www.elsevier.nl/locate/jelechem

Journal of Electroanalytical Chemistry 486 (2000) 9–15

Square wave voltammetry for a pseudo-first-order catalytic processat spherical electrodes

Angela Molina *, Carmen Serna, Francisco Martınez-OrtizDepartamento de Quımica Fısica, Facultad de Quımica, Uni6ersidad de Murcia, Espinardo, 30100 Murcia, Spain

Received 30 November 1999; received in revised form 20 March 2000; accepted 21 March 2000

Abstract

A theoretical equation corresponding to square wave voltammetry for a pseudo-first-order catalytic mechanism at sphericalelectrodes of any radius is presented. The effects of the electrode sphericity are analysed. Methods for calculating equilibrium andrate constants of the chemical reaction are proposed and applied to the system Fe(III)–triethylamine � Fe(II)–triethylamine in thepresence of hydroxylamine. The advantages of this equation for analytical purposes are laid out. © 2000 Elsevier Science S.A. Allrights reserved.

Keywords: Catalytic mechanism; Spherical electrodes; Microelectrodes; Multipotential step; Square wave voltammetry

1. Introduction

Theoretical expressions for square wave voltammetryhave been derived and tested experimentally by Zengand Osteryoung [1] for a pseudo-first-order catalyticprocess in planar electrodes, supposing irreversiblechemical behaviour, K=0.

In a recent paper we have deduced easy analyticalequations for the above mentioned mechanism whichare applicable to any multipotential step electrochemi-cal technique with planar and spherical electrodes andwhich consider that the chemical equilibrium constantcan take any value [2]. In this work we apply theseequations to square wave voltammetry at sphericalelectrodes of any size, including microelectrodes, andwe discuss the importance of considering the sphericityeffects when square wave voltammetry is used for ana-lytical purposes. The dependence of the forward, rever-sal and net square wave current on the equilibrium andrate constants of the chemical reaction is evaluated.

From equations given in this paper, the equilibriumand rate constants of the chemical reaction can beevaluated. When the kinetic steady state is reached, an

easily manageable analytical expression is presented,which shows that the effects of the electrode sphericityassume greater importance as the square wave potentialincreases in the determination of the rate constants.Analytical advantages of the use of spherical electrodesin square wave voltammetry are discussed and it is alsopointed out that the sphericity effects increase as thesquare wave frequency diminishes. Also it is preciselythe low square wave frequencies that are the mostdesirable in minimising the double layer effects.

The complex of Fe(III) with triethylamine (TEA) hasbeen used for the experimental verification of the theo-retical equations obtained in this work for the responseof a catalytic mechanism in square wave voltammetry.The reduced form of this complex (Fe(II)�TEA) isre-oxidized in the presence of hydroxylamine through apseudo-first-order reaction. We have also applied theabove methods for determining the rate constant to thissystem and we have compared the experimental and thetheoretical responses.

2. Experimental

The computer-driven potentiostat was designed andconstructed by Quiceltron (Spain). Square wave genera-tion and data acquisition were performed using i-

* Corresponding author. Tel.: +34-968-367524; fax: +34-968-364148.

E-mail address: amolina@fcu.um.es (A. Molina)

0022-0728/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved.PII: S 0 0 2 2 -0728 (00 )00115 -7

A. Molina et al. / Journal of Electroanalytical Chemistry 486 (2000) 9–1510

SBXDD4 and DAS16-330i boards (ComputerBoars,USA), respectively. Data acquisition and square wavegeneration were independent tasks carried out usingtwo different time scales and different but connectedroutines. For square wave generation we have used aninterrupt service routine programmed for this purposetriggered by the 8254 counter in the computer. Dataacquisition was under the control of the on-boardpacer, which usually ran at the maximum sampling rateand data were stored in the on-board FIFO buffer. Theinterrupt service routine was responsible for changes inthe potential values and for reading the acquisitioncounter and status. Hence, the use of appropriate pro-gramming techniques allows a large data oversampling(very useful for digital filtering) and almost completeversatility in the selection of amplitude and frequencyof the square wave along with a very good acquisitionsynchronisation.

A static mercury drop electrode (SMDE) was con-structed using a DME, EA 1019-1 (Metrohm) to whicha home-made valve was sealed. Different opening timesin this valve allowed the use of mercury drops of radiivarying from 0.01 to 0.05 cm. The electrode radius fora given opening time was determined by weighing alarge number of drops. The standard deviation in eachseries of radius determination was always lower than0.5%. The reference electrode was Ag � AgCl � 0.1 MKCl.

