The application of the concept of the steady-state reaction—diffusion layer to a study of the electrode processes with multistep reactions at microelectrodes under steady-state conditions

29

J. Electroad. Chem., 346 (1993) 29-51 Elsevier Sequoia S.A., Lausanne

JEC 02412

The application of the concept of the steady-state reaction-diffusion layer to a study of the electrode processes with multistep reactions at microelectrodes under steady-state conditions Qiankun Zhuang and Hongyuan Chen

l

Department of Chemistry, Nanjing University, 21OtM8, Nanjing (People’s Republic of China)

(Received 16 January 1992; in revised form 4 August 1992)

Abstract pUnder steady-state conditions, the general current equations for the first-order or pseudo-first-order EC, EC’, ECE and DISPl reactions and the second-order EC’ reactions at a spherical microelectrode are derived with the aid of the concept of the steady-state reaction-diffusion layer. The approach relies on the analogy between a spherical microelectrode and a rotating-disk electrode. The characteristics of the E,C, E,C, EiC, EC’, E,C’, E,C’, E,CE,, E,CE,, E,CE,, E&E, and DISPl mechanisms, where the electron transfer reaction is reversible, quasi-reversible or irreversible, are also discussed. Using these equations, some methods of determining kinetic parameters for the EC, EC’, ECE and DISPl reactions are presented.

INTRODUCTION

The development of electrodes of micrometer dimensions has expanded the scope of electrochemical studies in recent years [l-11]. The unique features of these microelectrodes ensure that they will continue to be important in electrochemical studies. One of their most striking characteristics is that they have a high mass transport rate. Therefore the time-independent current response is obtained a short time after applying a potential step to the microelectrode. The large mass transport coefficient makes it possible to study electrode reactions coupled to homogeneous processes under steady-state conditions. Examples of these are the analysis of voltammograms by Fleischmann et al. [9,101 who assumed that a disk

l

To whom correspondence

should be addressed.

0022-072t3/93/$06.00 0 1993 - Elsevier Sequoia S.A. All rights reserved

30

electrode was analogous to a sphere electrode and evaluated the kinetic parameters of first- and second-order chemical reactions at microdisk electrodes under steady-state conditions [9,10]. However, the sphere-disc equivalence is likely to be very inexact below the limiting current ill], and the methods used to determine the kinetic parameters in their paper may lose accuracy if the rate constant is extremely fast for the ECE and DISPl reactions. So far, there have been few studies of electrode reactions coupled with homogeneous processes at microelectrodes, and there are almost no reports of the general current-potential characteristics for electrode processes with multistep reactions. In this paper we derive a series of general current-potential expressions for firstor pseudo-first-order EC, EC’, ECE and DISPl reactions and second-order EC’ reactions at spherical microelectrodes under steady-state conditions using the concept of the steady-state reaction-diffusion layer. Obviously, these expressions describe the electrode processes more concisely. The kinetic parameters for these systems can easily be obtained from these equations. For comparison, we also derived the current equations for the reversible systems by solving the partial differential equations. Our results are reported in the Appendices, and some are the same as those in the literature [5,9,10]. Applications of these equations to the study of different electrochemical systems will be reported elsewhere. ANALYSIS OF TERMS OF k,

The limiting diffusion current that flows in response to a potential step under conditions of semi-infinite spherical diffusion can be described by the following well-known equation (ref. 12, Ch. 3): I=nFADc*

[&+b]

(1)

Here, rS and A are the radius and area respectively of the spherical microelectrode, D is the diffusion coefficient and t is the time of electrolysis. Equation (1) is made up of a transient and a steady-state component. The former is proportional to the electrode area, and its contribution to the current rapidly reduces with the increase of time. The latter is the steady-state current at the spherical microelectrode and is proportional to the radius. Moreover, as for the rotating-disk electrode (RDE), the steady-state current at the spherical microelectrode is observed for a short time and obeys the relationship I, = hnFDc

*rs

(2)

The limiting steady-state current is also given by the diffusion model as 1, = nFADc */S, where S is the thickness of the diffusion layer. Equating these two expressions, with A = 4rrS2, yields 6 = r,. Then the steady-state mass transport rate constant for the spherical microelectrode can be obtained: k, = D/6

