Chemical Physics Letters 387 (2004) 317–321

www.elsevier.com/locate/cplett

Current function for irreversible electron transfer processes inlinear sweep voltammetry for the reactions obeying Marcus kinetics

M. Arun Prasad, M.V. Sangaranarayanan *

Department of Chemistry, Indian Institute of Technology, Madras 600036, India

Received 15 December 2003; in final form 16 January 2004

Published online: 10 March 2004

Abstract

A simple analytical expression for the current function pertaining to the irreversible electron transfer processes in linear sweep

voltammetry is formulated for the systems obeying Marcus-like mechanisms. The influence of activation overpotential on the

current function has been pointed out. The variation of the current function with solvent reorganization energy has also been il-

lustrated which is in agreement with Marcus theory. The current function expression was verified experimentally using the reductive

cleavage of carbon tetrachloride in N ,N 0-dimethylformamide at glassy carbon electrodes.

� 2004 Elsevier B.V. All rights reserved.

1. Introduction

Stationary electrode polarography (linear sweep and

cyclic voltammetry) is one of the most powerful elec-

trochemical techniques which provides information on

thermodynamics of redox processes, kinetics of hetero-

geneous electron transfer reactions, coupled chemicalreactions, etc. [1]. Randles [2] and Sevcik [3] initiated the

theoretical development of the technique, while Delahay

[4] extended the theory to irreversible charge transfer

processes. Further investigations [5–8] subsequent to the

comprehensive treatment of the technique by Nicholson

and Shain [9] have essentially been confined to the the-

ory of reversible electron transfer processes; however

diverse numerical methods have been employed insolving a variety of electrochemical problems involving

complicated kinetic schemes [10]. These simulation

methods employ sophisticated discretization strategies

[11] and are valuable for mechanistic analysis. Auto-

matic simulation methods have also been developed,

which are known under the name of adaptive grid

* Corresponding author. Fax: +044-22570545.

E-mail address: mvs@chem.iitm.ac.in (M.V. Sangaranarayanan).

0009-2614/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.cplett.2004.01.126

strategies, for solving the partial differential equations

arising in electrochemical contexts [12]. Nevertheless,

the void arising from the lack of simple mathematical

relations involving present theories of electron transfer

at the electrode surface needs to be filled in so as to

obtain complimentary insights.

In this Letter, we report a simple analytical expres-sion for computing the current function pertaining to

the irreversible electrode reactions in linear sweep vol-

tammetry (LSV) incorporating the Marcus theory of

outer sphere electron transfer. The influence of solvent

reorganization factor and activation overpotential on

the current function has been illustrated. The analytical

expression has been verified experimentally using the

reductive cleavage of CCl4 in N ,N 0-dimethylformamide(DMF) at glassy carbon electrodes which follows the

quadratic-activation driving force relationship.

2. Formulation of the current function

The irreversible reduction of the oxidant (Ox) to the

product (Red) with a rate rðtÞ and corresponding rateconstant ðkÞ may be represented as

Oxþ ne� !k Red: ð1Þ

318 M. Arun Prasad, M.V. Sangaranarayanan / Chemical Physics Letters 387 (2004) 317–321

Assuming the heterogeneous electron transfer rate as

first order with respect to Ox, the rate rðtÞ is related to

the rate constant ðkÞ asrðtÞ ¼ kCs

OxðtÞ; ð2Þ

where CsOxðtÞ represents the surface concentration of Ox

as a function of time. The potential ðEÞ dependence of

the rate constant ðkÞ can be formulated using the ab-solute reaction rate theory [13] as

k ¼ ks exp�anFRT

ðE�

� E�0Þ�; ð3Þ

where E�0 denotes the formal potential. The dimension-

less transfer coefficient a, takes values between zero andunity; the absolute reaction rate theory does not impose

any restriction regarding its variation with respect to the

electrode potential. The symbols R, T , F , t and n assume

usual electrochemical significance [14]. The Marcus

theory of outer-sphere electrochemical electron transfer

[15] predicts a dependence of the transfer coefficient

upon the potential as

a ¼ 0:5þ F4k

ðE � E�0 � /2Þ ð4Þ

and yields the following expression for ks

ks ¼ Zel expð�k=4RT Þ; ð5Þwhere k is the solvent reorganization factor while Zel

represents the electrochemical collision factor (Zel ¼RTffiffiffiffiffiffiffi2pM

p , M: molar mass of the reactant). By assuming the

diffusion coefficient of Ox ðDOxÞ to be equal to that of

the reduced species (DRedÞ, the standard electrode po-

tential ðE�Þ becomes equal to that of the formal poten-

tial ðE�0Þ of the redox system. Further it can be assumed

that the reaction site is farther from the electrode than

the Helmholtz plane and hence /2 (the potential differ-

ence between the reaction site and the solution) can beneglected. The relation between the current and rate

