Theory of stationary current-voltage curves of redox-electrode reactions in hydrodynamic voltammetry

J. Electroanal. Chem., 79 (1977) 237--250

237

© Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands

THEORY OF STATIONARY CURRENT-VOLTAGE CURVES OF REDOX-ELECTRODE REACTIONS IN HYDRODYNAMIC VOLTAMMETRY PART X. CONVERGENT FLOW ELECTRODES

KOICHI TOKUDA and HIROAKI MATSUDA

Department of Electronic Chemistry, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo 152 (Japan) (Received 4th June 1976; in revised form 19th July 1976)

ABSTRACT Redox°electrode reactions at the two- and the three-dimensional convergent flow electrodes are treated theoretically considering both rates of convective diffusion and kinetic processes. A high current density is observed at both the leading edge and the back edge of these electrodes. This behavior of current density is quite different from that at other types of electrodes previously employed in hydrodynamic voltammetry. Simple approximate equations are presented, which represent the current-voltage curves of redox-electrode reactions without or with a preceding chemical reaction.

INTRODUCTION

This series of papers has been concerned with voltammetry at various types of electrodes placed in laminar streams, such as stagnation point electrodes (including disk and ring electrodes) [1], wedge electrodes [2], conical electrodes [3] and spherical electrodes [4] immersed in a uniform laminar flow; tubular and channel electrodes [ 5,6 ]; rotating ring [ 7 ] and ring-disk electrodes [ 8 ] ; and stationary disk and ring electrodes placed in a uniformly rotating fluid [9]. At most of these electrodes, the Prandtl boundary layer develops continuously with increasing distance from the forward end of the body, in which the electrode is embedded, in the downward direction, and the diffusion layer starts to develop from the leading edge of the electrode and increases in thickness with an increase in the distance from the leading edge. In some cases, however, the thickness of the Prandtl boundary layer decreases with increasing the distance. Examples of such flows are a two-dimensional convergent flow between non-parallel planes and a three-dimensional convergent flow like that in a funnel (see Fig. 1). An electrode forms a part of the wall of the two- or the three~limensional convergent channel, which we refer to as a "convergent flow electrode". At this electrode, the diffusion layer starts to develop, passes through a maximum of the thickness and then decreases in thickness, because the Prandtl boundary layer is markedly compressed in the neighborhood of the sink, which is situated at the vertex. Thus, it would be of great interest to examine the current density distribution at such an electrode.

238

In the present paper, the authors a t t e m p t a theoretical approach to this problem and present simple equations for the stationary current-voltage curves of redox-electrode reactions without and with a preceding chemical reaction at the two- and the three-dimensional convergent flow electrodes. S T A T I O N A R Y C UR R E N T - V O L T A G E CURVES OF REDOX-ELECTRODE REACTIONS IN THE ABSENCE OF ANY CHEMICAL COMPLICATION

According to the usual kinetic theory of the electrode processes [10], the current density for a simple redox-electrode reaction: Ox + ne ~ Red

(1)

involving only soluble species, is expressed as i= n {k(co)y=o--k(cR)y=0}

(2)

with = ko e x p [ - - ~ ( n F / R T ) ( E - - E°)]

= ko exp[(1 -- ~ ) ( n F / R T ) ( E - - E ° ) ]

(3)

where i is the current density, E the potential, E ° the standard potential of the reaction (1), a the cathodic transfer coefficient, k 0 the standard rate constant, F, R and T have their usual significance. (Co)y=0 and (cR)y=o are the concentraLions of Ox and Red at the electrode surface, respectively, which should be obtained for the two- and the three-dimensional convergent flow electrodes in what follows. Two-dimensional convergent flow electrode

The schematic diagram of the two-dimensional convergent flow electrode is shown in Fig. l a and the coordinate system is illustrated in Fig. 2, where x is the distance measured along the wall from an arbitrary point taken as the origin on the wall, x0, Xl and xs are the coordinates of the leading and the back edges of the electrode and the sink, respectively, and y is the distance measured from the wall. In a previous paper [11], the boundary value problem for two-dimensional convective diffusion has been solved to obtain the relation between the concentration and the flux, fi, of species j at the electrode surface. We have (c,)y=0 = c°

3-~/a P(~)

~

fj(x') dx'

D721apX 13 J

(4)

x x 0

(f [r(x")] 1/2 dx"}2/a X ~

where c o is the bulk concentration of species j, Dj the diffusion coefficient of species j,/a the viscosity of the fluid, and r the skin friction, which is defined by r

= t2(au/ay)y= 0

where u is the x - c o m p o n e n t of the flow velocity.

