The fusion reaction ${}^2\text{H}{+}^3\text{He}{\to }^4\text{He}+p$ has the following atomic masses: $m{(}^2\text{H})=2.014102\text{u}$, $m{(}^3\text{He})=3.016029\text{u}$, $m{(}^4\text{He})=4.002603\text{u}$, $m(p)=1.007276\text{u}$. (a) Calculate the energy released per reaction (Q‑value) in MeV. (b) If $2.0\text{g}$ of deuterium reacts completely with excess ${}^3$He, find the total energy released in joules. (Take $1\text{u}=931.5\text{MeV}/c^2$ and $1\text{MeV}=1.602\times 10^{-13}\text{J}$.)

Step-by-step solution

1. Write the mass balance: initial mass $=m{(}^2\text{H})+m{(}^3\text{He})$, final mass $=m{(}^4\text{He})+m(p)$.
2. Initial mass $$\begin{gathered} =2.014102+3.016029=5.030131 \\ , \\ textu \end{gathered}$$.
3. Final mass $$\begin{gathered} =4.002603+1.007276=5.009879 \\ , \\ textu \end{gathered}$$.
4. Mass defect $\Delta m=5.030131-5.009879=0.020252\text{u}$.
5. Convert to energy: $Q=\Delta m\times 931.5=0.020252\times 931.5\approx 18.87\text{MeV}\approx 18.9\text{MeV}$.
6. For part (b), find number of deuterium atoms in $2.0\text{g}$. Molar mass of ${}^2$H $\approx$ $2.014{\text{g mol}}^{-1}$.
7. Moles of D $=2.0/2.014=0.993\text{mol}$.
8. Number of D nuclei $=0.993\times 6.022\times 10^{23}\approx 5.98\times 10^{23}$ reactions (each reaction consumes one D).
9. Energy per reaction in joules: $18.87\text{MeV}\times 1.602\times 10^{-13}=3.02\times 10^{-12}\text{J}$.
10. Total energy $$\begin{gathered} =3.02 \\ times10^{-12} \\ times5.98 \\ times10^{23} \\ approx1.81 \\ times10^{12} \\ , \\ textJ \end{gathered}$$.
11. Rounded, $$\begin{gathered} E_{ \\ texttotal} \\ approx1.8 \\ times10^{12} \\ , \\ textJ \end{gathered}$$.

Concept tested: Fusion Q‑value and total energy calculation

Published 17 September 2026 · How this solution was prepared and checked