Characteristics of the dropping mercury electrode below the limiting current

J Ele~troanal Chem, 129 (1981) 53-65

53

Elsevier Sequom S A, Lausanne--Printed m The Netherlands

CHARACTERISTICS OF T H E D R O P P I N G M E R C U R Y E L E C T R O D E B E L O W THE LIMITING C U R R E N T

CLARENCE G LAW, Jr *, RICHARD POLLARD ** and JOHN NEWMAN Materials and Molecular Research Division, Lawrence Berkeley Laboratory and Department of Chenu~al Engineering, Umversttv of Cahforma, Berkeler', CA 94720 (U.S A )

(Received 8th September 1980, in revised form 17th April 1981)

ABSTRACT A model has been developed to describe the combined effect of electrochemical kinetics, ohmic potential drop and mass transfer for a metal-deposmon reaction at a dropping mercury electrode Current, potentml and surface concentration are expressed m terms of two parameters which characterize the results For certain parameters, even m the presence of an excess of supporting electrolyte, the current density from a metal-deposmon reaction can exceed the mass-transfer hm~t calculated by Ilkov~

INTRODUCTION P o l a r o g r a p h i c analysis with a d r o p p i n g m e r c u r y electrode ( D M E ) is usually carried out in the presence of a large excess of indifferent, n o n - r e a c t i n g electrolyte. This serves to reduce the o h m i c p o t e n t i a l d r o p in the solution a n d to reduce the effect of the electric field on the m o v e m e n t of reacting ionic species. F o r a sufficiently large a p p l i e d potential, the current to the d r o p is limited b y the rate of diffusion a n d convection a n d c o r r e s p o n d s to a zero c o n c e n t r a t i o n of r e a c t a n t at the surface. I n this situation, the c a t h o d i c c u r r e n t d e n s i t y is given b y the Ilkovi~ e q u a t i o n [1-3], p r o v i d e d that the volumetric flow rate of m e r c u r y t h r o u g h the c a p i l l a r y is constant. T h e o r e t i c a l e q u a t i o n s for p o l a r o g r a p h i c limiting c u r r e n t s have also b e e n develo p e d for a n u m b e r of systems involving specific c o m b i n a t i o n s of c h e m i c a l a n d electrochemical r e a c t i o n s [4-9]. These studies have focused a t t e n t i o n on the interactions b e t w e e n h o m o g e n e o u s a n d h e t e r o g e n e o u s processes and, in general, the i m p o r t a n c e of o h m i c p o t e n t i a l d r o p in the s o l u t i o n a n d surface o v e r p o t e n t i a l s for the electrochemical reactions has n o t been evaluated. However, early q u a l i t a t i v e studies i n d i c a t e d that, if there is insufficient s u p p o r t i n g electrolyte, the c u r r e n t d u e to one discharging species c o u l d p r o d u c e an electric field that e n h a n c e s the limiting c u r r e n t for o t h e r r e a c t a n t s [1,10]. * Present address Western Elecmc Co, Engineering Research Center, Princeton, NJ 08540, U S A ** Present address Chemical Englneenng Dept, Umverslty of Houston, Houston, TX 77004, U S A

54 The influence of ohmic potential drop on the distribution of current has been analyzed for disk, nng and ring-disk electrodes [11-13], as well as for planar [14], tubular [15] and spherical electrodes [16]. With smaller electrolyte conductivity, the distributions of current and concentration become more non-uniform and, under some circumstances, local current densities can exceed the local limiting current [17]. The instantaneous current and the average current to a DME in a binary salt solution have been calculated [18]. This analysis showed that ohmic potential drop can prevent the attainment of a limiting current during the initial stage of growth of the drop, particularly if the applied voltage is small. In this paper, a general model is presented for the DME below the limiting current. The analysis includes the effects of mass transfer, ohmic potential drop in the solution and electrode kinetics. Factors that govern the relative importance of these effects are identified for the example of a metal-deposition reaction. It is pointed out that the general approach presented here can be used to evaluate experimental situations different from those of traditional polarography. For example, the potential may not be constant throughout the life of the drop or the drop may not grow with the cube root of time. ANALYSIS At currents below the limiting current, it is necessary to consider the surface overpotential associated with the electrode reaction, the ohmic potential drop in the bulk of the solution and concentration variations near the drop surface. The analysis presented here is restricted to a single electrode reaction with stoichiometry represented by (1)