= D/r,

(3)

31

At the same time, the concentration of the electroactive species and the rate of the electrochemical reaction at the surface of the spherical microelectrode are equal at all points on the surface, and the rate of mass transport is independent of time. This allows the electrode processes to be studied on the basis of uniform surface concentration. We consider the reaction O+n

k, e FR k-,

that occurs at the interface between a spherical microelectrode with radius r, and a solution that contains a bulk concentration c$ of the reactant but from which species R is initially absent. The steady-state current controlled by both mass transport and charge transfer is given by Z=nFA(k,c”,-k_,c”,)

(4)

Z = nFAk,,(

(5)

cz - cOg)

Z = nFAk,,c”,

(6)

where cb and ci are the concentrations of species 0 and R at the electrode surface, k,, = D&r, and k,, = D&r, are the mass transport rate constants for species 0 and R (D, and D, are the diffusion coefficients for the species 0 and R), and k, and k _ 1 are the cathodic and anodic heterogeneous rate constants for the cathodic and anodic processes at a given potential E. These rate constants are given by

where k, is the standard heterogeneous rate constant, a is the charge-transfer coefficient of the cathodic reaction and E” is the standard potential. From eqns. (5) and (6), we have Z

c”o=cg--

nFAk DO Z

Co,=

(10)

nFAk DR

From eqns. (4), (9) and (lo), the steady-state current can be expressed as I=

nFAc;k, k,/km

+ k-,/k,,,

+ 1

(11)

When eqn. (11) is rewritten for a hemispherical microelectrode, eqn. (7) of ref. 13 will be obtained. As the potential becomes more negative, the current tends to its

32

diffusion-limited

steady-state value 1, which can be obtained from eqn. (11) as

Id = 4mzFD,c$,

(12)

because of k, +Q, and k_, +O as E-E”-, --00. Although the general steady-state current equation has been discussed by Oldham and Zoski 1131, they obtained the result by simplifying the DelmastroSmith relationship [14] for t --t m. The Delmastro-Smith relationship is the result of a complicated derivation which. involves treating potentiostatic chronoamperometry for a whole sphere of radius. Therefore the simple approach presented here is valuable, and may avoid the necessity of solving diffusion equations using a complicated mathematical technique. The currents for the reversible and irreversible waves can also be obtained from eqn. (11). THE EC MECHANISM

A simple model of EC reactions is given by A+n

e==$B1

k

c

where k is the rate constant for the irreversible chemical reaction, and k, and k _ 1 have the same definitions as the above. It is assumed that CA*is the bulk concentration of species A and that the diffusion coefficients for all species are equal (these assumptions will be used throughout this paper except for the second-order EC’ reactions). According to the concept of the steady-state reaction-diffusion layer [15], under steady-state conditions the flux at the spherical microelectrode can be expressed in different forms as follows: J=k,(c,*

-c;)

(14)

J=k&-k_&

(15)

J = k,c”, + pkc;

(16)

J = k,c”, + k,c”,

(17)

where fi is the dimension of the reaction layer (ref. 12, Ch. 5) which can be treated as an adjustable parameter (such as k,) and is equal to {(D/k). The current for EC reactions is given by I=nFAk,(c”,+c”,)

(18)

From eqns. (14)-(18) we have the general current equation for the EC reactions:

Id I=

l+k,/k,+k_,/k,+(pk/k,)(l+k,/k,)

(19)

The limiting current of EC reactions can be obtained from eqn. (19) and is equal to Zd’

33

We now introduce the following symbols: E,, the electron-transfer reaction is reversible; E,, the electron-transfer reaction is quasi-reversible; Ei, the electrontransfer reaction is irreversible. E,C mechanism If the rate of charge transfer is far larger than the rate of mass transport, i.e. k, >> hi,, the electrode reaction is reversible. From eqn. (19) we obtain the reversible current z=

Id

(20)

1 + k9+ i-,/m

where 0 = exp[(nF/RTXE -E”)], and eqn. (20) is the same as that obtained by using complicated mathematical techniques to solve the diffusion equations in Appendix A (see eqn. (A12)). Rearrangement of eqn. (20) gives

