Nuclear Binding Energy Calculator
Find how tightly any nucleus is held together. Get the mass defect, binding energy and binding energy per nucleon from measured AME2020 atomic masses, compare with the semi-empirical formula, and work out the energy released by fission, fusion or decay.
How to Use
- Pick Binding energy for one nucleus, or Reaction Q-value for fission, fusion or a decay.
- Type a nuclide as Fe-56, 56Fe, U 235 or iron-56. All 3,558 nuclides in the AME2020 table can be used; n, p, d, t and α work too.
- For a reaction, write it with an arrow, such as U-235 + n -> Ba-141 + Kr-92 + 3n. Put a number in front for several particles, and use e-, e+ and ν for beta decays.
- Read the binding energy, the energy per nucleon, the mass defect and the Weizsäcker estimate, or the Q-value and the energy per kilogram of fuel.
- See the nucleus on the curve of binding energy, and check Show Work for the masses and every term of the formula.
Worked Example
Iron-56. It has Z = 26 protons and N = 30 neutrons. From AME2020, m(¹H) = 1.007825031898 u, m(n) = 1.0086649159 u and m(⁵⁶Fe) = 55.934935537 u. The parts weigh 26 × 1.007825031898 + 30 × 1.0086649159 = 56.463398306 u, so the mass defect is 56.463398306 − 55.934935537 = 0.528462769 u. Times 931.49410372 MeV/u that is 492.26 MeV, or 492.26 ÷ 56 = 8.7904 MeV per nucleon, matching the 8,790.356 keV the AME2020 table lists.
Fission of uranium-235. U-235 + n weighs 235.043928117 + 1.0086649159 = 236.052593033 u; Ba-141 + Kr-92 + 3n weighs 140.914403653 + 91.926173092 + 3 × 1.0086649159 = 235.866571493 u. The 0.18602154 u difference is Q = 173.28 MeV released in every fission of this kind. (The electrons balance on both sides, 92 = 56 + 36, so atomic masses can be used directly.)
The common mistake: mixing atomic and nuclear masses. Tables list the masses of whole atoms, electrons included. Using the bare proton mass (1.007276467 u) with iron-56’s atomic mass leaves 26 electrons unaccounted for and gives 478.97 MeV, 13.29 MeV too low. Use the hydrogen atom’s mass for the protons, as here, so the electrons cancel.
Show Work
Formulas
Weighing Nuclei
Einstein showed in 1905 that energy has mass, E = mc², but the missing mass of nuclei only became measurable with Francis Aston’s mass spectrograph after 1919. Aston found that atoms weigh slightly less than whole numbers of hydrogen atoms and in 1927 plotted this “packing fraction” across the elements, an early version of the curve shown above. In 1935 Carl Friedrich von Weizsäcker wrote down the semi-empirical mass formula, treating the nucleus as a charged liquid drop, and Hans Bethe and Robert Bacher refined it in 1936.
The curve explained the two great discoveries that followed. When Otto Hahn and Fritz Strassmann found barium among the products of neutron-irradiated uranium in December 1938, Lise Meitner and Otto Frisch used the liquid-drop picture to show the nucleus had split and estimated that each split releases about 200 MeV. The same curve shows why fusing light nuclei, as in the Sun, also releases energy.
Masses here are from the Atomic Mass Evaluation 2020 (AME2020; W. J. Huang, M. Wang, F. G. Kondev, G. Audi and S. Naimi, Chinese Physics C 45, 2021), distributed by the IAEA Atomic Mass Data Center. The evaluation, begun by Aaldert Wapstra in the 1950s, combines thousands of measurements into one consistent table; values it marks with # are estimates from trends rather than measurements. The semi-empirical coefficients are those in Kenneth Krane’s Introductory Nuclear Physics (Wiley, 1988).
