the Gibbs paradox
A puzzle that quietly exposed the need for indistinguishability. Remove the partition between two halves of a box, both holding the same gas at the same temperature and pressure. Physically nothing happens, so the entropy should not change. Yet a naive calculation says mixing always raises entropy. Something is wrong with the counting.
Precisely, if you compute the entropy of an ideal gas treating the atoms as distinguishable, you get a non-extensive result, and removing the partition between two identical samples predicts a spurious entropy of mixing, Delta S = 2 N k_B ln 2 > 0, even though the final state is microscopically indistinguishable from the initial one. The resolution is that identical particles are genuinely indistinguishable, so you must divide the number of microstates by N!, the number of permutations. This 1/N! makes the entropy extensive and gives exactly zero mixing entropy for identical gases, yielding the Sackur-Tetrode result.
Historically it was a decisive hint that classical statistics was incomplete and that quantum indistinguishability is real, not a bookkeeping trick. Honestly, for genuinely different gases the mixing entropy 2 N k_B ln 2 is real and correct; the paradox concerns only identical gases. And the sharp, all-or-nothing distinction between identical and different is ultimately explained by quantum mechanics, not by classical physics.
Mixing two different ideal gases, say helium and argon, at equal T and P does raise the entropy by Delta S = 2 N k_B ln 2; mixing a gas with an identical sample of itself raises it by zero, and the 1/N! factor draws exactly this line.
The mixing entropy is real for distinct gases, and correctly zero for identical ones.
The mixing entropy is real and nonzero for distinct gases; the paradox is only that the naive formula wrongly predicts it for identical gases too, and indistinguishability (the 1/N!) is the fix.