(5)

239

.,J/

///

Fig. 1. Schematic diagram of (a) the two- and (b) the three-dimensional convergent flow electrodes. Shaded parts indleate the electrodes.

The flow velocity of the main stream in the two-dimensional convergent flow is given by U(x)

=

(6)

( Q / ~ d ) ( x s -- x ) - 1

where Q is the volume flow rate from the sink, d the width of the two-dimensional convergent channel and 0 the angle between the non-parallel planes. The problem of the boundary-layer equation of this flow has been worked out [12]. From the result, we have u = V~(~)

(7)

where ~7 = y ( Q f l ~ d ) l / 2

p-1/2(x s _ x)-I

(8)

is the kinematic viscosity and ~02 is a stream function which satisfies: ~02"(0) =

J

sink ×s

xl

electrode

zo

T

0

Fig. 2. Sectional view and the coordinate system of the convergent flow electrode.

240

1.1547. Thus we obtain 7" = pU~o'2'(O)(OT?/ay) = U(J2(x~ - - x ) -2

(9)

with

[32 = 1.1547(Q/Od) 312v-112

(10)

Substituting eqn. (9) into eqn. (4) yields

(cj),=o

3 - 1 1 z D~_21z{3_(x/z (~

Co

fj d x ' Jx 0 [ln(x s - - x ' ) --ln(xs --x)] 2t3

r(2)

(Ii)

Substituting eqn. (11) into eqn. (2) and using the relation: i = n F f o = - - n F f R , we obtain the following Volterra integral equation of second kind with respect to (i/nF): ( i / n F ) = ( ---> k c 0o - - k c ° )

1 X2 f (i/nF) dx' 1 F(~)( 2 X s-- X o) xo J F(~) [ln(xs - - x ' ) -- ln(x~ --x)] 2Is

(12)

with

X2 3z/3 ~/s

+

= 1.7705 (Q/Od)-l/2pl/6(xs --Xo)D -2/3 ko e-~r(1 + e r)

(13)

where D _- D IO- ~

D~R

(14)

- - E~/2)

= (nF/RT)(E

(15)

and E'I ]2 = E ° - - ( R T / n F ) l n ( D o / D R )2Is

(16)

Introducing a dimensionless function ~2 defined by (i/nF)

=

-~c° - - ~ c °

X2

"¢2(X, k2)

(17)

and dimensionless variables: X = (x - x o ) l ( x , - x o ) ,

X' = (x' - x o ) l ( x , - X o )

(18)

¢2 dX'

(19)

into eqn. (12), we have

= X2 - -

X2 1 F(~) 2 F(~)

~:

[In(1 -- X') -- In(l -- X)] 2la

When the electrode process (1) proceeds reversibly, eqn. (19) can be solved

241

analytically. Dividing eqn. (19) by k2 and letting k2 -~ = , we have x

q~2 dX'

oJ

1 2 F(~)F(~)

[ln(1 -- X') -- ln(1 -- X)] 2/a

2

(20)

7r/%/3

Writing X" instead of X in eqn. (20), multiplying both sides of the resulting equation by (1 - - X " ) -1 [ln(1 - - X " ) - - l n ( 1 - - X ) ] -1/s, then integrating with respect to X " from 0 to X and differentiating with respect to X, we obtain

~b~(X, oo) = (1 -- X)-1 [--In(1 -- X)] -11a

(21)

From eqns. (17), (21) ahd (13), we have the expressionsfor the cathodic and the anodic limitingcurrent densities: ~a ~ = 0.5648 nF(Q/Od) z12,'°~o~ongla--1161~'~ v~ - - x ) -1 In

(22)

and •a

$d

=

--0.5648 nF(Q/Od)l/2c°D213p-1/6(xs - - x ) -1 In L.

