Gibbs paradox

The apparent discontinuity in entropy of mixing as two gases become identical. It is resolved by treating permutations of indistinguishable particles as physically equivalent.

Entropy of mixing: The entropy change associated with mixing substances that were initially separated. Its classical value remains finite for identical gases, creating the paradox.
Statistical mechanics: The framework connecting microscopic states and dynamics to macroscopic thermodynamic properties. The paradox arises when microscopic arrangements are translated into thermodynamic entropy.
Josiah Willard Gibbs: An American physicist and mathematician who developed foundational ideas in thermodynamics and statistical mechanics. Gibbs formulated the particle-counting problem that bears his name.
Mixing entropy of identical gases: The entropy change when portions of the same gas are allowed to occupy a shared volume. It is zero in the idealized reversible process, unlike mixing distinguishable gases.
Sackur–Tetrode equation: The quantum-corrected entropy formula for a monatomic ideal gas. Its particle-number factorial correction yields extensive entropy and avoids the paradox.
Boltzmann entropy: A statistical-mechanical entropy defined from the number of microscopic states compatible with a macroscopic state. Counting particle arrangements without correcting for indistinguishability causes the excess entropy.
Microstate: A complete microscopic specification of a system’s state. The paradox turns on whether swapping identical particles creates a new microstate.
Elementary Principles in Statistical Mechanics: Gibbs’s 1902 book presenting a systematic framework for statistical mechanics. It develops the ensemble methods and counting conventions central to the paradox.
Mixing entropy of different gases: The positive entropy change when distinct ideal gases mix at fixed temperature and pressure. Its composition-dependent value contrasts with the zero change for identical gases.
Ideal gas: A model gas whose particles have negligible volume and interact only through elastic collisions. The paradox is most clearly derived using ideal gases.

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