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

Show Answer

Answer:

Correct Answer: 62. (a)

Solution:

  1. (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 $t=\frac{1}{7 \lambda}$