Modern Physics 3 Question 6
6. Half-lives of two radioactive elements $A$ and $B$ are $20 min$ and $40 min$, respectively. Initially, the samples have equal number of nuclei. After $80 min$, the ratio of decayed numbers of $A$ and $B$ nuclei will be
(2016 Main)
(a) $1: 16$
(b) $4: 1$
(c) $1: 4$
(d) $5: 4$
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Solution:
- A : Numbers left : $N \rightarrow \frac{N}{2} \rightarrow \frac{N}{4} \rightarrow \frac{N}{8} \rightarrow \frac{N}{16}$
$\therefore \quad$ Number decayed, $N _A=N-\frac{N}{16}=\frac{15}{16} N$
B : Numbers left : $N \rightarrow \frac{N}{2} \rightarrow \frac{N}{4}$ $\therefore \quad$ Numbers decayed, $N _B=N-\frac{N}{4}=\frac{3}{4} N$
Ratio :
$$ \frac{N _A}{N _B}=\frac{(15 / 16) N}{(3 / 4) N}=\frac{5}{4} $$