~ s l M ~ ' --->n e 1

A polarization equation of the form: i = 10[e"arn~/Rr-- e-~cFn,/Rr]

(2)

can be used to express the dependence of the reaction rate on the surface overpotential, ~/s = V - ~0 - U0- The exchange current density can be written as lo : io refII (cl o/C, ref) Y'H a~k

(3)

where i and k range over the ionic and neutral species, respectively. The theoretical open-circuit cell potential is given by [19] RT~,sIln U° : U° - Ur°ef - n---ff 1

c~,___&o+R T ~sl,refl n C~,r~f Po '/~"refF , I°0

(4)

provided that activity-coefficient corrections can be neglected. Furthermore, exponents 71 for ionic species in eqn. (3) are given the values: Y, : q, + a c s , / n

(5)

55 where ql = - s l for a reactant that is reduced (sl < 0) and is zero otherwise [20]. For a metal-deposition reaction, and with a reference electrode of the same kind, eqn. (4) reduces to Uo = - ( s l R T / n r )

ln(01/0m)

(6)

where the dimensionless concentrations of metal and metal ion are given by 01 = q,o /G,ref and 0m = Cm,o/Cm,re f respectively. The activity coefficients of metallic species are assumed to be unity. Consequently, substitution of eqns. (3), (5) and (6) into eqn. (2) gives, on rearrangement: '=/O.ref[0m e ~ " r ' v - o " ) / " r -- O, e - " ' r ( v - ° " ' / " r ]

(7)

Furthermore, the bulk solution potential ~0 can be evaluated at the drop surface from the resistance relationship for a spherical drop in a solution of uniform conductivity: (8)

dp0 -~ lro/K

For radial growth of a mercury drop without tangential surface motion, the reactant concentration obeys the equation of convective diffusion in the form: 0c 1 0c, D~ 0 ( 20c, 0l + 'or Or -- r 2 Or. r ~ ]

(9)

where the velocity v r is determined by the growth rate of the drop: (10)

vr = (ro/r)2(dro/dt)

Equation (9) can be expressed in terms of the normal distance y from the surface of the drop, provided that the diffusion layer is thin compared to the drop radius throughout the lifetime of the drop: Oc, 0t

2y dr 0 0c, 02c, r0 dt o y - D ' 0 y 2

(11)

A smfilar equation applies inside the sphere, but with the &ffusion coeffioent D~ of the discharged reactant in mercury. The diffusion equation can be solved subject to the conditions: cl----cl,o~

fort~<0

c~--,cl,o~

asr--,~

c,----C~,o(t )

fort>0

(12)

atr=r o fort>0

By superposition, the results can be used to express the concentration derivative at the surface in terms of an integral over the variation of surface concentration during the drop lifetime [21]: Oc~ y = O Oy

dt°

2r2 f f dc~'°

~l/2J 0 dt

(13)

t=to[4Dftrndt]l/2

/t

j,to

]

56 The surface fluxes, both inside and outside the drop, can be related to the instantaneous current density by an expression of the form: sli

ac 1 I

-'~ -- DI~y y~O

(14)

This equation is restricted not only to the large excess of supporting electrolyte, where the effects of migration can be neglected, but also to the absence of appreciable charging of the double layer, a process which does not follow Faraday's law. Concentration changes within the drop can be related to external changes by equating the superposltion integrals for the two regions through eqn. (14). This gives: 0m = 0m(0) -J- ( D , / D s ) ' / 2 ( 1

