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JEE Main & Advanced Physics Nuclear Physics And Radioactivity Question Bank Self Evaluation Test - Nuclei

  • question_answer
    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

    A)  \[\frac{1}{10\lambda }\]

    B)  \[\frac{1}{11\lambda }\]

    C)  \[\frac{11}{10\lambda }\]

    D)  \[\frac{1}{9\lambda }\]

    Correct Answer: D

    Solution :

    [d] \[{{N}_{1}}={{N}_{0}}{{e}^{-10\lambda t}}\] 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  \[t=\left( \frac{1}{9\lambda } \right)\]


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