =

- o

exp[(1-13)rO/RT]

[

(161)

How have we arrived at these equations? The argument follows the

273

INTERMEDIATES IN ELECI tODE KINETICS

0 lexp(rO/RT) = K oC.exp(EF/RT)

(121)

derived earlier from thermodynamic arguments.

reasoning that leads to the potential dependence of the rate constant.

It may be asked whether the symmetry factor in Eq. 171 is identical

If the standard free energy of the reaction is changed by rO, the

to the symmetry factor in Eq. 7D, as we have implied in our derivation. It is not unreasonable to make this assumption, since both depend on the

standard free energy of activation should be changed by a fraction of intermediate state between the initial and the final states. The

position of the same activated complex along the reaction coordinate, although it may be argued that the variation of the standard free energy

symmetry factor f3 was defined earlier as:

with potential and with coverage along this coordinate is not the same,

that quantity, namely by 13•r0, since the activated complex represents an

and that this leads to two different symmetry factors. Fortunately, it = [(SAG°"/84)]/[(SAG °/154)] = 8,6,0 °#/5.6,6°

(7D)

turns out that this does not affect the rate equations, as we shall show.

In the present context it can similarly be defined as

19.4 Calculating the Tafel Slopes and Reaction 13 = [(8A 6 - °#/r50)1/[(84W/r80)] = SAG"/84G °

(171)

If changing the potential increases the rate of charge transfer in one

Orders Under Temkin Conditions. Consider the reaction sequence

direction, it must retard it in the opposite direction. Similarly, if

k

k

adsorbed intermediate, its rate must decrease with increasing coverage, as shown in Eq. 151. If an adsorbed species is desorbed in the process, the rate will increase with increasing coverage, as represented by

Cl + e M

CI

the step considered in the reaction sequence entails the formation of an

ads

-I

followed by

+ads rd s

+ e

M

(201)

Eq. 161. Introducing the values of

and k 1 into the rate equation, and

which proceeds at intermediate values of the coverage, under Temkin conditions. The rate equation is:

assuming that the reaction is at equilibrium, we have

i 2 = Fk 2 C CI- 0•exp((3r0/RT)exp((3EF/RT)

k 1C, 0 (1cI— 0).exp(— (3rO/RT)exp((3EF/RT)

(21I)

We neglect the preexponential term in 0 and substitute for the exponenk 1,0 0.exp[(1 — 13),-0/RTlexp[— (1 — ()EF/RT) which leads to the Frumkin isotherm

(181)

tial term from the Temkin isotherm (i.e., the Frumkin isotherm for intermediate values of the coverage). For the first step at quasi equilibrium this leads to:

274

exp(rO/RT) = K i Cci:exp(EF/RT)

independent of and must be exactly 29.5 mV at room temperature. More important, the reaction order p 1 = 213 given in Eq. 271 does not have to be exactly unity. It is important to realize that an experimentally

(131) 1

The total current is given, under these conditions, as follows: i = 2Fk2 K 131 .Cc(11 +13) .exp [63 + (3)EF/RT]

275

I. INTERMEDIATES IN ELECTRODE KINETICS

ELEL i ti 0 DE KINETICS

observed value of, say, 0.90, could be the correct value and should not

(221)

be considered to be "effectively unity." This is a rather important point, since fractional reaction orders are characteristic of electrode

The corresponding Tafel slope and reaction order are

reactions occurring under Temkin conditions, at intermediate values of b = 2.3RT/(13 +13)F and p 1 = (alog i/alogCa ) E = (I +

(231) 1

We note that the exponential term in r0 in Eq. 211 is positive, since this is a desorption reaction. If instead of Eq. 201, the rate-determining step is assumed to be 2C1 —3—> Cl r ds

the coverage. The Tafel slopes and reaction orders for these two rate-determining steps at low, intermediate and high values of the fractional coverage are summarized in Table II. We need not consider here the case in which the first charge transfer is rate determining, since this tends to lead to low values of the coverage, as shown

k

ads

.

2

the rate equation will be:

(241) earlier. The derivation of the Tafel slope and reaction order can become more difficult when complex reactions are involved. If we consider a

I

3

=

k 02 .exp(2(3r0/RT) 3

(251)

Neglecting the preexponential term, and substituting for the exponential term in r0 from Eq. 131, we have in this case:

step in the oxygen evolution reaction, such as k

OH— + IC:C ads

2>ads +

H2O + e M

(65F)

the rate equation, under Temkin conditions, is i=

K211.C2 I32Fk •exp(213EF/RT)

31

CI

(261)

-

b = 2.3RT/2I3F and P t = (alog WalogCct ) E = 213 We could have substituted

C i2 = k__

2 UH e0H.

which yields:

13 =

(271)

0.5 in these equations, to obtain a

Tafel slope of b = 2.3RT/F for both mechanisms, as we did earlier. It

exp ((3EF/RT)exp [13(1-1 — r2)0./W11

(281)

where r and r2 refer to the rate of change of the standard free energy of adsorption with coverage of the respective species and OT is the total coverage by all types of adsorbed intermediates. If the parameter r is due to surface inhomogeneity, one would expect to have r 1 = r2,

is appropriate not to do that at this point, to emphasize that the Tafel

since the effect is inherent to the surface and does not depend on the

slope does not have to be exactly 59 mV at room temperature in this

species being adsorbed. In this case the exponential term in re would

case. In contrast, the value of b = 2.3RT/2F calculated earlier for step

disappear from the equation for step 281, although it would still affect

(241) as the rate-determining step at limitingly low coverage is

the dependence of 0 on potential in a preceding step at quasi-

• 276

ELECTRODE KINETICS

277

I INTERMEDIATES IN ELECT RODE KINETICS

If r is due to lateral interactions, then r 1 and r2 could be quite different. A detailed discussion of such cases would take us equilibrium.

into the realm of research and is clearly outside the scope of this book.

800

f = -F

o

> Table II Tafel Slopes and Reaction Orders Calculated for Two Different Mechanisms, Assuming Low, Intermediate, and High Coverage Eq.a

Parameter b

201 p1

0

0

RT/(1 + (3)F 2

0.2 < 0 < 0.8 RT/i3217 1+

10

0

20

400

E

O C

0 ----> 1

CI ads + e m

CI

0 a.

2C1 ads

RT/1317 —400 — 12

1

—8

—4

0

4

rds

CI 2

8

12

log i

b

RT/2F

P1

2

241

RT/213F 2

00

Fig. 51 Tafel plots calculated from the complete rate equation for different values of f. The curves for f = 10 and f = 0 are

3

displaced vertically by 200 mV each, to avoid overlap. There is a transition region, extending over two decades of current

a Equation number for the rate determining step

density or more, where the slope changes from (2.3RT/2F) for 19.5 Some Special Aspects of the Use of the Temkin Isotherm in Electrode Kinetics

low coverage to (2.3RT/F) for intermeidate values of the coverage.

In this section we shall discuss a few special aspects of the application of the Temkin isotherm to electrode kinetics.

coverage on the parameter f.

In the linear regions the lines are

parallel, however, independent of f, as expected. The transition from

(a) The transition from Langmuir to Temkin conditions

one linear Tafel region to the next occurs over two orders of magnitude

In Fig. 51 we show calculated Tafel lines for reaction 191 at

in current density in all cases.

quasi-equilibrium followed by reaction 201 as the rate-determining step.

(b) Nonlinearity of the dependence of AG ° on 0

All lines were calculated for 0 ranging from 1x10 -3 to (1 — 0) = 10 -3 fromtheulqain,woegcthprxonialems 0. The value of f used to calculate each line is shown. Note the dependence of the region of linearity of the Tafel lines at intermediate

An apparent weakness of the foregoing treatment is the assumed linear dependence of AG °d, on 0. While there is ample experimental evidence showing that the standard free energy of adsorption depends on

278

ELECT RODE KINETICS

coverage, this dependence is not shown experimentally to be exactly linear in most cases. Surface heterogeneity does not necessarily

1. INTERMEDIATES IN ELECTRODE KINETICS

'2,

permits us to neglect the preexponential terms in 0 in comparison with the exponential term, as required in the foregoing derivation.

produce a linear dependence, and lateral interactions can influence the dependence of AG o° on 0 in different ways, depending on the nature of the

(c) Equality of the symmetry factors

interaction. It is of interest, therefore, to consider the effect of a

Earlier we suggested the possibility that the symmetry factor used in relation to the variation of the standard free energy of adsorption

departure from such linearity on the kinetic equations. To do this, we

with coverage (cf. Eq. 171) may not be identical to the symmetry factor defined in terms of the variation of AG ° with potential (Eq. 5D). To

shall replace our first order approximation AG ° = AG° + r0 0 o

(8I)

see the consequence of a difference between the two, let us introduce for the moment a different symmetry factor, fp in Eq. 171. The rate

AG ° = AG° + rOn 0 o

(291)

equation for the atom-ion recombination step (Eq. 211) will be written as follows:

by the equation

in which n is a small number, between 0.5 and 2. We now repeat the derivation of one of the preceding rate equations, using this, instead of the linear relationship. For quasi-equilibrium in Eq. 191 we have, instead of Eq. 131: ex p (rOn/RT) = K i Cci ex p (EF/RT)

i

2

(301)

and for the rate of step 201 we have, instead of Eq. 211: 2

=

Fk

2

C

CI-

. exp

(PrO n/RT)exp(13EF/RT)

(321)

i = 2Fk 03 6 4') •exp[(13' + (3)EF/RT] 2 I C1

(331)

The numerical values of the Tafel slope and reaction order will depend (311)

on the value of 1:1' (just as they both depend on the value of f3), but the form of the rate equations is not changed. The same is true for any

Substituting rO n from Eq. 301 into 311 we obtain Eq. 221 as before: i = 2Fk21q•Cc(11 +13) .exp [03 + (3)EF/RT)

Fk C 0. exp(f3 ' rO/RT)expa3EF/RT) 2 CI-

and the resulting dependence of i on E for the overall reaction will be given by: -

i

=

mechanism, and the use of the same symmetry factor in Eq. 5D and Eq. 171 (221)

Thus, we reach the (happy) conclusion that the kinetic equations are independent of the shape of the variation of AG o° with 0, making the

does not restrict the validity or generality of the rate equations derived under Temkin conditions. (d) The likelihood of reactions occurring under Temkin conditions

treatment of electrode kinetics in terms of the Temkin isotherm much more general. In fact; we could have used any unspecified function f(0)

When we derived the kinetic equations for Langmuir conditions, at extreme values of the coverage (0 < 0.1 or 0 > 0.9), we raised the question of the probability of finding reactions for which the frac-

in Eq. 291. The result the same, as long as the function chosen

tional coverage will be in this extreme range. If the Langmuir isotherm

280

ELEC I RODE KINETICS

281

INTERMEDIATES IN ELECTRODE KINETICS

is applicable, the coverage changes from 0.1 to 0.9 in 0.11 Volt. Since the width of the voltage window available for measurement is of the BULK

order of 2-3 V, we estimated that there was a 95% probability that 0

-80-

would have an extreme value, for which the treatment according to the Langmuir isotherm is valid. If the Frumkin isotherm is applicable, the

-60 UPD -2

change of 0 from 0.1 to 0.9 occurs over a much wider range of potential, depending on the numerical value of the parameter f.

For f = 20 this

"` -40

corresponds to about 0.5 V. In this case, there is about a 17% chance for the coverage to be in the intermediate region, where the Temkin

z -20

isotherm applies. While this likelihood is significantly lower than

I—

that estimated for Langmuir conditions, the probability of observing

w

reactions occurring at intermediate values of the fractional surface coverage, in the Temkin region, is by no means negligible.

UPD-I

0

z ct it

0

20 40 0.4

I

0.2

19.6 Underpotential Deposition (a) Underpotential deposition of metals

When lead is placed in a solution containing lead ions (e.g., a Pb(C10

/HC10 solution), the potential measured with respect to a 4 )2 4 reference electrode is the reversible potential of the Pb 2+/Pb couple in that solution. If the potential is swept in the negative (cathodic) direction in a stirred solution, the current first rises and then reaches its mass-transport-limited value. If the lead electrode is replaced by a different metal, say gold, cathodic currents, corresponding to lead deposition, are observed at potentials that can be several hundred millivolts positive to the reversible potential, as shown in Fig. 61. The area under the peaks represents a charge of about 0.5 mC/cm 2 , corresponding to the deposition of a single monolayer of lead on gold. After this amount of lead has been deposited, the current decays to zero, and bulk deposition of the metal does not start until

I

I

1

-0.4 -0.2 0.0 POTENTIAL/volt vs SCE

I -0.6

Fig. 61 Underpotential deposition of lead on gold from a solution of in I .0 M 1100 . v = 10 mV/sec. Two underpo4 4 )2 tential (UPD) deposition and dissolution peaks are shown. Data 1.0 mM Pb(C10

from Deakin and Melroy, J. Electroanal. Chem. 239, 321 (1988). the reversible potential has been exceeded (i.e., until a negative overpotential required for metal deposition has been reached). This phenomenon, which is often observed when one metal is deposited on a different metal substrate, is called widerpotential deposition. This observation would at first seem to defy thermodynamics. To make an electrochemical reaction occur, one has to apply an overpotential. If a cathodic reaction such as metal deposition is involved, the overpotential should be negative (i.e., the potential applied should be negative with respect to the reversible potential). In Fig. 61, metal deposition is seen to occur at a potential positive with respect to the

ELECTRODE KINETICS

1. INTERMEDIATE.%

GOJ

reversible potential, at an underpotential instead of an overpotential.

crystal metal substrates.

A closer look at the reactions represented by the curves shown in

deposition peaks occur at different potentials on different crystal

Fig. 61 reveals that there is no contradiction with thermodynamics, of

faces of the same metal, which also possess different work functions and

course.

different values of the potential of zero charge.

The reversible potential for lead deposition refers to the free

As might be expected, the underpotential

It would be tempting to claim that underpotential deposition forms a uniform monolayer of, say, lead, covering the gold substrate and

energy change in the reaction

making it behave like bulk lead. Unfortunately, this is not borne out Pb2f

2e

Pb—Pb

ro

(341)

crys

where we write Pb—PbClyS to imply that the lead ion is discharged and becomes part of the lead crystal lattice. The free-energy change and the corresponding reversible potential refer to the formation of a bond between the deposited lead atom and atoms of the same metal on the surface.

ni

+ Au

crys

Pb—Au

deposited metal. In fact, it is usually less than that expected for a monolayer. Second, a metal covered with an underpotential-deposited layer of a second metal does not behave electrochemically as the second metal. Bulk deposition of lead on gold (which already has an underthan on a lead substrate. The difference, referred to as the crystalli-

(351)

crys

charge under the peak does not always correspond to a monolayer of the

potential deposited layer of lead) starts at a more negative potential

Underpotential deposition can be represented by the reaction Pb2t + 2e

by experiment, nor is it expected on theoretical grounds. First, the

zation overpotential, is related to the difference in crystal lattice dimensions, or perhaps to the difference in surface free energy of gold

which is a different reaction altogether and has, therefore, a different

and lead. Either way, a single layer of lead underpotential-deposited

reversible potential. The fact that lead can be deposited on gold at a

on gold does not make the interphase behave like the lead-solution

less negative potential than on lead shows that the Pb—Au

interphase. More likely it forms an epitaxial layer (taking on the

more stable than the Pb—Pb

crys

crys

bond is

bond.

crystal dimensions of the underlying substrate). Theory predicts that

Underpotential deposition of metals is a commonly observed pheno-

such a monolayer of one metal on top of another would modify its

menon, which has been found to occur for dozens of metal couples in both

electronic properties to some extent but would not exhibit the elect-

aqueous and nonaqueous media. It is characterized by the distance

ronic properties of the bulk metal forming the monolayer.

between the peak potential observed during a slow cathodic sweep, and

Scanning tunneling microscopy (STM) would seem like the ideal

the reversible potential for deposition of the metal in the same

technique to study underpotential deposition

solution. This potential difference has been related to the difference

monolayer as it is being formed and determining its nature and unifor-

in the electronic work function of the two metals concerned, although

mity. This relatively new technique in electrochemistry has not been

this correlation does not always hold strictly. Much research has been

developed to its full capability, however, hence has not yet yielded the

devoted to the study of underpotential deposition of metals on single

desired answers to these and similar questions.

Am.

in-situ, observing the

284

ELECTRODE KINETICS

285

I. INTERMEDIATES IN ELECTRODE KINETICS

An interesting aspect of UPD (underpotential deposition), which may

considering that platinum is known to be a good electrocatalyst for

be of great practical importance, is its effect on electrocatalysis.

organic oxidation, whereas the rate of the same reactions on a lead

Studies of the oxidation of organic molecules on platinum have shown a

electrode is extremely slow. The same type- of behavior is observed when

significant catalytic effect, caused by a UPD layer of lead, as shown in

other noncatalytic metals (such as tin or bismuth, for example) are

Fig. 71 for the oxidation of a 0.1 M solution of methanol in 1 M 11 2SO4.

deposited on platinum in a UPD layer.

The addition of a very small concentration of Pb 2+ ions is seen to

A truly satisfactory explanation of these observations, which can

enhance the anodic current peaks when sweeping in both the anodic and

be confirmed by independent experiments, has yet to be proposed.

the cathodic directions. This is a most surprising observation,

Tentatively it is argued that strongly adsorbed intermediates are formed during the oxidation of organic molecules. These intermediates block the surface and decrease its catalytic activity. Formation of a partial

N

layer of lead atoms hinders the adsorption of such intermediates and

5.0

thus allows the oxidation reaction to occur more readily on the remain-

E U

ing free platinum surface (or perhaps on parts of the surface covered by

E

a monolayer of lead, which may retain much of the catalytic properties

H

of the platinum substrate). This interpretation is attractive in that

(7) z

it is consistent with our general understanding of the electrocatalytic

LIJ 2.5

activity of metals (cf. Fig. 7F) but will have to he confirmed by other z

experiments before it can be accepted.

cc

0

(b) Underpotential deposition of halogen atoms A phenomenon similar to the underpotential deposition of metals is

0.0

I

-1.2

-0.8 -0.4 POTENTIAL/Volt vs MSE

0

also observed in the study of the anodic oxidation of halides. In Fig. 81 we show the dependence of current on potential during an anodic potential sweep obtained on a platinum electrode in a nona

Fig. 71 Catalytic effect of a UPD layer of lead on platinum, on the anodic oxidation of 0.1 M methanol in 0.1 M NaOH, v = 50 mV/sec. Solid lines: no Pb2+ ions. Dashed lines: 1x10 -6M Pb2+ Arrows

A current peak corresponding to the formation of atomic bromine on the surface is observed, about 0.4 V before the potential for formation of molecular bromine is reached. The two reactions considered:

show the direction of potential sweep. Data from Beden, Kadirgan, Lamy and Leger J. Electroanal. Chem. 142, 171 (1982).

Br + Pt and

) Br—Pt +

em

(361)

ICS

2Br

Br

2

2e

(371)

I.

1..ti.1

. ■ 415

i

IN ELECTRODE Ktiv

None of the halogen molecules is formed on the bare metal surface. This behavior may seem to be of little importance in, say, the industrial

are evidently different and are, therefore, expected to have different values of the reversible potential. It is well known that molecular bromine is a much more stable species than atomic bromine, and the formation of the latter at a less anodic potential (i.e., more easily), attests to the strong bond between atomic bromine and platinum on the

production of chlorine, but it could determine the catalytic activity and/or the stability of the electrodes used in this process. It is even more important to understand this behavior in the context of a fundamental study of such systems, and the interpretation of the transfer coefficient and other kinetic parameters observed experimentally.

surface. The interesting point to note is that bromine evolution does not occur on the bare platinum surface, but rather on a platinum surface covered by a monolayer of adsorbed bromine atoms. The same is true for the anodic formation of molecular chlorine and iodine (and probably fluorine, but there is not enough experimental evidence in this case).

(c) Underpotential deposition of atomic oxygen and hydrogen It was mentioned earlier, in the discussion of hydrogen evolution on platinum, that a layer of adsorbed hydrogen atoms is formed on the surface even before the reversible potential for hydrogen evolution has been reached. This is similar to the case of the oxidation of halides, and could be called a UPD layer of hydrogen. The phenomenon is best seen in cyclic voltammetry, as a peak current appearing at a charac-

c.) 0.30

>. F-

0.20

teristic potential, positive with respect to the reversible hydrogen electrode in the same solution, as shown in Fig. 91. picture depends on the type of electrode used.

The detailed

Figure 91(a) is the

in

typical curve obtained on a polycrystalline sample (e.g., a wire or a

o 0.10

foil). Figure 91(b) was obtained on a spherical single crystal, on

z

which different crystal orientations are exposed to the solution. The

z

ce cr 0.0 D 0

different peaks represent different energies of adsorption of atomic hydrogen. The area under all the peaks combined is equivalent to a charge of 0.22 mC/cm 2 of real surface area, namely to a monolayer of adsorbed hydrogen atoms. All the peaks are at a potential positive

Fig. 81 Under potential deposition of atomic bromine (large anodic peak) and the beginning of bromine evolution on platinum, in 1.0 M Al Br and 0.8 M KBr in ethyl benzene. v = 120 mV/sec. Data 2 6 from Elam and Gileadi, J. Electrochem. Soc. 126, 1474 (1979).

In cyclic voltammetry the voltage is proportional to time. Thus, a plot of i versus E is also a plot of i versus t, and the integral represented by the area under the peaks has the dimensions of charge.

288

ELEC. I RODE KINETICS

I. INTERMEDIATES IN ELECTRODE KINETICS

289

r-, with respect to the reversible hydrogen electrode and can be thought of as an underpotential-deposited layer of hydrogen atoms.

120 80

Cyclic voltammetry on noble metals has been studied extensively.

E

The shape of the curve shown in Fig. 9I(a), which is characteristic of

40

platinum in sulfuric acid, is often used as a test for the purity of the 0

system. There has been much discussion concerning the origin of the two

z 0 -40

peaks in this system and the type of bonding they represent. Extremely

I—

careful purification by Conway and his coworkers led to results indica-

z

Li Cr -80 1r

Fig. 91 Cyclic voltaminetry on Pt in 0.5 M H2 SO 4 (a) polycrystalline sample, 15 mV/s;

ting that there may be as many as five peaks (some of which merge together and can be detected only as "shoulders"). These are all

-120 -160

.

I

I

I

i

I

I

I

0

0.2

0.4

0.6

0.8

1.0

1.2

measured on platinum wires or foils, which are, of course, polycrys1.4

POTENTIAL/volt vs NHE

talline. More recent measurements on single-crystal platinum electrodes led to the realization that the various peaks correspond to adsorption

(b) spherical single-

on different crystal faces.

crystal Pt electrode, 50 mVlsec. Reprinted with permission from

peak for hydrogen adsorption on single-crystal substrates.

Clavilier and Armand, J. Electroanal. Chem.

This is

indeed observed experimentally. Actually one usually finds one or two additional small peaks, which are due to imperfections in the orientation of the single-crystal substrate.

E a 40

Coming back to Fig. 9I(a) we see a peak for adsorption of atomic

199 187. Copyright

1986 Elsevier Sequoia.

If this were so, one should see a single

oxygen (or formation of a "surface oxide", which is really the same).

cr)

z 0

As with hydrogen, formation of adsorbed oxygen atoms precedes oxygen

a

I—

evolution. The area under the peak corresponds to a monolayer of UPD

w -40 cr cc

oxygen atoms. Oxygen evolution does not take place on the bare platinum

z

surface, but on a surface modified by a layer of adsorbed oxygen atoms. It is interesting to compare the cyclic voltammogram for platinum

-80

to that of a gold electrode in the same solution, as seen in Fig. 101. The UPD layer of oxygen atoms is there, but hydrogen adsorption cannot

-120 ,

I1

0.2460.8 POTENTIAL/Volt vs RHE

be detected. Hydrogen evolution on gold seems to occur on the bare metal surface. The same is true for mercury, lead and other soft metals. It would seem that a low exchange current density is associated

29u

LLECIRODE KINETICS

L INTERMEDIATES IN ELECTRODE KIM:: I ICS

satisfactory understanding of the way in which the surface is modified by the UPD layer, which is essential to the elucidation of the mechanism

1

50

of the relevant reaction, is still lacking.

E 20. THE ADSORPTION PSEUDOCAPAC1TANCE C

>-

20.1 Formal Definition of C o and Its Physical Significance

(73

z

The adsorption isotherms discussed in Section 19 describe the

o - 50

potential dependence of the fractional coverage 0. For an intermediate

z w

formed in a charge-transfer process, as shown, for example, in Eq. 191,

cc -100 D

denote the charge required to form a complete monolayer of a monovalent

the fractional coverage is associated with a faradaic charge q p. If we

U

-

1 1 I 1 I r 1 1 1 1 1 0.6 - 0.4 - 0.2 0 0.2 0.4 0.6 0.8 1.0 1.2

POTENTIAL/volt vs SCE

species by q l , we have the simple relationship (381)

qF = q l O

Thus, the adsorption isotherm also yields the dependence of charge on

Fig. 101 Cyclic voltanunetry on a single-crystal (210) gold electrode in 10 mM NaF. v = 20 inVisec. Data from Hamelin, J. Electroanal.

potential. This allows us to define a new type of differential capacitance, which we call the adsorption pseudocapacitance C4) :

Chem. 138, 395, (1982). C4) a (aqFlaqi = q i (a0/aE) with the reaction taking place on the bare metal, whereas a high value of i

is found when the surface is modified by a layer of adsorbed 0 hydrogen atoms. In contrast, oxygen evolution does not occur on the bare surface of any metal electrode. We conclude this section by noting that underpotential deposition is a rather general phenomenon, occurring in both cathodic and anodic' reactions. The surface is modified by the UPD layer and its catalytic activity is altered, usually for the better. The UPD layer is "transparent" to electrons (even when it consists of a layer of halogen atoms), and probably should be considered to be an extension of the metal rather than a superficial layer of a foreign substance. A

(391) 1-1 1

This rather important concept in interfacial electrochemistry warrants discussion. Clearly, we are not dealing here with a pure

capacitor, such as C dr , because charge transfer is involved. Moreover, the very existence of the adsorption pseudocapacitance is linked to charge transfer. The double-layer capacitor is a pure capacitor. When charge is brought to one side (plate) of the capacitor, an equal charge is induced on the other side. An excess of electrons on the metal causes a rearrangement of the distribution of ions on the solution side of the interphase, yielding an excess of positively charged ions, and vice versa. There is no need for charge transfer across the interphase

292

ELECTRODE KINETICS

t INTERMEDIATES IN ELECTRODE KINETICS

293 +la

to occur. Admittedly, some charge transfer is always observed, since no Fr a ction a l co verag e/0

interphase is ideally polarizable, but this has nothing to do with the basic property of the double layer capacitor, and it is treated as an "error term", which must be separated and independently measured. Although not a pure capacitor, the adsorption pseudocapacitance exhibits many of the properties typical of capacitors. Whatever the type of isotherm applicable to the system, there is a singular relationship between the charge and the potential. Setting the potential determines the charge and vice versa. When the potential is changed from one value to another, a transient (faradaic) current is observed. 0

The current decays to zero when the charge passed is enough to bring the

600

fractional coverage from its initial value to the value corresponding to the new potential. At a fixed potential, the steady-state current is zero. This is exactly how a capacitor should behave. It allows the passage of transient currents but presents an infinite resistance to dc.

Fig. I II Variation of 0 around its equilibrium value as a result of a small ac perturbation of the potential. The standard potential

further illustrated by considering its ac response. Let us assume that

Ee° is defined as the potential where [0/(1 — 0)] = 0 for f = r/RT = 0 and the concentration is

a low amplitude sinusoidal voltage signal is applied to a system at

solution is unity.

The capacitive nature of the adsorption pseudocapacitance can be

for adsorption

equilibrium: E = Et

=o

+ (AE)sin(at)

(401)

which can also be written as: i F= (Ai )cos(oit) = (Ai )sin(oit — 71/2)

We assume here that the frequency v = cu/2n is small enough, so that the coverage at any moment is equal to its equilibrium value, corresponding to the momentary value of the potential, as shown in Fig. HI. The rate of change of coverage with time is proportional to the rate of change of potential: dO/dt oc dE/dt = (co.AE)cos(0).0

(41I)

The faradaic current is given by it = dq r/dt = q i (dO/dt)

(421)

(431)

where Ai is the amplitude of the ac current. The phase retardation of

— ir/2 between potential and current is the correct ac response of a pure capacitor. 20.2 The Equivalent Circuit Representation

What makes Co a pseudocapacitance? Obviously, it is the fact that it is intimately related to, and indeed dependent on, faradaic charge transfer across the interphase. The equivalent circuit that represents

294

ELECTRODE KINETICS

the adsorption pseudocapacitance itself is a resistor and a capacitor in series, as shown in Fig. 12I(a). Note that the resistance R 4 is an integral part of the physical phenomenon that gives rise to the formation of the adsorption pseudocapacitance; R 4) is a faradaic resistance, since C4) is due to a charge-transfer process. The association of this

c90

L INTERMEDIATES IN ELECTRODE KINETICS

How is the adsorption pseudocapacitance affected by frequency? Returning to Fig. 11I, we can see that as the frequency increases, the changes in partial coverage can no longer keep up with the changes in potential. At a sufficiently high frequency, the adsorbed intermediate

charge-transfer process with the formation of an adsorbed intermediate,

is effectively "frozen in" — the potential changes back and forth so fast that the coverage does not have a chance to follow. Since C 4) is

which can proceed only until the appropriate coverage has been reached, is manifested by placing the resistor in series with the capacitor. The

proportional to dO/dE, it tends to zero at high frequencies. The dependence of the impedance associated with C 4) on frequency can

circuit shown in Fig. 12I(a) does not allow the flow of a steady-state current. It should also be borne in mind that both C 4) and R 4) can depend

be visualized with the aid of the equivalent circuit shown in

on potential, as in other equivalent circuits representing the electrochemical interphase.

complete circuit, showing the other circuit elements we have already

Fig. 12I(b). Here the adsorption pseudocapacitance is part of a more

4

RS+R +R F

a

3

Fig. 121 Equivalent circuits for (a) the adsor-

0) 0

ption pseudocapacitance

2

C4) and the corresponding

b

resistance R4 and (b) an interphase containing an

C di

R5

adsorption pseudocapaci-

-4

-2

0

2

4

6

log W

tance. Fig. 131 Bode magnitude plot, showing the dependence of the absolute value of the impedance IZI on the angular velocity (1), (in units R

of Radians per second), plotted on a log-log scale. Numerical values chosen to calculate the curve: R F= 104 Q, R = 102 0, R = 20 0, C1= 400 j_LF and Cdl = 4 1.1F. s

296

ELECTRODE KINETICS

L

INTERMEDIATES IN ELECTRODE KINETICS

297

discussed. When a dc signal is applied, the interphase behaves like a resistor, having a total resistance of R = RF R R s . At very high frequencies it behaves like a capacitor, C e in series with the solution resistance R. Between the two extreme cases, the adsorption

where CLis the adsorption pseudocapacitance derived from the Langmuir isotherm. Taking the second derivative with respect to potential, we find that the maximum value of the adsorption pseudocapacitance is reached when 1 = KC•exp(EF/RT), corresponding to 0 = 0.5. Thus we can

pseudocapacitance is switched in and out again, at frequencies that depend on the numerical values of the various circuit elements shown.

write

This is seen in Fig. 131, where the absolute value of the impedance is plotted versus the frequency, on a log-log scale. 20.3 Calculation of C as a Function of 0 and E 431 The adsorption pseudocapacitance can be readily calculated from the appropriate isotherm, with the use of its definition, given in Eq. 391.

q,F/4RT and E.. = — (2.3RT/F)log(KC)

The dependence of CI, on coverage can be obtained by rewriting the Langmuir isotherm in the following form: E = (2.3RT/F)log[ 1 () 0] — (2.3RT/F)log(KC)

1_ ( 1 )( dE _ ( RT - k q i i‘ dO \ --47F-) o(1—o)

(a) The Langmuir isotherm The Langmuir isotherm describing the formation of an adsorbed intermediate by charge transfer can be written in the form

KC • exp(EF/RT) 1 + KC•exp(EF/RT)

(471)

(481)

Differentiating with respect to charge we have

This is shown next for the Langmuir and Frumkin isotherms.

0=

CL(max) =

(441)

(491)

which can be rearranged to CL = q i FIRT)[0(1 — C)]

(50I)

from which we obtain: dO/dE = (F/RT)

KC • exp(EF/RT) I [1 + KC•exp(EF/RT)} 2

Clearly, CLhas its maximum value, given by Eq. 471, at 0 = 0.50.

(451)

Setting the concentration in Eq. 47 equal to unity, we note that the potential at which CL is a maximum can be regarded as the standard

potential Ee° for the adsorption process. It is given by

Combining with Eq. 391 we have

E° = — (2.3RT/F)logK CL = q i (dO/dE) = (q i F/RT)

KC•exp(EF/RT) + KC•exp(EF/RT)1 2

(461)

(511)

The physical meaning of this choice is that the standard state of the system is represented by 0 = 0.5 and C = 1.0.

298

ELECTRODE KINETICS

It is interesting to evaluate the numerical value of C F(max). Using q 1 = 0.22 mC/cm 2 for a monolayer of single-charged species, we find ,

) = CF(max

0.22x10 3 x96.5x103

3

= 2.14x10 1.1F/cm

4x8.31 x298

2

1. INTERMEDIATES IN ELECTiwub. niNurics

We write the parameter r in dimensionless form as f = r/RT and obtain from Eq. 531 the following relation:

\ [0(1 1 dE/d0 = (RT/F)

(521)

This is about two orders of magnitude higher than typical values

= (RT/F)

observed for the double-layer capacitance. We can also evaluate the range of potential over which C L. is equal to or larger than Cdi . Substituting CL = 16 pF/cm 2 (i.e., a typical value of C dl) into Eq. 501

— )1

[1 + f0(1 0 0(1 — 0)

(541)

e adsorption pseudocapacitance under Frumkm conditions is thus: C

we find that the two values of 0 which satisfy this relationship are

0)i + (RTiqf

9_,

F=

0(1 — 0) 1 = C (401 F/10[ 1 + L [1 + f0(1 — 0)] 10(1— 0 )]

(551)

0.00187 and 0.99813. In other words, C L Cdl for values of 0 between

This equation yields a maximum value of:

about 0.2% and 99.8%. The corresponding range of potential (found by introducing these values of 0 into the Langmuir isotherm) is 0.16 V.

CF(max)

The important conclusion to be drawn from this numerical calculation is

= (q 1 F/4RT)[

1 1 + f/

C

(max)

1 j

(561)

+

that great care must be exercised in interpreting double-layer capaciThe dependence of C 4) on 0 is shown in Fig. 14I(a). We note that all curves are symmetrical around 0 = 0.5, irrespective of the value of the

tance measurements in systems in which an adsorbed intermediate can be formed. Even a minute fractional coverage can give rise to an adsorp-

parameter f. The maximum declines rapidly with increasing value of f. The dependence of the adsorption pseudocapacitance on potential is shown

tion pseudocapacitance comparable to or larger than the double-layer capacitance. When the Frumkin isotherm is applicable, the maximum of

in Fig. 14I(b). This function cannot be derived directly, since it is impossible to express 0 as an explicit function of E from Eq. 121.

the adsorption pseudocapacitance is smaller, but the effect is extended over a wider range of potential, as we shall see.

However, since the dependence of both E and (b) The Frumkin isotherm

CF

on 0 are known, the

curves shown in Fig. 141(b) can readily be calculated.

We proceed here to derive the expressions for the adsorption

One may be tempted to calculate the parameter f from the value of

pseudocapacitance under Frumkin conditions, which we shall denote CF

CF(max), as given by Eq. 561. The problem is that the maximum of the

TheFrumkinsot(Eq.12)wrienthfom:

pseudocapacitance is calculated per unit of real surface area, whereas what we measure is C (max) per unit of geometrical or apparent surface

F

E = (RT/F)114

0

1 — CI] + (r/F)0 — (RT/F)In(KC)

(531)

area. The ratio between the real and the apparent surface area is the so-called roughness factor, which is not known accurately. Moreover, even the value of the charge per monolayer, q 1 = 0.22 mC/cm 2 , which we

300

ELECTRODE KINETICS

I. INTERMEDIATES IN ELECTRODE KINETICS

301

a 2500 2000

0.3

E 1500 0.2

1000 500

O. 1

Fig. 141 The dependence of C o on (a) coverage and (b) potential, for different values of the parameter f, for q1 = 230 p.C1cm.

0

0.2

0.6

0.4

0.8

10

Fractional coverage/0 0

b

5

2000

iE

20

Fig. 151 The width of the plot of C F, the adsorption pseudocapacitance, calculated on the basis of the Frumkin isotherm, at halfheight AE , as a function of the parameter f = rIRT. 1/2

1500 1000 500

15

The Frumkin parameter f

2500 csi

10

What is the value of C 4) according to the Temkin isotherm? From

Eq. 141 0 —200

used to calculate C (max), is only in estimate, which can vary between substrates and between different a sorbed intermediates. On the other

0 = (RT/r)In(K C) + (F/r)E o we see that it should have a constant value of CT = (dOidE) =

hand we note that the curves in Fig. 14I(b) become wider with increasing value of the parameter f.

This provides a way to avoid the foregoing

uncertainty, by determining f from measurement of the width of these curves at half height. Taking the potential AE I/2 between two points on the curve where CF/CF(max) = 0.5, we have made the result independent of the surface area. The variation of AE 1r2 with the parameter f is shown in Fig. 151.

(141)

This is larger than C written as:

(max)

C (max) =

q i Fir = (q 1 F/RT)(1/f)

(571)

given in Eq. 561, which could also be

(q F/RT)[11(44-1)]

(581)

302

ELECHtuDE KINETICS

a

versus E and 0 versus E in Fig. 161(a) and 161(b), respectively, for f = 20. The lines for the Temkin isotherm are shown in both cases, for comparison. The discreTo clarify this point, we have plotted

1000 800 600

importance of the way in which data are presented. It should be noted

L.

■

that the lines in Fig. 16I(a) are the derivatives of those in

400

GC)

Fig. 16I(b), since C F is proportional to dO/dE. A small deviation from

-

200

linearity in the integral form of presenting the data, which can barely

Fig. 161 A comparison of

be detected, is magnified and emphasized when the same data are presen-

(a) CFversus 0 and (b) 0 versus E for the same

0

0.4

0.6

0.8

b

ted in differential form.

20.4 The Case of a Negative Value of the Parameter f We have already discussed qualitatively the case when the parameter

Frac tiona l co ve rag e/9

from the Temkin isotherm.

0.2

Fractional coverage/0

value of the parameter f = 20. Solid lines calisotherm, dashed lines

CF

pancy just alluded to is only apparent, of course, and it brings out the

E

culated from the Frumkin

303

I. INTERMEDIATES IN ELECTRODE KINETICS

r (or its equivalent in dimensionless form, .1) has a negative value, which can only be the result of attractive lateral interaction between

the adsorbed species. Let us treat this case quantitatively here. From the Frumkin isotherm we found dE/d0 =(RT/F)[ 1 + f0(1 — 0 ) 0(1 — 0)

(541)

Solving Eq. 541 for dE/dO = 0, we find Potential/mV vs E There is an apparent discrepancy between the treatment of electrode

0 = 0.5 ± 0.5(1 + 4/0 1/2 Since, by definition, 0

0

(591)

1, Eq. 591 has physically meaningful

kinetics under Temkin conditions, at intermediate values of the

solutions only for f — 4. A plot of the Frumkin isotherm for negative

coverage, and the results shown in Fig. 14I(b) for the adsorption

values of the parameter f is shown in Fig. 171. For f = — 12, the

pseudocapacitance in the same region. For the purpose of calculating

solution of Eq. 591 yields 0 = 0.11 and 0.89. Between these values, the

the kinetic parameters, we have assumed that 0 is a linear function of

coverage appears to increase with decreasing potential, which would

potential. This is a valid assumption, as we can see in Fig. 21. Yet

imply a negative value of the adsorption pseudocapacitance. This does

such a linear dependence of 0 on E should give rise to a constant value

not represent physical reality, of course. If we trace the potential in

of C4) , independent of E, which is not the case, as shown in Fig. 14I(b).

the positive direction, 0 will jump from 0.11 to 0.999. When the

304

ELEC I RODE KINETICS

I. INTERMEDIATES IN ELECTRODE KINETICS

305

derived if we drop the potential dependent term from the Frumkin Fra c tiona l coverag e /9

isotherm and use, instead of Eq. 541, the following equation: din C/d0 =[

- 0(f ? (I — 0y91] = 0

(601)

As in the electrochemical case, the critical value of the interaction parameter is f = — 4. For more negative values, a hysteresis loop indicative of two-dimensional phase formation will be observed.

0 -300

-200

-100

0

100

200

300

Potential/mV vs E c'e Fig. 171 The Frumkin isotherm with both negative and positive values of the parameter f. A negative value corresponds to attractive lateral interactions among adsorbed species, which may lead to two-dimensional phase formation. Dashed lines show the hysteresis loop which is expected in such cases. potential is swept back in the cathodic direction, it will decrease gradually to 0.89 and then change suddenly to 0.001, leading to a hysteresis loop, as shown by the dashed lines in Fig. 171. The sudden transition from low to high coverage is indicative of a two-dimensional phase formation. The adsorption pseudocapacitance is very high in this region. This behavior can he compared to the process of liquefaction of a gas. When the new phase is formed, both the isothermal compressibility (1/V)(3V/@P) Tand the heat capacity (aQ/aT)

tend to infinity.

A sudden change in coverage, of the type shown in Fig. 171, has been observed in several instances of adsorption of organic molecules on electrodes. It should be understood that such behavior is not necessarily related to charge transfer. The same kind of isotherm can be

J.

J.

ELECIK,..,b, ,,,t1.,

4

ELECTROSORPTION

21. PHENOMENOLOGY 21.1 What Is Electrosorption? Electrosorption is a replacement reaction.

We have already

discussed the role of the solvent in the interphase, in the context of its effect on the double-layer capacitance. It is most important for our present discussion to know that the electrode is always solvated and that the solvent molecules are held to the surface both by electrostatic and by chemical bonds. Adsorption of a molecule on such a surface requires the removal of the appropriate number of solvent molecules, to make place for the new occupant, so to speak. This is electrosorption. In this chapter we shall restrict our discussion to the electrosorption of neutral organic molecules from aqueous solutions, without charge transfer. Using the notation RH for an unspecified organic molecule, we can then represent electrosorption in general by the reaction

RH

sol

+ n(H 0) 2

RH + n(H o) 2 sol ads

ads

(1J)

Several important features of electrosorption follow from this simple equation. First it becomes clear that the thermodynamics of electrosorption depends not only on the properties of the organic molecule and its interactions with the surface, but also on the properties of water. In other words, the free energy of electrosorption is the difference between the free energy of adsorption of RH and that of n water molecules: AG ads

= (I RH

)

1-tRH ads

sol

nGt

Wads

— µw)

(2J)

sol

The same relationships also apply to the enthalpy and the entropy of

308

ELECTRODE KINETICS

1, ELE,CTROSORPTION

309

electrosorption. The enthalpy of electrosorption turns out to be less

constant for the electrosorption of this compound increases with

(in absolute value) than the enthalpy of chemisorption of the same

increasing temperature. Thus, raising the operating temperature of a

molecule on the same surface from the gas phase. On the other hand,

fuel cell, to enhance the reaction rate, does not necessarily have an

the entropy of chemisorption is, as a rule, negative, since the molecule

adverse effect on the extent of adsorption of the reactants.

RH is transferred from the gas phase to the surface and loses 3 degrees

The second point to note is that electrosorption depends on the

of freedom of translation. This is also true for electrosorption, but,

size of the molecule being adsorbed, vis-a-vis its dependence on the

in this case n molecules of water are transferred from the surface to

number of water molecules which have to be replaced for each RH molecule

the solution, leading to a net increase of 3(n — 1) degrees of freedom.

adsorbed. One may be led to think, on the basis of Eq. 2J, that large

As a result, the entropy of electrosorption is usually positive.

molecules cannot be electrosorbed. This is not true. As the size of

Remembering the well known thermodynamic relationship

the molecule increases, so does its interaction with the surface. Thus, both terms on the right-hand side of Eq. 2J increase with the parameter it, though not necessarily at the same rate.

OG = OH ads

ads

—

TOS

ads

(3J)

we conclude that in chemisorption a negative value of AG ads is a result of a negative value of AH ads, while the entropy term tends to drive the free energy in the positive direction. In electrosorption, the enthalpy term can be less negative or even positive, and it is the positive value of the entropy of electrosorption that renders the free energy negative, in most cases. It can be said that electrosorption is mostly entropy

driven, whereas chemisorption is mostly enthalpy driven. While the above conclusion is intellectually intriguing, it may also have some important practical consequences, particularly in the area of fuel cells and organic synthesis. Thus by "common wisdom" the extent of chemisorption decreases with increasing temperature. This follows formally from the well known equation dlogK ad s =— d(1/T)

ads

coverage is a function of potential, at constant concentration in solution. Thus, we can discuss two types of isotherms: those yielding 0 as a function of C and those describing the dependence of 0 on E.

This is not a result of faradaic charge transfer. Neither is it due to electrostatic interactions of the adsorbed species with the field inside

the compact part of the double layer, since a potential dependence is observed even for neutral organic species having no permanent dipole moment. As we shall see, it turns out that the potential dependence of

0 is due to the dependence of the free energy of adsorption of water molecules on potential. 21.2 Electrosorption of Neutral Organic Molecules

(4J)

2.3R

with negative values of Af-r ds for chemisorption.

An additional unique feature of electrosorption is that the

In studies of the

electrosorption of benzene on platinum electrodes from acid solutions, the enthalpy of adsorption was found to be positive. The equilibrium

Electrosorption has been studied on mercury more than on any other

metal, not because this is the most interesting system, either from the fundamental or the practical point of view, but because it is the easiest system to study and because the results obtained are not complicated by uncertainties resulting from different features of the

310

ELECTRODE KINETICS

surface, a problem common to the study of solid surfaces.

J. ELECThoSORMON

The depen-

1 .0

co

dence of 0 on potential for the adsorption of butanol on mercury was

L.L.1

shown in Fig. 12H above. In Fig. 1J we show plots of the fractional

0.8

coverage for the electrosorption of phenol with methanol or water as a > 0

solvent. Let us make a detour here for a moment, to discuss the question of

0.6

-J

the appropriate scale of concentration to be used when comparing

< 0.4 z

isotherms measured in different solvents. Chemists prefer to express

0.2

0

concentrations in units of moles per liter. This is fine for aqueous solutions (or at least for a fixed solvent), but it fails totally when

0.0

different solvents are compared. Thus, a 0.1 M solution in water

20 15 10 5 0 -5 CHARGE DENSITY q m /p.C.cm -2

25

- 10

corresponds to a mole fraction of 1.8x10 3 of the solute. If toluene is used as the solvent, the same molar concentration corresponds to a mole fraction of 12x10 3 ! Clearly the scale of mole fractions, which

co - 1.0

represents the ratio between the numbers of solute and solvent mole-

0

cules, is more relevant for the purpose of comparing solutions in

.1 0.8 m

different solvents.

0

In the case of electrosorption, it is best to use a dimensionless scale of C/C(sat) when comparing the adsorption of different solute molecules in the same solvent, or the same solute in different solvents. This scale permits us to compensate for the differences in the free

I.5M

• •

•

• 0.5

c.) 0.6 •

z

0 17: (-) 0.2

energy of interaction between the solvent and the solute, and the effects seen arise from the different interactions of the solutes with the surface. A good example is the adsorption of phenol on mercury from two different solvents, shown in Fig. 1J. The solubility of phenol in

25

20

15

10

5

0

-

-10

CHARGE DENSITY q m /p.0 cm-2

water is much lower than in methanol. It takes therefore a much higher

Fig. IJ The electrosorption of phenol from (a) aqueous and (b) methanolic solutions, as a function of the charge density on a mercury

concentration in methanol to reach a given value of the fractional

electrode. Supporting electrolyte: 0.1 M LiCl, the concentra-

coverage 0 than in an aqueous solution.

tions of phenol are marked on the curves. From Muller, Ph.D dissertation, Univ. of Pennsylvania, 1965.

ELECTRODE KINETICS

312

1. ELECTR OS ORPTION

313

The electrosorption of pyridine on mercury is shown in Fig. 2J. One should note that the dependence of 0 on E is roughly bell-shaped for

This is a satisfactory approximation in the present case, since the

most compounds, with the maximum of adsorption occurring at a potential

extend much beyond a monolayer.

that is slightly negative with respect to the potential of zero charge. It should be borne in mind that all the data reported in the

from the practical point of view. It represents an important factor in

literature for mercury are values of the relative surface excess F', not

many fields, including electrocatalysis, electroplating, corrosion and

the fractional coverage 0. In dilute solutions the relative surface

bioelectrochemistry, some of which are discussed later. Unfortunately,

excess is very nearly equal to the surface excess, in view of its

it is much more difficult to measure the surface coverage on solids, and

definition, given by Eq. 28H

the interpretation of data is complicated by lack of reproducibility of [ri-- Fw(x i/x01

(28H)

Also, if electrosorption is restricted to a monolayer, we can relate the

the surface, by the competing formation of adsorbed layers of oxygen and hydrogen, and by the possibility of faradaic reactions taking place during adsorption. The dependence of the surface excess on potential

(5J)

rirnlax

0.4 N 1 8 0.3 (.)

w 0

•ct I.0

IX 11.1

Electrosorption on solid electrodes is of much greater interest

for the electrosorption of n-decylamine on nickel is shown in Fig. 3J.

fractional surface coverage to the surface excess by simply writing:

o=

interaction of a neutral organic molecule with the surface does not

•

• - • 6--s,

•

ar 0 E

0. 2

0

0

0 _J

-3 o> "70......./0,,...._0 .......... — o -....... o o ----___07__----,:;.-. o o o--...2.-----0o---__,. ,_4,_ 8---"- .° o

0 .1

c:( 0.5 z O

I

0.0 -

Li- 0.0

1

1

I

1

-1.5 -1.0 -0.5 POTENTIAL/Volt vs N.C.E.

Fig. 2J Potential dependence of the electrosorption of pyridine (0.3 mM in 0.1 M KCl) on mercury. Data from Damaskin, Electrochirn. Acta, 9, 231, (1964).

0.4

t

i

I

'I

0

- 0.6 -0.8 -1.0 POTENTIAL/Volt vs NNE

Fig. 3J Adsorption of n-decylamine on nickel from 0.9 M NaCl0 4 , pH 12. Concentrations: (1) 7.5x10 -5 ; (2) 5x10 -5 ; (3) 2.5x10 -5 ; (4) 1x10 -5 ; (5) 0.5x10 -5 . The surface excess F is given per unit of geometrical surface area.

rmax = /moo 9 .

Data from Swinkles and Bockris, J. Electrochem. Soc. 111, 736, (1964).

314

ELEC

K ODE

KINETICS

J.

ELEC'TROS Of <

315

viv

H H

The adsorption of ethylene from acid solutions on platinized platinum electrodes is shown in Fig. 4J.

HH H——&H + H + H I I P t Pt Pt Pt

H—&&-H + 4Pt I H H

The form in which hydrocarbons are adsorbed on platinum was the

(6J)

subject of controversy at the time this work was done. To understand the difficulty involved, let us compare the electrosorption of ethane and ethylene. The only way to form covalent bonds between an ethane molecule and the surface is

to

break some C—H bonds, according to the

This is called dissociative adsorption.

