Derivations of Partial Molar Excess Gibbs Energy of Mixing Expressions for Common Thermodynamic Models

Modern computational thermodynamics methods rely on the use of numerical models that represent chemical systems. Typically, these models are formulated in terms of the Gibbs energy, which must be minimised to find the conditions of thermodynamic equilibrium. Numerous thermodynamic models have been developed to capture the behaviour of regular solid and liquid solutions, ionic ceramics, multi-sublattice metallic alloys, short and long range ordering, and much more. Some classes of commonly used thermodynamic models include substitutional solutions and compound energy formalism. The mathematical formulation of the Gibbs energy of a solution phase represented by any of the aforementioned models takes on a unique form, which requires special consideration for obtaining the partial derivatives in the Hessian matrix of a Gibbs energy minimiser. This paper provides derivations of the partial molar excess Gibbs energy of mixing of some common classes of thermodynamic models for use in a Gibbs energy minimiser.