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 of finding the mass number of the mother nucleus in an alpha decay reaction, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Reaction**: In alpha decay, a mother nucleus \( X \) with mass number \( A \) and atomic number \( Z \) transforms into a daughter nucleus \( Y \) with mass number \( A-4 \) and atomic number \( Z-2 \), while emitting an alpha particle \( \alpha \) which has a mass number of 4 and an atomic number of 2. The reaction can be expressed as:
\[ X_{Z}^{A} \rightarrow Y_{Z-2}^{A-4} + \alpha_{2}^{4} \]

2. **Use the Kinetic Energy and Q Value**: We know that the kinetic energy of the emitted alpha particle is given as \( 48 \, \text{MeV} \) and the Q value of the reaction is \( 50 \, \text{MeV} \). The relationship between the kinetic energy of the alpha particle, the mass of the daughter nucleus \( Y \), and the Q value is given by:
\[ KE_{\alpha} = \frac{m_Y}{m_Y + m_{\alpha}} \times Q \]
where \( m_{\alpha} \) is the mass of the alpha particle, which is approximately 4.

3. **Substituting Known Values**: We can substitute the known values into the equation. The kinetic energy of the alpha particle is \( 48 \, \text{MeV} \) and the Q value is \( 50 \, \text{MeV} \):
\[ 48 = \frac{m_Y}{m_Y + 4} \times 50 \]

4. **Rearranging the Equation**: Rearranging the equation to isolate \( m_Y \):
\[ 48(m_Y + 4) = 50m_Y \]
Expanding this gives:
\[ 48m_Y + 192 = 50m_Y \]

5. **Solving for \( m_Y \)**: Rearranging the terms gives:
\[ 50m_Y - 48m_Y = 192 \]
Simplifying this gives:
\[ 2m_Y = 192 \]
Therefore,
\[ m_Y = \frac{192}{2} = 96 \]

6. **Finding the Mass Number of the Mother Nucleus**: Since the mass number of the daughter nucleus \( Y \) is \( A - 4 \), we can express the mass number of the mother nucleus \( A \) as:
\[ A = m_Y + 4 = 96 + 4 = 100 \]

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