The original Gibbs paradox is the consequence of the erroneous identification of non-identical functions

TL;DR

This paper clarifies that the Gibbs paradox arises from confusing different entropy functions for ideal gases, showing the paradox is due to comparing non-identical functions rather than a true discontinuity.

Contribution

It demonstrates that the Gibbs paradox results from misidentifying non-identical entropy functions, resolving the paradox by clarifying their differences.

Findings

The entropy functions for different and identical gases are inherently different.

The perceived entropy jump is due to comparing different functions, not a real discontinuity.

The paradox is resolved by recognizing the functions' non-identity.

Abstract

This article presents the results of research into the causes of the Gibbs paradox in the formulation discussed by J. W. Gibbs himself. In this formulation, we are talking about an inexplicable (paradoxical) jump in the entropy of mixing of two ideal gases during the transition from mixing different to mixing identical gases. It is shown that the entropy of mixing of different ideal gases and the entropy of mixing of identical ideal gases are different (non-identical) functions of the same gas parameters. That, called a paradoxical jump in the entropy of mixing, is not a jump in the value of some function, but is the difference in the values of various functions, on condition that the variables and parameters on which these functions depend remain constant. Those who were looking for an explanation of the original Gibbs paradox did not notice this and tried to solve an unsolvable falsely posed problem: to find a parameter that change during the transition from different to identical gases caused the difference in the values of non-identical functions.