Rotational partition functions

There is no rotational term to consider for monoatomic molecules since the rotation about an axis passing through the nucleus, where practically the whole mass is concentrated, does not involve any etrergy. We will now consider three cases of polyatomic molecirles. 

These are diatomic molecules with two different atoms HCl. 

If the rotation about an axis passing through two nuclei does not involve any energy, there is therefore room to consider that the rotations occur about an axis perpendicular to this line. 

According to Schrodinger s equation, the energy of the rigid rotor of the moment of inertia 1 (see Appendix A. 2) is given by  

We call the characteristic rotational temperature, the quantity r defined 

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  

An additional constant factor enters into these expressions if the nuclei have nonzero spins. If i and are the two nuclear spins of a diatomic molecule, the complete rotational function is written as 

With the exception of the constant terms for nuclear spin and /i, this is again identical with the classical result. 

For complex, nonlinear molecules the rotational partition function may be written as 

The quantities /. cule about three principal axes through its center of gravity.  

We will assume the rigid-rotator approximation to obtain rr01. For the linear molecule we get 

However, for a symmetrical (AA) molecule, we find that only half the energy levels are occupied. Thus, in molecules such as H2 or N2, 7 = 0,2, 4, 6,. .. or 7= l, 3, 5, 7,. .. are allowed. To account for this difference, we correct gj by dividing by the symmetry number a so that 

The symmetry number is the number of indistinguishable rotated positions. For a homonuclear diatomic such as H2, a = 2 since H—H and H —H are indistinguishable. For a heteronuclear diatomic, such as CO, a — 1 since C—O and O—C are distinguishable. As another example, a = 2 for O—C—O and cr=l for O—N—N, both of which are linear molecules.s 

Substituting equations (10.92) and (10.93) into the equation for the partition function gives 

Generally, the energy levels are close together, and we can replace the sum by an integral unless the temperature is low.1 The result is 

---

The factor of 2 in the denominator of the H2 molecule s rotational partition function is the "symmetry number" that must be inserted because of the identity of the two H nuclei. 

Constant in rotational partition function of gases Constant relating wave number and moment of inertia Z = constant relating wave number and energy per mole... 

Here /, are the three moments of inertia. The symmetry index a is the order of the rotational subgroup in the molecular point group (i.e. the number of proper symmetry operations), for H2O it is 2, for NH3 it is 3, for benzene it is 12 etc. The rotational partition function requires only information about the atomic masses and positions (eq. (12.14)), i.e. the molecular geometry. 

But molecular gases also have rotation and vibration. We only make the correction for indistinguishability once. Thus, we do not divide by IV l to write the relationship between Zro[, the rotational partition function of N molecules, and rrol, the rotational partition function for an individual molecule, if we have already assigned the /N term to the translation. The same is true for the relationship between Zv,h and In general, we write for the total partition function Z for N units... 

Rotational Partition Function Corrections The rigid-rotator partition function given in equation (10.94) can be written as... 

MMl represents the mass and moment-of-inertia term that arises from the translational and rotational partition functions EXG, which may be approximated to unity at low temperatures, arises from excitation of vibrations, and finally ZPE is the vibrational zero-point-energy term. The relation between these terms and the isotopic enthalpy and entropy differences may be written... 

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

To calculate the rotational partition function for the molecule we need to be careful and check whether the assumption under which Eq. (56) has been derived is valid. 

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. 

Nonlinear polyatomic molecules require further consideration, depending on their classification, as given in Section 9.2.2. In the classical, high-temperature limit, the rotational partition function for a nonlinear molecule is given by... 

The degeneracies of the rotational energy levels are gj = 2J + 1. In terms of these quantities the rotational partition function becomes... 

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. 

Isotope (H (deuterium), discovered by Urey et al. (1932), is usually denoted by symbol D. The large relative mass difference between H and D induces significant fractionation ascribable to equilibrium, kinetic, and diffusional effects. The main difference in the calculation of equilibrium isotopic fractionation effects in hydrogen molecules with respect to oxygen arises from the fact that the rotational partition function of hydrogen is nonclassical. Rotational contributions to the isotopic fractionation do not cancel out at high T, as in the classical approximation, and must be accounted for in the estimates of the partition function ratio /. 

In a different approach, Bottinga (1969a) evaluated the nonclassical rotational partition function for water vapor, molecular hydrogen, and methane through the asymptotic expansion of Strip and Kirkwood (1951) ... 

---