Chronoamperometric behaviour of a CE process with fast chemical reactions at spherical electrodes and microelectrodes. Comparison with a catalytic reaction

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Electrochemistry Communications 8 (2006) 1062–1070

Chronoamperometric behaviour of a CE processwith fast chemical reactions at spherical electrodes andmicroelectrodes. Comparison with a catalytic reaction

Angela Molina *, Isabel Morales, Manuela Lopez-Tenes

Departamento de Quımica Fısica, Universidad de Murcia, Espinardo 30100, Murcia, Spain

Received 29 March 2006; received in revised form 5 April 2006; accepted 18 April 2006Available online 24 May 2006

Abstract

A simple approximated time-dependent equation for the limit current corresponding to fast chemical reaction preceding the elec-trochemical step (CE process under kinetic steady state conditions) applicable to spherical electrodes and microelectrodes is deduced.From this equation, a simple method to elucidate the homogeneous kinetic is proposed. The particular cases of planar, microsphericaland ultramicrospherical electrodes are obtained and the severe limitations for application of the time-independent equation for micro-electrodes, usually used in the literature, are demonstrated. The equation deduced in this paper has been shown to be fundamental inthe understanding of the reaction and diffusion layer thicknesses in steady state spherical diffusion, which have not been clearlydefined until now. The different types of steady states that CE and catalytic mechanisms can reach are compared and explained. Itis also shown that in the steady state, the CE mechanism becomes the catalytic one when reaction and diffusion layers thicknessesare equal.� 2006 Elsevier B.V. All rights reserved.

Keywords: CE mechanism; Catalytic mechanism; Fast chemical reaction; Spherical electrodes; Microelectrodes; Steady state; Diffusion layer thickness;Reaction layer thickness

1. Introduction

In a recent paper [1] we have studied the characteristicpathway that a catalytic process follows to reach the steadystate when using spherical electrodes of any radius, con-cluding that for this reaction mechanism the attainmentof the true steady state (i.e. independent of time response)is not limited by the electrode size, provided the values ofthe rate constants are sufficiently high. Thus, this steadystate can be achieved even with planar electrodes. However,in the same paper, we have drawn attention to the severerestrictions in the value of radius that have to be taken intoaccount when the true steady state is studied in a CE mech-anism and so it cannot be attained in a planar electrode.

1388-2481/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.elecom.2006.04.011

* Corresponding author. Tel.: +34 968 367524; fax: +34 968 364148.E-mail address: [email protected] (A. Molina).

Thus, in this paper, we analyze the behaviour of a CEprocess at spherical electrodes, starting from conditionsof kinetic steady state, which does not present an indepen-dent of time response, and we propose an approximationthat, as yet, has not been reported in the literature. Theapproximation leads to a very simple time-dependentexpression for the limit current of a CE mechanism atspherical electrodes, which, despite its simplicity, givesexcellent results for fast chemical reactions. From thisstudy, we have compared and explained the different typesof steady states that the CE and the catalytic mechanismscan reach concluding that the attainment of an independentof time response is more difficult and considerably morerestrictive in relation to the selection of microelectroderadius in the case of the CE mechanism. This differentbehaviour of the CE mechanism (and in general of pro-cesses with coupled non-catalytic homogeneous kinetics)

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�A. Molina et al. / Electrochemistry Communications 8 (2006) 1062–1070 1063

in comparison with a catalytic process had not been high-lighted to date [1], in spite of the extensive literature on thissubject [1–11].

From the simple expression of the current obtained inthis paper for a CE process with spherical electrodes weobtain as particular cases those corresponding to planarelectrodes [12,13] and to microelectrodes, being the currentin this last case a truly stationary one (i.e., independent oftime), coincident with that previously deduced in [4,5,9].We also establish the necessary conditions for the applica-tion of the last equation, showing that it can be used onlyfor a narrow interval of microelectrode radius. However,this limitation has not been duly considered in the litera-ture, where the equation for microelectrodes has beenwidely used without delimiting its range of applicability[4–11]. This is due without doubt to these authors solvingthis problem by starting from true steady state conditions,which make it impossible to detect the necessary require-ments under which the response obtained is valid.

Moreover, the equation for the current deduced in thispaper has been shown to be fundamental in the definitionof reaction and diffusion layers thicknesses in steady statespherical diffusion, not clearly defined until now, and fromwhich, the limit cases for planar, microspherical and ultrami-crospherical electrodes are obtained. In this respect, we havedemonstrated that, in the steady state, the catalytic mecha-nism can also be considered as a particular case of the CEprocess, with equal reaction and diffusion layers thicknesses.

