Clausius versus Sackur-Tetrode entropies

TL;DR

This paper derives the Clausius entropy for monoatomic ideal gases using extensivity, showing its relation to Sackur-Tetrode entropy, and discusses the limitations of classical thermodynamics in resolving the Gibbs paradox.

Contribution

It provides a mathematically consistent derivation of entropy without quantum or information theory, highlighting differences from statistical mechanics.

Findings

Clausius entropy coincides with Sackur-Tetrode entropy in the thermodynamic limit.

Classical thermodynamics does not fully resolve the Gibbs paradox.

Volume of phase space relates to thermodynamic observables without requiring the thermodynamic limit.

Abstract

Based on the property of extensivity (mathematically, homogeneity of first degree), we derive in a mathematically consistent manner the explicit expressions of the chemical potential $\mu$ and the Clausius entropy $S$ for the case of monoatomic ideal gases in open systems within phenomenological thermodynamics. Neither information theoretic nor quantum mechanical statistical concepts are invoked in this derivation. Considering a specific expression of the constant term of $S$, the derived entropy coincides with the Sackur-Tetrode entropy in the thermodynamic limit. We demonstrate however, that the former limit is not contained in the classical thermodynamic relations, implying that the usual resolutions of Gibbs paradox do not succeed in bridging the gap between the thermodynamics and statistical mechanics. We finally consider the volume of the phase space as an entropic measure, albeit, without invoking the thermodynamic limit to investigate its relation to the thermodynamic equation of state and observables.