Nuclei - Result Question 66
67. The half life of a radioactive nucleus is 50 days. The time interval $(t_2-t_1)$ between the time $t_2$ when $\frac{2}{3}$ of it has decayed and the time $t_1$ when $\frac{1}{3}$ of it had decayed is :
[2012M]
(a) 30 days
(b) 50 days
(c) 60 days
(d) 15 days
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Answer:
Correct Answer: 67. (b)
Solution:
$ \begin{align*} \text{ (b) } N_1 & =N_0 e^{-\lambda t} \quad N_1=\frac{1}{3} N_0 \\ \frac{N_0}{3} & =N_0 e^{-\lambda t_2} \\ \Rightarrow \quad \frac{1}{3} & =e^{-\lambda t^{2}} \tag{i}\\ N_2 & =\frac{2}{3} N_0 \end{align*} $
$ \begin{align*} \frac{2}{3} N_0 & =N_0 e^{-\lambda t_1} \\ \Rightarrow \quad \frac{2}{3} & =e^{-\lambda t_1} \tag{ii} \end{align*} $
Dividing equation (i) by equation (ii)
$ \begin{aligned} & \frac{1}{2}=e^{-\lambda(t_2-t_1)} \\ & \lambda(t_2-t_1)=\ln 2 \\ & t_2-t_1=\frac{\ln 2}{\lambda}=T _{1 / 2}=50 \text{ days } \end{aligned} $
