Energy and Entropy for Thermodynamical Systems

Abstract

In our treatment of homogeneous fluid bodies, we found that the laws of thermodynamics permitted us to augment the list of functions of state (𝓅, λ, 𝓈) in the description of such a body. The new functions of state, E and S—called energy and entropy functions, respectively—arise as potentials for vector fields on the state space, their existence is equivalent to the First and Second Laws, and their derivatives yield the original functions of state, 𝓅, λ, and 𝓈 in simple relations. In Chapters III and IV we have studied versions of the First and Second Laws which are meaningful for a class of systems far broader than the collection of homogeneous fluid bodies and which take the form of relations that restrict the behavior of a thermodynamical system in special cycles. For each of the two laws of thermodynamics, we found a condition that applies to all cycles and is equivalent to the original statement of the law. Theorem 3.1 of Chapter III and Theorem 6.1 of Chapter IV express these facts in precise mathematical language and tell us that Joule’s relation, (3.1) of Chapter III (with M = R/λ), is equivalent to the First Law, and that the Accumulation Inequality, (6.2) of Chapter IV, is equivalent to the Second Law. It is natural to ask whether analogues of energy and entropy can be obtained from Joule’s relation and the Accumulation Inequality, and the present chapter is devoted to providing an affirmative answer to this question.