Finally, we propose a method for obtaining the kineticparameters of the chemical reaction, based on the use ofthe equation deduced in this paper. The procedure is verysimple and can be used in a wide range of electrode radius,advantages which make this method a suitable and recom-mendable alternative.

2. CE mechanism under conditions of kinetic steady state

The scheme for a first or pseudo-first order CE processcan be written as [14–16]:

B ¢k1

k2

C

C þ ne� ¢kf

kb

DðScheme IÞ

where k1 and k2 are the homogeneous reaction rate con-stants and kf and kb are the rate constants of forward(reduction) and backward (oxidation) heterogeneous reac-tions, respectively.

We define the equilibrium constant K:

K ¼ k2

k1

¼ c�Bc�C

ð1Þ

and

j ¼ k1 þ k2 ð2Þc�i ði ¼ B or CÞ being the bulk concentrations of species Band C.

With the introduction of variables f and / defined as[12,13]:

fðr; tÞ ¼ cBðr; tÞ þ cCðr; tÞ; ð3Þ/ðr; tÞ ¼ cBðr; tÞ � KcCðr; tÞ; ð4Þ

where /(r,t) measures the perturbation of the chemicalequilibrium (see Eq. (1)), it is fulfilled, if a spherical elec-trode of radius r0 is used, that:

ofðr; tÞot

¼ Do2fðr; tÞ

or2þ 2

rofðr; tÞ

or

� �; ð5aÞ

o/ðr; tÞot

¼ Do

2/ðr; tÞor2

þ 2

ro/ðr; tÞ

or

� �� j/ðr; tÞ; ð5bÞ

where D is the diffusion coefficient, assumed equal for spe-cies B and C, and the boundary value problem is:

t ¼ 0; r P r0

t > 0; r !1

�fðr; tÞ ¼ f� ¼ c�B þ c�C; /ðr; tÞ ¼ 0

ð6Þt > 0, r = r0:

� Kofðr; tÞ

or

� �r¼r0

¼ o/ðr; tÞor

� �r¼r0

ð7Þ

fðr0; tÞ ¼ /ðr0; tÞ ð8Þ

with the current given by

I spherCE ¼ nFAD

ofðr; tÞor

� �r¼r0

; ð9Þ

where F and A have their usual electrochemical meaning.Note that conditions (7)–(9) imply that cC(r0, t) = 0 (limit

current) and ocBðr;tÞor

� �r¼r0

¼ 0 (electroinactivity of species B).

The kinetic steady state approximation only affects thevariable / (Eq. (4)) and consists in supposing that the pertur-bation of chemical equilibrium is independent of time, i.e.,

o/ðr; tÞot

¼ 0 ð10Þ

which is fulfilled in fast kinetic reactions (j� 1, K� 1).Under this condition, the solution for the variable / inEq. (5b) is immediately obtained and is given by:

/ðrÞ ¼ r0

r/ðr0Þe�

ffiffiffiffiffiffij=Dp

ðr�r0Þ ð11Þ

and then

d/ðrÞdr

� �r¼r0

¼ �/ðr0Þ1

r0

þffiffiffiffijD

r� �: ð12Þ

Thus, taking into account Eqs. (7), (9) and (12) we canwrite:

I spherCE

nFAD¼ ofðr; tÞ

or

� �r¼r0

¼ 1

K/ðr0Þ

1

r0

þffiffiffiffijD

r� �: ð13Þ

Eq. (12) is valid for any value of the electrode radius r0

(i.e., for 0 6 r0 61). However, in order to avoid thatthe presence of the homogeneous chemical reaction is

1064 �A. Molina et al. / Electrochemistry Communications 8 (2006) 1062–1070

masked by the high diffusive flux to a microelectrode, thefollowing condition should be accomplished [1]:

r0 P1

10

ffiffiffiffiDj

rð14Þ

in such a way that the response obtained differs by morethan 10% in reference to the situation in which the homo-geneous kinetic cannot be detected (see Eq. (36)).

The approximated solution for variable / given by Eq.(11) and the subsequent comments are general for anymechanism with coupled chemical reactions (CE, EC, cat-alytic, etc.) under kinetic steady state conditions.

