On calculating activity coefficients and other excess functions from the intracrystalline exchange properties of a double-site phase

For a phase at equilibrium in which two cation species are partitioned ideally between two sub-lattice sites, the excess functions of mixing (free energy, enthalpy and entropy) are directly related to the bulk composition of the phase and ΔG_E°(T, P), the standard-state intra- crystalline exchange free energy. If the phase is not at equilibrium internally, an additional ordering parameter is necessary to fix the excess free energy of mixing, G_mix^EX, unambiguously. Conversely, for any fixed G_mix^EX there exists an infinity of possible intracrystalline cation dis- tributions, only one of which is the equilibrium distribution for the specified temperature and pressure. As ideal intraphase cation ordering becomes more pronounced, G_mix^EX decreases. In response, the total free energy of mixing for the phase decreases progressively for non-end member compositions, approaching, at the limits of ordering, values appropriate for stabilizing compounds of intermediate composition.The model-dependent activity coefficient for component A in the phase, γ_A^T, can be calculated for any bulk composition, X_A^T, either from G_mix^EX directly or from more basic equations involving the interrelation of chemical potentials at equilibrium. A general form for γ_A^T is ln $\text{γ}{}_{\mathrm{A}}{}^{\mathrm{T}}\text{= 1n}\text{2(X}{}_{\mathrm{A}}{}^{\mathrm{\alpha }}\text{X}{}_{\mathrm{A}}{}^{\mathrm{\beta }}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{/}\text{(X}{}_{\mathrm{A}}{}^{\mathrm{\alpha }}\text{+X}{}_{\mathrm{A}}{}^{\mathrm{\beta }}\text{)}\text{]+Y}$, where Xj^κ denotes the mole fraction of species j in site κ. The first term on the right-hand side of this equation is the contribution to γ_A^T from ideal intracrystalline partitioning, and is common to the several theories lately presented to model intraphase cation partitioning. It can be shown rigorously that this term contributes to a negative deviation from ideality for the bulk phase. The second term is the contribution to the macroscopic activity coefficient from non-ideal intraphase partitioning, and is related to an enthalpy of mixing, H_mix^N in excess of that resulting from ideal inter-site cation ordering. While the expression represented by Y can take several functional forms, the additional enthalpy can be evaluated explicitly for specific non-ideal partitioning models from the relation $\text{H}{}_{\mathrm{mix}}{}^{\mathrm{N}}\text{= 2RT(1? X}{}_{\mathrm{A}}{}^{\mathrm{T}}\text{) ∝}\text{Y}\text{(1 ? X}{}_{\mathrm{A}}{}^{\mathrm{T}}\text{)}{}^2\text{dX}{}_{\mathrm{A}}{}^{\mathrm{T}}$.In those cases, G_mix^EX can also be determined exactly.