A radio isotope ‘ X ’ with a half life 1.4 × 10 9 years decays to ‘ Y ’ which is stable. A sample of the…

Given a radioactive isotope decays into a stable element,

$X\to Y$

At time $t=0$, $\left[x\right]= N_0$, $\left[y\right]=0$

At time $t=t$, $\left[x\right]=N_0-x$, $\left[y\right]=x$

$\frac{N_0-x}{x}=\frac{1}{7}\Rightarrow x=\frac{7N_0}{8}$
So, remaining nuclei of $X=N_0-x=\frac{N_0}{8}=\frac{N_0}{2^3}$
So, three half-life periods would have been passed,
$\Rightarrow t=3t_{\frac{1}{2}}=3\times 1.4\times {10}^9=4.2\times {10}^9$ years.