Partition functions of the activated complex

Thus, the partition function of the activated complex, with the special motion factored out, becomes... 

This relation indicates that the rate constant can be determined from a knowledge of the partition functions of the activated complex and the reactant species. For stable molecules or atoms... 

When we compare Eqs (6.8) and (6.25), we note that the TST expression for the rate constant would be obtained provided that the integral appearing in Eq. (6.25) was identical to the partition function of the activated complex. To that end, we introduce the approximation... 

Q is the usual partition function of the activated complex referred to the minimum in the potential of the normal molecule as the zero of energy, Q is the partition function qf the three rotations and three translations of the normal molecule, Ea IS the activation energy of the reaction as measured from the minimum of the normal molecule potential energy surface to the minimum of the activated complex, 0 is the zero-point energy of the activated complex, and the v( s are the vibrational frequencies, of the normal molecule. Moreover, A the rate of deactivation of active molecules to normal molecules, has been set equal to the collision number Z times an efficiency factor y, assumed to be isotope independent. 

We have neglected the vibrational partition function of D2, which is very close to unity at the temperature in question. The symmetry number cr is 2 for a homonuclear diatomic, and the rotational constant is 30.44 cm-1. Now, the partition function of the activated complex will have a translational piece that is the same regardless of the model ... 

Here is the ratio of the rotational partition function of the activated complex to that of the reactant molecule is the vibrational partition function of the reactant and involves vibrational-energy states only. This... 

This relation indicates that the rate constant can be determined from a knowledge of the partition functions of the activated complex and the reactant species. For stable molecules or atoms the partition functions can be calculated from experimental data that do not require kinetic measurements. However, they do require that molecular constants such as the vibrational frequencies and moments of inertia be evaluated from spectroscopic data. Evaluation of the partition function for the activated complex Gxyz presents a more difficult problem, since the moments of inertia and vibrational frequencies required cannot be determined experimentally. However, theoretical calculations permit one to determine moments of inertia from the various internuclear distances and vibrational frequencies from the curvature of the potential energy surface in directions normal to the reaction coordinate. In practice, one seldom has available a sufficiently accurate potential energy surface for the reaction whose rate constant is to be determined. 

In this situation use can be made of the adiabatic rate equation (11.IV) with the expression (12.IV) for the partition function of the "activated complex treated as a rigid rotator in a cen-... 

The connection between statistical mechanics and transition state theory is made by eliminating, from the partition function of the activated complex, the translational component due to the motion along the reaction coordinate (see (Glasstone et al. 1941, p. 189)). This step, however, is captured through the addition of a special law on statistical mechanics, and moreover, this special law characterises the transition state precisely in terms of its degrees of freedom. 

The contribution to the partition function of the activated complex corresponding to motion along the reaction coordinate, in this case translational motion, is factored out of the transition state partition function, <2ahb > to yield ... 

The partition function of the activated complex has a translational, a rotational, and a vibrational factor ... 

For our point at the top of the barrier, the degree of freedom of translation crossing the pass plays a specific role. Therefore by taking into account [A2.34] (see Appendix 2), the partition function of the activated complex can be written as ... 

To calculate the partition functions, we abandon our mass point and go back to molecules and atoms. The use of the equation [10.25] is required to calculate the partition functions of both the reactants and the activated complex. If the reactant partition functions can be calculated without problem using their characteristics (see Appendix 2), we must spend time calculating the partition function of the activated complex. 

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