Question #44078

The synthesis of a Schiff base via the condensation of benzaldehyde and a primary amine $\ce{RNH2}$ ($K_b = 1.0 \times 10^{-9}$) is highly pH-dependent. The reaction is conducted at $298\text{ K}$ in an aqueous solution initially containing only a weak acid $\ce{HA}$ ($K_a = 1.0 \times 10^{-5}$) at a concentration of $1.0 \times 10^{-5}\text{ M}$. A trace amount of the amine and benzaldehyde is added such that their presence does not perturb the bulk $[\ce{H+}]$ established by $\ce{HA}$.

The concentration of the active nucleophile depends strictly on the fraction of the amine in its unprotonated state. Calculate the true percentage of the amine available for nucleophilic attack, and the percentage a student would calculate if they erroneously applied the standard weak electrolyte approximation ($1 - \alpha \approx 1$) to determine the pH of the $\ce{HA}$ solution. (Given: $\sqrt{5} = 2.236$, $K_w = 1.0 \times 10^{-14}$)