Harmonic oscillator, diatomic gases

Under most circumstances the equations given in Table 10.4 accurately calculate the thermodynamic properties of the ideal gas. The most serious approximations involve the replacement of the summation with an integral [equations (10.94) and (10.95)] in calculating the partition function for the rigid rotator, and the approximation that the rotational and vibrational partition functions for a gas can be represented by those for a rigid rotator and harmonic oscillator. In general, the errors introduced by these approximations are most serious for the diatomic molecule." Fortunately, it is for the diatomic molecule that corrections are most easily calculated. It is also for these molecules that spectroscopic information is often available to make the corrections for anharmonicity and nonrigid rotator effects. We will summarize the relationships... 

Consider, for example, a single vibrational relaxation process, as in a gas of diatomic molecules exhibiting harmonic oscillations. Then for acoustic frequencies sound velocity becomes frequency dependent. To find F2(a>), the specific heat may be written in the complex form... 

Quantitative calculations of the rate constants ky y for vibrational transitions are relatively easy if the diatomic molecules are simulated by a harmonic oscillator interacting isotropically with the impinging atom (the so-called breathing sphere or the SSH (Schwartz-Slawsky-Herzfeld) model [3, 192, 339, 395] based on the one-dimensional Landau-Teller model [261]). Then the mean transition probability Py y per one gas-kinetic collision calculated to the first order of the semiclassical perturbation treatment is... 

The molecular partition function for a diatomic gas can also be corrected by going beyond the harmonic oscillator-rigid rotor approximation, including the correction terms in Eq. (22.2-45) for the energy levels. We do not discuss these corrections. 

Make a plot of Cy as a function of temperature for an ideal gas composed of diatomic molecules vibrating as a harmonic oscillator according to the energy level expression of E (kj mohi) = 23.0 (n + 1/2). 

A gas phase species with N atoms has 3N degrees of freedom associated with it. For a nonlinear molecule, three of these are translations and three are rotations (two, if linear). The remaining 3N - 6 (3N - 5, if linear) movements are the internal or normal modes of vibration for the species. The single vibration of a diatomic species can be modeled quantum mechanically in terms of a one-dimensional harmonic oscillator. Figure 3.4.1.1, giving the energy levels... 

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