Q155 / 160 · AP EAMCET 2009 · Physics

Two radioactive materials $X_1$ and $X_2$ have decay constants $10 \lambda$ and $\lambda$ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of $X_1$ to that of $X_2$ will be $1 / e$ after a time

  • A
    $(1 / 10 \lambda)$
  • B
    $1 /(11 \lambda)$
  • Correct answer
    C
    $11 /(10 \lambda)$
  • D
    $1 /(9 \lambda)$

Solution

$\begin{aligned} & N_1=N_0 e^{-10 \lambda t} \\ & N_2=N_0 e^{-\lambda t}\end{aligned}$ $\begin{aligned} & \Rightarrow \quad \frac{N_1}{N_2}=\frac{1}{e}=e^{-1}=e^{(-10 \lambda+\lambda) t} \\ & =e^{-9 \lambda t} \\ & \Rightarrow \quad t=\frac{1}{9 \lambda} \\ & \end{aligned}$

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

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