A modified coulostatic pulse method for fast electrode processes

J. Electroanal. Chem., 87 (1978) 155--163 © Elsevier Sequoia S.A., L a u s a n n e - Printed in The Netherlands

155

A MODIFIED COULOSTATIC PULSE METHOD FOR FAST ELECTRODE PROCESSES *

HIROAKI MATSUDA ** and SHIGERU AOYAGUI ***

Department of Electronic Chemistry ** Faculty of Engineering ***, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo 152 (Japan) (Received 2nd August 1977) ABSTRACT This paper deals with a generalized version of the galvanostatic reversed coulostatic pulse method proposed by van Leeuwen and Sluyters. The potential of the working electrode is stepped to an appropriate potential by any kind of d.c. perturbation and then allowed to relax. The relaxing potential is perturbed again at an arbitrary time by a coulostatic pulse with polarity opposite to that of the first perturbation. Tile charge of the coulostatic pulse is adjusted so that the potential vs. time curve may have a horizontal tangent at an appropriate time after the end of the pulse. The overpotential at that time is known from the potential vs. time curves obtained with and without the coulostatic pulse. Analysis of experimental results is made on the basis of the rigorous solution for the boundary value problem of diffusion. The proposed procedure is tested with success by experiments on the trisoxalatoferrate(III)/(II) electrode and the corresponding equivalent circuit of Randles type. Its applicability to such an electrode reaction as involves an unstable product is suggested.

INTRODUCTION A n e w r e l a x a t i o n m e t h o d n a m e d t h e g a l v a n o s t a t i c r e v e r s e d c o u l o s t a t i c (g.r.c.) p u l s e m e t h o d has b e e n p r e s e n t e d b y V a n L e e u w e n a n d S l u y t e r s [1]. In a g.r.c. p u l s e m e a s u r e m e n t , a g a l v a n o s t a t i c p r e - p u l s e is f o l l o w e d i m m e d i a t e l y b y a c o u l o s t a t i c p u l s e w i t h r e v e r s e d p o l a r i t y . T h e c h a r g e of t h e l a t t e r pulse is a d j u s t e d so t h a t t h e o v e r p o t e n t i a l vs. t i m e curve m a y s t a r t w i t h a h o r i z o n t a l t a n g e n t at t h e e n e d o f t h e pulse. A t t h i s m o m e n t , t h e r e f o r e , n e i t h e r t h e c a p a c i t y c u r r e n t n o r t h e f a r a d a i c c u r r e n t f l o w s t h r o u g h t h e c e l l . T h e g . r . c , p u l s e m e t h o d is q u i t e ingen i o u s b e c a u s e of this f e a t u r e . T h e c h a r g e t r a n s f e r o v e r p o t e n t i a l c a u s e d b y t h e prep u l s e can be m e a s u r e d d i r e c t l y b y a c o m p a r i s o n b e t w e e n t h e o v e r p o t e n t i a l vs. time curves o b t a i n e d by e x p e r i m e n t s with and w i t h o u t the reversed coulostatic pulse. This p a p e r deals w i t h t h e g e n e r a l i z a t i o n of t h e g.r.c, p u l s e m e t h o d . T h e prep u l s e n e e d n o t be a g a l v a n o s t a t i c p u l s e b u t it c a n be e i t h e r a p o t e n t i a l p u l s e or a c u r r e n t p u l s e w i t h an a r b i t r a r y wave f o r m . F u r t h e r m o r e , t h e r e v e r s e d c o u l o s t a t i c p u l s e n e e d n o t s t a r t i m m e d i a t e l y a f t e r t h e t e r m i n a t i o n o f t h e p r e - p u l s e . Thus, in t h e g e n e r a l i z e d v e r s i o n bf t h e r e v e r s e d c o u l o s t a t i c p u l s e m e t h o d , an a r b i t r a r y d.c. p e r t u r b a t i o n is a p p l i e d t o t h e w o r k i n g e l e c t r o d e i m m e r s e d in a s o l u t i o n con* Presented at the 19th Symposium on Polarography and Electroanalytical Chemistry, Sapporo, October 1973.

