SECOND EDITION

ELECTROCHEMICAL METHODS
Fundamentals and Applications

Allen J. Bard Larry R. Faulkner
Department of Chemistry and Biochemistry University of Texas at Austin

JOHN WILEY & SONS, INC.
New Yorke Chichester • Weinheim Brisbane e Singapore e Toronto

Acquisitions Editor David Harris Senior Production Editor Elizabeth Swain Senior Marketing Manager Charity Robey Illustration Editor Eugene Aiello This book was set in 10/12 Times Roman by University Graphics and printed and bound by Hamilton. The cover was printed by Phoenix. This book is printed on acid-free paper, oo

Copyright 2001 © John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: PERMREQ@WILEY.COM. To order books or for customer service, call 1 (800)-CALL-WILEY (225-5945). Library of Congress Cataloging in Publication Data: Bard, Allen J. Electrochemical methods : fundamentals and applications / Allen J. Bard, Larry R. Faulkner.— 2nd ed. p. cm. Includes index. ISBN 0-471-04372-9 (cloth : alk. paper) 1. Electrochemistry. I. Faulkner, Larry R., 1944- II. Title. QD553.B37 2000 541.3'7_dc21 00-038210 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

PREFACE
In the twenty years since the appearance of our first edition, the fields of electrochemistry and electroanalytical chemistry have evolved substantially. An improved understanding of phenomena, the further development of experimental tools already known in 1980, and the introduction of new methods have all been important to that evolution. In the preface to the 1980 edition, we indicated that the focus of electrochemical research seemed likely to shift from the development of methods toward their application in studies of chemical behavior. By and large, history has justified that view. There have also been important changes in practice, and our 1980 survey of methodology has become dated. In this new edition, we have sought to update the book in a way that will extend its value as a general introduction to electrochemical methods. We have maintained the philosophy and approach of the original edition, which is to provide comprehensive coverage of fundamentals for electrochemical methods now in widespread use. This volume is intended as a textbook and includes numerous problems and chemical examples. Illustrations have been employed to clarify presentations, and the style is pedagogical throughout. The book can be used in formal courses at the senior undergraduate and beginning graduate levels, but we have also tried to write in a way that enables self-study by interested individuals. A knowledge of basic physical chemistry is assumed, but the discussions generally begin at an elementary level and develop upward. We have sought to make the volume self-contained by developing almost all ideas of any importance to our subject from very basic principles of chemistry and physics. Because we stress foundations and limits of application, the book continues to emphasize the mathematical theory underlying methodology; however the key ideas are discussed consistently apart from the mathematical basis. Specialized mathematical background is covered as needed. The problems following each chapter have been devised as teaching tools. They often extend concepts introduced in the text or show how experimental data are reduced to fundamental results. The cited literature is extensive, but mainly includes only seminal papers and reviews. It is impossible to cover the huge body of primary literature in this field, so we have made no attempt in that direction. Our approach is first to give an overview of electrode processes (Chapter 1), showing the way in which the fundamental components of the subject come together in an electrochemical experiment. Then there are individual discussions of thermodynamics and potential, electron-transfer kinetics, and mass transfer (Chapters 2-4). Concepts from these basic areas are integrated together in treatments of the various methods (Chapters 5-11). The effects of homogeneous kinetics are treated separately in a way that provides a comparative view of the responses of different methods (Chapter 12). Next are discussions of interfacial structure, adsorption, and modified electrodes (Chapters 13 and 14); then there is a taste of electrochemical instrumentation (Chapter 15), which is followed by an extensive introduction to experiments in which electrochemistry is coupled with other tools (Chapters 16-18). Appendix A teaches the mathematical background; Appendix В provides an introduction to digital simulation; and Appendix С contains tables of useful data.

vi • Preface This structure is generally that of the 1980 edition, but important additions have been made to cover new topics or subjects that have evolved extensively. Among them are applications of ultramicroelectrodes, phenomena at well-defined surfaces, modified electrodes, modern electron-transfer theory, scanning probe methods, LCEC, impedance spectrometry, modern forms of pulse voltammetry, and various aspects of spectroelectrochemistry. Chapter 5 in the first edition ("Controlled Potential Microelectrode Techniques—Potential Step Methods") has been divided into the new Chapter 5 ("Basic Potential Step Methods") and the new Chapter 7 ("Polarography and Pulse Voltammetry"). Chapter 12 in the original edition ("Double Layer Structure and Adsorbed Intermediates in Electrode Processes") has become two chapters in the new edition: Chapter 12 ("Double-Layer Structure and Adsorption") and Chapter 13 ("Electroactive Layers and Modified Electrodes"). Whereas the original edition covered in a single chapter experiments in which other characterization methods are coupled to electrochemical systems (Chapter 14, "Spectrometric and Photochemical Experiments"), this edition features a wholly new chapter on "Scanning Probe Techniques" (Chapter 16), plus separate chapters on "Spectroelectrochemistry and Other Coupled Characterization Methods" (Chapter 17) and "Photoelectrochemistry and Electrogenerated Chemiluminescence" (Chapter 18). The remaining chapters and appendices of the new edition directly correspond with counterparts in the old, although in most there are quite significant revisions. The mathematical notation is uniform throughout the book and there is minimal duplication of symbols. The List of Major Symbols and the List of Abbreviations offer definitions, units, and section references. Usually we have adhered to the recommendations of the IUPAC Commission on Electrochemistry [R. Parsons et al., Pure Appl. С hem., 37, 503 (1974)]. Exceptions have been made where customary usage or clarity of notation seemed compelling. Of necessity, compromises have been made between depth, breadth of coverage, and reasonable size. "Classical" topics in electrochemistry, including many aspects of thermodynamics of cells, conductance, and potentiometry are not covered here. Similarly, we have not been able to accommodate discussions of many techniques that are useful but not widely practiced. The details of laboratory procedures, such as the design of cells, the construction of electrodes, and the purification of materials, are beyond our scope. In this edition, we have deleted some topics and have shortened the treatment of others. Often, we have achieved these changes by making reference to the corresponding passages in the first edition, so that interested readers can still gain access to a deleted or attenuated topic. As with the first edition, we owe thanks to many others who have helped with this project. We are especially grateful to Rose McCord and Susan Faulkner for their conscientious assistance with myriad details of preparation and production. Valuable comments have been provided by S. Amemiya, F. C. Anson, D. A. Buttry, R. M. Crooks, P. He, W. R. Heineman, R. A. Marcus, A. C. Michael, R. W. Murray, A. J. Nozik, R. A. Osteryoung, J.-M. Saveant, W. Schmickler, M. P. Soriaga, M. J. Weaver, H. S. White, R. M. Wightman, and C. G. Zoski. We thank them and our many other colleagues throughout the electrochemical community, who have taught us patiently over the years. Yet again, we also thank our families for affording us the time and freedom required to undertake such a large project. Allen /. Bard Larry R. Faulkner

CONTENTS
MAJOR SYMBOLS ix STANDARD ABBREVIATIONS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 xix 1

INTRODUCTION AND OVERVIEW OF ELECTRODE PROCESSES POTENTIALS AND THERMODYNAMICS OF CELLS KINETICS OF ELECTRODE REACTIONS BASIC POTENTIAL STEP METHODS POTENTIAL SWEEP METHODS 226 261 305 156 87 137 MASS TRANSFER BY MIGRATION AND DIFFUSION 44

POLAROGRAPHY AND PULSE VOLTAMMETRY CONTROLLED-CURRENT TECHNIQUES

METHODS INVOLVING FORCED CONVECTION—HYDRODYNAMIC METHODS 331 TECHNIQUES BASED ON CONCEPTS OF IMPEDANCE BULK ELECTROLYSIS METHODS 417 368

ELECTRODE REACTIONS WITH COUPLED HOMOGENEOUS CHEMICAL REACTIONS 471 DOUBLE-LAYER STRUCTURE AND ADSORPTION ELECTROCHEMICAL INSTRUMENTATION SCANNING PROBE TECHNIQUES 659 632 534 580 ELECTROACTIVE LAYERS AND MODIFIED ELECTRODES

SPECTROELECTROCHEMISTRY AND OTHER COUPLED CHARACTERIZATION METHODS 680 PHOTOELECTROCHEMISTRY AND ELECTROGENERATED CHEMILUMINESCENCE 736

APPENDICES A В С MATHEMATICAL METHODS REFERENCE TABLES INDEX 814 808 769 785 DIGITAL SIMULATIONS OF ELECTROCHEMICAL PROBLEMS

MAJOR SYMBOLS
Listed below are symbols used in several chapters or in large portions of a chapter. Symbols similar to some of these may have different local meanings. In most cases, the usage follows the recommendations of the IUPAC Commission on Electrochemistry [R. Parsons et al., Pure Appl. Chem., 37, 503 (1974).]; however there are exceptions. A bar over a concentration or a current [ej*., C o (x, s)] indicates the Laplace transform of the variable. The exception is when / indicates an average current in polarography.

STANDARD SUBSCRIPTS a с D d anodic (a) cathodic (b) charging disk diffusion dl double layer eq equilibrium f (a) forward (b) faradaic limiting / 0 P R r pertaining to species 0 in О + ne ±± R peak (a) pertaining to species R in О + ne ^ R (b) ring reverse

ROMAN SYMBOLS
Symbol Meaning (a) area (b) cross-sectional area of a porous electrode (c) frequency factor in a rate expression (d) open-loop gain of an amplifier absorbance (a) internal area of a porous electrode (b) tip radius in SECM activity of substance j in a phase a aFv/RT С CB capacitance series equivalent capacitance of a cell differential capacitance of the double layer integral capacitance of the double layer concentration of species; bulk concentration of species; concentration of species; at distance x Usual Units cm cm 2 depends on order none none cm 2 none s" 1 mol/cm2 F F F, F/cm2 F, F/cm2 M, mol/cm3 M, mol/cm3 M, mol/cm3 Section References 1.3.2 11.6.2 3.1.2 15.1.1 17.1.1 11.6.2 16.4.1 2.1.5 6.3.1 13.5.3 1.2.2, 10.1.2 10.4 1.2.2, 13.2.2 13.2.2 1.4.2, 4.4.3 1.4

cd

c't

Major Symbols
Usual Units M, mol/cm3 M, mol/cm3 M, mol/cm3 M, mol/cm3 M, mol/cm3 F/cm F cm/s cm /s cm2/s cm 3 eV~ ! none cm2/s Section References 1.4.2 4.4 4.4.3 9.3.3 9.3.4 18.2.2 10.1.3 17.1.2 14.4.2 1.4.1,4.4 3.6.3 B.1.3.B.1.8 14.4.2

Symbol CjCx = 0) Cj(x, t) Cj(O, f)

Meaning concentration of species j at the electrode surface concentration of species у at distance x at time t concentration of species у at the electrode surface at time t concentration of species у at distance у away from rotating electrode surface concentration of species у at a rotating electrode space charge capacitance pseudocapacity speed of light in vacuo diffusion coefficient for electrons within the film at a modified electrode diffusion coefficient of species у concentration density of states for species у model diffusion coefficient in simulation diffusion coefficient for the primary reactant within the film at a modified electrode distance of the tip from the substrate in SECM density of phase у (a) potential of an electrode versus a reference (b) emf of a reaction (c) amplitude of an ac voltage (a) pulse height in DPV (b) step height in tast or staircase voltammetry (c) amplitude (1/2 p-p) of ac excitation in ac voltammetry electron energy electric field strength vector electric field strength voltage or potential phasor (a) standard potential of an electrode or a couple (b) standard emf of a half-reaction difference in standard potentials for two couples electron energy corresponding to the standard potential of a couple formal potential of an electrode activation energy of a reaction ac component of potential base potential in NPV and RPV dc component of potential

Cj(y = 0) Csc
С

Dj(A, E) DM £s

d *\ E

/xm, nm g/cm3 V V V mV mV mV eV V/cm V/cm V V V V eV V kJ/mol mV V V

16.4.1

1.1,2.1 2.1 10.1.2 7.3.4 7.3.1 10.5.1 2.2.5, 3.6.3 2.2.1 2.2.1 10.1.2 2.1.4 2.1.4 6.6 3.6.3 2.1.6 3.1.2 10.1.1 7.3.2, 7.3.3 10.1.1

AE

E % % E

£°

AE°

E°
E0' EA Eac Eb Edc

Major Symbols Symbol Ещ EF Em
Eg

xi

Meaning equilibrium potential of an electrode Fermi level flat-band potential bandgap of a semiconductor initial potential junction potential membrane potential peak potential (a)|£pa-£pc|inCV (b) pulse height in SWV potential where / = /p/2 in LSV anodic peak potential cathodic peak potential staircase step height in SWV potential of zero charge switching potential for cyclic voltammetry quarter-wave potential in chronopotentiometry (a) measured or expected half-wave potential in voltammetry (b) in derivations, the "reversible" half-wave potential, Eo> + (RT/nF)\n(DR/D0)l/2 potential where i/i^ = 1 / 4 potential where ///d = 3/4 (a) electronic charge (b) voltage in an electric circuit input voltage output voltage voltage across the input terminals of an amplifier error function of x error function complement of x the Faraday constant; charge on one mole of electrons (a) F/RT (b) frequency of rotation (c) frequency of a sinusoidal oscillation (d) SWV frequency (e) fraction titrated Fermi function fractional concentration of species / in boxy after iteration к in a simulation Gibbs free energy Gibbs free energy change in a chemical process electrochemical free energy standard Gibbs free energy

Usual Units V eV V eV V mV mV V V mV V V V mV V V V V V

Section References 1.3.2,3.4.1 2.2.5, 3.6.3 18.2.2 18.2.2 6.2.1 2.3.4 2.4 6.2.2 6.5 7.3.5 6.2.2 6.5 6.5 7.3.5 13.2.2 6.5 8.3.1 1.4.2,5.4,5.5 5.4

E; Em

Щ

EP A£P Ep/2 £pa £pc £Z *л
Еф E\I2

Ещ
Е

Ъ1А

V V V V V /xV none none С V" 1 r/s s- 1 s- 1 none none none kJ, kJ/mol kJ, kJ/mol kJ, kJ/mol kJ, kJ/mol

5.4.1 5.4.1 10.1.1,15.1 15.2 15.1.1 15.1.1 A.3 A.3

e e\ e0

с

ег%) erfc(x) F

f

/(E)

fUk)
G AG G

9.3 10.1.2 7.3.5 11.5.2 3.6.3 B.1.3 2.2.4 2.1.2,2.1.3 2.2.4 3.1.2

G°

xii

Major Symbols Section References 2.1.2,2.1.3 3.1.2 2.3.6

Symbol

Meaning standard Gibbs free energy change in a chemical process standard Gibbs free energy of activation standard free energy of transfer for species j from phase a into phase /3 (a) gravitational acceleration (b) interaction parameter in adsorption isotherms (a) enthalpy enthalpy change in a chemical process standard enthalpy change in a chemical process standard enthalpy of activation Planck constant corrected mercury column height at a DME amplitude of an ac current convolutive transform of current; semi-integral of current current phasor diffusion current constant for average current diffusion current constant for maximum current peak value of ac current amplitude current difference current in SWV = if — ir difference current in DPV = /(r) - Z(r') initial current in bulk electrolysis characteristic current describing flux of the primary reactant to a modified RDE anodic component current (a) charging current (b) cathodic component current (a) current due to diffusive flux (b) diffusion-limited current average diffusion-limited current flow over a drop lifetime at a DME diffusion-limited current at tm.dX at a DME (maximum current) characteristic current describing diffusion of electrons within the film at a modified electrode (a) faradaic current (b) forward current kinetically limited current characteristic current describing cross-reaction within the film at a modified electrode

Usual Units kJ, kJ/mol kJ/mol kJ/mol cm/s 2 2 J-cm /mol kJ, kJ/mol s AG°

дс!transfer, j j 2

13.5.2 2.1.2 5.5.1 2.1.2 2.1.2 3.1.2 7.1.4 10.1.2 6.7.1 10.1.2 7.1.3 7.1.3 10.5.1 1.3.2 7.3.5 7.3.4 11.3.1 14.4.2 3.2 6.2.4 3.2 4.1 5.2.1 7.1.2 7.1.2 14.4.2

H Mi

-l/2

A#°

kJ, kJ/mol kJ, kJ/mol kJ/mol J-s cm A C/s1/2 A ^A-s1/2/(mg2/3-mM) M-s 1/2 /(mg 2/3 -mM) A A A A A A A A A A A A A A

/

/(0

7

/

А/ 8i /(0)
*А

Od)max

A A A A

5.7 9.3.4 14.4.2

Major Symbols
Symbol Ч Meaning limiting current limiting anodic current limiting cathodic current migration current characteristic current describing permeation of the primary reactant into the film at a modified electrode peak current anodic peak current cathodic peak current current during reversal step (a) characteristic current describing diffusion of the primary reactant through the film at a modified electrode (b) substrate current in SECM steady-state current tip current in SECM tip current in SECM far from the substrate exchange current true exchange current imaginary part of complex function w flux of species j at location x at time t (a) current density (b) box index in a simulation Usual Units A A A A A Section References 1.4.2 1.4.2 1.4.2 4.1 14.4.2

xui

k& kc h

>P
'pa *pc

'r 'S

A A A A A

6.2.2 6.5.1 6.5.1 5.7 14.4.2

4s

h
*T,oo

A A A A A A

•

16.4.4 5.3 16.4.2 16.4.1 3.4.1,3.5.4 13.7.1 A.5 1.4.1,4.1 1.3.2 B.1.2 A.5 3.4.1,3.5.4 3.6.1

h
*0,t

Im(w)

/jfe t) j (c)V^I h К exchange current density equilibrium constant precursor equilibrium constant for reactant j (a) rate constant for a homogeneous reaction (b) iteration number in a simulation (c) extinction coefficient Boltzmann constant standard heterogeneous rate constant (a) heterogeneous rate constant for oxidation (b) homogeneous rate constant for "backward" reaction (a) heterogeneous rate constant for reduction (b) homogeneous rate constant for "forward" reaction potentiometric selectivity coefficient of interferenty toward a measurement of species / true standard heterogeneous rate constant

mol c m " 2 s" 1 A/cm2 none none A/cm none depends on case depends on order none none J/K cm/s cm/s depends on order cm/s depends on order none

к

B.I 17.1.2 3.3, 3.4 3.2 3.1 3.2 3.1 2.4

k°

К

kf

*?? k° cm/s

13.7.1

xiv

Major Symbols
Usual Units Section References 11.6.2 A.I A.I 11.7.2 B.1.4 7.1.2 6.7.1 1.4.2 9.4.2 18.2.2 18.2.2 11.3.1 1.3.2 18.2.2 17.1.2 17.1.2 13.3.2 18.2.2 2.2.4, 13.1.1 13.3.2 13.3.2

Symbol L L{f(t)} L~]{f(s)} I € m m(t) m-} N NA ND iVj n

Meaning length of a porous electrode Laplace transform of/(0 = f(s) inverse Laplace transform of f(s) thickness of solution in a thin-layer cell number of iterations corresponding to t^ in a simulation mercury flow rate at a DME convolutive transform of current; semi-integral of current mass-transfer coefficient of species j collection efficiency at an RRDE (a) acceptor density (b) Avogadro's number donor density total number of moles of species j in a system (a) stoichiometric number of electrons involved in an electrode reaction (b) electron density in a semiconductor (c) refractive index complex refractive index number concentration of each ion in a z: z electrolyte electron density in an intrinsic semiconductor (a) number of moles of species у in a phase (b) number concentration of ion у in an electrolyte number concentration of ion у in the bulk electrolyte oxidized form of the standard system О + ne ^ R; often used as a subscript denoting quantities pertaining to species О pressure (a) hole density in a semiconductor (b) mjA/V hole density in an intrinsic semiconductor charge passed in electrolysis charge required for complete electrolysis of a component by Faraday's law chronocoulometric charge from a diffusing component charge devoted to double-layer capacitance excess charge on phase у reduced form of the standard system, О + ne i=^ R; often used as a subscript denoting quantities pertaining to species R

cm none mg/s 1/2 C/s cm/s none 3 cm" 1 тоГ 3 cm" mol none cm" 3 none none cm" 3 cm" 3 mol cm" 3 cm
-3

n n° щ щ

n® О

P p P\ Q o2 (c) dimensionless homogeneous kinetic parameter, specific to a method and mechanism (d) switching time in CV (e) wavelength of light in vacuo inner component of the reorganization energy equivalent ionic conductivity for ion у equivalent ionic conductivity of ion у extrapolated to infinite dilution

3.1.3 5.5.2 13.3.2 14.4.2 3.6 2.3.3 3.6 5.5.1 12.3

s nm eV cm 2 II 1 equiv ] cm 2 fl" 1 equiv" 1

6.5 17.1.2 3.6.2 2.3.3 2.3.3

xviii

Major Symbols Section References 3.6.2 1.5.2, 12.4.2 17.1.2 2.2.4, 2.2.5 2.2.4 2.2.4 2.2.4 9.2.2 2.1.5 3.6 5.4.1 4.2 5.2.3 3.6.3 6.2.1 10.2.3 1.2,2.2 13.3.4 8.2.2 5.1,7.3 5.7.1

Symbol

Meaning outer component of the reorganization energy (a) reaction layer thickness (b) magnetic permeability electrochemical potential of electrons in phase a electrochemical potential of species j in phase a chemical potential of species у in phase a standard chemical potential of species j in phase a (a) kinematic viscosity (b) frequency of light stoichiometric coefficient for species у in a chemical process nuclear frequency factor (D0/DR)112 (a) resistivity (b) roughness factor electronic density of states (a) nFv/RT (b) (1MFAV2)[/3O/£>O/2 " J3R/£>R2] excess charge density on phase у parameter describing potential dependence of adsorption energy (a) transition time in chronopotentiometry (b) sampling time in sampled-current voltammetry (c) forward step duration in a double-step experiment (d) generally, a characteristic time defined by the properties of an experiment (e) in treatments of UMEs, 4Dot/rl start of potential pulse in pulse voltammetry longitudinal relaxation time of a solvent work function of a phase (a) electrostatic potential (b) phase angle between two sinusoidal signals (c) phase angle between / a c and £ a c (d) film thickness in a modified electrode (a) electrostatic potential difference between two points or phases (b) potential drop in the space charge region of a semiconductor absolute electrostatic potential of phase j junction potential at a liquid-liquid interface

Usual Units eV cm none kJ/mol kJ/mol kJ/mol kJ/mol cm2/s none s" 1 none fl-cm none cm 2 eV" 1 s" 1 C/cm2 none s s

К

p(E)

Ф

Ф

none s s eV V degrees, radians degrees, radians cm V

5.3 7.3 3.6.2 3.6.4 2.2.1 10.1.2 10.1.2 14.4.2 2.2 18.2.2

V V

2.2.1 6.8

Major Symbols

xix

Symbol

Meaning standard Galvani potential of ion transfer for species j from phase a to phase /3 total potential drop across the solution side of the double layer potential at the OHP with respect to bulk solution (12/7)1/2£fT1/2/Do/2 dimensionless distance of box; in a simulation normalized current for a totally irreversible system in LSV and CV normalized current for a reversible system in LSV and CV rate constant for permeation of the primary reactant into the film at a modified electrode (a) ellipsometric parameter (b) dimensionless rate parameter in CV (a) angular frequency of rotation; 2тг X rotation rate (b) angular frequency of a sinusoidal oscillation; 2rrf

Usual Units V mV V none none none none cm/s

Section References 6.8 13.3.2 1.2.3, 13.3.3 7.2.2 B.1.5 6.3.1 6.2.1 14.4.2

Ф о ф2 X XU) x (bt) (at)

x

Xf

Ф

none none s" 1 s" 1

17.1.2 6.5.2 9.3 10.1.2

STANDARD ABBREVIATIONS
Abbreviation ADC AES AFM ASV BV CB CE CV CZE DAC DME DMF DMSO DPP DPV Meaning analog-to-digital converter Auger electron spectrometry atomic force microscopy anodic stripping voltammetry Butler- Volmer conduction band homogeneous chemical process preceding heterogeneous electron transfer1 cyclic voltammetry capillary zone electrophoresis digital-to-analog converter (a) dropping mercury electrode (b) 1,2-dimethoxyethane T , TV-dimethylformamide V Dimethylsulfoxide differential pulse polarography differential pulse voltammetry Section Reference 15.8 17.3.3 16.3 11.8 3.3 18.2.2 12.1.1 6.1,6.5 11.6.4 15.8 7.1.1

7.3.4 7.3.4

betters may be subscripted i, q, or r to indicate irreversible, quasi-reversible, or reversible reactions.

xx

Major Symbols
Section Reference 12.1.1 12.1.1 12.1.1 18.1 13.2.2 12.1.1 10.1.1 2.1.3 17.2.1 17.4.1 16.2 17.6.1 A.6 13.3.3 8.6 13.4.2 5.2.2 13.4.2 1.2.3, 13.3.3 1.2.1 17.2.1 17.2.1 2.4 6.8 18.2.5 14.2.1 11.6.4 17.3.3 6.1 11.8 1.1.1 7.3.2 7.3.2 1.2.3, 13.3.3 17.1.1 17.1.1 11.6.4 17.2.1 13.2.2 17.5

Abbreviation EC EC' ECE ECL ECM ЕЕ EIS emf EMIRS ESR ESTM EXAFS FFT GCS GDP HCP HMDE HOPG IHP IPE IRRAS IR-SEC ISE ITIES ITO LB LCEC LEED LSV MFE NHE NCE NPP NPV OHP OTE OTTLE PAD PC PDIRS PZC QCM
1

Meaning heterogeneous electron transfer followed by homogeneous 1 chemical reaction catalytic regeneration of the electroactive species in a following 1 homogeneous reaction heterogeneous electron transfer, homogeneous chemical reaction, and heterogeneous electron transfer, in sequence electrogenerated chemiluminescence electrocapillary maximum step wise heterogeneous electron transfers to accomplish a 2-electron reduction or oxidation of a species electrochemical impedance spectroscopy electromotive force electrochemically modulated infrared reflectance spectroscopy electron spin resonance electrochemical scanning tunneling microscopy extended X-ray absorption fine structure fast Fourier transform Gouy-Chapman-Stern galvanostatic double pulse hexagonal close-packed hanging mercury drop electrode highly oriented pyrolytic graphite inner Helmholtz plane ideal polarized electrode infrared reflection absorption spectroscopy infrared spectroelectrochemistry ion-selective electrode interface between two immiscible electrolyte solutions indium-tin oxide thin film Langmuir-Blodgett liquid chromatography with electrochemical detection low-energy electron diffraction linear sweep voltammetry mercury film electrode normal hydrogen electrode = SHE normal calomel electrode, Hg/Hg2Cl2/KCl (1.0M) normal pulse polarography normal pulse voltammetry outer Helmholtz plane optically transparent electrode optically transparent thin-layer electrode pulsed amperometric detection propylene carbonate potential difference infrared spectroscopy potential of zero charge quartz crystal microbalance

Letters may be subscripted /, q, or r to indicate irreversible, quasi-reversible, or reversible reactions.

