Question

Two radioactive samples $X$ and $Y$ having half life $3$ years and $2$ years, respectively, have been decaying for many years. Today both samples have equal number of atoms. After how many years the number of atoms in the sample $X$ will be twice of the number of atoms in the sample $Y$ ?

Answer 1

Answer: 6

$N_0{e}^{-{\lambda }_t}=2\left(N_0{e}^{-{\lambda }_{2}t}\right)$
$\Rightarrow e^{-{\lambda }_{1}t}=2e^{-{\lambda }_{2}t}$
$\Rightarrow e^{-\left({\lambda }_1-{\lambda }_2\right)t}=2$
$\Rightarrow \left({\lambda }_2-{\lambda }_1\right)t=\ln 2$
$\Rightarrow \left(\frac{\ln 2}{t_2}-\frac{\ln 2}{t_1}\right)t=\ln 2$
$\Rightarrow t=\left(\frac{t_1{t}_2}{t_1-t_2}\right)=6$ years