156

taining an oxidant and/or a reductant. When the perturbation ends, the perturbed electrode potential begins to relax towards the initial value. Then, a coulostatic pulse is applied to the cell at an appropriate time during the relaxation. The d.c. perturbation and the coulostatic pulse should be of opposite polarity. In the usual coulostatic pulse m e t h o d [2,3], the working electrode at the m o m e n t of pulse application is under an equilibrium potential, whereas in this method it is under a relaxing potential. The present method can thus be named the modified coulostatic pulse (m.c.p.) method. The modification requires a rigorous analysis based on the solution for the boundary value problem of diffusion. The theory is verified by the experiments on the trisoxalatoferrate (III)/(II) electrode and the corresponding equivalent circuit of Randles type. The pre-electrolysis is performed with a coulostatic pulse. Both the oxidant and the r e d u c t a n t are contained in the bulk of the solution. THEORETICAL

Consider a redox electrode reaction, O + n e ~- R, involving only soluble species, in which the rate-determining step and the over-all reaction require n electrons. The total current density, i t , is set equal to the algebraic sum of the faradaic and capacitive components, i t and ic: (1)

i t = i f + ic = it - - C d l ( d E / d t )

where E is the electrode potential and Cd~ the differential capacity of the double layer per unit area. A total current density vs. time profile and the corresponding potential vs. time profile are shown schematically in Fig. 1. The preelectrolysis is terminated at t -- to, after which the working electrode is allowed to relax to the initial potential. The broken curve in Fig. 1 shows the potential

t

q i I i i I I

"

~.

I

E3

i )

0

to

tl t~ t3

t

Fig. 1. A t o t a l c u r r e n t d e n s i t y vs. t i m e profile a n d t h e c o r r e s p o n d i n g p o t e n t i a l vs. t i m e profile in a m o d i f i e d c o u l o s t a t i c pulse m e a s u r e m e n t . The b r o k e n curve s h o w s t h e p o t e n t i a l v a r i a t i o n d u r i n g the r e l a x a t i o n process f o l l o w i n g t h e pre-electrolysis.

157

variation observed during such a r e l a x a t i o n process. In this case the following relations h o l d a m o n g the c u r r e n t densities: t > to:

(2) tf

--i'c = C d l ( d E ' / d t )

T h e p r i m e in the above and s u b s e q u e n t e q u a t i o n s designates the quantities concerning the e x p e r i m e n t w i t h o u t the reversed c o u l o s t a t i c pulse. When the coulostatic pulse with the p o l a r i t y o p p o s i t e to the pre-electrolysis is applied at t = tl, we o b t a i n the p o t e n t i a l vs. t i m e profile, r e p r e s e n t e d b y solid c u r v e in Fig. 1. T h e n we have it = 0

to ~< t ~< tl and

t2

~ t:

t

(3)

if = --ic = C d l ( d E / d t )

F u r t h e r m o r e , if the a m o u n t o f the injected charge, q, is s u f f i c i e n t l y large, the p o t e n t i a l - t i m e curve passed t h r o u g h a m a x i m u m at t = t3. T h u s t = t3

: it =

if = ic = 0

(4)

It follows f r o m eqn. (4) t h a t an e l e c t r o c h e m i c a l equilibrium state is established at t = t 3. This is a p s e u d o - e q u i l i b r i u m , because t h e surface c o n c e n t r a t i o n s o f O and R are n o t identical with the c o r r e s p o n d i n g bulk ones. T h e r e f o r e , the elect r o d e p o t e n t i a l at t = t 3, E h, can be given b y the N e r n s t e q u a t i o n : E L = E°+ (RT/nF)ln((cS)3/(cS)3

(5)

}

w h e r e E ° is the standard p o t e n t i a l and c s and c s are the surface c o n c e n t r a t i o n s o f O and R, respectively, and the o t h e r symbols have their usual significance. T h e subscript 3 d e n o t e s the q u a n t i t i e s at t = t3. It is assumed t h a t the faradaic c u r r e n t density is related t o the e l e c t r o d e potential b y the following e q u a t i o n o f B u t l e r - V o l m e r t y p e : if

=

nF ko(c s exp[--(anF/RT) (E -- E°)] -- c s e x p [ ( l -- a) (nF/RT) (E--E°)]}

(6) where k0 and a are the standard rate constant and the cathodic transfer coefficient of the charge transfer process, respectively. Now, consider only the potential region near E = E h. In this region, taking into account eqn. (5), eqn. (6) can be linearized into if `= io ( [c s