Major Symbols

xxi

Abbreviation QRE RDE RDS RPP RPV RRDE SAM SCE SECM SERS SHE SHG SMDE SNIFTIRS SPE SPR SSCE STM TBABF4 TBAI TBAP TEAP THF UHV UME UPD XPS VB

Meaning quasi-reference electrode rotating disk electrode rate-determining step reverse pulse polarography reverse pulse voltammetry rotating ring-disk electrode self-assembled monolayer saturated calomel electrode scanning electrochemical microscopy surface enhanced Raman spectroscopy standard hydrogen electrode = NHE second harmonic generation static mercury drop electrode subtractively normalized interfacial Fourier transform infrared spectroscopy solid polymer electrolyte surface plasmon resonance sodium saturated calomel electrode, Hg/Hg2Cl2/NaCl (sat'd) scanning tunneling microscopy square wave voltammetry tetra-/2-butylammonium fluoborate tetra-ft-butylammonium iodide tetra-w-butylammoniumperchlorate tetraethylammonium perchlorate tetrahydrofuran ultrahigh vacuum ultramicroelectrode underpotential deposition X-ray photoelectron spectrometry valence band

Section Reference 2.1.7 9.3 3.5 7.3.4 7.3.4 9.4.2 14.2.2 1.1.1 16.4 17.2.2 1.1.1 17.1.5 7.1.1 17.2.1 14.2.6 17.1.3 16.2 7.3.5

swv

17.3 5.3 11.2.1 17.3.2 18.2.2

CHAPTER

1
INTRODUCTION AND OVERVIEW OF ELECTRODE PROCESSES
1.1 INTRODUCTION
Electrochemistry is the branch of chemistry concerned with the interrelation of electrical and chemical effects. A large part of this field deals with the study of chemical changes caused by the passage of an electric current and the production of electrical energy by chemical reactions. In fact, the field of electrochemistry encompasses a huge array of different phenomena (e.g., electrophoresis and corrosion), devices (electrochromic displays, electro analytical sensors, batteries, and fuel cells), and technologies (the electroplating of metals and the large-scale production of aluminum and chlorine). While the basic principles of electrochemistry discussed in this text apply to all of these, the main emphasis here is on the application of electrochemical methods to the study of chemical systems. Scientists make electrochemical measurements on chemical systems for a variety of reasons. They may be interested in obtaining thermodynamic data about a reaction. They may want to generate an unstable intermediate such as a radical ion and study its rate of decay or its spectroscopic properties. They may seek to analyze a solution for trace amounts of metal ions or organic species. In these examples, electrochemical methods are employed as tools in the study of chemical systems in just the way that spectroscopic methods are frequently applied. There are also investigations in which the electrochemical properties of the systems themselves are of primary interest, for example, in the design of a new power source or for the electrosynthesis of some product. Many electrochemical methods have been devised. Their application requires an understanding of the fundamental principles of electrode reactions and the electrical properties of electrode-solution interfaces. In this chapter, the terms and concepts employed in describing electrode reactions are introduced. In addition, before embarking on a detailed consideration of methods for studying electrode processes and the rigorous solutions of the mathematical equations that govern them, we will consider approximate treatments of several different types of electrode reactions to illustrate their main features. The concepts and treatments described here will be considered in a more complete and rigorous way in later chapters.

2 • Chapter 1. Introduction and Overview of Electrode Processes

1.1.1

Electrochemical Cells and Reactions In electrochemical systems, we are concerned with the processes and factors that affect the transport of charge across the interface between chemical phases, for example, between an electronic conductor (an electrode) and an ionic conductor (an electrolyte). Throughout this book, we will be concerned with the electrode/electrolyte interface and the events that occur there when an electric potential is applied and current passes. Charge is transported through the electrode by the movement of electrons (and holes). Typical electrode materials include solid metals (e.g., Pt, Au), liquid metals (Hg, amalgams), carbon (graphite), and semiconductors (indium-tin oxide, Si). In the electrolyte phase, charge is carried by the movement of ions. The most frequently used electrolytes are liq+ + uid solutions containing ionic species, such as, H , Na , Cl~, in either water or a nonaqueous solvent. To be useful in an electrochemical cell, the solvent/electrolyte system must be of sufficiently low resistance (i.e., sufficiently conductive) for the electrochemical experiment envisioned. Less conventional electrolytes include fused salts (e.g., molten NaCl-KCl eutectic) and ionically conductive polymers (e.g., Nation, polyethylene oxide-LiClO4). Solid electrolytes also exist (e.g., sodium j8-alumina, where charge is carried by mobile sodium ions that move between the aluminum oxide sheets). It is natural to think about events at a single interface, but we will find that one cannot deal experimentally with such an isolated boundary. Instead, one must study the properties of collections of interfaces called electrochemical cells. These systems are defined most generally as two electrodes separated by at least one electrolyte phase. In general, a difference in electric potential can be measured between the electrodes in an electrochemical cell. Typically this is done with a high impedance voltmeter. This cell potential, measured in volts (V), where 1 V = 1 joule/coulomb (J/C), is a measure of the energy available to drive charge externally between the electrodes. It is a manifestation of the collected differences in electric potential between all of the various phases in the cell. We will find in Chapter 2 that the transition in electric potential in crossing from one conducting phase to another usually occurs almost entirely at the interface. The sharpness of the transition implies that a very high electric field exists at the interface, and one can expect it to exert effects on the behavior of charge carriers (electrons or ions) in the interfacial region. Also, the magnitude of the potential difference at an interface affects the relative energies of the carriers in the two phases; hence it controls the direction and the rate of charge transfer. Thus, the measurement and control of cell potential is one of the most important aspects of experimental electrochemistry. Before we consider how these operations are carried out, it is useful to set up a shorthand notation for expressing the structures of cells. For example, the cell pictured in Figure 1.1.1a is written compactly as Zn/Zn 2 + , СГ/AgCl/Ag (l.l.l)

In this notation, a slash represents a phase boundary, and a comma separates two components in the same phase. A double slash, not yet used here, represents a phase boundary whose potential is regarded as a negligible component of the overall cell potential. When a gaseous phase is involved, it is written adjacent to its corresponding conducting element. For example, the cell in Figure 1.1.1ft is written schematically as Pt/H2/H+, СГ/AgCl/Ag (1.1.2)

The overall chemical reaction taking place in a cell is made up of two independent half-reactions, which describe the real chemical changes at the two electrodes. Each halfreaction (and, consequently, the chemical composition of the system near the electrodes)

1.1 Introduction
Pt H2

3

Zn

Ag

СГ
Excess AgCI (а)

СГ j Excess AgCI (Ь)

Figure l.l.l Typical electrochemical cells, (a) Zn metal and Ag wire covered with AgCI immersed in a ZnCl2 solution, (b) Pt wire in a stream of H2 and Ag wire covered with AgCI in HC1 solution.

responds to the interfacial potential difference at the corresponding electrode. Most of the time, one is interested in only one of these reactions, and the electrode at which it occurs is called the working (or indicator) electrode. To focus on it, one standardizes the other half of the cell by using an electrode (called a reference electrode) made up of phases having essentially constant composition. The internationally accepted primary reference is the standard hydrogen electrode (SHE), or normal hydrogen electrode (NHE), which has all components at unit activity: Pt/H2(a - l)/H + (a = 1, aqueous) (1.1.3)

Potentials are often measured and quoted with respect to reference electrodes other than the NHE, which is not very convenient from an experimental standpoint. A common reference is the saturated calomel electrode (SCE), which is Hg/Hg2Cl2/KCl (saturated in water) Its potential is 0.242 V vs. NHE. Another is the silver-silver chloride electrode, Ag/AgCl/KCl (saturated in water) (1.1.5) (1.1.4)

with a potential of 0.197 V vs. NHE. It is common to see potentials identified in the literature as "vs. Ag/AgQ" when this electrode is used. Since the reference electrode has a constant makeup, its potential is fixed. Therefore, any changes in the cell are ascribable to the working electrode. We say that we observe or control the potential of the working electrode with respect to the reference, and that is equivalent to observing or controlling the energy of the electrons within the working electrode (1, 2). By driving the electrode to more negative potentials (e.g., by connecting a battery or power supply to the cell with its negative side attached to the working electrode), the energy of the electrons is raised. They can reach a level high enough to transfer into vacant electronic states on species in the electrolyte. In that case, a flow of electrons from electrode to solution (a reduction current) occurs (Figure 1.1.2a). Similarly, the energy of the electrons can be lowered by imposing a more positive potential, and at some point electrons on solutes in the electrolyte will find a more favorable energy on the electrode and will transfer there. Their flow, from solution to electrode, is an oxidation current (Figure 1.1.2b). The critical potentials at which these processes occur are related to the standard potentials, E°, for the specific chemical substances in the system.

4

Chapter 1. Introduction and Overview of Electrode Processes Electrode Solution
Electrode Solution

0
Potential Energy level of electrons

Vacant MO

0j

Occupied MO A + e— A > (a) Solution

Electrode

Solution

Electrode

0
Energy level of electrons Potential

Vacant MO

0l

Occupied MO A - e -^ A+ (b)

Figure 1.1.2 Representation of (a) reduction and (b) oxidation process of a species, A, in solution. The molecular orbitals (MO) of species A shown are the highest occupied MO and the lowest vacant MO. These correspond in an approximate way to the E°s of the A/A~ and A + /A couples, respectively. The illustrated system could represent an aromatic hydrocarbon (e.g., 9,10-diphenylanthracene) in an aprotic solvent (e.g., acetonitrile) at a platinum electrode.

Consider a typical electrochemical experiment where a working electrode and a reference electrode are immersed in a solution, and the potential difference between the electrodes is varied by means of an external power supply (Figure 1.1.3). This variation in potential, £, can produce a current flow in the external circuit, because electrons cross the electrode/solution interfaces as reactions occur. Recall that the number of electrons that cross an interface is related stoichiometrically to the extent of the chemical reaction (i.e., to the amounts of reactant consumed and product generated). The number of electrons is measured in terms of the total charge, Q, passed in the circuit. Charge is expressed in units of coulombs (C), where 1 С is equivalent to 6.24 X 10 1 8 electrons. The relationship between charge and amount of product formed is given by Faraday's law; that is, the passage of 96,485.4 С causes 1 equivalent of reaction (e.g., consumption of 1 mole of reactant or production of 1 mole of product in a one-electron reaction). The current, /, is the rate of flow of coulombs (or electrons), where a current of 1 ampere (A) is equivalent to 1 C/s. When one plots the current as a function of the potential, one obtains a current-potential (i vs. E) curve. Such curves can be quite informative about the nature of the solution and the electrodes and about the reactions that occur at the interfaces. Much of the remainder of this book deals with how one obtains and interprets such curves.

1.1 Introduction
Power supply

5

-Ag

Pt

-AgBr

1МНВГ

Figure 1.1.3 Schematic diagram of the electrochemical cell Pt/HBr(l M)/AgBr/Ag attached to power supply and meters for obtaining a currentpotential (i-E) curve.

Let us now consider the particular cell in Figure 1.1.3 and discuss in a qualitative way the current-potential curve that might be obtained with it. In Section 1.4 and in later chapters, we will be more quantitative. We first might consider simply the potential we would measure when a high impedance voltmeter (i.e., a voltmeter whose internal resistance is so high that no appreciable current flows through it during a measurement) is placed across the cell. This is called the open-circuit potential of the cell.1 For some electrochemical cells, like those in Figure 1.1.1, it is possible to calculate the open-circuit potential from thermodynamic data, that is, from the standard potentials of the half-reactions involved at both electrodes via the Nernst equation (see Chapter 2). The key point is that a true equilibrium is established, because a pair of redox forms linked by a given half-reaction (i.e., a redox couple) is present at each electrode. In Figure 1.1.1/?, for example, we have H + and H 2 at one electrode and Ag and AgCl at the other.2 The cell in Figure 1.1.3 is different, because an overall equilibrium cannot be established. At the Ag/AgBr electrode, a couple is present and the half-reaction is AgBr + e ±± Ag + Br = 0.0713 Vvs. NHE (1.1.6)

Since AgBr and Ag are both solids, their activities are unity. The activity of Br can be found from the concentration in solution; hence the potential of this electrode (with respect to NHE) could be calculated from the Nernst equation. This electrode is at equilibrium. However, we cannot calculate a thermodynamic potential for the Pt/H+,Br~ electrode, because we cannot identify a pair of chemical species coupled by a given halfreaction. The controlling pair clearly is not the H2,H+ couple, since no H 2 has been introduced into the cell. Similarly, it is not the O 2 ,H 2 O couple, because by leaving O 2 out of the cell formulation we imply that the solutions in the cell have been deaerated. Thus, the Pt electrode and the cell as a whole are not at equilibrium, and an equilibrium potential

*In the electrochemical literature, the open-circuit potential is also called the zero-current potential or the rest potential. When a redox couple is present at each electrode and there are no contributions from liquid junctions (yet to be discussed), the open-circuit potential is also the equilibrium potential. This is the situation for each cell in Figure 1.1.1.
2

Chapter 1. Introduction and Overview of Electrode Processes does not exist. Even though the open-circuit potential of the cell is not available from thermodynamic data, we can place it within a potential range, as shown below. Let us now consider what occurs when a power supply (e.g., a battery) and a microammeter are connected across the cell, and the potential of the Pt electrode is made more negative with respect to the Ag/AgBr reference electrode. The first electrode reaction that occurs at the Pt is the reduction of protons, 2H + 2 e - * H 2
+

(1.1.7)

The direction of electron flow is from the electrode to protons in solution, as in Figure 1.12a, so a reduction (cathodic) current flows. In the convention used in this book, ca3 thodic currents are taken as positive, and negative potentials are plotted to the right. As shown in Figure 1.1.4, the onset of current flow occurs when the potential of the Pt elec+ trode is near E° for the H /H 2 reaction (0 V vs. NHE or -0.07 V vs. the Ag/AgBr electrode). While this is occurring, the reaction at the Ag/AgBr (which we consider the reference electrode) is the oxidation of Ag in the presence of Br~ in solution to form AgBr. The concentration of Br~ in the solution near the electrode surface is not changed appreciably with respect to the original concentration (1 M), therefore the potential of the Ag/AgBr electrode will be almost the same as at open circuit. The conservation of charge requires that the rate of oxidation at the Ag electrode be equal to the rate of reduction at the Pt electrode. When the potential of the Pt electrode is made sufficiently positive with respect to the reference electrode, electrons cross from the solution phase into the electrode, and the oxPt/H+, ВГ(1 M)/AgBr/Ag Cathodic

1
1 1.5

1

I

Onset of H + reduction on Pt. \ i : /

\y
0 -0.5

/

/ /

1 \

\

L

Onset of Br" oxidation on Pt

Cell Potential

Anodic

Figure 1.1.4 Schematic current-potential curve for the cell Pt/H + , Br~(l M)/AgBr/Ag, showing the limiting proton reduction and bromide oxidation processes. The cell potential is given for the Pt electrode with respect to the Ag electrode, so it is equivalent to £ P t (V vs. AgBr). Since ^Ag/AgBr =

0.07 V vs. NHE, the potential axis could be converted to EPt (V vs. NHE) by adding 0.07 V to each value of potential.

The convention of taking / positive for a cathodic current stems from the early polarograhic studies, where reduction reactions were usually studied. This convention has continued among many analytical chemists and electrochemists, even though oxidation reactions are now studied with equal frequency. Other electrochemists prefer to take an anodic current as positive. When looking over a derivation in the literature or examining a published i-E curve, it is important to decide, first, which convention is being used (i.e., "Which way is up?").

3

1.1 Introduction

7

idation of Br~ to Br2 (and Br^~) occurs. An oxidation current, or anodic current, flows at potentials near the E° of the half-reaction, Br2 + 2 e ^ 2 B r ~ (1.1.8)

which is +1.09 V vs. NHE or +1.02 V vs. Ag/AgBr. While this reaction occurs (rightto-left) at the Pt electrode, AgBr in the reference electrode is reduced to Ag and Br~ is liberated into solution. Again, because the composition of the Ag/AgBr/Br~ interface (i.e., the activities of AgBr, Ag, and Br~) is almost unchanged with the passage of modest currents, the potential of the reference electrode is essentially constant. Indeed, the essential characteristic of a reference electrode is that its potential remains practically constant with the passage of small currents. When a potential is applied between Pt and Ag/AgBr, nearly all of the potential change occurs at the Pt/solution interface. The background limits are the potentials where the cathodic and anodic currents start to flow at a working electrode when it is immersed in a solution containing only an electrolyte added to decrease the solution resistance (a supporting electrolyte). Moving the potential to more extreme values than the background limits (i.e., more negative than the limit for H2 evolution or more positive than that for Br2 generation in the example above) simply causes the current to increase sharply with no additional electrode reactions, because the reactants are present at high concentrations. This discussion implies that one can often estimate the background limits of a given electrode-solution interface by considering the thermodynamics of the system (i.e., the standard potentials of the appropriate halfreactions). This is frequently, but not always, true, as we shall see in the next example. From Figure 1.1.4, one can see that the open-circuit potential is not well defined in the system under discussion. One can say only that the open-circuit potential lies somewhere between the background limits. The value found experimentally will depend upon trace impurities in the solution (e.g., oxygen) and the previous history of the Pt electrode. Let us now consider the same cell, but with the Pt replaced with a mercury electrode: Hg/H + ,Br-(l M)/AgBr/Ag (1.1.9)

We still cannot calculate an open-circuit potential for the cell, because we cannot define a redox couple for the Hg electrode. In examining the behavior of this cell with an applied external potential, we find that the electrode reactions and the observed current-potential behavior are very different from the earlier case. When the potential of the Hg is made negative, there is essentially no current flow in the region where thermodynamics predict that H2 evolution should occur. Indeed, the potential must be brought to considerably more negative values, as shown in Figure 1.1.5, before this reaction takes place. The thermodynamics have not changed, since the equilibrium potential of half-reaction 1.1.7 is independent of the metal electrode (see Section 2.2.4). However, when mercury serves as the locale for the hydrogen evolution reaction, the rate (characterized by a heterogeneous rate constant) is much lower than at Pt. Under these circumstances, the reaction does not occur at values one would predict from thermodynamics. Instead considerably higher electron energies (more negative potentials) must be applied to make the reaction occur at a measurable rate. The rate constant for a heterogeneous electron-transfer reaction is a function of applied potential, unlike one for a homogeneous reaction, which is a constant at a given temperature. The additional potential (beyond the thermodynamic requirement) needed to drive a reaction at a certain rate is called the overpotential. Thus, it is said that mercury shows "a high overpotential for the hydrogen evolution reaction." When the mercury is brought to more positive values, the anodic reaction and the potential for current flow also differ from those observed when Pt is used as the electrode.

8 • Chapter 1. Introduction and Overview of Electrode Processes
Hg/I-Г, ВГ(1 M)/AgBr/Ag Cathodic

Onset of H + reduction ,

0.5

-0.5 Onset of Hg oxidation

-1.5

Anodic Potential (V vs. NHE)

Figure 1.1.5 Schematic current-potential curve for the Hg electrode in the cell Hg/H + , Br (1 M)/AgBr/Ag, showing the limiting processes: proton reduction with a large negative overpotential and mercury oxidation. The potential axis is defined through the process outlined in the caption to Figure 1.1.4.

With Hg, the anodic background limit occurs when Hg is oxidized to Hg2Br2 at a potential near 0.14 V vs. NHE (0.07 V vs. Ag/AgBr), characteristic of the half-reaction Hg 2 Br 2 + 2e«±2Hg 2Br" (1.1.10)

In general, the background limits depend upon both the electrode material and the solution employed in the electrochemical cell. Finally let us consider the same cell with the addition of a small amount of Cd 2 + to the solution, Hg/H + ,Br"(l M), Cd 2+ (10" 3 M)/AgBr/Ag (1.1.11)

The qualitative current-potential curve for this cell is shown in Figure 1.1.6. Note the appearance of the reduction wave at about -0.4 V vs. NHE arising from the reduction reaction CdBr|~ + 2e S Cd(Hg) + 4Br~ (1.1.12) where Cd(Hg) denotes cadmium amalgam. The shape and size of such waves will be covered in Section 1.4.2. If Cd 2 + were added to the cell in Figure 1.1.3 and a current-potential curve taken, it would resemble that in Figure 1.1.4, found in the absence of Cd 2 + . At a Pt electrode, proton reduction occurs at less positive potentials than are required for the reduction of Cd(II), so the cathodic background limit occurs in 1 M HBr before the cadmium reduction wave can be seen. In general, when the potential of an electrode is moved from its open-circuit value toward more negative potentials, the substance that will be reduced first (assuming all possible electrode reactions are rapid) is the oxidant in the couple with the least negative (or most positive) E®. For example, for a platinum electrode immersed in an aqueous solution containing 0.01 M each of F e 3 + , Sn 4 + , and N i 2 + in 1 M HC1, the first substance reduced will be F e 3 + , since the E° of this couple is most positive (Figure 1.1.7a). When the poten-

1.1 Introduction
Hg/I-Г, ВГ(1 М), Cd2+(1mM)/AgBr/Ag Cathodic

Onset of Cd 2 ' reduction

Anodic l _ Potential (V vs. NHE)

Figure 1.1.6 Schematic current-potential curve for the Hg electrode in the cell Hg/H+, Br"(l M),Cd 2 + (l(T 3 M)/AgBr/Ag, showing reduction wave for Cd 2 + .

tial of the electrode is moved from its zero-current value toward more positive potentials, the substance that will be oxidized first is the reductant in the couple of least positive (or most negative) E°. Thus, for a gold electrode in an aqueous solution containing 0.01 M each of Sn 2 + and F e 2 + in 1 M HI, the Sn 2 + will be first oxidized, since the E° of this couple is least positive (Figure 1.1.7b). On the other hand, one must remember that these predictions are based on thermodynamic considerations (i.e., reaction energetics), and slow kinetics might prevent a reaction from occurring at a significant rate in a potential region where the E° would suggest the reaction was possible. Thus, for a mercury electrode immersed in a solution of 0.01 M each of Cr 3 + and Zn 2 + , in 1 M HC1, the first reduction process predicted is the evolution of H 2 from H + (Figure 1.1.7c). As discussed earlier, this reaction is very slow on mercury, so the first process actually observed is the reduc3+ tion of Cr .

1.1.2

Faradaic and Nonfaradaic Processes
Two types of processes occur at electrodes. One kind comprises reactions like those just discussed, in which charges (e.g., electrons) are transferred across the metal-solution interface. Electron transfer causes oxidation or reduction to occur. Since such reactions are governed by Faraday's law (i.e., the amount of chemical reaction caused by the flow of current is proportional to the amount of electricity passed), they are called faradaic processes. Electrodes at which faradaic processes occur are sometimes called chargetransfer electrodes. Under some conditions, a given electrode-solution interface will show a range of potentials where no charge-transfer reactions occur because such reactions are thermodynamically or kinetically unfavorable (e.g., the region in Figure 1.1.5 between 0 and —0.8 V vs. NHE). However, processes such as adsorption and desorption can occur, and the structure of the electrode-solution interface can change with changing potential or solution composition. These processes are called nonfaradaic processes. Although charge does not cross the interface, external currents can flow (at least transiently) when the potential, electrode area, or solution composition changes. Both faradaic and

10 • Chapter 1. Introduction and Overview of Electrode Processes

0
(V vs. NHE)

Possible reduction reactions

Possible oxidation reactions

0
E°
(V w. NHE)

-0.25 0 +0.15

N i 2 + + 2e -> Ni 2 H + + 2e - » H 9 S n 4 + + 2e -> S n 2 +

Approximate potential for zero current

-- 0 - - +0.15

(Au)

\2 + 2e | H 2 + OH" (1.2.3)

is thermodynamically possible, but occurs at a very low rate at a mercury surface unless quite negative potentials are reached. Thus, the only faradaic current that flows in this region is due to charge-transfer reactions of trace impurities (e.g., metal ions, oxygen, and organic species), and this current is quite small in clean systems. Another electrode that behaves as an IPE is a gold surface hosting an adsorbed self-assembled monolayer of alkane thiol (see Section 14.5.2). 1.2.2 Capacitance and Charge of an Electrode Since charge cannot cross the IPE interface when the potential across it is changed, the behavior of the electrode-solution interface is analogous to that of a capacitor. A capacitor is an electrical circuit element composed of two metal sheets separated by a dielectric material (Figure 1.2.1a). Its behavior is governed by the equation | =С (1.2.4)

e

^-

Battery —

e

_ Capacitor
++ ++

© e (b)

Figure 1.2.1 (a) A capacitor, (b) Charging a capacitor with a battery.

12

Chapter 1. Introduction and Overview of Electrode Processes
Metal Solution Metal Solution

(a)

(b)

Figure 1.2.2 The metal-solution interface as a capacitor with a charge on the metal, qM, (a) negative and (b) positive.

where q is the charge stored on the capacitor (in coulombs, С), Е is the potential across the capacitor (in volts, V), and С is the capacitance (in farads, F). When a potential is applied across a capacitor, charge will accumulate on its metal plates until q satisfies equation 1.2.4. During this charging process, a current (called the charging current) will flow. The charge on the capacitor consists of an excess of electrons on one plate and a deficiency of electrons on the other (Figure 1.2.1b). For example, if a 2-V battery is placed across a 10/л¥ capacitor, current will flow until 20 /лС has accumulated on the capacitor plates. The magnitude of the current depends on the resistance in the circuit (see also Section 1.2.4). The electrode-solution interface has been shown experimentally to behave like a capacitor, and a model of the interfacial region somewhat resembling a capacitor can be given. At a given potential, there will exist a charge on the metal electrode, qM, and a charge in the solution, qs (Figure 1.2.2). Whether the charge on the metal is negative or positive with respect to the solution depends on the potential across the interface and the composition of the solution. At all times, however, qM — -qs. (In an actual experimental arrangement, two metal electrodes, and thus two interfaces, would have to be considered; we concentrate our attention here on one of these and ignore what happens at the other.) The charge on the metal, qM, represents an excess or deficiency of electrons and resides in a very thin layer ( CQ(X = 0)], we obtain nFA = mo[C% - Co(x = 0)] (1.4.6)

Under the conditions of a net cathodic reaction, R is produced at the electrode surface, so that CR(x = 0) > CR (where CR is the bulk concentration of R). Therefore, = mR[CR(x = 0) - C*] (1A7)

11 While m0 is treated here as a phenomenological parameter, in more exact treatments the value of m0 can sometimes be specified in terms of measurable quantities. For example, for the rotating disk electrode, m 0 = 0.62Do/3(ol/2v~l/e, where со is the angular velocity of the disk (i.e., 2тг/, with/as the frequency in revolutions per second) and v is the kinematic viscosity (i.e., viscosity/density, with units of cm2/s) (see Section 9.3.2). Steady-state currents can also be obtained with a very small electrode (such as a Pt disk with a radius, r 0 , in the \xva range), called an ultramicroelectrode (UME, Section 5.3). At a disk UME, m0 = 4Do/7rr0.

1.4 Introduction to Mass-Transfer-Controlled Reactions or for the particular case when CR = 0 (no R in the bulk solution),

31

= 0)

(1.4.8)

The values of Co(x = 0) and CR(x = 0) are functions of electrode potential, E. The largest rate of mass transfer of О occurs when CQ(X — 0) = 0 (or more precisely, when C o (x = 0) < < Co, so that CQ - Co(x = 0) « CQ). The value of the current under these conditions is called the limiting current, //, where (1.4.9) When the limiting current flows, the electrode process is occurring at the maximum rate possible for a given set of mass-transfer conditions, because О is being reduced as fast as it can be brought to the electrode surface. Equations 1.4.6 and 1.4.9 can be used to obtain expressions for Co(x = 0): (1.4.10)

Co(x = 0) =

nFAmo

(1.4.11)

Thus, the concentration of species О at the electrode surface is linearly related to the current and varies from CQ, when / = 0, to a negligible value, when / = //. If the kinetics of electron transfer are rapid, the concentrations of О and R at the electrode surface can be assumed to be at equilibrium with the electrode potential, as governed by the Nernst equation for the half-reaction 12

= 0)

(1.4.12)

Such a process is called a nernstian reaction. We can derive the steady-state i-E curves for nernstian reactions under several different conditions. (a) R Initially Absent When C | = 0, CR(x = 0) can be obtained from (1.4.8): CR(x = 0) = i/nFAmR Then, combining equations 1.4.11 to 1.4.13, we obtain nF An i-E plot is shown in Figure 1 A.2a. Note that when / = ///2, nF (1.4.15) (1.4.14) (1.4.13)

12 Equation 1.4.12 is written in terms of E° , called the formal potential, rather than the standard potential E°. The formal potential is an adjusted form of the standard potential, manifesting activity coefficients and some chemical effects of the medium. In Section 2.1.6, it will be introduced in more detail. For the present it is not necessary to distinguish between E° and E°.

