Nuclear Binding Energy — Mass Defect, E = mc² and the Curve of Stability
Why a nucleus weighs less than its parts: mass defect, binding energy in MeV, the binding-energy-per-nucleon curve, and how it explains fusion and fission.
Quick answer
Binding energy is the energy you would have to supply to pull a nucleus apart into its individual protons and neutrons. The bound nucleus weighs slightly less than its separate parts — that missing mass, the mass defect, equals the binding energy divided by c².
Unit: megaelectronvolts (MeV)
Example: A helium-4 nucleus is about 0.0304 u lighter than 2 free protons + 2 free neutrons. Multiplying by 931.5 MeV/u gives a binding energy of about 28.3 MeV — roughly 7.1 MeV per nucleon.
What you will learn
- Define binding energy as the energy needed to break a nucleus into separate protons and neutrons
- Calculate mass defect from nuclear masses and convert it to energy with E = Δmc²
- Read the binding-energy-per-nucleon curve and identify the most stable nucleus
- Explain how fusion and fission both release energy by moving toward iron-56
Where the missing mass goes
Weigh a helium-4 nucleus and you get a surprise: it is lighter than two free protons plus two free neutrons. The difference is tiny — under one percent of the total — but mass spectrometers measure it precisely and it is unmistakably real.
That missing mass is called the mass defect. It did not vanish. The bound nucleus simply has less total energy than the separated particles, and mass is one form energy can take. Pulling the nucleus apart again would force you to supply that energy back — and the energy required is the binding energy.
In short: binding energy is the price of disassembly. The larger it is, the more tightly the nucleus holds together.
The formula: E = Δmc²
Einstein's mass–energy relation converts the mass defect directly into an energy. Nuclear masses are usually quoted in atomic mass units (u), where . Multiplying a mass defect in u by 931.5 gives the binding energy straight away in megaelectronvolts (MeV) — the natural currency of nuclear physics.
Note the sign convention: the mass defect is (mass of the separate nucleons) minus (mass of the nucleus), so it is positive for every bound nucleus.
Worked mini-example: helium-4
Two protons () plus two neutrons () total . The helium-4 nucleus itself weighs only (the atomic mass with its two electrons subtracted). The mass defect is therefore:
Converting to energy:
Per nucleon, that is — a per-nucleon figure that lets us fairly compare nuclei of very different sizes.
The binding-energy-per-nucleon curve
Total binding energy grows with nucleus size, so physicists compare nuclei by binding energy per nucleon — the average energy anchoring each proton or neutron. Plotted against mass number, this curve climbs steeply for light nuclei, peaks near iron-56 at about 8.8 MeV per nucleon, then declines gently for heavier nuclei.
Iron-56 sits at the summit: the most tightly bound common nucleus, the ash that fusion in massive stars can burn no further.
Every energy-releasing nuclear reaction moves nuclei toward iron-56 on this curve: light nuclei fuse their way up, heavy nuclei split their way down. The energy released equals the gain in binding energy.
Fusion and fission: two roads to iron
In the Sun, hydrogen nuclei fuse step by step toward helium-4. Helium's per nucleon far exceeds hydrogen's roughly per nucleon (deuterium binds at total for its two nucleons), so each fusion step climbs the curve and the surplus escapes as light and heat.
At the heavy end, uranium-235 manages only about per nucleon. Splitting it into two medium-weight fragments near per nucleon releases roughly 200 MeV per fission — the energy that heats nuclear reactors and, uncontrolled, drives nuclear weapons.
Same rule, opposite directions: the universe pays out energy whenever nuclei end up more tightly bound than they started.
Why the curve peaks where it does
Two forces wrestle inside every nucleus. The strong nuclear force attracts all nucleons but only reaches nearest neighbors, so its contribution per nucleon saturates as the nucleus grows. Electromagnetic repulsion between protons, by contrast, reaches across the whole nucleus and grows roughly as the square of the proton count.
For light nuclei, each added nucleon brings mostly new attraction, and the curve climbs. Past iron, every new proton adds more repulsion than the short-range strong force can offset, and binding per nucleon slides back down. That tug-of-war is the entire story of the curve's shape.
Variables and units
| Symbol | Name | Unit | Meaning |
|---|---|---|---|
| Binding energy | MeV (or J) | Energy needed to separate a nucleus into individual protons and neutrons | |
| Mass defect | u (or kg) | How much lighter the nucleus is than its separate nucleons | |
| Atomic mass unit | u | 1/12 the mass of a carbon-12 atom; 1 u = 931.5 MeV/c² | |
| Speed of light | m/s | 299792458 m/s (exact); converts mass to energy |
Common mistakes
Total energy is conserved in every reaction. The bound system simply has less total energy — and therefore less mass — than the separated parts. Mass is not a conserved substance; energy is.
Stability per nucleon peaks at iron-56 (about 8.8 MeV). Very heavy nuclei like uranium are less tightly bound per nucleon, which is exactly why splitting them releases energy.
Binding energy is the net result of strong-force attraction minus electromagnetic repulsion between protons. Both forces shape the curve.
Real-world applications
The Sun shines by fusing hydrogen toward helium-4; the mass defect of each fusion step becomes sunlight.
Fission reactors split uranium-235 into more tightly bound fragments, releasing about 200 MeV per fission as heat.
Hospital PET scanners detect gamma photons from positron–electron annihilation — mass converting directly into energy via E = mc².
Frequently asked questions
What is mass defect?
The mass defect is the difference between the total mass of a nucleus's separate protons and neutrons and the mass of the bound nucleus. For helium-4 it is about 0.0304 u. Multiply by c² — or by 931.5 MeV/u — and you get the binding energy: the energy released when the nucleus formed, or the energy needed to break it apart again.
How do I convert atomic mass units to MeV?
Multiply by 931.5. One atomic mass unit equals 931.5 MeV/c², so a mass defect of 0.0304 u becomes 0.0304 × 931.5 ≈ 28.3 MeV. In SI units, 1 u = 1.66053906660×10⁻²⁷ kg, so you can also apply E = Δmc² directly with c = 299792458 m/s to get joules.
What is the most stable nucleus?
Iron-56, with about 8.8 MeV of binding energy per nucleon — the highest of any common nucleus. 'Most stable' here means most tightly bound per nucleon. That is why fusion in dying stars stops at iron, and why both fusion of lighter elements and fission of heavier ones release energy.
If breaking nuclei apart costs energy, how can fusion release energy?
Breaking a nucleus into individual protons and neutrons always costs energy. But fusion does not do that — it rearranges light nuclei into a more tightly bound nucleus. The product's binding energy per nucleon is higher than the reactants', and that surplus binding energy is what escapes as kinetic energy and radiation.