--

(cS)3]/(cS)3

--

[ cS

--

(cS)3]/(cS)3

--

(nF/RT)

(E - - Eh)}

(7)

with io -- n F k 0 ( c os ) 31 - ~ (cS)~

(s)

If the d i f f e r e n c e b e t w e e n the p o t e n t i a l - t i m e curves o b t a i n e d w i t h o u t and with the reversed c o u l o s t a t i c pulse is w i t h i n few millivolts, the linearized relation (7) can be applied t o the f o r m e r case. Thus, we have i'f -- io([(cS) ' -- ( c S ) 3 ] / ( e S ) 3 - - [(cS) ' -- ( c S ) 3 ] / ( e S ) 3 - - ( n F / R T ) ( E ' - - E ~ ) }

(9)

158

Making the d i f f e r e n c e b e t w e e n eqns. (7) and (9), we o b t a i n if - - If ' = i0{ [C s --

( c oS) ] / '" (Coh

S

--

[c s_

s , ]/(c~)3 s (cR)

(nF/RT)

_

(E - - E') }

(10)

F o r a simple e l e c t r o d e r e a c t i o n , the c o n c e n t r a t i o n s o f species O and R at the surface o f a plane e l e c t r o d e can be e x p r e s s e d b y [4]

VDj ~

o (t--u) ll2 du (-: j=O'+: j= R)

(11)

w h e r e c ° 's are the bulk c o n c e n t r a t i o n s . This e q u a t i o n can also be used in the case o f the d r o p p i n g m e r c u r y e l e c t r o d e (DME), w h e n the d u r a t i o n o f observation is s h o r t e n o u g h f o r the change of the m e r c u r y d r o p area to be neglected. Since eqn. (11) holds regardless o f the electrolysis c o n d i t i o n e m p l o y e d , we can write t h e same relation as eqn. (11) f o r the case w i t h o u t the reversed coulostatic pulse. Thus, m a k i n g the d i f f e r e n c e c S - - (cS) ' and taking i n t o c o n s i d e r a t i o n the f a c t t h a t if = i~ for t < tl, we o b t a i n f o r t > tl

1 1 ~ (if-- i~)/nF cS--(cS)'=T-N/Di X/~ !, (t--u) 1/2-du

(12)

F u r t h e r m o r e , f r o m eqns. (2) and (3) we o b t a i n for t > t 1 (it -- i}) = i t + Cdl[d(E - -

E')/dt]

(13)

I n t r o d u c i n g eqns. (12) and (13) into eqn. (10) yields it + Cdl d ( E

--E')/dt

= io - - ~

d

it - - ,,~1/2

du --

(E - - E'

(14)

with L =~

1{ ( c S ) 31~ / ~ °

+ (c

sl}

(15)

E q u a t i o n (14) is the integro-differential e q u a t i o n with r e s p e c t t o (E - - E ' ) , - w h i c h has t h e f o r m similar t o those derived f o r the single and the d o u b l e pulse galvanostatic m e t h o d s [ 5,6]. T h u s eqn. (14) can easily be solved b y a p p l y i n g the m e t h o d o f Laplace t r a n s f o r m a t i o n . T h e result o b t a i n e d is 1

E - - E ' =---~dl~

t

it(u) F ( t - - u) du

(16)

with F(t) = {17 exp(72t) erfc(TV~) " 7

exp(/72t) e r f c ( f l ~ ) } / ( f l -- 7)

(17)

w h e r e fl is d e f i n e d b y the following e q u a t i o n :

13= (ioL)/2 + {(ioL/2) 2 -- (nF/RT) (io/Cdl)} 1/2

(18)

159

and 3' given b y the same e q u a t i o n as eqn. (18) e x c e p t f o r a minus sign in f r o n t o f t h e q u a n t i t y b e t w e e n brackets. When it is assumed t h a t the reversed c o u l o s t a t i c pulse is expressed in t e r m s o f the Dirac delta f u n c t i o n , the t o t a l c u r r e n t d e n s i t y i t can be given b y it

=

q ~(t - - tl)

(19)

