Atoms And Nuclei Question 374

Question: 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

Options:

A) $ \frac{1}{10\lambda } $

B) $ \frac{1}{11\lambda } $

C) $ \frac{11}{10\lambda } $

D) $ \frac{1}{9\lambda } $

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Answer:

Correct Answer: D

Solution:

  • $ N_{1}=N_{0}{{e}^{-10\lambda t}} $

    and $ N_{2}=N_{0}{{e}^{-\lambda t}}. $
    $ \therefore \frac{N_{1}}{N_{2}}=\frac{{{e}^{-10\lambda t}}}{{{e}^{-\lambda t}}}=\frac{1}{{{e}^{9\lambda t}}} $

    Given $ \frac{N_{1}}{N_{2}}=\frac{1}{e};\therefore \frac{1}{{{e}^{9\lambda t}}}=\frac{1}{e} $ or, $ 9\lambda t=1 $

    or $ t=( \frac{1}{9\lambda } ) $



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