Examination of the equations for calculation of chronopotentiometric transition time in membrane systems

Transition time, τ, is a very crucial characteristic in chronopotentiometry; it is important to have a simple equation to calculate this value. As early as in 1901, Sand has deduced his famous equation for calculating τ in electrode/solution systems with infinitely large diffusion layer. The exact analytical solution for the case of finite diffusion-layer thickness was obtained by Sheldeshov et al. in 1986. However, τ enters this solution as an implicit function of the current density, i. Recently, van Soestbergen and coauthors proposed an approximate formula where the transition time is an explicit function of i. In this paper, we examine the above equations along with our numerical solution by comparing the calculated values of τ with our experimental data for homogeneous (Fuji CEM Type I, Type II, Type X, Neosepta CMX) and heterogeneous (MK-40) membranes. A large gamma of current densities (from i = 1.0ilim to 2.5ilim, where ilim is the limiting current density) is applied. The most simple in use is the formula of van Soestbergen et al., which, however, gives at i > 1.0ilim the values of τ slightly (about 0.7%) lower than the exact analytical solution of Sheldeshov et al. We show that a better approximation to the exact solution is obtained when the first-order approximation of the analytical solution (obtained by van Soestbergen et al.) is applied in the range of i from 1.0 to 1.9ilim, while for i > 1.9ilim, the Sand equation is used. We find nevertheless that the experimental values of τ are higher than the theoretical ones for all the studied membranes. The causes of this deviation, which are mainly electroconvection and surface electric heterogeneity, are discussed.