Thus, the solution of equation system (5), with theapproximation given by Eq. (10) and the boundary condi-tions (6)–(8), leads to the following expression for the cur-rent (Eq. (13)) corresponding to a CE mechanism withrapid chemical reaction under limit conditions, I spher

CE , inspherical diffusion [2]:

I spherCE

nFADf�¼ h

1þ hr0

½1þ hr0 expðvspherÞ2erfcðvspherÞ�; ð15Þ

where

h ¼ 1

K1

r0

þffiffiffiffijD

r� �; ð16Þ

vspher ¼ ð1þ hr0Þn ¼1þ K

Knþ

ffiffiffivp

K; ð17Þ

n ¼ffiffiffiffiffiDtp

r0

; ð18Þ

v ¼ jt: ð19ÞNote that the imposition of kinetic steady state (Eq. (10))leads to a time-dependent current, i.e., to a non-stationaryresponse (see Eq. (15)). This is due to the variables / and fbeing related through the boundary value problem, and fdepending on time (see Eqs. (5a), (7)–(9)).

3. CE mechanism under conditions of kinetic steady state

supposing a purely diffusive behaviour for pseudo-species

f (=cB + cC)

We introduce in this section a new approximation whichwill be used for characterizing a CE process and which willbe shown to be fundamental for the general definition andunderstanding of the spherical reaction and diffusion layers(see below).

In this approximation, the variable / (Eq. (4)) retainsthe form given by Eq. (11) for the kinetic steady stateapproximation (o//ot = 0). In relation to variable f (Eq.(3)), we consider that of/ot 6¼ 0, as in Section 2, and there-fore f(r, t) must verify Eq. (5a). At this point, we assumethat the solution of Eq. (5a) has the same form as thatfor a pseudo-species that would only suffer spherical diffu-sion. This means that the solution for f has the form [17]:

fðr; tÞ ¼ f� � r0

rðf� � fðr0ÞÞerfc

r � r0

2ffiffiffiffiffiDtp

� �; ð20Þ

where f(r0) is the value of f at the electrode surface.

Note that the solution given by Eq. (20) is only rigor-ously applicable in spherical diffusion when f(r0) is time-independent [17] and hence it is approximated in thepresent case since from Eq. (20) we deduce that under theseconditions f(r0) depends on time and is given by the follow-ing expression (Eqs. (7), (8) and (12)):

fðr0; tÞ ¼ cBðr0; tÞ ¼Kf� 1

r0þ 1ffiffiffiffiffi

pDtp

� �

K 1r0þ 1ffiffiffiffiffi

pDtp

� �þ 1

r0þ

ffiffiffijD

p� � : ð21Þ

From Eqs. (9) and (20) it is deduced that:

I spherCE

nFAD¼ ofðr; tÞ

or

� �r¼r0

¼ ðf� � fðr0ÞÞ1

r0

þ 1ffiffiffiffiffiffiffiffipDtp

� �ð22Þ

and by inserting Eqs. (7), (8) and (12) in Eq. (22) we deducefor the current:

I spherCE

nFADf�¼

1r0þ

ffiffiffijD

p� �.K

1þ 1r0þ

ffiffiffijD

p� �.1r0þ 1ffiffiffiffiffi

pDtp

� �K

ð23Þ

which, in order to compare with Eq. (15) can also be writ-ten in the form:

I spherCE

nFADf�¼ h

1þ hr0

1þ hr0

1

1þ p1=2vspher

� : ð24Þ

Eqs. (23) or (24) are valid for any value of the electroderadius, r0, provided the kinetic is sufficiently fast, whichis logical since no constraint on the sphericity in the dif-fusional equation of f has been imposed. Indeed, theterms (exp(vspher)2erfc(vspher)) in Eq. (15) and 1

1þp1=2vspher

� �in Eq. (24) are coincident with a relative error of less than

5% for:

vspher P 9:7 ð25Þ

From Eqs. (17) and (25) it is clear that if the electrode ra-dius diminishes (n increases, see Eq. (18)) the value offfiffiffi

vp

=K required for Eq. (25) to be fulfilled diminishes withits maximum value being

ffiffiffivp

=K ¼ 9:7 for a planar elec-trode (r0!1,n! 0).

As far as we know, Eq. (23) has never been deduced.

3.1. Limit cases of Eq. (23)/(24)

� Planar electrode (r0!1,n! 0), Eq. (23) simplifies to:

IplanCE

nFADf�¼ 1=ðK

ffiffiffiffiffiffiffiffiffiD=j

pÞ

1þffiffiffiffiffiffiffiffipDtp

=ðKffiffiffiffiffiffiffiffiffiffiffiD=jÞ

p ; ð26Þ

which can also be written in the form:

IplanCE

IdðtÞ¼

ffiffiffipp

vplan

1þffiffiffipp

vplan; ð27Þ


where

IdðtÞ ¼ nFA

ffiffiffiffiffiDpt

rf� ð28Þ

and (see Eq. (17))

vplan ¼ffiffiffivp

K: ð29Þ

Eq. (27) for a planar electrode is similar to that obtainedin the literature for the expanding plane model of adropping mercury electrode [12–14].� Microelectrodes ðr0 6

