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  4. Two radioactive materials X1, and X2 have decay constants 5 λ and λ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of X1 to that X2 will be 1 / e after a time

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Q. Two radioactive materials $X_{1}$, and $X_{2}$ have decay constants $5 \lambda$ and $\lambda$ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of $X_{1}$ to that $X_{2}$ will be $1 / e$ after a time

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Solution:

$X_{1}=N_{0} e^{-\lambda 1 t} ; X_{2}=N_{0} e^{-\lambda 2 t}$
$\frac{X_{1}}{X_{2}}=e^{-1}=e^{\left(-\lambda_{1}+\lambda_{2}\right) t} ; e^{-1}=e^{\left(\lambda_{1}-\lambda_{2}\right) t}$
$\therefore t=\left|\frac{1}{\lambda_{1}-\lambda_{2}}\right|=\frac{1}{(5 \lambda-\lambda)}=\frac{1}{4 \lambda}$