32 * Chapter 1. Introduction and Overview of Electrode Processes Cathodic

( - ) •

H-

Anodic

Figure 1.4.2 (a) Current-potential curve for a nernstian reaction involving two soluble species with only oxidant present initially, (b) log[(// - /)//] vs. E for this system.

where Ещ is independent of the substrate concentration and is therefore characteristic of the O/R system. Thus, (1.4.16) When a system conforms to this equation, a plot of E vs. log[(// — /)//] is a straight line with a slope of 23RT/nF (or 59.1/л mV at 25°C). Alternatively (Figure 1.4.2b), log[(// i)li\ vs. E is linear with a slope of nF/23RT (or л/59.1 mV" 1 at 25°C) and has an ^-intercept of Ещ. When mo and mR have similar values, Ещ ~ E°'. (b) Both О and R Initially Present When both members of the redox couple exist in the bulk, we must distinguish between a cathodic limiting current, //c, when CQ(X = 0) ~ 0, and an anodic limiting current, z/a, when CR(x = 0) « 0. We still have CQ(X = 0) given by (1.4.11), but with // now specified as // c . The limiting anodic current naturally reflects the maximum rate at which R can be brought to the electrode surface for conversion to O. It is obtained from (1.4.7): (1.4.17) (The negative sign arises because of our convention that cathodic currents are taken as positive and anodic ones as negative.) Thus CR(X = 0) is given by CR(x = 0) = nFAm R CR(x = 0)_ CR The i-E curve is then
0

(1.4.18) (1.4.19)

- 1—

l

>

RT, ~n~Fln Щ

+ fjln(1

-W

(1.4.20)

A plot of this equation is shown in Figure 1.4.3. When / = 0, E = Eeq and the system is at equilibrium. Surface concentrations are then equal to the bulk values. When current flows,

1.4 Introduction to Mass-Transfer-Controlled Reactions

33

Figure 1.4.3 Current-potential curve for a nernstian system involving two soluble species with both forms initially present.

the potential deviates from £ e q , and the extent of this deviation is the concentration overpotential. (An equilibrium potential cannot be defined when CR = 0, of course.) (c) R Insoluble Suppose species R is a metal and can be considered to be at essentially unit activity as the electrode reaction takes place on bulk R.13 When aR = 1, the Nernst equation is (1.4.21) or, using the value of CQ(X = 0) from equation 1.4.11,

(1.4.22)
III 111
\

11

I

When i = О, Е = Ещ = Е0' + (RT/nF) In CQ (Figure 1.4.4). If we define the concentration overpotential, rjconc (or the mass-transfer overpotential, 7]mt), as n
'/cone

= F —F
^ ^eq

(\ 4 1 \ \ yLs-r.^o) then (1.4.24) When / = //, i7 c o n c — oo. Since 1 is a measure of polarization, this condition is sometimes > 7 called complete concentration polarization.

^ E

Figure 1.4.4 Current-potential curve for a nernstian system where the reduced form is insoluble.

13 This will not be the case for R plated onto an inert substrate in amounts less than a monolayer (e.g., the substrate electrode being Pt and R being Cu). Under those conditions, aR may be considerably less than unity (see Section 11.2.1).

34 • Chapter 1. Introduction and Overview of Electrode Processes Equation 1.4.24 can be written in exponential form: (1.4.25) The exponential can be expanded as a power series, and the higher-order terms can be dropped if the argument is kept small; that is, e*=l+jt + y - ! + . * (when* is small) (1.4.26)

Thus, under conditions of small deviations of potential from Eeq, the /-т?Сопс characteristic is linear: -RTi (1.4.27)

Since -r]/i has dimensions of resistance (ohms), we can define a "small signal" masstransfer resistance, Rmb as i? m t

=

RT

nF\it\

(1.4.28)

Here we see that the mass-transfer-limited electrode reaction resembles an actual resistance element only at small overpotentials.

1.4.3

Semiempirical Treatment of the Transient Response
The treatment in Section 1.4.2 can also be employed in an approximate way to timedependent (transient) phenomena, for example, the buildup of the diffusion layer, either in a stirred solution (before steady state is attained) or in an unstirred solution where the diffusion layer continues to grow with time. Equation 1.4.4 still applies, but in this case we consider the diffusion layer thickness to be a time-dependent quantity, so that UnFA = vmt = D o [ Cg - Co(x = 0)]/8 o (0 (1.4.29)

Consider what happens when a potential step of magnitude E is applied to an electrode immersed in a solution containing a species O. If the reaction is nernstian, the concentrations of О and R at x = 0 instantaneously adjust to the values governed by the Nernst equation, (1.4.12). The thickness of the approximately linear diffusion layer, > grows with time (Figure 1.4.5). At any time, the volume of the diffusion layer is

Co(x =

I
5(r2)

I
8(/ 3 )

Figure 1.4.5 Growth of the diffusion-layer thickness with time.

1.4 Introduction to Mass-Transfer-Controlled Reactions

35

No convection t Figure 1.4.6 Current-time transient for a potential step to a stationary electrode (no convection) and to an electrode in stirred solution (with convection) where a steady-state current is attained.

A8o(t). The current flow causes a depletion of O, where the amount of О electrolyzed is given by Moles of О electrolyzed in diffusion layer Г ( mi ^ ( Q f'idt fidt
(Л л

~m

By differentiation of (1.4.30) and use of (1.4.29), [Cg - Co(x = 0)] A d8(t) 2 dt or d8(t) flfr
= =

| nF

= 0)]

(1.4.31)

2D O 5(0

(1.4.32)

Since 5(0 = 0 at Г = 0, the solution of (1.4.32) is 8(t) = 2VD~t and (1.4.34) This approximate treatment predicts a diffusion layer that grows with tl/2 and a current that decays with t~l/2. In the absence of convection, the current continues to decay, but in a convective system, it ultimately approaches the steady-state value characterized by S(0 ~ ^o (Figure 1.4.6). Even this simplified approach approximates reality quite closely; equation 1.4.34 differs only by a factor of 2/тг1^2 from the rigorous description of current arising from a nernstian system during a potential step (see Section 5.2.1). (1.4.33)

1.5 SEMIEMPIRICAL TREATMENT OF NERNSTIAN REACTIONS WITH COUPLED CHEMICAL REACTIONS
The current-potential curves discussed so far can be used to measure concentrations, mass-transfer coefficients, and standard potentials. Under conditions where the electrontransfer rate at the interface is rate-determining, they can be employed to measure heterogeneous kinetic parameters as well (see Chapters 3 and 9). Often, however, one is interested in using electrochemical methods to find equilibrium constants and rate constants of homogeneous reactions that are coupled to the electron-transfer step. This section provides a brief introduction to these applications.

36 1,5.1

Chapter 1. Introduction and Overview of Electrode Processes Coupled Reversible Reactions If a homogeneous process, fast enough to be considered always in thermodynamic equilibrium (a reversible process), is coupled to a nernstian electron-transfer reaction, then one can use a simple extension of the steady-state treatment to derive the i-E curve. Consider, for example, a species О involved in an equilibrium that precedes the electron-transfer reaction14 A «± О + qY О + ne «± R (1.5.1) (1.5.2)

For example, A could be a metal complex, MYq+; О could be the free metal ion, M n + ; and Y could be the free, neutral ligand (see Section 5.4.4). For reaction 1.5.2, the Nernst equation still applies at the electrode surface, (1.5.3) and (1.5.1) is assumed to be at equilibrium everywhere: (1.5.4)

C

A

Hence KCA(x = 0) nF (1.5.5)

Assuming (1) that at t = 0, C A = C*> CY = C*, and C R = 0 (for all JC); (2) that C* is so large compared to C* that CY(x = 0) = C* at all times; and (3) that К « 1; then at steady state nFA - CA(x = 0)] (1.5.6) (1.5.7)

= 0)
Then, as previously, С Ax = 0) =

(1.5.8)

(// ~ 0 nFAmk CR(x = 0) =

(1.5.9)

(1.5.10)

E = El/2 + (0.059/л) log -4— where

(T = 25°)

(1.5.11)

14

To simplify notation, charges on all species are omitted.

1.5 Semiempirical Treatment of Nemstian Reactions with Coupled Chemical Reactions ^1 37 Thus, the i-E curve, (1.5.11), has the usual nernstian shape, but Ещ is shifted in a negative direction (since К « 1) from the position that would be found for process 1.5.2 unperturbed by the homogeneous equilibrium. From the shift of Ещ with log Cy, both q [= —(n/0.059)(dEi/2/d log C Y ] and К can be determined. Although these thermodynamic and stoichiometric quantities are available, no kinetic or mechanistic information can be obtained when both reactions are reversible.

Coupled Irreversible Chemical Reactions
When an irreversible chemical reaction is coupled to a nernstian electron transfer, the i-E curves can be used to provide kinetic information about the reaction in solution. Consider a nernstian charge-transfer reaction with a following first-order reaction: О + ne ^± R R->T l (1.5.13) (1.5.14)

k

where к is the rate constant (in s ) for the decomposition of R. (Note that к could be a pseudo-first-order constant, such as when R reacts with protons in a buffered solution and к = к'Сц+.) As an example of this sequence, consider the oxidation of p-aminophenol in acid solution.

2H + + 2e

(1.5.15)

+ NH 3

(1.5.16)

Reaction 1.5.16 does not affect the mass transfer and reduction of O, so (1.4.6) and (1.4.9) still apply (assuming CQ = CQ and C R = 0 at all x at t = 0). However, the reaction causes R to disappear from the electrode surface at a higher rate, and this difference affects the i-E curve. In the absence of the following reaction, we think of the concentration profile for R as decreasing linearly from a value CR(x = 0) at the surface to the point where C R = 0 at 8, the outer boundary of the Nernst diffusion layer. The coupled reaction adds a channel for disappearance of R, so the R profile in the presence of the reaction does not extend as far into the solution US 8. Thus, the added reaction steepens the profile and augments mass transfer away from the electrode surface. For steady-state behavior, such as at a rotating disk, we assume the rate at which R disappears from the surface to be the rate of diffusion in the absence of the reaction [(mRCR(;t = 0); see (1.4.8)] plus an increment proportional to the rate of reaction [/X£CR(JC = 0)]. Since the rate of formation of R, given by (1.4.6), equals its total rate of disappearance, we have nFA = mo\C% - Co(x = 0)] - mRCR(x = 0) + fxkCR(x = 0) (1.5.17)

38

Chapter 1. Introduction and Overview of Electrode Processes where /UL is a proportionality constant having units of cm, so that the product \xk has dimensions of cm/s as required. In the literature (3), \x is called the reaction layer thickness. For our purpose, it is best just to think of /UL as an adjustable parameter. From (1.5.17),

Substituting these values into the Nernst equation for (1.5.13) yields

(1.5.20) or

E = E[/2 + ° ^ i where O

gfc^

(at25°C)

(1.5.21)

•Ч^Ц^) or (1.5.22)

^

[

^

)

(1.5.23)

where Ещ is the half-wave potential for the kinetically unperturbed reaction. Two limiting cases can be defined: (a) When fik/m^ « 1, that is /лк « m R , the effect of the following reaction, (1.5.14), is negligible, and the unperturbed i-E curve results, (b) When fxk/mR » 1, the following reaction dominates the behavior and т т

^

(1.5.24)

The effect is to shift the reduction wave in & positive direction without a change in shape. For the rotating disk electrode, where m R = 0.62DR/3 + 2Ag + 2СГ: (a) Suppose zinc and silver chloride are mixed directly in a calorimeter at constant, atmospheric pressure and at 25°C. Assume also that the extent of reaction is so small that the activities of all species remain unchanged during the experiment. It is found that the amount of heat liberated when all substances are in their standard states is 233 kJ/mol of Zn reacted. Thus, AH0 = -233 kJ. 2 Zn/Zn 2+ (a = 1), CT(a = l)/AgCl/Ag

(b) Suppose we now construct the cell of Figure 1.1.1a, that is, (2.1.17)

and discharge it through a resistance R. Again assume that the extent of reaction is small enough to keep the activities essentially unchanged. During the discharge, heat will evolve from the resistor and from the cell, and we could measure the total heat change by placing the entire apparatus inside a calorimeter. We would find that the heat evolved is 233 kJ/mol of Zn, independent of R. That is, A#° = —233 kJ, regardless of the rate of cell discharge. (c) Let us now repeat the experiment with the cell and the resistor in separate calorimeters. Assume that the wires connecting them have no resistance and do not conduct any heat between the calorimeters. If we take Qc as the heat change in the cell and QR as that in the resistor, we find that Qc + QR = -233 kJ/mol of Zn reacted, independent ofR. However, the balance between these quantities does depend on the rate of discharge. As R increases, \QC\ decreases and |Q R | increases. In the limit of infinite R, QQ approaches —43 kJ (per mole of zinc) and QR tends toward -190 kJ.

In this example, the energy QR was dissipated as heat, but it was obtained as electrical energy, and it might have been converted to light or mechanical work. In contrast, Qc is an energy change that is inevitably thermal. Since discharge through R — °° corre» sponds to a thermodynamically reversible process, the energy that must appear as heat in traversing a reversible path, |A#° .

2.1.3

Free Energy and Cell emf We found just above that if we discharged the electrochemical cell (2.1.17) through an infinite load resistance, the discharge would be reversible. The potential difference is therefore always the equilibrium (open-circuit) value. Since the extent of reaction is supposed to be small enough that all activities remain constant, the potential also remains constant. Then, the energy dissipated in R is given by | AG\ = charge passed X reversible potential difference (2.1.20) (2.L21)

where n is the number of electrons passed per atom of zinc reacted (or the number of moles of electrons per mole of Zn reacted), and F is the charge on a mole of electrons, which is about 96,500 C. However, we also recognize that the free energy change has a sign associated with the direction of the net cell reaction. We can reverse the sign by reversing the direction. On the other hand, only an infinitesimal change in the overall cell potential is required to reverse the direction of the reaction; hence E is essentially constant and independent of the direction of a (reversible) transformation. We have a quandary. We want to relate a direction-sensitive quantity (AG) to a direction-insensitive observable (E). This desire is the origin of almost all of the confusion that exists over electrochemical sign conventions. Moreover the actual meaning of the signs — and + is different for free energy and potential. For free energy, — and + signify energy lost or gained from the system, a convention that traces back to the early days of thermodynamics. For potential, — and + signify the excess or deficiency of electronic charge, an electrostatic convention proposed by Benjamin Franklin even before the discovery of the electron. In most scientific discussions, this difference in meaning is not important, since the context, thermodynamic vs. electrostatic, is clear. But when one considers electrochemical cells, where both thermodynamic and electrostatic concepts are needed, it is necessary to distinguish clearly between these two conventions. When we are interested in thermodynamic aspects of electrochemical systems, we rationalize this difficulty by inventing a thermodynamic construct called the emf of the cell reaction. This quantity is assigned to the reaction (not to the physical cell); hence it has a directional aspect. In a formal way, we also associate a given chemical reaction with each cell schematic. For the one in (2.1.17), the reaction is Zn + 2AgCl -> Z n 2 + + 2Ag + 2 С Г (2.1.22)

The right electrode corresponds to reduction in the implied cell reaction, and the left electrode is identified with oxidation. Thus, the reverse of (2.1.22) would be associated with the opposite schematic: Ag/AgCl/Cl"(a = 1), Zn 2 + (« = 1)/Zn (2.1.23)

The cell reaction emf, £ rxn , is then defined as the electrostatic potential of the electrode written on the right in the cell schematic with respect to that on the left. For example, in the cell of (2.1.17), the measured potential difference is 0.985 V and the zinc electrode is negative; thus the emf of reaction 2.1.22, the spontaneous direction, is +0.985 V. Likewise, the emf corresponding to (2.1.23) and the reverse of (2.1.22) is —0.985 V. By adopting this convention, we have managed to rationalize an (observable) electrostatic quantity (the cell potential difference), which is not sensitive to the direction

2.1 Basic Electrochemical Thermodynamics

49

of the cell's operation, with a (defined) thermodynamic quantity (the Gibbs free energy), which is sensitive to that direction. One can avoid completely the common confusion about sign conventions of cell potentials if one understands this formal relationship between electrostatic measurements and thermodynamic concepts (3,4). Because our convention implies a positive emf when a reaction is spontaneous, AG = -nFEr, or as above, when all substances are at unit activity, AG° = (2.1.25) (2.1.24)

where E®xn is called the standard emf of the cell reaction. Other thermodynamic quantities can be derived from electrochemical measurements now that we have linked the potential difference across the cell to the free energy. For example, the entropy change in the cell reaction is given by the temperature dependence of AG:

дс hence

P

(2.1.26)

(2.1.27) and AH = AG + TAS = nF\ T The equilibrium constant of the reaction is given by (2.1.28)

RT\nKrxn=-AG°

=

(2.1.29)

Note that these relations are also useful for predicting electrochemical properties from thermochemical data. Several problems following this chapter illustrate the usefulness of that approach. Large tabulations of thermodynamic quantities exist (5-8).

2.1.4

Half-Reactions and Reduction Potentials
Just as the overall cell reaction comprises two independent half-reactions, one might think it reasonable that the cell potential could be broken into two individual electrode potentials. This view has experimental support, in that a self-consistent set of half-reaction emfs and half-cell potentials has been devised. To establish the absolute potential of any conducting phase according to definition, one must evaluate the work required to bring a unit positive charge, without associated matter, from the point at infinity to the interior of the phase. Although this quantity is not measurable by thermodynamically rigorous means, it can sometimes be estimated from a series of nonelectrochemical measurements and theoretical calculations, if the demand for thermodynamic rigor is relaxed. Even if we could determine these absolute phase potentials, they would have limited utility because they would

50

Chapter 2. Potentials and Thermodynamics of Cells depend on magnitudes of the adventitious fields in which the phase is immersed (see Section 2.2). Much more meaningful is the difference in absolute phase potentials between an electrode and its electrolyte, for this difference is the chief factor determining the state of an electrochemical equilibrium. Unfortunately, we will find that it also is not rigorously measurable. Experimentally, we can find only the absolute potential difference between two electronic conductors. Still, a useful scale results when one refers electrode potentials and half-reaction emfs to a standard reference electrode featuring a standard half-reaction. The primary reference, chosen by convention, is the normal hydrogen electrode (NHE), also called the standard hydrogen electrode (SHE): Pt/H2(a = 1)/Н (я - 1)
+

(2.1.30)

Its potential (the electrostatic standard) is taken as zero at all temperatures. Similarly, the standard emfs of the half-reactions: 2H + + 2 e * ± H 2 (2.1.31)

have also been assigned values of zero at all temperatures (the thermodynamic standard). We can record half-cell potentials by measuring them in whole cells against the NHE.3 For example, in the system Pt/H2(a = l)/H + (a = l)//Ag+(a = 1)/Ag (2.1.32)

the cell potential is 0.799 V and silver is positive. Thus, we say that the standard potential of the Ag+IAg couple is +0.799 V vs. NHE. Moreover, the standard emf of the Ag+ reduction is also +0.799 V vs. NHE, but that of the Ag oxidation is -0.799 V vs. NHE. Another valid expression is that the standard electrode potential ofAg+/Ag is +0.799 V vs. NHE. To sum all of this up, we write:4 Ag + + e
5

(2.1.45)

For a solute i, the activity is ax = yx (Cj/C°), where C\ is the concentration of the solute, C° is the standard concentration (usually 1 M), and y; is the activity coefficient, which is unitless. For a gas, av = y{ (PJP0), where P[ is the partial pressure of /, P° is the standard pressure, and yx is the activity coefficient, which is again unitless. For most of the published literature, including all before the late 1980s, the standard pressure was 1 atm (101,325 Pa). The new standard pressure adopted by the International Union of Pure and Applied Chemistry is 105 Pa. A consequence of this change is that the potential of the NHE now differs from that used historically. The "new NHE" is +0.169 mV vs. the "old NHE" (based on a standard state of 1 atm). This difference is rarely significant, and is never so in this book. Most tabulated standard potentials, including those in Table C.I are referred to the old NHE See reference 15.

2.1 Basic Electrochemical Thermodynamics

53

Because the ionic strength affects the activity coefficients, E0' will vary from medium to medium. Table C.2 contains values for this couple in 1 M HC1, 10 M HC1, 1 M HC1O4, 1 M H 2 SO 4 , and 2 M H3PO4. The values of standard potentials for half-reactions and cells are actually determined by measuring formal potentials values at different ionic strengths and extrapolating to zero ionic strength, where the activity coefficients approach unity. Often E° also contains factors related to complexation and ion pairing; as it does in fact for the Fe(III)/Fe(II) couple in HC1, H 2 SO 4 , and H3PO4 solutions. Both iron species are complexed in these media; hence (2.1.42) does not accurately describe the half-cell reaction. However, one can sidestep a full description of the complex competitive equilibria by using the empirical formal potentials. In such cases, E° contains terms involving equilibrium constants and concentrations of some species involved in the equilibria.

2.1.7

Reference Electrodes
Many reference electrodes other than the NHE and the SCE have been devised for electrochemical studies in aqueous and nonaqueous solvents. Several authors have provided discussions on the subject (16-18). Usually there are experimental reasons for the choice of a reference electrode. For example, the system Ag/AgCl/KCl (saturated, aqueous) (2.1.46)

has a smaller temperature coefficient of potential than an SCE and can be built more compactly. When chloride is not acceptable, the mercurous sulfate electrode may be used: Hg/Hg2SO4/K2SO4 (saturated, aqueous) (2.1.47)

With a nonaqueous solvent, one may be concerned with the leakage of water from an aqueous reference electrode; hence a system like Ag/Ag + (0.01 M in CH3CN) (2.1.48)

might be preferred. Because of the difficulty in finding a reference electrode for a nonaqueous solvent that does not contaminate the test solution with undesirable species, a quasireference electrode (QRE)6 is often employed. This is usually just a metal wire, Ag or Pt, used with the expectation that in experiments where there is essentially no change in the bulk solution, the potential of this wire, although unknown, will not change during a series of measurements. The actual potential of the quasireference electrode vs. a true reference electrode must be calibrated before reporting potentials with reference to the QRE. Typically the calibration is achieved simply by measuring (e.g., by voltammetry) the standard or formal potential vs. the QRE of a couple whose standard or formal potential is already known vs. a true reference under the same conditions. The ferrocene/ferrocenium (Fc/Fc+) couple is recommended as a calibrating redox couple, since both forms are soluble and stable in many solvents, and since the couple usually shows nernstian behavior (19). Voltammograms for ferrocene oxidation might be recorded to establish the value of £pc/Fc+ vs-tne QRE, so that the potentials of other reactions can be reported against £pc/Fc+- ^ *s unacceptable to report potentials vs. an uncalibrated quasireference electrode. Moreover a QRE is not suitable in experiments, such as bulk electrolysis, where changes in the composition of the bulk soluQuasi implies that it is "almost" or "essentially" a reference electrode. Sometimes such electrodes are also called pseudoreference electrodes (pseudo, meaning false); this terminology seems less appropriate.
6

54

Chapter 2. Potentials and Thermodynamics of Cells

£0(Zn2+/Zn)

-0.763

-1.00

3.7

-3.7

NHE SCE

0 0.242

-0.242 0

4.5 4.7

-4.5 -4.7

£0(Fe3+/Fe2+)

0.77

0.53

5.3

-5.3

E vs. NHE (volts)

E vs. SCE (volts)

E vs. vacuum (volts)

EF (Fermi energy) (eV)

Figure 2.1.1 Relationship between potentials on the NHE, SCE, and "absolute" scales. The potential on the absolute scale is the electrical work required to bring a unit positive test charge into the conducting phase of the electrode from a point in vacuo just outside the system (see Section 2.2.5). At right is the Fermi energy corresponding to each of the indicated potentials. The Fermi energy is the electrochemical potential of electrons on the electrode (see Section 2.2.4).

tion can cause concomitant variations in the potential of the QRE. A proposed alternative approach (20) is to employ a reference electrode in which Fc and Fc + are immobilized at a known concentration ratio in a polymer layer on the electrode surface (see Chapter 14). Since the potential of a reference electrode vs. NHE or SCE is typically specified in experimental papers, interconversion of scales can be accomplished easily. Figure 2.1.1 is a schematic representation of the relationship between the SCE and NHE scales. The inside back cover contains a tabulation of the potentials of the most common reference electrodes.

2.2 A MORE DETAILED VIEW OF INTERFACIAL POTENTIAL DIFFERENCES 2.2.1 The Physics of Phase Potentials
In the thermodynamic considerations of the previous section, we were not required to advance a mechanistic basis for the observable differences in potentials across certain phase boundaries. However, it is difficult to think chemically without a mechanistic model, and we now find it helpful to consider the kinds of interactions between phases that could create these interfacial differences. First, let us consider two prior questions: (1) Can we expect the potential within a phase to be uniform? (2) If so, what governs its value? One certainly can speak of the potential at any particular point within a phase. That quantity, ф(х, у, z), is defined as the work required to bring a unit positive charge, without material interactions, from an infinite distance to point (x, y, z). From electrostatics, we have assurance that ф(х, у, z) is independent of the path of the test charge (21). The work is done against a coulombic field; hence we can express the potential generally as

Ф(х, у, x)

rx,y,z
J oo

% -d\

(2.2.1)

2.2 A More Detailed View of Interfacial Potential Differences

55

where % is the electric field strength vector (i.e., the force exerted on a unit charge at any point), and d\ is an infinitesimal tangent to the path in the direction of movement. The integral is carried out over any path to (x, y, z). The difference in potential between points (У, / , z') and (x, y, z) is then

ф(х\ /, z') - ф(х, v, z) = f ' ' -

X

У Z

d1

(2.2.2)

In general, the electric field strength is not zero everywhere between two points and the integral does not vanish; hence some potential difference usually exists. Conducting phases have some special properties of great importance. Such a phase is one with mobile charge carriers, such as a metal, a semiconductor, or an electrolyte solution. When no current passes through a conducting phase, there is no net movement of charge carriers, so the electric field at all interior points must be zero. If it were not, the carriers would move in response to it to eliminate the field. From equation 2.2.2, one can see that the difference in potential between any two points in the interior of the phase must also be zero under these conditions; thus the entire phase is an equipotential volume. We designate its potential as ф, which is known as the inner potential (or Galvani potential) of the phase. Why does the inner potential have the value that it does? A very important factor is any excess charge that might exist on the phase itself, because a test charge would have to work against the coulombic field arising from that charge. Other components of the potential can arise from miscellaneous fields resulting from charged bodies outside the sample. As long as the charge distribution throughout the system is constant, the phase potential will remain constant, but alterations in charge distributions inside or outside the phase will change the phase potential. Thus, we have our first indication that differences in potential arising from chemical interactions between phases have some sort of charge separation as their basis. An interesting question concerns the location of any excess charge on a conducting phase. The Gauss law from elementary electrostatics is extremely helpful here (22). It states that if we enclose a volume with an imaginary surface (a Gaussian surface), we will find that the net charge q inside the surface is given by an integral of the electric field over the surface:

q = s 0 % • dS s a 7

(2.2.3)

where e 0 * proportionality constant, and dS is an infinitesimal vector normal outward from the surface. Now consider a Gaussian surface located within a conductor that is uniform in its interior (i.e., without voids or interior phases). If no current flows, % is zero at all points on the Gaussian surface, hence the net charge within the boundary is zero. The situation is depicted in Figure 2.2.1. This conclusion applies to any Gaussian surface, even one situated just inside the phase boundary; thus we must infer that the excess charge actually resides on the surface of the conducting phase.8
7

The parameter s0 is called the permittivity of free space or the electric constant and has the value 8.85419 X 10~ 12 C 2 N " 1 ггГ1. See the footnote in Section 13.3.1 for a fuller explanation of electrostatic conventions followed in this book.

8 There can be a finite thickness to this surface layer. The critical aspect is the size of the excess charge with respect to the bulk carrier concentration in the phase. If the charge is established by drawing carriers from a significant volume, thermal processes will impede the compact accumulation of the excess strictly on the surface. Then, the charged zone is called a space charge region, because it has three-dimensional character. Its thickness can range from a few angstroms to several thousand angstroms in electrolytes and semiconductiors. In metals, it is negligibly thick. See Chapters 13 and 18 for more detailed discussion along this line.

