Generalized derivation of an exact relationship linking different coefficients that characterize thermodynamic effects of preferential interactions

In solutions consisting of solvent water (component '1') and two solute components ('2' and '3'), various thermodynamic effects of differences between solute-solute and solute-solvent interactions are quantitatively characterized by state functions commonly called 'preferential interaction coefficients': gamma(mu(1),mu(3)) triple bond (delta(m3)/delta(m2))(T,mu(1),mu(3)) and gamma(mu(k)) triple bond (delta(m3)/delta(m2))(T,P,mu(k)), where k = 1,2 or 3. These different derivatives are not all directly accessible to experimental determination, nor are they entirely equivalent for analyses and interpretations of thermodynamic and molecular effects of preferential interactions. Consequently, various practical and theoretical considerations arise when, for a given system, different kinds of preferential interaction coefficients have significantly different numerical values. Previously we derived the exact relationship linking all three coefficients of the type gamma(mu(k), and hence identified the physical origins of the differences between gamma(mu(1)) and gamma(mu(3)) that have been experimentally determined for each of various common biochemical solutes interacting with a protein [J. Phys. Chem. B, 106 (2002) 418-433]. Continuing our investigation of exact thermodynamic linkages among different types of preferential interaction coefficients, we present here a generalized derivation of the relationship linking gamma(mu(1),mu(3)), gamma(mu(3)) and gamma(mu(1)), with no restrictions on m(2), m(3) or any physical characteristic of either solute component (such as partial molar volume). Hence, we show that (gamma(mu(1),mu(3)) - gamma(mu(3))) is related directly to (gamma(mu(3)) - gamma(mu(1))), for which the physical determinants have been considered in detail previously, and to a factor dependent on the ratio of the partial molar volumes V3/V1. Our generalized expression also provides a basis for calculating gamma(mu(1),mu(3)), even in situations where preferential interactions could not be investigated by equilibrium dialysis. To demonstrate this applicability, we analyze isopiestic distillation data for aqueous solutions containing urea and NaCl, two small solute components that cannot be selectively dialyzed.