Question

Two radioactive substances $X$ and $Y$ originally have $N_1$ and $N_2$ nuclei, respectively. The half-life of $X$ is half of the half-life of $Y$. After three half-lives of $Y$, numbers of nuclei of both are equal, the ratio $\frac{N_1}{N_2}$ will be equal to :

• A. $\frac{8}{1}$
• B. $\frac{1}{8}$
• C. $\frac{3}{1}$
• D. $\frac{1}{3}$

Answer 1

The correct option is A $\frac{8}{1}$

We know that, after $n$ half-life, number of nuclei undecayed,

$N=\frac{N_o}{2^n}$

Given:
$T_{1/2}\text{of X}=\frac{1}{2}{T}_{1/2}\text{of Y}$

$\Rightarrow T_{1/2}\text{of Y}=2T_{1/2}\text{of X}$

$\Rightarrow 3T_{1/2}\text{of Y}=6T_{1/2}\text{of X}$

$\therefore \frac{N_2}{2^3}=\frac{N_1}{2^6}$

$\Rightarrow \frac{N_1}{N_2}=\frac{8}{1}$