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 \[\frac{{{N}_{1}}}{{{N}_{2}}}={{\left( \frac{1}{e} \right)}^{2}}=\frac{1}{{{e}^{2}}}\] Hence, \[\frac{1}{{{e}^{2}}}=\frac{1}{{{e}^{4\lambda t}}}\] Comparing the powers, we get \[2=4\lambda t\] or \[t=\frac{2}{4\lambda }=\frac{1}{2\lambda }\]
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