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J. El ectroanal. Chem., 313 (1991) 3-16

Elsevier Sequoia S.A., Lausanne

JEC 01600

Steady-state microelectrode voltammetry as a route

to homogeneous kinetics

Keith B. Oldham

Electr~hemica~ ~bu~at~~‘es, Trent ~nj~ rsity, Peter~~~~gh ~Can~u~

(Received 13 November 1990; in revised form 3 April 1991)

Abstract

It is demonstrated that, under propitious circumstances, information concerning the kinetics of

homogeneous reactions can be gamed by studying how features of steady-state voltammograms depend

on the size of the voltammetric microelectrode. Applications to CE, EC, ECE and EC’ mechanisms are

discussed.

INTRODWCTION

The advantages of steady-state volta~et~ compared with traditional transient

voltammetries include much greater freedom from interferences arising from capaci-

tive currents, uncompensated resistance and instrumental imperfections [If. Micro-

electrodes [2-71 provide one of the simplest methods for achieving steady-state

voltammetry.

It has been demonstrated [8,9] that the kinetics of electron-transfer reactions are

accessible from steady-state voltammetric studies at electrodes of a variety of small

sizes. This article shows that the kinetics of preceding or succeeding homogeneous

reactions may be elucidated in a similar way. It is assumed that the microelectrode

is a hemisphere of radius a (or can be treated as if it were [lo]), and that a

heterogeneous electron-transfer reaction occurring at its surface is coupled, in some

way, to the homogeneous reaction.

Throughout this article the term

“isomer” will be used loosely. This word is

employed in the first three applications to refer to each of two species that may be

genuinely isomeric, e.g.

~22-0728/91/$03.50

0 1991 - Elsevier Sequoia S.A. All rights reserved

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HeC.COOH

HO0C.C.H

II

=

H-C-COOH

H COOH

(1)

. .

or they may be related to each other by solvation, e.g.

HCHO(aq) + H,O(l) e

H,C(OH),(aq)

or by a complexation exchange e.g.

FeCl’++ Cl-

= FeClT

(2)

(3)

involving ligands present in excess. In the final application, the meaning of “isomer”

is stretched even further to cover structurally similar species that

state.

diffe; in oxidation

Before considering the electrochemistry itself, a general result

will be derived.

CODIFFUSION OF INTERCONVERTING ISOMERS

Let P and Q be solute species that interconvert by a mechanism that involves

only first-order (or pseudo-first-order) reaction steps

P

k,

=

k-7

(4)

with homogeneous rate constants k, and k-i. In this section we derive the

steady-state concentration profiles of isomers P and Q when these are being

conveyed to or from a hemispherical (or spherical) electrode by diffusive transport

with diffusion coefficients D, and DQ.

Fick’s second law for the spherical diffusion of species P becomes modified to

a a 20, 3

pp =

Dpar2cp r =pp -

k,c, + k_,cQ

when isomeric interconversion occurs. Here cp and co are the concentrations of the

isomers and r denotes the radial coordinate. Under steady-state conditions the local

concentration of P does not change, so that ac,/& equals zero, and therefore

Similarly

2DQ

d

The solution to these simultaneous differential equations will be sought in terms of

the isomer concentrations c”p and c& at the electrode surface and the total bulk

concentration cb = cb + c& where we use superscripts “s” and “b” to denote

conditions at the surface of the electrode and in the bulk of the solution. Note that

rs = a.

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6

and the first right-hand term in eqns. (13) and (14) may be simplified accordingly.

For example, eqn. (13) becomes

Our prime interest is in the fluxes of the isomers at the electrode surface. These

are easily determined from Fiek’s first law j = - D(dc/dr) with r then set equal to

a. For isomer P we find

(18)

with a similar result for Q.

THE EQUIDIFFUSIYITY SIMPLIFICATION

In the last section the diffusion coefficients of P and Q were regarded as distinct.

This distinction could be maintained but the equations become simpler, while still

retaining all the essentials of the problem, when the isomers have similar diffusivi-

ties. Setting D, =

Do = D, eqn. (18) becomes

jS, =

-Dk_,cb

ka +

(G+k,j/-+-k-q& ;

where

k = k, + k _

.

Similarly

(19)

Moreover, we shall assume that other species, formed by electron transfer to or

from the codiffusing isomers, also share the same diffusion coefficient D.

THE CE MECHANISM

Consider species A to be electroinactive, whereas its isomer B is oxidized

(reduced) by an n-electron heterogeneous transfer to species Z, which diffuses away

from the electrode

k,

A-

‘k_,

B -ne_ Z

+n

e

(21)

The electroinacti~ty of isomer A implies that ji is zero. Replacing P and Q by A

and B, we then find from eqn. (19) that

s _ v%k_,cb + k_,av%c”,

cA

%k + k_,afi

Cw

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nFcb

t

8

1

,&

i,a

Fig. 1. How the dependence of the limiting current density

i

on the radius of a microelectrode at which

a CE mechanism occurs can be analyzed to obtain the rate constants. Note that k = k, + k_,.

