The Q-value of a nuclear reaction measures the net energy released or required by that reaction.
It comes directly from Einstein’s mass-energy relation \(E = mc^2\). If the total mass before the reaction is larger
than the total mass after the reaction, the difference appears as released kinetic energy or radiation. If the total
final mass is larger, the reaction needs external input energy.
Mass defect and Q-value
For a general reaction, the mass defect is defined by comparing the total initial and final masses:
Mass defect.
\[
\begin{aligned}
\Delta m &= \sum m_{\mathrm{initial}} - \sum m_{\mathrm{final}}
\end{aligned}
\]
The corresponding Q-value is
Q-value formula.
\[
\begin{aligned}
Q &= \Delta m\,c^2
\end{aligned}
\]
In practical nuclear calculations, masses are often entered in atomic mass units, so one uses the standard conversion
\[
\begin{aligned}
1\ \mathrm{u}\,c^2 &\approx 931.494\ \mathrm{MeV}
\end{aligned}
\]
which gives the very convenient working form
\[
\begin{aligned}
Q &\approx 931.494\,\Delta m\ \mathrm{MeV}
\end{aligned}
\]
How to interpret the sign of Q
The sign of the Q-value determines the energy character of the reaction:
| Condition |
Reaction type |
Interpretation |
| \(Q > 0\) |
Exoergic |
The reaction releases energy because the final mass is smaller |
| \(Q = 0\) |
Thermoneutral |
No net energy release or requirement in the simplified mass balance |
| \(Q < 0\) |
Endoergic |
The reaction requires input energy because the final mass is larger |
Sample reaction: D + T → He-4 + n
A standard fusion example is the deuterium-tritium reaction:
D + T → He-4 + n
Using atomic masses
\(m_{\mathrm{D}} = 2.0141018\ \mathrm{u}\),
\(m_{\mathrm{T}} = 3.0160493\ \mathrm{u}\),
\(m_{\mathrm{He4}} = 4.0026033\ \mathrm{u}\),
and
\(m_n = 1.0086649\ \mathrm{u}\),
the total initial and final masses are
Step 1. Sum the initial masses.
\[
\begin{aligned}
\sum m_i
&= m_{\mathrm{D}} + m_{\mathrm{T}} \\
&= 2.0141018 + 3.0160493 \\
&= 5.0301511\ \mathrm{u}
\end{aligned}
\]
Step 2. Sum the final masses.
\[
\begin{aligned}
\sum m_f
&= m_{\mathrm{He4}} + m_n \\
&= 4.0026033 + 1.0086649 \\
&= 5.0112682\ \mathrm{u}
\end{aligned}
\]
Step 3. Compute the mass defect.
\[
\begin{aligned}
\Delta m
&= \sum m_i - \sum m_f \\
&= 5.0301511 - 5.0112682 \\
&= 0.0188829\ \mathrm{u}
\end{aligned}
\]
Step 4. Convert to the Q-value.
\[
\begin{aligned}
Q
&= 931.494 \cdot \Delta m \\
&= 931.494 \cdot 0.0188829 \\
&\approx 17.6\ \mathrm{MeV}
\end{aligned}
\]
Because the Q-value is positive, this is an exoergic reaction. That is why D-T fusion is so important
in discussions of fusion energy: each reaction releases a substantial amount of energy.
Atomic masses and consistency
In many introductory Q-value calculations, one uses either nuclear masses or atomic masses. The important point is to be
consistent. If atomic masses are used on both sides of the reaction and the electron bookkeeping matches, the calculation
works correctly. That is why the D + T → He-4 + n example gives the standard result directly with atomic masses.
Physical meaning
A positive Q-value does not mean that all released energy must appear as one single particle’s kinetic energy. In general,
the released energy is shared among the reaction products as kinetic energy and sometimes radiation. The Q-value simply
tells you the total energy budget available from the reaction mass difference.
In stellar fusion, laboratory fusion, and many nuclear-decay problems, the Q-value is the first quantity one computes
because it immediately reveals whether the process can release energy and how large that energy scale is.
Advanced note: threshold energy
At university level, one goes beyond the Q-value alone. If a reaction is endoergic, it generally requires a nonzero
threshold energy in the incident channel. Even for exoergic reactions, momentum conservation determines
how the released energy is split among the products. Those refinements are important in accelerator physics, reaction
kinematics, and astrophysical modeling. This calculator deliberately focuses on the clean mass-balance core:
the reaction Q-value from the total initial and final masses.
| Concept |
Main relation |
Meaning |
| Mass defect |
\(\Delta m = \sum m_i - \sum m_f\) |
Difference between initial and final total masses |
| Q-value |
\(Q = \Delta m c^2\) |
Total reaction energy release or requirement |
| Working conversion |
\(Q \approx 931.494\,\Delta m\ \mathrm{MeV}\) |
Practical form when masses are entered in atomic mass units |
| Exoergic |
\(Q > 0\) |
The reaction releases energy |
| Endoergic |
\(Q < 0\) |
The reaction requires input energy |
| D-T fusion benchmark |
\(Q \approx 17.6\ \mathrm{MeV}\) |
Standard fusion example |