E=E”+E nF ln(l+rSE)

+$

(21)

In(y)

From eqn. (20, the half-wave potential and the potential difference between E1,4 and E3,4 are given by

(22)

E l/l=Eo+$ln

n(E,,,

- Ei,4) = - F

ln(9) = -56.5 mV

at 25°C

(23)

Equations (22) and 23) can be used to determine the electron number of the electrode reaction and the rate constant, and to differentiate the EC reaction from other mechanisms. If the rate constant is extremely large, p -SLr,. Equation (21) can then be simplified to

(24) and the half-wave potential is E 1,2 = E” + The characteristics of EC, EC’, ECE and DISPl reactions, where the chemical rate constant is extremely fast, have been discussed in an earlier paper [16] and will not be considered further here.

E,C mechanism If the rate of mass transport is much larger than the rate of charge transfer, i.e. and k, % k_,, the electrode reaction is totally irreversible. Then eqn. (19) can be simplified to

k,sk,

E=E”+

(26)

and the E,C reactions can be expressed as a single irreversible electron-transfer reaction. E,C mechanism If the electron-transfer reaction is quasi-reversible, and the current is controlled by both the electron-transfer reaction and the chemical reaction, eqn. (19) can be written as anF RT(E-E”)

I

Id - z =I-

l+rsiW

exP[g(E--E’)]

(27)

Substituting eqn. (20) into eqn. (27) and simplifying, we have E=E”+

5

,,(!$)+_!!I&

~n(!ff!-~)

(28)

where Z, is the current for the E,C reactions at potential E that can be obtained from eqn. (20) if the rate constant k is known. In this case, the rate constant can be obtained by fitting the plot of E -E” vs. In[& - I)/1 - (Id - Z,)/Z,] to a straight line based on eqns. (20) and (28). The electrode reaction parameters (Y and k, can be calculated from the slope and intercept of this line. THE PSEUDO-FIRST-ORDER

EC’ MECHANISM

The EC’ mechanism (catalytic reaction) can be expressed as O+ne+R

I

(29)

R+Zk,-O+P

(30) It is assumed that the bulk concentration cg of species Z is much larger than the bulk concentration cz of species 0, so that the concentration of species Z at the surface of the microelectrode is almost constant. Under steady-state conditions at the microelectrode, we can write the flux in different forms as follows: J = k,( c; - cOg)+ pk&c;

(31)

J= k,c”, - k_,c”,

(32)

35

J = k,c”,

+ pk,c”Rc;

(33)

J = k,c”,

+ k,c;

(34)

and the current due to EC’ reactions is given by

where p = \I( Z = nFAk D( c”R+ c;)

(35)

We can obtain the general current expression for eqns. (31)-(35): ‘d

’ = 1 +k-,/k,

+ (k,/k,)(

1+ pk&/kD)

(I+?)

(36)

Thus the limiting steady-state current for all EC’ reactions is (37)

This equation is in agreement with that obtained by Delmastro and Smith [14]. E,C’

mechanism

If the electron-transfer

reaction is reversible, eqn. (36) can be simplified to

which is the same as eqn. (BlO) in Appendix B. Substituting eqn. (37) into eqn. (38) we obtain I,-z

.=ZP+$

E,C’

In ( Z

(39)

1

mechanism

If the electron-transfer reaction is irreversible, and the current is controlled by both the chemical reaction and the electron-transfer reaction, from eqn. (36) we have k, -=-k,

1

‘d

(4.0)

I+ /&c,*/k,

Z

Combining eqns. (37) and (401, we obtain E=E”+

f$

In(%)+2

ln(:-4)

(41)

36

EqCf mechanism

If the electron-transfer k, -=-4

from eqn. (36) we have

i+e

&I

I

reaction is quasi-reversible,

(42)

1+ /-%B/k,

Combining eqns. (38) and (42) we have

(43) where 1, is the current due to the E,C’ reaction at potential E. The catalytic rate constant for all EC’ reactions can be determined from the slope of a plot of IL/Id vs. the spherical microelectrode radius (eqn. (37)). Application of the logarithmic plots to the voltammograms based on eqns. (39), (41) and (43) yield straight lines whose slopes and intercepts allow us to evaluate the electron reaction parameters for each type of EC’ reaction. This is impossible using conventional electrodes. SECOND-ORDER EC’ MECHANISM

