A Missing Deduction of the Clausius Equality and Inequality

Abstract

The two main statements of the second law of thermodynamics were given by Clausius, and Kelvin and Planck. Other statements followed well into the 20th century. It is known that the two main statements are not exactly equivalent, although their differences may never show outside academic environments or very specific areas of science. At negative absolute temperatures, a phenomenon studied mainly in connection to the constitution of matter and to its behavior under specific electromagnetic conditions, the non-equivalence of the two major statements, appears. Whereas the Kelvin−Planck statement must be modified to remain true, the Clausius statement remains true unaltered. This article proposes a deduction of the Clausius equality and inequality from the Clausius statement only, thus, inheriting its validity in the domain of negative absolute temperatures. The deduction developed here uses a simple arrangement of Carnot engines and a conceptualization of a system with very loose constraints. As a result, the level of generality achieved is high and includes both positive and negative absolute temperatures.