120

ffiffiffiffiffiffiffiffipDtp

; n P 20ffiffipp Þ. From Eq. (23),

by imposing the condition

1

r0

þ 1ffiffiffiffiffiffiffiffipDtp ’ 1

r0

; ð30Þ

we obtain:

ImicroCE

nFADf�¼

1r0þ

ffiffiffijD

p� �=K

1þ r01r0þ

ffiffiffijD

p� �=K

ð31Þ

or, taking into account Eq. (16) (see Eq. (24)):

ImicroCE

nFADf�¼ h

1þ hr0

: ð32Þ

Eq. (31) coincides with that deduced previously in [4,5,9],which was obtained under conditions of stationary behav-iour for variables / and f, i.e. when o/(r,t)/ot = 0 andof(r,t)/ot = 0. Thus, the time-independent response fora CE mechanism is only obtained if the electrode radius,r0, fulfils the condition ð1=r0 þ 1=


Þ ’ 1=r0 (Eq.(30)). In fact, Eqs. (23) and (31) coincide with an errorof less than 5% if r0 6


=20. Thus, if Eq. (31) is usedto determine the kinetic parameters of the chemical reac-tion, the following conditions must be fulfilled simulta-neously (see Eq. (14)):

1

10

ffiffiffiffiDj

r6 r0 6

1

20


; ð33Þ

i.e. (see Eqs. (18) and (19)):

10ffiffiffivp

P n P20ffiffiffipp : ð34Þ

Table 1Different types of steady states that the CE and the catalytic mechanisms can

Kinetic steady state (o//ot = 0), v� 1

r0 P 110

ffiffiffiDj

q

CE mechanism Response dependent

on time given by Eq. (15)Purely diffusive behaviofor fResponse dependent on

time given by Eq. (24)Catalytic mechanism Response independent

of time given by Eq. (37)Situations not possible ithat f(r,t) = f* "r,t

The condition r0 P 110

ffiffiffiDj

qis necessary in order to detect the kinetic of the che

The upper limit ðr0 6ffiffiffiffiffiffiffiffipDtp

=20Þ is necessary for an inde-pendent of time response to be reached (a true station-ary response), i.e., to get that of(r,t)/ot = 0, whereasthe lower limit ðr0 P


p=10Þ is required to avoid

the presence of coupled homogeneous chemical reac-tions being masked by the high diffusive flux at themicroelectrode.� Ultramicroelectrodes ðr0 6

110

ffiffiffiDj

q; n P 10

ffiffiffivp Þ. From Eq.

(23) it is obtained:

Iu-microCE

nFADf�¼ 1=ðr0KÞ

1þ r0=ðr0KÞ ð35Þ

or also

Iu-microCE ¼ Idð1Þ

1þ K¼ 4pr0nFDf�

1þ K¼ 4pr0nFDc�C ð36Þ

and hence the kinetic information about the chemicalreaction is lost. Note that in these conditions the currentis proportional to the bulk concentration of species C(see Scheme I) and so, by using electrodes of very small ra-dius we can immobilize the chemical reaction and detectthe free concentration of the electrochemical reactive, c�C.

4. Catalytic mechanism

The reaction scheme for a catalytic process can be writ-ten as [1]:

B ¢k1

k2

Cþ ne� ¢kf

kb

B ðCE0Þ

Cþ ne� ¢kf

kb

B ¢k1

k2

C ðEC0Þ

9>>=>>;

Catalytic mechanism:

ðScheme IIÞIn this case, taking into account that f(r, t) is not an un-known function since it is constant and equal to f* forany values of r and t, of(r,t)/ot = 0 is always satisfied, inde-pendently of whether the steady state has been reached ornot. Thus, it is only necessary to impose the condition o/(r,t)/ot = 0, accomplished for rapid kinetics in order to ob-tain a true stationary response for any value of the elec-trode radius [1], which is given by the expression:

I spherCAT

nFADf�¼

1r0þ

ffiffiffijD

p1þ K

: ð37Þ

reach

110

ffiffiffiDj

q6 r0 6

120


r0 <1

10

ffiffiffiDj

q

ur Diffusive steadystate (of/ot = 0)

Responses independent

of time and of the kinetic

of chemical reaction given by Eq. (36)Response independent

of time given by Eq. (32)n a catalytic process due to

mical reaction.


In Table 1, we summarize the different stages of steadystate that catalytic and CE mechanisms can present.

