0.514226
u
478.9984
MeV
8.5535
MeV/nucleon
46,216,322.96
kJ/mol
0.514226
u
478.9984
MeV
8.5535
MeV/nucleon
46,216,322.96
kJ/mol
The Nuclear Binding Energy Calculator computes the total binding energy and binding energy per nucleon of an atomic nucleus from its atomic number, mass number, and measured nuclear mass. Nuclear binding energy is the energy required to completely disassemble a nucleus into its constituent protons and neutrons, and is a measure of nuclear stability.
The binding energy arises from Einstein's mass-energy equivalence E = mc^2. The actual mass of a nucleus is always less than the sum of the masses of its constituent free nucleons — this difference is called the mass defect. The missing mass has been converted to binding energy: BE = delta_m * c^2 = delta_m(u) * 931.494 MeV, since 1 atomic mass unit = 931.494 MeV/c^2.
The binding energy per nucleon (BE/A) is the most important measure of nuclear stability. It peaks near iron-56 and nickel-62 at approximately 8.8 MeV/nucleon — the most tightly bound nuclei. Elements lighter than iron can release energy by fusion (combining light nuclei increases BE/A); elements heavier than iron can release energy by fission (splitting heavy nuclei increases BE/A toward the peak). This explains why the sun fuses hydrogen to helium and why uranium fission releases energy.
The liquid drop model of the nucleus (Weizsacker formula) describes the binding energy through five terms: volume energy (proportional to A), surface energy (proportional to A^2/3), Coulomb repulsion (proportional to Z^2/A^1/3), asymmetry energy (from neutron-proton asymmetry), and pairing energy (extra stability for even-even nuclei). This model successfully explains nuclear stability trends across the periodic table.
Mass defect: delta_m = Z*m_p + N*m_n - M_nucleus (in unified atomic mass units, u). Where m_p = 1.007276 u, m_n = 1.008665 u, N = A-Z. Binding energy: BE = delta_m * 931.494 MeV. BE per nucleon = BE/A. The mass_u input should be the actual nuclear mass (not atomic mass) for highest accuracy.
BE/A < 1 MeV/nucleon: very loosely bound (rare isotopes). BE/A ~ 5-7 MeV/nucleon: light nuclei (H, He through Si). BE/A ~ 7-9 MeV/nucleon: most stable medium nuclei. BE/A peaks at 8.8 MeV/nucleon near Fe-56/Ni-62. BE/A decreases above A=62: heavy nuclei release energy by fission. Nuclei with BE/A > 8 MeV/nucleon are highly stable (most common elements).
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Fe-56 has BE/nucleon = 8.80 MeV, near the maximum on the binding energy curve. Its stability explains why iron is the end product of stellar nucleosynthesis in massive stars — no further energy can be released by fusion or fission.
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He-4 (the alpha particle) has a binding energy of 28.3 MeV. Its high BE/nucleon (7.07 MeV) compared to nearby nuclei explains why alpha particles are emitted as a unit in radioactive decay and why helium-4 is the ash of stellar hydrogen fusion.
The energy required to completely separate a nucleus into its individual protons and neutrons. It equals the mass defect times c^2, where the mass defect is the difference between the sum of free nucleon masses and the actual nuclear mass.
delta_m = Z*m_p + N*m_n - M_nucleus. This missing mass has been converted to binding energy via E = delta_m*c^2. For Fe-56: delta_m ~ 0.529 u = 492 MeV of binding energy.
The peak reflects the balance between the attractive strong nuclear force (increases binding, proportional to A) and the repulsive Coulomb force between protons (decreases binding, proportional to Z^2). Fe/Ni nuclei have the optimal ratio of strong force benefit to Coulomb cost.
BE = a_V*A - a_S*A^(2/3) - a_C*Z(Z-1)/A^(1/3) - a_A*(A-2Z)^2/A + delta_pairing. The five terms represent: volume energy, surface energy, Coulomb energy, asymmetry energy, and pairing energy. It fits measured binding energies to within 1-2 MeV.
When uranium-235 fissions into two medium-mass fragments, the BE/nucleon increases by ~0.8 MeV/nucleon. For 235 nucleons: ~200 MeV per fission event. This is ~50 million times more energy per unit mass than a chemical explosion.
Fusing 4 protons (hydrogen) into one He-4 nucleus releases 26.7 MeV (the binding energy of He-4 minus 4 protons). This is the energy source of the sun — about 4 million tons of mass are converted to energy per second.
On a nuclear chart (Z vs N), stable nuclei follow a curved line (the valley of stability). Light nuclei have N ≈ Z; heavier nuclei require excess neutrons to dilute Coulomb repulsion. Nuclei off the valley decay by beta emission to approach stability.
Magic numbers (2, 8, 20, 28, 50, 82, 126) are numbers of protons or neutrons that correspond to closed nuclear shells, giving extra stability. Nuclei with magic numbers have higher BE/A than neighbors, analogous to noble gas stability in atomic physics.
Nuclear binding energies are millions of electron-volts (MeV), while chemical bond energies are a few electron-volts. Nuclear reactions release about 10^6 times more energy per atom than chemical reactions, explaining the power of nuclear fuel versus chemical fuel.
Mass spectrometry measures nuclear masses to sub-microgram precision. The difference between measured nuclear mass and sum of free nucleon masses gives the mass defect, from which BE = delta_m * 931.494 MeV is computed. Modern Penning trap mass spectrometers achieve relative precision of 10^-11.
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