(23) \

s

/_2

When the electrode process (1) is quasi-reversible or irreversible, i.e., when the value of the parameter X2 is finite, eqn. (19) cannot be solved analytically. If we apply to eqn. (19) the same procedure as used to obtain eqn. (21) from eqn. (20), we have

1

~

¢2(x,

d(~2(X', Xz)

. . . . . . .

d

°['nt1-~,]

~

]

,,1

ln(i"X)-ln(1-x)ji;

(24)

This can be solved numerically by the m e t h o d of Acrivos and Chambr~ [13] as used in the previous papers [6--9]. By subdividing the X-axis into N equal intervals of AX, we have N--1

X2H2(N, 1) ¢2( N A X , k2) =

+

p=l

¢2(pAX, k2){H2(N,p

+ 1) --H2(N, p)}

(2/3)k2(1 - - N A X ) + Hz(N, N)

(25)

with

1--pAX _ I l n ( 1 - [ p - 1] AX]'] 2/a I l n ( 1 ~ N A X ) I 2'3 i--NAX ~/J H2(N, p) = ln(1 - - p A X ) -- ln(1 -- [p -- 1] AX)

(26)

A tentative application of the m e t h o d of successive approximation to eqn. (19) indicates that the function ¢2 decreases linearly with X 1 [8 for small values of X. This fact suggests that the use of a variable ~ = X 1/3 instead of X for small values of X reduces greatly errors in numerical calculation. Introducing the variable t into eqn. (24) and subdividing t-axis into N equal intervals of At leads to

242 N--1

~,2H2(N, 1) + ¢2(NA}, X2) -

p=1

¢2(pA}, k2){H2(N, p + 1) -- Hz(N, p)} (27)

(213)k2 {1 -- (NA}) a ) + H2(N,N ) with

[ln[1 -(p,a;)"t-]

H2(N,P) = i_

[lnl -([p -1]

(1 -- (NA})a J3 1 ~iN~-)~ JJ ln(1 -- (pA}) 3} --ln{1 -- ( [ p - - 1]A}) 3}

(28)

Thus, eqns. (27) and (28) were used for 0 ~< X < 0.216 (0 < } ~< 0.6) and eqns. (25) and (26) for 0.192 ~< X ~< 0.98 by setting At = 0.002 and AX = 0.002. Numerical calculations were performed on a computer HITAC 8700 for various values of X2. The variations of ¢2 with X for several values of X2 are shown in Fig. 3. Since the current density is proportional to ~2 as can be seen from eqn. (17), Fig. 3 also shows the current density distributions. As has been expected from the characteristics of the convergent flow mentioned above, when the electrode process is reversible or the potential of the electrode is set on the limiting current region, the current density, which is infinite at the leading edge, falls rapidly with an increase in the distance, passes through the minimum, then increases gradually and becomes very high at the region of the electrode close to the sink. This current density profile is quite different from that observed at other types of electrodes, which have been treated in this series of papers. For small values of X2, for which the current is primarily controlled by the kinetics of the electrode process, rather uniform current density distributions are observed. The total current is obtained by .integrating the current density over the entire surface of the electrode. Thus we have I = w f _J

idx=

+ \l+e~

x0

~2(Xa,X2)

(29)

l+e-~

with (30)

Xl = (xl - Xo)/(xs - Xo)

I~ = 0.8472 nF(Q/Od) 1]2 -,,,o0 [ln( xs --X°~l - o ~D213~--1/6 o

2/3

(31)

k \xs--Xtl.]

2/3

I a = --0.8472 nF( Q/Od)* 12 wco D~m v-1/6 [In( xs -- x° ] l

I_ \ x s - - x l l . l

(32)

where w is the width of the electrode, I ¢ and I ~ are the cathodic and the anodic d = 3 2,a (1.1547) 1/a ]2 P(113). The f unchmltmg currents, respectwely, and 0.8472 tion ~2 is given by .

.

.

.

X~/3 q~2(X1, X2) = § [--ln(1 -- X 1)]-2/s / 0

3 ~b2(}, X2) ~2 d} for X1 ~< 0.192

(33)

243

!

10

6

I

4

O8 O6 "II 04 Q2 I

0

02

04

.

×

I

r

06

0.8

0

. . . , . . . ,

0

5

t

t

l

|

|

l

l

*

D

TO0

X2

Fig. 3. Variations of the function ~b2(X, ~2) with X. The numbers on the curves give the corresponding value of the parameter k2Fig. 4. Variations of the function ¢2(X1, ~2) with ~2 for a series of values of X 1 : (I) 0.01065, (II) 0.027, (III) 0.125, (IV) 0.30 and 0.70 (overlapping), (V) 0.50, and (VI) 0.90.

and

~2(X1, k2) = 2 [--ln(1 -- X1)] -2/a 0.1921/3

×
XI

3

2 at + f

0

¢2(X, k2)dX} for X1 > 0.192.

