Physics - Modern Physics Question with Solution | TestHub

After $\mathrm{n}$ half-lives $\begin{array}{c}\frac{\mathrm{N}}{{\mathrm{N}}_0}={\left(\frac{1}{2}\right)}^{\mathrm{n}}={\left(\frac{1}{2}\right)}^{\mathrm{t}/7}\\ \mathrm{N}=\frac{{\mathrm{N}}_0}{\mathrm{e}}=\frac{\mathrm{c}{\mathrm{N}}_0}{{\mathrm{cN}}_0}={\left(\frac{1}{2}\right)}^{5/7}\\ \Rightarrow \frac{1}{\mathrm{e}}={\left(\frac{1}{2}\right)}^{5/7}\end{array}$ Taking log on both sides, we get $\log 1-\log \mathrm{e}=\frac{5}{7}\log \frac{1}{2}$ $\begin{array}{l}-1=\frac{5}{7}(-\log 2)\\ \mathrm{T}=5{\log }_{\mathrm{e}}2\end{array}$ Now, let $\mathrm{t}$ ' be the time after which activity reduces to half $\begin{array}{l}\left(\frac{1}{2}\right)={\left(\frac{1}{2}\right)}^{1/5{\log }_{\mathrm{e}}2}\\ {\mathrm{t}}^{\prime }=5{\mathrm{olog}}_{\mathrm{e}}2\end{array}$