Brett Ch., Brett A. Electrochemistry.. principles, methods, and applications (Oxford, 1994)(T)(444s).pdf

ELECTROCHEMISTRY Principles, Methods, and Applications CHRISTOPHER M. A. BRETT and ANA MARIA OLIVEIRA BRETT Departamento de Quimica, Universidade de Coimbra, Portugal Oxford New York Tokyo OXFORD UNIVERSITY P...

ELECTROCHEMISTRY Principles, Methods, and Applications CHRISTOPHER M. A. BRETT and ANA MARIA OLIVEIRA BRETT Departamento de Quimica, Universidade de Coimbra, Portugal Oxford New York Tokyo OXFORD UNIVERSITY PRESS  Oxford University Press, Walton Street, Oxford OX2 6DP Oxford New York Athens Auckland Bangkok Bombay Calcutta Cape Town Dar es Salaam Delhi Florence Hong Kong Istanbul Karachi Kuala Lumpur Madras Madrid Melbourne Mexico City Nairobi Paris Singapore Taipei Tokyo Toronto and associated companies in Berlin Ibadan Oxford is a trade mark of Oxford University Press Published in the United States by Oxford University Press Inc., New York © Christopher M. A. Brett and Ana Maria Oliveira Brett, 1993 First published 1993 Reprinted 1994 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press. Within the UK, exceptions are allowed in respect of any fair dealing for the purpose of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act, 1988, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms and in other countries should be sent to the Rights Department, Oxford University Press, at the address above. This book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold, hired out, or otherwise circulated without the publisher's prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser. A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data Brett, Christopher M. A. Electrochemistry: principles, methods, and applications/ Christopher M. A. Brett and Ana Maria Oliveira Brett. Includes bibliographical references. 1. Electrochemistry. I. Brett, Ana Maria Oliveira. II. Title. QD553.B74 1993 541.3'7-dc20 92-29087 ISBN 0 19 855389 7 (Hbk) ISBN 0 19 855388 9 (Pbk) Printed in Great Britain by Bookcraft (Bath) Ltd., Midsomer Norton, Avon  PREFACE Electrochemistry has undergone significant transformations in the last few decades. It is not now the province of academics interested only in measuring thermodynamic properties of solutions or of industrialists using electrolysis or manufacturing batteries, with a huge gulf between them. It has become clear that these two, apparently distinct subjects, and others, have a common ground and they have grown towards each other, particularly as a result of research into the rates of electrochemical processes. Such an evolution is due to a number of factors, but principally the possibility of carrying out reproducible, dynamic experi- ments under an ever-increasing variety of conditions with reliable and sensitive instrumentation. This has enabled many studies of a fundamen- tal and applied nature to be carried out. The reasons for this book are twofold. First to show the all-pervasive and interdisciplinary nature of electrochemistry, and particularly of electrode reactions, through a description of modern electrochemistry. Secondly to show to the student and the non-specialist that this subject is not separated from the rest of chemistry, and how he or she can use it. Unfortunately, these necessities are, in our view, despite efforts over recent years, still very real. The book has been organized into three parts, after Chapter 1 as general introduction. We have begun at a non-specialized, undergraduate level and progressed through to a relatively specialized level in each topic. Our objective is to transmit the essence of electrochemistry and research therein. It is intended that the chapters should be as independ- ent of one another as possible. The sections are: Chapters 2-6 on the thermodynamics and kinetics of electrode reactions, Chapters 7-12 on experimental strategy and methods, and Chapters 13-17 on applications. Also included are several appendices to explain the mathematical basis in more detail. It is no accident that at least 80 per cent of the book deals with current-volt age relations, and not with equilibrium. The essence of any chemical process is change, and reality reflects this. We have not filled the text with lots of details which can be found in the references given, and, where appropriate, we make ample reference to recent research literature. This is designed to kindle the enthusiasm and interest of the reader in recent, often exciting, advances in the topics described. A major preoccupation was with notation, given the traditionally different type of language that electrochemists have used in relation to viii Preface other branches of chemistry, such as exchange current which measures rate constants, and given differences in usage of symbols between different branches of electrochemistry. Differences in sign conventions are another way of confusing the unwary beginner. We have decided broadly to follow IUPAC recommendations. Finally some words of thanks to those who have helped and influenced us throughout our life as electrochemists. First to Professor W. J. Albery FRS, who introduced us to the wonders of electrochemistry and to each other. Secondly to our many colleagues and students who, over the years, with their comments and questions, have aided us in deepening our understanding of electrochemistry and seeing it with different eyes. Thirdly to anonymous referees, who made useful comments based on a detailed outline for the book. And last, but not least, to Oxford University Press for its interest in our project and enabling us to bring it to fruition. Coimbra C.M.A.B. May 1992 A.M.O.B.  ACKNOWLEDGEMENTS Full bibliographical references to all material reproduced are to be found at the ends of the respective chapters. Figure 3.4 is reprinted with permission from D. C. Grahame, Chem. Rev.y 1947, 41, 441. Copyright 1947 American Chemical Society; Fig 7.1 is reprinted with permission from G. M. Jenkins and K. Kawamura, Nature, 1971, 231, 175. Copyright 1971 Macmillan Magazines Ltd; Fig. 8.2c is reprinted by permission of the publisher, The Electrochemical Society Inc., Fig. 9.10a is reprinted with permission from R. S. Nicholson and I. Shain, Anal. Chem., 1964, 36, 706. Copyright 1964 American Chemical Society; Fig. 12.3 is reprinted by permission of John Wiley & Sons Inc. from J. D. E. Macintyre, Advances in electrochemistry and electrochemical engineering, 1973, Vol. 9, ed. R. H. Muller, p. 122. Copyright © 1973 by John Wiley & Sons, Inc.; Fig. 12.15a is reprinted with permission by VCH Publishers © 1991; Fig. 12.15b is reprinted with permission from R. Yang, K. Naoz, D. F. Evans, W. H. Smyrl and W. A. Hendrickson, Langmuir, 1991, 7, 556. Copyright 1991 American Chemical Society; Fig. 15.9 is reproduced from J. P. Hoare and M. L. LaBoda, Comprehensive treatise of electrochemistry, 1981, Vol. 2, ed. J. O'M. Bockris et al., p. 448, by permission of the publisher, Plenum Publishing Corporation; Fig. 16.7 is reproduced by kind permission of the copyright holder, National Association of Corrosion Engineers; Fig. 17.3 is reproduced from S. Ohki, Comprehensive treatise of electrochemistry, 1985, Vol. 10, ed. S. Srinivasan et al, p. 94, by permission of the publisher, Plenum Publishing Corporation; Fig. 17.6 is reproduced from R. Pethig, Modern bioelectrochemistry, ed. F. Gutmann and H. Keyser, 1986, p. 201, by permission of the publisher, Plenum Publishing Corporation; Fig. 17.7 is reprinted with permission from M. J. Eddowes and H. A. O. Hill, /. Am. Chem. Soc, 1979, 101, 4461. Copyright 1979 American Chemical Society; Fig. 17.9 is reproduced from M. Tarasevich, Comprehensive treatise of electrochemistry, 1985, Vol. 10, ed. S. Sriniva- san et al., p. 260, by permission of the publisher, Plenum Publishing Corporation; Fig. 17.11 is reproduced with the kind permission of the Institute of Measurement and Control; Table 2.2 is reproduced by kind permission of Butterworth-Heinemann Ltd; Table 7.1 is reprinted from R. L. McCreery, Electroanalytical chemistry, 1991, Vol. 17, ed. A. J. Bard, p. 243, by courtesy of Marcel Dekker Inc.; Table 7.3 is reprinted by permission of John Wiley & Sons Inc. from D. T. Sawyer and J. L. Roberts, Experimental electrochemistry for chemists, 1974, Copyright © x Acknowledgements 191A by John Wiley & Sons, Inc.; Tables 9.1 and 9.2 are reprinted with permission from R. S. Nicholson and I. Shain, Anal. Chem., 1964, 36, 706. Copyright 1964 American Chemical Society; Table 9.3 is reprinted with permission from R. S. Nicholson, Anal. Chem.y 1965, 37, 1351, copyright 1965 American Chemical Society, and from S. P. Perone, Anal. Chem.y 1966, 38, 1158, copyright 1966 American Chemical Society; Table 15.2 is reprinted by permission of the publisher, The Electrochem- ical Society Inc.; Table 17.1 is reproduced from H. Berg, Comprehensive treatise of electrochemistry, 1985, Vol. 10, ed. S. Srinivasan et al., p. 192, by permission of the publisher, Plenum Publishing Corporation; Table 17.2 is reproduced from S. Srinivasan, Comprehensive treatise of electrochemistry, 1985, Vol. 10, ed. S. Srinivasan et al.y p. 476, by permission of the publisher, Plenum Publishing Corporation. The following are also thanked for permission to reproduce or reprint copyright material: Bioanalytical Systems Inc. for Fig. 14.8; Elsevier Science Publishers BV for Figs 8.3, 8.4, 8.6, 8.7, 11.7, Tables 8.1 and 8.2; Elsevier Sequoia SA for Figs 9.11, 9.12, 9.15, 12.4, 12.8, 12.20, and 14.3; Journal of Chemical Education for Fig. 9.13a; Kluwer Academic Publ- ishers for Fig. 3.10; R. Kotz for Fig. 12.1; Oxford University Press for Figs 2.11, 2.12, and 17.10; Royal Society of Chemistry for Table 14.2. Although every effort has been made to trace and contact copyright holders, in a few instances this has not been possible. If notified the publishers will be pleased to rectify any omission in future editions.  