)(

c z w a.. a.. <C 1 BIBLIOGRAPHY

A. Introductory Electrochemistry Texts

Al C. N. Reilley and R. W. Murray, Electroanalytical Principles, New York: Interscience, 1963.

A2 E. H. Lyons, Jr., Introduction to Electrochemistry, Boston: D. C. Heath, 1967.

A3 B. B. Damaskin, The Principles of Current Methods for the Study of Electrochemical Reactions, New York: McGraw-Hill, 1967.

A4 J. B. Headridge, Electrochemical Techniques for Inorganic Chemists, London: Academic Press, 1969. .

A5 J. Robbins, Ions in Solution (2): An Introduction to Electrochemistry, London: Oxford University Press, 1972.

A6 W. J. Albery, Electrode Kinetics, London: Oxford University Press, lW5.

A7 N. J. Selley, Experimental Approach to Electrochemistry, New York: John Wiley, 1976.

A8 D. R. Crow, Principles and Applications of Electrochemistry, 2nd ed, London: Chapman and Hall, 1979.

A9 D. Hibbert and A. M. James, Dictionary of Electrochemistry, 2nd ed, New York: John Wiley, 1985.

AlO T. Riley and C. Tomlinson, Principles of Electroanalytical Methods, New York: John Wiley, 1987.

All J. Koryta, Ions, Electrodes and Membranes, 2nd ed, New York: John Wiley, 1991.

B. More Advanced Texts

Bl P. Delahay, New Instrumental Methods in Electrochemistry, New York: Interscience, 1954.

B2 H. S. Hamed and B. B. Owen, Physical Chemistry of Electrolytic Solutions, New York: Reinhold,1958.

B'3 G. Charlot, J. Badoz-Lambling, and B. Tremillon, Electrochemical Reactions, Amsterdam: Elsevier, 1962.

427

428 Appendix 1

B4 D. A. MacInnes, The Principles of Electrochemistry, New York: Dover, 1966 (corrected reprint of 1947 edition).

B5 G. Kortum, Treatise on Electrochemistry, 2nd ed, Amsterdam: Elsevier, 1965.

136 B. E. Conway, Theory and Principles of Electrode Processes, New York: Ronald Press, 1965.

B7 K. J. Vetter, Electrochemical Kinetics, New York: Academic Press, 1967.

B8 J. O'M. Bockris and A. Reddy, Modern Electrochemistry, New York: Plenum Press, 1970.

139 R. A. Robinson and R. H. Stokes, Electrolyte Solutions, 2nd ed (rev), London: Butterworths, 1970.

B10 J. S. Newman, Electrochemical Systems, Englewood Cliffs, NJ: Prentice-Hall,1972.

Bll L. I. Antropov, Theoretical Electrochemistry, Moscow: Mir, 1972. B12 A. J. Bard and L. R. Faulkner, Electrochemical Methods, New

York: John Wiley, 1980. B13 R. Greef, R. Peat, L. M. Peter, D. Pletcher, and J. Robinson,

Instrumental Methods in Electrochemistry, Chicester: Ellis Horwood, 1985.

B14 J. Koryta and J. Dvorak, Principles of Electrochemistry, New York: John Wiley, 1987.

c. Specialized Books and Monographs

Cl W. Ostwald, Electrochemistry, History and Theory, Leipzig: Veit, 1896. Republished in English translation for the Smithsonian Institution, New Delhi: Amerind Publishing, 1980.

C2 V. G. Levich, Physiochemical Hydrodynamics, Englewood Cliffs, NJ: Prentice-Hall, 1962.

C3 R. N. Adams, Electrochemistry at Solid Electrodes, New York: Marcel Dekker, 1969.

C4 C. K. Mann and K. K. Barnes, Electrochemical Reactions in Nonaqueous Systems, New York: Marcel Dekker, 1970.

C5 A. Weissberger and B. W. Rossiter, eds, Physical Methods of Chemistry, Vol. 1 (Techniques of Chemistry), Parts IIA and lIB (Electrochemical Methods), New York: John Wiley, 1971.

C6 J. S. Mattson, H. B. Mark, Jr., and H. C. MacDonald, Jr., Electrochemistry; Calculations, Simulation, and Instrumentation (Computers in Chemistry and Instrumentation, Vol. 2), New York: Marcel Dekker, 1972.

C7 Yu. V. Pleskov and V. Yu. Filinovskii, The Rotating Disc Electrode, New York: Consultants Bureau, 1976.

CB J. O'M. Bockris and S. U. M. Khan, Quantum Electrochemistry, New York: Plenum Press, 1979.

Bibliography 429

C9 A. J. Fry and W. E. Britton, eds, Topics in Organic Electrochemistry, New York: Plenum Press, 1986.

C10 R. J. Gale, ed, Spectroelectrochemistry, Theory and Practice, New York: Plenum Press, 1988.

Cll H. D. Abrwia, ed, Electrochemical Interfaces: Modern Techniques for In-Situ Interface Characterization, New York: VCH Publishers, 1991.

C12 J. O'M. Bockris and S. U. M. Khan, Surface Electrochemistry, New York: Plenum Press, 1993.

D. Electroanalytical Methods

D1 1. M. Kolthoff and J. J. Lingane, Polarography, 2nd ed, New York: Interscience, 1952.

D2 J. J. Lingane, Electroanalytical Chemistry, 2nd ed, New York: Interscience, 1958.

D3 W. C. Purdy, Electroanalytical Methods in Biochemistry, New York: McGraw-Hill,1965.

D4 L. Meites, Polarographic Techniques, 2nd ed, New York: John Wiley, 1965.

D5 J. Heyrovsky and J. Kuta, Principles of Polarography, New York: Academic Press, 1966.

D6 H. Rossotti, Chemical Applications of Potentiometry, Princeton, NJ: Van Nostrand, 1969.

D7 R. G. Bates, Determination of pH: Theory and Practice, 2nd ed, New York: John Wiley, 1973.

DB Z. Galus, Fundamentals of Electrochemical Analysis, Chichester: Ellis Harwood, 1976.

D9 G. Dryhurst, Electrochemistry of Biological Molecules, New York: Academic Press, 1977.

D10 C. C. Westcott, pH Measurements, New York: Academic Press, 1978.

Dll J. Vesely, D. Weiss, and K. Stulik, Analysis with Ion-Selective Electrodes, Chichester: Ellis Horwood, 1978.

D12 A. M. Bond, Modern Polarographic Methods in Analytical Chemistry, New York: Marcel Dekker, 1980.

D13 J. A. Plambeck, Electroanalytical Chemistry, New York: John Wiley, 1982.

D14 J. Koryta and K. Stulik, Ion-Selective Electrodes, 2nd ed, London: Cambridge University Press, 1983.

E. Organic Electrosynthesis

El M. R. Rifi and F. H. Covitz, Introduction to Organic Electrochemistry, New York: Marcel Dekker, 1974.

430 Appendix 1

E2 N. L. Weinberg, ed, Technique of Electroorganic Synthesis (Technique of Chemistry, Vol. V), New York: John Wiley, 1974 (Part 1), 1975 (Part Il).

E3 D. K. Kyriacou, Basics of Electroorganic Synthesis, New York: John Wiley, 1981.

FA K. Yoshida, Electrooxidation in Organic Chemistry, New York: John Wiley, 1984.

E5 T. Shono, Electroorganic Chemistry as a New Tool in Organic Synthesis, Berlin: Springer-Verlag, 1984.

E6 A. J. Fry, Synthetic Organic Electrochemistry, 2nd ed, New York: John Wiley, 1989.

E7 H. Lund and M. M. Baizer, eds, Organic Electrochemistry, 3rd ed, New York: Marcel Dekker, 1991.

E8 T. Shono, Electroorganic Synthesis, San Diego: Academic Press, 1991.

F. Experimental Methods

F1 D. J. G. Ives and G. J. Janz, eds, Reference Electrodes, Theory and Practice, New York: Academic Press, 1961.

F2 W. J. Albery and M. L. Hitchman, Ring-Disc Electrodes, Oxford: Clarendon Press, 1971.

F3 D. T. Sawyer and J. L. Roberts, Jr., Experimental Electrochemistry for Chemists, New York: John Wiley, 1974.

F4 E. Gileadi, E. Kirowa-Eisner, and J. Penciner, Interfacial Electrochemistry--An Experimental Approach, Reading, MA: Addison-Wesley, 1975.

F5 D. D. MacDonald, Transient Techniques in Electrochemistry, New York: Plenum Press, 1977.

F6 P. T. Kissinger and W. R. Heineman, eds, Laboratory Techniques in Electroanalytical Chemistry, New York: Marcel Dekker, 1984.

G. Technological Applications of Electrochemistry

G1 J. O'M. Bockris and S. Srinivasan, Fuel Cells: Their Electrochemistry, New York: McGraw-Hill,1969.

G2 C. L. Mantell, Batteries and Energy Systems, New York: McGraw­Hill,1970.

G3 F. A. Lowenheim, ed, Modern Electroplating, New York: John Wiley, 1974.

G4 S. W. Angrist, Direct Energy Conversion, Boston: Allyn and Bacon, 1976.

G5 V. S. Bagotzky and A. M. Skundin, Chemical Power Sources, New York: Academic Press, 1980.

G6 D. Pletcher, Industrial Electrochemistry, London: Chapman and Hall,1982.

Bibliography 431

G7 N. L. Weinberg and B. V. Tilak, eds, Technique of Electroorganic Synthesis (Technique of Chemistry, Vol. V), Part III, New York: John Wiley, 1982.

G8 H. H. Uhlig and R. W. Revie, Corrosion and Corrosion Control, New York: John Wiley, 1984.

G9 H. V. Ventatasetty, ed, Lithium Battery Technology, New York: John Wiley, 1984.

G10 R. E. White, ed, Electrochemical Cell Design, New York: Plenum Press, 1984.

Gll Z. Nagy, Electrochemical Synthesis of Inorganic Compounds, New York: Plenum Press, 1985.

H. Electrochemical Data

HI W. M. Latimer, Oxidation Potentials, 2nd ed, Englewood Cliffs, NJ: Prentice-Hall, 1952.

H2 B. E. Conway, Electrochemical Data, Amsterdam: Elsevier, 1952. H3 R. Parsons, Handbook of Electrochemical Data, London:

Butterworths, 1959. H4 A. J. de Bethune and N. A. S. Loud, Standard Aqueous Electrode

Potentials and Temperature Coefficients at 25°C, Skokie, IL: Hampel, 1964.

H5 M. Pourbaix, Atlas of Electrochemical Equilibria, New York: Pergamon Press, 1966.

H6 G. J. Janz and R. P. T. Tomkins, Nonaqueous Electrolytes Handbook, New York: Academic Press, 1972.

H7 A. J. Bard and H. Lund, eds, The Encyclopedia of the Electrochemistry of the Elements, New York: Marcel Dekker, 1973.

H8 L. Meites and P. Zuman, Electrochemical Data. Part I. Organic, Organometallic, and Biochemical Systems, New York: John Wiley, 1974.

H9 D. Dobos, Electrochemical Data, Amsterdam: Elsevier, 1975. H10 G. Milazzo and S. Caroli, Tables of Standard Electrode Potentials,

New York: John Wiley, 1977. Hll L. Meites and P. Zuman, eds, CRC Handbook Series in Organic

Electrochemistry, Boca Raton, FL: CRC Press, 1977-. H12 L. Meites, P. Zuman, E. B. Rupp and A. Narayanan, eds, CRC

Handbook Series in Inorganic Electrochemistry, Boca Raton, FL: CRC Press, 1980-.

H13 A. J. Bard, R. Parsons, and J. Jordan, eds, Standard Potentials in Aqueous Solution, New York: Marcel Dekker, 1985.

H14 A. J. Bard, R. Parsons, and J. Jordan, eds, Oxidation-Reduction Potentials in Aqueous Solution, Oxford: Blackwell, 1986.

432 Appendix 1

1. Review Series

11 J. O'M. Bockris and B. E. Conway, eds, Modern Aspects of Electrochemistry, New York: Plenum Press, from 1954.

12 P. Delahay (Vols. 1-9), C. W. Tobias (Vols. 1- ), and H. Gerischer (Vols. 10- ), Advances in Electrochemistry and Electrochemical Engineering, New York: John Wiley, from 1961.

13 A. J. Bard, ed, Electroanalytical Chemistry, New York: Marcel Dekker, from 1966.

14 E. B. Yeager and A. J. Salkind, eds, Techniques of Electrochemistry, New York: John Wiley, from 1972.

15 G. J. Hills (Vols. 1-3) and H. R. Thirsk (Vol. 4- ), Senior Reporters, Electrochemistry, A Specialist Periodical Report, London: Royal Society of Chemistry, from 1971.