5. Transcendence of Eq. (23) deduced in this paper in the

definition of spherical reaction and diffusion layers

thicknesses

Under conditions of stationary kinetic and diffusion, thethickness of the reaction layer on the surface of a micro-electrode is less than that of the diffusion layer. Thus, theeffects of the electrode curvature on the diffusion can beignored. With this assumption the equation system (5) issimplified to the form:

d2fðrÞdr2

¼ 0; ð38aÞ

d2/ðrÞdr2

þ 2

ro/ðr; tÞ

or¼ j

D/ðrÞ; ð38bÞ

with the boundary value problem given by Eqs. (6)–(8).From Eq. (38a), which implies that the dependence of f

with the distance is linear, and supposing that its perturba-tion obeys only the diffusion of the species and extends to afinite distance from the electrode, we can define a ‘‘spheri-cal diffusion layer thickness’’, dspher, in such a way that:

dfðrÞdr¼ ctn ¼ dfðrÞ

dr

� �r¼r0

¼ f� � fðr0Þdspher

: ð39Þ

If it is also supposed that the dependence of the equilibriumperturbation function / with the distance is approximatelylinear and equal to its value at the electrode surface, andthis variation extends to a distance from the electrode thatwill be called ‘‘spherical reaction layer thickness’’, dspher

r , wecan write:

d/ðrÞdr’ ctn ’ d/ðrÞ

dr

� �r¼r0

¼ /ðdspherr Þ � /ðr0Þ

dspherr

¼ �/ðr0Þdspher

r

:

ð40ÞBy combining Eqs. (7)–(9), (39) and (40) we deduce the fol-lowing expression for the current:

I spherCE

nFADf�¼ ð1=dspher

r Þ=K

1þ ðdspher=dspherr Þ=K

: ð41Þ

Note that from the above results it is not possible to knowthe values of dspher and dspher

r . However, from comparisonof Eqs. (23) and (41) deduced in this paper, we can imme-diately identify the expressions of spherical reaction anddiffusion layers, which are then defined as:

dspherr ¼ 1

1r0þ

ffiffiffijD

p ¼ r0

ffiffiffiffiDp

ffiffiffiffiDpþ r0

ffiffiffijp ; ð42Þ

dspher ¼ 11r0þ 1ffiffiffiffiffi

pDtp¼ r0


r0 þffiffiffiffiffiffiffiffipDtp : ð43Þ

The same expressions for dspherr and dspher could be deduced

by comparing Eq. (40) with Eq. (12) and Eq. (39) withEq. (22), respectively.

5.1. Limit cases of Eq. (41)

5.1.1. CE mechanism

� Planar electrode (r0!1). In this case Eq. (41) can bewritten as:

IplanCE

nFADf�¼ ð1=dplan

r Þ=K

1þ ðdplan=dplanr Þ=K

; ð44Þ

where dplanr and dplan are the well known classical reac-

tion [14,18] and diffusion [14–16] layers thicknesses fora planar electrode, defined respectively as (see Eqs.(26), (42) and (43))

dplanr ¼


p; ð45Þ

dplan ¼ffiffiffiffiffiffiffiffipDtp

: ð46Þ

� Microelectrodes (r0 61

20


Þ. In this limit case, fromEq. (41) we have:

ImicroCE

nFADf�¼ ð1=dmicro

r Þ=K

1þ ðdmicro=dmicror Þ=K

: ð47Þ

Under these conditions by comparing Eq. (47) with Eqs.(23) and/or (31) we can obtain immediately theparticular values of reaction and diffusion layerthicknesses:

dmicror ¼ 1

1r0þ

ffiffiffijD

p ; ð48Þ

dmicro ¼ r0: ð49Þ

� Ultramicroelectrodes (r0 61

10

ffiffiffiDj

qÞ. In this last case Eq.

(41) simplifies to:

Iu-microCE

nFADf�¼ ð1=du-micro

r Þ=K

1þ ðdu-micro=du-micror Þ=K

; ð50Þ

where (see Eqs. (35), (42) and (43)):

du-micror ¼ du-micro ¼ r0 ð51Þ

and hence the thicknesses of the reaction and diffusionlayers are identical and coincident with the electrode ra-dius, r0, and therefore become independent of the kineticof the chemical reaction.

5.1.2. Catalytic mechanism

As we have explained in a recent publication [1], the dif-ferent behaviour between the CE and catalytic processes isdue to the fact that, for the CE mechanism (see Scheme I),C species is required by the chemical reaction, whose equi-librium is distorted in the reaction layer, and by the electro-chemical one, which is limited by the diffusion layer. For acatalytic mechanism (see Scheme II), C species is alsorequired for both the chemical and the electrochemicalreactions, but this last stage gives the same species, B,which is demanded by the chemical reaction in such away that only in the reaction layer do the concentrations


of B and C species take values different from those of thebulk of the solution (see Scheme III).

