Gibbs-Duhem-based relationships among derivatives expressing the concentration dependences of selected chemical...

For a two-component system, a derivative that specifies the concentration-dependence of one chemical potential can be calculated from the corresponding derivative of the other chemical potential by applying the Gibbs-Duhem Equation. To extend the practical utility of this binary thermodynamic linkage to systems having any number of components, we present a derivation based on a previously unrecognized recursive relationship. Thus, for each independently variable component, kappa, any derivative of its chemical potential, mukappa, with respect to one of the mole ratios {mkappa identical with nkappa/nomega} is related to as a characteristic series of progressively higher order derivatives of muomega for a single "probe" component, omega, with respect to certain of the {mkappa}. For aqueous solutions in which omega is solvent water and one or more of the solutes (kappa) is dilute, under typical conditions each sum of terms expressing a derivative of mukappa consists of at most a few numerically significant contributions, which can be quantified, or at least estimated, by analyzing osmometric data to determine how the single chemical potential muomega depends on the {mkappa} without neglecting any significant contributions from the other components. Expressions derived here also will provide explicit criteria for testing various approximations built into alternative analytic strategies for quantifying derivatives that specify the {mkappa} dependences of mukappa for selected components. Certain quotients of these derivatives are of particular interest in so far as they gauge important thermodynamic effects due to "preferential interactions".