ELECTROCHEMICAL SYSTEMS ELECTROCHEMICAL SYSTEMS Third Edition JOHN NEWMAN and KAREN E THOMAS-ALYEA University of California, Berkeley ELECTROCHEMICAL SOCIETY SERIES WILEYINTERSCIENCE A JOHN WILEY & SONS, INC PUBLICATION Copyright © 2004 by John Wiley & Sons, Inc All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada 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 Section 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, Inc 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-750-4470, or on the web at www.copyright.com Requests to the Publisher for permission should be addressed to the 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Care Department within the U.S at 877-762-2974, outside the U.S at 317-572-3993 or fax 317-572-4002 Wiley also publishes its books in a variety of electronic formats Some content that appears in print, however, may not be available in electronic format Library of Congress Cataloging-in-Publication Data: Newman, John S., 1938Electrochemical systems / John Newman and Karen E Thomas-Alyea.— 3rd ed p cm Includes bibliographical references and index ISBN 0-471-47756-7 (cloth) Electrochemistry, Industrial I Thomas-Alyea, Karen E II Title TP255.N48 2004 660" 297—dc22 2004005207 10 CONTENTS PREFACE TO THE THIRD EDITION XV PREFACE TO THE SECOND EDITION xvii PREFACE TO THE FIRST EDITION xix INTRODUCTION l.l Definitions / l Thermodynamics and Potential / 1.3 Kinetics and Rates of Reaction / 1.4 Transport / 1.5 Concentration Overpotential and the Diffusion Potential / 18 1.6 Overall Cell Potential / 21 Problems / 25 Notation / 25 v Vi CONTENTS PART A THERMODYNAMICS OF ELECTROCHEMICAL CELLS THERMODYNAMICS IN TERMS OF ELECTROCHEMICAL POTENTIALS 27 29 2.1 2.2 2.3 2.4 2.5 2.6 Phase Equilibrium / 29 Chemical Potential and Electrochemical Potential / 31 Definition of Some Thermodynamic Functions / 35 Cell with Solution of Uniform Concentration / 43 Transport Processes in Junction Regions / 47 Cell with a Single Electrolyte of Varying Concentration / 49 2.7 Cell with Two Electrolytes, One of Nearly Uniform Concentration / 53 2.8 Cell with Two Electrolytes, Both of Varying Concentration / 58 2.9 Standard Cell Potential and Activity Coefficients / 59 2.10 Pressure Dependence of Activity Coefficients / 69 2.11 Temperature Dependence of Cell Potentials / 70 Problems / 72 Notation / 82 References / 83 THE ELECTRIC POTENTIAL 85 3.1 The Electrostatic Potential / 85 3.2 Intermolecular Forces / 88 3.3 Outer and Inner Potentials / 91 3.4 Potentials of Reference Electrodes / 92 3.5 The Electric Potential in Thermodynamics / 94 Notation / 96 References / 97 ACTIVITY COEFFICIENTS 4.1 4.2 4.3 4.4 4.5 4.6 Ionic Distributions in Dilute Solutions / 99 Electrical Contribution to the Free Energy / 102 Shortcomings of the Debye-Hiickel Model / 107 Binary Solutions / 110 Multicomponent Solutions / 112 Measurement of Activity Coefficients / 116 99 CONTENTS 4.7 Weak Problems / Notation / References VÜ Electrolytes / 119 122 127 / 128 REFERENCE ELECTRODES 131 5.1 5.2 Criteria for Reference Electrodes / I3l Experimental Factors Affecting The Selection of Reference Electrodes / 133 5.3 The Hydrogen Electrode / 134 5.4 The Calomel Electrode and Other Mercury-Mercurous Salt Electrodes / 137 5.5 The Mercury-Mercuric Oxide Electrode / 140 5.6 Silver-Silver Halide Electrodes / 140 5.7 Potentials Relative to a Given Reference Electrode / 142 Notation / 147 References / 148 POTENTIALS OF CELLS WITH JUNCTIONS 149 6.1 Nernst Equation / 149 6.2 Types of Liquid Junctions / 150 6.3 Formulas for Liquid-Junction Potentials / 151 6.4 Determination of Concentration Profiles / 153 6.5 Numerical Results / 153 6.6 Cells with Liquid Junction / 154 6.7 Error in the Nernst Equation / 160 6.8 Potentials Across Membranes / 162 Problems / 163 Notation / 167 References / 167 PART B ELECTRODE KINETICS AND OTHER INTERFACIAL PHENOMENA STRUCTURE OF THE ELECTRIC DOUBLE LAYER 7.1 7.2 Qualitative Description of Double Layers / 172 Gibbs Adsorption Isotherm / 177 169 171 viii CONTENTS 7.3 7.4 7.5 The Lippmann Equation / 181 The Diffuse Part of the Double Layer / 185 Capacity of the Double Layer in the Absence of Specific Adsorption / 193 7.6 Specific Adsorption at an Electrode-Solution Interface / 195 Problems / 195 Notation / 199 References / 200 ELECTRODE KINETICS 203 8.1 8.2 Heterogeneous Electrode Reactions / 203 Dependence of Current Density on Surface Overpotential / 205 8.3 Models for Electrode Kinetics / 207 8.4 Effect of Double-Layer Structure / 225 8.5 The Oxygen Electrode / 227 8.6 Methods of Measurement / 229 8.7 Simultaneous Reactions / 230 Problems / 233 Notation / 236 References / 237 ELECTROKINETIC PHENOMENA 241 9.1 Discontinuous Velocity at an Interface / 241 9.2 Electro-Osmosis and the Streaming Potential / 244 9.3 Electrophoresis / 254 9.4 Sedimentation Potential / 256 Problems / 257 Notation / 260 References / 261 10 ELECTROCAPILLARY PHENOMENA 10.1 Dynamics of Interfaces / 263 10.2 Electrocapillary Motion of Mercury Drops / 264 10.3 Sedimentation Potentials for Falling Mercury Drops / 266 Notation / 267 References / 268 263 APPENDIX C 633 Figure C.2 gives a BANDmap showing how the governing equations and boundary conditions fit into a BAND structure This can be contrasted with that in Figure C.l Here first-order equations are programmed directly, with forward or backward differences being used so that the boundary conditions can be fit in where appropriate Figure C.3 gives concentration profiles calculated by the program This can be