Derivation of limiting ion mobility equation based on the application of solvation effect-incorporated Poisson-Boltzmann equation

Abstract

Interaction potential energy between ion and solvent molecular dipole moments in solvation shell around ion was derived on basis of the application of the solution of linear SEI-PB (Solvation Effects-Incorporated Poisson-Boltzmann) equation. Based on application of the solution of linear SEI-PB equation, distribution of solvent molecules in solvation shell around ion in electrolyte solution at thermodynamic equilibrium was derived. In case approaching to limit of infinite dilution, analytical expression for evaluating limiting ion mobility is derived from the boundary condition that the average number of solvent molecules accompanying with translational motion of ion is equal to the mean solvation number of ion. The derived limiting ion mobility formula is different from the Stokes–Einstein relation and lies in relation to direct proportion with the cube of dynamic size of ion in solution, depending not only on the viscous coefficient, dielectric constant, the density and molecular weight of solvent but also on temperature.By virtue of the equation, it was possible to estimate dynamic sizes of ions in solution. It was found the estimated dynamic sizes of ions in aqueous solution to be compatible with the average ion–water distances, and to be always greater than crystallographic ionic radii in size.A new relation between the limiting ion mobility and the solvent molecular dipole moment in solvation shell was established based on the interaction potential energy between ion and solvent molecular dipole moments in solvation shell around ion. By virtue of the proposed relation, we could find out the average water molecular dipole moment in hydration shell to be different more or less in size from that in bulk solution according to ionic species in aqueous solution.