SN1 or SN2, the kinetic isotope effect

Question

I know that you can't always trust Wikipedia, so I just want to check something regarding their kinetic isotope effect page.

The first example implies that $\sideset{^{12}}\ C$ and $\sideset{^{13}}\ C$ can be used to determine whether a reaction is $S_N1$ or $S_N2$:

...In the nucleophilic substitution reaction of methyl bromide with cyanide, the kinetic isotope effect of the methyl carbon, in this case defined as $k_{12}/k_{13}$, was found to be 1.082 ± 0.008 ... The observed methyl carbon kinetic isotope effect is indicative of an $S_N2$ mechanism."

No explanation appears to be offered for this inference. I know that the reaction will not proceed via $S_N1$ because of the unstable carbocation but (how) does the KIE datum predict an $S_N2$ mechanism?

Answer 1

The rate constant depends on the energy difference between the ground state of the reactants and the transition state. The ratio of the rate constants due to the kinetic isotope effect therefore depends on the difference in vibrational zero point energy between the initial state and the transition state. Vibrational ZPE depends on bond stiffness and reduced mass. The change in reduced mass from $\ce{^{12}C}$ to $\ce{^{13}C}$ causes the kinetic isotope effect, and the bond stiffness determines the size of the KIE. Bond stiffness is to a first approximation proportional to bond order.

In both $S_N1$ and $S_N2$, the initial state has a single bond between the methyl $\ce{C}$ and $\ce{Br}$ (as well as some $\ce{C-H}$ bonds which stay the same throughout the reaction). For $S_N2$, the transition state has partial bonds between the methyl $\ce{C}$ and both $\ce{Br}$ and the $\ce{C}$ of $\ce{CN}$. In the $S_N1$ transition state, the methyl $\ce{C}$ has only a partial bond to $\ce{Br}$.

Since the change in bond order is greater between initial state and $S_N1$ transition state than between ground state and $S_N2$ transition state, $S_N1$ will have a greater kinetic isotope effect. Measuring the KIE of an unknown reaction and comparing it to the KIEs of reactions with known mechanisms will therefore distinguish between $S_N1$ and $S_N2$.