Question

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

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

Answer 1

Answer: (C)

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

Let the half life be $t_{\frac{1}{2}}$.

Then, $\lambda =\frac{0.693}{t_{\frac{1}{2}}}=\frac{ln2}{t_{\frac{1}{2}}}$

$N=N_o{e}^{-\left(\frac{ln2}{t_{\frac{1}{2}}}\right)\times \frac{t_{\frac{1}{2}}}{2}}$

Ratio of decay $=\frac{N_o(1-e^{-\frac{ln2}{2}})}{N_o}=1-e^{ln\frac{1}{\sqrt{2}}}=1-\frac{1}{\sqrt{2}}=\frac{\sqrt{2}-1}{\sqrt{2}}$