Question

Two radioactive substances X and Y initially contain equal number of nuclei. X has a half - life of 1 hour and Y has a half - life of 2 hours. After two hours, the ratio of the activity of X to the activity of Y is:-

• A. $1:4$
• B. $1:2$
• C. $1:1$
• D. $2:1$

Answer 1

The correct option is B $1:2$

$\begin{array}{c}X=A_0{e}^{-{\lambda }_{1}t}\\ Y=A{e}^{-{\lambda }_{2}t}\\ \Rightarrow \frac{X}{Y}=e^{\left({\lambda }_2-\lambda 1\right)t}\\ Now,\\ {\lambda }_2=\frac{\ln 2}{2}\\ {\lambda }_1=\frac{\ln 2}{1}\\ \frac{X}{Y}=e^{{\left(\frac{\ln 2}{2}-\frac{\ln 2}{1}\right)}^2}\\ =e^{-\ln 2}\\ =\frac{1}{2}\\ Hence,optionBiscorrectanswer.\end{array}$