Question

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

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

Answer 1

The correct option is C $20min$

At a given instant, the number of undecayed nuclei in a given sample is, $N=N_0{e}^{\lambda t}$

$1^{st}$ case: $\frac{2}{3}{N}_0=N_0{e}^{-\lambda t_1}\Rightarrow e^{-\lambda t_1}=\frac{2}{3}$

$2^{nd}$ case: $\frac{1}{3}{N}_0=N_0{e}^{-\lambda t_2}\Rightarrow e^{-\lambda t_2}=\frac{1}{3}$

taking the ratio, we get,
$e^{\lambda [t_2-t_1]}=2$

taking logarithm on both sides,
$\lambda [t_2-t_1]=ln\left(2\right)\Rightarrow t_2-t_1=\frac{ln\left(2\right)}{\lambda }=T_{\frac{1}{2}}=20min$