56 • Chapter 2. Potentials and Thermodynamics of Cells

Charged conducting phase Interior Gaussian surface

Zero included charge

Figure 2.2.1 Cross-section of a three-dimensional conducting phase containing a Gaussian enclosure. Illustration that the excess charge resides on the surface of the phase.

A view of the way in which phase potentials are established is now beginning to emerge: 1. 2. Changes in the potential of a conducting phase can be effected by altering the charge distributions on or around the phase. If the phase undergoes a change in its excess charge, its charge carriers will adjust such that the excess becomes wholly distributed over an entire boundary of the phase. The surface distribution is such that the electric field strength within the phase is zero under null-current conditions. The interior of the phase features a constant potential, ф.

3. 4.

The excess charge needed to change the potential of a conductor by electrochemically significant amounts is often not very large. Consider, for example, a spherical mercury drop of 0.5 mm radius. Changing its potential requires only about 5 X 10~ 14 C/V (about 300,000 electrons/V), if it is suspended in air or in a vacuum (21).

2.2.2

Interactions Between Conducting Phases
When two conductors, for example, a metal and an electrolyte, are placed in contact, the situation becomes more complicated because of the coulombic interaction between the phases. Charging one phase to change its potential tends to alter the potential of the neighboring phase as well. This point is illustrated in the idealization of Figure 2.2.2, which portrays a situation where there is a charged metal sphere of macroscopic size, perhaps a mercury droplet 1 mm in diameter, surrounded by a layer of uncharged electrolyte a few millimeters in thickness. This assembly is suspended in a vacuum. We know that the
Electrolyte layer with no net charge

Surrounding vacuum

Metal with charge qu Gaussian surface

Figure 2.2.2 Cross-sectional view of the interacti56on between a metal sphere and a surrounding electrolyte layer. The Gaussian enclosure is a sphere containing the metal phase and part of the electrolyte.

2.2 A More Detailed View of Interfacial Potential Differences
M

57

charge on the metal, q , resides on its surface. This unbalanced charge (negative in the diagram) creates an excess cation concentration near the electrode in the solution. What can we say about the magnitudes and distributions of the obvious charge imbalances in solution? Consider the integral of equation 2.2.3 over the Gaussian surface shown in Figure 2.2.2. Since this surface is in a conducting phase where current is not flowing, % at every point is zero and the net enclosed charge is also zero. We could place the Gaussian surface just outside the surface region bounding the metal and solution, and we would reach the same conclusion. Thus, we know now that the excess positive charge in the solution, s q , resides at the metal-solution interface and exactly compensates the excess metal charge. That is, «7
S

= " <

(2.2.4)

This fact is very useful in the treatment of interfacial charge arrays, which we have al9 ready seen as electrical double layers (see Chapters 1 and 13). Alternatively, we might move the Gaussian surface to a location just inside the outer boundary of the electrolyte. The enclosed charge must still be zero, yet we know that the net charge on the whole system is 0м. A negative charge equal to 0м must therefore reside at the outer surface of the electrolyte. Figure 2.2.3 is a display of potential vs. distance from the center of this assembly, that is, the work done to bring a unit positive test charge from infinitely far away to a given distance from the center. As the test charge is brought from the right side of the diagram, it is attracted by the charge on the outer surface of the electrolyte; thus negative work is required to traverse any distance toward the electrolyte surface in the surrounding vacuum, and the potential steadily drops in that direction. Within the electrolyte, % is zero everywhere, so there is no work in moving the test charge, and the potential is constant at s. At the metal-solution interface, there is a strong field because of the double layer there, and it is oriented such that negative work is done in taking the positive test charge through the interface. Thus there is a sharp change in potential from ф^ to фм over the distance scale of the double layer.10 Since the metal is a field-free volume, the
Distance

Vacuum

Figure 2.2.3 Potential profile through the system shown in Figure 2.2.2. Distance is measured radially from the center of the metallic sphere.

9

Here we are considering the problem on a macroscopic distance scale, and it is accurate to think of qs as residing strictly at the metal—solution interface. On a scale of 1 fxva or finer, the picture is more detailed. One finds that g s is still near the metal-solution interface, but is distributed in one or more zones that can be as thick as 1000 A (Section 13.3).
10 The diagram is drawn on a macroscopic scale, so the transition from фБ to фм appears vertical. The theory of the double layer (Section 13.3) indicates that most of the change occurs over a distance equivalent to one to several solvent monolayers, with a smaller portion being manifested over the diffuse layer in solution.

58

Chapter 2. Potentials and Thermodynamics of Cells potential is constant in its interior. If we were to increase the negative charge on the metal, we would naturally lower ф м , but we would also lower s, called the interfacial potential difference, depends on the charge imbalance at the interface and the physical size of the interface. That is, it depends on the charge density (C/cm2) at the interface. Making a change in this interfacial potential difference requires sizable alterations in charge density. For the spherical mercury drop considered above (A = 0.03 cm 2 ), now surrounded by 0.1 M strong electrolyte, one would need about 10~6 С (or 6 X 10 12 electrons) for a 1-V change. These numbers are more than 107 larger than for the case where the electrolyte is absent. The difference appears because the coulombic field of any surface charge is counterbalanced to a very large degree by polarization in the adjacent electrolyte. In practical electrochemistry, metallic electrodes are partially exposed to an electrolyte and partially insulated. For example, one might use a 0.1 cm 2 platinum disk electrode attached to a platinum lead that is almost fully sealed in glass. It is interesting to consider the location of excess charge used in altering the potential of such a phase. Of course, the charge must be distributed over the entire surface, including both the insulated and the electrochemically active area. However, we have seen that the coulombic interaction with the electrolyte is so strong that essentially all of the charge at any potential will lie adjacent to the solution, unless the percentage of the phase area in contact with electrolyte is really minuscule.11 What real mechanisms are there for charging a phase at all? An important one is simply to pump electrons into or out of a metal or semiconductor with a power supply of some sort. In fact, we will make great use of this approach as the basis for control over the kinetics of electrode processes. In addition, there are chemical mechanisms. For example, we know from experience that a platinum wire dipped into a solution containing ferricyanide and ferrocyanide will have its potential shift toward a predictable equilibrium value given by the Nernst equation. This process occurs because the electron affinities of the two phases initially differ; hence there is a transfer of electrons from the metal to the solution or vice versa. Ferricyanide is reduced or ferrocyanide is oxidized. The transfer of charge continues until the resulting change in potential reaches the equilibrium point, where the electron affinities of the solution and the metal are equal. Compared to the total charge that could be transferred to or from ferri- and ferrocyanide in a typical system, only a tiny charge is needed to establish the equilibrium at Pt; consequently, the net chemical effects on the solution are unnoticeable. By this mechanism, the metal adapts to the solution and reflects its composition. Electrochemistry is full of situations like this one, in which charged species (electrons or ions) cross interfacial boundaries. These processes generally create a net transfer of charge that sets up the equilibrium or steady-state potential differences that we observe. Considering them in more detail must, however, await the development of additional concepts (see Section 2.3 and Chapter 3). Actually, interfacial potential differences can develop without an excess charge on either phase. Consider an aqueous electrolyte in contact with an electrode. Since the electrolyte interacts with the metal surface (e.g., wetting it), the water dipoles in contact with the metal generally have some preferential orientation. From a coulombic standpoint, this situation is equivalent to charge separation across the interface, because the dipoles are
1]

As it can be with an ultramicroelectrode. See Section 5.3.

2.2 A More Detailed View of Interfacial Potential Differences

* 59

not randomized with time. Since moving a test charge through the interface requires work, the interfacial potential difference is not zero (23-26). 12

2.2.3

Measurement of Potential Differences
We have already noted that the difference in the inner potentials, Аф, of two phases in contact is a factor of primary importance to electrochemical processes occurring at the interface between them. Part of its influence comes from the local electric fields reflecting the large changes in potential in the boundary region. These fields can reach values as high as 107 V/cm. They are large enough to distort electroreactants so as to alter reactivity, and they can affect the kinetics of charge transport across the interface. Another aspect of Аф is its direct influence over the relative energies of charged species on either side of the interface. In this way, Аф controls the relative electron affinities of the two phases; hence it controls the direction of reaction. Unfortunately, Аф cannot be measured for a single interface, because one cannot sample the electrical properties of the solution without introducing at least one more interface. It is characteristic of devices for measuring potential differences (e.g., potentiometers, voltmeters, or electrometers) that they can be calibrated only to register potential differences between two phases of the same composition, such as the two metal contacts available at most instruments. Consider Аф at the interface Zn/Zn 2 + , Cl~. Shown in Figure 2.2.4a is the simplest approach one could make to Аф using a potentiometric instrument with copper contacts. The measurable potential difference between the copper phases clearly includes interfacial potential differences at the Zn/Cu interface and the Cu/electrolyte interface in addition to Аф. We might simplify matters by constructing a voltmeter wholly from zinc but, as shown in Figure 2.2.4b, the measurable voltage would still contain contributions from two separate interfacial potential differences. By now we realize that a measured cell potential is a sum of several interfacial differences, none of which we can evaluate independently. For example, one could sketch the potential profile through the cell Cu/Zn/Zn2+,Cr/AgCl/Ag/Cu;
13

(2.2.5)

according to Vetter's representation (24) in the manner of Figure 2.2.5. Even with these complications, it is still possible to focus on a single interfacial potential difference, such as that between zinc and the electrolyte in (2.2.5). If we can maintain constant interfacial potentials at all of the other junctions in the cell, then any change in E must be wholly attributed to a change in Аф at the zinc/electrolyte boundary. Keeping the other junctions at a constant potential difference is not so difficult, for the metalZn f^\ Zn
Figure 2.2.4 Two devices for measuring the potential of a cell containing the Zn/Zn2 interface.
12 Sometimes it is useful to break the inner potential into two components called the outer (or Volta) potential, ф, and the surface potential, x- Thus, ф = ф + х- There is a large, detailed literature on the establishment, the meaning, and the measurement of interfacial potential differences and their components. See references 23—26. 13 Although silver chloride is a separate phase, it does not contribute to the cell potential, because it does not physically separate silver from the electrolyte. In fact, it need not even be present; one merely requires a solution saturated in silver chloride to measure the same cell potential.

60

Chapter 2. Potentials and Thermodynamics of Cells

Cu'
Electrolyte Zn
J

= £

Cu

Distance across cell

Figure 2.2.5 Potential profile across a whole cell at equilibrium.

metal junctions always remain constant (at constant temperature) without attention, and the silver/electrolyte junction can be fixed if the activities of the participants in its half-reaction remain fixed. When this idea is realized, the whole rationale behind half-cell potentials and the choice of reference electrodes becomes much clearer.

2.2.4 Electrochemical Potentials
Let us consider again the interface Zn/Zn 2+ , Cl~ (aqueous) and focus on zinc ions in metallic zinc and in solution. In the metal, Zn 2+ is fixed in a lattice of positive zinc ions, with free electrons permeating the structure. In solution, zinc ion is hydrated and may interact with Cl~. The energy state of Zn 2 + in any location clearly depends on the chemical environment, which manifests itself through short-range forces that are mostly electrical in nature. In addition, there is the energy required simply to bring the +2 charge, disregarding the chemical effects, to the location in question. This second energy is clearly proportional to the potential ф at the location; hence it depends on the electrical properties of an environment very much larger than the ion itself. Although one cannot experimentally separate these two components for a single species, the differences in the scales of the two environments responsible for them makes it plausible to separate them mathematically (23-26). Butler (27) and Guggenheim (28) developed the conceptual separation and introduced the electrochemical potential, Jlf, for species / with charge zx in phase a:

z? = tf +
The term fxf is the familiar chemical potential

(2.2.6)

(2.2.7) where щ is the number of moles of / in phase a. Thus, the electrochemical potential would be

я (ж)

=

(2

- *

2

8)

where the electrochemical free energy, G, differs from the chemical free energy, G, by the inclusion of effects from the large-scale electrical environment.

2.2 A More Detailed View of Interfacial Potential Differences - 61 > (a) Properties of the Electrochemical Potential 1. 2. 3. 4. 5. For an uncharged species: Jif = fjbf. For any substance: [xf = ix®a + RT In af, where /if" is the standard chemical potential, and a? is the activity of species / in phase a.

Hf = rf -

For a pure phase at unit activity (e.g., solid Zn, AgCl, Ag, or H 2 at unit fugacity): a For electrons in a metal (z = — 1): ~jx% = /л®а - ¥фа. Activity effects can be disregarded because the electron concentration never changes appreciably. For equilibrium of species / between phases a and /3: JLf = jitf.

(b) Reactions in a Single Phase Within a single conducting phase, ф is constant everywhere and exerts no effect on a chemical equilibrium. The ф terms drop out of relations involving electrochemical potentials, and only chemical potentials will remain. Consider, for example, the acid-base equilibrium: HO Ac < ± H + + OAc~ = This requires that Дн + + ДолеМн+ + рФ + Доле- - РФ А*Н+ + МОАс(c) Reactions Involving Two Phases Without Charge Transfer Let us now examine the solubility equilibrium AgCl (crystal, c) +± Ag + + С Г (solution, 5), (2.2.13) (2.2.10) (2.2.11) (2.2.12) (2.2.9)

which can be treated in two ways. First, one can consider separate equilibria involving Ag + and Cl~ in solution and in the solid. Thus 1 ^cr Recognizing that ^f one has from the sum of (2.2.14) and (2.2.15), MAgl?' = M V Expanding, we obtain A^cf 1 = MAg+ + RT and rearrangement gives Й (2.2.19) where ^ s p is the solubility product. A quicker route to this well-known result is to write down (2.2.17) directly from the chemical equation, (2.2.13). Note that the terms canceled in (2.2.18), and that an implicit cancellation of AgC1 /s terms occurred in (2.2.16). Since the final result depends only on chemical potentials, the ln a + C1

g = Ma-

(2-2.14) (2- 2 - 1 5 ) (2-2.16)

^ r

(2-2.17)

A g + + F a^ trode,

16

When the cell operates galvanically, an oxidation occurs at the left elecH 2 -> 2H (a) + 2e(Pt)
+

(2.3.4)

and a reduction happens on the right, 2H+(j3) + 2e(Pt') -> H 2 (2.3.5)

Therefore, there is a tendency to build up a positive charge in the a phase and a negative charge in p. This tendency is overcome by the movement of ions: H + to the right and Cl~ to the left. For each mole of electrons passed, 1 mole of H + is produced in a, and 1 mole of H + is consumed in /3. The total amount of H + and Cl~ migrating across the boundary between a and /3 must equal 1 mole. The fractions of the current carried by H + and Cl~ are called their transference numbers (or transport numbers). If we let t+ be the transference number for H + and t- be that for Cl~, then clearly, t+ + t- = 1 In general, for an electrolyte containing many ions, /, (2.3.7) Schematically, the process can be represented as shown in Figure 2.3.3. The cell initially features a higher activity of hydrochloric acid (+ as H + , — as Cl~) on the right (Figure (2.3.6)

(«)

и/н 2 /! 1

/ t i l l

1 _/H 2 /R

(с)

/!

+

_ !

! !/H2/PI

Figure 2.3.3 Schematic diagram showing the redistribution of charge during electrolysis of a system featuring a high concentration of HCl on the right and a low concentration on the left. A cell like (2.3.3), having electrodes of the same type on both sides, but with differing activities of one or both of the redox forms, is called a concentration cell.
16

66

Chapter 2. Potentials and Thermodynamics of Cells 2.3.3a); hence discharging it spontaneously produces H + on the left and consumes it on the right. Assume that five units of H + are reacted as shown in Figure 233b. For hydrochloric acid, t+ ~ 0.8 and t- ~ 0.2; therefore, four units of H + must migrate to the right and one unit of Cl~ to the left to maintain electroneutrality. This process is depicted in Figure 2.3.3c, and the final state of the solution is represented in Figure 2.3.3d. A charge imbalance like that suggested in Figure 233b could not actually occur, because a very large electric field would be established, and it would work to erase the imbalance. On a macroscopic scale, electroneutrality is always maintained throughout the solution. The migration represented in Figure 2.3.3c occurs simultaneously with the electron-transfer reactions. Transference numbers are determined by the details of ionic conduction, which are understood mainly through measurements of either the resistance to current flow in solution or its reciprocal, the conductance, L (31, 32). The value of L for a segment of solution immersed in an electric field is directly proportional to the cross-sectional area perpendicular to the field vector and is inversely proportional to the length of the segment along the field. The proportionality constant is the conductivity, к, which is an intrinsic property of the solution:
L = KA/1 (2.3.8)

The conductance, L, is given in units of Siemens (S = fl" 1 ), and к is expressed in S cm" 1 or ft"1 cm" 1 . Since the passage of current through the solution is accomplished by the independent movement of different species, к is the sum of contributions from all ionic species, /. It is intuitive that each component of к is proportional to the concentration of the ion, the magnitude of its charge |ZJ|, and some index of its migration velocity. That index is the mobility, щ, which is the limiting velocity of the ion in an electric field of unit strength. Mobility usually carries dimensions of cm 2 V" 1 s" 1 (i.e., cm/s per V/cm). When a field of strength % is applied to an ion, it will accelerate under the force imposed by the field until the frictional drag exactly counterbalances the electric force. Then, the ion continues its motion at that terminal velocity. This balance is represented in Figure 2.3.4. The magnitude of the force exerted by the field is \z-\ e%, where e is the electronic charge. The frictional drag can be approximated from the Stokes law as 6ттг]ги, where rj is the viscosity of the medium, r is the radius of the ion, and v is the velocity. When the terminal velocity is reached, we have by equation and rearrangement,

The proportionality factor relating an individual ionic conductivity to charge, mobility, and concentration turns out to be the Faraday constant; thus (2.3.10)

Direction of movement

Drag force

ч

—У

Electric force

Figure 2.3.4 Forces on a charged particle moving in solution under the influence of an electric field. The forces balance at the terminal velocity.

2.3 Liquid Junction Potentials

67

The transference number for species / is merely the contribution to conductivity made by that species divided by the total conductivity: (2.3.11)

For solutions of simple, pure electrolytes (i.e., one positive and one negative ionic species), such as KC1, CaCl2, and HNO 3 , conductance is often quantified in terms of the equivalent conductivity, Л, which is defined by (2.3.12) where C e q is the concentration of positive (or negative) charges. Thus, Л expresses the conductivity per unit concentration of charge. Since C\z\ = C e q for either ionic species in these systems, one finds from (2.3.10) and (2.3.12) that

Л = F(u+ + u-)

(2.3.13)

where u+ refers to the cation and u- to the anion. This relation suggests that Л could be regarded as the sum of individual equivalent ionic conductivities, Л = Л+ + A_ hence we find Ai = Fu{ In these simple solutions, then, the transference number tx is given by (2.3.15) (2.3.14)

л or, alternatively,

(2.3.16)

(2.3.17) Transference numbers can be measured by several approaches (31, 32), and numerous data for pure solutions appear in the literature. Frequently, transference numbers are measured by noting concentration changes caused by electrolysis, as in the experiment shown in Figure 2.3.3 (see Problem 2.11). Table 2.3.1 displays a few values for aqueous solutions at 25°C. From results of this sort, one can evaluate the individual ionic conductivities, Aj. Both Aj and t-x depend on the concentration of the pure electrolyte, because interactions between ions tend to alter the mobilities (31-33). Lists of A values, like Table 2.3.2, usually give figures for AOi, which are obtained by extrapolation to infinite dilution. In the absence of measured transference numbers, it is convenient to use these to estimate t\ for pure solutions by (2.3.16), or for mixed electrolytes by the following equivalent to (2.3.11), (2.3.18)

In addition to the liquid electrolytes that we have been considering, solid electrolytes, such as sodium /3-alumina, the silver halides, and polymers like polyethylene

68

Chapter 2. Potentials and Thermodynamics of Cells TABLE 2.3.1 Cation Transference Numbers a for Aqueous Solutions at 25°C Concentration, Ceq Electrolyte HC1 NaCl KC1 NH4C1 KNO3 Na 2 SO 4 K 2 SO 4 a b

0.01 0.8251 0.3918 0.4902 0.4907 0.5084 0.3848 0.4829

0.05 0.8292 0.3876 0.4899 0.4905 0.5093 0.3829 0.4870

0.1 0.8314 0.3854 0.4898 0.4907 0.5103 0.3828 0.4890

0.2 0.8337 0.3821 0.4894 0.4911 0.5120 0.3828 0.4910

From D. A. Maclnnes, "The Principles of Electrochemistry," Dover, New York, 1961, p. 85 and references cited therein. ^Moles of positive (or negative) charge per liter. oxide/LiClO4 (34, 35), are sometimes used in electrochemical cells. In these materials, ions move under the influence of an electric field, even in the absence of solvent. For example, the conductivity of a single crystal of sodium /3-alumina at room temperature is 0.035 S/cm, a value similar to that of aqueous solutions. Solid electrolytes are technologically important in the fabrication of batteries and electrochemical devices. In some of these materials (e.g., a-Ag2S and AgBr), and unlike essentially all liquid electrolytes,

TABLE 2.3.2 Ionic Properties at Infinite Dilution in Aqueous Solutions at 25°C Ion H+ K+ Na + Li + NH^ A0,cm2n"1equiv"lfl 349.82 73.52 50.11 38.69 73.4 59.50 198 76.34 78.4 76.85 71.44 40.9 68.0 79.8 44.48 101.0 110.5 и, cm 2 sec" 1 V" 1 * 3.625 7.619 5.193 4.010 7.61 6.166 2.05 7.912 8.13 7.96 7.404 4.24 7.05 8.27 4.610 1.047 1.145 X 10" 3 X 10" 4 X 10" 4 X 10" 4 X 10~4 X 10~4 X 10" 3 X 10~4 X 10" 4 X 10" 4 X 10" 4 X 10" 4 X 10" 4 X 10" 4 X 10" 4 X 10~3 X 10" 3

ka

2+

OH~ СГ Br" I" NO3OAc" СЮ4

kstit

HCO3|Fe(CN)^ |Fe(CN)^ a From D. A. Maclnnes, "The Principles of Electrochemistry," Dover, New York, 1961, p. 342 ^Calculated from AQ.

2.3 Liquid Junction Potentials

69

WWWWWWWWWWWWVWNA-J Figure 2.3.5 Experimental system for demonstrating reversible flow of charge through a cell with a liquid junction. there is electronic conductivity as well as ionic conductivity. The relative contribution of electronic conduction through the solid electrolyte can be found by applying a potential to a cell that is too small to drive electrochemical reactions and noting the magnitude of the (nonfaradaic) current. Alternatively, an electrolysis can be carried out and the faradaic contribution determined separately (see Problem 2.12).

2.3.4

Calculation of Liquid Junction Potentials
Imagine the concentration cell (2.3.3) connected to a power supply as shown in Figure 2.3.5. The voltage from the supply opposes that from the cell, and one finds experimentally that it is possible to oppose the cell voltage exactly, so that no current flows through the galvanometer, G. If the magnitude of the opposing voltage is reduced very slightly, the cell operates spontaneously as described above, and electrons flow from Pt to Pt' in the external circuit. The process occurring at the liquid junction is the passage of an equivalent negative charge from right to left. If the opposing voltage is increased from the null point, the entire process reverses, including charge transfer through the interface between the electrolytes. The fact that an infinitesimal change in the driving force can reverse the direction of charge passage implies that the electrochemical free energy change for the whole process is zero. These events can be divided into those involving the chemical transformations at the metal-solution interfaces: (2.3.19) Н + (/3) + e(Pt') ^± ^Н 2 and that effecting charge transport at the liquid junction depicted in Figure 2.3.6: t+H+(a) + Г-СГ08) ^± t+ H + 08) + /- С Г (a) (2.3.21) (2.3.20)

Note that (2.3.19) and (2.3.20) are at strict equilibrium under the null-current condition; hence the electrochemical free energy change for each of them individually is zero. Of course, this is also true for their sum: H + 08) + e(Pt') H + (a) (2.3.22)

(a2)

LCI-

Figure 2.3.6 Reversible charge transfer through the liquid junction in Figure 2.3.5.

70

Chapter 2. Potentials and Thermodynamics of Cells which describes the chemical change in the system. The sum of this equation and the charge transport relation, (2.3.21), describes the overall cell operation. However, since we have just learned that the electrochemical free energy changes for both the overall process and (2.3.22) are zero, we must conclude that the electrochemical free energy change for (2.3.21) is also zero. In other words, charge transport across the junction occurs in such a way that the electrochemical free energy change vanishes, even though it cannot be considered as a process at equilibrium. This important conclusion permits an approach to the calculation of junction potentials. Let us focus first on the net chemical reaction, (2.3.22). Since the electrochemical free energy change is zero, ^H+ + Jg = /Zg+ + jS* FE = F(0 P t ' - i \F \

(2.3.35)

(2.3.36) which is the general expression for the junction potential. It is easy to see now that (2.3.30) is a special case for type 1 junctions between 1:1 electrolytes having constant tv Note that £j is a strong function of t+ and t-, and that it actually vanishes if t+ = t~. The value of Щ as a function of t+ for a 1:1 electrolyte with ai/a2 = 10 is Ei = 59.1 at 25°C. For example, the cell Ag/AgCl/KCl (0.1 M)/KC1 (0.01 M)/AgCl/Ag (2.3.38) - 1) mV (2.3.37)

has t+ = 0.49; hence E-} = -1.2 mV. While type 1 junctions can be treated with some rigor and are independent of the method of forming the junction, type 2 and type 3 junctions have potentials that depend on the technique of junction formation (e.g., static or flowing) and can be treated only in an approximate manner. Different approaches to junction formation apparently lead to

- f/zj mole of each cation -fj/z, mole ofeach anion

Location Electrochemical potential

x + dx

Figure 2.3.7 Transfer of net positive charge from left to right through an infinitesimal segment of a junction region. Each species must contribute ty moles of charge per mole of overall charge transported; hence t-\z-\ moles of that species must migrate.

72

Chapter 2. Potentials and Thermodynamics of Cells different profiles of tx through the junction, which in turn lead to different integrals for (2.3.36). Approximate values for E-} can be obtained by assuming (a) that concentrations of ions everywhere in the junction are equivalent to activities and (b) that the concentration of each ion follows a linear transition between the two phases. Then, (2.3.36) can be integrated to give the Henderson equation (24, 30): — [COS) - C;(a)] 1
F * i У. \z{ «AW RT, i F v i MiCiO3) i (2.3.39)

where щ is the mobility of species /, and C\ is its molar concentration. For type 2 junctions between 1:1 electrolytes, this equation collapses to the Lewis-Sargent relation'. (2.3.40) where the positive sign corresponds to a junction with a common cation in the two phases, and the negative sign applies to the case with a common anion. As an example, consider the cell Ag/AgCl/HCl (0.1M)/KCl (0.1 M)/AgCl/Ag (2.3.41)

for which Есец is essentially E-y The measured value at 25°C is 28 ± 1 mV, depending on the technique of junction formation (30), while the estimated value from (2.3.40) and the data of Table 2.3.2 is 26.8 mV.

