Sketch and interpret the binding energy per nucleon curve
π₯
State that iron-56 has the highest binding energy per nucleon
π₯
Explain how fusion of light nuclei releases energy
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Explain how fission of heavy nuclei releases energy
π’
Calculate energy released in fusion and fission from the curve
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Compare energy yields of fusion and fission
The Binding Energy Per Nucleon Curve
The binding energy per nucleon (E_B/A) is the average energy needed to remove one nucleon from a nucleus. Plotting this against nucleon number A gives the characteristic curve:
Region
A
E_B/A (approx)
Features
Light nuclei
1β20
1β7 MeV
Rising steeply; large variation
Peak (iron)
~56
~8.8 MeV
Most stable region
Medium-heavy
20β100
~8.0β8.8 MeV
Fairly flat plateau
Heavy nuclei
>100
Decreasing to ~7.6 MeV
Decreasing due to Coulomb repulsion
A higher binding energy per nucleon means a more stable nucleus. The nucleus is in a deeper energy well and harder to break apart.
Iron-56 (β΅βΆFe) sits at the peak of the curve at approximately 8.8 MeV/nucleon β it is the most energetically stable nucleus in nature.
Why the Curve Has This Shape
The shape arises from competing forces:
Rising part (light nuclei, A < 56):
Each nucleon added gains neighbours to bond with (strong force is short-range)
Volume binding energy increases faster than surface energy loss
Coulomb repulsion is small for light nuclei
Note: β΄He (8 nucleons... 4 nucleons), ΒΉΒ²C and ΒΉβΆO lie slightly above the smooth curve β these have especially stable shell configurations
Falling part (heavy nuclei, A > 56):
Proton-proton Coulomb repulsion grows as ZΒ² β affects entire nucleus
Strong force only acts between nearest neighbours (short range)
Net effect: each additional proton is less tightly bound
Fusion Releases Energy for Light Nuclei
Nuclear fusion is the combining of light nuclei to form a heavier nucleus. Since light nuclei (low A) have a lower E_B/A than the fused product, the product is more tightly bound:
Energy released = (E_B/A of product β E_B/A of reactants) Γ A_product
Fusion moves nuclei up the left side of the binding energy curve β each nucleon ends up more tightly bound, releasing energy. This powers stars.
Fusion only releases energy for nuclei lighter than iron. Fusing nuclei heavier than iron would absorb energy (endothermic).
Fission Releases Energy for Heavy Nuclei
Nuclear fission is the splitting of a heavy nucleus into two medium-mass fragments. Since heavy nuclei (high A) have a lower E_B/A than the fragments, the products are more tightly bound:
Energy released per fission β (E_B/A)_products Γ A_products β (E_B/A)_parent Γ A_parent
Example: Uranium-235 fission releases approximately 200 MeV per fission event.
Fission moves nuclei up the right side of the curve β products are lighter with higher E_B/A, so energy is released.
Both fusion and fission release energy by moving nuclei towards iron-56 on the binding energy per nucleon curve. Iron is the "energy minimum" β you cannot extract energy by moving away from it.
Comparing Fusion and Fission Energy Yields
Property
Fusion
Fission
Typical energy per reaction
~10β20 MeV
~200 MeV
Energy per kg of fuel
~10ΒΉβ΄ J/kg
~8 Γ 10ΒΉΒ³ J/kg
Fuel
Hydrogen isotopes (abundant)
U-235 (limited)
Products
Helium (inert)
Radioactive fragments
Difficulty
Requires extreme T and P
Achievable with neutrons
Per nucleon, fusion releases more energy than fission β but per event, fission releases more because so many more nucleons are involved (A β 235 vs A β 5).
Example 1: Energy released in fusion from the curve
Two deuterium nuclei (A=2, E_B/A β 1.1 MeV) fuse to form helium-4 (A=4, E_B/A β 7.1 MeV). Estimate the energy released.
1 Total binding energy of products: 4 Γ 7.1 = 28.4 MeV
2 Total binding energy of reactants: 2 Γ (2 Γ 1.1) = 4.4 MeV
3 Energy released = 28.4 β 4.4 = 24.0 MeV
Approximately 24 MeV released per fusion event (actual value ~24 MeV, consistent)
Example 2: Energy released in fission from the curve
Uranium-235 (A=235, E_B/A β 7.6 MeV) fissions into two fragments each with Aβ118 (E_B/A β 8.5 MeV). Estimate the energy released.
1 Total binding energy of products: 235 Γ 8.5 = 1997.5 MeV
2 Total binding energy of parent: 235 Γ 7.6 = 1786 MeV
3 Energy released = 1997.5 β 1786 = 211.5 MeV
Approximately 210 MeV per fission event (actual ~200 MeV β consistent with realistic E_B/A values)
Example 3: Binding energy per nucleon comparison
Compare the stability of ΒΉΒ²C (E_B = 92.2 MeV) and Β²Β³βΈU (E_B = 1802 MeV).