(15)

-- O,)/O s

where 0s = Cm,ref/Cl,ref. The model presented here is more general than the approach taken by Ilkovi~ since two basic constraints made in his development can be removed. Namely, the potential can vary throughout the hfe of the drop and can be expressed, for example,

as V : Vmt -~ Bt where/3 is the scan rate of the applied potential. Also, it is not necessary to maintain a constant flow rate of mercury through the capillary. Removal of the last constraint is particularly important in evaluating the characteristics of modem polarographic equipment. Although the results presented here do not evaluate the importance of scan rate and constant flow rate, it is appropriate to indicate the general utility of this model. When the volumetric flow rate of mercury is constant, the growth rate is given as

r0 = T/l/3

(16)

With this growth rate, the governing equations (7) and (13) for the DME below the limiting current can be expressed in dimensionless form as 801

d(N'-6)

N:NI

~PO~ -gfoN-6d(g_6)

(17)

(1--[(N1-6)/(N-6)]7/3) 1/2

q~oN2 ----g[o m e -aa(E+q'°)/ac -01 e e+~'o]

(18)

where % = a c F r o t / g R T , E = - a c F V / R T

and N -6 is a dimensionless time given by

N_6:

I

KRTs, , acnF2c,,o./]

]2 3~]3

(19)

7D 1 ] t

The dimensionless parameter K represents a combination of quantities associated

57 with kinetic, ohmic and mass-transfer effects: 7io ref

K= ~

[ acF'Y ~3[ Fn

~2

~'--~1 I T cl'O0) D1

(20)

In eqns. (17) and (18), (#o and 81 are dependent variables and N -6 is the independent variable. The parameters E and K are expected to have a slgmficant impact on the system behavior, whereas aa/a~, D J D s, 0s, and 0m(0) are of relatively minor importance and should not markedly influence the results. The governing equations (17) and (18) are solved by a stepwise numerical procedure that involves discretization of the integral equation and a NewtonRaphson technique to obtain values for qb and 01 at each time step [22,23]. Since the variables may vary very rapidly at short times, it is necessary to vary the step size to ensure accurate results. In addition, the initial singularity in eqn. (17) is avoided by using a short-time series expansion for the concentration derivative over the first time interval. RESULTS AND DISCUSSION The time dependence of the dimensionless potential ¢0 is presented in Fig. 1 for several values of the parameters E and K. This diagram also depicts changes in the instantaneous current density through the relationship, ¢0 = acFroi/~RT" Lines of slope 1//3, 0 and - 1 / 6 represent the kinetic, ohmic and mass-transfer limits respectively. Figure 1 illustrates that it is not possible to generalize the results 10z E=20-E=IO--

/ E-IO~ f J r

f JJ

j/~

i0-~

~ 10-16

I 10-12

I

I

I

I0 - 8

I0 - 4

I

103

N-6

Fig 1 Time dependenceof dimensionless,instantaneouscurrent denmty,~o =acFroz/xRT, for a metal deposmon reacUon at a growing mercury drop Parameter values. D,/Ds=l 0; aa/ac=l O, Os=l.0, .... )K=10 4 Om(O)=O O' ¢],ref =¢1,c¢" ( ) K= 10 -10, ( - - - - - - ) K = 1 0 - 3 , (

