Product partition function, corrections

To arrive at K and k, our task is to express the following terms appearing in eqns. (5.15) and 5.16) the partition functions (Q) of reactants, products and of the activated complex, the heat of reaction at absolute zero, AHq, the enthalpy of activation at absolute zero, Hq, and the tunnelling correction factor, P. For an ideal gas the total partition function can be expressed within the rigid-rotor and harmonic oscillator (RRHO) approximation as a product... 

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... 

The total vibrational partition function is a product of the Wigner-Kirkwood partition functions for each corrected mode and the harmonic partition function of the remaining vibrations ... 

The first assumption is that the partition function Q is the product of the individual partition functions for the component parts, corrected for indistinguishability. 

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. 

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