Supplement 2: Relation of Apparent Chemical Potential to Species Potentials

To understand the origin of the models for the mean activity coefficient, some discussion of the chemical potentials is necessary. Many of the concepts are extensions of the methods used for reaction engineering. For example, consider the general case of an electrolyte dissociating in solvent. When an electrolyte dissociates, the material balance gives the molality of positive and negative charges in solution,

where the subscript u represents the amount of electrolyte that is un-ionized. The change in Gibbs energy at fixed temperature and pressure is given by the changes in composition and the chemical potentials,

which rearranges to

The quantities μ₊ and μ_– are analogous to the partial molar Gibbs energies of other components and the subscript w represents water or solvent. Since the positive and negative charges cannot change independently in a physical mixture as required by the rigorous definition of the partial molar quantity where all but one species is constrained when the derivative is evaluated, they are nonphysical, but they can be calculated by theory. Recall that the quantity dn_i relates to changes in the apparent composition. As a solution of fixed apparent concentration equilibrates (e.g., dn_i = 0), the un-ionized concentration changes until Gibbs energy is minimized and dG = 0. Thus, we conclude that at the equilibrium ionization the first term in parentheses is zero. Thus,

This relationship between the “products” and “reactants” of the ionization is analogous to the relations developed for molecular reacting systems. This equality can be inserted into the second term, at equilibrium ionization resulting in

On an apparent basis, we also must satisfy

where μ_i is the apparent chemical potential. By comparison of the last two equations we conclude that the apparent chemical potential must equal the chemical potential of the un-ionized species, which also must equal the weighted sum of the chemical potentials of the ions:

Thus, the approach for developing a model for the apparent chemical potential and apparent activity coefficient is based on developing models for the ions and then using the weighted sum.