The energy that holds nuclei together
Protons and neutrons are held together in the nucleus by the strong nuclear force. Pulling them apart takes energy, the binding energy, and the same energy is released when a nucleus forms. Because of Einstein’s E = mc², it shows up as missing mass: a nucleus always weighs less than its separate parts.
The calculation
With Z protons, N = A − Z neutrons and an atomic mass M in unified atomic mass units (u):
- Mass defect: Δm = Z × m(¹H) + N × mₙ − M
- Binding energy: E = Δm × 931.494 MeV per u
- Binding energy per nucleon: E ÷ A
The calculator uses m(¹H) = 1.00782503 u for the hydrogen-1 atom, which includes one electron, and mₙ = 1.00866492 u for the neutron, so the electrons in the atomic mass cancel out.
Example
Iron-56 has 26 protons, 30 neutrons and an atomic mass of 55.934936 u:
- Mass defect: 26 × 1.00782503 + 30 × 1.00866492 − 55.934936 = 0.528462 u
- Binding energy: 492.26 MeV, or 7.89 × 10⁻¹¹ J
- Per nucleon: 8.790 MeV
The binding energy curve
Binding energy per nucleon rises steeply from hydrogen to helium-4, about 7.07 MeV, then more slowly to a peak of about 8.79 MeV around iron-56 and nickel-62, before falling gently to about 7.6 MeV for uranium-238. This curve explains:
- Fusion: joining light nuclei moves up the curve and releases energy, which powers the Sun.
- Fission: splitting heavy nuclei such as uranium-235 also moves toward the peak, releasing about 200 MeV per fission.
- Stellar nucleosynthesis: stars can fuse elements up to iron for energy, but not beyond.
Semi-empirical estimate
For comparison, the calculator applies the Bethe-Weizsäcker formula with typical coefficients: a volume term of 15.75 MeV per nucleon, minus surface, Coulomb and asymmetry terms, plus or minus a pairing term. For iron-56, it gives about 495 MeV, within 1% of the measured value. Find accurate atomic masses in the Atomic Mass Evaluation tables published by the IAEA.