considered a solution to Problem 11.4(b), and the student can see how the profiles relate to electroneutrality, migration and diffusion of each species, and production or consumption, if any, in the electrode reaction C.7 DISCUSSION AND CONCLUSIONS The procedure outlined here for solving coupled, nonlinear, difference equations by linearization and subsequent iteration is quite general and flexible and has proved useful for a number of problems For special problems it may be possible to devise more efficient methods, but with a loss of generality and an expense of effort Two other methods might occur to one faced with a problem of the type treated here One is to linearize and decouple the equations by taking the coefficients of the derivatives to be given by a trial solution, for example, approximate c\dci/dx by cfdcj/dx Then the decoupled equations are solved one after another in a cyclic process, producing new functions to be used as a trial solution In general, the convergence behavior is poorer than for the present method, although there are special problems where the coupling is not strong and the method works A second method would be to treat the problem as an initial-value problem and to fabricate the needed initial conditions This method requires little storage space, but - l 1.0002 - X 1.0O0O +„ 0.9998 J3 o c /Cl 0.6 - O B 0.4 - -\ \OH" o o2 - i i _g 0.9996 0.9994 i Ü c u o a o U 0.9992 Distance from electrode Figure C.3 Concentration profiles for reduction of from a solution of NaCl, as calculated by the transient multicomponent-diffusion program 634 NUMERICAL SOLUTION OF COUPLED, ORDINARY DIFFERENTIAL EQUATIONS the adjustment of the added initial conditions so as to satisfy the boundary conditions atx = L can be tricky or impossible The errors in the present method arise from three sources, as discussed earlier: convergence errors for the nonlinear problem (which can be made negligibly small here), errors in the difference approximations to the differential equations (which decrease with the mesh interval h), and round-off errors in the computer (which increase as the mesh distance is decreased) Convergence may not be possible if there are sharp variations of the unknowns in some region of x; in such a case a singular-perturbation method may be appropriate REFERENCES John Newman, "Numerical Solution of Coupled, Ordinary Differential Equations," Industrial and Engineering Chemistry Fundamentals, (1968), 514-517 John Newman, Numerical Solution of Coupled, Ordinary Differential Equations, (Berkeley: Lawrence Radiation Laboratory, University of California, August, 1967) (UCRL-17739) Bernard Tribollet and John Newman, "Impedance Model for a Concentrated Solution Application to the Electrodissolution of Copper in Chloride Solutions," Journal of the Electrochemical Society, 131 (1984), 2780-2785 INDEX absorption coefficient, 580, 587 acceptor atom, 572, 574 activation energy, 210-211 activity in a kinetic expression, 590-592 relative, 61 activity, absolute, 36, 66, 594 of an alloy, 42-43, 150 activity coefficient, 35-38, 50, 52-53, 55, 57, 59-60, 64, 67, 78, 99-129, 142, 152, 154, 157, 160, 162, 185,336,519,568,571, 582 of binary solution, 110, 113 composition dependence, 105 Debye-Hiickel limiting law, 106-108 from Debye-Hiickel theory, 55, 105-106, 163 effect of dissociation, 40-41, 74, 99, 119-121, 126 of electrons, 570, 571 of holes, 570, 571 ionic, 115, 149-150, 269, 290-291 mean ionic, 69 mean molal, 39, 42, 110,300 mean molar, 40, 41, 73 measurement of, 61, 116-119 molar and molal relationships, 36, 40, 123 of multicomponent solution, 112-116 pressure dependence of, 69-70 and standard cell potential, 59-68 adsorption, 172, 177-184, 193-198, 241, 517,551-552 adsorption-desorption reactions, 235, 516, 597 alloy, 44, 61, 150,547 see also electrode, alloy amalgam, 44-46, 59-61, 132-133 annulus, 11, 10-25, 389-393, 395, 397, 406-409 anode, anode, sacrificial, 436, 445-447 anodic protection, 231, 430, 447 apparent transfer coefficient, 8, 529 arsenic, 136, 553 AUTOBAND, 622-623 see also BAND(J) Avogadro's number, 103, 569, 574 backward-difference formula, 613 Electrochemical Systems, Third Edition, by John Newman and Karen E Thomas-Alyea ISBN 0-471 -47756-7 © 2004 John Wiley & Sons, Inc 635 636 INDEX band, 568 bending, 575, 584-587, 597 conduction, 568-570, 578-580 valence, 568-570, 578-580 BAND(J), 613, 616, 619-621 see also coupled, linear, difference equations battery, 1, 75, 78, 80, 292, 517-518, 520-521, 524, 537, 543, 545, 594 optimization, 537-544 reaction mechanisms, 535-536 simulation, 535-550 binary electrolyte, see electrolyte, binary bisulfate ions, 119-121, 126,471-476 see also dissociation Boltzmann approximation, 571, 577 Boltzmann distribution, 100,186,306,572, 580-582 boundary layer, 612 boundary-value problem, 611 Br0nsted's principle, i l l , 114, 115 Bruggeman equation, 540 Butler-Volmer equation, 7, 205, 212, 223, 427, 522 see also kinetics cadmium sulfide, 572, 596 calomel electrode, see reference electrode capacitance at semiconductor-solution interface, 580-583 see also double-layer capacity capacity, 540, 543, 539 see double-layer capacity capillary action, 138 capillary tube, 175, 776, 244, 399, 406 carbon dioxide, 133, 136 catalyst, 403-404 for hydrogen dissociation, 135-136 poisoning, 136 cathode, 6, 22-25 cathodic protection, 433, 434, 435-436, 439, 440, 554 design graphs for, 440-441, 442, 443, 445 design spreadsheet for, 448 principles of, 430-449 cavity potential, see potential, cavity