The big idea: Weigh all the protons and neutrons of a nucleus separately, then weigh the finished nucleus. The nucleus is always a tiny bit lighter.
What the missing mass means
- That missing mass is the mass defect
- Through E = mc² it equals the binding energy
- Binding energy = the energy you would need to pull the nucleus completely apart
Loose protons + neutrons
- All the nucleons separate, far apart
- Their masses simply add up — the larger total
- Higher energy: it took work to pull them apart
The bound nucleus
- The nucleons stuck together by the strong force
- Slightly smaller total mass — the mass defect Δm
- Lower energy: building it released the binding energy
New words, plainly: Mass defect (Δm) = (mass of the separate nucleons) − (mass of the bound nucleus).
Binding energy = the energy that mass defect is worth (E = mc²); also the energy needed to split the nucleus back into loose nucleons.
Binding energy per nucleon = binding energy ÷ number of nucleons — the fair way to compare different nuclei.
| Mass added up | Mass / u |
|---|---|
| 2 protons (2 × 1.007276) | 2.014552 |
| 2 neutrons (2 × 1.008665) | 2.017330 |
| Separate nucleons, total | 4.031882 |
| Actual helium-4 nucleus | 4.001506 |
| Mass that went missing (Δm) | 0.030376 |
Why 'per nucleon' matters: A big nucleus has a big total binding energy just because it has more nucleons.
To compare how stable two nuclei are, divide by the number of nucleons. The higher the binding energy per nucleon, the more stable the nucleus.
What holds a nucleus together
The strong nuclear force: Protons push each other apart electrically, yet nuclei hold together. Something stronger must pull the nucleons together: the strong nuclear force, attractive and short-range (about the size of a nucleon). The binding energy is the work needed to pull the nucleons apart against it.
HL: evidence, neutrons and the flat curve: Evidence: nuclei exist and stay together although their protons repel; scattering at high energy also shows an attractive force at short range. Neutron-to-proton ratio: neutrons add strong attraction without electric repulsion, so heavier nuclei need more neutrons than protons to be stable (N/Z rises to about 1.5). Flat curve above A ≈ 60: the binding energy per nucleon stays about 8 MeV because each nucleon is held only by its nearest neighbours, which is evidence that the strong force is short-range.
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Three steps: find the mass defect (separate nucleons − nucleus), turn it into energy with E = mc², then divide by the number of nucleons to get the binding energy per nucleon.
- energy equivalent of the mass — here the binding energy (J or MeV)
- mass converted — here the mass defect Δm of the nucleus (kg or u)
- speed of light, 3.00 × 10⁸ m s⁻¹ (given constant)
Two units, one idea: In joules: put Δm in kilograms (1 u = 1.661 × 10⁻²⁷ kg) and multiply by c².
In MeV (faster): keep Δm in u and multiply by 931.5 (because 1 u = 931.5 MeV c⁻²).
Both give the same energy — pick whichever the question asks for.
Three units for one mass: Nuclear masses come in kg, in u (1 u = 1.661 × 10⁻²⁷ kg) or in MeV c⁻² (1 u = 931.5 MeV c⁻²). A mass in MeV c⁻² turns into an energy in MeV just by dropping the c⁻².
A helium-4 nucleus is made of 2 protons and 2 neutrons. Proton mass = 1.007276 u, neutron mass = 1.008665 u, and the helium-4 nucleus has mass 4.001506 u. (a) Find the total binding energy of the nucleus. (b) Find its binding energy per nucleon. (1 u = 931.5 MeV c⁻².)
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Plot the binding energy per nucleon against the nucleon number A for every nucleus and you get one of the most important graphs in physics. It explains where nuclear energy comes from.
Animated graph
Watch the graph build step by step in study mode.
Reading the curve: Higher up = more tightly bound = more stable. The curve peaks near iron (A ≈ 56) — the most stable nuclei of all.
Any reaction that moves a nucleus toward the peak ends up more tightly bound, so it releases energy.
How this is tested — binding energy shows up in both papers:
Paper 1A
- A quick MCQ — read off which nucleus is most stable.
- Or compare the energy released per unit mass in fusion vs fission.
Paper 2
- A calculation — find a mass defect, the binding energy, and the binding energy per nucleon from given masses.
The classic trap: Thinking only fission releases energy. Both fusion (light nuclei) and fission (heavy nuclei) release energy, because both move toward iron.
Fusion (light fuel)
- Joins light nuclei, e.g. hydrogen isotopes
- Big jump UP the steep part of the curve
- Releases several MeV per nucleon → most energy per kilogram
Fission (heavy fuel)
- Splits a heavy nucleus, e.g. uranium
- Smaller climb up the gentle part of the curve
- Releases under 1 MeV per nucleon → less energy per kilogram
A fusion reaction releases 17.6 MeV from a total of 5 nucleons (deuterium + tritium). A fission reaction releases 200 MeV from a total of 236 nucleons (a uranium nucleus). Find the ratio of the energy released per nucleon in fusion to that in fission.
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