1 E_B/A for C-12: 92.2/12 = 7.68 MeV/nucleon
2 E_B/A for U-238: 1802/238 = 7.57 MeV/nucleon
Carbon-12 has slightly higher E_B/A (7.68 vs 7.57 MeV/nucleon) and is more stable per nucleon. Although U-238 has much higher total binding energy, uranium nuclei are more susceptible to decay due to Coulomb repulsion of its 92 protons.
Example 4: Why iron cannot release energy by fission or fusion
Iron-56 has E_B/A β 8.79 MeV/nucleon. Explain using the binding energy curve why neither fission nor fusion of iron can release energy.
1 Iron sits at the peak of the binding energy per nucleon curve.
2 Fission would produce lighter fragments moving to the left on the curve β lower E_B/A β less stable. Energy must be absorbed, not released.
3 Fusion would move to heavier nuclei (right of peak) β also lower E_B/A. Again, energy must be absorbed.
Iron-56 is at the global minimum of nuclear energy β all reactions moving away from Fe-56 are endothermic. This is why stars stop generating energy when their core becomes iron, leading to supernova.
Q1. Which nucleus has the highest binding energy per nucleon?
Q2. Moving from left to right on the binding energy per nucleon curve beyond iron-56, what happens to E_B/A?
Q3. Why does fusion of light nuclei release energy?
Q4. A nucleus has total binding energy 500 MeV and A = 62. What is its binding energy per nucleon?
Q5. Helium-4 lies slightly above the smooth binding energy curve. What does this indicate?
Challenge 1. Using average binding energies per nucleon from the curve (D-T: Β²H β 1.1, Β³H β 2.8, β΄He β 7.1 MeV/nucleon), estimate the energy released in the D-T fusion reaction: Β²βH + Β³βH β β΄βHe + ΒΉβn. Compare with the exact value of 17.6 MeV.
Total binding energy of reactants:
Β²H: 2 Γ 1.1 = 2.2 MeV
Β³H: 3 Γ 2.8 = 8.4 MeV
Total reactant BE = 2.2 + 8.4 = 10.6 MeV
Total binding energy of products:
β΄He: 4 Γ 7.1 = 28.4 MeV
ΒΉn: 0 MeV (free neutron, no binding)
Total product BE = 28.4 MeV
Energy released = 28.4 β 10.6 = 17.8 MeV
This agrees very well with the exact value of 17.6 MeV (within 1%), demonstrating the power of the binding energy curve for energy estimates.
Challenge 2. Explain why stars burn hydrogen first, then helium, then heavier elements up to iron β and why no further energy-releasing fusion is possible beyond iron. What happens to a star whose core becomes iron?
Each fusion stage moves nuclei up the binding energy per nucleon curve towards iron-56:
H β He: moves from A=1β4 (low E_B/A) to higher E_B/A β large energy release
He β C, O: continues climbing the curve β energy released but less per unit mass
C, O β heavier elements: still moving towards Fe-56 peak
At iron: the nucleus is at the peak of the curve. Any further fusion would produce nuclei with LOWER E_B/A β endothermic.
The star can no longer generate energy from nuclear reactions.
Radiation pressure drops; gravity overwhelms the star.
The core collapses, causing a supernova explosion.
The remaining core becomes a neutron star or black hole depending on mass.
Elements heavier than iron are synthesised in supernova explosions, using the energy of the collapse.
Challenge 3. A student claims "fission releases more energy per reaction than fusion, so fission is a better energy source." Evaluate this claim quantitatively, considering energy per nucleon and per unit mass of fuel.
The student is partially correct but misleading:
Per reaction: Fission releases ~200 MeV vs fusion ~17 MeV β so fission wins per event.
Per nucleon: Fission ~200/235 β 0.85 MeV/nucleon; D-T fusion ~17.6/5 β 3.5 MeV/nucleon. Fusion releases ~4Γ more energy per nucleon.
Per unit mass of fuel: Fusion wins significantly β hydrogen is much lighter than uranium.
D-T: ~3.4 Γ 10ΒΉβ΄ J/kg; U-235 fission: ~8 Γ 10ΒΉΒ³ J/kg
Fusion releases ~4Γ more energy per kg of fuel.
Other considerations: Fusion fuel (H isotopes) is far more abundant; fusion products are not radioactive; but fusion requires extreme temperatures (~10βΈ K) not yet achieved in a sustained commercial reactor. Overall, fusion is the superior energy source in principle, but fission is currently the only proven nuclear energy technology.