A Poisson-Boltzmann Description for the Double-Layer Capacitance of an Electolytic Cell

A Poisson-Boltzmann approach is used to determine the double-layer integral and differential capacitances in a finite-length situation for an electrolytic cell. By means of simple analytical calculations, it is shown how these quantities exhibit the bell-like and the camel-like shapes as a function of the applied voltage, when the thickness of the sample or the surface charge are varied. The problem is formulated treating the bulk liquid as a member of a grand canonical ensemble in contact with blocking electrodes. As a consequence, an applied voltage dependent Debye's screening length arises as a fundamental length governing the electrical behavior of the system.