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PS11Solutions - Chemistry 120B SP11 Problem Set 11...

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Chemistry 120B SP11 Problem Set 11 Solutions 1. (i) The transition state theory rate constant is k TST A B = k B T h Q vib Q A vib e - β Δ E The total vibrational partition function is the product of the q vib ’s of all the modes. Ignoring quantum mechanical effects, q vib = 1 ~ ω . So, Q vib = 1 β ~ n 1 ω 1 ω 2 · · · ω n ! and Q A vib = 1 β ~ n +1 1 ω 1 ω 2 · · · ω n ω ξ The ratio of these partition functions is Q vib Q A vib = β ~ × ω 1 ω 2 · · · ω n ω ξ ω 1 ω 2 · · · ω n and the rate constant is k TST A B = 1 2 π ω 1 ω 2 · · · ω n ω ξ ω 1 ω 2 · · · ω n ! e - β Δ E = ω ξ 2 π Y j ω j ω j e - β Δ E (ii) Assume the isotopic substitution affects only fluctuations along ξ , so all the ω j ’s and ω j ’s cancel except j = ξ . Since Δ E is the same for both isotopes, k H k D = ω H ξ ω D ξ = r μ D μ H = s m D m C m D + m C m H + m C m H m C = 1 . 36
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