Statistical Formulation of Reaction Rates

POLANYI /3/, in which a thermal equilibrium between reactants and the activated complexes is postulated. In general,in any statistical theory one accepts that the thermal equilibrium is maintained during the reaction, while in the usual collisional treatment such an equilibrium is assumed only for reactants. 

We will not here make any hypothesis about the existence of the activated complex however, we may use this concept in the sense of a possible state which could be realized only under certain extreme conditions already discussed in the introduction. This approach is similar to that used in the theory of real gases which is based on the notion of the perfect gas as a state under the limiting conditions of very low presure or very high temperature. 

The equations (56.111) (a) and (b) correspond to the cases of a relative translation and vibration of reactants along the reaction coordinate.  

These equations are valid for any quantum state of reactants (n) and activated complexes(n ).We may restrict, however, the choice of n by setting n = n so that 

We consider both equations (60.Ill) and (61.Ill) solely as a definition of the activated complex thought of as a virtual state which does not at all reveal its real existence. Using (60.Ill) we 

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The relation (184 111) could be considered a definition of a characteristic temperature Tj corresponding to Eyring s formulation of activated complex theory /20b/. However, when the non-adiabatic condition (72.Ill) is really fulfilled, equation (184.III) actually represents only a relation of equivalency of the collision and statistical formulations of reaction rate theory in the high temperature region T >T /2 to which both the formulas (177.Ill) and (183.Ill) refer. This means that if T < T, the formula (183.III) cannot be used in the temperature range TjJ/2 < T < Tj /2 for physical reasons. 

Our treatment, based on both the collision and the statistical formulations of reaction rate theory, shows that there exist two possibilities for an interpretation of the experimental facts concerning the Arrhenius parameter K for unimolecular reactions. These possibilities correspond to either an adiabatic or a non-adiabatic separation of the overall rotation from the internal molecular motions. The adiabatic separability is accepted in the usual treatment of unimolecular reactions /136/ which rests on transition state theory. To all appearances this assumption is, however, not adequate to the real situation in most unimolecular reactions.The nonadiabatic separation of the reaction coordinate from the overall rotation presents a new, perhaps more reasonable approach to this problem which avoids all unnecessary assumptions concerning the definition of the activated complex and its properties. Thus, for instance, it yields in a simple way the rate equations (7.IV), corresponding to the "normal Arrhenius parameters (6.IV), which are both direct consequences of the general rate equation (2.IV). It also predicts deviations from the normal values of the apparent frequency factor K without any additional assumptions, such that the transition state (AB)" (if there is one) differs more or less from the initial state of the activated molecule (AB). ... 

We conclude that the current interpretations of isotope effects in proton transfer reactions in terms of proton tunneling should be accepted with caution. It is first necessary to take into account that the assumption of a separable reaction coordinate is more justified in the collisional formulation than in the statistical formulation of reaction rates. Moreover, the motion along the reaction coordinate... 

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