Activation energy relation with Transition-state theory

A more general, and for the moment, less detailed description of the progress of chemical reactions, was developed in the transition state theory of kinetics. This approach considers tire reacting molecules at the point of collision to form a complex intermediate molecule before the final products are formed. This molecular species is assumed to be in thermodynamic equilibrium with the reactant species. An equilibrium constant can therefore be described for the activation process, and this, in turn, can be related to a Gibbs energy of activation ... 

Several attempts to relate the rate for bond scission (kc) with the molecular stress ( jr) have been reported over the years, most of them could be formally traced back to de Boer s model of a stressed bond [92] and Eyring s formulation of the transition state theory [94]. Yew and Davidson [99], in their shearing experiment with DNA, considered the hydrodynamic drag contribution to the tensile force exerted on the bond when the reactant molecule enters the activated state. If this force is exerted along the reaction coordinate over a distance 81, the activation energy for bond dissociation would be reduced by the amount ... 

For some reactions, especially those involving large molecules, it might be difficult to determine the precise structure and energy levels of the activated complex. In such cases, it can be useful to phrase the transition-state theory result for the rate constant in thermodynamic terms. It does not bring any new information but an alternative way of interpreting the result. This formulation leads to an expression where the preexponential factor is related to an entropy of activation that, at least qualitatively, can be related to the structure of the activated complex. We will encounter the thermodynamic formulation again in Chapter 10, in connection with chemical reactions in solution, where this formulation is particularly useful. 

In terms of traditional Transition State Theory (TST) for solution reactions [40,41], in which e.g. the activation free energy AG can be estimated with equilibrium solvation dielectric continuum theories [42-46], the nonequilibrium or dynamical solvation aspects enter the prefactor of the rate constant k, or in terms of the ratio of k to its TST approximation kTST, k, the transmission coefficient, k and kTST are related by [41]... 

The transition state theory gives us a framework to relate the kinetics of a reaction with the thermodynamic properties of the activated complex (Brezonik, 1990). In kinetics, one attempts to interpret the stoichiometric reaction in terms of elementary reaction steps and their free energies, to assess breaking and formation of new bonds, and to evaluate the characteristics of activated complexes. If, in a series of related reactions, we know the rate-determining ele-mentaiy reaction steps, a relationship between the rate constant of the reaction, k (or of the free energies of activation, AG ), and the equilibrium constant of the reaction step, K (or the free energy, AG°), can often be obtained. For two related reactions. 

We use the constructs of transition state theory in order to define the Br0nsted-Evans-Polanyi (BEP) relationship, which relates the equilibrium thermodynamics (reaction enthalpy or free energy) with non-equilibrium thermodynamic features, namely the activation energy and activation entropy. A small value of the proportionality parameter in the BEP relationship, a, is identified with an early transition state, whereas values of a that are close to 1 relate to a late transition state. Microscopic reversibility ensures that if the forward reaction is an early transition state then the backward reaction must be a late transition state and vice versa. 

Meanwhile, the pre-exponential factor A in the Arrhenius Eq. (2.39) is the temperature independent factor related to reaction frequency. Comparing the Eq. (2.33) for the collision theory and Eq. (2.38) with the transition state theory, the pre-exponential factors in these theories contain temperature dependences of T and T respectively. Experimentally, for most of reactions for which the activation energy is not close to zero, the temperature dependence of the reaction rate constants are known to be determined almost solely by exponential factor, and the Arrhenius expression holds as a good approximation. Only for the reaction with near-zero activation energy, the temperature dependence of the pre-exponential factor appears explicitly, and the deviation from the Arrhenius expression can be validated. In this case, an approximated equation modifying the Arrhenius expression can be used. 

It is useful to consider to what extent qualitative interpretations of isotope effects are backed up by calculations. The most fundamental calculations using transition-state theory are those based on a calculated potential-energy surface for the reaction. We have seen how transition-state force constants measure the curvature of a surface at its saddle point. Numerical or analytical evaluation of the appropriate derivatives d V/dAr for a calculated surface permits calculation of force constants and thence of isotope effects. Moreover the surface also yields an activation energy and energy of reaction, and there is no need for additional assumptions about the relation between these properties and force constants before making comparisons with experiment. This advantage over less fundamental treatments where force constants are assigned directly has been emphasized by Bell [2, 82]. 

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