2.3.5

Minimizing Liquid Junction Potentials In most electrochemical experiments, the junction potential is an additional troublesome factor, so attempts are often made to minimize it. Alternatively, one hopes that it is small or that it at least remains constant. A familiar method for minimizing £j is to replace the junction, for example, HC1 (Ci)/NaCl (C2) (2.3.42)

with a system featuring a concentrated solution in an intermediate salt bridge, where the solution in the bridge has ions of nearly equal mobility. Such a system is HC1 (Ci)/KCl (C)/NaCl (C2) Table 2.3.3 lists some measured junction potentials for the cell, Hg/Hg2Cl2/HCl (0.1 M)/KC1 (С)/Ка (0 (2.3.44) (2.3.43)

As С increases, £j falls markedly, because ionic transport at the two junctions is dominated more and more extensively by the massive amounts of KC1. The series junctions become more similar in magnitude and have opposite polarities; hence they tend to cancel. Solutions used in aqueous salt bridges usually contain KC1 (t+ = 0.49, t- = 0.51) or, where Cl" is deleterious, KNO3 (t+ = 0.51, t- = 0.49). Other concentrated solutions with equitransferent ions that have been suggested (39) for salt bridges include CsCl (f+ = 0.5025), RbBr (t+ = 0.4958), and NH4I (t+ = 0.4906). In many measurements, such as the determination of pH, it is sufficient if the junction potential remains constant between calibra-

2.3 Liquid Junction Potentials
TABLE 2.3.3 Effect of a Salt Bridge on Measured Junction Potentials" Concentration of KC1, C(M) 0.1 0.2 0.5 1.0 2.5 3.5 4.2 (saturated) £j,mV 27 20 13 8.4 3.4 1.1 > N a + > K + , Rb + , Cs + > > Ca 2 + , (b) sodium-sensitive electrodes with the order Ag + > H + > Na + > > K + , Li + > > Ca 2 + , and (c) a more general cation-sensitive electrode with a narrower range of selectivities in the order H + > K + > Na + > N H | , Li + » Ca 2 + . There is a large literature on the design, performance, and theory of glass electrodes (37, 46-55). The interested reader is referred to it for more advanced discussions.

2.4 Selective Electrodes

79

Other Ion-Selective Electrodes
The principles that we have just reviewed also apply to other types of selective membranes (48, 50-59). They fall generally into two categories. (a) Solid-State Membranes Like the glass membrane, which is a member of this group, the remaining common solidstate membranes are electrolytes having tendencies toward the preferential adsorption of certain ions on their surfaces. Consider, for example, the single-crystal LaF3 membrane, which is doped with EuF2 to create fluoride vacancies that allow ionic conduction by fluoride. Its surface selectively accommodates F~ to the virtual exclusion of other species except OH~. Other devices are made from precipitates of insoluble salts, such as AgCl, AgBr, Agl, Ag2S, CuS, CdS, and PbS. The precipitates are usually pressed into pellets or are suspended in polymer matrices. The silver salts conduct by mobile Ag + ions, but the heavy metal sulfides are usually mixed with Ag2S, since they are not very conductive. The surfaces of these membranes are generally sensitive to the ions comprising the salts, as well as to other species that tend to form very insoluble precipitates with a constituent ion. For example, the Ag2S membrane responds to Ag + , S2~~, and Hg 2 + . Likewise, the AgCl membrane is sensitive to Ag + , Cl~, Br~, I~, CN~, and OH~. (b) Liquid and Polymer Membranes An alternative structure utilizes a hydrophobic liquid membrane as the sensing element. The liquid is stabilized physically between an aqueous internal filling solution and an aqueous test solution by allowing it to permeate a porous, lipophilic diaphragm. A reservoir contacting the outer edges of the diaphragm contains this liquid. Chelating agents with selectivity toward ions of interest are dissolved in it, and they provide the mechanism for selective charge transport across the boundaries of the membrane. A device based on these principles is a calcium-selective electrode. The hydrophobic solvent might be dioctylphenylphosphonate, and the chelating agent might be the sodium salt of an alkyl phosphate ester, (RO)2 PO^Na + , where R is an aliphatic chain having 8-18 carbons. The membrane is sensitive to Ca 2+ , Zn 2+ , Fe 2 + , Pb 2 + , Cu 2+ , tetra-alkylammonium ions, and still other species to lesser degrees. "Water hardness" electrodes are based on similar agents, but are designed to show virtually equal responses to Ca 2+ and Mg 2+ . Other systems featuring liquid ion-exchangers are available for anions, such as NO^~, ClO^, and Cl~. Nitrate and perchlorate are sensed by membranes including alkylated 1,10-phenanthroline complexes of Ni 2 + and Fe 2 + , respectively. All three ions are active at other membranes based on quaternary ammonium salts. In commercial electrodes, the liquid ion-exchanger is in a form in which the chelating agent is immobilized in a hydrophobic polymer membrane like poly(vinylchloride) (Figure 2.4.4). Electrodes based on this design (called polymer or plastic membrane ISEs) are more rugged and generally offer superior performance. Liquid ion-exchangers all feature charged chelating agents, and various ion-exchange equilibria play a role in their operation. A related type of device, also featuring a stabilized liquid membrane, involves uncharged chelating agents that enable the transport of charge by selectively complexing certain ions. These agents are sometimes called neutral carriers. Systems based on them typically also involve the presence of some anionic sites in the membrane, either naturally occurring or added in the form of hydrophobic ions, and these anionic sites contribute to the ion-exchange process (56-58). It has also been proposed that electrodes based on neutral carriers operate by a phase-boundary (i.e., adsorption), rather than a carrier mechanism (59).

80

:J

Chapter 2. Potentials and Thermodynamics of Cells
Electrical contact

Module housing Aqueous reference solution Reference element (AgCI) Ion selective membrane

Figure 2.4.4 A typical plastic membrane ISE. [Courtesy of Orion Research, Inc.]

For example, potassium-selective electrodes can be constructed with the natural macrocycle valinomycin as a neutral carrier in diphenyl ether. This membrane has a much higher sensitivity to K + than to Na + , Li + , Mg 2 + , Ca 2 + , or H + ; but Rb + and Cs + are sensed to much the same degree as K + . The selectivity seems to rest mostly on the molecular recognition of the target ion by the complexing site of the carrier. (c) Commercial Devices Table 2.4.1 is a listing of typical commercial ion-selective electrodes, the pH and concentration ranges over which they operate, and typical interferences. Selectivity coefficients for many of these electrodes are available (55, 57). TABLE 2.4.1 Typical Commercially Available Ion-Selective Electrodes Species Ammonium (NH4") Barium (Ba 2 + ) Bromide (Br~) Cadmium (Cd 2 + ) Calcium (Ca 2 + ) Chloride (СГ) Copper (Cu 2 + ) Cyanide (CN") Fluoride (F~) Iodide ( Г ) Lead (Pb 2 + ) Nitrate (NO3") Nitrite (NO2) Potassium (K + ) Silver (Ag + ) Sodium (Na + ) Sulfide (S2~) a Type* L L S S L S

Concentration Range(M)

рн
Range 5-8 5-9 2-12 3-7 4-9 2-11 0-7 10-14 5-8 3-12 0-9 3-10 3-10 4-9 2-9 9-12 12-14 Interferences K + ,Na + ,Mg 2 + K + ,Na + ,Ca 2 + I", S 2 ", CN" Ag + ,Hg 2 + ,Cu 2 + ,Pb 2 + ,Fe 3 + Ba 2 + ,Mg 2 + ,Na + ,Pb 2 + I", S2~, CN~, Br" Ag + ,Hg 2+ ,S 2 ~,CT,Br~ S2~ OH" S2" Ag + ,Hg 2 + ,S 2 -,Cd 2 + ,Cu 2 + ,Fe 3 + Cl",Br",NO2,F",SO5~ Cl",Br"",NOJ,F",SO5~ Na + , Ca 2 + , Mg 2 + S2",Hg2+ Li + , K + , NH^ Ag + ,Hg 2 +

КГЧоКГ КГЧоКГ 5

6

s s s s s

L L L S G S

1 to 10~5 10"1 to 10~7 1 to 10~7 1 to 5 X 10~5 КГЧоКГ7 10" 2 to 10" 6 1 to 10" 7 1 to 10~7 1(ГЧо1(Г 6 1 to 5 X 10~6 1 to 10~6 1 to 10" 6 1 to 10~7 Sat'd to 10~6 1 to 10~7

G = glass; L = liquid membrane; S = solid-state. Typical temperature ranges are 0-50°C for liquidmembrane and 0-80°C for solid-state electrodes.

2.4 Selective Electrodes • 81 « (d) Detection Limits As shown in Table 2.4.1, the lower limit for detection of an ion with an ISE is generally 10~ 6 to 10~ 7 M. This limit is largely governed by the leaching of ions from the internal electrolyte into the sample solution (60). The leakage can be prevented by using a lower concentration of the ion of interest in the internal electrolyte, so that the concentration gradient established in the membrane causes an ion flux from the sample to the inner electrolyte. This low concentration can be maintained with an ion buffer, that is, a mixture of the metal ion with an excess of a strong complexing agent. In addition, a high concentration of a second potential-determining ion is added to the internal solution. Under these conditions, the lower detection limit can be considerably improved. For example, for a conventional liquid-membrane Pb 2 + electrode with an internal filling solution of 5 X 10 4 M Pb z + and 5 X 10 l M M g z \ the detection limit for Pb: was 4 X 10~6 M. When the internal solution was changed to 10~3 M Pb 2 + and 5 X 10~ z M Na2EDTA (yielding a free [Pb z+ ] = 10" i 2 M), the detection limit decreased to 5 X 10~ 12 M (61). In the internal solution, the dominant potential-determining ion isNa+at0.1M.

2.4.4 Gas-Sensing Electrodes
Figure 2.4.5 depicts the structure of a typical potentiometric gas-sensing electrode (62). In general, such a device involves a glass pH electrode that is protected from the test solution by a polymer diaphragm. Between the glass membrane and the diaphragm is a small volume of electrolyte. Small molecules, such as SO2, NH3, and CO2, can penetrate the membrane and interact with the trapped electrolyte by reactions that produce changes in pH. The glass electrode responds to the alterations in acidity. Electrochemical cells that use a solid electrolyte composed of zirconium dioxide containing Y2O3 (yttria-stabilized zirconia) are available to measure the oxygen content of gases at high temperature. In fact, sensors of this type are widely used to monitor the exhaust gas from the internal combustion engines of motor vehicles, so that the airto-fuel mixture can be controlled to minimize the emission of pollutants such as CO and NOX. This solid electrolyte shows good conductivity only at high temperatures (500-1,000°C), where the conduction process is the migration of oxide ions. A typical sensor is composed of a tube of zirconia with Pt electrodes deposited on the inside and outside of the tube. The outside electrode contacts air with a known partial pressure of

Outer body Inner body

O-ring Spacer — Bottom cap Sensing element Membrane

Figure 2.4.5 Structure of a gas-sensing electrode. [Courtesy of Orion Research, Inc/

82

Chapter 2. Potentials and Thermodynamics of Cells oxygen, p a , and serves as the reference electrode. The inside of the tube is exposed to the hot exhaust gas with a lower oxygen partial pressure, p e g . The cell configuration can thus be written Pt/O2 (exhaust gas, peg)/Zr02 + Y 2 O 3 /O 2 (air,/?a)/Pt (2.4.21)

and the potential of this oxygen concentration cell can be used to measure pQg (Problem 2.19). We note here that the widely employed Clark oxygen electrode differs fundamentally from these devices (18, 63). The Clark device is similar in construction to the apparatus of Figure 2.4.5, in that a polymer membrane traps an electrolyte against a sensing surface. However, the sensor is a platinum electrode, and the analytical signal is the steady-state current flow due to the faradaic reduction of molecular oxygen.

2.4.5

Enzyme-Coupled Devices
The natural specificity of enzyme-catalyzed reactions can be used as the basis for selective detection of analytes (49, 64-68). One fruitful approach has featured potentiometric sensors with a structure similar to that of Figure 2.4.5, with the difference that the gap between the ion-selective electrode and the polymer diaphragm is filled with a matrix in which an enzyme is immobilized. For example, urease, together with a buffered electrolyte, might be held in a crosslinked polyacrylamide gel. When the electrode is immersed in a test solution, there will be a selective response toward urea, which diffuses through the diaphragm into the gel. The response comes about because the urease catalyzes the process:

О
N H 2 — С—NH 2 + H + + 2H 2 O
Urease

нсо;

(2.4.22)

The resulting ammonium ions can be detected with a cation-sensitive glass membrane. Alternatively, one could use a gas-sensing electrode for ammonia in place of the glass electrode, so that interferences from H + , Na + , and K + are reduced. The research literature features many examples of this basic strategy. Different enzymes allow selective determinations of single species, such as glucose (with glucose oxidase), or groups of substances such as the L-amino acids (with L-amino acid oxidase). Recent reviews should be consulted for a more complete view of the field (66-68). Amperometric enzyme electrodes are discussed in Sections 14.2.5 and 14.4.2(c).

2.5 REFERENCES
1. The arguments presented here follow those given earlier by D. A. Maclnnes ("The Principles of Electrochemistry," Dover, New York, 1961, pp. 110-113) and by J. J. Lingane ("Electroanalytical Chemistry," 2nd ed., Wiley Interscience, New York, 1958, pp. 40-45). Experiments like those described here were actually carried out by H. Jahn (Z. Physik. С hem., 18, 399 (1895). 2. I. M. Klotz and R. M. Rosenberg, "Chemical Thermodynamics," 4th ed., Benjamin/Cummings, Menlo Park, CA, 1986. 3. J. J. Lingane, "Electroanalytical Chemistry," 2nd ed., Wiley-Interscience, New York, 1958, Chap. 3. 4. F. C. Anson, /. Chem. Educ., 36, 394 (1959). 5. A. J. Bard, R. Parsons, and J. Jordan, Eds., "Standard Potentials in Aqueous Solutions," Marcel Dekker, New York, 1985. 6. http://webbook.nist.gov/, National Institute of Standards and Technology. 7. A. J. Bard and H. Lund, Eds., "Encyclopedia of Electrochemistry of the Elements," Marcel Dekker, New York, 1973-1980.

2.5 References 8. M. W. Chase, Jr., "NIST-JANAF Thermochemical Tables," 4th ed., American Chemical Society, Washington, and American Institute of Physics, New York, for the National Institute of Standards and Technology, 1998. 9. L. R. Faulkner, /. Chem. Educ, 60, 262 (1983). 10. R. Parsons in A. J. Bard, R. Parsons, and J. Jordan, Eds., op.cit., Chap. 1. 11. A. Henglein, Ber. Bunsenges. Phys. Chem., 94, 600 (1990). 12. A. Henglein, Top. Curr. Chem., 1988, 113. 13. A. Henglein, Accts. Chem. Res., 9, 1861 (1989). 14. D. W. Suggs and A. J. Bard, /. Am. Chem. Soc, 116, 10725 (1994). 15. R. Parsons, op. cit., p. 5. 16. D. J. G. Ives and G. J. Janz, Eds., "Reference Electrodes," Academic, New York, 1961. 17. J. N. Butler, Adv. Electrochem. Electrochem. Engr.,7, 77 (1970). 18. D. T. Sawyer, A. Sobkowiak, and J. L. Roberts, Jr., "Electrochemistry for Chemists," 2nd ed., Wiley, New York, 1995. 19. G. Gritzner and J. Kuta, Pure Appl. Chem., 56, 461 (1984). 20. (a) P. Peerce and A. J. Bard, /. Electroanal Chem., 108, 121 (1980); (b) R. M. Kannuck, J. M. Bellama, E. A Blubaugh, and R. A. Durst, Anal. Chem., 59, 1473 (1987). 21. D. Halliday and R. Resnick, "Physics," 3rd ed., Wiley, New York, 1978, Chap. 29. 22. Ibid., Chap. 28. 23. J. O'M. Bockris and A. K. N. Reddy, "Modern Electrochemistry," Vol. 2, Plenum, New York, 1970, Chap. 7. 24. K. J. Vetter, "Electrochemical Kinetics," Academic, New York, 1967. 25. В. Е. Conway, "Theory and Principles of Electrode Processes," Ronald, New York, 1965, Chap. 13. 26. R. Parsons, Mod. Asp. Electrochem., 1, 103 (1954). 27. J. A. V. Butler, Proc. Roy. Soc, London, 112A, 129 (1926). 28. E. A. Guggenheim, /. Phys. Chem., 33, 842 (1929); 34, 1540 (1930). 29. S. Trasatti, Pure Appl. Chem., 58, 955 (1986). 30. D. A. Maclnnes, "Principles of Electrochemistry," Dover, New York, 1961, Chap. 13. 31. J. O'M. Bockris and A. K. N. Reddy, op. cit., Vol. 1, Chap. 4. 32. D. A. Maclnnes, op. cit., Chap. 4. 33. /Ш.,Спар. 18.

83

34. D. O. Raleigh, Electroanal. Chem., 6, 87 (1973). 35. G. Holzapfel, "Solid State Electrochemistry" in Encycl. Phys. Sci. TechnoL, R. A. Meyers, Ed., Academic, New York, 1992, Vol. 15, p. 471. 36. J. O'M. Bockris and A. K. N. Reddy, op. cit., Chap. 3. 37. R. G. Bates, "Determination of pH," 2nd ed., Wiley-Interscience, New York, 1973. 38. R. M. Garrels in "Glass Electrodes for Hydrogen and Other Cations," G. Eisenman, Ed., Marcel Dekker, New York, 1967, Chap. 13. 39. P. R. Mussini, S. Rondinini, A. Cipolli, R. Manenti and M. Mauretti, Ber. Bunsenges. Phys. Chem., 97, 1034(1993). 40. H. H. J. Girault and D. J. Schiffrin, Electroanal. Chem., 15, 1 (1989). 41. H. H. J. Girault, Mod. Asp. Electrochem., 25, 1 (1993). 42. P. Vany sek, "Electrochemistry on Liquid/Liquid Interfaces," Springer, Berlin, 1985. 43. A. G. Volkov and D. W. Deamer, Eds., "LiquidLiquid Interfaces," CRC, Boca Raton, FL, 1996. 44. A. G. Volkov, D. W. Deamer, D. L. Tanelian, and V. S Markin, "Liquid Interfaces in Chemistry and Biology," Wiley-Interscience, New York, 1997. 45. E. Grunwald, G. Baughman, and G. Kohnstam, /. Am. Chem. Soc, 82, 5801 (1960). 46. M. Dole, "The Glass Electrode," Wiley, New York, 1941. 47. G. Eisenman, Ed., "Glass Electrodes for Hydrogen and Other Cations," Marcel Dekker, New York, 1967. 48. R. A. Durst, Ed., "Ion Selective Electrodes," Nat. Bur. Stand. Spec. Pub. 314, U.S. Government Printing Office, Washington, 1969. 49. N. Lakshminarayanaiah in "Electrochemistry," (A Specialist Periodical Report), Vols. 2, 4, 5, and 7; G. J. Hills (Vol. 2); and H. R. Thirsk (Vols, 4, 5, and 7); Senior Reporters, Chemical Society, London, 1972, 1974, 1975, and 1980. 50. H. Freiser, "Ion-Selective Electrodes in Analytical Chemistry," Plenum, New York, Vol. 1, 1979; Vol. 2, 1980. 51. J. Koryta and K. Stulik, 'Ion-Selective Electrodes," 2nd ed., Cambridge University Press, Cambridge, 1983. 52. A. Evans, "Potentiometry and Ion Selective Electrodes," Wiley, New York, 1987.

84 Э Chapter 2. Potentials and Thermodynamics of Cells 53. D. Ammann, "Ion-Selective Microelectrodes: Principles, Design, and Application," Springer, Berlin, 1986. 54. E. Lindner, K. Toth, and E. Pungor, "Dynamic Characteristics of Ion-Sensitive Electrodes," CRC, Boca Raton, FL, 1988. 55. Y. Umezawa, Ed., "CRC Handbook of Ion-Selective Electrodes," CRC, Boca Raton, FL 1990. 56. E. Bakker, P. Buhlmann, and E. Pretsch, Chem. Rev., 97, 3083 (1997). 57. P. Buhlmann, E. Pretsch, and E. Bakker, Chem. Rev., 98, 1593 (1998). 58. R. P. Buck and E. Lindner, Accts. Chem. Res., 31, 257 (1998). 59. E. Pungor, Pure Appl. Chem., 64, 503 (1992). 60. S. Mathison and E. Bakker, Anal. Chem., 70, 303 (1998). 61. T. Sokalski, A. Ceresa, T. Zwicki, and E. Pretsch, /. Am. Chem. Soc, 119, 11347 (1997). 62. J. W. Ross, J. H. Riseman, and J. A. Krueger, Pure Appi Chem., 36, 473 (1973). 63. L. С Clark, Jr., Trans. Am. Soc. Artif. Intern. Organs, 2, 41 (1956). 64. G. G. Guilbault, Pure Appl. Chem., 25, 727 (1971). 65. G. A. Rechnitz, Chem. Engr. News, 53 (4), 29 {1915). 66. E. A. H. Hall, "Biosensors," Prentice Hall, Englewood Cliffs, NJ, 1991, Chap. 9. 67. A. J. Cunningham, "Introduction to Bioanalytic a l Sensors," Wiley, New York, 1998, Chap. 4. 68. H. S. Yim, C. E. Kibbey, S. C. Ma, D. M. Kliza, D. Liu, S. B. Park, C. E. Torre, and M. E. Meyerhoff, Biosens. Bioelectron., 8, 1 (1993).

2.6 PROBLEMS
2.1 Devise electrochemical cells in which the following reactions could be made to occur. If liquid junctions are necessary, note them in the cell schematic appropriately, but neglect their effects. (a) H 2 O ^± H + + OH~ (b) 2H 2 + O 2 ^± H 2 O (c) 2PbSO4 + 2H 2 O ±± PbO 2 + Pb + 4H + + 2 S O f (d) An T 4- TMPD^ ±± An + TMPD (in acetonitrile, where An and An T are anthracene and its anion radical, and TMPD and TMPD^ are AW^'^-tetramethyl-p-phenylenediamine and its cation radical. Use anthracene potentials for DMF solutions given in Appendix C.3). (e) 2Ce 3 + + 2H + + BQ ^± 2Ce 4 + + H 2 Q (aqueous, where BQ is p-benzoquinone and H 2 Q is phydroquinone) (f) Ag + + I~ / _
CQ(0,

t)

C R (0, t)

FT)

(3.4.30)

which can be substituted as above and rearranged to give
V l F

v'o

kc

W

(3.4.31)

In terms of the charge- and mass-transfer pseudoresistances defined in equations 1.4.28 and 3.4.13, this equation is (3.4.32) Here we see very clearly that when /0 is much greater than the limiting currents, Rct « RmtiC + i?mt,a a n d the overpotential, even near £ e q , is a concentration over-

3.5 Multistep Mechanisms

107

-200

-300

-400 , mV

Figure 3.4.6 Relationship between the activation overpotential and net current demand relative to the exchange current. The reaction is О + e ^ R with a = 0.5, T = 298 K, and / c = -//a = //. Numbers by curves show /0///. /

potential. On the other hand, if IQ is much less than the limiting currents, then /?mt,c + ^mt,a < < ^ct» a n c * the overpotential near Ещ is due to activation of charge transfer. This argument is simply another way of looking at the points made earlier in Section 3.4.3(a). In the Tafel regions, other useful forms of (3.4.29) can be obtained. For the cathodic branch at high r\ values, the anodic contribution is insignificant, and (3.4.29) becomes

= or h-JL

(3.4.33)

RT, *o , RT, (*'/,c ~ 0 (3.4.34) 7] = —~ In — + —~ In at iic aF This equation can be useful for obtaining kinetic parameters for systems in which the normal Tafel plots are complicated by mass-transfer effects.

3.5 MULTISTEP MECHANISMS (11, 13, 14, 25, 26, 35)
The foregoing sections have concentrated on the potential dependences of the forward and reverse rate constants governing the simple one-step, one-electron electrode reaction. By restricting our view in this way, we have achieved a qualitative and quantitative understanding of the major features of electrode kinetics. Also, we have developed a set of relations that we can expect to fit a number of real chemical systems, for example, Fe(CN)^" + e Fe(CN)^~ (3.5.1) (3.5.2) Anthracene + e ^ Anthracene" (3.5.3)

108

Chapter 3. Kinetics of Electrode Reactions But we must now recognize that most electrode processes are mechanisms of several steps. For example, the important reaction 2H + + 2е±±Щ (3.5.4)

clearly must involve several elementary reactions. The hydrogen nuclei are separated in the oxidized form, but are combined by reduction. Somehow, during reduction, there must be a pair of charge transfers and some chemical means for linking the two nuclei. Consider also the reduction Sn 4 + + 2e *± Sn 2 + (3.5.5)

Is it realistic to regard two electrons as tunneling simultaneously through the interface? Or must we consider the reduction and oxidation sequences as two one-electron processes proceeding through the ephemeral intermediate Sn 3 + ? Another case that looks simple at first glance is the deposition of silver from aqueous potassium nitrate: Ag + + e ^ A g (3.5.6)

However, there is evidence that this reduction involves at least a charge-transfer step, creating an adsorbed silver atom (adatom), and a crystallization step, in which the adatom migrates across the surface until it finds a vacant lattice site. Electrode processes may also involve adsorption and desorption kinetics of primary reactants, intermediates, and products. Thus, electrode reactions generally can be expected to show complex behavior, and for each mechanistic sequence, one would obtain a distinct theoretical linkage between current and potential. That relation would have to take into account the potential dependences of all steps and the surface concentrations of all intermediates, in addition to the concentrations of the primary reactants and products. A great deal of effort has been spent in studying the mechanisms of complex electrode reactions. One general approach is based on steady-state current-potential curves. Theoretical responses are derived on the basis of mechanistic alternatives, then one compares predicted behavior, such as the variation of exchange current with reactant concentration, with the behavior found experimentally. A number of excellent expositions of this approach are available in the literature (8-14, 25, 26, 35). We will not delve into specific cases in this chapter, except in Problems 3.7 and 3.10. More commonly, complex behavior is elucidated by studies of transient responses, such as cyclic voltammetry at different scan rates. The experimental study of multistep reactions by such techniques is covered in Chapter 12.

3.5.1

Rate-Determining Electron Transfer
In the study of chemical kinetics, one can often simplify the prediction and analysis of behavior by recognizing that a single step of a mechanism is much more sluggish than all the others, so that it controls the rate of the overall reaction. If the mechanism is an electrode process, this rate-determining step (RDS) can be a heterogeneous electron-transfer reaction. A widely held concept in electrochemistry is that truly elementary electron-transfer reactions always involve the exchange of one electron, so that an overall process involving a change of n electrons must involve n distinct electron-transfer steps. Of course, it may also involve other elementary reactions, such as adsorption, desorption, or various chemical reactions away from the interface. Within this view, a rate-determining electrontransfer is always a one-electron-process, and the results that we derived above for the

3.5 Multistep Mechanisms

109

one-step, one-electron process can be used to describe the RDS, although the concentrations must often be understood as applying to intermediates, rather than to starting species or final products. For example, consider an overall process in which О and R are coupled in an overall multielectron process О + ne *± R by a mechanism having the following general character: О + n'e +± O' O'+e^R' К R' + ri'e > & %& P P relation is (3.3.11) written for the RDS and multiplied by n, because each net conversion of O' to R' results in the flow of n electrons, not just one electron, across the interface. The concentrations CQ>(0, t) and C R '(0, i) are controlled not only by the interplay between mass transfer and the kinetics of heterogeneous electron transfer, as we found in Section 3.4, but also by the properties of the preceding and following reactions. The situation can become quite complicated, so we will make no attempt to discuss the general problem. However, a few important simple cases exist, and we will develop them briefly now. 11

3.5.2

Multistep Processes at Equilibrium
If a true equilibrium exists for the overall process, all steps in the mechanism are individually at equilibrium. Thus, the surface concentrations of O' and R' are the values in equilibrium with the bulk concentrations of О and R, respectively. We designate them as (C(y)eq and (C R ') e q - Recognizing that / = 0, we can proceed through the treatment leading to (3.4.2) to obtain the analogous relation

?ds) = < £ ф

(3.5.12)

For the mechanism in (3.5.8)-(3.5.10), nernstian relationships define the equilibria for the pre- and postreactions, and they can be written in the following forms: C* е (C )
^Y(£eq--is post) _ ^ R ; ^eq ^ ^ ^ч

п'/(Ещ-Е°рТ&)

_

Q

10 1

The discussions that follow hold if either or both of n' or n" are zero.

^n the first edition and in much of the literature, one finds n.d used as the n value of the rate-determining step. As a consequnce n.d appears in many kinetic expressions. Since n.d is probably always 1, it is a redundant symbol and has been dropped in this edition. The current-potential characteristic for a multistep process has often been expressed as i = nFAk0 [C o (0, ^-«"аЯЯ-я 0 ') _ C R ( 0 ? ^(i-«)«a/(£-£ 0 ')] This is rarely, if ever, an accurate form of the i-E characteristic for multistep mechanisms.

