Question

The half life of a radioactive substance is $20$ minutes. The approximate time interval $(t_2-t_1)$ between the time $t_1$when $\frac{1}{3}$ of it had decayed and $t_2$ when $\frac{2}{3}$ of it had decayed is :

• A. $7min$
• B. $14min$
• C. $20min$
• D. $28min$

Answer 1

The correct option is C $20min$

Using first order decay equation,
Number of remaining nuclei $N=N_o{e}^{-\lambda t}$
We get $\Delta t=t_2-t_1=\frac{1}{\lambda }{\log }_e\frac{N_1}{N_2}$
Where $N_1=\frac{2}{3}{N}_o$ $N_2=\frac{1}{3}{N}_o$
Also $\lambda =\frac{{\log }_{e}2}{T_{1/2}}=\frac{{\log }_e}{20}$
$\therefore$ $t_2-t_1=\frac{T_{1/2}}{{\log }_{e}2}{\log }_{e}2=20min$