T h e n , a f t e r i n t r o d u c i n g eqn. (19) into eqn. (16) and p e r f o r m i n g t h e i n d i c a t e d integration, eqn. (16) can be r e d u c e d t o E - - E ' = - - ( q / C d l ) F(t - - tl)

(20)

F r e q u e n t l y it occurs t h a t the pulse shape b e c o m e s r o u g h l y Gaussian because the rise time o f the pulse is c o m p a r a b l e w i t h the pulse w i d t h A. Even in such a case, eqn. (20) holds w i t h i n an e r r o r o f O ( [ A / ( t - - Q)]2). E q u a t i o n (20) with t = t 1 provides t h e following e x p r e s s i o n f o r CdV Cd, = [ ( - - q ) / ( E ~ - - E~)] F(t 3 -- ti)

(21)

F u r t h e r m o r e , since d E / d t = 0 at t = t3, the following relation is o b t a i n e d :

--(dE'/dt)a =(--q/Cdl)(nF/RT)(io/Cdl) G(t3 -- tl)

(22)

with G(t) = (1//~3') dF(t)/dt

(23)

Equations (21) and (22) together yield (Eh--E3) 2 (dE'/dt)3

(--qi

=[.

'~1

[F(t3--tl)] 2

~ n F ] i0 [--G(t3 -- Q)]

(24)

By e x p a n d i n g the f u n c t i o n s F and G in eqn. (24) w i t h respect to (t 3 - - tl), we obtain

(dE'/dt)3 (--q) = ~

io-

+ [(4--1)(ioL)2--(~F)(io/Cdl)](t3--tl)...)

(25)

It follows f r o m the above e q u a t i o n that, f o r sufficiently small values o f (t3 -- tl), a p l o t of the t e r m o n the left-hand side o f eqn. (25) against (t3 -- tl) 1/2 yields a straight line w h o s e i n t e r c e p t at t3 - - tl = 0 is ( R T / n F ) (1/io). This p r o v i d e s a simple e x p e r i m e n t a l m e t h o d f o r d e t e r m i n i n g the e x c h a n g e c u r r e n t density. It should be here n o t i c e d t h a t the e x c h a n g e c u r r e n t d e n s i t y thus o b t a i n e d does n o t corres p o n d to the bulk c o n c e n t r a t i o n s , c~, b u t to the surface c o n c e n t r a t i o n s at t = t3, (cS)3, which are n o t d i r e c t l y measurable. H o w e v e r , t h e expressions f o r these surface c o n c e n t r a t i o n s can readily be derived, as follows: c o m b i n i n g t h e equations o b t a i n e d b y writing eqn. (11) at t = t3 f o r j = O and R, we o b t a i n ,/Vo(cS)3 +

--

+

(26)

160 On the o t h e r hand, we have f r o m the N e r n s t e q u a t i o n (5) V ~ o ( c S ) 3 -- x/~R(cS)3 exp(~ h) = 0

(27)

with ~ = ( n F / R T ) ( E h -- E~1/2)

(28)

E~1/2 = E ° -- ( R T / n F ) In V ~ o / D R

(29)

w h e r e E~1/2 d e n o t e s the reversible half-wave potential. T h u s c 5 + X / ~ R / D o C~

(C~))3 =

1 + exp(--~ "h)

(30)

+

(cS)3 =

(31) 1 + exp(~)

Introducing eqns. (30) and (31) into eqn. (8) yields the following expression of io in terms of the bulk concentrations:

io = nF V~oC~) + ~/DRC~ ko e x p ( - - ~ ) v/D 1 + exp(--~)

(32)

with D = Do 1-~DR ~

(33)

Finally, consider the case, in which the solution contains both substances O and R in the bulk and the working electrode is initially under the equilibrium potential corresponding to the bulk concentrations of O and R. Further assume that the perturbation is so small that Eh and E~ are apart from the initial equilibrium potential only by few millivolts, as in usual electrochemical relaxation methods. In such a case, the surface concentrations at E = Eh are, as a first approximation, equal to the corresponding bulk concentrations, so that the exchange current density determined from eqn. (25) approximates to the expression with respect to the bulk concentrations. EXPERIMENTAL Apparatus