CONTENTS Notation and Units xxi Main Symbols xxii Subscripts xxvi Abbreviations xxvii Fundamental physical constants xxix Mathematical constants xxix Useful relations at 25°C (298.15 K) involving fundamental constants xxix 1 INTRODUCTION 1 1.1 The scope of electrochemistry 1 1.2 The nature of electrode reactions 1 1.3 Thermodynamics and kinetics 2 1.4 Methods for studying electrode reactions 5 1.5 Applications of electrochemistry 5 1.6 Structure of the book 6 1.7 Electrochemical literature 7 PART I Principles 2 ELECTROCHEMICAL CELLS: THERMODYNAMIC PROPERTIES AND ELECTRODE POTENTIALS 13 2.1 Introduction 13 2.2 The cell potential of an electrochemical cell 14 2.3 Calculation of cell potential: activities or concentrations? 16 2.4 Calculation of cell potential: electrochemical potential. 18 2.5 Galvanic and electrolytic cells 20 2.6 Electrode classification 21 2.7 Reference electrodes 22 2.8 Movement of ions in solution: diffusion and migration. 25 2.9 Conductivity and mobility 26 2.10 Liquid junction potentials 32 2.11 Liquid junction potentials, ion-selective electrodes, and biomembranes 33 2.12 Electrode potentials and oxidation state diagrams... 34 References 38 xii Contents 3 THE INTERFACIAL REGION 39 3.1 Introduction 39 3.2 The electrolyte double layer: surface tension, charge density, and capacity 39 3.3 Double layer models 44 the first models: Helmholtz, Gouy-Chapman, Stern, and Grahame 45 Bockris, Devanathan, and Muller model 51 'chemical' models 52 3.4 Specific adsorption 54 3.5 The solid metallic electrode: some remarks 56 3.6 The semiconductor electrode: the space-charge region. 58 3.7 Electrokinetic phenomena and colloids: the zeta potential 64 electrophoresis 66 sedimentation potential 67 electroosmosis 67 streaming potential 68 limitations in the calculation of the zeta potential.. 68 References 68 4 FUNDAMENTALS OF KINETICS AND MECHANISM OF ELECTRODE REACTIONS 70 4.1 Introduction 70 4.2 The mechanism of electron transfer at an electrode.. 70 4.3 The mechanism of electron transfer in homogeneous solution 71 4.4 An expression for the rate of electrode reactions... 72 4.5 The relation between current and reaction rate: the exchange current 76 4.6 Microscopic interpretation of electron transfer.... 77 References 81 5 MASS T R A N S P O R T 82 5.1 Introduction 82 5.2 Diffusion control 83 5.3 Diffusion-limited current: planar and spherical electrodes 85 5.4 Constant current: planar electrodes 90 5.5 Microelectrodes 92 5.6 Diffusion layer 94  Contents xiii 5.7 C o n v e c t i o n a n d diffusion: h y d r o d y n a m i c s y s t e m s... 95 5.8 H y d r o d y n a m i c systems: s o m e useful p a r a m e t e r s.... 97 5.9 A ne x a m p l e o f a c o n v e c t i v e - d i f f u s i o n s y s t e m : t h e r o t a t i n g disc e l e c t r o d e 98 References 102 K I N E T I C S A N D T R A N S P O R T IN ELECTRODE REACTIONS 103 6.1 Introduction 103 6.2 The global electrode process: kinetics and transport.. 103 6.3 Reversible reactions 106 6.4 Irreversible reactions 109 6.5 The general case Ill 6.6 TheTafellaw 113 6.7 The Tafel law corrected for transport 115 6.8 Kinetic treatment based on exchange current 115 6.9 The effect of the electrolyte double layer on electrode kinetics 116 6.10 Electrode processes involving multiple electron transfer 119 6.11 Electrode processes involving coupled homogeneous reactions 122 References 126 PART II Methods E L E C T R O C H E M I C A L E X P E R I M E N T S.... 129 7.1 Introduction 129 7.2 Electrode materials for voltammetry 129 metals 130 carbon 130 other solid materials 133 mercury 133 7.3 The working electrode: preparation and cleaning... 134 7.4 The cell: measurements at equilibrium 136 7.5 The cell: measurements away from equilibrium.... 137 electrodes 137 supporting electrolyte 138 removal of oxygen 140 7.6 Calibration of electrodes and cells 142 7.7 Instrumentation: general 142 xiv Contents 7.8 Analogue instrumentation 143 potentiostat 146 galvanostat 147 compensation of cell solution resistance 148 7.9 Digital instrumentation 148 References 149 8 HYDRODYNAMIC ELECTRODES 151 8.1 Introduction 151 8.2 Limiting currents at hydrodynamic electrodes 155 8.3 A special electrode: the dropping mercury electrode.. 157 8.4 Hydrodynamic electrodes in the study of electrode processes 163 reversible reaction 163 the general case 164 8.5 Double hydrodynamic electrodes 165 8.6 Multiple electron transfer: the use of the RRDE... 167 consecutive reactions 168 parallel reactions 168 consecutive and parallel reactions 169 8.7 Hydrodynamic electrodes in the investigation of coupled homogeneous reactions 169 8.8 Hydrodynamic electrodes and non-stationary techniques 171 References 172 9 CYCLIC VOLTAMMETRY AND LINEAR SWEEP TECHNIQUES 174 9.1 Introduction 174 9.2 Experimental basis 175 9.3 Cyclic voltammetry at planar electrodes 176 reversible system 177 irreversible system 181 quasi-reversible system 183 adsorbed species 185 9.4 Spherical electrodes 187 9.5 Microelectrodes 188 9.6 Systems containing more than one component 188 9.7 Systems involving coupled homogeneous reactions... 189 9.8 Convolution linear sweep voltammetry 191 9.9 Linear potential sweep with hydrodynamic electrodes. 193 9.10 Linear potential sweep in thin-layer cells 194 References 197  Contents xv 10 S T E P A N D P U L S E T E C H N I Q U E S 199 10.1 Introduction 199 10.2 Potential step: chronoamperometry 200 reversible system 202 quasi-reversible and irreversible systems 203 more complex mechanisms 205 10.3 Double potential step 205 10.4 Chronocoulometry 206 10.5 Current step: chronopotentiometry 208 reversible system 209 quasi-reversible and irreversible systems 211 10.6 Double current step 212 10.7 Methods using derivatives of chronopotentiograms... 213 10.8 Coulostatic pulses 214 10.9 Pulse voltammetry 214 tast polarography 215 normal pulse voltammetry (NPV) 216 differential pulse voltammetry (DPV) 217 square wave voltammetry (SWV) 219 other pulse techniques 221 applications of pulse techniques 222 References 222 11 IMPEDANCE METHODS 224 11.1 Introduction 224 11.2 Detection and measurement of impedance 225 a.c. bridges 225 phase-sensitive detectors and transfer function analysers 227 direct methods 228 11.3 Equivalent circuit of an electrochemical cell 229 11.4 The faradaic impedance for a simple electrode process. 230 11.5 The faradaic impedance, Z f , and the total impedance: how to calculate Zf from experimental measurements. 232 11.6 Impedance plots in the complex plane 233 11.7 Admittance and its use 236 11.8 A.c. voltammetry 238 11.9 Second-order effects 240 higher harmonics 240 other second-order methods 241 faradaic rectification 242 demodulation 242 xvi Contents 11.10 More complex systems, porous electrodes, and fractals. 244 11.11 Нуdrodynamic electrodes and impedance 248 11.12 Transforms and impedance 249 References 251 12 NON-ELECTROCHEMICAL PROBES OF ELECTRODES AND ELECTRODE PROCESSES 253 12.1 Introduction 253 12.2 In situ spectroscopic techniques 254 transmission 254 reflectance, electroreflectance and ellipsometry... 255 internal reflection 258 Raman spectroscopy 259 electron spin resonance (ESR) spectroscopy.... 260 X-ray absorption spectroscopy 261 second harmonic generation (SHG) 263 12.3 Ex situ spectroscopic techniques 263 photoelectron spectroscopy (XPS) 263 Auger electron spectroscopy (AES) 264 electron energy loss spectroscopy (EELS) 266 electrochemical mass spectrometry (ECMS) and secondary ion mass spectrometry (SIMS)... 266 low-energy and reflection high-energy electron diffraction (LEED and RHEED) 267 12.4 In situ microscopic techniques 268 scanning tunnelling microscopy (STM) 269 atomic force microscopy (AFM) 270 scanning electrochemical microscopy (SECM).... 272 scanning ion conductance microscopy (SICM).... 273 12.5 Ex situ microscopic techniques: electron microscopy.. 