Question

There are two radio nuclei $A$ and $B$. $A$ is a $\alpha$-emitter and $B$ is $\beta$-emitter. Their disintegration constants are in the ratio $1:2$. What should be the number of atoms ratio of $A$ and $B$ at the time $t = 0$, so that the initial probability of getting $\alpha$ and $\beta$ particles are the same.

Answer 1

Hint: The particle which decays more readily as compared to the other particle should be present in more numbers than the other particle so that the probability of getting both the particles are equal. Here the rate of disintegration of one nucleus is double as compared to the other. Therefore, its particles must also be present in double amounts.

Complete step by step solution:
According to the radioactive decay law, “The probability per unit time of the disintegration of the nucleus is independent of time and would be constant.”
We know that,
$\dfrac{\partial N} {\partial t} = - \lambda N$
Where $\lambda$ is the constant of proportionality, which is also known as radioactive decay constant or disintegration constant
$\partial N$ is the change in the number of nuclei,
$\partial t$ is the time taken
$N$ is the initial number of nuclei present
Let the disintegration constant of nuclei $A$ and $B$ be ${\lambda _A}$ and ${\lambda _B}$, as given in the question:
$\Rightarrow \dfrac{{{\lambda _A}{N_A}}}{{{\lambda _B}{N_B}}} = \dfrac{1}{2}$
And as given in the question, we need to find out the number of atoms ratio of $A$ and $B$ at the time $t = 0$, so that initially probability of getting of $\alpha$ and $\beta$ particles are same, thus the rate of disintegration of both the particles must be same:
$\Rightarrow {\lambda _A}{N_A} = {\lambda _B}{N_B}$
Hence, we get:
$\Rightarrow \dfrac{{{N_A}}}{{{N_B}}} = \dfrac{{{\lambda _B}}}{{{\lambda _A}}}$
Therefore,
$\Rightarrow \dfrac{{{N_A}}}{{{N_B}}} = \dfrac{2}{1} = 2:1$
Therefore, the number of atoms ratio of $A$ and $B$ at the time $t = 0$, so that initially, the probability of getting $\alpha$ and $\beta$ particles should be $2:1$.

Note: Since the time is zero, time will not be a factor on which the radioactive decay of the two particles depends. The question asks the number of atoms ratio at the time $t = 0$. We have used the radioactive decay law to calculate the ratio of atoms.