• # question_answer The half lives of radioisotopes ${{P}^{32}}$ and ${{P}^{33}}$ are 20 days and 30 days respectively. The radioisotopes are mixed in the ratio of 4 :1 of their atoms. If the initial activity of the mixed sample is $7\,mCi,$ then the activity of the mixed isotopes after 60 days. A)  $2.5\,mCi$                 B)  $1\,mCi$ C)  $3.2\,mCi$                 D)  $3\,mCi$

As,      $\frac{{{N}_{32}}}{{{N}_{33}}}=\frac{4}{1}$             $[{{N}_{33}}={{N}_{0}}]$ $\Rightarrow$            $\frac{dN}{N}={{\lambda }_{1}}{{N}_{32}}+{{\lambda }_{2}}{{N}_{33}}$ $\Rightarrow$            $7\times {{10}^{-3}}={{\lambda }_{1}}4{{N}_{0}}+{{\lambda }_{2}}{{N}_{0}}$ $\Rightarrow$            $7\times {{10}^{-3}}={{N}_{0}}\,\left[ \frac{In\,(2)}{5}+\frac{In\,(2)}{30} \right]$ $\Rightarrow$            ${{N}_{0}}=\frac{30\times {{10}^{-3}}}{In\,(2)}$ After 60 days, $\frac{dN}{dt}={{\lambda }_{1}}\frac{4{{N}_{0}}}{8}+\frac{{{\lambda }_{2}}{{N}_{0}}}{4}=1\,mCi$