Thermodynamic Expressions Relating Different Types of Preferential Interaction Coefficients in Solutions Containing Two Solute Components

In aqueous solutions that contain a macromolecular solute (charged or uncharged) and molecules or ions of a smaller solute, differences between solute−macromolecule and water−macromolecule interactions have solute-concentration-dependent thermodynamic effects that are characterized quantitatively by “preferential interaction” coefficients. Of primary interest in this paper are Γ μ 1 and Γ μ 3, defined as partial derivatives that specify the dependence of the molality of the smaller solute on the macromolecular molality at fixed temperature and pressure and the chemical potential indicated by the subscript (μ 1 for water and μ 3 for the smaller solute). Coefficients of the type Γ μ 3 (but not Γ μ 1 ) are direct gauges of thermodynamic effects due to the preferential interactions of a small solute with a macromolecule. Although individual values of Γ μ 3 cannot generally be obtained directly by any experimental method, corresponding values of Γ μ 1 are directly measurable for involatile solutes (0.01 ≲ m 3 ≲ 3 m ) by an accurate, efficient method based on water vapor pressure osmometry [Courtenay et al. Biochemistry 2000, 39, 4455]. In that study, alternative approximate expressions, for which we present generalized derivations here, were used to calculate Γ μ 3 . The resulting values differ significantly from the corresponding values of Γ μ 1 for the interactions of various small biochemical solutes with a common globular protein. To identify the general physical origins and thermodynamic implications of numerical differences between Γ μ 3 and Γ μ 1, we derive an exact thermodynamic relationship linking these coefficients and examine how it is affected, for various types of systems and ranges of conditions, by contributions from the ideal mixing entropy of components 2 and 3 and from the nonideality due (primarily) to interactions of the macromolecule with the smaller solute. This analysis shows why, in general, Γ μ 3 differs significantly from Γ μ 1 and why Γ μ 3 ≅ Γ μ 1 in the exceptional cases where components 3 and 2 have a common ion present in large excess over the macromolecular species.