Vibrational-rotational partition functions atom + diatom

For diatomic gases, the vibrational statistical weight s = 1. Moreover, for the rigid rotator model, the rotational partition function is independent of the vibrational state Z = Z . In this case, the ratio of the backward and forward reaction rate coefficients takes the reduced form. The expression (74) can be simplified if the collision partner is an atom. 

The total partition function may be approximated to the product of the partition function for each contribution to the heat capacity, that from the translational energy for atomic species, and translation plus rotation plus vibration for the diatomic and more complex species. Defining the partition function, PF, tlrrough the equation... 

The classical value is attained by most molecules at temperatures above 300 K for die translation and rotation components, but for some molecules, those which have high heats of formation from die constituent atoms such as H2, die classical value for die vibrational component is only reached above room temperature. Consideration of the vibrational partition function for a diatomic gas leads to the relation... 

The partition function of the atomic species consists of the electronic and translational contributions only, but for the diatomic molecule A2 the partition function involves the electronic, translational and rotational factors, and also the contribution of one vibrational mode. The translational partition function is given by equation (16.16) as... 

Next, consider a reaction between an atom and a diatomic molecule, A + BC. Reactions can differ also in their energetic requirements, but to focus attention on the steric requirements with the energetic effects being equal we take the barrier height Eq to be the same as in the previous reaction. To have minimal steric requirements let us take the transition state, ABC, to be bent. This choice allows A to approach BC within a cone. Because BC has an internal structure, the partition function for the reactants becomes Q = Q Q QvQ - The transition state is a bent triatomic. It has three vibrations, one of which is the reaction coordinate. (As we saw in Section 5.1, this is the asymmetric stretch vibration.) The bent transition state has three planes of rotation, = Q QIQ - Accordingly, for reasons that will become immediately apparent, we write k T) as... 

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