Step-function method

When the electrode reaction (2.30) is electrochemically reversible, (2.37) and (2.38) are combined with the Nemst equation (1.8) yielding an integral eqnation that relates the current with time and the electrode potential. The nnmerical solntion derived by the step function method [52] is given by the following recursive formulae ... 

The physical meaning of condition (2.233) is that the diffusion of the electroactive species is blocked at the distance x = Z, i.e., where L is the thickness of the film. This bormdary condition complicates significantly the mathematical procedure compared to the semi-infinite diffusion case. To resolve the mathematical complexity, recently a novel mathematical approach has been developed which is based on the modification of the step function method [162], as elaborated in more detail in the Appendix. The numerical solution for a reversible electrode reaction is given by [155] ... 

According to the step-function method of Nicholson and Olmstead, the integral equation (A.24) can be transformed into the following approximate expression ... 

The step-function method is based on incrementalization of the total time of the voltammetric experiment t by dividing it into finite equal time increments of width J, and assuming that the unknown functions, (cr)x=o andl(t) can be regarded as constants within each time interval. In (A.27), m is the serial number of time increments, ranging from 1 to M, where M is the total number of time increments, i.e., Md = t. Furthermore, and / are the discrete values of the unknown... 

The applicability of the foregoing procednre has been tested by modeling simple reaction under semi-infinite diffusion conditions (reaction 1.1) and EC mechanism coupled to adsorption of the redox couple (reaction (2.177)) [2]. The solutions derived by the original and modified step-function method have been compared in order to evaluate the error involved by the proposed modification. As expected, the precision of the modified step-function method depends solely on the value of p, i.e., the number of time subintervals. For instance, for the complex EC mechanism, the error was less than 2% for p>20. This slight modification of the mathematical procedure has opened the gate toward modeling of very complex electrode mechanisms such as those coupled to adsorption equilibria and regenerative catalytic reactions [2] and various mechanisms in thin-film voltammetry [5-7]. 

For electron transfer processes with finite kinetics, the time dependence of the surface concentrations does not allow the application of the superposition principle, so it has not been possible to deduce explicit analytical solutions for multipulse techniques. In this case, numerical methods for the simulation of the response need to be used. In the case of SWV, a semi-analytical method based on the use of recursive formulae derived with the aid of the step-function method [26] for solving integral equations has been extensively used [6, 17, 27]. 

The relationship between the transient and stationary approaches to the relaxation times has been considered by Eigen and de Maeyer. For any chemical equilibrium a system of nonhomogeneous differential equations which represent the rates of concentration change may be set up. The complete solution of the system is the sum of two solutions. One of these depends on the initial conditions of the dependent variables and upon the forcing function (the transient solution), while the other depends on the differential equation system and on the forcing function (the forced solution). The latter does not depend on the initial conditions of concentration, etc. The step-function methods for studying chemical relaxation experimentally determine the transient behaviour, while the stationary methods determine the steady-state behaviour. 

Mathematical Modeling of Electrode Reaction in a Thin-Layer Cell with the Modified Step-Function Method... 

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