cell, see electrochemical cell cell potential, 21-23, 31, 43-44, 47, 52-53, 58, 74, 149, 154, 156, 162-164,325,437,539,548 calculation of, 65-66 computation with transference, 47-53 computation without transference, 43-46 and equilibrium constants, 66-67 relationship to electrochemical potential, 44 single electrolyte with transference, 52 temperature dependence of, 70-74 two electrolytes, 53-59 cell-sandwich model, 544-546 central difference formula, 612 central ion, 99 centrifugal field, 330 channel flow cell, 512 charge density, 86,100,173,215,243,246, 286-288, 356-358 in a semiconductor, 572, 576, 580-581 charge number, 3, 30 see also valence charge separation, charging processes, 103-104, 108-109, 122,551-552 chemical formula, 31, 51 chemical potential, 31-35, 47, 57, 66, 69, 149-150, 165 of a component species, 114 of an electrolyte, 38-39, 51, 300 of a gaseous species, 42 relationship to electrochemical potential, 30,38 thermodynamic definition of, 33 see also electrochemical potential colligative properties, 37 collision, 89-90 complexes, 133, 139, 142 concentrated solution theory, 18, 297-315, 404-406, 545 binary electrolyte, 299-300 connection to dilute-solution theory, 305-307 in porous electrodes, 524 transport laws, 297-299 concentration cell, 19, 49-50, 52, 60, 75, 151, 302-305 see also transference, cell with concentration profile, 48-50, 54, 57-59, 153-154, 166-167, 378, 393, 395, 409, 459, 466, 471, 547-548, 586 concentration, superficial, 519 conduction band, see band, conduction conductivity, 47, 245, 274-276, 285, 302, 335,420,427,450,472,478,519,540 effective, 523, 528-529, 540, 557 electronic, 2, INDEX ionic, 2, 11 metallic, 568-569, 572 mixed, in multicomponent solution, 308 semiconductor, 568-569, 572 surface, 253, 258 conductor, see conductivity conformai mapping, 421 conservation of charge, 276-277 in a porous electrode, 521, 524, 527 conservation of mass, 347 see also continuity equation conservation of momentum, 348 continuity equation, 254, 312, 352, 360, 374 see also conservation of mass continuous-mixture junction, 151-152, 155-159 control-volume approach, 614 convection, 9, 15-18, 208, 271, 292, 373, 377-413, 422, 480, 545, 553-558 see also convective-diffusion equation see also convective-transport problem see also convective transport, simplifications for forced, 500, 612 free, 400-403, 405, 411, 465, 476, free and forced, 403 free, limiting currents for, 480-486 convective-diffusion equation, 278-280, 323, 363, 377, 383, 404, 406, 410, 459 see also convection in porous electrodes, 525 convective-transport problem, 370, 377-418, 459-460 see also convection convective transport, simplifications for, 377-378, 403-404 convective velocity, 273 copper electrode, / / , 63, 204 207-208, 214-215, 229, 233,495, 509-510 corrosion, 203, 206, 230, 430-431, 433, 567, 596 corrosion potential, see potential, corrosion corrosion, localized, 447-449 corrosion, pitting, 448, 475 coulombic forces, 32 and electroneutrality, 91 see also electrical forces Coulomb's law, 76-79, 85 coupled, 612 coupled, linear, difference equations program for, 619-623 637 solution of, 616-618 see also BAND(J) covalent forces, see specific interactions cupric sulfate, 214-215, 326, 342, 463, 471-473,481-483,509-510 curl, 356, 606 current, 297 anodic, cathodic, diffusion limiting, 463, 473 electric, see also current density current, limiting, 288, 389, 394, 396, 400, 402-403, 405-406, 411, 430, 434, 459, 495, 509, 536, 555-556, 558, 587 correction factor for, 463-465 currents below, 499-515 effect of migration on, 459-490 effect of migration on, program for, 623-627 for free convection, 480-486 current collector, 527, 539 current density, 4, 47, 49, 214, 244, 272, 373, 404, 421, 438, 447, 461, 504, 528, 538-539, 551 anodic, 147 and electrochemical potential gradient, 47, 308 superficial, 519 surface, 252-255, 263-264 and surface overpotential, 7-8, 9, 10, 132,205-207 current distribution, 12, 404, 419-420, 422, 479, 507, 508, 528 below limiting, 499-512 limiting, 378-413, 459-465 and mass transfer, 399-601 porous electrodes, 526-532 see also primary current distribution see also secondary current distribution current efficiency, 25 cylinder, rotating, 10-25, 229, 359, 397-399,411,479-480 Debye charging process, 104 Debye length, 101,172,192,246,575, 584 Debye-Hiickel approximation, 101, 198, 243, 245 Debye-Hiickel limiting law, 55, 106 substantiation by singular-perturbation, 108, 122 Debye-Hiickel parameters, 106 638 INDEX Debye-Hückel theory, 91, 96, 99-102, 185, 306, 336, 595 activity coefficient from, 55, 105-106 ionic distributions from, 102 osmotic coefficient from, 107 potential due to central ion from, 101 shortcomings of, 107-109 decomposition, 596 defects, see trap states degeneracy, 573 density of states, 569, 570 density profile, 483, 485, 486 depletion, 573 diaphragm cell, 335-336, 342 dielectric constant, 86, 243 differential equations, 153 diffuse double layer, see double layer, diffuse diffuse layer, see double layer, diffuse diffuse layer capacity, 193, 198 diffusion, 9, 13-15, 46, 49, 72, 151, 153, 208,271,311,422,534,545 multicomponent, 282-283 in semiconductors, 578-580 thermal, 318-320 diffusion coefficient, 278, 300, 404, 459-460,503,519,578 see also friction coefficient binary electrolyte diffusion coefficient, 14, 280, 300 correction in porous electrodes, 523-524 integral, 340-343 measurement of, 335-336, 381, 400, 412 and mobility, 283-285 multicomponent, 337 polarographic, 341 species diffusion coefficient, 