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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.

The integer molecular mass of the stated isotopologue isTake $p^{\circ }=1\mathrm{bar}=10^5\mathrm{Pa}$, the recommended standard pressure.^[ref]

At $298\mathrm{K}$, the thermal de Broglie wavelength is $\Lambda \simeq 9.3100\times 10^{-12}\mathrm{m}$. The result of part (a) givesIf $1\mathrm{atm}$ is used instead, the result decreases by $R\ln (1.01325)\simeq 0.1094\mathrm{J}{\mathrm{K}}^{-1}\mathrm{mo}{\mathrm{l}}^{-1}$, giving $168.23$ in the same units. Specifying the pressure removes that small convention dependence.