1. 15.4 Rotational modes of diatomic molecules
The moment of inertia, where μ is the reduced mass r0 is the equilibrium value of the distance between the nuclei.

2. From quantum mechanics, the allowed angular momentum states are where l = 0, 1, 2, 3… … From classical mechanics, the rotational energy equals with w is angular velocity. The angular momentum therefore, the energy

3. Define a characteristic temperature for rotation θrot can be found from infrared spectroscopy experiments, in which the energies required to excite the molecules to higher rotational states are measured. Different from vibrational motion, the energy levels of the above equation are degenerate. for level

4. Now, one can get the partition function For , virtually all the molecules are in the few lowest rotational states. As a result, the series of can be truncated with negligible errors after the first two or three terms!

5. For all diatomic gases, except hydrogen, the rotational characteristics temperature is of the order of 10 k (Kelvin degree). At ordinary temperature, Therefore, many closely spaced energy states are excited. The sum of may be replaced by an integral. Define:

6. Note that the above result is too large for homonuclear molecules such as H2, O2 and N2 by a factor of 2… why? The slight modification has no effect on the thermodynamics properties of the system such as the internal energy and the heat capacity!

7. Using (Note: ) again At low temperature Keeping the first two terms

8. Using the relationship (for ) And

9. Characteristic Temperatures
Characteristic temperature of vibration of diatomic molecules
Substance θvib(K) H O N
HCl CO NO Cl
Characteristic temperature of rotation of diatomic molecules
Substance θrot(K) H O N
HCl CO NO Cl

10. 15.5 Electronic Excitation
The electronic partition function is where g0 and g1, are, respectively, the degeneracies of the ground state and the first excited state. E1 is the energy separation of the two lowest states. Introducing

11. For most gases, the higher electronic states are not excited (θe ~ 120, 000k for hydrogen). therefore, At practical temperature, electronic excitation makes no contribution to the external energy or heat capacity!

12. 15.6 The total heat capacity
For a diatomic molecule system
Since
Discussing the relationship of T and Cv (p )

13. Heat capacity for diatomic molecules