Physics - Modern Physics Question with Solution | TestHub

Fraction remains after $n$ half lives $$\frac{\mathrm{N}}{{\mathrm{N}}_0}={\left(\frac{1}{2}\right)}^{\mathrm{n}}={\left(\frac{1}{2}\right)}^{\mathrm{t}/\mathrm{T}}$$ Given $\mathrm{N}=\frac{{\mathrm{N}}_0}{\mathrm{e}}\Rightarrow \frac{{\mathrm{N}}_0}{{\mathrm{eN}}_0}={\left(\frac{1}{2}\right)}^{5/\mathrm{T}}$ or $\frac{1}{\mathrm{e}}={\left(\frac{1}{2}\right)}^{5/T}$ Taking $\log$ on both sides, we get $$\begin{array}{rl}\log 1-\log \mathrm{e}& =\frac{5}{\mathrm{T}}\log \frac{1}{2}\\ -1& =\frac{5}{\mathrm{T}}(-\log 2)\\ \Rightarrow \mathrm{T}& =5{\log }_{\mathrm{e}}2\end{array}$$ Now, let ${\mathrm{t}}^{\prime }$ be the time after which activity reduces to half $$\begin{array}{rl}\Rightarrow \left(\frac{1}{2}\right)& ={\left(\frac{1}{2}\right)}^{{\mathrm{t}}^{\prime }/5{\log }_{\mathrm{e}}2}\\ \Rightarrow {\mathrm{t}}^{\prime }& =5{\log }_{\mathrm{e}}2\end{array}$$