The general expression for second-order

catalytic reactions is

O+ne----_‘R

(44)

R+Zk-O+P mR+P-

(45) mO+S

(46) in which the electron exchange step is reversible, the first chemical reaction (eqn. (45)) is the rate-determining step for the whole electrode process and the second chemical reaction (eqn. (46)) is rather fast. This means that the concentration of species P at the microelectrode surface tends to zero. We assume that the bulk concentrations of 0 and Z are cz and cg respectively, and those of R and S are zero. Under steady-state conditions, the current due to second-order EC’ reactions at the microelectrode is given by

I=nFA[k,,c”,+

(1 +m)k,,&]

(47)

According to the concept of the reaction layer (ref. 12, Ch. 5), under steady-state conditions the flux of species Z at a spherical microelectrode can be written in different forms as follows: J=k,(c,*

-c”z).

(4)

J = pkc;c”R

(49)

J = k,,c”,

(50)

where p =

[ D,/(

1 + m)kcg]

is the dimension of the reaction layer.

From eqns. (4%(50)

(‘::

- ‘“z) *+

and the definition of EL,we have

kD,c=‘:, (1 + “)k&

kD & (‘z* -‘“z)

-

(52)

(l+m)k~,C’=o

Equation (52) has the general solution kD,c”; CZ*-&=

-

2(1 +m)k&,

+ (1 +m)k;zc’

* !i

1 1 l/2

4kD,c”:,

1

Because ck - c; > 0, we can write

(53)

l/2

4kD,c”:,

+ (1 + m)kfjzcz*

(54)

From eqns. (44)-(46) we obtain the following equations under all conditions: D,c”,

+ D,c”,

= D&

(55)

(56) Combining eqns. (55) and (56), we obtain co = DocW, R 1 + Doe/D,

(57)

Substituting eqns. (51), (54) and (57) into eqn. (47) gives z=

kr;D,c;

Zd ’ - 2D,Dz[

1 + (Do/D,)@

+ ti

2

kr;D,cT,

1 (i

D,D,[l

+ ( Do/D,)e]

where Id = 4nnFD,r&. obtained:

1 + (Do/D,P] 4k& + (’ +m)

Consequently,

D,

the steady-state

+(1+m)r

w 11 I

(58)

limiting current can be

4kr
(59)

Obviously eqn. (13) of ref. 17, which was obtained by using complicated mathematical techniques to solve the differential diffusion equations, is regained.

38

It should be emphasized that the results obtained using the concept of the reaction layer are valid only under the conditions in which p is less than 6/3 [18]. In the case of a steady state at a spherical microelectrode, 6 is equal to the radius of the sphere. Thus the kinetics considered within the approach used here must be rather fast for the second-order EC’ reactions, i.e. when p < r,/3 all the results obtained above are valid. Although the reaction layer is located in solution [19], it is not far from the electrode surface because of the very large value of k. THE ECE MECHANISM

A simple model of ECE reactions is kl A+n, e- =B k-1 Bk’C C+n,

(61) kz

e-e

D

k-2

E;

(62)

Here k,, k_, and k,, k_, are the cathodic and anodic heterogeneous constants for the cathodic and anodic processes at a given potential E: k, = k,, exp

-a,n,F

1

(63)

(E _E;) 1 [(l-ignlF 1 1 RI7

(E--ET)

-(Y,?l,F RT

(E-E2

rate

k-, = k,, exp

k, = k,, exp

k-, = k,, exp

(’ -ff2T)n2F

(E _E;)

(66)

I

According to the concept of the steady-state reaction-diffusion equations for the flux J at the microelectrode surface are J= k,(c;E

(65)

layer, the

-c;)

(67)

J=k,c;-k_,c”,

(68)

J = k,c”, + pkc;

(69)

J = k&

+ k,c;

- k_,c”,

(70)

J = k,c”, + k,c;

