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A sample of radioactive element contains $4 \times 10^{16}$ active nuclei. The half-life of the element is 10 decays, then number of decayed nuclei after 30 days is:
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Verified Answer
The correct answer is:
$3.5 \times 10^{16}$
Half life number is given by $n=\frac{1}{T}$ $=\frac{30}{10}=3$
Nuclear of active nuclei left after $\mathrm{n}$ half-life is
$$
\begin{aligned}
& N=N_0\left(\frac{1}{2}\right)^n \\
= & \frac{4 \times 10^{16}}{(2)^3}=0.5 \times 10^{16} \\
\therefore & \text { Decay nuclei }=N_0-N \\
= & 4 \times 10^{16}-0.5 \times 10^{16} \\
= & 3.5 \times 10^{16}
\end{aligned}
$$
Nuclear of active nuclei left after $\mathrm{n}$ half-life is
$$
\begin{aligned}
& N=N_0\left(\frac{1}{2}\right)^n \\
= & \frac{4 \times 10^{16}}{(2)^3}=0.5 \times 10^{16} \\
\therefore & \text { Decay nuclei }=N_0-N \\
= & 4 \times 10^{16}-0.5 \times 10^{16} \\
= & 3.5 \times 10^{16}
\end{aligned}
$$
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