Question

Half-life of a radioactive species is $2a$ year. What fraction of this sample will remain undecayed after $a$ year?

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

Answer 1

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

The integrated rate law expression for first order decay process is
$t=\frac{2.303}{\lambda }\log \frac{N_0}{N}$
or
$t=\frac{2.303\times \text{half life period}}{0.693}\log \frac{N_0}{N}$
Substitute values in the above expression.
$a=\frac{2.303\times 2\times a}{0.693}\log \frac{N_0}{N}$
$\therefore \frac{N}{N_0}=0.707=\frac{1}{\sqrt{2}}$
Hence, the fraction of this sample will remain undecayed after a year is
$\frac{1}{\sqrt{2}}$