Consider a classical gas of diatomic molecules whose orientation is fixed by a strong magnetic field. The molecules are not free to rotate, but they are free to vibrate. Assuming that the vibrations are approximately harmonic, calculate the contribution to the partition function due to vibrations.

Evaluate the free energy $F=-k T \ln Z$, where $Z$ is the total partition function for the gas, and hence calculate the entropy.

$\left[\right.$ Note that $\int_{-\infty}^{\infty} \exp \left(-a u^{2}\right) d u=\sqrt{\pi / a}$ and $\int_{0}^{\infty} u^{2} \exp \left(-a u^{2}\right) d u=\sqrt{\pi} / 4 a^{3 / 2} . \quad$ You may approximate $\ln N$ ! by $N \ln N-N$.]