Question

The fraction of a radioactive sample 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 $\frac{\sqrt{2}-1}{\sqrt{2}}$

For time, $t=\frac{T_{1/2}}{2}$

Number of half times,

$n=\frac{t}{T_{1/2}}=\frac{\frac{T_{1/2}}{2}}{T_{1/2}}$

$\therefore n=1/2$

The fraction remaining,

$\frac{N}{N_0}=\frac{1}{2^n}=\frac{1}{2^{1/2}}=\frac{1}{\sqrt{2}}$

The fraction decayed,

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

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

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