A Cell Model of an Ion-Exchange Membrane. Capillary-Osmosis and Reverse-Osmosis Coefficients

The capillary-osmosis and reverse-osmosis coefficients of an ion-exchange membrane have been calculated as the kinetic coefficients of the Onsager matrix within the thermodynamics of nonequilibrium processes and on the basis of the cell model proposed previously by the author for charged porous layers. The membrane is assumed to consist of an ordered set of spherical completely porous charged particles placed into spherical shells filled with a binary electrolyte solution. Boundary value problems have been analytically solved to determine the capillary-osmosis and reverse-osmosis coefficients of such a membrane under the Kuwabara boundary condition imposed on the cell surface. The consideration has been implemented within the framework of small deviations of system parameters from their equilibrium values under the action of external fields. Different particular cases of the obtained exact analytical equations have been studied including the case of a binary symmetric electrolyte and an ideally selective membrane. It has been shown that, for the considered cell model of an ion-exchange membrane, the Onsager reciprocity theorem is violated; i.e., the found kinetic cross coefficients are unequal to each other. The violation is explained by the fact that the reciprocity theorem is valid only for systems implying the linear thermodynamics of irreversible processes, for which generalized flows are equal to zero at nonzero thermodynamic forces.