Computation of chemical equilibrium properties of material-balance constrained, multi-phase and non-ideal systems by Gibbs free energy minimization is a reputable method which is extensively used in materials science, chemical and process engineering as well as in energy and environmental technology. However, in many prospective applications, materials and processes are influenced by other constraining factors, such as electrochemical, -magnetic, charge transport or surface phenomena. For those situations, an extended Gibbs energy method can be applied. In this technique, the supplementary work-related condition is introduced to the Gibbs energy calculation as an additional undetermined Lagrange multiplier, which represents the constraint potential. Similar technique can be used for a fixed amount of a system component, which then serves to externally limit the extent of selected chemical reactions in terms of their affinities. A number of new phenomena can thus be included in Gibbs'ian calculations and both equilibrium and non-equilibrium properties of complex systems can be solved.

In this presentation we delineate a Gibbs free energy technique based on the extension of the mass and energy exchange (stoichiometric) matrix of the Lagrange method. As the Lagrange multipliers represent the chemical potential of the Gibbs'ian system components, the exchange matrix is developed to include other contributions than those directly involved in the (elemental) mass balance of the system. For example, interfacial energy between immiscible equilibrium phases is a feature explained by classical thermodynamics, yet it can not be included in a multi-phase calculation with mere mass balance constraints. By introducing an extended matrix with a surface area constraint, the ‘surface phase' can be consistently incorporated. Similarly, the electrochemical potential difference in an aqueous multi-phase system with two solution phases which are separated by a membrane can be taken into account by utilizing the matrix extension. In an analogous fashion, a further extension of the matrix brings about the possibility to include reaction kinetic restrictions to control the extents of chemical reactions in terms of their affinity.