+ k,c”,

(71)

where p = /D/k), I =AFk,[

and the general current due to the ECE reaction is

n&, + n,co, + ( n1 + n,)c;]

(72)

39

If we assume n, = n2 = n, the current is given by

(73) From eqns. (67)-(73) we obtain the general form of the reaction current due to the ECE reaction:

z=

1+ l&/k,

Zd + k_,/k,

+ (/.LLk/k,)(1+

k&r)

pk/k.-b/k, l+(k,+k-d/k, 1 (74)

The limiting steady-state current I, due to all types of ECE reactions can easily be obtained from eqn. (74):

(75) This result is the same as that obtained by Fleischmann et al. [lo]. If Ey -x E;, the second electron transfer reaction (eqn. (62)) is very fast compared with the first one (eqn. (60)); this means that every molecule of C formed immediately takes part in the second electron transfer reaction, i.e. k, B- k,, k, >>k_, and c: = 0. Under these conditions, the current due to the ECE reaction can be obtained from eqn. (74) as follows:

Zd ‘= 1 + k,/k,

+ k-,/k,

+ (pk/k,,)(l

+ k,/k,)

P-6)

If E; = E;, the first and second electrode reactions take place simultaneously and c: # 0. The current due to the ECE reaction in this case is given by eqn. (74). If E; x- E”,, the second n-electron wave is completely separated from the first n-electron wave. This type of ECE reaction can be expressed as an EC reaction and an n-electron electrode reaction. Thus the currents for these two waves can easily be obtained as described above. This is not discussed further here. There are many different mechanisms for ECE reactions such as E,CE,, E,CE,, E,CE,, E&E,, E,CE,, E&E,, E,CE,, EiCE, and E,CE,. However, we discuss only four mechanisms here. E,CE, mechanism If the first and second electron-transfer reactions are reversible and the chemical reaction is the rate:determining step, we can obtain the current due to the E,CE, reaction from eqns. (74) and (76) with k, ZF=+ k, and k, s- k,.

(77) where

and eqn. (77) is the same as that obtained by solving the diffusion equations in Appendix C [see eqn. (C2811. Combining eqns. (75) and (771, we have

E=E;+gln(l+rSE)

+$

(78)

In(q)

If E:=E;, z=

Id 1 + 0 + r,/m

(79

[l+r~/(x+)]

where 8=exp [

$(E-EL)]

and E;+E; Ey2 = 2 Equation (79) is as the same as eqn. ((32) in Appendix C. The rate constant for E,CE, reactions can be obtained not only by using the limiting currents at different microelectrode radii but also by fitting the experimental data to numerical calculations of Z(E, k, rs) based on eqn. (79). E,CE, mechanism The current due to E,CE, eqn. (74):

z=

reaction can be obtained by assuming k, =E,k, r,!m

Id

-D/r&,

1 + 8 + D/r,k,

1 + 8 + r&/m

Rearranging eqn. (80) and defining D -= ‘Sk,

1

in

(8’3)

1 + e + r,/m (1/1,)[1+e+r,fik/D)]

-r,/m

-(1+e)=9t

(81)

41

we obtain (82) E,CE, mechanism The current for the E,CEi reaction can be obtained by assuming k, zs- k_,, k, B k, and k, z+ k, in eqn. (74)

Zd

z=

rs)lo 1

1 + 8 + r,/m

+

- D/rsk, D/r,k,

Rearranging eqn. (83) and defining

1

(83)

(84) obtain

we

(85) E&E,

mechanism

If E; -K E;, the current due to E,CE, rewriting eqn. (76): ‘d

z= 4 + [1+

r,/m

(1 + D/r,kd

reactions can be expressed as follows by

(1+2&)

(86)

By combining eqns. (75) and (77), eqn. (86) can be simplified to E=E;+

-!!&n(!$t)+AL~n(!!$-!!+)

(87)

where Z, is the reversible current due to E,CE, reactions which can be obtained from eqn. (77). If Ey = E”,, we can obtain the following expression for the current from eqn. (74):

(88) Rearranging eqn. (88) and defining -=

D

r&l

(zd/z)(i+r~~t(2+e)/(i+e)1)-e

_l=~ 3

l+~sllo

(89)