16 Analytical Chemistry, Fundamental Annual Reviews (April issue of even-numbered years), Washington: American Chemical Society.

>< -c z w a.. a.. til( 2 SYMBOLS AND UNITS

Table A.t The International System (SI) of Units

Physical Quantity Unit Symbol

Fundamental units: Length meter m mass kilogram kg time second s electric current ampere A temperature kelvin K amount of substance mole mol luminous intensity candela cd Derived units: force newton N (kg m s-2)

energy joule J (Nm)

power watt W (J s-l)

pressure pascal Pa (N m-2)

electric charge coulomb C (A s)

electric potential volt V (J C-l)

electric resistaunce ohm n (V A-l)

electric conductance siemens S (A V-I)

electric capacitance farad F (C V-I)

frequency hertz Hz (s-l)

433

434 Appendix 2

Table A.2 Values of Physical Constants

Constant Symbol Value

pennittivity of free space €() 8.8541878 x 10-12 C2J-1m-1

electronic charge e 1.602189 x 10-19 C

Avogadro's number NA 6.02204 x 1023 mol-1

Faraday constant F 96484.6 C mol-l

gas constant R 8.3144 J moI-lK-1

Boltzmann constant k 1.38066 x 10-23 J K-l

Planck constant h 6.62618 x 10-34 J s

gravitational acceleration g 9.80665 m s-2

Table A.3 List of Symbols

Symbol Name Units

a activity none

a radius m A area m2

Ci molar concentration, species i mol L-1 (M) mol m-3 (mM)

C differential capacity F m-2

d density kgm-3

D· I diffusion coefficient, species i m2s-1

E energy J E cell potential V

E1I2 half-wave potential V E electric field strength V m- l

Ii frictional coefficient, species i kg s-1

F force N

G Gibbs free energy J mol- l

H enthalpy J mol- l

electric current A

Symbols and Units 435

Symbol Name Units

iD diffusion-limited current A iL limiting current A 1 ionic strength mol L-1

1 a.c. current amplitude A ID diffusion current constant rnA mM-1

(mg s-1)-213s -1I6

j volume flux m3s-1

j current density A m-2

J molar flux density mol m-2s-1

k rate constant variable

kD mass-transport rate constant m s-l

kij potentiometric selectivity coefficient none K equilibrium constant none L length m m mass kg mi molal concentration, species i mol kg-I

M molecular weight g mol-1

n number of moles mol ni kinetic order, species i none N number of molecules none p pressure bar (105 Pa) q heat J Q electric charge C

r,R radial distance m

ro microelectrode radius m R resistance n S entropy J mol-1K-l

t time s ti transference number, species i none T temperature K

436 Appendix 2

Table A.3 List of Symbols (continued)

Symbol Name Units

Ui mobility, species i m2V-1s-1

U mass flow rate kg s-l

U internal energy J mol-1

v velocity m s-l

V volume m3

v potential scan rate V s-l

w work J

x distance m

XA ion atmosphere thickness m (Debye length)

XD diffusion layer thickness m XH hydrodynamic distance parameter m XR reaction layer thickness m Xi mole fraction, species i none Zi charge, species i none Z impedance n (l cathodic transfer coefficient none (l degree of dissociation none

C1.i electrokinetic coefficient variable (l,~,y,o stoichiometric coefficients none (l,~ phase labels none

~ anodic transfer coefficient none y surface tension Nm-1

'W activity coefficient, species i none (molar scale)

Yim activity coefficient, species i none (molal scale)

y,.x activity coefficient, species i none (mole fraction scale)

Y± mean ionic activity coefficient none

Symbols and Units 437

Symbol Name Units

£ dielectric constant none

Tl coefficient of viscosity kg m-1s-1 (Pa-s)

Tl overpotential V l'} polar angle rad e exp[F(E - EO)IRTJ none 1C conductivity Sm-1

A reaction zone parameter none A molar conductivity S m2mol-1

Il chemical potential J mol-1

v kinematic viscosity m2s-1

Vi moles of ion i per mole of salt none Vi stoichiometric coefficient, species i none

X (DoIDR) 112 none p resistivity Om p space charge density C m-3

a surface charge density C m-2

't characteristic time s

4> current efficiency none q> azimuthal angle, phase angle rad <l> electric potential V CJ) angular frequency rad s-1

~ zeta potential V

ELECTROCHEMICAL DATA

Table A.4 Standard Reduction Potentials at 25°C.

Main group elements:

Half-Cell Reaction

2 H + + 2 e- -+ H2(g)

Li+ + e- -+ Li(s)

Na+ + e- -+ Na(s)

K+ + e- -+ K(s) Rb+ + e- -+ Rb(s)

Cs+ + e- -+ Cs(s)

Be2+ + 2 e- -+ Be(s)

Mg2+ + 2e- -+ Mg(s)

Ca2+ + 2 e- -+ Ca(s)

Ba2+ + 2 e- -+ Ba(s)

Al3+ + 3 e- -+ Al(s)

C02(g) + 2 H+ + 2 e- -+ CO(g) + H20

C02(g) + 2 H+ + 2 e- -+ HCOOH

2 C02<g) + 2 H+ + 2 e- -+ H2C204

Pb2+ + 2 e- -+ Pb(s)

Pb02(S) + 4 H+ + 2 e- -+ Pb2+ + 2 H20

N03-+3H++2e- -+ HN02+H20

N03- + 4 H+ + 3 e- -+ NO(g) + 2 H20

N03- + 10 H+ + 8 e- -+ NH4+ + 3 H20

PO(OH)3 + 2 H+ + 2 e- -+ HPO(OHh + H20

HPO(OH)2 + 2 H+ + 2 e- -+ H2PO(OH) + H20

438

0.0000

-8.045 -2.714 -2.925 -2.925 -2.923 -1.97

-2.356 -2.84 -2.92

-1.67

~.106

~.199

~.475

~.1251

1.468

0.94

0.96

0.875 ~.276

~.499

Electrochemical Data 439

Half-Cell Reaction EON

HPO(OH)2 + 3 H+ + 3 e- ~ P(s) + 3 H2O -0.454 P(s)+3H++3e' ~ PH3 -0.111 AsO(OH)s + 2 H+ + 2 e- ~ As(OH)a + H2O 0.560 As(OH)a + 3 H+ + 3 e- ~ As(s) + 3 H2O 0.240 As(s) + 3 H + + 3 e- ~ AsHS(g) -0.225 02(g) + H+ + e- ~ H02 -0.125 02(g) + 2 H+ + 2 e- ~ H202 0.695 H202 + H+ + e- ~ HO· + H2O 0.714 H202 + 2 H+ + 2 e- ~ 2 H2O 1.763 &.2082- + 2 e- ~ 2 8042- 1.96 8042- + H20 + 2 e- ~ 8Os2- + 2 OH- -0.94 28042- +4H++ 2 e- ~ 82062- + 2 H20 -0.25 2 802(aq) + 2 H+ + 4 e- ~ 820S2- + H2O 0.40 802(aq) + 4 H+ + 4 e- ~ 8(s) + 2 H2O 0.50 84062- + 2 e- ~ 2 &.20a2- 0.08 8(s) + 2 H+ + 2 e- ~ H2S(aq) 0.14 F2(g) + 2 e- ~ 2 F- 2.866 CI04- + 2 H+ + 2 e- ~ CIOs- + H2O 1.201 CIOs- + 3 H+ + 2 e- ~ HCI02 + H2O 1.181 CIOs- + 2 H+ + e- ~ CI02 + H2O 1.175 HCI02+2H++2e- ~ HOCI+H20 1.701 2 HOCI + 2 H+ + 2 e- ~ CI2(g) + 2 H2O 1.630 CI2(g) + 2 e- ~ 2 CI- 1.35828 CI2(aq) + 2 e- ~ 2 CI- 1.396 Br04- + 2 H+ + 2 e- ~ BrOs- + H2O 1.853 2 BrOs- + 12 H+ + 10 e- ~ Br2(l) + 6 H2O 1.478 2 HOBr + 2 H+ + 2 e- ~ Br2(l) + 2 H2O 1.004 Br2(l) + 2 e- ~ 2 Br 1.0652 Br2(aq) + 2 e- ~ 2 Br 1.0874 10(OH)5 + H+ + e- ~ 103- + 3 H2O 1.60 2 lOs- + 12 H+ + 10 e- ~ 12(S) + 6 H2O 1.20

440 Appendix 3

Table A.4 Standard Reduction Potentials at 25°C (continued)

Main group elements:

Half-Cell Reaction

2HOI+2H++2e- ~ lis)+2H20

lis) + 2 e- ~ 2 l-

Is- + 2 e- ~ 31-liaq)+2e- ~ 21-

Transition and post·transition elements:

Half-Cell Reaction

V02+ + 2 H+ + e- ~ V02+ + H20

V02+ + 2 H+ + e- ~ \13+ + H20

\13+ + e- ~ V2+

V2+ + 2 e- ~ V(s)

Cl"2072- + 14 H+ + 6 e- ~ 2 Cr3+ + 7 H~ Cr3+ + e- ~ Cr2+

Cr2+ + 2 e- ~ Cr(s)

Mn04- + e- ~ Mn042-

Mn04- + 8 H+ + 5 e- ~ Mn2+ + 4 H20

Mn02(S) + 4 H+ + 2 e- ~ Mn2+ + 2 H20

Mn3+ + e- ~ Mn2+

Mn2+ + 2 e- ~ Mn(s)

Fe3+ + e- ~ Fe2+

Fe(phen)3+ + e- ~ Fe(phen)2+

Fe(CN)63- + e- ~ Fe(CN)64-

Fe(CN)64- + 2 e- ~ Fe(s) + 6 CN­

Fe2+ + 2e- ~ Fe(s)

Co3+ + e- ~ Co2+

CO(NH3)63+ + e- ~ CO(NH3)62+

Co(phen)33+ + e· ~ Co(phen)32+

Co(C204)33- + e- ~ CO(C204)34-

1.44

0.5355 0.536 0.621

1.000 0.337

-0.255 -1.13

1.38 -0.424 -0.90 0.56 1.51

1.23

1.5

-1.18 0.711

1.13 0.361

-1.16 -0.44 1.92

0.058 0.327 0.57

Electrochemical Data

Half-Cell Reaction

C02+ + 2 e- --+ Co(s)

NiOis) + 4 H+ + 2 e- --+ Ni2+ + 2 H20

Ni2+ + 2 e- --+ Ni(s)

Ni(OH>2(s) + 2 e- --+ Ni(s) + 2 OH­

Cu2+ + e- --+ Cu+

CuCI(s) + e- --+ Cu(s) + CI­

Cu2+ + 2 e- --+ Cu(s)

Cu(NHa)42+ + 2 e- --+ Cu(s) + 4 NHa

Ag2+ + e- --+ Ag+

Ag+ + e- --+ Ag(s)

AgCI(s) + e- --+ Ag(s) + CI­

Zn2+ + 2 e- --+ Zn(s)

Zn(OH)42- + 2 e- --+ Zn(s) + 4 OH­

Cd2+ + 2 e- --+ Cd(s)

2 Hg2+ + 2 e- --+ Hg22+

Hg22+ + 2 e- --+ 2 Hg(l)

Hg2CI2(S) + 2 e- --+ 2 Rg(l) + 2 CI-

Data from Bard, Parsons, and Jordan (RI3).

