JEE CHEMISTRY MCQ: CHEMICAL KINETICS

The acid catalysed reaction of acetic acid with ethanol:

$\mathrm{CH}_3 \mathrm{COOH}+\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH} \rightarrow \mathrm{CH}_3 \mathrm{COOC}_2 \mathrm{H}_5 +\mathrm{H}_2 \mathrm{O}$

follows the rate law $-\frac{\mathrm{d}\left[\mathrm{CH}_3 \mathrm{COOH}\right]}{\mathrm{d} t}=K\left[\mathrm{H}^{+}\right]\left[\mathrm{CH}_3 \mathrm{COOH}\right] \left[\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right]$ $=K^{\prime} \left[\mathrm{CH}_3 \mathrm{COOH}\right] \left[\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right]$. When $\left[\mathrm{CH}_3 \mathrm{COOH}\right]_0=\left[\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right]_0=0.2 \mathrm{~M}$ and $\mathrm{pH}=3,$ the half-life for the reaction is 50 minutes. The value of true rate constant, $K$, of the reaction is