The rate constant for the radioactive decay of C-11 is 0.0341 ${\min }^{-1}$. How long will it take for a sample of C-11 to decrease to $\frac{1}{4}$ of its original activity? 

All radioactive decay follows $1^{\mathrm{st}}$ order kinetic $\Rightarrow \mathrm{C}={\mathrm{C}}_0{\mathrm{e}}^{-\mathrm{kt}}$ 

$\mathrm{A}={\mathrm{A}}_0{\mathrm{e}}^{-\mathrm{\lambda t}}$ $\mathrm{\lambda }$: decay constant

A_o: Initial Activity/Number of nuclei present

${\mathrm{A}}_0$ : Hime token
initially
A: Activity after time " t "/Number of nuclei present After time “t”

$\mathrm{\lambda }=0.0341{\min }^{-1}$

$A\mathit{=}\frac{{\mathit{A}}_{\mathit{0}}}{\mathit{4}}$

$\frac{{\mathrm{A}}_0}{4}={\mathrm{f}}_0{\mathrm{e}}^{-\mathrm{\lambda t}}\Rightarrow \frac{1}{4}={\mathrm{e}}^{-\mathrm{\lambda t}}$

$\ln \left(\frac{1}{4}\right)=-\lambda t;-1.3862=-\lambda t$

$$\begin{gathered} \mathrm{t}=\frac{1.3862}{\mathrm{\lambda }} \\ =\frac{1.3862}{0.0341}\min \\ =40.6\min \end{gathered}$$