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BHU PMT BHU PMT (Screening) Solved Paper-2008

  • question_answer
    The radioactivity of a sample is\[{{R}_{1}}\]at a time\[{{T}_{1}}\]and\[{{R}_{2}}\]at times\[{{T}_{2}}\]. the half-life of the specimen is T, the number of atoms that have disintegrated at the time\[({{T}_{2}}-{{T}_{1}})\]is proportional to

    A)  \[{{R}_{1}}{{T}_{1}}-{{R}_{2}}{{T}_{2}}\]                               

    B)  \[{{R}_{1}}-{{R}_{2}}\]

    C)  \[\frac{{{R}_{1}}-{{R}_{2}}}{T}\]                               

    D)  \[({{R}_{1}}-{{R}_{2}})T\]

    Correct Answer: D

    Solution :

                     \[{{R}_{1}}={{N}_{1}}\lambda \]and\[{{R}_{2}}={{N}_{2}}\lambda \] Also       \[T=\frac{{{\log }_{e}}2}{\lambda }\] Or           \[\lambda =\frac{{{\log }_{e}}2}{T}\] \[\therefore \]  \[{{R}_{1}}-{{R}_{2}}=({{N}_{1}}-{{N}_{2}})\lambda \]                 \[=({{N}_{1}}-{{N}_{2}})\frac{{{\log }_{e}}2}{T}\] \[\therefore \]  \[({{N}_{1}}-{{N}_{2}})=\frac{({{R}_{1}}-{{R}_{2}})T}{{{\log }_{e}}2}\] ie,           \[({{N}_{1}}-{{N}_{2}})\propto ({{R}_{1}}-{{R}_{2}})T\]


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