Arrhenius equation comparison with collision

Comparison with Arrhenius Equation and Collision Theory... 

A comparison of equations (4.54) and (4.55) shows that the rate constant for a complex reaction differs from that obtained in simple atomic reaction by a factor of (qjqr)5. Since qv is nearly unity, while qr varies from 10 to 100 for a complex molecule, the ratio qv/qr, therefore, varies from 10 I to 10 2 and (qv/qT)5 varies from 10 5 to 10 10. This factor may link to steric factor p. On comparing equation (4.55) with collision theory and Arrhenius equation, we get... 

By comparison of (M) and (F), it can be seen that the preexponential factor A in the Arrhenius equation can be identified with PaA]i(8kT/Tr/ji,y/2 and the activation energy, a, with the threshold energy Eu. It is important to note that collision theory predicts that the preexponential factor should indeed be dependent on temperature (Tl/2). The reason so many reactions appear to follow the Arrhenius equation with A being temperature independent is that the temperature dependence contained in the exponential term normally swamps the smaller Tl/Z dependence. However, for reactions where E.t approaches zero, the temperature dependence of the preexponential factor can be significant. 

Comparison of this equation with the Arrhenius form of the reaction rate constant reveals a slight difference in the temperature dependences of the rate constant, and this fact must be explained if one is to have faith in the consistency of the collision theory. Taking the derivative of the natural logarithm of the rate constant in equation 4.3.7 with respect to temperature, one finds that... 

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