Chemical and Ionic Equilibrium 1 Question 29
30. The initial rate of hydrolysis of methyl acetate $(1 \mathrm{M})$ by a weak acid $(\mathrm{H} A, 1 \mathrm{M})$ is $1 / 100$ th of that of a strong acid $(\mathrm{H} X, 1 \mathrm{M})$, at $25^{\circ} \mathrm{C}$. The $K_{a}(\mathrm{H} A)$ is
(2013 Adv.)
(a) $1 \times 10^{-4}$
(b) $1 \times 10^{-5}$
(c) $1 \times 10^{-6}$
(d) $1 \times 10^{-3}$
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Solution:
- PLAN $R \mathrm{COO} R^{\prime}+\mathrm{H}_{2} \mathrm{O} \xrightarrow{\mathrm{H}^{+}} R \mathrm{COOH}+R^{\prime} \mathrm{OH}$
Acid hydrolysis of ester is follows first order kinetics.
For same concentration of ester in each case, rate is dependent on $\left[\mathrm{H}^{+}\right]$from acid.
$$ \text { Rate }=k\left[R \mathrm{COOR}^{\prime}\right] $$
Also for weak acid, $\mathrm{H} A \rightleftharpoons \mathrm{H}^{+}+A^{-}$
$$ \begin{aligned} K_{a} & =\frac{\left[\mathrm{H}^{+}\right]\left[A^{-}\right]}{[\mathrm{H} A]} \ (\text { Rate }){\mathrm{H} A} & =k\left[\mathrm{H}^{+}\right]{\mathrm{H} A} \ (\text { Rate }){\mathrm{H} X} & =k\left[\mathrm{H}^{+}\right]{\mathrm{H} X} \ (\text { Rate }){\mathrm{H} X} & =100(\text { Rate }){\mathrm{H} A} \end{aligned} $$
$\therefore$ Also in strong acid, $\left[\mathrm{H}^{+}\right]=[\mathrm{H} X]=1 \mathrm{M}$
$$ \begin{aligned} & \frac{(\text { Rate }){\mathrm{H} X}}{(\text { Rate }){\mathrm{H} A}}=100=\frac{\left[\mathrm{H}^{+}\right]{\mathrm{H} X}}{\left[\mathrm{H}^{+}\right]{\mathrm{H} A}}=\frac{1}{\left[\mathrm{H}^{+}\right]{\mathrm{H} A}} \ & \therefore {\left[\mathrm{H}^{+}\right]{\mathrm{H} A}=\frac{1}{100} } \ & \mathrm{H} A \rightleftharpoons \mathrm{H}^{+}+A^{-} \ & 1 \quad 0 \quad 0 \ &(1-x) \quad x \quad x \ & \therefore \quad \quad K_{a}=\frac{\left[\mathrm{H}^{+}\right]\left[A^{-}\right]}{[\mathrm{H} A]}=\frac{0.01 \times 0.01}{0.99}=1.01 \times 10^{-4} \end{aligned} $$
