Two radioactive materials X₁ and X₂ have decay constant 5λ and λ respectively inti

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13.Nuclei

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Two radioactive materials $X_1$ and $X_2$ have decay constant $5\lambda$ and $\lambda$ respectively intially they have the saame number of nuclei, then the ratio of the number of nuclei of $X_1$ to that $X_2$ will be $\frac{1}{e}$ after a time

A

$4λ$

B

$2λ$

C

$\frac{1}{{2\lambda }}$

D

$\;\frac{1}{{4\lambda }}$

(AIPMT-2008)

Solution

$X_{1}=N_{0} e^{-\lambda_{1} t}$ ; $X_{2}=N_{0} e^{-\lambda_{2} t}$

$\frac{X_{1}}{X_{2}}=e^{-1}$ $=e^{\left(-\lambda_{1}+\lambda_{2}\right) t}$ ; ${e^{ – 1}} = {e^{ – (\lambda 1 – \lambda 2)t}}$

$\therefore t=\left|\frac{1}{\lambda_{1}-\lambda_{2}}\right|=\frac{1}{(5 \lambda-\lambda)}=\frac{1}{4 \lambda}$

Standard 12

Physics

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