There are two radio nuceli `A` and `B. A` is an `alpha` emitter and `B` a `beta` emitter. Their disintegration constant 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?

Options

A. `2:1`
B. `1:2`
C. e
D. `e^(-1)`

Answer

To solve the problem, we need to find the ratio of the number of atoms of radionuclei A (alpha emitter) and B (beta emitter) such that the probabilities of emitting alpha and beta particles are the same at any time \( t \).

### Step-by-Step Solution:

1. **Understanding the Disintegration Constants**:
- Let the disintegration constant (decay constant) for nucleus A be \( \lambda_A \).
- Let the disintegration constant for nucleus B be \( \lambda_B \).
- According to the problem, the ratio of their disintegration constants is given as:
\[ \frac{\lambda_A}{\lambda_B} = \frac{1}{2} \]
- This implies:
\[ \lambda_B = 2 \lambda_A \]

2. **Defining the Number of Atoms**:
- Let \( N_A \) be the number of atoms of nucleus A.
- Let \( N_B \) be the number of atoms of nucleus B.

3. **Setting Up the Emission Rates**:
- The rate of emission of alpha particles from nucleus A is given by:
\[ R_A = \lambda_A N_A \]
- The rate of emission of beta particles from nucleus B is given by:
\[ R_B = \lambda_B N_B \]

4. **Equating the Emission Rates**:
- For the probabilities of getting alpha and beta particles to be the same, we need the emission rates to be equal:
\[ R_A = R_B \]
- Substituting the expressions for \( R_A \) and \( R_B \):
\[ \lambda_A N_A = \lambda_B N_B \]

5. **Substituting the Value of \( \lambda_B \)**:
- Replace \( \lambda_B \) with \( 2 \lambda_A \):
\[ \lambda_A N_A = 2 \lambda_A N_B \]

6. **Simplifying the Equation**:
- Since \( \lambda_A \) is common on both sides, we can divide both sides by \( \lambda_A \) (assuming \( \lambda_A \neq 0 \)):
\[ N_A = 2 N_B \]

7. **Finding the Ratio**:
- Rearranging gives us the ratio of the number of atoms:
\[ \frac{N_A}{N_B} = 2 \]

### Final Answer:
The ratio of the number of atoms of A to B should be:
\[ \frac{N_A}{N_B} = 2:1 \]