Question

Two radioactive materials $A_1$ and $A_2$ have decay constants of $10{\lambda }_0$ and ${\lambda }_0$ . If initially they have same number of nuclei, the ratio of number of their undecayed nuclei after a time $\left(\frac{1}{e}\right)$will be

• A. $\frac{1}{{\lambda }_0}$
• B. $\frac{1}{9{\lambda }_0}$
• C. $\frac{1}{10{\lambda }_0}$
• D. 1

Answer 1

The correct option is B $\frac{1}{9{\lambda }_0}$

considering first order decay process, and same no. of initial no. of nuclei:
$N_1=N_0{e}^{-10{\lambda }_{0}t}$
$N_2=N_0{e}^{-{\lambda }_{0}t}$
$\frac{N_1}{N_2}=\frac{e^{N_0{e}^{-10{\lambda }_{0}t}}}{e^{N_0{e}^{-{\lambda }_{0}t}}}$
$\frac{N_1}{N_2}=e^{-9{\lambda }_{0}t}$
When, $\frac{N_1}{N_2}=\frac{1}{e}$
$\frac{1}{e}=e^{-9{\lambda }_{0}t}\Rightarrow t=\frac{1}{9{\lambda }_0}$