Journal of Electroanalytical Chemistry 464 (1999) 1 – 13
Convolutive modelling of electrochemical processes based on the
relationship between the current and the surface concentration
Peter J. Mahon, Keith B. Oldham *
Chemistry Department, Trent Uniersity, Peterborough, Ont., K9J 7B8, Canada
Received 22 January 1998; received in revised form 5 November 1998
Abstract
Three of the fundamental variables in voltammetry are the faradaic current I (t ) and the surface concentrations c
s
R
(t ) and c
s
P
(t )
of the reactant and product. This article explores the relationships between these variables for cases in which the surface
concentrations are uniform. Convolutions are derived which permit interconversion between the current and the surface
concentrations. Examples are presented that apply to various electrode geometries, with and without the participation of
homogeneous kinetics. An intriguingly simple connection is established between the convolution required for the I (t ) c
s
(t )
conversion and that applicable to the converse c
s
(t ) I (t ) transformation. These operations may be exploited in modelling
voltammetric experiments and several exemplars are developed. © 1999 Elsevier Science S.A. All rights reserved.
Keywords: Voltammetric modelling; Convolution integral; Semiintegration; Cyclic voltammetry; Chronoamperometry; Chrono-
coulometry
1. Introduction
For more than a quarter century it has been recog-
nized that, for the electron-transfer reaction
R
soln
-n e
-
P
soln
(1)
there is a general relationship linking the concentra-
tions, c
s
R
(t ) and c
s
P
(t ), of the reactant R and product P
at the electrode surface to the faradaic current I (t ).
This classical relationship is ‘general’ in the sense that it
applies to any voltammetric technique and is indepen-
dent of the reversibility or mechanism of reaction (1).
However the validity of the relationship requires that
no homogeneous reactions are experienced by R or P
and that transport to and from the electrode is by
planar semiinfinite diffusion.
The relationships in question may be written as
D
R
c
R
(t ) =D
P
c
P
(t ) =
M(t )
nFA
(2)
where A is the electrode area, F is Faraday’s constant,
and the c terms represent the absolute values of the
concentration excursions at the electrode surface,
namely
c
i
(t ) =c
i
s
(t ) -c
i
b
for i =R or P (3)
with c
b
R
and c
b
P
being the bulk concentrations, the latter
usually equalling zero. D
i
is the diffusivity (diffusion
coefficient) of species i. Note that, to match IUPAC’s
sign convention for current, n is positive for an oxida-
tion, negative for a reduction.
The quantity M(t ) was described as the ‘faradaic
semiintegral’ by Oldham and coworkers [1 – 5]
M(t ) =
d
-1/2
I (t )
dt
-1/2
(4a)
Save ´ant et al. ([6,7], using the symbolism I (t ) to replace
our M(t ) and i (t ) to replace our I (t )), employed the
term ‘convolved current’
* Corresponding author. Tel.: +1-705-748-1336; fax: +1-705-748-
1625; e-mail: keith.oldham@sympatico.ca.
0022-0728/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved.
PII:S0022-0728(98)00450-1