JEE Main & Advanced Sample Paper JEE Main Sample Paper-35

  • question_answer
    The time during which three-fourth of a sample will decay if decaying both by \[\alpha \] and \[\beta \] -emission simultaneously is 312 year. The mean life of this sample is 900 years for \[\alpha \] - emission. Find the mean life of this sample for \[\beta \] - emission.

    A)  550 years           

    B)  300 years

    C)  615 years                           

    D)  655 years

    Correct Answer: B

    Solution :

    \[N={{N}_{0}}{{e}^{-\lambda t}}\]             \[\frac{{{N}_{0}}}{4}\,={{N}_{0}}{{e}^{-\lambda t}}\]             \[\lambda t=\,\ln 4\]             \[t=\frac{1}{\lambda }\,\ln \,4=\frac{1}{\lambda }\ln \,{{2}^{2}}=\frac{2}{\lambda }In\,2=312\]             \[\lambda =\frac{2\ln 2}{312}\,\times 4.443\,\times {{10}^{-3}}\]             \[=\frac{1}{225}\]             \[\lambda ={{\lambda }_{\alpha }}\,+{{\lambda }_{B}}\]             \[\frac{1}{225}\,=\frac{1}{900}\,+{{\lambda }_{\beta }}\]             \[{{\lambda }_{\beta }}\,=\frac{1}{225}\,-\frac{1}{900}=\frac{1}{300}\] Decay constant is reciprocal of mean life. So, mean life for \[\beta \] - emission = 300 years.

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