Modern Physics 3 Question 15
15. 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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Solution:
- $\frac{N _{x _1}(t)}{N _{x _2}(t)}=\frac{1}{e}$ or $\frac{N _0 e^{-10 \lambda t}}{N _0 e^{-\lambda t}}=\frac{1}{e}$
(Initially, both have same number of nuclei say $N _0$ )
$$ \begin{array}{rlrl} \text { or } & & e & =e^{-\lambda t} / e^{-10 \lambda t} \\ \text { or } & e & =e^{9 \lambda t} \\ \text { or } & & 9 \lambda t & =1 \\ \text { or } & & t & =\frac{1}{9 \lambda} \end{array} $$
