Here it is, your final!
Due: May 18, 2001-ish.

#1) Show that the fluctuation in molecular energy, e is a lot larger than the fluctuation in ensemble average energy E by deriving the following relation. You can start your derivation from Eqns. (8) and (9) of Lecture 5.

#2) It can be shown that the partition function of an ideal gas of diatomic molecules, using canonical ensemble, in an external electric field E is

where

Here we have m is molecular dipole moment. It can be shown by using the partition function

where

and where m with bar is the average dipole moment of a molecule in the direction of the external field E.

Show that

Sketch this result versus E from 0 to infinity, and interpret the result.

#3) The model we used in the diatomic molecules for rotation and vibration was the rigid-rotor harmonic oscillator moldel. Sometimes, such approximation doesn't give a good result. The harmonic oscillator approximation can be improved by perturbation theory, and the vibrational energy is given by

where x_e is the anharmonicity constant. The rigid rotor approximation can be improved also by using perturbation theory, and the rotational energy is given by

where D is the centrifugal distortion constant. There is also a cross term, which gives rise to coupling between rotation and vibration; as you can imagine, the moments of inertia changes during a period of vibration. The coupling term is given by

where a is the rotation-vibration coupling constant.

Show that the molecular partition function is separable and derive the following.

where the last term in q_corr is a higher order terms. Calculate the effect of q_corr on E and C_v for O₂ at 300K, given the following values of the spectroscopic parameters: x_e = 0.0076, D = 4.8 x 10^-6cm^-1, and a = 0.016cm^-1.

#4) Calculate S^o₂₉₈ for nitromethane given the following information.