the Sackur-Tetrode equation
/ SAHK-oor TE-troh-duh /
This is the absolute entropy of a monatomic ideal gas, written out explicitly, a triumph of early quantum statistical mechanics. It answers a bold question: exactly how much entropy does, say, a mole of argon possess in absolute terms, not merely how much it changes? Remarkably, getting a sensible, extensive answer requires both Planck's constant and the indistinguishability of the atoms.
Precisely, S = N k_B [ ln( V / (N lambda^3) ) + 5/2 ], where lambda = h / sqrt(2 pi m k_B T) is the thermal de Broglie wavelength. Equivalently S = N k_B [ ln( (V/N) (4 pi m E / 3 N h^2)^(3/2) ) + 5/2 ]. The V/N rather than V, and the additive 5/2, both come from Stirling-approximating the 1/N! that indistinguishability demands; the h enters because phase space must be diced into cells of size h^3 per degree of freedom to count states at all.
It matches the measured entropies of the noble gases beautifully and resolves the Gibbs paradox by making the entropy extensive, so that S(2V, 2N, 2E) = 2 S(V, N, E). Honestly, it is valid only in the classical, non-degenerate regime N lambda^3 / V << 1. At low temperature it wrongly predicts S going to minus infinity, violating the third law; the true fix is the full quantum, Fermi or Bose, treatment, which the classical formula simply cannot capture.
For one mole of argon at 298 K and 1 atm, the Sackur-Tetrode formula gives S about 155 J/(mol K), matching calorimetric measurements to within experimental error.
A classical formula that nonetheless needs h and 1/N! to get the number right.
The equation fails at low temperature, where it gives negative entropy, a signal that quantum statistics (Fermi-Dirac or Bose-Einstein) take over and the classical ideal-gas picture breaks down.