42

we obtain

(90)

3

Therefore it is clear that the kinetic parameters can easily be found for ECE reactions at the spherical microelectrode by using eqns. (75), (781, (821, (85), (87) or (90). THE DISPl MECHANISM

DISPl reactions can be expressed as follows: A+n

e-&B k-1

(91)

Bk’C B+C-

(92) k’

A+D

(93)

Under steady-state conditions, we have J=k&cz--cl)

+pk’c$c;

(94)

J = k,c; - k_ &

(95)

J = k,c”, + pkc; + pk’c;c;

(96)

J= k,c”, + k,c”,

(97) and cc = 0 under steady-state condi-

where F = \i( because kc; = k’cic: tions. According to the general expression for DISPl reactions I = dFK,(

cOg + c;)

(98) Combining eqns. (94)-(971, we have the following equation for the current due to DISPl reactions:

2p.k

Zd ‘=

1 + k-,/k,

+ k,/k,

+ (pk/k,)(l

+ 2k,/k,)

( -1 ’ + k,

(99)

From eqn. (99), the limiting steady-state current for this reaction system is given by

This result is in agreement with eqn. (7) of ref. 10. If the electron-transfer reaction is reversible and the first chemical reaction is the rate-determining step of the complete electrode process, then the current due to DISPl reactions can be obtained from eqn. (99): Z

=l+e+r,/m

Id

(ltlr,$&)

(101)

43

Equation (101) is the same as that obtained in Appendix D. Combining eqns. (100) and (lOl), we obtain

( 102) If the electron-transfer both the electron-transfer obtained from eqn. (99):

reaction is irreversible and the current is controlled by reaction and the chemical reaction, the current can be

Substituting eqn. (100) into eqn. (103) we obtain the following Z-E equation: 1+

’ 1+ wrshmm

I+-$

in(q)

(104)

If the electron-transfer reaction is quasi-reversible and the current is controlled by both the electron-transfer reaction and the chemical reaction, the quasi-reversible current is given by eqn. (99). Combining eqns. (1001, (101) and (991, we obtain the following I-E equation: E=E”-

RT ln$ l+ anF ss ( )[ ( 105)

where Z, is the current due to DISPl reactions with a reversible electron-transfer reaction. Thus it is possible to evaluate a and k, from the logarithmic plot using eqns. (104) or (105) with a known value of k. DISCUSSION

A method based on the concept of steady-state reaction-diffusion layer has been established for studying an electrode reaction coupled with a homogeneous chemical reaction. Although Oldham [5] has carried out a theoretical study by solving differential equations based on the codiffusion of interconverting isomers for the E,C, E,CE, and pseudo-first-order E,C’ reactions, it is necessary to develop a new approach for studying some complex systems. The method presented in this paper is powerful since it yields a simplified treatment of otherwise complicated reaction-diffusion problems. All the results presented in this paper for the reversible electrode are in good agreement with those in the literature [5]; results for quasi-reversible and irreversible systems have not yet been reported. It must be emphasized here that the method is suitable for those cases in which the dimension of the reaction layer is not larger than that of the diffusion layer

44

(p G 61, i.e. k 2 O/r,’ because the conditions of steady-state mass transport and equal fluxes only exist in the reaction-diffusion layer. The results obtained by Oldham [5] support this conclusion. ACKNOWLEDGEMENT This

