Nuclear Binding Energy Calculator

Calculate nuclear binding energy and binding energy per nucleon for any isotope.

Iron: Z=26, Carbon: Z=6, Uranium: Z=92
Iron-56: N=30, Carbon-12: N=6
1 u = 931.5 MeV/c²
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Mass Defect and Nuclear Binding Energy

A bound nucleus has less mass-energy than the same protons and neutrons separated to infinity, and that difference is the nuclear binding energy. The missing mass is called the mass defect. Multiplying it by c2 gives the energy required to separate the nucleus completely, or equivalently the energy released when the bound nucleus forms.

The mass convention matters. If a tabulated atomic mass is used, the clean formula uses the mass of a neutral hydrogen atom, mH, rather than a bare proton, because the isotope's atomic mass includes Z electrons. Then Δm=ZmH+Nmn−matom. If a true nuclear mass with electrons removed is used instead, use the bare proton mass and account consistently for electron masses. Mixing the two conventions creates a systematic error.

Δm=ZmH+Nmn−matom,   BE=Δm×931.494MeV,   BE/A=BE/(Z+N)
SymbolMeaningWhy it appears / units
ZNumber of protonsDetermines the element and the number of hydrogen-atom masses in the atomic-mass convention.
NNumber of neutronsNeutron count; A=Z+N.
ΔmMass defectu; positive for a bound nucleus relative to separated constituents.
BETotal binding energyMeV; energy needed to separate all nucleons.
BE/ABinding energy per nucleonMeV/nucleon; useful for comparing average binding across nuclei of different sizes.

Binding energy per nucleon rises rapidly for light nuclei, reaches a broad maximum near iron and nickel, then decreases gradually for heavier nuclei. That shape explains why fusion of light nuclei and fission of very heavy nuclei can both release energy: the products move toward more tightly bound configurations. It does not mean the nucleus with the largest BE/A is automatically the longest-lived against every decay mode.

Worked Examples

Example 1: Iron-56: Z=26, N=30, mass=55.9349
Δm=0.5287 u
Result: 492.3 MeV total, 8.79 MeV/nucleon
Near the peak of the binding-energy-per-nucleon curve
Example 2: Helium-4: Z=2, N=2, mass=4.00260
Δm=0.03038 u
Result: 28.3 MeV, 7.07 MeV/nucleon
Alpha particle stability explains alpha decay
Example 3: Deuterium using atomic masses
Z=1, N=1, matom=2.0141018u → Δm≈0.0023882u
Result: BE≈2.225MeV, BE/A≈1.112MeV/nucleon
Using hydrogen-atom mass keeps the electron accounting consistent with a tabulated neutral-atom mass.
Example 4: Carbon-12
Z=6, N=6, matom=12u → Δm≈0.09894u
Result: BE≈92.16MeV, BE/A≈7.680MeV/nucleon
Total binding energy grows with nucleon count, so BE/A is usually the better quantity for comparing different nuclei.

Common Mistakes

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Mixing atomic mass with bare-proton mass

Atomic masses include electron masses. Use hydrogen-atom mass with tabulated atomic isotope masses, or convert consistently to nuclear masses before using bare-proton mass.

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Comparing total binding energy instead of binding energy per nucleon

A larger nucleus often has a larger total BE directly because it contains more nucleons. BE/A is the useful average measure for comparing how tightly different nuclei are bound.

⚠️
Using 931.5 MeV as though it were energy per kilogram

The conversion is approximately 931.494MeV per atomic mass unit of mass defect. If mass defect is entered in kilograms, use E=Δmc2 directly in SI units.

Frequently Asked Questions

Why does binding energy matter?
Binding energy is released when nucleons bind together. Nuclear fission (splitting heavy atoms) and fusion (joining light atoms) both release energy by moving toward the iron peak.
What is mass defect?
The difference between the mass of the separated constituents and the bound system; when neutral atomic masses are used, hydrogen-atom mass keeps electron accounting consistent. This mass difference (Δm) converts to energy via E=Δmc².
Why can both fusion and fission release energy?
Light nuclei can release energy by fusing into products with greater binding energy per nucleon, while very heavy nuclei can release energy by splitting into more tightly bound medium-mass products. In both cases, the final total mass-energy is lower and the difference appears as released energy.
Which nucleus has the highest binding energy per nucleon?
The broad maximum is in the iron-nickel region. Nickel-62 is commonly cited among the nuclei with the highest measured binding energy per nucleon, while iron-56 is especially important in stellar nucleosynthesis and lies very near the peak.
Is mass defect actually missing matter?
No matter mysteriously disappears. The bound system has lower total internal energy than the separated constituents, and mass is one form of energy through E=mc2. The lower rest mass of the bound nucleus reflects that lower total energy.
Why must electron masses be handled carefully?
Mass tables normally report neutral atomic masses, which include electrons. A calculation based on bare proton masses is therefore inconsistent unless electron masses are removed from the tabulated atomic mass or otherwise accounted for. Using hydrogen-atom masses avoids that mismatch for neutral atoms.

Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

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