Salt Flux and Electromotive Force in Concentration Cells with Asymmetric Ion Exchange Membranes and Ideal 2:1 Electrolytes

Abstract

The previously proposed Nernst-Planck-Donnan description (J. Phys. Chem. 1996, 100, 7623) of the salt flux and the emf in concentration cells with asymmetric ion exchange membranes is generalized to encompass ideal 2:1 electrolytes (doubly charged cation and singly charged anion). Any point in the membrane may be considered to be in Donnan equilibrium with a given external salt concentration. The profile of this salt concentration through the membrane determines the ion concentrations, the local salt flux, the profile of electric field strength, and the immediate value of the emf. The Donnan potential and ion distribution are found as the unique, positive and real root of a third-degree polynomial. In this paper we focus on the stationary state rather than the initial state. For this purpose, the stationary state nonlinear differential equation for the salt concentration profile is solved numerically by the "shooting method". We consider salt concentration profiles, ion concentration profiles, and field strength profiles in three different cases: (1) a very weak cation exchange membrane (VWC), (2) a weak anion exchange membrane (WA), and (3) a strong anion exchange membrane (SA). The membranes are asymmetric with spatial dependence of the Nernst distribution coefficient for the salt, of the fixed charge density, and of the ion diffusion coefficients. We study both directions of stationary flow through the membranes. The emf is a functional of the salt concentration profile and is found by numerical integration. The overall behavior of the VWC is almost Fickian with respect to diffusion of salt in both direction, and there is practically no diffusion asymmetry. However, there may be considerable differences in the stationary state emf values for the two directions of diffusion. The WA is close to Fickian for the diffusion in one direction, but strongly non-Fickian with reversed diffusion flux. There is a large diffusion and emf asymmetry for stationary state diffusion in the two directions for large differences of concentration. The order of magnitude found for the calculated stationary state emf asymmetry corresponds to observed values for various membranes. The SA is strongly non-Fickian, but there is practically no diffusion asymmetry. The SA is almost an ideal anion exchange membrane because of the Donnan exclusion from the membrane of the doubly charged cation. Thus, the emf measured with electrodes reversible to the anion should be zero, and so it is found within numerical uncertainty.

Modeling of Amino Acid Electrodiffusion through Fixed Charge Membranes

“…However, we must emphasize that solution-diffusion models usually neglect the influence of the electrodiffusion electric field through the membrane, which is not correct for ionic species. Also the lack of flux saturation could be a consequence of the breakdown of the GCF assumption (25).…”

“…[24] and [25] below), and µ 0 i is the reference chemical potential. Equations [20] and [21] arise from the constancy of the electrochemical potentialμ i = µ 0 i + RT ln c i + z i Fφ through the respective membrane/solution interfaces, where φ is the local electric potential.…”

Estimation of Diffusion Coefficients in Biomembranes

“…This model assumes that the chemical potential gradient across the membrane is only due to a concentration gradient. The observed overall membrane diffusion in weak-acid cation-exchange asymmetric membranes is almost Fickian (17). Therefore, the molar flux of a cation across the membrane is usually described and analyzed in terms of Fick's law,…”

“…Different theories have been proposed to understand ion transport across membranes with heterogeneous structures: the percolation theory (12,13), theory based on nonequilibrium thermodynamics (14), and theory beginning with the Nernst-Planck equation (15,16) in which the transport properties are generally interpreted by means of an asymmetric potential (17)(18)(19)(20)(21). The theory for asymmetric membranes, which assumes only electrical double layers and exclusion of ions in the membrane, seems to be applicable to bilipid layer membranes.…”

Nernst–Planckian Electrodynamics, the Excess Interfacial Impedance, and the Complex Permittivity of a Lamellar Membrane with Overlapping Dynamic Electric Double Layers II. The Case of Negligible End Effects, Identical Ionic Diffusion Coefficients in Each Phase, and Identical Nernst Distribution Coefficients

“…[6] We have the important characteristic inverse lengths with the previously introduced abbreviations dition are simply logically equivalent ( there is only one condition ) . We may also consider a 2É1 interface, but the result is the same as that above ( because of the symmetry involved ) .…”