DIFFUSIONAL MODEL FOR INTRAPARTICLE ION EXCHANGE KINETICS IN NONLINEAR SELECTIVE SYSTEMS

Abstract

ABSTRACT The diffusional model for the intraparticle ion exchange (IE) kinetics in selective ion exchangers is presented. The model develops the phenomenological formulation based on the irreversible thermodynamics and includes mass balance equations with description of ion fluxes by the Nernst-Plank relations. The model is based on the supposition that during the exchange RA+B = RB + A involving a chemical reaction of counter-ions (B and A) with the resin site R, diey can exist in the exchanger in two states, namely “free” and as the species formed by their combination with fixed groups R. Counter-ions B and A are considered to be immobile while “bound” and to move unimpeded while “free”. The mathematical formulation of the IE kinetic process in this case is described by a system of two nonlinear diffusion equations in partial derivatives where the effective self diffusion coefficients Dii and the cross interdiffusion coefficients Dij depend on the concentration of the diffusing counter-ions (B and A) and coions Y, and also on the values of equilibrium constants KRB and KRA describing the stability of RB and RA complexes in the ion exchanger phase. As the result, the ion fluxes are described within the generalized Fick formulation. Fickian diffusivities Dij compose 2×2 diffusional matrix with determining each of them through the individual diffusivity Di of the ions B, A and F. The set of diffusional equations along with the appropriate initial and boundary conditions solved by using a computer gives transient concentration profiles which compose concentration waves in the ion exchanger bead. For various combination of diffusional and selectivity factors the concentration waves in the bead have been presented in the pseudo 3dimensional space {concentration - radius - time}. This presentation simplifies the interpretation of the process and makes it visual. It is shown that the difference both in concentration waves profiles and rate between forward and reverse exchange of two counter-ions may arise not only from unequal diffusion coefficients but also from unequal dissociation constants of the complexes formed with the fixed groups.