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13.Nuclei
medium
Two radioactive materials $A$ and $B$ have decay constants $25 \lambda$ and $16 \lambda$ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of $B$ to that of $A$ will be "$e$" after a time $\frac{1}{a \lambda}$. The value of $a$ is $......$
A
$9$
B
$8$
C
$5$
D
$6$
(JEE MAIN-2022)
Solution
$N = N _{0} e ^{-\lambda t }$
$\frac{ N _{ B }}{ N _{ A }}=\frac{ e ^{-\lambda_{2} t }}{ e ^{-\lambda_{1} t }}= e ^{-\lambda_{2} t } \cdot e ^{\lambda_{1} t }$
$e ^{1}= e ^{\left(\lambda_{1}-\lambda_{2}\right) t }$
$\left(\lambda_{1}-\lambda_{2}\right) t =1$
$t =\frac{1}{\lambda_{1}-\lambda_{2}}=\frac{1}{25 \lambda-16 \lambda}=\frac{1}{9 \lambda}$
Standard 12
Physics
