Question

Two samples X and Y contain equal amount of radioactive substances. If $\frac{1^{th}}{16}$ of the sample X and $\frac{1^{th}}{256}$ of the sample Y, remain after 8 hours, then the ratio of half life periods of X and Y is:

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

Answer 1

The correct option is A 2 : 1

$\begin{array}{c}\text{Given-}\ast 1/{16}^{\text{th}}\text{of sample}X\text{and}1/{256}^{\text{th}}\text{of sample}Y\\ \text{Remains after}8\text{hours.}\\ \text{we know that by law of Disintegration-}\\ [N=N_0\cdot e^{-\lambda t}]N=\text{Num of Nuclei left}\end{array}$

$\begin{array}{c}\text{for sample}X\to \frac{N_0}{16}=N_0\cdot e^{-8{\lambda }_X}\Rightarrow {\lambda }_x=\frac{\ln 16}{8}=\frac{ln2}{2}\\ \text{for sample}Y\to \frac{N_0}{256}=N_0\cdot e^{-8{\lambda }_Y}\Rightarrow {\lambda }_y=\frac{\ln 256}{8}=\ln 2\\ \text{we know that}{t}_{1/2}=\frac{\ln 2}{\lambda }\\ \Rightarrow \frac{t_x}{t_y}=\frac{{\lambda }_Y}{{\lambda }_X}=\frac{2}{1}\Rightarrow \text{Ratio}=2:1\end{array}$