If the potential is suffici-

ently positive with respect to the reversible hydrogen electrode in the same solution, the adsorbed hydrogen atoms will be ionized, and Eq. 6J should be rewritten in the following form:

equation:

H H I I H H

HH H—&6—H + 2H + + 2e m I P t Pt

+ 2Pt

(7J)

If we consider ethylene instead of ethane, we could have the same type of dissociative electrosorption, namely: H—C=C-11 + 2Pt --> H—C=C—H + 2H + + 2e 1 1 H 14 t

(8J)

or non-dissociative electrosorption, which can be represented by the equation HH

HH 0

0.1

0.3

0.5

+ 2Pt

07

H4-&H

(9J)

Pt Ft

POTENTIAL/volt vs NHE As it turns out, it is very easy to distinguish between these two modes

Fig. 4J The surface concentration of ethylene, per unit of geometrical surface area, as a function of potential, on a platinized platinum electrode in 0.5 M H SO . (1) I.7x10 5 ; (2) 9x10 -6; 2 4 (3) 4.6x10 6 ; (4) 4x10 6 ; (5) 2.1x10 6M. Reprinted with permission from Gileadi, Rubin and Bockris, J. Phys. Chem. 67, 3335, (1965). Copyright 1965, the American Chemical Society.

of electrosorption. To do this, one injects a solution containing the hydrocarbon into a cell in which a platinum electrode is held at a potential of about + 0.5 V versus RIM, where adsorbed hydrogen is rapidly ionized. will be observed.

If dissociative adsorption occurs, a transient current The total charge during this transient should

correspond to the ionization of two moles of hydrogen atoms for each mole of ethylene adsorbed. If adsorption is non-dissociative, a very

ELECTRODE KINETICS

316

317

J. FI PCTROSORPTION

small transient current is expected. The charge in this case is only

hydrogen.

that which is released from the double-layer capacitor at constant

therefore been studied extensively. Typical results obtained on bright

potential, as a result of the decrease in its capacitance, which is

platinum are shown in Fig. 5J. Although the dependence on potential is

caused by electrosorption. As a rule, saturated hydrocarbons are

"bell-shaped," just like that observed for the adsorption of organic

The electrosorption of methanol on platinum electrodes has

electrosorbed dissociatively, while unsaturated hydrocarbons tend to be

molecules on mercury, one should be careful in interpreting these data.

adsorbed without dissociation. This may depend, however, on experi-

Thus, the decrease in coverage on the anodic side may be due, at least

mental conditions, and particularly on the temperature and the type of

in part, to the anodic oxidation of methanol in this range of poten-

electrode used. Methanol is a potentially attractive fuel for electrochemical

tials. On the cathodic branch, competition with adsorbed hydrogen may modify the form of the potential dependence.

energy conversion devices (which is just another name for fuel cells) for two reasons: first, because it is relatively easily oxidized electrochemically, and secondly because it can be cheaply manufactured from hydrogen and can, in effect, serve as a chemical means of storing

Fig. 6J Electrosorption of naphthalene on gold from 0.5 M H 2SO 4 . 0.6 0.4 0.2 POTENTIAL/Volt vs RHE

Concentration of adsorbate: (I) 5x10-5 ; (2) 10-6 ; (3) 2x10 -7M.

0.8

Fig. 5J Electrosorption of methanol on bright Pt electrodes from 1.0 M 1-1 SO . Concentration of methanol: (1) 1x10 -3 ; (2) 1x10 -2; (3) 2 4 1x10 -1 ; (4) 1.0 M. Data from Bagotzky and Vassiliev, Electrochim. Acta 11, 1439, (1966).

Data from Swinkles, Bockris and Green, J. Electrochem. Soc. 110, 1075, (1963).

■

*

Methanol contains 12.5% hydrogen by weight, compared to less than 2% for metal-hydrides, such as TiFeH 1.8 or LaNi5H6.8.

318

ELi•CTRODE KINETICS

319

ELECTROSORPTION

Spectroscopic evidence obtained recently indicates that the

allows us to extend the measurement of capacitance to low concentrations

electrosorption of methanol on platinum is probably a very complex

of the electrolyte (cf. Section 16.8), increasing the accuracy of the

process, in which several different partially oxidized species may be formed, in ratios that depend both on the potential and on the bulk

determination of Ez on solid electrodes. One should bear in mind, however, that the minimum in capacitance coincides with Ez only if a

concentration of methanol.

symmetrical electrolyte is used.

A much simpler case is presented by the electrosorption of naphtha-

Although the instrumental aspects of measuring Ez can yield quite

lene on gold, shown in Fig. 6J. The higher stability of this compound

accurate results, chemistry is lagging behind. The most difficult

combines with the lower catalytic activity of gold, to ensure that

problem, as always with solid electrodes, is the lack of reproducibility

partial oxidation does not occur in the range of potential shown. Also,

of surface preparation. Best results can be obtained on single crystals

hydrogen adsorption does not occur on the cathodic branch of the curve.

of noble metals, where it is observed that E depends on the particular

Thus, the potential dependence of 0 i4 this case can be interpreted in

crystal face exposed to the solution. An exceptional example of this is

terms of the properties of the double layer, just as in the case of

shown in Fig. 7J, where E is plotted for a large number of crystal faces

mercury.

of gold. We need not go into the details of these different crystal faces; Fig. 7J simply illustrates that

21.3 The Potential of Zero Charge and Its Importance in Electrosorption The concept of the potential of zero charge (PZC or E), has

E

can be measured rather

accurately, even on solids, and that it is clearly a function of the crystallographic orientation. On a polycrystalline sample (e.g., a wire or a foil) certain

already been discussed in the context of electrocapillary thermodyna-

crystal faces may dominate, depending on the mechanical, thermal and

mics, where we showed that, for an ideally polarizable interphase, the

electrochemical pre-treatment of the sample, giving rise to different

PZC coincides with the electrocapillary maximum. In view of the very

values of the PZC. Thus, it was observed that platinum, from which all

high accuracy attainable with the electrocapillary electrometer, it is

traces of absorbed hydrogen have been removed by careful annealing in

possible to measure Ez for liquid metals near room temperature to within about 1 mV. This accuracy is limited, however, to mercury, some dilute

argon, yields EL = 0.55 V versus RHE. If, on the other hand, hydrogen is evolved on the metal for even a short time, allowing some penetration

amalgams, and gallium.

of atomic hydrogen into it, the value of E shifts to about 0.25 V on

•

solid electrodes. The one that seems to be most reliable, and rela-

the same scale. More recently it was shown that cycling the potential of a platinum electrode between oxygen and hydrogen evolution for a long

tively easy to perform, is based on diffuse-double-layer theory.

time causes faceting, with the (111) crystal face becoming predominant.

Measurement of the capacitance in dilute solutions (C 5. 0.01 M) should

Determination of E on base metals such as copper, nickel or iron is complicated by the formation of oxide layers. The value of Ez

Many methods have been used to determine the value of the PZC on

show a minimum at Ez , as seen in Eq. 15G and Fig. 4G. Lowering the concentration yields better defined minima. Modern instrumentation

320

ELECTRODE KINETICS

321

J. ELEC I ROSORPTION

Table 1.1 The Potential of Zero Charge

o

N

Lc)

N

=

Wi Z7) I

(r)

I

I

I

Cd

— 0.90

5 mM

MCI

TI

— 0.82

1 mM

I
Pb

— 0.67

1 mM

II SO

Zn

— 0.63

1

M

IICI

0.1

Ga

— 0.60

1

M

11C1

0

Bi

— 0.40

1

M

I ICI

Fe

— 0.37

1 mM

II

Sn

— 0.35

1 mM

11 SO

Hg

—0.19

0.1

Ag

0.05

0.1 M

KNO

Pt

0.25

50 mM

H SO

0.3

> 0.2 0

> _J

z

for Different Metals/V, NILE

0 0 0 0 0 cv rn =

LL.1

to -0.1 a_ I

I

I

1

I

I

CRYSTALLOGRAPHIC ORIENTATION Fig. 7J The potential of zero charge for single-crystal gold, plotted as a function of crystal orientation. Data from Lecoeur, Andro

M

2

2

4

SO

2

4 4

NaP

2

3 4

and Parsons, Surf. Sci. 114, 320, (1982). Values of the PZC are shown in Table 1J, for a number of solid metals. The value for mercury is also included for comparison. measured may then correspond to an oxide-covered surface, rather than to the bare metal.

Electrosorption depelids primarily on the excess charge density qm . Coverage by neutral organic species decreases at both negative and

The occurrence of faradaic reactions of any kind, and particularly

positive values, with a maximum of coverage at g m = — 2 ttC/cm2 . On the

those leading to the formation of adsorbed intermediates, can severely

potential scale, the region of significant coverage extends over about

interfere with the determination of E , when based on measurement of the

0.8 V, from — 0.6 to + 0.2 V on the rational scale. For aromatic

capacitance minimum. The high values of the adsorption pseudo-

compounds the coverage declines more slowly on the anodic side, probably

capacitance, C4) , which extends over a significant range of potential,

due to interaction of the it-electrons with the metal. For charged

can distort the measurements of double-layer capacitance in dilute solutions, as discussed in Section 20.3.

32:2

ELICI :!...)E KINETICS

r..LECTROSORPTION

.%

(a) Radiotracer methods species the situation is more straightforward.

Positively charged

molecules are adsorbed mostly at negative rational potentials and vice versa. This effect is superimposed on other factors controlling electrosorption, so that a negatively charged molecule may be specifically adsorbed on a negatively charged surface, as a result of the chemical energy of interaction between the molecule and the surface. Naturally, the dependence of 0 on potential will not be symmetrical in this case, as it is for neutral molecules. It should be noted here that the adsorption of intermediates formed by charge transfer is not controlled by the potential of zero charge. When have as of the Cl + M

M—C1 + e

m

(7I)

Radiotracer techniques are ideally suited for the detection of very small amounts of a chemical species, such as are found in a monolayer or a fraction of it, on the surface. The electrode is placed on the window of a suitable counter (in some cases the electrode constitutes the window) so that close to half the radiation emitted is directed toward the counter. The relative effect of the background from the environment can be suppressed by the use of a sufficiently high concentration of the radioisotope, but the background from solute molecules near the surface which are not adsorbed is increased proportionally. This effect depends on the penetrating power of the radiation through the solution. If an isotope emitting y radiation or hard 3 radiation is used, the background from the solution is high, since it comes from a relatively large volume of the solution, and measurement of 0 is difficult or impossible.

which leads to an adsorption isotherm of the form

Although this is a constant contribution that can, in principle, be subtracted, we recall that nuclear disintegration is a random process,

1.

.1

(12I)

and a signal can be detected only if it is larger than the fluctuations in the background.

the region of potential over which 0 is significant depends on the equilibrium constant K

The following numerical example will serve to clarify the effect of

o' which is related to the free energy of adsorption (cf. Eq. 111). The value of K o depends on the metal through its

background radiation from solution. The thickness of a layer of

dependence on the free energy of the M—Cl bond, but it is not directly

the surface can readily be calculated. For the case of ethylene, which

dependent on E.

The case in which electrosorption occurs with charMe

has been studied by this method (cf. Fig. 4J), the maximum surface

transfer, so that both types of interactions have to be considered

coverage is calculated to be Fmax = 5.7X10 1° mol/cm 2 (of real surface area), and the highest concentration in solution was 2x10 -5 M. The

simultaneously, is discussed in detail later. 21.4 Methods of Measurement of Coverage on Solid Electrodes

solution that contains the same number of molecules as a monolayer on

corresponding thickness of this layer is found to be 285 tam. Since the range of the most energetic 13 particles emitted from "C in water is

In this section we shall discuss briefly, the underlying principles

only 32 p.m, the background radiation cannot exceed about 11% of that

of some of the methods by which adsorption on solid electrodes can be measured.

derived from a monolayer and is, in effect, lower because most electrons

Ls

324

ELECTRODE KINETICS

1. ELECTROS OR PTION

325

are emitted with a lower energy. Moreover, the background can readily

volume of the solution and the surface area of the electrode. A typical

be determined with an accuracy of ± 10% or better, making the error in

electrochemical cell has about 10 cm 3 of solution per square centimeter of surface area. Using again the example of ethylene, about 6x10 11

the determination of coverage due to the radioactive adsorbate in solution well below 1%.

molewibrqudtchangesrfovby6,0=.1This

The very rudimentary calculation just presented ignores several

represents a change of concentration of only 0.03%, if the initial

secondary effects, but it should serve to clarify the main factors

concentration is 2x10 -5 M. The sensitivity can be increased by using a

involved. Using a higher energy (3 emitter or a higher concentration in

high-surface-area electrode, such as platinized platinum, and a lower

solution decreases the sensitivity of the method. On the other hand, to

concentration in solution. If we increase the roughness factor by as

increase the sensitivity, one can use a large-surface-area electrode

much as 100 and decrease the concentration in solution by an order of

(platinized platinum having a roughness factor of about 50 was employed

magnitude, the change in concentration resulting from monolayer adsorp-

to obtain the data in Fig. 4J). Several techniques have been used to

tion will be 30%, and values of AEI = 0.1 can readily be determined.

overcome the problem of background radiation from solution, by using the

Larger surface-to-volume ratios can he achieved by using porous elec-

equivalent of a thin-layer cell, or by "squeezing" most of the solution

trodes or by rolling up an electrode in a minimum volume of solution.

out just before measurement. It must be remembered, however, that the

Care must be taken, however, to ensure that potential control and

coverage depends on potential, and control of the potential must be

uniformity are maintained in this type of measurement. This is rela-

maintained during such measurements.

tively simple if electrosorption is measured in the range where the

One of the advantages of the radiotracer method is that it can be

electrode is highly polarizable. If a faradaic process takes place, the

used to follow adsorption as a function of time, namely to study the

current distribution may be non-uniform, leading to different metal-

kinetics of adsorption.

solution potential differences on different parts of the electrode. The change in concentration in solution can be measured by the

(b) Methods based on the change in bulk concentration

analytical technique most suitable for the particular substance being

An obvious way to determine the amount of a substance adsorbed on

studied. Under favorable conditions, if the change in concentration is

the surface is to measure the resulting change in its bulk concentra-

monitored continuously (e.g., by a spectrophotometric method), it may be

tion. This is equivalent to measuring adsorption from the gas phase by

possible to follow the kinetics of adsorption, but this is rarely

determining the decrease in partial pressure of the relevant gas.

possible.

The sensitivity of such methods depends on the ratio between the (c) Electrochemical methods of determining the coverage

* This is of particular importance for the study of the adsorption of inorganic ions, such as SO 42 ; from concentrated solutions.

The adsorption of potential fuels on platinum electrodes can be measured electrochemically, as is illustrated in Fig. 8J, where the i/E relationship during a potential sweep on platinum in a pure sulfuric

326

ELECTRODE KINETICS

acid solution is represented by the solid line.

J. ELEcTROsoRP noN

327

Fig. 8J also shows two correction terms, represented by much

Upon addition of benzene and after allowing sufficient time for adsorption equilibrium to

smaller shaded areas marked B and C. These are associated with the

be attained, the dashed curve is obtained. The fractional coverage is

changes in the double-layer capacitance and the amount of oxide formed

calculated from the shaded area (marked A) between the curves on the anodic sweep.

as a result of adsorption of the organic molecules, and are not essential for the understanding of this method. The main assumption upon which this method is based is that the organic molecule is completely

I

I

I

oxidized to CO during the fast anodic sweep. Lesser assumptions are 2 that there is not enough time for readsorption of the molecule from solution during the transient, or for the molecules to be desorbed

CU RREN T

without charge transfer. The last two assumptions can be verified by studying the coverage as a function of the sweep rate used to "burn off" the adsorbed molecules, and subsequently operating in a region where 0 is independent of sweep rate. In the example shown in Fig. 8J, a value of v = 5 V/s

0

:15 0

was found to be sufficient, but for the study of methanol adsorption, sweep rates as high as 800 V/s were necessary, because of the much

U

higher bulk concentrations of adsorbate employed. In spite of several uncertainties, this method was found to be 0.3 0.7 1.1 1.5 POTENTIAL/ Volt vs RHE

Fig. 8J Electrochemical method for the measurement of the electrosorption of benzene (2 AM in 0.5 M 11 SO ) on platinized platinum. 2 4 V = 1.0 Vls. The area A yields the charged consumed in oxidizing the benzene initially adsorbed on the surface. B and C represent the change in charge associated with the formation and the reduction of the oxide, respectively, resulting from the adsorption of benzene. Reprinted . with permission from Duic, Bockris and Gileadi, Electrochim. Acta, 13, 1915, (1968). Copyright 1968, Pergamon Press.

useful for the study of the adsorption of potential fuels on platinum electrodes. A comparative study of the adsorption of benzene on platinum by the radiotracer and the electrochemical technique showed reasonably good agreement, as seen in Fig. 9J. In practice it is not necessary to start each measurement in a new solution. The application of a series of potential pulses, as shown in Fig. 10J, cleans the surface and prepares it for the next experiment. Both the potential at which adsorption takes place and the time allowed for adsorption to occur can be controlled in this way, making it possible to study both equilibrium coverage and the kinetics of adsorption as a function of potential.

328

ELECTRODE KINETICS

329

ELECIROSORPTION

1.8V 2 sec

ct 0.4 0

cc 0

V/sec

0.4V 120sec

w LLI

1.2V

0.3 '

0.12V

0.12V

15sec

10sec

F

C Z 0.2 0

Fig. 10J Potential pulses used to clean the electrode surface and pre(`;ct 0.1 Lt.

pare it in-situ for adsorption measurement. Data from Brieter and Gilman, Trans. Faraday Soc. 61, 2546, (1965). 5 10 15 CONCENTRATION/NM

be set at any desired value. Finally, in stage F a fast anodic linear

Fig. 9J Comparison of the radiotracer and the electrochemical methods for the adsorption of benzene on Pt. v = 1 Vls. Radiotracer method • • e, electrochemical method p o

G..

Data from Duic,

Bockris and Gileadi, Electrochim. Acta, 13, 1915, (1968).

sweep is applied, producing a current-potential plot of the type shown in Fig. 8J. The background current is measured by skipping stage E and going directly from D to F (i.e., by not allowing any time for adsorption to occur). An additional method for the determination of adsorption on solid electrodes by capacitance measurements, based on the theory of electro-

During stage A the solution is saturated with the hydrocarbon. Application of a high anodic potential in stage B cleans the surface of any organic matter which may have been adsorbed and leaves it covered with an oxide layer. The purpose of stage C is to remove molecular oxygen and allow the solution to reach equilibrium with the hydrocarbon. At the potential of 1.2 V, RHE, the surface oxide remains intact, and adsorption of the organic molecules cannot occur. In stage D the oxide is reduced; adsorption of the organic species is prevented in this stage by the relatively negative value of the potential. Stage

E is the adsorption stage. Both the voltage and the time during this stage can

sorption developed by Frumkin, is discussed in Section 22.2. 22. 1SOTIIERMS 22.1 General Comments An isotherm describing electrosorption can be written in general form as: f(0) = KC°g(E)

(I0J)

where f(0) and g(E) represent unspecified functions of the fractional

330

ELECTRODE KINETICS

33,

J. ELECTROSORPTION

X

coverage and the potential, respectively. Up to now we have been

0 + (1 — 7 RH,ads In

0)n

and X

W,ads

(1 — 0)n 0 + (1 — 0)n

(15J)

concerned with the form of the function f(0). The potential dependence of 0 for an adsorbed species formed by charge transfer was obtained by

Substituting these expressions for the mole fractions into Eq. 11J, we

simple kinetic or thermodynamic considerations and had the form of the

have:

Nernst equation. Here we shall discuss first the form of the function

K(E)C ° /55.4 = RH

f(0) for a large adsorbed species, and then proceed to investigate the

[[0 + n(1 — n

— 0)n

(16J)

11

shape of the potential dependence of the isotherm in the absence of The right-hand side of this equation is the function f(0) in Eq. 10J.

charge transfer.

The dependence of the equilibrium constant on the potential is discussed

For the electrosorption equilibrium described by Eq. IJ

later.

n(H 2 0) ads <

121-l

n(H 0)

ads

2

sol

(1J)

we can write the equilibrium constant in terms of the mole fractions of the species involved as: K(E) —

(X RH,ads )(X W,soI ) n (X

RH,sol

)(X

)n

How can we relate this rather complex isotherm to the Langmuir and Frumkin isotherms discussed earlier? It is easy to see that if we set n = 1 in Eq. 16J, we retrieve the Langmuir isotherm. This should not be surprising, because we have not introduced any coverage dependence of

(11J)

W,ads

In dilute solutions we can write:

1.0

X W,S01 ----- I and X

RII,sol

RH

/55.4

(12J)

a) 0.a N at cfa 0

w 0.6 0

and on the surface we have: X

RH

RH,ads

rRH+

(13J) W

0.4 0

° 0.2 LA-

The surface concentrations can be related to the fractional coverage as: 0

RH

=

OF1211,max and

rW,max = nrRH,max

Substituting Eq. 14J into 13J we obtain

(14J)

10-2

io - ' cimm

10

Fig. I IJ Langmuir isotherm for the electrosorption of a large neutral species, each adsorbate molecule replacing n water molecules from the surface.

332

ELECTRODE KINETICS

333

1. ELECTR OS OR PTION

qo = CoE and q I = C I (E— EN)

(17J)

the free energy of electrosorption in its derivation. Thus, Eq. 16J can be viewed as the Langmuir isotherm, as applied to the electrosorption of a large molecule, which takes up several sites on the surface. The dependence of 0 on logC RN is shown in Fig. 11J for three values of n. The shape of the isotherm depends significantly on the size factor n.

where C and C are the values of the double-layer capacitance at 0 = 0 1 o and 0 = 1, respectively, and E is the potential on the rational scale. The parameter EN represents the shift in the potential of zero charge caused by a full monolayer of adsorbed species.

The fractional coverage changes more slowly with concentration for

higher values of n.

The Frumkin isotherm is based on the assumption that at any value

This makes sense, since as the coverage is

of the coverage, the interface can be viewed as two capacitors connected

increased, it becomes more difficult to find clusters of empty site

in parallel. It follows immediately from this assumption that the

consisting of n elementary sites on the surface, which are needed to

charge q 0 corresponding to a given value of 0 can he written as:

accommodate the large molecules. go = q0(1 — 0) + q 1 0 = CoE(1 — 0) + C I (E — E N)0

(18J)

22.2 The Parallel Plate Model of Frumkin -

The derivation of the Frumkin isotherm is rather involved and is not We now turn to the potential dependence of electrosorption of neutral molecules, considering first the model developed by Frumkin. This is a phenomenological model, which depends on considerations of the

given here. We note only that it is based on calculating the difference in electrostatic energy of charging the double-layer capacitor with and without the adsorbed species. The final result is as follows:

changes in the electrostatic energy of the interphase caused by adsorption. Assuming that measurements are taken in concentrated solutions of Kti) = K(i=o)-exp[ — 0.5(C

a supporting electrolyte, we can neglect diffuse-double-layer effects

0

— C )E 2 + C I EE N I RTF

and focus our attention on the Helmholtz part of the double layer,

(19J)

max

considered as a parallel-plate capacitor. In the pure solvent the

where K(i=0) is the value of the equilibrium constant at the PZC. If we

capacitance C o and the corresponding charge q o are determined by the

ignore for a moment the term containing EN , we find that this isotherm

properties of water in the interphase, mainly its effective dielectric

predicts a maximum of adsorption at the PZC, with 0 declining symmetri-

constant and its dimensions (which determine the thickness of the

cally on either side. The term C I EEN accounts for the failure of the

capacitor). We have already seen that adsorption of an organic molecule

potential of maximum adsorption observed experimentally to coincide with

tends to decrease the capacitance. This property can be associated with

E.

the combined effects of a lower dielectric constant and an increase in thickness of the parallel-plate capacitor. For the charge densities q o q 1 , at 0 = 0 and 0 = 1, respectively, Frumkin proposed the equation: and

All the quantities that appear in the exponent of Eq.19J can be measured, at least in principle, thus permitting the isotherm to be tested by comparison with experiment. This is a great advantage of any theory, inasmuch as it does not contain any adjustable parameters. The

334

ELECIRODE KINETICS

J.

W-ECIMOSORPTION

disadvantage of the Frumkin model is that it makes no attempt to explain the observed phenomena on the molecular level. Thus, the values of

22.3 The Water-Replacement Model of Bockris, Devanathan, and Muller

C — C and ENare taken as such; it is not explained why they have o their observed values or how these values depend on molecular size, on the orientation of the molecules in the interphase and on the nature of

The isotherm derived by Bockris, Devanathan, and Muller (BDM) is centered around the role of the water in the interphase.

the interactions with the surface.

Electrosorption is a replacement reaction, and its standard free

The assumption underlying the derivation of the Frumkin isotherm is tantamount to assuming that the surface charge density is a linear

energy is the difference between the standard free energies of adsorption of the organic species and of the water molecules it replaces

function of coverage at constant potential, as seen in Eq. 18J. This is

from the surface (cf. Eqs. 1J and 2J). Thus, the dependence of 0 on

by no means generally correct, although it may constitute a fairly good

potential is associated with the variation with potential of the

approximation in many cases.

standard free energy of adsorption of water. The electrostatic energy

Equation 18J can be rewritten in the following form: 0 = (q — q 0)/(q i — q0)

of interaction of the water dipole t with the field F in the Helmholtz (20J)

part of the double layer is given by: U1 = i.tFcos cp

(23J)

which may be used to determine the coverage from differential capacitance measurements, if the potential of zero charge is known with

A fundamental premise of the BDM isotherm is that water molecules can take up only one of two positions in the interphase: with the dipole

sufficient accuracy to obtain the charge by integration. An approximate form of Eq. 20J is sometimes used to determine coverage on solid electrodes. Differentiating Eq. 18J with respect to

vector either in the direction of the field or in the opposite direction In other words, the angle cp between the direction of the field and the orientation of the dipole can either be zero or rc, and the energy of

potential, one has

interaction between water dipoles and the field is either 1.1F or — 1.1F, dq/dE = (dq0/dE)(1 — 0) + (dq 1 /dE)0 — (q0— q i )(dO/dE)

(21J)

depending on its orientation. From considerations of energy alone, all the water molecules should

If one neglects the potential dependence of 0, it follows from this

have their positive ends facing the surface when it is negative and vice

equation that

versa, with a sharp transition at the PZC. There are, however, several

e = (c - Co )/(C I - co )

(22J)

factors that modify this transition, making it gradual. 1.

First there is the entropy term. A situation in which all dipoles

This is a poor approximation, particularly when the purpose of the

are oriented in one direction represents the minimum entropy for this

experiment is to determine the potential dependence of 0!

system. Completely random orientation corresponds to maximum entropy. Thus, enthalpy and entropy affect orientation in opposite directions, as is often found in chemistry, and a compromise, determined by the

336

ELECTRODE KINETICS

condition of lowest free energy, is reached.

This is similar, for

example, to the equilibrium between N 204 and NO2 in the gas phase. The requirement of minimum energy would shift the equilibrium completely in the direction of the dimers, while the requirement of maximum entropy would shift it completely in the direction of the monomers. The ratio between the two types of species at equilibrium represents the compromise between these conflicting tendencies. In the electrochemical

-44111 -114141‘

example, the resulting orientation of water molecules in the interphase depends both on temperature and on potential (or charge). The parameter Z, is defined as: Z—

I\14 — Nt

(24J)

NT

337

L

1. FLECTR OS OR PT1ON

1

-16 -12

I

I

1

L_____

-8 -4 0 4 8 CHARGE DENSITY q m /FLC.cm -2

1

J

12

16

where N1' and NI' are the numbers of water molecules per square centimeter in the two allowed orientation, depends on charge density, as shown

in the interphase with charge density. (I) only the energy of

schematically in Fig. 12J. 2.

Next there is a lateral interaction term.

Fig. 12J Schematic representation of the variation of water orientation

If left alone (i.e., in

the absence of an external field), dipoles will pair up to minimize their total energy of interaction, yielding Z = 0.

This opposes the

tendency of the field to orient all the dipoles in one direction, making

interaction between the dipole and the field in the d.l. is considered (2) an entropy term is added (3) lateral interactions considered (4) the difference in chemical energy of adsorption in the two orientations is also taken into account.

the transition from one orientation to the other even slower. 3.

Finally there is the chemical interaction of water with the

surface.

A water molecule cannot be regarded as a structureless

electric dipole. It is more strongly adsorbed when oriented with the positive end of the dipole (i.e., the hydrogen atoms) facing the

We shall again skip the tedious part of deriving the isotherm and write the final result in simplified form as follows: K(E) =

K(E=o).exp[— nZ(

11F kT Za )]

(25J)

surface. Thus, at q m = 0 the parameter Z is positive; to force Z to be

It is interesting to consider the physical origin of the various terms

zero, one must apply a negative charge to the electrode to overcome the

in this equation. If we remove two oppositely oriented water molecules

small difference in the chemical energy of adsorption of water in the two orientations. This charge is q m = — 2 1.1C/cm 2 , which corresponds to the potential of maximum adsorption, as we shall see.

from the surface, the net change in electrical energy will be zero. What has been gained by removing one molecule will be lost by removing the other. The term nZ is the number of water molecules removed from

338

ELECIRODE KINETICS

J. ELECTROSORPTION

the surface for each organic molecule adsorbed, multiplied by the

This is the BDM isotherm, not taking into account the small shift

fractional excess of water molecules oriented one way or the other. The

between the potential of zero charge and the potential of maximum

parameter Z is defined in such a way that the product Z(pF) is always

adsorption, which arises from

positive, leading to a symmetrical decrease of the equilibrium constant

interaction of water with the surface in its two allowed orientations.

the different chemical energies of

(hence of the coverage) on both sides of Z = 0. The parameter 6 is the

Equation 27J is not in a form that can be conveniently used by

lateral interaction coefficient. It is the total electrical energy of

electrochemists. We would like to express the surface coverage in terms

interaction of one water dipole with all other dipoles surrounding it.

of the charge or the potential, rather than the field. The field is

It is also multiplied by Z, since interactions with oppositely oriented

related to the charge through the Gauss theorem, namely

dipoles cancel each other.

F = — 4rrq mie

The complete isotherm may have the form:

o

(10G)

If we prefer to write the BDM isotherm in terms of potential, we can use the relationship

+ no -

Lo -

F = (E — Ez)/5 = E/5

= K(i=o)-C ° -exp EnZ( IF — ZG RH kT

(26J)

where we have included the numerical factor of 55.4 in the equilibrium

(28J)

in which 5 is the thickness of the Helmholtz double layer. This leads to the BDM isotherm in which 0 is expressed as a function of potential, namely:

constant. Although this may look rather complex, it is a simplified — ZG ) 1 f(0) = K(E=o).C ° •expE nZ( /1E/8 kT RH

form of the isotherm. Thus, the function f(0) on the left-hand side of

(29J)

Eq. 26J does not take into account changes in the standard free energy of adsorption due to surface heterogeneity or to lateral interactions * between the adsorbed molecules. Since the unique feature of this isotherm is the potential dependence of the coverage, we prefer to write it in the following form: f(0) = K(.,-=o)•C: 14•exp

Equation 28J is not quite correct, since it implies that the absolute metal-solution potential difference at the PZC is zero. In other words, it implies that we can measure the absolute metal-solution potential difference, which is not the case, as we showed in detail at the outset

nZ(

k— T ZG ))

(27J)

of this book. The error is, however, only a constant, which can be lumped into the equilibrium constant and has no effect on the potential dependence of 0. The Erunikin isotherm (cf. Eq. 19J), can be written in the following

* Ihe parameter G on the right-hand side relates to lateral interactions between the water dipoles, not between molecules of the adsorbate.

form:

340

ELECTRODE KINETICS

f(0) = KtE=o)•C R° H .exp[—

0.5(C — 0

I

E2

RTr

Both isotherms can predict

qualitatively

I

-E'E N

341

J. ELELI tOSORPTION

00FI

sol

> (1)0Hads

1101 0) 2 ads

(31J)

11(1i o) 2 sol

(30J)

max

the results obtained experimen-

0011

where

sol

Il(11 0) 2 ads

>

ads

+ n(H2o)sol

H+ +

M

(32J)

It is evident that the

represents the benzyl radical

tally. A detailed study, which would allow us to decide which is the

calculation of the dependence of 0 on potential for the process shown in

"better." isotherm — namely, which fits the experimental results more

Eq. 32J, should take account of both charge transfer and the effect of

closely and for a larger number of systems — has unfortunately not been

competition with water. This leads to the

performed. It should be noted that a study of the potential dependence

derived by Gileadi.

combined adsorption isotherm

of 0, such as shown, for example, in one of the curves in Fig. 1J is not

A simple equation results if we use the BDM isotherm for potentials

enough, as long as we do not know the exact fcu'm of the fUnction f(0).

far removed from the potential of maximum adsorption. In this case the

This problem can be avoided if we measure the isotherm as a function of

high field aligns all water molecules in one orientation, leading to

both concentration and potential, and determine the partial derivative

Z = ±1. The term Zu in the exponent of Eq. 27J will be nearly constant

(a log C: i ja In other words, these isotherms can best be tested by

and small compared to pf, so that this equation can be simplified to:

determining the concentration in solution required to reach a certain value of 0, as a function of the applied potential. We may not know the

f(0) = KC ° •exp(— n.tF/kT)

(33J)

000

exact form of the function f(0), but we can be sure that it has a fixed value for each value of 0 chosen. Thus, the fact that f(0) is not known does not hamper our ability to test the validity of Eqs. 29J and 30J.

0 ° ods

22.4 The Combined Adsorption Isotherm of Gileadi In Section 19 we discussed the isotherms applicable to adsorbed intermediates formed in charge-transfer processes. In Sections 22.2 and 22.3 we focused our attention on the potential dependence of electrosorption of neutral species on an ideally polarizable surface. What happens if both processes occur simultaneously? In Fig. 13J we show schematically the electrosorption of phenol as such and as a phenoxy radical on a platinum electrode. The two reactions can be written, respectively, as follows:

Fig. 13J Schematic representation of the potential dependence of the electrosorption of phenol and of the phenoxy radical on Pt, at three concentrations in solution.

342.

ELECTRODE KINETICS

where the equilibrium constant K includes the factor exp(ZG/kT).

The

exponent in this equation can be rewritten (cf. Eq. 28J) as: n. [t F _ (n• 1.1)( kT k

_

(tilt ve • Nv vETS-

J. ELECTROSORPTION

1.

343

In Eqs. 31J and 32J we used the same value of the size parameter n.

This assumption may not be generally correct, since the species being adsorbed are not identical. Even if they do not differ in actual size (34J)

hence

(the removal of one proton does not change the size of a phenol molecule significantly), their orientation on the surface could be quite diffe-

n.pt v nkT F — e.ti

(35J)

rent, and the number of water molecules replaced may not be the same. This is not relevant to the derivation of the combined adsorption

where e is the unit charge of the electron and N is Avogadro's number. Using 1.4. = 1.8x10 18 esu and 5 = 2.7x10 -8 cm for the dipole moment of

isotherm, since we are dealing with the process shown in Eq.32J only.

water and its diameter, respectively, and e = 4.8x10 1° esu, we have:

of water, taken from gas-phase measurements, may be criticized. Mutual

2.

The numerical value of vt = 1.8x10 18 esu used for the dipole moment

depolarization of the closely packed dipoles on the surface may lead to f(0) = KC ° •exp

0.14n(EF/RT)J

(36J)

Note that we have replaced here the rational potential E by the potential E, measured versus some reference electrode. The difference between them is a constant, which we can lump into the equilibrium

a smaller effective value of g. Also, taking the diameter of a water molecule to represent 6, the thickness of the Helmholtz double layer, probably constitutes an underestimate. The use of more accurate values for these parameters may decrease the numerical parameter in Eq. 36L from 0.14 to perhaps 0.10. Since n is an adjustable parameter (within

constant. The other equation we need to use in this case is the isotherm

certain limits), this does not affect our considerations. It may be argued that the field required to make all water dipoles

describing charge-transfer equilibrium (cf. Eq. 19F), which can be

3.

written as:

turn one way is too high and may not be encountered in practice in most cases. Fortunately, it is enough to assume that the variation of Z with f(0) = KC; 011 (CHo i ) 1 •exp(EF/RT)

(37J)

potential is negligible, even if its absolute value is less than unity, to arrive at the correct form of the combined isotherm. Considering

Combining Eqs. 35J and 36J we have

Fig. 12J we note that this occurs well before full orientation is reached. Having I Z I < 1 may modify he numerical value of 0.14 used in f(0) = KC OH(C° ) 1 •exp[(EF/RT)(1 — 0.14n)) y H+

(38J)

Eq. 38J but, as we have noted above, this is not very important. Also, if the assumption that liF » Za used to arrive at Eq. 33J does not quite hold, the equation is still correct, as long as the variation of Z with

This is the combined adsorption isotherm. As usual, we have made some approximations in arriving at this result. A few points of clarification are therefore in place.

potential is negligible. It is interesting to note how the two separate regions in Fig. 13J For electrosorption of the depend on concentration in solution.

344

ELECTRODE KINETICS

unreacted phenol molecule, the curves shift vertically, almost in

345

J. ELECTROSORPTION

Consider a reaction sequence similar to Eqs. 8F and 9F, such as

parallel, as seen also in some of the experimental results presented Brs

earlier (cf. Fig. 3J to 6J) The maximum value of 0 depends on concentration, but the position of the maximum along the potential axis is

ot

+ n(H 0) 2

shown. The curves shift with concentration in parallel, in the horizon-

ads

+ n(H 0) 2

sot

+e

(39J)

M

followed by

essentially independent o it. For the adsorption of a phenoxy radical, the opposite behavior is

Br

ads E

Br

ads

+

Br

sot

+

l 0 2 sot

rds

) Br

2,sot

rl(H 0) 2 ads

e

M

(40J)

For the first step at quasi-equilibrium, we can write:

tal direction. Full coverage is reached at each concentration, but the potential corresponding to any given coverage changes with concentration according to a Nernst type equation (cf. Eq. 19F). The difference between the two types of behavior shows clearly that the potential

[ 1

0

xp(EF/RT)(1 — 0.1411)) 01 = KC ° (e Br-

(41J)

For the rate determining step, we have:

dependence of 0 observed in the two regions is due to entirely different i = kC° 0.exp((3FE/RT)exp[0.14n((3FE/RT) )

physical phenomena.

Br-

(42J)

The potential dependence of the coverage by adsorbed intermediates should, as a rule, be discussed in terms of the combined adsorption isotherm. The importance of the use of this isotherm grows with the size of the adsorbed species. Thus, the effect is small and may be negligible when we consider small species, occupying only one site on the surface (e.g.,

H ads

and °H ad). It becomes predominant when n is

larger than 2 or 3.

In step (40J) an adsorbed

from the specirmovd surface.

The

standard free energy of adsorption depends on potential as a result of the effect of competition with water (in addition to the direct dependence of 0 on potential due to charge transfer). The standard free energy of activation depends on a fraction I3 of the same potential. This is the reason for the introduction of the term exp[0.14n((3FE/RT)] in Eq. 42J.

22.5 Application of the Gileadi Combined Adsorption Isotherm to Electrode Kinetics

Assuming 0 « 1, we can substitute 0 from Eq. 41J into Eq. 42J, to obtain, for the overall current density i

We saw earlier (cf. Section 19) that the potential dependence of adsorption of intermediates formed by charge transfer affects the

i = 2FKk(C B° r ) 2exp((3FE/RT)exp [0.14n ((3FE/RT)] exp [(EE/RT)(1-0.14n))

kinetics of electrode reactions. We have worked out the kinetic parameters for a few mechanisms under so-called Langmuir and Temkin conditions (i.e., when the Langmuir and the Temkin isotherms are applicable, respectively). Here we shall derive the appropriate kinetic equations for the combined adsorption isotherm.

= 21-71(k(C0r ) 2 .exp(EE/RT) [( 1 + (3) - (1 - 13)0.14n) The Tafel slope, taking 13 = 0.5 (as in all earlier calculations) is:

(43J)

346

ELECTRODE KINETICS

b = (2.3RT/F)(

1 ) 1.5 — 0.07n

(44J)

Table 2J Tafel Slope (mV) as a Function of the Number of Water Molecules Replaced from the Surface, for Two Different Mechanisms.

If an atom-atom recombination step is rate determining, namely Br

ads

+

Br

ads

+

2n(II 0) 2 sol

s

Br

2,sol

2n(H 0) 2 ads

(45J)

the rate equation is i = k02.exp [0.14n(2(3FE/RT)]

34/

.1. ELECTROSORPTION

(46J)

0

1

2

3

4

5

401

39.3

41.3

43.4

45.7

48.4

51.3

45J

29.5

31.7

34.3

37.3

41.0

45.4

rds

Here again the rate depends on potential, although no charge transfer is involved in the rate-determining step, through the dependence of 0 on

universally recognized. For larger molecules the effect is much more

potential, resulting both from charge transfer and from the effect of

significant. For example, if n = 4, we find a slope of 41.0 mV for

competition with water. Assuming that 0 « 1, we can substitute from

atom-atom recombination as the rate-determining stem, which could easily

Eq. 41J and obtain, for the overall reaction:

be mistaken for the 39.3 mV slope predicted for atom-ion recombination

i = 2FK 2k(C° 2 .exp(2FE/RT)[1 — (1 — 13)0.14n] Br )

(47J)

The best example reported so far, showing the effect of the size of

The corresponding Tafel slope is: b = (2.3RT/F) [

step as the rds for n = 0. the molecule on the resulting Tafel slope is presented in Fig. 141, for

1 2 — 0.14n

(48J)

the reduction of a series of aliphatic nitroalkanes, starting from nitromethane. The mechanism of this reaction is rather complex and need

Some values of the Tafel slopes calculated for these two mechanisms for

not concern us here. The important thing to notice is that b increases

different values of the size parameter n are shown in Table 2J.

systematically by about 5 mV for each additional carbon atom (which

It should be noted that the Tafel slopes just given were calculated

probably corresponds to an increase of n by unity). The difference is

for the combined isotherm under Langmuir conditions, namely at very low

small, but well outside the limits of experimental error. One must

coverage. The same type of calculation can be repeated to obtain the

assume in this type of study that the mechanism is the same for all the

kinetic parameters for different mechanisms both at low and at interme-

molecules being compared in a given homologous series. This does not

diate values of the coverage. The effect on the Tafel slope of competi-

seem unreasonable when comparing CH 3 NO2 to C2H5 NO2 to (CH3) 2CHNO2.

tion with water is rather small for small molecules. Thus, for n = 1 the Tafel slope changes only by about 2 mV for the two mechanisms just

In any event, a change in mechanism is expected to be accompanied by a large change in the kinetic parameters (e.g., with b changing from 60 mV

discussed. This is within experimental error in most cases, perhaps

to 120 mV) and should be easy to detect.

explaining why the need to use the combined adsorption isotherm is not

348

ELECTRODE KINETICS

349

K. EXPERIMENTAL TECHNIQUES: I

DIMEN SI ONL E SSRATECON S TA N T X

K. EXPERIMENTAL TECIINIQUES: rl 23. FAST TRANSIENTS

lo °

23.1 The Need for Fast Transients In this chapter we shall focus our attention on the use of transients for two purposes: for the separation between activation and

to -'

mass-transport-controlled processes and for the study of short-lived intermediates formed in a reaction sequence. The need to enhance the rate of mass transport is obvious. Looking again at Eq. 3A we note that the measured current density becomes equal to the activation-controlled current density if the latter is small 10-3

o

-50

-100

-150

- 200

compared to the mass-transport-limited current density.

POTENTIAL,E-E(i/idr10 -3 )/mV

l/i = 1/i

Fig. 14J Tafel plots for the reduction of nitroalkanes at the dropping

ac

+ 1/i

L

(3A)

mercury electrode. pH 6.65. (I) CH 3NO 2, b = 53 mV; (2) C H NO ,b = 58 mV; (3) (CH3)2CHNO 2 ,b = 62 mV. Reprinted 2 5 2 with permission from Kirowa-Eisner, Kasha and Gileadi,

The problem is aggravated when the reaction is fast, as shown in

Electrochim. Acta, 32, 221, (1987). Copyright 1987, Pergamon Press.

the reverse reaction can be neglected. This condition corresponds

Fig. 1K. To understand this figure we recall that a linear Tafel plot is expected only at some distance from the equilibrium potential, where roughly to (0)

1 or to (0 0)

10. At the high end, mass transport

becomes significant when (i/i L) 0.1. Thus, the linear Tafel region is expected to extend roughly between these limits, namely when 10i i 0.1i . A favorable case is shown in Fig. 1K(a). In 0 plotting this figure it was assumed that i L/i. = 105 , thus leaving about three orders of magnitude of current density over which the plot of

versus log i is linear. In Fig. 1K(b) we show a more difficult case, from the experimental point of view. Assuming a value of io which is

These limits are somewhat arbitrary, since they depend on the accuracy desired.

....(7TKODE KINETICS

K. EXPERIMENT,

ions, and is relevant only when the reacting species is charged. It is Slow reaction linear Tafel region

a -6

-5

-4

-3

-2

-1

0

1

customary to minimize this mode of mass transport in research, by the use of a large concentration of supporting electrolyte. This reduces the electrical field in the bulk of the solution and diminishes the

Fast reaction

fraction of the electricity carried by the reacting ions (i.e., their transference number). It should be remembered, though, that this is not the only reason for using a supporting electrolyte. It is also impor-

linear Tafel region

b -6

-5

-4

-3 l

-6

-5

' -4

-2

-1

layer effects. Furthermore, the presence of an excess of supporting

linear ; i Tafel 'region

0

-3

-2

tant in reducing the iR s potential drop in suppressing diffuse double electrolyte ensures a constant ionic strength in solution and particu-

0

1

log i/A•crn

Fig. I K The linear Tafel region as related to the ratio between i and i 0 for (a) slow and (b) fast electrode reactions.

larly in the Nernst diffusion layer at the electrode surface, where a * concentration gradient of the reactant exists. Mass transport by convection is associated with gross movement of the solution, usually caused by the input of mechanical energy (stirring, pumping) or by gravity (as a result of density gradients). Convective mass transport is of utmost importance in industrial pro-

two orders of magnitude higher than in Fig. 1K(a), we note that there is

cesses and is also used in research, notably in the case of the rotating

at most one order of magnitude of current density over which the Tafel

disc electrode. Unfortunately, the rate of mass transport cannot be

plot is linear. This is not sufficient, as a rule, to obtain reliable

calculated in most cases and is not easy to reproduce experimentally.

values of the kinetic parameters.

All one can achieve by stirring the solution, for example, is "good

The linear Tafel region can be extended by increasing the limiting

mixing" which eliminates gross concentration changes in solution but

current, as indicated in Fig. 1K(b). This can he done by better

does not provide a well-defined and reproducible limiting current.

stirring, or by employing short pulses, as discussed later. One may be tempted to increase the ratio by increasing the concentration,

Diffiision is caused by a gradient in chemical potential (which in most cases is approximated by a concentration gradient). It is predo-

since the limiting current is always proportional to the concentration.

minant when the migration and convection modes of mass transport are

This does not necessarily work, however, since the exchange current density also increases with concentration, often to the same extent. Now, mass transport can occur in solution by three principal mechanisms: migration, convection and diffusion.

Migration is caused by the effect of the electrical field on the

*

This is essential for the application of the diffusion equation in simple form, ignoring the effect of concentration on the diffusion coefficient D.

352

ELECTRODE KINETICS

K. EXPERIMENTAL TECHNIQUES: I

353

• a. •• -

carefully eliminated, by proper choice of the experimental conditions.

in which S is the Nernst diffusion layer thickness. We have discussed

One commonly thinks of diffusion in a quiescent solution, at relatively

this quantity in Section 9.4 For mass transport controlled by diffu-

short times (of the order of tens of seconds), but the time can be

sion, S is proportional to the square root of time and can be written in

longer when microelectrodes are used. Moreover, diffusion can be the

the following form:

overwhelming factor in mass transport even in stirred solutions, if measurements are taken at a sufficiently short time, of the order of 10 3 s or less. The equivalent circuit for a system in which diffusion can play a

Thus, the diffusion-limited current density i d decreases with the square root of time:

i

significant role is shown in Fig. 2K. The symbol —W— is the Warburg Impedance, which accounts for mass transport limitation by diffusion. It is advantageous to conduct measurements under conditions such that diffusion is the sole mode of mass transport, because the diffusionlimited current density idcan be rigorously calculated, and a proper

(9D)

8 = (nD0 112

nFDC ° d

—

t

(1K)

I/2

Taking measurements at shorter times (i.e., using fast transients), has more the same effect as increasing the rate of mass transport by efficient stirring. Table 1K shows the time required to reach a certain

correction can be applied, to obtain the activation-controlled current density as a function of potential. What is to be gained by using fast transients?

The limiting

current density i L is given by i = n FD C°/8

Table I K . Comparison of Values of S Obtained by Different Methods to Those Obtained by Fast Transients

5 (pm)

Natural convection

150 — 250

7 — 20

50 — 100

0.8 — 3.2

RDE at 400 rpm

25

0.2

RDE at 104 rpm

5

(4A)

Magnetic stirrer Fig. 2K Equivalent cir-

C dl

Corresponding time for transient (s)

Type of stirring

cuit including a Warburg impedance Z , in w series with the fara-

RS

Fast impinging jet

daic resistance R , to account for mass transport limitation by diffusion.

R

F

zw

2—5

8x10 -3 (1.3 — 8)x10 3

Graphite fiber microelectrode-

3.5

4x10-3

Ultramicro electrode

0.25

0.02x10 3

354

ELECTRODE KINETICS

value of S (according to Eq. 9D) as compared with typical values observed under different conditions of stirring. We note that the best

355

K. EXPERIMENTAL TECHNIQUES: I

If the system is perturbed by a small signal Art, the resulting current can be expressed by the relation:

method of mass transport by convection can be equalled by a pulse of about 1 ms duration, which is very easy to implement experimentally.

i + Ai = io.exp[a(i + An)F/RT)(

3K)

Table 1K shows the strength of fast-transient methods in the study of electrode reactions. Their limitations, both from the experimental and the theoretical points of view, are discussed shortly. Table 1 also

Subtracting Eq. 2K from 3K one has: Ai = io .exp(ocriR/RT) [exp(a4F/RT) — 1)(

4K)

includes a comparison with microelectrodes, to show the potential of transient techniques. The basis for this comparison is discussed in

For a sufficiently small perturbation, ocAnF/RT « 1, and the above

detail in Section 23.5.

equation can be linearized to yield

Ai =

23.2 Small-Amplitude Transients Transient measurements can he of two types: small-amplitude transients, which give rise to a linear response and large-amplitude transients, which result in a nonlinear, often exponential, response.

which is similar to Eq. 51F.

aF ) .An

(5K)

The relative sensitivity Ai/i is propor-

tional to the perturbation An and is independent of the steady-state current density or the overpotential.

We have already seen (cf. Sections 12.4and 14.7) that a system at

The result shown in Eq. 5K should not be surprising. The physical

equilibrium responds linearly to a small perturbation in potential or in

meaning of this equation is simply that even when the relationship

current, according to the equation

is exponential, a small interval of this curve, near any given steady

i/i = (n/v)(11F/RT) 0

(51F)

state value, can be linearized. A small current or voltage perturbation also implies that the

A "small" perturbation in this context is one for which miF/vRT « 1 or

changes in concentration of the reactants and products near the elect-

i/i « 1. The linearity of the response allows easier and more rigorous 0 mathematical treatment and is, therefore, often preferred. It is

rode surface are small and the associated equations can be linearized

interesting to note that a linear response is also obtained when a small

Large perturbations of the potential or the current are treated

perturbation is applied to a system far away from equilibrium. To show

quite differently. The most common example of a large perturbation

when appropriate, to simplify the mathematical treatment.

this, we write the usual rate equation for an activation controlled process in the linear Tafel region (cf. Eq. 7F) namely: = ioexp(om F/RT)

* Note that r1 and i in Eq. 5IF have exactly the same meaning as (2K)

Eq. 2K represents the current-potential relationship at steady state.