....().2:l7

1.593

....().257

....().72

0.159

0.121

0.340 ....().oo 1.980

0.7991

02223 ....().7626

-1.285

""().4025

0.9110

0.7960

0.26816

441

442

Table A.S Biochemical Reduction Potentials

Half-Cell Reaction

Reduction of a carboxyl group to an aldehyde: 1,3-diphosphoglycerate + 2 e-

-+ 3-phosphoglyceraldehyde + HP042-acetyl-CoA + 2 H+ + 2 e- ~ acetaldehyde + coenzyme A oxalate + 3 H+ + 2 e- ~ glyoxal ate

gluconate + 3 H+ + 2 e- ~ glucose acetate + 3 H + + 2 e- ~ acetaldehyde

Reduction of a carbonyl group to an alcohol: dehydroascorbic acid + H+ + 2 e- ~ ascorbate glyoxylate + 2 H + + 2 e- ~ glycolate hydroxypyruvate + 2 H + + 2 e- ~ glycerate

oxaloacetate + 2 H + + 2 e- ~ malate pyruvate + 2 H+ + 2 e- ~ lactate acetaldehyde + 2 H + + 2 e- ~ ethanol acetoacetate + 2 H+ 2 e- ~ ~-hydroxybutyrate

Carboxylation: pyruvate + C02 (g) + H+ + 2 e- ~ malate a-ketoglutarate + C02(g) + H+ + 2 e- ~ iso-citrate succinate + C02(g) + 2 H+ + 2 e- ~ a-ketoglutarate + H20 acetate + C02(g) + 2 H+ + 2 e- ~ pyruvate + H20

Appendix 3

E'/V

-0.286

-0.412

-0.462 -0.47

-0.598

0.077 -0.090 -0.158 -0.166 -0.190 -0.197 -0.349

-0.330 -0.363 -0.673 -0.699

Reduction of a carbonyl group with formation of an amino group: oxaloacetate + NH4+ + 2 H+ + 2 e- ~ aspartate + H20 -0.107 pyruvate + Nl4+ + 2 H+ + 2 e- ~ alanine + H20 -0.132 a-ketoglutarate + NH4+ + 2 H+ + 2 e- ~ glutamate + H20

Reduction of a carbon-carbon double bond: crotonyl-CoA + 2 H+ + 2 e- ~ butyryl-CoA fumarate + 2 H+ + 2 e- ~ succinate

-0.133

0.187 0.031

Electrochemical Data

Half-Cell Reaction

Reduction of disulfide: cystine + 2 H+ + 2 e- ~ 2 cysteine

glutathione climer + 2 H+ + 2 e- ~ 2 glutathione

Other reductions of biochemical interest: 02(g)+4H++4e- ~ 2 H20 cytochrome c (Fe3+) + e- ~ cytochrome c (Fe2+) FAD+ + H+ + 2 e- ~ FADH NAD++H++2e- ~ NADH 2 H + + 2 e- ~ H2(g)

443

E'/V

-D.340 -D.340

0.816 0.25

-D.20 -D.320 -D.414

Reduction potentials at 25°C, pH 7 standard state; data from H. A. Krebs, H. L. Kornberg, and K. Burton, Erg. Physiol. 1957,49,212.

444 Appendix 3

Table A.6 Some Formal Reduction Potentials

Couple 1MHCI04

Ag(l)l Ag( 0) 0.792 As(V)1 As(III) 0.577 Ce(IV)/Ce(III) 1.70 Fe(IIl)lFe(II) 0.732 Ag(l)l Ag( 0) 0.792 As(V)1 As(III) 0.577 Ce(IV)/Ce(III) 1.70 Fe(IIl)lFe(II) 0.732 H(I)IH(O) -0.005 Hg(II)IHg(1) 0.907 Hg(l)IHg(O) 0.776 Mn(IV)/Mn(II) 1.24 Pb(II)lPb(O) -0.14 Sn(lV)/Sn(II) Sn(Il)/Sn( 0) -0.16

EO'/V

1MHCl

0.228 0.577 1.28

0.700 0.228 0.577 1.28

0.700

-0.005

0.274

0.14

0.77

1.44

0.68 0.77

1.44

0.68

0.674

-0.29

Data from E. H. Swift and E. A. Butler, Quantitative Measurements and Chemical Equilibria, San Francisco: Freeman, 1972.

Table A. 7 Reference Electrode Potentials

Electrode EO'IV (dE/dT)/mV K-l

Calomel (0.1 M KCD 0.336 -0.08 Calomel (1.0 M KCD 0.283 -0.29 Calomel (satd. KCl) 0.244 -0.67 Ag/AgCl (3.5 M KCl) 0.205 -0.73 Ag/AgCI (satd. KCl) 0.199 -1.01

Electrochemical Data 445

Table A.8 Molar Ionic Conductivities

Ion AD Ion AD Ion AD

H+ 349.8 Pb2+ 139.0 103- 40.5 Li+ 38.7 Mn2+ 107. 104- 54.6

Na+ 50.1 Fe2+ 107. Mn04- 62.8 K+ 73.5 Co2+ 110. HC03- 44.5 Cs+ 77.3 Ni2+ lOB. H2P04- 36.

Nli4+ 73.6 Al3+ 189. HC02- 54.6

(CH3)4N+ 44.9 Cr3+ ~1. CH3C02- 40.9

(C2HS)4N+ 32_7 Fe3+ ~. C2HSC02- 35.8

(C3H7)4N+ .23.4 OH- 199.2 C6HSC0 2- 32.4 Ag+ 61.9 F- 55.4 C032- 138.6 Mg2+ 106.1 Cl- 76.3 8042- 160.0 Ca2+ 119.0 Br 78.1 &P32- 174.B Sr2+ 118.9 I- 76.B Cr042- 170. Ba2+ 127.3 CN- 7B. HP042- 114. Cu2+ 107.2 N02- 72. C20 42- 148.3 Zn2+ 105.6 N03- 71.5 P30s3- 25O.B Cd2+ 108.0 ClO3- 64.6 Fe(CN)63- 302.7 Hg2+ 127.2 ClO4- 67.4 Fe(CN)64- 442.

Conductivities from Robinson and Stokes (B9) and Dobos (H9) in units of 10-4 S m2mol-1, aqueous solutions at infinite dilution, 25DC.

446 Appendix 3

Table A.9 Solvent Properties

Liquid Vapor Dielectric Solvent Range/oC Pressureb Constant Viscosity

water 0 3.2 78.4 0.89 propylene carbonate (PC) -49 to 242 0.0 64.4 2.5 dimethylsulfoxide 19 to 189 0.1 46.7 2.00 (DMSO)

N ,N -dimethylformamide -60 to 153 0.5 36.7 0.80 (DMF)

acetonitrile -44 to 82 11.8 37.5 0.34 nitromethane -29 to 101 4.9 35.9 0.61 methanol -98 to 65 16.7 32.7 0.54 hexamethylphosphor- 7to233 0.01 30.d 3.47d amide (HMPA)

ethanol -114 to 78 8.0 24.6 1.08

acetone -95 to 56 24.2 20.7 0.30 dichloromethane -95 to 40 58.1 8.9 0.41 trifluoroacetic acid -15 to 72 14.4 8.6 0.86 tetrahydrofuran (THF) -108 to 66 26.3 7.6 0.46 1,2-dimethoxyethane -58 to 93 10.0 72 0.46 (glyme, DME)

acetic acid 17 to 118 2.0 6.2d 1.13

p-dioxane 12 to 101 4.9 22 1.2

a Data at 25°C from J. A. Riddick and W. B. Bunger, Organic Solvents, 3rd ed, New York: Wiley-Interscience, 1970. b Vapor pressure in units of kPa. C Viscosity in units of 10-3 kg m-ls-l. d 20°C.

Electrochemical Data 447

Table A.1O Potential Range for Some Solutions

Potential RangeN (us. s.c.e.)

Solvent Electrolyte Pt Hg

Propylene carbonate E4NCI04 1.7 to-1.9 0.5to-2.5 Dimethylsulfoxide NaC104 0.7 to-1.8 0.6to-2.9

E4NCI04 0.7 to-1.8 0.2to-2.8 BU4NI -0.4to-2.8

Dimethylformamide NaCI04 1.6to-1.6 0.5to-2.0

E4NCI4 1.6to-2.1 0.5to-3.0

E4NBF4 -to-2.7

BU4CI04 1.5 to-2.5 0.5to-3.0

BU41 -0.4 to-3.0 Acetonitrile NaCI04 1.8to-1.5 0.6to-1.7

E4NCI04 0.6to-2.8

E4NBF4 2.3 to- -to-2.7 BU4NI -O.6to-2.8

BU4PF6 3.4to-2.9

Acetone NaCI04 1.6 to-

E4NCI04 -to-2.4

Dichloromethane BU4NCI04 1.8 to-1.7 0.8to-1.9 BU4NI 0.2to-1.7 -O.5to-1.7

1,2-Dimethoxyethane BU4NCI04 0.6to-2.9

Data from C. K. Mann, Electroanal. Chem. 1969,3, 57.

>< -c z W Q. Q. cc 4 LAPLACE TRANSFORM

METHODS

The method of Laplace transforms provides a powerful aid to the solution of differential equations. l The method is particularly useful in solving the coupled partial differential equations which are encountered in electrochemical diffusion problems. Here we will introduce the technique and demonstrate the method by deriving a few of the results quoted in the text.

Laplace Transformations

The Laplace transform of a function F(t) is defined by

f(s) = L-F(t) exp(-st) dt (A. I)

Not all functions possess a Laplace transform. Clearly, F(t) must be finite for finite t and F(t) exp(-st) must go to zero as t ~ 00. The Laplace transformation can be thought of as an operation in linear algebra:

f(s) = L[F(t)]

which is reversible by the inverse operation

F(t) = L-llfls)]

The Laplace transformation is a linear operation, that is, sums or differences of functions are transformed as

L[F(t) + G(t)] = f(s) + g(s) (A.2)

Multiplicative constants are unaffected by Laplace transformation:

L[a F(t)] = a f(s) (A.3)

1 F. E. Nixon, Handbook of Laplace Transformations: Tables and Examples, Englewood Cliffs, NJ: Prentice-Hall, 1960; R. V. Churchill, Modern Operational Mathematics in Engineering, 2nd ed, New York: McGraw-Hill, 1963; P. A. McCollum and B. F. Brown, Laplace Transform Tables and Theorems, New York: Holt, Rinehart, and Winston, 1965; M. G. Smith, Laplace Transform Theory, London, D. Van Nostrand, 1966.

448

Laplace Transform Methods

The Laplace tranform of a constant is

L(a) = als

449

Functions of variables other than t behave as constants in the transformation

L[H(x)] = H(x)ls

The utility of Lapace transforms in the solution of differential equations is that the transform of a derivative is a simple function

L[dF(t)/dt] = s f{s) - F(O)

L[d2F(t)/dt2] = s2f{s) - s F(D) - (dFldt)o

(AAa)

(A4b)

A short selection of Laplace transforms are found in Table All. Two properties of the Laplace transformation are sometimes useful

in finding the inverse transform. The shift theorem allows the zero of s to be displaced by a constant:

L-llfts + a)) = F(t) exp(~t) (A5)

The convolution theorem is useful when the inverse transformation of f(s) cannot be found, but f(s) can be written as the product of two functions, f{s) = g(s)h(s), the inverse transforms of which can be found. If

then

G(t) = L-lfg(s)]

H(t) = L-l[h(s)]

(A.6)

The specific solution to a differential equation depends on the initial and boundary conditions on the problem. The solution to a differential equation using Laplace transform methods in general follows the steps: (1) Transform the differential equation to remove derivatives with

respect to one of the variables. An ordinary differential equation will then be an algebraic equation and a partial differential equation with two independent variables will become an ordinary differential equation.

(2) Transform the initial and boundary conditions. (3) Solve the resulting system of algebraic equations or ordinary

differential equations, using the transformed boundary conditions to evaluate constants.

(4) Take the inverse transform to obtain the solution to the original differential equation.

450 Appendix 4

Table A.ll Some Laplace Transforms

F(t) f{s) F(t) f{s)

a (a constant) als sin at a a 2+ s2

t s-2

t n- 1 s-n

(n -I)!

l/'fii s-1I2

2fiTiC s-3/2

F(t)

exp(at) - exp(bt) a-b

exp(at) erfW

exp(at)[l- erfYat]

1- exp(at)[l- erfYat]

~ exp(-l/4at) 1tat3

.b exp(-l/4at) ,1tt

1-erf(lJ2W)

cos at s. a 2+ s2

sinh at _a_ s2-a2

cosh at ~

s2-a2

exp at _I_ s-a

f{s)

1 (s-a)(s-b)

ifI -IS(s - a)

1

s (-IS + ifI)

exp (-Vs/a)

(l/-IS) exp (-Vs/a)

(1/s) exp (-V sial

Consider as an example the ordinary differential equation

d2F(t) = a 2F(t) dx 2

with boundary conditions F(O) = 0, (dF/dx)o = a. Taking the Laplace transform, using eqs (A.2) - (A.4), we have

s2f{s) - a + a2f(s) = 0

from which we obtain

Laplace Transform Methods

(ts) = a a 2 + s2

The inverse transform from Table A,11 gives

F(t) = sin at

451

We really didn't need a fancy method to solve this problem, but other cases arise which are not quite so simple.

Solutions of the Diffusion Equation

Now let us apply Laplace transform methods to the solution of the one-dimensional diffusion equation

aC(X,t)

at

2 = D a C(X,t)

ax2 (A.7)

We can do step (1) of the solution procedure in general. Writing the Laplace transform of C(x,t) as c(x,s), the transformed diffusion equation is

2 s C(X,s) _ C(X,O) = D a c(x,s)

ax2 (A,B)

We need initial and boundary conditions to solve eq (A,B), and these differ from one problem to another.