Figure 2 shows the b l o c k diagram o f the a p p a r a t u s e m p l o y e d . The circuit f o r d e t e c t i n g the fall o f m e r c u r y d r o p has been described elsewhere [ 7]. T h e cont r o l l e d - c u r r e n t pulse o f the t y p e s s h o w n in Fig. 2 were fed t o t h e cell. T h e widths and heights o f the pulses as well as their interval were variable: 40 to 2 0 0 ns in w i d t h and 0.6 t o 3.6 ps in interval. T h e m a x i m u m o u t p u t c u r r e n t was 140 m A ; it was held c o n s t a n t to a precision o f 5% f o r a change o f load resistance f r o m 0 to 70 f~. T h e rise and the fall times of each pulse were less t h a n 5 ns. In this e x p e r i m e n t , the widths were respectively 100 and 40 ns for the first and the s e c o n d pulses. T h e pulse interval was 2.5 ps. T h e charge o f the second pulse was k n o w n b y measuring the voltage d r o p across the 2 0 2 0 p F c a p a c i t o r inserted i n t o the c o n t r o l l e d - c u r r e n t circuit in place o f the cell.

161 ~DROP-FALL DETECTOR

]

TRIGGER-PULSE GENERATOR

X-AXIS OF CRO I

I IST PULSEI GENERATOR

2ND PULSE I GENERATOR

ATTENUATORI

ATTENUATORl

i

CONTROLLEDCURRENTCIRCUIT

i

1

CONTROLLEDI CURRENTCIRCUIT J

I

I '~

w~1I Y-AXIS OF CRO ]

/ TYPE 0~ PULSE: (I) t - - l - - ,

(2)

I

Fig. 2. The block diagram of the apparatus used in modified coulostatic pulse measurements.

Cell T h e electrolysis cell u s e d was c o m p o s e d o f t w o c o m p a r t m e n t s s e p a r a t e d b y a glass frit. T h e D M E a n d a m e r c u r y p o o l e l e c t r o d e w e r e in one of t h e m and t h e s o l u t i o n in the o t h e r c o m p a r t m e n t was c o n n e c t e d w i t h a s a t u r a t e d c a l o m e l elect r o d e (SCE) via an a q u e o u s agar bridge. T h e c o u l o s t a t i c pre-pulse was a p p l i e d b e t w e e n the D M E and t h e m e r c u r y p o o l e l e c t r o d e at a t i m e 2.9 s later t h a n t h e b i r t h of a m e r c u r y d r o p w h o s e life was a b o u t 8.2 s. T h e d r o p area at the m o m e n t o f pulse a p p l i c a t i o n was 2.5 X 10 - 2 c m 2. T h e w o r k i n g e l e c t r o d e was p o l a r i z e d c a t h o d i c a l l y b y this pulse a n d t h e n a n o d i c a l l y b y the f o l l o w i n g pulse w i t h reversed sign. Oscilloscope tracings of potential vs. t i m e curves w e r e p h o t o g r a p h e d with a P o l a r o i d L a n d c a m e r a .

Solution An a q u e o u s s o l u t i o n c o n t a i n i n g 6.9 m M ferric s u l p h a t e , 1 M p o t a s s i u m oxalate and 0.05 M oxalic acid was e l e c t r o l y z e d p o t e n t i o s t a t i c a l l y in a d v a n c e to each m.c.p, m e a s u r e m e n t so t h a t t h e b u l k c o n c e n t r a t i o n s of t r i s o x a l a t o c o m plexes o f i r o n ( I I ) a n d i r o n ( I I I ) m i g h t be 2.9 a n d 4.0 mM, respectively. T h e y were d e t e r m i n e d p o l a r o g r a p h i c a l l y using the k n o w n d i f f u s i o n coefficients: D O = D R = 4.8 × 10 - 8 c m 2 s --1 [8]. RESULTS AND DISCUSSION T h e t r a n s i e n t c h a r a c t e r i s t i c s of t h e a p p a r a t u s w e r e e x a m i n e d b y m e a s u r i n g the CR circuit s h o w n in Fig. 3. I t is n e a r l y e q u i v a l e n t to t h e r e d o x s y s t e m to be

