Question #44815

Consider the alkylation of 2-methylcyclohexanone ($K$) with methyl iodide (\ce{CH3I}) in a protic solvent $BH$ using a base $B^-$ at 298 K. The reaction proceeds via a fast, reversible proton abstraction to form two enolate intermediates: $E_1$ (kinetic, less substituted) and $E_2$ (thermodynamic, more substituted). This is followed by a slow, irreversible alkylation step to yield products $P_1$ and $P_2$, respectively.

Given the following thermodynamic and kinetic parameters at 298 K:

• Standard free energy difference between the enolates: $G^\circ(E_2) - G^\circ(E_1) = -11.42 \text{ kJ/mol}$
• Activation free energy difference for the alkylation step: $\Delta G^\ddagger_2 - \Delta G^\ddagger_1 = 17.13 \text{ kJ/mol}$ (Assume identical pre-exponential factors for the alkylation steps, and use $2.303 RT = 5.71 \text{ kJ/mol}$ at 298 K)

Assuming $[E_1], [E_2] \ll [K]$, let $K_{eq1}$ and $K_{eq2}$ be the equilibrium constants for the formation of $E_1$ and $E_2$, and $k_1$ and $k_2$ be their respective alkylation rate constants. Which of the following correctly represents the overall rate law for the consumption of \ce{CH3I} and the product ratio $[P_1]/[P_2]$?