110

Chapter 3. Kinetics of Electrode Reactions where E^Q and £p0St apply to (3.5.8) and (3.5.10), respectively. Substitution for the equilibrium concentrations of O' and R' in (3.5.12) gives

CR

Recognizing that n = n' + n" + 1 and that E 2.10)

o>

for the overall process is (see Problem

.(v _ ^rds + "'£pre + ""^post

we can distill (3.5.14) into ^/(£ e q-£°') = _2.
CR

(3.5.16)

which is the exponential form of the Nernst equation for the overall reaction, (3.5.17) Of course, this is a required result if the kinetic model has any pretense to validity, and it is important that the В V model attains it for the limit of / = 0, not only for the simple onestep, one-electron process, but also in the context of an arbitrary multistep mechanism. The derivation here was carried out for a mechanism in which the prereactions and postreactions involve net charge transfer; however the same outcome can be obtained by a similar method for any reaction sequence, as long as it is chemically reversible and a true equilibrium can be established. 3.5.3 Nernstian Multistep Processes If all steps in the mechanism are facile, so that the exchange velocities of all steps are large compared to the net reaction rate, the concentrations of all species participating in them are always essentially at equilibrium in a local context, even though a net current flows. The result for the RDS in this nernstian (reversible) limit has already been obtained as (3.4.27), which we now rewrite in exponential form:

°' ' 1 = ef{E~E^

(3.5.18)

Equilibrium expressions for the pre- and post-reactions link the surface concentrations of O' and R' to the surface concentrations of О and R. If these processes involve interfacial charge transfer, as in the mechanism of (3.5.8)—(3.5.10), the expressions are of the Nernst form: e n'f(E-E%e)

C =

O(°> 0

e

n"f(E-E%st)

=

Q(00

C R (0,0

By steps analogous to those leading from (3.5.12) to (3.5.16), one finds that for the reversible system
(3 5 2 0 )

С RVU, t)

3.5 Multistep Mechanisms which can be rearranged to J?T Cr\(0. t)

111

(3.5.21) This relationship is a very important general rinding. It says that, for a kinetically facile system, the electrode potential and the surface concentrations of the initial reactant and the final product are in local nernstian balance at all times, regardless of the details of the mechanism linking these species and regardless of current flow. Like (3.5.17), (3.5.21) was derived for pre- and postreactions that involve net charge transfer, but one can easily generalize the derivation to include other patterns. The essential requirement is 12 that all steps be chemically reversible and possess facile kinetics. A great many real systems satisfy these conditions, and electrochemical examination of them can yield a rich variety of chemical information (see Section 5.4.4). A good example is the reduction of the ethylenediamine (en) complex of Cd(II) at a mercury electrode: Cd(en)^+ + 2e ^ Cd(Hg) + 3en (3.5.22)

3*5.4 Quasireversible and Irreversible Multistep Processes
If a multistep process is neither nernstian nor at equilibrium, the details of the kinetics influence its behavior in electrochemical experiments, and one can use the results to diagnose the mechanism and to quantify kinetic parameters. As in the study of homogenous kinetics, one proceeds by devising a hypothesis about the mechanism, predicting experimental behavior on the basis of the hypothesis, and comparing the predictions against results. In the electrochemical sphere, an important part of predicting behavior is developing the current-potential characteristic in terms of controllable parameters, such as the concentrations of participating species. If the RDS is a heterogeneous electron-transfer step, then the current-potential characteristic has the form of (3.5.11). For most mechanisms, this equation is of limited direct utility, because O' and R' are intermediates, whose concentration cannot be controlled directly. Still, (3.5.11) can serve as the basis for a more practical current-potential relationship, because one can use the presumed mechanism to reexpress Qy(0, i) and CR/(Q, t) in terms of the concentrations of more controllable species, such as О and R (36). Unfortunately, the results can easily become too complex for practical application. For example, consider the simple mechanism in (3.5.8)—(3.5.10), where the pre- and postreactions are assumed to be kinetically facile enough to remain in local equilibrium. The overall nernstian relationships, (3.5.19), connect the surface concentrations of О and R to those of O' and R'. Thus, the current-potential characteristic, (3.5.11), can be expressed in terms of the surface concentrations of the initial reactant, O, and the final product, R. i = nFAk°rdsCo(0, t)e-n'f{E-EQv^e-af{E-E^ (3.5.23)

This relationship can be rewritten as i = nFA[kfCo(0, t) - kbCR(0, t)] (3.5.24)

12 In the reversible limit, it is no longer appropriate to speak of an RDS, because the kinetics are not ratecontrolling. We retain the nomenclature, because we are considering how a mechanism that does have an RDS begins to behave as the kinetics become more facile.

112 P Chapter 3. Kinetics of Electrode Reactions where kf

=

к

О^е

The point of these results is to illustrate some of the difficulties in dealing with a multistep mechanism involving an embedded RDS. No longer is the potential dependence of the rate constant expressible in two parameters, one of which is interpretable as a measure of intrinsic kinetic facility. Instead, k° becomes obscured by the first exponential factors in (3.5.25) and (3.5.26), which express thermodynamic relationships in the mechanism. One must have ways to find out the individual values of n\ n", £p re , E®osV and E®ds before one can evaluate the kinetics of the RDS in a fully quantitative way. This is normally a difficult requirement. More readily usable results arise from some simpler situations: (a) One-Electron Process Coupled Only to Chemical Equilibria Many of the complications in the foregoing case arise from the fact that the pre- and postreactions involve heterogeneous electron transfer, so that their equilibria depend on E. Consider instead a mechanism that involves only chemical equilibria aside from the ratedetermining interfacial electron transfer:

0 + Y;
0'
•f e

(net result of steps precedinggRDS)

(3 .5.27) (3 .5.28) (3 .5.29)

X> ku D

(RDS) (net result of steps following RDS)

I'

Because equilibrium is established, the Nernst equation for the overall process is applicable. Taking it in the form of (3.5.16) and raising both sides to the power — (nr + a)/n, we have I0 = r^AI^^'^U-^'^^Tds-E^c* [!-(„'+*)/,,] c * [Co(NH 3 ) 2 + -.

Cr(bpy)33+

Inner-sphere Co(NH 3 ) 5 CI 2 +

> (NH3)5Co Cl Cr(H2O)*

Homogenous Electron Transfer Outer-sphere Inner-sphere

Solvent

Figure 3.6.1 Outer-sphere and inner-sphere reactions. The inner sphere homogeneous reaction produces, with loss of H2O, a ligand-bridged complex (shown above), which decomposes to CrCl(H2O)^+ and Co(NH 3 ) 5 (H 2 O) 2+ . In the heterogeneous reactions, the diagram shows a metal ion (M) surrounded by ligands. In the inner sphere reaction, a ligand that adsorbs on the electrode and bridges to the metal is indicated in a darker color. An example of the latter is the + oxidation of Сг(Н 2 О)з at a mercury electrode in the presence of Cl~ or Br~.

Even if there is not a strong interaction with the electrode, an outer-sphere reaction can depend on the electrode material, because of (a) double-layer effects (Section 13.7), (b) the effect of the metal on the structure of the Helmholtz layer, or (c) the effect of the energy and distribution of electronic states in the electrode.

13

3.6 Microscopic Theories of Charge Transfer

117

Outer-sphere electron transfers can be treated in a more general way than innersphere processes, where specific chemistry and interactions are important. For this reason, the theory of outer-sphere electron transfer is much more highly developed, and the discussion that follows pertains to these kinds of reactions. However, in practical applications, such as in fuel cells and batteries, the more complicated inner-sphere reactions are important. A theory of these requires consideration of specific adsorption effects, as described in Chapter 13, as well as many of the factors important in heterogeneous catalytic reactions (56).

3.6.1

The Marcus Microscopic Model
Consider an outer-sphere, single electron transfer from an electrode to species O, to form the product R. This heterogeneous process is closely related to the homogeneous reduction of О to R by reaction with a suitable reductant, R\ 0 + R ' ^ R + O' (3.6.1)

We will find it convenient to consider the two situations in the same theoretical context. Electron-transfer reactions, whether homogeneous or heterogeneous, are radiationless electronic rearrangements of reacting species. Accordingly, there are many common elements between theories of electron transfer and treatments of radiationless deactivation in excited molecules (57). Since the transfer is radiationless, the electron must move from an initial state (on the electrode or in the reductant, R') to a receiving state (in species О or on the electrode) of the same energy. This demand for isoenergetic electron transfer is a fundamental aspect with extensive consequences. A second important aspect of most microscopic theories of electron transfer is the assumption that the reactants and products do not change their configurations during the actual act of transfer. This idea is based essentially on the Franck-Condon principle, which says, in part, that nuclear momenta and positions do not change on the time scale of electronic transitions. Thus, the reactant and product, О and R, share a common nuclear configuration at the moment of transfer. Let us consider again a plot of the standard free energy14 of species О and R as a function of reaction coordinate (see Figure 3.3.2), but we now give more careful consideration to the nature of the reaction coordinate and the computation of the standard free energy. Our goal is to obtain an expression for the standard free energy of activation, AG^\ as a function of structural parameters of the reactant, so that equation 3.1.17 (or a closely related form) can be used to calculate the rate constant. In earlier theoretical work, the pre-exponential factor for the rate constant was written in terms of a collision number (37, 38, 58, 59), but the formalism now used leads to expressions like: kf = K?,ovnKelexp(-AG}/RT) (3.6.2)

where AGjf is the activation energy for reduction of О; К? о is a precursor equilibrium constant, representing the ratio of the reactant concentration in the reactive position at the electrode (the precursor state) to the concentration in bulk solution; vn is the nuclear frequency factor (s" 1 ), which represents the frequency of attempts on the energy barrier (generally associated with bond vibrations and solvent motion); and к е 1 is the electronic transmission coefficient (related to the probability of electron tunneling; see Section 3.6.4). Often, #cel is taken as unity for a reaction where the reactant is close to the electrode, so that there is strong coupling between the reactant and the electrode
14

See the footnote relating to the use of standard thermodynamic quantities in Section 3.1.2.

118 • Chapter 3. Kinetics of Electrode Reactions (see Section 3.6.4).15 Methods for estimating the various factors are available (48), but there is considerable uncertainty in their values. Actually, equation 3.6.2 can be used for either a heterogeneous reduction at an electrode or a homogeneous electron transfer in which О is reduced to R by another reactant in solution. For a heterogeneous electron transfer, the precursor state can be considered to be a reactant molecule situated near the electrode at a distance where electron transfer is possible. Thus KFO = Co,Surf/Co> where C0,SUrfis a surface concentration having units of mol/cm2. Consequently Kp o has units of cm, and kf has units of cm/s, as required. For a homogeneous electron transfer between О and R\ one can think of the precursor state as a reactive unit, OR', where the two species are close enough to allow transfer of an electron. Then KpO = [OR']/[O][R'], which has units of M~x if the concentrations are expressed conventionally. This result gives a rate constant, kf, in units of M ^ s " 1 , again as required. In either case, we consider the reaction as occurring on a multidimensional surface defining the standard free energy of the system in terms of the nuclear coordinates (i.e., the relative positions of the atoms) of the reactant, product, and solvent. Changes in nuclear coordinates come about from vibrational and rotational motion in О and R, and from fluctuations in the position and orientation of the solvent molecules. As usual, we focus on the energetically favored path between reactants and products, and we measure progress in terms of a reaction coordinate, q. Two general assumptions are (a) that the reactant, O, is centered at some fixed position with respect to the electrode (or in a bimolecular homogeneous reaction, that the reactants are at a fixed distance from each other) and (b) that the standard free energies of О and R, GQ and GR, depend quadratically on the reaction coordinate, q (49): G°0(q) = (k/2)(q - q0)2 & = (k/2)(q - qR)
2

(3.6.3) (3.6.4)

+ AG°

where qo and qR are the values of the coordinate for the equilibrium atomic configurations in О and R, and к is a proportionality constant (e.g., a force constant for a change in bond length). Depending on the case under consideration, AG° is either the free energy of reaction for a homogeneous electron transfer or F(E - E°) for an electrode reaction. Let us consider a particularly simple case to give a physical picture of what is implied here. Suppose the reactant is A-B, a diatomic molecule, and the product is A-B~. To a first approximation the nuclear coordinate could be the bond length in A-B (qo) and A-B~ (gR), and the equations for the free energy could represent the energy for lengthening or contraction of the bond within the usual harmonic oscillator approximation. This picture is oversimplified in that the solvent molecules would also make a contribution to the free energy of activation (sometimes the dominant one). In the discussion that follows, they are assumed to contribute in a quadratic relationship involving coordinates of the solvent dipole. Figure 3.6.2 shows a typical free energy plot based on (3.6.3) and (3.6.4). The molecules shown at the top of the figure are meant to represent the stable configurations of the reactants, for example, Ru(NH 3 )6 + and Ru(NH 3 )6 + as О and R, as well as to provide a view of the change in nuclear configuration upon reduction. The transition state is the position where О and R have the same configuration, denoted by the reaction coordinate
The pre-exponential term sometimes also includes a nuclear tunneling factor, Г п . This arises from a quantum mechanical treatment that accounts for electron transfer for nuclear configurations with energies below the transition state (48, 60).
15

3.6 Microscopic Theories of Charge Transfer « 119 0(3+)

Figure 3.6.2 Standard free energy, G , as a function of reaction coordinate, q, for an electron transfer reaction, such as Ru(NH3)6 + e — • RU(NH 3 )J5 + . This diagram applies either to a > heterogeneous reaction in which О and R react at an electrode or a homogeneous reaction in which О and R react with members of another redox couple as shown in (3.6.1). For the heterogeneous case, the curve for О is actually the sum of energies for species О and for an electron on the electrode at the Fermi level corresponding to potential E. Then, AG = F(E — E°). For the homogeneous case, the curve for О is the sum of energies for О and its reactant partner, R', while the curve for R is a sum for R and O'. Then, AG° is the standard free energy change for the reaction. The picture at the top is a general representation of structural changes that might accompany electron transfer. The changes in spacing of the six surrounding dots could represent, for example, changes in bond lengths within the electroactive species or the restructuring of the surrounding solvent shell.

qx. In keeping with the Franck-Condon principle, electron transfer only occurs at this position. The free energies at the transition state are thus given by - qof - qR) x x 2

(3.6.5) (3.6.6) x + AG°

Since G%(q ) = GR(q ), (3.6.5) and (3.6.6) can be solved for q with the result, 2 ' k(qR -

AG°

qo)

(3.6.7)

The free energy of activation for reduction of О is given by (3.6.8)

Chapter 3. Kinetics of Electrode Reactions where we have noted that G%(q0) = 0, as defined in (3.6.3). Substitution for cf from (3.6.7) into (3.6.5) then yields
:0 2AG° 12

(3.6.9)

Defining A = (k/2)(qR — q0)2, we have (3.6.10a) or, for an electrode reaction (3.6.10b) There can be free energy contributions beyond those considered in the derivation just described. In general, they are energy changes involved in bringing the reactants and products from the average environment in the medium to the special environment where electron transfer occurs. Among them are the energy of ion pairing and the electrostatic work needed to reach the reactive position (e.g., to bring a positively charged reactant to a position near a positively charged electrode). Such effects are usually treated by the inclusion of work terms, WQ and WR, which are adjustments to AG° or F(E — E ). For simplicity, they were omitted above. The complete equations, including the work terms, are 1 6 A/
4

\

AG°- -WQ A FiE-E

+ wRy /
V )

(3.6.11a)

\ G\-

4 1Л A\

f

°)" WQ +
A

(3.6.11b)

The critical parameter is Л, the reorganization energy, which represents the energy necessary to transform the nuclear configurations in the reactant and the solvent to those of the product state. It is usually separated into inner, Aj, and outer, Ao, components: A = Ai + Ao (3.6.12)

where Aj represents the contribution from reorganization of species O, and Ao that from reorganization of the solvent.17

The convention is to define vv0 and wR as the work required to establish the reactive position from the average environment of reactants and products in the medium. The signs in (3.6.1 la,b) follow from this. In many circumstances, the work terms are also the free energy changes for the precursor equilibria. When that is true, w0 17

16

= -RT In Kpt0 and wR = -RT

In

KPR.

One should not confuse the inner and outer components of Л with the concept of inner- and outer-sphere reaction. In the treatment under consideration, we are dealing with an outer-sphere reaction, and Aj and Ao simply apportion the energy to terms applying to changes in bond lengths (e.g., of a metal-ligand bond) and changes in solvation, respectively.

3.6 Microscopic Theories of Charge Transfer < 121 To the extent that the normal modes of the reactant remain harmonic over the range of distortion needed, one can, in principle, calculate Aj by summing over the normal vibrational modes of the reactant, that is,
A

i -E

(3.6.13)

where the k's are force constants, and the g's are displacements in the normal mode coordinates. Typically, Ao is computed by assuming that the solvent is a dielectric continuum, and the reactant is a sphere of radius ao. For an electrode reaction, л е1 ло — - ( 1

Л (1

\£оР

1)

(3.6.14a)

where e o p and e s are the optical and static dielectric constants, respectively, and R is taken as twice the distance from the center of the molecule to the electrode (i.e., 2JC0, which is the distance between the reactant and its image charge in the electrode).18 For a homogeneous electron-transfer reaction: (3.6.14b) where a\ and #2 are the radii of the reactants (O and R' in equation 3.6.1) and d = a\ + #2Typical values of A are in the range of 0.5 to 1 eV. Predictions from Marcus Theory While it is possible, in principle, to estimate the rate constant for an electrode reaction by computation of the pre-exponential terms and the A values, this is rarely done in practice. The theory's greater value is the chemical and physical insight that it affords, which arises from its capacity for prediction and generalization about electron-transfer reactions. For example, one can obtain the predicted a-value from (3.6.10b): tt= 1 kex by labeling О isotopically and measuring the rate at which the isotope appears in R, or sometimes by other methods like ESR or NMR. A comparison of (3.6.14a) and (3.6.14b), where a0 = a\ = a2 = a and R = d = 2a, yields Aei = Aex/2 (3.6.17)

where Ael and Aex are the values of Ao for the electrode reaction and the self-exchange reaction, respectively. For the self-exchange reaction, AG° = 0, so (3.6.10a) gives AGf = Aex/4, as long as Ao dominates Aj in the reorganization energy. For the electrode reaction, k° corresponds to E = E°, so (3.6.10b) gives AG* = Aei/4, again with the condition that Aj is negligible. From (3.6.17), one can express AGf for the homogeneous and heterogeneous reactions in common terms, and one finds that kex is related to k° by the expression (£exMex)1/2 = *°/A,l (3-6.18)

where Aex and AQ\ are the pre-exponential factors for self-exchange and the electrode reaction. (Roughly, A d is Ю4 to 105 cm/s and A ex is 10 11 to 10 1 2 Af"1 s" 1 .) 1 9 The theory also leads to useful qualitative predictions about reaction kinetics. For example, equation 3.6.10b gives AG^ ~ X/4 at E°, where kf = k^ = £°. Thus, k° will be larger when the internal reorganization is smaller, that is, in reactions where О and R have similar structures. Electron transfers involving large structural alterations (such as sizable changes in bond lengths or bond angles) tend to be slower. Solvation also has an impact through its contribution to A. Large molecules (large ao) tend to show lower solvation energies, and smaller changes in solvation upon reaction, by comparison with smaller species. On this basis, one would expect electron transfers to small molecules, such as, the reduction of O 2 to O2~ in 2~aprotic media, to be slower than the reduction of Ar to Агт, where Ar is a large aromatic molecule like anthracene. The effect of solvent in an electron transfer is larger than simply through its energetic contribution to Ao. There is evidence that the dynamics of solvent reorganization, often represented in terms of a solvent longitudinal relaxation time, TL, contribute to the preexponential factor in (3.6.2) (47, 62-65), e.g., vn °c T~l. Since r L is roughly proportional to the viscosity, an inverse proportionality of this kind implies that the heterogeneous rate constant would decrease as the solution viscosity increases (i.e., as the diffusion coefficient of the reactant decreases). This behavior is actually seen in the decrease of k° for electrode reactions in water upon adding sucrose to increase the viscosity (presumably without changing Ao in a significant way) (66, 67). This effect was especially pronounced in other studies involving Co(IIIAI)tris(bipyridine) complexes modified by the addition of
19 This equation also applies when the X{ terms are included (but work terms are neglected). This is the case because the total contribution to Л j is summed over two reactants in the homogeneous self-exchange reaction, but only over one in the electrode reaction (61).

3.6 Microscopic Theories of Charge Transfer < 123

long polyethylene or polypropylene oxide chains to the ligands, which cause large changes in diffusion coefficient in undiluted, highly viscous, ionic melts (68). A particularly interesting prediction from this theory is the existence of an "inverted region" for homogeneous electron-transfer reactions. Figure 3.6.3 shows how equation 3.6.10a predicts AGjf to vary with the thermodynamic driving force for the electron transfer, AG°. Curves are shown for several different values of A, but the basic pattern of behavior is the same for all, in that there is a predicted minimum in the standard free energy of activation. On the right-hand side of the minimum, there is a normal region, where AGjf decreases, hence the rate constant increases, as AG° gets larger in magnitude (i.e., becomes more negative). When AG° = -A, AGf is zero, and the rate constant is predicted to be at a maximum. At more negative AG° values, that is for very strongly driven reactions, the activation energy becomes larger, and the rate constant smaller. This is the inverted region, where an increase in the thermodynamic driving force leads to a decrease in the rate of electron transfer. There are two physical reasons for this effect. First, a large negative free energy of reaction implies that the products are required to accept the liberated energy very quickly in vibrational modes, and the probability for doing so declines as - AG° exceeds A (see Chapter 18). Second, one can develop a situation in the inverted region where the energy surfaces no longer allow for adiabatic electron transfer (see Section 3.6.4). The existence of the inverted region accounts for the phenomenon of electrogenerated chemiluminescence (Chapter 18) and has also been seen by other means for several electron-transfer reactions in solution. Even though (3.6.10b) also has a minimum, an inverted-region effect should not occur for an electrode reaction at a metal electrode. The reason is that (3.6.10b) was derived with the implicit idea that electrons always react from a narrow range of states on the electrode corresponding to the Fermi energy (see the caption to Figure 3.6.2). Even though the reaction rate at this energy is predicted to show inversion at very negative overpotentials, there are always occupied states in the metal below the Fermi energy, and they can transfer an electron to О without inversion. Any low-level vacancy created in the metal by heterogeneous reaction is filled ultimately with an electron from the Fermi energy, with dissipation of the difference in energy as heat; thus the overall energy change is as expected from thermodynamics. A similar argument holds for oxidations at metals, where unoccupied states are always available. The ideas behind this discussion are developed much more fully in the next section.
3.5 -

3 2.5 -

2 —
AG* 1.5 —

1 —

1.5 eV,
0.5

-3.5

^ l 0
-0.5 -

e V

0.5

Figure 3.6.3 Effect of AG° for a homogeneous electron-transfer reaction on LG\ at several different values of A.

124 • Chapter 3. Kinetics of Electrode Reactions An inverted region should be seen for interfacial electron transfer at the interface between immiscible electrolyte solutions, with an oxidant, O, in one phase, and a reductant, R', in the other (69). Experimental studies bearing on this issue have just been reported (70). 3.6.3 A Model Based on Distributions of Energy States An alternative theoretical approach to heterogeneous kinetics is based on the overlap between electronic states of the electrode and those of the reactants in solution (41, 42, 46, 47, 71, 72). The concept is presented graphically in Figure 3.6.4, which will be discussed extensively in this section. This model is rooted in contributions from Gerischer (71, 72) and is particularly useful for treating electron transfer at semiconductor electrodes (Section 18.2.3), where the electronic structure of the electrode is important. The main idea is that an electron transfer can take place from any occupied energy state that is matched in energy, E, with an unoccupied receiving state. If the process is a reduction, the occupied state is on the electrode and the receiving state is on an electroreactant, O. For an oxidation, the occupied state is on species R in solution, and the receiving state is on the electrode. In general, the eligible states extend over a range of energies, and the total rate is an integral of the rates at each energy.

-2

ш

I

Unoccupied States

(2)UT)V2

DO(\,E)

ш —4

1 Electrode States Reactant States

Figure 3.6.4 Relationships among electronic states at an interface between a metal electrode and a solution containing species О and R at equal concentrations. The vertical axis is electron energy, E, on the absolute scale. Indicated on the electrode side is a zone A4T wide centered on the Fermi level, Ep, where/(E) makes the transition from a value of nearly 1 below the zone to a value of virtually zero above. See the graph of/(E) in the area of solid shading on the left. On the solution side, the state density distributions are shown for О and R. These are gaussians having the same shapes as the probability density functions, W0(X, E) and WR(\, E). The electron energy corresponding to the standard potential, E°, is -3.8 eV, and Л = 0.3 eV. The Fermi energy corresponds here to an electrode potential of -250 mV vs. E°. Filled states are denoted on both sides of the interface by dark shading. Since filled electrode states overlap with (empty) О states, reduction can proceed. Since the (filled) R states overlap only with filled electrode states, oxidation is blocked.

