JEE PYQ: Nuclear Physics Question 9
Question 9 - 2021 (26 Feb Shift 2)
A radioactive sample is undergoing $\alpha$ decay. At any time $t_1$, its activity is $A$ and another time $t_2$ the activity is $\frac{A}{5}$. What is the average life time for the sample?
(1) $\frac{t_2 - t_1}{\ln 5}$
(2) $\frac{\ln(t_2 + t_1)}{\ln 5}$
(3) $\frac{t_1 - t_2}{\ln 5}$
(4) $\frac{(t_2 - t_1)}{\ln 5 - \ln 4}$
Show Answer
Answer: (1)
Solution
For activity of radioactive sample: $A = A_0 e^{-\lambda t_1}$ …(1)
$\frac{A}{5} = A_0 e^{-\lambda t_2}$ …(2)
From (1)/(2):
$5 = e^{-\lambda(t_1 - t_2)}$ $\ln(5) = (t_2 - t_1)\lambda \Rightarrow \lambda = \frac{\ln(5)}{t_2 - t_1}$
avg. life $= \frac{1}{\lambda} = \frac{t_2 - t_1}{\ln(5)}$
BETA
