Question #44824

In a specialized non-aqueous acidic buffer, aniline undergoes N-alkylation with methyl iodide. The active nucleophile is the unprotonated aniline ($\ce{PhNH2}$), while the protonated form ($\ce{PhNH3+}$) is completely unreactive. The buffer maintains a constant $\ce{H+}$ concentration, and aniline exists predominantly ($>99\%$) in its protonated form.

The true activation energy for the bimolecular nucleophilic substitution ($S_N2$) between free aniline and methyl iodide is $E_{a,true} = 39.4 \text{ kJ/mol}$. Due to severe solvent restructuring, the protonation equilibrium $\ce{PhNH2 + H+ <=> PhNH3+}$ is anomalously endothermic with a standard enthalpy change $\Delta H^\circ_{prot} = +93.0 \text{ kJ/mol}$.

Assuming the Arrhenius pre-exponential factor and standard enthalpies are temperature-independent, calculate the observed temperature coefficient ($Q_{10}$) for the overall reaction rate when the temperature is raised from $300 \text{ K}$ to $310 \text{ K}$.

(Given: $R = 8.314 \text{ J K}^{-1} \text{ mol}^{-1}$, $\ln 2 = 0.693$)