Afl

and Ai in Eq. 5K, except that in the former thte overpotential and the current density before the transient are zero.

356

ELEC I RODE KINETICS

signal is the linear potential sweep or cyclic volta discussed in Section 25.

►

metry, which is

23.3 The Sluggish Response of the Electrochemical Interphase

K. EXPERIMENTAL TECHNIQUES: I

time.

357

The linear Tafel region, in which the condition that

10 i i 0.1i applies, is marked by the shaded area. The steady° state limiting current density, which is measured for this reaction on a rotating disc electrode operated at 1x10 4 rpm is also shown, for

Consider the electrochemical oxidation of H , a crucial reaction in 2 many types of fuel cells. The solubility of H2 in aqueous solutions is

comparison. It is clear that the limiting current density for this

rather low, (of the order of 0.1 mM), and the diffusion coefficient is D = 1.6x10 5 cm2/s. The diffusion limited current density for this

can be performed only if very short transients are employed.

reaction, calculated from Eq. 1K is shown in Fig. 3K, as a function of

nano-seconds to study this reaction. This is unfortunately not possible

reaction is generally low, and measurements in the linear Tafel region It would be nice to use very short pulses, in the range of because of the sluggishness of the interphase, which is due to the need to charge the double-layer capacitor. If we wish to change the potential by, say, 10 mV, we need to add a charge of q = CdI Orl = 20 1.1F/cm2x10 mV = 200 nC/cm2

(6K)

U

For this change of potential to occur in 1 ns, an average current -U

density of 200 A/cm2 is required. This is clearly impractical. Thus,

0)

the slow response of the electrochemical interphase is due to the large

0

value of the double-layer capacitance. Extending the pulse duration to 10 Its reduces the average current density to 0.02 A/cm 2 . A current step of this magnitude and duration can readily be applied. Unfortunately, —8

—6

—4

—2

0

log t /sec

Fig. 3K A plot of log id for the oxidation of molecular hydrogen, as a function of log t. C° = 0.1 mM; D = 1.6x10 -5 cm 21s, i = 2x10 -5 A/cm 2 . Shaded area shows the region where a linear

the limiting current will have decayed in 10 i_ts, in accordance with Eq. 1K, to about 0.014 A/cm 2, limiting the linear Tafel region to less that a decade, as seen in Fig. 3K.

23.4 How to Overcome the Slow Response of the Interphase

Tafel relationship can be expected. Due to the low solubility of H 2, the linear Tafel region is very limited in this case,

(a) Galvanostatic transient

since it is not practical to conduct potential-step measurement with transients shorter than about 10 PS.

ship for a given reaction, over a wide range. We can apply a series of

Imagine that we wish to determine the current-potential relationgalvanostatic steps and observe the steady-state potential corresponding

ELECTRODE KINETICS

359

K. EXPERIMENTAL TECHNIQUES: I

to each current density. The trick, as we can see from Fig. 3K, is to

Let us consider several ways to separate the activation from the

do the measurement rapidly, before diffusion limitation starts to play a

diffusion-controlled process, and to evaluate the kinetic parameters of

role. On the other hand we must wait long enough for the double layer

an electrode reaction.

to be charged up to its steady-state value, when the potential across it is given by iR F.

The results of this type of experiment, for different

(b) The double - pulse galvanostatic method

kinetic parameters, are shown in Fig. 4K. The parameter ettd T in

Consider the equivalent circuit shown in Fig. 2K, ignoring, for the

Fig. 4K is discussed later. Suffice it to note here that the relaxation

moment, the Warburg impedance. When a galvanostatic pulse is applied to

time for the activation-controlled process to the exchange current density while

T

is inversely proportional

such a circuit, the response is that shown in Fig. 5M. The equation

I'd ,

the diffusional relaxation

describing the change of overpotential with time during the transient is

time, is independent of i . Thus, a large value of T /t d indicates that

0

the electrode reaction is slow, and vice versa. 1.0 6 C dl

8

Rs

iRF

0.5

5

Fig. 4K Variation of the

—AAWAA--

overpotential with time

R

4

during a galvanostatic transient, for different

0 iR s

8 3 3

values of the parameter T /T . c

d

When this ratio

4

2

reaction can be conside-

Fig. 5K The response of the overpotential to a galvanostatic transient under purely activation controlled conditions. Tc = RFC,H is

red to be under purely

the characteristic relaxation time for the activation

activation control.

controlled process. Note that this figure is presented in

exceeds about 1x10 3 , the

0

2

6

4 / -Tc

8

10

dimensionless form, as TIM . versus HT . Consequently, it is independent of the values of the circuit elements of the equivalent circuit shown.

-a

360

ELECTRODE KINETICS

= 1141 — exp(— t/t)] + iR in which

T

s

361

K. EXPERIMENTAL TECHNIQUES: I

(7K)

is the relaxation time for charging the double layer, given

iP

by 'C

c

R Cd1 F

(8K)

a

and the term iR

is the ohmic overpotential (i.e., the residual ohmic s potential drop between the working electrode and the point in solution 0

where the reference electrodes, or the tip of the Luggin capillary

1.0

t/T c

leading to it, is located). There is nothing to be gained by simply increasing the applied current, since as we have seen, the relaxation

1.2

1 1 /i2

time is independent of current. In fact, all transients taken at any 1.0

however, to charge the interphase faster, by applying two consecutive Fig. 6K.

100 90

It is possible,

pulses, the first substantially larger than the second, as shown in

Double galvanostatic

110

current density are described by the single line shown in Fig. 5K, which is plotted in dimensionless form, as TV% versus ?Pr .

30

2.0

0.8

8

gs. N 0.6

b

Galvanostatic

The idea behind this method is to charge the double-layer capacitance rapidly with a large current pulse

i

and interrupt this pulse

0.4

just as the overpotential has reached the correct value, corresponding to steady state at the lower current i, namely when 11 = = iR F . One

0.2

does not know this "correct" value of the overpotential, of course, since it is the quantity being measured. This disadvantage can be

0

overcome by trial and error, as shown in Fig. 6K. An overshoot or an

0.5

1.0

1.5

2.0

2.5

30

t/T c

undershoot can be detected and the correct value of the ratio i /i can be found. The reader should perhaps be warned here that life is not as easy as might be inferred upon viewing the calculated transients shown in

Fig. 6K Graphical representation of the double-pulse galvanostatic method. (a) the pulse shape (b) the response, calculated for ,Tc /I d =

100. A single pulse galvanostatic transient,

between the instruments and the electrochemical cell and saturation of

calculated for the same value of ti /'td, is also shown for c comparison. The optimal ratio of currents in this particular

the input amplifier as a result of a large iR s potential drop, may

case is i li = 100.

Fig. 6K. Factors such as electronic noise, poor impedance matching

362

ELECTRODE KINETICS

363

K. EXPERIMENTAL TECHNIQUES: I

1.0

distort the pulse and make the determination more difficult and sometimes impossible. Chemical factors, such as modification of the surface during a pulse, may also make it hard to chose the correct value of

0.8

i /i. Most of these problems can, however, be overcome by following correct experimental procedures, and the double pulse method can yield 8 g-

very useful kinetic data for fast reactions. (c) The coulostatic (or charge-injection) method

0.6

0.4

Consider the application of a very short current pulse to the interphase. The charge q = i t injected during the pulse changes the

0.2

potential across the double-layer capacitance by

`6'E = qpiCar Starting from the equilibrium potential, this will be equal to the overpotential

no, as seen in Fig. 7K. We use the subscript zero in this figure, since

6

4/

this is the initial overpotential (corresponding to t = 0) in the decay

-

8

10

1- c

transient studied in a coulostatic experiment. We shall treat the simplest case, in which the pulse duration is very short compared to the time constant for charging the double-layer capacitance

(tp /tic « 1), and diffusion limitation can be ignored.

Under such conditions no faradaic reaction takes place during the

Fig. 7K Charge injection followed by open-circuit decay (the coulostatic method). Diffusion limitation (low values of c lid) slows down the decay transient, as expected from the equivalent circuit shown in Fig. 2K.

charging pulse. Once on open circuit, the capacitor will be discharged through the faradaic resistor, R F.

It is easy to derive the form of the

Remember that this is an internal current, since the the decay transient is followed at open circuit. It is interesting to note, in this

decay transient. On the one hand, the current is given by:

context, that during the charging pulse, Col and R F are effectively i = — Cdi (di/dt)

(9K)

circuit elements must be considered to be connected in series, and the

and on the other hand it is also given by

same (internal) current is flowing through both.

i nF • n i=

= n/R F v • RT

connected in parallel, while during open-circuit decay the same two

(10K)

The above equation can be combined and written in the form

ELECTRODE KINETICS

364

K. EXPERIMENTAL TECHNIQUES: I

365

4.

11

2.

t

(11K)

dlnri = — (1/x c )1 0dt

Electrode reactions can be studied in poorly conducting solutions, since there is no error due to the iR potential drop in the course of

s

o

an open-circuit measurement. This feature may he particularly useful

which shows that the overpotential decays

exponentially with time,

following the equation:

for studies in nonaqueous solutions and at low temperatures. Although this is fundamentally correct, there are practical limitations to its

= 11 0-exp(— t/T e)

(12K)

applicability. To give an extreme example, one cannot follow the open-circuit decay of potential over a range of 10 mV, if the

iRs

The relaxation time ti has been given above as R FC d I . Substituting the value of R into this expression we can write in terms of the

potential during the pulse is, say, 10 V.

exchange current density as follows:

relaxation time of the reaction that can be studied can perhaps he

The relationship between the solution resistance and the shortest

F

(13K)

s IZT/F,

t o = (vin)(

/(

The relaxation time can be obtained according to Eq. 12K from the slope =q , of a plot of Ion versus t. The interceptat t = 0 yields 11 0 P is tile experimentally controlled q is obtained, since C from which dl

P

'S C. parameter. With C di known, R F and io can readily be obtained from If diffusion limitation is considered, the overpotential decays

clarified by the following numerical example. Consider a small electrode of 0.05 cm 2 , for which Cdi = 1.0 i_tF, and assume that the charge injected is 0.01 .tC/cm 2 , yielding a value of n

o = 10 mV. employing high quality instrumentation one can measure the decay of overpotential with sufficient accuracy if iR s is not more than, say, 100r1 0 . All this can

be expressed by the inequality (Cdl11 0/tp)Rs S 100.n 0 or tp

more slowly, as shown in Fig. 7K. This should be evident, since the

Cdi Rs/100

(14K)

Warburg impedance —W— is added in series with the faradaic resistance In this case the plot of logn versus t is not linear and a much .

in which the expression in parenthesis is the charge injected, divided by the pulse duration, namely the current during the pulse. The choice of 100 Ti is appropriate to present day instrumentation. We might have

equations, must be applied to calculate the kinetic parameters.

used 10 rl a decade ago and perhaps 500 i will be more appropriate 10 o o years hence. The reasoning will endure, however.

RF more complex mathematical treatment, taking into account the diffusion

The unique feature of the coulostatic method is that measurement is

o

made at open circuit. This leads to two important consequences. Since the charge is injected in a very short time (< 1 gs),

1. measurement often can be completed before diffusion limitation has In this respect the charge-injection (coulostatic): become significant.

Equation 14K sets the lower limit of the pulse time, for a given solution resistance. The upper limit is set by the requirement that it

be

very short with respect to the specific relaxation time for the reaction being studied. We may choose this limit as

method is similar to the double-pulse galvanostatic method, except that

t

one has more freedom in the choice of the parameters of the pulse, since there is no need to match it to the second pulse.

The

0.01T

last two inequalities can be combined to yield

(15K)

E.LECFRODE KINt i ICS

(C dl R1100 ) S

tp

(t c/100)

(16K)

X. EXPERIMENTAL its.. i ..QUL

determines the time at which diffusion limitation will become important. This is given by

Inserting the preceding numerical values, we can show the effect of the solution resistance on the limits of applicability of the coulostatic method, as given, for example, in Table 2K.

T

1/2

la C =

dI

\

n F)

2

V

0x C ° D "2 O x Ox

R

(17K)

C ° D I/2 R R

Table 2K The effect of solution resistance on the minimum pulse duration and the maximum value of the exchange current density measurable in a coulostatic experiment. R s (f)

t (s)?.

10

10-7 10 -4

10 4

i (A)5 10 -

ti ° (s)—

io(A/cm 2 )

10 5

1 .0

10.. 2

where the subscripts "Ox" and "R" refer to the stoichiometric coefficients v, the concentrations C ° and the diffusion coefficients D of the oxidized and reduced forms, respectively. This rather cumbersome equation can be simplified if we assume that the concentrations and the diffusion coefficients of the oxidized and the reduced form are equal and vox = v = 1. The diffusional relaxation time then takes the R following simplified form:

1 0- 2

Thus, even though measurements are taken at open circuit, it is clearly advantageous to use highly conducting solutions whenever possible, and this becomes essential if we wish to study the rate of fast reactions. These considerations become even more critical when the effects of diffusion limitation are included, as we shall see.

1/2

ti^=

2C

RT (n F) 2

(18K)

C ° 1) 1/2

If, on the other hand, one of the two species is predominant in solu-

tion, the concentration of the species "in short supply" will appear in Eq. 18K and the the factor 2 in the numerator will be deleted.

The rate of diffusion depends primarily on the product C °D 1/2 .

23.5 Analysis of the Information Content of Fast Transients

The main purpose of using fast transients is to deal with very fast reactions, for which diffusion limitation is significant, even at short times. We have discussed the time required to charge the double layer, in terms of the relaxation time for charge transfer ti °, given by

= C o RF =

(v/n)(RT/F)(C dl/io

Since the diffusion coefficient in simple solutions does not usually vary by more than an order of magnitude, ‘ve reach the rather obvious conclusion that the diffusional relaxation time depends primarily on the concentrations of the reactant and the product. If the product C °D 1/2 is small. Introducing typical values of Cdl and D into Eq. 18K, we find Td = 11 AS for a 1 mM solution. Note that this relaxation time depends on the square of the concentration. Thus, in 0.1 M solution its value is only about 1 ns, and diffusion limitation will be minimal, except for very fast reactions. is large, diffusion is rapid and

(13K)

It is convenient to define also a diffusional relaxation time r which d

Id

368

ELECTRODE KINETICS

The ratio between the two relaxation times is the critical para-

369

K. EXPERIMENTAL TECHNIQUES: I

becomes important early on during the pulse. The dependence on concentration is less clear-cut, since i o is also a function of concentration.

meter determining the behavior of the interphase during a transient.

Usually the exchange current density increases with the first order of the concentration. Thus, the ratio of relaxation times increases with

tetcd—[(%),3][(721 o d I

(19K)

concentration, making it easier to evaluate the kinetic parameters in more concentrated solutions, but this is not always the case.

exceeds about 1x10 3 , the reaction can be said to be activation cd controlled. If it is less than unity, the reaction is largely diffusion

These considerations can he put into quantitative form by analyzing

If

controlled. For intermediate values of T it the reaction may be d activation controlled at short times during a transient and will approach diffusion control as time goes on, and the solution near the electrode surface is gradually depleted. We have seen that one may be able to separate the two processes and determine the faradaic resistance in a graphical manner, by employing the double-pulse galvanostatic method. The same goal can be achieved with the use of any other small-amplitude perturbation technique. To do this, we solve the diffusion equation with the appropriate initial and boundary condi* tions, and then obtain the kinetic parameters by a suitable parameterfitting method.

If Tird < 1, it will be very difficult if not impos-

sible to evaluate the kinetic parameters, because the response of the interphase to a small perturbation is dominated by diffusion limitation, even before the double layer has been charged. Considering Eq. 19K, we note that the ratio of relaxation times

the information content of the measurement. To understand this concept, let us first consider the choice of the optimum time scale for making a measurement. We already know that by using a very fast transient we can minimize the effects of diffusion, so we would first be inclined to use the fastest transients allowed by instrumentation. But if the measurement is conducted on a time scale that is short compared to

,

all we

can observe is a linear variation of potential with time, which depends on the double-layer capacitance (dri/dt

=

We cannot calculate i o

from this part of the transient. The information content

with respect

to i , which we denote I(i ), has a very low value, close to zero. At

0

long times, the information content with respect to i again approaches 0 zero, since the shape of the transient is controlled by diffusion. Some intermediate time scale must he found, for which 1(10) is a maximum. The information content can be defined by the equation I(i ) 0

Anki

(20K)

Di /i

0 0

T

decreases with increasing i for a given concentration in solu-

0

iCd

tion. This is another way of saying that it is hard to determine the

It is the ratio between the relative error in the measured quantity 3 .1

kinetic parameters of fast reactions, because diffusion limitation

and the relative error in the value of i

*

In many cases the solution of the diffusion equation is already given in the literature.

calculated from it. When 0 1(i ) has its maximum value of 1, an error of, say, 2% in the measure° ment of ri leads to exactly the same 2% error in the calculated value of

i . If, on the other hand, the value of 1(i) is, say, 0.2, Eq. 20K 0 implies that an error of 2% in II leads to an error of 10% in io.

370

ELL:_ 1(00E KINETICS

3

EXPERIMEN 1n , i ECILN 1QUES: 1

Equation 20K defines the instantaneous information content, which is what one has if the information is obtained from a single experi-

are

mental point. This is not the way to do an experiment, of course. Instead, one measures the change of overpotential over a period of time

plotted as a function of the duration of measurement T, in dimensionless form, for different values of i 0/'td . The important thing to note is

T, and determines the exchange current density from a parameter fitting

that, for the coulostatic method, the maximum value of 7(i )0 always

of the result to the theoretical curve. The relevant quantity to

occurs at about T/c = 1.8, irrespective of the value of the ratio of

consider in this case is the average information content, 7(i), which is related to the instantaneous information content defined in Eq. 20K by:

relaxation times. It is also evident that this maximum decreases as

T

—

I

—

Idt

(21K)

The information content depends on the time scale used and on the type of perturbation employed. The calculations are complex, but the results

fortunately rather simple. An example is given in Fig. 8K, calcu-

lated for the coulostatic method. The average information content is

diffusion becomes more important. It would seem that we have an easy way to chose the optimal conditions for measurement, which is about 2'r for the coulostatic method. Unfortunately, the value of ti 0 is not known a priori. In fact, this is the quantity we are trying to measure (cf. Eqs. 8K and 10K). The problem can be solved by resorting to a method of trial and error. Let us assume that we know Cdl from an independent measurement. We can then run the experiment on a number of time scales, and try to obtain an approximate value of i o . This approximate value is used to choose a

0.3

better time scale for the next measurement. From this result, a more accurate value of i is obtained and is used to choose an even more suitable value of T. This procedure is repeated until the results

0.2

converge, and there is no further improvement in accuracy. Note that the choice of Th e = 2 as the optimum time scale is

0. 1

specific to the coulostatic method. For galvanostatic measurements the optimal time scale depends on the ratio of i 0/id and is around T/t e = 6 1 Optimum time

—

10

scale for measurement, T/T c

Fig. 8K The average information content, 7(i ), in a coulostatic experi0 merit (in which C has been determined independently), as a dl function of the time of measurement T.

Reprinted with per-

mission from Reller and Kirowa-Eisner, J. Electrochem. Soc.

127, 1725, (1980). Copyright 1980, the Electrochemical Society.

for a value of

t[I

d

= 0.1

(corresponding to substantial control by

diffusion), rising to about 12 for

T

cd

= 100,

as seen in Fig. 9M.

Although the calculated optimum time scale rises further with increasing value of T c it d , this range is of less interest, since it represents conditions under which there is little interference by diffusion, and one can probably use a steady state method to measure io.

372

ELECTRODE KINETICS

K. EXPERIMENTAL TECHNIQUES: I

373

their maximum values at the same time. Choosing a value of T which is good for one will he had for the other, and vice versa. A compromise value is bad for both! We could, of course, measure Cdi at the shortest

30

possible time allowed by the available instrumentation and use this value to calculate i from measurements taken at an optimum time scale, 0 where 7(i ) has its maximum value. This would, however, be equivalent

k2-' 20 N

I-

0

in an independent experiment. Cdl With increasing computing power, there is a tendency to try to

to having obtained

10

obtain all the information from a minimum number of experiments. It may 0 0.1

1

10

be argued that since computing power is almost unlimited (in relation to

1000

100

the type of calculations involved), all the experimental parameters

Tc d

Fig. 9K The optimum time scale for the determination of i o as a function of the ratio 'C /id for galvanostatic measurements. Data from c

,

Relief. and Kirowa-Eisner, J. Electrochem. Soc. (1982).

129, 1473

(i.,

Cdl, R S and D) can be obtained by parameter fitting of the data to the most general equation. Whereas this approach is possible in principle (in the sense that the computer will indeed produce a set of results) it is not recommended, since the information content with respect to the different measured quantities peaks under different experimental conditions. There is no time scale that can produce the highest

The reason for the different time scales needed for measurement in the coulostatic and the galvanostatic methods is simple. During coulostatic measurement, the overpotential decays with time (on open circuit) to zero, so that the information content decreases with time, even under pure activation control (t c/td —3

00).

possible accuracy for all four quantities listed above. Each quantity that is determined in an independent experiment will not only be known more accurately, but the use of its value in the parameter fitting program will enhance the accuracy of all other quantities determined.

During a galvanostatic

pulse the overpotential rises. Under pure activation control it reaches steady state at long times, and the information content approaches unity. When diffusion is taken into account, the information content decreases again at longer times. The optimum time scale for measurement is hence longer in galvanostatic than in coulostatic transients. What happens if we try to obtain both io and

Cdi

from the same

measurement? This is possible but not recommended. It turns out that the information contents for the measurement of C

dl

and i do not have

Consider a series of steady state current-potential measurements with, say, a rotating disc electrode, supplemented with determination of

R

and C from the sudden jump and the following linear rise of s dl potential with time, observed after application of a very short currentstep pulse. If we consider this from the point of view of the information content, we realize that in these experiments we have, in effect, measured each quantity when its information content was unity, or very close to it. This procedure yields the best results, but it is limited to relatively slow reactions. Thus, we could say that the concept of

'ROUE

.

information content is implicit in all measurements, but it becomes crucial when we try to push our techniques to the very limit allowed by

K. EXPER1MEN ,

a real current, associated with the discharge of the capacitor, although it cannot be detected in the external circuit. Integrating Eq. 24K we arrive at an expression of the form

instrumentation.

11 =

24 LARGE-AMPLITUDE TRANSIENTS

a — b.ln(t +

(25K)

in which 24.1 Open-Circuit-Decay Transients

a = — b.ln(i/bCdi) and ti

Whereas the charge-injection method is a small-amplitude perturbation method in which measurement is conducted during open-circuit decay, we now discuss a different open-circuit measurement, in which the initial overpotential is high, in the linear Tafel region. The equations we need to solve are similar to Eqs. 9K and 10K, except that the value of the current in Eq. 10K is that corresponding to the linear Tafel region, namely

=

(bCd/idexp(-1/b)

(26K)

Thus, the Tafel slope can be determined from the slope of the opencircuit decay curve, once the parameter ti is known. The latter is found by trial and error, as the number that gives the best straight line in a semilogarithmic plot of it versus ln(t + 'r). This line must be consistent with the plot of TI versus In t at long times, when 't « t. If the current-potential relationship can be determined experimentally and compared to the open-circuit-decay behavior, the validity of

(22K)

i = io.exp(ri/b)

the assumption that the capacitance is independent of potential can be tested. Indeed, under these conditions the capacitance can readily be

Combining with Eq. 9K yields

found from the open-circuit-decay curve, with the use of Eq. 9K, by (23K)

i = io•exp(fi/b) = — Cdi(ftri/dt) If we assume that

Cdt

is independent of potential in the range of

determination of the slope diVdt as a function of rl, and with the value of i corresponding to each overpotential. We might ask ourselves why the results in the two cases of open-

interest, we can write

circuit decay are so different. For small transients we found that lnri "11

\

exp(-ri/b)d=—0C

dt

(24K)

11 0 The justification for writing these equations is that one assumes that the overpotential depends on the current in the same manner during external polarization and on open circuit. This must be so, since all one is really saying is that the potential developed across the faradaic resistance depends only on the current flowing through it, not on the source driving this current. During open circuit the current flowing is

is proportional to t (cf. Eq. 12K), whereas for large transients we find ri proportional to ln(t+T). The answer is very simple. In the former case, the faradaic resistance is taken to be a constant, independent of overpotential, while in the latter it is an exponential function of potential. The situation can become even more complicated if there is a strong dependence of the capacity on potential. This is the case when the coverage by adsorbed intermediates is high and a large adsorption pseudocapacitance is involved. A large pseudocapacitance gives rise to a slow decay of potential with time on open circuit, as seen in Fig. 10K.

376

ELECTRODE KINETICS

377

K. EXPERIMENTAL TECIIN IQUES: I

[a C(x,t)/a

(27K)

VIDC(x,t))

0.8

OVER POTE NTI AL(Vo lts )

where V 2 is the Laplacian operator, correspOnding to the second derivative of the concentration with respect to distance, in the appropriate 0.7

coordinates. The diffusion coefficient must be considered, in the general case, to be a function of concentration. To simplify Eq. 27K two assumptions are commonly made in electrochemistry: D is taken as a

0.6

constant, independent of concentration and

semi-infinite linear diffu-

sion is assumed. With these assumptions Eq. 27K is simplified to 0.5

a C(x,t)/a t = D [3 2 C(x,t)/a x 2] 0.4

1.2

(28K)

How serious are these assumptions? The concentration of the electroactive species in the Nernst diffusion layer can vary from zero (at

Fig. 10K Open-circuit decay of overpotential. (1) constant capacitance (2) potential dependent adsorption pseudocapacitance, (3)

x = 0 and for i = i ) to the bulk concentration, which is typically a few millimoles per liter. Since measurements are conducted in the

corresponding variation of C4) with n. From Gileadi, Kirowa-Eisner and Penciner, "Interfacial Electrochemistry - An

presence of a large excess of supporting electrolyte, this represents a

Experimental Approach" Addison Wesley, Publishers 1975, with

negligible. What is semi infinite in this context? We consider planar

permission.

diffusion, in the direction perpendicular to the surface into the

very small change in the total concentration, and the variation in D is

solution, not into the electrode, namely, only half the space. As for In fact, such a plateau on the open-circuit-decay plot is a good indication of significant coverage by adsorbed intermediates.

infinity, we already know that it is not very far from the electrode surface. Employing Eq. 9D we find that at 100 seconds, the Nernst diffusion layer thickness is 8 = 0.06 cm, which means that infinity

24.2 The Diffusion Equation and Its Boundary Conditions

lies less than 1 cm away. We do not even have to use a planar electrode to achieve planar diffusion. A cylindrical electrode (i.e., a wire) or

In the study of the diffusion of species to and from the electrode surface, we use the notation C i(t,x) to describe the concentration of the ith species as a function of time and distance from the electrode. The time-dependent diffusion equation in its general form is written as:

a spherical electrode will also look "planar" as long as the Nernst diffusion layer thickness is small compared to the radius of curvature. On the other hand, a small planar electrode will not follow the equations for semi-infinite linear diffusion if its radius (assuming the electrode is circular) is of the order of the diffusion layer thickness.

•18

ELECTRODE KINETICS

379

K. EXPERIMENTAL TECHNIQUES: I

The equation to be solved (Eq. 28K) is a second-order differential

If we set the condition of applicability of the equations for planar diffusion as r 20(nD01 /2, if — in other words, we limit the Nernst

equation in two variables. It requires, therefore, three initial and/or

diffusion layer thickness to 5% of the radius and introduce a typical

boundary conditions. Consider a simple reaction of the type

value of D, we arrive at an inequality which is easy to remember, namely

t < too.r2

(29K)

Ox + ne

R

tvi

(30K)

where both the oxidized and the reduced form are in solution. One

t is the

actually must solve two similar diffusion equations simultaneously, one

longest time for which the equations of semi-infinite linear diffusion

for the reactants and one for the products. Thus, there is a total of

are valid.

six initial and boundary conditions that must be define.

where r is the radius of a wire or a disc electrode, and

It is often stated that the diffusion equation does not apply for longer than 20-50 seconds, because convective mass transport becomes important at longer times. We note that for a planar disc electrode embedded in an flat insulator (i.e., an RDE configuration) the above

(a) Potential step, reversible case The initial conditions for this and all other cases we are going to discuss are as follows:

inequality holds for r = 0.5 cm up to about 25 seconds. For smaller electrodes the length of the experiment will be limited by departure from the equations for semi-infinite linear diffusion, before they become limited by convective mass transport. There are two aspects to solving the diffusion equation. One must first set the initial and boundary conditions, then find a mathematical procedure for solving the equations. Here we shall concentrate on the former aspect. Mathematical techniques for solving the diffusion equation are discussed in many texts, since this problem is not unique to electrochemistry. The initial and boundary conditions under which the diffusion equation is solved define in mathematical form the kind of experiment being performed and the initial conditions of the experiment. It is important to realize that the resulting equations hold true only

if

these initial and boundary conditions have been maintained. This can be explained with the use of a few examples.

C 0 x (x,0) = C ° x and C (x,0) = C ° = 0 O

(31K)

In words, these equations state that the concentrations of both reactant and product are uniform everywhere in solution at t = 0 (i.e., before we have applied the potential pulse). For simplicity we assume here that there is no product initially in solution, but this is not essential. These initial conditions may seem self evident, but they really are not. In particular, when the experiment is conducted by applying a series of pulses (e.g., each at a different potential) care must be taken to make the solution homogeneous before each pulse — for instance, by stirring for a short time and then allowing the solution to become completely quiescent. The next two equations arise from the condition of infinity.

They

are written as follows: C

x

(.3,0 Cox ; C

R

( 00 ,0 = C ° = 0

(32K)

380

ELECTRODE KINETICS

381

K. EXPERIMENTAL TECHNIQUES: I PP-

Since, as we have shown, "infinity" is less than 1 cm away in these experiments, these conditions generally apply, except when a conscious

E = E° + (RT/nF)In

[C (0,1) 1 Ox

(34K)

C R (0,1) j

effort is made to place the working and counter electrodes very close to Often this equation is written in a different form as

each other, as in thin-layer cells. Then we have an equation of mass balance, written as follows:

0= C (0,t) ox

D ox [aC ox (0,0/ad + D R [aC R (0,0/ed

=

0

(33K)

- exp [()(E —

(35K)

CR (0,t)

For the reaction assumed here, one molecule of R is produced for

Note that we have ignored the ratio of activity coefficients in the last

each molecule of Ox that has been consumed; hence the flux of the

two equation. This is a very good approximation, since a large excess

oxidized species reaching the surface must equal the flux of the reduced

of supporting electrolyte is used, but an appropriate correction factor

species leaving it. This boundary condition leads to some restrictions

can be readily introduced, if deemed necessary.

on the use of the resulting diffusion equation. Equation 33K is valid

Having established the physical conditions and the six initial and

only if both species are soluble. It does not apply, for example, to an

boundary conditions, one can proceed to solve the diffusion equation.

electroplating process, because the product stays on the surface, it is

We shall skip the tedious process and proceed directly to the solution,

applicable, however, to the deposition of, say, cadmium on mercury,

applicable to a potential step under reversible conditions, which is

where the product is soluble in the metal phase, forming an amalgam. Even for a reaction such as bromine evolution, where both reactants and nFDC °

products are soluble, Eq. 33K would have to be slightly modified to take into account the fact that two reacting species combine to form a single

112 (TEDt)

[

1

(36K)

1 4- 0

molecule of the product. The diffusion equation can, of course, be solved for these and other cases as well. The point we want to emphasize is that before using a solution of the diffusion equation given in a book, it is important to know the boundary conditions under which this solution was obtained, to ensure that they apply to the experiment being analyzed. The sixth and last boundary condition follows from the assumption

In this equation D and C° correspond to the diffusion coefficient and bulk concentration of the reactant, and we have made the simplifying assumption that D ox/D R = 1. As the diffusion-limited current density is reached, the concentration of reactant at the surface is reduced to zero, and therefore 0 .= 0. Substituting in Eq. 36K we have

of reversibility. If the reaction rate is assumed to he very high, the reactants and products at the electrode surface will be at equilibrium

1

C]

-

nFDC°

(37K)

(nD 0 1/2

at all times, and their concentrations will conform to the Nernst equation. This boundary condition can be written as follows:

Combining Eqs. 36K and 37K, we can relate the current density to

382

ELEC I RODE KINETICS

not negligible in comparison with the size of the potential pulse

potential and time by the simple equation i(t) =

i dd ( t)

383

K. EXPERIMENTAL TECHNIQUES: 1

applied, as shown in Fig. 6D. (38K)

But where exactly has this implicit assumption been introduced? If we had known exactly how the potential changes with time during the

The potential dependence, which is "hidden" in 0, can be introduced

transient, where might we have introduced this variation in the boundary

explicitly, employing Eq. 35K, to yield

conditions of the differential equation? Examining our six initial and

1+0

boundary conditions we find that the one that would have been affected is Eq. 35K, since the function 0 would become time dependent. This =E

1/2

+ (RT/nF)1n(i d/i — I)

(39K)

where E

is the polarographic half-wave potential, at which i = 0.5i . 1/2 d To be exact, Eq. 39K does not follow from introducing the potential dependence of 0 from Eq. 35K into Eq. 38K. What one obtains is the same equation with

E la

problem has been dealt with in the literature: the result is significantly more complicated than the equations given here and is not of general interest. We raised the point only to show that one must be careful of hidden assumptions, which can lead to erroneous results under certain experimental conditions.

replaced by E° . These quantities are related to each

other through Eq. 40K 1.0

)(D RIDOY /21 I/2 = E ° + (RT/nF)14(yox/yR

(40K)

0.8 0

N U 0.6

In the presence of a large excess of supporting electrolyte, the difference between them is very small and may be ignored. We have discussed the assumptions under which Eq. 36K is valid.

0.4

0.2

But we have made an implicit assumption, that has not been stated, relating to the shape of the potential step transient. In fact, it has been implicitly assumed that the potential changes from zero to a preset value E instantaneously. This is never the case in practice. The assumption is justified if the rise time of the pulse is short compared to the duration of the experiment. This assumption can also be a major source of error if the uncompensated iR s potential drop in solution is

0

2

3

4

Distance /p,m

Fig. I IK Evolution of the concentration profile with time near an electrode stuface, just after the potential has been stepped to the limiting current region.

384

ELECTRODE KINETICS

Solving the diffusion equation under the foregoing conditions yields the concentration profile near the electrode surface, as a

K. EXPERIMENTAL TECHNIQUES: I

solution of the diffusion equation yields the current density as a function of time and potential, as:

function of time.

i(t) = nFk h C°.exp(A,2)erfc(A,) C(x,t) = C.erf[

x

(41K)

385

(43K)

in which the dimensionless parameter ? is given by

(4D0 1/2

X = k h (t/D) 112

Plots of the dimensionless concentration C/C° as a function of distance

(44K)

at different times are shown in Fig. 11K The gradual development of the Nernst diffusion layer with time can be clearly seen. The concent-

The heterogeneous rate constant k h depends on potential exponentially,

ration profile near the surface is linear, but a deviation from lineari-

following a Tafel-like relationship:

ty is observed farther away, as the concentration approaches its bulk kh = k h° -exp[— cxF(E — E ° )/RT]

value.

(45K)

Combining Eq. 43K with the expression for the limiting current density,

(b) Potential step, linear Tafel region

which we have obtained earlier (cf. Eq. 37K), we have: Our second example is also a potential step experiment. Here, however, it is assumed that kinetic limitation exists. Moreover, we i/i d = F I (X) = rc 1/2 (X)exp(X 2 )erfc(X)

assume that the potential range studied is far from equilibrium, so that

(46K)

only the forward reaction need be considered. This is the condition under which a linear Tafel plot can normally be observed. We refer to it therefore as the linear Tafel region. In the original literature this condition was referred to as the totally irreversible case, a term which we consider to be rather misleading. The first five initial and boundary conditions of the differential equation remain unchanged, only the sixth (Eq. 35K) is different. This boundary condition is obtained by relating the current density to the specific rate constant of the forward reaction k h and to the flux of reactant at the electrode surface: i/nF = k C(0,t) = D(aC(0,t)/ax) h

The function F (A), which has been tabulated in detail in the literal ture, can be used to obtain X for any given ratio of i/i d . In this way k, which is proportional to the activation-controlled current density, can be evaluated as a function of potential. A plot of log k h or of log ? versus E is equivalent to the traditional Tafel plot, in which log i

tiC

is plotted versus E or versus 11.

What are the limitations imposed on the validity of Eqs. 43K and 46K? In writing the sixth boundary condition (Eq. 42K) we made the assumption that the reaction is first order with respect to the reac-

(42K)

This equation replaces Eq. 35K, as the sixth boundary condition. The

tant. This is a serious limitation, because in a study of the mechanism of electrode reactions, the reaction order is one of the quantities we wish to determine experimentally. Obviously the values of k h obtained

386

ELEGIROI., ii KINETICS

3t3/

K. EXPERIMENTAL TECHNIQUES:

x/,u,m (for T =

from Eq. 46K in solutions containing different concentrations of the

40

msec)

reactant cannot be used to evaluate the reaction order, since this equation is valid only if the reaction order is unity. (c) Current step (chronopotentiometry) Our third example, in which a current step is applied, could be called a galvanostatic experiment, in the sense that the current, rather than the potential, is the externally controlled parameter. The first five initial and boundary conditions of the diffusion equation remain unaltered, and it is again the sixth that must be changed, to make the result applicable to this particular experimental 0

technique. Since the current is externally controlled, one controls, in

=D

C(, t) 1

[a ax

J

(47K) ,

0.4

0.6

1.0

0.8

Dimensionless distance, x/(2D

effect, the flux at the electrode surface. This is expressed mathematically by:

0.2

-T 1/2 )

Fig. 12K Development of the concentration profile with time during a constant current (chronopotentiometric) transient. ti = 40 ms, D = 6x10-6cm2Is. The distance, x, is given in dimensionless form in the bottom scale and in micrometers in the top scale.

Note that the specific heterogeneous rate constant k h is not part of this equation, because the reaction is forced to proceed at a rate

The transition time ti corresponds to the time taken for the concentration at the surface to be reduced to zero.

determined by the applied current. Solving the diffusion equation, one obtains the concentration as a

but the galvanostatic experiment the flux at the surface is constant In contrast, in a potentiostatic concentration decreases with time.

function of time C/C° = 1 — (ttr) I/2 [exp(—X2) — ir inX.erfc(X))

(48K)

experiment, the surface concentration is held constant and the flux (i.e., the current density, which is proportional to it) decreases with

where 't is the transition time, defined by Eq. 49K below, and the dimensionless parameter X is equal to x/(4D0 1/2 . Concentration profiles calculated from Eq. 48K for different times are shown in Fig. 12K. The important thing to note, in comparing Figs. 11K and 12K is that in a

time.

* The flux is proportional to the gradient of concentration

at the

electrode surface, namely to (aClax) x=o which is constant in this type of experiment, as seen in Fig. 12K.

388

ELECTRODE KINETICS

389

K. EXPERIMENTAL TECHNIQUES. I •

How long will it take for the concentration at the surface to reach in Eq. 48K. The result is known as the Sand equation, which had already been derived in 1901:

IT

1/2

=

[

nF(Tc1)) 1/2 } co

2

(49K)

Poten tia l/ V vs SCE

zero? This can be calculated by setting x = 0 and C(x,t) = C(0,'r) = 0

Here (and in Fig. 12K) T is the transition time required to reduce the concentration of the reactant at the electrode surface to zero. Note that this experiment must be performed in a quiescent solution. Thus,

0

while the concentration at the surface declines to zero, the bulk concentration is essentially unchanged, as prescribed by the boundary conditions (Eq. 32K). The meaning of the transition time, T , may be clarified by considering Fig. 13K, which displays a typical transient. The shape of the transient shown in Fig. 13K depends on electrode kinetics, although the transition time 'C is independent of it. For the reversible case, this can be obtained by introducing the time-dependent

30

20

10

40

Time /s Fig. 13K Chronopotentiometric transient, showing the meaning of T and 2+ Reduction of 2 mM Cd in 1.0 M KNO3 , at a mercury of E 1/4

electrode. i = 0.16 mA/cm. From Gileadi, Kirowa-Eisner and Penciner, "Interfacial Electrochemistry — An Experimental Approach" Addison Wesley, Publishers 1975, with permission.

concentrations of reactants and products at the electrode surface, C (0,t) and C (0,t), respectively, into the Nernst equation. The ox R result is

to correlate the current with the specific rate constant.

Combining

this relation with Eq. 45K, which describes the potential dependence of k , we obtain

„c 1/2 — t 1/2

E

E

1/4

+ (RT/nF)In

1/2

(50K) E = E°+

RTr ,In[2k /(1tD) I/2

RT In /1 1/2_ t 1/2 —TF C,

(51K)

The quarter-wave potential E

used in Eq. 50K is equal to the 114 polarographic half-wave potential and is therefore characteristic of the electroactive species in solution. If the reaction rate is slow and the transient is measured in the linear Tafel region, Eq. 42K must be used

Thus, while the transition time is independent of electrode kinetics (because we force the reaction to occur at a given rate), the variation of potential with time depends on kinetics, as seen by comparing Eqs. 50K and 51K.

ELECTRODE KINETICS

390

391

K. EXPERIMENTAL TECHNIQUES: I

We return now for a moment to examine the concentration profiles

obtained by applying larger and larger potential steps, until further

shown in Fig. 12K. The data are presented in dimensionless form, as the

increase of the overpotential has no effect on the shape of the resul-

relative change in concentration, C(x,t)/C ° versus the dimensionless

ting current transient.

distance x/(4DT) I/2 . This is a very useful way of presenting simulated

Typical potentiostatic transients are shown in Fig. 14K. Such data

data, since it allows us to provide much information in a single figure.

can be employed in two ways to evaluate the activation-controlled rate

On the other hand, one may lose sight of physical reality, to some

as a function of overpotential. We have already seen that the measured

extent. Thus, the "distance" scale used in Fig. 12K actually depends on

current density is related to the diffusion-limited current density by

the diffusion coefficient and on time, which may look like an odd way to

the equation

define a distance. To bridge the mathematical formality and real space,

(46K)

i/i d = F l (k) = rc ia (k)exp(X2)erfc(X)

we added the top scale in Fig. 12K, where the distance is given in micrometers for a given set of conditions, which are described in the figure legends. Taking values of C° and i that yield a transition time

from which X = k (t/D) 1/2 can be evaluated, yielding the heterogeneous rate constant k , as a function of potential. The two currents i and id

of T = 40 ms, we note that unity on the dimensionless time scale corresponds to an actual distance of 10 ktm. This provides a feel for

0.6

the distance from the electrode surface over which the concentration is disturbed in a given time. It must not be forgotten, however, that

E

these distances depend on the value of T, which itself is determined by the concentration chosen and the current density applied.

in 0.3

z

24.3 Single-Pulse Techniques When the specific rate of an electrode reaction is not very high, large-amplitude potentiostatic or galvanostatic transients can be used to obtain the current-potential relationship by appropriate corrections for partial diffusion limitation. This can he achieved either by using the solutions of the diffusion equation discussed in Section 24.2, or by extrapolating the measured current or potential back to zero time, where

t-- 0.2 z (r 0.1 U

0.0

0

2

6 4 TIME/sec

8

10

Fig. 14K Potentiostatic transients at high ovelpotentials. n = 2; D =

the rate of diffusion tends to infinity.

= 50 mM. From Gileadi, Kirowa-Eisner and 6x10 -6 cm 21s; Penciner, "Interfacial Electrochemistry - An Experimental

(a) Potentiostatic transients The curve corresponding to

E 0.4

as a function of time is readily as

Approach" Addison Wesley, Publishers 1975, with permission.

392

ELECTRODE KINETICS

393

K. EXPERIMENTAL TECHNIQUES: I

must, of course, be measured at the same time after application of the

electrode, as discussed in Section 10.4. In both cases the extrapolated

transient. Obtaining k h from the value of i/id at a single time is a

value of the current corresponds to an "infinite" rate of mass transport

rather wasteful way of analyzing the experiment, since only one experi-

(at t = 0, in the present case and at oi -1/2 = 0 for the RDE). The

mental point is taken from each transient. Better accuracy can be

constant K in Eq. 53K is given by

achieved by using the data from almost the entire transient and determining X from Eq. 46K, employing a suitable parameter-fitting program. Alternatively, one could determine i/id at different times during the transient, using the average value of k h, obtained in the range in which it does not change systematically with time. The second method may

K=

1+

(54K)

nF(7rD) 1/2 C°

in which 0 is the ratio of concentrations of reactants and products at the electrode surface, as defined in Eq. 35K. At high overpotentials,

appear to be less sophisticated, but it does have an advantage, from the

in the linear Tafel region, 0 approaches zero, and K is independent of

experimental point of view. A systematic variation of k t: implies that

potential. At lower overpotentials 0, and hence also K, depend on potential.

Eq. 46K is not valid. This could be caused, for instance, by a relatively large contribution of double-layer charging current. Instrumental

Consider now the effect of uncompensated iR s on the shape of the

limitation could cause a deviation on the short time scale and convec-

potentiostatic transients. This was shown in Fig. 6D. The point to

tive mass transport could lead to an error at long times. The region in

remember is that although the potentiostat may put out an excellent step

which Eq. 46K applies can thus be readily identified, and a better value

function — one with a rise time that is very short compared to the time of the transient measured — the actual potential applied to the inter-

of the rate constant can be obtained. An alternative method, based on the notion that mass transport and

phase changes during the whole transient, as the current changes with

charge transfer occur consecutively, may also be employed. The current

time (cf. Section 10.2). This effect is not taken into account in the

density during a transient is related to the activation-controlled

boundary conditions used to solve the diffusion equation, and the

current density by

solution obtained is, therefore, not valid. The resulting error depends 1/i = 1/i + ac

d

(52K)

on the value of R , and it is very important to minimize this resiss tance, by proper cell design and by electronic iR s compensation.

Now, we already know (cf. Eq. 1K) that i d varies with (" 2 so that the above equation can be rewritten in the form The error resulting from an uncompensated iR

1/i = 1/i + Kt ia ac

(53K)

can be decreased by s making measurements at lower concentrations of the reactant, which will

should give a straight line, with an

lead to lower current density, On the other hand, diffusion limitation

intercept at t = 0, corresponding to Hi .. This treatment is similar

becomes more severe at lower concentrations, and an optimum concentra-

to the method of evaluating i

tion for conducting the experiment must be found.

and a plot of

Ili versus t

I12

ac

from measurements at a rotating disc

ELECTRODE KINETICS

'MENTAL 1 EL IQUES: K. EXPERIMENTAL

(b) Galvanostatic transients

The behavior shown in Fig. 15K is readily understood by considering

Figure 15K represents a typical response of the electrochemical

Fig. 3K. The hatched area in Fig. 3K shows the longest time for which

interphase to constant current pulses of different magnitude. For a

interference by diffusion can still be considered to be negligible. As

sufficiently low current density, a true steady state is reached before

the current applied is increased, the electroactive species at the

the effect of diffusion limitation can be observed.

interface is depleted more rapidly and the time for which the transient

As the current

density is increased, diffusion limitation sets in earlier.

Extrapola-

tion of the potential to zero time can sometimes be employed to estimate

can be considered to be essentially activation-controlled becomes shorter.

a correct value of the activation overpotential, as shown in this

a

Fig. 15K. If such extrapolation is not practical, one may use a higher concentration of the

reactant, to increase the ratio ti /'t d (i.e., to enhance the relative rate of diffusion) or employ the double-pulse galvanostatic method

too E

described earlier to separate the regions of

.

activation and diffusion control.

-

0 E a)

0 IL 0.3

Fig. I6K Galvanostatic transients. (a) without iR

0

compensation (b)

0

10

s with 95% iR s compensa-

0 0.2 -J

40

50

60

50

60

b

rence in the scale of

H

30

20

Time/ms

lion. Note the diffe-

LL.1

15

potential.

0

Pote ntia l/mV

a_

CC Lu

1R 5

50

0. 1

0

0.0

0

I

2

RF

10

5

3

TIME/sec 0

Fig. 15K Response of the interface to galvanostatic pulses of different height. From Gileadi, Kirowa-Eisner and Penciner, "Interfacial Electrochemistry - An Experimental Approach" Addison Wesley, Publishers -1975, with permission.

10

20

30

Time/ms

40

396

ELEL I RODE KINETICS

397

K. EXPERIMENTAL TECHNIQUES: I

this type of measurement, since it is a constant. In principle, it is

the number of electrons taking part in the overall reaction, since the accuracy required, considering that n must be an integer, is much less.

possible to perform the experiment without any iR s compensation, measure

One could use the equation of chronopotentiometry under irrever-

The ohmic potential drop should not be a source of major error in

this correction term independently, and apply an appropriate correction to the result. Better sensitivity and accuracy can be achieved, however, if iR s is measured first and its value subtracted from the measur-

sible conditions (Eq. -51 K) to determine the transfer coefficient from a plot of E versus 14r 112— t ia), but again, the uncertainty in the determination of ti makes this method less reliable than other similar

ed potential electronically, particularly when it is large compared to

techniques, such as the linear potential sweep method which will be

the measured activation overpotential. The reason for this should be apparent from a comparison of the curves obtained with and without elec-

discussed below. Equation 49K is valid only if the reaction considered is a simple

tronic compensation, as seen in Fig. 16K.

charge-transfer process. If there is a preceding or following chemical

Chronopotentiometry is a special case of a galvano'static transient

step, the response of the interphase changes and this equation no longer

in which one allows sufficient time for the diffusion-limited current to

applies. This is commonly used in practice to test the complexity of

fall below the applied current.

the process being studied, by conducting the experiment at a series of

What information can be extracted from an experiment conducted in this manner? From Eq. 4K (the Sand equation) we note that the transition time ti is proportional to the square of the bulk concentration in solution: ill/2

[ nF(TED) 2

112

l c°

(49K)

different current densities and/or concentrations and plotting the so-called current function 0'1 11210 versus i or C° . A systematic dependence of the current function on either i or C° is a definite indication of the complexity of the reaction taking place. 24.4 Reverse Pulse Techniques

This would suggest that chronopotentiometry could be a sensitive electroanalytical technique. It is rarely used in this context, however, since it is often difficult to determine the transition time

during the transient can, in general, yield information on the kinetics

accurately, because of double-layer charging at short times and com-

intermediate formed in the course of the reaction). The physical

peting reactions at long times. The same limitations apply when one

reasoning behind such techniques is simple. For a reaction sequence of

attempts to use Eq. 49K to measure the diffusion coefficient. On the

the

other hand this equation can be used as a quick method of obtaining n, * This is equivalent to allowing the concentration at the surface to be reduced to zero, which happens at t = 'r, as seen in Fig. 12K.

Reversing the direction of the applied pulse at a preset time of reactions following charge transfer (i.e., on the stability of the

type Ox + ne M

k R ----> product

(55K)

the reduction product R is produced at the surface at a certain rate. If k is very small (on the time scale of the experiment), reversing the current or the potential will cause the reoxidation of R. If k is very

K.

31:ri

EXYLKIIMENTAL

on either of these parameters would indicate that the kinetics of the

large, the reduction product will react further almost simultaneously, leaving nothing to be reoxidized. Between the two extremes, the shape

reaction is different from that assumed here.

of the reverse transient can be analyzed to determine the value of kr The mathematical analysis is usually complex, because the boundary

An actual example of a reaction of the type given by Eq. 55K is the oxidation of p-aminophenol (PAP) to p-quinoneimide (PQI), which is

conditions represented by the concentration of reactants and products at

subsequently hydrolyzed to p-benzoquinone (PBQ):

the surface can in themselves be complex functions of time and the rate constant k .

