Determine the standard molar entropy of $\mathrm{Ne}$ and $\mathrm{Kr}$ under standard thermodynamic conditions.
Beginning with the expression for entropy [see Equation (15.63)]:
$$
\begin{aligned}
S & =\frac{5}{2} R+R \ln \left(\frac{V}{\Lambda^3}\right)-R \ln N_A \\
& =\frac{5}{2} R+R \ln \left(\frac{V}{\Lambda^3}\right)-54.75 R \\
& =R \ln \left(\frac{V}{\Lambda^3}\right)-52.25 R
\end{aligned}
$$
The conventional standard state is defined by $T=298 \mathrm{~K}$ and $V_m=24.4 \mathrm{~L}\left(0.0244 \mathrm{~m}^3\right)$. The thermal wavelength for $\mathrm{Ne}$ is
$$
\begin{aligned}
\Lambda & =\left(\frac{h^2}{2 \pi m k T}\right)^{1 / 2} \\
& =\left(\frac{\left(6.626 \times 10^{-34} \mathrm{~J} \mathrm{~s}\right)^2}{2 \pi\left(\frac{0.02018 \mathrm{~kg} \mathrm{~mol}^{-1}}{N_A}\right)\left(1.38 \times 10^{-23} \mathrm{~J} \mathrm{~K}^{-1}\right)(298 \mathrm{~K})}\right)^{1 / 2} \\
& =2.25 \times 10^{-11} \mathrm{~m}
\end{aligned}
$$
Using this value for the thermal wavelength, the entropy becomes
$$
\begin{aligned}
S & =R \ln \left(\frac{0.0244 \mathrm{~m}^3}{\left(2.25 \times 10^{-11} \mathrm{~m}\right)^3}\right)-52.25 R \\
& =69.84 R-52.25 R=17.59 R=146 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}
\end{aligned}
$$
The experimental value is $146.48 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ ! Rather than determining the entropy of $\mathrm{Kr}$ directly, it is easier to determine the difference in entropy relative to Ne:
$$
\begin{aligned}
\Delta S & =S_{K r}-S_{N e}=S=R \ln \left(\frac{V}{\Lambda_{K r}^3}\right)-R \ln \left(\frac{V}{\Lambda_{N e}^3}\right) \\
& =R \ln \left(\frac{\Lambda_{N e}}{\Lambda_{K r}}\right)^3 \\
& =3 R \ln \left(\frac{\Lambda_{N e}}{\Lambda_{K r}}\right) \\
& =3 R \ln \left(\frac{m_{K r}}{m_{N e}}\right)^{1 / 2} \\
& =\frac{3}{2} R \ln \left(\frac{m_{K r}}{m_{N e}}\right)=\frac{3}{2} R \ln (4.15) \\
& =17.7 \mathrm{Jmol} ^{-1}\mathrm{K}^{-1}
\end{aligned}
$$
Using this difference, the standard molar entropy of $\mathrm{Kr}$ becomes
$$
S_{K r}=\Delta S+S_{N e}=164 \mathrm{Jmol}^{-1} \mathrm{~K}^{-1}
$$
The experimental value is $163.89 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$, and the calculated value is again in excellent agreement!