Gibbs' paradox highlights an apparent contradiction in statistical mechanics regarding the entropy of mixing identical particles or gases. It arises when considering the entropy change associated with mixing two gases or ensembles of particles that are indistinguishable. In classical thermodynamics, when two different gases are mixed, the entropy of the system increases due to the increased number of available microstates.
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Counting permutations of identical classical particles as different states produces a spurious entropy of mixing when otherwise identical gases are combined. The dilute ideal gas partition function removes that label multiplicity and gives an extensive entropy in the thermodynamic limit. For different species, genuinely distinct composition information still produces a entropy of mixing. quantum statistics supplies the underlying identical-particle counting.