A) \[\text{lo}{{\text{g}}_{\text{e}}}\text{2/5}\]
B) \[\frac{5}{{{\log }_{e}}2}\]
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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