(57K)

PAP t—=) PQI + 2H+ + 2eM

Solutions have been given in the literature, but these

should be used with great caution and only if one is fully aware of the

k PQI + H 2O

limiting assumption made in obtaining any given solution or numerical result, as we have pointed out several times.

f

(58K)

PBQ + NH 3

Some results of measurements taken in this system are shown in Fig. 18K. For example, the second reduction wave seen in Fig. 18K corresponds to

(a) Chronopotentiornetry with current reversal

the reduction of p-benzoquinone (PBQ) to p-hydroquinone (PHQ), The simplest case to consider is that in which the product of reaction is stable, namely, when k r = 0. In this case the transition time on the reverse pulse Tr is just one third the time tr of the forward pulse, as long as t r Tf. This is shown in Fig. 17K. If the rate of decomposition of R is finite, the transition time of the reverse pulse is obviously shorter, and its value depends on the

0

_J

H z

) 1/2 = erf [1( (1. + T f f r

> > - 0. 6

rate constant k . This is expressed by 1 2.erf(k ft )

,) - 0.7

U (

(56K)

From this equation one can construct a table of values of kir as a function of the ratio Tr/f Taking measurements at different times f. during the forward current pulse, one can obtain the product k rtr as a function of tr, from which k is calculated. Note that Eq. 56K was derived with the assumption of first-order kinetics for the decomposition of R. The validity of this assumption can be tested by repeating

0. 5

L"-- Tr

10

30 20 TIME /msec

40

Fig. 17K Current reversal chronopotentiometry. 2 mM Cd 2+ in 1 mM KNO 3 i=0.16tnAlcmTheduraiosftwndrevs pulses are T f and r , respectively. From Gileadi, KirowaEisner and Pencitzer, "Interfacial Electrochemistry — An Experimental Approach" Addison Wesley, Publishers, 1975, with

the experiment at different current densities and at different bulk concentrations. A systematic dependence of the calculated value of k

0

a_

permission. f

400

ELECTRODE KINETICS

K. EXPERIMENTAL TECHNIQUES: I

externally controlled parameter. 0.8 w cr)

401

' 66

The potentials both in the forward

step and the reversed step are set in their respective limiting current regions, making the result independent of the specific rate constant for

0. 6

charge transfer. The ratio of currents during the forward and reversed steps depends on the homogeneous rate constant kf for the chemical

> 0.4

reaction following charge transfer. A typical current-time plot is shown in Fig. 19K.

I- 0.2

z

0.0 0

10

20

30

40

Fig. 18K Current—reversal chronopotentiometry, for the oxidation of I mM PAP to PQI in 0.1 M H SO , at a platinum electrode, 2 4 followed by hydrolysis to PBQ. i = 0.10 mAlcm 2 . Tr is the transition time on the reverse pulse, following a forward pulse of duration t. From Gileadi, Kirowa-Eisner and Penciller, "Interfacial Electrochemistry - An Experimental

CU RRE NT D EN SI T

TIME/msec

tf

if

Approach" Addison Wesley, Publishers, 1975, with permission.

namely, the reaction PBQ + 21-1 + + 2em

i PHQ

(59K)

Fortunately, this reaction occurs at a much more negative potential, so

TIME Fig. 19K Variation of the current with time during reverse step volta-

that it does not interfere with measurement of the transition time

mmetry. (t - total time; t f - switching time. From Gileadi,

for the reduction of PQI.

Kirowa-Eisner and Penciller, "Interfacial Electrochemistry - An Experimental Approach" Addison Wesley, Publishers 1975, with

(b) Reversed step voltammetry Reversed step voltammetry is similar to the technique just described except that the potential rather than the current is the

permission.

402

)DE KINETICS

As in most of these cases, the mathematics is rather involved and is not given here. The results appear in the original literature in the form of tables or as "working curves," allowing the calculation of the rate constant k1 of the chemical step following charge transfer, from the ratio of the currents in the anodic and the cathodic directions, at

L EXPERIMENTAL TCt ..,.,■ IQUtLs • ..

L. EXPERIMENTAL TECHNIQUES: 2 25. LINEAR POTENTIAL SWEEP AND CYCLIC VOLTAMMETRY 25.1 Three Types of Linear Potential Sweep

a given time. We might be tempted to measure the currents at a very short time

Linear potential sweep is a potentiostatic technique, in the sense

after switching, since this leads to the highest sensitivity and allows

that the potential is the externally controlled parameter. The poten-

measurement of the highest rate constants. On the other hand, it is in

tial is changed at a constant rate

this time period that interference by double-layer charging and by distortion of the pulse shape by an uncompensated solution resistance is

(1L)

v = dE/dt and the resulting current is followed as a function of time.

most severe. This is why the ratio ia/ic is commonly measured over a

In most cases the current is plotted as a function of potential on

wide range of experimental conditions and the average value of kr ,

an X-Y recorder or on a plotter. This becomes particularly useful when

obtained in a range where it is least affected by experimental artifacts is evaluated.

the potential is swept forward and backward between two fixed values, a In this way the current technique referred to as cyclic voltammetry. measured at a particular potential on the anodic sweep can readily be

b

11

0.2 0 -0.2 Potentiol,E-E 1 , 2/Volt

Fig. 11, Plots of current versus time and versus potential for the same data obtained in cyclic voltammetry, recorded (a) on a stripchart recorder and (b) on an X-Y recorder.

404

ELECTRODE KINETICS

compared with the current measured at the same potential on the cathodic sweep. Almost all literature data are presented in this form. Figure IL presents the same data in the form of an plot and an i/E plot, for comparison. Linear potential sweep measurement are generally of three types: (a)

Very slow sweeps

I. EXPERIMENTAL TECHNIQUES: 2

405

activation controlled processes, which may be performed in stirred solutions. The typical sweep rates are also 0.01-100 V/s, but here the lower limit is determined by background currents from residual impurities in solution (and perhaps by the desire of the experimenter to collect more data in a given time) while the upper limit is determined by uncompensated solution resistance and by instrumentation. The relative current for double-layer charging is independent of sweep rate,

When the sweep rate is very low, in the range of v = 0.1-2 mV/s, the system can be considered to be almost at equilibrium and measurement is conducted under quasi-steady-state conditions. The sweep rate v

as we shall see. In this chapter we shall discuss only the second and third cases,

plays no role in this case, except that it must be slow enough to ensure

potential automatically, under conditions in which the sweep has

that the reaction is effectively at steady state at each potential in the course of the sweep. This type of measurement is used in corrosion

practically no effect on the current observed.

and passivation studies, as we shall see, and also in the study of some fuel cell reactions in stirred solutions. Reversing the direction of the sweep should have no effect on the ilE relationship, if the sweep is

25.2 Double-Layer-Charging Currents

slow enough. This is rarely the case, however, because the surface changes during the sweep. In most cases an oxide is formed, and its

type of experiment. We use the simple equation:

reduction occurs at a more negative potential than the potential of its formation, as seen, for example, in Fig. 101. (b) Studies of oxidation or reduction of species in the bulk In the second case, the sweep rate is usually in the range of 0.01-100 V/s. The lower limit is determined by the need to maintain the total time of the experiment below 10-50 seconds (i.e., before mass transport by convection becomes important). The upper limit is determined by the double-layer charging current and by the uncompensated solution resistance, as discussed in Section 25.2. (c)

Studies of oxidation or reduction of species on the surface The redox behavior of species which are adsorbed on the surface are

since very slow sweeps are just a convenient method for scanning the

We cannot discuss the linear sweep method properly without appreciating the importance of the double-layer-charging current in this

1 d1

= C dl (dE/dt) = C di .v

(2L)

If double-layer charging is the only process taking place in a given potential region (this would be the case for an ideally polarizable interphase) and one cycles the potential between two fixed values, the results should be such as shown in Fig. 2L(a). Plotting Ai = i s — ie = 21i1 as a function of v, as shown by line 1 in Fig. 2L(b), one can obtain the value of the double-layer capacitance from the slope. If a faradaic reaction is taking place, a result such as shown by line 2, from which Cat can still be obtained (cf. Fig. 14G), might be observed.

ELECTRODE KINETICS

a

I

L EXPERIMENTAL TECHNIQUES: _

in which accurate measurements can be made, on the basis of the following assumptions: (a) the peak current of the faradaic process being

b

4v

studied should be at least ten times the double-layer-charging current

3v

to allow reliable correction for the latter; (b) the sweep rate should

2v

10 mV/s, chosen so as to limit errors due to convective mass be v 2 20 mA/cm , in transport; (c) the peak faradaic current should be i

1

order to limit the error due to uncompensated solution resistance; and (d) the bulk concentration should be C ° 100 mM, to ensure that it

0

will always be possible to have an excess of supporting electrolyte, to <1

suppress mass transport by migration. /

1

1

3

Potential 0

Fig. 2L (a) Cyclic voltamnzetry for an ideally polarizable interphase. (b) The plot of Ai = i i as a function of the sweep rate v —

a

c

for an ideally polarizable interphase (line 1) and for the case where there is some residual faradaic current iF(line 2).

—2 cs, 0

,c —4

It is important to evaluate the numerical value of i di . For a very slow sweep experiment, using v = 1 mV/s and i

dl

•_ rn 0

Cat =

20 pF/cm2, one has

—6

= 20 nA/cm2 . This is negligible, even with respect to the small

currents observed on passivated electrodes, and can hence be ignored.

—8

The interplay between double layer charging and oxidation or reduction of an electroactive material in the bulk of the solution is

—4

—3

—2

0

—1

2

3

log v

illustrated in Fig. 3L. The faradaic currents shown by the solid lines are the peak currents i calculated according to Eq. 11L or 12L, which are discussed in Section 25.4. It is important to note that these currents change with V I / 2 , while is proportional to v. As a result, double-layer charging becomes dl increasingly more important with increasing sweep rate. The hatched area in Fig. 3L represents the region of sweep rates and concentrations

Fig. 3L The optimum range of concentration and sweep rate (hatched area) for measurements in cyclic voltammetry on a smooth electrode. The double-layer-charging current iathas been calculated for Cat

=

20 1.tFlcm.

408

ELECTRODE KINETICS

409

L EXPERIMENTAL TECHNIQUES: 2

The numerical values used to construct Fig. 3L are somewhat

employed, the diffusion rate is higher and the uncompensated solution

arbitrary. They do represent, however, the right order of magnitude and

resistance is significantly lower, increasing the range of measurement

are probably correct within a factor of two or three. The important

by an extent which depends on the radius of the microelectrodes, as we

thing to learn from Fig. 3L is the type of factors which must be

shall see in Section 27..

considered in setting up an experiment. One could readily construct a similar diagram using somewhat different assumptions, but

the

conclusions would not be fundamentally different. Thus, one notes that

25.3 The Form of the Current-Potential Relationship We have seen a typical

curve obtained in a linear potential

it is difficult to make measurements at concentrations below 0.1 mM or

sweep experiment in Fig. 1L. This particular curve was observed in a

at sweep rates above 100 V/s. The best concentration to use is in the

solution containing 3 mM Fe 2+ ions in 1 M H 2 SO4 . Thus, the anodic peak corresponds to the simple reaction

range of 1.0-10 mM. Interestingly, this conclusion does not depend

2+ Fe ---->

strongly on the four assumption we have made above, since it refers to the "middle of the field" of applicability of the method. An interesting point to consider is the effect of surface roughness on the range of applicability of the linear potential sweep method.

The

double-layer-charging current i dl in Fig. 3L is calculated for C

dI

= 20 pF/cm 2, which is a relatively low value for solid electrodes,

implying a highly polished surface with a roughness factor of 1.5 or less. If the roughness factor is increased, for example, by using platinized platinum instead of bright platinum, the charging current could increase by a factor of typically 20-100, while the diffusion current remains essentially unchanged. Using corresponding values of in Fig. 3L, we note that the range of applicability is seriously dl limited — to concentrations above 1 mM and to sweep rates below a few i

volts per second. Thus, linear-sweep voltammetry should be conducted, whenever possible, on smooth electrodes. The application of this method to the study of porous electrodes, of the type used in fuel cells and in metal-air batteries, is very limited.

I+

+e

rvt

(3L)

in which both reactant and product are stable species and both are soluble. The initial concentration of Fe 3+ is zero, but its surface

concentration is very close to the bulk concentration of Fe 2+ . As a result, the cathodic reduction peak is nearly equal to the anodic is a good . ( anodic)/I (cathodic) oxidation peak. In fact, the ratio( indication of the chemical stability of the product formed by charge transfer. By studying the dependence of this ratio on sweep rate, one can determine the homogeneous rate constant in a reaction sequence such as shown in Eq. 55K. Why is a peak observed in this type of measurement? In the experiment shown in Fig. IL the sweep is started at a potential of — 0.2 V versus the standard potential for the Fe 2+/Fe3+ couple, where no faradaic reaction takes place. At about — 0.05 V the anodic current starts to increase with potential. Initially the current is activation controlled, but as the potential becomes more anodic, diffusion

We should perhaps conclude this section by noting that the above considerations are valid for large electrodes, operating under semi-infinite linear diffusion conditions. If ultramicro electrodes are

*

In fact, there is a small cathodic current flowing, which must be

due to the reduction of some impurity, but this is of no interest here.

410

ELtu. a ODE KINETICS

L EXPERIMENTi.. I

i ECHNIQUES; 2

4

The observed current is the inverse sum of the

linear potential sweep are really the same as those written for the

As time

potential step experiment, as discussed in Section 24.2, because in both

goes on, the activation controlled current increases (due to the linear

cases the potential is the externally controlled parameter. As before,

increase of potential with time) but the diffusion controlled current

we can distinguish between "the reversible case", in which we assume

decreases. As a result, the observed current increases first, passes

that the concentrations at the surface are determined by the potential

through a maximum and then decreases. At higher sweep rates each

via the Nernst equation (cf. Eq. 35K) and "the linear Tafel region",

limitation sets in.

activation and the diffusion limited currents, (cf. Eq. 3A).

*

potential is reached at a shorter time, when the effect of diffusion

whertspcifaon reltdhsufaconerti

limitation is less, hence the peak current is found to increase with

C(0,t) and to the flux at the surface, as given by Eq. 42K. There is

sweep rate.

one important difference, however. During a potential-step experiment,

In cyclic voltammetry, the potential is made to change linearly

the potential is assumed to be constant throughout the transient.

with time between two set values. Often the current during the first

During potential sweep it is assumed to change linearly with time,

cycle is quite different from that in the second cycle, but after 5-10

following the simple equation

cycles the system settles down, and the current traces the same line as

E(t) = E i ± v•t

(4L)

a function of potential, independent of the number of cycles. This is often referred to as a "steady-state voltammogram", which is an odd name

where E is the initial potential at t = 0, which is chosen in a range

to use, considering that the current keeps changing periodically with

where no faradaic process takes place. Incorporating this equation into

potential and time. The term is used here in the sense that the

the appropriate boundary condition (Eq. 35K or 42K) will then produce a

voltammograin as a whole is independent of time. This is a very nice

result which is applicable to the the linear potential-sweep experiment.

experiment to perform, because it tends to be highly reproducible.

The change in the boundary conditions complicates the mathematics, to

Moreover the voltammograms are stable over long periods of time,

the point that an explicit algebraic solution for the whole i/E curve

sometimes in the range of hours. Interpretation of the data is another matter. Using the results for qualitative detection of reactions taking place in a given range of potential is fine, but quantitative treatment,

has not been found. Numerical solutions are, however, available for the different cases. The discussion here is limited to the coordinates of the peak, namely to the values of the peak current

i

and the peak

using the equations developed for a single linear potential sweep is wrong because (a) the initial and boundary conditions of the diffusion equation have change and (b) convective mass transport may already play an important role. 25.4 Solution of the Diffusion Equations

*

This is usually called the "totally irreversible case", but we

consider the term to he misleading because, as seen in Fig. IL, the reaction can, in fact, be reversed. Referring to it as the "linear Tafel region" implies that the reaction occurs at high overpotentials, where the rate of the reverse reaction can be neglected.

The boundary conditions for solving the diffusion equation for

412

ELECTRODE KINETICS

413

L EXPERIMENTAL TECHNIQUES: 2

potential E , as a function of sweep rate and the kinetic parameters of

(8L)

ip(rev) = (0.446nF(nF/RT) 1/2 D 1/2 1 -v 1/2

the reaction involved. which can be written, for room temperature, as (a) Reversible region I p(rev) =

The peak potential in the case of a reversible linear potential sweep is given by the simple equation Ep(rev)E

1 /2

where the units are as follows: i

+ 1.1(RT/nF)

(5L)

where E

is the polarographic half-wave potential, which is very close 1/2 to the standard potential E ° (cf. Eq. 40K). The positive sign in Eq. 5L is applicable to an anodic sweep and the negative sign ; applies to a cathodic peak. In either case, the peak appears about (28/n) mV after E

,

[2.72x105 n1/2 D 1/2Cl•v 1/2 (rev),

A/cm2 ; D,

(9L) cm2/s; C° , mol/cm 3 ;

and v, V/s. For a 3 mM solution of Cl this would yield a peak current density of about 2 mA/cm 2 at a sweep rate of 0.10 V/s. (b) Linear Tafel region In the linear Tafel region the peak potential depends logarithmically on sweep rate, following the equation

in the direction of the sweep. The peak potential is independent

la of the sweep rate in the reversible case. This characteristic can, in

E

(irrev) =

E

1/2

+ (b/2)[1.04 — log(b/D) — 2log k + log v]

(10L)

fact, be used as a criterion for reversibility. It is also independent of concentration. This, however, is correct only if one considers a simple reaction, such as represented by Eq. 3L. For a slightly more complex stoichiometry, such as that represented by the oxidation of CI to CI

From a plot of Ep(irrev) versus log v one can obtain the Tafel slope, and from the intercept at. E

(irrev) =

E

(rev)

the specific rate constant

can be calculated. Note that the value of k obtained in this way is that corresponding to E = EP(rev) only; but since the Tafel slope is

2 2C1

CI + 2e 2

M

(6L)

known, k r at any other potential is readily calculated. The peak current density in the linear Tafel region is given by

the half-wave potential, and with it E (\ rev), depend logarithmically on concentration. E =

i pOrrev) =

+ (RT/nF)In ((y0x/yR)(DR/D0x)1/2] + (RT/nOnC

(7L)

5 I 2 — I /2 c o v1/2 [3.01x10 n.a u

(11L)

In this equation a is the transfer coefficient, which is obtained directly from the Tafel slope. The ratio of the peak current densities

For other stoichiometries the dependence on concentration will always be of the form (RT/nF)ln

nc

where

nc

exact form of which depends on the specific stoichiometry of the reaction being considered. The peak current density for a reversible linear potential sweep is given by

in the two regions is given by

is a product of concentrations, the i(irrev)/ i p (rev) =

1.107(a/n) I /2

(12L)

Since (a/n) is usually smaller than unity, the peak current in the linear Tafel region is, as a rule, smaller than in the reversible region. The difference is not very large, however. For a = 0.5 and

▪

ELLC:i '10 DE KINETICS

415

1LN tyllES: 2

L EXPERIMENTAL

p-nitrosophenol to p-phenolhydroxylamine 0

>

0.2

OH

O

CI

0.1

LI

OH (13 L)

+2H -I- 2e

CI

_J • 00 z w

—

On the return sweep the reverse reaction is clearly seen as a peak at

E 112

vc

0 0_ -

- 0.05 V, but another anodic peak is observed at about 0.22 V. We did

0.I

not observe a corresponding cathodic peak in the first sweep, but such a

0_ 10 -3

10 -2

10 -1

I

10

10 2

peak is clearly seen in the second and subsequent sweeps. A chemical reaction following charge transfer must have taken place, producing a

SWEEP RATE/Volt sec -1

Fig. 4L Variation of the peak potential E for an anodic process with sweep rate over a wide range, covering both the reversible and

new redox couple. This has been identified as the decomposition of p-phenolhydroxylamine to p-imidequinone and water

the linear Tafel regions. The "critical" sweep rate v is the one to be used in Eq. IOL to calculate k f.

OH + H 2O

(14L)

n = 1 the above ratio is equal to 0.78 and for a = 0.5 and n = 2 it is HNOH

reduced to 0.55.

NH

The dependence of the peak potential on sweep rate over a wide which forms a redox couple with p-aminophenol, as shown by Eq. 15L

range is shown in Fig. 4L.

OH

25.5 Uses and Limitations of the Linear Potential Sweep Method + +2H +

Linear potential sweep and cyclic voltammetry are at their best for

qualitative studies of the reactions occurring in a certain range of potential. In Fig. 5L, for example, we see the cyclic voltammogram

NH

2e

(15L)

M H2

obtain on a mercury-drop electrode in a solution of p-nitrosophenol in acetate buffer. Starting at a potential of 0.3 V

versus SCE, and

sweeping in the cathodic direction, one observes the first reduction peak at about - 0.1 V. This potential corresponds to the reduction of

Since the two redox couples have widely different redox potentials, they can be easily detected on the cyclic voltammogram. This is an example of a reaction sequence commonly referred to as an

ece mechanism,

416

ELECTRODE KINETICS

Another example of the usefulness of cyclic voltammetry, shown in

indicating that an electrochemical step is followed by a chemical step; which is again followed by an electrochemical step. It should be obvious that the relative peak heights in Fig. 51, depend on the sweep rate and on the homogeneous rate constant for hydrolysis. Increasing the sweep rate causes an increase in the first

417

L EXPERIMENTAL TECHNIQUES: 2

Fig. 6L features the reduction of Ti 4+ in a low temperature molten salt bath of the chloroaluminate type, consisting of a 1:2 mixture of NaC1 and AICI . Two reduction steps, corresponding to the Ti 4+/Ti3+ and the 3 Ti3+/Ti2+ couples, separated by about 0.5 V, are clearly seen.

oxidation peak and a decrease of the second, and vice versa. Methods of evaluating the rate constants of reactions preceding or following charge transfer, from the dependence of the peak currents on sweep rate, have been described in the literature in detail and will not be discussed here. +3 +4

(.1

Ti /I)

E 2 +3 Ti+ / Ti

I— E I-

z

0

CU RRENT

Lu t-

w ct -2 -0.5

0.2

0.1

0.0

-0.1

- 0.2

POTENTIAL/Volt vs SCE

Fig. 5L Cyclic voltaminogram in a solution of p-nitrosophenol in acetate buffer, on a mercury-drop electrode. Note the appearance of the second cathodic peak in the second sweep only. Reprinted with permission from "Instrumental Methods in Electrochemistry" page 199. The Southampton Group, copyright 1985, Ellis Norwood.

0

0.5 POTENTIAL/Volt vs RAIE

Fig. 6L Cyclic voltammogram in 1:2 NaCIIAlCl 3 molten salt bath contain-

ing 4.8mM Ti 2+ (added by anodic dissolution of titanium), at 150° C. Working electrode: polished tungsten wire sealed in glass, 0.005 cm 2. An aluminum wire placed in a separate compartment containing no titanium served as the reference electrode. v = 100 mV/s. Data from G. Stafford, NIST, USA,. private communication.

ELL.' i RODE KINETICS

L EXPERIMENTAL 'IL.L'itNIQUL,s: 2

Determining the peak current density in cyclic voltammetry can

Perhaps the greatest drawback in the quantitative use of linear

sometimes be problematic, particularly for the reverse sweep, or when

potential sweep is due to the uncompensated solution resistance. We

there are several peaks, which are not totally separated on the axis of

have already discussed this point in some detail with respect to

potential. The usual way to determine the peak currents is shown in

galvanostatic and potentiostatic measurements. It was shown that if

Fig. 7L. For the forward peak, the correction for the baseline is small

this uncompensated resistance cannot be reduced to a negligible level,

and does not substantially affect the result. For the two reverse

(either by proper cell design or by electronic means, or preferably by

peaks, however, the baseline correction is quite large and may introduce

both), galvanostatic measurements are, as a rule, more reliable. The

a substantial uncertainty in the value of the peak current density. In

reason, as we have demonstrated, is that in a galvanostatic measurement

fact, there is no theory behind the linear extrapolation of the base-

the experiment is conducted correctly, even if iR s is substantial. One

lines shown in Fig. 7L, and this leaves room for some degree of

is then left with the problem of measuring a small activation overpoten-

"imaginative extrapolation." This is one of the weaknesses of cyclic

tial on top of a large (but constant) signal due to solution resistance,

voltammetry, when used as a quantitative tool, in the determination of

which is a matter of using high quality measuring instruments (cf.

rate constants and reaction mechanisms.

Fig. 16K). In a potentiostatic experiment, the uncompensated solution resistance distorts the shape of the pulse, so that the experiment itself is not conducted under the presumed conditions (the potential

1. 0

step is no longer a sharp step), a problem that cannot be corrected `‘1

post factual by increasing the sensitivity and accuracy of the measuring

0.8

equipment. Linear potential sweep, being a potentiostatic technique,

E

0.6

Fig. 7L Commonly used

to measure the peak current densities in cyclic voltammetry.

worse, as we shall see in a moment. 0.4

graphical method of extrapolating the baseline,

suffers from the same drawback, but the problem in this case is even

E

In Fig. 8D, we compared the potential applied to the interface

z

during linear potential sweep with and without an uncompensated solution

0.2

resistance. Clearly, the error is a maximum at the peak, where the

H z

cc

current has its highest value. Just before and during the peak, the

0.0

effective sweep rate imposed on the interphase is much less than that -0 2

applied by the instrument. The assumption that v = constant, which has been used as one of the boundary conditions for solving the diffusion

-0.4

0.4 0.6 0.8 POTENTIAL/Volt

0.0 0.2

10

equation, does not apply. In this sense the experiment is no longer conducted "correctly". The problem, in the case of linear potential sweep experiments, is

420

ELBA ODE KINETICS

421

L EXPERIMENTAL TECHNIQUES: 2

a-_ aggravated by the common practice of extracting the kinetic information

any value in this range, the current decays to zero, since the coverage

from the coordinates of the peak namely, from the values of i p and EP,

is a function of potential and does not continue to change with time at

and their dependence on sweep rate. In other words, the information is obtained from the point at which the error is a maximum. This is by no

constant potential. This is the behavior characteristic of a capacitor

means a constant error! As the sweep rate is increased, the peak

is that required to charge and discharge the adsorption pseudocapaci-

and, in fact, the current•measurecl during the sweep, which we denote i 4,

becomes worse. The effect

tance C4) , discussed in Section 20. Assuming that bulk faradaic pro-

can be quite dramatic, as seen by comparing the two cyclic voltammograms

cesses (such as discussed in Section 25.5) and double-layer charging are

shown in Fig. 7D, which were obtained in the same solution, with and

negligible, we can write

current increases and the error due to iR

s

without electronic iR compensation. Admittedly we have used a rather s

ig) = q i (dO/dt) = q i (dO/dE)(dE/dt) = C 4) -v

extreme case, in which the voltammogram is visibly distorted, but it should be evident that even much smaller values of iR s can make the results of a quantitative analysis questionable. We conclude this section by noting that linear potential sweep and cyclic voltammetry are excellent

qualitative

(17L)

This makes life easy, because we already know how C 4) depends on potential for different isotherms. If adsorption follows the Langmuir isotherm, the current during cyclic voltammetry is given by

tools in the study of

electrode reactions. However, their value for obtaining

quantitative

information is rather limited. The best advice to the novice in the field is that cyclic voltammetry should always be the first experiment

i4)

[[ q Pi RT

K C • exp(--EF/RT) [1 + K 0 C•exp(—EF/RT)]

2

•v

(18L)

performed in a new system, but never the last.

25.6 Cyclic Voltammetry for Monolayer Adsorption (a)

which we obtained by substituting the value of C I) from Eq. 461 into Eq. 17L. We cannot derive an explicit form of the dependence of C I) on

Reversible region

potential for the Frumkin isotherm, but the shape of the curve can In this section we discuss a simple charge-transfer process, leading to the formation of an adsorbed intermediate, such as H0+

3

+

e

tvi

-4

H + H0 ads

2

readily be obtained numerically from the dependence of C q) on 0, on the one hand, and from the dependence of 0 on E on the other hand. The peak

(16L)

As we sweep the potential from an initial value where 0 = 0 to a final value where 0 is essentially unity and back to the initial value, we observe a faradaic current, associated with the formation and removal of a monolayer of adsorbed species. If we hold the potential constant at

current density is obtained, by combining Eq. 561 with 17L. i l) (p) = (qiF/4RT)[

+ (f/4)

v

(19L)

in which f = r/RT is the dimensionless parameter defining the rate of change of the standard free energy of adsorption with coverage. It is important to note that the peak current density, and indeed the current

42.2

FCI RUDE KINETICS

L

TECHNIQUES: 2

density at any value of the potential, is proportional to the sweep rate v. This makes it relatively easy to distinguish between activation controlled surface processes and diffusion controlled bulk processes, for which the peak current is proportional to v 1/2 . 0.2

The peak potential can readily be obtained from the Frumkin isotherm [

exp(rO/RT) = K 0C•exp(—EF/RT)

(1210

if we recall that the maximum pseudocapacitance always occurs at = 0.5, irrespective of the value of the parameter f (cf. Fig. 141). This yields E = (RT/F)1n(K C) — (RT/F)(f/2) o

(20L)

We have defined above (cf. Eq. 51I) the standard potential for 0 adsorption, EG , as E°

(2.3RT/F)logK

o

(51I)

Substituting this into Eq. 20L, we then have E = E ° + (2.3RT/F)logC — (RT/F)(02) P

-0.2

(21L) 200

which shows that the peak potential is shifted from its value for the

0

-200

-400

-600

Potential/mV vs E °0

Langmuir case by an extent which is proportional to the interaction parameter f.

Fig. 8L Cyclic voltammogram for monolayer adsorption and desorption of

Cyclic voltammograms calculated for different values of the

species formed by charge transfer. v = 0.1 Vls, q I = 230

parameter f are shown in Fig. 8L. One should note that in the present

[tClcm 2 . The value of the parameter f = rIRT of the Frumkin is the standard potential 0 for an adsorbed species, defined as the potential at isotherm is shown on each curve. E

In chapter I the standard potential for adsorption go was defined for an anodic process. Here we deal with a cathodic process, hence the difference in sign.

(01(1 — 0)) = 1.0 and C = 1, for the Langmuir isotherm (f = 0).

424

ELECTRODE KINETICS

425

L. EXPERIMENTAL TECHNIQUES: 2

4 case the peak current is independent of concentration, while the peak potential depends on it through a Nernst-type equation. This is in contrast with the similar equations for reaction of a bulk species, where i

is proportional to the bulk concentration (Eq. 8L and 11L),

while E is independent of it, for a simple stoichiometry (Eq. 5L and 10L). Also, the anodic and cathodic peaks for formation and removal of

E

—3

an adsorbed species occur at exactly the same potential, unlike the case of reaction of a bulk species. As before, we would like to have an estimate of the peak current,

—5

0)

0

for comparison to the current resulting from other processes, which may

—7

take place simultaneously. For the so-called Langmuir case, when f = 0, we calculated a

9

\

1 = 2.3x103 j1F/cm2 , which gives rise to a peak maximum value of Co(max,

log v

current density of 0.23 mA/cm 2 at v = 0.1 V/s, about 100 times the typical values of id1 . Consequently, double-layer charging does not interfere seriously with measurement of the cyclic voltammogram. Since depend linearly on sweep rate, their ratio is independ1 dent of it. For finite values of the parameter f, the current is, of course, smaller. In Fig. 9L we show the values of io (max) as a function of

both i and i

Fig. 9L Comparison of the peak current density for formation of an adsorbed monolayer io (max) with the peak current density observed for the diffusion-limited oxidation of an impurity (assumed concentration of 0.01 mM) i , as a function of sweep rate. Hatched area represents the range of sweep rates where measurement is recommended.

sweep rate, for f ranging from zero to 35. Assuming that the only other material that could react electro* chemically is some unknown impurity, at a concentration of 0.01 mM, we obtain the range of applicability of cyclic voltammetry for the study of adsorbed intermediates formed by charge transfer.

Figure 9L has several features in common with Fig. 3L; but there are also some differences. In both cases we have defined the "useful range" as one in which the peak current is at least 10 times the background current. The top of both diagrams is bounded by the maximum current density that can be used without causing a severe error due to

iR , as discussed earlier. There is no need to choose a lower limit of s sweep rate in Fig. 9L, since this process is not influenced by mass *

transport. The lowest sweep rate is, in fact, determined by inter-

This is a rather pure solution, with an impurity level of about

I ppm.

ference of the residual faradaic current, hence it depends on the purity of the system.

426

ELECT RODE KINETICS

The values of f used in Fig. 9L cover the range of values of this parameter reported in the literature. It is not a parameter one can

4L.

L. ILX k IMENTAL TECHNIQUES.

considered. For the linear Tafel region we can write

control experimentally; rather it is a property of the system. The

(22L)

— 0)exp(—(3EF/RT)

i4) =

upper sweep rate recommended is determined by the highest current that can be measured without substantial distortion of the linear waveform

which is simply the expression for the forward rate of the reaction

due to uncompensated solution resistance. For the value of 20 mA/cm 2 we

shown in Eq. 16L. The peak current follows from the condition that

have chosen here, this is in the range of 9-90 V/s, depending on the

di /dt = 0, namely

value of f.

This limit can be significantly increased by the use of

ultramicro electrodes in which case iR decrease. s Last but not least, increasing the roughness factor is advantageous in this case, for the same reason that it is a disadvantage in the case

1.6 E LL

E

1.2

shown in Fig. 3L, since i o is proportional to the real surface area while i is proportional to the geometrical surface area (i.e., it is

U

essentially independent of the roughness factor).

F - 0. 8

GO

Linear Tafel region In the foregoing discussion we have tacitly referred to the

reversible case, in which the rates of adsorption and desorption are so fast that the value of 0 at any moment during the transient is equal to its equilibrium value. Having 0 controlled totally by the potential,

z

a_ (.)

o 0.4 D

—

U) 0_

0.0

0.5

0.6

linear potential sweep. For the case of formation of atomic hydrogen on the surface of a platinum electrode, (Eq. 16L) this assumption holds up to a sweep rate of about 1 V/s. For the formation of OH species on the same surface, it does not hold even for a slow sweep rate of 1 mV/s. The transition from reversible to irreversible conditions is, of course, gradual. Here we shall discuss the equivalent of the linear Tafel region, namely the case in which only the forward reaction needs to be

0.8

0.9

POTENTIAL/Volt

through the appropriate adsorption isotherm, is equivalent to having the surface concentrations of reactants and products totally controlled by the Nernst equation, which is the assumption made for the reversible

0.7

plots during linear potential sweep for monoFig. IOL Calculated layer adsorption, as a function of sweep rate, for q i = 160 1.tClcm. The range of sweep rates and rate constants have been chosen so as to show both the reversible and the linear Tafel regions. The Y — axis is given in units of pseudocapacitance, C = i/v. Reprinted with permission from Srinivasan and Gileadi, Electrochim. Acta, 11, 321, (1966). Copyright 1966, Pergamon Press.

428

ELEC I RODE KINETICS

0 = diet = k.exp(—(3EF/RT)[(1 — 0)((3F/RT)•v — (dO/dt)] (23L)

L

429

EXPERIMENTAL TECHNIQUES: 2

which makes it easy to probe the interphase over a wide range of frequencies, and to record and analyze the data. Modern instrumentation, which is commercially available, covers a frequency range of about

hence i4) (max) is given by i1(max) = (IP — 0)((3F/RT)•v

(24L)

12 orders of magnitude; from l0 5 FIZ to 107 Hz. This is a very wide range of frequencies indeed, when compared to other fields of spectros-

It can be shown that in this case the peak current occurs when

copy. We recall, for instance, that visible light scarcely extends over

0 = (1 — e-1 ) = 0.63. Hence the peak current is given by

a factor of 2 in frequency, and the whole range from vacuum UV to the far infrared covers no more than three orders of magnitude in frequency.

i 1 (max) =

((3/e)(q 1 F/RT)•v =

5.44 (qi F/RT)•v

(25L)

This value of ymax) is applicable to the Langmuir case, and should therefore be compared to Eq. 19L with f = 0. The irreversible peak is hence somewhat lower, as shown in Fig. 10L. The peak potential is shifted with sweep rate, following the equation

In fact, the range of frequencies which can be used in EIS measurements is limited more by the electrochemical aspects of the system than by instrumentation. Thus, measurements at very low frequencies take a long time, during which the interphase may change chemically. While it is technically possible to make measurements at, say, 10 -5 Hz, this would take longer than a day, and the changes in the interphase during the measurement at a single frequency could make the result meaningless. At

E = — (RT/13014(3•q 1 F/RT) — (RT/1301n(v/k)

(26L)

and the symmetry factor 13 can be obtained from the slope of a plot of Ep versus log v.

the high frequency end, stray capacitances and inductances combine with possible nonuniformity of current distribution at the electrode surface, to make the results unreliable. For these reasons, EIS experiments are usually conducted in the range of 10 -3 to 105 Hz.

26. ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY (EIS)

For a circuit containing both capacitors and resistors, the ratio between the applied voltage signal and the resulting current signal is

26.1 Introduction The use of a phase-sensitive voltmeter for the study of the electrical response of the interphase was mentioned in Section 16.8 as

the impedance Z(w), which is a function of frequency. The impedance of a pure resistor is simply its resistance R, while the impedance of a pure capacitor is given by Z = — (j/coC) c

an accurate method for the measurement of the double-layer capacitance. But this instrument has far more important uses in electrochemistry than

(27L)

The total impedance can be written in concise form as:

just the measurement of capacitance. by combining a phase-sensitive voltmeter (also called a lock-in amplifier) with a variable frequency sine-wave generator, one obtains an electrochemical impedance spectrometer. Such instruments are commonly combined with a microcomputer,

*

The minimum time needed to make a measurement at any frequency is

the inverse of the frequency (i.e., the period) of the perturbing wave.

430

ELEC "I RODE KINETICS

Z(w) = ReZ — j(ImZ)

(31G)

where ReZ and ImZ refer to the real and the imaginary parts of the impedance, respectively, and j = (-1) 112. It follows that the absolute value of the impedance vector is given by

L EXPERIMENTAL ECHN1QUES: 2

26.2 Graphical Representations The results of electrochemical impedance spectroscopy (EIS) can be displayed in a number of different forms. We discuss first the one Similar commonly referred to as the complex-plane impedance plot. representations, in which the coordinates are the real and imaginary

I Z(w) I = [(ReZ) 2 + (ImZ)1 1/2 (28L)

admittance (ReY and ImY) or capacitances (ReC and ImC) are referred to

The angle (f) is the phase shift between the applied sine-wave voltage and

as the complex-plane admittance and the complex-plane capacitance plots, respectively. Other ways of displaying the results are the so-called

the resulting sine wave current. E = AE•sin(w•t) and i = Ai•sin(w•t + (p)

(29L)

where AE and Ai are the amplitudes of the voltage and current waves, respectively.

For a pure resistor, the phase shift is zero and for a

pure capacitor it is — 1t/2.

For any actual interphase the value is

somewhere in between, depending on the chemical nature of the interface and on the frequency employed. It is most convenient to represent Z(w) as a vector in the complex plane, in which the X-axis is ReZ and the Y-axis is ImZ, as shown in Fig. 11L. Both the absolute value of the impedance vector and the phase angle vary with frequency, of course, for any given equivalent circuit.

Bode magnitude plot, in which log I Z(w)I is shown as a function of log(w) and the Bode angle plot, in which the phase-angle, cp, is plotted versus log(w). There has been some argument in the literature as to the best method of displaying EIS data. Nowadays, it is easy to display the data in all the above ways, to compare and choose the one that best suits the particular system being studied, making the above argument somewhat obsolete. The perfect semicircle shown in Fig. 12L is constructed by connecting the tips of the impedance vectors at different frequencies. The frequency itself is not shown, and this is one of the disadvantages of this form of presentation. This is sometimes corrected by marking the frequencies at which specific measurements were taken on the semicircle (cf. Fig 13L). The diameter of the semicircle is equal to the resis-

Fig. I IL Vector represenfV

tance and is independent of the capacitance. As a result, plots measured for a fixed value of R Fbut different values of C cannot be

E

distinguished in this type of presentation, even though corresponding

tation of the impedance Z(w) in the complex plane. ReZ and ImZ are the real

dl

point on the semicircle have been measured at different frequencies.

and the imaginary components of the impedance, respectively.

Recall (cf. Fig. 10G) that at the highest point on the semicircle ReZ

one has R C F d1

= 11(0

max

(30L)

432

ELECTRODE KINETICS

ReZ ReY = [„ (ReZ) 2 + (ImZ) 21

C di

--I IImY ---

ImC = Fig. 12L Complex-plane representation of the impedance vector as a

If we use values of R F= 103 SI and C = 1 pF, this point will be dI observed at co = 103 Rad/s. If C is increased to 10 pF without changing RF , the same plot will be observed, but the maximum will now occur at co = 102 Rad/s. This is not a real disadvantage from the point of view of calculating the numerical values of various components in an equivalent circuit, but some visual information is lost, since changes in capacitance do not show up at all.

(31L)

10 2 I mZ ImZ (

(32L)

z )2

and for the complex-plane capacitance plot one has:

ReC =

function of frequency for a simple circuit, consisting of a capacitor and resistor in parallel.

ReZ

ImZ

(ReZ) 2 + (ImZ) 21

AAMAMR

433

L EXPERIMENTAL TECHNIQUES: 2

ImZ (ReZ) 2 + (ImZ) 2 1 °1 —

ImZ

(IZ

(33L)

i) 2

ReZ

1

ReZ

(ReZ) 2 + (ImZ) 2

(I)

(1Z1) 2

1

(34L)

In Fig. 13L, four different ways of presenting the response of the same circuit are compared. The values of the components constituting the 20 p.F. equivalent circuit were chosen as RF = 10 kO, Rs = 1 la) and Cdl =

Each way of presentation has its advantages and its disadvantages. From the impedance plot RF and R s can be read directly and the double-layer capacitance can be calculated, employing Eq. 30L. The relevant time = 5 Rad/s. The constant in this case is T = R C = 0.2 s, hence co F di max complex-plane admittance plot yields the same information. One should note, however, that the relevant time constant in this type of presenta-

obtains the complex-plane impedance plot shown in Fig. 13L(a), which is

= RC where R = (R R )/(R + RF) is the parallel combination S dl S F In the present example i s = 0.018 s and of the two resistors.

similar to Fig. 12L, except that the semicircle is displaced to the right on the ReZ axis.

= 55 Rad/s. The same information can be obtained from these two ways of presentation, but the complex-plane impedance plot is mostly

For the complex-plane admittance plot the real and the imaginary parts of the admittance are defined as

preferred, because the two resistances are shown in it directly. On the complex-plane capacitance plot the semicircle results from the series combination of R s and Cdt and the intercept with the real

Adding a solution resistance to the circuit shown in Fig. 12L, one

tion is

max

ELECTRODE KINETICS

434

L EXPERIMENTAL TECHNIQUE.:i. 2

of the capacitance and its variation with the different parameters of a

the experiment.

8

Plotting the same data in the Bode-type representation, one notes

C 6 10,•

,••

5 .....

two points: 1.

•„ • • 2.5

4

Although the values of the two resistors are easily discerned,

there is no region in which the circuit behaves as a pure capacitor. 22

2

The slope never reaches a value of — 1, and y never even comes close to

50 0

— 90°, which one would have for a pure capacitor.

2

2.

resistive behavior of the system than is the plot of log I Z(co)I

d

C

16

• 7.1 • • 8.9 •

12

—80 6 -

E

60\

N o) 2

• • •••140

N 8 o

- —

0

3.2:

40

to. 89

0

4 500 0

Vi 0

280 4

8

12

20

16

-3 -2

ReC/p,F

-1

0

1

2

3

versus

log(w). The detailed shapes of these curves depend, of course, on the numerical values chosen for the various circuit elements. Had we used a

Rs

16

40 22 • •-•-•...,•••

7

The phase angle is a more sensitive test of the capacitive or

4

a.

log W /Rad sec - I

value of R = 10 SI instead of 1,000 C2, the two horizontal lines in s Fig. 13L(d) would have been much farther apart and an (almost) pure capacitive behavior would have been observed in the intermediate region. It should be remembered that the curves shown in Fig. 13L are all simulated and therefore "ideal" in the sense that they follow exactly the equations derived for the given equivalent circuit. In practice,

Fig. 13L Comparison of (a) complex-plane impedance, (b) complex-plane

the points are always scattered as a result of experimental error.

admittance, (c) complex-plane capacitance and (d) Bode

Also, the frequency range over which reliable data can be collected does

magnitude and Bode angle plots for the same equivalent

not necessarily correspond to the time constant which one wishes to

circuit. C = 20 p.F; R = 10 kfl; R = 1 M. Values of 0) s (Radls) at which some of the points were calculated are shown.

measure. For the case shown in Fig. 13L(a) the semicircle can be constructed from measurements in the range of 1

20.

In

Fig. 13N(b) one would have to use data in the range of about 10 co 200 axis yields the value of the capacitance. The vertical line is due to the parallel combination of

RF

and Cd! .

It would seem then that the complex-plane impedance plot is the better way of presenting the data, if one is mainly interested in the value R Fand its variation with time or potential. The complex-plane capacitance plot, on the other hand, brings out more directly the value

••.

to evaluate the numerical values of the circuit elements. From the Bode magnitude plots, Rs can he evaluated from high-frequency measurements (co 100), while R Fcan be obtained from low frequency data (co 1). The capacitance can he obtained approximately as

Cdl = 1A0 I Z I

at the

inflection point (which coincides with the maximum on the Bode angle plot), but this would he correct only if cp =

—

90°, that is, if the

436

ELECTRODE KINETICS

circuit behaved as a capacitor at this frequency.

In the present

in which the parameter cr is defined as follows:

example Amax = — 58 ° and the value of Cdl calculated in this manner is 18.4 p.F, compared to the value of 20 j_IF used to obtain the curves.

RT 1 1 F 2 2 1/2 c o D I/2 Ox ox )

[

26.3 The Effect of Diffusion Limitation

437

L. EXPERIMENTAL TECHNIQUES: 2

(n

(37L) 1 CD R

" 2

R

This equation can be simplified if we assume that the two diffusion

So far in this chapter we have discussed only equivalent circuits

coefficients are approximately equal, and consider a solution in which

that correspond to charge transfer: that is, the situation in which the

the concentrations of the oxidized and the reduced species are also

faradaic resistance R F is high and diffusion limitation is negligible. We might note in passing that EIS is inherently a small-amplitude

equal, yielding

technique, in which the ac component of the i/E relationship is in the

RT

G=

21 "

(38L)

D I /2 C°

linear region. This is most readily realized by having the dc level at zero — in other words, by maintaining the system at its equilibrium potential — and applying a low amplitude perturbation. It should be recalled, though, that the response to a small perturbation can be linear, even if the system is in the nonlinear region, as discussed in Section 23.2 (cf. Eq. 5K). The faradaic resistance measured by EIS is

the differential faradaic resistance also depends exponentially on potential. tl = b•log(i/i .) hence

RF =

b/i

(35L)

Turning now to the case in which diffusion control must be considered, we have already seen that the equivalent circuit takes the form shown in Fig. 2K, in which the symbol —W— represents the so-called

Warburg impedance which accounts for diffusion limitation. The diffusion equations have been solved for the case of a low amplitude sine wave applied to the interphase. The Warburg impedance is given by

Z

w = a.co

- 1 /2

— j• CY• (0

- 1 /2

(36L)

6 —Im Z /0. cm 2

In the linear the differential resistance, defined as R = On/30 c . Tafel region i is an exponential function of the overpotential, hence

4

*

*- -s -

-

2

•a

`,* ,*

0

0

2

4

6

8

10

ReZ /0.cm 2 Fig. I4L Complex-plane-impedance plot for an equivalent circuit with some diffusion limitation at low frequencies. Rs = I Cl.cm 2; R F= 5 f•on2; C = 20 [iF1cm2; Cox ° = C° = 10 mM; D = di R Ox = DR 2 M /x/0 -5 cm 21s; (c = 12 f2 on A , IT = 900, i = 5 mA1cm 2 c 5 and k = 5x10 3 culls). Calculated for 10 -3 CO 5_ 10 Radls. s,h

438

ELL.11.:

I I,

1,3 IL)

If the two concentrations differ widely, the lower concentration determines the value of a. One notes that the real and the imaginary

L

I-Li ,. I

AL

•IECHN IQ LI

chosen here correspond to a moderately slow reaction, having a standard heterogeneous rate constant of ks,h

= 2 .7 X 10-3 cm/s.

parts of the Warburg impedance in Eq. 36L depend on frequency in the same way. For this reason the Warburg impedance is sometimes referred to as a "constant phase element", implying that the phase shift generaimpedance format, this leads to a straight line with a slope of unity,

a

C=100mM

1.2

N

E

_ 1 0=26mA/cm 2

—iM Z/kn. c M 2

ted by it is independent of frequency. Plotted in the complex-plane

•

6 0 .8

as shown in Fig. 14L. We note that the Warburg impedance, which is proportional to to -1/2 , is in series with the faradaic resistance,

RF .

N

E

0.4

At high frequencies one 0 10.0

obtains the usual semicircle, whereas the Warburg impedance becomes

- c= tomm

12

1 0=2.6mA/cm 2 •

8

4 -

••

•-•

•

•

•

•

" t.

0 10.4

predominant at low frequencies. The frequency at which the transition

11.6

11.2

10.8

10

12 0

14

18

22

ReZ / kf)

ReZ /kn•cm 2

26

30

.cm 2

occurs depends on the concentrations of reactants and products, (which 3000

overpotential, which determine the value of the faradaic resistance,

N

RF .

In Fig. 15L(a—d) we show four complex-plane-impedance plots calculated for the concentrations of 100, 10, 1 and 0.1 mM, respectively. Note that

RF,

which is proportional to W ., also depends on

concentration, therefore the scale of both ImZ and ReZ must be changed from Fig. 15(a) to 15L(d). For these calculations we assumed that i is a linear function of concentration. At the highest concentration shown

E C

—IM Z /k Q• CM 2

determine the value of 0), on the exchange current density and on the 120 - C.= 1mM i 0=0.26mA/cm 2 80

N

40

C=0.1mM 1 0=26 p.A/cm 2 2000 •

1000 ...........

0

0 0

40

80

160 200

120

ReZ /k()-cm 2

0

1000

2000 3000 4000

ReZ /kf). CM 2

here a semicircle characteristic of a charge-transfer limited process is clearly seen, with diffusion limitation becoming important only at low frequencies. As the concentration is decreased, diffusion limitation becomes gradually more important. In a 1 mM solution the initial part of the semicircle is barely seen and in a 0.1 mM solution the process is mostly diffusion controlled. It may be noted that the values of

RF

Fig. 15L Complex-plane impedance plots showing the gradual change from charge-transfer to mass-transport limitation with decreasing concentration. 10 -3 < 03 < 10 5 Radls. Cdt = 20 pFlcm 2 , R s = 10 •cm. Values of the parameters for a, b, c and d, respectively:

RF =

1.0; 10; 100 and 1,000 •cm 2; G = 1.2; 12; 120

and 1,200 acm 21s 112 ; C = 100; 10; 1 and 0.1 mM.

Note that a =

"2

0.112 -C ), dl

where Td is the relaxation time

for diffusion, as defined in Eq. 17M. Hence Cc /T

cd

Am.

2

R /2a - C F

dl

.

