Calculate the molar heat capacity at constant volume

Question

Response to the following problem:

Given a gaseous system of NA indistinguishable, weakly interacting diatomic molecules:

(a) Each molecule may vibrate with the same frequency v, but with an energy ?_i, given by

?_i = (1/2 + i)hv      (i=0,1,2,...).

Show that the vibrational partition function Z_v is

Z_v = e^-hv/2kT/1 - e^-hv/kT

(b) Each molecule may rotate, and the rotational partition function Z, has the same form as that for translation, except that the volume V is replaced by the total solid angle 47π,the mass is replaced by the moment of inertia I, and the exponent 3/2 (referring to three translational degrees of freedom) is replaced by 2/2, since there are only two rotational degrees of freedom. Write the rotational partition function.

(c) Taking into account translation, vibration, and rotation, calculate the Helmholtz function.

(d) Calculate the pressure.

(e) Calculate the internal energy.

(f) Calculate the molar heat capacity at constant volume.