Thermodynamics based on the Hahn-Banach Theorem: The Clausius inequality

The theoretical basis of the Clausius inequality (CI) is analyzed to determine the extent to which it is a consequence of the Second Law. An applicable version of the Hahn-Banach theorem is presented and used to examine the criteria for the existence and uniqueness of a positive-valued temperature scale which gives temperature as a function of state and satisfies the CI for cyclic processes. The argument includes discussion of cyclic heating systems and Kelvin-Planck systems, the numerical representation of hotness, the ordering of hotness levels, and the role of Carnot elements. It is shown that at least one Clausius temperature scale exists for any Kelvin-Planck system, and that this existence follows directly from the Second Law without the intervention of Carnot cycles or other conceptual apparatus, but that proof of the uniqueness of a scale requires the presence of reversible processes in general and Carnot cycles in particular.