Gibbs paradox and thermodynamics of colloids

Abstract

Using the “correct Boltzmann counting”, the statistical-mechanical entropy is given by $S^{\text{stat}}=k_B\text{ln}{\Omega }_B-k_B\text{ln}N!$. The factor $-k_B\text{ln}N!$ has been recognized as essential for understanding the collective behavior of colloids, which are classically distinguishable particles. To account for the thermodynamics of colloids, we introduce a method to define thermodynamic entropy using the statistical-mechanical entropy $S^{\text{stat}}$. Our definition is formulated as a partial differential equation that links internal energy, thermodynamic entropy, and particle number. This equation inherently defines the dependence of thermodynamic entropy on particle number, thereby resolving the Gibbs paradox within classical thermodynamics. For extensive systems, this equation yields the ideal gas law, while for non-extensive systems, it may provide new physical insights. In contrast, when the Boltzmann entropy $S^{\text{Boltz}}=k_B\text{ln}{\Omega }_B$ is used to define thermodynamic entropy, extensiveness and the ideal gas law become incompatible.