Question

Two radioactive material $A_1$ and $A_2$ have decay constants of $10{\lambda }_0$ and ${\lambda }_0.$ If initially they have same number of nuclei, then after time $\frac{1}{9{\lambda }_0}$ the ratio of number of their undecayed nuclei will be:

• A. $\frac{1}{e}$
• B. $\frac{1}{e^2}$
• C. $\frac{1}{e^3}$
• D. $\frac{\sqrt{e}}{1}$

Answer 1

Answer: (A)

$\frac{1}{e}$

The integrated rate law for the first order reaction is $\left[N\right]=[N{]}_0{e}^{-kt}$.
For the first material, the decay equation becomes $[N{]}_1=[N{]}_0{e}^{-{\lambda }_0\times \frac{1}{9{\lambda }_0}}$.
For the second material, the decay equation becomes $[N{]}_2=[N{]}_0{e}^{-10{\lambda }_0\times \frac{1}{9{\lambda }_0}}$.
The ratio of number of their undecayed nuclei will be $\frac{[N{]}_1}{[N{]}_2}=\frac{[N{]}_0{e}^{-{\lambda }_0\times \frac{1}{9{\lambda }_0}}}{[N{]}_0{e}^{-10{\lambda }_0\times \frac{1}{9{\lambda }_0}}}=\frac{1}{e}$