273 12.6 Other in situ techniques 276 measurement of mass change: the quartz crystal microbalance (QCM) 276 measurement of absorbed radiation: thermal changes 277 12.7 Photoelectrochemistry 278 12.8 Electrochemiluminescence 282 References 282 P A R T III Applications 13 POTENTIOMETRIC SENSORS 289 13.1 Introduction 289  Contents xvii 13.2 Potentiometric titrations 290 13.3 Functioning of ion-selective electrodes 294 13.4 Glass electrodes and pH sensors 295 13.5 Electrodes with solid state membranes 297 13.6 Ion-exchange membrane and neutral carrier membrane electrodes 301 13.7 Sensors selective to dissolved gases 303 13.8 Enzyme-selective electrodes 303 13.9 Some practical aspects 304 13.10 Recent developments: miniaturization 305 ISFETs 305 coated wire electrodes 306 hybrid sensors 307 13.11 Potentiometric sensors in flow systems 307 13.12 Electroanalysis with potentiometric sensors 308 References 309 14 AMPEROMETRIC AND VOLTAMMETRIC SENSORS 310 14.1 Introduction 310 14.2 Amperometric titrations 311 simple amperometric titrations 311 biamperometric titrations 312 amperometric titrations with double hydrodynamic electrodes 313 14.3 Membrane and membrane-covered electrodes 314 14.4 Modified electrodes.316 14.5 Increase of sensitivity: pre-concentration techniques.. 318 14.6 Amperometric and voltammetric electroanalysis.... 322 References 324 15 ELECTROCHEMISTRY IN I N D U S T R Y.... 326 15.1 Introduction 326 15.2 Electrolysis: fundamental considerations 327 15.3 Electrochemical reactors 328 15.4 Porous and packed-bed electrodes 331 15.5 Examples of industrial electrolysis and electrosynthesis. 332 the chlor-alkali industry 332 metal extraction: aluminium 336 water electrolysis 338 organic electrosynthesis: the Monsanto process... 339 15.6 Electrodeposition and metal finishing 341 15.7 Metal processing 345 x v jjj Contents 15.8 Batteries 346 15.9 Fuel cells 349 15.10 Electrochemistry in water and effluent treatment... 350 References 351 16 C O R R O S I O N 353 16.1 Introduction 353 16.2 Fundamentals 353 thermodynamic aspects 354 kinetic aspects 354 16.3 Types of metallic corrosion 361 16.4 Electrochemical methods of avoiding corrosion.... 363 electrochemically produced protective barriers... 364 sacrificial anodes 364 methods of impressed current/potential 365 corrosion inhibitors 365 References 366 17 B I O E L E C T R O C H E M I S T R Y 367 17.1 Introduction 367 17.2 The electrochemical interface between biomolecules: cellular membranes, transmembrane potentials, bilayer lipid membranes, electroporation 368 17.3 Nerve impulse and cardiovascular electrochemistry.. 373 the nerve impulse 374 cardiovascular problems 376 17.4 Oxidative phosphorylation 378 17.5 Bioenergetics 379 17.6 Bioelectrocatalysis 381 17.7 Bioelectroanalysis 387 17.8 Future perspectives 391 References 391 Appendices Al U S E F U L M A T H E M A T I C A L R E L A T I O N S.. 395 Al.l The Laplace transform 395 introduction 395 the transform 395 important properties 397 A1.2 The Fourier transform 398  Contents xix A1.3 Other useful functions and mathematical expressions. 399 the Airy function 399 the gamma function 399 the error function 400 Taylor and Maclaurin series 401 hyperbolic functions 403 Reference 404 A2 PRINCIPLES OF A. C. CIRCUITS 405 A2.1 Introduction 405 A2.2 Resistance 406 A2.3 Capacitance 406 A2.4 Representation in the complex plane 406 A2.5 Resistance and capacitance in series 407 A2.6 Resistance and capacitance in parallel 408 A2.7 Impedances in series and in parallel 410 A2.8 Admittance 410 A2.9 The Kramers-Kronig relations 410 References 411 A3 DIGITAL SIMULATION 412 A3.1 Introduction 412 A3.2 Simulation models 412 A3.3 Implicit methods 414 References 414 A4 S T A N D A R D E L E C T R O D E P O T E N T I A L S.. 416 INDEX 419 Notation and Units As far as possible without straying too far from common usage, the guidelines of IUPAC have been followed, described in Quantities, units and symbols in physical chemistry (Blackwell, Oxford, 1988). Other, more detailed information has been taken from the following sources in the IUPAC journal, Pure and Applied Chemistry: 'Electrode reaction orders, transfer coefficients and rate constants. Amplification of definitions and recommendations for publication of parameters', 1979, 52, 233. Tnterphases in systems of conducting phases', 1986, 58, 454. 'Electrochemical corrosion nomenclature', 1989, 61, 19. Terminology in semiconductor electrochemistry and photo- electrochemical energy conversion', 1991, 63, 569. 'Nomenclature, symbols, definitions and measurements for electrified interfaces in aqueous dispersions of solids', 1991, 63, 896. The units quoted are those recommended. In practice, in electrochem- istry, much use is made of sub-multiples: for example, cm instead of m and JUA or mA instead of A, for obvious reasons. The text tends to use the commonly employed units. In the list of symbols, those used at only one specific point in the text are mostly omitted, in order to try and reduce the length of the list, since explanation of their meaning can be found next to the relevant equation. We have also provided a list of frequently used subscripts, a list of abbreviations, and values of important constants and relations derived from these. Following recommended usage, loge is written as In and log10 is written as lg. Notation: main symbols Units а activity — а nozzle diameter of impinging jet m а radius of colloidal particle m А area m2 А 'constant' varies Ь Tafel slope V" 1 с concentration mol m~3 c() concentration at electrode surface Coo bulk concentration С capacity F Cd differential capacity of double layer C; integral capacity of double layer Cs capacity in RC series combination C sc capacity of semiconductor space-charge layer D diffusion coefficient m2s-1 е electron charge С Е electric field strength Vm" 1 Е electrode potential V Z?"0" standard electrode potential E^r formal potential £cel, cell potential (electromotive force) Ecor corrosion potential EV2 half-wave potential £j liquid junction potential Em membrane potential Ep peak potential Ez potential of zero charge Ex inversion potential in cyclic voltammetry Ес lowest energy of semiconductor conduction band eV Eg bandgap energy in semiconductor eV Еу highest energy of semiconductor valence band eV EF Fermi energy eV L-* rcdox energy of redox couple eV f frequency Hz  Notation: main symbols xxiii /DL Frumkin double layer correction — F force N g acceleration due to gravity ms"2 g constant in Temkin and Frumkin isotherms — G Gibbs free energy JmoP1 h height m H enthalpy at constant pressure JmoP1 I electric current A /c capacitative current /f faradaic current /L diffusion limited current /p peak current I ionic strength molm~ j electric current density Am"2 J volume flux s" 1 к rate constant: homogeneous first order rate constant: heterogeneous ms"1 ka rate constant for oxidation at electrode kc rate constant for reduction at electrode kd mass transfer coefficient potentiometric selectivity coefficient — К equilibrium constant — I length of electrode m m mass kg mt mass flux of liquid kgs" 1 m molality kgm- 3 n number of electrons transferred n' number of electrons transferred in rate determining step 3 и, number density of species i m~ S P (DO/DR) where s = 1/2 (stationary electrodes and DMEs), s = 2/3 (hydrodynamic electrodes), s = 1 (microelectrodes) — Pi partial pressure of i Pa P pressure (total) Pa Pe Peclet number (Pe = vl/D) — Q electric charge С r radial variable m r0 radius of (hemi-)spherical e l e c t r o d e rx radius of disc electrode r2 inner radius of ring electrode r3 o u t e r radius of ring electrode rc capillary radius xxiv Notation: main symbols R resistance Q Rct charge transfer resistance Rs resistance in RC series combination JRQ cell solution resistance R radius of tube m Re Reynolds number (Re = vl/v) — S entropy J т о Г l K" 1 Sc Schmidt number (5c = v/D) — Sh Sherwood number (Sh = kJ/D) — t time s ti transport number of species / — T temperature К щ mobility of species i m 2 V" l s~l ue electrophoretic mobility U potential (same meaning as E, used in photo- and semiconductor electrochemistry) V Ufb flat-band potential v velocity ms"1 v potential scan rate V s" 1 V voltage (in operational amplifiers, etc.) V V volume m3 V{ volume flow rate m 3 s~l W rotation speed Hz x distance m X reactance Q V admittance S z ion charge — Z impedance Q Zs impedance of R C series combination Z' real part of impedance Z" imaginary part of impedance Zf Faradaic impedance Zw Warburg impedance oc electrochemical charge transfer coefficient — a& anodic ac cathodic a electrode roughness parameter — a double hydrodynamic electrode geometric constant — /3 double hydrodynamic electrode geometric constant — P Esin-Markov coefficient — /3 energetic proportionality coefficient —  Notation: main symbols xxv У activity coefficient — У surface tension Nm" 1 У dimensionless concentration variable — г surface excess concentration 2 mol m~ diffusion layer thickness m hydrodynamic boundary layer thickness m molar absorption coefficient n^mol" 1 permittivity Fin" 1 1 permittivity of vacuum Fm" relative permittivity — € porosity of material — zeta (electrokinetic) potential V (nF/RT)(E-E1/2) — overpotential ч viscosity V Pas contact angle fractional surface coverage — в exp [(nF/RT)(E - E^)] — в 1 к conductivity Sm" А value of t where sweep is inverted in cyclic voltammetry s 2 Л molar conductivity S m