There are 10¹⁰ radioactive nuclei in a given radioactive element. Its half-life time is 1min. How many nuclei will remain after 30s? $(\sqrt{2}=1.414)$

Initial number of radioactive nuclei, ${\mathrm{N}}_0={10}^{10}$

Half-life, t_1/2 = 1 min = 60 s

Time, t = 30 s

We know that, number of nuclei remaining in radioactive decay after time t is given by

    $\frac{\mathrm{N}}{{\mathrm{N}}_0}={\left(\frac{1}{2}\right)}^{\mathrm{n}}={\left(\frac{1}{2}\right)}^{\frac{\mathrm{t}}{{\mathrm{t}}_{1/2}}}$

Substituting the values in above relation, we get

$\begin{array}{l} \frac{\mathrm{N}}{{10}^{10}}={\left(\frac{1}{2}\right)}^{30/60}={\left(\frac{1}{2}\right)}^{1/2}=\frac{1}{\sqrt{2}}\\ \Rightarrow \mathrm{N}=\frac{1}{\sqrt{2}}\times {10}^{10}=0.707\times {10}^{10}\\ \Rightarrow \mathrm{N}\approx 7\times {10}^9\end{array}$

Thus, the radioactive nuclei remaining after 30 s is $7\times {10}^9$.