162 m e a s u r e d in this w o r k e x c e p t t h a t the Warburg i m p e d a n c e is neglected. Figure 3 also shows the p l o t o f the l o g a r i t h m o f o v e r p o t e n t i a l vs. time o b t a i n e d with a pulse o f t y p e 2 in Fig. 2. T h e zero o f the time was set at the end o f the pulse. The analysis o f the results was p e r f o r m e d a c c o r d i n g to the usual p r o c e d u r e o f c o u l o s t a t i c pulse m e t h o d using the relation log ~(t) = log 7(0) - - t / 2 . 3 0 C ~ R f

(34)

with C d l = q/r~(O)

(35)

w h e r e 77 is the o v e r p o t e n t i a l and R~ the charge transfer resistance. T h e C~ and Rf values were d e t e r m i n e d to be 1.08 p F and 2.51 ~2, respectively. It is n o t e d t h a t the e x p e r i m e n t a l p o i n t at t = 0.1 ps lies on the straight line in Fig. 3. This m a y p r o v e t h a t the transient ring at the end o f the c o u l o s t a t i c pulse has been d a m p e d a w a y b e f o r e this time. The e q u i v a l e n t circuit was t h e n m e a s u r e d with the aid of the m.c.p, m e t h o d t o test eqn. (25). A pair o f r e s p o n s e curves to the pulses to t y p e s I a n d 2 were p h o t o g r a p h e d on the same film. D a t a for a run o f e x p e r i m e n t s are s h o w n in Table 1. The a g r e e m e n t b e t w e e n the results o f the t w o e x p e r i m e n t s with d i f f e r e n t q values was n o t quite s a t i s f a c t o r y b o t h in Cdl and R~. F u r t h e r , the results seem to be less reliable t h a n t h o s e o b t a i n e d b y the usual c o u l o s t a t i c pulse m e t h o d . The modified m e t h o d , h o w e v e r , has an advantage t h a t the c o e x i s t e n c e o f substances O and R is n o t necessary. In the case w h e r e either O or R is unstable, this m a y be m o r e t h a n e n o u g h t o c o m p e n s a t e f o r the d r a w b a c k . In a run o f e x p e r i m e n t s on t r i s o x a l a t o f e r r a t e ( I I I ) / ( I I ) electrode, the charge o f the reversed c o u l o s t a t i c pulse was varied, with o t h e r c o n d i t i o n s held u n c h a n g e d .

14 +0.9

,

,

,

,

,

,

,

I

I

I

I

I

I

I

I

12

,

Cdl

+ 0.8 * 0.7 > +0.6 ~: +0.5 ~-- +0.4 + 0.3 + 0.2 + 0.1

-

I

10 "''S

t ~ Lt

8

6 4

0 0.1

I

o

I

1

I

I

2 t/ps

,

,

3

'

'

4

2

I

0

I

1

I

I

2

I

3

( t3- h ) v 2 / p s v~

Fig. 3. Plot of logarithm of overpotential ~'7vs. time for the equivalent circuit of Randles type. Rf = 2.5 ~, Cdl= 1.11 pF and R~Z = 20 £~. Fig. 4. Plot of charge transfer resistance (E h -- E3)2/[(dE'/dt)3 (--q)] vs. (t 3 --/1) 1/2 for trisoxalatoferrate(III)/(II) electrode in 1 M potassium oxalate +0.05 M oxalac acid with c~) = 4.0 mM and c~ = 2.9 mM. (c)) Observed values, ( . . . . . . ) calculated values with k 0 = 1.16 cm s-1 and ~ = 0.86 [8].

163 TABLE

1

Results of modified coulostatic pulse measurements w i t h R f = 2 . 5 ~'~, Cdl = 1 . 1 1 p F a n d R t ~ = 2 0

for the equivalent

circuit given in Fig. 3

q/nC

(E h -- E 3 ) / m V

(dE'/dt)a/mV p s - 1

Rf/~

Cdl/PF

3.33 4.34

2.8 3.7

1.05 1.32

2.24 2.40

1.19 1.20

C o n s e q u e n t l y , t h e p s e u d o - e q u i l i b r i u m surface c o n c e n t r a t i o n s of O a n d R s h o u l d n o t be c o n s t a n t t h r o u g h o u t t h e run, t h o u g h the r e s u l t a n t change in e l e c t r o d e potential was f o u n d to be smaller t h a n 1 m V . T h u s , the charge t r a n s f e r resistance was c a l c u l a t e d on the basis o f the m e a n surface c o n c e n t r a t i o n s in an e x p e r i m e n t a l run.