(34)

0.192

The indicated integration was performed with the aid of Simpson's 1/3 rule. The variations of the function ~2(X1, k2) with X2 are shown in Fig. 4 for several values of X1. Fortunately, it was found that the function ~2 can be expressed

by

),2 ~2(X1, k2) = A 2 ( X 1) + X2

(35)

within a maximum error of + A. The values of A2(X1) and A are tabulated in Table 1 for a series of values of X1. From Table 1, it can be seen that this approximate equation is valid for X1 ~< 0.8. For X1 > 0.8, eqn. (35) contains a relatively large error, especially for small values of ?~2, which can be seen, for example, from the curves VI in Fig. 4 for X1 = 0.9. To obtain the value of A2 for any electrode geometry parameter X1, an adequate interpolation technique would be necessary. Substituting eqns. (35) and (13) into eqn. (29), we obtain the final expression

244 TABLE 1 Values of A 2 and A for a series of values of X 1

X1

A2

A

X1

A2

A

0.000008 0.000216 0.001000 0.002744 0.005832 0.010648 0.017576 0.027000 0.039304 0.054872 0.074088 0.097336 0.125000 0.157464 0.195112 0.22 0.26 0.30

71.550 23.865 14.380 10.274 8.007 6.557 5.564 4.840 4.287 3.857 3.514 3.236 3.009 2.821 2.666 2.588 2.491 2.420

0.0046 0.0050 0.0048 0.0045 0.0045 0.0045 0.0045 0.0044 0.0044 0.0044 0.0044 0.0044 0.0043 0.0043 0.0042 0.0041 0.0040 0.0039

0.34 0.38 0.42 0.46 0.50 0.54 0.58 0.62 0.66 0.70 0.74 0.78 0.82 0.86 0.90 0.94 0.98

2.368 2.330 2.307 2.294 2.291 2.298 2.315 2.343 2.385 2.445 2.525 2.632 2.780 2.994 3.330 3.957 5.820

0.0037 0.0034 0.0032 0.0028 0.0024 0.0019 0.0014 0.0008 0.0008 0.0019 0.0034 0.0054 0.0081 0.0118 0.0172 0.0268 0.0504

for the cu~ent-voltage curves: +

1 + e~

1 + e -~! (Q[Od)-i/2/~1/6(X

X

s

_ _

xo)D-213koe - ~ ( 1

+ e [)

0.565 A2(X1) + (Q/Od)-ll2pll6(xs --xo)D-213ko e - ~ ( 1 + e ~)

(36)

Three-dimensional convergent flow electrode The three-dimensional convergent flow electrode is depicted schematically in Fig. l b and Fig. 2 is again used here to show the coordinate system and the flow profile. In this case, Fig. 2 represents a cross section of the convergent conical channel by a plane including the symmetrical axis. Only one of the cross sections of the electrode is shown and the other is omitted in Fig. 2. 0 is the vertical angle and 0 = r corresponds to a plane, ro is the distance of an arbitrary point on the surface of the wall from the symmetrical axis. Thus, r0 = sin(0/2)(xs -- x)

(37)

From the solution of the b o u n d a r y value problem for axisymmetrical convective diffusion [1], we have

(cj)y=o = c o

3 -113 D~-2/3 pl/3 f r(~)

x

f~(x') ro(X') dx'

x0 ( f ro(X,,)3/2r(x,,)l/2 dx,,} ~/3 X t

(38)

245 The flow velocity of the main stream and that in the boundary layer in the threedimensional convergent flow are given by [14] U(x) = [Q/2 7r(1 -- cos(0/2)) ](Xs -- x) - 2

(39)

and u = U~(~)

(40)

respectively, with 77 = y [ Q/2 7r( 1 - cos(Ü/2)} ] 1/2(2 p)-I/2(x s __ x)-3/2

where ~3 is a stream function which satisfies: ~3( tt/O~)

(41) 2.273. Thus we have (42)

~ = #fl3(xs - - x ) -712

with f13 = 0.568 [ Q/~ { 1 -- cos(0/2) } ] 3/2 v-1/2

(43)

Substituting eqns. (37) and (42) into eqn. (38) leads to --

31/3 4_2/3/3~ 1/3 Di -2/3

/'x

(xs --x')/~

x0 J

-x')

3/4 -

dx'

(xs - x ) 3 / 4 ]

