Question

A radioactive element $X$ converts into another stable element $Y$. Half life of $X$ is $2hrs$. Initially only $X$ is present. After time $t$, the ratio of atoms of $X$ and $Y$ is found to be $1:4$, then $t$ in hours:

• A. $2$
• B. $4$
• C. Between $4$ and $6$
• D. $6$

Answer 1

Answer: (C)

Between $4$ and $6$

Let $N_0$ be the number of atoms of $X$ at time $t=0$.
Then at $t=4hrs$ (two half lives)
$N_x=\frac{N_0}{4}$ and $N_y=\frac{3N_0}{4}$
$\therefore N_x/N_y=1/3$
and at $t=6$ hrs (three half lives)
$N_x=\frac{N_0}{8}$ and $N_y=\frac{7N_0}{8}$ or $\frac{N_x}{N_y}=\frac{1}{7}$
The given ratio $\frac{1}{4}$ lies between $\frac{1}{3}$ and $\frac{1}{7}$.
Therefore, $t$ lies between $4$ hrs and $6$ hrs.