Q. The activity of a sample of a radioactive material is $A$ at time $t_{1}$ and $A_{2}$ at time $t_{2}(t_{2} > t_{1})$. If its mean life is $T$, then
A) $A_{1}t_{1} = A_{2}t_{2}$
B) $A_{1} - A_{2} = t_{2} - t_{1}$
C) $A_{2} = A_{1}e^{(t_{1} - t_{2})/T}$
D) $A_{2} = A_{1}e^{(t_{1}/t_{2})/T}$
Explanation
(c) $A = A_{0}e^{- \lambda t} = A_{0}e^{- t/\tau}$; where $\tau =$ mean life
So $A_{1} = A_{0}e^{- t_{1}/T}$
$\Rightarrow A_{0} = \frac{A_{1}}{e^{- t_{1}/T}} = A_{1}e^{t_{1}/T}$
$\therefore A_{2} = A_{0}e^{- t/T} = \left( A_{1}e^{t_{1}/T} \right)e^{- t_{2}/T}$
$\Rightarrow A_{2} = A_{1}e^{(t_{1} - t_{2})/T}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.