PYQ NEET- Atoms And Nuclei L-3

Question: At any instant, two elements $X_1$ and $X_2$ have same number of radioactive atoms. If the decay constant of $X_1$ and $\mathrm{X}_2$ are $10 \lambda$ and $\lambda$ respectively, then the time when the ratio of their atoms becomes $\frac{1}{e}$ respectively will be :#

A) $\frac{1}{5 \lambda}$

B) $\frac{1}{11 \lambda}$

C) $\frac{1}{9 \lambda}$

D) $\frac{1}{6 \lambda}$

Answer: (C) $\frac{1}{9 \lambda}$#

Sol:#

Number of radioactive nuclei at any time is $$ N=N_0 e^{-\lambda t} $$

Initial number of nuclei is same for sample $X_1 & X_2$. Let after time ’ $t$ ’ the ratio of their atoms become $\frac{1}{e}$ $$ \Rightarrow \frac{N_0 e^{-10 \lambda \times t}}{N_0 e^{-\lambda \times t}}=\frac{1}{e} \Rightarrow e^{-9 \lambda t}=e^{-1} \Rightarrow t=\frac{1}{9 \lambda} $$

Hence, $t=\frac{1}{9 \lambda}$