In an `alpha` -decay, the kinetic energy of `alpha`-particles is `48 MeV` and `Q` value of the reaction is `50 MeV`. The mass number of the mother nucleus is (assume that daughter nucleus is in ground state)

Options

A. 96
B. 100
C. 104
D. none of these

Explanation

To solve the problem, we will use the principles of alpha decay and the relationship between the kinetic energy of the emitted alpha particle, the Q value of the reaction, and the mass numbers involved.

### Step-by-Step Solution:

1. **Understand the Reaction**:
In an alpha decay, a parent nucleus (let's denote it as \( X \)) decays into a daughter nucleus (denote as \( Y \)) and an alpha particle (which is a helium nucleus). The reaction can be represented as:
\[ X \rightarrow Y + \alpha \]
Here, \( Y \) has an atomic number \( Z-2 \) and a mass number \( A-4 \) (where \( A \) is the mass number of the parent nucleus \( X \)), and the alpha particle has an atomic number of 2 and a mass number of 4.

2. **Given Values**:
- Kinetic energy of the alpha particle, \( KE_{\alpha} = 48 \, \text{MeV} \)
- Q value of the reaction, \( Q = 50 \, \text{MeV} \)

3. **Relate Kinetic Energy and Q Value**:
The kinetic energy of the alpha particle can be related to the Q value of the reaction using the formula:
\[ KE_{\alpha} = \frac{A - 4}{A} Q \]
Where \( A \) is the mass number of the parent nucleus \( X \).

4. **Substituting the Known Values**:
Substitute the known values into the equation:
\[ 48 = \frac{A - 4}{A} \times 50 \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ 48A = 50(A - 4) \]
Expanding the right side:
\[ 48A = 50A - 200 \]

6. **Solving for A**:
Rearranging to isolate \( A \):
\[ 50A - 48A = 200 \]
\[ 2A = 200 \]
\[ A = 100 \]

7. **Conclusion**:
The mass number of the mother nucleus \( X \) is \( 100 \).

### Final Answer:
The mass number of the mother nucleus is \( 100 \).

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