Manipulation of Partial Derivatives and Sample Problems

Abstract

The material presented in this lecture is adapted from Chapter 5 in T&M. Continuing with the material introduced in Lecture 13, first, we will discuss two additional restrictions which need to be imposed on the internal energy fundamental equation, U, including presenting the Corollary to Postulate I. Second, we will discuss how to reconstruct the fundamental equation, U, if we know the (n + 2) first-order partial derivatives of U. Third, we will motivate the need to manipulate partial derivatives of thermodynamic functions in order to calculate changes in these thermodynamic functions when the system evolves from an initial state i to a final state f. To this end, we will discuss how to devise useful integration paths which connect state i with state f. Fourth, we will discuss various useful methods to relate an unknown partial derivative to other partial derivatives which can be calculated or determined experimentally. Fifth, among the useful methods to relate a desired partial derivative to others, we will discuss the triple product rule then add another variable rule, the derivative inversion rule, and Maxwell’s reciprocity rule. In addition, we will discuss Jacobian transformations. Finally, we will solve Sample Problems 14.1 and 14.2 to illustrate the use of some of the methods presented to calculate partial derivatives.