Question #44825

The keto-enol tautomerization of Uracil ($U$) in a mixed solvent system containing water ($W$) and an inert diluent proceeds via a solvent-bridged intermediate ($I$). The proposed mechanism is:

Step 1: $\ce{U + W <=>[k_1][k_{-1}] I}$ Step 2: $\ce{I + W ->[k_2] E + 2W}$ (where $E$ is the enol form)

The Arrhenius parameters for the elementary steps at $300\text{ K}$ are given below: $A_1 = 10^{11}\text{ M}^{-1}\text{s}^{-1}$, $E_{a1} = 15.0\text{ kJ/mol}$ $A_{-1} = 10^{12}\text{ s}^{-1}$, $E_{a,-1} = 28.5\text{ kJ/mol}$ $A_2 = 10^{14}\text{ M}^{-1}\text{s}^{-1}$, $E_{a2} = 40.0\text{ kJ/mol}$

Applying the steady-state approximation to $[I]$, the reaction exhibits a change in kinetic order with respect to $[W]$ depending on the solvent composition. Determine the transition concentration $[W]_{trans}$ at which the rate of dissociation of $I$ equals its rate of forward reaction to $E$. Furthermore, calculate the apparent activation energy ($E_{a,obs}$) of the overall tautomerization when $[W] \ll [W]_{trans}$.

(Given: $2.303 RT = 5.75\text{ kJ/mol}$ at $300\text{ K}$)