Legendre transforms and other potentials

Abstract

New thermodynamic potentials from baths We began our discussion of equilibrium by considering how the entropy emerges in isolated systems, finding that it is maximized under constant E , V , N conditions. In practical settings, however, systems are not often isolated and it is difficult to control their energy and volume in the presence of outside forces. Instead, it is easier to control so-called field parameters like temperature and pressure. In this chapter, we will discuss the proper procedure for switching the independent variables of the fundamental equation for other thermodynamic quantities. In doing so, we will consider non-isolated systems that are held at constant temperature, pressure, and/or chemical potential through coupling to various kinds of baths. In such cases, we find that entropy maximization requires us to consider the entropy of both the system and its surroundings. Moreover, new thermodynamic quantities will naturally emerge in this analysis: additional so-called thermodynamic potentials . To achieve conditions of constant temperature, pressure, or chemical potential, one couples a system to a bath. As discussed before, a bath is a large reservoir that can exchange energy, volume, or particles with the system of interest. While exchanging volume/energy/particles alters the system state, such changes are so minuscule for the bath that it is essentially always at the same equilibrium condition.