Derivation of eq (3.22). Let us start with the problem posed in §3.3. We considered a solution layered on pure solvent so that the initial condition was C = C* for x < 0, C = 0 for x > O. The boundary condition is C -7 C* as x -7 -00, C -7 0 as x -7 +00. We will divide the problem into two regimes, -00 < X < 0 and 0 < x < +00, with the requirement that C(x,t) and J(x,t) be continuous at x = 0 for t > O. Thus for x > 0, we have

2 D a c(x,S) () 0 sex,s =

ax 2

the general solution to which is

c(x,s) =A(s) exp(-YslD) + B(s) exp(+YslDx) (A.9a)

where A(s) and B(s) are to be determined from the boundary conditions. One of the boundary conditions requires c(x,s) -7 0 as x -7 00 so that B(s) = O.

For x < 0, eq (A,B) gives

D ic'(x,s) , C* s C (x,s) + = °

dX 2

452 Appendix 4

The general solution to this differential equation is

c'(x,s) = C*ls +A'(s) exp(-VsIDx) +B'(s) exp(+VsIDx) (A.9b)

The boundary condition requires c'(x,s) ~ C*ls as x ~ --00 so thatA'(s) = 0. We now apply the continuity restraints to determine A(s) and B'(s). If C(O,t) = C'(O,t), then C(O,S) = c'(O,s). Thus we have

so that

C(O,S) =A(s)

c'(O,s) = C*ls + B'(s)

A(s) - B'(s) = C*ls

The equal fluxes at x = ° means that

dC(O,s) dC'(O,s) ---=

dX dX

Differentiating eqs (A.9) and setting x = 0, we have

-VsIDA(s) = +VsIDB'(s)

Thus

A(s) = -B'(s) = C*/2s

Equations (A.9) then become

C(x,s) = ~ exp (- fk Ixl)

c'(x,s) = c;- - ~; eXP(-fklxl) Taking the inverse transforms, we have

C(x,t) =~[1-erf~] 2 2Wt

C'(x,t) = C* - ~[1- erf---L-] 2 2Wt

x>o

x<o

x>o

x<o

Since the error function is an odd function of the argument, i.e., erf(-'I') = -erf('I'), we see that these two functions are in fact identical:

C(x,t) = C* [1- erf .~] 2 2,Dt (3.22)

Derivation of eqs.(4.2). In typical electrochemical applications of the diffusion equation, the concentrations of the diffusing species are uniform at the beginning of the experiment, C(x,O) = C* and, at later

Laplace Transform Methods 453

times, approach the initial concentration at sufficient distance from the electrode, C(x,t) ~ C* as x ~ 00. With the electrode as x = 0, we need not consider negative values for x. Thus eq (AB) is

2 D a c(x,s) () C* - 0 s C x,s + -

ax 2

The solution consistent with the boundary condition (at x ~ 00) is

c(x,s) = 9- + A(s) exp (- VIi x) (A10)

where A(s) must be determined by the boundary condition at x = O. Ifwe have two species, 0 and R, which are involved in an electrode process, each transformed concentration will have the form of eq (A.10). If the initial concentrations are Co(x,O) = Co*, CR(O,t) = 0, then eq (A.10) gives

Co(x,s) = Cf +A(s)exp(-ffox) (Alla)

CR(X,S) = B(s) exp (-~ DR x) (Al1b)

For a reversible electrode process, the surface boundary conditions are: (1) the concentration ratio atx = 0, governed by the Nernst equation

Co(O,t) = e = exp nF(E - EO) (A.12a) CR(O,t) RT

and (2) the continuity restriction

Jo(O,t) = - JR(O,t)

or

-Do aCo(O,t) = DR aCR(O,t) ax ax

The boundary conditions transform to

coCO,s) = e CR(O,S)

and

-Do o<:o(O,s) =DR aCR(O,S) ax ax

(A12b)

(A.13a)

(A13b)

Differentiating co(x,s) and CR(X,S) with respect to x and setting x = 0, we have on substitution in eq (A13b),

Y sDo A(s) = - YSDR B(s)

454

or

B(s) = - ~A(s)

where

~= "IDo/DR Setting x = 0 in eqs (All) and substituting in eq (A.I3a) gives

Co*/s + A(s) = eB(s) = - ~e A(s)

so that

A(s) =_ Co* s(1 + ~e)

Thus the transformed concentrations are

( ) Co* [1 exP{-Ys7i50x}] cox,s = -s-1 + ~e

CR(X,S) = ~~o* exp{-Vs7l5'Rx} 1 + ~e

Taking the inverse transform, we have

C ( ) -C *~e+erf(x/2YDot) ox,t - 0 1 + ~e

C ( ) - C * ~ [1- erf(x/2Y DRt }] R x,t - 0

1 + ~e

Appendix 4

(AI4)

(4.2a)

(4.2b)

Derivation of eq (4.20). The Laplace transform of eq (4.19) is

( ) C* D [d2C(r,S) .2. dC(r,S)] scr,s - = + r--dr2 dr

This differential equation can be converted to a more familiar form by the substitution

vCr,s) = r c(r,s) 2

s vCr,s) _ rC* = D d vCr,s) dr2

Remembering that the range of r is ro to 00, the solution analogous to eq (AIO) is

Laplace Transform Methods

Reverting to the transformed concentration function, we have

c(r,s) = ~ + A~) exp [-VI (r-rol]

This expression is consistent with the boundary condition

C(r,t) -+ C*, c(r,s) -+ C*/s as r -+ 00

The surface boundary condition,

C(ro,t) = 0, t > 0

transforms to c(ro,s) = O. Thus we find A(s) = -roC*/s and have

c(r,s) = ~ II-~exp [-VI (r-rol]) Taking the reverse Laplace transform, we have

C(r,t) = C* [I-E.Q(I- erf r- ro)~ r 2Wt~

455

(4.20)

Derivation of eq (4.26). In double potential step chronoamperometry, the electrode is polarized for a time 't at a sufficiently negative potential that C o(O,t) = 0; the potential is then stepped to a positive potential so that CR(O, t - 't) = O. This problem is easily solved using Laplace transforms by noting that eq (4.2b) gives the initial concentration distribution of R for the second potential step. Thus substituting eq (4.2b) with t = 't and e = 0 into eq (A.IO), we have

~C * CR(X,S) = +[I-e1~)] +A(S)ex~-ff.x)

where the transform variable s corresponds to (t - 't). The boundary condition CR(O,t) = 0 for t > 't implies that CR(O,S) = O. Thus

(cCo*ls) + A(s) = 0

so that

A(s) = - ~Co*ls

We need the flux of R at the electrode in order to calculate the current. Thus we compute the derivative with respect to x

CkR(O,S) ~Co* ~Co* ---=- +--

ax s "1tDR't Y DRS

and take the inverse transform to obtain

aCR(O,t) ~Co* ~Co* -a-x- = - "1tDR't + J"=1tD~R(;::;:t=-='t=<=)

456

Since the current is

we get

i = nFACo*fl5Qiii(-L-..l..) t > t vt-t Yt

Appendix 4

(4.26)

Derivation of eq (4.27). In a constant current experiment such as chronopotentiometry, the flux of 0 at the electrode surface is constant up to the transition time when Co(O,t) -+ 0 and the potential swings negative. Thus the boundary condition is

Do OCo(O,t) ---.L ax nFA

Taking the Laplace transform, we have

Do aco(O,s) = ~ ax nFAs

Differentiating eq (A.10) and setting x = 0, we have

-Doh/Do A(s) = nJAs

Solving for A(s) and substituting in eq (A. 10) with x = 0: C * . cO<O,s) = 0 I

S nFAsYsDo Taking the inverse transform, we get

CO<O,t) = Co* _ 2ifi nFAVreDo

Apparently, Co(O,t) goes to zero when

nFA VreDo Co* = 2ifi

so that the transition time is given by

fC = nFA VreDo Co*/2 i (4.27)

Derivation of eq (4.29). For a linear potential scan experiment, the boundary conditions are similar to those used in deriving eq (4.2) except that the potential is time-dependent. Thus we can proceed as before up through eq (A.14), but A(s) cannot be so simply evaluated. We can write the transformed current as

Laplace Transform Methods

i(s) = nFADo (aco(x,s)) ax %=0

Substituting eq (A.lla) for CO(x,s), we have

i(s) = - nFAV sDo A(s)

457

Solving for A(s), substituting into eqs (A.ll), and setting x = 0, we have

(0) Co· i(s) co ,s =-----==

s nFA VsDo

(0 ) i(s) CR ,s = ---'--===-

nFA VSDR We now use the convolution theorm, eq (A.6), with h(s) = i(s), g(s) = s-1/2 to obtain

Co(O,t) = Co. - 1 t i(t) dt nFA Y7r.Do 10 yt-t

For a nernstian process, the ratio of the surface concentrations is

Co(O,t) = 9(t) = exp nF(Ei - vt _EO) CR(O,t) RT

Substituting the surface concentrations into this expression and rearranging, we have

nFA fit1JO Co· = t ~dt 1 + ~9(t) 10 yt-t

(4.29)

where ~ = VDo/DR.

Derivation of eq (5.31). We are concerned here with the flux of 0 at the electrode surface when Co(O,t) = 0 and 0 is formed from Y in an equilibrium prior to electron transfer. We start with diffusion equations in Y and 0, similar to eqs (5.2) but including the time derivatives. With the new functions of eqs (5.3), we obtain the differential equations

2 aC(x,t) = D a C(X,t)

at ax 2

458 Appendix 4

dC'(X,t) = D iC'(x,t) _ (k i + k_l ) C'(X,t)

dt dx 2

We will assume that the equilibrium strongly favors Y so that K = kl/k_1 «1. Taking Laplace transforms, we obtain simple differential equations which are readily solved to give

c(x,s) = C*/s + A(s) exp -Ysil5 x

c'(x,s) = A'(s) exp -yes + k_I)ID x

With eq (5.6b), the boundary condition, CO(O,t) = 0, gives

CO(O,s) = 6[-9-+ A(S)] -l;KA'(s)=O

Since Y is not electroactive, Jy(O,t) = 0 and eq (5.6a) gives

(dCO) =-A(s)~rr _A'(S).yS + k-I =0 dXx=o VD D

Solving the simultaneous equations, we have

A(s) = C* = _ A'(s) A / K 2s + Kki S(l + • / S ) K·V s

'V K% + Kki Neglecting K compared with 1 and assuming that K 2s «Kkl,l the transformed flux of 0 is

;o(O,s) = -C* w fKki. s + YKklS

Taking the inverse Laplace transform and converting to current, we have

(5.31)

Derivation of eqs (6.36). Allowing for a finite electron transfer rate, the surface boundary conditions are

D dCo(O,t) - D dCR(O,t) - k C (0) k C (0 ) o - - R - c 0 ,t - a R ,t dx dx

Transforming the boundary conditions, we again have eq (A.13b), but instead of eq CA.13a), we get

1 This approximation is equivalent to neglecting the very small amount of 0 initially present near the electrode; thus the result if only approximate at short times.

Laplace Transform Methods

Do oco(O,s) = kceo(O,s) - kaCR(O,S) ax

459

Thus, with co(x,s) and CR(X,S) given by eqs (A.ll), and B(s) = -x A(s), we have

- VsDo A(s) = kc[Co*ls +A(s)] + ka~A(s)

or

A(s) = _ keCo* VDos(A +-IS)

where

A = kc!V Do + ka/V DR

Substituting A(s) into eqs (A.ll) with x = 0, we have

coCO,s) = C~* [1- Wo ke ] Do (A +-IS)

CR(O,s) = keCo* V DR seA + -IS)

Taking the inverse transforms, we have

Defining

we have

Co(O,t) = Co* - kcCo* (1- exp(A2t}[1- erf(A Vt))) 'A.VDo

CR(O,t) = keCo* !1-eXP('A.2t}[1-erf('A.Vt)J) 'A.VDR

(('A.Vt) = 'A.fi[i exp('A.2t)[1- erf('A.Vt)]

Co(O,t) = Co* _~[l_t{'A.Vt)] 1 + ~9 'A.fi[i

CR(O,t) = ~Co* [l_t{'A.Vt)] 1+~9 Afi[i

where we have used the relation kalke = 9.

