Nuclear Energy: Mass Defect, Binding Energy, Fission and Fusion

Energy Locked in the Nucleus

The Sun has been shining for about 4.6 billion years. Nuclear power stations generate electricity from small amounts of fuel. Both get their energy from the nucleus, where enormous amounts of energy are stored in the forces that hold protons and neutrons together. This tutorial explains nuclear energy: mass defect and binding energy, the atomic mass unit and MeV, the binding energy per nucleon curve, nuclear fission and chain reactions, nuclear reactors, and nuclear fusion, with worked calculations. It builds on the tutorial on radioactivity.

Mass Defect

A nucleus is made of protons and neutrons (nucleons). Surprisingly, the mass of a nucleus is always slightly less than the total mass of its separate protons and neutrons. The difference is the mass defect:

Δm = (Z × m(p) + N × m(n)) − m(nucleus)

Here Z is the number of protons and N the number of neutrons. The "missing" mass has been converted into the binding energy that holds the nucleus together, following Einstein's equation E = mc².

Units: the Atomic Mass Unit and the MeV

Nuclear masses are tiny, so they are measured in atomic mass units (u): 1 u = 1.66054 × 10⁻²⁷ kg, one-twelfth of the mass of a carbon-12 atom. Nuclear energies are measured in mega-electronvolts: 1 MeV = 1.602 × 10⁻¹³ J.

A very useful conversion follows from E = mc²:

1 u of mass is equivalent to 931.5 MeV of energy

Useful masses: proton m(p) = 1.007276 u, and neutron m(n) = 1.008665 u.

Binding Energy

The binding energy of a nucleus is the energy needed to separate it completely into its individual protons and neutrons. Equivalently, it is the energy released when the nucleus forms from them:

Binding energy = Δm × 931.5 MeV (with Δm in u)

Worked example 1: helium-4

A helium-4 nucleus (2 protons and 2 neutrons) has a mass of 4.001506 u. Find its mass defect, binding energy and binding energy per nucleon.

Mass of the separate nucleons = 2(1.007276) + 2(1.008665) = 4.031882 u.

Mass defect Δm = 4.031882 − 4.001506 = 0.030376 u.

Binding energy = 0.030376 × 931.5 ≈ 28.3 MeV.

Binding energy per nucleon = 28.3 ÷ 4 ≈ 7.07 MeV.

For comparison, a chemical reaction such as burning releases only a few eV per atom. Nuclear binding energies are millions of times larger.

The Binding Energy per Nucleon Curve

The binding energy per nucleon measures how tightly bound, and so how stable, a nucleus is. Plotted against mass number:

It rises steeply for light nuclei.
It reaches a peak around iron-56, at about 8.8 MeV per nucleon, the most stable nucleus.
It falls slowly for heavier nuclei, reaching about 7.6 MeV per nucleon for uranium.

This curve explains both kinds of nuclear energy. Moving towards iron from either side increases the binding energy per nucleon and so releases energy:

Heavy nuclei release energy by splitting: fission.
Light nuclei release energy by joining: fusion.

Nuclear Fission

In fission, a heavy nucleus such as uranium-235 absorbs a slow neutron, becomes unstable, and splits into two medium-sized nuclei, releasing 2 or 3 neutrons and a large amount of energy. A typical reaction is:

U-235 + n → Ba-141 + Kr-92 + 3n + about 200 MeV

The total mass of the products is about 0.2 u less than that of the reactants, and this mass appears as the kinetic energy of the fragments and as gamma radiation.

Chain reactions

Each fission releases 2 or 3 neutrons, which can cause further fissions. This is a chain reaction. If the amount of uranium is above the critical mass, enough neutrons are captured for the reaction to sustain itself. Uncontrolled, it grows explosively, as in an atomic bomb. Controlled, it releases energy steadily, as in a nuclear reactor.

Worked example 2: energy from 1 kg of uranium-235

Estimate the energy released if every nucleus in 1 kg of uranium-235 undergoes fission, releasing 200 MeV each.

Number of nuclei = (1000 g ÷ 235 g/mol) × 6.022 × 10²³ ≈ 2.56 × 10²⁴.

Energy = 2.56 × 10²⁴ × 200 × 1.602 × 10⁻¹³ J ≈ 8.2 × 10¹³ J.