14, 272, 284, 339, 340, 341 of supporting electrolyte, 281 thermal, 318 with thermodynamic driving force, 72-74, 299-300 diffusion layer, 16, 80, 174,175, 209, 231, 358, 377, 378, 386, 400-401, 404, 406, 409, 460-461, 464, 476-478, 482, 500-501, 502-503 see also Nernst diffusion layer axisymmetric, 395-396, 502 in cylindrical geometries, 386, 390 equations, 503 in laminar forced convection, 393-394 two-dimensional, 393-394, 502, 504 diffusion limiting current, see current, diffusion limiting diffusion potential, see potential, diffusion diffusivity ionic, 14 thermal, 323 dilute solution theory, 18, 54-55, 95, 269, 271-274, 335, 374, 406, 493, 496, 573 connection to concentrated-solution theory, 305-307 in semiconductors, 573-576, 578, 595 see also transport laws, dilute dipole moment, 172, 198 discharge curve, 548, 549 discharge time, see time of discharge disk electrode, 423, 424, 425-427, 429, 486 see also rotating disk electrode dispersion, axial, 523 dispersion coefficient, 524, 554 disproportionation reaction, 67, 139, 208 dissociation, 39, 74-75, 119-120, 277, 471-472, 483 see also bisulfate ions divergence, 348, 605-606, 608 divergence theorem, 87, 608 donor atom, 572 dopant, 572, 575, 582, 587 doping, 572-573, 583, 587 double layer, 171, 203, 273, 534, 574-575, 580,612 effect on kinetics, 225-227 in porous electrodes, 517-518, 522, 551-552 potential difference due to, 208 qualitative description of, 172-177 structure of, 171-201 double-layer capacity, 183, 189-191, 193-194, 225, 455, 523, 551-552 in absence of specific adsorption, 193 double-layer capacitor, 552 double layer, diffuse, 171, 174,175, 185-193, 203, 209, 226, 241-243, 256, 306, 583, 585, 588 dropping mercury electrode, see growing mercury drops Dufour effect, 318, 321 dust trap, 136 dyadic, 608 eddy diffusivity, 364, 365 INDEX eddy kinematic viscosity, 364 eddy viscosity, 361-362 efficiency, see energy efficiency electric field, 8, 86, 188, 243-244, 272, 356, 459, 463, 586 electric force, 85-88, 99-102, 172, 177, 241-242, 263, 286, 348, 355-358 magnitude of, 355-358 electrical state, 33, 48, 92, 94-96, 573-576,591,594 variable, 573-576 electrocapillary curve, 147, 184, 193 electrocapillary maximum, 184 see also potential of zero charge electrocapillary phenomena, 169, 263-268, 351 electrochemical cell, 3, 54, 59, 81, 165 electrochemical potential, 4, 5, 29-84, 90, 94, 149, 154, 165, 204, 289, 297, 318, 574 of a component species, 48, 59, 95, 289 of electrons and holes, 577-578, 593 of electrons in cell leads, 44-45 of electrons in metals, 574 gradient of, as driving force, 289, 297, 578-580 Guggenheim definition of, 33 of an ion in a junction region, 51, 59 in quantum mechanical terms, 32 relationship to chemical potential, 30,38 relationships for chemical reactions, 44-45 in a semiconductor, 580 thermodynamics in terms of, 29-84 variation in a junction region, 48 see also chemical potential electrochemical reaction, electrode, 2, 43, 44, 47, 52, 420, 423, 478 see also ideally polarizable electrode see also reference electrode alloy, 536 see also alloy intercalation, 535 see also intercalation semiconductor, 567-601 semiconductor, overview and applications of, 567-568 see also semiconductor solution, precipitation, 535 electrode, porous see porous electrode electrode of the second kind, 132,140, 151, 154 electrodes with films, 535 electrodialysis, 163 639 electrokinetic phenomena, 241-261, 349, 357 see also kinetics electrolyte, 2, 38, 47, 59, 99 chemical potential of an, 39 concentrated, 404-406 potential of cell with single, 43-53 potential of cell with two, 53-59 weak, 119-122 electrolyte, binary, 9,50,60,322,325,357, 545 see also solution of a single salt activity coefficient of, 110-112 analogy to semiconductors, 577-578 concentration overpotential in, 493-495 thermal effects, 318-319, 329-330 thermodynamics, 38-43, 49-53, 110-112 transport, concentrated, 299-300 transport, dilute, 277-280, 404-406, 410-411,482,503,506 electrolyte, indifferent, see electrolyte, supporting electrolyte, supporting, 18, 75, 163, 280-282, 401, 411, 480, 503, 556 concentration overpotential with, 495 concentration variation of, 465-471 paradoxes with, 476-480 reasons for use, 479 electrolytic cell, electrolytic solution, 45 electromotive force, 594 electron, 568, 575, 578, 583, 592-593 as species, 569-572, 589 electroneutrality, 3, 91, 113, 124, 182, 194, 277, 282, 288, 297, 308, 348, 373, 420, 460, 469, 473, 503, 575 accuracy of, 273 and Laplace's equation, 286-288 see also electric forces, magnitude of in a porous electrode, 521 electro-osmosis, 244, 248, 249 electrophoresis, 243, 254-256, 259, 263-266 electrophoretic velocity, 256,259,263,266 electroplating, 412, 425, 449-450 electropolishing, 134 electrostatic potential, see potential, electrostatic elementary step, 207, 211, 213 energy, 32 see also free energy see also internal energy see also thermal energy 640 INDEX energy efficiency, 24 energy gap, - , 579 see also gap energy energy states, - enthalpy, 62, 72, 591 and chemical potential, 33 of mixing, 115 partial molar, 116, 322 enthalpy potential, see potential, enthalpy entropy, 33, 62, 71-72, 180-181, 591 partial molar, 318, 324 equation of motion, 352, 360 see also Navier-Stokes equation equation of state, 77 equilibrium, see phase equilibrium equilibrium constant, 66-67, 120-121, 221,591 equilibrium potential, see potential, equilibrium equivalent conductance, 284, 285, 337 exchange current density, 8,206,212-214, 426, 450, 504, 506, 529, 533, 560 faradaic, 523 