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)

$5 : 4$

Answer 2

$80$ minutes $= 4$ half-lives of $A = 2$ half-lives of $B$
Let the initial number of nuclei in each sample be $N$
$N_A$ after $80$ minutes $= \frac{N}{2^4}$
$\Rightarrow$ Number of $A$ nuclides decayed $= \frac{15}{16} N$
$N_B$ after $80$ minutes $= \frac{N}{2^2}$
$\Rightarrow$ Number of $B$ nuclides decayed $= \frac{3}{4} N$
Required ratio $= \frac{15/16}{3/4} = \frac{5}{4}$