A) \[\text{lo}{{\text{g}}_{\text{e}}}\text{2/5}\]
B) \[\frac{\text{5}}{\text{lo}{{\text{g}}_{\text{e}}}\text{2}}\]
C) \[5{{\log }_{10}}2\]
D) \[5{{\log }_{e}}2\]
Correct Answer: D
Solution :
Fraction remains after n half lives \[\frac{N}{{{N}_{0}}}={{\left( \frac{1}{2} \right)}^{n}}={{\left( \frac{1}{2} \right)}^{t/T}}\] Given \[N=\frac{{{N}_{0}}}{e}\Rightarrow \frac{{{N}_{0}}}{e{{N}_{0}}}={{\left( \frac{1}{2} \right)}^{5/T}}\] or \[\frac{1}{e}={{\left( \frac{1}{2} \right)}^{5/T}}\] Taking log on both sides, we get \[\log 1-\log e=\frac{5}{T}\log \frac{1}{2}\] \[-1=\frac{5}{T}(-\log 2)\] \[\Rightarrow \] \[T=5{{\log }_{e}}2\] Now, let t' be the time after which activity reduces to half \[\left( \frac{1}{2} \right)={{\left( \frac{1}{2} \right)}^{t'/5{{\log }_{e}}2}}\] \[\Rightarrow \] \[t'=5{{\log }_{e}}2\]
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