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  4. Two radioactive materials X1 and X2 have decay constants 10 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 of X2 will be 1 / e after a time

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Q. Two radioactive materials $X_{1}$ and $X_{2}$ have decay constants $10 \lambda$ 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 of $X_{2}$ will be $1 / e$ after a time

EAMCETEAMCET 2009

Solution:

$N_{1}=N_{0} e^{-10 \lambda t}$
and $ N_{2}=N_{0} e^{-\lambda t} $
$ \Rightarrow \frac{N_{1}}{N_{2}}=\frac{1}{e}=e^{-1} =e^{(-10 \lambda+\lambda) t} $
$ =e^{-9 \lambda t} $
$ \Rightarrow t=\frac{1}{9 \lambda} $