This result, when substituted into the analogue of eqn. (20) reveals that

(23)

after some algebra.

The current density (positive when anodic) is - nFjg (n is negative for a cathodic

reaction), i.e.

nFD

i = -

a

b k s

Jr I

-Kc”

(24

and has its maximum value (the limiting current density i,) when the electrode is

polarized to such an extreme value that cS, = 0. Thus

(25)

after rearrangement. Figure 1 shows how this equation may be further rearranged to

provide a linear graph from measurements of the limiting current density as a

function of electrode radius. The slope and intercept permit the homogeneous

kinetics to be elucidated.

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8

The most useful situation arises when the equilibrium constant between A and B

favours the electroinert isomer A so that

Then, provided the microelectrode radius a is such that, approximately

then the limiting current density is independent of a, whereas it is inversely

proportional to a outside this range.

This result is true irrespective of the degree of reversibility of the electron-trans-

fer process. However, if the process is reversible, the Nernst law

applies. One shows easily that the ncentration profile of product Z is cz = a&/r

if Z is absent from the bulk solution. Therefore

i=nFjg=-nFD

If one subtracts eqn. (24) from eqn. (25) and divides by eqn. (29), one finds

which may be combined with the Nernst eqn. (28) to produce

Hence the preceding isomerization does not affect the shape of the reversible

voltammogram but does influence its position on the potential axis.

THE ErC

MECH NISM

Now transfer attention to the mechanism

-rte-

Y

k,

A

V

e

Z

+n

e-

k-2

in which the product Y of

(32)

a reversible electrooxidation (or electroreduction, in

which case n is negative) isomerizes to Z. Both Y and Z are absent from the bulk

solution. These two species are the codiffusing pair so, by analogy with eqns. (19)

and (20)

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where now k = k, + k_,, and

Likewise, from the concentration profile

one finds that

The fact that jg = 0 establishes the result

c; =

ak&

ak_, + JDk

with the help of eqn. (35); then substitution into eqn. (33) shows that

j = [ alr:+z]

The current density may be expressed either as

so that

-i=

nFD

iL

- cs

a A

(34)

(35)

(36)

(37)

(38)

(39)

(40)

or as

i = nFj =

(41)

The Nernst equation

may now be combined with eqns. (41) and (40) into

(42)

(43)

The succeeding homogeneous reaction evidently does not affect either the height

or the shape of the reversible steady-state voltammogram, but it does influence its

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position on the potential axis. The halfwave potential (where

i =

iL/2) is

E

where K, = k,/k_,. The “constant” in this equation is the hdfwave potential when

a is large, and corresponds to the classical macroelectrode value of E1,,2, namely

E,” = Rllj/nF) ln 1 + K,). Even with a microelectrode, we see that the voltammetry

is scarcely affected by the succeeding isomerization when K, is small. When this

equilibrium constant is large, however, there exists a wide range of electrode radii

that satisfy the inequality

In this range eqn. (44) shows that E,,, is dependent on a according to

whereas outside this range the halfwave potential is virtu~ly independent of

electrode radius. Figure 2 illustrates the behaviour graphically and shows how E,,,.

Fig. 2. Dependence of the halfwave potential of an EC ekctron transfer on the k?garithmof the

microelectrode radius.

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versus a data may be

their quotient, K,.

THE EFE, MECHANISM

11

analyzed to determine the rate constants k , and k _, and

Next consider the mechanism

k,

A (E;) & P m

+n e-

k-2

Q G’) + z

47)

in which the product P of a reversible electron-transfer reaction must isomerize to Q

before a further reversible oxidation (or reduction, n then being negative) to Z is

possible. For simplicity, the same number of electrons is assumed to be transferred

in each electrochemical step. Only species A is present in the bulk solution.

Using the same approach that has been used in prior sections, the surface fluxes

of the four species are found to be

(48)

where, once more, k = k 2 + k _,, and

ji_; SC- (50

The diffusion away from the electrode of each P or Q corresponds to the transfer

of n electrons, whereas 2n electrons are transferred for every Z that leaves the

electrode. It follows that the current density is given by

i = nF ( j g + j ;S ) + 2nF j i = nF ( j g - j i )

(52)

the second equality following from the conservation requirement that ji + j$ +jS, +

jg be zero.

Because the interconversion of P and Q is a homogeneous reaction, the amount

of this isomerization taking place at the electrode surface will be negligible. This

means that j: + j; = 0 and j; + ji = 0, whereby eqns. (48)-(51) reveal that

es\+ 1 k ~~/~~c~ - (k _*~ /~~c~ = c;

(53)

and

c;+ (l+ k _,a/& )cs, - (k & )c”p==O

(54)

Next, note that because both electron transfers are reversible, each may be

allocated a “potential-dependent equilibrium constant” via Nernst’s law:

K,(E) =exp{ -$(E-SF)) = 2

(55)

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b)

Fig. 3. For an ECE mechanism, similar information is available a) from the limiting current density i,

and b) from the halfwave potential E,,,, when the radius LI of the microelectrode is varied.

current density reduces to

i l F

j = ___

i

1+ 2k,a/m

a

1+

[~/K,(E)] + k,a/m

I

(61)

which describes a single wave of normal sigmoidal shape but with a halfwave

potential that shifts with microelectrode size according to the equation

g(E,,,

- Ef) = ln[ K,( E1/2)] = -ln[l +

k,a/m]

(62)

which is illustrated in Fig. 3(b) and shows how kinetic information may be

extracted.

