Question

The half life of a radioactive nucleus is $50days.$ The time interval $(t_2-t_1)$ between the time $t_2$ when $\frac{2}{3}$ of it has decayed and the time $t_1$ when $\frac{1}{3}$ of it had decayed is: -

• A. $60days$
• B. $15days$
• C. $30days$
• D. $50days$

Answer 1

The correct option is D $50days$

From radio active decay law, $N=N_0{e}^{-\lambda t}$ where $N_0$ be the amount of nuclei at $t=0$ and $N$ be the left nuclei under decay at any time t.

At $t_2\Rightarrow N=N_0-2N_0/3=N_0/3$ and at $t_1,N=N_0-N_0/3=2N_0/3$

Now , $N_0/3=N_0{e}^{-\lambda t_2}...\left(1\right)$

and $2N_0/3=N_0{e}^{-\lambda t_1}...\left(2\right)$

$\left(1\right)/\left(2\right)\Rightarrow e^{\lambda (t_2-t_1)}=2$

$\Rightarrow t_2-t_1=ln2/\lambda =t_{1/2}=50$ days