Abstract
In this contribution, we carry on with the research program initiated in J. Math. Chem., 58(6), 2020. Using the methods from geometric thermodynamics, we formally derive and analyze different conditions for thermodynamic stability and determine the limits of their use. In particular, we study, in detail, several versions of the Le Chatelier—Brown principle and demonstrate their application to the analysis of thermodynamic stability.
Highlights
The conditions of thermodynamic stability belong to the basic principles of general and chemical thermodynamics and are of the utmost importance for various fundamental and applied problems.Recall, for instance, the quotation from Herbert Callen in [1]: “Considerations of stability lead to some of the most interesting and significant predictions of thermodynamics”
The main difference between these formulations is the choice of the initial thermodynamic potentials, as well as the set of additional conditions imposed on the thermodynamic system
While most criteria are formulated in terms of the stability matrix, that is, the matrix formed by the second derivatives of thermodynamic potentials, there are several different approaches based upon expressing stability conditions in terms of inequalities involving partial derivatives of different thermodynamic potentials under different fixations of variables
Summary
The conditions of thermodynamic stability belong to the basic principles of general and chemical thermodynamics and are of the utmost importance for various fundamental and applied problems. We take a different route and apply the developed geometric approach to the rigorous mathematical analysis of stability conditions for an equilibrium thermodynamic system. Legendre manifold of a particular thermodynamic differential 1-form, [27], and apply the tools from linear algebra and multivariable calculus, in particular the implicit function theorem and its ramifications, [28,29] Along these lines, we formally derive some classical relations such as the Le Chatelier—Brown principle, and carry out a rigorous mathematical analysis of these relations and determine the limits of their applicability.
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