In a companion article it was shown in a certain precise sense that, for any
thermodynamical theory that respects the Kelvin-Planck Second Law, the
Hahn-Banach Theorem immediately ensures the existence of a pair of continuous
functions of the local material state -- a specific entropy (entropy per mass)
and a thermodynamic temperature -- that together satisfy the Clausius-Duhem
inequality for every process. There was no requirement that the local states
considered be states of equilibrium. This article addresses questions about
properties of the entropy and thermodynamic temperature functions so obtained:
To what extent do such temperature functions provide a faithful reflection of
``hotness"? In precisely which Kelvin-Planck theories is such a temperature
function essentially unique, and, among those theories, for which is the
entropy function also essentially unique? What is a thermometer for a
Kelvin-Planck theory, and, for the theory, what properties does the existence
of a thermometer confer? In all of these questions, the Hahn-Banach Theorem
again plays a crucial role