Question

The fraction of a radioactive sample that will decay during half of its half-life period, is,

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

Answer 1

The correct option is D $\frac{\sqrt{2}-1}{\sqrt{2}}$


Here, $t=\frac{T}{2}$
We Know that,
$t=nT$
$\frac{T}{2}=nT$
$\therefore n=\frac{1}{2}$
The fraction remaining is given by, $\frac{N}{N_0}=\frac{1}{2^n}=\frac{1}{2^{\frac{1}{2}}}=\frac{1}{\sqrt{2}}$

So, the fraction decayed is,

$=\frac{N_0-N}{N_0}=1-\frac{N}{N_0}$

$=1-\frac{1}{\sqrt{2}}=\frac{\sqrt{2}-1}{\sqrt{2}}$

Hence, $\left(D\right)$ is the correct answer.