Question

Half-lives of two radioactive elements $A$ and $B$ are $20\, $ minutes and $40\,$ minutes, respectively. Initially, the samples have equal number of nuclei. After $80\,$ minutes, the ratio of decayed numbers of $A$ and $B$ nuclei will be :

• A. $1 : 16$
• B. $4 : 1$
• C. $1 : 4$
• D. $5 : 4$

Answer 1

Answer: (D)

$
\begin{array}{l}
n = n _{0} e ^{-\alpha t } \\ \alpha_{ A }=\ln 2 / 20 \\ \alpha_{ B }=\ln 2 / 40 \end{array}
$
After $80 min$, remaining number of nuclei for $A , n _{ A }= n _{ o } / 16$
After $80 min$, remaining number of nuclei for $B, n_{B}=n_{0} / 4$
Decayed number of nuclei for $A , \Delta n _{ A }=\frac{15}{16} n _{ O }$
Decayed number of nuclei for $B , \Delta n _{ B }=\frac{3}{4} n _{ O }$
Ratio of decayed number of nuclei is
$
\frac{\Delta n _{ A }}{\Delta n _{ B }}=\frac{15 / 16}{3 / 4}=5: 4
$