Determine the Q -value for the fusion reaction: ^ 2 _ 1 H + ^ 2 _ 1...

The given fusion reaction is:
^{2}_{1}\text{H} + ^{2}_{1}\text{H} \to ^{3}_{1}\text{H} + ^{1}_{1}\text{H}$<br><br>To find the$Q$-value of the reaction, we first calculate the mass defect ($\Delta m$). The standard atomic mass of hydrogen (^{1}_{1}\text{H}$) is approximately $1.007825 \text{ u}$.

The total mass of the reactants is:
$m_{\text{reactants}} = 2 \times m(^{2}_{1}\text{H}) = 2 \times 2.014102 \text{ u} = 4.028204 \text{ u}$

The total mass of the products is:
$m_{\text{products}} = m(^{3}_{1}\text{H}) + m(^{1}_{1}\text{H}) = 3.016049 \text{ u} + 1.007825 \text{ u} = 4.023874 \text{ u}$

The mass defect $\Delta m$ is:
$\Delta m = m_{\text{reactants}} - m_{\text{products}}$
$\Delta m = 4.028204 \text{ u} - 4.023874 \text{ u} = 0.004330 \text{ u}$

The $Q$-value is calculated by multiplying the mass defect by $931.5 \text{ MeV/u}$:
$Q = \Delta m \times 931.5 \text{ MeV/u}$
$Q = 0.004330 \times 931.5 \approx 4.033 \text{ MeV}$

Depending on the exact rounded value of the mass of ^{1}_{1}\text{H}$used in standard textbook problems (often taken as$1.0078 \text{ u}$), the calculated$Q$-value comes out to be approximately$4.05 \text{ MeV}$. The closest given option is$4.05 \text{ MeV}$.<br><br>Answer:$4.05 \text{ MeV}