Rotational Partition Function Corrections

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

A more elaborate treatment, which does not involve replacing the summation in equation (10.113) with an integral, gives 

Substitution into the equations relating Zmro[( = r ) to the thermodynamic properties gives 

Using Table 10.4, which neglects the correction terms ( 0.033 and -0.0002) to calculate (Hm - Hm 0). would introduce an error of 3.3% in the calculation. 

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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... 

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... 

A formula for the rotational partition function of a diatomic substance that gives corrections to the formula ofEq. (25.4-13) is ... 

Each of the partition functions is now regarded as a product of independent translational, rotational, and vibrational partition functions—the implication being that vibration-rotation interaction is negligible, and it is then assumed that the rotations are classical and the vibrations harmonic. If the structure of each molecular species is known, the moments of inertia can be calculated, and if necessary— as may well be the case for hydrogen isotopes—a correction can be applied to account for the fact that the rotational partition function has not reached its classical value. If complete vibrational analjrses of all the molecules are also available, the vibrational partition functions can be set up, and an approximate correction for neglect of anharmonicity can also be made. Having done all this, we can calculate the isotope effect. 

C) The error in AE" /AEq is 0.1 kcal/mol. Corrections from vibrations, rotations and translation are clearly necessary. Explicit calculation of the partition functions for anharmonic vibrations and internal rotations may be considered. However, at this point other factors also become important for the activation energy. These include for example ... 

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... 

By starting with this partition function and going through considerable mathematical manipulation, one arrives at the following equations for calculating the corrections to the rigid rotator and harmonic oscillator values calculated from Table 10,4, U... 

The simplest QCE model incorporates environmental effects of cluster-cluster interactions by (1) approximate evaluation of the excluded-volume effect on the translational partition function >trans (neglected in Section 13.3.3) and (2) explicit inclusion of a correction A oenv) for environmental interactions in the electronic partition function qiQiec. Secondary environmental corrections on rotational and vibrational partition functions may also be considered, but are beyond the scope of the present treatment. 

The vibrational qv) and rotational (q r) partition functions may be calculated within the harmonic and rigid rotor approximations, respectively. In addition, in this work, the qy values were corrected by replacing some of the large amplitude vibrations by the corresponding hindered internal rotations, when necessary. 

In order to illustrate the consequences of equation (70), it will be assumed that the partition functions for the reactants and the complex can be expressed as products of the appropriate numbers of translational, rotational and vibrational partition functions. For simplicity we shall also neglect factors associated with nuclear spin and electronic excitation. If = total number of atoms in a molecule of species i and = 0 for nonlinear molecules, 1 for linear molecules, and 3 for monatomic molecules, then the correct numbers of the various kinds of degrees of freedom are obtained in equation (70) by letting... 

Electronic levels (T ) and vibrational-rotational constants of the observed states are from the optical study of Barrow et al. (J ) and the microwave work of Tiemann et al. (2). Other low-lying electronic states and their vibrational-rotational constants are estimated in isoconfIgurational groups by analogy with BaO (8) and from trends observed in the known states of the other alkaline-earth oxides and sulfides. Thermodynamic functions are calculated using first-order anharmonlc corrections to and in the partition function Q = exp(-c ej /T). Uncertainty in the energy and molecular constants for the... 

The vibrational and rotational constants of the respective electronic levels were taken from Rosen (2 ). The thermodynamic functions are calculated using first-order anharmonic corrections to and 0 in the partition function Q = Q,j.EQ Q gj exp(-... 

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