Question 8.E.K: A solution with an ionic strength of 0.10 M containing 0.010...
A solution with an ionic strength of 0.10 M containing 0.010 0 M phenylhydrazine has a pH of 8.13. Using activity coefficients correctly, find pK_{a} for the phenylhydrazinium ion found in phenylhydrazine hydrochloride. Assume that \gamma_{BH^{+}} = 0.80.
The reaction of phenylhydrazine with water is
B + H_{2}O \rightleftharpoons BH^{+} + OH^{−} K_{b}
We know that pH = 8.13, so we can find [OH^{−}].
[OH^{−}] = \frac{\mathcal{A}_{OH^{−}}}{\gamma_{OH^{−}}} = \frac{K_{w}/10^{−pH}}{\gamma_{OH^{−}}} = 1.78 × 10^{−6} M
(using \gamma_{OH^{−}} = 0.76 for μ = 0.10 M)
K_{b} = \frac{[BH^{+}]\gamma_{BH^{+}}[OH^{−}]\gamma_{OH^{−}}}{[B]\gamma_{B}}= \frac{(1.78 × 10^{−6})(0.80)(1.78 × 10^{−6})(0.76)}{[0.010 − (1.78 × 10^{−6})](1.00)}
= 1.93 × 10^{−10}
K_{a} = \frac{K_{w}}{K_{b}} = 5.19 × 10^{−5} ⇒ pK_{a} = 4.28
To find K_{b}, we made use of the equality [BH^{+}] = [OH^{−}].