The working solutions of Fe(III) complexed withtriethylamine (TEA) and different amounts of NH2OHwere prepared following the procedure given by Zengand Osteryoung [1].

The diffusion coefficient of complexed Fe(III) wasdetermined by chronoamperometric measurements in asolution which did not contain NH2OH. A value of(5.0090.05)×10−6 cm2 s−1 was obtained. The E°%value of this experimental system was determined fromthe peak potential in the square wave voltammograms,giving E°%= −102591 mV. All experiments begin at apotential (usually −0.75 V) at which no faradaic pro-cesses take place. The same potential was used toensure mechanical stability of the drops over 2 s beforethe voltammetric scan.

All chemicals were from Merck reagent grade andwere used without further purification.

In all the experiments the temperature was keptconstant at (2090.2)°C.

3. Theory

In a previous paper we deduced the general voltam-metric theory corresponding to a pseudo-first-order cat-alytic mechanism in spherical electrodes of any size [2].The scheme of this mechanism is given by

A+ne− ? B

B+Z =k%1

k %2A+P (I)

where A and B are the electroactive species and Z andP are electroinactive species in the whole range ofapplied potentials, which should be in high excess withrespect to A and B species, and in such a way thatk1=k %1cZ* and k2=k %2cP* are both pseudo-first-order rateconstants (with c*i being the bulk concentration of thei species).

For the above mechanism, the equation for the cur-rent Im measured at the end of the mth potentialcorresponding to any sequence of potential stepsE1, E2, …, Em, is given by

Im=nFA0Dj*� %

m

j=1

f(xjm)'k1+k2

DZj

+1−KJm

r0(1+K)(1+Jm)n

(1)

where DA=DB=D is the diffusion coefficient of spe-cies A and B, K is the equilibrium constant of thechemical reaction (K=k2/k1=cB* /cA* ), r0 and A0 are theelectrode radius and area, respectively, and

j*=cA*+cB* (2)

Z1=1−KJ1

(1+K)(1+J1)

Zj=1

1+Jj

−1

1+Jj−1

; j\1

Jj=exp[nF(Ej−E0%)/RT ]

ÂÃÃÌÃÃÅ

(3)

xjm= (k1+k2)tjm (4)

tjm= %m

k= j

tk (5)

f(xjm)=e−xjm

pxjm

+erf(xjm) (6)

It should be noted that the electrode sphericity af-fects only the potential pulse considered (mth), such asoccurs in the case of a reversible E mechanism [3] underthe same conditions (i.e. DA=DB=D).

It is of interest to point out that, for any value of theequilibrium constant, when k1+k2�0 (xjm�0) a cata-lytic process (Eq. (1)) behaves as an E process in whichboth electroactive species are initially present in thesolution with c*A and c*B values in a ratio (c*B/c*A) equalto the equilibrium constant of the catalytic process, K(see Eq. (20) in Ref. [3]).

3.1. Square wa6e 6oltammetry at spherical electrodes

Eq. (1) can be easily applied to square wave voltam-metry by taking into account that, in these techniques,a sequence of potential pulses of the same duration, tp,given by [4,5]

A. Molina et al. / Journal of Electroanalytical Chemistry 486 (2000) 9–15 11

Em=Ein−DEs

2m− [1− (−1)m]

�Esw+

DEs

4�

m=1, 2, 3, … (7)

is applied, where DEs is the staircase potential andEsw the square wave amplitude, in agreement with thescheme given by Osteryoung [4,5] (Fig. 1). The pulsewidth, tp, coincides with the half period of the stair-case (t/2).

The current is measured at the end of each poten-tial pulse, i.e. at nominal times tp and 2tp in eachcycle and the net current for the mth cycle of thesquare wave is given by

DIm=I2m−1−I2m (8)

In general, the currents on the forward half-cycles,m=1, 3, 5, …, are referred to as forward currents, If,and those on reverse half-cycles, m=2, 4, 6, …, as re-versal currents, Ir, such that Eq. (8) can also be writ-ten as

DI=If−Ir (9)

for each cycle of the square wave.The index potential for the voltammogram, in line

with Refs. [4,6], is chosen at the midpoint of eachsquare wave cycle in order to ensure that, for a cata-lytic process with a reversible charge transfer reaction,the experimental peak potential occurs at Ep E°% (aswe are supposing DA=DB=D). Thus, the index po-tential, E, is given by

E=E2m−1+E2m

2=Ein−mDEs−Esw (10)

