Electrochemical Impedance of Ion-Exchange Membranes in Ternary Solutions with Two Counterions

This paper aims to study the electrochemical impedance of a system consisting of an ion-exchange membrane and two diffusion boundary layers on both sides of the membrane bathed in ternary electrolyte solutions with two counterions. An approximate analytical expression has been arranged for the impedance of an ideal ion-exchange membrane system for counterions with identical charge number on the basis of the Nernst–Planck–Donnan equations under electroneutrality. The electrochemical impedance for different values of the direct electric current of ion-exchange membrane systems with univalent ions has been numerically obtained from the Nernst–Planck and Poisson equations. The results have been compared with the approximate analytical expressions in different arrangements of practical interest, including conventional electrodialysis, bi-ionic conditions, and reverse electrodialysis. Our results provide new insights into the role played by the generalized expressions of the transport numbers, electrolyte diffusion coefficients, limiting electric currents, and membrane resistance in the interpretation of electrochemical impedances of ion-exchange membranes in ternary electrolyte solutions with two counterions. Moreover, new physical insights emerge from the analysis of the conductive contribution to the total impedance of the system.