Molecular Diffusion Phenomena of Ion Exchange in the Bulk Phase

The most complex component of the mathematical model of ion exchange is the description of the kinetics of the process, for which the mass-transfer equations for each phase are used in engineering practice. Calculation of the mass-transfer coefficients requires knowledge of the coefficients of molecular diffusion relative to the phase boundary. While for the ion exchanger the expressions relating these coefficients to the diffusion coefficients of individual ions are given in the literature, for the continuous phase there are no substantiated expressions. In this paper, by solving the system of equations for the flows of each type of ion and the condition of electroneutrality of the solution, expressions are obtained for the coefficients of molecular diffusion relative to the phase boundary in the continuous phase during ion exchange. Multicomponent and binary cases are considered. The difference between these expressions and those known both for the ion-exchanger phase and for the solution when a substance is dissolved in it is shown. A significant dependence is revealed for the coefficients of molecular diffusion relative to the phase boundary in the continuous phase on the concentrations and charges of the ions participating in the ion exchange, as well as on their numbering. A number of examples for the two-component case are used to examine the limiting changes in the diffusion coefficients when the mole fractions of the components tend to zero and one. For the three-component case, the graphs show the dependence of the elements of the matrix of the multicomponent diffusion coefficients on the ratio of the component fractions; it is demonstrated that the off-diagonal elements of the matrix can be either positive or negative and either smaller or larger than the diagonal elements. A method for numbering components is determined that allows minimizing the off-diagonal elements of the matrix of multicomponent diffusion coefficients.