Question

If $T$ is the half life of a radioactive material, then the fraction that decay after a time $\frac{T}{2}$is

• A. $\frac{1}{2}$
• B. $\frac{3}{4}$
• C. $\frac{1}{\sqrt{2}}$
• D. $\frac{\sqrt{2}-1}{\sqrt{2}}$

Answer 1

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

$N=N_O{e}^{-\lambda t}$

$\Rightarrow N=N_O{e}^{\frac{-\ln 2}{T}\times \frac{t}{2}}$

$\Rightarrow N=N_O{e}^{-\ln 2^{\frac{1}{2}}}$

$\Rightarrow N=\frac{N_O}{\sqrt{2}}$

$\therefore$ remaining $=N_O-N$

$N_O-\frac{N_O}{\sqrt{2}}=\left(\frac{\sqrt{2}-1}{\sqrt{2}}\right)$

Hence,

option $\left(D\right)$ is correct answer.