(6.36a)

(6.36b)

Derivation of eqs (6.47). When the boundary condition is determined by a sinusoidal current

let) = 10 sin rot

460 Appendix 4

the flux of 0 at the electrode surface is

- Do dCO(O,t) = -~ sin rot dX FA

Taking the Laplace transform, we have

Do dCO(O,S) = 10 0)

dX FA s2+ 0)2

Substituting the first derivative of eq (A.ll), evaluated at x = 0, we get

_ VsDoAs = 10 0) FA s2+ 0)2

Solving for A(s) and substituting in eq (A.ll) with x = 0, we have

co(O,s) = Co* _ 10 [ 0) ] s FAVDo VS(s2 + 0)2)

The inverse transform of this function cannot be found in tables, so we have recourse to the convolution theorem, eq (A.6), taking

g(s) = _0)_ => G(t) = sin rot s2+c02

h(s) = l => H(t) = _1_ VS f1ti

Thus

L-1 [ 0) ] = t_1- sinroCt-'t)d't VS(s 2 + co2) Jo iii

U sing the trigonometric identity

sin roCt - 't) = sin rot cos CJ)'t - cos rot sin CJ)'t

the integral becomes

sin rot t ~ cos CJ)'t d't -~ t ~ sin CJ)'t d't fit Jo ft fit Jo ft

The factor of 11ft in the integrand represents a transient response to the application of the sinusoidal current which dies off to give a steady-state sinusoidal variation in the concentrations. Since we are interested only in the steady state, the limits on the integrals can be extended to infinity, obtaining

Laplace Transform Methods

Thus we have

CO<O,t) = Co. _ 10 {sin rot - cos rot) FAY2roDo

Equation (6.47b) for CR(O,t) results from a similar development.

461

(6.47a)

>< -c z W Il. Il. < 5 DIGITAL SIMULATION

METHODS

The theoretical description of an electrochemical experiment usually requires the solution of a set of coupled partial differential equations based on the diffusion equation. In experiments which include forced convection, such as r.d.e. voltammetry, a driving term is added to each equation as in eq (4.43). When a diffusing species is involved in a chemical reaction, reaction rate terms must be added to the equation describing its concentration. The set of equations often has time-dependent boundary conditions and can be devilishly difficult to solve. In some fortunate cases, such as those treated in Appendix 4, the use of Laplace transforms leads to closed-form analytical solutions, but more often solutions are obtained in terms of infinite series or intractable integrals which must be evaluated numerically. In the 1950's and 1960's, a great deal of effort was expended by theorists in obtaining mathematical descriptions of electrochemical experiments. While the results provide an invaluable aid to understanding (we have quoted many of these results), one often finds that the theoretical results available in the literature do not quite cover the experimental case at hand. If the problem seems to be of sufficient generality and interest, it may be worthwhile attempting an analytical approach. More often, however, electrochemists have turned to the digital computer to simulate experiments. The details of digital simulation are beyond the scope of this text, but a brief outline of the strategy is in order. For further details see reviews by Feldberg,l Maloy,2 or Britz.3

In general the problem to be solved involves equations of the form 2 ae D a C k· . d/ d·· - = -- + InetIc an or nvmg terms

at ax2

together with a set of initial and boundary conditions. The concentrations are functions of time and distance from the electrode and

1 s. W. Feldberg, Electroanalytical Chemistry 1969, 3, 199. 2 J. T. Maloy in Laboratory Techniques in Electroanalytical Chemistry, P. T. Kissinger and W. R. Heineman, eds, Kew York: Marcel Dekker, 1984. 3 D. Britz, Digital Simulation in Electrochemistry, Lecture Notes in Chemistry, Vol. 23, Heidelberg: Springer·Verlag, 1981.

462

Digital Simulation Methods 463

the general strategy in a digital simulation is to divide the time and distance axes into discrete elements of size ot and Ox, respectively. The first step in the development of a simulation program is to convert the differential equations into finite difference equations.

Thus Fick's first and second laws,

J(x,t) = -D aC(x,t) ax

aC(x,t) aJ(x,t) ---=----

at ax which describe the diffusional part of the problem, can be written in approximate form in terms of the finite differences, Ox and ot:

J(x,t) '" -D C(x + 0xI2,t) - C(X - 0xI2,t)

Ox C(X,t + ot) - C(x,t)

ot

J(X + 0xI2,t) - J(x - &X/2,t)

Ox Combining the two expressions, we have

C(x,t + Ot) = C(x,t) + .D.aL[C(x + Ox,t) - 2C(x,t) + C(X - Ox,t)] (A.15) (axf

If the spatial boxes, of equal width Ox, are labeled 1,2,3 .. .j. .. and the time boxes of width ot, are labeled 1,2,3 ... k ... we can rewrite eq (A.15) as

C(j,k+l) = C(j,k) + D[C(j+I,k) - 2C(j,k) + C(j-l,k)] (A. 16)

where D = Dotl(ox)2 is a dimensionless diffusion coefficient. Thus eq (A.16) models the diffusion process as follows: during the time interval ot, an amount DC(j,k) moves from boxj to each of the adjacent boxes. But meanwhile DC(j-I,k) and DC(j+l,k) move from the adjacent boxes into box j; the change in concentration in box j is the sum of these contributions. Clearly, we can't take more out of a box than was there to begin with, so that we require D s; 0.5.

The net flux at the electrode (related to the current) is determined by the changes in concentrations in the j = 1 box (adjacent to the electrode) which are required to satisfy the surface boundary condition. For example, for a reversible electrode process, the boundary condition corresponds to the surface concentration ratio CoiCR = e, specified by the Nernst equation. The equilibrium surface concentration of 0 can be written as

Co(eq) =_e_[Co(I,k) + CR(I,k)] 1+8

464 Appendix 5

The net flux then corresponds to the difference between Co(1,k) and the equilibrium surface concentration

SCo(k) = Co(1,k) - 9CR(1,k) = - SGR(k) 1+9

The surface concentrations then are corrected to

Co(1,k + 1) = Co(1,k) - SGo(k)

CR(1,k + 1) = CR(l,k) + SCo(k)

Since the current is given by

i =nFAJo(O)

the simulation current is proportional to SCo(k).

(A.17a)

(A. 17b)

In an experiment where the electrode potential is time-dependent (e.g., cyclic voltammetry), the potential (and thus the concentration ratio 9) will be different for each time increment. If the simulated experiment involves two or more electron-transfer processes, these are handled independently and the current contributions added to get the total current. Experiments with slow electron-transfer kinetics can be simulated by including Butler-Volmer electron-transfer rate equations in place of the above expression which assumes that equilibrium is attained instantaneously.

If species A is consumed in a first-order chemical reaction,

A~B

the contribution to the rate equations is

aCB(X,t) = _ aCA(X,t) = kCA(X,t)

at at In terms of the finite differences, these expression are

CB(X,t + at) - CB(X,t) =_ CA(X,t + at) - CA(X,t) = kCA(x,t)

at at or, using the indicesj and k,

CAV,k+1) = CAV,k) - kCAV,k)

CBV,k+1) = CBV,k) + kCAV,k)

(A.1Ba)

(A.1Bb)

where k = kSt is the dimensionless rate constant. In practice, k must be small (~ 0.1) in order to accurately model the system; for a given value of k, this places a restriction on 8t. This restriction can be somewhat relaxed (k ~ 1) by using an analytical solution for the extent of reaction during a time increment, e.g., for a first-order process

(A.19a)

Digital Simulation Methods

CB(j,k+l) = CB(j,k) + CA(j,k) [1- exp (-k)]

465

(A. 19b)

Analytical expressions can be used for more complex reaction schemes, or, for still more complex schemes, the extent of reaction can be estimated using the modified Euler method. 1

For a cyclic voltammetry simulation, the time increment is determined either by the scan rate with v ot '" 1 mV, or, when coupled chemical reactions are considered, by the rate constant. The number of time increments to be used in the simulation is determined by

nt = tlot (A.20)

where t is the total time of the experiment to be modeled. The size of the spatial increment is determined by

&c = yr-D- o-tlD- (A.21)

If the solution is isotropic at the beginninuf the experiment, the diffusion layer grows to a thickness of about 6Y Dt during the time of the experiment. Thus the number of spatial boxes required in the simulation is determined by

or

or nx '" 4 fflt ifD = 0.45.

nx&c = erlDnt Ot

nx = 6V Dnt &X&cf nx = 6VDnt (A.22)

Even with the use of analytical expressions such as eq (A.19) to model the effects of chemical reactions, nt can be very large for schemes with fast reactions. Since computer execution time increases as the product of nt and nx, there is a practical upper limit to rate constants. One solution to this problem is to use variable-width increments in the simulation. For very fast reactions, the reaction layer, XR, is thin and, in the simulation, most of the action takes place in the first few spatial boxes adjacent to the electrode. Thus some saving in execution time can be realized if the size of the spatial boxes is allowed to increase with increasing distance from the electrode.2 This approach leads to a different D for each box and a somewhat more complex diffusion algorithm. A still more efficient approach is to expand the time grid as well for boxes far from the electrode.3 Thus boxes in which not much is happening are sampled less frequently.

The overall structure of the simulation program then is as follows:

1 D. K. Gosser and P. H. Rieger, Anal. Chern. 1988,60, 1159. 2 T. Joslin and D. Pletcher, J. Electroanal. Chern. 1974,49, 171. 3 R. Seeber and S. Stefani, Anal. Chern. 1981,53, 1011.

466 Appendix 5

(1) Set concentrations to initial values. (2) Correct each concentration in each box for the results of the

chemical reactions using eqs (A.1B) or (A.19). (3) Correct each concentration in each box for the results of diffusion

using eq (A.16). (4) Change the concentrations in box 1 to satisfy the surface boundary

condition, e.g., using eqs (A.17) for a nernstian process; the changes correspond to the flux and thus to the current.

(5) Go to the next time increment and adjust time-dependent parameters such as the electrode potential.

(6) Repeat steps (2) - (5) for the required number of time increments.

>< -c z W Il. Il. <C 6 ANSWERS TO

SELECTED PROBLEMS

U (a) PtIFe2+, Fe3+) I I Mn04-, Mn2+, H+IPt (b) Fe3+ + e- ~ Fe2+ Mn04- + 8 H+ + 5 e- ~ Mn2+ +4 H20 (c) 5 Fe2+ + Mn04- + 8 H+ ~ 5 Fe3+ + Mn2+ + 4 H20 (d) EO = 1.51- 0.771 = 0.74 V (e)E = 0.56 V (f) !lGo = -357 kJ mol-I, K = 3 x 1062

1.2 (a) 1.0662 V (b)-O.607V (c)-O.76V

1.4 Ksp = 1.6 x 10-8

1.5 Ksp = 4.5 x 10-18, pH = 7.00

1.7 E = 0.077, 0.085, 0.107, 0.145, 0.302,0.458,0.491 V

1.8 1.27 x 10-4 M

1.9 K = 9.8 x 1018

LlO (b) pH 5.0

lJ.1 flux = 2.2 x 10-3 mole Na+ s-l; firing rate = 2.2 s-l

L12 (a) E' = +0.043, +0.353, -0.154 V (b) !lG' ;;; -8.3, -68.1, +29.7 kJ moP (c) K = 28, 8.6 x 1011, 6.2 x 10-8

L13 (a) !lGo ;;; -130 kJ mol-1 (b) !lGo = -76 kJ mol-1 (c) !lGo = -206 kJ mol-1

Ll4 (a) !lGo = -35 kJ mol-1

467

468

(b) !l.GO = +234 kJ mol-1 (e) !l.Go = -458 kJ mol-1

1.15 pH = 10.386 ± 0.008

1.16 [Na+] = (1.12 ± 0.03) x 10-4 M

1.17 (a) kH Na = 7.8 x 10-12

(b) pH 11.19 solution would give apparent pH 11.14

1.18 (a) Zn(s) + 2 OH- ~ ZnO(s) + H20 + 2 e­Ag20(S) + H20 + 2 e- ~ 2 Ag(s) + 20H­Ag20(S) + Zn(s) ~ 2 Ag(s) + ZnO(s) (b) 1100 J g-1

2.8 (a) Qoo = 2.0 IlC (b) io = 1.0 rnA (e) t = 0.092 s

2.9 (a) y±(NaCl) (b) y±(NaF)fy±(NaCl) (e) y±(NaF)

2.11 Y±(exptl) = 0.905, 0.875, 0.854, 0.826, 0.807, 0.786 Y±(eq 57) = 0.889, 0.847, 0.816, 0.769, 0.733, 0.690 Y±(eq 56) = 0.903, 0.872, 0.851, 0.821,0.800,0.776

