A) \[\frac{1}{4\lambda }\]
B) \[4\,\lambda \]
C) \[2\,\lambda \]
D) \[\frac{1}{2\lambda }\]
Correct Answer: D
Solution :
| Number of nuclei remained after time t can be written as |
| \[N={{N}_{0}}{{e}^{-\lambda t}}\] |
| where \[{{N}_{0}}\] is initial number of nuclei of both the substances. |
| \[{{N}_{1}}={{N}_{0}}{{e}^{-5\lambda t}}\] (i) |
| and \[{{N}_{2}}={{N}_{0}}{{e}^{-\lambda t}}\] (ii) |
| Dividing Eq. (i) by Eq. (ii), we obtain |
| \[\frac{{{N}_{1}}}{{{N}_{2}}}={{e}^{(-5\lambda +\lambda )t}}={{e}^{-4\lambda \,t}}=\frac{1}{{{e}^{4\lambda \,t}}}\] |
| But, we have given |
| \[\frac{{{N}_{1}}}{{{N}_{2}}}={{\left( \frac{1}{e} \right)}^{2}}=\frac{1}{{{e}^{2}}}\] |
| Comparing the powers, we get |
| \[2=4\lambda t\] |
| or \[t=\frac{2}{4\lambda }=\frac{1}{2\lambda }\] |
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