Last updated: 2026-06-16

Nuclear Binding Energy

Nuclear binding energy is the energy that holds an atomic nucleus together. It equals the energy that would be needed to separate the nucleus into its individual protons and neutrons, and it comes from a remarkable fact: a bound nucleus weighs slightly less than the sum of its parts. That missing mass — the mass defect — is converted to energy through Einstein's $E = mc^2$.

This calculator takes the number of protons $Z$, the number of neutrons $N$ and the neutral atomic mass $M$ (in atomic mass units), and returns the mass defect, the total binding energy, and the binding energy per nucleon.

How the model works

To assemble a nucleus from free nucleons, energy is released; to take it apart again, that same energy must be supplied. Conservation of energy then requires the bound nucleus to be lighter than its separate constituents by exactly the binding energy divided by $c^2$.

The figures here use the atomic-mass convention: you enter the mass of the neutral atom, not the bare nucleus. The advantage is that the $Z$ electrons of the atom cancel against the electrons already counted in the $Z$ hydrogen atoms used for the protons, so the electron masses drop out of the arithmetic automatically.

Formula

The mass defect $\Delta m$ is reported in atomic mass units (u). One atomic mass unit of mass corresponds to about 931.5 MeV of energy.

Worked example

Helium-4 has Z = 2 protons, N = 2 neutrons and a neutral atomic mass of $M = 4.002602\ \text{u}$.

Step 1 — Mass defect:

Step 2 — Binding energy:

Step 3 — Per nucleon:

The unusually high binding energy of helium-4 is why it is so stable and why it is emitted whole, as an alpha particle, in radioactive decay.

Binding energy per nucleon for selected nuclei

Real-world relevance

The binding-energy-per-nucleon curve explains where nuclear energy comes from. It rises steeply for light nuclei, peaks near iron-56 at about 8.79 MeV per nucleon, and falls off slowly for the heaviest elements. Light nuclei release energy by fusing toward the peak, which powers the Sun and other stars; heavy nuclei release energy by splitting toward it, which is the basis of nuclear fission in reactors and weapons. Both processes move nucleons toward the most tightly bound region of the curve.

Limitation: idealized masses

The calculation assumes accurate input masses and the atomic-mass convention. Tabulated atomic masses already include the electron binding energies, which are negligible compared with nuclear binding energy and are ignored here. For precise nuclear-data work, use measured atomic masses from a current evaluation; for understanding the physics and producing textbook-level results, the formula above is sufficient.