Question #44061

The formation of the Wheland intermediate ($\sigma$-complex) during the electrophilic aromatic substitution of 1,3,5-trimethoxybenzene (TMB) with tropylium tetrafluoroborate ($\ce{C7H7+ BF4-}$) is an equilibrium-controlled process:

$\ce{TMB + C7H7+ <=> \sigma-complex+}$

In 1,2-dichloroethane (DCE, a non-polar solvent), the equilibrium constant is $K_1 = 0.1$ at 298 K. When the solvent is changed to acetonitrile (MeCN, a polar aprotic solvent), the $\sigma$-complex and the transition state ($TS^\ddagger$) are significantly stabilized by solvation. The transfer free energies from DCE to MeCN are $\Delta G_{tr}(\sigma^+) = -22.8 \text{ kJ/mol}$ and $\Delta G_{tr}(TS^\ddagger) = -11.4 \text{ kJ/mol}$.

Assuming the transfer free energies of the neutral TMB and the large, diffuse $\ce{C7H7+}$ cation are negligible ($\approx 0$), calculate the equilibrium constant $K_2$ and the ratio of the forward rate constants ($k_{f2}/k_{f1}$) in MeCN at 298 K.

(Given: $2.303 RT = 5.7 \text{ kJ/mol}$ at 298 K)