We revisit Gibbs' famous paradox: identical gases appear to gain entropy when mixed in classical counting, yet no macroscopic change should occur. We'll trace the flaw to assuming distinguishable particles, show how dividing by N! fixes the counting, and connect this to the Sackur–Tetrode equation, entropy extensivity, and the quantum twist of indistinguishability. We'll also discuss the mixing paradox, Jaynes's perspective on entropy as a measure of distinguishability, and what this tells us about the very meaning of a thermodynamic state.

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