Binding Energy Per Nucleon Curve: Shape and Key Insights

If we divide a nucleus's total binding energy by its number of nucleons, we get binding energy per nucleon. Plotting this against mass number A for all elements gives a famous curve that rises steeply, peaks near iron, and then slowly falls. This single graph explains both fission and fusion.

The curve matters because nuclei always move toward higher binding energy per nucleon when they can. Light nuclei gain by fusing together; heavy nuclei gain by splitting apart. Both release energy.

Key Concepts

• Binding energy per nucleon (BE/A): average energy holding each nucleon, in MeV per nucleon.
• Peak: near A = 56 (iron-56) and 62 (nickel-62) at about 8.8 MeV per nucleon — the most stable nuclei.
• Light end: hydrogen and helium have low values near 1 to 7 MeV per nucleon.
• Heavy end: uranium-238 sits near 7.6 MeV per nucleon, less bound per nucleon than iron.

Formulas

BE/A = (Δm × 931.5) / A MeV per nucleon, where Δm is mass defect in u and A is mass number. Energy released in a reaction: Q = (BE/A of products − BE/A of reactants) × total A, in MeV.

Worked Example

Estimate energy when uranium-235 (BE/A ≈ 7.6 MeV) fissions into two fragments near A = 117 with BE/A ≈ 8.3 MeV each.

1. Gain per nucleon = 8.3 − 7.6 = 0.7 MeV.
2. Total nucleons ≈ 235.
3. Q ≈ 0.7 × 235 = 164.5 MeV, close to the real value near 200 MeV when neutron contributions are included precisely.

Common Misconceptions

• Students think the peak is at the heaviest nucleus. It is near iron; heavier nuclei are less tightly bound per nucleon.
• Many believe fusion only happens in the Sun. The curve shows fusion releases energy for all light nuclei up to iron, which is why stars stop fusing at iron.
• Some read the curve as total binding energy. It is per nucleon, so comparisons are fair across sizes.

Real-World Applications

The curve guides nuclear reactor fuel choice (splitting heavy nuclei), fusion research with hydrogen isotopes, understanding why stars collapse after forming iron cores, and dating which nuclei can exist naturally.

One more insight: the small dips and bumps on the curve come from especially stable arrangements called magic numbers of protons or neutrons, such as helium-4, which sits above its neighbours because it is doubly magic and unusually tightly bound for its size.

Remember the shape: steep rise, peak near iron-56 at about 8.8 MeV per nucleon, gentle fall — fuse on the left of the peak, split on the right, and energy comes out both ways.