When is quantum mechanics required to describe the thermodynamics of the ideal gas?

Abstract

A persistent conceptual puzzle in statistical mechanics is that the entropy S of a classical ideal gas depends on quantum mechanical quantities such as Planck's constant h and the factor 1/N!, which is commonly attributed to the indistinguishability of quantum particles. We resolve this puzzle by distinguishing entropy differences ΔS computed within the gas phase from those computed across a phase transition. For thermodynamic transformations within the gas phase, ΔS depends exclusively on classical macroscopic variables. In such processes, both h and 1/N! contribute only additive constants that cancel in ΔS, rendering them irrelevant for classical measurements. However, computing the latent heat released during a phase transition requires the entropy difference between two distinct phases with fundamentally different microscopic structures. Since the additive contributions do not cancel across phase transitions, a microscopic theory that correctly counts the states of each phase becomes essential, and quantum mechanics is required.