A Guide to Entropy and the Second Law of Thermodynamics

T his article is intended for readers who, like us, were told that the second law of thermodynamics is one of the major achievements of the nineteenth century—that it is a logical, perfect, and unbreakable law—but who were unsatisfied with the “derivations” of the entropy principle as found in textbooks and in popular writings. A glance at the books will inform the reader that the law has “various formulations” (which is a bit odd, as if to say the Ten Commandments have various formulations), but they all lead to the existence of an entropy function whose reason for existence is to tell us which processes can occur and which cannot. We shall abuse language (or reformulate it) by referring to the existence of entropy as the second law. This, at least, is unambiguous. The entropy we are talking about is that defined by thermodynamics (and not some analytic quantity, usually involving expressions such as −p lnp, that appears in information theory, probability theory, and statistical mechanical models). There are three laws of thermodynamics (plus one more, due to Nernst, which is mainly used in low-temperature physics and is not immutable— as are the others). In brief, these are: The Zeroth Law, which expresses the transitivity of thermal equilibrium and which is often said to imply the existence of temperature as a parametrization of equilibrium states. We use it below but formulate it without mentioning temperature. In fact, temperature makes no appearance here until almost the very end. The First Law, which is conservation of energy. It is a concept from mechanics and provides the connection between mechanics (and things like falling weights) and thermodynamics. We discuss this later on when we introduce simple systems; the crucial usage of this law is that it allows energy to be used as one of the parameters describing the states of a simple system. The Second Law. Three popular formulations of this law are: