Question

A radioactive isotope $X$, having a half life $1.4\times {10}^9$ years, decays to $Y$ which is stable. A rock sample from a cave was found to contain $X$ and $Y$ in the ratio $1:7$. The age of the rock is,

• A. $1.9\times {10}^9$ years
• B. $8.40\times {10}^9$ years
• C. $3.92\times {10}^9$ years
• D. $4.20\times {10}^9$ years

Answer 1

The correct option is D $4.20\times {10}^9$ years

The ratio of $X$ and $Y$, after time $t$, is $1:7$

$\therefore \frac{\text{no. of atoms of X after time t}}{\text{total no. of atoms initially}}=\frac{1}{8}$

$\therefore$ The fraction of the original sample remaining is,
$=\frac{1}{8}=\frac{1}{2^3}$

$\Rightarrow \frac{N}{N_0}=\frac{1}{8}=\frac{1}{2^n}$

$\Rightarrow n=3$

Now, $t=nT=3\times 1.4\times {10}^9$
$\therefore t=4.2\times {10}^9$ years

Hence, $\left(D\right)$ is the correct answer.