mol chemical potential Jmol"1 £ electrochemical potential Jmol"1 V frequency of electromagnetic radiation s" 1 V stoichiometric number — V kinematic viscosity m 2 s" x р resistivity Qm density kgm" 3 2 а surface charge density Cm" а v(nF/RT) s" 1 а mass-transport dependent expression (Table 8.2) т characteristic time in experiment s Ф electrostatic potential V 0 inner electric potential V 0 phase angle surface electric potential V \р outer electric potential V (0 angular velocity, rotation speed rads"1 (0 circular frequency rads"1 Subscripts a anodic max maximum value с cathodic min minimum value С capacitive О oxidized species det detector electrode p peak value D disc electrode R reduced species f faradaic R ring electrode f final value 0 at zero distance (electrode gen generator electrode surface) i species i at infinite distance (bulk i initial value solution) L diffusion-limited value * at OHP Abbreviations AES Auger electron spectroscopy AFM atomic force microscopy ASV anodic stripping voltammetry AdSV adsorptive stripping voltammetry BLM bilayer lipid membrane CDE channel double electrode CE electrode process involving chemical followed by electrochemical step C'E catalytic electrode process involving chemical followed by electrochemical step CV cyclic voltammetry DDPV differential double pulse voltammetry DISP electrode process involving electrochemical followed by chemical, followed by disproportionation step to regenerate reagent DME dropping mercury electrode DNPV differential normal pulse voltammetry DPV differential pulse voltammetry DSA dimensionally stable anode EC electrode process involving electrochemical followed by chemical step ECE electrode process involving electrochemical followed by chemical, followed by electrochemical step ECL electrochemiluminescence ECMS electrochemical mass spectroscopy EELS electron energy loss spectroscopy EMIRS electrochemically modulated infrared spectroscopy EQCM electrochemical quartz crystal microbalance ESR electron spin resonance EXAFS extended X-ray absorption fine structure FFT fast Fourier transform GC glassy carbon HMDE hanging mercury drop electrode HOPG highly oriented pyrolytic graphite HPLC high-performance liquid chromatography IHP inner Helmholtz plane XXV111 Abbreviations IRRAS infrared reflection absorption spectroscopy ISE ion-selective electrode ISFET ion-selective field effect transistor ISM ion-selective membrane LEED low-energy electron diffraction LSV linear sweep voltammetry MCFC molten carbonate fuel cell MS mass spectrometry NHE normal hydrogen electrode NPV normal pulse voltammetry OA operational amplifier OHP outer Helmholtz plane OTE optically transparent electrode OTTLE optically transparent thin-layer electrode PAFC phosphoric acid fuel cell PAS photoacoustic spectroscopy PSA potentiometric stripping analysis QCM quartz crystal microbalance RDE rotating disc electrode RHEED reflection high-energy electron diffraction RRDE rotating ring-disc electrode SCC stress corrosion cracking SCE saturated calomel electrode SCM surface compartment model SECM scanning electrochemical microscopy SEM scanning electron microscopy SHG second harmonic generation SICM scanning ion conductance microscopy SIMS secondary ion mass spectroscopy SMDE static mercury drop electrode SNIFTRS subtractively normalized interfacial Fourier transi infrared spectroscopy SOFC solid oxide fuel cell STM scanning tunnelling microscopy SWV square wave voltammetry TDE tube double electrode ТЕМ transmission electron microscopy WJRDE wall-jet ring-disc electrode XANES X-ray absorption near edge structure XPS X-ray photoelectron spectroscopy Fundamental physical constants с speed of light in vacuum 2.99792458 x l O ^ s " 1 e unit of electron charge 1.602177 х Н Г С 19 F Faraday constant 9.6485 xlO С т о Г 1 4 kB Boltzmann constant 1.38066 x l ( T 2 3 J К" 1 R gas constant 8.31451 J КГ 1 т о Г 1 h Planck constant 6.62608 x ИГ 3 4 Js NA Avogadro constant 6.02214 х К Р т о Г 1 e0 permittivity of vacuum 8.85419 x 10~12 Г 1 С 2 т " 1 g acceleration due to gravity 9.80665 ms~ 2 Mathematical constants 3.14159265359 e 2.71828182846 In 10 2.302585 Useful relations at 25°C (298.15 K) involving fundamental constants RT/F 25.693 mV (RTУ F) In 10 59.160 mV kBT 25.7 meV (4.12 xlO~ 2 1 J)  INTRODUCTION 1.1 The scope of electrochemistry 1.2 The nature of electrode reactions 1.3 Thermodynamics and kinetics 1.4 Methods for studying electrode reactions 1.5 Applications of electrochemistry 1.6 Structure of the book 1.7 Electrochemical literature 1.1 The scope of electrochemistry Electrochemistry involves chemical phenomena associated with charge separation. Often this charge separation leads to charge transfer, which can occur homogeneously in solution, or heterogeneously on electrode surfaces. In reality, to assure electroneutrality, two or more charge transfer half-reactions take place, in opposing directions. Except in the case of homogeneous redox reactions, these are separated in space, usually occurring at different electrodes immersed in solution in a cell. These electrodes are linked by conducting paths both in solution (via ionic transport) and externally (via electric wires etc.) so that charge can be transported. If the cell configuration permits, the products of the two electrode reactions can be separated. When the sum of the free energy changes at both electrodes is negative the electrical energy released can be harnessed (batteries). If it is positive, external electrical energy can be supplied to oblige electrode reactions to take place and convert chemical substances (electrolysis). In this chapter, a brief overview of electrochemistry, and particularly of electrode reactions, is given in order to show the interdisciplinary nature and versatility of electrochemistry and to introduce a few of the important fundamental concepts. Before discussing these it is worth looking briefly at the nature of electrode reactions. 1.2 The nature of electrode reactions Electrode reactions are heterogeneous and take place in the interfacial region between electrode and solution, the region where charge distribu- 2 Introduction tion differs from that of the bulk phases. The electrode process is affected by the structure of this region. However, we first assume that there is no effect apart from charge separation. At each electrode, charge separation can be represented by a capacitance and the difficulty of charge transfer by a resistance. For the rest of this and the ensuing sections we consider only one of the electrodes. The electrode can act as only a source (for reduction) or a sink (for oxidation) of electrons transferred to or from species in solution, as in where О and R are the oxidized and reduced species, respectively. Alternatively, it can take part in the electrode reaction, as in dissolution of a metal M: М-н>М"+ + ие- In order for electron transfer to occur, there must be a correspondence between the energies of the electron orbitals where transfer takes place in the donor and acceptor. In the electrode this level is the highest filled orbital, which in a metal is the Fermi energy level, EF. In soluble species it is simply the orbital of the valence electron to be given or received. Thus: for a reduction, there is a minimum energy that the transferable electrons from the electrode must have before transfer can occur, which corresponds to a sufficiently negative potential (in volts) for an oxidation, there is a maximum energy that the lowest unoccupied level in the electrode can have in order to receive electrons from species in solution, corresponding to a sufficiently positive potential (in volts). The values of the potentials can be controlled externally. In this way we can control which way an electrode reaction occurs and to what extent. The thermodynamics and kinetics of electrode processes are sum- marized in the following section. However, before this we return to the structure of the interfacial region. The change in charge distribution from the bulk in this region means that the relevant energy levels in reacting species and in the electrode are not the same as in the bulk phases, and soluble species need to adjust their conformation for electron transfer to occur. These effects should be corrected for in a treatment of kinetics of electrode processes—the thinner the interfacial region the better, and this can be achieved by addition of a large concentration of inert electrolyte. 