4. The translational, rotational and vibrational canonical partition functions of a diatomic molecule are given, respectively, by:
$z_{tras} = V(\frac{2\pi mkT}{h^2})^{3/2}$,
$z_{rot} = \sum_{J=0}^{\infty} (2J+1)e^{-\frac{h^2 J(J+1)}{8\pi^2 IkT}}$,
$z_{vib} = \frac{e^{-\frac{hv}{2kT}}}{1 - e^{-\frac{hv}{kT}}}$,
for which J is the angular momentum and v is the frequency of vibration.
(a) Obtain the canonical partition function of a molecule $Z_{sp}$ of an ideal diatomic gas for a temperature T, where $kT >>> hv$
(b) Show that for an adiabatic expansion at temperatures $kT >> hv$, the ideal gas equation of state remains valid.
(c) and for the same case, that the isentropic line $pV^\lambda$ is constant and determine the value of the constant $\lambda$.
