Rotation, partition function for

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... 

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... 

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... 

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) ... 

To calculate the molecular rotational partition function for an asymmetric, linear molecule, we use Eq. 8.16 for the energy level of rotational state /, and Eq. 8.18 for its degeneracy. As discussed in Section 8.2, rotational energy levels are very closely spaced compared to k/jT unless the molecule s moment of inertia is very small. Therefore, for most molecules, replacing the summation in Eq. 8.50 with an integral introduces little error. Thus the... 

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... 

Although we will not provide a derivation, the molecular rotational partition function for a general polyatomic molecule is also simple in form,... 

The activated complex partition function has contributions from translation (with total mass w,4 +mj) and from rotation of the (linear) activated molecule. Assuming that the bond length of C is the sum of the atomic radii r a and rg, the rotational partition function for the activated complex can be calculated from Eq. 8.65, the moment of inertia / = m 2(rA + re)2, where m 2 is the A-B reduced mass (Eq. 10.38). 

All terms have been defined previously except for Z r, which here stands for the vibration-rotation partition function for active degrees of freedom of the molecule. 

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 rotational partition function for an N2 molecule bound to a tungsten surface is more in doubt, but it would probably have very little temperature dependence if the molecule could turn or vibrate on the surface only with difficulty. 

TlntermedtaleTiomJalizetfwave lncfl6n7 t2 Internal energy, 298, 374 Internal rotation, partition function for, 306 Intersecting potential energy surfaces model, 48... 

The classical approximation for the rotational density of states of a molecule is familiar from elementary statistical mechanics, where it is common to assume that the rotational states form a continuum in calculating the rotational partition function. For the external rotations of most molecules this approximation is very good. For example the classical approximation for the rotational partition function of an asymmetric top is... 

Rotational partition function for I2 substituting into the equation of Section 9.8 we obtain... 

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 ... 

Vibration-rotation partition function for HC1 obtained via standard Rayleigh-Ritz variational (var) basis-set methods from Topper et al. [46]. 

Vibration-rotation partition function for HC1 obtained via Fourier path-integral AOSS-U Monte Carlo calculations from Topper et al. [46]. Error bars are given at 95% confidence level (2w ). Unless otherwise noted, all calculations used = 128 Fourier coefficients per degree of freedom and n = 100000 Monte Carlo samples. 

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. 

D17.1 An approximation involved in the derivation of all of these expressions is the assumption that the contributions from the different modes of motion are separable. The expression = kT/hcB is the high temperature approximation to the rotational partition function for nonsymmetrical linear rotors. The expression q = kT/hcv is the high temperature form of the partition function for one vibrational mode of the molecule in the haimonic approximation. The expression (f- =g for the electronic partition function applies at normal temperatures to atoms and molecules with no low lying excited electronic energy levels. 

A. Rotational Partition Function for a Rigid Asymmetric Molecule 441... 

A. ROTATIONAL PARTITION FUNCTION FOR A RIGID ASYMMETRIC MOLECULE... 

---