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Unformatted text preview: CE 561 Lecture Notes Fall 2009 p. 1 of 9 Day 11: Transition State Theory In the last set of lecture notes, we arrived at the conclusion that we know how to, in principle, calculate rate parameters from first principles by deriving a potential surface from the solution of the Schrdinger equation for the electrons at different positions of the nuclei and then solving the Schrdinger equation for motion of the nuclei on that potential surface. However, except for the simplest of molecules and reactions, it is not practical to do so. So, here we discuss statistical theories of reaction rates (transition state theory) that require much less information about the potential surface, but involve several significant approximations. Transition state theory allows us to replace the detailed trajectory or scattering calculations with a simpler, less detailed model. Instead of calculating individual trajectories from reactants in a particular state to products in a particular state, we use the tools of statistical mechanics to calculate the total rate at which molecules cross through some surface that divides reactants from products. Transition state theory was first introduced in the mid-1930s by Eyring and by Evans and Polanyi. There are a number of approximations and assumptions about the system that are made in transition state theory. The basic ones are: (1) The Born-Oppenheimer approximation (or its classical equivalent) is valid, so that the motion of the electrons can be separated form the motions of the nuclei. (2) The reactant molecules are distributed in their energetic states in accordance with the Maxwell-Boltzmann distribution. This means that the translational, vibrational, and rotational degrees of freedom of the molecules are in thermal equilibrium. The number of molecules in a state with energy i is proportional to exp(- i /( kT )). (3) Molecular systems that have crossed the transition state (the dividing surface) in the direction of products do not turn around to form reactants. (4) In the transition state, motion along the reaction coordinate may be separated from the other motions and treated classically as a translation. (5) Even in the absence of equilibrium between reactant and product molecules, the transition states that are becoming products are distributed among their states according to the Maxwell-Boltzmann distribution. It can be shown that this assumption is not really necessary, as it is implied by the second and third assumptions. We now present the standard or quasi-equilibrium derivation of transition-state theory. There is also a more rigorous, dynamical derivation (which we will not present here) that gives the same result with fewer assumptions....
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- Fall '09