representing the flux at the electrode surface is given as

i ¼ rðtÞnFA: ð6ÞThe current for the irreversible electron transfer re-

actions [16] follows as

i ¼ nFACbulk

ffiffiffiffiffiffiffiffiffipbD

pv; ð7Þ

where b ¼ anF t=RT and v is the current function, whichmay be interpreted as the dimensionless part of the cur-

rent response in the voltammetric technique containing

all the mechanistic and thermodynamic information of

the electrode reaction. The first step in this study involves

the formulation of a simple analytical expression for the

current function in the case of irreversible electron

transfer processes with subsequent incorporation of the

characteristic features of the Marcus kinetics.Using the classical Euler transformation, the current

function for the irreversible charge transfer was given in

the form of an infinite series by Reinmuth [17,18], viz,

vðEÞ ¼ 1ffiffiffip

pX/j¼1

ð�1Þjþ1 p0:5j

½ðj� 1Þ!�1=2

� exp�jnaF ðE � E�Þ

RT

� j ln

ffiffiffiffiffiffiffiffiffipDb

p

ks

!: ð8Þ

A rational function approximation has been at-

tempted in our laboratory using the series solution of

Reinmuth. The procedure was implemented starting

from the lowest order of the exponential function inMatlab [8]. The lowest possible order of the function

approximation for the series solution in the entire po-

tential range is deduced as

ffiffiffip

pv ¼ a1hþ a2h

2

b1 þ b2hþ b3h2

¼ 1:7807hþ 0:3361h2

1:0000þ 2:0492hþ 1:2705h2: ð9Þ

The above equation may be regarded as a [2/2] Pade’

approximant, wherein the first term of the numerator

ða0Þ is zero and h is a simple exponential function rep-

resented as

h ¼ exp½�Fx=RT �; ð10Þx being the potential scale given by Nicholson and Shain[9].

x ¼ naðE � E�Þ þ RTF

ln

ffiffiffiffiffiffiffiffiffipbD

p

ks: ð11Þ

In this context, two equivalent procedures attempted

previously [6] using n – convergence algorithm for the

acceleration of convergence of the original series and the

Pade’ approximation scheme are worth mentioning.Eq. (9) is the simplest expression for the current

function pertaining to the irreversible electron transfer

reactions and the coefficients are accurate to �0.0001.

The potential scale can be simplified by referring the

potential axis with respect to the peak potential of the

wave instead of the Nicholson and Shain potential scale

denoted as ‘(Potential scale)N=S’. This can be achieved

by subtracting the peak potential from the Nicholsonand Shain potential scale, viz,

ðPotential scaleÞN=S � ðPotentialÞpeak

¼ naðE

� E�Þ þ RTF

ln

ffiffiffiffiffiffiffiffiffipbD

p

ks

!

� naðEp

� E�Þ þ RT

Fln

ffiffiffiffiffiffiffiffiffipbD

p

ks

!: ð12Þ

Since the peak potential in the Nicholson and Shain

potential scale is )5.34 mV, Eq. (12) is simplified as

ðPotential scaleÞN=S � ð�5:34 mVÞ ¼ naðE � EpÞ: ð13Þ

As a next step, the characteristic features of the

Marcus mechanism can be introduced into the approx-

Fig. 2. Variation of the dimensionless current function with electrode

potential and activation overpotential obtained from Eqs. (9) and (16).

The ranges of values chosen for the mesh plot are )350:5:350 for po-

tential and 0:20:1000 for activation overpotential. Other parameters

are as follows: n ¼ 1; da=dE ¼ 0:5 V�1.

M. Arun Prasad, M.V. Sangaranarayanan / Chemical Physics Letters 387 (2004) 317–321 319

imation function for the irreversible electron transfer

processes (Eq. (9)) through the potential dependence of

the transfer coefficient (Eq. (4)) occurring in the poten-

tial scale of the function. Further, the potential variable

in the expression for a can also be converted to (E � Ep)by adding the activation overpotential ðgactÞ of the

electrode reaction, viz,

a ¼ 0:5þ F4k

ðE � Ep þ gactÞ ð14Þ

and gact can be defined as

gact ¼ Ep � E�: ð15ÞEq. (13) can now be given as

ðPotential scaleÞN=S � ðPotentialÞpeak

¼ nðE � EpÞ 0:5

�þ F4k

ðE � Ep þ gactÞ�; ð16Þ

where F =4k is a constant and is obtained from experi-

mental studies as the slope ðda=dEÞ using a vs E data.