1.3 Thermodynamics and kinetics Electrode reactions are half-reactions and are, by convention, expressed as reductions. Each has associated with it a standard electrode potential,  1.3 Thermodynamics and kinetics 3 & E y measured relative to the normal hydrogen electrode (NHE) with all species at unit activity (я, = 1). For half-reactions at equilibrium, the potential, E, can be related to the standard electrode potential through the Nernst equation ^ (1.1) where v, are the stoichiometric numbers, positive for products (reduced species) and negative for reagents (oxidized species). The tendency for the reduction to occur, relative to the NHE reference, is thus given by AG^=-nFE^ (1.2) under standard conditions. Thus, for example, Group IA metals which have very negative values of £ ° , tend to oxidize (see Appendix 4). It is often useful to be able to employ concentrations, ch instead of activities, where at = у{с{ with y, the activity coefficient of species i. The Nernst equation (1.1) is rewritten as RT £ = £^'- —Ev/lnc, (1.3) in which £ ° is the formal potential, dependent on the medium since it includes the logarithmic activity coefficient terms as well as £ °. If the oxidized and reduced species involved in an electrode reaction are in equilibrium at the electrode surface, the Nernst equation can be applied. The electrode reaction is then known as a reversible reaction since it obeys the condition of thermodynamic reversibility. Clearly the applicability of the Nernst equation, and therefore reversibility, has to do with the time allowed for the electrode reaction to reach equilibrium. The concentrations of species at the interface depend on the mass transport of these species from bulk solution, often described by the mass transfer coefficient kd. A reversible reaction corresponds to the case where the kinetics of the electrode reaction is much faster than the transport. The kinetics is expressed by a standard rate constant, k0, which is the rate constant when E = E^'. So the criterion for a reversible reaction is ko»kd By contrast, an irreversible reaction is one where the electrode reaction cannot be reversed. A high kinetic barrier has to be overcome, which is achieved by application of an extra potential (extra energy) called the overpotentialy r\y and in this case ko«kd Quasi-reversible reactions exhibit behaviour intermediate between 4 Introduction reversible and irreversible reactions, the overpotential having a relatively small value, so that with this extra potential reactions can be reversed. The potential-dependent expression for the rate constant of an electrode reaction is, for a reduction, kc = k0 exp [-acnF(E - E^')/RT] (1.4) and for an oxidation k.A = k0 exp [aanF(E - E^')/RT] (1.5) In these equations occ and ara are the cathodic and anodic charge transfer coefficients and are a measure of the symmetry of the activation barrier, being close to 0.5 for a metallic electrode and a simple electron transfer process. As mentioned above, the standard rate constant is the rate constant at E = E^f. An alternative way used to express the rates of electrode reactions is through the exchange current, / 0. This is the magnitude of the anodic or cathodic partial current at the equilibrium potential, Eeq. It is equivalent to measuring the standard rate constant, kQ. Experimentally, rates of electrode reactions are measured as the current passed, to which they are directly proportional. The dependence of current, /, on potential is exponential, suggesting a linear relation between lg / and potential—this is the Tafel relation. However, the rate (product of rate constant and reagent concentration) cannot rise indefin- itely because the supply of reactants begins to diminish and becomes transport-limited. Whereas for reversible reactions only thermodynamic and mass- transport parameters can be determined, for quasi-reversible and irre- versible reactions both kinetic and thermodynamic parameters can be measured. It should also be noted that the electrode material can affect the kinetics of electrode processes. The rate constant of an electrode reaction does not measure the rate of electron transfer itself, as this is an adiabatic process, following the Franck-Condon principle, and occurs in approximately 10~16s. What it does measure is the time needed for the species, once they have reached the interfacial region, to arrange themselves and their ionic atmospheres into position for electron transfer to be able to occur. More complex electrode processes than those described above involve consecutive electron transfer or coupled homogeneous reactions. The theory of these reactions is also more complicated, but they correspond to a class of real, important reactions, particularly involving organic and biological compounds.  1.5 Applications of electrochemistry 5 1.4 Methods for studying electrode reactions In order to study electrode reactions, reproducible experimental condi- tions must be created which enable minimization of all unwanted factors that can contribute to the measurements and diminish their accuracy. Normally we wish to suppress migration effects, confine the interfacial region as close as possible to the electrode, and minimize solution resistance. These objectives are usually achieved by addition of a large quantity of inert electrolyte (around lmoldm" 3 ), the electroactive species being at a concentration of 5 т м or less. A complete study of an electrode process requires measurement of kinetic as well as thermodynamic parameters. This means that conditions in which the system is not reversible must be used. Since the standard rate constant, k0, cannot be changed, then the mass transfer coefficient, kd, may have to be increased until the reaction becomes at least quasi-reversible. This can be done in various ways in various types of experiment: steady state methods: hydrodynamic electrodes, increasing convec- tion; microelectrodes, decreasing size linear sweep methods: increasing sweep rate step and pulse techniques: increasing amplitude and/or frequency impedance methods: increasing perturbation frequency, registering higher harmonics, etc. The type of technique chosen will depend very much on the timescale of the electrode reaction. Non-electrochemical methods can and should be used for studying electrode surfaces and the interfacial region structure, particularly in situ in real time where this is possible. 1.5 Applications of electrochemistry Once electrode reactions and electrode processes are understood, this knowledge can be used for: tailoring electrode reactions so as to enhance required and inhibit unwanted electrode reactions, perhaps by changing electrode material or developing new electrode materials studying complex systems in which many electrode reactions occur simultaneously or consecutively, as in bioelectrochemistry 6 Introduction measuring concentrations of electroactive species, making use of the selectivity of the potential and of the electrode material at or outside equilibrium (as in potentiometric, amperometric, voltammetric, and enzyme sensors). Thus the range of applications is vast. Electroanalysis, potentiometric and voltammetric; industrial electrolysis, electroplating, batteries, fuel cells, electrochemical machining, and many other related applications, including minimization of corrosion; biosensors and bioelectrochemistry. 1.6 Structure of the book This book is organized into three main sections, as its subtitle suggests. In the first part, Chapters 2-6, some fundamentals of electrode processes and of electrochemical and charge transfer phenomena are described. Thermodynamics of electrochemical cells and ion transport through solution and through membrane phases are discussed in Chapter 2. In Chapter 3 the thermodynamics and properties of the interfacial region at electrodes are addressed, together with electrical properties of colloids. Chapters 4-6 treat the rates of electrode processes, Chapter 4 looking at fundamentals of kinetics, Chapter 5 at mass transport in solution, and Chapter 6 at their combined effect in leading to the observed rate of electrode processes. The second part of the book discusses ways in which information concerning electrode processes can be obtained experimentally, and the analysis of these results. Chapter 7 presents some of the important requirements in setting up electrochemical experiments. In Chapters 8-11, the theory and practice of different types of technique are presented: hydrodynamic electrodes, using forced convection to increase mass transport and increase reproducibility; linear sweep, step and pulse, and impedance methods respectively. Finally in Chapter 12, we give an idea of the vast range of surface analysis techniques that can be employed to aid in investigating electrode processes, some of which can be used in situ, together with photochemical effects on electrode reactions— photoelectrochemistry. In the third part of the book areas in which there are important applications of electrochemistry are described. Chapters 13 and 14 look at potentiometric and amperometric/voltammetric sensors respectively, focusing particularly on recent developments such as new electrode materials and miniaturization. Electrochemistry in industry, which prod- uces many materials used directly or indirectly in everyday life, as well as batteries, is described in Chapter 15. The electrochemical phenomenon  1.7 Electrochemical literature 7 of corrosion, economically prejudicial, is described in Chapter 16. Finally, since many biochemical processes involve charge transfer reac- tions, in Chapter 17 the many possibilities that arise from their study by electrochemical methods, bioelectrochemistry, are presented. 