Two radioactive materials A and B have decay constants


Two radioactive materials $\mathrm{A}$ and $\mathrm{B}$ 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 A to that of B will be $1 / \mathrm{e}$ after a time :

  1. (1) $\frac{1}{9 \lambda}$

  2. (2) $\frac{1}{11 \lambda}$

  3. (3) $\frac{11}{10 \lambda}$

  4. (4) $\frac{1}{10 \lambda}$

Correct Option: 1


(1) $\mathrm{As}, \mathrm{N}=\mathrm{N}_{0} \mathrm{e}^{-\lambda \mathrm{t}}$

so, $\frac{N_{A}}{N_{B}}=e^{\left(\lambda_{B}-\lambda_{A}\right) t}=\frac{1}{e} \Rightarrow\left(\lambda_{B}-\lambda_{A}\right) t=-1$

$\Rightarrow\left(\lambda_{\mathrm{A}}-\lambda_{\mathrm{B}}\right) \cdot \mathrm{t}=1$

$\Rightarrow \mathrm{t}=\frac{1}{\left(\lambda_{\mathrm{B}}-\lambda_{\mathrm{A}}\right)} \mathrm{t}=\frac{1}{10 \lambda-\lambda}=\frac{1}{9 \lambda}$

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