Calculate the Q-values of the following fusion reactions:
(a)`1^2H+1^2H rarr 1^3H+1^1H`.
`1^2H+1^2H rarr 2^3(He)+n`
`1^2H+1^3H rarr 2^4(He)+n`.
Atomic masses are `m(1^2H)=2.014102 u`, `m(1^3H)=3.016049 u`, `m(2^3(He))=3.016029 u`, `m(2^4(He))=4.002603 u`.

Explanation

To calculate the Q-values for the given fusion reactions, we will use the formula:

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

Where:
- \( \Delta m \) is the difference in mass between the reactants and products.
- \( c^2 \) is the conversion factor from mass to energy, which is approximately \( 931 \, \text{MeV/u} \).

### (a) Reaction: \( ^2_1H + ^2_1H \rightarrow ^3_1H + ^1_1H \)

1. **Identify the masses:**
- Mass of \( ^2_1H \) (Deuterium) = \( 2.014102 \, u \)
- Mass of \( ^3_1H \) (Tritium) = \( 3.016049 \, u \)
- Mass of \( ^1_1H \) (Proton) = \( 1.007276 \, u \)

2. **Calculate total mass of reactants:**
\[ \text{Total mass of reactants} = 2 \times m(^2_1H) = 2 \times 2.014102 \, u = 4.028204 \, u \]

3. **Calculate total mass of products:**
\[ \text{Total mass of products} = m(^3_1H) + m(^1_1H) = 3.016049 \, u + 1.007276 \, u = 4.023325 \, u \]

4. **Calculate \( \Delta m \):**
\[ \Delta m = \text{Total mass of reactants} - \text{Total mass of products} = 4.028204 \, u - 4.023325 \, u = 0.004879 \, u \]

5. **Calculate Q-value:**
\[ Q = \Delta m \cdot c^2 = 0.004879 \, u \cdot 931 \, \text{MeV/u} \approx 4.55 \, \text{MeV} \]

### (b) Reaction: \( ^2_1H + ^2_1H \rightarrow ^3_2He + n \)

1. **Identify the masses:**
- Mass of \( ^2_1H \) = \( 2.014102 \, u \)
- Mass of \( ^3_2He \) = \( 3.016029 \, u \)
- Mass of neutron \( n \) = \( 1.008665 \, u \)

2. **Calculate total mass of reactants:**
\[ \text{Total mass of reactants} = 2 \times m(^2_1H) = 4.028204 \, u \]

3. **Calculate total mass of products:**
\[ \text{Total mass of products} = m(^3_2He) + m(n) = 3.016029 \, u + 1.008665 \, u = 4.024694 \, u \]

4. **Calculate \( \Delta m \):**
\[ \Delta m = 4.028204 \, u - 4.024694 \, u = 0.003510 \, u \]

5. **Calculate Q-value:**
\[ Q = 0.003510 \, u \cdot 931 \, \text{MeV/u} \approx 3.26 \, \text{MeV} \]

### (c) Reaction: \( ^2_1H + ^3_1H \rightarrow ^4_2He + n \)

1. **Identify the masses:**
- Mass of \( ^2_1H \) = \( 2.014102 \, u \)
- Mass of \( ^3_1H \) = \( 3.016049 \, u \)
- Mass of \( ^4_2He \) = \( 4.002603 \, u \)

2. **Calculate total mass of reactants:**
\[ \text{Total mass of reactants} = m(^2_1H) + m(^3_1H) = 2.014102 \, u + 3.016049 \, u = 5.030151 \, u \]

3. **Calculate total mass of products:**
\[ \text{Total mass of products} = m(^4_2He) + m(n) = 4.002603 \, u + 1.008665 \, u = 5.011268 \, u \]

4. **Calculate \( \Delta m \):**
\[ \Delta m = 5.030151 \, u - 5.011268 \, u = 0.018883 \, u \]

5. **Calculate Q-value:**
\[ Q = 0.018883 \, u \cdot 931 \, \text{MeV/u} \approx 17.57 \, \text{MeV} \]

### Summary of Q-values:
- (a) \( Q \approx 4.55 \, \text{MeV} \)
- (b) \( Q \approx 3.26 \, \text{MeV} \)
- (c) \( Q \approx 17.57 \, \text{MeV} \)