Nuclei - Result Question 74
75. Two radioactive materials $X_1$ and $X_2$ have decay constants $5 \lambda$ and $\lambda$ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of $X_1$ to that of $X_2$ will be $\frac{1}{e}$ after a time
[2008]
(a) $\lambda$
(b) $\frac{1}{2} 1$
(c) $\frac{1}{41}$
(d) $\frac{e}{1}$
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Answer:
Correct Answer: 75. (c)
Solution:
- (c) Let the required time be t. Then
$N_1=N_0 e^{-111^{t}} ; N_2=N_0 e^{-12 t}$
Where
$N_1=$ number of nuclei of $X_1$ after timet
$N_2=$ number of nuclei of $X_2$ after timet
$N_0=$ initial number of nuclei of $X_1$ and $X_2$ each.
Now, $\frac{N_1}{N_2}=\frac{N_0 e^{-11 t}}{N_0 e^{-12 t}}$ Here $\frac{N_1}{N_2}=\frac{1}{e}$
$\lambda_1=5 \lambda ; \quad \lambda_2=\lambda$
$\therefore \frac{1}{e}=\frac{e^{-51 t}}{e^{-1 t}} \Rightarrow e^{-1}=e^{-4 \lambda t} \Rightarrow 4 \lambda t=1$
$\therefore t=\frac{1}{41}$
