Coordination state probabilities and the solvation free energy of Zn ² in aqueous methanol solutions

Coordination state probabilities for the [Zn(H ₂O) _n(CH ₃OH) _m] ² complex in aqueous methanol solutions are calculated as a function of the bulk solution concentration, and the number of methanol ligands, m = 0, 1, ⋯ , 6 with n+m = 6. Zinc ion solvation free energies, which serve to normalize these probabilities, also reproduce the methanol concentration dependence of the experimentally derived free energy of zinc ion transfer from water to aqueous methanol solutions. Coordination state probabilities, p(n, m), are derived by extending quasi-chemical theory of ion hydration to solvent mixtures and mixed ligands. Free energy contributions to p(n, m) include the free energy of forming the mixed-ligand complex in the ideal gas, obtained by quantum chemical calculations, and the solvation free energy of the complex, approximated by a dielectric continuum model. We find that replacing water ligands with methanol ligands preferentially stabilizes methanol-rich complexes in the ideal gas. Conversely, water-rich complexes are stabilized by the solvation free energy contribution, such that the [Zn(H ₂O) ₆] ²⁺ complex is the dominant species in solution for all methanol concentrations considered. Stabilization of the methanol-rich complexes is a consequence of the local coordination chemistry, dominated by the delocalization of charge on the zinc ion, while the stabilization of water-rich complexes is a consequence of favorable ion-solvent electrostatic interactions and smaller dielectric cavities for the water-rich complexes at fixed total charge in the dielectric continuum model. Our analysis also highlights an entropic contribution associated with the reversible work required to remove n water and m methanol molecules from bulk solution to form the [Zn(H ₂O) _n(CH ₃OH) _m] ²⁺ complex, which captures the methanol concentration dependence of the solvation free energy of the zinc ion.