Swelling Phenomena and the Sorption of Solvents

The electrochemical potentials of freely permeant components, ions and solvent, in chemical equilibrium are equal in each phase. The total electrochemical potential or partial molar free energy of a component is made up of contributions from its activity a, pressure P, and its ionic, electrical potential ip, all referred to the same standard and reference states of unit activity a = 1), and infinite dilution (activity coefficient = 1) respectively at 1 atmosphere pressure. The electrochemical potential of a component i in a given phase is given by the equation  

Some further nomenclature is now necessary to describe absorption equilibria in ion exchange systems. For a species i, m and C, represent the molal and molar concentrations respectively, whilst A and Xi denote the mole fraction and equivalent ionic fraction of i respectively. Single ion activity coefficients are denoted yj and mean ionic activity coefficients by yj . Whether the latter quantities refer to the molar or molal concentration scales is decided by the choice of units defining concentration. Thermodynamic activities and activity coefficients for the resin phase using the equivalent or mole fraction concentration scale (rational scale) are sometimes defined differently and are discussed in a later section. Finally, the exchanger and external solution phases are differentiated by subscripts r and s respectively. 

Alternatively the molecular models for linear and crosslinked polyelectrolytes (including ion exchange resins) interpret the swelling 

Gregor and Pepper and their co-workers have carried out detailed studies on the swelling properties of ion exchange resins from which the following general conclusions may be drawn  

The thermodynamic relationship describing the osmotic equilibrium between a resin and pure liquid water may be derived as a result of reconsidering equation 5.2. Equilibrium, with pure water only, means that ion exchange need not be considered and therefore the electrical potential term Z Fip can be ignored, whereby the chemical potential /t of component i is given by  

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