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NEET Sample Paper NEET Sample Test Paper-64

  • question_answer
    Two radioactive materials \[{{X}_{1}}\,\,and\,\,{{X}_{2}}\] have decay constants \[5\lambda \,\,and\,\,\lambda \] 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 \[\frac{1}{e}\] after a time.

    A) \[\lambda \]                               

    B) \[\frac{1}{2}\,\,\lambda \]

    C) \[\frac{1}{4\lambda }\]              

    D) \[\frac{e}{\lambda }\]

    Correct Answer: C

    Solution :

    If N is the number of radioactive nuclei present at some instant, then \[N\,\,=\,\,{{N}_{0}}{{e}^{-\lambda t}}\] The constant \[{{N}_{0}}\] represents the number of radioactive nuclei at \[t=0\] \[\frac{{{N}_{1}}}{{{N}_{2}}}\,\,=\,\,\frac{{{e}^{-{{\lambda }_{1}}t}}}{{{e}^{-{{\lambda }_{2}}t}}}\,\,\,or\,\,\frac{{{N}_{1}}}{{{N}_{2}}}\,\,=\,\frac{{{e}^{-5\lambda t}}}{{{e}^{-\lambda t}}}\,\,=\,\,{{e}^{-4\lambda t}}\] \[but\,\,\frac{{{N}_{1}}}{{{N}_{2}}}\,=\,\,\frac{1}{e}\,\,(as\,\,provided)\] Therefore, \[\frac{1}{e}=\frac{1}{{{e}^{4\lambda t}}}\,\,\,or\,\,4\lambda t=1\] \[or\,\,\,t=\frac{1}{4\lambda }\]


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