1.7 Electrochemical literature The electrochemical literature is very widespread. Some indication of its breadth is given below. The references at the end of each chapter complement this list. General books Many books on electrochemistry have been published in recent decades. Mostly the more general ones are not cited throughout the text, but this does not reflect on their quality. A list of them is given below, in chronological order. P. Delahay, New instrumental methods in electrochemistry, Interscience, New York, 1954. K. J. Vetter, Electrochemical kinetics. Academic Press, New York, 1967. R. N. Adams, Electrochemistry at solid electrodes, Dekker, New York, 1969. J. O'M. Bockris and A. N. Reddy, Modern electrochemistry, Plenum, New York, 1970. J. Newman, Electrochemical systems, Prentice Hall, Englewood Cliffs, NJ, 1973. D. T. Sawyer and J. L. Roberts, Experimental electrochemistry for chemists, Wiley, New York, 1974. E. Gileadi, E. Kirowna-Eisner, and J. Penciner, Interfacial electrochem- istry. An experimental approach, Addison-Wesley, Reading, MA, 1975. W. J. Albery, Electrode kinetics, Clarendon Press, Oxford, 1975. A. J. Bard and L. R. Faulkner, Electrochemical methods, fundamentals and applications, Wiley, New York, 1980. A. M. Bond, Modern polarographic methods in analytical chemistry, Dekker, New York, 1980. Southampton Electrochemistry Group, New instrumental methods in electrochemistry, Ellis Horwood, Chichester, 1985. J. Goodisman, Electrochemistry: theoretical foundations, Wiley- Interscience, New York, 1987. 8 Introduction J. Koryta, Principles of electrochemistry, Wiley, Chichester, 1987. P. H. Rieger, Electrochemistry, Prentice-Hall International, Englewood Cliffs, NJ, 1987. D. R. Crow, Principles and applications of electrochemistry, 3rd edn, Chapman and Hall, London, 1988. P. W. Atkins, Physical chemistry, 4th edn., Oxford University Press, 1990, Chapters 10, 25, and 30. D. Pletcher, A first course in electrode processes. The Electrochemical Consultancy, Romsey, UK, 1991. J. Koryta, Ionsy electrodes, and membranes, Wiley, Chichester, 1991. Series A number of series of volumes dealing with electrochemistry have been published. Those recently issued or currently being published are listed below. Advances in electrochemistry and electrochemical engineering, Wiley, New York. Volumes 1-9, ed. P. Delahay and C. W. Tobias; Volumes 10-13, ed. H. Gerischer and C. W. Tobias. Advances in electrochemical science and engineering, ed. H. Gerischer and С W. Tobias, VCH, Weinheim (continuation of Adv. Electro- chem. Electrochem. Eng.\ 1 volume until end 1991). Comprehensive treatise of electrochemistry, ed. J. O'M. Bockris, В. Е. Conway, E. Yeager et al., Plenum, New York, Volumes 1-10. Comprehensive chemical kinetics, section 10; electrode kinetics, ed. R. G. Compton et al., Elsevier, Amsterdam, Volumes 26-29. Electroanalytical chemistry: a series of advances, ed. A. J. Bard, Dekker, New York (17 volumes until end 1991). Modern aspects of electrochemistry, ed. J. O'M. Bockris, В. Е. Conway et al, Plenum, New York (21 volumes until end 1991). International journals devoted to electrochemistry There are a number of international journals devoted primarily to electrochemistry: Bioelectrochemistry and Bioenergetics (an independent section of /. Electroanal. Chem.) Corrosion Corrosion Science Electroanalysis Electrochimica Acta  1.7 Electrochemical literature 9 Elektrokhimiya {Soviet Electrochemistry) Journal of Applied Electrochemistry Journal of Electroanalytical and Interfacial Electrochemistry Journal of the Electrochemical Society Selective Electrode Reviews (formerly Ion Selective Electrode Reviews, until 1988) Articles with electrochemical themes also regularly appear in a large number of other journals. PART I Principles  ELECTROCHEMICAL CELLS: THERMODYNAMIC PROPERTIES AND ELECTRODE POTENTIALS 2.1 Introduction 2.2 The cell potential of an electrochemical cell 2.3 Calculation of cell potential: activities or concentrations? 2.4 Calculation of cell potential: electrochemical potential 2.5 Galvanic and electrolytic cells 2.6 Electrode classification 2.7 Reference electrodes 2.8 The movement of ions in solution: diffusion and migration 2.9 Conductivity and mobility 2.10 Liquid junction potentials 2.11 Liquid junction potentials, ion-selective electrodes and biomembranes 2.12 Electrode potentials and oxidation state diagrams 2.1 Introduction An understanding of thermodynamic properties associated with electrode processes is fundamental in order to answer questions such as: Why is it that half-reactions in electrochemical cells proceed spon- taneously in one direction and furnish current? What is the effect of the salt bridge? What is the effect of ion migration? In this chapter we attempt to reply to these and to other related questions. To treat the topic in a concrete way, we consider two electrochemical cells: Zn|Zn 2 + (aq)|Cu 2 + (aq)|Cu and Hg | Hg2Cl2 | Cl-(aq) ji Zn2+(aq) | Zn where we represent only the species of interest. In these cells the symbol 14 Electrochemical cells | denotes a phase boundary, | a junction between miscible liquids, and jj a salt bridge (liquid junction) whose function is to provide an electrically conducting link between two spatially separated components of the cell in the liquid phase. It should be stressed that, according to the internation- ally accepted IUPAC convention, the half-reactions are considered in the way the cell is depicted on paper, that is oxidation in the left half-cell (the electrode is the anode) and reduction in the right half-cell (the electrode is the cathode)1. 2.2 The cell potential of an electrochemical cell The cell potential of an electrochemical cell is calculated from the electrode potentials (reduction potentials) of the respective half- reactions1. Given that, by convention, the half-reaction on the left is considered to be an oxidation and that on the right a reduction we have = ^cell bright ~ ^left (2-1) where £right and £left are the potentials of each half-cell, obtained from the Nernst equation. The Nernst equation relates the activities of the species involved with the electrode potential, Ey of the half-reaction and its standard electrode potential, E"0", which is the value of the potential relative to the standard hydrogen electrode when the activities of all species are unity. For the generic half-reaction where n is the stoichiometric number of electrons transferred for each species, the Nernst equation is £ = £^-^2>,1па,. (2.2) nt in which V/ has positive values for products (reduced species) and negative values for reagents (oxidized species). This can be written as nF Пай, For example, for MnO2 + 4H + + 2e~ -* Mn 2 + + H 2 O  2.2 The cell potential of an electrochemical cell 15 the logarithmic term is RT \n IF IF а М п 2 + йн 2 о ан2о i s approximately constant and is neglected in the Nernst equation except in the case of a mixture with another solvent or in very concentrated solutions. The cell potential tells us the maximum work (maximum energy) that the cell can supply2. This value is AG = -nFEcell (2.4) It is evident that on removing energy (in the form of current or converted chemical substances) the amount of unconverted substances remaining is diminished, reflecting the changes in the concentrations of the species in the liquid phase. In the solid phase, however, there is no alteration of activity, which is normally accepted as being unity. We now calculate the cell potential for the two cases mentioned above. Case 1 Zn|Zn 2 + (aq)iCu 2 + (aq)|Cu which means we consider the cell reaction as Zn + Cu 2 + -*Zn 2 + + Cu The half-reactions are represented by right: Cu2 + 2e~ - Cu £ ° = +0.34 V 2+ left: Zn + 2e" -> Zn £ ° = -0.76 V If the aqueous species have unit activity, then E^ values may be used and E£n = +0.34 - (-0.76) = +1.10 V The corresponding A G 0 value is ° = -2.20F = -212 kJ mol"1 which is negative. This result shows that the reaction proceeds spon- taneously as written. The equivalent of the Nernst equation for the whole cell is (2.5) 16 Electrochemical cells It can be seen that if the ratio (acu2+/aZn2+) is sufficiently small, Ecell becomes negative and the direction of spontaneous reaction is changed. Case 2 Hg | Hg2Cl2 | СГ(а Ч ) jj Zn 2+ (aq) | Zn The stoichiometric cell reaction to consider is 2Hg + 2СГ + Zn 2 + - Hg2Cl2 + Zn and the half-reactions are represented by right: Zn 2 + + 2e"-> Zn £ e = -0.76 V e left: Hg2Cl2 + 2e~ -> 2Hg 4- 2СГ £ = +0.27 V For unit activities, Etn = -0.76 - 0.27 = -1.03 V and AG°=+199kJmor1 The negative value of E^n (and positive AG°) means that at unit activities the cell functions spontaneously in the direction opposite to that written above. Thus the spontaneous cell reaction is Hg2Cl2 + Zn2+ -> 2Hg 4- Zn 2+ + 2СГ The half-cell on the left is an example of a reference electrode (Section 2.7) so called since, as Hg2Cl2 is a sparingly soluble salt, the activities of Hg and Hg2Cl2 can be taken as unity. The potential of the half-cell is altered solely by the chloride ion activity according to the expression RT Ясен = £ £ „ - — I n a a - (2-6) This electrode is known as the calomel electrode. 