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A sample of a radioactive element whose half-life is 30 s contains a million nuclei at a certain instant of time. How many nuclei will be left after $10 \mathrm{~s}$ ?
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Verified Answer
The correct answer is:
$7.96 \times 10^{5}$
Half-life of radioactive sample,
$$
T_{1 / 2}=30 \mathrm{~s}, N_{0}=10^{6}
$$
If $N$ be the remaining number of nuclei after $t=10 \mathrm{~s}$, then according to radioactive decay law,
$$
\begin{aligned}
N &=N_{0}\left(\frac{1}{2}\right)^{n}=10^{6}\left(\frac{1}{2}\right)^{t / T_{1 / 2}} \\
&=10^{6}\left(\frac{1}{2}\right)^{\frac{10}{30}}=10^{6}\left(\frac{1}{2}\right)^{1 / 3} \\
&=10^{6}(0.796)=7.96 \times 10^{5}
\end{aligned}
$$
$$
T_{1 / 2}=30 \mathrm{~s}, N_{0}=10^{6}
$$
If $N$ be the remaining number of nuclei after $t=10 \mathrm{~s}$, then according to radioactive decay law,
$$
\begin{aligned}
N &=N_{0}\left(\frac{1}{2}\right)^{n}=10^{6}\left(\frac{1}{2}\right)^{t / T_{1 / 2}} \\
&=10^{6}\left(\frac{1}{2}\right)^{\frac{10}{30}}=10^{6}\left(\frac{1}{2}\right)^{1 / 3} \\
&=10^{6}(0.796)=7.96 \times 10^{5}
\end{aligned}
$$
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