project was supported

by the National Natural

Science Foundation

of

China. REFERENCES 1 M. FIeischmann, S. Pons, D.R. Rolision and P.P. Schmidt (Eds.), Ultramicroelectrodes, Datatech Systems Inc., Morganton, NC, 1987, Ch. 2. 2 R.M. Wightman, D.O. Wipf and A.J. Bard (Ed.), Electroanalytical Chemistry, Vol. 15, Marcel Dekker, New York, 1989, p. 267. 3 A.M. Bond, K.B. Oldham and C.G. Zoski, Anal. Chim. Acta, 216 (1989) 177. 4 S. Pons and M. Fleischmann, Anal. Chem., 59 (1987) 1391A. 5 K.B. Oldham, J. Electroanal. Chem., 313 (1991) 3. 6 K.B. Oldham, J.C. Myland, C.G. Zoski and A.M. Bond, J. Electroanal. Chem., 270 (1989) 79. 7 R. Brina, S. Pons and M. FIeischmann, J. Electroanal. Chem., 244 (1988) 81. 8 CA. Widrig, M.D. Porter, M.D. Ryan, T.G. Strein and A.G. Ewing, Anal. Chem., 62 (1990) 1R. 9 M. Fleischmann, F. Lasserre, J. Robinson and D. Swan, J. Electroanal. Chem., 177 (1984) 97. 10 M. Fleischmann, F. Lasserre and J. Robinson, J. Electroanal. Chem., 177 (1984J.115. 11 C.G. Phillips, J. Electroanal. Chem., 2% (1990) 255. 12 P. Delahay, New Instrumental Methods in Electrochemistry, Interscience, New York, 1954. 13 K.B. Oldham and C.G. Zoski, J. Electroanal. Chem., 256 (1988) 11. 14 J.R. Delmastro and D.E. Smith, J. Phys. Chem., 71 (1967) 2138. 15 A.J. Bard and L.R. FauIkner, Electrochemical Methods: Fundamentals and Applications, Wiley, 1980, Ch. 1. 16 Qiankun Zhuang and Hongyuan Chen, J. Electroanal. Chem., 346 (1993) 471. 17 G. Denuault and D. Pletcher, J. Electroanal. Chem., 305 (1991) 131. 18 J. Heyrovskjr and J. Kuta, Zdkland Polarografie, Nakladatelstivi &skosIovenske Akademie V6d, Prague, 1962. 19 D. Britz, Digital Simulation in Electrochemistry (2nd edn.), Springer-Verlag, Berlin, 1988, p. 158.

APPENDIX A

Under steady-state conditions in a spherical diffusion field the flux of species A and B for the system of eqn. (1% is described by a2c,

D -++ar2 a2c,

D -+

ar2

2 ac,

=O

r ar ) 2 ac,

--

r ar

-kc,=0

(AlI

45

with the initial and boundary conditions t=o,

r,
CA=&

t > 0,

r=co:

t > 0,

r=r,:

CA=C;, a~, ac, -= ar

--

c,=c,=o c,=c,=o

ar

$=exp[g(E-E’)]=e Equations (Al) and (A2) have the general solution G c,=-+H r

The boundary conditions require that H = CA*and P = 0; hence the concentrations and fluxes-of A and B at r = rs can be written as

(4

cA I r=Tg= ci + G/r,

(A6) (A71 (w From the boundary conditions and eqns. (AS)-(AS) we obtain G -=rs

cz[1 +rSDPV]

Q

cz

(A%

1 + e + r,{(iQEj-

(AlO)’

-=l+e+r,/(k/D) rS

exp

Substituting eqn. (A9) into eqn. (A7) gives Wz[l

+r,/WW]

(All)

T=TS= rs 1 + e + r,{m The total current at a spherical electrode Z* I=,. = l+e+r,/m

where Id = 4mtFDr,c:.

is

(fw

46 APPENDIX B

Under steady-state conditions in a spherical diffusion field the flux of species 0 and R in eqns. (29) and (30) is described by

W) W) with the initial and boundary conditions r, 0,

r=co:

t > 0,

r=r,:

CR--0 CR- -0

cg =c;,

ac,

ac,

ar=-ar CO

-=exp CR

Ig(E I

- E”) = 8

Equation (B2) has the general solution

The boundary conditions require that P = 0; hence the concentration flux of R at r = rs can be written as cairCrS=t D: i

1 r=rs

exp( -rs/F)

and the

(Ml

=_!? r,” exp( -rs/F)

-F/m

exp( -rs/y) w

Substituting eqn. (B3) (with P = 0) into eqn. (Bl), we obtain the general solution co=4+H-fexp(-rJ~)

w

** hence the concentration From the boundary conditions, H = co, at r- rs can be written coi.S,=,,.:-E

exp(-rs/F)

and the flw of 0

W)

47

From the boundary conditions and eqns. (B4), (B5), (B7) and (B8), we obtain G=O

Q=-

and

es

W)

