Question

A radioactive element $X$ converts into another stable element $Y$. Half-life of $X$ is $2h$. 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 is

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

Answer 1

The correct option is C between $4$ and $6$

Assume initial composition to contain only $X\left(N\right)$.
Given that one half life is $2$ hours.
After one half life.
Amount of $X$ remaining is $=\frac{1}{2}N$
Amount of $Y$ is $=\frac{1}{2}N$
Hence ratio is $1:1$
After two half lives.
Amount of $X$ remaining is $=\frac{1}{4}N$
Amount of $Y$ is $=\frac{3}{4}N$
Hence ratio is $1:3$
After three half lives.
Amount of $X$ remaining is $=\frac{1}{8}N$
Amount of $Y$ is $=\frac{7}{8}N$
Hence ratio is $1:7$
The ratio of $1:4$ is obtained somewhere between two and three half lives. Since each half life is $2$ hours long. Some where between $4$ to $6$ hours the ratio of $1:4$ is obtained.