The dimensionless square wave currents, cf, cr andDc, are defined as

cf=If

nFA0j*'tp

D

cr=Ir

nFA0j*'tp

D

Dc=DI

nFA0j*'tp

D

ÂÃÃÃÌÃÃÃÅ

(11)

such that, from Eqs. (1) and (8) or (9) and takinginto account that

tjm= (m− j+1)tp (12)

we write Dc for a spherical electrode as

Dc=Dcpl+Dcsph (13)

with Dcpl and Dcsph being the planar and sphericalcontribution to the total net current Dc, which aregiven by

Dcpl=xt p

! %2m−1

j=1

( f [(2m− j )xt p]

− f [(2m− j+1)xt p])Zj− f(xt p

)Z2m

"Dcsph= −

Dtp

r0

Z2m

(14)

where

xt p= (k1+k2)tp (15)

From Eqs. (13) and (14) we deduce that thesphericity (Dtp/r0) is operative only at the two po-tential pulses which intervene in the mth square wavecycle (i.e. E2m−1 and E2m, through Z2m). So, for pla-nar electrodes (r0��) Dcsph=0 and Eqs. (13) and(14) coincide when K=0 with that previously de-duced by Zeng and Osteryoung [1].

From Eqs. (1) and (8) or (9) we can distinguishtwo different steady states: a first kinetic steady statewhich is reached when f(xt p

)�1 (xt p]1.9), for which

we obtain

DI ss=nFA0Dj*�'k1+k2

D+

1r0

n� 11+J2m−1

−1

1+J2m

n(16)

This equation is formally analogous to that previ-ously obtained by us in differential pulse voltammetry(Eq. (15) in Ref. [8]) since only two pulses are in-volved.

The second steady state (non kinetic) is reachedwhen 1/r0]50f(xt p

)(k1+k2)/D (microelectrodes orultramicroelectrodes) [2]. In this case, the catalyticprocess loses its identity and behaves like a reversibleE process with the two electroactive species initiallypresent in the solution and it is fulfilled that

DI ss=nFA0Dj*

r0

� 11+J2m−1

−1

1+J2m

n(17)

ÌÃ

Ã

Ã

Ã

Ã

Ã

Â

Å

Fig. 1. Potential–time wave form corresponding to the SWV tech-nique.

A. Molina et al. / Journal of Electroanalytical Chemistry 486 (2000) 9–1512

Fig. 2. Influence of the electrode sphericity on the theoretical squarewave voltammograms calculated from Eqs. (13) and (14) with Esw=50 mV, DEs=5 mV, K=0, Dtp/r0: (solid line) 0.354, (broken line)0 (plane electrode). The values of xt p

are on the curves. (a) Forwardand reversal dimensionless square wave currents, (b) dimensionlessnet square wave current.

4. Results and discussion

The sphericity effects on the dimensionless forward,cf, and reversal, cr, currents and on the net current,Dc, when conventional spherical electrodes are used(i.e. electrodes of radius r0\10−3 cm at time pulsetpB0.1 s for which the sphericity Dtp/r0B1 by sup-posing D$10−5 cm2 s−1) are shown in Fig. 2 forK=0 for Dtp/r0=0 (planar electrodes) and Dtp/r0=0.354 at two different values of xt p

. These effectsfor cf and cr, are greatest in the zone of the voltam-mograms corresponding to the cathodic limit current(Fig. 2(a)). The influence of these effects on thevoltammograms corresponding to the net current Dc

(Fig. 2(b)) is highest in the peak current region, whichis that of greatest analytical use. The relative influenceof the sphericity decreases when k1 (and therefore xt p

)increases (31.4% for xt p

=0.5 and 13.6% for xt p=5 in

Fig. 2(b)). Evidently, this influence is greater whenspherical electrodes of smaller radii are used.

Fig. 3 shows that the effects of sphericity increase inabsolute form when the square wave amplitude, Esw,increases.

In our experimental study we have used workingsolutions of the complex Fe(III)�TEA. The reducedform of this complex (Fe(II)�TEA) is oxidized chemi-cally by hydroxylamine in a pseudo-first-order reaction(if the appropriate experimental conditions areachieved) as has been indicated by Koryta [7]. Thisexperimental system has also been studied by Zengand Osteryoung [1].

Fig. 4 shows the influence of the square wave fre-quency (1/2tp) on the sphericity effects when r0=4.87×10−2 and 8.25×10−3 cm, for this experimentalsystem for which k1:2.85 s−1 and K=0 (see below).In this case it may be of interest to increase the tp

value in order to increase the kinetic effects, which ishighly desirable from an analytical point of view since,for values of k1 of this order, a catalytic process be-haves like an E process for small values of time [8]. Inthis situation, this figure clearly shows that sphericityeffects increase with tp and these must be taken intoaccount.

