Toward Formal Analysis of Thermodynamic Stability: Le Chatelier—Brown Principle

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.

Geometric Analysis of a System with Chemical Interactions

“…In [5,6], the authors proposed a systematic approach to the analysis of thermodynamic systems based upon the apparatus of contact geometry. Below, we will present a brief overview thereof.…”

“…It allows one to compute the partial derivatives of some dependent parameters with respect to certain independent parameters for different choices of independent parameters, thus providing a detailed characterization of the energy manifold curvature. In [6], such calculations were used to derive equilibrium conditions for a thermodynamic system not undergoing a chemical reaction.…”

“…We thus restrict ourselves to considering 2n thermodynamic variables Y and X. The problem of determining which variables can be taken as dependent boils down to finding n + k linearly independent columns of the [2n × 2n] matrix (6). The first n columns correspond to the extensive variables Y, while the remaining columns correspond to the intensive variables X.…”

“…the extensive variables at different fixings of the sets of intensive and extensive parameters [1], (p. 114). A thorough analysis of these conditions was performed in the papers [5,6] based upon a contact geometric formulation. The geometric framework has proved to be rather helpful for analyzing complex thermodynamic relations.…”

“…This matrix possesses a number of nontrivial properties and has been subject to a thorough study for decades; see [18,19] for an overview of recent results. Quite remarkably, within the framework developed in [5,6], the matrix theoretic analysis of chemical reactions turns out to be well connected to the geometric study of the isoaffine submanifolds. This connection will be demonstrated below.…”