Binding Energy per Nucleon Calculator
Find binding energy (BE) and binding energy per nucleon (BE/A) from a nuclide’s atomic mass (u), or from mass defect (Δm), or from binding energy directly. Includes quick picks, step-by-step, a mini stability gauge, a binding-curve mini chart, plus a nucleus + mass-defect visual.
Background
Nuclei weigh a tiny bit less than the total mass of their separate nucleons. That “missing mass” is the mass defect (Δm), and it becomes energy: BE = Δm·c². Dividing by A gives BE/A, a handy “how tightly bound?” comparison.
How to use this calculator
- Pick a mode: Nuclide (Z, A, atomic mass), Δm, or BE.
- Enter the values you have, then click Calculate.
- Use BE/A to compare stability across nuclei (higher usually = more tightly bound).
- Use the visuals to build intuition: gauge + curve + nucleus dots + mass-defect bar.
How this calculator works
- Neutrons: N = A − Z.
- Mass defect (atomic masses): Δm = Z·m(¹H) + N·mₙ − matom (hydrogen mass helps the electron masses “cancel out” when using atomic mass tables).
- Energy conversion: 1 u = 931.494 MeV/c², so BE(MeV) = Δm(u)·931.494.
- Per nucleon: BE/A = BE ÷ A.
Formulas & Equations Used
Neutrons: N = A − Z
Mass defect: Δm = Z·m(¹H) + N·mₙ − matom
Binding energy: BE = Δm·c² = Δm(u)·931.494 MeV
Binding energy per nucleon: BE/A = BE ÷ A
Example Problem & Step-by-Step Solution
Example 1 — Iron-56 (near the peak)
Given Z=26, A=56, and matom=55.93493633 u, find BE and BE/A.
- Neutrons: N = 56 − 26 = 30.
- Mass defect: Δm = 26·m(¹H) + 30·mₙ − 55.93493633.
- Binding energy: BE = Δm·931.494 MeV.
- Per nucleon: BE/A = BE ÷ 56.
Example 2 — Helium-4 (light but tightly bound)
Use Z=2, A=4, m=4.00260325413 u.
- N = 4 − 2 = 2.
- Δm = 2·m(¹H) + 2·mₙ − m.
- BE = Δm·931.494, then BE/A = BE ÷ 4.
Example 3 — From mass defect
A worksheet gives Δm = 0.528 u and A=56. Find BE and BE/A.
- BE = 0.528 × 931.494 MeV.
- BE/A = BE ÷ 56.
Frequently Asked Questions
Q: What does binding energy per nucleon tell me?
It’s a quick stability comparison. Higher BE/A usually means nucleons are more tightly bound.
Q: Why does the curve peak near iron?
Mid-mass nuclei are the “sweet spot” between the strong nuclear force (short range) and proton-proton repulsion (grows with Z).
Q: Why use hydrogen mass in nuclide mode?
Because isotope tables typically give atomic masses. Using m(¹H) makes the electron masses cancel neatly.
Q: What units does the calculator output?
Binding energy is shown in MeV (and also in J). Mass defect is shown in u, MeV/c², and kg.
