Question

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. $\frac{ t _{2}- t _{1}}{\ln 5}$
• D. $\frac{\ell n \left( t _{2}+ t _{1}\right)}{2}$

Answer 1

Answer: (C)

Let initial activity be $A _{0}$
$A=A_{0} e^{-\lambda t_{1}} \,\,\,\,\,\, (i)$
$\frac{ A }{5}= A _{0} e ^{-\lambda t _{2}} \,\,\,\, ....(ii)$
(i) $\div$ (ii)
$5= e ^{\lambda\left(t_{2}-t_{1}\right)}$
$\lambda=\frac{\ell n 5}{ t _{2}- t _{1}}=\frac{1}{\tau}$
$\tau=\frac{ t _{2}- t _{1}}{\ell n 5}$