58

for large and small values of E and K. Clearly, both parameters are influential in determining ~'0- For large times and moderate to large values of E, t0 is independent of K and E and a single line of slope - 1/6 is obtained in Fig. 1, in accordance with the Ilkovi~ equation. However, at short times, kinetic factors and, subsequently, ohmic factors can prevent attainment of the mass-transfer limit. These effects are particularly important for small values of K or E. For E = 0.2 and K = 10 -l° or 10-3, kinetic control, with a slope of 1/3, is predicted over the complete time range considered. A reduction in K corresponds to a small exchange current density, bulk reactant concentration, diffusion coefficient or drop growth rate, or a large electrolyte conductivity. A larger electrolyte conductivity will also reduce the ohmic potential drop in the solution, and consequently ohmic limitations are less prevalent with small values of K, for a specified magnitude of E. Furthermore, the effect of K is more pronounced at small values of E. The parameter E is a dimensionless applied potential which includes the cathodic transfer coefficient for the deposition reaction. As E is increased, ohmic factors have progressively more impact upon the short-time behavior. The ohmic limit is given by q'o = E

(21)

For E = 80, the three curves in Fig. 1 are almost horizontal, corresponding to ohmic limitations, and they are almost superimposed upon each other. However, even under these conditions, the curves are not precisely horizontal due to the finite rate of the electrochemical reaction. The mass-transfer limit is represented by epo / N = 1 - exp(nE/sla c)

(22)

where q , o / N = i / i h m = 1 -- Ca,o/C~,oo

(23)

At large applied potentials, the intersection of the ohmic and mass-transfer limits can be identified from eqns. (21) and (22) as q~0 = N = E. Figure 2 shows the time dependence of the average current density defined by iavg = (1/Td)f0Td/dl

(24)

for fixed values of E and K. This average current is made dimensionless in the same manner as Fig. 1. Total currents can be obtained directly from the relation I = 4rrro2i. Figures 1 and 2 are analogous, except that the magnitude of the current densities in Fig. 2 have been altered in accordance with eqn. (24). The time dependence of the surface concentration is presented in Fig. 3. Rapid reductions in composition are observed for large values of E and K, in keeping with the early onset of mass-transfer limitations predicted in Fig. 1, for similar conditions. With small applied potentials, and particularly for small values of K, kinetic factors can control the deposition rate, and the corresponding variations in concentration are less marked. Figure 4 shows the variations in instantaneous current density normalized with

59 102 E:20, E=IO. -

--

- -

__ __ ~

d---------:

--

I

~E:02

~

~

~

~

jzj

i0-4

//. /

iO-e

[0-12

I0 -16

I0-12

I0-8

IO-4

I

I0 5

N-6

Fig 2 Time dependence of dimensionless average current density, etc FroiaVg/K R T, for a metal deposition reaction at a growing mercury drop Parameters as m FIg 1

the mass-transfer limiting current density defined by eqn. (23). Values in excess of the mass-transfer limit of Ilkovi~ are obtained. This is similar to results obtained with disk [11], ring [12] and plane [14] electrodes. In transient stagnant-diffusion-cell

I 001~-

I-

- - ~ ~

\

-~.

~

\

°8°I-\\\

I

? 0

x

~

\

'),\~-~o\ \\ \

\

~

,

\1

/

\~

H

\

/

\ \

E=o 2I I

\-i

\

\ \

\

E=O 2

E:2O\

\\

\ \

2, IO E•O

~

~

E:o 2 -., ~ ~- -.. \E:,o

\

~ \-

o6oi\\,.~I.... ~ \

I. . . .

~-..~.

~

\

/

\l \1

040

°- I

°2°F

0 0010-16

',',,I , ' , , "'~/

IO-12

\

I0 - 8

I

',

' \

\\

I0 - 4

I

I03

N-6

Fig 3 Time dependence of dimensionless surface concentrat]on, 0, Parameters as m Fig 1

60 I 20

," [\xEx,,.

0 80

//

E:80~ 060

/ //

E

<

040

~ ..

