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STD 12 - 13. Nuclei — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 13. Nuclei4 Marks
MCQ
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
    $\frac{ t _{1}- t _{2}}{\ell n 5}$
  • ✓
    $\frac{ t _{2}- t _{1}}{\ell n 5}$
  • D
    $\frac{\ell n \left( t _{2}+ t _{1}\right)}{2}$
✓

Answer

Correct option: C.
$\frac{ t _{2}- t _{1}}{\ell n 5}$
c
Let initial activity be $A _{0}$

$A = A _{0} e ^{-\lambda t_{2}}....(i)$

$\frac{ A }{5}= A _{0} e ^{-\lambda t_{2}}....(ii)$

$( i ) \div ( ii )$

$5= e ^{\lambda\left(t_{2}-t_{1}\right)}$

$\lambda=\frac{\ell n 5}{t_{2}-t_{1}}=\frac{1}{\tau}$

$\tau=\frac{t_{2}-t_{1}}{\ell n \cdot 5}$

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