440

ELECTRODE KINETICS

26.4 Some Experimental Results

L. EXPERIMENTAL TECHNIQUES: 2

441

3.

interface, and may sometimes be understood by correlating to some other,

The plots we have shown, which are all based on simulated data,

independently measured, property.

serve the purpose of illustrating the principles involved. The results

The semicircle may also be distorted by experimental errors, which

of real experiments are rarely so simple and easy to interpret. This is

arise mainly from three sources: (a) nonuniform current distribution,

caused by two types of factors: on the one hand, the reaction may not be

caused by the geometry of the cell as a whole, or by screening of part

as simple as is assumed in the model. The formation of adsorbed

of the working electrode with the Luggin capillary. (b) solution

intermediates, for example, can lead to an adsorption pseudocapacitance.

creeping in the crevice formed between the electrode and its nonconduc-

The corresponding equivalent circuit usually has two widely different

ting holder, and (c) changes occurring at the surface during measure-

time constants that show up as two semicircles, which could be partially

ment. It should be remembered in this context that the equations for

overlapping. Surface heterogeneities are tantamount to different values

EIS are based on the tacit assumption that the surface is invariant

of R F(and to a lesser extent also of C dl ). Such differences can lead to a whole range of time constants that are close to each other (since

during measurement, as the frequency is scanned. This is particularly

R Ftends to change gradually from site to site on a heterogeneous surface). Rather than having many semicircles, the result is often a so-called depressed semicircle, namely one with its center lying below

problematic when one wishes to extend the data to the very low frequency

tJ

E

the ReZ axis, as shown in Fig. 16L. How should one calculate the faradaic resistance from such a plot? Surely one cannot just take the distance between the points A and B on the ReZ axis as being equal to

RF,

since this is not the diameter of the

semicircle. The distance between the points A' and B' is equal to the diameter of the semicircle, but this line does not lie on the ReZ axis.

A' ■

ReZ

What is the physical meaning of the angle of depression? A result such as shown in Fig. 16L indicates clearly that the

B

'

system cannot be described correctly by a simple equivalent circuit of the types discussed so far. Sometimes, if the depressed angle

a is

small (say, less than 10°) the problem may perhaps be ignored, and one may obtain

RF

Fig. 16L Complex-plane impedance plot with depressed semicircle. A'—B' is the diameter of the semicircle, depressed by an angle a.

either as the distance from A to B or from A' to B', which

will differ in this case by a few 10%. In any event, changes in the angle of depression or in the radius of the depressed semicircle still can be taken as an indication of variation in the properties of the

This may be the case when the Luggin capillary is brought too close to the stuface, in an effort to minimize the iR potential drop. s

442

F(9T(ODE KINETICS

443

L. EXPERIMENTAL TECHNIQUES: 2

range since, as we have noted, the time taken to make each measurement

component of the interface) represent the "capacitive loop": they are

is inversely proportional to the frequency.

due to some combination of capacitors and resistors. Points below this

In Fig. 17L some results obtained for a colToding iron electrode

axis (positive values of ImZ) belong to the "inductive loop," which can

are shown. Two well-separated time constants appear in Fig. 17L(a).

be simulated by some combination of inductors and resistors. Now, the

The corresponding faradaic resistances are due to two different charge-

physical meaning of capacitance (either

transfer processes.

(R

Cdt

or C(1)) and of resistance

Data of the type shown in Fig. 17L(b) are often observed but are

R or R ) in the equivalent circuit describing the electrodeF s' (13 electrolyte interface is well understood. On the other hand, it is not

not easy to interpret. From the formal electrical point of view, all

clear what an inductance (or a pseudoinductance) stands for, in terms of

points above the ReZ axis (i.e., all negative values of the imaginary

the physical reality at the interface, although some attempts to explain this behavior have been made.

27. MICROELECTRODES 27.1 The Unique Features of Microelectrodes So far we have restricted our discussion of diffusion-controlled processes to the case of semi infinite linear diffusion, -

which corres-

ponds to a planar electrode of infinite dimensions, in a cell where the

ReZ/kR

solution extends to infinity. It has already been pointed out that the b

word "infinity" should not frighten us, since the dimension should be "infinitely large" only compared to the thickness 8 of the diffusion layer. Since diffusion-controlled experiments are restricted to about

E

50 seconds, by which time mass transport by natural convection becomes

-

A

I

4

I 8

I

1 12

ReZ/k5-1

significant, 6 = (nD0 1/2 0.04 cm. Thus, the condition of semiinfinite linear diffusion is fulfilled to a good approximation for an electrode diameter of 1 cm, in a cell of similar dimensions. There are

Fig. 17L Complex-plane impedance plots for the corrosion of iron at two

always edge effects at the periphery of the electrode, but their

different potentials. (a) two well-separated time constants

influence on the observed relationship is usually negligible. As one

are .shown. (b) An inductive loop' is observed. Data from

decreases the size of the electrode, edge effects become more pronoun-

Epelboin, Gabrielli, Keddam and Takenouti, Electrochim. Acta,

ced. Eventually, when a microelectrode is considered, edge effects

20, 913, (1975).

become predominant. A microelectrode can be considered to be "all

444

ELECTRODE KINETICS

445

L. EXPERIMENTAL TECHNIQUES: 2

This depends, of course, on the actual size of the micro-

material diffuses to each segment on the surface from a cone of given

electrode and on the time scale used. For an electrode of 10 pm, radius

solid angle. Thus, while the diffusion.length increases with time, the

the diffusion layer thickness will be equal to the radius after about

cross section for diffusion also increases. The two effects happen to

32 ms. For an ultra-microelectrode of r = 0.25 pm, the same is true in

compensate each other. exactly in this particular geometry, with the

less than 20 ps.

result that the diffusion current density becomes independent of time,

edge."

The situation at a miniature disc microelectrode embedded in a flat

as we shall see:

insulator surface (such as an RDE of very small size) can be approxi-

The effect of geometry on the resistivity can be understood in a

mated by spherical symmetry, obtained for a small sphere situated at the

similar way. The real bottleneck is very close to the surface, where

center of a much larger (infinitely large, in the present context)

the cross section for conduction is small. Farther out, the cross sec-

spherical counter electrode. How will the change of geometry influence

tion increases with the square of the distance. Hence the contribution

the diffusion-limited current density? This is shown qualitatively in

of this region to the total resistance soon becomes negligible. It is

Fig. 18L. As time progresses, the diffusion layer thickness increases,

indeed found, as we showed in Figs. 3C and 4C, that the total resistance

causing, in the planar case, a proportional decrease in the diffusion

between the working and counter electrodes is independent of the

current density. In the spherical configuration the electroactive

distance between them, provided this distance is large compared to the diameter of the smaller sphere. This statement is very similar to what we have said about spherical diffusion, where the current becomes independent of time while the diffusion layer thickness keeps growing.

Fig. ]8L Schematic repre-

This happens when 5 has become very large compared to the radius of the

sentation of planar and

working electrode.

spherical geometry for diffusion (the latter is shown in two dimensions, for simplicity). Hatched

27.2 Enhancement of Diffusion at a Microelectrode The response of a spherical electrode to a potential-step function

in the limiting current region is given by

areas represent lamina and cones from which electroactive material can diffuse to the surface.

=

1

).cl)t)

I/2

(39L)

At long times, spherical diffusion is predominant, the current becomes independent of time, and Eq. 39L takes the form i = nFDC°/r

(40L)

446

1LCHNIQUES: 2

Thus, in a spherical field of diffusion (which is achieved for a

flow, which typically creates a diffusion layer thickness of the order

microelectrode after a time determined by its radius), one obtains an

of 50-100 pm. This can be a great asset for on-line analysis in

equation similar to that given for semi-infinite linear diffusion,

industrial applications, where the flow rate may fluctuate and would

except that the radius of the electrode plays the role of the Nernst

otherwise have to be measured and corrected for.

diffusion layer thickness.

The validity of Eq. 40L is one of the

incentives for fabricating ultramicro electrodes.

Thus, for example,

27.3 Reduction of Solution Resistance

equaled by the current at a microelectrode of r = 5 pm, in a quiescent

The iR potential drop due the uncompensated solution resistance s associated with different geometries was discussed in Section 8.3. For

solution. Such a device is relatively easy to fabricate and would not

a spherical electrode, which is of interest here, we can write (cf.

even be considered to be an ultra-microelectrode. Electrodes having a

Eq. 8C)

the limiting current obtained on an RDE operated at 10 4 rpm can be

radius of 0.25 pm have been prepared in several laboratories. Using Eq. 40L we note that the limiting current density at such an electrode is about 0.8 A/cm 2 when n = 2 and C ° = 10 mM. Such limiting current densities, which cannot be reached at steady state by any other method, substantially increase the range over which the current-potential

E

iR

= i.d.p s

[

1 r+dj

(41 L)

where p is the specific resistivity and d is the distance of the probe from the electrode surface. If the current density i is replaced by the total current I, Eq. 41L takes the form I•

relationship can be obtained under activation-controlled conditions, as

d • p2 1( r ±r d

(42L)

4nr

discussed in Section 23.1. It should also be noted that the time of experiment for a micro-

Of interest to us here is the limiting form of this equation, for large

electrode is not limited by natural convection, as found in the case of

values of d (d/r » 1), namely, far away from the electrode. Equation

semi-infinite linear diffusion. Natural convection gives rise to a

41L yields a resistance of

diffusion layer thickness of the order of 0.02 cm, and its effect is not felt, if the radius of the microelectrode is less than about 10 pm.

R = p•r s

(43L)

Another advantage of having a very large limiting current density

showing that the resistance (in units of •cm 2 i.e., normalized for unit

at steady state is in the analysis of trace elements. Using Eq. 40L

surface area), is proportional to the radius of the ultramicro elec-

again for the same sized electrode, we obtain a current density of about

trode. As usual, a numerical example will help to illustrate the advantage

8 pA/cm 2 for a concentration of 0.01 ppm, (assuming a molecular weight of 100). Thus, measurements in the part per billion range should be possible with ultramicro electrodes.

of ultramicro electrodes, from the point of view of solution resistance. In Section 27.2 we obtained a limiting current density of 0.8 A/cm 2 for

When an ultramicro electrode is used as an electroanalytical tool,

an electrode having a radius of 0.25 pm, in a 10 mM solution. If we

the diffusion-limited current density is not affected by the rate of

assume a specific resistivity p of 40 S•cm, the solution resistance Rs,

448

ELECTRODE KINETICS

449

L EXPERIMENTAL TECHNIQUES: 2

a-

medium conductivity, and yet arrived at an ohmic potential drop of less

12 density of 1 mA/cm 2 the total current observed is hence only 2x10 A. although this is measurable in the advanced laboratory environment, the

than 1 mV at the very large current density of 0.8 A/cm 2. Thus, using

measurement is by no means easy, and accuracy is limited. The need to

an ultramicro electrode extends the range of measurable current densi-

measure such small currents would make the use of ultramicro electrodes

ties because (a) the limiting current density is inversely proportional

for any routine measurements, particularly in an industrial environment,

to the radius and (b) the resistivity is proportional to the radius.

impractical. The second disadvantage entails the extremely high volume

More concisely, we could say that both the diffusion-limited current

to surface ratio and the impossibility of purifying solutions to a level

density and the conductivity are inversely proportional to the radius.

that would ensures that impurities could not accumulate on the surface

according to Eq. 43L, is 1 x10 3 f•cm 2 . We assumed here a solution of

Another simple calculation that will emphasize the advantage of

during measurement, as discussed in Section 14.9. Thus, for a regular

using ultramicro electrodes refers to poorly conducting solutions. The

electrode one may have a VIA ratio of 10 cm. For a thin-layer cell this

specific resistivity of deionized water, for example, is about 1x106 •cm. For an electrode of the size just discussed, this would give

ratio could be as low as 10 -3 cm, while for an ultramicro electrode the ratio is of the order of 10 8 cm. Even if used in a thin-layer cell

rise to a resistance of 25 •cm 2 , allowing measurements up to several

configuration, an ultramicro electrode would have a volume-to-surface

milliamperes per square centimeter with reasonable iRs compensation.

ratio on the order of 10 6 cm. This is an inherent difficulty, which

There is something odd about Eq. 43L which needs to be clarified.

cannot be overcome by improved instrumentation. To a first approxima-

The resistance is proportional to the radius, hence in the limit of

tion these numbers mean that, given a desired level of purity of the

r 0, it should approach zero. Intuitively, one feels that this is

surface, the allowed level of impurities in solution would have to be

wrong, since the resistance should increase with decreasing size of the

seven orders of magnitude lower for the ultramicro electrode than for a

electrode. Note, however, that this resistance was given in units of

regular electrode. Bearing in mind that proper electrochemical measure-

[acm 2 ], and should be multiplied by the current density to yield the

ments must be conducted in highly purified solutions, even when elect-

potential drop. The total resistance, in Ohms, is obtained from Eq. 42L

rodes of macroscopic dimensions are employed, the level of purity needed

as p/4rcr.

for work with ultramicro electrodes is clearly not achievable.

Thus, the resistance in Ohms is found to be inversely

proportional to the radius, tending to infinity as r --> 0, as expected.

One way around this problem, which retains most of the advantages of ultramicro electrodes while largely overcoming their disadvantages,

27.4 Single Microelectrodes versus Ensembles

is to use ensembles (sometimes also referred to as arrays) of ultramicro

Microelectrodes have many advantages over regular-sized electrodes,

electrodes. Suppose that the surface of an insulator is dotted with a

but they also have two major disadvantages. First, since the electrode

regular array of conducting spots, serving as the ultramicro electrodes,

is very small, the total current flowing in the circuit is minute and

which are all connected at the back to a common current collector, as

may be difficult to measure accurately. The ultramicro electrode just

shown in Fig. 19L.

discussed has a total surface area of about 2x10 -9 cm 2 . For a current

450

ELECIRODE KINETICS

Side view

Top view

----- Insulator

Electrodes---

451

L EXPERIMENTAL TECHNIQUES: 2

The solution of the diffusion equation is best obtained by digital simulation. Fortunately, one can understand the behavior of such ensembles qualitatively, and the conclusions reached in this way are in quite good agreement with the results of (rather tedious) numerical calculations. We can discuss this problem in terms of the ratio between the Nernst diffusion layer thickness 8, given by (700 112, and the radius of the electrode on the one hand, and between S and the distance between

Current collector

two electrodes, on the other. To do this, we shall list the various possibilities, and derive the corresponding behavior qualitatively.

Fig. 19L An ensemble of ultramicro electrodes. Note that the distance between the electrodes is large compared to their diameter.

1. For 8/r 5. 0.3 the system is in the range of semi-infinite linear diffusion. The current, per unit of total surface area, is given by:

The fraction of the surface that is active is given by (d/L) 2, where d is the diameter of each electrode and L is the distance between their centers. Designing such an ensemble of ultramicro electrodes, one must compromise between the desire to make the ratio d/L as small as possible (to decrease the overlap between the diffusion fields of the individual electrodes), and the desire to make d/L as large as possible (to increase the total active area). Values of d/L in the range of 0.03-0.1, corresponding to 0.1-1% of active area, seem to be a reasonable choice, as we shall see. Interestingly, the problem just described was first solved for an entirely different physical situation, referred to as a partially blocked electrode. In this case one assumes that the surface of a

(44L)

2. For 8/r 3 but 8/L 5 0.3, the diffusion field around each electrode is spherical and the overlap between the diffusion fields of The diffusion limited neighboring electrodes is still negligible. current, per unit of total surface area, is given approximately by I

d

=

(nFDC/r)(r/L) 2

(45L)

3. For 8/L 3, complete overlap between the diffusion fields of the individual micro electrodes can be assumed. The total current is given in this case by I—

regular macroscopic electrode is partially covered by some nonconducting

d

material, which could be an oxide formed electrochemically or an impurity sticking to the surface. The physical situation is the same, whether a large part of the surface has become inactive accidentally, by the accumulation of some impurity, or whether it was made inactive by design.

(nFDC°/(700 1/2)(r/L) 2

Id =

nFDC° (TED t

(46L)

"2

Note that the current density in this case is larger by a factor of (L/r) 2 than at short times, when Eq. 44L applies. The development of

the diffusion field at an ensemble of microelectrodes is shown schematically in Fig. 20L.

• 452

ELEC. I RODE KINETICS

a

b

L. EXPERIMENTAL TECHNIQUES: 2

451

7m.

ratio of r/L, since for this equation to hold it is required that the ratio r « 5 « L. 27.5 Shapes of Micro Electrodes and Ensembles We have considered so far only disc-shaped microelectrodes, for which spherical diffusion can be applied, to a good approximation. Other forms have been used, mainly because they might be easier to fabricate. Most noted among these is the linear or strip microelectrode, which is macroscopic in length but microscopic in width. The. diffusion field at such electrodes can be approximated satisfactorily by

•

Electrode

O

Solution

•

Insulator

diffusion to a cylinder. The enhancement of diffusion is less than that

10 3

\\ 46L

N

Fig. 20L Development of the diffusion .field near the swface of an ensemble of micro electrodes. (a) planar diffusion; (b) spherical diffusion with no overlap; (c) spherical diffusion

E102

with substantial overlap; (d) total overlap, equivalent to

45L

planar diffusion to the whole surface. 1—

10

1

z

The response of an ensemble of ultramicro electrodes to a potential step is shown in Fig. 21L. The equation that controls the current in each segment of the curve is marked. Note that the current in Fig. 21L and in Eqs. 44L-46L is the total diffusion-limited current. The current

44L\ \\

0

U

1

10

10 -6

I

10-4

I

1

1

10 -2

1

10°

\

I

10 2

TIME/sec

density is always obtained by multiplying by the factor (L/r) 2 . Equation 45L corresponds to spherical diffusion to each ultramicro electrode, with negligible overlap between the individual electrodes. This is the region in which the total limiting current is nearly independent of time, as seen in Fig. 21L. Whether such a region is observed in practice depends on the design of the ensemble, that is, on

Fig. 21L Dependence of the limiting current on time following a potential step, at an ensemble of ultramicro electrodes. The equations governing different sections of the curve are marked. Data from Reller

,

Kirowa-Eisner and Gileadi, J.

Electroanal. Chem. 138, 65, (1982).

kiECI RODE. KINKI

for a disc microelectrode, of course, but the increase in surface area

M. APPLICATIONS

alleviates, to some extent, the problems of very low currents to be measured and the exceedingly large volume-to-surface ratio. The best configuration of ensembles of microelectrodes is a collection of micro discs of equal size, arranged on a uniform grid, at

28. BATTERIES AND FUEL CELLS 28.1 General Considerations

equal distances from each other. Sometimes this is not feasible, and

A battery is an energy storage device. It stores chemical energy

ensembles containing electrodes of nonuniform size and/or distance have

and releases it on demand as electrical energy. This is not necessarily

been considered. Ensembles of strip microelectrodes have also been

an efficient way of storing energy. For example, the total energy

constructed, and the variation of current with time for such configura-

consumed in the manufacturing of a commercial battery may be ten times

tions has been evaluated.

as much as the energy that can be retrieved from it, yet batteries offer

An entirely different class of microelectrodes consists of elec-

by far the best way to store energy in small packages, and in many

trodes used in biological and medical research, mostly for application

applications they constitute the only way to store energy. In recharge-

in situ.

In this case the small size enables the researcher to

able batteries the efficiency of storage and retrieval of energy, which

introduce the electrode into the living organism with minimum damage and

is often called the electric-to-electric (ETE) efficiency, can be quite

to study local effects on the scale of living cells or even smaller.

high. In a lead-acid battery used by the automobile industry the ETE

The electrochemical properties of microelectrodes, namely enhancement of

efficiency could be as high as 80%, depending on the way the battery is

the rate of diffusion and decrease of the resistance, are not the main

used. Competing methods of storing energy include chemical energy,

issue in such uses, although they should by no means be ignored in the

thermal storage, compressed gas, fast-turning flywheels (usually

design of the microelectrode and in the evaluation of its response.

operated in vacuum, to minimize losses due to friction), and pumped

We conclude this section by noting that ensembles of ultramicro

water in hydroelectric stations. We shall not discuss the advantages

electrodes hold more promise for future use in research and in industry

and shortcomings of these methods except to note that they depend on the

than single microelectrodes. Unfortunately, neither type is yet avail-

end use required and on the amount of energy being stored. For all

able commercially as a standard tool, which can be employed as needed.

portable electrical and electronic devices, batteries are practically the only solution. The unique feature of batteries is that in them electrical energy is produced directly from chemical energy, bypassing the need to construct a so-called heat engine of one type or another, which is

456

ELECIRODE KINETICS

457

M. APPLICATIONS

a-

limited by the efficiency of the Carnot cycle.

This follows from the

simple thermodynamic relationship

and on the equivalent weight of the electrochemically active ingredients, ignoring the casing, the current collector etc. As an example,

AG = — nFE

(1M)

rev

consider the Ni/Cd rechargeable battery, consisting of a nickel hydroxide positive electrode and a cadmium negative electrode, in a

which shows that, at the limit of reversibility, the Gibbs free energy released by the system can be converted to electrical energy with 100% efficiency. In fact, in certain cases it appears as though the effici-

concentrated solution of KOH. The electrode reactions during discharge are NiOOH + H2O + em --> Ni(OH) 2 + (OH)

(3M)

ency could be greater than 100%, as for the electrolysis of water: 2H

2

0

-4

21-1

2

+ 0

2

Cd + 2(OH) —p Cd(OH) 2 + 2em

(2M)

(4M)

The reversible potential for this reaction is 1.23 V, corresponding to a

The reversible cell potential in 30% KOH is 1.29 V. The electrical

free-energy change of 237 kJ/mol. Yet, when we burn the hydrogen formed

energy produced per mole is given by

in electrolysis in a calorimeter, we find that the total heat released

nFE = 2x96,485x1.29 = 2.5x105 W•s = 69 W•h rev

is 286 kJ/mol. It would appear that the efficiency of production of

(5M)

The overall reaction is

thermal energy by water electrolysis could be as high as 120%. The laws of thermodynamics are not violated, of course. The reversible potential

2NiOOH + Cd + 2H20

2Ni(OH) 2 + Cd(OH) 2

(6M)

is related to the free energy, while the heat measured in burning the

The sum of the molecular weights of the reactants is 332, leading to a

product (at constant pressure) is equal to the enthalpy. The difference

theoretical energy density of

is the entropy of the reaction, which increases in this case, since two moles of liquid are converted to three moles of gas.

energy density = 69/0.332 = 208 W•h/kg

(7M)

This should be compared to a practical value of about 40 W•h/lcg. What 28.2 The Maximum Energy Density of Batteries

is the purpose of this kind of calculation? In the development of a

Thermodynamics allows us to calculate the maximum energy density of

device it is important to know how far we are from its theoretical

a battery, which may be approached, but never reached in real batte. ** ries. This type of calculation is based on the reversible potential

limit, since the practical limit, which is roughly half the theoretical limit, can be reached asymptotically, and the effort in approaching it grows exponentially. Thus, if current technology represents only 5% of the theoretical limit, there is a very good probability that it can be

* An internal combustion engine, as well as a major electrical power

developed to, say, 50%, namely by a factor of ten. If, on the other

station are both "heat engines" in the thermodynamic sense, and their

hand, current technology has reached 60% of the theoretical limit, it is

theoretical maximum efficiency is that of the Carnot cycle. ** The practical limit is typically 50-60% of the theoretical value.

probably close to its practical limit, and the wisdom of attempting to develop it further may be questioned.

-08

ELEL-1 )DE KINETICS

28.3 Types of Batteries Primary batteries are manufactured for a single use. Best known among them is the Zn/M ► 10, dry cell, developed by Leclanche more than 100 years ago and manufactured since then in huge numbers. Primary batteries, born in a world obsessed by the development of new technologies, may die soon in a world obsessed by ecology and recycling. It it almost unbelievable that in the last decade of the twentieth century, we still use and throw out hundreds of millions of batteries every year, while the technology to produce rechargeable batteries, which could be reused a thousand times, already exists. Considered from the point of view of recycling, the transition from primary to secondary batteries is

459

M. APPLICATIONS

before use. Cycle life, which is the number of times the battery can be charged and discharged, is important. The efficiency of energy storage in secondary batteries is quite high, on the order of 60-80% and a good battery can be charged and discharged a thousand times with little loss in performance. Energy density (W•h/kg), power density (W/kg) and the temperature range over which the battery can be operated are important. The energy density of a battery depends on the rate at which it is discharged, and it must be defined for a specific rate. Two factors are involved here. First, the overpotential depends on the current density during discharge (i.e., on the rate of discharge). Second, the amount of available charge depends on the discharge rate. In Fig. 1M we show

equivalent to recycling paper or aluminum or refilling beer bottles hundreds of times. The important properties of primary batteries are energy and power density, temperature range of operation, shelf life (which is a measure

' cp 1000

of the rate of self discharge) and cost. Self-discharge is an inherent property of batteries. The fact that energy can be stored in a battery implies that it is in a thermodynamically unstable state. In its normal mode of operation the excess free energy is transformed into electrical

>F-

in

100

work. During storage, free energy can be slowly released when the same (or some other) process occurs through a chemical pathway, in which all

0

the free energy released is wasted as heat.

tZ

10

Primary batteries are extremely expensive, when the cost is calculated per unit of energy. Thus, 1 kWh • may cost anywhere between $100 and $10,000, compared to about $0.05 paid to the electric utility for the same amount of energy. As a result, primary batteries are used

O tl

1000 100 10 ENERGY DENSITY/Wh kg -1

when the amount of energy required is small, mostly in portable devices, where the cost is not critical. Secondary batteries are rechargeable. The rate of self-discharge is less critical in this case, since the battery can always be recharged

Fig. 1M The dependence of the energy density on the power density for three types of secondary batteries. Data from Kordesch, "Brennstoffbatterien", Springer-Verlag, 1984.

460

ELECTRODE KINETICS

461

M. APPLICATIONS

such dependences for three types of batteries. In each case a range,

as we shall show. The rate of self-discharge is important mainly for

rather than a single line, is shown because the characteristics of any

primary batteries. A battery that loses less than 20% of its capacity

type of battery depend on design. The energy density of the lead-acid

per year may be acceptable in most applications, but in lithium-thionyl

battery is seen to depend substantially on the power level, while that

chloride and other nonaqueous lithium batteries, a much higher degree of

of the Ni/Fe and the Na/S batteries is dependent on the power to a much

stability has been achieved. The current-voltage characteristic is very important for all

lesser extent. A Fuel cells is a different type of secondary battery, in which the

batteries. Ideally one would like to have a flat discharge curve,

chemical energy is stored in an outside container, rather than in the

namely a potential that is almost constant throughout the discharge

electrode material. This can be a great advantage from the point of

stage and falls fairly sharply to zero when the battery has been

view of energy density since, for extended operation, the weight of the

exhausted. This is shown in Fig. 2M(a), which presents discharge curves

cell itself becomes insignificant and the power density depends mainly

for two types of lithium batteries. In comparison, the voltage of a

on the weight of the fuel and its containers. Hydrogen is an ideal

simple Leclanch6 cell declines rapidly with time during discharge, as

fuel cell the theoretical energy density is 3.66x10 H2/02 W-h/kg. This should be compared to the value of 208 W.h/kg calculated fuel. In an

3

shown in Fig. 2M(h).

above for the Ni/Cd battery, or to the practical values shown in Fig. 1M Very high energy densities have indeed been realized in fuel cells built for space application, but the cost and safety aspects of storing

b

2.2

hydrogen have so far prevented this type of fuel cell from becoming

1.01.1 1.8

widely used. 28.4 Design Requirements and Characteristics of Batteries The specifications of batteries are many and varied.

First one 1

considers the energy density and the power density. These are usually given per unit weight, but in certain applications the volume may be more important than the weight. This is the case for very small

I

20 40 60 80 100 OF DISCHARGE

% OF DISCHARGE

batteries used for wrist watches and for electronic calculators, for

Fig. 2M The discharge curves of (a) a LilSOCl 2 and a Li/SO 2 battery and

hearing aids and for implantable devices. Weight is the critical factor

(b) a Leclanche cell. Note the constant voltage during most of

for portable electronic devices for both military and civilian applica-

the discharge of the lithium-based batteries. Data from Linden

tions. The future of electric vehicles depends predominantly on the

in "Handbook of Batteries and Fuel Cells" Chap.3. Linden,

energy and the power densities of secondary batteries per unit weight,

editor, McGraw Hill, 1984.

462

POTRui)L: KINETICS

463

M. APPLICATIONS

Reliability and cost are closely related, and the best compromise

active material in the cathode and carbon powder is added to make it

depends on the end use. Manufacturers of electronic home appliances

electronically conductive. The electrolyte is an aqueous solution of

such as portable radios often warn us to remove the batteries when the device is not in use for a long time, because of possible leakage. This

ZnC1 and NH 4 Cl, which is immobilized in a suitable absorbing material. 2 It is referred to as a "dry" cell in the sense that it does not contain

is an indication that the manufacturers of batteries for such devices

free liquid. There are several other materials added, the most impor-

may be producing cheap, but not very reliable, products. At the other

tant of which is HgO, which serves to decrease the rate of self-

extreme, one is willing to pay a high price for the very high reliabi-

discharge, as discussed later. The actual composition of this and all

lity required of a battery used in an heart pacemaker. Aerospace and

other batteries is a commercial secret, and different formulations are

military applications also require a high degree of reliability, but not

used by different manufacturers. Here we shall discuss only the main

quite as high as for implanted devices, probably because a larger degree

ingredients. The reactions taking place in batteries may be quite complex. In

redundancy can be built into systems of the former types. A battery stack consists of a number of cells connected in series

the case of the Leclanche cell the most important reactions are:

and packaged as a unit. It is invariably observed that the reliability

Zn + 2H20

in performance and the lifetime of stacks of batteries is less than that

2Mn0 + 2H+ + 2e ivt 2

of single cells. This can be understood on the basis of very simple statistical reasoning. If the probability of failure of a single cell

resulting in the overall reaction

during the intended service life of a stack is 1%, the probability for

Zn + 2Mn0 2 + 2H20

its flawless operation is 0.99. The probability of trouble-free operation of a stack consisting of 12 cells of this type connected in series is (0.99) 12 = 0.89. Thus, a reliability of 99% for the individual cells translates to a lower reliability of only 89% for the stack! 28.5 Primary Batteries In Sections 28.5-28.7 we shall describe a few energy storage devices. This is not meant to be a comprehensive listing of all batteries of a given kind, but rather a short review of some batteries, which are either in wide use or have been the subject of extensive research and development efforts in the past few decades. The Leclanch6 cell has a cathode consisting of a mixture of Mn0

2 and carbon powder and a zinc anode. Manganese dioxide is the electro-

Zn(OH) 2 + 2H+ + 2e N4

(8M)

2MnOOH

(9M)

Zn(OH) 2 + 2MnOOH

(10M)

Note that water is consumed in this process and enough of it must be provided in the immobilized electrolyte to allow the reaction to proceed. Zinc is an active metal (E ° = — 0.763 V, NHE) that corrodes in aqueous solutions, giving off molecular hydrogen, according to the reaction Zn + 2H20

Zn(OH) 2 +

H2

(11M)

This reaction, which takes place at open circuit, or as a side reaction during discharge of the battery, is detrimental in two ways: it consumes one of the active materials in the cell, and it produces a gas that can build up pressure and eventually rupture the cell. This is the purpose of adding HgO to the zinc anode. In contact with metallic zinc, it is reduced to mercury, which amalgamates the zinc. The exchange

464

ELECTRODE KINETICS

465

M. APPLICATIONS

current density for hydrogen evolution is much lower on mercury and its * amalgam than on zinc, thus reducing the rate of self-discharge via the hydrogen evolution reaction. Other inhibitors are also used, but

the zinc anode, and is less likely to leak.

mercury is most effective in this respect. In early designs the mercury

Lithium batteries • employing nonaqueous solvents constitute a relatively new class of primary batteries, developed in the past two decades. Here we shall discuss the Li/SOC1 2 battery, which is representative of this class.

content of the anode in the final product was several percent. The amount has been gradually reduced because of toxicity, but mercury has not been totally eliminated yet.

It is also more expen-

sive. As usual, one can buy higher reliability and better performance for a higher price, and the choice depends on end use.

Leclanche cells have several disadvantages. They have a relative-

The battery consists of a lithium anode and a carbon paste cathode.

ly short shelf life, and they must be refrigerated for long-time

The electrolyte consists of a solution of LiAIC1 4 in thionyl chloride.

storage. The energy density is about 75 W.h/kg, which is relatively low

The anode reaction is metal dissolution:

for a primary battery, and the power density is also low. The voltage

Li

Li + + e

at constant load declines during discharge, which is a disadvantage for

ni

(12M)

most application, although it affords a convenient way to monitor the

The cathode reaction is more complicated, since in this battery there is

state of charge of the battery. The greatest advantage of Leclanclid

no reducible material in the cathode itself, and it is the solvent that

cells is their low price. They are therefore widely used for simple

serves as the active cathode material. The cathode reaction is usually

applications, where reliability and performance are not of critical

written in the literature as follows:

importance.

4Li+ + 2SOCI + 4e 2

A similar system, containing the same active materials, is the so-called alkaline battery.

M

SO + S + 4LiCl 2

(13M)

This type of battery differs from the

The reversible cell voltage, which is higher than in any aqueous

Leclanchd cell in that the electrolyte is concentrated KOH, the zinc is

battery, has been estimated at 3.65 V. Assuming the cell reaction given

in the form of a high-surface-area paste and the casing is made of

in Eqs. 12M and 13M, this leads to a theoretical energy density of

nickel-coated steel. It has a much higher power density, because of the

1.48x103 W•i/kg. Values as high as 700 W•h/kg have been realized in

higher conductivity of the electrolyte and the larger surface area of

commercial cells. Unlike the Leclanche cell, lithium batteries are designed to be either anode limited (i.e., to have a stoichiometric

*

One can "engineer around" this problem by designing the battery with a stoichiometric excess of zinc. This may seetn like a wasteful approach, but it works. In fact, most batteries are designed with an excess of either anodic or cathodic material, for one reason or another.

*

In Leclanche cells the casing is made of zinc and serves as the

anode. This design increases to some extent the chances of leakage but makes manufacturing very cheap.

-+06

ELECTRODE KINETICS

deficiency of lithium), or to have about equal stoichiometric concentrations of the anode and cathode materials, depending on application.

M. APPLICATIONS

467

use of the system as a battery. In addition to the high energy density of Li/SOC1 2, the operating

Lithium batteries have many advantages compared to Leclanche cells

cell voltage in low-rate batteries is around 3.4 V, about twice that of

and other aqueous batteries, but they are also interesting from the

aqueous batteries. As a result, only half as many cells are needed in a

fundamental point of view. To begin with, the free energy of interac-

stack, which improves reliability, as we have noted. The rate of self

tion of lithium with the solvent is so high that they would be expected

discharge is very low, about 10 times lower than that of Leclanche

to react violently, leading to a very high rate of self-discharge at

cells, allowing storage without refrigeration for periods up to 10

best, and to dangerous explosions at worst. This was realized by Peled

years. Another welcome feature is the stability of the voltage during

and others, who found that as soon as contact between the metal and the

discharge, which is shown in Fig. 2M. This is an obvious advantage for

solvent is made, a protective layer is formed, which prevents further

the operation of any electronic device. The only drawback is that the

chemical reaction. In the Li/SOCI cell this layer consists of LiC1, 2 which is not soluble in the solvent. If propylene carbonate is used as

voltage at open circuit or under load cannot be used as a measure of the

the solvent, the protective layer consists of Li 2CO3 . Fortunately these

special devices had to be developed to determine the state of charge of

layers are permeable to Li + ions, but not to electrons. In fact, the

lithium-thionyl chloride and other lithium-based nonaqueous batteries.

state of charge of the battery after a certain period of use, and

protective layer on the surface of the metal serves as an electrolyte,

Figure 3M shows the voltage of an Li/SOC1 2 cell during discharge at

having a transference number of unity with respect to the positive ion.

different rates. These data were obtained for a recent design of this

It is referred to in the literature as the solid-electrolyte interface

type of battery, in which an energy density of 740 W•h/kg was achieved

(SEI).

for a D-type cell, the highest energy density reported so far for any

A layer that forms on top of a metal upon contact with the solution

battery. The decline in energy density with the rate of discharge is

can be of three different types. If it is dense and nonconducting, it

shown in the inset. The data in Fig. 3M were obtained with a smaller,

can protect the metal from further corrosion, but the system cannot be

AA-type cell, which has a somewhat lower energy density. Based on

used as a battery, since the metal is totally isolated from the solu-

chemical analysis of the amount of Li, this cell should have a total

tion. If it is electronically and ionically conducting, reduction of

charge of 2.65 A.h. The charge measured at a low rate of 1 mA is

the solvent at the film-electrolyte interface and oxidation of the metal at the metal-film interface can proceed freely, leading to a fast rate of self discharge. It is only when the film is both an ionic conductor and an electronic insulator that the chemical pathway of spontaneous

*

In particular, computer memories based on complementary metal

reduction of the solvent at the anode is prevented, whereas the electro-

oxide semiconductor devices (C-MOS) require a little more than 3 V to

chemical pathway of oxidation of the metal at the anode and reduction of

operate. The LilSOCl 2 battery is the only battery that can provide this

the solvent at the cathode can proceed at a sufficient rate to allow the

voltage from a single cell.

468

ELEL. I RODE KINETICS

469

M. APPLICATIONS

There are also disadvantages.

4.0

1

I

I

I

►

1

1

First, lithium batteries are very

expensive. Compared on the basis of equal energy content, they may cost

1

three to five times more than Leclanch6 cells. More serious is the matter of safely. For low power applications, these batteries are quite safe, but high power lithium batteries have been known to explode. For 0

2

3.0

example, accidental heating may melt the lithium (m.p. 180.5 °C). This

1

can rupture the protective SEI layer, leading to a violent reaction

0 <=t 1_J 0

between the metal and the solvent, and eventually to explosion. The use of high power lithium batteries has so far been limited 10 mA

---I _J 2.0 '

LLJ

mostly to military applications and to expensive appliances, such as

20 i/mA

15 5

sophisticated cameras and computers. Low rate lithium batteries are widely used where long service life is important (wrist watches,

0.0

0.5

2.0 1.5 1.0 CHARGE CAPACITY/A•h

25

calculators, memory backup in computers, etc). The lithium-iodine solid—state battery

is used exclusively for

heart pacemaker. The electrode reactions in this case are very simple,

Fig. 3M Discharge curves of a 2.65 A•h AA-size LilSOCl 2 battery designed for low-rate applications, at different rates of discharge.

leading to the overall cell reaction: 2Li + I

Inset shows the dependence of the charge capacity on discharge

2

2Lil

(14M)

rate. Data from Yamin, Pallivathikal and Zak, 5th Int. Seminar

The electrolyte is solid LiI, which is formed in situ, during operation

on Li Battery Technology and Applications, Florida, 1991.

of the cell. This solid electrolyte is conducting for Li + ions but nonconducting for electrons. The cathode consists of a compound such as

2.41 A•h, which constitutes 91% of the calculated capacity, showing very

poly-2-vinylpyridine, mixed with molecular iodine and melted together

efficient utilization of the active material. Some of this efficiency

until a homogeneous material has been formed.

is lost when the discharge rate is increased, as seen in the inset in

charge-transfer complex, allowing easy transfer of the iodine.

Fig. 3M. At a rate of 15 mA the charge capacity is 1.64 A•h, correspon-

thickness of the solid LiI layer grows during discharge of the battery

ding to only 62% utilization of the active material. The high energy

and its resistivity is quite high, but not too high, considering the

density of this type of Li/SOC1 2 battery combines with the low rate of

very low currents (in the range of 5 10 vtA) needed to operate the

self-discharge and the high operating voltage to make such devices

pacemaker.

ideally suited to power the memory backup in computers. In this use the battery is designed to outlast the useful lifetime of the computer itself and probably will never have to be replaced.

This constitutes a The

4/U

b. LL

The cell voltage is 2.8 V,

KINETICS

4„

M. APPL1CA l IONS

*

and the energy density is about 250 W•h/kg. The cost is extremely high: about $10,000/kW•h, compare to about $250 for the Li/SOC1, battery and $70 for the Leclanchd cell, for the same amount of energy stored. The advantages of this type of battery are its very low rate of self-discharge and its extremely high reliability. Although it has been used in tens of thousands of cardiac patients over more than a decade,

Pb + S024 ---> PbSO 4

+

2e

(15M)

M

while at the cathode lead dioxide is being reduced, yielding the same product Pb0

2

+

2H

2

SO

4

+

2e ro --> PbSO 4

+

2H

2

(16M)

0 + S02 4

The overall cell reaction is given by Pb0

2

+

Pb + 2H

2

SO

4

—4

(17M)

2PbSO +2H 2 0 4

there has not been a single case of failure causing internal injury of

We note that sulfuric acid is being consumed in the process, and an

any kind. For the very special requirement of heart pacemakers the

equivalent amount of water is formed. The resulting decrease in density

foregoing advantages make this type of battery commercially viable in ** spite of its very high cost. The internal resistance of the battery

is the basis for the common method used in many garages to test the

is quite high, however, limiting its use to low power application.

28.6 Secondary Batteries

state of charge of a car battery. The cell voltage is 2.1 V, the highest voltage for any

aqueous

battery. In fact, we would not expect a battery having such a high voltage to hold charge at all, since the voltage is high enough to

The best known and most widely used secondary battery is the

electrolyze water, providing an efficient way for rapid self-discharge.

lead—acid battery, which consists of a lead anode and a lead dioxide

Indeed, if we connect the terminals of a lead-acid battery to two

cathode in a 25% solution of sulfuric acid. The reaction at the anode during discharge is

platinum electrodes placed in a cell containing the same solution, copious evolution of hydrogen and oxygen will be observed.

But lead is

not platinum. The kinetics of hydrogen evolution on lead is slow, leading to a very high overpotential for this reaction. Thus, the

* According to tables of standard potentials, the open-circuit

operation of the lead-acid battery depends, to a large extent, on the

potential of this cell should be 3.58 V. Note, however, that standard

sluggishness of the cathode with respect to a side reaction. This is a

potentials are given for aqueous solutions, whereas here the reaction

case in which having a bad electrocatalyst is an asset rather than a

occurs in the solid state, where the free energy of the different

liability! The successful operation of the lead-acid battery depends,

species is quite different.

** Most of the cost is due to the extremely high reliability

of course, on the rate of reactions 16M and 17M, which are fast enough to allow acceptable rates of charge and discharge.

demanded in this kind of application. For other uses, such as memory

The theoretical energy density of lead-acid batteries is only

backup in computers, the same battery could be manufactured at a much

171 W.h/kg, as a result of the high atomic weight of lead. The prac-

lower cost.

tical energy density depends on the rate of discharge, as seen in Fig. 1M, but even at low rates it does not exceed about 40 W-h/kg. This

472

ELEC I KODE KINETICS

473

M. APPLICATIONS

represents about 23% of the theoretical value, despite massive invest-

sulfate formed on the surface of these particles during discharge is

ments in engineering over the past two decades, aimed at increasing the

not.

energy density (and power density) of this type of batteries.

Another common mode of failure of batteries is loss of electrical

The rate of self discharge of secondary batteries is not as

contact between the active material and the current collector. There

important as it is for primary batteries. One does not expect a car

are many other ways in which batteries can fail, such as the aging of

battery to be fully charged after the car has been idle for several

separators and accidental contact between anode and cathode. These

months. The quality of a secondary battery is measured, instead, in

problems are not discussed here.

terms of its service life, which is determined by the number of times it

The largest asset of lead-acid batteries is their low price,

can be charged and discharged. To understand this limitation, we must

compared to any other secondary battery currently available. The energy

consider the structure of the electrodes in the cell and the way in

density is inherently low, but in its main application as a car battery

which they are charged and discharged. As a rule, the active material

this is tolerable. For application as the main power source of electric

is in the form of a high-surface-area powder, pressed onto the surface

vehicles, an energy density of at least 100 W•h/kg is necessary. This

of a metallic grid, which serves as the current collector. The anode in

corresponds to 58% of the theoretical energy density — a very difficult

the lead-acid battery consists of small particles of lead bonded to a

goal. Power density is another limitation, particularly since increas-

lead screen. During extended use, and in particular as a result of

ing the power decreases the energy density substantially.

abuse, the particles may agglomerate, causing a loss of surface area.

The nickel—cadmium battery consists of a Cd anode and an Ni0OH

Now, we note that during discharge lead is converted into PbSO 4, which

cathode in a concentrated solution of KOH. The exact composition of

is an insulator. If a lead particle is coated with its oxidation

the fully charged cathode is not entirely clear. Some prefer to write

product, it becomes isolated from the solution and can no longer take

it as NiOx in which 3 S x 4, implying that there might be some Ni 4+

part in the discharge process. The larger the particles, the more

34". There seems to he little doubt that the cathode can be mixedwthN

active material is lost in this manner, decreasing the effective charge

charged beyond the level corresponding to the conversion of all the

capacity of the battery. The same mechanism can cause deterioration of

active material to trivalent nickel, but the existence of tetravalent

the cathode as well, since Pb0 2 is electronically conducting, while the

nickel has not been confirmed. We shall not discuss this problem here; we simply write the cathode reaction in terms of the well-known species

*

The real situation is invariably more complex. The screen is not made of pure Pb but may contain some Ca or Sb. A binding material is

*

This again is an oversimplification. The cadmium electrode may

used to increase adhesion of the lead particles to the screen. Each

contain a few percent each of iron, nickel, and graphite. The NiOOH

manufacturer has its own secret formulations, but we do not need to know

cathode may contain some cobalt and also graphite. The electrolyte is

these to understand the behavior of such batteries.

25-30% KOH, but some LiOH is added to improve perfornzance.

474

ELECTRODE KiNE I

M. Al -' 1.• ■

•

The main advantages of Ni/Cd over lead-acid batteries is the higher

NiOOH as follows: NiOOH + H2O + em --> Ni(OH) 2 + (OH)

(3M)

discharge rate possible and the longer cycle life. At an 8-hour discharge rate (C/8), the two batteries may be nearly equal in energy

The reaction taking place during discharge at the anode is

density, but at a 30 minute rate (2C) the Ni/Cd battery still performs

Cd(OH) 2 + 2e m

Cd + 2(011)

(4M)

well whereas the lead-acid battery can barely work, losing 80% of its capacity or more.

and the overall reaction is

In Fig. 5M we show the deterioration of high quality Ni/Cd batte2Ni0OH + Cd + 2H20

2Ni(OH) 2 + Cd(OH)2

(6M)

ries with prolonged cycling. About 10% of the capacity is lost in the

The cell voltage is 1.29 V and the theoretical energy density is

first thousand cycles, but no further decline in capacity is observed up

208 W.h/kg. Discharge curves for a Ni/Cd battery are shown in Fig. 4M.

to about 2,500 cycles. Performance deteriorates rapidly beyond 3,500 cycles, and this can be considered to be the limit of the useful cycle life of such batteries. In terms of real time, Ni/Cd batteries can last

1.2

100

0> 0 0.8

80

_J 0 10C 5C

_j 0.4

1C 0.1C

60

_J U

40 1

0

40 80 % CHARGE

120

3000 1000 2000 NUMBER OF CYCLES

Fig. 5M The decline in charge capacity with cycle life for a high quali-

Fig. 4M Typical discharge curves of a Ni/Cd battery at different rates.

ty sealed Ni/Cd battery. The battery, which has a nominal

A discharge rate of 5C implies that the cell is discharged in

charge of 1.2 A•h, was charged for S hours and discharged for 3

0.2 hours while a 0.1 C rate indicates that it is discharged in

hours at 0.4 A. Discharge was stopped when the potential

10 hours. The charge measured at IC is defined as 100%. Data

reached 1.0 V. Data from Wiseman in "Handbook of Batteries and

from Evjen and Catotti in "Handbook of Batteries and Fuel

Fuel Cells" Chapter 18, Linden, editor, McGraw-Hill, 1984.

Cells" Chap.17, Linden, editor, McGraw-Hill, 1984.

476

ELECTRODE KINETICS

477

M. APPLICATIONS

5-15 years, depending on design, quality, and type of application. In

significant.

If developed into a mature technology, it could have

comparison, lead-acid batteries typically do not last more than about

replaced conventional methods of electrical power generation. A fuel

500 cycles or 5 years.

cell, sometimes referred to as an electrochemical energy conversion

The worst drawback of the Ni/Cd battery is its cost, but progress

device, is not limited by the Carnot cycle, hence it has an inherently

has been made in recent years and Ni/Cd batteries are gradually rep-

higher efficiency than the thermal route of converting chemical energy

lacing Leclanche cells and alkaline Zn/Mn0 2 primary batteries in many

to electricity. For mobile applications, such as space missions or

simple applications, such as mechanical toys and even flashlights. This

electric vehicles, fuel cells have the added advantage of a very high

is a very welcome development from the ecological point of view, since

energy density, which asymptotically approaches the energy density of

each Ni/Cd battery can replace hundreds of primary cells before it is

the fuel and its container, as the amount of energy that needs to be

thrown out.

stored is increased.

The nickel oxide cathode has been used in combination with several

Thirty five years later it must be admitted that fuel cell tech-

other anodes to form different batteries. It was Edison who invented

nology has not lived up to many of the early expectations. The most

the Ni/Fe battery, which, at the time, competed well with Ni/Cd and has

important stumbling block has been the development of an inexpensive

drawn renewed interest in recent years. The most recent application of

electrocatalyst, suitable for work in acid solution over long periods of

this cathode is in combination with a metal hydride anode (such as

time. Catalysts consisting of large-surface-area graphite, coated with

which was mentioned in Section 15.3, in the context of LaNi H 5 6.7' hydrogen storage) as an Ni/H 2 battery. This system is somewhat inferior

small amounts of platinum and bonded with Teflon, have been developed

to Ni/Cd from the engineering point of view, but it has the ecological

amount of platinum needed per unit surface area (from about 10 mg/cm 2

advantage of eliminating the use of cadmium, which is a highly toxic

2 in recent versions), but the activity for usedintalyo<0.5mg/c

pollutant.

and improved over the years, to increase activity and decrease the

direct oxidation of hydrocarbons is still too low for such fuel cells to be technologically viable. In this section we shall discuss the

28.7 Fuel Cells

properties of some of the fuel cells that have been developed so far.

Fuel cells can be regarded as secondary batteries in which the

The solid polymer electrolyte (SPE) fuel cell makes use of the high

chemical energy is stored in an outside container, rather than in the

stability and the cation selectivity of Nafion, a Teflon-like material

electrode material. Although the basic concept of a hydrogen-oxygen

that has been modified by the incorporation of sulfonic groups. The

fuel cell had been discussed more than a century ago, it was only in

membrane is coated with a porous catalyst on both sides. Hydrogen is

1959 that an extensive research and development effort in this area

oxidized on one side of the membrane and the H 30+ formed in the process

began, following the observation that saturated hydrocarbons could be

is transported across the membrane to the other side, where it interacts

oxidized electrochemically on a platinum electrode in acid solutions at

with (OH) ions formed by the reduction of oxygen, to form water. This

moderate temperatures. The implications of this finding were quite

water is removed from the cell by capillary action with the use of a

ELECIRODE KINETICS

M. APPLICATION

wick. The SPE fuel cell technology was developed for aerospace applica-

fuel stream depends on temperature. At 200 °C a concentration of 2% or

tions, and such devices have been successfully employed on several space

less does not affect the performance in a major way.

missions. The membranes, which serve as the solid electrolyte, are thin

The boiling point of concentrated phosphoric acid is above 250 °C, allowing operation of this type of fuel cell at temperatures up to 225 °C without the need to pressurize the cell. High temperature helps the

(< 0.1 cm). This leads to a very compact design, occupying about 50 liters for a 1 kW unit. Hydrogen and oxygen were stored cryogenically in the spacecraft, and as an added advantage the cell produced about 0.5 1 of pure water per kW-11 of electrical energy generated. The efficiency of energy conversion in the SPE fuel cells used on space

kinetics, particularly at the cathode, where oxygen is reduced. It also diminishes the sensitivity of the electrocatalyst to poisoning. Both electrodes contain noble metals, dispersed on high-surface-

85% of its maximum theoretical Carnot efficiency would have to be heated

area graphite. Since the amount of catalyst has been reduced to about 0.2 mg/cm2 at the anode and 0.4 mg/cm2 at the cathode, the cost of the

to 975°C to achieve the same energy conversion efficiency.

noble metal is no longer prohibitive, although it still constitutes a

missions was as high as 65%. For comparison, a heat engine operating at

Although the SPE fuel cell can be considered to be a technological

major factor in the cost of the fuel cell assembly, as we shall see.

success, it could not be introduced into everyday commercial use because

The cells are operated at a potential of about 0.70 V, corresponding to

of its high cost, which is due mainly to the high cost of the ion-

an efficiency of 0.70/1.48 = 0.47, which is high compared to conven-

selective polymer membrane and the heavy loading of the noble metal

tional power stations. The overall system efficiency is about 40%, since the energy used to operate all the auxiliary equipment, such as pumps and various control units must be accounted for.

catalysts needed for satisfactory operation. The phosphoric acid fuel cell (PAFC) uses a hydrocarbon as the original fuel, but a steam reformer is used first to convert the

The literature contains different estimates of the cost of PAFCs.

hydrocarbon to hydrogen and CO 2 before the gas is pumped into the cell.