1+e

From eqns. (B8) and (B9), the current is given by I-n&D

(5

@lo)

ar )._,,=i+ii(l+rs/%?)

where Id = 47mFDr,c;F. APPENDIX C

Under steady-state conditions the flux of species A, B, C and D in eqns. (60)-(62) for the spherical diffusion field is described by

(Cl) (C2)

w (Cd) with the initial and boundary conditions t > 0,

r,
t > 0,

r = r,:

t = 0,

cA=c;,

cg=cc=cD=o

cA=ci,

c,=c,=c,=o

ac,

ac,

T=-

ar

ac, -=--

ac,

ar

ar

(C5) (C6) (CT (C8) ((3 (ClO)

Equations (Cl), (C2) and (C4) have the general solution G CA= --+H r

(C11)

(Cl21 L

C,=-+J

(Cl3

r

The boundary conditions require that H = c,*, P = 0 and J = 0; hence the concentrations and fluxes of A, B and D at r = r, can be written as c,,._I r=rs = cz + G/r,

( C14) (CW (Cl6) (C17)

acJ3 ar

-1 ,=ls

=

(CW

-

(cw From eqns. (C7), (C% (C141, (Cl%, (C17) and (CM) we obtain

(cm Q

-= rs

c2 1 + 8, + r,/m

exp rs

(al)

Substituting eqn. (C20) into eqn. (Cl71 gives

c,*D[l+r,ii’(k/D)] =r,[l+8,+r,i(k/D)]

r-r5

(C22)

Substituting eqn. (C12) (with P = 0) into eqn. (C3) gives the general solution as follows: M c,=;+N-;exp

Q

(C23)

The boundary conditions require N = 0; then the concentration r = rs can be written as

M

Q

s

s

cCIrsr,=~--;

exp

and flux of C at

(ad)

If E’; * Ez, cc = 0 at the surface of microelectrode. (C21) we have

From eqns. (C24) and

(C26) Substituting eqn. (C26) into eqn. (C25), we obtain c;D

(C27)

r=rs = l+fYI+rSJm

The total current at a spherical microelectrode

is given by

where Id = 4mzFDr,c,. If Ey = Ez, cc + 0 at the surface of microelectrode. (C161, (C19), (C21), (C24) and (C25) we have

Z

c: r,M

From eqns. (CB), (ClO),

(C29)

-q

W30)

where

and

I

E;+E; Ey2== 2 From eqns. (C21), (C25) and (00)

we have (C31)

50

The total current for the case of ECE reactions at a spherical microelectrode given by

is

cc33 where id = 4pnFDr,c,*. APPENDIX D

In the case of steady-state conditions the flux of species A and B in eqns. (911, (92) and (93) for a spherical diffusion field is described by

a2c,

2 ac,

+Tar Di ar2

a2c,

D -++( ar2

)

2 ac, r

ar )

+k’cBcC=O

P)

-kc,

W)

- k’cBcC = 0

with the initial and boundary conditions t = 0,

r,
t > 0,

r=co:

t > 0,

r=r,:

cA=cA,

*

c,=c,=o

* CA=CA,

ac, V=-

c,=c,=o

ac,

-

ar

!$=exp[g(E-E’)]=tl Under steady-state solution

conditions,

kc, = k’cBcC. Then eqn. (D2) has the general

c.=fexp(rg)+Fexp(-rE)

The boundary conditions require that P = 0; hence the concentration of B at r = rs can be written as c~I~=,,=F

Q s

exp

P) and the flux

51

Substituting eqn. (D3) (with P = 0) into eqn. (Dl), we obtain the general solution G CA---+H-2rexp r

Q

VW

** thus the concentration From the boundary conditions, H = CA, r = rs can be written as

cAIr_.,=c;+~--2,

G

Q

s

exp

and flwr of A at

(w

s

From the boundary conditions and eqns. (D4), (D5), (D7) and (D8), we obtain

Q -=

rs

G

-=rs

w9 P 10)

From eqns. (D8), (D9) and (DlO) the total current is given by

Pll) where Id = 47rnFDr,c,*.