Question #44449

The absolute difference in activation energies, $|\Delta E_a| = |E_{a,\ce{F^-}} - E_{a,\ce{I^-}}|$, for the $S_N2$ reaction of methyl tosylate (\ce{CH3OTs}) with \ce{F^-} and \ce{I^-} is measured in two different solvents at $298\text{ K}$.

In Methanol (\ce{CH3OH}), $|\Delta E_a| = 11.42\text{ kJ/mol}$. In Dimethyl Sulfoxide (DMSO), $|\Delta E_a| = 5.71\text{ kJ/mol}$.

Assume the Arrhenius pre-exponential factor ($A$) is identical for all four reactions. It is given that the rate constant for the reaction with \ce{I^-} in DMSO is $10^3$ times the rate constant for the reaction with \ce{F^-} in Methanol ($k_{\ce{I^-}, \text{DMSO}} = 10^3 \times k_{\ce{F^-}, \text{MeOH}}$).

Calculate the ratio $\frac{k_{\ce{I^-}, \text{MeOH}}}{k_{\ce{F^-}, \text{DMSO}}}$.

(Given: $2.303 \times R \times 298\text{ K} = 5.71\text{ kJ/mol}$)