Atoms And Nuclei Question 366

Question: A radioactive element X converts into another stable element Y. Half-life of X is 2 hrs. Initially only X is present. After time t, the ratio of atoms of X and Y is found to be 1 : 4, then t in hours is

Options:

A) 2

B) 4

C) between 4 and 6

D) 6

Show Answer

Answer:

Correct Answer: C

Solution:

  • Let $ N_{0} $ be the number of atoms of X at time $ t=0. $

    Then at $ t=4 $ hrs (two half-lives) $ N_{x}=\frac{N_{0}}{4} $

    and $ N_{y}=\frac{3N_{0}}{4} $
    $ \therefore N_{x}/N_{y}=1/3 $

    and at $ t=6 $ hrs (three half-lives) $ N_{x}=\frac{N_{0}}{8} $

    and $ N_{y}=\frac{7N_{0}}{8} $ or $ \frac{N_{x}}{N_{y}}=\frac{1}{7} $

    The given ratio $ \frac{1}{4} $ lies between $ \frac{1}{3} $

    and $ \frac{1}{7} $ .

    Therefore, t lies between 4 hrs and 6 hrs.



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