KAHIBARO
Table of Contents
Why fission releases energy
When a heavy nucleus undergoes fission, it splits into two medium mass nuclei, usually with a few neutrons emitted as well. The striking result is that the total mass of the products is slightly less than the mass of the original nucleus plus any absorbed neutron. That missing mass appears as released energy.
This energy release comes from nuclear binding. Very heavy nuclei are not bound as tightly, per nucleon, as medium mass nuclei. When a heavy nucleus such as uranium splits into fragments closer to the middle of the binding energy curve, the nucleons end up in a more tightly bound arrangement. The difference in binding energy is released.
The key idea is that fission releases energy because the fission products have a larger total binding energy than the original heavy nucleus.
This means the final total mass is smaller, and the mass difference becomes energy through
$$E = \Delta m c^2$$
Mass defect and Q value
The energy released in a nuclear reaction is described by the reaction Q value. For fission, the Q value is positive if energy is released.
For a general reaction,
$$Q = \left(m_{\text{initial}} - m_{\text{final}}\right)c^2$$
If $Q > 0$, the reaction is exothermic, meaning it releases energy.
In fission, a typical example is
$$^{235}\mathrm{U} + n \rightarrow ^{141}\mathrm{Ba} + ^{92}\mathrm{Kr} + 3n + Q$$
The exact fragments can vary, but the principle is always the same. The sum of the final masses is smaller than the sum of the initial masses, and the difference gives the released energy.
For energy release in fission,
$$Q = \Delta m c^2$$
where $\Delta m = m_{\text{initial}} - m_{\text{final}}$.
A positive $\Delta m$ means energy is released.
Connection to the binding energy curve
Heavy nuclei such as uranium and plutonium lie on the high mass side of the binding energy per nucleon curve. Medium mass nuclei, roughly around iron and nearby elements, have higher binding energy per nucleon. Fission moves the system from a less tightly bound state to a more tightly bound state.
This is why splitting a heavy nucleus can release energy, while splitting a light nucleus usually does not.
A simple comparison is shown below.
| Type of nucleus | Binding energy per nucleon | Tendency in energy terms |
|---|---|---|
| Very light nuclei | Low to moderate | Fusion can release energy |
| Medium mass nuclei | High | Most stable region |
| Very heavy nuclei | Lower than medium mass | Fission can release energy |
Where the released energy goes
The released energy does not appear in just one form. It is shared among several products of the fission event.
In a typical fission reaction, most of the energy appears as kinetic energy of the two fission fragments. Smaller amounts go into the kinetic energy of emitted neutrons, gamma radiation, and later radioactive decay of the fragments.
A rough distribution for one fission event is:
| Form of energy | Typical amount |
|---|---|
| Kinetic energy of fission fragments | about 160 to 170 MeV |
| Kinetic energy of emitted neutrons | about 5 MeV |
| Prompt gamma rays | about 5 to 10 MeV |
| Beta decay and associated radiation from fragments | several MeV |
| Total | about 200 MeV |
The exact numbers depend on the nucleus and the fission products, but about $200 \,\text{MeV}$ per fission is a useful typical value.
A useful average value is
$$E_{\text{fission}} \approx 200 \,\text{MeV per fission}$$
Most of this becomes kinetic energy of the fission fragments.
Why the fragments carry most of the energy
After the nucleus splits, the two positively charged fragments repel each other strongly because of the electric force between their protons. They fly apart at high speed. This electrostatic repulsion is a major reason the fragments gain so much kinetic energy.
The emitted neutrons also carry kinetic energy, but much less than the heavy fragments because the fragments receive most of the push from the Coulomb repulsion during separation.
Energy release per atom and per amount of fuel
The energy from one fission event is tiny in everyday units, but enormous when many atoms are involved.
To convert from mega electron volts to joules,
$$1 \,\text{MeV} = 1.602 \times 10^{-13} \,\text{J}$$
So for one fission,
$$200 \,\text{MeV} \approx 200 \times 1.602 \times 10^{-13} \,\text{J}$$
$$\approx 3.2 \times 10^{-11} \,\text{J}$$
That seems small, but one mole contains Avogadro's number of nuclei, so the total becomes very large.
If one mole of nuclei undergoes fission,
$$E \approx \left(6.02 \times 10^{23}\right)\left(3.2 \times 10^{-11}\right) \,\text{J}$$
$$\approx 1.9 \times 10^{13} \,\text{J}$$
This is why nuclear fuel can produce so much energy from a small mass.
Useful conversions:
$$1 \,\text{MeV} = 1.602 \times 10^{-13} \,\text{J}$$
Typical fission energy:
$$200 \,\text{MeV} \approx 3.2 \times 10^{-11} \,\text{J}$$
Example of a Q value calculation
Suppose the total initial mass of a fission reaction is slightly larger than the total final mass by
$$\Delta m = 0.215 \,\text{u}$$
Using
$$1 \,\text{u}c^2 \approx 931.5 \,\text{MeV}$$
the released energy is
$$Q = 0.215 \times 931.5 \,\text{MeV} \approx 200 \,\text{MeV}$$
This is the right scale for a typical fission event.
Prompt and delayed energy
Not all the released energy appears immediately. Some appears promptly, right at the fission event, mainly as fragment kinetic energy, neutron kinetic energy, and prompt gamma rays. Some appears later because the fission fragments are usually radioactive and undergo beta decay and further gamma emission.
This means the total energy release includes both immediate and delayed contributions. In reactors, this matters because part of the heat continues to be produced even after fission stops.
Comparison with chemical energy
Fission energy is far greater than chemical energy per atom. Chemical reactions involve electron arrangements and typically release energies of a few electron volts per atom or molecule. Nuclear fission involves changes inside the nucleus and releases about $200$ million electron volts per event.
| Process | Typical energy scale per event |
|---|---|
| Chemical reaction | a few eV |
| Nuclear fission | about $200 \,\text{MeV}$ |
Since
$$1 \,\text{MeV} = 10^6 \,\text{eV}$$
fission is enormously more energetic on a per event basis.
Nuclear fission releases energy on the scale of millions of electron volts per event, much larger than chemical reactions, which are typically only a few electron volts per event.
Summary of the physical picture
A heavy nucleus can lower its total energy by splitting into medium mass nuclei. Because the products are more tightly bound, the total mass decreases. That mass difference is released as energy, mostly as kinetic energy of the fission fragments, with additional energy carried by neutrons, gamma rays, and later radioactive decay.
The central quantitative idea is the Q value,
$$Q = \left(m_{\text{initial}} - m_{\text{final}}\right)c^2$$
and for a typical fission event,
$$Q \approx 200 \,\text{MeV}$$
This is the origin of the large energy output of nuclear fission.