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13.Nuclei
hard
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?
A
$\frac{\ell n 5}{ t _{2}- t _{1}}$
B
$\frac{ t _{1}- t _{2}}{\ell n 5}$
C
$\frac{ t _{2}- t _{1}}{\ell n 5}$
D
$\frac{\ell n \left( t _{2}+ t _{1}\right)}{2}$
(JEE MAIN-2021)
Solution
Let initial activity be $A _{0}$
$A = A _{0} e ^{-\lambda t_{2}}….(i)$
$\frac{ A }{5}= A _{0} e ^{-\lambda t_{2}}….(ii)$
$( i ) \div ( ii )$
$5= e ^{\lambda\left(t_{2}-t_{1}\right)}$
$\lambda=\frac{\ell n 5}{t_{2}-t_{1}}=\frac{1}{\tau}$
$\tau=\frac{t_{2}-t_{1}}{\ell n \cdot 5}$
Standard 12
Physics
