Q125 / 180 · AIIMS 2001 · Physics

A certain radioactive element has a half-life of 20 years. If we have a block with $10 \mathrm{~g}$ of the element in it, after how many years will there be just $2.5 \mathrm{~g}$ of the element in the block?

  • A
    80 years
  • Correct answer
    B
    40 years
  • C
    100 years
  • D
    60 years.

Solution

$$ \begin{aligned} & T_{1 / 2}=20 \text { years } \\ & N_0=10 \mathrm{~g} \\ & N=2.5 \mathrm{~g} \\ N= & N_0 e^{-\lambda t} \quad \lambda=\frac{0.693}{T_{1 / 2}} \\ \Rightarrow \quad e^{-\lambda t}= & \ln \left(\frac{N}{N_0}\right) \\ \Rightarrow \quad t= & \frac{1}{\lambda} \ln \left(\frac{N_0}{N}\right)=\frac{T_{1 / 2}}{0.693} \times \ln \left(\frac{N_0}{N}\right) \\ = & \frac{20}{0.693} \times \ln \left(\frac{10}{2.5}\right)=40 \text { years } \end{aligned} $$

Hence, the correct answer is option: 'B'.

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