2.3 Calculation of cell potential: activities or concentrations? Although the use of activities in the Nernst equation is undoubtedly correct, it is worth considering whether it is necessary and what is the difference between activities and concentrations in general. In the context of this book, a detailed discussion of activities and concentrations is not justified. However, it is clear that in relatively concentrated solutions there will be interionic interactions that do not  2. 3 Calculation of cell potential: activities or concentrations ? 17 occur in very dilute solutions because of the large interionic distances in the latter. Consequently the velocity of ion migration (i.e. the momen- tum of each ion) will be altered, and this can reduce, or possibly increase, ionic activity. Thus we write the relations a = Ymm (2.1a) a = ycc (2.1b) where ym is the activity coefficient for concentrations in relation to molality (molkg" 1 ), and yc in relation to molarity (moldm" 3 ). Thus, these coefficients are proportionality factors between activity and con- centration, whose values vary with concentration. It is often useful to employ concentrations instead of activities in electrochemical experiments: for example, in preparing solutions we use masses and volumes, that is we determine the concentration of a solution. Thus, the Nernst equation, instead of being written as (2.3) can be formulated as In this last equation, E^' is the formal potential. It is related to the standard electrode potential, E^y by ^ ' =^ +^ 1 1 1 ^ (2.9) nF П Ус,'к, Experimentally we measure the formal potential, ZT0"', relative to a reference electrode (Section 2.7). However, by performing measure- ments at different concentrations and extrapolating to zero concentration, values of E^~ can be obtained. Another factor that can enter into the values of /Г0"' is perturbations caused by other reactions, normally due to complexation. An example of the difference between values of IT0"' and E^ is the values obtained in the potentiometric titration of Fe 2 + with Ce 4 + in 0. 5 M H 2 S O 4. These are, relative to the normal hydrogen electrode (NHE): Standard electrode „... ^ A. t Formal potential r potential E^/V £°7V 3+ 2+ Fe | Fe +0.77 +0.68 Ce4+ Ce3+ +1.61 +1.44 18 Electrochemical cells The differences reflect not only the activities of the ions involved in the half-reactions but also the fact that 0. 5 M H 2 S O 4 does not have pHO (in fact the second ionization is only partially effected). 2.4 Calculation of cell potential: electrochemical potential Although the calculation of EceU in the previous section appears satisfac- tory, it is not very rigorous. In this section we show how a rigorous thermodynamic argument4 leads to the same result. For this we need the concept of the electrochemical potential Д, that obeys the same criteria at equilibrium as the chemical potential \i. Its definition for component / in phase oc is ji?= tf + zfQ" (2.10) a = JU°" + RT Inui + ztF (2.11) which is the sum of a term due to the chemical potential and another that represents the contribution from charged species described by the electrostatic potential ф in phase oc. Since " f = (fr) (2Л2) \drii/.. ТгРгП ф then #] (4.2) then к = A' exp [-(AH* - T AS*)/RT] = A' exp [-AGVRT] (4.3) We now see how the potential applied to the electrode is reflected in the values of AG*. Consider a half-reaction of first order occurring at an 'inert' metallic electrode; species О and R both being soluble. The О | R couple has an associated energy that can be related to the electrode potential (see Section 4.6). We call this energy £redox- By applying a potential to the electrode, we influence the highest occupied electronic level in the electrode. This level is the Fermi level, EF - electrons are always transferred to and from this level. The situation is shown schematically in Fig. 4.2, where one sees how different potentials applied to the electrode can change the direction of electron transfer. The energy level £redox is fixed: by altering the applied potential, and thence EF, we oblige the electrode to supply electrons to species О (reduction) or remove electrons from species R (oxidation). What is, then, the energy profile describing electron transfer? In a similar fashion to the description of the kinetics of homogeneous reactions, in the development of a model for electron transfer parabolic energy profiles have been used for reagents and products. Nevertheless, the region where the profiles intersect is of paramount interest since this corresponds to the activated complex: in this region the energy variation is almost linear—its variation far from the intersection is not important. Figure 4.3 shows a typical profile. A change x in the free energy of О will result in a change acx in the activation energy, assuming a linear  4.4 An expression for the rate of electrode reactions Reduction Oxidation ('negative' electrode potential) ('positive' electrode potential) (a) (b) Fig. 4.2 Electron transfer at an inert metallic electrode. The potential applied to the electrode alters the highest occupied electronic energy level, EF, facilitating (a) reduction or (b) oxidation. intersection. So for a reduction we can write (4.4a) In a similar way, for an oxidation AG* = AG*,o - ocjiFE (4.46) E = E (negative) о Reaction coordinate Fig. 4.3 Effect of a change in applied electrode potential on the reduction of О to R (R considered absent in bulk solution and in the electrode material). 74 Fundamentals of kinetics and mechanism of electrode reactions where E is the potential applied to the electrode and or is a measure of the slope of the energy profiles in the transition state zone and, therefore, of barrier symmetry. Values of aa and ac can vary between 0 and 1, but for metals are around 0.5. A value of 0.5 means that the activated complex is exactly halfway between reagents and products on the reaction coordinate, its structure reflecting reagent and product equally. In this simple case of a one-step transfer of n electrons between О and R, it is easily deduced that (аа + ac) = 1. We now substitute the expressions for AG* from (4.4a) and (4.46). We obtain for a reduction kc = A' exp [-AGiJRT] exp [-acnFE/RT] (4.5a) and for an oxidation ka = A' exp [-AGlo/RT] exp [aanFE/RT] (4.5b) These equations can be rewritten in the form К = kc,o exp [-acnFE/RT] (4.6a) and ka = kOt0 exp [ shown schematically.  References 81 If X ~ 0 then EF = Eredox (4.24) Thus EF is associated with the electrode potential and £ redox with the redox potential of the species: since in general ХФ0, we cannot assume their equivalence. A measurement of potential gives values of electrode potentials and never redox potentials. The crucial point is that the difference of potential available to effect electrode reactions and surmount activation barriers is not simply the difference between the Galvani potential (i.e. the Fermi energy) and the potential in solution. On the side of the solid it is the Volta potential and on the side of the solution it is the potential at the inner Helmholtz plane, where species have to reach to in order for electron transfer to be possible. Corrections to rate constants for the latter are commonly carried out using the Gouy-Chapman model of the electrolyte double layer and will be described in Section 6.9. References 1. K. J. Vetter, Electrochemical kinetics, Academic Press, New York, 1967. 2. W. J. Albery, Electrode kinetics, Clarendon Press, Oxford, 1975. 3. R. A. Marcus, J. Phys. Chem., 1963, 67, 853. 4. J. A. V. Butler, Trans. Faraday Soc, 1924, 19, 729 and 734; T. Erdey-Gruz and M. Volmer, Z. Physik. Chem., 1930, 150A, 203. 5. R. Marcus, Ann. Rev. Phys. Chem., 1964, 15, 155 and references therein. 6. V. G. Levich, Advances in electrochemistry and electrochemical engineering, ed. P. Delahay and C. W. Tobias, Wiley, New York, Vol. 4, 1966, pp. 249-371. 7. R. R. Dogonadze, Reactions of molecules at electrodes, ed. N. H. Hush, Wiley-Interscience, New York, 1971, Chapter 3. 8. A. M. Kuznetsov, Faraday Disc. Chem. Soc, 1982, 74, 49. 9. В. Е. Conway, Modern aspects of electrochemistry, Plenum, New York, Vol. 16, 1985, ed. B. E. Conway, R. E. White, and J. O'M. Bockris, pp. 103-188. 10. P. J. Holmes ed., The electrochemistry of semiconductors, Academic Press, London, 1962. 11. S. R. Morrison, Electrochemistry at semiconductor and oxidised metal electrodes, Plenum, New York, 1980. 12. K. Uosaki and H. Kita, Modern aspects of electrochemistry, Plenum, New York, Vol. 18, 1986, ed. R. E. White, J. O'M. Bockris, and В. Е. Conway, pp. 1-60. 13. A. Hamnett, Comprehensive chemical kinetics, Elsevier, Amsterdam, Vol. 27, 1987, ed. R. G. Compton, Chapter 2. 14. Ref. 11 p. 5. 15. H. Reiss, /. Phys. Chem., 1985, 89, 3873; S. U. M. Khan, R. С Kainthla, and J. O'M. Bockris, /. Phys. Chem., 1987, 91, 5974; H. Reiss, /. Electrochem. Soc, 1988, 135, 247C.  