Question

The half life of a radioactive isotope $X$ is $20$ years. It decays to another element $Y$ which is stable. The two elements $X$ and $Y$ were found to be in the ratio $1:7$ in a sample of a given rock. The age of the rock is estimated to be

• A. $100\text{years}$
• B. $40\text{years}$
• C. $60\text{years}$
• D. $80\text{years}$

Answer 1

The correct option is C $60\text{years}$

Given that,

$X\to Y\text{(stable)}$

Let $N_x$ be the number of active nuclei of $X$ and $N_y$ be the number of active nuclei of $Y$.

Given that, $\frac{N_x}{N_y}=\frac{1}{7}$

$\Rightarrow \frac{N_x}{N_x+N_y}=\frac{N}{N_0}=\frac{1}{8}$

$\Rightarrow \frac{N}{N_0}=\frac{1}{2^3}=\frac{1}{2^n}$

$\therefore n=3$

So, age of the rock will be,

$t=n{t}_{1/2}=3\times 20=60\text{years}$

Hence, option $\left(\text{C}\right)$ is correct.