faradaic reaction, 203, 225 Faraday's constant, 3-4 Faraday's law, 3, 203, 374, 434, 554, 556 in a porous electrode, 521 Fermi energy, see Fermi level Fermi level, 569, 571-572, 574, 580 Fermi-Dirac distribution, 569, 571-573, 577 in solution, 593 ferricyanide-ferrocyanide, see redox couple finite-difference methods, 430-449 convergence over nonlinearities, 614-616 errors in, 612-614 fixed charge, 572, 575 flat band potential, 580-581, 583 flat plate, see plane electrode flow-by system, 553 flowing junction, 151 flow-through electrochemical reactors, 553-558 see also electrode, porous, flow-through design concerns and principles, 553-554 design equation, 555, 557 schematic, 557 fluctuations, 359-360, 364, 391-393 fluid mechanics, 255, 297, 347-368, 605 flux, 9, 271 flux density, 9, 11, 14-15, 271-272, 299, 347, 461, 554 of a dilute solute in a pore, 523 due to convection, 15 due to diffusion, 14, 271 due to migration, 11 net flux density, 15, 271-272 pore-wall, 519 superficial, 523 forced convection, see convection, forced forward-difference formula, 613 free and forced convection, see convection, free and forced free convection, see convection, free free diffusion, 153 free-diffusion junction, 151, 155-159 free energy, 32 electrical contribution to, 102-107 Gibbs, - , 33, 72, 96 Gibbs, for multicomponent dilute solutions, 113, 124-126 Gibbs, of formation, 71 Helmholtz, 33, 103 relationship to chemical potential, 104 standard Gibbs, 591 freezing-point depression, 39-40, 116 friction coefficient, 297 see also interaction coefficient Frumkin correction, 233, 593 fuel cell, 70-72, 517-518, 594 fugacity, 42, 62, 76, 150, 157, 431 fundamental equations, 373-374 gallium arsenide, 572-573, 578, 595-596 Galvani potential, see inner potential galvanic cell, galvanized steel, 446 galvanostat, gap energy, see energy gap gaseous species, 42, 150 Gauss's law, 87, 100, 574, 590, 609 generation, 578-580, 585, 587 Gibbs adsorption isotherm, 177-181 Gibbs free energy, see free energy Gibbs interface, 199 see also Gibbs surface Gibbs invariant, 178 Gibbs surface, 178 see also Gibbs interface Gibbs-Duhem equation, 37, 42, 51, 106, 126, 179-180,298,303 INDEX Goldman constant-field equation, 152 Graetz functions, 385 Graetz problem, 382-388, 390 Graetz problem, Lévêque solution, 386-388, 390, 396 Grashof number, 401, 403, 481-482 growing mercury drops, 399-400, 405-406, 461-462, 471, 471, 474, 476, 486 see also polarography see also reference electrode, mercurycontaining Guggenheim condition, 38 Giintelberg charging process, 104 half-cell, 3, 65 heat capacity, 71-72 heat conduction, 320-323, 421 heat conservation, 320-323 heat flux, 321,421 see also thermal flux heat generation, 72, 320-323 at an interface, 323-325, 326 irreversible, 526 reversible, 526 heat of transfer, 325, 332 heat transfer, 317, 320-323 Helmholtz free energy, see free energy Helmholtz plane, inner, 173,175, 209, 576, 585, 588-589 Helmholtz plane, outer, 173,175, 195, 209, 585, 589 Henderson formula, 152 heterogeneous reaction, 3, 203-205 Heyrovsky reaction, 217, 224 Heyrovsky-Volmer mechanism, 220-222, 225 Hittorf measurement, 60, 335, 339 hole, 568, 570, 575, 578, 583, 592-593 as species, 569-572 homogeneous reaction, 272-273, 472, 578-580 Hull cell, 452-453 hydrodynamic flow, 241 hydrogen electrode, see reference electrode see Heyrovsky reaction see Heyrovsky-Volmer mechanism see Tafel reaction see Volmer reaction see Volmer-Tafel mechanism hydrogen evolution, 230, 464, 510 hydrogen fluoride, 139 641 ideal-gas state, 76 ideally polarizable electrode, 144, 173-174, 175, 181-185, 243, 254, 263, 265, 576, 580, 583 see also Lippmann equation llkovic equation, 400, 462 illumination, 575, 579, 586 image point, 614 immobile charge, 581 impurities, 61, 133, 136, 141-142, 227, 466, 568 indifferent electrolyte, see electrolyte, supporting infinite dilution, 118, 577 inner potential, see potential, inner inner shell electrons, 568 insertion, 520 see also intercalation insulator, 420, 423, 501,568 interaction coefficient, 297 see also friction coefficient intercalation, 520 see also electrode, intercalation see also insertion interface dynamics, 263-264 interfacial force balance, 350-351 interfacial tension, 177-181, 199, 263-264, 350 see also surface tension interfacial velocity, 342, 612 intermolecular forces, 88-91 internal energy, 33 interrupter, 21, 229,492 inversion, 582-583, 585 iodate, reduction of, 405 ion cloud, 108 ionic distributions in dilute solutions, 99-102 from Debye-Hiickel theory, 102 ionic size, 100-102, 109, 187 ionic species, 36, 89 see also species ionic strength, 55, 105, 473 "true," 121,475 ionization level, 572 isopiestic measurement, 117 Joule heating, 322-323 junction region, 46, 49, 149-168, 310 between metals, 573 between semiconductors, 574 electrochemical potential gradient of an ion in a, 48, 51 642 INDEX junction region (Continued) potentials of cells with a, 149-168 semiconductor-electrolyte, 580 transport processes in, 47-49 types of, 151 see also liquid junction see also transition region kinetics, 7, 169, 203-239, 432 see also kinetics, electrode see also polarization equation anodic and cathodic rate, 8, 210 generalized interfacial, 588-592 linear, 132,205,425,531 of semiconductor electrodes, 592-593 Tafel, 8,10,205,425-426,454,50,508, 529-531 see also Tafel approximation see also Tafel plot see also Tafel slope Butler-Volmer, 7, 205 see also Butler-Volmer equation models for, 207-225 see also polarization kinetics, electrode, 374-375, 424-425, 