Present costs are very high, but this is attributed to the low volume of

The reaction involved, taking propane as an example, is

production. Typical estimates, assuming mass production, are around

C

3

H

8

+

6H2 0

3C0

2

+

10H

2

(18M)

The fuel cell itself contains concentrated phosphoric acid (85-100%) and is operated at 180-200 °C. This electrolyte rejects CO 2 , therefore the

$500 per kilowatt of installed power. Conventional power stations cost about twice as much. It should be remembered, though, that the cost of PAFCs are estimates whereas the figures for conventional power stations are based on proven technology.

gas mixture produced in the steam reformer can be pumped directly into the fuel cell, where hydrogen is oxidized at the anode. Actually the reaction does not proceed quantitatively according to Eq. 18M, and some

* For proper comparison with the efficiency of a heat engine, the

CO is produced as a side product. This is removed with a so-called shift reactor, in which - CO is selectively oxidized to CO 2. The tolerance of the electrocatalyst at the anode toward residual CO in the

efficiency of a fuel cell must be calculated on the basis of the heat content of the fuel (A1-112F = 1.48 V), not on the basis of the free energy change (AG/2F = 1.23 V).

it

480

ELECTRODE KINETICS

481

M. APPLICATIONS

am: One of the arguments in favor of fuel cells in this kind of

An interesting possibility is to operate phosphoric acid fuel cells

comparison is their modular design. We might think of a 1,000 MW power

for the combined purpose of producing electricity and heat. Since the

station based on 100 fuel cell units, each delivering 10 MW of power. It may take 8 years to build such a station, at a rate of one unit per

waste heat is produced in the cell at about 200 °C, it can readily be used for space heating or air-conditioning. The efficiency of phos-

month, but each unit can be put in operation as soon as it is completed.

phoric acid fuel cells used in this manner has been estimated to be

In comparison, a nuclear power station also may take 8 years to const-

about 85%. The advantage of fuel cells over conventional power stations

ruct, but it does not start to produce electricity until it is complete.

(which also produce large amounts of waste heat) is that fuel cells can

Thus, even if both stations can be built for the same cost, it will be

be efficient in relatively small units and can be located on site (say

less expensive to build the fuel cell plant, because partial production

in the basement of an apartment building), eliminating the need to

can start much earlier. On the other hand, thermal power stations have

transport the heat over long distances.

operated for 25 years and longer, whereas the lifetime of PAFCs has not been tested for periods of more than 5 years.

The technology of phosphoric acid fuel cells is already fairly advanced. Large (4.3 MW) units have been built and tested in several

Let us calculate now the cost of the noble metal itself in the

countries for a number of years. Smaller units (40 kW) have been

estimated cost of a PAFC. All we need to know to make this calculation

operated continuously for periods of up to 40,000 hours, with little

is the cost of platinum metal, which we shall take as $600/oz ($19/g),

decline in performance. Since these units are rather compact (having a

the loading of platinum on the electrodes, taken as 0.6 mg/cm 2 for the

footprint of only 5 111 2), several of them could be accommodated in the

two electrodes combined, and the power output of the cell per unit area.

basement of a typical apartment building, providing on-site electricity

Current technology allows operation at about 0.4 A/cm 2 at a cell

and heat for the whole building.

potential of 0.70 V, yielding a power output of 0.28 W/cm 2 . These

The third type of fuel cell we are going to discuss here is the

high-temperature solid oxide (HTSO) fuel cell.

numbers lead to

This is a very difficult

0.6 mg/cm 2/0.28 W/cm2 = 2.14 g/kW

technology, which is much farther from commercial realization than the

which costs about $41/kW, namely about 8% of the estimated cost of the

fuel cells discussed so far, but it is interesting enough to warrant a

whole fuel cell.

short discussion and may, in the long run, be the winning technology. The heart of this system is an yttria-stabilized zirconia, Zr(Y)0 2 film,whcatsodelry,awinghcodutvyfr

*

Actually the cost is higher, since the power per unit area

02— ions at about 1,000 ° C. The fuel electrode is a porous Ni/Zr0 2 cermt,whisvboaelctrysndaheut

calculated here assumes an efficiency of 47%, based on the cell voltage of 0.70 V, whereas the system efficiency is only about 40%. Taking this

collector. Actually, electrocatalysis is no longer a problem at such

into account increases the cost per kilowatt installed by a factor of

elevated temperatures, and different fuels, including H 2, regular fossil

0.47/0.40 = 1.175, to $48/kW.

fuels, and even CO, can be used. The air (oxygen) electrode is a

462

ELECTRODE KINETICS

483

NI, APPLICATIONS

discontinuous layer of strontium-doped lanthanum manganite catalyst,

type of electrochemical energy conversion devices.

La(Sr)Mn03, coated with a porous doped indium oxide current collector.

An interesting recent design of the HTSO fuel cell, developed at Argonne National Laboratory, is the so-called monolithic fuel cell shown

Cells are connected in series in a so-called bipolar configuration, in which the anode of one cell is internally connected to the cathode of the next cell and only the terminal anode and cathode are connected to

in Fig. 6M. The unique features of this design are its compactness and suitability for mass production of stacks of cells, eliminating the need

the external terminals of the stack. The interconnecting material,

for the costly assembly process. The layers of solid electrolyte,

which must be a good electronic conductor but at the same time imper-

anode, cathode and interconnecting materials are all very thin

vious to ions, is manganese-doped lanthanum chromite, La(Mn)Cr0 3 .

(25-50 ilm), allowing very compact design and low internal resistance.

The HTSO fuel cell has many advantages over other types of fuel cells. Because of the high temperature it is not sensitive to impurities and can be used with different fuels or with a mixture of fuels. The electrolyte is solid, but unlike the SPE, it is dry. This property eliminates many engineering problems of water management, which tend to

It is appropriate to end this section with the reminder that fuel cells are still the best source of electrical energy for mobile applications and that they have inherently the highest energy density of all batteries, which approaches asymptotically the energy density of the fuel itself and its container. The high efficiency, particularly when

complicate the design and operation of other types of fuel cells. Waste heat is produced at high temperature and can be used for different purposes, making the overall efficiency of electricity and heat production very high. In fact, the waste heat produced in this cell can even be used to produce electricity in a conventional heat engine. In this way fuel can be converted to electricity at an overall efficiency exceeding 60%.

• Interconnection

The difficulties in the development of HTSO fuel cells are in the area of stability

Anode

of materials rather than in catalysis. Different Electrolyte

materials, some of them ionic conductors with no electronic conductivity and others electronic conductors with no ionic conductivity, must be

0 Cathode

compatible with each other chemically at a high temperature and mechanically during temperature cycling. Improvements in materials are steadily made, but the more sophisticated materials developed for this purpose tend to increase the cost. Once the materials problems have been overcome, the inherent simplicity of the design and operation of high temperature solid oxide fuel cells may make them the most useful

Fig. 6M High-temperature solid-oxide (HTSO) fuel cell. The monolithic design, by Argonne National Laboratory.

484

ELECTRODE KINETICS

M. APPLICATIONS

485

used in the so-called co-generation mode (i.e., when both the electrical energy and the waste heat are being utilized), leads to a reduction in

On the other hand, development of porous electrodes having a large

total fuel consumption. This can result in decreased global pollution,

density has been increased by many orders of magnitude, while the amount

in addition to beneficial local effect due to the lower emissions of

of noble metal needed -has declined steadily, from about 10 mg/cm 2 in the

noxious fumes from fuel cells, compared to internal combustion engines.

early designs to less than 0.5 mg/cm 2 in state-of-the-art (1993) technology. Although the need to develop cheap and stable electrocata-

The three major areas in which progress is essential are the development of better, more stable, and cheaper electrocatalysts, preferably not based on noble metals; improvements in materials for use in high temperature fuel cells; and reduction in costs. Although fuel cells are not commercially available on a large scale, some of the research in this area has led to other related products. The so-called metal-air

batteries use air electrodes developed originally for fuel

surface area has come a long way. The effective exchange current

lysts is still crucial for the future development and commercialization of fuel cells, a temporary solution has been achieved and is being used in the phosphoric acid fuel cells currently under development and testing. But catalysis is not the only reason for using porous electrodes. Neither is it the main reason. For a fuel cell to be economically

cells. The Ni/H 2 secondary battery alluded to earlier makes use of hydrogen electrodes and metal hydride storage systems which have also

higher. Such current densities are hard to reach at planar electrodes

been developed in the framework of fuel cell research. Moreover, a

even in well-stirred solutions and at substantial concentrations of the

great deal of understanding of the theory and practice of preparing

electroactive material. In fuel cells the reactants are usually gases

large-surface-area catalysts and on the operation of porous electrodes

that have low solubilities, and vigorous stirring or pumping of the

has been gained in the course of research in this field. Some of this

solution is not acceptable, since too much of the energy produced by the

knowledge has already been put to use in improving the design and

fuel cell would have to he consumed for this purpose. Porous electrodes

performance of existing primary and secondary batteries.

are therefore used mainly to enhance the rate of mass transport.

viable, the current density should be at least 0.2 A/cm 2 and preferably

The way a porous electrode works can best be understood in refe28.8 Porous Gas Diffusion Electrodes

rence to single pores, which are shown in Fig. 7M. If one assumes that

There are two ways of increasing the catalytic activity of an

the pores are conical and are made of a material that is wetted by the

electrode in a fuel cell: (a) by finding a better catalyst, which

solution to some extent, a meniscus will be formed, as shown in

yields a higher value of the exchange current density, hence a lower

Fig. 7M(a). An enlarged view of the region of contact between the

overpotential at any chosen current density, and (b) by increasing the surface area of the electrode per unit

apparent or geometrical

surface

area. Achievements in the former direction have not been spectacular so

* This is true for any industrial electrolytic process, since the

far, and platinum, perhaps combined with some other noble metals, is

cost of constructing a plant of given output increases with decreasing

still the best electrocatalyst for oxygen reduction in acid solutions.

current density.

486

ELECTR(.0E KINETICS

M. APPLICATIC:::

solution and the gas phase inside a pore is seen in Fig. 7M(b). Two factors must be taken into account: the rate of diffusion of the gas through the thin layer of the liquid to the side of the pore, which contains the electrocatalyst, and the resistance of the thin layer of the solution, which is a measure of the rate at which the ions formed at

a

b

the surface can be removed to the bulk of the solution. Near the iR s limited current

three-phase boundary the liquid film is very thin, allowing fast

Region of highest current density

diffusion of the gas, but the ohmic resistance- through the solution is high. Reaction at the surface of the pore in this region is therefore limited by a high ohmic overpotential. Far away from the edge of the

Gas

meniscus, the ohmic overpotential is low, but here the rate of diffusion is also low, since the layer of liquid through which the reacting gas must diffuse to reach the electrode surface is relatively thick. The reaction will be diffusion limited in this region. Clearly the current density must increase first and then decrease as a function of the distance from the edge of the pore, reaching a maximum somewhere in between.. This is the active area of the pore. A great saving in expensive catalyst material can be realized if this area is identified and controlled and the catalyst material is applied only

Fig. 7M (a) The meniscus formed in conical pores of a gas-diffusion electrodes and (b) the current distribution near the threephase boundary in one of the pores.

to it. The single-pore model bears only a distant resemblance to the

larger pores) on the gas phase side of the electrode; this serves to

structure of a porous electrode, but the principles discussed above are

achieve and maintain better stability of the three-phase regions inside

valid. Porous electrodes are usually made by mixing small graphite particles, coated with the catalyst, with a suitable binder. The latter

the porous structure. It is important to understand that increasing the roughness factor

also determines the degree of wettability of the matrix and serves to

of a planar electrode increases the rate of charge transfer but has

bind the electrode material to a suitable current collector. The pores

little effect on the rate of mass transport. On the other hand, the use

are not conical, but neither are they uniform. This design is necessary

of correctly designed porous electrodes can increase the rates of both

to stabilize the gas-liquid interface, preventing flooding of the

processes. Thus the use of porous electrodes is essential whenever

electrode on the one hand, and drying out on the other. Often a

gaseous reactants (e.g., H 2 or 02 ) are employed, even if an ideal

double-pore structure is used, with the larger particles (hence the

electrocatalyst could be found.

488

ELECTRODE KINETICS

489

M. APPLICATIONS

in electroplating, in the electrolytic industry, and in most experiments

28.9 The Polarity of Batteries The polarity of batteries may sometimes lead to confusion, which we would like to dispel. The anode and the cathode are defined as the

in the research laboratory, where a current is imposed on the system and the potential is measured, or vice versa. •

electrodes at which oxidation and reduction occurs, respectively. But

3.6

is the anode the positive or the negative terminal of a battery? This 0.4 -cu+ 2 /c u

question is particularly relevant for the case of secondary batteries, where the electrode serving as the anode during discharge becomes the

Lu

cathode during charging and vice versa.

z

Consider the lead-acid battery. The standard reversible potentials for the two half-cells are as follows:

3.4 w 3.2 2

cn

> 0.0

3.0 > 0 2.8

O

Pb0 2 /PbSO 4

E° = 1.682 V

PbSO4/Pb

E° = — 0.359 V

It is clear that the Pb0 2 /PbSO 4 electrode, which serves as the cathode during discharge, is the positive terminal of the battery and the anode is the negative terminal. When the battery is being charged, the

_J

_J

—

0.4

2.4 IA

Po a_

2.6

Zr1+ 2 /Zn -0.8

---••00011111111

O

2.2 2.0

CURRENT

electrodes change role, but not polarity. The Pb02/PhSO4 electrode is now the anode (since PbSO is being oxidized to Pb0 2 ), but it is still 4 the positive terminal of the battery. The way in which the potential at each electrode is changed during charge and discharge is shown in Fig. 8M, with a Cu/Zn battery taken as an example. We must distinguish between two cases. When the battery is in the driving mode — that is, when the battery is the source of energy — the positive terminal is the cathode and the negative terminal is the anode. The same applies also to corrosion and to any other electrochemical process that occurs spontaneously.

When the battery is in the driven

mode — that is, when it is being charged — the positive terminal is the anode and the negative terminal is the cathode. The latter is the case

Fig. 8M The potential of the two electrodes in a CulZn battery during charge and discharge. Note that the polarity of the cell is not changed, although the potential it delivers is always smaller than the potential needed to charge it. Reprinted with permission from Moran and Gileadi, J. Chem. Education, 66, 912. (1989). Copyright 1989, Division of Chemical Education of the American Chemical Society.

490

EL Lt 1:(0DE KINETICS

M. APPLICATIONS

L

eventually replaced because of corrosion.

29. CORROSION

In most cases corrosion can be effectively prevented (i.e., slowed 29.1 Scope and Economics of Corrosion

down to the point that the device, be it a piece of machinery or a

Corrosion is a common phenomenon, observed all around us. Wherever

structure, will have to be replaced for some other reason, before

there is a metal there is hound to be, sooner or later, corrosion. This

corrosion has become severe), by investment in the construction mate-

is hardly surprising, since all metals except gold are thermodynamically

rial. Jewelry and coinage are extreme examples, but even in a chemical

unstable with respect to their oxides in air and in water. This is

plant one has the choice of designing for minimum maintenance and long

manifested by the observation that metals are not found in nature in their "native" or metallic form, but rather in the form of some com-

periods between overhauls at a high initial cost, or frequent main* tenance at a lower initial investment.

pound, an oxide, a sulfide a silicate and so on. The history of mankind

There are "corrosive" environments and those which are considered

is closely linked with the technology of reducing ores to the correspon-

benign. The combination of high humidity and high temperature favors

ding metals or alloys.

corrosion, but above all the presence of chloride ions is detrimental to

This requires the input of energy, and the Corrosion can be

almost all metals and interferes with many methods of corrosion protec-

regarded as the natural tendency of metals to revert to a more stable

tion, as we shall see. Chloride is not the only ion that enhances

state as a chemical compound of one kind or another, depending on the

corrosion, but it is the one most commonly found all around us, in sea

environment. Our technology therefore depends on our ability to slow

water and even in fresh-water, in the ground and in the human body.

the corrosion process to an acceptable level.

Salt spray carried by the wind from the sea is a major cause of corro-

resulting product is unstable thermodynamically.

The cost to society of corrosion and its prevention is staggering:

sion, and it is easy to see how the importance of this factor diminishes

it has been estimated to be about $200 billion per annum in the United States alone, corresponding to $800 per capita per year. Much of this

with the distance inland. As a rule, corrosion is not uniformly distributed on the exposed

amount could be saved by proper design and choice of materials, and by

surface. An average rate of corrosion of 0.05 cm/year may be concent-

the use of existing prevention methods. The problem cannot be eliminated, however, since as pointed out above, corrosion represents the

rated in spots, leading to holes in a piece of metal (e.g., a pipeline) that is 0.5 cm thick or more. Perhaps the worst design error, from the

natural tendency of all systems toward a state of minimum free energy.

corrosion point of view, is to combine two different metals without

The damage caused by corrosion is of two general kinds: aesthetic

isolating them electrically. For example, connecting copper plates with

and engineering. An example of the former is the development of rust spots on so-called stainless steel cutlery. Although rust, which is just a mixture of oxides of iron, is not harmful in any way, one would not like to eat with a rusted fork or spoon. Examples of engineering damage due to corrosion are countless. From car bodies to pipelines to electronic components; almost everything must be protected and -•

In a favorite anecdote among corrosion engineers, the chief engineer of a chemical plant is asked about corrosion problems. He responds with great confidence: "Never have any problems. We replace all piping in our plant every 2 months".

492

ELECTRODE KINETICS

493

M. APPLICATIONS

potential.

*

copper is the cathode. Moreover, since the two metals are in intimate

The accompanying reaction in aqueous solutions is usually hydrogen evolution or oxygen reduction.

contact with each other, this is equivalent to having the terminals of

When we place a piece of iron in 1.0 M HCI it dissolves readily,

this battery effectively shorted, leading to a very high rate of

with simultaneous evolution of hydrogen. We can measure the rate of

discharge (i.e., corrosion). As long as the structure is dry, nothing

dissolution by determining the weight loss of iron, or by analysis of

will happen, but if water accumulates on the surface, corrosion of the

the solution for its ions, but we could also determine this quantity by

rivets may occur, leading to critical structural damage.

measuring the volume of hydrogen evolved. The rates of anodic metal

steel rivets in effect creates a battery in which steel is the anode and

In the following sections we shall discuss the basic electro-

dissolution and of cathodic hydrogen evolution must be equal, since

chemistry of corrosion and some of the more common methods of corrosion

there can he no accumulation of charge in the metal or the solution phase.

protection.

At what potential will this process occur? The reversible poten29.2 Fundamental Electrochemistry of Corrosion

tials for the two reactions depend on the composition of the solution in

The first and most fundamental step in corrosion is the oxidation

calculation we shall assume that after a short time the solution is

of the metal to its lowest stable valence state, for example Fe

Fe

2+

2e

ivt

contact with the electrode surface. For the purpose of the present

(19M)

saturated with molecular hydrogen and the concentration of iron is 1.0 1.t.M. With these assumptions we have, for the anodic reaction:

This is most often followed by the formation of insoluble products, the exact nature of which depends on the metal and on the environment in which it is corroding. The ion formed in the initial step may be oxidized further, producing oxides or other compounds of mixed valency. In some cases (e.g., Al Ti, Cr) the corrosion products form dense

E

re v

=

— (2.3RT/nE)log(C

FeFe++ ) = — 0.617 V, NHE

(20M)

and for the cathodic reaction, by definition: E = E° = 0.00 V, NHE re v

(21M)

insulating layers, which prevent further corrosion. In other cases

Clearly, for anodic dissolution to occur, the potential must be anodic

(mostly iron and low-carbon steel), the layer is porous, allowing the corrosion process to occur until the whole metal piece has been con-

with respect to the reversible potential for the Fe 2+/Fe couple and cathodic with respect to the reversible potential for the h.e.r. Thus,

sumed. When a protective layer does exist, it does not have to be very

all we can predict from this thermodynamic argument is that the poten-

thick. About 5 nm in the case of aluminum and 3 nm in the case of

tial must be somewhere between — 0.617 and 0.00 V versus NHE. The rest

stainless steel. Thus, as little as 10-20 molecular layers of the protective oxide film can provide excellent protection for long periods of time. Anodic dissolution of a metal cannot occur by itself for any length of time, since it would lead to charging of the metal to a high negative

If one assumes a double-layer capacitance of 20 1.tFIcin2 , it would take 20 viC to change the potential across the interface by 1.0 V. This corresponds to about 10-10 mu! or 6 ng of i ron.

ELEC I RODE KIM; I JCS

495

M. APPLICATIONS

depends on kinetics. To proceed we need to know the exchange current

example shown in Fig. 9M this occurs at about 0.52 V, NHE. The corres-

densities and the Tafel slopes for the two reactions concerned. In

ponding current is the corrosion current i . In the present example

Fig. 9M we have plotted the two partial currents as a function of potential, assuming the following kinetic parameters: for the cathodic reaction i = 10 6 A/cm2 and b = 0.118 V and for the anodic reaction

it is about 27 mA/cm2. This is quite a high current density, corresponding to fast dissolution of the metal and vigorous hydrogen evolution,

i = 10 A/cm2 and b = 0.039 V. The potential must settle at the 0 a point at which the anodic and cathodic currents are equal, which is

HCI.

called the corrosion potential E or mixed potential. tort

in a straightforward manner by writing the Tafel equation for the two

For the

COIT

which are indeed observed when iron is dipped into a 1.0 M solution of We can calculate the corrosion potential and the corrosion current partial reactions and solving for the potential at which the currents are equal:

0 (a).exp

E cor r

E rev

(a)

1

• = (0.exp -

Po te n tia l/ V vs RH E

a

This yields E

—0.2

COIT

= - 0.524 V, NHE and i

corr

E

Cor r

-E

be

rev

(c)

(22M)

= 27.4 mA/cm 2 , close to

the values estimated from Fig. 9M. This type of representation, very common in corrosion studies, is referred to as an Evans diagram. We shall, therefore, discuss it a —0.4

little further. Consider first the effect of pH. Assuming that all the kinetic parameters are unchanged and only the reversible potential for hydrogen evolution is affected, we obtain the curves shown in Fig. 10M.

—0 . 6

—7 —6

—5

—4

—3

—2

log i/A•cm -2

Fig. 9M Evans Diagram showing the currents for iron dissolution and hydrogen evolution, and the resulting values of E. and iCOIT . Parameters Pr the anodic -and cathodic reactions, respectively are: i = 10-4 Alcm 2 and 10 6 Alcm 2; b = 0.039 V and 0.118 V; E rev = - 0.617 and 0.0 V, NHE.

As the pH is increased, the corrosion potential becomes more negative, approaching the reversible potential for the Fe 2f/Fe redox couple, and the corrosion current decreases, from about 27 mA/cm 2 at pH = 0 to 0.14 mA/cm 2 at pH = 6. This is hardly surprising. It simply

*

This assumption is made here only for convenience of presentation. Usually the exchange current density for the h.e.r. is found to be lower at intermediate pH values than in strong acid or strong alkaline media.

496

ELECTRODE KINETICS

497

M. APPLICATIONS

In practice, there is little interest in the corrosion of iron in 1 M HCI. The pH in typical environments (e.g., in the ground and in

0

natural bodies of water) is in the range of 5-9_ Under these conditions the rate of hydrogen evolution may be very slow, and oxygen reduction

> —0.2

can become the main cathodic process, controlling the rate of corrosion.

N

We recall that the standard potential for oxygen reduction' is 1.23 V,

—0.4

NHE, which leads to a reversible potential of 0.817 V, NHE at p11 7.

0

Thus, corrosion occurs at a very high (negative) overpotential with

a-

-0.6

respect to oxygen reduction, and the current is mass-transport limited. Figure 11M presents the Evans diagram for iron in neutral, aerated solutions. The exchange current density for oxygen reduction was taken

Fig. 10M Evans Diagram showing the current densities for iron dissolution and hydrogen evolution at different pH values. The kinetic parameters, which are the same as in Fig. 9M, have been arbitrarily assumed to be independent of pH. confirms the common observation that iron dissolves faster in concentrated than in dilute acid. The interesting new insight we can gain from Fig. 10M is that this difference is not directly related to the rate of metal dissolution in the different media. It is, if fact, determined by the different rates of hydrogen evolution, which is necessary to use up the electrons released in the process of oxidizing the metal to its ions. The process is evidently cathode limited, and the corrosion potential is close to the reversible potential for the anodic process. Since the process is cathode limited, it is possible to slow it down by inhibiting the rate of hydrogen evolution. Many commercial corrosion inhibitors function in this manner. Considering Fig. 10M, it is easy to see that decreasing the exchange current density of hydrogen Lr

iF

tl

evolution by the addition of a suitable corrosion inhibitor is equivalent to increasing the pH, in terms of its effect on E

corn

and i

corr

log 1/A•cm -2

Fig. 11114 Evans diagram in neutral solution. Two values of the limiting current for oxygen reduction (10 -3 and 10-4 Alcm 2) are shown, yielding two different values for E and i . Corr

COtT

Ell.:121RODb KIN E I La

as 10- toA/cm 2 and for the h.e.r. a value of 10 8 A/cm 2 , two orders of magnitude lower than in acid solutions, was assumed. The contribution of the h.e.r. to the measured corrosion current is, therefore, quite negligible in neutral aerated solutions.

M. AC' i'LiCA riONS

The next question to consider in this context is the way a metal corrodes when two cathodic reactions of comparable magnitude occur in parallel on the same surface. This is shown in Fig. 12M. The total cathodic current is given by

The mass-transport-limited current density for oxygen reduction is

i = i toexp (E — E rev )/b e + i 0 ]

independent of the kinetic parameters for this reaction; rather it

(23M)

depends on factors such as the concentration and the diffusion coeffi-

and the open circuit corrosion potential is the intersection of the line

cient of oxygen in the medium. It depends on the rate of flow of the

given by this equation with the line for anodic oxidation of iron.

liquid in a pipe or around a sailing ship or a structure immersed in a river.

reactions most often involved in environmental corrosion, other electroactive materials may take part in the process, enhancing or retarding it, depending on whether they can be reduced or oxidized, respectively, in the range of potential in which corrosion takes place.

—0.45

Po ten tia l/ V vs NHE

Although hydrogen evolution and oxygen reduction are the two

29.3 Micropolarization Measurements —0.50

The understanding gained by considering the Evans diagrams allows us to measure the corrosion current in a straightforward manner. First we must realize that the corrosion potential is in fact the open-circuit

—0.55

potential of a system undergoing corrosion. It represents steady state, but not equilibrium. It resembles the reversible potential in that it

—0.60

can be very stable. Following a small perturbation, the system will return to the open-circuit corrosion potential just as it returns to the

—0.65 —2.0

log i/A•cm -2 Fig. 12M Evans diagram for the corrosion of iron in the presence of two simultaneous cathodic reactions (hydrogen evolution and oxygen reduction). The dotted line represents the sum of both cathodic currents. pH = 4, i = 10 ° A/cm 2 , be= 0.118 V and iL= 0 0.63 mA/cm 2 for the oxygen reduction reaction. Other kinetic parameters are as in Fig. 10M.

reversible potential. It differs from the equilibrium potential in that it does not follow the Nernst equation for any redox couple and there is both a net oxidation of one species and a net reduction of another. Consider now the current-potential behavior of a system close to E . Assuming that the two partial currents are in their respective COIT

linear Tafel region, we can write i = i (c).exp[— (E — E (c))/b o

rev

(24M)

500

ELECTRODE KINETICS

The potential E near the corrosion potential can be written as: E = E + AE

The net current density observed at a potential E, close to the open(25M)

corn

circuit corrosion potential, is hence

Substituting in Eq. 24M we have ie = ioto.exp[-- (E

COIT

—E

501

M. APPLICATIONS

i = ia — i = i rev

(OA •exp(— AE/b c ) e

(26M)

corr

For small values of I AE/b

[exp(AE/be) — exp(— AE/be)]

1,

(31M)

we can linearize the exponents (cf. Section

12.4) and obtain The cathodic current density at the corrosion potential is equal to the corrosion current density i/i iCOtT

iotei.ex p

(E. — E r e v (c))/bei

+

(32M)

This equation is usually written in the form: (28M)

b•b

be

tort

This is very similar to the Tafel equation, written for a cathodic process as:

where R ic =i -exp(— TIM ) 0

= AE [

(27M)

Hence, Eq. 26M can be written in the simple form i = i COTT.exp(— AE/be) C

cor r

(29M)

The corrosion current density, like the exchange current density, is an

The potential difference AE is the difference between the applied potential internal current, which is not observed in the external circuit.

and the open-circuit potential, just as is the difference between the

b

a

a+

[

(33M)

is the polarization resistance, given by (AEli).

These equations allow us to determine the corrosion current by making current-potential measurements in the range of about ± 20 mV ** around the open-circuit corrosion potential. If the cathodic reaction is mass-transport-controlled, we can derive a similar expression for the micropolarization region. For the

applied potential and the reversible potential. The big difference is that i

con-

is equal to the anodic and cathodic currents of two entirely

different processes, whereas io represents the equal anodic and cathodic currents of the same reaction at the equilibrium potential. Following the same arguments we can derive for the anodic current an expression equivalent to Eq. 28M, namely i = i •exp(AE/b ) a

tort

a

(30M)

The polarization resistance R define here is just another name, commonly used in corrosion studies, for the faradaic resistance Rr, which has been defined in Section 2.2. ** One needs to know the values of the anodic and the cathodic Tafel slopes, to evaluatecurt i from Eq. 32M or 33M. When these slopes are not known, a value of 0.12 V is often used for both, as a rough approximation.

5U2

LL: I

RODE KJNETIC1

anodic current Eq. 30M holds, and for the cathodic reaction one has i c i = The total current is hence L

corn

i = ia — ie = ie..exp(AE/b — i a

tTLICA'cioNs

Pourbaix. These equilibrium diagrams relate the reversible potentials of reactions of interest in corrosion studies to the pH and the concentration of different ionic species in solution. We shall use a number of

COIT

= i [exp(AE/b a) — 1] corn

(34M)

examples to illustrate the principles involved, starting with the most basic diagram relating to water and some of the ionic and molecular

When we linearize the exponent, this gives rise to (35M)

species at equilibrium with it. To construct such diagrams, one has to identify the chemical and

which is similar to Eqs. 32M and 33M. Equation 35M could be derived

electrochemical reactions of interest and write the appropriate chemical

directly from Eq. 32M or 33M by setting b e - cc , which is the appropriate value for a mass-transport-controlled process, for which the current is independent of potential.

equilibria and Nernst equations, respectively. The three most important equilibria for water are given below:

i/i

COIT

= AE/b

a

or

i

con = b a/R p

1.

Self-ionization

Experimental studies usually yield good agreement between the rates of corrosion obtained from polarization resistance measurements and those derived from weight-loss data, particularly if we recall that the

2H20

K 2.

be the most critical aspect when localized corrosion occurs. In particular it should be noted that at the open-circuit corrosion potential, the total anodic and cathodic currents must be equal, while

+

(OH)

(36M)

for which the equilibrium constant is given, at 25 °C, by

Tafel slopes for the anodic and the cathodic processes may not be known very accurately. It cannot be overemphasized, however, that both methods yield the average rate of corrosion of the sample, which may not

H 0+ 3

w

= (C(i 01)) (C(OH)) = 1.00x10 14 3

(37M)

Hydrogen evolution, for which the Nernst equation is E = 0.00 — (2.3RT/2F)log(P(4 2)/C2(H30 +))

(38M)

rev

This equation can also be written in the form

the local current densities on the surface can be quite different. This E = — (2.3RT/2F)logPoi — (2.3RT/F).pH

could be a serious problem when most of the surface acts as the cathode

rev

and small spots (e.g., pits or crevices) act as the anodic regions. The rate of anodic dissolution inside a pit can, under these circumstances, be hundreds or even thousands of times faster than the average corrosion rate obtained from micro polarization or weight-loss measurements. 29.4 Potential/p11 Diagrams

3.

(39M)

2

The oxygen evolution reaction, for which E = 1.229 + (2.3RT/4F)logP(o — (2.3RT/F)•pH 2

(40M)

rev

A very useful method of describing the stability of metals in

* Unless otherwise stated, concentrations and partial pressures,

different environments is the potential/pH diagrams introduced by

instead of activities and fugacities, respectively, are used here and in all following equations, for simplicity.

504

ELECTRODE KINETICS

505

M. APPLICATIONS it

A very simple potential/pH diagram, showing only these three equilibria,

potentials above 1.229 V, whereas an 1-1 2/02 fuel cell must operate at a

appears in Fig. 13M. The shaded area represents the region of thermo-

potential lower than this value. Typical values are 1.6-2.0 V for the

dynamic stability of water. Water electrolysis cannot occur inside this

former and 0.6-0.8 V for the latter. From the point of view of energy

region. Above and below it, oxygen and hydrogen evolution are thermo-

consumption or production, all we need to know is the cell voltage.

dynamically possible. Whether these reactions will in fact occur at a

From the point of view of corrosion of the electrodes and of any metal

measurable rate depends on their kinetic parameters.

in contact with the solution (current collectors, terminal bus etc.),

The region of stability of water is 1.229 V independent of pH, since the reversible potentials for hydrogen and oxygen evolution change

however, we shall see that the potential with respect to the NHE is very important.

with pH in the same manner, This, incidentally, is the potential region

The lines in Fig. 13M represent equilibria. The dashed vertical

in which the H /0 fuel cell can operate. Thermodynamic considerations 2 2 lead us to the conclusion that a water electrolyzer must Operate at

line corresponds to equal concentrations of the two ions. The lines bounding the shaded area are the reversible potentials for oxygen and hydrogen evolution as functions of pH. The effect of partial pressure of oxygen and hydrogen on the region of stability of water is rather

2

I

I

I

I

I

I

I

I

small (cf. Eqs. 39M and 40M). Increasing the partial pressure of both

Po te n tia l / V vs N HE

H30 +

gases from 1 atm to 10 atm will change the potential by 44 mV, from 1.229 V to 1.273 V. This, incidentally, is the basis for the technology of production of hydrogen or oxygen at high pressure by electrolysis of water, without the need to use a compressor. The region of stability of water is of central importance for the understanding of corrosion and of metal deposition, as we shall see. The two lines bounding this region in Fig. 13M are therefore included in all potential/pH diagrams.

-2 0

8

4

12

16

pH

Next we take a look at Fig. 14M, the simple potential/pH diagram representing the behavior of magnesium in aqueous solutions. For

Fig. 13M Potential/pH diagrams for water. Only the equilibria for water

equilibrium between a solid and a soluble species, the concentration of

electrolysis and for self ionization are shown. The partial

the latter must be specified. In Fig. 14M(a) lines are shown for

pressures of oxygen and hydrogen are taken as unity. The

concentrations of Mg 2+ of 1 p.M, 1 mM and 1 M. It is customary to

shaded area is the region of thermodynamic stability of water.

simplify potential/pH diagrams by showing only the lines corresponding

Data from Pourhaix in "Atlas of Electrochemical Equilibria in

to a concentration of 1 ltM of each soluble species. This convention is

Aqueous Solutions", Pergamon Press, 1966.

followed in all further potential/pH diagrams shown in this book. This is reasonable in view of the fact that a moderate rate of corrosion may

506

ELECTRODE KINETICS

50/

M. APPLICATIONS

2 correspond to about 10 RA/cm or less, which cannot cause a significant accumulation of soluble corrosion products near the electrode surface,

region depends on the nature of the oxide formed on its surface, in particular on its porosity and its ionic and electronic conductivity.

except in confined areas, such as pits or crevices. Fig. 14M(b) is a simplified form of Fig. 14M(a). The region of immunity is cathodic to

The potential/pH diagram for magnesium is relatively simple because there is only one stable oxidation state of the metal ions and because

the reversible potential of the metal, where it cannot be oxidized.

magnesium is not amphoteric, namely, the oxide and hydroxide are not

Farther to the right, in alkaline

soluble in strong alkaline solutions. The corrosion of aluminum represents a somewhat more complicated

Above it is the region of corrosion.

media, there is a region in which the metal can be oxidized anodically, but the product is an insoluble oxide or hydroxide, in this case Mg(OH)2 . This is called the region of passivation, which is discussed

situation, since this metal is soluble both in acid and in alkaline media. The potential/pH diagram is shown in Fig. 15M. The most

in Section 29.5. Whether the metal will actually be passivated in this 2

a

b

2

2

Po te n tia l/ V vs NHE

2+

0

-2

3

Ma A _I_

0

1

1.1111

8

4

pH

12

3 16

pH

Fig. 14M The potential/pH diagram for magnesium. (a) the detailed diagram. Lines correspond to different concentrations of 2+ Mg , as marked. (b) simplified form, defining regions of immunity, passivity and corrosion. Data from Pourbaix in "Atlas of ElectrocheMical Equilibria in Aqueous Solutions", Pergamon Press, 1966.

Fig. 15M Potential/pH diagram for aluminum. The solid phase is assumed to be hydrargillite (A1,9 3 •3H20). Filled areas represent regions where soluble species are stable and therefore corrosion can thermodynamically occur. Data from Pourbaix in "Atlas of Electrochemical Equilibria in Aqueous Solutions", Pergamon Press, 1966.

ELECTRODE_ KINETICS

508

509

M. APPLICATIONS

Al 0 + 61-1 + + 6e m 2 3

important feature in this diagram is a passivation region at interme-

2AI + 3H 0 2

(45M)

diate pH values, with corrosion possible at both higher and lower pH. The two soluble species are A1 3+ and A10 2 , and the lines representing their equilibria with the various solid phases correspond to a concent-

for which the reversible potential depends on pH according to the equation

ration of 1 p.M, as explained earlier. The equilibrium between them is

Erev

=—

1.550 — (2.3RT/F).p il

(46M)

In alkaline media the equilibrium to be considered is

given by: Al3+ + 2H 0 t Al0; + 4H+

(41 M)

2

and the equilibrium constant is written as:

Al + 4(OH)

(47M)

and the appropriate Nernst equation is

(C(AIO ))(C ( FI +)) 4 = 10 20.3 2 C(A

A10 + 2H 0 + 3eM 2 2

E (42M)

rev

= — 1.262 — (4/3)(2.3RT/F)TH + (2.3RT/3F)logC(mo ) 2

(48M)

All potentials in these equations are given in volts versus NHE.

I 3+ )

There is a very strong dependence on pH, and the two species should be found at equal concentration at pH = 5.07. This should be represented by a vertical line, like the equilibrium between H 30+ and OH in Fig. 13M. In the case of aluminum this is irrelevant because in the range of about 4.0 pH 8.6 only the solid phase is thermodynamically stable. There are two chemical equilibria, represented by the vertical lines in this figure, between the hydrated oxide hydrargillite and the two ions, and three electrochemical equilibria between metallic aluminum, the two ions, and the oxide. We shall write here only the electro-

Aluminum represents an interesting case, which warrants further discussion. We note that the limit of the region of thermodynamic immunity lies at very negative potentials with respect to the lower limit of stability of water (which is the reversible potential for hydrogen evolution) at all pH values. Thus, one would expect rapid * dissolution of this metal in any aqueous medium. This is indeed found in acid and alkaline solutions, but around neutral pH the oxide formed is very dense and nonconducting, and oxidation is effectively stopped after a thin layer of about 5 nm has been spontaneously formed in contact with air or moisture. This thin layer of oxide permits aluminum

chemical equilibria and the corresponding Nernst equations. The equilibrium with Al3+ is simple, and its reversible potential is independent of pH, since there are no protons or hydroxyl ions

to be used as a construction material and in many other day-to-day applications. There are, of course, additional ways (e.g., anodizing

involved: Al3+ + 3e

m

(43M)

<==> — Al

The Nernst eq uation is

*

This statement is a little careless, since we cannot deduce the

rate of a reaction from thermodynamic data. Yet, when there is a very 3+

E = — 1.663 — (2.3RT/3F)logC(Ai ) rev

(44M)

large driving force (i.e., when the system is far from equilibrium), the reaction will tend to be fast, unless some special mechanism prevents it

At intermediate pH values we have the equilibrium

or slows it down.

510

ELECTRODE KINETICS

511

M. APPLICATIONS

and painting), to protect aluminum that is exposed to harsh environ-

rather than the hydroxides, postulated as the solid phases. The result

ments, beyond the protection afforded by the spontaneously formed oxide

would be changes in the details, but the general features of the diagram

film. However, the unique feature of this metal (and several others:

would be retained. Passivity can be expected where solid species are

e.g., titanium, tantalum and niobium) is that it repassivates sponta-

predominant, and corrosion can occur where soluble ionic species are

neously when the protective layer is removed mechanically or otherwise,

thermodynamically stable.

as long as the pH of the medium in contact with it is in the appropriate

On the basis of Fig. 16M we can conclude that corrosion of iron can occur for pH values of 9 or less or for pH of 12.5 or more. This

range shown in Fig. 15M. Next we consider the Pourbaix diagram for iron, which is, of

includes most natural environments with which structural materials are

course, of paramount importance for the understanding of corrosion of

commonly in contact, making iron and many of its alloys highly vulner-

ferrous alloys such as the many types of steel and stainless steel.

able to corrosion. No soluble species is shown between pH 9 and 12.5,

This is a rather complex diagram, since two oxidation states of iron exist both in the liquid and the solid state and the metal is amphoteric shown in the original work of Pourbaix. The two soluble species in acid solutions are Fe2+ and Fe 3+ . The relevant equilibria are Fe3+

eM

Fe2+

, E° =

0.771 V versus NHE

(49M)

and Fe2+ + 2e

Fe,

m

E° = - 0.440 V versus NHE

(50M)

2

Poten t ia l/ V vs NH E

to some extent. Figure 16M is a simplified version of the diagrams

0

At high pH, the anion FIFe0, exists at equilibrium with the metal, the divalent and the trivalent hydroxides. Two electrochemical and one

16

chemical equilibria are involved: Fe(OH), + H B O E HFe02 + H3 0+

(51M)

Fig. 16M Simplified potentiallpH diagram for iron. Vertical lines represent chemical equilibria, in which the state of oxidation does

liFe0

2

+

3H+ + 2e

rs,4

Fe + 2H

2

Fe(011) + e --=-) HFe0 + H2 O 3 M 2

0

(52M) (53M)

.

The lines representing equilibria for Fe 2+/Fe(011) 3 and Fe 3+/Fe(OH) 3 , as well as for Fe/Fe(OH) and Fe(OH) ') /Fe(OH) also appear in Fig. 16M. 2 3' This diagram could be drawn in another way: with the different oxides,

not change. Horizontal lines correspond to electrochemical equilibria in which H30 + and OH ions do not participate. Lines between a solid and a liquid phase apply to equilibria with a 1.0 pt.M solution of the soluble species. Data from Pourbaix in "Atlas of Electrochemical Equilibria in Aqueous Solutions", Pergamon Press, 1966.

508

ELECTRODE KINETICS

509

M. APPLICATIONS

Al 0 2 3

important feature in this diagram is a passivation region at interme-

+

6H + + 6e

rvi

2Al + 3H

2

(45M)

0

diate pH values, with corrosion possible at both higher and lower pH. The two soluble species are Al3+ and Al0 2 , and the lines representing their equilibria with the various solid phases correspond to a concent-

for which the reversible potential depends on pH according to the equation (46M)

E = — 1.550 — (2.3RT/F)•pH

ration of 1 as explained earlier. The equilibrium between them is

rev

In alkaline media the equilibrium to be considered is

given by: A13+ + 2H 0 2

(41M)

A10 + 4H+

and the equilibrium constant is written as: (C(A10 ))(C(H + )) 4 2

A10 + 2H 0 2 2

+

3eM

(47M)

Al + 4(OH)

and the appropriate Nernst equation is = 10-

203

(48M)

E = — 1.262 — (4/3)(2.3RT/F)•pli + (2.3RT/3F)logC(mo 2 (42M)

rev

)

All potentials in these equations are given in volts versus NHE.

C(A I 3+ )

There is a very strong dependence on pH, and the two species should be found at equal concentration at p1-1 = 5.07. This should be represented by a vertical line, like the equilibrium between H 30+ and OH in Fig. 13M. In the case of aluminum this is irrelevant because in the range of about 4.0 pH 8.6 only the solid phase is thermodynamically stable. There are two chemical equilibria, represented by the vertical lines in this figure, between the hydrated oxide hydrargillite and the two ions, and three electrochemical equilibria between metallic aluminum, the two ions, and the oxide. We shall write here only the electro-

Aluminum represents an interesting case, which warrants

further

discussion. We note that the limit of the region of thermodynamic immunity lies at very negative potentials with respect to the lower limit of stability of water (which is the reversible potential for hydrogen evolution) at all pH values. Thus, one would expect rapid * dissolution of this metal in any aqueous medium. This is indeed found in acid and alkaline solutions, but around neutral pH the oxide formed is very dense and nonconducting, and oxidation is effectively stopped after a thin layer of about 5 rim has been spontaneously formed in contact with air or moisture. This thin layer of oxide permits aluminum

chemical equilibria and the corresponding Nernst equations. The equilibrium with A1 3+ is simple, and its reversible potential is independent of pH, since there are no protons or hydroxyl ions

to be used as a construction material and in many other day-to-day applications. There are, of course, additional ways (e.g., anodizing

involved: Al3+ + 3e m (> Al

(43M)

This statement is a little careless, since we cannot deduce the

The Nernst equation is E = — 1.663 — (2.3RT/3F)logOA( 3+) rev

* rate of a reaction from thermodynamic data. Yet, when there is a very

(44M)

large driving force (i.e., when the system is far from equilibrium), the reaction will tend to be fast, unless some special mechanism prevents it

At intermediate pH values we have the equilibrium

or slows it down.

510

ELECTRODE KINETICS

511

M. APPLICATIONS

and painting), to protect aluminum that is exposed to harsh environ-

rather than the hydroxides, postulated as the solid phases. The result

ments, beyond the protection afforded by the spontaneously formed oxide

would be changes in the details, but the general features of the diagram

film. However, the unique feature of this metal (and several others:

would be retained. Passivity can be expected where solid species are

e.g., titanium, tantalum and niobium) is that it repassivates sponta-

predominant, and corrosion can occur where soluble ionic species are

neously when the protective layer is removed mechanically or otherwise,

thermodynamically stable.

as long as the pH of the medium in contact with it is in the appropriate

On the basis of Fig. 16M we can conclude that corrosion of iron can occur for pH values of 9 or less or for pH of 12.5 or more. This

range shown in Fig. 15M. Next we consider the Pourbaix diagram for iron, which is, of

includes most natural environments with which structural materials are

course, of paramount importance for the understanding of corrosion of

commonly in contact, making iron and many of its alloys highly vulner-

ferrous alloys such as the many types of steel and stainless steel.

able to corrosion. No soluble species is shown between pH 9 and 12.5,

This is a rather complex diagram, since two oxidation states of iron exist both in the liquid and the solid state and the metal is amphoteric shown in the original work of Pourbaix. The two soluble species in acid solutions are Fe 2+ and Fe 3+ . The relevant equilibria are Fe3+ + e

Fe2+ ,

m

= 0.771 V versus NIIE

(49M)

and Fee' + 2e M

Fe,

E° = - 0.440 V versus NHE

(50M)

2

Pote n tia l/ V vs N H E

to some extent. Figure 16M is a simplified version of the diagrams

1

0

At high pH, the anion HFeO 2 exists at equilibrium with the metal, the divalent and the trivalent hydroxides. Two electrochemical and one chemical equilibria are involved:

I 0

I 8

4

12

16

pH

Fe(OH), + H B O

HFe02 + H3 0+

(51M)

Fig. 16M Simplified potential/pH diagram for iron. Vertical lines represent chemical equilibria, in which the state of oxidation does

HFe0

2

+

3H+ + 2e

Fe(OH) + e 3 M

m

Fe + 2H HFe0

2

+

2

H2 O

0

(52M) (53M)

not change. Horizontal lines correspond to electrochemical equilibria in which H301- and OH ions do not participate.

The lines representing equilibria for Fe 2f/Fe(01-1) 3 and Fe3+/Fe(OH) 3 , as well as for Fe/Fe(OH) 2 and Fe(OH),/Fe(OH) 3 , also appear in Fig. 16M.

Lines between a solid and a liquid phase apply to equilibria with a 1.0 1.1114 solution of the soluble species. Data from Pourbaix in "Atlas of Electrochemical Equilibria in Aqueous

This diagram could be drawn in another way: with the different oxides,

Solutions", Pergamon Press, 1966.

