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Q.
The initial rate of hydrolysis of methyl acetate $(1 M)$ by a weak acid $(H A, 1 M)$ is $1 / 100$ th of that of a strong acid $(H X, 1 M)$, at $25^{\circ} C .$ The $K_{a}(H A)$ is
$R C O O R'+H _2O \xrightarrow{H^{+}} R C O O H+R'O H$
Acid hydrolysis of ester is follows first order kinetics. For same concentration of ester in each case, rate is dependent on $\left[H^{+}\right]$from acid.
Rate $= k [RCOOR' ]$
Also for weak acid, $HA \rightleftharpoons H^{+} + A^{-}$
$K_{a}=\frac{\left[H^{+}\right]\left[A^{-}\right]}{[H A]}$
$(\text { Rate })_{H A}=k\left[H^{+}\right]_{H A}$
$(\text { Rate })_{H x}=k\left[H^{+}\right]_{H X}$
$(\text { Rate })_{H X}=100($ Rate $)_{H A}$
$\therefore $ Also in strong acid, $\left[H^{+}\right]=[H X]=1 M$
$\frac{(\text { Rate })_{H X}}{(\text { Rate })_{H A}}=100=\frac{\left[H^{+}\right]_{H X}}{\left[H^{+}\right]_{H A}}=\frac{1}{\left[H^{+}\right]_{H A}}$
$\therefore \left[H^{+}\right]_{H A}=\frac{1}{100}$
$\therefore K_{a}=\frac{\left[H^{+}\right]\left[A^{-}\right]}{[H A]}=\frac{0.01 \times 0.01}{0.99}$
$1 \times 10^{-4}$
