Transition State Theory and Microscopic Reversibility

In earlier chapters, we have seen how kinetic phenomena can be interpreted, for example, to provide a reaction scheme consisting of a set of elementary reactions. Over the years, several models have been devised to explain and sometimes to predict the rates of elementary reactions. It is these that we now wish to examine on a more fundamental basis in this chapter, plus Chapters 9 and 10. 

The experimental side of the subject explores the effects of certain variables on the rate constant, especially temperature and pressure. Their variations provide values of the activation parameters. They are the previously mentioned energy of activation, entropy of activation, and so forth. The chemical interpretations that can be realized from the values of the activation parameters will be explored in general terms, but readers must consult the original literature for information about those chemical systems that particularly interest them. On the theoretical side, there is the tremendously powerful transition state theory (TST). We shall consider its origins and some of its implications. 

we shall explore a conceptual relation between kinetics and thermodynamics that allows one to draw certain conclusions about the kinetics of the reverse reaction, even when it has itself not been studied. Second, we shall show how the thermodynamic state functions for the transition state can be defined from kinetic data. These are the previously mentioned activation parameters. If their values for the reaction in one direction have been determined, then the values in the other can be calculated from them as well as the standard thermodynamic functions. The implications of this calculation will be explored. Third, we shall consider a fundamental principle that requires that the 

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There is one path between reactants and products that has a lower energy maximum than any other this is the pathway that the reaction will follow. The line in a two-dimensional potential energy plot represents this lowest-energy pathway. It represents a path across an energy surface describing energy as a function of the spatial arrangement of the atoms involved in the reaction. The principle of microscopic reversibility arises directly from transition. state theory. The same pathway that is traveled in the forward direction of a reaction will be traveled in the reverse direction ... 

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. 

Here, Ri f and Rf i are the rates (per moleeule) of transitions for the i ==> f and f ==> i transitions respeetively. As noted above, these rates are proportional to the intensity of the light souree (i.e., the photon intensity) at the resonant frequeney and to the square of a matrix element eonneeting the respeetive states. This matrix element square is oti fp in the former ease and otf ip in the latter. Beeause the perturbation operator whose matrix elements are ai f and af i is Hermitian (this is true through all orders of perturbation theory and for all terms in the long-wavelength expansion), these two quantities are eomplex eonjugates of one another, and, henee ai fp = af ip, from whieh it follows that Ri f = Rf i. This means that the state-to-state absorption and stimulated emission rate eoeffieients (i.e., the rate per moleeule undergoing the transition) are identieal. This result is referred to as the prineiple of microscopic reversibility. 

Partly ionized gas or vapour is called a plasma. It contains atoms, molecules, and ions from which some fraction may be in excited states, and free electrons. Several theoretical models have been presented to describe a plasma. One of these is the so called Thermal Equilibrium Theory, which is based on the micro reversible principle. According to this principle, each energy process is in equilibrium with a reverse process. For example, the number of transitions per time unit from the state f to the state (absorption) is exactly the same as the number of the reverse transitions (emission). According to Maxwell, the microscopic states of the plasma at thermal equilibrium may be calculated on the basis of the temperature, which is the only variable. The number of particles (dA) with the speed between v dv is ... 

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