When does mixing produce entropy? The quantum resolution of an extensivity paradox.
The Gibbs Paradox: If two volumes of gas at the same T, P are separated and then mixed, the entropy of mixing is
ΔS = −nR Σ xᵢ ln xᵢ > 0 (distinguishable gases)
This is positive — mixing is irreversible. But what if the gases are
identical? Classically, ΔS should still be positive (the formula doesn't know about identity). Yet physically, removing and replacing the partition changes nothing — ΔS must be 0.
Paradox!
Resolution: Quantum mechanics: identical particles are truly indistinguishable. Boltzmann's 1/N! correction (Gibbs correction) removes the overcounting of microstates.
S_Sackur-Tetrode includes 1/N! → ΔS_mix = 0 for identical gases
This makes entropy
extensive. The paradox reveals that extensivity is fundamentally quantum mechanical.
Maximum mixing entropy occurs at equal proportions (x₁ = x₂ = ... = 1/n): ΔS_max = R ln n per mole.