A) \[1.96\times {{10}^{9}}\,yr\]
B) \[3.92\times {{10}^{9}}\,yr\]
C) \[4.20\times {{10}^{9}}yr\]
D) \[8.40\times {{10}^{9}}\,yr\]
Correct Answer: C
Solution :
Ratio of X : Y is given =1:7 \[\frac{{{m}_{x}}}{{{m}_{y}}}=\frac{1}{7}\] \[\Rightarrow \] \[7{{m}_{x}}={{m}_{y}}\] Let the initial total mass is m. \[\Rightarrow \,\,\,{{m}_{x}}+{{m}_{y}}=m\Rightarrow \frac{{{m}^{y}}}{7}+{{m}_{y}}=m\] Þ \[\frac{8{{m}_{y}}}{7}=m\] \[\Rightarrow {{m}_{y}}=\frac{7}{8}m\] only \[\frac{1}{8}\] part remains Þ \[1\xrightarrow{\begin{smallmatrix} {{T}_{1/2}} \\ \end{smallmatrix}}\,\frac{1}{2}\,\,\xrightarrow{\begin{smallmatrix} {{T}_{1/2}} \\ \end{smallmatrix}}\,\frac{1}{4}\,\xrightarrow{\begin{smallmatrix} {{T}_{1/2}} \\ \end{smallmatrix}}\frac{1}{8}\] So, time taken to become \[\frac{1}{8}\] unstable part \[=3\times {{T}_{1/2}}=3\times 1.4\times {{10}^{9}}\] \[=4.2\times {{10}^{9}}y\]
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