Equivalence of energy, entropy, and thermodynamic potentials in relation to the thermodynamic equilibrium of multitemperature gas mixtures

The central theme of this study is the thermodynamic equilibrium of multitemperature gas mixtures. The presented material is meant to complement and, for certain aspects, to complete a previous contribution of the author on the subject matter. The analysis begins with a brief introductory survey of the main theoretical approaches pursued to characterize quantitatively multitemperature equilibria with the intent to emphasize the discordant findings of these approaches and the diverging opinions they have originated in the literature. The equilibrium problem is then confronted within the framework of axiomatic thermodynamics. The general equilibrium principle in its axiomatic form is recalled and the importance of the physical constraints imposed on the gas mixture in connection with the application of the principle is recognized. A rigorous proof is given of the equivalence between energy minimization and entropy maximization for the purpose of determining the equilibrium conditions in multitemperature circumstances and regardless of the active internal constraints. Moreover, the influence of the kind of internal constraints in establishing the mathematical form of the equilibrium equations is pointed out and the divergence among the findings of other approaches is thus explained. The equivalence feature is also considered in relation to the thermodynamic potentials. Evidence is given that not all thermodynamic potentials possess the equivalence property, i.e., attainment of an extremum, in conditions of thermodynamic equilibrium. Consistently, mathematical properties relevant to the search of the extrema of the Legendre transforms are recalled and elaborated upon. A selection rule is formulated that permits the identification of the thermodynamic potentials possessing the equivalence property. The essential role played by the internal constraints in the selection procedure is described and fully evidenced in the subsequent application of the method to two representative cases of equilibrium that occur often in the applications, namely, in the absence of internal constraints and when energetic freezing prevails.