_

II/ I E:lO/I //" // ii I / I/ I ii /

I00

/fP / / /

/

/

I

i

/ /

/

I

E:2o/ /

//E:2o

/

/

Jl 00C 10-16

/E:lo

~

/.I

10-12

/ /

E:o~. ¢

/

/ / ~/J /

/

/. //

/

i

///

/// 0 20

/

I

-/

//

=~i io--8

' E:20//

, ___.~= E=o2 i /io 3

io-~

E:O2,10

N-6

Fig 4 Tzme dependence of the instantaneous current density normahzed w~th the instantaneous current density in the mass-transfer h n u t Parameters as m F~g l

,2o] I 001-

-/-~

/

".'F

I

080

~:,o/

.,L,"I,',; / /

~:~o~-/

/ / / • //

, ~

/

////,/

~" 060 "o, ~

/

/./ /

I

,~

/~=2o /

//

/

E:,O// /

/ /

/,/ o~o ' / / " '/~/

/

10-16

//

/

,/

/

I0- I z

/

//

/

/

;" / /

I0 -8

N-6

.-/

ii ii

/

~'/ ~ .' / /

-

/

IIl//E:20/I I ii

040

,'

~-

/

i/ i/ /

/

/

"

-/

-y //

/E=IO

10- 4

/

/E:02

/

/ /-

E=20/

/

I /10

3

E=O2, io

Fzg 5 Time dependence of the average current denszty normahzed wzth the average current denszty prechcted with the Ilkov]~ equation. Parameters as m Fig. l

61

I

,oo

i

I

I

~

;

-

~

I0

iI

O8O

EffiI0

//i

/

O6O

2o o20

ooo

,~

11

// .__.

I0-16

iO-e N-6

Fig 6 Time dependence of the instantaneous current density n o r m a h z e d with the instantaneous current density m the mass transfer hmlt Parameters' t o.r~f = 10 - 3 A c m - 2 , DI = 10 - 5 cm 2 s - ' , cl, oo = q,ref = 10 - 5 mol cm -3, n = 1.0, a a = a c = 0 5 , T = 2 5 ° C , D1/Ds=I.O; O s = l 0; 0m(0)=0 0. ( - - - - - - ) K = I 0 - 3 (1 e x/'r=O 371 s 1/3 f~- I c m - 2 ) ; ( . . . . ) K = 104 (Le ~ / ' / = 1 721 × 10 --3 St/3 fl -- l crn 2), ( ) classical polarograpluc theory [25,26] (A,C) K = 104, (B) g = 10 -3.

experiments, current densities measured and calculated by Hsueh and Newman were found to overshoot the mass-transfer limit [24]. Material adjacent to the drop surface that does not react at short times can do so subsequently, when kinetic and ohmic factors no longer limit the reaction rate. In contrast, Fig. 5 illustrates the average current density obtained from eqn. (24) which rises monotonically to the average limiting current density calculated with the Ilkovl~ equation. The average current density cannot exceed the average limiting current density. Figure 6 shows the influence of ohmic potential drop on the time dependence of the instantaneous current density. Values for the exchange current density, diffusion coefficient and bulk reactant concentration are specified to allow comparison with classical polarographic theory [25,26]. Consequently, changes in the parameter K correspond directly to changes in bulk electrolyte conductivity, in accordance with eqn. (20). With K = 104 and E = 10, there is a significant difference between the classical result (curve A), which does not include ohmic effects, and the present model. The potential drop in the solution reduces the effective driving force for the electrochemical reaction and a reduction in instantaneous current density results. Curve B is identical to curve A, but it is shifted on the dimensionless time scale to correspond to K--- 10 -3. In this case, the difference between the theories is less marked because there is less potential drop in the electrolyte. Additional calculations show that, with K = 10 -I° and E - - 10, ohmic effects are negligible and the two approaches correspond exactly. Figure 6 also includes results for E = 0.2 to emphasize that the importance of ohmic drop is also dependent on the applied voltage.

