Binding energy per nucleon
Key idea: Binding energy per nucleon compares how tightly nuclei are bound. Fusion or fission releases energy when the products have greater total binding energy than the reactants.
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Guided lesson
Guided steps
- Read why comparing different-sized nuclei needs an energy per nucleon.
- Calculate both averages, then compare totals and averages separately.
- Read how nuclear attraction and electric repulsion shape the curve.
- Predict iron and uranium totals before using the explorer.
- Compare iron and uranium using the total and average readings.
- Sketch the curve on paper, then compare its key features.
- Connect greater total binding with lower rest mass and energy release.
- Find the energy release when the fragments have unequal sizes.
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ContinueA uranium nucleus has a far larger binding energy than a helium nucleus, simply because it contains far more nucleons. To compare how tightly different nuclei are held together, you need the binding energy per nucleon. Plotted against nucleon number, it gives one of the most important graphs in nuclear physics: it shows which nuclei are most tightly bound and why both fusion and fission can release energy.
Binding energy per nucleon
The binding energy per nucleon is the total binding energy divided by the nucleon number, Eb/A. It is the average energy needed per nucleon to separate a nucleus completely. The larger it is, the more energy is needed on average to separate the nucleons completely. This compares how tightly nuclei are bound; radioactive decay depends on the possible daughter nuclei too.
| Nucleus | Total binding energy | Binding energy per nucleon |
|---|---|---|
| helium-4, A = 4 | 28.3 MeV | 28.3/4 = 7.08 MeV |
| lithium-7, A = 7 | 39.2 MeV | 39.2/7 = 5.60 MeV |
Lithium-7 needs more energy to pull it completely apart, because it has more nucleons. But helium-4 has the larger average binding energy per nucleon. This comparison describes complete separation, rather than the energy needed to remove one particular nucleon.
Check your understanding 1
Calcium-40 has a total binding energy of 342 MeV. Tin-120 has a total binding energy of 1021 MeV. Find each binding energy per nucleon to three significant figures. Then choose which has the larger total binding energy and which has the larger value per nucleon.
Show answer
Tin-120 has the larger total. Per nucleon, calcium-40 has 342/40 = 8.55 MeV and tin-120 has 1021/120 = 8.51 MeV, so calcium-40 is slightly more tightly bound.
The shape of the curve
- For light nuclei, the binding energy per nucleon rises steeply, from 1.1 MeV for hydrogen-2 to 7.7 MeV for carbon-12. Helium-4, at 7.1 MeV, lies well above its neighbours.
- It reaches a broad maximum of about 8.8 MeV per nucleon near iron-56 and nickel-62. These are the most tightly bound nuclei.
- For heavier nuclei it falls slowly, to about 7.6 MeV per nucleon for uranium-238.
The shape comes from two forces. The strong nuclear force attracts each nucleon only to its nearest neighbours. In a small nucleus, a large fraction of the nucleons sit at the surface with fewer neighbours, so the average binding is low; adding nucleons raises it. The electric repulsion between protons acts across the whole nucleus, so in a large nucleus every proton repels every other proton. As nuclei get heavier, this repulsion grows faster than the attraction, and the binding energy per nucleon slowly falls.
Check your understanding 2
Before using the explorer, estimate the total binding energies of iron-56 and uranium-238 using 8.8 and 7.6 MeV per nucleon respectively. Does the lower uranium value per nucleon mean it has the smaller total?
Show answer
No. The curve shows an average. The total binding energy is A × (Eb/A): uranium-238 has 238 × 7.6 ≈ 1800 MeV, far more than iron-56’s 56 × 8.8 ≈ 490 MeV.
Compare total and average binding
In Nuclide view, select Iron-56, then Uranium-238. Compare the readings for Binding energy and E_B per nucleon with your estimates. The table uses more precise masses, so small numerical differences from the rounded curve values are expected.
Iron-56: 26 protons and 30 neutrons. Mass defect 0.5285 u, binding energy 492.3 MeV, 8.79 MeV per nucleon.
- Make-up
- Z = 26, N = 30, A = 56
- Z ¹H atoms + N neutrons
- 56.463398 u
- Mass of the atom
- 55.934936 u
- Mass defect, Δm
- 0.528463 u
- Binding energy, EB = Δm c²
- 492.3 MeV
- EB per nucleon
- 8.790 MeV
- Mass before
- 5.030151 u
- Mass after
- 5.011268 u
- Mass lost, Δm
- 3.14 × 10⁻²⁹ kg
- Energy released, Q = Δm c²
- 17.59 MeV
- Q per nucleon of fuel
- 3.52 MeV
Try this
0 of 4 doneCompare iron-56 and nickel-62 at the top of the curve. (not done yet)
Nickel-62 (8.795 MeV per nucleon) just beats iron-56 (8.790 MeV). Nuclei near A = 60 are the most stable.
Pull a helium-4 nucleus apart. How much energy does it take? (not done yet)
28.3 MeV, which is Δm c². The same energy is released when two protons and two neutrons bind into helium-4.
Split uranium-235. Where do the fragments sit on the curve? (not done yet)
Both fragments are higher on the curve than uranium: they are more tightly bound, and the extra binding energy, about 180 MeV, is released.
Run D–T fusion too. Which releases more energy per nucleon of fuel? (not done yet)
Fusion: 17.6 MeV from 5 nucleons, 3.5 MeV each. Fission: about 180 MeV from 236 nucleons, under 1 MeV each.
You’ve tried every task.
Check your understanding 3
You may draw on paper. Without looking at the figure, sketch the binding energy per nucleon against nucleon number. Label both axes, and mark the peak and the values at the peak and for uranium-238.
Show answer
The vertical axis is binding energy per nucleon in MeV, and the horizontal axis is nucleon number A. The curve rises steeply from near zero for the lightest nuclei, with helium-4 standing out above its neighbours, reaches a broad peak of about 8.8 MeV near A = 56 to 62 (iron and nickel), then falls gradually to about 7.6 MeV at A = 238. The vertical axis must be the value per nucleon: a total-binding-energy plot is not this curve. Check both axis labels, the steep rise, the broad peak and the gradual heavy-nucleus decline on your sketch.
What the curve tells you about energy release
Nuclei near the peak are the most tightly bound. A reaction releases energy when the total binding energy of all products is greater than that of all reactants, with the same total numbers of protons and neutrons. Moving bound nucleons towards the peak often increases that total. Light nuclei can move towards the peak by joining together; very heavy nuclei can move towards it by splitting. These are fusion and fission.
The curve shows a trend, not a rule for every nucleus. A larger binding energy per nucleon does not mean a nucleus cannot decay, and it does not predict a half-life. It is also an average: the energy needed to remove one particular nucleon is different, because removing it leaves a different nucleus.
Check your understanding 4
In a simplified fission, a nucleus with A = 240 and Eb/A = 7.60 MeV splits into two nuclei: one with A = 90, Eb/A = 8.70 MeV, and one with A = 150, Eb/A = 8.40 MeV. No free neutrons are emitted and the total numbers of protons and neutrons are unchanged. Find the energy released by the reaction.
Show the steps Optional working
Show answer
Initial binding energy: 240(7.60) = 1824 MeV. Final binding energy: 90(8.70) + 150(8.40) = 783 + 1260 = 2043 MeV. The increase is 2043-1824 = 219 MeV, which is released. A simple mean of 8.70 and 8.40 would give equal weight to unequal numbers of nucleons and the wrong energy. Check the weighted totals and the sign of the release.
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Syllabus and review details
- GCE A-Level H2 Physics 2027 · 2027
Content Overview, PDF pages 9–10; Subject Content, PDF pages 11–30
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