Question #44117

A $100 \text{ mL}$ aliquot of $0.1 \text{ M}$ diprotic acid $H_2A$ ($pK_{a1} = 4.0$, $pK_{a2} = 8.0$) is titrated by adding drops of concentrated $10 \text{ M } NaOH$ at $298 \text{ K}$ (assume negligible volume change). The solution also contains $0.1 \text{ M } AgNO_3$. To detect the first equivalence point (formation of the amphiprotic ion $HA^-$), an acidic indicator $HIn$ ($pK_{In} = 6.0$) is added, making its total analytical concentration $1.0 \times 10^{-4} \text{ M}$. The indicator anion $In^-$ forms a sparingly soluble red precipitate with $Ag^+$, $\ce{AgIn(s)}$, which acts as the visual endpoint. Given $K_{sp}(\ce{AgIn}) = 1.0 \times 10^{-8}$. Assume $Ag^+$ does not precipitate or complex with any species of $A$. Calculate the ratio of $[HA^-]/[H_2A]$ at the moment the red precipitate first appears (apparent endpoint), and the theoretical pH of the first equivalence point.