Nuclei - Result Question 61
62. Radioactive material ’ $A$ ’ has decay constant ’ $8 \lambda$ ’ and material ’ $B$ ’ has decay constant ’ $\lambda$ ‘. Initially they have same number of nuclei. After what time, the ratio of number of nuclei of material ‘B’ to that ‘A’ will be $\frac{1}{e}$ ?
[2017]
(a) $\frac{1}{7 \lambda}$
(b) $\frac{1}{8 \lambda}$
(c) $\frac{1}{9 \lambda}$
(d) $\frac{1}{\lambda}$
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Answer:
Correct Answer: 62. (a)
Solution:
- (a) Given, $\lambda_A=8 \lambda, \lambda_B=\lambda$
$N_B=\frac{N_A}{e}$
$\Rightarrow \quad N_o e^{-\lambda_B t}=N_o \frac{e^{-\lambda A^{t}}}{e}$
$e^{-\lambda t}=e^{-8 \lambda t} e^{-1}$
$e^{-\lambda t}=e^{-8 \lambda t-1}$
Comparing both side powers
$-\lambda t=-8 \lambda t-1$
$-1=7 \lambda t$
$t=-\frac{1}{7 \lambda}$
The best possible answer is $t=\frac{1}{7 \lambda}$