62 The analysis considered above does not account for the capacitive current needed to charge the mercury-solution interface. To assess the effects of the capacitive t:urrent the total current can be expressed as I -- If + Inf

(25)

where, for linear kinetics,

If = 4rrr2to,ret(F/RT)(aa + a c ) ( V - ~0 -- Vo)

(26)

The capacitive term is d Inf = ~-~ [4~rr2(qo +

C[V-tb o - Uo])]

(27)

where qo is the charge on the interface when V - ~o = Uo- Substitution of eqns. (26) and (27) into eqn. (25) with use of eqn. (8) yields: 4~-xy/1/3~o _--

4~y2tz/3tO,ref-~T

( a a + a c ) ( V - - qb0 -- Uo)

3tW3 tqo +C(V_dPo

+8¢ryz[

Uo)]-4~ry 2t 2"'3 / C d(I)o ~t

(28)

This equation can be rearranged to show the importance of the faradaic and non-faradaic contribution to the total current:

(o0)

tc-

1

(o0)

V - Wo

c 1

+t°'ref(aa +ac)Ft 1 __

2

q0

V-""~og° )

-~ xdp°t2/3

3 (v-

V - Wo

(29)

v(V-to)

The last term on the left-hand side of eqn. (29) is the result of the faradalc process, the last term on the right-hand side represents the total current, and the remaining terms account for the non-faradaic process. The charging current, represented by the second term in eqn. (29), should be considered for t<-

2

cgv +ac)F

(30)

3 lO,ref(O/a

For typical values of the parameters C = 30 # F cm -2, t0,re~ = 1 0 - 4 A c m - 2 , ( a a + ac) = 1, the non-faradaic current is eqmvalent to the faradaic current at about 5 ms. The criterion expressed by eqn. (30) only applies for i0,ref(Ota "at-Otc) F7 [ C~I ] '/1 2• ~--~ ~-~- ] < 1

(31)

(the usual case). In the contrary case, the charging current (represented by the first

63 term in eqn. 29) would need to be considered for

t<(CT/3x) 3/2

(32)

One should be reminded that eqn. (29) has not considered mass-transfer effects. S U M M A R Y A N D CONCLUSIONS

A model is presented for the current and potential distributions of a D M E below the limiting current. Results are dependent upon a potential parameter E and an additional parameter K which reflects the relative importance of the kinetic, ohmic and mass-transfer resistances. For relatively large values of these parameters, the instantaneous current density of a metal deposition reaction in the presence of an excess of supporting electrolyte can exceed the mass-transfer limiting value given by Ilkovi~. ACKNOWLEDGEMENT

This work was supported by the Assistant Secretary of Conservation and Renewable Energy, Office of Advanced Conservation Technology, Electrochemical Systems Research Division of the U.S. department of Energy under Contract No. W-7405ENG-48. LIST OF SYMBOLS ak

C ¢1 C1,0 Cl,ref ¢1,o0 ¢m,0 Crn,ref

D~ E F l l avg lhm I hm,avg 10 l 0,ref

relative activity of speoes k capacity of the double layer farad, cm-2 concentration of species i, mol cm -3 surface concentration of species i, mol cm -3 reference concentration of species i, mol cm -3 bulk concentration of species 1, mol cm-3 surface concentration of discharged metal, mol cm -3 reference concentration of discharged metal, tool cm-3 diffusion of species i in the electrolyte, cm 2 s diffusion coefficient of discharged metal in mercury, cm 2 s - i dimensionless applied voltage, E = -acFV/RT Faraday's constant (96487 C mol - i) instantaneous current density, A cm -2 average current density defined by eqn. (24), A cm -2 current density in mass-transfer limit, as predicted with the Ilkovi~ equation, A cm -2 average current density in mass transfer limit, A cm -2 exchange current density, A cm -2 exchange current density at reference concentrations of reactants and products, A cm -2

