Vibrational motion polyatomic gases

The harmonic approximation reduces to assuming the PES to be a hyperparaboloid in the vicinity of each of the local minima of the molecular potential energy. Under this assumption the thermodynamical quantities (and some other properties) can be obtained in the close form. Indeed, for the ideal gas of polyatomic molecules the partition function Q is a product of the partition functions corresponding to the translational, rotational, and vibrational motions of the nuclei and to that describing electronic degrees of freedom of an individual molecule ... 

For a diatomic or polyatomic ideal gas, Cy is greater than j R, because energy can be stored in rotational and vibrational motions of the molecules a greater amount of heat must be transferred to achieve a given temperature change. Even so, it is still true that... 

As a polyatomic gas is heated, the gas molecules absorb energy to increase their rotational and vibrational motions as well as to move through space (translate) at higher speeds. Recall from our previous discussions that... 

Since rotational motion can occur only about the common center of mass of a polyatomic system, its absence is an indication that the molecules in the gas are not polyatomic. Isolated atoms are perfectly spherical, so any rotational motion they might exhibit is undetectable. Similarly, vibrational motion can exist only between any two or more atoms in a polyatomic molecule, so its absence would also accord with monoatomicity. 

We will apply this expression only to a monatomic gas. For a diatomic or polyatomic gas the thermal conductivity is more complicated, since it is not an adequate approximation to assume that rotational and vibrational motions are equilibrated at the upper and lower planes. 

In the case of polyatomic molecules the radial distribution curve as deduced from electron-diffraction gas experiments may also be considered as a kind of a weighted sum of p ip curves for the internal motion in the molecule, but here all intemuclear distances are inseparably mixed together in a one-dimensional representation. For a rigid molecule, such as carbon tetrachloride or benzene, electron diffraction may produce quite accurate information as to the geometry of the molecule. As to the internal motion of the molecule, vibrational amplitudes may be deduced and compared with the corresponding data, differently but usually considerably more accurately, obtained by spectroscopic methods. How this is actually done in practice is perhaps most elegantly described by S. Cyvin4). 

Clearly U denotes the mean kinetic energy per mole, and in monatomic gases is identical with the total energy of the molecules in a mole. In polyatomic molecules the relations are more complicated, on account of the occurrence of rotations of the molecules, and vibrations of the atoms within a molecule. It can be shown, however, that the preceding formula for the gas pressure holds in this case also U denotes as before the mean kinetic energy of the translational motion of the molecules, per mole, but it is no longer the same as the total energy. 

Polyatomic molecules in the gas phase undergo three types of motion translation, vibration, and rotation. The latter two are quantized, giving rise to spectra those corresponding to vibration are usually observed in the infrared, whereas those for rotation are normally observed in the microwave region. Electronic spectra, when carefully analyzed, also contain information about both the vibration and rotation of small molecules. 

Many of the ideas that are essential to understanding polyatomic electronic spectra have already been developed in the three preceding chapters. As in diatomics, the Born-Oppenheimer separation between electronic and nuclear motions is a useful organizing principle for treating electronic transitions in polyatomics. Vibrational band intensities in polyatomic electronic spectra are frequently (but not always) governed by Franck-Condon factors in the vibrational modes. The rotational fine structure in gas-phase electronic transitions parallels that in polyatomic vibration-rotation spectra (Section 6.6), except that the rotational selection rules in symmetric and asymmetric tops now depend on the relative orientations of the electronic transition moment and the principal axes. Analyses of rotational contours in polyatomic band spectra thus provide valuable clues about the symmetry and assignment of the electronic states involved. 

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