For a first order reaction A 5 → 5 B 2 , the concentrations vs time plot is as shown. The half-life of…

The half-life of a reaction is the time required for the reactant concentration to decrease to one-half its initial value. The half-life of a first-order reaction is a constant that is related to the rate constant for the reaction: ${\mathrm{t}}_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} = 0.693/\mathrm{k}.$

$\mathrm{k}=\frac{2.303}{\mathrm{t}}\log \frac{\mathrm{a}}{\mathrm{a}-\mathrm{x}}$

$$\begin{gathered} {\mathrm{A}}_5\to 5{\mathrm{B}}_2 \\ \mathrm{initial}: \mathrm{a} 0 \\ \mathrm{at} \mathrm{time}=\mathrm{t} \mathrm{a}-\mathrm{x} 5\mathrm{x} \end{gathered}$$

$\mathrm{a}=6\mathrm{x}$

$$\begin{gathered} \mathrm{k}=\frac{2.303}{30}\log \frac{6\mathrm{x}}{5\mathrm{x}} \\ =0.006 {\min }^{-1} \end{gathered}$$

${\mathrm{t}}_{1/2}=\frac{0.693}{0.006}\simeq 114 \min$

Note: In the question paper 140 minutes given as the 4th option. but answer is 114 minutes.