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A count rate meter shows a count of 240 per min from a given radioactive source. One hour later the meter shows a count rate of 30 per min. The half-life of the source is
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$20 \mathrm{~min}$
Let $N$ be the amount of substance left,
$N_{0}$ be the total amount of substance.
The amount of substance left after $N$ half-life is given by $\quad \frac{N}{N_{0}}=\frac{1}{2^{n}}$
Hence, $\quad \frac{30}{240}=\frac{1}{2^{n}}$
(Since, decay rate proportional to amount of substance)
$$
\Rightarrow \quad 2^{n}=8 \Rightarrow n=3
$$
Number of half-life passed $=3$
Time taken for the substance to decay from 240 count rate to 30 count rate $=1 \mathrm{~h}=60 \mathrm{~min}$
$$
\begin{aligned}
\text { Half-life } &=\frac{\text { Total time }}{\text { Number of half - life }} \\
t_{1 / 2} &=\frac{60}{3}=20 \mathrm{~min}
\end{aligned}
$$
$N_{0}$ be the total amount of substance.
The amount of substance left after $N$ half-life is given by $\quad \frac{N}{N_{0}}=\frac{1}{2^{n}}$
Hence, $\quad \frac{30}{240}=\frac{1}{2^{n}}$
(Since, decay rate proportional to amount of substance)
$$
\Rightarrow \quad 2^{n}=8 \Rightarrow n=3
$$
Number of half-life passed $=3$
Time taken for the substance to decay from 240 count rate to 30 count rate $=1 \mathrm{~h}=60 \mathrm{~min}$
$$
\begin{aligned}
\text { Half-life } &=\frac{\text { Total time }}{\text { Number of half - life }} \\
t_{1 / 2} &=\frac{60}{3}=20 \mathrm{~min}
\end{aligned}
$$
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