512

ELECTRODE KINETICS

M. APPLICATIONS 4+ ,

513

at

and iron should pass directly from the immune to the passive region as

In spite of the foregoing limitations, the corrosion scientist and

the potential is increased. Also, it should be possible to passivate

engineer can derive a wealth of information by consulting the relevant

iron in solutions where the pH exceeds about 4, by oxidizing the

potential/pH diagrams. The regions of immunity, passivity, and possible

surface, either chemically or electrochemically. We shall conclude this section by making some general remarks on

corrosion are demarcated,' and the most common corrosion products are

the advantages and limitations of potential/pH diagrams. It has already been stated that these are equilibrium diagrams. Hence we can learn

corrosion study, but it should never be the only tool used to solve the problem.

from them what cannot happen (e.g., a metal cannot be anodically dissolved in the region in which it is "immune", and water cannot be

29.5 Passivation and Its Breakdown

shown. Studying the relevant diagram is an excellent way to start a new

electrolyzed in the region of its stability, which is indicated in all

Chemical passivation was discovered about 200 years ago. A piece

such diagrams). We cannot deduce which reaction will happen at a

of iron placed in concentrated nitric acid was found to be passive,

measurable rate. The fact that at a certain pH and potential a metal

while the metal dissolved readily in dilute HNO 3 , with copious evolution

can corrode according to its Pourbaix diagram is no proof that it

of hydrogen. This type of behavior can be demonstrated in a very

actually will do so. The regions marked as "passivation" only indicate that a solid

simple, yet quite spectacular, experiment. Nitric acid of various concentrations, from 1 mM to 70%, is introduced into a series of test

(usually an oxide or a hydroxide) is the thermodynamically stable

tubes, and an aluminum wire is placed in each solution. No reaction is

corrosion product, whether it can passivate the metal depends on the

observed in the most dilute solutions. As the concentration is inc-

nature of the oxide and on the environment in which corrosion occurs.

reased, however, hydrogen evolution becomes visible. At even higher

An important point to remember is that potential/pH diagrams are

concentrations, reduction of the acid takes place, in addition to

usually given for the pure elements. Now, high purity metals belong to

of

hydrogen evolution. This is evidenced by the liberation of a brown gas, NO , which is one of the reduction products. When the concentration has 2 reached 35%, the reaction suddenly stops. There is no gas evolution and

metallurgy have been in the area of new and improved alloys design to

the surface of he metal is not attacked. Accurate measurements show no

fit specific engineering requirements. The corrosion behavior of an

weight loss when aluminum is kept in these solutions for months.

alloy is rarely, if ever, a linear combination of the corrosion of its

Aluminum is passivated in concentrated IINO . A thin oxide film is 3 formed on the surface and further attack is prevented.

the research laboratory, but nothing is ever constructed of a pure metal. In fact, the most important developments in the field

components. Even for a given composition, the corrosion of an alloy usually depends on metallurgical factors such as the grain size and heat

Electrochemical passivation is in many ways similar to chemical

treatment of the material. An extreme example is the high corrosion resistance of so-called glassy metals or amorphous alloys, compared to

passivation. As the potential of an iron sample is increased in the

alloys of the same composition in their usual crystalline form.

and then decreases almost to zero. Further increase of the potential

anodic direction, the rate of dissolution increases, reaches a maximum, has little effect on the current in the passive region until passivity

ELECI 1( out: K1 i

J 1.1

M. APPL1CA'FIONS

breaks down, whereupon the current rises rapidly with potential. The 2.0

sequence of events observed on an iron electrode when its potential is swept very slowly in the anodic direction is shown schematically in Fig. 17M, where currents are plotted on a logarithmic scale. The peak current density may be as high as 10 mA/cm 2 , whereas the current density

z 1.0

in the passive region is on the order of 1 IAA/cm 2 . The potential at which the anodic dissolution current has its maximum value, and beyond

0

crit

which it starts to decline rapidly, is called. the primary passivation potential E . The corresponding current is referred to as the critical PP

Epp

s,

corrosion current i

In the passive region, which may extend over Grit half a volt or more, the current is nearly constant. It starts to rise

I-

0

Ecorr

0

again at the so-called breakdown potential, above which pitting corrosion occurs, along with oxygen evolution and electrochemical dissolution * of the passive film. On so-called "valve metals" (e.g., Ti and Ta) the

40 -1

0

I

2

3

log i/i.LA cm -2

oxide continues to thicken as the potential is increased. If there are no aggressive anions in solution, this anodization process can lead to very thick oxide films, up to hundreds of micrometers, on which oxygen evolution cannot occur. Curves of the type shown in Fig. 17M are obtained by sweeping the potential in the anodic direction very slowly, at a rate of 0.1-1.0 mV/s. Even so, steady state is not quite reached and the values of E PP and Ebdepend to some extent on sweep rate.

Fig. 17M Schematic representation of the corrosion and passivation of iron in sulfuric acid. The primary passivation potential E PP

and the corresponding critical current density for corrosion i , are shown. Breakdown of the passive film occurs at crit

potentials more positive than

Eb .

protective oxide is removed by scratching the surface, a large transient

repassivation potential, which is best explained in terms of a simple experiment. If the

current is observed, but this current decays rapidly back to its value

potential is held in the lower part of the passive region and the

vates spontaneously at this potential. As the experiment is repeated at

Another important quantity is the so-called

before the oxide has been mechanically removed. The surface repassiincreasing anodic potentials, it is found that repassivation takes

*

It may appear odd that an oxide formed anodically on the surface can dissolve anodically. The anodic process taking place at such high potentials is transformation of the oxide to a higher oxidation state (e.g., C•3t to chromate) which is generally more soluble.

longer and longer, until a potential is reached beyond which the surface can no longer repassivate. This so-called repassivation potential is ordinarily found to be less anodic than the breakdown potential. Thus, it would seem that there is a potential region in which an anodic passive film cannot be formed on a bare metal surface, although an

516

ELECTRODE KINETICS

existing film is chemically and electrochemically stable.

517

M. APPLICATIONS

A true

that the breakdown potential and the repassivation potential are one and the same, and the apparent difference observed between them is just a manifestation of the long induction period needed for breakdown at potentials close to E b . One of the unique features of a corroding metal undergoing passivation is a region of apparent negative resistance. Looking at Fig. 17M, we note that at potentials anodic to the primary passivation potential E , the current density decreases with increasing anodic potential PP

until it reaches the passive region.

This is an unstable region in

PO TENTIAL /vo lt vs NNE

hysteresis of this type may indeed occur, although it has been argued

which the current keeps decreasing with time, even at constant potential as a result of the formation and growth of the passive film. It is interesting to consider how changes in the rate of the cathodic reaction can influence the open-circuit corrosion potential and the corresponding corrosion current. This behavior is shown schematically in Fig. 18M.

0

I 2 3 log i/p.A crn- 2

Fig. 18M The effect of changing the rate of the cathodic reaction on the corrosion potential and the corrosion current in a system

Several cases can be distinguished.

Line 1 crosses the anodic

dissolution curve in the active region, leading to a substantial rate of corrosion. The situation is very similar to that shown in Fig. 10M: an increase in the rate of the cathodic reaction leads to a shift of the

undergoing passivation. Inhibiting the cathodic current can have the adverse effect of shifting the corrosion potential from the passive region (point E on line 3) to the active corrosion region (point A on line I).

corrosion potential in the anodic direction and an increase in the corrosion current. three places.

Line 2 crosses the line for metal dissolution in

The crossing point C is unimportant, since it is

unstable, as pointed out earlier. stable corrosion potentials.

This leaves the system with two

Point B represents the usual situation

found for an actively corroding metal. Increasing the cathodic current from line 1 to line 2 raises the corrosion rate from point A to B. But line 2 also crosses the anodic dissolution curve at point D, in the passive region. As a result, a different corrosion potential, corresponding to point D, can also be established. Note that in the present

example the corrosion potential at point B represents a corrosion rate about 300 times that represented by point D, and in practice the ratio could be substantially higher. Where will the system actually settle? This depends on the initial conditions. If the metal is initially passivated (by oxidizing it chemically or by increasing the potential in the anodic direction), it can remain passivated with the corrosion potential at D. If it is initially in the active region, it can establish its corrosion potential

IS

ELECTRODE KINETICS

M. APPLICATiu. ,i5

at B.

This is referred to in the literature as unstable passivation, because passivation can be lost by transition from the corrosion poten-

I

tial at point D to an equally stable corrosion potential at point B. Line 3 represents stable passivation. The anodic and cathodic lines cross at a single point and a corrosion potential is set up at point E, well inside the passive region. Increasing the cathodic current even more can move the corrosion potential to point F in the transpassive region, where corrosion can again occur at a substantial rate.

"Yo N o C I

u-)

0.5

. • .,*

I.0

•

_J 0.5

z

►

•. .

Passive films formed in aqueous solutions consist of an oxide or a mixture of oxides, usually in hydrated form. The oxide formed on some

0.1 `1../

0 .0

1.5

U)

1-c; -

I

\

3.5 ° 0.0

•

metals (e.g., Al, Ti, Ta, Nb) is an electronic insulator, while on other 10 -1 10-2 10 -3 CURRENT DENSITY/A.cm -2

metals the passivating oxide film behaves like a semiconductor. Nickel, chromium, and their alloys with iron (notably the various kinds of stainless steel) can be readily passivated and, in fact, tend to be spontaneously passivated upon contact with water or moist air. It should be noted that passivation does not occur when chloride ions are introduced into the solution; indeed a preexisting passive film may be destroyed. Many other ions are detrimental to passivity, such as Br, -

I , SO4 ,

and CIO

-

4'

10 °

Fig. 19M The effect of increasing the concentration of chloride ions on the passive current and on the range of potential over which passivity can be observed, for nickel in 0.5 M H 2SO4 . Data from Piron, Koutsoukos and Nobe, Corrosion, 25, 151, (1969).

but chloride is the worst offender, because of its

omnipresence in the environment. When an experiment such as shown in Fig. 17M is conducted in solutions of increasing concentration of NaC1, the behavior shown in Fig. 19M is observed. The passivation current density is increased and the breakdown potential becomes less anodic until, at sufficiently high concentration, passivation can no longer be observed. Conversely, if NaCI is introduced into a solution in which iron is held in the passive region, nothing will happen at first, but after an induction period that depends, among others, on the concentration of Cl ions, the current will start to increase, and breakdown of passivity will be evidenced.

29.6 Localized Corrosion Measurement of the average corrosion rate, per square centimeter of the sample, yields only part of the pertinent information, often a small part. Consider a piece of metal corroding in a given environment at a rate of 0.1 mm/year. (In corrosion engineering, the British units of mpy are often used: 1 mpy = lx10 3 inch/year) This is a rather low rate, which would not worry the designer too much, if corrosion were uniform. Adding 1 mm to the wall thickness of a pipe, for example, would provide an additional service life of 10 years. On the other hand, if corrosion is localized to, say, 1% of the surface, the same average corrosion rate

520

ELECTRODE KINETICS

521

M. APPLICATIONS I-

would correspond to a penetration of 10 mm/year and one could not very

current density at the mouth of the pit is the highest, and the rate of

well increase the wall thickness to 100 mm to provide a service life of

anodic dissolution declines with depth inside the pore. The end result

10 years.

is an accumulation of AlC1

Two important forms of localized corrosion are pitting and crevice corrosion.

Although the causes of these phenomena may be quite diffe-

rent, the chemistry involved is similar and the following discussion is

3 near the mouth of the pore. The local concentration can exceed the solubility, leading to a precipitate of AlC1 , which can partially block the pore. These probably are the 3 conditions needed for the next stage — lowering the pH inside the pit.

pertinent to both. Consider a pit formed in a piece of aluminum that is in contact with seawater. As we shall show, the pH of the solution inside a pit

NaCI

can become quite low, leading to an increased rate of corrosion, which further lowers the pH, and so on. Thus, pitting corrosion can be

02 CI

OH

AICI 3 (solid)

/

considered to be an autocatalytic process, with its rate increasing with time. The processes that take place in the pit and in its vicinity are shown in Fig. 20M(a). At the concentration of chloride ions found in seawater, the passive layer on aluminum breaks down, and anodic dissolution of the metal can occur. This happens mostly inside the pit, where

20Am

a

b

the supply of oxygen is slow. On the other hand, oxygen reduction can readily take place on the surface of the metal outside the pits, where its diffusion path is short. Thus, the cathodic area is typically hundreds of times greater than the anodic area; consequently the anodic current density inside the pit is hundreds of times higher. Inside the

pit, aluminum is being dissolved anodically, to form Al 3+ ions. To compensate for the excess positive charge, chloride ions must be transferred into the pit. Since the current density inside the pit can

d

be quite high, the ohmic potential drop becomes substantial. The Fig. 20M Schematic representation of early stages in the formation of

*

a

pit. (a) the reactions taking place in and around a pit. (b)

Remember that at the steady-state corrosion potential, the total

formation of a deposit of corrosion

product, partially

anodic and cathodic currents must be equal, but the current densities

blocking the exit of the pit. (c) and (d) propagation of the

may be quite different.

pit, which is almost filled with solid corrosion products.

ELE CI RODE KINETICS

A1C13 + 3H20

The important point to remember in the context of pitting corrosion is that the volume of the solution inside the pit is very small. More low. For a deep cylindrical pore, this is given approximately by TER 2

=

-2- C1113/C111 2

Al(OH)3 + 3HC1

(55M)

A similar reaction occurs during pitting corrosion of iron and its alloys. Partial hydrolysis, leading to the formation of Al(OH) +2 and

accurately stated, the volume of liquid per unit surface area is very

-2Tr RTh

M. APPLICA f IoNS

A1(OH) 2+ may also occur, but all such reactions lead to the formation of acid, making the solution inside the pit much more aggressive than (54M)

outside. Measurement of the pH inside a pit is not an easy matter, but estimates based on various calculations and on measurements in model

For a typical radius of 10 tm this leads to a volume of about 5x10 4cm3 persquacntimofear—butodsfmagnie

pits lead to values as low as 1-2 for chromium-containing ferrous alloys

less than in a regular electrochemical measurement. Inside the pits

and about 3.5 for aluminum-based alloys, depending on experimental

this can lead to rather unusual chemical phenomena that are not commonly encountered otherwise.

conditions. While hydrolysis of the reaction product can explain qualitatively

A numerical example might help to illustrate this point. Assume

the lowering of the pH in pitting and crevice corrosion, attempts to

that the current density inside the pit is 0.1 A/cm 2 . The rate of production of Al 3+ is then given by

calculate this quantitatively do not always give a straightforward answer. In fact, simple calculation would indicate that the H 30+ ion (which has a diffusion coefficient 5-7 times higher than that of most

(0.1 C/cm 2.$)/(5x10 -4cin3/cm2 ) = 200 C/cm 3 •s At this rate it takes only 15 seconds to increase the concentration of Al3+ inside the pit from zero to 10 M, enough to precipitate even the most soluble salts that may be formed in the process. Actually it may take longer for the concentration to grow, since diffusion of Al 3+ ions out of the pit occurs. On the other hand, the current density inside the pit could be higher. In either case this simple calculation shows that precipitation of a salt near the mouth of a pit is a likely event, in view of the small volumes of the solutions involved. The situation a short time after the pit has started to grow may be represented schematically by Fig. 20M(b). Blocking of the entrance to the pit by the corrosion product is an important factor in determining the pH developed inside the pit, as we shall see in a moment. Inside the pit, hydrolysis of AlC1 3 occurs, yielding HO.

other ions) can diffuse readily out of a pit of typical depth (< 0.1 mm), and the pH should not decrease very much, unless one assumes an extremely high current density inside the pit. Blocking of the mouth of the pit with corrosion products is probably the main factor slowing down the escape of 11 30+ ions, allowing the development of a strongly acid medium inside the crevice. The increased viscosity, due to the high concentration of corrosion products in the electrolyte inside a pit, may also play a role in retarding the escape of H 3 0+ ions. The situation inside a pit at a later stage may be represented by Fig. 20M(c). The walls of the pit are roughened by corrosion. The electrolyte in the pit is saturated with respect to A1C1 3, and there is a great deal of solid AlC1 3 inside the pit, slowing down the movement of ions and allowing the pH to decrease. Eventually a stable pH is reached inside the pit. Its numerical value depends on the equilibrium constant in Eq. 55M or similar equations corresponding to other ions, depending

524

ELECTRODE KINETICS

on the alloy composition.

525

M. APPLICATIONS

taking into account the solution resistance as a function of depth in a

The foregoing description is of necessity oversimplified. For one

pit or a crevice may be rather complex, but the trends are very simple.

thing, one is always dealing with an alloy rather than the pure metal,

We can view a corroding system as a battery with its terminals shorted.

as pointed out earlier. During corrosion, the components of the alloy

The current is then determined by the free energy of the reaction, which

do not, as a rule, dissolve at a rate that is proportional to their

controls the potential, and by the resistance which in this case is the

respective concentrations. The medium in which the sample is corroding

sum of the solution resistance and the Faradaic resistances at the

may also be complex, containing, for example, several different anions.

various interphases. Clearly, some of the driving force for corrosion

All this leads to the formation of many simultaneous hydrolysis equi-

is dissipated by the solution resistance, and the corrosion rate is

libria. Nevertheless, a detailed analysis can be made, at least in

accordingly decreased.

principle, for any specific case of interest, and the observed behavior can be interpreted in terms of the phenomena just discussed.

Differential aeration is also important whenever corrosion in a confined region is considered. The outer surface is accessible to

Crevice corrosion, as the name indicates, occurs in narrow spaces

oxygen and therefore becomes the site for the cathodic reaction. Inside

where an electrolyte can creep, usually by capillary forces, between two

a pit, the solution is rapidly depleted of oxygen, and this area becomes

pieces of metal or between a metal and an insulator. This type of

the site for the anodic reaction, namely, the dissolution of the metal.

corrosion is usually found where two metals have been riveted together,

Since the outer surface is, in such cases, many orders of magnitude

under the head of a bolt, or under an insulating 0-ring. It should be

larger than the surface inside a pit, very high

evident at this point that the chemistry and electrochemistry of crevice

can be observed. Differential aeration is not limited to crevices and

corrosion must be dominated by the same property that dominates pitting

pits, of course. One of the most common manifestations of this pheno-

corrosion, namely, the very small volume of the solution per unit of

menon is observed on partially immersed structures. Corrosion is

surface area. But there are also two differences: first, we know

commonly found to be most severe a short distance under the waterline,

exactly how a crevice is formed, while the phenomenon of pit initiation

just below the oxygen-rich cathodic region, which is protected from

is not very well understood. The other difference entails the relative

corrosion. Farther down, the rate of corrosion is smaller because of

dimensions in pits and in crevices. The width and depth of a pit are of

the iR potential drop in solution, as discussed earlier.

local corrosion rates

s

the same order of magnitude. This is certainly true in the initial stages of pit formation, but it holds true even during more advanced stages of its propagation. In comparison, the width of a crevice is of the order of micrometers, while its depth can be on the order of millimeters. This brings into the forefront the importance of the iR s

Unlike the case of batteries, the loss of driving force is, of

local corrosion potential potenialdrsu,etmingh at different points in the crevice. In the Evans diagrams discussed so

course, a welcome effect in corrosion. It is well known, for example, that the corrosion rate in poorly conducting solutions is lower than that in highly conducting media.

far (cf. Fig. 9-12M) this effect has been ignored. A detailed analysis,

ICS

I.

I r

M. APPLICATION.s

r

29.7 Corrosion Protection

scale, gold is the most stable and magnesium is the most active metal

The best method of corrosion protection is proper design. Unfortunately, corrosion engineers are not usually members of design teams, and

(excluding the alkali metals, which are of no interest in this context).

are often left with the task of stopping the spread of corrosion, where

and copper is more noble than niobium. Unfortunately, this thermo-

it should not have occurred in the first place. Proper design, it must

dynamic scale has little to do with reality, for a number of reasons.

be admitted, is no minor feat. It is the best compromise among a number

Metals such as niobium, tantalum, zirconium and aluminum form protective anodic films spontaneously, when brought into contact with air or water. This makes them more noble than iron and even copper on a practical

of factors, notably high strength, low weight, pleasing appearance and acceptable cost.

Iron and zinc are more noble than aluminum and titanium on this scale,

scale,

(a) Bimetallic (galvanic) corrosion Perhaps the worst (and most common) result of poor design is bimetallic corrosion. For example, if a copper faucet is connected to a

which is what matters in engineering design. Thus, in an

aluminum structure connected with steel rivets, the rivets will act as the anodes in the bimetallic couple formed, although the respective standard potentials of aluminum and iron are — 1.66 and — 0.42 V versus

effect, formed, leading to rapid corrosion of iron, which is the more

NHE. In terms of the Pourbaix potential/pH diagrams, the theoretical

active of the two metals. Corrosion will be worst near the contact

scale compares the potentials of immunity of the different metals, while

between the two metals, because the driving force farther along the tube

the practical scale compares the potentials of passivation. But this is

is diminished by the potential drop on the solution resistance. This

not enough either. The real scale depends on the environment with which

type of galvanic (bimetallic) corrosion is prevalent not only in

the structure will be in contact during service. Passivity, as we have

immersed structures or in pipelines carrying an ionically conducting

seen, depends on pH. It also depends on the ionic composition of the

liquid, but also in metals exposed to humid atmospheres, where the ionic

electrolyte, particularly the concentration of chloride ions or other

path of conductivity is established through a thin film of moisture

species that are detrimental to passivity. Finally, one must remember

accumulating on the surface. If the use of two or more metals is

that construction materials are always alloys, never the pure metals. The tendency of a metal to be passivated spontaneously can depend

water pipe made of low-carbon steel, a copper-iron battery is, in

unavoidable, the best way of combating galvanic corrosion is to isolate the different metals electrically. If this is not possible, one should

dramatically on alloying elements. For example, an alloy of iron with

attempt to use metals that have very similar corrosion potentials, to

8% nickel and 18% chromium (known as 304 stainless steel) is commonly

ensure that the potential developed between them, upon immersion in the same solution, will he minimal.

used for kitchen utensils. This alloy passivates spontaneously and should be ranked, on the practical scale of potentials, near copper. If

How can we tell which will he the more active metal, or what metals will be about equal? Theory provides a very simple answer. The higher the standard potential, the more noble will the metal be. On this

* The "practical scale" is also referred to as "the galvanic series for the solution of interest".

528

ELECTRODE KINETICS

M. APPLICATIONS

529

we increase the concentration of chromium by 10%, its position with

thin layer of zinc, functions in the same way. The coating provides a

respect to copper will not change significantly. If, on the other hand,

certain degree of protection as long as it is intact. When it is

we decrease the concentration of chromium by the same amount, the alloy will no longer act as a stainless steel. Its potential in solution will

partially removed, either by corrosion or abrasion, the exposed surface

Similarly,

Ni/Cr coating may provide much better protection as long as it is

alloying aluminum with a few percent of tin or gallium affects its

intact, but there will be severe galvanic corrosion in the area of a

ability to form a passive film and causes its open-circuit potential to

scratch, since the coating will act as a large-area cathode and the

shift to much more negative values. The effect of different alloying

exposed steel surface as a small-area anode — a very bad combination.

be much more negative, in the vicinity of iron itself.

is still protected cathodically and does not corrode. In contrast, a

elements is nontrivial and cannot, as a rule, be predicted from theory.

The theory of cathodic protection is simple and straightforward.

The best way to determine whether two metals are going to be compatible

The real engineering challenge is to design the anodes and position them

from the point of view of galvanic corrosion is to measure their

so that they provide uniform current distribution on the part being

potentials for extended periods of time in a medium as close as possible

protected. There are two aspects to this problem: parts of the struc-

to that which will be encountered in service.

ture that are too far away from the anode or screened from it may not have sufficient protection, and parts that are too close to the anode

(b) Cathodic protection

may be "overprotected". We recall that the cathodic reaction in most

Cathodic protection can be viewed as a form of galvanic corrosion,

cases is hydrogen evolution or oxygen reduction. Both reactions lead to

put to good use. In this case an active metal (most often zinc, but

the formation of (01-I) ions near the surface. This can weaken the

under special circumstances magnesium or aluminum) is employed as a

bonding of paints and other nonmetallic coatings to the surface, causing

It is attached to the steel structure being protec-

delamination. Thus, under certain circumstances, overprotection can be

ted in one or several locations and does not constitute part of the

worse than no protection at all. Also, excessive hydrogen evolution can

structure itself. The steel structure becomes the site of the cathodic

lead to penetration of atomic hydrogen into steel, causing embrittle-

reaction, and its potential is driven in the negative direction until

ment.

sacrificial anode.

corrosion either stops or is slowed to an acceptable level. The

There are no simple equations from which the current distribution

sacrificial anode is, of course, corroding at a relatively high rate and

on complex structures may be obtained. A number of numerical methods

must be replaced periodically, but damage to the structure by corrosion

can be used, however, and there is a wealth of practical experience,

can be minimized.

which can serve at least as a very good "first guess" in a calculation

Sacrificial anodes are most commonly employed to protect the hulls

involving many successive iterations. The real problem lies in the fact

of ships and smaller boats, for offshore oil rigs and underground

that conditions during service life change, not always in a predictable

pipelines. They are designed to be replaced, if necessary, during

manner. Thus, the protective coating on the hull of a ship or on a

routine maintenance. But these are not their only uses. Galvanized

pipeline may be damaged with time, changing the effective area that

steel, which is usually a low-carbon steel plate or wire coated with a

needs to be protected. Rain or drought will change the conductivity of

530

ELECTRODE KINETICS

M. . t•LICA'FIONS

integral efficiency =

the soil in which a pipeline is buried. The supply of oxygen to the immersed parts of a boat changes dramatically when it lifts anchor to sail at full speed. The conductivity of the water changes by several orders of magnitude when a ship sails from a river into the open sea. All these factors change the current distribution. If designed for optimum protection at sea, a ship will be overprotected in fresh water

corr

—i

a

(56M)

imp

is the impressed-current density, given by (i c — i2 ) and the absolute values of the currents are used. Values of 54% and 39% are where i

,

found for lines 1 and 2 in Fig. 21M, respectively. We can define similarly a differential efficiency for impressed current cathodic protection as:

and vice versa. The best way to overcome the limitations - of cathodic protection related to changes in the environment is to monitor the potential and adjust the cathodic currents accordingly. This cannot be readily done with sacrificial anodes, and the method of impressed-current cathodic

differential efficiency = [

ia I

—6 1

imp2

—

(57M)

a2j impl

where the numbers in the subscripts refer to the two lines marked in Fig. 21M.

protection is sometimes preferred, in spite of its higher cost.

Impressed-current cathodic protection

entails the use of an

external power source in combination with a stable anode. The potential respect to its open-circuit corrosion potential, and its rate of anodic dissolution is consequently reduced.

The result of impressing a

cathodic current on the structure is shown in Fig. 21M: parameters used to draw this figure we obtain i

cort

for the

= 48.8 i.tA/cm2 and

E

= — 0.554 V, NHE. Applying a cathodic current density of 72 con2 i_tA/cm shifts the potential to — 0.58 V, NHE and the anodic current density is reduced to 10 IAA/al -1 2 , as shown by line 1. The current flowing in the external circuit is the difference between the cathodic

Poten tia l/V vs N HE

of the specimen being protected is forced to negative values with

0.3

-0.4

-0.5

2.9

-0.6

-7

and the anodic currents flowing at this potential. Application of a

-6

10 -/gb■ 82 ■INFE■ 121

-5

-4

-3

log i/A•cnn -2

current of 72 i.tA/cm 2 causes a decrease of the rate of corrosion by about a factor of 5, from 48.8 .1A/cin 2 to 10 j.tA/cm 2 . To decrease the potential to — 0.60 V, NHE, the current density must he increased to 118 1,tA/cm 2 . The anodic corrosion rate will decrease to 2.9 1.1A/cm 2 , as shown by line 2. We can define the integral efficiency for impressed current cathodic

Fig. 21M Evans diagram for iron at pH = 6, showing the principle of impressed-current cathodic protection. The two horizontal lines show two levels of cathodic protection. The impressed cut- rent is the difference between the cathodic and the anodic currents shown.

protection as: -at

532

ELECTRODE KINETICS

533

M. APPLICATIONS

In the example shown here this differential efficiency is:

10 - 2.9 118 - 72

- 15.4%

a 80

80

E

U

The anodic current and the two types of efficiency just defined are

60

shown in Fig. 22M(a). In Fig. 22M(b) the range of almost complete corrosion rate can be reduced to negligible levels by applying a

integral

40

protection is shown. It is important to note in Fig. 22M that (a) the

•

•••••... .....

-r---

‘•

0

sufficiently high cathodic current to the structure being protected and

6 20

(b) the efficiency drops sharply in the region of almost complete

L

s

0

s..differential

0

..........

0

protection. It takes 151 vtA/cm 2 to slow the corrosion from 48.8 to 1 2 2 gA/cm , and it will take another 25 pA/cm to decrease it further to

0

30

60

90

120

150

0 180

Impressed c.d./,uA•cm -2

0.1 pA/cm 2 . The choice is between higher protection at the risk of protection at the risk of corrosion, particularly in regions farther

2.0

from the anodes. It is also a choice between the high cost of electric power on the one hand and the need for more frequent maintenance due to incomplete protection from corrosion, on the other. Cathodic protection is usually not used by itself. A pipe buried in the ground is painted or coated, for added protection against corrosion. Ideally such coatings should provide complete protection,

6

E 1.5

N -6 d

---------

3

1.0

O

0.5 L O

but in service they never do. It is hard to tell how much of a coating

0 140

is initially damaged and it is usual to determine the potential at which

1-2 tA/cm 2 . If 1% of the coating is initially damaged, this corresponds to 100-200 pA/cm 2 on the small regions where the coating has been damaged and the underlying metal needs protection. The potential can be monitored continuously at different locations, and the impressed current can be adjusted automatically to maintain the desired level of corrosion protection.

160

170

0 180

Impressed c.d./pA•cm -2

one wishes to operate, adjusting the impressed current accordingly. An average cathodic current density for protection may typically be

150

Dif feren tia l e ffic iency, %

b

local overprotection (and possible hydrogen embrittlement) and less

Fig. 22M (a) The corrosion-current density under cathodic protection, as a function of the impressed current. Fig. 21M. i

tort

Parameters as in

= 48.8 µA/cm. The differential and integral

efficiencies are shown. (b) Detail of (a) for the range of almost complete cathodic protection, showing the low values of the differential efficiency of the impressed current in this region.

534

ELE.C114. 0 OE KINETICS

M. APPLICATIONS

Impressed-current cathodic protection requires a little more

2.0

sophistication than the use of sacrificial anodes, but it also lends itself to periodic adjustment and provides higher flexibility, particularly when structures having rather intricate shapes are considered.

z (c) Anodic protection

1.0

>

Anodic protection makes use of the ability of iron and many of its

0

alloys to become passive in the absence of CI - and other aggressive

_J

1 onodic protection

ions, as discussed in Section 29.5. If we extend Fig. 17M to show the

Epp

current during one cycle of the potential from the active to the 0

transpassive region and back to the passive region, the behavior observed will be that shown schematically in Fig. 23M. Setting the potential anywhere between E , the primary passivation potential, and

-1.0

PP

E , the repassivation potential, causes the metal to be passivated. Between E and the breakdown potential E , the system can be unstable, rp as indicated by the bursts of current.

Ecorr

0

0

I

3

2

4

log i/FLA cm -2 Fig. 23M Current-potential characteristic of a system undergoing passi-

anodically. It is important to use the term "systems" rather than

vation. The optimum potential region for anodic protection is shown. Eb - breakdown potential; E - repassivation poten-

"metals" or "alloys" because the ability to form a stable passive film

tial;

Systems behaving in the manner shown in Fig. 23M can be protected

depends on both the metal and the electrolyte in contact with it, as pointed out earlier (cf. Fig. 19M). As a result, anodic protection cannot be used as universally as cathodic protection. In particular, its use is limited to situations where chloride ions are absent, a limitation that excludes it from all applications in or near the sea.

rp

E - primary passivation PP

potential;

E corr

open

circuit corrosion potential. Where applicable, anodic protection has great advantages over cathodic protection for a number of reasons. First, it requires typically only 1-2 1.1.A/cm 2 , about three orders of magnitude less than cathodic protection. In addition to the saving in energy, the negative side effects of cathodic protection — namely hydrogen embrittlement and

The bursts of current result from metastable pits, where breakdown

delamination of nonmetallic coatings, resulting from the high pH

of the film, followed by rapid repassivation, occurs. As the potential

generated at the cathodic sites — are eliminated as well. Since anodic

is increased, breakdown is more likely and repassivation is slower, until the potential Ebis reached, beyond which breakdown is the predo-

passivation is performed potentiostatically, and since the currents

minant process.

involved are very small, uniform current distribution is easier to

536

ELECTRODE KINETICS

maintain and overprotection is not likely to occur.

M. APPLICATIONS

537

inhibitor, it is important to know the corrosion potential with respect

afforded by such paints can be the result of a number of mechanisms

E. If the corrosion potential is anodic to the PZC, a negatively charged inhibitor may be the better choice. If E occurs at a negative rational potential, a cationic corr inhibitor may be preferable. Neutral molecules can best serve as inhibitor if E = E . It should be borne in mind that many of the COtT 7. commercial inhibitors are weak acids or bases, and their charge depends

operating in parallel. One of them is the high positive potential set

on pH. Thus, an inhibitor that acts well in a medium of low pH may be

up by the oxidizing agent, bringing the metal into the passive region.

quite useless in a medium of high pH, or vice versa. When the usual

Unfortunately, anodic protection is limited to certain environment, where the liquid in contact with the protected structure is well defined and known to allow passivation. There is, however, a widely used chemical form of anodic protection, entailing paints that contain strong oxidizing agents such as K2Cr207 and Pb304 . The corrosion protection

to the potential of zero charge

aqueous medium is replaced by a nonaqueous or mixed solvent, the (d) Coatings and inhibitors

situation can change dramatically. The charge on the inhibitor molecule

Finally we shall discuss very briefly coatings and inhibitors used

may be quite different, and Ez also depends on the solvent. The

to prevent or slow corrosion. Coating can be considered in two groups:

solubility of the inhibitor also depends on the solvent composition.

active coatings such as zinc, which act as sacrificial anodes even after parts of the underlying metal have been exposed to the environment, and barrier coatings such as paints of all sorts and protection by a more

This changes the surface coverage corresponding to a given bulk concent-

noble metal, such as nickel on iron or silver and gold on copper. Such

scale of concentration C/C(sat), obtained by dividing the concentration

coatings prevent corrosion by simply isolating the metal from the

in each solvent by its saturation value. Thus a different range of

environment. They can be excellent as long as they are intact; once

concentrations of inhibitor must be used for each solvent, assuming that

damaged, however, galvanic corrosion (in the case of more noble metals)

the inhibitor works at all.

and differential aeration (in the case of nonmetallic coatings) may lead to an increase in the rate of corrosion on the exposed parts. In this context we might mention that surface preparation is a major factor in obtaining good, adherent coatings of any type. Degreasing, chemical cleaning and, in some cases, mechanical treatment are essential steps in the preparation of the surface for coating, which often consists of several layers, for optimum protection.

Corrosion inhibitors are commonly used to prevent corrosion. There are many hundreds of different inhibitors in commercial use. Some act by slowing the cathodic reaction and others inhibit the anodic reaction. Some are ionic and some are neutral. In choosing a suitable corrosion

ration. To a first approximation, the surface coverage should be similar in different solvents, when comparison is made on a normalized

.538

ELECTRODE' KINETICS

30. ELECTROPLATING

M. APPLICATIONS

and are therefore thermodynamically more stable, is considered to be the rate-determining step for metal deposition in some cases. Impurities or

30.1 General Observations

additives adsorbed on the surface can hinder such diffusion and can control the surface morphology of the resulting deposit.

Metal deposition appears to be a very simple electrochemical process: an ion in solution accepts one or more electrons and becomes a neutral atom, to be incorporated into the metal lattice. In reality,

It is interesting to consider the metal deposition process from a microscopic point of view. At a rate of, say, 20 mA/cm 2, we have as

the situation is infinitely more complex. We have already noted that

many as 6x10 16 divalent ions/cm2 •s being deposited on the surface, corresponding to the formation of about 40 atomic layers per second.

the ion is always solvated (cf. Eq. 2F in Section 14.1). It is unlikely * that the hydration shell is removed in a single step, since this

This may be too fast for the adatoms to reach their equilibrium posi-

requires a very high energy of activation, which is not consistent with

at high temperature, can be formed by electrodeposition at room tempe-

the high exchange current densities observed for many metal deposition

tions. It is indeed observed that unusual alloys, which are stable only

processes. We have to ask ourselves what is the course of discharge of

rature. The interaction between the substrate and the metal being deposited

a multivalent ion like Cu 2+ or Fe2+ ? Do we postulate the existence of a

can also play an important role in determining the quality of a plated

monovalent intermediate ionic species, or must we assume that two

product. When the crystal parameters of the two metals are different,

charges are transferred in a single step? Cuprous ions do exist, but

one of two situations may be observed. The metal being plated may

they are unstable in aqueous solutions. On the other hand, monovalent

initially attain the crystal structure of the substrate, although this

iron (and other ionic species having an intermediate valency, such as Pb+ or Ga2+ ) have not been observed as such, although they could exist

is not its most stable form. This is referred to as epitaxial growth.

as adsorbed intermediates, stabilized by their bond to the metal.

stable crystal structure. The stress created by the epitaxial growth

As the thickness of the deposit grows, it gradually reverts to its more

Farther down the reaction sequence we must consider the fate of a

can be relaxed by impurity atoms and by dislocations in the metal.

so-called adatom formed on the surface. This term is used to indicate

Large differences between the crystal structures of the two metals do

that although the ion has landed on the surface and has been discharged, it has not reached a position of lowest free energy and in this sense

not favor epitaxial growth. In such cases a so-called crystallization overpotential is observed, followed by two-dimensional nucleation on the

has not yet been incorporated into the metal lattice. Diffusion of

surface. The effect is similar to the formation of small crystals in a

adatoms on the surface, from their initial landing site to edges, kinks

supersaturated solution or droplets in a vapor.

or vacancies on the surface, where they can be more highly coordinated * * Since we are dealing mainly with aqueous solutions, we can replace the general term "solvation" by "hydration."

Note that a fivefold supersaturation is equivalent, in terms of

free energy, to an overpotential of only 21 mV for the deposition of a divalent ion such as copper.

540

ELECTRODE KINETICS

541

M. APPLICATIONS

Side reactions, mostly hydrogen evolution, play an important role

(taken as 5) and M is the ratio of coating thicknesses on the two

in electroplating. As a rule, their effect is detrimental to the

cathodes. This is a little awkward, because ideal throwing power,

process, because of the loss of energy and possible hydrogen embrittle-

defined as uniform plating thickness irrespective of geometry, corres-

ment. But side reactions can also be beneficial, improving the uni-

ponds to M = 1, yielding a value of T.P. = 80%, rather than 100%, which

formity of plating, as we shall see. Finally, there is the question of uniformity. Parts to be plated

one would expect for the upper limit of such a quantity. Still, it provides a very useful quantitative scale describing one of the most

are rarely flat. They have grooves, edges, corners, protrusions and so

important properties of plating baths. If there is no throwing power at

on. A good plating bath, which covers a surface uniformly irrespective

all, the thickness will simply be inversely proportional to the dis-

of its shape, is said to have a good throwing power.

The factors

tance. This yields a value of M = K and T.P. = 0, as expected.

controlling the throwing power of plating baths are discussed in detail

Having defined the throwing power quantitatively, we can now

in Sections 30.2 and 30.3, where we discuss some of the practical

proceed to discuss the physical reasons for the dependence of the

aspects of electroplating.

throwing power on geometry and the methods available to increase the

Although it must be admitted that most electroplating baths were originally developed by methods of trial and error, current understanding allows us to set the theoretical basis for their operation. This can lead to improvements in the operation of existing plating baths and

value of the T.P. to acceptable levels. This discussion refers to the so-called macro throwing power.

In Section 30.3 we shall discuss the micro throwing power, which controls the smoothness of the deposit and depends on quite different factors.

to the development of new ones, to meet the challenges of evolving new technologies. 30.2 Macro Throwing Power

Cathode Anode

Cathode

The throwing power of a bath is a measure of its ability to produce electroplated coatings of uniform thickness on samples having complex geometries. A quantitative measure of this property can be defined in terms of the so-called Haring and Blum cell, which is used to determine it. In this cell, two cathodes are positioned at unequal distances from two sides of an anode, as shown in Fig. 24M. The throwing power (T.P.) is defined as: T.P. = [ K

M x100

(58M)

where K is the ratio of distances between the anode and the two cathodes

Fig. 24M Top view of the "Haring and Blum cell" for the determination of the throwing power of plating solutions.

542

ELEC I RUDE KiNb.

M. APPLICATIONS

As long as metal deposition is the only reaction taking place, the

observed. What this means in a practical sense is that, under condi-

variations of the thickness of the deposit are an expression of the

tions of secondary current distribution, the resistance between the

current distribution pertaining to the specific conditions of plating.

working electrode (which is the part being plated) and the counter

We have two limiting cases: primary current distribution, which is

electrode is determined primarily by the resistance of the interface to

determined exclusively by the conductivity of the solution and the

charge transfer. Changes in the distance between the electrodes play a

geometry of the cell, and secondary current distribution, which is also

negligible role. Consequently, the throwing power tends to its maximum

influenced by the kinetic parameters of the deposition process. The

value and uniform coatings are observed. Primary current distributions,

transition from one regime to another is characterized by a dimension-

on the other hand, leads to low values of the throwing power. In

less number called the Wagner number, defined as:

practice a system is said to be under primary current distribution if

Wa = K(an/ai) "

4

L

Wa < 0.1 and under secondary current distribution if Wa > 10. The choice of the so-called characteristic length L is not always (59M)

obvious. It may be the length of the electrode being plated, the distance between the working and the counter electrodes or "the dimen-

where K = 1/p is the specific conductivity of the solution and L is a

sion of the irregularity" — for example, the difference between the

characteristic length. The partial derivative, taken at constant

shortest and the longest distance between the two electrodes. The last

concentration (and, of course, constant temperature and pressure), is

is the best choice in most cases, since it reflects the differences in

the differential faradaic resistance, in units of U•cm 2 . The solution

the solution resistance on different areas of the piece being plated.

resistance, expressed in the same units, can be written as follows:

The exact value taken is not very important, however, since the Wagner number must be used only as a guideline, and the actual current distri-

R = pL s

(60M)

experimentally or by obtaining an accurate numerical solution for the

Hence the Wagner number can be expressed simply as Wa

(an/a i) = RR p•L S

bution (or variation of thickness of plating) can be found either specific geometry and the kinetic parameters considered.

(61M)

In the absence of mass transport limitations, the local current density at a given potential is determined by the sum of two resistors in

But is the Wagner number, as defined here, the real criterion for transition from primary to secondary current distribution? For a reaction occurring in the linear Tafel region one has

=

series: the faradaic resistance and the solution resistance. For values of Wa much less than unity the solution resistance is dominant and the

hence RF = TIM i)c =

current distributions depends primarily on geometry. This is the realm of primary current distribution. For Wa much greater than unity the faradaic resistance is predominant and secondary current distribution is

(62M)

b•log(i/i.) b/i

(63M)

and Wa = (b/i)(1/p•L)

(64M)

Fr FCTRODE KINETICS

544

This would lead to the (unreasonable) conclusion that the exchange

545

M. APPLICATIONS

0.0V

0.44V

current density has no effect on the current distribution at high overpotentials. According to Eq. 64M, the Wagner number applicable in a

l o = 10 4 A/cm 2

given solution depends only on the current density used for plating and on the Tafel slope b, and is independent of the specific rate of the

-2

reaction. The correct expression is obtained if we replace the differential

0.0V

0.96V

A/CM 2=10

1. 16V

faradaic resistance (ari/a0 c in the definition of the Wagner number (Eq. 59M) by the integral resistance (TO). The modified Wagner number, i =10 — 10 A/ c m 2

is then given by 96 0 cm 2

20 0 cm 2

I = 10

-2

A/CM 2

(65M)

-

Fig. 25M The relative roles of the solution resistance and the integral

pL

faradaic resistance in determining the current distribution, In the linear Tafel region we can write

ri/i = (b/i)log(i/i. o)

(66M)

for different values of i . Parameters used: b = 0.12 V; p 0 = 10 Slcm; L = 2 cm.

To illustrate the importance of using the integral faradaic resistance, consider two reactions taking place under identical conditions, one having an exchange current density of i o = 10-4 A/cm2 and the other a value of i = 10 1°A/cm 2 . Using values of p•L = 20 SI•cm 2 , b = 0.12 V and i = 10 mA/cm 2 , we obtain from Eq. 63M a value of WA = 0.60 for both

current density, the closer will the system be to conditions of secondary current distribution. Figure 25M illustrates the values of the integral faradaic resistance and the potential drop across the cell and across each resistor for the two cases discussed. For i = 10 4A/cm2 theponialdrcstheoubayr,dtheplicaonf

reactions, implying that the current distribution is largely primary. a current density of 10 mA/cm 2 , constitutes about 55% of the total cell

For the modified Wagner number we have, -

log(i/i ) - 0

voltage. For io= 10-10Akm2 this fraction grows to 83%. (67M)

The integral faradaic resistance decreases with increasing current density and the (modified) Wagner number changes in the same direction.

This leads to the well-established

This is shown in Fig. 26M, calculated for a moderately fast reaction, having an exchange current density of 5x10 5 A/cm 2 . The specific resis-

observation that, other things being equal, the lower the exchange

tivity was taken as 10 acm and L = 2 cm. The regions of primary and

Equation 67M yields values of `Ill of 1.2 and 4.8 for the faster and the slower reaction, respectively.

secondary current distribution are shown. For the parameters chosen,

D46

Fara da ic res istance /0- cm 2

ELECTRODE KINETICS

10 4

10 3

10 2

secondary

541

M. APPLICATIONS

w" _a E

---------

10 2 o-

10 3 a) cr)

10 2

secondary

mixed -a

10

mixed

1 10 -1

•—

10— o

10

1 0 -2

10 —

primary primary 1

10 —s

10 -3 -6 10 10 -4

10 -2

10 -4

10 -2 log i /A•cm -2

Current density/A •cm -2 Fig. 26M The integral faradaic resistance and the modified Wagner number, as a function of the current density. Parameters used: i0 = 5x10-5 Alcm 2 ; p = 10 acm; L = 2 cm; b = b = 0.118 V.

Fig. 27M The modified Wagner number, calculated for different values of the exchange current density i o , as a function of the applied current density. p = 50 acm; L = 2 cm.

a

Regions of primary, secondary and mixed current distribution are shown. secondary current distribution is observed only for i 5_ 0.7 mA/cm 2 . For a reasonable plating rate of 10 mA/cm 2 , the modified Wagner number %id is about 1.4, leading to medium values of the throwing power. Next we show the dependence of the modified Wagner number on the applied current density for different values of the exchange current density of the reaction being studied. At low current densities is inversely proportional to i , as seen in Fig. 27M. In the linear Tafel region is proportional to - log i and decreases with increasing current density as seen from Eq. 67M. We note that for the specific resistivity p = 50 u•cm chosen to calculate these curves, it will be difficult to obtain good throwing power at reasonable plating rates if

the exchange current density exceeds 10 4A/cm2 , unless some means of increasing the throwing power is implemented. This leads us to the discussion of the methods by which the throwing power can be increased. Increasing the conductivity of the solution is an obvious approach, but is limited in scope. A specific resistivity of 5 acm, found it the case of the so-called acid copper bath, which contains CuSO 4 and H2SO4' is about as low as one can get in aqueous solutions. The other approach is to decrease i o. The kinetics of metal deposition from the simple ions is usually fast, but when the ion is complexed, much lower values of the exchange current density can be realized. This is one of the reasons for using cyanide baths for the electrodeposition of many metals. Copper, for example, can be deposited from an alkaline bath containing KCN. Instead of the usual aquo-complex 28 2+ [Cu(H 2 0) 4 ] one has the much more stable (K = 5.6x10 ) cyanide

548

ELECTRODE KINETICS

549

M. APPLICATIONS

complex of monovalent copper [Cu(CN) 3j2— from which copper is deposited

current density is associated with a lower fraction of the total current

at a lower rate, leading to improved throwing power, albeit at the cost

consumed to discharge the metal ions, and vice versa.

of lower rates of plating, to say nothing of the severe environmental

A decrease in F.E. with increasing current density is commonly

problems related to the toxicity of the CN ion. Alkaline cyanide baths

observed in many cases, although the optimal conditions represented by

are commonly used for the plating of many other metals, including

Eq. 69M do not usually hold. This can be regarded as a negative feed-

nickel, zinc, silver, and gold. Other complexing anions such as

back effect, in that there is a tendency to "stabilize" the process, so

pyrophosphate and tartrate are also used for the same purpose.

to speak. The decrease in F.E. counteracts the uneven current distribu-

When a metal ion is complexed, its standard potential is shifted cathodically by (2.3RT/nF)logK, where K is the stability constant of the

tion, yielding much more uniform coatings than might have been expected for any given value of the Wagner number.

complex formed. As a result, hydrogen evolution can occur along with

The dependence of the F.E. on current density is governed by the

metal deposition. The faradaic efficiency, which is the fraction of the

kinetic parameters of the two reactions involved. On paper it is easy

current consumed to deposit the metal, may decrease. This quantity is

to produce conditions under which the F.E. will either decrease or

defined by

increase with current density, or be independent of it. For example, in F.E. =

1

M

M +1

(68M) H

and iHrefer to the partial current densities for metal irs,4 deposition and hydrogen evolution, respectively. where

Although hydrogen evolution occurring as a side reaction during metal deposition is, in a general sense, detrimental, it can improve the

the commonly encountered situation in which hydrogen evolution is activation controlled and metal deposition is partially controlled by mass transport (0.05 5_ Wi L 5_ 0.7), the F.E. decreases with increasing current density. If both reactions are in the linear Tafel region and the slopes are different, the partial current for the reaction having the lower Tafel slope will increase faster. Thus, if b H = 0.12 V and b m

increase rapidly with increasing total current =0.4V,theFEwil

throwing power, under certain favorable conditions. The key to this effect lies in the dependence of the faradaic efficiency on applied current density. If it decreases with increasing current density, the

density. A well known case in which the F.E. increases with current density is the deposition of chromium from solutions containing chromic acid

throwing power will be improved. Consider a hypothetical case in which the faradaic efficiency is inversely proportional to the current efficiency. We could then write

(Cr0 ) and sulfuric acid. Somewhat surprisingly, the throwing power in 3 this system is not as low as might have been expected on the basis of the variation of the F.E. with total current (although it is far from

F.E. = i rvi/i = K/i,

hence i

m

=K

(69M)

What this means, in simple terms, is that even though a low Wagner number may give rise to an uneven current distribution on the surface, the rate of metal deposition is equal everywhere, since a higher local

Metal deposition is not conducted very close to the limiting current density, since this tends to produce low quality deposits.

J JU

ELECIROoL KINETICS

M. APPLICATIONS

being ideal). We must bear in mind, however, that the variation of the faradaic efficiency is just one of several factors that can control the

5.6x10-28 . The concentration of free cuprous ions * is hence 1.7x10 25 . If the electron is transferred to this ion, the rate must be controlled

observed throwing power. Furthermore, the electrodeposition of chromium

by the rate of decomposition of the complex, since the concentration of

from the hexavalent state is a particularly complex process, which has

free Cu+ ions is too low to support any measurable current. It is more likely that electrons are transferred directly from the

not yet been fully understood, and one cannot expect its behavior to follow the simple regularities discussed above.

metal to the negatively charged complex ions, which have a concentration

Electrodeposition of a metal from a negative complex ion can influence the throwing power and the morphology of the deposit in other

of 0.3 M in the same solution. One might expect this to be a slower

ways as well. Where the local current density is higher, the potential

copper bath. This is one of the reasons for the improved throwing power

on the solution side of the interphase is more negative. This causes a local decrease in the concentration of the negative ions, which slows

found in cyanide baths. Adsorption of the cyanide ion on the surface is also very likely,

the reaction. In other words, a negative feedback mechanism is again

in view of its high local concentration. This is an added factor which

operative, counteracting the variation of local current density caused by the primary current distribution.

is expected to reduce the rate of metal deposition, leading to higher

In the case of the alkaline copper cyanide bath, the overall electrode reaction is 2—

[CU(CN) 3 ]

em

Cu + 3(CNY

(70M)

For each atom of copper deposited, three cyanide ions are released at the electrode surface. The concentration of free (CN) ions at the electrode surface is thus higher than its bulk concentration. This shifts the potential on the solution side of the double layer in the negative direction, lowering the concentration of the complex ions, hence lowering the rate of reaction. A typical copper cyanide bath is composed of 0.3 M CuCN and 0.7 M KCN. The equilibrium constant in the reaction

\ 2— 1

ECU(CN) 3

Cu + + 3(CN)

(71M)

(which is the so-called instability constant of the complex) is

process than transferring the electron to a positive ion in an acid

values of the Wagner number. Finally we recall that the effect of the diffuse double layer is quite significant in the reduction of anions, as discussed in Sections 16.6 and 16.7. Although the total ionic strength in typical plating baths is high, the effect of the diffuse-double-layer potential 4) 2 on the observed exchange current density can amount to several orders of magnitude, depending on the potential at which metal deposition takes place, compared to the potential of zero charge. Cyanide is the most commonly used complexing ion in electroplating. The above discussion applies, however, equally well to any other ligand,

The cuprous ion is not stable in solution and undergoes disproportionation. This, however, does not preclude the possibility that it might exist at very low concentration at the surface, long enough to be discharged.

