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 } $
Show Answer
Answer:
Correct Answer: D
Solution:
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$ 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 } ) $
