Fundamentals of Thermodynamic Stability

Abstract

The second law of thermodynamics states that the entropy change in any spontaneous adiabatic process is greater than or equal to zero. It is a disarmingly simple statement but one that is a cornerstone of scientific theories. It is instrumental in describing the extent and direction of all physical and chemical transformations and contains within it the essential ideas for developing thermodynamic stability theory. Stability theory concerns itself with answering questions such as (1) What is a stable thermodynamic state? (2) Which conditions define the limit to this state beyond which the system becomes unstable? (3) How does the instability manifest itself ? In a real sense, stability theory provides the underlying framework for a macroscopic understanding of phase transitions and critical phenomena, the subject of this text. Many of the results of stability theory related to phase equilibria are well known; an example is the condition that, for a pure fluid in a stable state, the quantity −(∂P/∂V )T,N must be greater than or equal to zero, with the equality condition holding at the limit of stability. Many other facets of thermodynamic stability theory, however, are relatively unfamiliar. For example, any of the well-known thermodynamic potentials E, H, A, and G can be used to develop stability criteria for a given system. Are these criteria always equivalent, or do some take precedence over others? If so, what are the implications of this for understanding phase transformations in physicochemical systems? It is questions of this sort that we take up in this chapter, where we lay the macroscopic foundations for the material developed throughout the rest of the text. In this analysis, we rely heavily upon results taken from linear algebra, a branch of mathematics that provides an ideal tool for developing a comprehensive description of thermodynamic stability concepts. The combination of the first and second law of thermodynamics for a closed system leads to the well-known equation: . . . dE = T dS − PdV . . . . . . (1.1) . . . where E(S, V ) represents the system energy as a function of the independent variables S and V.