Physics - Modern Physics Question with Solution | TestHub

$\text{At time }{t}_2\text{, }\frac{2}{3}\text{ of the sample had decayed}$

$\therefore N=\frac{1}{3}{N}_0$

$\therefore \frac{1}{3}{N}_0=N_0{e}^{-\lambda t_2}$.....(i)

$\text{At time }{t}_1,\frac{1}{3}\text{ of the sample had decayed}$

$\therefore N=\frac{2}{3}{N}_0$

$\therefore \frac{2}{3}{N}_0=N_0{e}^{-\lambda t_1}$.....(ii)

Divide (i) by (ii), we get

$\frac{1}{2}=\frac{e^{-\lambda t_2}}{e^{-\lambda t_1}}$

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

$\text{λ}\left(t_2-t_1\right)=$ln2

$t_2-t_1=\frac{\text{ln}2}{\lambda }=\frac{\text{ln}2}{\left(\frac{\text{ln}2}{T_{1/2}}\right)}\left(\because \lambda =\frac{\text{ln}2}{T_{1/2}}\right)$

$=T{}_{1/2}$$=50$days