Fig. 5 shows the square wave voltammograms ob-tained by assuming that the chemical reaction is re-versible (this case has not been considered in Refs.[1,4–6]). The curves on this figure show that for agiven value of k1, reversibility and sphericity amplifythe Dc response notably (Fig. 5(b)). Another interest-ing consequence of the chemical reversibility on theforward and reversal voltammograms is that whenK"0, the cf and cr currents (Fig. 5(a)) reach a valueother than zero at the beginning of the experiment(E�E0%). This is due to the initial presence of Bspecies in the bulk solution.

Fig. 3. Influence of the electrode sphericity on the theoretical squarewave voltammograms for the dimensionless net current calculatedfrom Eqs. (13) and (14) with DEs=2.5 mV, K=0, xt p

=0.05, Dtp/r0: (solid line) 0.141, (broken line) 0 (plane electrode). The values ofEsw/V are on the curves.

A. Molina et al. / Journal of Electroanalytical Chemistry 486 (2000) 9–15 13

4.1. Methods for the experimental determination of theequilibrium and the rate constants

In accordance with the above discussion, a catalyticirreversible process (K=0) can be easily characterisedthrough a simple visual inspection of the anodicplateaux of the forward and reverse curves, cf and cr,which always take the constant value of zero. There-fore, according to Fig. 4, the value of the equilibriumconstant for the system Fe(III)�TEA � Fe(II)�TEA isK=0.

The equilibrium constant, K, can be easily deter-mined from measurements of the cathodic and anodiclimit plateaux of the direct and/or reversal currents If

and Ir when the kinetic steady state has been reached,as

(If,r)cathodic plateauss

nFA0Dj*=�'k1+k2

D+

1r0

� 11+K

(18)

Fig. 5. Influence of the electrode sphericity on the theoretical squarewave voltammograms calculated from Eqs. (13) and (14) with Esw=50 mV, DEs=5 mV, k1=1 s−1, tp=0.1 s, Dtp/r0: (solid line) 0.2,(broken line) 0 (plane electrode). The values of K are on the curves.(a) Forward and reversal dimensionless square wave currents, (b)dimensionless net square wave current.

Fig. 4. Influence of the electrode sphericity on the experimentalsquare wave voltammograms for 0.37 mM Fe(III)+0.1 M TEA+0.25 M NaOH+0.04 M NH2OH. Esw=50 mV, DEs=2.5 mV. Thevalues of the electrode radii, r0/cm: () 8.25×10−3, (�) 4.87×10−2. The values of the square wave frequencies are on the curves.(a) Forward and reversal dimensionless square wave currents, (b)dimensionless net square wave current.

(If,r)anodic plateauss

nFA0Dj*= −

�'k1+k2

D+

1r0

� K1+K

(19)

The determination of the equilibrium constant isimmediate by dividing Eqs. (18) and (19).

For the determination of the rate constants we candistinguish two situations:1. The kinetic steady state is not reached.

In this case, we can determine the rate constants byfitting the experimental curves to the theoretical onesobtained from Eq. (13). This procedure has beenapplied to the system Fe(III)�TEA � Fe(II)�TEA inthe presence of 0.04 M hydroxylamine. Under theseconditions, we have obtained a value of the pseudo-first-order rate constant of k1=2.8590.05 s−1.

Fig. 6 shows the experimental If, Ir and DI curves(points) obtained for the above system for threedifferent electrode radii and the theoretical curves(solid lines) calculated from Eqs. (11)–(14) with k1=2.85 s−1. The excellent agreement between the exper-imental and theoretical curves underlines the goodapplicability of the theory developed in this paper.

A. Molina et al. / Journal of Electroanalytical Chemistry 486 (2000) 9–1514

2. The kinetic steady state is reached.Under these conditions, the rate constants can be

determined directly from measurement of the peakcurrent, since from Eq. (16) it can easily be deducedthat

DIpeakss =nFA0Dj*

�'k1+k2

D+

1r0

�tanh

�nFEsw

2RT�

(20)

This equation is identical to Eq. (34) in Ref. [8],corresponding to differential pulse voltammetry, ifwe change Esw for 9 �DE �.

