PYQ NEET- Atoms And Nuclei L-10

Question: The half life of a radioactive isotope ’ $\mathrm{X}$ ’ 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) 80 years

B) 100 years

C) 40 years

D) 60 years

Answer: (D) 60 years#

Sol:#

Explanation Initial number of atoms of $\mathrm{X}$ is $\mathrm{N}_0$ and Initial number of atoms of $Y$ is 0

Number of atoms after time $\mathrm{t}$, for $\mathrm{X}$ is $\mathrm{N}$ and for $\mathrm{Y}$ is $\mathrm{N}_0-\mathrm{N}$

According to the question, $$ \begin{aligned} & \frac{N}{N_0-N}=\frac{1}{7} \ & \Rightarrow \frac{N}{N_0}=\frac{1}{8} \end{aligned} $$

As $\frac{N}{N_0}=\left(\frac{1}{2}\right)^n$ where $\mathrm{n}$ is the no. of half lives $$ \begin{aligned} & \therefore\left(\frac{1}{2}\right)^n=\frac{1}{8} \ & \Rightarrow \mathrm{n}=3 \ & \mathrm{t}=\mathrm{nT}_{1 / 2}=3 \times 20=60 \text { years } \end{aligned} $$

Hence, the age of rock is 60 years.