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)`

Explanation

To solve the problem, we need to calculate the energy released (Q) in the reaction of two deuterons forming an alpha particle. We will use the binding energy per nucleon for both deuterons and alpha particles.

### Step-by-Step Solution:

1. **Identify the Reaction**:
The reaction given is:
\[ \, _1H^2 + \, _1H^2 \rightarrow \, _2He^4 + Q \]
Here, we have two deuterons (each represented as \( _1H^2 \)) combining to form one alpha particle (represented as \( _2He^4 \)) and releasing energy \( Q \).

2. **Determine the Number of Nucleons**:
- Each deuteron has 2 nucleons (1 proton and 1 neutron).
- Therefore, for 2 deuterons:
\[ \text{Total nucleons from deuterons} = 2 \times 2 = 4 \]
- The alpha particle also has 4 nucleons (2 protons and 2 neutrons).

3. **Binding Energies**:
- 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.
- The total binding energy for 2 deuterons is:
\[ \text{Total Binding Energy from Deuterons} = 2 \text{ deuterons} \times 2 \text{ nucleons/deuteron} \times x_1 = 4x_1 \]
- The total binding energy for the alpha particle is:
\[ \text{Total Binding Energy for Alpha Particle} = 1 \text{ alpha particle} \times 4 \text{ nucleons} \times x_2 = 4x_2 \]

4. **Calculate the Energy Released (Q)**:
The energy released in the reaction can be calculated by finding the difference between the binding energies of the products and reactants:
\[ Q = \text{Total Binding Energy of Products} - \text{Total Binding Energy of Reactants} \]
Substituting the values we found:
\[ Q = 4x_2 - 4x_1 \]
This simplifies to:
\[ Q = 4(x_2 - x_1) \]

5. **Final Result**:
The energy released in the reaction is:
\[ Q = 4(x_2 - x_1) \]