Quasi-Lattice Model of Reciprocal Salt Systems. A Generalized Calculation

A generalized statistical‐mechanical calculation of the solution thermodynamics of the reciprocal molten salt system A⁺, B⁺, C⁻, and D⁻ dilute in A⁺ ions has been made. The calculation is based on the quasi‐lattice model of reciprocal salt systems and treats the problem as an order‐disorder problem. Associations of the A⁺ and C⁻ ions as $$A^{+}{\mathrm{+C}}^{-}\mathrm{\rightleftharpoons AC}$$ $${\mathrm{AC+C}}^{-}{\mathrm{\rightleftharpoons AC}}_2^{-}$$ $${\mathrm{AC}}_2^{-}{\mathrm{+C}}^{-}{\mathrm{\rightleftharpoons AC}}_3^{=}$$ are taken into account. In terms of the conventional equilibrium constants K₁, K₂, and K₃ for (A), (B), and (C), respectively, $$K_1{\mathrm{=Z(\beta }}_1\text{−1)}$$ $$K_1{K}_2=\frac{\text{Z(Z−1)}}{\text{2!}}{\mathrm{(\beta }}_1{\beta }_2{\text{−2β}}_1\text{+1)}$$ $$K_1{K}_2{K}_3=\frac{\text{Z(Z−1)(Z−2)}}{\text{3!}}{\mathrm{(\beta }}_1{\beta }_2{\beta }_3{\text{−3β}}_1{\beta }_2{\text{+3β}}_1\text{−1),}$$ where Z is the quasi‐lattice coordination number, β_j=exp(—ΔA_j/kT) and ΔA_j may be termed a specific bond free energy of the jth C⁻ ion and is the specific Helmholtz free energy change for the formation of the jth bond. Equations (1), (2), and (3) demonstrate the surprising fact that the higher order association constants are dependent on the magnitude of the lower order association constants not only through the factor containing Z but also through the values of the β_i for the lower association constants. If β₁=β₂=β₃=··· etc., the equations reduce to the statistical ratios of Bjerrum. The higher association constants can be shown to be smaller for a case in which the bonding is directional (e.g., linear AC₂⁻), than if they are nondirectional, as in Eqs. (2) and (3), even if the bond free energies are equal in the two cases.