Question

There are two radio nuclei A and B. A is an $\alpha$-emitter and B a $\beta$- emitter. Their disintegration constants are in the ratio of $1:2$. What should be the ratio of number of atoms of A and B at any time $t$ so that probabilities of getting alpha and beta particles are same at that instant?

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

Answer 1

The correct option is A $2:1$

Probability of getting $\alpha$ and $\beta$ particles is directly proportional to the rate of decay
$\frac{dN}{dt}=\lambda N$
$\Rightarrow \frac{P_A}{P_B}=\frac{{\lambda }_A{N}_A}{{\lambda }_B{N}_B}$
$\Rightarrow \frac{{\lambda }_A{N}_A}{{\lambda }_B{N}_B}=1$
$\Rightarrow \frac{N_A}{N_B}=\frac{{\lambda }_B}{{\lambda }_A}$
$\Rightarrow \frac{N_A}{N_B}=\frac{2}{1}$