Question
A first order reaction takes $$40\,\min $$ for $${\text{30% }}$$ decomposition. What will be $${{\text{t}}_{\frac{1}{2}}}?$$
A.
77.7 $$\min .$$
B.
52.5 $$\min .$$
C.
46.2 $$\min .$$
D.
22.7 $$\min .$$
Answer :
77.7 $$\min .$$
Solution :
$$30\% \,\,{\text{decomposition means}}$$ $$X = 30\% \,\,{\text{of}}\,\,{R_0}$$
\[\begin{align}
& \text{or}\,\,R={{R}_{0}}-0.3{{R}_{0}}=0.7{{R}_{0}} \\
& \text{For first order,} \\
& k=\frac{2.303}{t}\log \frac{\left[ {{R}_{0}} \right]}{\left[ R \right]} \\
& \,\,\,=\frac{2.303}{40}\log \frac{10}{7}{{\min }^{-1}} \\
& \,\,\,=8.918\times {{10}^{-3}}\,\min {{.}^{-1}} \\
& {{t}_{\frac{1}{2}}}=\frac{0.693}{k} \\
& \,\,\,\,\,\,=\frac{0.693}{8.918\times {{10}^{-3}}\min {{.}^{-1}}} \\
& \,\,\,\,\,\,=77.7\,\min . \\
\end{align}\]