Question

Which of the following describes the correct relation for $pOH$ for an aqueous salt solution of weak acid and weak base at ${25}^{\circ }C?$

• A. $pOH=14-\frac{1}{2}(p{K}_a-p{K}_b)$
• B. $pOH=7+\frac{1}{2}(p{K}_a-p{K}_b)$
• C. $pOH=7-\frac{1}{2}(p{K}_a-p{K}_b)$
• D. $pOH=14+\frac{1}{2}(p{K}_a-p{K}_b)$

Answer 1

The correct option is C $pOH=7-\frac{1}{2}(p{K}_a-p{K}_b)$

Theory:
Salt of weak acid and weak base (WA) and (WB) :
$C{H}_{3}COOH+N{H}_{4}OH⟶C{H}_{3}COON{H}_4$
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 undissociated form.
$K_h=\frac{\left[N{H}_{4}OH\right]\left[C{H}_{3}COOH\right]}{\left[N{H}_4^{+}\right]\left[C{H}_{3}CO{O}^{-}\right]}$
$\frac{K_w}{K_a{K}_b}=K_h=\frac{\left[H_3{O}^{+}\right]\left[O{H}^{-}\right]}{\frac{\left[C{H}_{3}CO{O}^{-}\right]\left[H_3{O}^{+}\right]}{\left[C{H}_{3}COOH\right]}\times \frac{\left[N{H}_4^{+}\right]\left[O{H}^{-}\right]}{\left[N{H}_{4}OH\right]}}$

$K_h=\frac{\left(Ch\right)\left(Ch\right)}{C^2(1-h{)}^2}=\frac{h^2}{(1-h{)}^2}$
$\frac{h}{1-h}=\sqrt{K_h}$
Finding pH:
consider weak acid dissociation,
$K_a=\frac{\left[C{H}_{3}CO{O}^{-}\right]\left[H^{+}\right]}{\left[C{H}_{3}COOH\right]}⟶\frac{C(1-h)\left[H^{+}\right]}{Ch}$
$\left[H^{+}\right]=K_a\frac{h}{1-h}=K_a\sqrt{K_h}$
$\left[H^{+}\right]=K_a\sqrt{\frac{K_w}{K_a{K}_b}}$

$\left[H^{+}\right]=\sqrt{\frac{K_w{K}_a}{K_b}}$
$log\left[H^{+}\right]=\frac{1}{2}log{K}_w+\frac{1}{2}log{K}_a-\frac{1}{2}log{K}_b$
$-log\left[H^{+}\right]=-\frac{1}{2}log{K}_w-\frac{1}{2}log{K}_a+\frac{1}{2}log{K}_b$
$pH=\frac{1}{2}(p{K}_w+p{K}_a-p{K}_b)$
valid only if h<0.1 or 10% or when $C/K_h>100$
so, $pH=7+\frac{1}{2}(p{K}_a-p{K}_b)pOH=14-pHpOH=14-(7+\frac{1}{2}(p{K}_a-p{K}_b)pOH=7-\frac{1}{2}(p{K}_a-p{K}_b)at{25}^{\circ }C$