Binding Energy
Why nuclei weigh less than their parts.
Lesson at a glanceShow the key points
- B = [Z m(¹H) + N mₙ − m(X)]c² (46.2).
- B/A peaks at ~8.8 MeV near A ≈ 56.
- Liquid drop: volume − surface − Coulomb (46.3).
- Moving towards the peak releases energy.
Weigh a helium nucleus and you find it is lighter than the two protons and two neutrons it is made of. The missing mass is the energy that binds them, and how that energy varies across the elements explains why both splitting and fusing nuclei can release power.
Mass defect and binding energy
To pull a hydrogen atom apart you must supply 13.6 eV; bringing a proton and electron together releases the same 13.6 eV. Nuclei are similar, on a scale a million times larger. The energy needed to separate a nucleus completely into free protons and neutrons is its binding energy B. Since energy is released when the nucleus forms, the nucleus has less rest-mass energy than its parts:
Atomic masses: ¹H 1.007825 u, neutron 1.008665 u, ¹²C exactly 12 u. Find the binding energy of ¹²C and the binding energy per nucleon (1 u = 931 MeV/c²).
The atomic mass of ¹⁶O is 15.994915 u. Using the masses above, find its binding energy.
Binding energy per nucleon
Dividing B by A tells you how tightly, on average, each nucleon is held. For the deuteron (one proton, one neutron) it is only 2.22/2 = 1.11 MeV. It rises quickly with A, reaches a broad maximum of about 8.7–8.8 MeV for A ≈ 50–80 (iron and nickel), then falls slowly to about 7.5 MeV for uranium. Middle-weight nuclei are the most tightly bound.
Explaining the shape
- Volume term (+a₁A): the nuclear force is short-ranged, so a nucleon inside interacts only with its immediate neighbours (about 12, packed like spheres). The binding energy from all those bonds is proportional to A.
- Surface term (−a₂A2/3): nucleons at the surface have fewer neighbours, so binding is lost in proportion to the surface area R² ∝ A2/3. This matters most for small nuclei, so the curve starts low.
- Coulomb term (−a₃Z(Z − 1)/A1/3): every pair of protons repels, with energy ∝ 1/R. This grows fastest for heavy nuclei, making the curve droop at large A.
Mass excess
The atomic mass of ⁴He is 4.002603 u. Find its mass excess.
Releasing nuclear energy
Whenever nucleons regroup into nuclei with a higher total binding energy, the rest-mass energy drops and the difference is released (the Q-value). Both ends of the curve slope up towards the middle:
- Fusion: joining light nuclei into a heavier one moves up the steep left side.
- Fission: splitting a heavy nucleus into two middle-weight ones moves up the gentle right side.
How much energy is released if four ⁴He nuclei (atomic mass 4.002603 u) fuse into ¹⁶O (15.994915 u)?
- 1B = [Z m(¹H) + N mₙ − m(X)]c²; a nucleus weighs less than its free nucleons.
- 2B/A rises to ~8.8 MeV near A ≈ 56 and falls to ~7.5 MeV for uranium.
- 3Liquid-drop model: volume − surface − Coulomb (+ a Pauli term).
- 4Energy is released when nucleons move to more tightly bound nuclei: fusion of light, fission of heavy.
Check yourself
Keep exploring
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