Eq. (20) provides a simple way to determine the rateconstants for spherical electrodes or microelectrodes ofany radius, as

k1+k2=D� DIpeak

ss

nFA0Dj*tanh(nFEsw/2RT)−

1r0

�2

(21)

Note that the determination of the rate constantsunder these conditions is independent of the staircase

Fig. 7. Influence of the electrode sphericity on the experimentalkinetic steady state square wave voltammograms for 0.37 mMFe(III)+0.1 M TEA+0.25 M NaOH+0.1 M NH2OH. Frequency2.5 Hz. The electrode radii, r0/cm are: (1) 8.25×10−3, (2) 1.61×10−2, (3) 4.87×10−2. Other conditions as in Fig. 4.

Fig. 6. Comparison of theoretical and experimental: (a) forward andreversal square wave currents, (b) dimensionless net square wavecurrent, for different values of the electrode radius. Frequency 20 Hz.In the theoretical calculations we have used Eqs. (13) and (14) withn=1, D=5×10−6 cm2 s−1, K=0, k1=2.85 s−1. The electroderadii, r0/cm are: (1) 3.37×10−2, (2) 2.26×10−2, (3) 1.61×10−2.Other conditions as in Fig. 4.

potential and clearly shows that the effects of theelectrode sphericity on the value of the rate constantsdetermined are greater when Esw increases.

The experimental system Fe(III)�TEA � Fe(II)�TEAreaches a stationary state if the hydroxylamine concen-tration is increased and the square wave frequency isdecreased. Hence, for working solutions containing 0.1M hydroxylamine, the net square wave current becomesindependent of the square wave frequency for frequen-cies lower than 5 Hz, showing that a kinetic steady stateis reached. Fig. 7 shows an example of these steadystate square wave voltammograms obtained for threedifferent values of the electrode radius. By applying Eq.(21) we have obtained k1=7.5090.05 s−1. From thevalues of k1 obtained in this paper, values of k %1 in therange 143–150 s−1 M−1 are obtained, which are invery reasonable agreement with the values reported at25°C by Zeng and Osteryoung [1] (151 s−1 M−1) andKoryta [7] (190 s−1 M−1).

5. Conclusions

The theoretical equations derived in this paper forspherical electrodes of any radius present, among oth-ers, the following advantages when analysing a catalyticprocess: Eqs. (14) and (16) and Figs. 2–7 show that theeffects of the electrode sphericity are, in general, impor-tant in SWV. It is, therefore, highly desirable to haveequations that are applicable to an SMDE of anyradius instead of those corresponding to planar geome-

A. Molina et al. / Journal of Electroanalytical Chemistry 486 (2000) 9–15 15

try, which have been used in Ref. [1] to analyse thebehaviour of a catalytic mechanism in an SMDE.

The sphericity effects are highest in the peak currentregion, which is that of greater analytical interest.These effects increase as the square wave frequency(1/t) decreases.

For analysis, fast catalytic reactions are of greatinterest since in this case the kinetic steady state isreached and the easy analytical equations proposed inthis paper (Eqs. (20) and (21)) can be applied.

In the case of catalytic reactions that are not so fast,for the enhancement of kinetic effects and to avoid thedouble layer effects it is necessary to work at the lowestsquare wave frequencies possible. Evidently, the consid-eration of the sphericity effects (Dtp/r0) at these fre-quencies (B10 Hz) is imperative.

Acknowledgements

The authors greatly appreciate the financial sup-port provided by the Direccion General de Investi-

gacion Cientıfica y Tecnica (Project No. PB96-1095)and the Fundacion Seneca (Expedient No. 00696/CV/99).

References

[1] J. Zeng, R.A. Osteryoung, Anal. Chem. 58 (1986) 2766.[2] A. Molina, C. Serna, J. Gonzalez, J. Electroanal. Chem. 466

(1999) 15.[3] C. Serna, A. Molina, J. Electroanal. Chem. 454 (1998)

8.[4] J. Osteryoung, J.J. O’Dea, in: A.J. Bard (Ed.), Electroanalytical

Chemistry, vol. 14, Marcel Dekker, New York, 1986, pp. 209–308.

[5] J. Osteryoung, in: M.I. Montenegro, et al. (Eds.), Microelec-trodes: Theory and Applications, Kluwer, Dordrecht, 1991, pp.139–152.

[6] J.J. O’Dea, J. Osteryoung, R.A. Osteryoung, J. Phys. Chem. 87(1983) 3911.

[7] J. Heyrovsky, J. Kuta, Principles of Polarography, AcademicPress, New York, 1966, p. 386.

[8] C. Serna, A. Molina, F. Martınez-Ortiz, J. Gonzalez, J. Elec-troanal. Chem. 468 (1999) 158.

.