Question

Two samples $X$ and $Y$ contain equal amount of radioactive substances. If $\left(\frac{1}{16}\right)^{\text {th }}$ of the sample $X$ and $\left(\frac{1}{256}\right)^{\text {th }}$ of the sample $Y$, remain after $8$ hours, then the ratio of half periods of $X$ and $Y$ is

• A. 2 : 1
• B. 1 : 2
• C. 1 : 4
• D. 4 : 1

Answer 1

Answer: (A)

As $\frac{N}{N_{0}}=\left(\frac{1}{2}\right)^{n}$
where, Number of half lives, $n=\frac{t}{T}$
$T$ is the half life period.
For $X$ sample
$\frac{1}{16}=\left(\frac{1}{2}\right)^{8 / T_{X}}$ or
$\left(\frac{1}{2}\right)^{4}=\left(\frac{1}{2}\right)^{8 / T_{X}}$ or
$4=\frac{8}{T_{X}} \dots$(i)
For $Y$ sample
$\left(\frac{1}{256}\right)=\left(\frac{1}{2}\right)^{8 / T_{Y}}$ or
$\left(\frac{1}{2}\right)^{8}=\left(\frac{1}{2}\right)^{8 / T_{Y}}$ or
$8=\frac{8}{T_{Y}} \dots$(ii)
Divide (i) by (ii) we get
$\frac{4}{8}=\frac{8}{T_{X}} \times \frac{T_{Y}}{8} $
$\Rightarrow \frac{1}{2}=\frac{T_{Y}}{T_{X}}$ or
$\frac{T_{X}}{T_{Y}}=\frac{2}{1}$