Equilibrium Thermodynamics

The ideas which we shall present in the remainder of this book are intimately connected with thermodynamics. In order to describe the various transport processes in structured continua, one must first build a solid foundation of equilibrium thermodynamics upon which to base further development. In approaching transport phenomena from an energetic viewpoint, one must first define what is meant by various thermodynamic variables as, for example, the temperature and pressure, in terms of the primitive variables used to characterize the system under investigation. In this chapter, we present a brief overview of equilibrium thermodynamics tailored to the needs of this book. Specifically, we want to define explicitly the thermodynamic quantities which are used in subsequent chapters. For clarity and completeness, we shall re-derive, rather than merely state, some of the standard thermodynamic relationships. Of course, the experienced reader may proceed directly to the next chapter and use this chapter as a reference to notation as the need arises. The starting point for our discussion of equilibrium thermodynamics is the axiomatic foundation of the description of macroscopic equilibria on certain fundamental principles. First, the macroscopic equilibrium of a closed system is completely described through the specification of a number of extensive (i.e., proportional to the total mass or volume of the system, and additive between systems) or intensive (i.e., independent of the total mass or volume of the system) parameters. This is a very important point which is usually overlooked in the traditional thermodynamic development. The extensive nature of the primary variables of the system introduces an additional relationship which acts on the allowed variations of the differentials, which, as we shall see, is tantamount to the Gibbs/Duhem relation. This implies, as we shall demonstrate in §4.3, that the density formalism, where every extensive quantity is reported on a unit volume basis, is a much more natural framework for describing the system in that it avoids a number of pitfalls of the traditional formalism.