(44) 2/3

Combining eqn. (44) with eqn. (2) yields "-* o0 -- "( i / n F ) =- (kc kc ° )

314

~3 f (xs - - x ' ) ( i / n F ) dx' r ( 1) r ( 2 ) (Xs--XO) 3/2 J [ ( X s - - X ' ) 3 / 4 - ( x s - x ) 3 / 4 ] XO

2/3

(45)

with

41/3F(½)(Xs--XO)3/2 ( -~ +

)

= 2.469[Q/v{1 -- c o s ( O / 2 ) ) ] - l t 2 p l m ( x s --Xo)312D-213ko e-~¢(1 + e ¢)

(46)

Introducing a dimensionless function Ca defined by -+

(i/nF) =

~-- 0 k c ° -- k c a X ~ ~b3( , )~3)

(~7)

and dimensionless variables, X and X' defined by eqn. (18) into eqn. (45), we obtain

¢3(X, k3) = ~3

(3/4)~31 2 /

F(~) r ( g )

X' (1 -- X' )(~3(__ , X3)dX'

[(1 - - X ' ) 314

(1 - - X ) 3/4] 2/3

(48)

Following the same procedure as used in the above section, we can solve eqn.

246

(48) analytically for reversible processes to obtain ¢3(X, oo) = (1

-- X)-5/4[1

--

(1 - " X)3/4] -1/3

(49)

From eqns. (46), (47) and (49), the cathodic and the anodic limiting current densities are given by ~d = 0.405 nF[Q/~(1 -- cos(0/2)} ] 112 ,.o m ~ O ~n2ta,-1 O v X (X s - - X ) - 5 / 4 [(X s - - X 0 ) 3/4 - - (X s - - X ) 3 / 4 ] - 1 / 3

(50)

and ld'a = --0.405

nF[ Q/fr { 1 -- cos[0~/2~,}], 1/2

C 01¢D2R/31)-I/6

(51)

X (X s - - X) - 5 / 4 [(X s - - X 0 ) 3 / 4 __ (X s __ X ) 3 / 4 ] - - 1 / 3

respectively. For quasi-reversible and kreversible processes, we obtain the following equation similar to eqn. (24): de' 3 (X', k3) 1 ~b3(X, k 3 ) =

k3(1

x d[l

- - X ) 5/4 /

--X')3/4]d[(l--X')3/4]

[(1 - - X ' ) 3j4 - - ( 1 - - X ) 3 / 4 ] ~/3

(52)

Similarly, we have N--1

~3H3(N, 1) +

~3 (NAX, k3) =

¢3(p~,

k3){H3(N,

p + 1) --Ha(N,

p=l

p)} (53)

( 2 / 3 ) k 3 (1 -- NAX) 5/4 + H3(N,N)

with H3(N,

p) =

{(1 -- [p -- 1]AX) 314 -- (1 -- N A X ) 3/4 } 2/3 __ {(1

- - p A X ) 314 - -

(1

- - N A X ) 3/4

} 213

( 1 - - [ 2 - - 1 ] A X ) 3/* - - ( 1 - - p A X ) 3/4

(54) For the same reason as mentioned in the above section, numerical calculations of ~ba were carried o u t b y the use of the variables, ~ = X 1/3 for 0 ~< X ~< 0.216 (0 ~< ~ ~< 0.6) and X for 0.192 ~< X ~< 0.98. Equations for ¢3(NA~, k3) and H3(N, p) similar to eqns. (53) and (54), respectively, were employed. Figure 5 shows the variations of the function ~b3 with X for several values of ka. The current density distributions obtained are similar to those for the two-dimensional case given in Fig. 3. The total current is expressed b y

I =2~r f x0

iro d x =

1 + e~

+

1

q~3(XI, X3)

(55)

+ e -~I

where X1 is defined by eqn. (30), I~ and 12 are the cathodic and the anodic limiting currents given b y

247

lO

8 2O

6 I0

4

2

T I

0

02

i

I

06

OB

..

04

X Fig. 5. Variations of the f u n c t i o n ~3(X, ~t3) w i t h X. The numbers on the curves give the corresponding value o f t h e p a r a m e t e r ~3.

