Nuclear Binding Energy Calculator
Calculate the nuclear binding energy and mass defect of an atomic nucleus from proton and neutron counts using Einstein mass-energy equivalence.
Nuclear Binding Energy
The nuclear binding energy is the energy required to completely disassemble an atomic nucleus into its constituent protons and neutrons. It arises from the strong nuclear force that holds nucleons (protons and neutrons) together, one of the four fundamental forces of nature.
Mass Defect and E = mc²
When protons and neutrons come together to form a nucleus, the resulting nucleus is actually slightly lighter than the sum of its parts. This difference in mass is called the mass defect (Δm).
Einstein’s famous equation connects mass and energy:
E = Δm × c²
Where:
- E = binding energy in Joules
- Δm = mass defect in kilograms
- c = speed of light = 2.998 × 10⁸ m/s
The Mass Defect Calculation
Δm = Z × m_H + N × m_n − M_atomic
Where:
- Z = number of protons (atomic number)
- N = number of neutrons
- m_H = mass of a hydrogen-1 atom = 1.007825 u
- m_n = neutron mass = 1.008665 u
- M_atomic = the tabulated atomic mass of the nuclide, in atomic mass units (u)
1 u (atomic mass unit) = 1.66054 × 10⁻²⁷ kg
Why hydrogen atoms and not bare protons. This trips up almost everyone the first time. Every mass table you will find, including the 55.9349 u for iron-56 suggested above, lists the atomic mass, which includes the Z electrons orbiting the nucleus. If you subtract that from Z bare protons plus N neutrons, you have quietly thrown away Z electron masses and your binding energy comes out too small. Use the hydrogen atom mass instead and the electrons cancel on both sides: Z hydrogen atoms carry exactly the Z electrons the neutral nuclide has. The electron binding energies that survive are a few eV against a few MeV, far below anything visible here.
Get this wrong on iron-56 and you get 8.55 MeV/nucleon instead of 8.79. On deuterium, where there is only one electron to lose but very little binding energy to begin with, you get 0.86 instead of 1.11, which is 23% out.
Binding Energy per Nucleon
The binding energy per nucleon (BE/A) is one of the most important quantities in nuclear physics. It tells you how tightly each nucleon is bound on average.
| Nuclide | BE/Nucleon (MeV) |
|---|---|
| Deuterium (²H) | 1.11 |
| Iron-56 (⁵⁶Fe) | 8.79 (maximum, the most stable nucleus) |
| Uranium-238 (²³⁸U) | 7.57 |
Iron-56 has the highest binding energy per nucleon, making it the most tightly bound nucleus. Nuclei lighter than iron can release energy by fusion; heavier nuclei release energy by fission.
Conversion: eV and MeV
- 1 MeV = 10⁶ eV = 1.602 × 10⁻¹³ J
- 1 u of mass defect = 931.5 MeV of energy
This convenient conversion (931.5 MeV/u) makes nuclear energy calculations straightforward.
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This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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