• # question_answer The radium and uranium atoms in a sample of uranium mineral are in the ratio of $1:2.8\times {{10}^{6}}$. If half-life period of radium is 1620 years, the half-life period of uranium will be [MP PMT 1999] A)            $45.3\times {{10}^{9}}$ years                                         B)            $45.3\times {{10}^{10}}$ years C)            $4.53\times {{10}^{9}}$ years                                         D)            $4.53\times {{10}^{10}}$ years

According to radioactive equilibrium ${{\lambda }_{A}}{{N}_{A}}={{\lambda }_{B}}{{N}_{B}}$                    or $\frac{0.693\times {{N}_{A}}}{{{t}_{1/2}}(A)}=\frac{0.693\times {{N}_{B}}}{{{t}_{1/2}}\,(B)}\left[ \lambda =\frac{0.693}{{{t}_{1/2}}} \right]$            Where ${{t}_{1/2}}(A)$ and ${{t}_{1/2}}(B)$ are half periods of A and B respectively                    $\therefore \frac{{{N}_{A}}}{{{t}_{1/2}}(A)}=\frac{{{N}_{B}}}{{{t}_{1/2}}(B)}\,\,\text{or}\,\,\frac{{{N}_{A}}}{{{N}_{B}}}=\frac{{{t}_{1/2}}(A)}{{{t}_{1/2}}(B)}$                    $\therefore$ At equilibrium A and B are present in the ratio of their half lives $\frac{1}{2.8\times {{10}^{6}}}=\frac{1620}{\text{Half}\,\text{life}\,\,\text{of}\,\text{uranium}}$                    $\therefore$Half-life of uranium                    = $2.8\times {{10}^{6}}\times 1620=4.53\times {{10}^{9}}$years.