JEE Main & Advanced Chemistry Nuclear chemistry Question Bank Critical Thinking Nuclear chemistry

  • 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

    Correct Answer: C

    Solution :

           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.

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