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 mu ltitemperature gas mixtures. The presented material is meant to comple ment and, for certain aspects, to complete a previous contribution of the author on the subject matter. The analysis begins with a brief int roductory survey of the main theoretical approaches pursued to charact erize quantitatively multitemperature equilibria with the intent to em phasize the discordant findings of these approaches and the diverging opinions they have originated in the literature. The equilibrium probl em 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 mixt ure in connection with the application of the principle is recognized. A rigorous proof is given of the equivalence between energy minimizat ion and entropy maximization for the purpose of determining the equili brium conditions in multitemperature circumstances and regardless of t he active internal constraints. Moreover, the influence of the kind of internal constraints in establishing the mathematical form of the equ ilibrium equations is pointed out and the divergence among the finding s of other approaches is thus explained. The equivalence feature is al so considered in relation to the thermodynamic potentials. Evidence is given that not all thermodynamic potentials possess the equivalence p roperty, i.e., attainment of an extremum, in conditions of thermodynam ic equilibrium. Consistently, mathematical properties relevant to the search of the extrema of the Legendre transforms are recalled and elab orated upon. A selection rule is formulated that permits the identific ation of the thermodynamic potentials possessing the equivalence prope rty. The essential role played by the internal constraints in the sele ction procedure is described and fully evidenced in the subsequent app lication of the method to two representative cases of equilibrium that occur often in the applications, namely, in the absence of internal c onstraints and when energetic freezing prevails.