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

$N_{0} e^{-\lambda_{t}}=2\left(N_{0} e^{-\lambda_{2} t}\right)$
$\Rightarrow e^{-\lambda_{1} t}=2 e^{-\lambda_{2} t}$
$\Rightarrow e^{-\left(\lambda_{1}-\lambda_{2}\right) t}=2$
$\Rightarrow\left(\lambda_{2}-\lambda_{1}\right) t=\ln 2$
$\Rightarrow\left(\frac{\ln 2}{t_{2}}-\frac{\ln 2}{t_{1}}\right) t=\ln 2$
$\Rightarrow t=\left(\frac{t_{1} t_{2}}{t_{1}-t_{2}}\right)=6$ years