Analytical Results on the Behavior of a Liquid Junction across a Porous Diaphragm or a Charged Porous Membrane between Two Solutions According to the Nernst-Planck Equation

We model the behavior of an ideal liquid junction, across a porous and possibly charged medium between two ion-containing solutions, by means of the Nernst–Planck equation for the stationary state, in conditions of local electroneutrality. An analytical solution of the equation was found long ago by Planck for the uncharged junction with only ions of valences +1 and −1. Other analytical results, which have later been obtained also for more general situations, seem impractical for performing calculations. In this paper, we obtain analytical solutions for systems with up to three valence classes, which can be applied to perform numerical calculations in a straightforward way. Our method provides a much larger amount of information on the behavior of the system than the well-known Henderson’s approximation. At the same time, it is more simple and reliable, and much less demanding in terms of computational effort, than the nowadays commonly employed numerical methods, typically based on discrete integration and trial-and-error numerical inversions. We present some examples of practical applications of our results. We study in particular the uphill transport (i.e., the transport from the lower-concentration to the higher-concentration region) of a divalent cation in a liquid junction containing also other univalent anions and cations.