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Assume a dilute classical ideal gas of $N$ indistinguishable, noninteracting particles, with translational states dense enough to replace their sum by a continuum integral. Separate translation from all internal degrees of freedom. ThenUsing Stirling's approximation in the Helmholtz free energy,Differentiate at fixed $N,V$. Since ${\Lambda }^{-3}\propto T^{3/2}$,Only after this derivative set $N=N_A$ and $V=RT/p^{\circ }$ for one mole at standard pressure:This is the translational Sackur-Tetrode equation. The $N!$ factor is essential for the correct extensive entropy; replacing $V$ by a temperature-dependent standard molar volume before taking the fixed-volume derivative would give the wrong constant term.

For one molecule whose translational motion separates from its internal degrees of freedom, the classical one-particle partition function iswhere $q_{\mathrm{int}}$ contains rotational, vibrational, and other internal states. For $N$ indistinguishable noninteracting molecules, the classical ideal-gas partition function is $Z_N=z_1^N/N!$. The thermodynamic pressure therefore obeyssoThe internal structure changes the energy, entropy, and heat capacity through $q_{\mathrm{int}}(T)$, but it does not change the ideal gas law while the internal partition function is independent of volume and the molecules have negligible interactions.

For a dilute monatomic gas $q_{\mathrm{int}}=1$. Using the Stirling formula in $Z_N=(V/{\lambda }^3{)}^N/N!$ and differentiating the Helmholtz free energy $F=-k_{B}T\log Z_N$ gives the Sackur-Tetrode equationWith $N$ fixed, an isentropic process therefore satisfies $V{T}^{3/2}=\text{constant}$, or $T{V}^{2/3}=\text{constant}$. Eliminating $T$ with the ideal-gas law yields the monatomic reversible ideal-gas adiabat

Particles within each species are indistinguishable, while the two species are physically distinguishable and their numbers are fixed. Independence makes the canonical partition function factor by species:Since $Z_1\propto T^{3/2}$ at fixed volume, $U=-{\partial }_{\beta }\ln Z=3N{k}_{B}T/2$. The canonical entropy is thereforeFor large species populations, Stirling's approximation gives the equivalent Sackur-Tetrode equation formFor structureless particles with the same translational partition function, $Z_1=V/{\lambda }^3$, where $\lambda$ is their common thermal de Broglie wavelength. More generally the supplied common constant factor in $Z_1$ should be retained in the entropy.

Writing $x_i=N_i/N$ separates the total-number contribution from the composition term:The last term is the entropy of mixing relative to separated ideal components at the same temperature and pressure. This is the entropy of a two-species classical ideal gas. Artificially labeling otherwise identical chemical atoms would not justify dividing by separate factorials or adding a mixing entropy. The pure-species limit is obtained by $x\ln x\to 0$.