About This Tool
This calculator looks up any of the 3,558 nuclides in the AME2020 mass table and works out the mass defect, the binding energy in MeV and joules with its uncertainty, and the binding energy per nucleon, then compares it term by term with the semi-empirical mass formula and marks it on the curve of binding energy. For a reaction or decay it checks that mass number and charge balance, then gives the Q-value, the share of the mass converted and the energy per kilogram of fuel.
Everything runs in your browser; the mass table is downloaded once and nothing you enter is sent anywhere.
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Frequently Asked Questions
What is nuclear binding energy?
It is the energy needed to pull a nucleus apart into separate protons and neutrons, and it shows up as missing mass. Twenty-six hydrogen atoms and 30 neutrons weigh 56.4633983 u, but an iron-56 atom weighs 55.9349355 u. The 0.5284628 u mass defect times 931.494 MeV/u is a binding energy of 492.26 MeV, or 8.7904 MeV per nucleon.
Which nucleus is the most tightly bound?
Per nucleon, nickel-62 at 8.7946 MeV, just ahead of iron-58 (8.7923 MeV) and iron-56 (8.7904 MeV), using AME2020 masses. Iron-56 is often named instead because it is far more common: it is the end point of the nuclear burning in massive stars, and the most tightly bound per unit of mass rather than per nucleon.
Why do both fission and fusion release energy?
Both move nucleons up the curve towards its peak near iron. Uranium-235 has 7.5909 MeV per nucleon, while its fission fragments barium-141 and krypton-92 have 8.326 and 8.513, so U-235 + n → Ba-141 + Kr-92 + 3n releases 173.28 MeV. Deuterium (1.112) and tritium (2.827) fuse into helium-4 (7.074), releasing 17.589 MeV.
How much energy is in a kilogram of nuclear fuel?
For the fission above, 173.28 MeV per 235.04 u of uranium is 7.113 × 10¹³ J/kg, the energy of about 17,000 tonnes of TNT; only 0.0788% of the mass becomes energy. D–T fusion turns 0.375% of the fuel mass into energy: 3.374 × 10¹⁴ J/kg, or about 80,640 tonnes of TNT per kilogram. The figure of about 200 MeV often quoted for fission also counts the energy the radioactive fragments release later as they decay.
What is the semi-empirical mass formula?
Weizsäcker’s 1935 formula treats the nucleus like a drop of liquid: a volume term, minus surface, Coulomb and asymmetry terms, plus a pairing term. With Krane’s coefficients (15.5, 16.8, 0.72, 23 and 34 MeV) it gives 494.86 MeV for iron-56, 0.53% above the measured 492.26 MeV, and 1,789.9 MeV for uranium-238 against 1,801.69. It fails for the lightest nuclei: for the deuteron it gives −15.89 MeV instead of 2.2246 MeV.
How do I use the Nuclear Binding Energy Calculator?
Simply type your numbers and read the result, which refreshes the instant you change something. There is nothing to submit and nothing to wait for.
Is it free? Does it work without internet?
Yes to both. It is free with no sign-up, and once the page has loaded it keeps working even with no internet.
Where does my data go?
Nowhere — every calculation runs on your own device. Nothing you enter is uploaded, logged, or stored.
Common Use Cases
Reactor physics
U-235 + n → Ba-141 + Kr-92 + 3n converts 0.18602 u of mass into 173.28 MeV, 7.113 × 10¹³ J per kilogram of uranium-235.
Fusion energy
D + T → He-4 + n releases 17.589 MeV, 3.374 × 10¹⁴ J per kilogram of fuel, almost five times as much per kilogram as fission.
Radioactive decay
Uranium-238 alpha decay to thorium-234 releases 4.270 MeV; tritium beta decay to helium-3 releases just 18.59 keV.
Nuclear stability
Helium-4 has 7.074 MeV per nucleon against 5.332 for lithium-6, which is why heavy nuclei shed alpha particles rather than other small pieces.
Physics coursework
The deuteron is bound by only 2.2246 MeV: a gamma ray of a little more than that splits it into a proton and a neutron.
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