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NEET Sample Paper NEET Sample Test Paper-78

  • 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 after a time.

    A)  \[\lambda \]                              

    B)  \[\frac{1}{2}\lambda \]

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

    D)  \[\frac{e}{\lambda }\,\]

    Correct Answer: C

    Solution :

    We know that \[N={{N}_{0}}{{e}^{-\,\lambda t}}\] N = number of radioactive nuclei present at some instant \[{{N}_{0}}=nuclei at\,\,t = 0\] \[\Rightarrow \,\,\,\,\,\,\,\,\frac{{{N}_{1}}}{{{N}_{2}}}=\frac{{{e}^{-\,{{\lambda }_{1}}t}}}{{{e}^{-\,{{\lambda }_{2}}t}}}=\frac{{{e}^{-\,{{\lambda }_{5}}t}}}{{{e}^{-\,\lambda t}}}={{e}^{-\,4\lambda t}}\] \[\Rightarrow \,\,\,\,\,\,\,As,\,\,\,\,\,\,\,\,\,\,\frac{{{N}_{1}}}{{{N}_{2}}}=\frac{1}{e}\]                    (given) \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\frac{1}{e}={{e}^{-\,4\lambda t}}\,\,\Rightarrow \,\,\,{{e}^{-\,4\lambda t}}=e\] \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,-4\lambda t=1\,\,\,\Rightarrow \,\,\,t=\frac{1}{4\lambda }\]


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