# ps11 - Chem120B Problem Set 11 Due 1 Expressed in terms of...

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Chem120B Problem Set 11 Due: April 29, 2011 1. Expressed in terms of partition functions, the transition state theory estimate of a first order rate constant reads k TST A B = k B T h Q Q A This expression can be evaluated in detail if the potential energy U is sufficiently simple in the reac- tant ( A ) and transition state regions. Here you will carry out such a calculation by representing the fluctuations described by Q and Q A as harmonic oscillations. This situation is sketched below. The reactant state possesses n +1 normal modes with vibrational frequencies ω 1 2 ,... ,ω n ξ . The ( n + 1) th mode corresponds to motion along the reaction coordinate ξ . The transition state possesses n normal modes with frequencies ω 1 2 ,... ,ω n . The minimum energy for this state lies higher than that of the reactant state by an amount Δ E . (i) Calculate k TST A B for this harmonic model in terms of temperature, Δ E , and the vibrational fre- quencies listed above. For the moment, do not worry about quantum mechanical effects, i.e., make use of the classical partition function q vib = 1 / ( β p ω ) for a single oscillator with frequency ω . (ii) The vibrational frequency ω = r k/μ of a chemical bond between two atoms is determined by its restoring force constant k and by the reduced mass μ = m 1 m 2 / ( m 1 + m 2 ) , where m 1 and m 2 are the individual atomic masses. Using your result from part (i), compare the classical rate constant k H for a reaction that breaks a C–H bond with the rate constant k D for an equivalent reaction that breaks a C–D bond. Assume that all vibrational frequencies are insensitive to this isotopic substitution

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## This note was uploaded on 09/12/2011 for the course CHEM 107B taught by Professor Jamesames during the Spring '09 term at UC Davis.

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ps11 - Chem120B Problem Set 11 Due 1 Expressed in terms of...

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