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Acid hydrolysis of ester is first order reaction and rate constant is given by $k=\frac{2.303}{t} \log \frac{V_{\infty}-V_{0}}{V_{\infty}-V_{t}}$ where, $V_{0}, V_{t}$ and $V_{\infty}$ are the volumle of standard $\mathrm{NaOH}$ required to neutralise acid present at a given time, if ester is $50 \%$ neutralised then
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$V_{\infty}=2 V_{t}-V_{0}$
$\mathrm{RCOOR}^{\prime}+\mathrm{H}_{2} \mathrm{O} \stackrel{\mathrm{H}^{+}}{\longrightarrow} \mathrm{RCOOH}+\mathrm{R}^{\prime} \mathrm{OH}$
$\begin{array}{lccc}\text { At } \mathrm{t}=0, & \text { a } & 0 & 0 \\ \text { At timet, } & \mathrm{a}-\mathrm{x} & \mathrm{x} & \mathrm{x} \\ \text { At time } \infty, & \mathrm{a}-\mathrm{a} & \mathrm{a} & \mathrm{a} \\ \text { At } \mathrm{t}=0, \mathrm{~V}_{0}=\text { volume } & \mathrm{Va}_{\mathrm{t}}=\mathrm{x}+\mathrm{V}_{0} & & \text { (catalyst) }\end{array}$
$\mathrm{V}_{\infty}=\mathrm{a}+\mathrm{V}_{0}$
If ester is $50 \%$ hydrolysed then, $x=\frac{a}{2}$
$\begin{array}{l}
\text { or } \mathrm{V}_{\mathrm{t}}=\frac{\mathrm{a}}{2}+\mathrm{V}_{0} \\
\text { or } \mathrm{a}=2 \mathrm{~V}_{\mathrm{t}}-2 \mathrm{~V}_{0} \\
\therefore \mathrm{V}_{\infty}=2 \mathrm{~V}_{\mathrm{t}}-2 \mathrm{~V}_{0}+\mathrm{V}_{0} \\
=2 \mathrm{~V}_{\mathrm{t}}-\mathrm{V}_{0}
\end{array}$
$\begin{array}{lccc}\text { At } \mathrm{t}=0, & \text { a } & 0 & 0 \\ \text { At timet, } & \mathrm{a}-\mathrm{x} & \mathrm{x} & \mathrm{x} \\ \text { At time } \infty, & \mathrm{a}-\mathrm{a} & \mathrm{a} & \mathrm{a} \\ \text { At } \mathrm{t}=0, \mathrm{~V}_{0}=\text { volume } & \mathrm{Va}_{\mathrm{t}}=\mathrm{x}+\mathrm{V}_{0} & & \text { (catalyst) }\end{array}$
$\mathrm{V}_{\infty}=\mathrm{a}+\mathrm{V}_{0}$
If ester is $50 \%$ hydrolysed then, $x=\frac{a}{2}$
$\begin{array}{l}
\text { or } \mathrm{V}_{\mathrm{t}}=\frac{\mathrm{a}}{2}+\mathrm{V}_{0} \\
\text { or } \mathrm{a}=2 \mathrm{~V}_{\mathrm{t}}-2 \mathrm{~V}_{0} \\
\therefore \mathrm{V}_{\infty}=2 \mathrm{~V}_{\mathrm{t}}-2 \mathrm{~V}_{0}+\mathrm{V}_{0} \\
=2 \mathrm{~V}_{\mathrm{t}}-\mathrm{V}_{0}
\end{array}$
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