What is the Q of the following fusion process?
`._(1)^(2)H+._(1)^(1)Hto._(2)^(3)He+` photon
Here are some at atomic masses.
`""_(1)^(2)H 2.014102u ""_(1)^(1)H1.007825u`
`""_(2)^(3)He 3.016029u`

Answer

To find the Q value of the fusion process given by the equation:

\[ _{1}^{2}H + _{1}^{1}H \rightarrow _{2}^{3}He + \text{photon} \]

we need to calculate the mass defect (Δm) and then convert that mass defect into energy using Einstein's equation \( E = mc^2 \).

### Step-by-Step Solution:

1. **Identify the masses involved in the reaction:**
- Mass of deuterium (\( _{1}^{2}H \)): \( m_{D} = 2.014102 \, u \)
- Mass of protium (\( _{1}^{1}H \)): \( m_{p} = 1.007825 \, u \)
- Mass of helium-3 (\( _{2}^{3}He \)): \( m_{He} = 3.016029 \, u \)

2. **Write the mass balance equation:**
The total mass before the reaction is the sum of the masses of the reactants:
\[ m_{\text{initial}} = m_{D} + m_{p} \]
The total mass after the reaction is the mass of the product (helium-3) plus the mass of the photon (which is negligible):
\[ m_{\text{final}} = m_{He} \]

3. **Calculate the initial and final masses:**
\[ m_{\text{initial}} = 2.014102 \, u + 1.007825 \, u = 3.021927 \, u \]
\[ m_{\text{final}} = 3.016029 \, u \]

4. **Calculate the mass defect (Δm):**
\[ \Delta m = m_{\text{initial}} - m_{\text{final}} = 3.021927 \, u - 3.016029 \, u = 0.005898 \, u \]

5. **Convert the mass defect to energy (Q value):**
We use the conversion factor \( 1 \, u \approx 931.5 \, MeV/c^2 \):
\[ Q = \Delta m \times 931.5 \, \text{MeV/u} \]
\[ Q = 0.005898 \, u \times 931.5 \, \text{MeV/u} \approx 5.49 \, \text{MeV} \]

### Final Answer:
The Q value for the fusion process is approximately \( 5.49 \, \text{MeV} \).

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