Burning 1 kg of coal releases only about 3 × 10⁷ J, so uranium fission releases millions of times more energy per kilogram.

Nuclear Reactors

A nuclear power station uses controlled fission to boil water and drive turbines. Its key parts are:

Fuel rods: enriched uranium (or plutonium).
Moderator: graphite or water, which slows down the fast neutrons released by fission, because slow neutrons are much more likely to cause further fission of U-235.
Control rods: boron or cadmium, which absorb neutrons. They are raised or lowered to control the rate of reaction, and dropped fully in to shut the reactor down.
Coolant: water, gas or liquid metal, which carries heat away from the core to make steam.
Shielding: thick concrete and steel, which absorbs radiation and protects workers.

Nuclear power produces no carbon dioxide while running, but it creates long-lived radioactive waste that must be stored safely for thousands of years, and accidents (such as Chernobyl in 1986 and Fukushima in 2011) can be serious.

Nuclear Fusion

In fusion, light nuclei combine to form a heavier nucleus. For example, deuterium and tritium (two isotopes of hydrogen) fuse:

H-2 + H-3 → He-4 + n + about 17.6 MeV

Worked example 3

Using atomic masses (H-2 = 2.014102 u, H-3 = 3.016049 u, He-4 = 4.002602 u, n = 1.008665 u), find the energy released by this fusion reaction.

Mass before = 5.030151 u. Mass after = 5.011267 u. Δm = 0.018884 u.

Energy = 0.018884 × 931.5 ≈ 17.6 MeV.

Fusion powers the Sun and the stars, where hydrogen fuses into helium. On Earth, it requires extremely high temperatures, over 100 million °C, so that the positively charged nuclei move fast enough to overcome their electrical repulsion. Scientists are building experimental reactors (such as ITER in France) because fusion promises abundant fuel (deuterium from seawater) and much less long-lived waste than fission.

Fission and Fusion Compared

Fission splits heavy nuclei (uranium, plutonium). It is used today in power stations, and it produces long-lived radioactive waste.
Fusion joins light nuclei (hydrogen isotopes). It needs extreme temperatures and is still experimental for power generation. It releases more energy per kilogram of fuel, and it produces much less long-lived waste.
Both release energy because the products have higher binding energy per nucleon, and so less total mass, than the reactants.

Common Mistakes with Nuclear Energy

Thinking the nucleus weighs more than its parts. It weighs less, and the difference is the mass defect.
Confusing total binding energy with binding energy per nucleon. Stability is judged by the per-nucleon value.
Forgetting unit conversions: 1 u = 931.5 MeV and 1 MeV = 1.602 × 10⁻¹³ J.
Mixing up the moderator and the control rods. The moderator slows neutrons. Control rods absorb them.
Saying fusion happens at room temperature. It needs extremely high temperatures.

Nuclear Energy: Practice Questions

How much energy, in MeV, is equivalent to a mass of 1 u?
A nuclear reaction has a mass defect of 0.002 u. How much energy is released?
A nucleus with 238 nucleons has a binding energy of 1784 MeV. Find its binding energy per nucleon.
What is the function of the moderator in a nuclear reactor?
What is the function of the control rods?
Why does nuclear fusion need very high temperatures?

Answers: 1. 931.5 MeV. 2. 0.002 × 931.5 ≈ 1.86 MeV. 3. 1784 ÷ 238 ≈ 7.5 MeV. 4. It slows down fast neutrons so they are more likely to cause fission. 5. They absorb neutrons, controlling the rate of the chain reaction or stopping it. 6. The nuclei are positively charged and repel each other. Only at very high temperatures do they move fast enough to get close enough to fuse.

Conclusion

Nuclear energy comes from the mass defect. A nucleus weighs less than its separate nucleons, and the missing mass, Δm × 931.5 MeV, is its binding energy. The binding energy per nucleon peaks near iron, so heavy nuclei release energy by fission (about 200 MeV per uranium nucleus, sustained as a chain reaction in reactors with moderators and control rods), and light nuclei release energy by fusion (about 17.6 MeV per deuterium–tritium reaction, the power source of the Sun). Both release millions of times more energy per kilogram than burning fuel.