Question

If the half-life of any sample of radioactive substance is $4$ days, then the fraction of sample will remain undecayed after $2$ days, will be?

• A. $ \sqrt{2} $
• B. $ \frac{1}{\sqrt{2}} $
• C. $ \frac{\sqrt{2}-1}{\sqrt{2}} $
• D. $ \frac{1}{2} $

Answer 1

Answer: (B)

From Rutherford-Soddy law, the fraction left after $n$ half-lives is
$N=N_{0}\left(\frac{1}{2}\right)^{n}$
where $n$ is number of half-lives.
$n=\frac{\text { time }(t)}{\text { half-life }\left(T_{1 / 2}\right)}=\frac{2}{4}=\frac{1}{2} $
$\therefore \frac{N}{N_{0}}=\left(\frac{1}{2}\right)^{1 / 2} $
$\Rightarrow \frac{N}{N_{0}}=\frac{1}{\sqrt{2}}$