In fact, if we make the following formal identity:

dspher ¼ dspherr ð52Þ

in Eq. (41), which gives the current for a CE mechanism asfunction of spherical diffusion and reaction layers thick-nesses, we find immediately:

I spherCAT

nFADf�¼ 1=dspher

r

1þ Kð53Þ

which is the stationary current for a catalytic mechanism(Eq. (37)), given as a function of the spherical reactionlayer thickness, dspher

r (Eq. (42)).By proceeding in a similar way as for a CE mechanism,

the limit cases of planar, microspherical and ultramicro-spherical electrodes can be deduced.

The above, interesting, conclusions are shown in Table 2.

0

01r

r

r D=

+

Reaction layer

Electrode Bulk ofthe solution

CECE mechanismmechanism

D D

0

01

r

r Dt=

+

Diffusion layer

C B C B

* * *B C D B C Dc c c c c c r+ + = + +

Scheme III. Steady state spherical diffusion and reaction

Table 2Expressions for the current and diffusion and reaction layers thicknesses for dmechanisms

Expression

CE mechanism Spherical electrode Eq. (41) inI spher

CE

nFADf� ¼ 1þð

Microelectrode:(D/j)1/2/10 6 r0 6 (pDt)1/2/20

ImicroCE

nFADf� ¼ 1þð

Ultra-microelectrode: r0 < (D/j)1/2/10Iu-micro

CE

nFADf� ¼ 1þð

Planar electrode: r0!1Iplan

CE

nFADf� ¼ð

1þð

Catalytic mechanism Spherical electrode and microelectrodewith (D/j)1/2/10 6 r0

Eq. (41) in

dspher ¼ dspr

Ultra-microelectrode: r0 < (D/j)1/2/10Iu-micro

CAT

nFADf� ¼1=d

1

Planar electrode: r0!1Iplan

CAT

nFADf� ¼1=d1þ

6. Results and discussion

In order to compare the results for the current obtainedfrom Eqs. (15) and (24), corresponding to the kineticsteady state approximation (Section 2) and to the approx-imation introduced in this paper (Section 3), respectively,

in Fig. 1 we have plottedIspher

CE

Id ðtÞþId ð1Þ versus n for several val-

ues of K andffiffiffivp

(Id(t) and Id(1) are given by Eqs. (28) and(36), respectively). In this figure, solid and dashed lines cor-respond to the kinetic steady state approximation (Eq.(15)) and the approximation introduced in this paper(Eq. (24)), respectively. The current has been normalizedin such a way that for small values of n, the reference isthe current for an E mechanism at a planar electrode,

Id(t). Thus, the relationIplan

CE

Id ðtÞ is always less than the unitand it tends to this limit as

ffiffiffivp

increases and K decreases(see Eqs. (27) and (29)). Conversely, for high values of n,

Bulk ofthe solution

CatalyticCatalytic mechanismmechanism

B C

Electrode

0

01r

r

r D=

+

Reaction layer

Diffusion layer

* *B C B Cc +c c c+ =

C B

r

layers in the cases of CE and catalytic mechanisms.

ifferent sizes of the electrode radius in both the cases of CE and catalytic

for the current Diffusion layer thickness Reaction layer thickness

this paper:ð1=dspher

r Þ=Kdspher=dspher

r Þ=K

dspher ¼ 11

r0þ 1ffiffiffiffi

pDtp dspher

r ¼ 11

r0þffiffijD

p

ð1=dmicror Þ=K

dmicro=dmicror Þ=K

dmicro = r0 dmicror ¼ 1

1r0þffiffijD

p

ð1=du-micror Þ=K

du-micro=du-micror Þ=K

du-micro ¼ du-micror ¼ r0

1=dplanr Þ=K

dplan=dplanr Þ=K

dplan ¼ffiffiffiffiffiffiffiffipDtp

dplanr ¼


pthis paper with

her:I spher

CAT

nFADf� ¼1=dspher

r1þK

dspher ¼ dspherr ¼ 1

1r0þffiffijD

p

ru-micro

þK du-micro ¼ du-micror ¼ r0

planrK dplan ¼ drplan ¼


p

Isp

her C

E/(

I d(t)

+I d(

))

0.0

0.2

0.4

0.6

0.8

1.0

5

25

50

100

a

8

0.0

0.2

0.4

0.6

0.8

Isp

her C

E/(

I d(t)

+I d(

))

