The Gibbs paradox

TL;DR

This paper explores how molecular collisions and spatial accessibility influence the entropy of an ideal gas, proposing a new perspective that naturally accounts for extensivity without the traditional N! correction.

Contribution

It introduces a novel approach linking spatial volume distribution to entropy, eliminating the need for the N! correction in ideal gas entropy calculations.

Findings

Derives a probability distribution for molecular volume states proportional to exp(-vi/v0).

Shows that incorporating this distribution makes the entropy extensive without N! correction.

Connects the volume distribution factor to the Boltzmann factor, unifying spatial and energetic considerations.

Abstract

Molecular collision within an ideal gas originates from an intrinsic short-range repulsive interaction. The collision reduces the average accessible physical space for a single molecule and this has a direct consequence on the entropy of the gas. The accessibility of a molecule to a spatial coordinate (x, y, z) inside the system depends on the local molecular density. By considering mechanical equilibrium between a system and a reservoir, the probability of the system in state i with volume vi is shown to be proportional to exp(-vi/v0) where v0 is the average volume per molecule. Incorporating this factor into the single particle partition function automatically leads to an N-particle entropy that is extensive without applying the N! correction factor. The exp(-vi/v0) factor plays a similar role in describing the volume distribution as the Boltzmann factor which governs the energy distribution.