Question #44821

The acid-catalyzed Fischer esterification of acetic acid (\ce{CH3COOH}) is studied with two different alcohols under identical conditions:\Reaction X: Esterification with ethanol.\Reaction Y: Esterification with 2,4-dimethyl-3-pentanol.\At $300\text{ K}$, the rate constant for Reaction X ($k_X$) is $204,800$ times the rate constant for Reaction Y ($k_Y$). According to Collision Theory, the Arrhenius pre-exponential factor for Reaction X ($A_X$) is $100$ times that of Reaction Y ($A_Y$) due to severe steric hindrance restricting the effective collision orientations in Reaction Y.\When the temperature is raised from $300\text{ K}$ to $330\text{ K}$, the rate constant $k_X$ increases by a factor of $f_X$, and $k_Y$ increases by a factor of $f_Y$. Assuming activation energies and pre-exponential factors are temperature-independent, and no side reactions occur, what is the value of the ratio $\frac{f_Y}{f_X}$?