2.12 [KOH] = 0.988 M, y± = 0.742

2.14 E = 0.767 V

3.1 (a) L fA = 29.05 m-1

(b) A = 14.66 x 10-4 S m2mol-1

(e) A ° = 390.7 x 10-4 S m 2mol-1 (d) ex = 0.0375, K = 1.5 x 10-5

3.2 N = 133.4 x 10-4 S m2mol-1

3.3 s(calc) = 2.88 x 10-4 S m1l2mol-3/2 s(expt) = 3.03 x 10-4

3.4 s = 1.94 x 10-3 S m1l2mol-3/2

3.5 0.203

Appendix 6

Answers to Selected Problems

3.6 1.4 x 10-4 M

3.7 K = 1.6 x 10-4

3.8 u = 1.046 x 10-7, 1.145 x 10-7 m2V-1s-1 r = 274, 334 pm f = 4.60 x 10-12, 5.60 x 10-12 kg s-l D = 8.96 x 10-10, 7.36 x 10-10 m2s-1

3.12 D = 2.45 x 10-11 m2s-1, t '" 6.5 years

3.13 D = 1.59 x 10-9 m2s-1

3.14 teu = 0.366

3.16 [Ba(OH)2] = 0.0422 M, A = 411 x 10-4 S m2mol-1

3.18 [Na+]a = 0.0319 M, [Na+]~ = 0.0281 M, ~<l> = 3.3 mV

4.4 E1I2 = 0.764 V

4.5 A square wave signal is required.

4.7 (a) Gain> 104 (b) Gain> 4 x 104

(c) Scan rate is 2 ppm smaller when output voltage is -1 V. (d) i = 100 j.4A, RcelJ < 10 kn, nominal current correct to 0.001%

4.8 Co = 0.14 mM

4.12 4.75 j.4A

4.14 (a) El/2 = -0.693 V (b)R = 1480 n

4.15 Diameter greater than 0.5 mm

4.16 Electrode radius less than about 2.3 ~

4.20 13 '" 2.4 eV (Figure 4.42a), 13 '" 2.1 eV (Figure 4.42b)

4.26 13 '" 4 x 1012

4.27 (a) p = 2, El/2 = -0.258 V (b) 13 "" 1035

469

470 Appendix 6

5.1 k1K = 0.008 s-l

5.2 (a) CA> 10-6 M (b) A = 1 cm2

5.7 (a) (iL - i)/i = 82(1 + 281)/(2 + 82) (b) E1I2 = (E1° + E2°)/2 (c) The Heyrovsky-Ilkovic equation is obtained when E2° »E1° (d) E3/4 -E1/4 = 42.9,33.8,30.3,28.5 mV for E2° -E1° = 0, 50,100, and 200 mY. The Tomes criterion for a reversible two-electron wave gives 28.2 mY.

5.8 k = 0.8 s-l

5.10 (b) k = 1.0 s-l

5.11 (a) 60 (b) 1.3 (c) 5.1

5.12 (a) DEl/2 = 17.8,61.6 mV (b) DEl/2 = 101.9,41.8 mV

6.2

6.5

6.9

i - nFA ( kcCo* - kaCR* ) - 1 + kalkn + kclkn

Jo = 0.0079 A m-2, (Xapp = 0.58, ko = 2.2 x 10-10 m s-l

(a) E1I2 _EO = -24.7 mV (b) kolkn = 0.141

6.10 (b) ko = 2.1 x 10-4 m s-l

6.12 ko = 5.8 x 10-5 m s-l

6.13 Resistive component: Rs = I Zrl cos cp Capacitive component: 1/roCs = I Zrl sin cp

6.14 (b) Width at half height = 90.6 mY.

6.15 (b) Width between extrema = 67.6 mY.

6.16 (a) ko s 10-5 m s-l (b) ko s 0.0002 m s-l (c) ko s 0.002 m s-l

Answers to Selected Problems

7.2 t = 27.5 days

7.3 (a) 22 min (b) GB(t) = 0.10, 0.14, and 0.12 mM at t = 5, 10, and 25 min

7.6 126C

7.7 (a)0.234C (b) 48.5 J.!M

7.8 (a) R = 1000 Q (b) Rcell < 9 kQ

7.9 3.51 roM

7.10 5-mL sample, 2 rnA current give t = 241 s, uncertainty is ±0.6%

471

7.12 Hydrogen reduction current: j = 0.27 A m-2 on Pt (cp = 0.997),j = 1.9 rnA m-2 on Cu (cp = 0.9992)

7.13 Optimum current = 133 A Cost = (raw materials + capital and labor + energy) Cost = (1.00 + 0.84 + 0.91) = $2.75 per kilogram With doubled energy costs, optimum current = 91 A Cost = $3.57 per kilogram

7.15 (a) Ec = -0.73 V, Corrosion rate = 0.5 mg Zn m-2s- l , Hydrogen evolution rate = 0.18 mL m-2s-l (b) Corrosion rate = 2.2 mm yearl

AUTHOR INDEX

An author's work is mentioned on page numbers given in roman type, biographical sketches on page numbers given in bold, and literature citations on page numbers in italics.

Abruiia, H. D. 429 Adams, R. N. 171,237,286,309, 428 Alberts, G. S. 293, 309 Albery, VV.J.236,239,277,309,42~430 Alder, R. 253,308 Allendoerfer, R. D. 261,309,322,367 Amatore, C. 303, 309 Anderson, T. N. 89, 105 Angrist, S. VV. 430 Antropov, L. I. 428 Arrhenius, S. A 126 Austen, D. E. G. 259, 308 Badoz-Lambling, J. 427 Baer, C. D. 215,238,342,367 Bagotzky, V. S. 430 Baizer, M. M. 430 Bard, A J. 428, 431, 432, 441 Barendrecht, E. 388, 389,422 Barker, G. C. 202, 2M, 238 Barnes, K K 428 Bates, R. G. 429 Bauer, H. H. 200,238 Bell, R. P. 277, 309 Belleau, B. 395, 422 Bernal, I. 259, 261,308 Bezems, G. J. 308, 309 Bier, M. 85, 105 Birke, R. L. 235,239, 281,309 Bockris, J. O'M. 333, 367, 369, 415, 428,

430,431 Bond,A. M. 207,215,232,238,239,253,

308,341,366,367,429 Brdi&a, R. 198,238,278,280,309,287 Brezina, M. 236, 239 Britton, vv. E. 169,237,428

472

Britz, D. 462 Brooks, M. A 236, 239 Brown, B. F. 448 Bronsted, J. N. 98, 105 Bruckenstein S. 261,309 Buck, R. P. 40, 54 Bungenberg de Jong, H. G. 81, 105 Bunger, VV. B. 446 Burbank, J. 51,54 Burnett, R. VV. 236, 239 Burton, K 443 Butler, E. A 444 Butler, J. A V. 318, 367 Butler, J. N. 107 Cairns, E. J. 45, 50, 54 Camaioni-Neto, C. 306,309, 342,367 Carlisle, A 4, 113, 371 Caroli, S. 431 Carroll, J. B. 261,309 Casanova,J. 391, 422 Cawley, L. P. 85, 105 Chapman, D. L. 60, 61, 105 Charlot, G. 427 Chateau-Gosselin, M. 236, 239 Christie, J. H. 206, 238 Chung, Y. K 306, 309 Churchill, R. V. 448 Colton, R. 253, 308 Compton, R. G. 258,308 Connelly, N. G. 247, 263,308,309 Conway,B.E.42~431 Cooke, W. D. 175,237 Cottrell, F. G. 156, 237 Covitz, F. H. 429 Crow, D. R. 232,239,427

Author Index

Cruikshank, W. 371, 372, 377, 421 Curran, D. J. 387, 422 Dalrymple-Alford, 193, 238 Damaskin, B. B. 427 Daniell, J. F. 4 Davis, D. G. 181,238 Davy, H. 4 de Bethune, A J. 431 de Montauzon, D. 226,238,247,308 de Smet, M. 79, 80, 105 Debye,P.J. W.90,91,105, 124,147 DeFord, D. D. 383,421 Delahay, P. 26, 54, 179, 181,237,292,

309, 427, 432 Deryagin, B. V. 73, 105 Despic, A R. 333,367, 369,415 Dietz, R. 361, 367 Dobos, D. 431, 445 Donnan, F. G. 142, 147, 158 Dordesch, K V. 45, 54 Drazic, D. 333,367,369,415 Dryhurst, G. 171, 237, 429 Dukhin, S. S. 73, 105 Dvorak, J. 428 Ebsworth, E. A V. 20, 54 Edison, T. A. 52 Ehlers, R. W. 97, 105 Eisenberg, M. 50, 54 Engles, R. 394, 422 Enke, C. G. 113, 146 Erdey-Gruz, T. 318,367 Erman,P.113,115,146,371 Falkenhagen, H. 124, 147 Faraday, M. 4,52,371,421 Faulkner,L. R. 204,238,266,309,428 Feldberg,S. W.286,293,294,303,304,

309,462 Fick, A E. 130 Filinovskii, V. Yu. 428 Fillenz, M. 236, 239 Flato, J. D. 232,239 Fraenkel, G. K 259, 261, 308 Frost, A. A 20, 54 Frumkin, A N. 324, 367 Fry, A J. 169,237,428,430 Fuoss, R. M. 122, 146 Furman, N. H. 39,54,175,237 Furstenau, D. W. 70, 105 Gale, R. J. 429 Galus, Z. 171, 237, 429 Galvani, L. 4 Gardner, A. W. 202, 238 Geary, C. G. 107 Geiger, W. E. 247,308,363,365,367

Gerischer, H. 432 Geske, D. H. 259,308 Gibbs, O. W. 377,421 Gileadi, E. 430 Given, P. H. 259,308 Goldberg, I. B. 258, 308 Gosser, D. K 266,309,465 Goto, M. 193,238 Gouy, L.-G. 60,61, 105 Grahame, D. C. 60,87,88,89,105 Greef, R. 428 Gregory, D. P. 45,54 Gritzer, G. 171,237 Gross, M. 247,308 Grove, W. R. 48, 49, 54 Grunwald, R. A 236, 239 Grzeszczuk, M. 365, 367 Guggenheim, E. A 142, 147 Haber, F. 36, 54 Hajdu, J. 391,422 Hall, C. M. 402 Hamaker, H. C. 69, 105 Hamed, H. S. 97, 105, lO7,427 Harrar, J. E. 383, 421 Harris, M. D. 236, 239 Hawkridge, F. M. 173, 237 Hawley, M. D. 247, 293,308,309 Headridge,J.B.427 Healy, T. W. 70, 105 Heineman, W. R. 389,422,430 Heinze,J.218,220,222,238 Helmholtz, H. von 60, 61, 105 Henderson, P. 139, 147 Hershberger, J. W. 305, 309 Heubert, B. J. 322,367 Heydweiller, A 126, 147 Heyrovsky, J. 157,158, 165, 194,232,

237,239,429 Heroult, P. L. T. 402 Hibbert, D. 427 Hills, G. J. 432 Hitchman, M. L. 430 Hittorf, J. W. 116, 117, 146 Hogg, R. 70, 105 Hoijtink, G. J. 225,238 Holler, F. J. 113, 146 Howell, J. O. 222,238 Huston, R. 107 Huckel, E. 90, 91, 105 Ilkovic, D. 157, 158, 196,237,238 Ingram, D. J. E. 259,308 Ishii, D. 238 Israel, Y. 348,367 Ives, D. J. G. 430

473

474

James, A. M. 427 Janata, J. 39,54 Janz,G.J.43~431 Jefti~L.293,294,303,309 Jensen, B. S.299,309 Jones, R. D. 207,238 Jordan,J.431,441 Joslin, T. 465 KekuIe, A. 396 Kern, D. M. H. 279,309 Khan, S. U. M. 428 Kim, S. 306, 309 Kirowa-Eisner, E. 204, 234, 238, 239,

