Partition function diatomic molecule

The classical value is attained by most molecules at temperatures above 300 K for die translation and rotation components, but for some molecules, those which have high heats of formation from die constituent atoms such as H2, die classical value for die vibrational component is only reached above room temperature. Consideration of the vibrational partition function for a diatomic gas leads to the relation... 

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... 

Note that a diatomic molecule in the gas phase has only one vibration, but as soon as it adsorbs on the surface it acquires several more modes, some of which may have quite low frequencies. The total partition function of vibration then becomes the product of the individual partition functions ... 

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

Partition functions of a diatomic molecule (per degree of freedom)... 

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. 

Berlin showed that a diatomic molecule can be partitioned into a binding region and an antibinding region, as shown in Figure 6.1. These regions are separated by two surfaces of revolution given by the function B ... 

From the quantum mechanical standpoint the appearance of the factor 1/2 = 1/s for the diatomic case means the configurations generated by a rotation of 180° are identical, so the number of distinguishable states is only one-half the classical total. Thus the classical value of the partition function must be divided by the symmetry number which is 1 for a heteronuclear diatomic and 2 for a homonuclear diatomic molecule. 

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. 

For practical purposes the rules for diatomic molecules concerning even and odd J reduce to the statement that for homonuclear diatomic molecules the molecular partition function must be divided by two (s = 2), while for heteronuclear diatomic molecules no division is necessary (s = 1). The idea of the symmetry number, s,... 

This study is one of the earliest attempts to calculate equilibrium fractionation factors using measured vibrational spectra and simple reduced-mass calculations for diatomic molecules. For the sake of consistency I have converted reported single-molecule partition function ratios to units. 

The adsorption of diatomic or dimeric molecules on a suitable cold crystalline surface can be quite realistically considered in terms of the dimer model in which dimers are represented by rigid rods which occupy the bonds (and associated terminal sites) of a plane lattice to the exclusion of other dimers. The partition function of a planar lattice of AT sites filled with jV dimers can be calculated exactly.7 Now if a single dimer is removed from the lattice, one is left with two monomers or holes which may separate. The equilibrium correlation between the two monomers, however, is appreciable. As in the case of Ising models, the correlation functions for particular directions of monomer-monomer separation can be expressed exactly in terms of a Toeplitz determinant.8 Although the structure of the basic generating functions is more complex than Eq. (12), the corresponding determinant for one direction has been reduced to an equally simple form.9 One discovers that the correlations decay asymptotically only as 1 /r1/2. 

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... 

Evaluation of the partition function in the general case of an asymmetric rotor molecule is rather tedious. However, for rigid diatomic or linear molecules characterized by the rotational constant B (in MHz), the partition function can be calculated by replacing the summation over the total quantum number J by an integral... 

Vibrational Partition Function/ The thermodynamic quantities for an ideal gas can usually be expressed as a sum of translational, rotational, and vibrational contributions (see Exp. 3). We shall consider here the heat capacity at constant volume. At room temperature and above, the translational and rotational contributions to are constants that are independent of temperature. For HCl and DCl (diatomic and thus linear molecules), the molar quantities are... 

For AT diatomic molecules that have only a single internal vibration the partition function is easily determined from the approximate formula for the energy levels... 

For a diatomic molecule the allowed rotational states will depend on whether or not the two nuclei are identical. At temperatures at which the energy difference of adjacent rotational states is small compared to kT we can write the approximate partition function for N molecules ... 

In order to use the procedure described above to calculate the partition function of a given molecule as a function of T and V, one must know the appropriate I and v values for the various degrees of freedom of the molecule. A great deal of this type of data is tabulated for diatomic molecules in [11] and for polyatomic molecules in [12]. Updated information may be found in [13], [14], and [15]. 

Tn calculations of the vibrational contribution to isotope effects on partition functions for diatomic molecules it is usual to employ the expression... 

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