Question #44827

The acid-catalyzed hydration of 1-methylcyclohexene to 1-methylcyclohexanol proceeds via a rate-determining protonation step. In pure water at $300\text{ K}$ (viscosity $\eta_1 = 1.0\text{ cP}$), the reaction exhibits an activation energy $E_a = 63.19\text{ kJ mol}^{-1}$ and a rate constant $k_1$.

To study solvent-solute dynamics, the reaction is transferred to a highly viscous glycerol-water mixture. In this medium, the dielectric constant remains identical to water, but the kinetics enter the Kramers high-friction regime. According to Kramers' theory, the Arrhenius pre-exponential factor $A$ becomes inversely proportional to the solvent viscosity ($\eta$) due to the Franck-Rabinowitch caging effect.

At $330\text{ K}$, the viscosity of the glycerol-water mixture is $\eta_2 = 2.0\text{ cP}$. Assuming $E_a$ remains strictly constant across both media, what is the ratio of the rate constant in the viscous mixture at $330\text{ K}$ ($k_2$) to $k_1$?

(Given: $R = 8.314\text{ J K}^{-1}\text{mol}^{-1}$, $\ln(10) = 2.303$)