Question

Which of the following describes the correct relation for hydrolysis constant $\left(K_h\right)$ for a salt of weak acid and weak base at ${25}^{\circ }C?$

• A. $K_h=\frac{K_w}{K_a\times K_b}$
• B. $K_h=\frac{K_a}{K_w\times K_b}$
• C. $K_h=\frac{K_a\times K_b}{K_w}$
• D. $K_b=\frac{K_h}{K_a\times K_w}$

Answer 1

The correct option is A $K_h=\frac{K_w}{K_a\times K_b}$

Salt of weak acid and weak base (WA) and (WB) :
$C{H}_{3}COOH$ weak acid + $N{H}_{4}OH$ weak base $⟶C{H}_{3}COON{H}_4$
Partial dissociation (Close to 100% but not exact)
$C{H}_{3}COON{H}_4\left(aq\right)⟶N{H}_4^{+}\left(aq\right)+C{H}_{3}CO{O}^{-}\left(aq\right)$
Hydrolysis :
$N{H}_4^{+}\left(aq\right)+C{H}_{3}CO{O}^{-}\left(aq\right)+H_{2}O\left(l\right)\rightleftharpoons C{H}_{3}COOH\left(aq\right)+N{H}_{4}OH\left(aq\right)$
$CC00$
$C-ChC-ChChCh$
Since both the acid and the base are weak, they will exist in dissociated form.
$K_h=\frac{\left[N{H}_{4}OH\right]\left[C{H}_{3}COOH\right]}{\left[N{H}_4^{+}\right]\left[C{H}_{3}CO{O}^{-}\right]}$

Finding $\frac{K_w}{K_a{K}_b}$

$\frac{K_w}{K_a{K}_b}=\frac{\left[H_3{O}^{+}\right]\left[O{H}^{-}\right]}{\frac{\left[C{H}_{3}CO{O}^{-}\right]\left[H_3{O}^{+}\right]}{\left[C{H}_{3}COOH\right]}\times \frac{\left[N{H}_4^{+}\right]\left[O{H}^{-}\right]}{\left[N{H}_{4}OH\right]}}\frac{K_w}{K_a{K}_b}=\frac{\left[N{H}_{4}OH\right]\left[C{H}_{3}COOH\right]}{\left[N{H}_4^{+}\right]\left[C{H}_{3}CO{O}^{-}\right]}$
so, $\frac{K_w}{K_a{K}_b}=K_h$