Question #44081

Azo coupling is highly sensitive to the pH of the reaction medium due to the competitive ionization of the coupling partners. Consider an equimolar aqueous mixture of a phenol derivative $\textbf{X}$ ($pK_a = 10.0$) and an aniline derivative $\textbf{Y}$ ($pK_b = 9.0$). A small amount of a diazonium salt, $\ce{ArN2+ Cl-}$, is added to the mixture at $298\text{ K}$.

The intrinsic second-order rate constant for the electrophilic attack of $\ce{ArN2+}$ on the phenoxide ion of $\textbf{X}$ is $k_X = 1.0 \times 10^9 \text{ M}^{-1}\text{s}^{-1}$. The intrinsic second-order rate constant for the attack on the neutral free amine form of $\textbf{Y}$ is $k_Y = 2.0 \times 10^4 \text{ M}^{-1}\text{s}^{-1}$.

Assuming that coupling with neutral $\textbf{X}$ and protonated $\textbf{Y}$ are negligible, at what pH will the initial rate of formation of the azo product from $\textbf{X}$ be exactly equal to that from $\textbf{Y}$?

(Given: $K_w = 10^{-14}$ at $298\text{ K}$, $\log 2 = 0.30$, $\log 5 = 0.70$)