In the foregoing it has been assumed that neither the disproportionation

P+Q-+A+Z

(63)

nor the converse conproportionation reactions occur homogeneously. The likelihood

of such bimolecular reactions diminishes as ck is lowered.

THE EC’ MECHANISM

The “EC’ ” symbol is given [12] to a mechanistic scheme known alternatively as

“catalytic regeneration”.

If the product Z of an electron transfer reaction is

reconverted to the reactant A by a homogeneous pseudo-first-order reaction of rate

constant k _

1

(64)

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Fig. 4. The dependence on l/a, the reciprocal of the electrode radius, of the limiting current density due

to catalytic regeneration.

then analogues of eqns. (19) and (20) apply with k, set to zero. That is

j;= -;(

-C;)-pJC;

and

(65)

and since these two surface fluxes must sum to zero to satisfy conservation, it

follows that cS = cb - cS

The currentzdensity i;‘easily found to be

The limiting current density, which corresponds to CA = 0, is therefore

i, =

n

(67)

and varies with electrode size as illustrated in Fig. 4. Evidently the catalytic rate

constant k _, is easily found by measuring the current density for several electrode

sizes.

CONCLUSIONS

The kinetics of all the mechanisms that we have considered may be elucidated, at

least in part, by studying how the steady-state voltammograms respond to changes

in electrode size. The technique is most successful for rate constants whose magni-

tude is comparable with D/a’ . Radii of microelectrodes are limited to values

greater than about 0.1 pm by fabrication difficulties (but see ref. 13). There is no

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15

corresponding upper limit, but the time required to reach a steady state [14]

becomes excessive for microelectrodes much larger than about 10 pm. Hence these

methods should be ideal for reactions with first order rate constants in the 102-lo4

~ ’ range.

Of course, there exist several other electrochemical methods for investigating

homogeneous reactions. These are discussed in several reviews [12,15-171, where

also will be found examples of systems that conform to the various mechanisms. In

addition, the literature [e.g. 18, 191 contains derivations with objectives similar to

those reported here.

GLOSSARY

i,

B

cx

c”x

Cb

csc

D

Dx

e-

E

EZ

E

l/2

F

g, h

i

i,

_i

A

k

km

k-m

K,

K,(E)

n

P, Q

r

R

t

T

y, z

A, II

radius of (hemi)spherical microelectrode, m

reactant species

local concentration of species X, mol mP3

concentration of X in the bulk of the solution, mol me3

total bulk concentration, mol me3

concentration of X at the electrode surface, mol me3

common diffusion coefficient of all relevant species, m2 s-’

diffusion coefficient of species X, m2 s-r

an electron

electrode potential, V

conditional potential of the mth reaction, V

halfwave potential, V

Faraday’s constant, 96485 C mol-’

D,c, + DQcQ, mol m-l s-r

current density, A m-’

limiting current density, A mP2

local flux, mol me2 s-’

flux of species X at electrode surface, mol me2 s-i

sum of the rate constants (e.g. k, + k_,) for the relevant reaction, s-l

rate constant for the forward direction of the mth reaction, s-r

rate constant for the backward direction of the mth reaction, s-i

equilibrium constant of the mth reaction, = k,/k_,

“potential dependent equilibrium constant” of mth reaction

number of electrons transferred (negative for a reduction)

arbitrary or intermediate species

radial coordinate, m

gas constant, 8.3144 J K-i mol-’

time, s

temperature, K

product species

(WW f (L/Do), me2

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REFERENCES

1 A.M. Bond, K.B. Oldham and C.G. Zoski, Anal. Chim. Acta, 216 (1989) 177.

2 R.M. Wi~tman and D.O. Wipf, in A.J. Bard (Ed.), El~~oan~ytic~ Chemistry, Vol. 15, Marcel

Dekker, New York, 1989, p. 267.

3 G.A. Brydon and K.B. Oldham, J. Electroanal. Chem., 122 (1981) 353.

4 J.O. Howell, Curr. Sep., 8 (1987) 2.

5 S. Ports and M. Fleischmann, Anal. Chem., 59 (1987) 1391A.

6 R.M. Wightman, Science, 240 (1988) 415.

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1960.

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13 R.M. Penner, M.J. Heben, T.L. Longin and N.S. Lewis, Science, 250 (1990) 1118.

14 C.G. Zoski, A.M. Bond, E.T. Allinson and K.B. Oldham, Anal. Chem., 62 (1990) 37.

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Chemical Kinetics, Vol. 26, Elsevier, Amsterdam, 1986, pp. 317-349.

17 C.M.A. Brett and A.M.F.C. Ohveira-Brett, in ref. 16, pp. 411-421.

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