Physics - Modern Physics Question with Solution | TestHub

Fraction remains after n half-lives

$\frac{N}{N_0}={\left(\frac{1}{2}\right)}^n={\left(\frac{1}{2}\right)}^{\frac{\mathrm{+t}}{\mathrm{T}}}$

$\mathrm{}\mathrm{t}\mathrm{ }=5$

$\mathrm{N}=\frac{{\mathrm{N}}_0}{\mathrm{e}}$

$\frac{1}{\mathrm{e}}={\left(\frac{1}{2}\right)}^{\frac{+5}{\mathrm{T}}}$

Taking log on both sides, we get

$\log 1-\log e=\frac{5}{\mathrm{T}}\log \frac{1}{2}$

$-1=\frac{5}{\mathrm{T}}\left(-\log 2\right)$

$T=5{\log }_{\mathrm{e}}2$

Now, let$t^{\prime }$be the time after which activity reduces to half

$\left(\frac{1}{2}\right)={\left(\frac{1}{2}\right)}^{t^{\prime }/5{\log }_{e}2}$

$t^{\prime }=5{\log }_{e}2$