15.4 Rotational modes of diatomic molecules The moment of inertia, where μ is the reduced mass r 0 is the equilibrium value of the distance between the nuclei.

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

Define a characteristic temperature for rotation θ rot 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

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!

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

Note that the above result is too large for homonuclear molecules such as H 2, O 2 and N 2 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!

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

Using the relationship (for ) And (for )

Characteristic Temperatures Characteristic temperature of rotation of diatomic molecules Substance θ rot (K) H O N HCl 15.2 CO 2.8 NO 2.4 Cl Characteristic temperature of vibration of diatomic molecules Substance θ vib (K) H O N HCl 4150 CO 3080 NO 2690 Cl 2 810

15.5 Electronic Excitation The electronic partition function is where g 0 and g 1, are, respectively, the degeneracies of the ground state and the first excited state. E 1 is the energy separation of the two lowest states. Introducing

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

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

Heat capacity for diatomic molecules

Example I (problem 15.7) Consider a diatomic gas near room temperature. Show that the entropy is Solution: For diatomic molecules

For translational motion, the molecules are treated as non-distinguishable assemblies

For rotational motion (they are distinguishable in terms of kinetic energy)

Example II (problem 15-8) For a kilomole of nitrogen (N 2 ) at standard temperature and pressure, compute (a) the internal energy U; (b) the Helmholtz function F; and (c) the entropy S. Solution: calculate the characteristic temperature first!