A p l o t o f (E~- E~)2/[(dE'/dt)3 (--q) vs. (t3 - - tl) 1/2 is s h o w n in Fig. 4. T h e e x p e r i m e n t a l p o i n t s lay o n a straight line in a c c o r d a n c e w i t h t h e r e q u i r e m e n t o f eqn. (25). T h e e x c h a n g e c u r r e n t d e n s i t y free f r o m t h e e f f e c t of c o n c e n t r a t i o n p o l a r i z a t i o n was o b t a i n e d t o be 354, m A c m - 2 f r o m t h e i n t e r c e p t o n t h e axis o f o r d i n a t e s . T h e b r o k e n line in Fig. 4 was d r a w n a f t e r eqn. (25) w i t h t h e k n o w n values for t h e kinetic p a r a m e t e r s a n d t h e d i f f u s i o n c o e f f i c i e n t s [8], i.e., k0 = 1.16 c m s-1, a = 0.86 a n d D o = D R = 4.8 × 10 - 6 a m 2 s - 1 . A g r e e m e n t b e t w e e n these t w o p l o t s is excellent. A c c o r d i n g to the t h e o r y , the p r e - e l e c t r o l y s i s can be of a r b i t r a r y kind. B u t t h e use o f a sharp c o u l o s t a t i c pulse will m a k e it possible t o m e a s u r e t h e kinetic p a r a m eters o f a charge t r a n s f e r r e a c t i o n w h o s e p r o d u c t is q u i t e u n s t a b l e ; t h e r e i n m a y be t h e m e r i t of this t e c h n i q u e . In such a m e a s u r e m e n t , h o w e v e r , a s o l u t i o n cont a i n i n g a stable r e a c t a n t is u s e d a n d the u n s t a b l e r e a c t a n t m u s t be g e n e r a t e d at t h e e l e c t r o d e surface b y t h e first c o u l o s t a t i c pulse o f c o n s i d e r a b l e a m o u n t o f charge. T h e p o t e n t i a l o f the w o r k i n g e l e c t r o d e w o u l d t h e n be largely shifted f r o m the initial one; i.e., t h e p o t e n t i a l vs. t i m e curve like t h a t o f Fig. 1 o b s e r v e d on t h e o s c i l l o s c o p e screen w o u l d be biassed b y several h u n d r e d s of m V . T h e bias v o l t a g e m u s t be r e a d to a precision o f 1 m V or b e t t e r . T h e technical difficulties in it, h o w e v e r , can be o v e r c o m e b y use o f e l e c t r o n i c devices r e c e n t l y d e v e l o p e d . An e x t e n s i o n o f t h e p r e s e n t w o r k in this d i r e c t i o n is n o w in progress. REFERENCES 1 H.P. van L e e u w e n a n d J.H. S l u y t e r s , J. E l e c t r o a n a l . Chem., 39 ( 1 9 7 2 ) 25, 233. 2 P. D e l a h a y , J. Phys. Chem., 66 ( 1 9 6 2 ) 2 2 0 4 ; Anal. Chem., 34 ( 1 9 6 2 ) 1 1 6 1 ; Anal. Chim, A c t a , 27 ( 1 9 6 2 ) 9O. 3 W.H. R e i n m u t h a n d C.E. Wilson, Anal. Chem., 34 ( 1 9 6 2 ) 1 1 5 9 ; W.H. R e i n m u t h , Anal. Chem., 34 (1962) 1272. 4 H. M a t s u d a a n d Y. A y a b e , Z. E l e k t r o c h e m . , 59 ( 1 9 5 5 ) 494. 5 T. Berzins a n d P. D e l a h a y , J. Amer. Chem. Soc., 77 ( 1 9 5 5 ) 6 4 4 8 . 6 H. Matsuda, S. Oka a n d P. D e l a h a y , J. A m e r . C h e m . Soc., 81 ( 1 9 5 9 ) 5077. 7 M. K o g o m a , Y. K a n z a k i and S. A o y a g u i , C h e m . Instr., 7 ( 1 9 7 6 ) 193. 8 T. R o h k o , M. K o g o m a a n d S. A o y a g u i , J. E l e c t r o a n a l . Chem., 38 ( 1 9 7 2 ) 45.