I~ = 1.621 r1/2 nF[Q/{1 --cos(0/2)}] 1/2 sin(O/2)c°D21Sv -1Is × ((x s - - x 0 ) s/4 -- (xs - - x l ) s/4 }21s

(56)

I~ = --1.621 7d/2 nF[Q/(1 -- cos(0/2)}] 1/2 sin(O/2)c°D~lav-ll6 × {(x~ --Xo) s/4 -- (x~ - - x l ) s/4 }2/s

(57)

respectively, and

X~13

3(1 --}s)}Z@s(}, ks) d} forX1 ~<0.192 (5S)

~s(X~, ks) = 0.511 -- (1-- x1)s/4]-2/s f 0

and • s(X,, ks) = 0.511 -- (1 --X1)a/~] -2/s 0.192113

X {f

Xl

3(1 -- }3)}2¢3(~, ks) d} + f

0

(1 --X)¢s(X, ka)dX}

0.192

for 0.192 < X ~< 0.98.

(59)

It was also found that the function cPa can be approximated by ks (ha(X1, ks) = As(XI) + ks

(60)

within a m a x i m u m error of + A. The values of As and A are listed in Table 2 for a series of values of X 1. This equation is valid for X1 ~ 0.8. Substituting eqns. (60) and (46) into eqn. (55) yields the expression for the

248 TABLE 2 Values of A 3 and A for a series of values of X 1 XI

A3

A

X1

A3

A

0.000008 0.000216 0.001000 0.002744 0.005832 0.010648 0.017576 0.027000 0.039304 0.054872 0.074088 0.097336 0.125000 0.157464 0.195112 0.22 0.26 0.30

77.732 26.283 15.826 11.316 8.822 7.239 6.152 5.360 4.762 4.303 3.942 3.653 3.421 3.234 3.088 3.017 2.936 2.882

0.0071 0.0050 0.0046 0.0046 0.0045 0.0045 0.0045 0.0044 0.0044 0.0044 0.0044 0.0043 0.0042 0.0042 0.0041 0.0040 0.0038 0.0036

0.34 0.38 0.42 0.46 0.50 0.54 0.58 0.62 0.66 0.70 0.74 0.78 0.82 0.86 0.90 0.94 0.98

2.851 2.836 2.837 2.850 2.877 2.912 2.965 3.033 3.117 3.218 3.341 3.493 3.683 3.922 4.239 4.684 5.438

0.0032 0.0029 0.0026 0.0020 0.0012 0.0008 0.0010 0.0024 0.0038 0.0052 0.0073 0.0095 0.0140 0.0190 0.0248 0.0350 0.0512

current-voltage curves:

1 + e~

1 + e- ~ !

[ Q/~{1 -- c o s ( 0 / 2 ) } ] - 1 ]2vl ] 6 ( X 0.405

s __

Xo)312D-2]3 ko e - a ¢ ( 1 + e ¢)

A3(X1) + [ Q / ~ { 1 - - cos(O/2)}]-l12vl16(xs --Xo)312D-213ko e - ~ ( 1 + e ~)

I t s h o u l d be n o t e d t h a t t h e e q u a t i o n s derived here, f o r e x a m p l e , eqns. (36) and (61), are n o t valid f o r values o f X1 or X ' close t o u n i t y , since t h e b o u n d a r y layer a p p r o x i m a t i o n , o n w h i c h t h e t h e o r e t i c a l t r e a t m e n t d e s c r i b e d here is based, is n o t valid in t h e i m m e d i a t e n e i g h b o r h o o d o f t h e sink where, in a n y case, t h e ass u m e d m a i n s t r e a m f l o w given b y eqns. (6) or (39) c a n n o t r e p r e s e n t an actual flow t h r o u g h a c h a n n e l or a hole o f small b u t finite size. F r o m an e x p e r i m e n t a l v i e w p o i n t , it is i m p o r t a n t t o e s t i m a t e the m i n i m u m distance b e t w e e n t h e sink and t h e b a c k edge o f t h e e l e c t r o d e f o r w h i c h t h e e q u a t i o n s derived here are valid. H o w e v e r , t h e t h e o r e t i c a l e s t i m a t i o n o f this distance seems q u i t e difficult and an e x p e r i m e n t a l e x a m i n a t i o n m u s t be awaited. STATIONARY CURRENT-VOLTAGE CURVES OF REDOX-ELECTRODE REACTIONS PRECEDED BY A FAST CHEMICAL REACTION Consider the e l e c t r o d e process r e p r e s e n t e d b y t h e f o l l o w i n g s c h e m e : Y ~ Ox

(62)

249 Ox + ne ~ Red

(63)

where Y is electro-inactive at the potential at which the reduction of Ox occurs. Furthermore, we assume that the chemical reaction (62) proceeds so fast that the thickness of the reaction layer is much smaller than that of the diffusion layer. Under this assumption, we can solve the boundary value problem for the above reaction scheme by the same procedure as used in the previous papers [1--5,7,9] to obtain the following results.

