Question

Decay constants of two radio-active samples $A$ and $B$ are $15x$ and $3x$ respectively. They have equal number of initial nuclei. The ratio of the number of nuclei left in $A$ and $B$ after a time $\frac{1}{6x}$ is

• A. $e$
• B. $e^2$
• C. $e^{-1}$
• D. $e^{-2}$

Answer 1

The correct option is D $e^{-2}$

Given : ${\lambda }_A=15x$ ${\lambda }_B=3x$ $t=\frac{1}{6x}$

Number of nuclei of a sample left $N=N_o{e}^{-\lambda t}$ where $N_o$ is the initial number of nuclei

$\therefore$ $\frac{N_A}{N_B}=\frac{e^{-{\lambda }_{A}t}}{e^{-{\lambda }_{B}t}}=e^{-({\lambda }_A-{\lambda }_B)t}$ where ${\lambda }_A-{\lambda }_B=15x-3x=12x$

$\therefore$ $\frac{N_A}{N_B}=e^{-\frac{12x}{6x}}$ $⟹$ $\frac{N_A}{N_B}=e^{-2}$