5

25

50

100

10K =

8

b

5K =

0 2 4 6 8 10 12 140.0

0.2

0.4

0.6

0.8

ξ

Isp

her C

E/(

I d(t)

+I d(

))

2550

100

χ =

50K =

5

8

c

χ =

χ =

( )1 1 K+

( )1 1 K+

( )1 1 K+

0.63

Fig. 1. Normalized currentIspher

CE

Id ðtÞþId ð1Þ versus n ð¼ffiffiffiffiffiDtp

=r0Þ for severalvalues of K and

ffiffiffivp

(Id(t) and Id(1) given by Eqs. (28) and (36),respectively). In this figure, solid and dashed lines correspond to thekinetic steady state approximation (Eq. (15)) and the approximationintroduced in this paper (Eq. (24)), respectively. (a) K = 5, (b) K = 10, (c)K = 50. Values of

ffiffiffivp

are on the curves. The value of n = 0.63 is shown,from which Eq. (23), deduced in this paper, can always be used.

Isp

her C

E/I

mic

roC

E1.0

1.1

1.2

1.3

1.4

1.5

1.6

5K =

χ =

5

50100

25

a

Isp

her C

E/I

mic

roC

E

1.0

1.1

1.2

1.3

1.4

1.5

1.6

10K =

525

10050

b

ξ0 5 10 15 20 25

Isp

her C

E/I

mic

roC

E

1.0

1.1

1.2

1.3

1.4

1.5

1.6

50K =

525

100

c

50

χ =

χ =

0.63 11.3

Fig. 2. Normalized currentIspher

CE

ImicroCE

versus n ð¼ffiffiffiffiffiDtp

=r0Þ for several values of

K andffiffiffivp

(Eqs. (23) and (31)). Solid lines correspond to the normalizedcurrent for the approximation introduced in this paper (Eq. (23)) andhorizontal dashed lines correspond to the normalized current formicroelectrodes (Eq. (31)). (a) K = 5, (b) K = 10, (c) K = 50. Values offfiffiffi

vp

are on the curves. The values of nmin = 0.63 from which Eq. (23),deduced in this paper, can always be used and nmin = 11.3 from which Eq.(31) for microelectrodes can always be used, are shown.


the reference is the current for an ultramicroelectrode,where, according to Eq. (36), the kinetic information has

been lost. In this case the relationIspher

CE

Id ð1Þ tends to 11þK (see

Eq. (36)). The excellent coincidence of both approxima-tions, better as n increases, is clearly observed. We haveshown in this figure the value of nmin = 0.63 from whichEq. (24), deduced in this paper, can be used with a relativeerror in the current of less than 5%, for any value of

ffiffiffivp

and K. This result implies (see Eq. (18)), that forD = 10�5 cm2/s and t = 1 s, for example, the electroderadius should be less than 50 lm. Therefore, the use of elec-trodes of small radius is clearly advantageous for the studyof a CE mechanism, although the electrodes of very smallradius present serious problems, which must be conve-niently taken into account (see below).


In Fig. 2 the current from Eq. (23) has been normalizedin relation to the current given by Eq. (31) and plottedversus n. Thus, the dashed horizontal lines in this figurecorrespond to results from Eq. (31). According to Eq.(34), we have highlighted the value of nmin ðnmin ¼20ffiffipp ¼ 11:3Þ, required for the condition of/ot = 0 to be ful-

filled. Note that this value of the sphericity parameter nmin

is 18 times greater than that necessary for Eq. (23) to be ful-filled (nmin = 0.63) and due to this, the electrode radius nec-essary for reaching a true steady state and for Eq. (31) to beapplied is very restricted, as is shown in Table 3.

In Table 3, we illustrate the severe constraints (Eq.(33)) in the electrode radius necessary to obtain a true sta-tionary response (Eq. (31)) useful for obtaining the homo-geneous rate constants. As can be observed, for smalltimes (0.01 s) only very rapid kinetics could be detectedat small electrodes, in the order of 100–300 nm. Increasingthe time of the experiment (0.1–1 s), the detection of lessrapid kinetics becomes possible, although for a narrowinterval of electrode radius, in the range of lm. Obvi-ously, greater values of time are not realistic in electro-chemical measurements.

Therefore, it is clear that Eq. (23) deduced in this papershould be applied instead of Eq. (31) when spherical elec-trodes of small size are used to study a CE process with fastchemical kinetic since the radius interval is sufficiently wideto vary the radius, working at times in the order of 1 s. Theprocedure to obtain kinetic parameters by using Eq. (23) isindicated below.

