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 |
| \[\frac{{{m}_{x}}}{{{m}_{y}}}=\frac{1}{7}\] |
| \[\Rightarrow \] \[7{{m}_{x}}={{m}_{y}}\] |
| \[\Rightarrow \] Let the initial total mass is m. |
| \[\Rightarrow \] \[{{m}_{x}}+{{m}_{y}}=m\Rightarrow \frac{{{m}^{y}}}{7}+{{m}_{y}}=m\] |
| \[\Rightarrow \] \[\frac{8{{m}_{y}}}{7}=m\] |
| \[\Rightarrow \] \[{{m}_{y}}=\frac{7}{8}m\] |
| only \[\frac{1}{8}\] part remains |
| \[\Rightarrow \] \[1\xrightarrow{\begin{smallmatrix} {{T}_{1/2}} \\ \end{smallmatrix}}\,\frac{1}{2}\,\,\xrightarrow{\begin{smallmatrix} {{T}_{1/2}} \\ \end{smallmatrix}}\,\frac{1}{4}\,\xrightarrow{\begin{smallmatrix} {{T}_{1/2}} \\ \end{smallmatrix}}\frac{1}{8}\] |
| 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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