Hence, by referring the potential scale of the linear

sweep voltammogram with respect to the peak potential,

the potential scale for the current function can be rep-

resented in terms of ðE � EpÞ for the irreversible electrontransfer processes following the Marcus mechanism.

Fig. 1 shows the comparison between the current func-tions computed using Eq. (9) and that reported by

Nicholson and Shain [9]. Excellent agreement can be

noticed between the waves and in this context, the ad-

dition of 5.34 mV to the Nicholson and Shain potential

scale is especially convenient. The elegance of this

methodology is demonstrated in the experimental veri-

fication of the current function.

Using the expression for the current function in con-junction with Eq. (16), the influence of the activation

Fig. 1. Dimensionless current function for the irreversible electron

transfer processes in linear sweep voltammetry. Circles denote the

values computed using Eq. (9) while the line is drawn from the tabular

compilation of Nicholson and Shain [9] wherein a value of 5.34 mV has

been added to the potential axis (T ¼ 298 K).

overpotential or the solvent reorganization factor on the

current function can be analyzed. Fig. 2 illustrates the

variation of the current function with electrode potential

and gact. As can be noticed from Fig. 2, the wave becomes

broad with increase in the activation overpotential. Thisarises on account of the fact that the electrode process

tends to become more sluggish with increase in the acti-

vation overpotential and hence a concomitant broaden-

ing of the wave ensues. A similar trend, but less

pronounced, can be noticed in the case of the influence of

da=dE on the current function (Fig. 3). In Fig. 3, the wave

tends to be sharper with increase in da=dE or decrease in

kð*da=dE / 1=kÞ. As per the Marcus theory, the smallerthe solvent reorganization factor, the faster is the electron

transfer and hence, sharper is the current function.

Fig. 3. Variation of the current function with potential at various

ðda=dEÞ values pertaining to the irreversible electron transfer scheme

computed using Eqs. (9) and (16). Other parameters are as follows:

n ¼ 1; gact ¼ 0 mV. (a) 0.1 V�1; (b) 0.5 V�1 and (c) 1.0 V�1.

Fig. 4. Comparison between the experimental and theoretical current

functions. Circles (this work) and stars (from [20]) denote, respectively,

the experimental current functions pertaining to the reductive cleavage

of CCl4, while the solid line is drawn using the Eqs. (9) and (16) with

gact ¼ �902 mV and da=dE ¼ 1:706� 10�4 mV�1.

320 M. Arun Prasad, M.V. Sangaranarayanan / Chemical Physics Letters 387 (2004) 317–321

3. Experimental verification of the current function

The reductive cleavage of CCl4 in DMF at glassy

carbon electrodes has been selected as an illustrative

example so as to verify the current function expression.Since the double-layer effects increase with the charge of

the reactants, the neutral CCl4 is expected to be most

useful in the verification of the current function. Carbon

tetrachloride undergoes dissociative two electron re-

duction following the quadratic activation-driving force

relationship [19–21]. The theory for the adiabatic dis-

sociative electron transfer processes was developed by

Saveant [22], wherein the activation energy–drivingforce relationship has been described by a Marcus-like

quadratic equation, with the intrinsic barrier ðDG�0Þ now

including the contribution from the bond dissociation

energy of the cleaving bond, viz,

DG� ¼ DG�0 1

�þ DG0

4DG�0

�2

; ð17Þ

where DG� is the activation free energy and DG0 repre-

sents standard Gibbs free energy of the process. Con-

sequently, da=dE is given as [22],

dadE

¼ F2ðDC–Cl þ kÞ ð18Þ

and DC–Cl denotes the bond dissociation energy of the

carbon–chlorine bond.

The voltammetric studies were carried out in the BAS

100A Electrochemical workstation using a single com-

partment cell thermostatted at 298 K. The working

electrode was a glassy carbon (Bioanalytical systems,BAS) of 3 mm diameter and polished with the alumina

solution (BAS) prior to the use. Tetra n-butylammo-

nium bromide (0.1 M, Fluka) was the supporting elec-

trolyte and used as received. The silver/silver ion (1 mM)

electrode (BAS) was used as the quasi-reference elec-

trode, while a platinum foil (2 cm2) served as the counter

electrode. Carbon tetrachloride (SRL, India) was of

spectra grade and used without further purification. Thevoltammetric current devoid of the effects of charging

current was transformed by convolution with the linear

diffusion characteristic function 1=ffiffiffiffiffipt

pinto the convo-

luted current. Further analysis of the convoluted vol-

tammogram, in conjunction with the linear sweep

voltammogram, yielded da=dE and activation overpo-

tential for the reduction of CCl4 in DMF. The experi-

mental and computational details of the voltammetricand convolution analyses have been previously de-

scribed [21].

