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_2-x_1)`
D. `2(x_1+x_2)`

Answer

To find the energy \( Q \) released in the reaction \( _1H^2 + _1H^2 \rightarrow _2He^4 + Q \), we will follow these steps:

### Step 1: Understand the reaction
The reaction involves two deuterium nuclei (\( _1H^2 \)) fusing to form an alpha particle (\( _2He^4 \)). We need to calculate the energy released during this fusion process.

### Step 2: Define binding energy per nucleon
Binding energy per nucleon is the energy required to remove a nucleon from a nucleus. For the deuteron, the binding energy per nucleon is denoted as \( x_1 \), and for the alpha particle, it is denoted as \( x_2 \).

### Step 3: Calculate total binding energy for reactants and products
- The total binding energy of the two deuterons (reactants) can be calculated as:
\[ \text{Total Binding Energy of Reactants} = 2 \times x_1 \]
(since each deuteron has 2 nucleons).

- The total binding energy of the alpha particle (product) is:
\[ \text{Total Binding Energy of Products} = 4 \times x_2 \]
(since the alpha particle has 4 nucleons).

### Step 4: Calculate the energy released \( Q \)
The energy released in the reaction is equal to the difference in binding energy between the reactants and the products:
\[ Q = \text{Total Binding Energy of Products} - \text{Total Binding Energy of Reactants} \]
Substituting the values we calculated:
\[ Q = (4 \times x_2) - (2 \times x_1) \]
Rearranging gives:
\[ Q = 4x_2 - 2x_1 \]

### Step 5: Final expression for \( Q \)
Since energy is released, we can express this as:
\[ Q = 2(x_2 - x_1) \]

### Conclusion
The energy \( Q \) released in the fusion of two deuterons to form an alpha particle is given by:
\[ Q = 2(x_2 - x_1) \]