The binding energies per nucleon for a deuteron and an `alpha-`particle are `x_1` and `x_2` respectively. What will be the energy `Q` released in the following reaction ?
`._1H^2 + ._1H^2 rarr ._2He^4 + Q`.

Options

A. `4(x_(1)+x_(2))`
B. `4(x_(2)=x_(1))`
C. `2(x_(1)+x_(2))`
D. `2(x_(2)=x_(1))`

Answer

To solve the problem, we need to determine the energy \( Q \) released in the reaction:

\[ _1H^2 + _1H^2 \rightarrow _2He^4 + Q \]

### Step-by-Step Solution:

1. **Identify the Components**:
- The reactants are two deuterons (\( _1H^2 \)), each having 2 nucleons.
- The product is an alpha particle (\( _2He^4 \)), which has 4 nucleons.

2. **Binding Energy Definitions**:
- Let \( x_1 \) be the binding energy per nucleon for a deuteron.
- Let \( x_2 \) be the binding energy per nucleon for an alpha particle.

3. **Calculate Total Binding Energy**:
- The total binding energy of the products (alpha particle) is:
\[ \text{Total Binding Energy of } _2He^4 = \text{Number of nucleons} \times \text{Binding energy per nucleon} = 4 \times x_2 \]
- The total binding energy of the reactants (two deuterons) is:
\[ \text{Total Binding Energy of } 2 \times _1H^2 = \text{Number of nucleons} \times \text{Binding energy per nucleon} = 4 \times x_1 \]

4. **Energy Released (Q)**:
- The energy \( Q \) released in the reaction can be calculated as the difference between the total binding energy of the products and the total binding energy of the reactants:
\[ Q = \text{Total Binding Energy of Products} - \text{Total Binding Energy of Reactants} \]
- Substituting the values we calculated:
\[ Q = (4 \times x_2) - (4 \times x_1) \]

5. **Final Expression for Q**:
- Factor out the common term:
\[ Q = 4(x_2 - x_1) \]

### Conclusion:
Thus, the energy \( Q \) released in the reaction is given by:

\[ Q = 4(x_2 - x_1) \]