Modern Physics Ques 113

  1. 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 $ $\lambda$

(c) $11 / 10 $ $\lambda$

(d) $1 / 9 $ $\lambda$

$(2000,2 M)$

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Answer:

Correct Answer: 113.(d)

Solution:

Formula:

Radioactive Decay:

  1. $\frac{N _{x _1}(t)}{N _{x _2}(t)}=\frac{1}{e}$ or $\frac{N _0 e^{-10 \lambda t}}{N _0 e^{-\lambda t}}=\frac{1}{e}$

(Initially, both have same number of nuclei say $N _0$ )

$ \begin{array}{rlrl} \text { or } & & e & =e^{-\lambda t} / e^{-10 \lambda t} \\ \text { or } & & e & =e^{9 \lambda t} \\ \text { or } & & 9 \lambda t & =1 \\ \text { or } & & t & =\frac{1}{9 \lambda} \end{array} $



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