The activation overpotential for the reduction of

CCl4 and the slope of the variation of the apparent

transfer coefficient with electrode potential are evaluated

as )902 mV and 1:706� 10�4 mV�1, respectively. The

experimental voltammogram was made dimensionless

using Eq. (7) and referenced with respect to the peak

potential. Fig. 4 illustrates the comparison between the

experimental current function and that predicted by

Eq. (9) based on the values of other electrode kinetic

parameters mentioned above and a good agreement can

be noticed between the current function waves. The

current function expression (9) has also been comparedwith the experimental current function obtained by

Saveant and coworkers [20]. Despite the satisfactory

agreement between the waves, it can be noted that the

experimental current function of [20] is slightly broader

at the diffusional tail. This is due to the fact that gact is 9mV more negative in the experimental data of [20] in

comparison with the present analysis. As anticipated

from Fig. 2, a higher value of gact results in a broaderwave, which may also be due to the different supporting

electrolyte (nBu4NBF4) employed in [20]. Based on the

above discussion, it may be inferred that the formulated

approximation function is valid to the systems obeying

Marcus-type mechanisms and especially useful in the

practical contexts.

4. Summary

The formulation of the current function, incorporat-

ing the features of the Marcus theory in the case of ir-

reversible electron transfer processes, consists essentially

of two parts: (i) approximation of the series solution

pertaining to the irreversible electron transfer processes

and (ii) introduction of the potential dependence of thetransfer coefficient using Marcus theory. The depen-

dence of the current function on solvent reorganization

energy is illustrated. The influence of the activation

overpotential on the current function has also been in-

dicated which assumes importance since a majority of

M. Arun Prasad, M.V. Sangaranarayanan / Chemical Physics Letters 387 (2004) 317–321 321

the electron transfer reactions are associated with high

activation overpotentials. The current function expres-

sion has been verified experimentally using the reductive

cleavage of CCl4 in DMF.

Acknowledgements

M.A.P. thanks the CSIR, Government of India, for

the award of a Junior Research Fellowship. This work

was supported by the DST, Government of India.

References

[1] J. Heinze, Angew. Chem. Int. Ed. Engl. 23 (1984) 831.

[2] J.E.B. Randles, Trans. Faraday Soc. 44 (1948) 327.

[3] A. Sevcik, Collect. Czech. Chem. Commun. 13 (1948) 349.

[4] P. Delahay, J. Am. Chem. Soc. 75 (1953) 1190.

[5] J.C. Myland, K.B. Oldham, J. Electroanal. Chem. 153 (1983) 43.

[6] C.A. Basha, M.V. Sangaranarayanan, J. Electroanal. Chem. 261

(1989) 431.

[7] T. Suzuki, I. Mori, H. Nagamoto, M.M. Ito, H. Inoue,

J. Electroanal. Chem. 324 (1992) 397.

[8] M. Arun Prasad, M.V. Sangaranarayanan, Electrochim. Acta. 49

(2004) 445.

[9] R.S. Nicholson, I. Shain, Anal. Chem. 36 (1964) 706.

[10] See for example, M. Rudolph, in: I. Rubinstein (Ed.), Physical

Electrochemistry – Principles, Methods and Applications, Marcel

Dekker, New York, 1995 (Chapter 3).

[11] B. Speiser, Electroanal. Chem. 19 (1996) 1.

[12] L.K. Bieniasz, J. Electroanal. Chem. 529 (2002) 51.

[13] H. Eyring, S. Glasstone, K.J. Laidler, J. Chem. Phys. 7 (1939)

1053.

[14] A.J. Bard, L.R. Faulkner, Electrochemical Methods – Funda-

mentals and Applications, second edn., Wiley, New York,

2001.

[15] R.A. Marcus, J. Chem. Phys. 43 (1965) 679.

[16] P. Delahay, J. Am. Chem. Soc. 75 (1953) 1190.

[17] W.H. Reinmuth, Anal. Chem. 33 (1961) 1793.

[18] W.H. Reinmuth, Anal. Chem. 34 (1962) 144.

[19] J.M. Saveant, Adv. Phys. Org. Chem. 26 (1990) 1.

[20] L. Pause, M. Robert, J.M. Saveant, J. Am. Chem. Soc. 122 (2000)

9829.

[21] M. Arun Prasad, M.V. Sangaranarayanan, Chem. Phys. Lett. 382

(2003) 325.

[22] J.M. Saveant, J. Am. Chem. Soc. 109 (1987) 6788.