245,430 Kissinger, P. T. 430 Klemenciewicz, Z. 36, 54 Klingler, R. J. 305, 309 Kochi, J. K 305,309 Kohlrausch, F. 114, 115, 126, 146, 147 Kolbe, H. 395, 396,422 Kolthotf, I. M. 207, 234,238, 429 Kornberg, H. L. 443 Kortum, G. 428 Koryta, J. 236,239, 427, 428, 429 Kosaka, T. 394, 422 Kosower, E. M. 391, 422 Koutecky, J. 278,309 Krebs, H. A. 443 Kublik, Z. 389, 422 Kuta, J. 171,237, 429 Kyriacou, D. K 430 Laitinen, H. A. 207,238,392,422 LaMer, V. K 98, 105 Latimer, W. M. 18,431 Le Bel, J. A. 396 Le Chatelier, H. 401 Leclanche, G. 45 Lehnhotf,N.S.215,238,266,309 Lemoine, P. 247,308 Leon, L. E. 233,239 Levich, V. G. 212, 238, 428 Lewis, G. N. 18 Lingane,J.J.230,234,238,384,421,429 Lingane,P.J.217,238 Loud, N. A. S. 431 Loveland, J. W. 128, 147 Lowenheim, F. A. 430 Lund, H. 430, 431 Lyons, E. H., Jr. 427 MacDonald, D. D. 430 MacDonald, H. C., Jr. 428 MacInnes, D. A. 427 Maki, A. H. 259, 308 Maloy, J. T. 462

Author Index

Manecke, G. 327, 367 Mann, C. K 169,237, 428, 447 Mantell, C. L. 430 Marcoux, L. S. 286, 309 Marcus, R. A. 316, 319, 367 Mark, H. B., Jr. 389, 422, 428 Martinchek, G. A. 261,309 Marzlutf, W. F., Jr. 281, 309 Mattson, J. S. 428 Mazorra,M.235,239 McAllister, D. L. 171,237 McBreen, J. 45, 54 McCollum, P. A. 448 McCormick, M. J. 253, 308 McKinney, T. M. 258,308 Meites, J. 384, 421 Meites, L. 194, 233, 238, 239, 348, 367,

384,421,429,431 Meng, Q. 306, 309 Milazzo, G. 431 Miller, J. W. 383, 421 Milner, P. C. 53, 54 Mohammad, M. 391, 422 Mohilner, D. M. 60, 105 Moraczewski, J. 363,367 Morris J. R. 204,238 Murray, R. W. 39, 54, 427 Myland, J. C. 341,367 Nagy, Z. 431 Narayanan, A. 431 Nernst, W. 11, 158,318 Neto, C. C. see Camaioni-Neto, C. Newman,J.220,238 Newman, J. S. 428 Nicholson, R. S. 183, 187, 238, 296, 298,

309, 348, 366, 367 Nicholson, W. 4, 113, 371 Nishiguchi, I. 395, 422 Nixon, F. E. 448 O'Brien, P. 342, 367 O'Halloran, R. J. 366, 367 O'Neill, R. D. 236, 239 Ohkawa, M. 395, 422 Oldham, K B. 193,215,217,238,274,

309,341,367 Onsager,L. 77,105,122, 123, 146 Orpen, A. G. 263, 309 Ostapczuk, P. 389, 422 Osteryoung, J. 204,234,238,239,245 Osteryoung, R. A. 206,238, 390, 422 Ostwald, F. W. 5,48,54, 156,428 Overman, R. F. 56 Owen, B. B.427 Parker, V. D. 191,238,299,309

Author Index

Parsons,R.60,105,324,367,431,441 Patriarche, G. J. 236,239 Peat, R. 428 Penciner, J. 430 Peover,~.E.259,308,361,367 Perkins, R. S. 89, 105 Peter, L.~. 428 Pinson,J.303,309 Plambeck, J. A. 429 Plante, G. 52 Pleskov, Yu. V. 428 Pletcher, D. 428,430,465 Poilblanc, R. 226,238,247,308 Pourbaix,~.26,54,431 Prada!; J. 236, 239 Prade~va,J.236,239 Purdy, W. C. 429 Reddy, A. 428 Reeves, R. ~. 60, 105 Reilley, C. N. 175,237,427 Reinmuth, W. H. 179,237, 259, 261, 290,

308 Revie, R. W. 431 Riddick, J. A. 446 Riddiford, A. C. 213,238 Rieger, A. L. 263, 309 Rieger,P. H. 258,259, 261,263,266,308,

308,309,322,367,465 Rifi, ~. R. 392, 422, 429 Riley, T. 427 Robbins, J. 427 Roberts, J. L., Jr. 430 Robinson, J. 428 Robinson, R. A. 428, 445 Roe, D. K. 167,238 Rogers, J. R. 391, 422 Rosair, G. ~. 263, 309 Rossiter, B. W. 428 Rossotti, H. 429 Roston, D. A. 389, 422 Rupp, E. B. 431 Rutgers, A. J. 79, 80, 105 Ruzic, I. 366, 367 Rysselberghe, P. van 26, 54 Sack, H. 124,147 Safford, L. K. 350, 367 Salkind, A. J. 432 Sand,H.T.S.182,238 Saveant,J.~.303,308,309 Sawyer,D.T.233,239,430 Schafer, R. 394, 422 Schwartz, W. M. 289, 309 Scott, C. J. 263,309 Seeber, R. 465

475

Selley, N. J. 427 Shain, I. 183, 187, 190,238,289,293,296,

298,309,389,422 Shannon, R. D. 121, 146 Shaw, D. J. 69,70,85,105 Shedlovsky, L. 113, 146 Shedlovsky, T. 113, 146 Shimizu, K 390, 422 Shono,T. 394, 395,422,430 Siegerman, H. 236, 239 Simon, A. C. 51, 54 Skundin, A. ~. 430 Smith, D. E. 322, 356, 365, 366, 367 Smith, ~. G. 448 Somo~,Z.380,421 Soos, Z. G. 217,238 Spiro, ~. 116, 146 Srinivasan, S. 430 Steckhan, E. 394, 422 Stefani, S. 465 Steihl, G. L. 292, 309 Stem, O. 60,61, 105 Stock, J. T. 177,237 Stokes, G. G. 121 Stokes, R. H. 428, 445 Stone, N. J., see N. S. Lehnhoff SWrzbach, ~. 220, 222, 238 Streitwieser, A. 226, 238 Stulik, K. 429 Sweigart, D. A. 215, 238, 266,306,309,

342,367 Swift, E. H. 444 Szebelledy, L.380,421 Tachikawa, H. 266, 309 Tafel, J. 325, 326,367 Talmor, D. 244 Tanaka, N. 377,421 Testa, A. C. 290, 309 Thiebault, A. 303, 309 Thirsk, H. R. 432 Thomas, U. B. 53,54 Tilak, B. V. 430 Tilset, ~. 299, 309 Tiselius, A. 84, 105, 117 Tobias, C. W. 432 Tome'§, J. 158,237 Tomkins, R. P. T. 431 Tomlinson, C. 427 Trasatti, S. 337, 367 Tremillon, B. 427 Tropp, C. 280,309 Turner, J. A. 206,238 Uhlig, H. H. 418,422,431 Underwood, A. L. 237,239

476

van't Hoff, J. H. 396 Vandenbalck, J. L 236,239 Verhoef, J. C. 388,422 Vesely, J. 429 Vetter, K J. 327, 330, 367, 428 Visco, S. J. 308,309 Volmer, M. 318,367 Volta, A 4, 371 Vukovic, M. 206, 238 Waller, A G. 258, 308 Wawzonek,S.39,54,392,422 Weaver,M.J.350,367 Weinberg, N. L. 395, 422, 429,430 Weiss, D. 429 Weissberger, A. 428 Westcott, C. C. 429 Weston, E. 7 Whewell, W. 4 White, R. E. 431 Wien, M. 124, 147 Wiese, G. R. 70,105 Wiesner, K 278, 280, 309 Wightman, R. M. 215, 223,238 Willihnganz, E. 51,54 Winograd, N. 171, 237 Wipf, D. O. 215,238 Wise, J. A. 389, 422 Wopschall, R. H. 190, 238 Yeager, E. B. 124, 147,432 Yoshida, K 430 Zana, R. 124, 147 Zhan~ Y.266,309,342,367 Zoski, C. G. 215, 217,238,341,367 Zuman, P. 236,239,247,308,431

Author Index

SUBJECT INDEX

a.c. cyclic voltammetry 366 a.c. polarography 322, 356-366

EC mechanism 362-366 irreversible process 360-362 second-harmonic 358

activation free energy 316 activity 9

of solvent 100-102 activity coefficients 9, 17

experimental determination 96-98 from Debye-Hiickel theory 91-95 mean ionic activity coefficients 93 molal & mole fraction scales 99-100

adsorption effect on cyclic voltammetry 189-190 effect on polarography 198-199

aluminum production 401-2 anodic stripping voltammetry 389-390 anodization 400 band theory of solids 1 biamperometric titrations 388 biochemical half-cell potentials

table 442 Butler-Volmer equation 318 calomel electrode 31, 170 catalytic current 280 cathodic stripping voltammetry 390 CE mechanism 256, 274-278, 287, 296 charge-transfer resistance 326-327, 354 chemical analysis

a.c. polarography 358 amperometric titration 177-178 biamperometric titration 388 conductometric titration 127-128 constant current coulometry 384 constant potential coulometry 380 coulometric titration 385 dead-stop titration 388 electrogravimetry 377-379 electrolysis methods 376-390 ion-selective electrodes 40-42

477

Karl Fischer titration 387 polarography 231-236, 281 potentiometric titration 42-44 stripping voltammetry 389-390 of cyanide ion 245 of hydroxide ion 234 of metal ions 223 ofthiols 235

chemical kinetics 247-249 chemical potential 91

electrochemical potential 134 chlor-alkaIi process 404-407 chronoamperometry 179-181

CE mechanism 287 double potential step 180, 289 EC mechanism 288-289 EC' mechanism 292-293 ECE mechanisms 293, 294, 295 capacitive charging current 180, 220 irreversible process 346-348 time scale 250

chronocoulometry 241, 369 chronopotentiometry 181-183

current reversal 182, 290-291 EC mechanism 290-291 time scale 250

colloids 68-73, 81-85 electrophoresis 82-85 electroviscous effect 81 sedimentation potential 82 stability of 69-71 surface pH 72 surface potential 64, 72

conductance 109 measurement 110-113 relation to conductivity 109 relation to resistance 109

conductivity 109-124 electrolyte charge type from 127 equilibrium constants from 125 at high electric field strength 124

478

at high frequency 124 ionic conductivity 115 of pure water 126 relation to diffusion 134 relation to mobility 119 theory of 122-124 units 109

conductometric titrations 128 convection 152

r.d.e. forced convection 208 conventions

potential scale zero 12 sign of current 151, 195 sign of potential 7 sign of work 5 writing half-cell reactions 11

corrosion 412-421 corrosion inhibitors 420 differential aeration 419 passivation 417-418, 421 prevention of corrosion 420-421 reaction of metal with air 416 reaction of metal with water 413-416 sacrificial anodes 420

corrosion potential 414 Cottrell equation 156 coulometric titrations 387 coulometry 380-388

constant potential coulometry 380 constant current coulometry 384

current-potential curve 155-162 cyclic voltammetry 183-194

adsorption effects 189-190 capacitive charging current 185, 223 CE mechanism 296 EC mechanism 296-300 EC' mechanism 300 ECE mechanisms 301-308 derivative presentation 190-192 irreversible,quasi-reversible 348 ohmic potential drop 185 semiderivative presentation 192-194 time scale 250

Daniell cell 2 dead-stop titration 388 Debye length 66, 92 Debye-Falkenhagen effect 124 Debye-Huckel theory 90-98

comparison with experiment 95-98 extensions from 102-104 limiting law 95 relation to conductivity 122

dielectric constant 61, 169 table 446

Subject Index

diffusion 128-136 Fick's first law 130 Fick's second law 131 flux 130 random walk model 128-130 to a microdisk electrode 216-219 to a planar electrode 153-162 to a spherical electrode 162-165

diffusion coefficient 130 relation to conductivity 135 relation to frictional coefficient 134 relation to mobility 133-135

diffusion layer thickness 158, 195, 211 diffusion-limited current 152-164, 196 digital simulation 462-466 Donnan membrane potential 141-143 double layer

effect on colloid stability 68 electron-transfer rate 323-324 interfacial tension 86-88

Gouy-Chapman theory 59-68 surface pH 72, 410 thickness of 66, 92

double layer capacitance 88-90, 110,351 effect on

chronoamperometry 180, 220 cyclic voltammetry 185, 223 polarography 197, 202-206

dropping Hg electrode 172, 194-201 effect of potential on drop time 86 homogeneous kinetics at 272 see polarography

dry cell 45 Ebsworth diagrams 20 EC mechanism 256, 279, 288-291, 296-

300, 362-366 EC' mechanism 256, 280-281, 292-293,

300 ECE mechanisms 256, 282-287,293-295,

301-308 Edison cell 52 EE mechanism 332-334, 342-345 efficiency of fuel cells 47 electric migration 152 electrical circuits

a.c. bridge 112 cell equivalent circuit 110, 351 current source 168 electrometer 30 constant current source 386 for measuring a cell potential 28, 29 potentiometer 29 operational amplifier 30, 166