Two-dimensional convergent flow electrode i=(/~' + I~ 1 \ 1 + e¢' 1 + e- ~ ' !

lOd\ll2 .

.

.

k'o e_a¢,(1 + e~.,) (64)

.

(O~)liu

0.565 Ae + - -

k'o I(I: ') e_a~, +e(l--a)~'}

vlm(xs --Xo) D~-~2]a (\I~ /

(Dyo/Dg)(Do/Dy )1/2/(-1 [k(l + K)] 1/2 0.565A2 [____~ D y o 1/6 (Q/Od)112+ (Dvo] (Do~ 1'2 [k(1 + K)] 1:2

\,]

Xs - - X o

\ Dy ] ~'~y]

(65)

K

with

n2/3,,--1/6 [ In (Xs--XOll

I~' = 0.8472 nF(Q/Od) 112 wc ° ,-'YO v

2/3

(66)

L \ x s - - x l ]_1

~' = (nF/RT)(E -- W{']2) k~ = k o / ( 1 + K) 1-a -- JJYO

c o -- c °

(69) In(l +

(KOy + Do)/(1 + K) +

(68)

~-~R

Eli 2 = E ° -- ( R T / n F )

Oyo =

(67)

K ) -- ( R T / n F )

l n { O y o / O R)213

"~

(70) (71) (72)

where c ° is the concentration of Y in the bulk of the solution, D y the diffusion coefficient of Y, k and K the rate and the equilibrium constants of the preceding chemical reaction, respectively, k~ the rate constant of the charge transfer reaction at the standard potential of the overall reaction, E~'/2 the reversible halfwave potential, I4' the cathodic limiting current, which would be obtained if the preceding chemical reaction were so fast as to preclude any kinetic effect and Ii¢ the cathodic limiting current controlled by convective diffusion and the preceding chemical reaction.

250

Three-dimensional convergent flow electrode

1 + e r'

l + e -~'1

[,{1 - CQ(0/2)}]112 1,6(x,_x0) 3,2-k~)

D'213

×

0.405 Aa + I~r{1--QS(O /2) ).1112 vl/6(X --Xo)3/2

e _ ~ , (1 + e [')

D '2/3 [ \ g / (73)

(Dyo/Dy)(Do/Dy )112 K---1 [k(1 + K)] 1/2

i

0.405 As

()i0

()

[Q/~r{1 - cos(0/2)}] 1/2 + / D y o ] Do 1/2 [k(1 +K)] 1/2 (x -xo) s/2 \ Dy / K (74)

where I~' = 1.621 ~1/2 nF[Q/{1 -- cos(0/2)}] 1/2 sin(O/2)c°D2y/~ v-1/6 × {(xs --Xo) 3/4 -- (Xs - - x l ) 3/4 }2/a and other symbols are defined above. REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14

H. Matsuda, J. Electroanal. Chem., 15 (1967) 109. H. Matsuda, J. Electroanal. Chem., 21 (1969) 433. H. Matsuda, J. Eleetroanal. Chem., 22 (1969) 413. H. Matsuda, J. Electroanal. Chem., 25 (1970) 461. H. Matsuda, J. ElectroanaL Chem., 15 (1967) 325. K. T o k u d a and H. Matsuda, J. Electroanal. Chem., 44 (1973) 199. H. Matsuda, J. Electroanal. Chem., 35 (1972) 77. K. T o k u d a and H. Matsuda, J. Electroanal. Chem., 52 (1974) 421. H. Matsuda, J. Electroanal. Chem., 38 (1972) 159. K.J. Vetter, Electroehemmal Kmetics, Academic Press, N e w York, 1967, p. III. H. Matsuda, J. Electroanal. Chem., 16 (1968) 153. H. Schlichtmg, Boundary-Layer Theory, McGraw-Hill, N e w York, 1968, p. 152. A. Acrivos and P.L. Chambrd, Ind. Eng. Chem., 49 (1957) 1025. L. Rosenhead (Ed.), Laminar Boundary Layers, Oxford University Press, London, 1963, P. 427.

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