Question

If the half life of a radioactive nucleus is $3$ days, nearly what fraction of the initial number of nuclei will decay on the $3$rd day ? (Given that $\sqrt[3]{0.25} \approx 0.63)$)

• A. 0.63
• B. 0.5
• C. 0.37
• D. 0.13

Answer 1

The Correct Option is D

Given, $t_{1 / 2}=3$ days
Number of active nuclei remaining after time $t$,
$N=N_{0}\left(\frac{1}{2}\right)^{\frac{t}{t_1/2}}$
After $t=2$ days.
$N_{1} =N_{0}\left(\frac{1}{2}\right)^{2 / 3}=\frac{N_{0}}{2^{2 / 3}}=\frac{N_{0}}{4^{1 / 3}}=0.63 N _{ 0 } $
$\therefore N_{1} =0.63\, N _{0}$
After $t=3$ days,
$N_{2}=\frac{N_{0}}{2}=0.5 \,N_{0}$
Number of nuclei that will decay on the $3$ rd day,
$N_{3}=N_{2}-N_{1}=0.63 N_{0}-0.5 N_{0}=0.13 N_{0}$
In the term, fraction is $0.13$