6.1. Determination of kinetic parameters of the chemical

reaction

Due to its simple form, Eq. (23) can be easily rearrangedand rewritten in the following form:

I spherCE

Idð1Þr0


r0 þffiffiffiffiffiffiffiffipDtp � r0 ¼

ffiffiffiffiDj

r� K

ffiffiffiffiDj

r

� I spherCE

Idð1Þ1þ 1

K


r0 þffiffiffiffiffiffiffiffipDtp

� �:

ð54Þ

Thus, a plot ofIspher

CE

Id ð1Þr0

ffiffiffiffiffipDtp

r0þffiffiffiffiffipDtp � r0 versus

IspherCE

Id ð1Þ 1þ 1K

ffiffiffiffiffipDtp

r0þffiffiffiffiffipDtp

� �,

at fixed time and varying the electrode radius provides a

Table 3Radius interval for detecting the homogeneous kinetic in a CE processfrom the stationary response given by Eq. (31), for different values of timeand j

t/s j = 10 s�1,r0 (lm) interval

j = 100 s�1,r0 (lm) interval

j = 1000 s�1,r0 (lm) interval

110

ffiffiffiDj

q6 r0 6

120


0.01 1 6 r0 6 0.28 0.32 6 r0 6 0.28 0.1 6 r0 6 0.28Not possible! Not possible!

0.1 1 6 r0 6 0.89 0.32 6 r0 6 0.89 0.1 6 r0 6 0.89Not possible!

1 1 6 r0 6 2.80 0.32 6 r0 6 2.80 0.1 6 r0 6 2.8010 1 6 r0 6 8.86 0.32 6 r0 6 8.86 0.1 6 r0 6 8.86

linear graph from whose slope �KffiffiffiDj

q� �and interceptffiffiffi

Dj

q� �the characteristic parameters of the chemical reac-

tion can be determined. Alternatively, we can use the chro-noamperometric response (I � t curve), working at a fixedradius.

From a practical point of view, it is recommended toapply Eq. (54) working at electrode radius that fulfils0.63 < n < 11.3 since in this interval a time-dependentresponse is obtained. Obviously, Eq. (54) remains applica-ble in the interval 11.3 < n < 31.5. In this case, the responsebecomes independent of time [5], and the severe constraintsin Table 3 have to be taken into account.

7. Conclusions

� We have proposed a new approximation for the study ofCE processes with fast chemical reactions at sphericalelectrodes. The use of this approximation gives a verysimple time-dependent analytical equation for the cur-rent. However, despite its simplicity, we demonstratein this paper the excellent results obtained with theequation.� From the above equation for spherical electrodes of any

radius (Eq. (23)), we have deduced those for the partic-ular cases of planar, microspherical and ultramicro-spherical electrodes.� The comparison between CE and catalytic mechanisms

reveals that, whereas a true steady state (i.e., indepen-dent of time response) is obtained in the catalytic casewith the only condition of kinetic steady state (o//ot = 0), in the case of the CE process the imposition ofthe above condition does not lead to a time-independentresponse and it allows us to introduce other additionalassumption of purely diffusive behaviour of pseudo-spe-cies f before the true steady state is reached (o//ot = 0and of/ot = 0). This essential difference means that,whereas in the catalytic mechanism the time-indepen-dent response can be obtained even with planar elec-trodes, in the case of the CE process it is onlyobtained at microelectrodes.� We demonstrate in this paper that the equation

obtained in the particular case of microelectrodes(Eq. (31)), which is identical to that previously deducedin the literature [4,5,9], is only applicable in a very nar-row interval of the electrode radius. Thus, it must beused with care and this has not been duly consideredin previous works.� The transcendence of the Eq. (23), deduced in this paper

is apparent on account of the fact that it allows thedefinition of spherical reaction and diffusion layersthicknesses, a topic not resolved adequately in the elec-trochemical literature until now.� We have demonstrated that in the steady state the cata-

lytic mechanism can be considered as a particular case ofthe CE process, when reaction and diffusion layer thick-nesses are equal.


� We have proposed a method for obtaining the kineticparameters of the chemical reaction, based on the useof the equation deduced in this paper. The procedureis very simple and can be used in a wide range of elec-trode radius, advantages that make this method a suit-able and recommendable alternative.

Acknowledgements

The authors greatly appreciate the financial supportprovided by the Direccion General de InvestigacionCientıfica y Tecnica (Project Number BQU2003-04172)and by the Fundacion SENECA (Expedient number PB/53/FS/02). Also, I.M. thanks the Ministerio de Educaciony Ciencia for the grant received.

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Documents

Chronoamperometric behaviour of a CE process with fast chemical reactions at spherical electrodes and microelectrodes. Comparison with a catalytic reaction