532 see also kinetics laminar flow, 359, 397, 403 Laplace's equation, 87,254-256,265,277, 419-421, 425, 430, 438, 465, 491, 499-501 and electroneutrality, 286-288 Laplacian, 608 lead-acid cell, 304-305, 517, 535, 544-545, 594 see also battery lead sulfate, 65-66, 75, 144-145, 292, 496-497 Lévêque solution, see Graetz problem, Lévêque solution Lighthill transformation, 404, 462 limitations of surface reactions, 403-404 limiting current, see current, limiting Lippmann equation, 181-185, 186, 196, 265 liquid junction, cells with, 49, 56, 92, 119, 132-133, 146, 151, 154-159,166, 307 see also junction region see also potential, liquid-junction see also transition region liquid-junction photovoltaic cell, 584 see also solar cell liquid-junction potential, 50, 150, 154, 155-158, 157-165, 283, 310-311, 526 correction for in reference electrodes, 142-143 error due to neglect of, 132 formulas for, 151-152 see also potential of a cell with liquid junction lithium alloy, iron sulfide cell, 544, 545, 546-551 lithium cell, 521, 535, 539, 544 lithium-ion cell, 535, 544 macroscopic theory, 96,177, 204,227,243, 274,518-519 magnetic force, 88, 348 majority charge carrier, 583-584 Marangoni effect, 264 mass and momentum balances, 347-348 mass transfer, 280, 340-343 and current distribution, 399-601 in turbulent flow, 363-366 mass-transfer coefficient, 555 mass transfer rates, 231, 363-366, 377-406 see also current, limiting material balance, 272, 297, 300, 555 for a solute species at an interface, 522 Maxwell relations, 196 Maxwell's equations, 86 membranes, 142, 162-163, 312 mercuric oxide, 53, 74, 118, 140 mercurous salts, 137-140 mercury-containing electrodes, see reference electrodes microscopic theory, 96, 177, 204, 227, 274 migration, 9, 10-13, 271, 405-406, 419, 459-460, 463-464, 478, 545 effect on limiting currents, 459-490, 503 effect on limiting currents, program for, 623-627 in semiconductors, 578-580 migration velocity, 272 minority charge carrier, 579, 582 mixed potential, see potential, mixed mobility, 11,272,374 and diffusion coefficient, 283-285 moderately dilute solutions, 289-291 INDEX molal ionic strength, see ionic strength molar ionic strength, see ionic strength molality, 36, 300 molarity, 36 momentum density, 347 Mott-Schottky plot, 582, 583 moving boundary measurement, 60, 335 multicomponent diffusion, see multicomponent transport multicomponent diffusion equation, 297, 318 multicomponent solution, 338-340 activity coefficients in, 112-116 Gibbs free energy of, 113 multicomponent transport, 307-310 program for, 637-634 Navier-Stokes equation, 242, 254-255, 265, 349, 374 see also equation of motion «-doped semiconductor, 572 Nernst diffusion layer, 231, 287-288, 462, 476-479, 486 see also diffusion layer Nernst equation, 5, 57, 76, 78, 149-150, 163-164, 212, 216, 218, 221, 591 error in, 160-162 Nernst-Einstein relation, 25,283,289, 298, 305-307,311,467,494 nerve impulses, 163 Newton-Raphson method, 614-615 Newton's law of viscosity, 359 nickel/metal hydride cell, 520 nonaqueous solutions, 45, 55, 131, 137, 139, 142 numerical solution of differential equations, 611-634 see also finite-difference methods Nusselt number, 384, 390-393, 396, 399, 401, 409, 480 ohmic drop, 18, 20, 391, 404-405, 459, 479, 491-492, 499-503, 509, 517, 529, 538, 555-556 Ohm's law, 9, 12, 47-48, 275, 286, 320, 500, 523, 527, 530, 533, 554, 556 Onsager reciprocal relation, 245, 299 open-circuit potential, 5, 47, 79-80, 165, 204-205, 217, 230, 328, 446, 522, 535, 539-540, 587, 592, 594 and activity coefficient, 117-119 643 and current density, 48 see also potential, equilibrium osmotic coefficient, 41, 74 of binary solution, 110 from Debye-Hiickel theory, 107 measurement of, by vapor pressure, 117-118 of multicomponent solution, 115 of single electrolyte solution, 41-42 outer potential, see potential, outer outer shell electrons, 568 overpotential, 4, 432, 446, 509 overpotential, concentration, 18, 20, 404, 491-498,500,547 of a binary salt, 25 calculated values of, 495-496 definition of, 491-492 overpotential, surface, - , 9,10, 132, 147, 204-207, 212, 218, 324, 375, 404, 421, 425, 499-500, 522, 594 and Gibbs energy change, 96 sign of, 22, 204 overpotential, total, 216,404,501,504,510 oxidation, 6, 433 oxidizing agent, 431 oxygen, 133, 136, 138, 141-142,206, 230-233, 292 oxygen electrode, 63, 74, 206, 227-228, 230-233 partial molar volume, 69-70 of electrolyte in infinitely dilute solution, 123 relationship to density, 603-604 passivation, 206, 207, 234, 447-449 p-doped semiconductor, 572, 575 Péclet number, 383-384, 500-501, 554 Peltier coefficient, 324 Peltier heat, 329 penetration depth, 533-534, 557-558 penetration model, 460, 611 permittivity, 85-86, 286 perturbation, 281, 499-500, 634 pH, 67-68, 81, 163, 232, 431-434, 435, 436, 479 and hydrogen electrodes, 136-137 and reference electrodes, 133 phase equilibrium, 29-31, 43, 47, 49, 57-58 general mathematical description of, 51 photoresponse, 579 pill box, 87 644 INDEX pipe, 382-388, 396, 409 see also Graetz problem see also Poiseuille flow pitting corrosion, see corrosion, pitting plane electrode effect of migration, 460-465 forced convection, 389-392, 396-397, 409 free convection, 400-403, 405, 480-486 primary distribution, 421-423 secondary distribution, 428 stagnant solution, 460-465 point of zero charge, 175, 776, 196 see also potential of zero charge Poiseuille flow, 359, 382, 511 Poisson-Boltzmann equation, 108, 122 Poisson's equation, 86, 100, 109, 186, 242, 286-287, 574-575, 581 polarization, 