Binding energy calculator: mass defect to MeV and joules

A nucleus weighs less than the protons and neutrons it is made from. The missing mass, the mass defect, is the binding energy, by E = mc²: the energy needed to pull the nucleus apart into separate nucleons. In atomic mass units, each 1 u of mass defect is 931.5 MeV, so helium-4, with a mass defect of 0.030377 u, has a binding energy of 0.030377 × 931.5 = 28.30 MeV. This binding energy calculator turns a mass defect into a binding energy in MeV or joules, with the working.

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Binding Energy Calculator

Calculate nuclear binding energy per nucleon

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How to use the binding energy calculator

1. Open the Binding Energy Calculator and choose MeV (mass defect in atomic mass units) or Joules (mass defect in kilograms).
2. For MeV, type the mass defect in u. For Joules, type it as a number and a power of ten, such as 5.044 and −29 for 5.044 × 10⁻²⁹ kg.
3. Read the binding energy, with the steps, and the same energy in MeV, eV, J, kJ and MJ under Also equal to.
4. For practice questions, press Random problem.

The calculator starts from the mass defect: you work that out first, as below. It doesn't look up nuclear masses or divide by the number of nucleons.

How to find the mass defect

Using whole-atom masses with hydrogen-1 atoms, rather than bare nuclei with protons, means the electrons cancel. The masses from NIST are:

Then E = Δm × 931.5 MeV, and dividing by the number of nucleons A gives the binding energy per nucleon, which says how tightly each nucleon is held.

Worked examples you can check in the calculator

Example 1: helium-4

Helium-4 has 2 protons and 2 neutrons. Δm = 2 × 1.007825 + 2 × 1.008665 − 4.002603 = 0.030377 u. Choose MeV and type 0.030377:

Per nucleon, 28.30 ÷ 4 = 7.07 MeV.

Example 2: carbon-12

6 protons and 6 neutrons: Δm = 6 × 1.007825 + 6 × 1.008665 − 12 = 0.098940 u. The calculator gives 92.16 MeV, or 92.16 ÷ 12 = 7.68 MeV per nucleon.

Example 3: iron-56

26 protons and 30 neutrons: Δm = 26 × 1.007825 + 30 × 1.008665 − 55.934936 = 0.528462 u, so E = 492.3 MeV and 492.3 ÷ 56 = 8.79 MeV per nucleon, far more tightly bound than helium.

Example 4: deuterium, the lightest

One proton and one neutron: Δm = 1.007825 + 1.008665 − 2.014102 = 0.002388 u, so E = 2.22 MeV, just 1.11 MeV per nucleon. Deuterium is easily broken apart.

Example 5: in kilograms, with E = mc²

The same helium-4 mass defect in kilograms is 0.030377 × 1.6605 × 10⁻²⁷ = 5.044 × 10⁻²⁹ kg. Choose Joules and type 5.044 and −29: E = mc² = 5.044 × 10⁻²⁹ × (2.998 × 10⁸)² = 4.533 × 10⁻¹² J, which the calculator shows is 28.29 MeV, matching Example 1 to rounding.

Try turning mass into energy yourself, then press Mark to check your answer in the Nuclear Energy course.

The binding energy per nucleon curve

As the table shows, the lightest nuclei hold each nucleon weakly, the figure climbs fast up to a peak near iron, and beyond iron it drops off gently for the heaviest nuclei. That is why both fusion and fission release energy: joining light nuclei, or splitting heavy ones, makes nuclei nearer the peak, which are more tightly bound, and the difference in binding energy is given out.

Common mistakes

• Mixing nuclear and atomic masses. Use atomic masses with hydrogen-1 atoms, or nuclear masses with protons. Mixing them leaves electron masses in the answer.
• Rounding masses too early. The mass defect is a small difference between big numbers. Keep at least 6 decimal places.
• Using 931.5 with kilograms. 931.5 MeV goes with u. For kg, use E = mc² and get joules.
• Thinking binding energy is stored in the nucleus. It is the energy you would have to supply to separate the nucleons, released when the nucleus formed.
• Comparing total binding energies. Compare per nucleon to see which nucleus is more stable.

Read next

Physics · Atoms and particlesMass defect and binding energy: formula and examplesWhat mass defect and binding energy are, how to calculate them in MeV and joules, binding energy per nucleon, and worked A-level examples.