MASS TRANSPORT 5.1 Introduction 5.2 Diffusion control 5.3 Diffusion-limited current: planar and spherical electrodes 5.4 Constant current: planar electrodes 5.5 Microelectrodes 5.6 Diffusion layer 5.7 Convection and diffusion: hydrodynamic systems 5.8 Hydrodynamic systems: some useful parameters 5.9 An example of a convective-diffusion system: the rotating disc electrode 5.1 Introduction In the last chapter it became clear that in the expression for the rate of an electrode reaction the values of ka and kc (electrode kinetics) and of [O]* and [R]* are both of extreme importance. These, in turn, are affected not only by the electrode reaction itself but also by the transport of species to and from bulk solution. This transport can occur by diffusion, convection, or migration (Section 2.8). Normally, conditions are chosen in which migration effects can be neglected, this is the effects of the electrode's electric field are limited to very small distances from the electrode, as described in Chapter 3. These conditions correspond to the presence of a large quantity (>0.1м) of an inert electrolyte (supporting electrolyte), which does not interfere in the electrode reaction. Using a high 3 concentration of inert electrolyte, and concentrations of 10~ м or less of electroactive species, the electrolyte also transports almost all the current in the cell, removing problems of solution resistance and contributions to the total cell potential—an exception to this is ultramicroelectrodes, where the currents are so low that higher solution resistances can be tolerated. In these conditions we need to consider only diffusion and convection.  5.2 Diffusion control 83 Diffusion is due to the thermal movement of charged and neutral species in solution, without electric field effects. Forced convection considerably increases the transport of species, as will be demonstrated, and in many cases can be described mathematically. Natural convection, due to thermal gradients, also exists, but conditions where this movement is negligible are generally used. In this chapter we consider systems under conditions in which the kinetics of the electrode reaction is sufficiently fast that the control of the electrode process is totally by mass transport. This situation can, in principle, always be achieved if the applied potential is sufficiently positive (oxidation) or negative (reduction). First we consider the case of pure diffusion control, and secondly systems where there is a convection component. 5.2 Diffusion control As mentioned previously, diffusion is the natural movement of species in solution, without the effects of the electric field. Thus the species can be charged or neutral. The rate of diffusion depends on the concentration gradients. Fick's first law expresses this: (5.., where / is the flux of species, дс/дх the concentration gradient in direction x—a plane surface is assumed—and D is the proportionality constant known as the diffusion coefficient. Its value in aqueous solution 5 6 2 1 normally varies between 10~ and 10~ cm s~ , and can, in general, be determined through application of the equations for the current-voltage profiles of the various electrochemical methods. Alternatively, the Nernst-Einstein or Stokes-Einstein relations discussed in Chapter 2 may be used to estimate values of D. The next question is: what is the variation of concentration with time due to diffusion? The variation is described by Fick's second law which, for a one-dimensional system, is дс д2с —4 (5.8) with the boundary conditions corresponding to our experiment, which are t = 0, c 0 = Coo (no electrode reaction) (5.9a) 155 0 lim с = Coo (bulk solution) (5.96) >0 1 t>0 x \ co = 0 (diffusion-limited current,/ d ) (5.9c) in which c() represents the concentration at the electrode and c^ the concentration in bulk solution. The mathematical solution to this problem is facilitated by using a dimensionless concentration y =^ - ^ (5.10)  5.3 Diffusion-limited current 87 and then (5.8) is transformed with respect to t using the Laplace transform, leading to *y = £ > 0 (5.11) The general solution to this equation is well known: Y = A'(s) exp [-(s/D)mx] + B'(s) exp [(s/D)1/2x] (5.12) Since the second term of the right-hand side does not satisfy the second boundary condition (*->0), then B'(s) = 0. The third boundary condition in Laplace space is x =0 f=-l/s (5.13) and we obtain from (5.12) for x = 0, A'(s) = -l/s (5.14) Substituting, х] (5.15) and differentiating, ( | £ ) = (*£>Г1/2ехр [-(s/D)V2x] (5.16) Inversion of (5.15) leads to the variation of concentration with distance from the electrode according to the value of t: ( 5 Л 7 ) which is represented in Fig. 5.4 for various values of t. From (5.16) we obtain the current by putting x = 0 and inverting the transform (see Appendix 1, Table 1): Ш (5-18) кдх/о i and so ,ч nFADv\ w (5Л9) — {ntf12 This is known as the Cottrell equation2 (Fig. 5.5). The current decreases with tll2y which means that after a certain time we cannot have confidence in the measured currents owing to the  Mass transport х/цт Fig. 5.4. Variation of concentration with distance at a planar electrode for various values of t after the application of a potential step, following (5.17). contribution of natural convection, etc., that perturbs the concentration gradients. This critical time can vary between some seconds and several minutes depending on the system's experimental arrangement. It should also not be forgotten that, from a practical point of view, for small values of t there is a capacitive contribution to the current, due to double layer charging, that has to be subtracted. This contribution arises Fig. 5.5. Variation of current with time according to the Cottrell equation.  5.3 Diffusion-limited current 89 from the attraction between the electrode and the charges and dipoles in solution, and differs according to the applied potential; a rapid change in applied potential causes a very fast change in the distribution of species on the electrode surface and a large current during up to 0.1s (see Chapter 3). At a spherical electrode, of radius r0, the relevant equation to solve is 2 эс 2 + with boundary conditions t =0 r^r0 с— Co, (no electrode reaction) (5.21a) t^ 0 lim с = Coo (bulk solution) (5.21ft) Г—>°o t>0 r = r0 c =0 (diffusion-limited current, /d) (5.21c) corresponding to (5.8) and (5.9). Defining a dimensionless concentration as before, C ^ (5.10) Coo and putting v = ry, (5.20) becomes dv 32v (5-22) This equation is the same as for a planar electrode and the boundary conditions are of the same form. Thus the method of solution is the same. The result is This is the Cottrell equation, (5.19), plus a spherical correction term. Two extreme cases can be considered: Small t. The second term in (5.23) can be neglected, in other words the spherical nature of the electrode is unimportant. Diffusion at a sphere can be treated as linear diffusion. This is very important for the dropping mercury electrode (Section 8.3): for typical values of drop radius of 0.1 cm and D = 10~5 cm2 s" 1 , after t = 3 s there is only a 10 per cent error in using (5.19). 90 Mass transport Table 5.2. Diffusion currents for planar and spherical electrodes: assuming Do = DR. ro = sphere radius; в = [О Type of electrode Equation Comments ia Oxidation or Planar nFAD c Cottrell equation (5.19) reduction (*0 1/2 Spherical nFADl/2c^ nFADcx Cottrell equation plus ("0" 2 ' spherical correction (5.23) Oxidation and Planar nFADV2cx reduction (close (1 + в)(М)1'2 to equilibrium Spherical nFADl/2cx nFADcx potential) (1 + в){м)ш (l + 0)ro Large t. The spherical term dominates, which represents a steady- state current. However, due to the effects of natural convection this steady state is never reached at conventionally-sized electrodes. The smaller the electrode radius, the faster the steady state is achieved. It is possible to achieve a steady state at microelectrodes. These are described further in Section 5.5. The equations for the diffusion-limited current at planar and spherical electrodes are shown in Table 5.2 together with the expressions for the diffusion currents when the potential is not far from the equilibrium potential so that oxidation and reduction occur at the same time. 5.4 Constant current: planar electrodes Starting at t = 0 a constant current is applied to the electrode in order to cause oxidation or reduction of electroactive species, and the variation of the potential of the electrode with time is measured (chronopoten- tiometry). Fick's second law is solved using the Laplace transform as in the previous section; the first two boundary conditions are the same, but the third is different: t =0 c 0 = Coo (no electrode reaction) (5.9a) t^O lim с = сх (bulk solution) (5.9b) Jt—»QO t>0 x =0 I = nFAD(dc/dx)0 (5.24) The third condition expresses the fact that a concentration gradient is being imposed at the electrode surface. As in the last section, and following the same arguments, we reach Y=A'(s) exp [-(s/D)1/2x]