4, 420, 424-428, 500, 534, 550 polarization equation, see kinetics in a porous electrode, 522, 528, 546, 554 polarography, 233, 406, 463 see also growing mercury drops pore-wall flux density, see flux density, pore-wall porosity, 518-519, 536-538, 543, 546-547, 550 porous electrode, 517-565 advantages of, 517-518 capacitive processes in, 522-523 continuity equation, 525 convective-diffusion equation, 525 double-layer charging and adsorption, 551-552 flow-through, 518, 553-558 see also flow-through electrochemical reactor macroscopic description of, 518-526 mass transfer, steady, 532-534 mass transfer, transient, 534 material balance for insertion electrodes, 520-521 material balance for solutes in, 520 nonuniform reaction rates, 527-532 structural changes, 536-537 thermal behavior of, 526 thickness and porosity optimization, 537 transport processes in, 523-524 potential, 4, 11, 60, 153, 204, 278, 302-305, 304, 420 see also overpotential of batteries, 537 cavity, 574, 592, 595 see also potential, outer cell, see cell potential chemical, see chemical potential of a cell with liquid junction, 50-59 of cells with junction regions, 50-59, 149-168 corrosion, 230 cutoff, 538, 541 diffusion, 18, 20, 274-276, 278, 282, 471,487,492,496,533 due to central ion, from Debye-Hiickel theory, 101 electric, 30, 32-34, 85-97, 320, 323 electrochemical, see electrochemical potential electrostatic, 85-88, 165, 167, 177, 577, 595 electrostatic, measurability of, 94 enthalpy, 71,329 equilibrium, 5, 147, 204-205, 212, 230, 433, 548 see also open-circuit potential equilibrium, and reference electrodes, 132 Galvani, see potential, inner half-wave, 233 inner, 91-92 liquid-junction, see liquid-junction potential of mean force, 108 mixed, 230 outer, 91-92, 143 in a porous electrode, 525, 531 proper definition of, 94, 591-592 quasi-electrostatic, 95, 150, 152, 157, 196,289,304,311,594 reference electrode, 92-93, 131, 142-147,302-305 relative to a given reference electrode, 142-147 sedimentation, 256-257, 259, 263, 266-267 standard cell, 55, 60-65, 74-76, 79, 118, 150, 163,219 standard cell, and activity coefficients, 59-68 standard electrode, 63, 64, 77, 81, 145, 164 streaming, 244, 250, 252, 259 surface, 92 thermal neutral, 72, 329 thermodynamic, see potential, equilibrium INDEX Volta, see potential, cavity of zero charge, 175, 176, 184, 197 zeta, 242, 243, 251,256 potential distribution, 286, 419-449, 531-532 across ideally polarizable electrodes, 175 across phases, 594-596 metal-metal, 573 n-p semiconductor, 574, 594-596 semiconductor under illumination, 583, 595-596 potential theory, applications of, 419-458 potential-theory problem, 370 simplifications for, 420-421 potentiostat, Pourbaix diagram, 67-68,68,81,431,432, 433-434 pressure, 348 effective pressure of hydrogen gas, 137 pressure, dynamic, 348, 356 pressure, hydrostatic, 348 pressure, thermodynamic, 348 primary current distribution, 229, 399, 419-424, 427-429 see also current distribution see also primary potential distribution primary potential distribution, 419, 421-424,426 see also primary current distribution primary reference state, 35, 39, 591 probability of occupancy, 571 pulsed plating, 412 quantum-mechanical energy states, 568, 585 quasielectrostatic potential, see potential, quasielectrostatic quasi-Fermi level, 577, 596 Ragone plot, 542, 544 Raman spectra, 120 rate constant, 210, 222, 591 reaction, 31, 51, 213 see also elementary step see also simple reaction reaction order, 210, 213 reaction rate, see kinetics see rate constant reaction zone, 536-537, 538, 542, 547 reactions, simultaneous, 67-68, 230-233 recombination, 578, 587 rectifier, 436, 438, 440-441, 443, 445 645 redox couple Fermi-level, 523 ferricyanide-ferrocyanide, 64, 284, 465, 466-467, 484, 485, 593-594, 596 semiconductor kinetics, 592-593 redox reaction, 209, 464-471, 482-486, 497,510 reduction, reference electrode, 6,23,43, 51,131-148, 204,302,420-421,491-492 calomel, 137-140, 164, 196 calomel, role of dissolved oxygen in, 138 copper/copper sulfate, 434, 436, 437 criteria for, 131-133 experimental factors affecting selection of, 133-134 of a given kind, 234, 525 hydrogen electrode, 53,62,68,134-137, 166, 217-218 impurities in, 132 liquid-junction potentials and, 133 mercury-containing, 53, 74, 118, 137-140, 171, 264, 305, 341-342 see also growing mercury drops see also reference electrode, calomel non-idealities in, 131-134 potential of, 92-93, 302 potentials relative to, 142-147 of the second kind, 132, 140, 151, 153 silver-silver halide, 53, 140-142, 166 reference state, 33, 55, 61 see also primary reference state see also secondary reference state resistance, 423 kinetic, 22 ohmic, 12-13 resistivity, resistivity, volume-average, 518 restricted diffusion, 156 restricted-diffusion junction, 151,155-158 reversible heat, 324-325 Reynolds number, 355,383,390,392-393, 397, 398, 399, 403, 554 Reynolds stress, 360-362 see also turbulent momentum flux rotating cylinder, see cylinder, rotating rotating disk electrode, 341-342, 351-355, 378-382, 395, 405-406, 410-411, 460, 463-464, 467, 474, 475,476,496,506-510 see also disk electrode dimensionless mass-transfer rates, 381 rotating ring-disk electrode, 450 roundoff error, 614 646 INDEX sacrificial anode, see anode, sacrificial Schmidt number, 342, 358, 378, 382, 390, 393, 396, 399, 401-402, 409, 460, 481, 502, 554 Schwarz-Christoffel transformation, 421, 423 second virial coefficient, 77 secondary current distribution, 424-430 see also current distribution secondary