Subject Index

current follower 165 ramp generator 167 voltage follower 165 coulometer 382 potentiostat 165 galvanostat 386 voltage integrator 167, 382 Wheatstone bridge 112

electrocapillarity 86-88 electrochemical cell potentials

measurement of 27 electrochemical cells

conductance cells 113 for coulometry 380 Daniell cell 2, 151 Edison cell 52 for electrogravimetry 379 electrolysis cells 5, 151 equivalent circuit 110,351 fuel cells 47-50 galvanic cells 2 lead-acid cell 51 Leclanche cell (dry cell) 45 mercury cell 46 nickel-cadmium cell 53 production of aluminum 402 production of Cl2 & NaOH 405 production of lead tetraalkyls 411 silver cell 47 silver-zinc cell 58 sodium-sulfur cell 53 storage batteries 50-54 thermodynamics 5 three-electrode configuration 165 for voltammetry 173 Weston cell 7-12, 29

electrochemical potential 134 electrodes

anode 4 auxiliary electrode 165 cathode 4 in conductance cell 113 for stripping analysis 389 see indicator electrodes see reference electrodes

electrogravimetric analysis 377-379 electrokinetic phenomena 73-80

electroosmosis 75 electroosmotic pressure 75 streaming current 73 streaming potential 74 theory of 76-79 zeta potential 75, 79-80, 83

electrolysis 391 analytical applications 376-390 current efficiency 374 electroseparation 374-375 electrosynthesis 390-396 industrial processes 396-412

electrometer 30 electromotive force 6

479

electron spin resonance 258-264 electron-transfer rate 316-319, 321-324,

360,366 Frumkin effect 324 Marcus theory 319-322

electroosmosis 75 electroosmotic pressure 75 electrophoresis 82-85 electrophoretic painting 400 electroplating 398-400 electrorefining 403 electrosynthesis 390-396

Kolbe hydrocarbon synthesis 395-396 oxidation of olefins 394-395 reduction of aromatics 392 reductive elimination reactions 391

equation Boltzmann distribution law 62 Butler-Volmer equation 318 Cottrell equation 156 Debye-Huckel limiting law 95 Einstein relation 135 Fick's first law 130 Fick's second law 131 Gibbs-Duhem equation 86, 101 Henderson equation 140 Heyrovsky-Ilcovic equation 157 Ilkovic equation 196 Kohlrausch equation 114 Levich equation 212 Lingane equation 230 Nernst equation 10-13 Nernst-Einstein equation 135 Nernst-Planck equation 134 Ohm's law 76 Ostwald's dilution law 126 Poiseuille's equation 76 Poisson equation 61 Poisson-Boltzmann equation 63 Sand equation 182 Stokes law 82 Tafel equation 325

equilibrium constants from cell potential data 14 from conductance data 125 from polarographic data 227-231

480

error function 131-132 exchange current 318 faradaic impedance 110, 351-356 Faraday's laws of electrolysis 371, 380 Fermi level 1 ferrocene

as a potential reference 171 rate of oxidation 350

Fick's laws of diffusion 128 Flade potential 417 formal potentials

table 444 Franck-Condon principle 319 free energy-oxidation state diagrams

19-21 frictional coefficient 120

relation to diffusion coefficient 134 Frost diagrams 19-21 Frumkin effect 324 fuel cells 47-50 galvanic cells 2 Gibbs free energy

of activation 316-317 from cell potential data 14, 224-227 relation to electrical work 5

Gouy layer 60 Gouy-Chapman theory 59-68 half-cell potentials

table 438 half-cell reactions 8 half-wave potential 157, 212

correlation with IR frequencies 226 correlation with MO theory 224 stability constants from 227-231 table 233

Hall-Heroult process 401 Helmholtz layer 60 heterogeneous rate constants 316 Heyrovsky-IlcovU: equation 157 history of electrochemistry

double layer theory 60 conductivity 114-115 electrolysis & Faraday's laws 371 origins 3-4 polarography 194

hydrodynamic layer thickness 209 hydrogen evolution kinetics 334-337 indicator electrodes 171-172

dropping Hg electrode 172, 194-201 glass electrode 35 ion-selective electrode 35-39, 143-146 microelectrodes 215-223 quinhydrone electrode 56 rotating-disk electrode 172, 207-215

Subject Index

rotating platinum electrode 207 rotating ring-disk electrode 214 static Hg drop electrode 206-207

industrial processes 396-412 anodization 400 electrophoretic painting 400 electroplating 398-399 electrorefining 403 hydrometallurgical processes 403 organic syntheses 412 production of

adiponitrile 409-410 alkalies & alkaline earths 403 aluminum 401-402 chlorates and bromates 407 Cl2 and NaOH 404-407 fluorine 407 lead tetraalkyls 410-412 manganese dioxide 408 perchlorates 407 potassium dichromate 408 potassium permanganate 408

infrared spectroscopy 265-268 ion-selective electrode 35-39, 143-146 ionic conductivity

table 445 ionic radii

crystal radii 121 Stokes law radii 120-121

ionic strength 63 kinetic current 274 kinetic zones

CE mechanism 275-276 EC' mechanism 292-293

kinetics of electron transfer 315-324 Kohlrausch law of independent ionic

migration 115 Kolbe hydrocarbon synthesis 395 Laplace transforms 448-461

table 450 Latimer diagrams 18 lead-acid cell 51 Lec1anche cell 45 liquid junction potentials 2, 136-141 London force 69 lyotropic series 71 Marcus theory 319-322 mass transport rate constants 159, 211,

218 maximum suppressor 200-201 mechanisms 247

CE 256, 274-278, 287, 296 EC 256, 279, 288-291, 296-300, 362-366 Ee' 256, 280-281, 292-293, 300

Subject Index

ECE 256, 282-287, 293-295, 301-308 EE 332-334, 342-345 bond cleavage 251 electrophilic attack 252 multi-electron processes 249 multi-step processes 328-334 reactions of olefin radical cations

394 rearrangement 253 reductive elimination reactions 391 reduction of aromatic

hydrocarbons, nitro and carbonyl compounds 293

mechanistic data on hydrogen evolution 334-337 oxidation of

p-aminophenol291 ArCr(CO)2(alkyne) 263 iron 333 (mesitylene)W(CO)3 266 Mn(CO)3(dppm)CI 253 CpMn(CO>2L 305 9-phenylanthracene 299

reduction of azobenzene 289 (COT)CoCp 363 cyclooctatetraene 322 triiodide ion 330 Mn3+ 327 CpMn(NO)(CO)2 306 p-nitrosophenol 295 1,1,2,3,3-pentacyanopropenide 261

mercury cell 46 microelectrode voltammetry

see steady-state voltammetry, cyclic voltammetry

microelectrodes 215-223 electron transfer kinetics 338, 350 homogeneous reactions at 272

mobility electrophoretic mobility 82-84 ionic mobility 118-119 relation to conductivity 119 relation to diffusion coefficient 133

molecular orbital theory 224 N ernst diffusion layer 158 Nernst equation 10-13 nickel-cadmium cell 53 ohmic potential drop

in cyclic voltammetry 185, 191, 193 in steady· state voltammetry 213,

220,222 Onsager reciprocal relations 77 osmotic pressure 142

overpotential (overvoltage) 317 oxidation state diagrams 19 peak polarogram 183 pH measurements 40 phase angle 352 planar diffusion 153-162 polarography 176, 194-201,231-234

a.c. polarography 322, 356-366 adsorption effects 198-199 analytical applications 231-234 anodic waves 235 capacitive charging current 197,

ID2-200 CE mechanism 274 criteria for reversibility 345 current maxima 201 differential pulse 204 EC mechanism 279 EC' mechanism 281 ECE mechanisms 282 instrumentation 165-168 irreversible waves 348 maximum suppressors 200 peak polarography 183 polarographic wave 157, 161 pulse polargraphy 202 resolution 162, 205

481

reverse pulse polarography 204 sampled d.c. (tast) polarography 201 sensitivity 197,201-206 square wave 206 stability constants from 227-231 static Hg drop electrode 206 time scale 250 total electrolysis in 198

potential corrosion potential 414 between dissimilar conductors 1 Donnan membrane potential 141-

143 electrochemical cell potential 6 ferrocene as a potential standard

171 Flade potential 417 formal potentials 17, 154 half-cell potentials 11-16 half-wave potential 157,212 ion-selective membrane 37, 143·146 Latimer diagrams 18 liquid junction potential 2, 136-141 measurement 27 potential of zero charge 87-90, 324 sedimentation potential 82 standard reference half-cell 12

482

surface potential 64, 66 ultrasonic vibration potential 124 zeta potential 75, 79-80, 83

potential range for solvents 447 potential-pH diagrams 25-26 potentiometric titration 42-44 potentiostat 165-166 predominance area diagrams

(Pourbaix diagrams) 26-27, 413 r.d.e. voltammetry

see steady-state voltammetry stripping analysis 389

rate laws 247 rate of electron transfer in

oxidation of ferrocene 351 reduction of

(COT)CoCp 363 cyclooctatetraene 322 Mn porphyrin complex 342 substituted stilbenes 361

reaction layer thickness· 251, 270-273 reference electrodes 31, 170-171

Ag/AgCI electrode 32, 170 calomel electrode 31, 170 hydrogen electrode 12 Luggin probe 168 table of potentials 444

residence time 250, 338 resistance 109 resistivity 109 reversibility

operational definition 28 reversibility, criteria for

a.c. polarography 360 cyclic voltammetry 188, 348 steady-state voltammetry 158, 345

rotating platinum electrode 207 rotating ring-disk electrode 214 rotating-disk electrode 172, 207-215

homogeneous reactions at 270 Sack effect 124 salting-out effect 69-71 sedimentation potential 82 silver cell 47 silver-zinc cell 58 silver/silver chloride electrode 32, 170 sodium-sulfur cell 53 solvents

choice of 168-169 properties 169 table of properties 446,447

spectroelectrochemistry 257 electron spin resonance 258-264 infrared spectroscopy 265-268

Subject Index

spherical diffusion 162-165,216-219 square scheme 263, 305, 307, 364 standard states 9,17,21,91 static Hg drop electrode 206 steady-state voltammetry 176, 212, 220

CE mechanism 274-278 coupled homogeneous reactions 269 criteria for reversibility 345 EC mechanism 279 EC' mechanism 280-281 ECE mechanisms 282-287 EE process 342-345 irreversible/quasi-reversible 338 time scale 250

Stem model for double layer 60 Stokes' law 120 Stokes' law radii 120-121 storage batteries 50-54 streaming current 73 streaming potential 74 stripping voltammetry 389-390 supporting electrolyte 153, 170 surface charge density 66, 86, 89 surface tension

effect of potential on 86-88 symmetry factor 319, 333 table

biochemical half-cells 442-443 conductivities, mobilities and

diffusion coefficients 136 formal potentials 444 half-cell potentials 438-441 ion atmosphere thickness 66 ionic conductivity 445 Laplace transforms 450 list of symbols 434-437 parameters for extended Debye-

Huckel theory 104 physical constants 434 polarographic data 234 potential range 447 reference electrode potentials 444 SI units 433 solvent properties 446 Stokes law and crystal radii 121

Tafel plot 325-326 tast polarography 201 thermodynamics 5 thickness

ion atmosphere thickness 66 diffusion layer thickness 158, 195,

210 hydrodynamic layer 209 reaction layer thickness 251, 270

Subject Index

time scales for experiments 250 titration

amperometric titration 177 biamperometric titration 388 conductometric titration 128 coulometric titration 385 dead-stop titration 388 Karl Fischer titration 387 potentiometric 42, 176

Tomes:riteria for reversibility 158, 345 transfer coefficient 317-323, 324, 333 transference numbers 116-118

Hifforf method 116-117 moving boundary method 117

transition state theory 316 transition time 182 transport impedance 355 transport processes 128 ultrasonic vibration potential 124 units

cell potential and free energy 7 concentration 62 conductivity 109

483

diffusion coefficient 130 for variables used in text 434-437 heterogeneous rate constants 159,

316 kinematic viscosity 208 resistivity 109 SI units 433

van der Waals attraction 69 vapor pressure 169

table 446 viscosity 169

table 446 voltammetry

differential pulse 204 linear potential sweep 183 square wave 206 see steady-state voltammetry, cyclic

voltammetry Warburg impedance 355 Wheatstone bridge 112 Wien effect 124 work

electrical work 5-6 surface tension - area work 86

zeta potential 75, 79-80, 83