3.6 Microscopic Theories of Charge Transfer < 125 On the electrode, the number of electronic states in the energy range between E and E 4- dE is given by Ap(E)dE, where A is the area exposed to the solution, and p(E) is the density of states [having units of (area-energy)" 1 , such as с т ~ 2 е У - 1 ] . The total number of states in a broad energy range is, of course, the integral of Ap(E) over the range. If the electrode is a metal, the density of states is large and continuous, but if it is a semiconductor, there is a sizable energy range, called the band gap, where the density of states is very small. (See Section 18.2 for a fuller discussion of the electronic properties of materials.) Electrons fill states on the electrode from lower energies to higher ones until all electrons are accommodated. Any material has more states than are required for the electrons, so there are always empty states above the filled ones. If the material were at absolute zero in temperature, the highest filled state would correspond to the Fermi level (or the Fermi energy), Ep, and all states above the Fermi level would be empty. At any higher temperature, thermal energy elevates some of the electrons into states above Ep and creates vacancies below. The filling of the states at thermal equilibrium is described by the Fermi function, /(E), /(E) = {1 + exp[(E - E F )/47]}" l (3.6.19)

which is the probability that a state of energy E is occupied by an electron. It is easy to see that for energies much lower than the Fermi level, the occupancy is virtually unity, and for energies much higher than the Fermi level, the occupancy is practically zero (see Figure 3.6.4). States within a few 4T of Ep have intermediate occupancy, graded from unity to zero as the energy rises through Ep, where the occupancy is 0.5. This intermediate zone is shown in Figure 3.6.4 as a band 44T wide (about 100 meV at 25°C). The number of electrons in the energy range between E and E + dE is the number of occupied states, A/V0CC(E)dE, where Afocc(E) is the density function AW(E)=/(E)p(E) (3.6.20)

Like p(E), 7V0CC(E) has units of (area-energy)"1, typically c m " 2 eV~\ while/(E) is dimensionless. In a similar manner, we can define the density of unoccupied states as AU>cc(E) = [1 ~/(E)]p(E) (3.6.21)

As the potential is changed, the Fermi level moves, with the change being toward higher energies at more negative potentials and vice versa. On a metal electrode, these changes occur not by the filling or emptying of many additional states, but mostly by charging the metal, so that all states are shifted by the effect of potential (Section 2.2). While charging does involve a change in the total electron population on the metal, the change is a tiny fraction of the total (Section 2.2.2). Consequently, the same set of states exists near the Fermi level at all potentials. For this reason, it is more appropriate to think of p(E) as a consistent function of E — Ep, nearly independent of the value of Ep. Since /(E) behaves in the same way, so do N0CC(E) and Nunocc(E). The picture is more complicated at a semiconductor, as discussed in Section 18.2. States in solution are described by similar concepts, except that filled and empty states correspond to different chemical species, namely the two components of a redox couple, R and O, respectively. These states differ from those of the metal in being localized. The R and О species cannot communicate with the electrode without first approaching it closely. Since R and О can exist in the solution inhomogeneously and our concern is with the mix of states near the electrode surface, it is better to express the density of states

126 • Chapter 3. Kinetics of Electrode Reactions in terms of concentration, rather than total number. At any moment, the removable electrons on R species in solution in the vicinity of the electrode20 are distributed over an energy range according to a concentration density function, £>R(A, E), having units of (volume-energy)"1, such as cm" 3 eV" 1 . Thus, the number concentration of R species near the electrode in the range between E and E + dE is £>R(A, E)dE. Because this small element of the R population should be proportional to the overall surface concentration of R, CR(0, t), we can factor Z)R(A, E) in the following way: DR(A, E) = WACR(0, t)WR(\, E) (3.6.22)

where NA is Avogadro's number, and WR(A, E) is a probability density function with units of (energy)"1. Since the integral of DR(A, E) over all energies must yield the total number concentration of all states, which is iVACR(0, t), we see that VFR(A, E) is a normalized function WR(\,E)dE = 1
0

(3.6.23)

Similarly, the distribution of vacant states represented by О species is given by £>o(A, E) = NACo(0, t)Wo(\, E) (3.6.24)

where W0(X, E) is normalized, as indicated for its counterpart in (3.6.23). In Figure 3.6.4, the state distributions for О and R are depicted as gaussians for reasons that we will discover below. Now let us consider the rate at which О is reduced from occupied states on the electrode in the energy range between E and E + dE. This is only a part of the total rate of reduction, so we call it a local rate for energy E. In a time interval Дг, electrons from occupied states on the electrode can make the transition to states on species О in the same energy range, and the rate of reduction is the number that succeed divided by Д*. This rate is the instantaneous rate, if At is short enough (a) that the reduction does not appreciably alter the number of unoccupied states on the solution side and (b) that individual О molecules do not appreciably change the energy of their unoccupied levels by internal vibrational and rotational motion. Thus At is at or below the time scale of vibration. The local rate of reduction can be written as Local Rate(E) =
Pred(E)AN0CC(E)dE red осс д ^

—

(3.6.25)

where AN0CCdE is the number of electrons available for the transition and Pred(E) is the probability of transition to an unoccupied state on O. It is intuitive that /\ed(E) is proportional to the density of states DO(A, E). Defining e r e d (E) as the proportionality function, we have Local Rate(E) = 6rred(E)£>o(A, E)ANocc(E)dE red ° '—^^-^— (3.6.26)

In this discussion, the phrases "concentration in the vicinity of the electrode" and "concentration near the electrode" are used interchangeably to denote concentrations that are given by C(0, i) in most mass-transfer and heterogeneous rate equations in this book. However, C(0, t) is not the same as the concentration in the reactive position at an electrode (i.e., in the precursor state), but is the concentration just outside the diffuse layer. We are now considering events on a much finer distance scale than in most contexts in this book, and this distinction is needed. The same point is made in Section 13.7.

20

3.6 Microscopic Theories of Charge Transfer

127

where ered(E) has units of volume-energy (e.g., cm3 eV). The total rate of reduction is the sum of the local rates in all infinitesimal energy ranges; thus it is given by the integral Rate = v \
J —00 ered (E)D o (A, E)AN0CC(E)dE

(3.6.27)

where, in accord with custom, we have expressed Ar in terms of a frequency, v = 1/Дг. The limits on the integral cover all energies, but the integrand has a significant value only where there is overlap between occupied states on the electrode and states of О in the solution. In Figure 3.6.4, the relevant range is roughly -4.0 to -3.5 eV. Substitution from (3.6.20) and (3.6.24) gives Rate = vANKC'o(0, t) J
J —00

s red (E)W 0 (A, E)/(E)p(E)dE

(3.6.28)

This rate is expressed in molecules or electrons per second. Division by ANA gives the rate more conventionally in mol c m " 2 s" 1 , and further division by CQ(0, t) provides the rate constant,
Г 00

(3.6.29) In an analogous way, one can easily derive the rate constant for the oxidation of R. On the electrode side, the empty states are candidates to receive an electron; hence Nunocc(E) is the distribution of interest. The density of filled states on the solution side is £>R(A, E), and the probability for electron transfer in the time interval Ar is P 0X (E) = 80X(E)Z)R(A, E). Proceeding exactly as in the derivation of (3.6.29), we arrive at (3.6.30) In Figure 3.6.4, the distribution of states for species R does not overlap the zone of unoccupied states on the electrode, so the integrand in (3.6.30) is practically zero everywhere, and & is negligible compared to kf. The electrode is in a reducing condition with ь respect to the O/R couple. By changing the electrode potential to a more positive value, we shift the position of the Fermi level downward, and we can reach a position where the R states begin to overlap unoccupied electrode states, so that the integral in (3.6.30) becomes significant, and къ is enhanced. The literature contains many versions of equations 3.6.29 and 3.6.30 manifesting different notation and involving wide variations in the interpretation applied to the integral prefactors and the proportionality functions e r e d (E) and e o x (E). For example, it is common to see a tunneling probability, Kej, or a precursor equilibrium constant, A^o or /T R, extracted from the e-functions and placed in the integral prefactor. Often the sp frequency v is identified with vn in (3.6.2). Sometimes the prefactor encompasses things other than the frequency parameter, but is still expressed as a single symbol. These variations in representation reflect the fact that basic ideas are still evolving. The treatment offered here is general and can be accommodated to any of the extant views about how the fundamental properties of the system determine v, e r e d (E), and e o x (E). With (3.6.29) and (3.6.30), it is apparently possible to account for kinetic effects of the electronic structure of the electrode by using an appropriate density of states, p(E), for

128

Chapter 3. Kinetics of Electrode Reactions the electrode material. Efforts in that direction have been reported. However, one must be on guard for the possibility that ered(E) and eOx(E) also depend on p(E). 2 1 The Marcus theory can be used to define the probability densities W0(A, E) and WR(A, E). The key is to recognize that the derivation leading to (3.6.10b) is based implicitly on the idea that electron transfer occurs entirely from the Fermi level. In the context that we are now considering, the rate constant corresponding to the activation energy in (3.6.10b) is therefore proportional to the local rate at the Fermi level, wherever it might be situated relative to the state distributions for О and R. We can rewrite (3.6.10b) in terms of electron energy as
E

~ ^

»

(3.6.31)

where E° is the energy corresponding to the standard potential of the O/R couple. One can easily show that AG| reaches a minimum at E = E° + A, where AGj = 0. Thus the maximum local rate of reduction at the Fermi level is found where E F = E° + Л. When the Fermi level is at any other energy, E, the local rate of reduction at the Fermi level can be expressed, according to (3.6.2), (3.6.26), and (3.6.31), in terms of the following ratios

Local Rate (E F = E)
Local Rate (E F = E° + A) gred А Л ^^ЫУ
VnKel

Е- E

JJ

(3

-63

(E)Po(A,E)/(EF)p(EF)

ered (E° + A) Do (A, E° + A)/(E F ) p (E F ) Assuming that e r e d does not depend on the position of E F , we can simplify this to

Z) O (A,E°

Z)0(A,E)

_ = exp Г- (Е-Е°-А) 1

2

(3.6.33)

This is a gaussian distribution having a mean at E = E° + Л and a standard deviation of (2A^T) 1/2 , as shown in Figure 3.6.4 (see also Section A.3). From (3.6.24), £>O(A, E)/D0(A, E° + Л) = Wo(\, E)/Wo(\, E° + A). Also, since Wo(\, E) is normalized, the exponential prefactor, WO(X, E° + A), is quickly identified (Section A.3) as (2тг)~ / times the reciprocal of the standard deviation; therefore
Wo(\, E) = (4TT\£T)~1/2

exp

-

(3.6.34)

Consider, for example, a simple model based on the idea that, in the time interval Ar, all of the electrons in the energy range between E and E + dE redistribute themselves among all available states with equal probability. A refinement allows for the possibility that the states on species О participate with different weight from those on the electrode. If the states on the electrode are given unit weight and those in solution are given weight /cred(E), then /cred (E)D0(A, E)5 p(E) + /cred(E)£>0(A, E)d where 8 is the average distance across which electron transfer occurs, and /cred(E) is dimensionless and can be identified with the tunneling probability, /cel, used in other representations of kf. If the electrode is a metal, p(E) is orders of magnitude greater than Kred(E)Z)0(A, E)8; hence the rate constant becomes

21

which has no dependence on the electronic structure of the electrode.

3.6 Microscopic Theories of Charge Transfer *i 129 In a similar manner, one can show that (3.6.35) thus the distribution for R has the same shape as that for O, but is centered on E° - Л, as depicted in Figure 3.6.4. Any model of electrode kinetics is constrained by the requirement that =е (3.6.36)

which is easily derived from the need for convergence to the Nernst equation at equilibrium (Problem 3.16). The development of the Gerischer model up through equations 3.6.29 and 3.6.30 is general, and one can imagine that the various component functions in those two equations might come together in different ways to fulfill this requirement. By later including results from the Marcus theory without work terms, we were able to define the distribution functions, WO(X, E) and WR(A, E). Another feature of this simple Gerischer-Marcus model is that sox(E) and ered(E) turn out to be identical functions and need no longer be distinguished. However, this will not necessarily be true for related models including work terms and a precursor equilibrium. The reorganization energy, A, has a large effect on the predicted current-potential response, as shown in Figure 3.6.5. The top frame illustrates the situation for A = 0.3 eV, a value near the lower limit found experimentally. For this reorganization energy, an overpotential of —300 mV (case a) places the Fermi level opposite the peak of the state
-2 -3 -4 -5 -6 . ~

_ERe° •

d—ажк=а

Electrode States

Solution States

Figure 3.6.5 Effect of A on kinetics in the Gerischer-Marcus representation. Top: A = 0.3 eV. Bottom: A = 1.5 eV. Both diagrams are for species О and R at equal concentrations, so that the Fermi level corresponding to the equilibrium potential, E F eq , is equal to the electron energy at the standard potential, E° (dashed line). For both frames, E° = -4.5 eV. Also shown in each frame is the way in which the Fermi level shifts with electrode potential. The different Fermi levels are for (a) r] = -300 mV, (b) г] = +300 mV, (c) rj = -1000 mV, and (d) r) = +1000 mV. On the solution side, Wo(\, E) and WR(X, E) are shown with lighter and darker shading, respectively.

130 • Chapter 3. Kinetics of Electrode Reactions distribution for O; hence rapid reduction would be seen. Likewise, an overpotential of + 300 mV (case b) brings the Fermi level down to match the peak in the state distribution for R and enables rapid oxidation. An overpotential of -1000 mV (case c) creates a situation in which WO(A, E) overlaps entirely with filled states on the electrode, and for 7] = +1000 mV (case d), WR(\, E) overlaps only empty states on the electrode. These latter two cases correspond to very strongly enabled reduction and oxidation, respectively. The lower frame of Figure 3.6.5 shows the very different situation for the fairly large reorganization energy of 1.5 eV. In this case, an overpotential of —300 mV is not enough to elevate the Fermi level into a condition where filled states on the electrode overlap Wo(\, E), nor is an overpotential of +300 mV enough to lower the Fermi level into a condition where empty states on the electrode overlap WR(A, E). It takes 7 ~ -1000 mV 7 to enable reduction very effectively, and 7 ~ +1000 mV to do the same for oxidation. 7 For this reorganization energy, the anodic and cathodic branches of the i-E curve would be widely separated, much as shown in Figure 3.4.2c. Since this formulation of heterogeneous kinetics in terms of overlapping state distributions is linked directly to the basic Marcus theory, it is not surprising that many of its predictions are compatible with those of the previous two sections. The principal difference is that this formulation allows explicitly for contributions from states far from the Fermi level, which can be important in reactions at semiconductor electrodes (Section 18.2) or involving bound monolayers on metals (Section 14.5.2). 3.6.4 Tunneling and Extended Charge Transfer In the treatments discussed above, the reactant was assumed to be held at a fixed, short distance, JC0, from the electrode. It is also of interest to consider whether a solution species can undergo electron transfer at different distances from the electrode and how the electron-transfer rate might depend on distance and on the nature of the intervening medium. The act of electron transfer is usually considered as tunneling of the electron between states in the electrode and those on the reactant. Electron tunneling typically follows an expression of the form: Probability of tunneling °c ехр(-Дх) (3.6.37)

where x is the distance over which tunneling occurs, and /3 is a factor that depends upon the height of the energy barrier and the nature of the medium between the states. For example, for tunneling between two pieces of metal through vacuum (73) /3 « 4тг(2тФ) //г « 1.02 A " eV~
28 1/2 1 1/2

X Ф

1/2

(3.6.38)

where m is the mass of the electron, 9.1 X 10~ g, and Ф is the work function of the 1 metal, typically given in eV. Thus for Pt, where Ф = 5.7 eV, /3 is about 2.4 A " . Within the electron-transfer theory, tunneling effects are usually incorporated by taking the transmission coefficient, / 1 when x is at the distance where the interaction of reactant with the electrode is sufficiently strong for the reaction to be adiabatic (48, 49). In electron-transfer theory, the extent of interaction or electronic coupling between two reactants (or between a reactant and the electrode) is often described in terms of adiabaticity. If the interaction is strong, there is a splitting larger than 6T in the energy curves at the point of intersection (e.g., Figure 3.6.6a). It leads to a lower curve (or surface) pro-

3.6 Microscopic Theories of Charge Transfer -I 131

G°(q)

(a)

(b)

Figure 3.6.6 Splitting of energy curves (energy surfaces) in the intersection region, (a) A strong interaction between О and the electrode leads to a well-defined, continuous curve (surface) connecting О + e with R. If the reacting system reaches the transition state, the probability is high that it will proceed into the valley corresponding to R, as indicated by the curved arrow. (b) A weak interaction leads to a splitting less than 4T. When the reacting system approaches the transition state from the left, it has a tendency to remain on the О + е curve, as indicated by the straight arrow. The probability of crossover to the R curve can be small. These curves are drawn for an electrode reaction, but the principle is the same for a homogeneous reaction, where the reactants and products might be О + R' and R + O', respectively.

ceeding continuously from О to R and an upper curve (or surface) representing an excited state. In this situation of strong coupling, a system will nearly always stay on the lower surface passing from О to R, and the reaction is said to be adiabatic. The probability of reaction per passage approaches unity for an adiabatic reaction. If the interaction is small (e.g,. when the reactants are far apart), the splitting of the potential energy curves at the point of intersection is less than &T (Figure 3.6.6/?). In this case, there is a smaller likelihood that the system will proceed from О to R. The reaction is said to be nonadiabatic, because the system tends to stay on the original "reactant" surface (or, actually, to cross from the ground-state surface to the excited-state surface). The probability of reaction per passage through the intersection region is taken into account by KQi < 1 (47, 48). For example, к е 1 could be 10~5, meaning that the reactants would, on the average, pass through the intersection region (i.e., reach the transition state) 100,000 times for every successful reaction. In considering dissolved reactants participating in a heterogeneous reaction, one can treat the reaction as occurring over a range of distances, where the rate constant falls off exponentially with distance. The result of such a treatment (48, 74) is that electron transfer occurs over a region near the electrode, rather than only at a single position, such as the outer Helmholtz plane. However, the effect for dissolved reactants should be observable experimentally only under rather restricted circumstances (e.g., D < 10~ 10 cm2/s), and is thus usually not important. On the other hand, it is possible to study electron transfer to an electroactive species held at a fixed distance (10-30 A) from the electrode surface by a suitable spacer, such as an adsorbed monolayer (Section 14.5.2) (75, 76). One approach is based on the use of a blocking monolayer, such as a self-assembled monolayer of an alkane thiol or an insulating oxide film, to define the distance of closest approach of a dissolved reactant to the electrode. This strategy requires knowledge of the precise thickness of the blocking layer and assurance that the layer is free of pinholes and defects, through which solution species might penetrate (Section 14.5). Alternatively, the adsorbed monolayer may itself

132

Chapter 3. Kinetics of Electrode Reactions
Electroactive Centers

Alkylthiol chains

Gold Electrode

Figure 3.6.7 Schematic diagram of an adsorbed monolayer of alkane thiol containing similar molecules with attached electroactive groups held by the film at a fixed distance from the electrode surface.

contain electroactive groups. A typical layer of this kind (75) involves an alkane thiol (RSH) with a terminal ferrocene group (-Fc), that is, HS(CH2)nOOCFc (often written as HSCnOOCFc; typically n = 8 to 18) (Figure 3.6.7). These molecules are often diluted in the monolayer film with similar nonelectroactive molecules (e.g., HSC^CH3). The rate constant is measured as a function of the length of the alkyl chain, and the slope of the plot of ln(£) vs. nor x allows determination of /3. For saturated chains, /3 is typically in the range 1 to 1.2 A" 1 . The difference between this through-bond value and that for vacuum (through-space), ~ 2 A~ , reflects the contribution of the molecular bonds to tunneling. Even smaller /3 values (0.4 to 0.6 A" 1 ) have been seen with тг-conjugated molecules [e.g., those with phenyleneethynyl
(-Ph-C=C-) units] as spacers (77, 78). Confidence in the /3 values found in these electrochemical studies is reinforced by the fact that they generally agree with those found for long-range intramolecular electron transfer, such as in proteins.

3.7 REFERENCES
1. W. C. Gardiner, Jr., "Rates and Mechanisms of Chemical Reactions," Benjamin, New York, 1969. 2. H. S. Johnston, "Gas Phase Reaction Rate Theory," Ronald, New York, 1966. 3. S. Glasstone, K. J. Laidler, and H. Eyring, "Theory of Rate Processes," McGraw-Hill, New York, 1941. 4. H. Eyring, S. H. Lin, and S. M. Lin, "Basic Chemical Kinetics," Wiley, New York, 1980, Chap. 4. 5. R. S. Berry, S. A. Rice, J. Ross, "Physical Chemistry," Wiley, New York, 1980, pp. 931-932. 6. J. Tafel, Z. Physik. Chem., 50A, 641 (1905). 7. P. Delahay, "New Instrumental Methods in Electrochemistry," Wiley-Interscience, New York, 1954, Chap. 2. 8. P. Delahay, "Double Layer and Electrode Kinetics," Wiley-Interscience, New York, 1965, Chap. 7. 9. В. Е. Conway, "Theory and Principles of Electrode Processes," Ronald, New York, 1965, Chap. 6. 10. K. J. Vetter, "Electrochemical Kinetics," Academic, New York, 1967, Chap. 2. 11. J. O'M. Bockris and A. K. N. Reddy, "Modern Electrochemistry," Vol. 2, Plenum, New York, 1970, Chap. 8. 12. T. Erdey-Gruz, "Kinetics of Electrode Processes," Wiley-Interscience, New York, 1972, Chap. 1. 13. H. R. Thirsk, "A Guide to the Study of Electrode Kinetics," Academic, New York, 1972, Chap. 1. 14. W. J. Albery, "Electrode Kinetics," Clarendon, Oxford, 1975. 15. J. E. B. Randies, Trans. Faraday Soc, 48, 828 (1952). 16. С N. Reilley in "Treatise on Analytical Chemistry," Part I, Vol. 4,1. M. Kolthoff and P. J. Elying, Eds., Wiley-Interscience, 1963, Chap. 42.

3.7 References 17. J. A. V. Butler, Trans. Faraday Soc, 19, 729, 734 (1924). 18. T. Erdey-Gruz and M. Volmer, Z. Physik. Chem., 150A, 203 (1930). 19. R. Parsons, Trans. Faraday Soc, (1951). 47, 1332

133

43. J. O'M. Bockris, Mod. Asp. Electrochem., 1, 180 (1954). 44. P. P. Schmidt in "Electrochemistry," A Specialist Periodical Report, Vols. 5 and 6, H. R. Thirsk, Senior Reporter, The Chemical Society, London, 1977 and 1978. 45. A. M. Kuznetsov, Mod. Asp. Electrochem., 20, 95 (1989). 46. W. Schmickler, "Interfacial Electrochemistry," Oxford University Press, New York, 1996. 47. C. J. Miller in "Physical Electrochemistry. Principles, Methods, and Applications," I. Rubinstein, Ed., Marcel Dekker, New York, 1995, Chap. 2. 48. M. J. Weaver in "Comprehensive Chemical Kinetics,"' R. G. Compton, Ed., Elsevier, Amsterdam, Vol. 27, 1987. Chap. 1. 49. R. A. Marcus and P. Siddarth, "Photoprocesses in Transition Metal Complexes, Biosystems and Other Molecules," E. Kochanski, Ed., Kluwer, Amsterdam, 1992. 50. N. S. Hush, /. Electroanal. Chem., 470, 170 (1999). 51. L. Eberson, "Electron Transfer Reactions in Organic Chemistry," Springer-Verlag, Berlin, 1987. 52. N. Sutin, Accts. Chem. Res., 15, 275 (1982). 53. R. A. Marcus and N. Sutin, Biochim. Biophys. Acta, 811, 265 (1985). 54. H. Taube, "Electron Transfer Reactions of Complex Ions in Solution," Academic, New York, 1970, p. 27. 55. J. J. Ulrich and F. C. Anson, Inorg. Chem., 8, 195 (1969). 56. G. A. Somorjai, "Introduction to Surface Chemistry and Catalysis," Wiley, New York, 1994. 57. S. F. Fischer and R. P. Van Duyne, Chem. Phys., 26,9(1977). 58. R. A. Marcus, /. Chem. Phys., 43, 679 (1965). 59. R. A. Marcus, Annu. Rev. Phys. Chem., 15, 155 (1964). 60. B. S. Brunschwig, J. Logan, M. D. Newton, and N. Sutin, /. Am. Chem. Soc, 102, 5798 (1980). 61. R. A. Marcus, /. Phys. Chem., 67, 853 (1963). 62. D. F. Calef and P. G. Wolynes. /. Phys. Chem., 87, 3387 (1983). 63. J. T. Hynes in "Theory of Chemical Reaction Dynamics," M. Baer, Editor, CRC, Boca Raton, FL, 1985, Chap. 4. 64. H. Sumi and R. A. Marcus, /. Chem. Phys., 84, 4894(1986).

20. J. O'M Bockris, Mod. Asp. Electrochem., 1, 180 (1954). 21. D. M. Mohilner and P. Delahay, /. Phys. Chem., 67, 588 (1963). 22. M. E. Peover, Electroanal. Chem., 2, 1 (1967). 23. N. Koizumi and S. Aoyagui, /. Electroanal. Chem., 55, 452 (1974). 24. H. Kojima and A. J. Bard, /. Am. Chem. Soc, 97,6317(1975). 25. K. J. Vetter, op. cit., Chap. 4. 26. T. Erdey-Gruz, op. cit., Chap. 4. 27. P. Delahay, "Double Layer and Electrode Kinetics," op. cit., Chap. 10. 28. N. Tanaka and R. Tamamushi, Electrochim. Acta, 9, 963 (1964). 29. В. Е. Conway, "Electrochemical Data," Elsevier, Amsterdam, 1952. 30. R. Parsons, "Handbook of Electrochemical Data," Butterworths, London, 1959. 31. A. J. Bard and H. Lund, "Encyclopedia of the Electrochemistry of the Elements," Marcel Dekker, New York, 1973-1986. 32. E. Gileadi, E. Kirowa-Eisner, and J. Penciner, "Interfacial Electrochemistry," Addison-Wesley, Reading, MA, 1975, pp. 60-75. 33. K. J. Vetter and G. Manecke, Z. Physik. Chem. (Leipzig)., 195, 337 (1950). 34. P. A. Allen and A. Hickling, Trans. Faraday Soc, 53, 1626 (1957). 35. P. Delahay, "Double Layer and Electrode Kinetics," op. cit., Chaps. 8-10. 36. К. В. Oldham, /. Am. Chem. Soc, 77, 4697 (1955). 37. R. A. Marcus, /. Chem Phys., 24, 4966 (1956). 38. R. A. Marcus, Electrochim. Acta, 13, 955 (1968). 39. N. S. Hush, /. Chem. Phys., 28, 962 (1958). 40. N. S. Hush, Electrochim. Acta, 13, 1005 (1968). 41. V. G. Levich, Adv. Electrochem. Electrochem. Engr., 4, 249 (1966) and references cited therein. 42. R. R. Dogonadze in "Reactions of Molecules at Electrodes," N. S. Hush, Ed., Wiley-Interscience, New York, 1971, Chap. 3 and references cited therein.

134 • Chapter 3. Kinetics of Electrode Reactions 65. M. J. Weaver, Chem. Rev., 92, 463 (1992). 66. X. Zhang, J. Leddy, and A. J. Bard, /. Am. Chem. Soc, 107, 3719 (1985). 67. X. Zhang, H. Yang, and A. J. Bard, /. Am. Chem. Soc, 109, 1916 (1987). 68. M. E. Williams, J. С Crooker, R. Pyati, L. J. Lyons, and R. W. Murray, /. Am. Chem. Soc, 119, 10249 (1997). 69. R. A. Marcus, /. Phys. Chem., 94, 1050 (1990); 95, 2010 (1991). 70. M. Tsionsky, A. J. Bard, and M. V. Mirkin, /. Am. Chem. Soc, 119, 10785 (1997). 71. H. Gerischer, Adv. Electrochem. Electrochem. Eng., 1, 139 (1961). 72. H. Gerischer in "Physical Chemistry: An Advanced Treatise," Vol. 9A, H. Eyring, D. Henderson, and W. Jost, Eds., Academic, New York, 1970. 73. С J. Chen, "Introduction to Scanning Tunneling Microscopy," Oxford University Press, New ' 1 9 9 3 ' P- 5 74. S. W. Feldberg, /. Electroanal. Chem., 198, 1 (1986). 75. H. O. Finklea, Electroanal. Chem., 19, 109 (1996).
7 6 Y o r k

j F . Smalley, S. W. Feldberg, С E. D. Chidsey, M. R. Linford, M. D. Newton, and Y.-P. Liu, /. Phys. Chem., 99, 13141 (1995).

77. S. B. Sachs, S. P. Dudek, R. P. Hsung, L. R. Sita, J. F. Smalley, M. D. Newton, S. W. Feldberg, and С. Е. D. Chidsey, /. Am. Chem. Soc, 119, 10563 (1997). 78. S. Creager, S. J. Yu, D. Bamdad, S. O'Conner, T. MacLean, E. Lam, Y. Chong, G. T. Olsen, J Luo, M. Gozin, and J. F. Kayyem, /. Am. Chem. Soc, 111, 1059 (1999).

