ABSTRACT
Cyclic voltammetry is without any doubt the most universal electrochemical technique, used either to elucidate reaction mechanisms or to carry out quantitative analysis. In fact, it should be more accurately called cyclic voltamperometry, but cyclic voltammetry has become the accepted term. The technique consists of varying the electrode potential in a linear fashion between two limits: the initial electrode potential Ei and the final electrode potential Ef so as to probe the reactivity of the electrochemical system over a large range of potentials in a single sweep. By varying the sweep rate, we can also probe the kinetics of the reactions and/or the mass transfer process. For an oxidation, we usually start from an electrode potential value where no oxidation takes place and sweep, on the forward scan, the electrode potential to more positive values
E(t) = Ei + vt (10.1)
After reaching the final value usually set at electrode potential values just before the oxidation of the solvent or that of the supporting electrolyte, the electrode potential is scanned back to the initial value
E(t) = Ef –vt (10.2)
On the reverse scan, we reduce part of the species oxidised on the forward scan. is called the sweep rate (or scan rate) and can vary from a few millivolts per second to a few million volts per second, according to the application and the size of the electrode. The differential equations for cyclic voltammetry are the differential diffusion equations for the oxidised or reduced species
(10.3)
and
(10.4)
The boundary conditions for an oxidation on the forward scan usually assume that we start with only reduced species present in solution, and the initial conditions read
The equality of the diffusion fluxes at the interface creates an additional boundary condition
(10.7)
To solve the Fick differential equations with the Nernst equation as a boundary condition, it is preferable to define the following dimensionless parameters:
(10.8)
with
0 (10.9)
and
RT (10.10)
To calculate the current-potential response, we shall first calculate the interfacial concentrations of O and R to substitute in the Nernst equation (10.8). As seen in Chapter 8, the Laplace transform for the concentration of the reduced species is obtained by solving the Laplace transform of equation (10.4)
(10.11)
In turn, the Laplace transform for the current is
(10.12)
Thus, the constant A(s) can be determined as :
(10.13)
The Laplace transform of the interfacial concentration of the reduced species is then
(10.14)
In order to calculate the inverse Laplace tranform of equation (10.14), we use the convolution theorem that provides the inverse transform of a product
By introducing f(), the interfacial mass flux defined by
(10.17)
equation (10.16) becomes
The integral of equation (10.18) is what is commonly called the convoluted current,
(10.20)
From the two expressions for the interfacial concentrations of O and R, and the Nernst equation, we obtain a relationship between the convoluted current and the potential
(10.21)
from which we can express the convoluted current as
1 (10.22)
with being a dimensionless number, previously defined by
To be able to integrate equation (10.22), it is handy to make the following change of variable
the function is dimensionless being defined as
This function can be calculated numerically step by step after segmenting the variable z in n intervals.