64

I K n r/ref

N ql qo r

r0 R S1 Sl,ref

t T

r. uo

Uj0 V Vlnt

Vr Y

total current, A dimensionless parameter defined by eqn. (20) symbol for the chemical formula of species i number of electrons transferred in electrode reaction number of electrons transferred in reference electrode reaction dimensionless parameter defmed by eqn. (19) reaction order for cathodic reactants charge on the mercury-solution interface at open circuit, C cm -2 radial coordinate, cm drop radius, cm universal gas constant (8.3143 J mol - l K -1) stoichiometric coefficient of species i in electrode reaction stoichiometric coefficient of species i in reference electrode reaction time, s absolute temperature, K drop lifetime, s theoretical open-circuit potential for electrode reaction at the composition prevailing locally at the drop surface, relative to a reference electrode of a given kind, V standard electrode potential for reaction j, V apphed voltage, V initial applied voltage, V radial velocity, cm s - ] distance from drop surface, y = r - r0, cm

Greek letters 0~ a

O/c

Y~ Yk

~s O~

Om Ore(O) K

Po q% '/'o

transfer coefficient in anodic direction transfer coefficient in cathodic direction constant defined by eqn. (16) exponent in eqn. (3) and defined by eqn. (6) exponent in eqn. (3) surface overpotential, V dimensionless reactant concentration, G,o/C~,ref dimensionless product concentration, Cm.o/Cm,ref initial dimensionless product concentration ratio of reference concentrations, Cm,ref/Cx,ref solution conductivity, mho cm-1 density of pure solvent, g c m - 3 electric potential in the solution, immediately adjacent to the drop surface, V dimensionless electric potential, ~b0 = acFrol/xRT

65 REFERENCES 1 D Ilkovi~, Collect Czech Chem, Commun, 6 (1934) 498 2 D Ilkovl~, J Cham Phys Physlcochxm B]ol, 35 (1938) 129 3 D MacGallavry and E K Radeal, Rec Trav Chlm, 56 (1937)1013 4 C Nlshxhara and H Matsuda, J Electroanal Chem, 51 (1974) 287 5 I Ruzlc, D E Smith and S W Feldberg, J Electroanal Chem, 52 (1974) 157 6 I Ruzlc, H Sobel and D E Smith, J Electroanal Chem, 65 (1975) 21 7 C Nxshthara and H Matsuda, J Electroanal Chem, 73 (1976) 261 8 C Nash]hara and H Matsuda, J Electroanal Chem, 85 (1977) 17 9 J Galvez, A Serna, A Mohna and D Mann, J Electroanal Chem, 102 (19"79) 277 10 I Slendyk, Collect Czech Chem Commun., 3 (1931)385 11 J Newman, J Electrochem Soc, 113 (1966)1235 12 P Plenm and J Newman, J Electrochem. Soc, 125 (1978) 79 13 P Plenm and J Newman, J Electrochem Soc, 124 (1977) 701 14 W R Parnsh and J. Newman, J Electrochem Soc, 116 (1969) 169 15 R Alkxre and A M]rafi, J Electrochem Soc, 120 (1973) 1507 16 K Nlsanooglu and J Newman, J Electrochem Soc, 121 (1974) 241 17 J Newman, Electrocham Acta, 22 (1977) 903 18 T W Chapman and J Newman, J Phys Chem, 71 (1967) 241 19 R Whate, Ph D Dissertation, Umverslty of Cahforma, Berkeley, 1977 20 J Newman, Electrochemical Systems, Prennce-Hall, Englewood Chffs, N J , 1973, p 174 21 J Newman m A J Bard (Ed), Electroanalyncal Chemistry, Vol 6, Marcel Dekker, New York, 1973, p 187 22 C Wagner, J Math Phys, 32 (1954) 289 23 A Acnvos and P L Chambr6, Ind Eng Chem, 49 (1957)1025 24 L Hsueh and J Newman, J Electrochem Soc, 117 (1970) 1242 25 J Kouteck2~, Collect Czech Chem Commun, 18 (1953) 597 26 H Matsuda and Y Ayabe, Bull Chem Soc Jap, 28 (1955) 422