A complex between a methane molecule and a metal atom (RH-M) serves as an intermediate in the dissociation of CH_{4} into CH_{3} and H. The rate of methane dissociation is determined by a kinetic competition between detachment of methane from the metal atom (rxn 1) and methane dissociation (rxn 2) as represented by the following reactions,

RH-M ® RH + M (*k*_{1})

RH-M ® R + H-M _{} (*k*_{2})

In this model, CH_{4} is treated as a pseudo-diatomic R-H where R = CH_{3} and M represents a metal atom.

Assume that the initial state and the transition states for both reactions are linear structures and that the vibrational frequencies are the same in the initial state and the transition states for all modes other than those associated with motion along the reaction coordinates. You may also assume that the moments of inertia are approximately the same for the initial state and the transition states.

The following data is available,

Mass of RH = 16 amu; Mass of M = 195 amu

R-H stretching frequency: n_{hi} = 3000 cm^{-1}; RH-M stretching frequency: n_{lo} = 300 cm^{-1}

Electronic barrier heights: *V*_{1} = 40 kJ/mol, *V*_{2} = 55 kJ/mol

Statistical factors: *l*_{1} = 1, *l*_{2} = 2

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**Part a. **Calculate the zero-point corrected energy barriers (*DE*_{1} and *DE*_{2}) for reactions 1) and 2).

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**Part b. **Using transition state theory, derive an expression for the branching ratio *k*_{2}/*k*_{1}. Write your equation in terms of known quantities and clearly state your approximations. In this problem, the only known vibrational frequencies are those given above. Calculate the branching ratio *k*_{2}/*k*_{1} for reaction at 150 K.

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**Part c.** Calculate the branching ratio *k*_{2}/*k*_{1} for the dissociation of a CD_{4}-M complex at 150 K. For the CD_{4}-M system, n_{lo} = 271 cm^{-1} and n_{hi} = 2165 cm^{-1}, and the mass of CD_{4} is 20 amu.

**Part d.** The model predicts a difference in the magnitudes of the low-temperature branching ratios for CH_{4} vs. CD_{4}. According to the model, what is an even more significant difference in these branching ratios that should be experimentally observable?

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