Application to Rate Processes

It is interesting to see how the statistical treatment of equilibrium sys-tem may be applied to the calculation of kinetic data, at least to the same accuracy as was obtained when the assumption of a Maxwellian distribution was employed. Let us assume that we have a mixed gas of hard sphere molecules A and B, capable of forming a weakly bound complex AB. Let us further assume that the molecules A and B possess no internal energy. We can then write the molecular partition functions for A, B, and AB  

If now we admit that the forces between A and B even at small distances are small, so that hv C kT and further that Eq RT, we can expand the two exponential terms and find 

In this form the units of K arc molecules/cc and V/N = 1 in the standard state. Thus 

But this can be interpreted quite readily because v is now the limiting value (as the forces between A and B approach zero) of the vibration frequency of A and B in the molecule AB. Since in the limiting case each such vibration leads to a dissociation into free species A and B, the quantity vNab must represent the rate at which the species AB is decomposing to produce A and B. Finally since there is an equilibrium, this must also equal the rate at which A and B are combining to form AB. We can recognize that the right-hand side of Eq. (IX.13.7) is precisely the rate at which Na and Nb make collisions [Eq. (VII.8F.5)], so that we see that our collision formula is also contained in the equilibrium constant. However, this is not strange, since both were derived from the same postulates. We shall see later how the partition function may be used to deduce formulas for various collision processes. 

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The next section shows that binding polynomials are just as applicable to rate processes as to binding thermodynamics. 

A qualitative description of solvent-solute effects involving ionic and dipolar interactions is given by Kirkwood theory [83], which considers the free energy change for transfer of an ion- or dipole-possessing sphere from a vacuum of unit dielectric constant to a medium of dielectric constant e. The application to rate processes is due to Laidler [84]. Contributions to the volume of activation from dipolar interactions are. 

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