Binding Energy Per Nucleon Calculator

Binding energy per nucleon (BE/A) measures how tightly bound each nucleon is. Iron-56 has the highest BE/A (~8.8 MeV/nucleon) — fusion of light elements and fission of heavy ones both move toward iron.

☢️ Nuclear Physics📐 BE/A = Δmc²/A⚛️ Nuclear Stability
Proton number Z
Neutron number N
Nuclear mass M (AMU)
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Binding EnergyBEBE = (Zmp + Nmn − M)c²MeV
Per NucleonBE/ABE/(Z+N)MeV/nucleon
Proton massmp1.007276 AMUAMU
Neutron massmn1.008665 AMUAMU
Iron-56 BE/A—8.790 MeV/nucleon (maximum)MeV

Step-by-Step Examples

Example 1
Iron-56

Z=26, N=30, M=55.9349 AMU.

  • dm = 26*1.007276 + 30*1.008665 - 55.9349 = 0.52882 AMU
  • BE = 0.52882*931.494 = 492.3 MeV
  • BE/A = 492.3/56 = 8.790 MeV/nucleon
✓ 8.790 MeV/nucleon — peak of binding curve
Example 2
Helium-4

Z=2, N=2, M=4.00260 AMU.

  • dm = 2*1.007276 + 2*1.008665 - 4.00260 = 0.03038 AMU
  • BE = 0.03038*931.494 = 28.30 MeV
  • BE/A = 28.30/4 = 7.07 MeV/nucleon
✓ 7.07 MeV/nucleon — alpha particle is especially stable
Example 3
Hydrogen-2 (Deuteron)

Z=1, N=1, M=2.01355 AMU.

  • dm = 1.007276+1.008665-2.01355 = 0.00239 AMU
  • BE = 2.224 MeV
  • BE/A = 1.112 MeV/nucleon
✓ 1.112 MeV/nucleon — loosely bound

Real-World Applications

☢️
Nuclear Fission
U-235 (7.6 MeV/nucleon) splits to products near Fe (~8.5 MeV/nucleon), releasing ~0.9 MeV/nucleon.
☀️
Stellar Nucleosynthesis
Stars fuse hydrogen to helium to carbon... up to iron. Iron is the end point — fusion beyond iron requires energy.
💥
Fusion Reactors
D+T fusion: BE increases dramatically from ~1 to ~8 MeV/nucleon, releasing ~17.6 MeV per reaction.
🧪
Nuclear Stability
High BE/A means more stable nucleus. Magic numbers (2,8,20,28,50,82,126 nucleons) give extra stability.

Common Mistakes to Avoid

⚠️
Use nuclear mass, not atomic mass

Nuclear mass = atomic mass - Z*electron_mass. For quick calculations, atomic mass tables are used with correction.

⚠️
A is mass number, not mass

A = Z + N = total nucleons (integer). Do not confuse with atomic mass M (in AMU, not integer due to binding).

⚠️
Higher BE/A means more stable

Greater binding per nucleon = harder to disassemble. Fe-56 at 8.79 MeV/nucleon is the most stable nuclide.

Frequently Asked Questions

Why is iron the most stable nucleus? ▾
At Fe-56, the balance between the attractive strong nuclear force and repulsive proton Coulomb force reaches optimum. Both fusion beyond iron and fission below iron release less energy.
What are magic numbers? ▾
Nuclei with Z or N = 2,8,20,28,50,82,126 have unusually high binding energy per nucleon. They correspond to filled nuclear shells, analogous to noble gases in chemistry.
Why does fusion release energy? ▾
Light nuclei (H, He) have low BE/A ~1-7 MeV/nucleon. Fusing them to larger nuclei increases BE/A, releasing the difference as energy.
Why does fission release energy? ▾
Heavy nuclei (U, Th) have BE/A ~7.6 MeV/nucleon. Splitting to medium nuclei (near iron) at ~8.5 MeV/nucleon releases ~0.9 MeV/nucleon.
What is the liquid drop model? ▾
Semi-empirical mass formula models BE as volume + surface + Coulomb + asymmetry + pairing terms. Predicts BE/A for all nuclei with ~1% accuracy.

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☢️ Nuclear Reaction Q-Value☢️ Nuclear Binding Energy⛛ Mass-Energy☢️ Half-Life☢️ Radioactive Decay

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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