Diatomic molecule rotational partition function

For a hetero-diatomic molecule, the partition function of rotational motion equals... 

Finally, the rotational partition function of a diatomic molecule follows from the quantum mechanical energy level scheme ... 

As the rotational partition function for a diatomic molecule such as CO is... 

The rotational microwave spectrum of a diatomic molecule has absorption lines (expressed as reciprocal wavenumbers cm ) at 20, 40, 60, 80 and 100 cm . Calculate the rotational partition function at 100 K from its fundamental definition, using kT/h= 69.5 cm" at 100 K. 

Proceeding in the spirit above it seems reasonable to inquire why s is equal to the number of equivalent rotations, rather than to the total number of symmetry operations for the molecule of interest. Rotational partition functions of the diatomic molecule were discussed immediately above. It was pointed out that symmetry requirements mandate that homonuclear diatomics occupy rotational states with either even or odd values of the rotational quantum number J depending on the nuclear spin quantum number I. Heteronuclear diatomics populate both even and odd J states. Similar behaviors are expected for polyatomic molecules but the analysis of polyatomic rotational wave functions is far more complex than it is for diatomics. Moreover the spacing between polyatomic rotational energy levels is small compared to kT and classical analysis is appropriate. These factors appreciated there is little motivation to study the quantum rules applying to individual rotational states of polyatomic molecules. 

We noted in Section 8.2 that only half the values of j are allowed for homonuclear diatomics or symmetric linear polyatomic molecules (only the even-y states or only the odd- y states, depending on the nuclear symmetries of the atoms). The evaluation of qmt would be the same as above, except that only half of the j s contribute. The result of the integration is exactly half the value in Eq. 8.64. Thus a general formula for the rotational partition function for a linear molecule is... 

Consider the molecular rotational partition function for the CO molecule, a linear diatomic molecule. The moment of inertia of CO is / = 1.4498 x 10-46 kg-m2, and its rotational symmetry number is a = 1. Thus, evaluating Eq. 8.65 at T = 300 K, we find the rotational partition function to be... 

For diatomic molecules, corrections can be made for the assumption used in the derivation of the rotational partition function that the rotational energy levels are so closely spaced that they can be considered to be continuous. The equations to be used in making these corrections are given in Appendix 6. Also given are the equations to use in correcting for vibrational anharmonicity and nonrigid rotator effects. These corrections are usually small.22... 

This is the correct expression for the rotational partition function of a heteronuclear diatomic molecule. For a homonuclear diatomic molecule, however, it must be taken into account that the total wave function must be either symmetric or antisymmetric under the interchange of the two identical nuclei symmetric if the nuclei have integral spins or antisymmetric if they have half-integral spins. The effect on Qrot is that it should be replaced by Qrot/u, where a is a symmetry number that represents the number of indistinguishable orientations that the molecule can have (i.e., the number of ways the molecule can be rotated into itself ). Thus, Qrot in Eq. (A.19) should be replaced by Qrot/u, where a = 1 for a heteronuclear diatomic molecule and a = 2... 

It is important to note in Figure 1 that both curves show a decrease with temperature, and it should be possible to fit B smoothly onto A by multiplying by a suitable scale factor, possibly as shown by the dashed line. To explain the data shown in Figure 1 the temperature dependence of fs/fg is needed. The rotational partition function for a diatomic molecule that is free is... 

The moment of inertia can be derived from spectroscopic data or it can be calculated from the dimensions of the molecule, so that the rotational partition function can be determined. The values of the moments of inertia of a number of diatomic molecules in their ground states are given in Table VIII. ... 

At all reasonable temperatures the rotational levels of a molecule containing more than two atoms, like those of diatomic molecules, are occupied sufficiently for the behavior to be virtually classical in character. Assuming that the molecule can be represented as a rigid rotator, the rotational partition function, excluding the nuclear spin factor, for a nonlinear molecule is given by... 

A linear molecule containing more than two atoms is analogous to a diatomic molecule. It has two identical moments of inertia, and the rotational partition function is given by the same equation (16.24) as for a diatomic molecule. 

Derive the value of the universal constant a in the expression Qr aa IT for the rotational partition function of any diatomic (or any linear) molecule I is the moment of inertia in e.g.s. units and T is the absolute temperature. Calculate the rotational partition function of carbon dioxide (a linear symmetrical molecule) at 25 C. 

Similarly, most terms cancel in the calculation of the ratio of rotational partition functions. For diatomic molecules and linear polyatomic molecules, this ratio is given by fjl ... 

For calculations of rotational partition functions the moment of inertia (e.g. the molecule structure) should be known. In case of a diatomic moleule the rotation partition function is... 

This is the value of the rotational partition function for unsymmetrical linear molecules (for example, heteronuclear diatomic molecules). Using this value of we can calculate the values of the thermodynamic functions attributable to rotation. 

The same interpretation results from the above approximate treatment of radical recombination reactions, using the "diatomic" model, where the collision diameter d is related to the high temperature approximation of expression OOliV) for the rotational partition function of product molecule AB, being in a state which should be considered a.non-stationary transition state. 

Since cr = 2 for a homonuclear diatomic molecule, the rotational partition function at room temperature is (see Equation (11.30))... 

The rotational partition function of an unsymmetrical diatomic molecule is derived from the energy levels of a linear rigid rotator. These are... 

For homonuclear diatomic molecules, then, the nuclear and rotational partition functions must be considered together. The only real difference is that the nuclear partition function introduces an additional degeneracy to the overall partition... 

In most cases, for polyatomic molecules the nuclear partition function is again neglected, because it usually has a very small effect on the overall thermodynamic properties of polyatomic molecules. (Indeed, the only reason why we had to consider it for diatomic molecules is because it imposes an obvious, measurable effect on various observations, like spectra and thermodynamic properties to be considered in section 18.8.) In the high-temperature limit, a linear polyatomic molecule has the same rotational partition function as a homonuclear diatomic molecule ... 

Rgure 25.4 A Graphical Representation of the Rotational Partition Function (Drawn for CO at 298.15 K). This figure is analogous to Figure 25.2 for the translational partition function. The area under the bar graph is equal to the rotational partition function for a diatomic molecule, and the area under the curve is equal to the Integral approximation to the rotational partition function. 

For a linear polyatomic molecule like acetylene or cyanogen, the rotational energy levels are the same as those of diatomic molecules in Eq. (22.2-18). Equation (25.4-13) can be used for the rotational partition function with the appropriate symmetry number and moment of inertia. The rotational energy levels of nonlinear polyatomic molecules are more complicated than those of diatomic molecules. The derivation of the rotational partition function for nonlinear molecules is complicated, and we merely cite the result ... 

The quantum rotational partition function for a diatomic or linear polyatomic molecule is equal to the classical version divided by and also divided by the symmetry number a, equal to 1 for a heteronuclear diatomic molecule and equal to 2 for a homonuclear diatomic molecule. 

We next consider the rotational partition functions for diatomic molecules. According to section 14c, there is no restriction on the allowed values of the rotational quantum number J if the nuclei are different. If the two nuclei have spins si and S2 the nuclear spin statistical weifeht is (2si + l)(2s2 + 1). The rotational energy levels are... 

In any homonuclear diatomic molecule, with nuclear spin s, the statistical weight of the ortho states is s + 1) 2s + 1) that of the para states is s(2s -b 1). The rotational partition function will thus in general be... 

For all diatomic molecules, we may therefore write the rotational partition function as... 

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