The rotational partition function

Thus as we would expect the gas follows the perfect-gas equation of state. 

We stated earlier (Section 9.1) that for the rotational motion of a linear molecule the energy levels are given by 

Each rotational level has a degeneracy of (2J + 1) so there are (2J 4 1) states at each level. For most molecules other than hydrogen the rotational energy levels are sufficiently close compared with RT see Table 9.1) that in order to evaluate the partition functions we can replace the summation by an integration over J. 

This is the result for a heteronuclear diatomic molecule. In order to generalize it we must include a factor a that takes into account that for a homonuclear diatomic, A2, a full rotation gives rise to two indistinguishable orientations A — A and A - A. This reduces the number of different terms contributing to the partition function by two. Thus 

More generally for polyatomic molecules with three rotations we obtain m n2kTfl2 Ixlyltyi2 

To discuss the heat capacity at intermediate and low temperatures requires some additional assumption about the 3 AT frequencies. The simplest approach is that of Einstein who assumed that all the frequencies have the same value, v. The partition function then takes the form 

The Debye theory assumes that there is a continuous distribution of frequencies from V = 0 to a certain maximum value v = Vj). The final expression obtained for the heat capacity is complicated, but succeeds in interpreting the heat capacity of many solids over the entire temperature range rather more accurately than the Einstein expression. At low temperatures, the Debye theory yields the simple result 

Since the z component of the angular momentum may have any of the values 0, 1, 2. J, there are 2J + 1 orientations of the angular momentum vector the degeneracy, Qj = 2J + 1. The rotational partition function is therefore given by 

Unfortunately, the sum on the right side cannot be evaluated in closed form. Nonetheless, using Eqs. (29.27) and (29.32) we find that the heat capacity, due to the rotation of 

We can evaluate this expression by performing the required differentiations on the series in Eq. (29.57), inserting the numerical values, and summing term by term until the desired 

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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... 

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... 

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 /. 

As was also previously noted in Section 9.3.1, the completely general rigid-rotor Schrodinger equation for a molecule characterized by three unique axes and associated moments of inertia does not lend itself to easy solution. However, by pursuing a generalization of the classical mechanical rigid-rotor problem, one can derive a quantum mechanical approximation that is typically quite good. Within that approximation, the rotational partition function becomes... 

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 a linear molecule, the rotational partition function qmx was given by Eq. 8.65. From Eq. 8.103, the rotational contribution to the entropy is... 

Note that Eqs. 8.127 and 8.128 only hold when the rotational partition functions are described by Eq. 8.65 or 8.67. When the rotational energy spacing is large, as for molecules with small moments of inertia, use Eq. 8.69 in 8.123. 

The translational contribution to the molecular partition function, which is calculated using Eq. 8.59, clearly makes the largest contribution. (In obtaining this value, we also made use of the ideal gas law to calculate the volume V = 0.02479 m3 of a mole of gas at this temperature and pressure.) The rotational partition function is evaluated via Eq. 8.67, and the vibrational partition function for each mode is found via Eq. 8.71. Only the very... 

For a linear molecule with a small moment of inertia (e.g., H2), Eq. 8.81 will not be valid. Starting with Eq. 8.78, derive an expression for (E — Eq)v01 f°r the case where the high-temperature limit is not valid, that is, when an explicit term-by-term summation is needed to evaluate the rotational partition function. Use the derived formula to evaluate (E — Eq )rot for IV = A = 1 mole of H2 at 298 K. Compare the result with the high-temperature limit prediction. Find the percent difference in the two results. 

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). 

The rotational partition function qimt can be similarly evaluated in the continuum limit as... 

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... 

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