552

ELECTRODE KINETICS

553

M. APPLICATIONS

and ions such as pyrophosphate and tartrate are also used. The quality

distribution should not have any effect on micro throwing power.

of the plated product depends on the stability constant of the complex

Another way to look at it is to consider the appropriate Wagner number

formed and on the interaction of the free ligand ions with the surface,

for this situation. We used a characteristic length of 2 cm to calcu-

but the throwing power is, as a rule, improved.

late the Wagner numbers for macro throwing power in Figs. 26M and 27M. If we replace this by a typical roughness parameter of, say, 0.1 p.m, the

30.3 Micro Throwing Power

same curves will be obtained, but with the Wagner number multiplied by a

Scale is very important in electrode processes. In the case of macro throwing power the irregularities of the shape of the electrode are on the same scale as the cell itself. The distance between the anode and the cathode may be of the order of 10 cm, and the characteris-

factor of 2x10 5 , bringing it deep into the realm of secondary current distribution. There must be a different mechanism controlling micro throwing power and leveling. This is tertiary current distribution, which is

tic length used in Section 30.2 to calculate the Wagner numbers was

mass transport limited. The important parameter to consider in such

2 cm. When we are considering the appearance of the surface, particu-

cases is the ratio between the roughness parameter, or the amplitude of

larly its brightness, the scale of interest is of the order of magnitude

the roughness, and the thickness of the Nernst diffusion layer. The

of the wavelength of visible light. It follows from electromagnetic

latter grows initially with the square root of time and reaches steady

theory that the ratio between the light scattered from a surface and

state after a short time compared to the time of plating. During the

that reflected from it depends on (L/X) 2, where L is the amplitude of

initial stages, S is given by

the roughness and X is the wavelength of light. If (L/X) 2 approaches

5 = (rtDt) 1/2

(9D)

zero, one has specular reflection; that is, the surface reflects light like a mirror. As this ratio grows, the surface first looks dull and eventually becomes black as (LA) exceeds unity. Thus the scale of interest, from the point of view of brightness of electrodeposits, is of the order of X for visible light, namely in the range of 0.4-0.8 i_tm. The ability of a plating bath to form uniform coatings on this scale of roughness is called micro throwing power.

It is also referred to as

leveling. We can see intuitively that cell geometry has little to do with micro throwing power. If the amplitude of roughness is of the order of 0.1 p.m or less and the distance between the anode and the cathode is a few centimeters, the variation of solution resistance at crests and valleys on the surface must be negligible. Hence primary current

The value of S at steady state depends on the rate of stirring and/or agitation of the solution. Typical values may range from 10 to 50 [tm. Thus it will take well under a minute for steady state to be established — that is, for S determined by diffusion to exceed its steady state value, controlled by convection. A common observation in electroplating is that the roughness of the deposit increases with thickness. It is quite easy to produce a smooth deposit of 0.1 tm thickness, but keeping it smooth when the thickness has grown to 25 tm requires very special measures. Such an observation implies that a positive feedback mechanism is operative, with the local current density higher at protrusion than in recessed areas. It is easy to understand this behavior, if the plating process is assumed to be at least partially controlled by mass transport. We recall that the

J54

ELECTR (./ I )E. KINETICS

and producing a bright metal luster.

current density can be written as follows: _ nFD(C ° — C(s)) 6

M. APPLICATIONS

J

D

Although the properties of

additives differ widely, the mechanism by which they operate is common (72M)

The Nernst diffusion layer thickness is larger in a recessed area than at a crest, hence the local current density is smaller. As a result, recessed areas grow more slowly than crests, and the amplitude of roughness increases with time during plating. It is not difficult to see how a rough surface will grow even rougher by the foregoing mechanism, but how is roughness initiated? Experiments show that even when plating is conducted on a highly polished surface, the deposit will gradually increase in roughness. We may expect that plating on an atomically flat, single crystal-surface in a highly purified solution will not produce a rough deposit, but this is of little practical interest. A likely mechanism of roughening in real plating baths is the adhesion of foreign particles to the surface during plating. These contaminants may be dust particles, or solid grains of metal that fell from the anode during plating. We should remember in this context that impurity particles of micrometer dimensions, which are often difficult to filter out efficiently, are fairly large on the scale of importance here. Another mechanism of roughness initiation may be associated with the nonuniformity of the substrate. The activation-controlled current density at sites of inclusions (such as graphite or sulfur), at grain boundaries, and even at different crystal faces may be different, causing of uneven growth of the deposit. Needless to say that incomplete cleaning of the surface or residual patches of oxide left on it can cause uneven growth and lead to the formation of rough deposits. There is great commercial incentive to produce smooth and bright deposits. Consequently, there is a vast choice of additives for use to improve micro throwing power, making deposits smoother and more uniform,

and easy to understand. Molecules of the additive adsorbed on the surface prevent or inhibit metal deposition. To a first approximation it can be said that the rate of metal deposition is simply proportional to (1 — 0), where 0 is the fractional surface coverage by molecules of the additive. A more detailed analysis shows that adsorption on part of the surface has an effect on the rate of metal deposition on the bare sites, but this refinement need not concern us now. As a rule the concentration of the additive is small compared to that of the metal ion being plated. Consequently, the rate of adsorption is diffusion controlled. This helps to produce a smooth surface for the same reason that a rough surface is formed in the absence of a suitable additive. In recessed areas, the rate of mass transport by diffusion is lower than on protruding parts on the surface, and the fractional coverage is consequently lower. The current density is therefore higher in the recessed regions, and leveling occurs. Under favorable conditions this effect can actually be strong enough to reverse the trend, namely to yield a smooth deposit on an initially rough surface. The additive adsorbed on the surface may be buried as such. Alternatively, an additive is first reduced, whereupon fragments of it are buried under the layers of metal being deposited. Thus the additive is consumed in a plating bath and must be periodically replaced. The incorporation of foreign molecules in the metal deposit affects its mechanical properties, as well as its corrosion resistance and here the art of finding the right additive for each plating bath comes in. There is an optimum range of concentration over which each additive is most active. This is also easy to understand, in terms of the mechanism just discussed. At low concentrations, the activity of each

556

ELECTRODE KINETICS

557

M. APPLICATIONS

0 S 0.8, because this is the range in which a0/aC is

additive grows with concentration, because there just is not enough

where 0.2

material in solution to do the job: that is, coverage on the protruding

greatest, and we can expect to obtain the highest difference of adsorp-

areas cannot reach sufficiently high values to induce significant

tion on different regions on the surface. On the other hand, if the

leveling. In the best concentration range, coverage on protruding areas

adsorption isotherm is not known, it is probably easier to determine the

is high but in recessed areas it is relatively low, yielding the desired

optimum range of concentration directly than to measure the isotherm and

leveling effect. As the concentration of the additive in solution is

deduce the desired range of concentration from it.

increased, the coverage on protruding areas reaches a limiting value and

The choice of a good leveling agent depends, among other things, on

can grow no longer; the coverage on other areas keeps growing, however,

the position of the potential of zero charge with respect to the

until a high coverage is reached everywhere on the surface. The

potential at which deposition is taking place in a given bath. To

leveling effect of additives depends on the difference of coverage on

clarify this point, let us compare the deposition of lead to that of

different areas, caused by the different rates of diffusion. This does

zinc. The standard potentials for these metals are — 0.126 V and

not work at very low concentrations, when there is hardly any coverage

—0.763 V, NHE, respectively, and their potentials of zero charge are

anywhere. Neither does it work in concentrated solutions, where the

— 0.67 V and — 0.63 V on the same scale. Thus the standard potentials

coverage everywhere is at its saturation value during plating. The effect of stirring on the rate of adsorption can be utilized to

on the rational scale are 0.54 V for lead and — 0.13 V for zinc. The potential at which the metal is actually deposited depends on the

identify the optimum range of concentration for any particular system.

composition of the bath, but we may conclude that lead is probably

This can best be performed with the use of a rotating disc electrode,

deposited at a positive rational potential and zinc at a negative

for which the rate of rotation can be scanned linearly by controlling

rational potential. Noting that the potential of maximum adsorption

the voltage to the linear motor. If the concentration of the additive

occurs a little negative to E (cf. Section 21), we may conclude that

is below the optimum value, the measured current will decrease with

lead is deposited in a region where the adsorption of a neutral molecule

increasing rate of rotation, because the rate of supply of additive to

increases if the potential is made more negative.

the surface is higher, leading to increased values of 0. If the

favorable for leveling, because the potential is more negative during

concentration of the additive is too high, surface saturation is already

metal deposition where the local current density is higher.

reached at low rotation rates and increasing the rate of mass transport

other hand, zinc is deposited on the negative side of the maximum in the

will have the effect of increasing the current for metal reduction.

adsorption curve for neutral species.

This behavior is On the

Here adsorption of a neutral

Finally we might ask what determines the suitable range of concent-

additive will be lower where the current density is higher, a situation

ration of an additive. It is clear that the answer is different for

that is unfavorable for leveling. From this point of view, a positively

different additives and depends on the metal being deposited. If we

charged additive is better for both metals, but in the case of lead the

know the adsorption isotherm for the additive on the same metal, we

optimum concentration needed may be rather high, since a positive ion is

might guess that the optimum concentration for leveling is in the range

not readily adsorbed at positive rational potential unless its free

ELECTRODE KINETICS

energy of electrosorption is very high.

M. APPLICATK,

copious hydrogen evolution, but no detectable metal deposition. On the

We conclude this section by noting that, although the mechanism by

other hand, sodium and other alkali metals can be deposited on mercury

which different additives operate is fairly well understood, we have

from an alkaline solution, probably because of the very low exchange

certainly not reached the point at which the choice of an additive can

current density for hydrogen evolution on this metal, and because an

be based on its known molecular structure or even on measurement of its

amalgam is formed, so that the active surface is always mercury or its

adsorption isotherm under equilibrium conditions. Such knowledge can be used to advantage for preliminary screening and intelligent guessing,

amalgam, not the metal being deposited. When deposition from an aqueous solvent is impossible, one must

but it cannot substitute for some degree of trial and error in identify-

resort to nonaqueous systems. These present a number of technical

ing a good additive for a given purpose.

difficulties and have been used in practice only when there has been no alternative. With evolving technological development, it is anticipated

30.4 Plating from Nonaqueous Solutions Many metals can be plated from aqueous solutions, even though their

that plating from nonaqueous systems will nevertheless be adopted for commercial use, and a short discussion is therefore warranted.

reversible potential is cathodic to the region of stability of water

The most obvious way to proceed would seem to be with the use of an

(c.f Fig. 13M). Hydrogen evolution can occur in such cases as a side reaction, but as long as the faradaic efficiency is not too low, plating

appropriate molten salt. Magnesium can be deposited from anhydrous molten MgCl 2 , and aluminum can be deposited from a cryolite bath, since

can be conducted on an industrial scale. One of the important reasons

in these baths metal deposition is the only cathodic reaction that can

for this is that the exchange current density for metal deposition is

take place. The quality of the deposits in these baths is usually poor,

usually much higher than that for hydrogen evolution, with the result

however, and they are used for metal winning rather than electroplating.

that the rates of these reactions are comparable, even where the

Refractory metals, such as tantalum and zirconium, can be deposited

reversible potential for metal deposition is significantly more cathodic. In a cyanide bath the rate of metal deposition is slowed down (to

from their fluorides in a molten salt bath. In the case of zirconium, for example, the bath consists of ZrF 4 or ZrF26 in a KF/NaF/LiF mixture.

shifted cathodically, but the high concentration of (CN) ions at the

The alkali fluorides are employed to increase conductivity and decrease the melting point. Even so, these baths are operated at about 800 °C.

metal surface during plating lowers the rate of hydrogen evolution,

Good deposits have been reported as long as the right valency was chosen

allowing the process to occur at a reasonable faradaic efficiency.

for each metal (3 for Mo and V, 4 for Nb and Zr, 5 for Ta). The bath

Also, cyanide plating baths operate at high pH, and the reversible

must be operated in a pure argon atmosphere, and impurities must be

potential for hydrogen evolution is lowered. ChromiuM is deposited from

strictly excluded. It should be obvious that the operation of such

aqueous solutions with a faradaic efficiency that'can be as low as 15%.

baths is expensive and control is difficult. Thus their use is limited

More active metals, such as aluminum, titanium, and magnesium, cannot be

either for research purposes or for highly specialized applications,

deposited from an aqueous medium at all. An attempt to do so leads to

where cost is of secondary importance.

achieve better macro throwing power) and the potential of deposition is

560

ELECTRODE KINETICS

The search for a room temperature plating bath for aluminum has

M. APPLICATIONS

561

'Ma

to provide electrolytic conductivity.

This bath, operated at about

been conducted for many years, in view of the excellent corrosion

100°C, has excellent conductivity and good throwing power. Although it

resistance of this metal. An early technological success is the

has made some inroads to engineering applications, its widespread

so-called hydride bath, which consists of a solution of AlC1 3 and LiA1H4

application has been limited, probably because of the need to use an

3/LiAIH4 =inethrs.Alagxcofuminhlrdes(AIC1

expensive and dangerous metal-organic compound that ignites spontaneous-

7/1) and AIHC1 is believed to be formed in the following equilibrium: 2

ly in air. A third plating bath is based on the use of Al 2Br6 and KBr in

LiA1H + 3A1C1 4 3

4A1HC1 + LiC1 2

(73M)

The exact mechanism of metal deposition from this bath is not known, but there can be no doubt that the hydride plays a crucial role, since the bath cannot be operated after it has been depleted of LiA1H 4 , even if the concentration of AIC1 is kept constant. This technology has been 3 used on one or two occasions for highly specialized purposes, mainly for the production of aluminum mirrors for space missions. It has not gained

toluene, ethylbenzene, or similar aromatic solvents. The chlorides and bromides of aluminum are covalent compounds and are highly soluble in aromatic hydrocarbons. An ionic compound such as KBr is not soluble in an aromatic hydrocarbon but is readily dissolved in a solution containing Al2Br6 , forming a compound according to the equation Al Br + KBr 2 6

[K1- (Al Br ) 7

(74M)

widespread commercial application because it requires the use of a

We have not written an equilibrium here, unlike Eq. 73M, because KBr

highly flammable and toxic solvent as well as chemicals that are very

cannot exist in solution in an aromatic solvent. Ionization of this

sensitive to water and oxygen.

species still leaves us with a potassium ion, which is unstable in

The most successful molten salt plating systems are those employing a mixture of A1C1

and KCI, the so-called low temperature molten salts. 3 The actual melting point depends on composition and the bath can be

nonpolar solvents. The real ionic species must therefore be somewhat more complex. Experiments indicate that these are formed in the following equilibrium

operated in the range of 200-300 °C. Two anions can exist in this melt: A1C1

and Al 2C17. Their relative concentrations depend on the ratio of 4 the two salts used. The great advantage of this system is that it dissolves salts of other metals, such as titanium and manganese, and allows the deposition of alloys of these metals with aluminum. The

3 [K+ (Al2Br7)

[K2 (Al 2 BT7 )]

[K(Al 2Br2 ) 2 1

(75M)

The conductivity of solutions of Al 2Br6 and KBr in aromatic solvents behaves anomarously, increasing exponentially with increasing concentration of the electrolytes, as shown in Fig. 28M. This observa-

greatest disadvantage is that the bath must be operated under strictly

tion is not consistent with regular hydrodynamic movement of the ionic

anhydrous conditions, since AlC1 3 is highly hygroscopic, releasing HCI

species in a viscous fluid under the influence of the electric field

when in contact with water or humid air.

(often referred to as the Stokesian mechanism).

Another near-room-temperature bath for aluminum plating contains a metal-organic compound, Al(C 2 H5)3 dissolved in toluene, with AlC1 3 added

It is believed to be

due to a kind of hopping mechanism somewhat similar to that of proton conductivity in aqueous solutions. Whatever the mechanism may be, the

.702.

ELECTRODE KINETICS

M. APPLICATIONS

563

cannot be deposited from any polar solvent. For example, a solution of

Fig. 28M Variation of the molar conductivity of a solution of Al Br

6

and KBr in toluene, with the concentration of KAI 2 Br 7 . Data from Reger, Peled and Gileadi, J. Phys. Chem. 83, 873, (1979).

MO L A R CO NDU C TIVITY /S. c m 2 mo le - 1

LiC1 and A1C1 3 in acetonitrile or propylene carbonate yields a deposit of metallic lithium, but no aluminum, even though thermodynamically 10 2

aluminum should be deposited first. The reason evidently lies in the

10

kinetics of the process. In any polar solvent the energy of solvation of the small Al3+ ion is so high that the first step in the reaction

'

10 0

sequence — the removal of a single solvent molecule from the inner solvation shell — requires a very high energy of activation. It is only when the solvent is nonpolar that this process can proceed at a signifi-

jo -2 -3 10 -410 10-5

10-6 10 -4 10-3 10-2 10-i 10 0 10 1 CONCENTRATION [K(AI 2 Br7)]

cant rate. It is not easy to find a suitable nonaqueous electrolyte for plating aluminum, titanium, and other active metals. Operating such a bath may be even more difficult. First, water and often oxygen must be excluded. This can be done rather easily in continuous processes, such as plating a wire or a metal sheet. In most applications, however, electroplating is typically a batch process — parts are introduced and removed from the bath regularly. Whereas the technology to perform such operations exists, it is more expensive and much less convenient than operation in an open aqueous bath. Most nonaqueous solvents are either flammable or toxic or both. Most salts used to make up the bath are expensive, and some are quite unstable. Even a relatively inexpensive salt such as KBr can become very expensive when it must be very dry. A faradaic efficiency that is a little below 100%, which may be a minor irritation in aqueous solution, can turn out to be a major problem

conductivity is found to be much larger than would be expected in a nonpolar solvent, and it is this anomalous behavior that makes this system a promising plating bath for aluminum. It is interesting to note that all three aluminum plating baths discussed here employ a solvent of low polarity. In fact, aluminum

in nonaqueous media, because the side reactions can lead to the accumulation of products that are detrimental to the operation of the bath, to say nothing of the health of the operator. Waste disposal, a problem even in aqueous plating baths, can he much more costly in nonaqueous baths.

564

ELECTRODE KINETICS

These are some of the reasons for the failure of nonaqueous plating

BIBLIOGRAPHY

565

BIBLIOGRAPHY

baths to come into general use, in spite of some clear technological The list of review articles and books given below should be

advantages they can offer, in particular in making products that cannot be made by other means. There is little doubt, however, that exacting demands of emerging new technologies, accompanied by research and development in this field, will eventually lead to the introduction of nonaqueous plating technologies into industrial applications.

considered as primary Keferences. Many of them are "classical" reviews which represent the best source for the fundamental theory or practice of the specific subject concerned, even though they may have been published many years ago. Specific references to papers are not given here on purpose, since that is considered outside of the scope of this book. On the other hand, the review articles cited all contain long lists of references for those who wish to study the subject in greater depth. Chapter B 1.

S. Trasatti, The electrode potential, in Comprehensive Treatise of Electrochemistry, Vol. 1, J. O'M. Bockris, B. E. Conway and E. Yeager, editors, Plenum Press, New York, 1980, pp. 45-82.

2.

R. Parsons, The structure of the electrical double layer and its influence on the rates of electrode reactions, in Advances in Electrochemistry and Electrochemical Engineering, Vol. 1, P. Delahay, editor, Wiley-Interscience, New York, 1961, pp. 1-64.

3.

J. S. Newman, Electrochemical Systems, Prentice Hall, Englewood Cliffs, NJ, 1991, pp. 73-85.

Chapter C 1.

J. S. Newman, The fundamental principles of current distribution and mass transport in electrochemical cells, in Electroanalytical Chemistry — A Series of Advances, Vol. 6, A. J. Bard, editor, Marcel Dekker, New York, 1972, pp. 187-352.

2.

N. Ibl, Current distribution, in Comprehensive Treatise of Electrochemistry, Vol. 6, E. Yeager, J. O'M. Bockris, B. E. Conway and S. Sarangapany, editors, Plenum Press, New York, 1983, pp. 239-316.

:

KINETICS

3.

J. S. Newman, Electrochemical Systems, Prentice Hall, Englewood Cliffs, NJ, 1991, pp. 378-396.

4.

H. Angerstein-Kozlowska, Surfaces, cells and solutions for kinetic studies, in Comprehensive Treatise of Electrochemistry, Vol. 9, E.

B 1B L1UG

Chapter E 1.

L. I. Krishtalik, Kinetics of electrochemical reactions at metalsolution interfaces, in Comprehensive Treatise of Electrochemistry, Vol.7,BECnwayJO'M.ockris,EYeagSU.MKhnd

Yeager, J. O'M. Bockris, B. E. Conway and S. Sarangapany, editors, Plenum Press, New York, 1984, pp. 15-61. Chapter D

R. E. White, editors, Plenum Press, New York, 1983, pp. 87-172. B. G. Levich, Present state of the theory of oxidation-reduction in solution (bulk and electrode reactions), in Advances in Electro-

1.

chemistry and Electrochemical Engineering, Vol. 4, P. Delahay,

2.

A. C. Riddiford, The rotating disk system, in Advances in Electrochemistry and Electrochemical Engineering, Vol. 4, P. Delahay, editor, Wiley-Interscience, New York, 1966, pp. 47-116.

2.

3. 4.

V. Yu. Filinovsky and Yu. V. Pleskov, Rotating ring and ring-disk electrodes, in Comprehensive Treatise of Electrochemistry, Vol. 9, E. Yeager, J. O'M. Bockris, B. E. Conway and S. Sarangapany, editors, Plenum Press, New York, 1984, pp. 293-352. W. J. Albery, Ring-Disk Electrodes, Clarendon Press, Oxford, 1971.

S. L. Marchiano and A. L. Arvia, Diffusion in the absence of convection: steady state and non-steady state, in Comprehensive Treatise of Electrochemistry, Vol. 6, E. Yeager, J. O'M. Bockris, B. E. Conway and S. Sarangapany, editors, Plenum Press, New York, 1983, pp. 65-132.

6.

7.

chemistry, Vol. 7, B. E. Conway, J. O'M. Bockris, E. Yeager, S. U.

M. Khan and R. E. White, editors, Plenum Press, New York, 1983, 4.

N. Ibl, Fundamentals of transport phenomena in electrolytic systems, in Comprehensive Treatise of Electrochemistry, Vol. 6 E. Yeager, J. O'M. Bockris, B. E. Conway and S. Sarangapany, editors, Plenum Press, New York, 1983, pp. 1-64.

5.

3.

J. Ktita, Polarography, in Comprehensive Treatise of Electrochemistry, Vol. 8, R. E. White, J. O'M. Bockris, B. E. Conway and E. Yeager, editors, Plenum Press, New York, 1984, pp. 249-338, D. E. Smith, AC polarography and related techniques, in Electroanalytical Chemistry — A Series of Advances, Vol. 1, A. J. Bard, editor, Marcel Dekker, New York, 1966, pp. 1-156.

editor, Wiley-Interscience, New York, 1966, pp. 249-372. R. R. Dogonadze and A. M. Kuznetsov, Quantum electrochemical kinetics: continuum theory, in Comprehensive Treatise of Electro-

pp. 87-172. S. U. M. Khan, Some fundamental aspects of electrode processes, in Modern Aspects of Electrochemistry, Vol. 15, R. E. White, J. O'M. Bockris, and B. E. Conway, editors, Plenum Press, New York, 1983,

5.

pp. 305-350. J. Albery, Electrode Kinetics, Clarendon Press, Oxford, 1975, pp. 92-124.

Chapter F 1. A. N. Frumkin, Hydrogen overvoltage and adsorption phenomena: Part I: Mercury, in Advances in Electrochemistry and Electrochemical Engineering, P. Delahay, editor, Vol. 1, Wiley-Interscience, New

York, 1961, pp. 65-122. 2. A. N. Frumkin, Hydrogen overvoltage and adsorption phenomena: Part II: Solid metals, in Advances in Electrochemistry and Electrochemical Engineering, Vol. 3, P. Delahay, editor, WileyInterscience, New York, 1963, pp. 287-392.

568

ELECTRODE KINETICS

569

BIBLIOGRAPHY

fr.;

3.

4.

L. I. Krishtalik, Hydrogen overvoltage and adsorption phenomena:

Plenum Press, New York, 1974, pp. 369-470.

Part III: Effect of the adsorption energy of hydrogen on over-

11. K. Kinoshita, Small-particle effects and structural considerations

voltage and the mechanism of the cathodic process, in Advances in

for electrocatalysis, in Modern Aspects of Electrochemistry,

Electrochemistry and Electrochemical Engineering, P. Delahay,

Vol. 14, J. O'M. Bockris, B. E. Conway and R. E. White, editors,

editor, Vol. 7, Wiley-Interscience, New York, 1963, pp. 283-340.

Plenum Press, New York, 1982, pp. 557-638.

M. Enyo, Hydrogen electrode reaction on electrocatalytically active

12. J. Albery, Electrode Kinetics, Clarendon Press, Oxford, 1975,

metals, in Comprehensive Treatise of Electrochemistry, Vol. 7, J.

pp. 125-163.

O'M. Bockris, B. E. Conway, E. Yeager, and R. E. White, editors, 5.

6.

7.

8.

Plenum Press, New York, 1983, pp. 241-300.

Chapter G

P. K. Subramanian, Electrochemical aspects of hydrogen in metals,

1.

double-Layer theory, in Electroanalytical Chemistry — A Series of

Bockris, B. E. Conway, E. Yeager, and R. E. White, editors, Plenum

Advances, Vol. 1, A. J. Bard, editor, Marcel Dekker, New York,

Press, New York, 1981, pp. 441-462.

1966, pp. 241-410.

M. Breiter, Some problems in the study of oxygen overvoltage, in

2.

R. Parsons, The structure of the electrical double layer and its

Advances in Electrochemistry and Electrochemical Engineering,

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D/4

6.

ELEC1 k ,;

N.,

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KINETICS

Corrosion

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8.

9.

N. Marincic, Lithium batteries with liquid depolarizers, in Modem Aspects of Electrochemistry, Vol. 15, J. O'M. Bockris, and B. E. Conway, editors, Plenum Press, New York, 1983, pp. 167-234. J. McBreen, Secondary batteries — introduction, in Comprehensive Treatise of Electrochemistry, Vol. 3, J. O'M. Bockris, B. E. Con-

Solutions, Pergamon Press, Oxford, 1966, (also available from

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10. F. von Sturm, Secondary batteries, nickel/cadmium battery, in Comprehensive Treatise of Electrochemistry, Vol. 3, J. O'M. Bockris, B.

6.

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E. White, editors, Plenum Press, New York, 1981, pp. 97-150.

11. K. V. Kordesch, Power sources for electric vehicles, in Modem Aspects of Electrochemistry, Vol. 10 J. O'M. Bockris and B. E.

7.

I. Epelboin, C. Gabrielli, M. Keddam and H. Takenouti, The study of

Conway, editors, Plenum Press, New York, 1975, pp. 339-444.

the passivation process by the electrode impedance analysis, in Comprehensive Treatise of Electrochemistry, Vol. 4, J. O'M.

12. K. Kinoshita and P. Stonehart, Preparation and characterization of

Bockris, B. E. Conway, E. Yeager and R. E. White, editors, Plenum

highly dispersed electrocatalytic materials, in Modem Aspects of Electrochemistry, Vol. 12, J. O'M. Bockris, and B. E. Conway,

8.

editors, Plenum Press, New York, 1977, pp. 183-266. 13. K. Kinoshita, Small-particle effects and structural considerations for electrocatalysis, in Modem Aspects of Electrochemistry,

9. Vol.14,JO'MBckrisE.ConwaydRWhite,Eors

pp. 557-638, Plenum Press, New York, 1982 Vol. 14.

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FIFCTRODE KINETICS

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Electroplating

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pp. 67-148. M. Fleishmann and H. R. Thirsk, Metal deposition and electrocrys-

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1.

577

K. M. Gorbunova and Yu. M. Polukarov, Electrodeposition of alloys, in Advances in Electrochemistry and Electrochemical Engineering, Vol. 5, P. Delahay, editor, Wiley-Interscience, New York, 1963, pp. 205-248.

5.

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6.

chemistry, Vol. 2, J. O'M. Bockris, B. E. Conway, E. Yeager, and R. E. White, editors, Plenum Press, New York, 1980, pp. 381-398. F. Beck, Electrodeposition of paint, in Comprehensive Treatise of

10. A. M. Pesco and H. Y. Cheh, Theory and applications of periodic electrolysis, in Modern Aspects of Electrochemistry, Vol. 19, J. O'M. Bockris and B. E. Conway, editors, Plenum Press, New York, 1989, pp. 251-294.

LIST OF ACRONYMS

LIST OF ACRONYMS BDM

Bockris, Devanathan and Muller

C.E.

counter electrode

DME

dropping mercury electrode

EIS

electrochemical impedance spectroscopy

ETE

electric-to-electric (efficiency)

F.E.

faradaic efficiency

FFTIR

fast Fourier transform infrared (spectroscopy)

h.e.r.

hydrogen evolution reaction

HTSO

high-temperature solid-oxide (fuel cell)

ImZ

"imaginary" component of the impedance

1HP

inner Helmholtz plane

MNHE

modified normal hydrogen electrode

NHE

normal hydrogen electrode

NCE

normal calomel electrode

0I-IP

outer Helmholtz plane

PAFC

phosphoric acid fuel cell

PZC

potential of zero charge

RAZE

reversible aluminum electrode

RDE

rotating-disc electrode

R.E.

reference electrode

ReZ

"rear component of the impedance

RI-fE

reversible hydrogen electrode

RRDE

rotating-ring-disc electrode

rds

rate-determining step

SCE

saturated calomel electrode

SEI

solid electrolyte interphase

SPE

solid-polymer electrolyte

STM

scanning tunneling microscopy

T.P.

throwing power

UPD

underpolential deposition

W.E.

working electrode

LIST OF SYMBOLS

LIST OF SYMBOLS A

surface area (cm 2)

A

chemical affinity (J/mol)

A

Helmholtz free energy (J/mol)

a, a b

activity of the i th species in solution Tafel constant (mV) Tafel slope (mV/decade)

b;b a

C

c

dl M2 2S

C

o

CI

Tafel slopes for the anodic and the cathodic process double-layer capacitance (.1F/cm 2) capacitance of the Helmholtz double layer capacitance of the diffuse double layer double-layer capacitance at 0 = 0 double-layer capacitance at 0 = 1 adsorption pseudocapacitance (µF/cm 2)

CL

adsorption pseudocapacitance derived from the Langmuir isotherm

C

F

adsorption pseudocapacitance derived from the Frumkin isotherm

CT

adsorption pseudocapacitance derived from the Temkin isotherm

I

C°

bulk concentration (mol/cm 3)

C(s)

concentration at the electrode surface (mol/cm 3)

C(sat)

saturation concentration

C,

concentration of the i th species

C,(t,x)

concentration at time t at a distance

x from the

surface diffusion coefficient (cm 2/s) effective distance between the working electrode and the tip of the Luggin capillary leading to the reference electrode

ELECTRODE KINETICS

E

potential (V)

E° E

standard potential breakdown potential (of passive film)

b

Corr

E

pp

rev E

standard potential for a surface species formed by charge transfer.

1/4

E lt2 E E e

quarter-wave potential in chronopotentiometry polarographic half-wave potential peak potential, in cyclic voltammetry potential of zero charge (PZC)

z

E

J

standard electrochemical enthalpy of activation

AH°4

T(i)

total current (A) mass-transport-limited current imaginary component of the impedance (0.cm 2) instantaneous information content for i 0 (dimensionless) average information content for i o

i

current density (A/cm 2) anodic and cathodic current density

I IL ImZ

repassivation potential

T E 0 E

corrosion potential (also called mixed potential) primary passivation potential reversible potential

LIS

N

s4

F

,

c

a

activation-controlled current density corrosion current density (at open circuit)

ac Corr

crit

diffusion-limited current density

ld

rational potential ( = E — E )

double-layer-charging current density total current on the disc (in RRDE configuration)

shift in E resulting from change in 0 from zero to unity electron in the metal

mass-transport-limited current density total current on the ring (in RRDE configuration)

di

exchange current density peak current density, in cyclic voltammetry faradaic current density passed in the formation of

Faraday's constant (96,484.6 C) O

F

field in the compact double layer (V/cm)

f

rate of change of the free energy of adsorption with coverage, in dimensionless form ( = r/RT)

AG

change in Gibbs free energy (J/mol)

AG°

change in standard free energy

1F

an adsorbed intermediate

0#AG

standard free energy of activation

K.

°4 Aa

standard electrochemical free energy of activation standard free energy of adsorption

kB kf

AG °

standard free energy of adsorption at 0 = 0

kh

AG °

k°

AH

standard free energy of adsorption at a coverage 0 acceleration due to gravity (9.86 m/s 2) enthalpy change (J/mol)

AH°#

standard enthalpy of activation

° AG

ads

critical current density for passivation

k

s,h

(- 1 ) 14 equilibrium constant for the activated complex equilibrium constant in the i th step in a reaction sequence Boltzmann's constant (1.38x10 23 J/deg) rate constant for the forward reaction potential-dependent heterogeneous rate constant (cm/s) the value of khat 04) = 0 rate constant for the i-th step in a reaction sequence standard heterogeneous rate constant (at E = E°)

584

ELEC. I RODE KINETICS

distance between the centers of two adjacent microelectrodes

L L

LIST OF SYMBOLS

R

s

residual (uncompensated) solution resistance (S•cm 2)

in an ensemble of microelectrodes

faradaic resistance in series with adsorption

characteristic length in the calculation of the Wagner number

pseudocapacitance (S2•cm 2)

or 94 for current distribution

Re

Reynolds number (dimensionless)

characteristic length for the calculation of the

ReZ

real component of the impedance (•cm 2)

Reynolds number, Re

r

rate of change of the free energy of adsorption

WA

M

mass of a drop of liquid (e.g. mercury in a DME)

m

rate of flow of mercury in a DME (mg/sec)

S

entropy (J/mol•deg)

N

rotation rate in rpm (for RDE)

AS °It

standard entropy of activation

N

collection efficiency (for RRDE studies)

S

hydrogen-deuterium separation factor

N''; N4'

number of water molecules (per cm 2) in the "up" and '!down"

S

position in the double-layer

T T

with coverage (J/mol)

EAD

Hrr

hydrogen-tritium separation factor temperature (degree, absolute scale)

n

number of electrons taking part in the overall reaction

n

number of water molecules replaced from the surface for

time (seconds)

each organic molecule adsorbed

duration of pulse

pressure (atm)

P

partial pressure of the i-th species

U U

Q

gm gs go gl q1 qF

potential energy (J) electrostatic interaction of the water dipoles ll

ratio of concentrations of products and reactants in a chemical reaction (Q = U C . )

period of time over which data are collected

with the field in the double layer V

volume

charge (coulombs)

reaction rate (mol/s or mol/cm 2.$)

charge density on the metal side of the interface (µC/cm2)

linear velocity (cm/s)

charge density on the solution side of the interface

potential sweep rate (V/s)

charge density at 0 = 0

eq

charge density at 0 = 1 faradaic charge required to form a monolayer (= 230 µC/cm 2)

exchange rate (rate at which reaction proceeds back and forth at equilibrium)

V r ,V z ,V 0

faradaic charge during formation of adsorbed species

radial, perpendicular and tangential velocities at the surface of an RDE or RRDE.

qp R

charge passed during a pulse

W

mechanical work (J)

faradaic resistance (acm 2)

W

Warburg impedance

R

polarization resistance (used in corrosion studies to

We

Wagner number

F

P

signify the faradaic resistance)

Modified Wagner number

5 So

ELECTRODE KINETICS

X#

the activated complex

Xi X

mole fraction of the i-th species

IZI Z(co)

8

ratio of concentrations of reactant and products at the electrode surface during current flow

0

fractional coverage (dimensionless)

absolute value of the impedance vector (•cm 2)

0

total coverage by different adsorbed species

T K

specific conductivity (S/cm) Debye reciprocal length (cm -1 )

K

formal charge on a species (dimensionless)

la dimensionless rate constant [ = k h (t/D) ] chemical potential of the i th species in

transfer coefficient (dimensionless)

the j th phase (J/mol) standard chemical potential of the i-th species

Warburg impedance (•cm 2)

a

a;

e

dimensionless distance [ = x/(4D0 112] frequency-dependent impedance of the interface impedance of a capacitor

Z c Z w z.

LIST OF SYMBOLS

c

anodic and cathodic transfer coefficients

electrochemical potential of the i-th species

;

surface concentration or surface excess (mol/cm 2)

V

dipole moment kinematic viscosity (= fl/p; cm2/s)

maximum surface concentration relative surface excess

V

stoichiometric number (dimensionless)

symmetry factor (dimensionless)

max

7,

max

?

A v2

surface pressure (N/m or dyne/cm) C

activity coefficient of the i-th species surface tension, (N/m or dyne/cm) surface tension at the electrocapillary maximum finite difference (e.g., AG) the Laplacian operator

P,

P2 p(x)

S S

Nernst diffusion layer thickness (cm)

a C

partial derivative dielectric constant ti

T1

overpotential (V) viscosity (poise)

t ax

activation overpotential

ti ti

1R

concentration overpotential resistance overpotential

small but finite increment (e.g., SE)

a

product of concentrations density (g/cm3) reaction order at constant potential (dimensionless) reaction order at constant overpotential volume charge density, at a distance x from the interface (.tC/cm 3) lateral interaction parameter in the BDM theory of electrosorption of organic molecules concentration-dependent parameter of the Warburg impedance

T

c d

4)

relaxation time for charge transfer (seconds) relaxation time for diffusion drop-time (of mercury in polarography) transition time in chronopotentiometry inner potential of a phase (V) potential in the bulk of the solution

588

(134 x

4) 1 4/ 2 j Ai(1) A41 rev

ELECTRODE KINETICS

potential at a distance x from the interface

SUBJECT INDEX

potential at the inner Helmholtz plane

Absolute rate theory, 109 Activated complex, 110 Activation overpotential, 106

potential at the outer Helmholtz plane potential difference between two phases

Active coating, 536 activity, 16 Activity coefficient, 16, I 1 1

absolute metal-solution potential difference at the reversible potential phase angle between the real and the imaginary

(1)

Adsorption-desorption peaks, 258

dimensionless rate constant calculated for an expanding mercury drop [ = (12/7) In(t/D) lnk

Adsorbed impurity, 157 Adsorbed intermediates, 177 Adsorption, 225

components of the electrochemical impedance X

SUBJECT INDEX

Adsorption energy, 182 Adsorption isotherm, 179, 228, 261

h

angular velocity (rad/s)

Adsorption of ions, 252 Adsorption of neutral molecules, 184, 257

preexponential frequency term in the absolute rate theory

Adsorption pseudocapacitance, 291, 296, 299, 375

]

Adsorption, rate of, 158 Affinity, 102 Alkaline batteries, 464 Anodic protection, 534 Argand plot, 215 Arrhenius plot, 152 Auxiliary reference electrode, 38 Average information content, 370 Barrier coating, 536 Batteries, 455, 460 BDM isotherm, for adsorption of neutral species, 335, 339 Bimetallic corrosion, 526 Binary alloys, 181, 179 bipolar configuration, 482 Bode magnitude plots, 295 Boltzmann equation, 191 Boundary conditions (for solving the diffusion equation), 376 Breakdown potential (for passive layer), 514 Capacitive loop, 443 Capillary depression, 242 Capillary rise, 241 Catalytic activity, 179, 182 Cathodic protection, 172, 528, 530 Cell geometry, 37 Charge density on the surface, 185, 189, 193 Charge injection method, 362

ELECTRODE KINETICS

SUBJECT INDEX

Charge transfer, 4, 55

Double-layer capacitance, measurement of, 213, 216, 222

Chemical passivation, 513

Double-layer capacitance on single crystal gold, 187

Chlorine evolution, 131, 134

Double-layer capacitance on mercury, 187

Chronopotentiometry, 386, 396, 398 Co-generation, 484

Double-layer charging current, 405

Cole-tole plot, 215 Collection efficiency, 95 Combined adsorption isotherm of gileadi, 340, 342, 344 Combined isotherm, application to electrode kinetics, 344, 347 Complex-plane admittance plot, 215 Complex-plane capacitance plot, 215 Complex-plane impedance plot, 215 Concentration overpotential, 108

Double potentiostat, 94 Double-pulse galvanostatic transient, 359 Driving force, 15 Dropping mercury electrode, 72, 155, 162 Drop-time method, of measuring surface tension, 248 Electrocapillarity, 225 Electrocapillary curve, 241

Constant-phase element, 438

Electrocapillary curve for KBr, 254 Electrocapillary curve for thallium nitrate, 254

Contact adsorption, 201

Electrocapillary electrometer, 241

Convection, 351

Electrocapillary equation, 230, 238 Electrocapillary equation, reversible interphase, 238

Copper electroplating, 211 Corrosion, 494 Corrosion, economics of, 490 Corrosion, electrochemistry of, 492 Corrosion inhibitors, 536 Corrosion protection, 526 Coulostatic method, 362 Coverage, measurement of, 322, 325 Crevice corrosion, 520, 524 Critical corrosion current, 514

Electrocapillarity maximum, 232 Eiectrocatalysis, 284 Electrochemical energy conversion device, 477 Electrochemical impedance spectroscopy, 213, 428, 434, 439 Electrochemical passivation, 513 Electrochemical potential, 16 Electrochemical rate equation (single step), 111

Crystallization overpotential, 283

Electrochemical timer, 12 Electron spin resonance, 179 Electrosorption, 159, 258, 307, 309

Current distribution, 46, 91

Electrosorption, of pyridine on mercury, 312

Cyclic voliammetry, 289, 403, 410, 416, 420 Cylindrical configuration, 40

Electrosorption, of n-decylamine on nickel, 313

Debye reciprocal length, 226

Electrosorption, of methanol on platinum, 316 Electrosorption, of naphthalene on gold, 317

Depressed semicircle, (in EIS), 440 Differential aeration, 525

Electrostriction, 202 Energy density, (of batteries), 456

Diffuse double layer, 225 Diffuse double layer theory, 190, 193, 206, 318

Energy of activation, 152 Enhancement of diffusion (at microelectrodes), 445

Diffuse double layer correction in electrode kinetics, 205, 210 Diffusion equation, 351, 376, 410

Ensembles of microelectrodes, 448

Diffusion layer thickness, 5, 57, 98, 352 Diffusion limitation, 436

Equivalent circuit, 10, 68, 186, 293

Diffusion limited current density, 73,353 Dimensionless representation, 59 Dimensionless rate constant, 77, 163, 385 Discharge curves (of batteries), 461 Dissociative adsorption, 264, 315 Double-layer capacitance, 7, 185

Electrosorption, of ethylene on platinum, 314

Enthalpy of activation, 151 Evans diagram, 495 Excess surface charge density, 185, 189, 193 Excess surface free energy, 229 Exchange current density, 103, 114, 183 Exchange rate, 103

592

ELECTRODE KINETICS

SUBJECT INDEX

Faradaic efficiency, 154 Faradaic resistance, 11, 106, 185

Impurity, of electroactive species, 155

Fast fourier transform infrared, 179

Indicator electrodes, 37

Fast transients, 38, 64, 349, 349 Fractional coverage, measurement of, 322, 235 Free energy of activation, 53, 110

Inductive loop, 443 Information content, 366, 369

Free energy of adsorption, dependence on coverage, 277 Frumkin isotherm, for adsorption of neutral species, 332, 339 Frumkin isotherm, for adsorption with charge transfer, 266, 267, 298 Fuel cells, 455, 460, 476, 504 Galvanic corrosion, 526 Galvanostatic measurements, 61 Galvanostatic transients, 66, 357, 359, 394 Gas-diffusion electrodes, 484 Gauss theorem, 192, 339

Impurity, rate of adsorption of, 157

Inhibitors, 536 Inner Helmholtz plane, 196 Inner potential, 17 Instantaneous information content, 370 Interface, 7 Intermediates in electrode reactions, 261 Interphase, 6, 225 Iodide, oxidation of, 167 Ionic double layer capacitance, 185, 213 Ionic hydration, 200

Gibbs adsorption isotherm, 228

Iron-titanium alloy, 170 Isothermal enthalpy of activation, 153

Gibbs-Duhem equation, 235

Isotherms, for large species, 329

Gileadi combined adsorption isotherm, 340, 342, 344

Isotope effects, 148

Gileadi isotherm, application to electrode kinetics, 344, 347 Gouy-Chapman theory, 190, 193, 200 Graphical representation, (of impedance data), 431

Kinematic viscosity, 83 Kinetic parameters, 173 Koutecky correction, 77

Half-wave potential, (polarography), 73, 74, 382 Helmholtz model (of the double layer), 188, 200 High temperature solid electrolyte, 174

Laminar flow, 82 Langmuir isotherm, 179, 261, 264, 296, 331

High temperature solid oxide fuel cell, 481

Lanthanum-nickel alloy, 170

Hydrodynamic layer thickness, (RDE), 98

Laplacian operator, 377

Hydrogen adsorption, 263

Large amplitude transients, 374

Hydrogen absorption in metals, 169

Lead-acid battery, 470

Hydrogen embrittlement, 171, 169

Leclanche cells, 458, 462

Hydrogen evolution, 146, 149

Levich equation, 85 Limitations of the linear potential sweep method, 414

Hydrogen evolution on mercury, 123, 161, 209 Hydrogen evolution on platinum, 164

Limiting current density, 60

Hydrogen oxidation, 356

Linear current-potential region, 103, 116

Hydrogen storage, 169

Linear potential sweep, 69, 403,414

Hydroxylamine, reduction of, 76

Linear response (to a perturbation), 354 Linear Tafel region, 350, 384, 350, 384, 413, 426

Ideally polarizable interphase, 186

Lithium batteries, 465

Ilkovic equation, 73

Lithium-iodine solid-state battery, 469

Immunity (of metals to corrosion), 506

Lockin amplifier, 428

Impedance, real and imaginary, 215, 430 Impedance vector, 213

Luggin capillary, 40, 45

Impressed current cathodic protection, 530

Mass transport, 4, 55, 350

Impurity, adsorbed on the electrode, 156 Impurity, allowed level of, 155, 158

Maxwell equations, 238, 350, Mercury content (of primary batteries), 464

ELECTRODE K iNETICS

SUBJECT INDEX

Metal-air batteries, 484

Pit initiation, 521

Microelectrodes, 158, 353, 443, 445, 448

Pitting corrosion, 520

Micropolarization, 116, 150, 167, 499

Planar configuration, 39

Migration, 350

Poisson equation, 191

Mixed control, 87

Polarity of batteries, 488

Mixed potential, 494

Polarizable interphase, 9

Mode of failure (of batteries), 473

Polarography, 72, 80

Modified normal hydrogen electrode scale, 30

Polarographic half-wave potential, 74, 382

Mole fraction, 235, 330

Porous electrodes, 484

Molten salt bath 174

Positive feedback, 247

Monolayer adsorption, 261, 280, 312, 420,

Potential dependence of b, 125

Multi-step electrode reactions, 130

Potential of maximum adsorption, 336 Potential of zero charge, 160, 318, 537

Negative feedback, 246

Potential/pH diagram, 502

Nernst diffusion layer, 5, 57, 86, 98, 352, 451

Potential/pH diagram, advantages and limitations of, 512

Nernst equation, 8, 21, 101, 138, 147, 381

Potential step, 379

Nickel-cadmium batteries,473

Potentiostatic measurements, 61

Nonisothermal enthalpy of activation, 153

Potentiostatic transients, 67, 390

Nonpolarizable interphase, 8 33

Pourbaix diagrams, 503, 512

Normal hydrogen electrode scale, 27

Power density (of batteries), 458

Numerical value of b, 123

Practical scale of potential (for galvanic corrosion), 527

Nyquist plot, 215

Preelectrolysis, 160 Primary batteries, 458, 462

Open-circuit decay of potential, 363, 374

Primary current distribution, 47

Outer Helmholtz plane, 196

Primary passivation potential, 514

Overpotential, 101, 106

Pseudocapacitance, 291, 296, 299, 375

Oxygen evolution, 172

Purity (of solutions), 155

Oxygen, solubility in water, 155 Oxygen reduction, 175

Quarter-wave potential, 388 Quasi-equilibrium, 131,

Parallel-plate model (for adsorption isotherm), 332

Quasi-zero order kinetics, 142,

Parallel-plate model (of the double layer), 188 Partially blocked electrode, 450

Radial velocity,(RDE), 96

Partial surface coverage, 131

Radiotracer method, 323

Passivation, 506

Rate constant, dimensionless, 77, 163, 385

Passivation and its breakdown, 513

Rate-determining step, 131

Peak current density, determination of, 418, 421

Rate of adsorption, 158

Perpendicular velocity, (RDE), 96

Rational potential, 207

Phase formation, two dimensional, 304

Reaction order, 140

Phase-sensitive voltmeter, 213, 428

Reciprocal Debye length, 226

Phase shift, 213

Recycling, 458

pH - effect on reaction rates, 144

Redox reactions, 26

p1-1 - in different solvents, 145

Redox electrodes, 24

Phosphoric acid fuel cell, 478

Reduced carbon dioxide, 178

Phthalocyanine, 176

Reduction of resistance (at microelectrodes), 447

Pickling, 172

Reduction of anions, 216

596

ELECTRODE KINETICS

SUBJECT INDEX

Reference electrodes, 37

Surface pressure, 229

Relative surface excess, 236

Surface tension, 72, 229 Symmetry factor, 53, 114, 120,

Relaxation time, 358, 366, 369

127

Symmetry factor, potential dependence, 162

Repassivation potential, 514 Resistance overpotential, 107 Reverse-pulse techniques, 397

Tafel equation, 108, 118,

Reverse-step voltammetry, 400

Tafel slope, 108, 118, 128, 134, 273

Reversibility, 78

Temkin isotherm, 266, 271, 273

Reversible interphase, electrocapillary equation of, 238

Thin-layer cell, 157, 167

Reversible region, 412

Three-electrode measurement, 34

Reynolds number, 83

Titanium, reduction of, 417

Rotating cone electrode, 92

Totally irreversible case, 384

Rotating cylinder electrode, 93

Transfer coefficient, 127, 147, 174 Transients, 38, 64, 349, 349, 349, 349, 354, 357, 359, 374

Rotating disc electrode, 82

Transition time (in chronopotentiometry) 386

Rotating ring-disc electrode. 83. 93

Turbulent flow, 84

Rotating ring electrode, 92 Roughness factor, 299

Underpotential deposition, 280, 285, 287, Ultramicro electrodes, 353, 408, 449

Sacrificial anodes, 528 Sand equation, 388

Uncompensated solution resistance, 39, 419

Scanning tunneling microscopy, 283

Unstable passivity, 518

Secondary batteries, 458, 470 Valve metals, 175

Secondary current distribution, 47 Semi-infinite linear diffusion, 377 Separation factor, 148, 167

Warburg impedance, 352, 436

Service life (of batteries), 472

Water electrolyzer, 504

Small amplitude transients, 354

Water replacement model, for adsorption of neutral species, 335

Solid electrolyte interphase, 56, 466 Young-Laplace equation, 241

Solid polymer electrolyte, 477 Solvent, role of in the intephase, 200 Solution resistance, 10, 33, 35, 39, 66,

185, 419

Specific adsorption, 198, 200 Specific adsorption of cations, 253 Spherical configuration, 41 Standard free energy of adsorption, 262 Standard potential, 139, 297 State-of-charge meter (for lithium batteries), 467 Steady state, 131 Stern model (of the double layer), 195 Stoichiometric number, 149 Strip microelectrode, 453 Supporting electrolyte, 208, 351 Surface concentration, 131 Surface excess, 225, 229 Surface excess of anions, 256 Surface excess, relative, 236