reference state, 35, 61, 69, 76, 118,577 for an alloy, 42-43 relationship between molar and molal, 37, 145 for a single electrolyte, 38-40 for a species in solution, 37, 61, 69 sedimentation potential, see potential, sedimentation Seebeck coefficient, 324 semiconductor, 568-569, 573 analogy to binary electrolyte, 577-578 nature of, 568-580 semiconductor electrodes, see electrodes, semiconductor shear stress, 264, 481 short-range specific interaction force, 89 side reaction, 204-205, 535, 554, 556, 558 similarity transformation, 153 simple reaction, 207-208, 214-217 simultaneous reactions, see reactions, simultaneous slip velocity, 241-243, 242, 257 sodium/sulfur cell, 545 solar cell, 567, 583-588 solid electrolyte interphase, 535 solubility product, 54, 56, 67, 76, 154 solution of a single salt, 335-338 see also electrolyte, binary Soret coefficient, 319-320, 322 Soret coefficient, practical, 326 Soret effect, 318 space charge due to surface states, 576 modified by polarization, 576 in a solar cell, 585 sparingly soluble salt, 45, 53-54, 57 species, 2, 30, 569 chemical potential of a, 114 electrochemical potential of a, 95 electrons and holes as, 569-572 ionic, 39 specific adsorption, 195 see also Gibbs adsorption isotherm specific energy, 539-540, 541 specific interactions, 177 see also specific adsorption specific interfacial area, 519 specific power, 544 sphere, 395, 399-402 stability, hydrodynamic, 359 standard cell potential, see potential, standard cell standard electrode potential, see potential, standard electrode Stanton number, 364 Stefan-Maxwell equation, 299 stoichiometric coefficient, 4, 31, 51 Stokes-Einstein relationship, 285 Stokes's law, 267 Stokes's theorem, 609 streaming current, 251-252, 259 streaming potential, see potential, streaming stress, 348 see also Reynolds stress see also shear stress boundary conditions on, 349-351 in a Newtonian fluid, 349 Sturm-Liouville system, 385 sulfate electrodes, 65-66, 75, 139, 144, 292, 304, 305, 497 sulfur compounds, 136, 141 superficial concentration, see concentration, superficial supporting electrolyte, see electrolyte, supporting surface charge density, 87, 182-183, 227, 522 surface concentration, 178, 184, 196, 466, 467, 468, 475, 476, 478, 484, 505, 507, 522 see also surface excess surface conductivity, see conductivity surface current density, see current density, surface surface excess, 197, 522 see also surface concentration surface state, 569, 585, 588, 589 surface tension, 171, 178-179, 181,184, 199, 264 see also interfacial tension surface transport properties, 253 surface treatment, 133-134 surface viscosity, 264 symmetry factor, 210, 593 INDEX Tafel approximation, 529, 532 Tafel kinetics, see kinetics, Tafel Tafel plot, 8,10, 214, 219, 220, 223, 531 Tafel reaction, 217, 224 Tafel slope, 8, 219, 233, 531, 533, 593 double, 531,533, 560 Taylor vortices, 397, 397, 398 tensors, see vectors and tensors thermal conductivity, 321-322, 421 thermal diffusion, see diffusion, thermal thermal energy balance, 323 in a porous electrode, 526 thermal flux, 323 see also heat flux thermal generation-recombination, 578 thermocouple, 330-331 thermodynamic data, 454 thermodynamic functions, definition of, 35-43 thermodynamic measurement, see activity coefficient, measurement of thermodynamics, 4, 32-33, 35, 591 electric potential in, 94-96 of interfaces, 177-181 in terms of electrochemical potential, 29-83, 149 thermoelectric coefficient, 327 thermoelectric effect, 319-320, 327 thermogalvanic cells, 326-329 time of discharge, 538-540, 543 tortuosity, 523-524 transfer coefficient, see apparent transfer coefficient transference, 43 cell with, 49 see also concentration cell cell without, 29, 43-47 transference number, 13, 46-47, 50, 52, 54-55, 60, 157, 163, 274-276, 335, 478, 506 binary electrolyte, 14, 279, 283, 300 dependence on reference velocity, 301, 311-312 multicomponent, 308-309 transition region, 46, 49, 54, 57, 161 see also liquid junction see also junction region transpassive region, 206, 207 transport, 8, 18, 37, 47, 49, 60, 269-368 transport laws, dilute, 271-274, 373-374 transport number, see transference number transport properties, 298-299, 335-345, 406 see also conductivity 647 see also diffusion coefficient see also diffusivity see also mobility see also Soret coefficient see also surface transport properties see also transference number see also viscosity trap state, 569, 576, 579, 590 tridiagonal matrices, 153, 615-616 turbulent flow, 358-363, 391, 397, 403 turbulent momentum flux, 359 see also Reynolds stress turbulent natural convection, 401 universal velocity profile, 361 vacant sites, 590 valence, 103 see also charge number valence band, see band, valence vapor pressure, 39, 60, 65, 72, 117 varying concentration, cell with, 49-59 vectors and tensors, 605-609 velocity fluctuation, 360 velocity, mass-average, 300-301 velocity, molar-average, 300-301 velocity, reference, 300-301 velocity, slip, 243 velocity, volume-average, 301, 524-525 viscosity, 243, 285, 338, 349, 607, 612 kinematic, 349 viscous dissipation, 321, 323 viscous momentum flux, 359 Volmer reaction, 217-218, 224 Volmer-Tafel mechanism, 218-220, 225 Voltapile, 1-2 Volta potential, see potential, outer volumetric flow rate, 244, 252 von Kârmdn transformation, 353, 357 vorticity, 606 Wagner number, 428 wedge effect, 138 work function, 573-575 work to move charge, 33, 103 working electrode, 43, 204, 421 zero charge, point of, 175, 183 see also potential of zero charge zeta potential, see potental, zeta