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A first order reaction, which is $30 \%$ complete in 30 minutes has a half-life period of
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Verified Answer
The correct answer is:
$58.2 \mathrm{~min}$
For first order reaction,
$\begin{aligned} & t=\frac{2.303}{k} \log \frac{a}{a-x} \Rightarrow \frac{t_{30 \%}}{t_{50 \%}}=\frac{\frac{2.303}{k} \log \frac{100}{100-30}}{\frac{2.303}{k} \log \frac{100}{100-50}} \\ & \Rightarrow \frac{30}{t_{1 / 2}}=\frac{\log \frac{10}{7}}{\log 2} \Rightarrow \frac{30}{t_{1 / 2}}=\frac{0.1549}{0.3010} \\ & \Rightarrow t_{1 / 2}=\frac{0.3010 \times 30}{0.1549}=58.29 \mathrm{~min}\end{aligned}$
$\begin{aligned} & t=\frac{2.303}{k} \log \frac{a}{a-x} \Rightarrow \frac{t_{30 \%}}{t_{50 \%}}=\frac{\frac{2.303}{k} \log \frac{100}{100-30}}{\frac{2.303}{k} \log \frac{100}{100-50}} \\ & \Rightarrow \frac{30}{t_{1 / 2}}=\frac{\log \frac{10}{7}}{\log 2} \Rightarrow \frac{30}{t_{1 / 2}}=\frac{0.1549}{0.3010} \\ & \Rightarrow t_{1 / 2}=\frac{0.3010 \times 30}{0.1549}=58.29 \mathrm{~min}\end{aligned}$
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