3.8 PROBLEMS
3.1 Consider the electrode reaction О + ne «^ R. Under the conditions that CR = C o = 1 mM, R k° = 10~7 cm/s, a = 0.3, and n = 1: (a) Calculate the exchange current density, jo = IQ/A, in fiA/cm2. (b) Draw a current density-overpotential curve for this reaction for currents up to 600 /лА/ст2 anodic and cathodic. Neglect mass-transfer effects. (c) Draw log \j\ vs. 7) curves (Tafel plots) for the current ranges in (b). 3.2 A general expression for the current as a function of overpotential, including mass-transfer effects, can be obtained from (3.4.29) and yields
1

~ i

expt-a/rj] - exp[(l - a)fr]] exp[(l - a)frj\ | exp[- afq]

(a) Derive this expression. (b) Use a spreadsheet program to repeat the calculation of Problem 3.1, parts (b) and (c), including the effects of mass transfer. Assume m0 - mR= 10~3 cm/s. 3.3 Use a spreadsheet program to calculate and plot current vs. potential and ln(current) vs. potential for the general i-iq equation given in Problem 3.2. (a) Show a table of results [potential, current, ln(current), overpotential] and graphs of / vs. 1 and 7 ln|/| vs. 7] for the following parameters: A = 1 cm 2 ; C o = 1.0 X 10~3 mol/cm3; C R = 1.0 X 10" 5 mol/cm3; n = 1; a = 0.5; k° = 1.0 X 10" 4 cm/s; m o = 0.01 cm/s; mR= 0.01 cm/s; E° = -0.5 V vs. NHE. (b) Show the i vs. E curves for a range of k° values with the other parameters as in (a). At what values of к0 are the curves indistinguishable from nernstian ones? (c) Show the / vs. E curves for a range of a values with the other parameters as in (a). 3.4 In most cases, the currents for individual processes are additive, that is, the total current, /t, is given as the sum of the currents for different electrode reactions (/1? 12, /3,.. . ) . Consider a solution with a Pt working electrode immersed in a solution of 1.0 M HBr and 1 mM K 3 Fe(CN) 6 . Assume the following exchange current densities: H + /H 2 Br2/Br~ 70 = 10~3 A/cm2 j 0 = 10~2 A/cm2 j 0 = 4 X 10~5 A/cm2

Fe(CN)^/Fe(CN)£-

3.8 Problems

135

Use a spreadsheet program to calculate and plot the current-potential curve for this system scanning from the anodic background limit to the cathodic background limit. Take the appropriate standard potentials from Table C.I and values for other parameters (ra 0 , a, . . . ) from Problem 3.3. 3.5 Consider one-electron electrode reactions for which a = 0.50 and a = 0.10. Calculate the relative error in current resulting from the use in each case of: (a) The linear i—r) characteristic for overpotentials of 10, 20, and 50 mV. (b) The Tafel (totally irreversible) relationship for overpotentials of 50, 100, and 200 mV. 3.6 According to G. Scherer and F. Willig [/. Electroanal Chem., 85, 77 (1977)] the exchange current 2 density, j 0 , for Pt/Fe(CN)^~ (2.0 mM), Fe(CN)^" (2.0 mM), NaCl (1.0 M) at 25°C is 2.0 mA/cm . The transfer coefficient, a, for this system is about 0.50. Calculate (a) the value of k°; (b)y'o for a solution 1 M each in the two complexes; (c) the charge-transfer resistance of a 0.1 cm electrode in a solution 10~4 M each in ferricyanide and ferrocyanide. 3.7 Berzins and Delahay [/. Am. Chem. Soc, 11, 6448 (1955)] studied the reaction Cd 2 + + 2e ^ Cd(Hg) and obtained the following data with Ccd(Hg) = ®'^® ^ CCd2+(mM) ;o (mA/cm2) 1.0 30.0 17.3
:

0.50

0.25 10.1

0.10 4.94

(a) Assume that the general mechanism in (3.5.8)-(3.5.10) applies. Calculate n' + a, and suggest values for n', n'\ and a individually. Write out a specific chemical mechanism for the process. (b) Calculate k%p. (c) Compare the outcome with the analysis provided by Berzins and Delahay in their original paper. 3.8 (a) Show that for a first-order homogeneous reaction, A-IB the average lifetime of A is l/kf. (b) Derive an expression for the average lifetime of the species О when it undergoes the heterogeneous reaction,

Note that only species within distance d of the surface can react. Consider a hypothetical system in which the solution phase extends only d (perhaps 10 A) from the surface, (c) What value of kf would be needed for a lifetime of 1 ms? Are lifetimes as short as 1 ns possible? 3.9 Discuss the mechanism by which the potential of a platinum electrode becomes poised by immersion into a solution of Fe(II) and Fe(III) in 1 M HCl. Approximately how much charge is required to shift the electrode potential by 100 mV? Why does the potential become uncertain at low concentrations of Fe(II) and Fe(III), even if the ratio of their concentrations is held near unity? Does this experimental fact reflect thermodynamic considerations? How well do your answers to these issues apply to the establishment of potential at an ion-selective electrode? 3.10 In ammoniacal solutions ([NH 3 ] ~ 0.05 M), Zn(II) is primarily in the form of the complex ion Zn(NH 3 ) 3 (OH) + [hereafter referred to as Zn(II)]. In studying the electroreduction of this compound to zinc amalgam at a mercury cathode, Gerischer [Z. Physik. Chem., 202, 302 (1953)] found that д log L 0 0.41 0.03 2,NH NH 3 + F 2 , O H ~
OH

(fast pre-reactions) (rate-determining step)

where Zn(I) stands for a zinc species of unknown composition in the + 1 oxidation state, and the v's are stoichiometric coefficients. Derive an expression for the exchange current analogous to (3.5.40), and find explicit relationships for the logarithmic derivatives given above. (b) Calculate a and all stoichiometric coefficients. (c) Identify Zn(I) and write chemical equations to give a mechanism consistent with the data. (d) Consider an alternative mechanism having the pattern above, but with the first step being ratedetermining. Is such a mechanism consistent with the observations? 3.11 The following data were obtained for the reduction of species R to R~ in a stirred solution at a 0.1 cm 2 electrode; the solution contained 0.01 M R and 0.01 M R~. 7j(mV): i(jiA): Calculate: /0, k°, a, Rcb //, ra0, Rmt 3.12 From results in Figure 3.4.5 for 10~2 M Mn(III) and 10~2 M Mn(IV), estimate j 0 and k°. What is the predicted j 0 for a solution 1 M in both Mn(III) and Mn(IV)? 3.13 The magnitude of the solvent term (1/гор - l/ss) is about 0.5 for most solvents. Calculate the value of Лo and the free energy of activation (in eV) due only to solvation for a molecule of radius 4.0 A spaced 7 A from an electrode surface. 3.14 Derive (3.6.30). 3.15 Show from the equations for D O (E, Л) and D R (E, Л) that the equilibrium energy of a system, E e q , is related to the bulk concentrations, CQ and C R and E° by an expression resembling the Nernst equation. How does this expression differ from the Nernst equation written in terms of potentials, Ещ and £°? How do you account for the difference? 3.16 Derive (3.6.36) by considering the reaction О + e ±± R at equilibrium in a system with bulk concentrations CQ and C R . -100 45.9 -120 62.6 -150 100 -500 965 -600 965

CHAPTER

4
MASS TRANSFER BY MIGRATION AND DIFFUSION
4.1 DERIVATION OF A GENERAL MASS TRANSFER EQUATION
In this section, we discuss the general partial differential equations governing mass transfer; these will be used frequently in subsequent chapters for the derivation of equations appropriate to different electrochemical techniques. As discussed in Section 1.4, mass transfer in solution occurs by diffusion, migration, and convection. Diffusion and migration result from a gradient in electrochemical potential, JL. Convection results from an imbalance of forces on the solution. Consider an infinitesimal element of solution (Figure 4.1.1) connecting two points in the solution, r and s, where, for a certain species j , Jip) Ф ^(s). This difference of ^ over a distance (a gradient of electrochemical potential) can arise because there is a difference of concentration (or activity) of species у (a concentration gradient), or because there is a difference of ф (an electric field or potential gradient). In general, a flux of species j will occur to alleviate any difference of /Zj. The flux, Jj (mol s ^ c m " 2 ) , is proportional to the gradient of /xj: Jj oc grad^uj or Jj oc V)itj (4.1.1)

where grad or V is a vector operator. For linear (one-dimensional) mass transfer, V = i(d/dx)9 where i is the unit vector along the axis and x is distance. For mass transfer in a three-dimensional Cartesian space, V = i | - + j | - + k|dx
J

(4.1.2)

dy

dz

Point r

Point s
#

(a)

(b)

Figure 4.1.1

A gradient of electrochemical potential.

137

138 • Chapter 4. Mass Transfer by Migration and Diffusion The constant of proportionality in (4.1.1) turns out to be —CJO>^IRT\ thus, (4.1.3) For linear mass transfer, this is (4.1.4) The minus sign arises in these equations because the direction of the flux opposes the direction of increasing ~jlIf, in addition to this ~jx gradient, the solution is moving, so that an element of solution [with a concentration C}(s)\ shifts from s with a velocity v, then an additional term is added to the flux equation: (4.1.5) For linear mass transfer, CD Taking cij ~ Cj, we obtain the Nernst-Planck equations, which can be written as ^-(RT In С д + ~-(z:F 6CU(II)

diffusion 5Cu(II) 7Cu(I) ЗСГ

diffusion 5Cu(II) 7Cu(I) ЗСГ

Figure 4.3.3 Balance sheet for electrolysis of the Си(П), Cu(I), NH3 system, (a) Cell schematic. (b) Various contributions to the current when 6e are passed in the external circuit per unit time; / = 6, n = 1. For Cu(II) at the cathode, |i m | = (l/2)(l/3)(6) = 1 (equation 4.3.3), /d = 6 - 1 = 5 (equation 4.3.4). For Cu(I) at the anode, |/J = (l/l)(l/6)(6) = 1, /d = 6 + 1 = 7. tribute to the carrying of the current, but serves only to stabilize the copper species in the +1 and + 2 states. The resistance of this cell would be relatively large, since the total concentration of ions in the solution is small.

4.3.2 Effect of Adding Excess Electrolyte
Example 43
Let us consider the same cell as in Example 4.2, except with the solution containing 0.10 M NaClO 4 as an excess electrolyte (Figure 4.3.4a). Assuming that ANa+ = А с ю ^ = ^» we obtain the following transference numbers: tNa+ = ^107 » = 0.485, fcu(n) = 0.0097, *Cu(D = °- 0 0 4 8 5 > 'ci- = 0.0146. The N a + and СЮ4 do not participate in the electrontransfer reactions; but because their concentrations are high, they carry 97% of the current in the bulk solution. The balance sheet for this cell (Figure 4.3.4Z?) shows that most of the Cu(II) now reaches the cathode by diffusion, and only 0.5% of the total flux is by migration. Thus, the addition of an excess of nonelectroactive ions (a supporting electrolyte) nearly eliminates the contribution of migration to the mass transfer of the electroactive species. In general, it simplifies the mathematical treatment of electrochemical systems by elimination of the Чф or дф/дх term in the mass transport equations (e.g., equations 4.1.10 and 4.1.11). In addition to minimizing the contribution of migration, the supporting electrolyte serves other important functions. The presence of a high concentration of ions decreases

144

Chapter 4. Mass Transfer by Migration and Diffusion

• Hg/Cu(NH3)4CI2(10~3 M), Cu(NH3)2 CI(10~3 M)/Hg NH 3 (0.1 M), NaCIO4 (0.10 M) Cu(NH 3 )f (10- 3 M), Cu(NH3)2 (10- 3 M), СГ(3 x 10~3 M), Na + (0.1 M), CIO4 (0.1 M)

Э
Q (Anode)

Ions in cell:

(Cathode)
6e

Г\

6Cu(II) + 6e -» 6Cu(I)

6Cu(II) 6Cu(I) 2.91 Na + 2.91 CIO7 0.0291 Cu(II) 0.0291 Cu(I) 0.0873 СГ diffusion 5.97Cu(II) 2.92Na + 2.92CIO4 diffusion 5.97Cu(II) 6.029Cu(I) 2.92Na+ 2.92CIO4

6Cu(I) - 6e -> 6Cu(II)

(b)

Figure 4.3.4 Balance sheet for the system in Figure 4.3.3, but with excess NaC104 as a supporting electrolyte, (a) Cell schematic, (b) Various contributions to the current when 6e are passed in the external circuit per unit time (/ = 6, n = 1). fCu(II) = [(2 X 10 ~ 3 ) A/(2 X 1(Г 3 + 1(Г 3 + 3 X 1(Г 3 0.2)Л] = 0.0097. For Си(П) at the cathode, |/J = (l/2)(0.0097)(6) = 0.03, i d = 6 - 0.03 = 5.97. the solution resistance, and hence the uncompensated resistance drop, between the working and reference electrodes (Section 1.3.4). Consequently, the supporting electrolyte allows an improvement in the accuracy with which the working electrode's potential is controlled or measured (Chapter 15). Improved conductivity in the bulk of the solution also reduces the electrical power dissipated in the cell and can lead to important simplifications in apparatus (Chapters 11 and 15). Beyond these physical benefits are chemical contributions by the supporting electrolyte, for it frequently establishes the solution composition (pH, ionic strength, ligand concentration) that controls the reaction conditions (Chapters 5, 7, 11, and 12). In analytical applications, the presence of a high concentration of electrolyte, which is often also a buffer, serves to decrease or eliminate sample matrix effects. Finally, the supporting electrolyte ensures that the double layer remains thin with respect to the diffusion layer (Chapter 13), and it establishes a uniform ionic strength throughout the solution, even when ions are produced or consumed at the electrodes. Supporting electrolytes also bring some disadvantages. Because they are used in such large concentrations, their impurities can present serious interferences, for example,

4.3 Mixed Migration and Diffusion Near an Active Electrode
25 20 I i I a _

145

15 -10 О

5 -

/JJ
I -0.8 I -1.0 E/V

/

7 /

/7

b с _ d

i

-0.6

-1.2

-1.4

Figure 4.3.5 Voltammograms for reduction of 0.65 mM T12SO4 at a mercury film on a silver ultramicroelectrode (radius, 15 jam) in the presence of (a) 0, (b) 0.1, (c) 1, and (d) 100 mM LiClO4. The potential was controlled vs. a Pt wire QRE whose potential was a function of solution composition. This variability is the basis for the shifts in wave position along the potential axis. [Reprinted with permission from M. Ciszkowska and J. G. Osteryoung, Anal. С hem., 67, 1125 (1995). Copyright 1995, American Chemical Society.] by giving rise to faradaic responses of their own, by reacting with the intended product of an electrode process, or by adsorbing on the electrode surface and altering kinetics. Also, a supporting electrolyte significantly alters the medium in the cell, so that its properties differ from those of the pure solvent. The difference can complicate the comparison of results obtained in electrochemical experiments (e.g., thermodynamic data) with data from other kinds of experiments where pure solvents are typically employed. Most electrochemical studies are carried out in the presence of a supporting electrolyte selected for the solvent and electrode process of interest. Many acids, bases, and salts are available for aqueous solutions. For organic solvents with high dielectric constants, like acetonitrile and N,N-dimethylformamide, normal practice is to employ tetraл-alkylammonium salts, such as, Bu 4 NBF 4 and Et 4 NClO 4 (Bu = «-butyl, Et = ethyl). Studies in low-dielectric solvents like benzene inevitably involve solutions of high resistance, because most ionic salts do not dissolve in them to an appreciable extent. In solutions of salts that do dissolve in apolar media, such as Hx 4 NC10 4 (where Hx = nhexyl), ion pairing is extensive. Studies in very resistive solutions require the use of UMEs, which usually pass low currents that do not give rise to appreciable resistive drops (see Section 5.9.2). The effect of supporting electrolyte concentration on the limiting steady-state current at UMEs has been treated (12-14). Typical results, shown in Figure 4.3.5, illustrate how the limiting current for reduction of Tl + to the amalgam at a mercury film decreases with an increase in LiClO 4 concentration (15). The current in the absence of LiClO4, or at very low concentrations, is appreciably larger than at high concentrations, because migration of the positively charged T1(I) species to the cathode enhances the current. At high LiClO 4 concentrations, Li + migration replaces that of Tl + , and the observed current is essentially a pure diffusion current. A similar example involving the polarography of Pb(II) with KNO 3 supporting electrolyte was given in the first edition.1
1

First edition, p. 127.

146

Chapter 4. Mass Transfer by Migration and Diffusion

P 4.4 DIFFUSION
As we have just seen, it is possible to restrict mass transfer of an electroactive species near the electrode to the diffusive mode by using a supporting electrolyte and operating in a quiescent solution. Most electrochemical methods are built on the assumption that such conditions prevail; thus diffusion is a process of central importance. It is appropriate that we now take a closer look at the phenomenon of diffusion and the mathematical models describing it (16-19).

4.4.1 A Microscopic View—Discontinuous Source Model
Diffusion, which normally leads to the homogenization of a mixture, occurs by a "random walk" process. A simple picture can be obtained by considering a one-dimensional random walk. Consider a molecule constrained to a linear path and, buffeted by solvent molecules undergoing Brownian motion, moving in steps of length, /, with one step being made per unit time, r. We can ask, "Where will the molecule be after a time, ft" We can answer only by giving the probability that the molecule will be found at different locations. Equivalently, we can envision a large number of molecules concentrated in a line at t = 0 and ask what the distribution of molecules will be at time t. This is sometimes called the "drunken sailor problem," where we envision a very drunk sailor emerging from a bar (Figure 4.4.1) and staggering randomly left and right (with a stagger-step size, /, one step every т seconds). What is the probability that the sailor will get down the street a certain distance after a certain time tl In a random walk, all paths that can be traversed in any elapsed period are equally likely; hence the probability that the molecule has arrived at any particular point is simply the number of paths leading to that point divided by the total of possible paths to all accessible points. This idea is developed in Figure 4.4.2. At time т, it is equally likely that the molecule is at +/ and -/; and at time 2т, the relative probabilities of being at +2/, 0, and —2/, are 1, 2, and 1, respectively. The probability, P(m, r), that the molecule is at a given location after m time units (m = t/т) is given by the binomial coefficient

^h$"

(4A1)

where the set of locations is defined by x = (—m + 2r)/, with r = 0, 1 , . . . m. The mean square displacement of the molecule, Д 1 , can be calculated by summing the squares of the displacements and dividing by the total number of possibilities (2 m ). The squares of the displacements are used, just as when one obtains the standard deviation in statistics, because movement is possible in both the positive and negative directions, and the sum of the displacements is always zero. This procedure is shown in Table 4.4.1. In general, Д 2 is given by ~~ * (4.4.2)

-4/

-3/

-2/

-i

о

I +/

| +2/

\ +3/

| +4/

|

Figure 4.4.1 The onedimensional random-walk or "drunken sailor problem."

4.4 Diffusion
-5/ -41 -31 -21

i 147

J
От 1т 1 1 1 1 I 1 1 1 1
Y/

L i i I

+/ I

+21 I

+31 I

+41 I

+51 I

I I I

1 >

1 у

3T

1 у

A i A i if

1 у

A i A i i

1 V 1

i i
V 1

I I I V 1
J

i i i i V 1

I I I I I Figure 4.4.2 (a) Probability distribution for a one-dimensional random walk over zero to four time units. The number printed over each allowed arrival point is the number of paths to that point, (b) Bar graph showing distribution at t = AT. At this time, probability of being at x = 0 is 6/16, at x = ± 2/ is 4/16, and at JC = ± 4/is 1/16.

4 2 у

Ч З у

4 4 у

A i A i 4 A i 4 6 у

V A i 4 A i \ 4

4 A i 4.

-4/

-2/

+2/

+41

where the diffusion coefficient, D, identified as /2/2r, is a constant related to the step size and step frequency.2 It has units of Iength2/time, usually cm2/s. The root-mean-square displacement at time t is thus = V2Dt (4.4.3)

This equation provides a handy rule of thumb for estimating the thickness of a diffusion layer (e.g., how far product molecules have moved, on the average, from an electrode in a certain time). A typical value of D for aqueous solutions is 5 X 10~6 cm2/s, so that a diffusion layer thickness of 10~4 cm is built up in 1 ms, 10~3 cm in 0.1 s, and 10~2 cm in 10 s. (See also Section 5.2.1.) As m becomes large, a continuous form of equation 4.4.1 arises. For No molecules located at the origin at t = 0, a Gaussian curve will describe the distribution at some later TABLE 4.4.1 Distributions for a Random Walk Process t 0

n»

Д

с

SД 0

F = ISA2 0 /2
2/2 3/2 4/2

От
IT 2T 3T 4T

к = 2°)
2( 4( 8( 16(
2
m

= = = =

2') 22) 23) 24)

0 ±/(1) 0(2), ± 2/(1) ±/(3), :t 3/(1) 0(6), ± 2/(4), ±4/(1)

г/

2

S/2 24/2 64/2 m 2 mnl\ = m2 / )

тт a b

ml2

l = step size, 1/r = step frequency, t = mr = time interval. n = total number of possibilities. A = possible positions; relative probabilities are parenthesized.

C

This concept of D was derived by Einstein in another way in 1905. Sometimes D is given as//2/2, where/is the number of displacements per unit time (= 1/r).

2

148

Chapter 4. Mass Transfer by Migration and Diffusion time, t. The number of molecules, N(x, t), in a segment Ax wide centered on position x is (20)

«*•»-

**,(=£)

.4.4.4,

A similar treatment can be applied to two- and three-dimensional random walks, where the root-mean-square displacements are (4Dt)l/2 and (6Dt)l/2, respectively (19, 21). It may be instructive to develop a more molecular picture of diffusion in a liquid by considering the concepts of molecular and diffusional velocity (21). In a Maxwellian gas, a particle of mass m and average one-dimensional velocity, vx, has an average kinetic energy of l/2rnv2. This energy can also be shown to be 4T/2, (22, 23); thus the average molecular velocity is vx = (6T/m)m. For an O2 molecule (m = 5 X 10~ 23 g) at 300 K, one finds that vx = 3 X 104 cm/s. In a liquid solution, a velocity distribution similar to that of a Maxwellian gas may apply; however, a dissolved O 2 molecule can make progress in a given direction at this high velocity only over a short distance before it collides with a molecule of solvent and changes direction. The net movement through the solution by the random walk produced by repeated collisions is much slower than vx and is governed by the process described above. A "diffusional velocity," u^, can be extracted from equation 4.4.3 as vd = A/t = (2D/t)l/2 (4.4.5)

There is a time dependence in this velocity because a random walk greatly favors small displacements from a starting point vs. large ones. The relative importance of migration and diffusion can be gauged by comparing ud with the steady-state migrational velocity, v, for an ion of mobility щ in an electric field (Section 2.3.3). By definition, v = uY%, where % is the electric field strength felt by the ion. From the Einstein-Smoluchowski equation, (4.2.2), v = \z{\ FDfflRT (4.4.6)

When v « L?d, diffusion of a species dominates over migration at a given position and time. From (4.4.5) and (4.4.6), we find that this condition holds when 2D\112 RTl\z{\F which can be rearranged to (2Д/)
1/2

%«

2 -Щ-

(4.4.8)

where the left side is the diffusion length times the field strength, which is also the voltage drop in the solution over the length scale of diffusion. To ensure that migration is negligible compared to diffusion, this voltage drop must be smaller than about 2RTl\z-^F, which is 51.4/|z}| mV at 25°C. This is the same as saying that the difference in electrical potential energy for the diffusing ion must be smaller than a few 4T over the length scale of diffusion. 4.4.2 Fick's Laws of Diffusion Fick's laws are differential equations describing the flux of a substance and its concentration as functions of time and position. Consider the case of linear (one-dimensional) diffusion. The flux of a substance О at a given location x at a time t, written as JQ{X, t), is the

4.4 Diffusion

149

net mass-transfer rate of O, expressed as amount per unit time per unit area (e.g., mol s l cm~ 2 ). Thus Jo(x, i) represents the number of moles of О that pass a given location per second per cm 2 of area normal to the axis of diffusion. Fick's first law states that the flux is proportional to the concentration gradient, дСо/дх: -Jo(x, t) = Do dC0(x, t) dx (4.4.9)

This equation can be derived from the microscopic model as follows. Consider location x, and assume NQ(X) molecules are immediately to left of x, and NQ(X + Дл:) molecules are immediately to the right, at time t (Figure 4.4.3). All of the molecules are understood to be within one step-length, Ax, of location x. During the time increment, Д*, half of them move Ax in either direction by the random walk process, so that the net flux through an area A at x is given by the difference between the number of molecules moving from left to right and the number moving from right to left: No(x) No(x + Ax) At (4.4.10)

Jo(x,t) = ±Multiplying by Дх /Ax rive

and noting that the concentration of О is C o = No/AAx, we de-

-Jo(x, i) =

2At

Ax

(4.4.11)

From the definition of the diffusion coefficient, (4.4.2), Do = Ax2/2At, and allowing Ax and Д По approach zero, we obtain (4.4.9). Fick's second law pertains to the change in concentration of О with time: (4.4.12) This equation is derived from the first law as follows. The change in concentration at a location x is given by the difference in flux into and flux out of an element of width dx (Figure 4.4.4). dCo(x, i) dt J(x, t) - J(x + dx, t) dx (4.4.13)

Note that J/dx has units of (mol s l cm 2)/cm or change in concentration per unit time, as required. The flux at x + dx can be given in terms of that at x by the general equation J(x + dx, t) = J(x, t) + dx (4.4.14)

No (x + Ax) No (x + Ax)
N0(x)
2

^~~

— ^

2

Figure 4.4.3 in solution.

Fluxes at plane x

150

Chapter 4. Mass Transfer by Migration and Diffusion dx J0

(X, t)

JQ (x + dx, t)

x + dx

Figure 4.4.4 Fluxes into and out of an element at x.

and from equation 4.4.9 we obtain _ dJ(x, t) dx d dx'

x, t) dx (4.4.15)

Combination of equations 4.4.13 to 4.4.15 yields (4.4.16) When Do is not a function of x, (4.4.12) results. In most electrochemical systems, the changes in solution composition caused by electrolysis are sufficiently small that variations in the diffusion coefficient with x can be neglected. However when the electroactive component is present at a high concentration, large changes in solution properties, such as the local viscosity, can occur during electrolysis. For such systems, (4.4.12) is no longer appropriate, and more complicated treatments are necessary (24, 25). Under these conditions, migrational effects can also become important. We will have many occasions in future chapters to solve (4.4.12) under a variety of boundary conditions. Solutions of this equation yield concentration profiles, CQ(X, t). The general formulation of Fick's second law for any geometry is dt
2

=

DOV2CC
2

(4.4.17)

where V is the Laplacian operator. Forms of V for different geometries are given in Table 4.4.2. Thus, for problems involving a planar electrode (Figure 4.4.5a), the linear diffusion equation, (4.4.12), is appropriate. For problems involving a spherical electrode

TABLE 4.4.2 Type

Forms of the Laplacian Operator for Different Geometries'* Variables x r г r, z x, z V2
2 2

Example

Linear Spherical Cylindrical (axial) Disk Band a Shielded disk electrode д /дх 2 2 Hanging drop electrode d /dr ^- (2/r)(d/dr) Wire electrode д2/дг2 Нh (l/r)(d/dr) 2 2 д2/дг2 Н- (l/r)(d/dr) H d /dz Inlaid disk ultramicroelectrode^ 6 2 2 Inlaid band electrode д2/дх2 -f a /az

See also J. Crank, "The Mathematics of Diffusion," Clarendon, Oxford, 1976. r = radial distance measured from the center of the disk; z = distance normal to the disk surface. c x — distance in the plane of the band; z = distance normal to the band surface.

4.4 Diffusion

151

I
X

i i i

/

А \
\ \ \ \
\

Т

\ \ \ \

ifl)

Figure 4.4.5 Types of diffusion occurring at different electrodes. (a) Linear diffusion to a planar electrode, (b) Spherical diffusion to a hanging drop electrode.

(Figure 4.4.5b), such as the hanging mercury drop electrode (HMDE), the spherical form of the diffusion equation must be employed: (4.4.18) The difference between the linear and spherical equations arises because spherical diffusion takes place through an increasing area as r increases. Consider the situation where О is an electroactive species transported purely by diffusion to an electrode, where it undergoes the electrode reaction О + ne °°) the concentration reaches a constant value, typically the initial concentration, so that, for example, lim Co(x9 t) =C% lim CR(JC, t) = 0

(at all 0
(at all t)

(4.4.25)
(4.4.26)

For thin-layer electrochemical cells (Section 11.7), where the cell wall is at a distance, /, of the order of the diffusion length, one must use boundary conditions at x = I instead of those for лс—> °°. (c) Electrode Surface Boundary Conditions Additional boundary conditions usually relate to concentrations or concentration gradients at the electrode surface. For example, if the potential is controlled in an experiment, one might have Co(0,t)=f(E) (4.4.27)

where f(E) is some function of the electrode potential derived from the general currentpotential characteristic or one of its special cases (e.g., the Nernst equation). If the current is the controlled quantity, the boundary condition is expressed in terms of the flux at x = 0; for example,

The conservation of matter in an electrode reaction is also important. For example, when О is converted to R at the electrode and both О and R are soluble in the solution phase, then for each О that undergoes electron transfer at the electrode, an R must be produced. Consequently, /o(0, t) = - / R ( 0 , t), and

FHL
r