Hard-sphere Collision Theory

There are a few cases where the rate of one reaction relative to another is needed, but the absolute rate is not required. One such example is predicting the regioselectivity of reactions. Relative rates can be predicted from a ratio of Arrhenius equations if the relative activation energies are known. Reasonably accurate relative activation energies can often be computed with HF wave functions using moderate-size basis sets. 

Since the reactions are very similar, we will assume that the pre-exponential factors A are the same, thus giving. 

Substituting T = 298 K and the gas constant gives a ratio of about 81. Thus, we expect there will be 80 times as much para product as ortho product, assuming that the kinetic product is obtained. 

The simplest approach to computing the pre-exponential factor is to assume that molecules are hard spheres. It is also necessary to assume that a reaction will occur when two such spheres collide in order to obtain a rate constant k for the reactants B and C as follows  

To a first approximation, the activation energy can be obtained by subtracting the energies of the reactants and transition structure. The hard-sphere theory gives an intuitive description of reaction mechanisms however, the predicted rate constants are quite poor for many reactions. 

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We are concerned with bimolecular reactions between reactants A and B. It is evident that the two reactants must approach each other rather closely on a molecular scale before significant interaction between them can take place. The simplest situation is that of two spherical reactants having radii Ta and tb, reaction being possible only if these two particles collide, which we take to mean that the distance between their centers is equal to the sum of their radii. This is the basis of the hard-sphere collision theory of kinetics. We therefore wish to find the frequency of such bimolecular collisions. For this purpose we consider the relatively simple case of dilute gases. 

On the basis of the preceding assumptions, Hinshelwood s theory proceeds as follows. Let c be the concentration of reactant, W the equilibrium fraction of activated molecules, i.e., that at high pressures, where the rate of reaction is extremely small relative to the rate of deactivation, and c the concentration of activated molecules at any pressure. The formal expression for the number of deactivating collisions at high pressures, namely, accW, where a was found from elastic-hard-sphere collision theory to be equal to 2nd kT/ (= Z jc represents the rate of activation at all pressures. Since at the steady state at any pressure this rate of activation will be equal to the rate of deactivation plus the rate of reaction,... 

A comparison of the rate constants computed from transition-state theory (4.151) and that of hard-sphere collision theory (4.127), given in Table 4.5, clearly... 

This section and the following sections will explore the transition-state expressions for the rate constants of surface reactions. We start with the adsorption of atoms. The expression for the rate of adsorption according to hard sphere collision theory was covered. Expression (4.112) will be demonstrated as to how it can be rederived within the transition-state theory. 

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