In a nuclear fusion reaction, two nuclei, A & B , fuse to produce a nucleus C , releasing an amount of energy `DeltaE` in the process.If the mass defects of the three nuclei are `DeltaM_A, DeltaM_B` & `DeltaM_C` respectively , then which of the following relations holds ? Here, c is the speed of light.

Options

A. `DeltaM_A + DeltaM_B = DeltaM_C - DeltaE//c^2`
B. `DeltaM_A + DeltaM_B = DeltaM_C + DeltaE//c^2`
C. `DeltaM_A - DeltaM_B = DeltaM_C - DeltaE//c^2`
D. `DeltaM_A - DeltaM_B = DeltaM_C + DeltaE//c^2`

Answer

To solve the problem, we need to understand the relationship between mass defect and energy in nuclear reactions. The mass defect is the difference between the mass of the reactants and the mass of the products. In a fusion reaction, the mass defect can be related to the energy released using Einstein's mass-energy equivalence principle, which is given by the equation:

\[ E = \Delta m \cdot c^2 \]

Where:
- \( E \) is the energy released,
- \( \Delta m \) is the mass defect,
- \( c \) is the speed of light.

### Step-by-Step Solution:

1. **Identify the Mass Defects**:
- Let \( \Delta M_A \) be the mass defect of nucleus A.
- Let \( \Delta M_B \) be the mass defect of nucleus B.
- Let \( \Delta M_C \) be the mass defect of nucleus C.

2. **Write the Fusion Reaction**:
- The fusion reaction can be represented as:
\[ A + B \rightarrow C \]

3. **Apply the Mass-Energy Equivalence**:
- The total mass defect before the reaction (for nuclei A and B) is:
\[ \Delta M_{total} = \Delta M_A + \Delta M_B \]
- The mass defect after the reaction (for nucleus C) is:
\[ \Delta M_C \]

4. **Relate the Energy Released to Mass Defect**:
- The energy released in the fusion reaction is given by the difference in mass defects:
\[ \Delta E = (\Delta M_A + \Delta M_B - \Delta M_C) \cdot c^2 \]

5. **Rearranging the Equation**:
- From the above equation, we can derive the relation:
\[ \Delta M_A + \Delta M_B = \Delta M_C + \frac{\Delta E}{c^2} \]

### Conclusion:
Thus, the relation that holds in this nuclear fusion reaction is:
\[ \Delta M_A + \Delta M_B = \Delta M_C + \frac{\Delta E}{c^2} \]