A framework to investigate the immittance responses for finite length-situations: Fractional diffusion equation, reaction term, and boundary conditions

The Poisson–Nernst–Planck (PNP) diffusional model for the immittance or impedance spectroscopy response of an electrolytic cell in a finite-length situation is extended to a general framework. In this new formalism, the bulk behavior of the mobile charges is governed by a fractional diffusion equation in the presence of a reaction term. The solutions have to satisfy a general boundary condition embodying, in a single expression, most of the surface effects commonly encountered in experimental situations. Among these effects, we specifically consider the charge transfer process from an electrolytic cell to the external circuit and the adsorption–desorption phenomenon at the interfaces. The equations are exactly solved in the small AC signal approximation and are used to obtain an exact expression for the electrical impedance as a function of the frequency. The predictions of the model are compared to and found to be in good agreement with the experimental data obtained for an electrolytic solution of CdCl 2 H 2 O .