Atomic Mass Calculator

A nucleus always weighs less than the sum of its parts, and that shortfall is the whole point of this calculation. Adding free-particle masses gives 1.007276 u per proton, 1.008665 u per neutron and 0.000549 u per electron, but the assembled atom measures less. The missing mass, called the mass defect, is the binding energy holding the nucleus together, converted at 931.494 MeV per unified atomic mass unit. Helium-4 illustrates the scale: the parts sum to about 4.0330 u while the atom measures 4.002602 u, a defect near 0.0304 u corresponding to roughly 28.3 MeV, or about 7.07 MeV per nucleon. Binding energy per nucleon rises steeply through the light elements, peaks in the iron and nickel region at about 8.79 to 8.80 MeV, with nickel-62 marginally the most tightly bound nuclide, and then declines slowly, which is why fusion releases energy below iron and fission releases it above. When a measured binding energy is not supplied, the value has to be estimated from the semi-empirical mass formula, which sums a volume term proportional to A, a surface term in A to the two-thirds power, a Coulomb repulsion term in Z times Z minus one over the cube root of A, and an asymmetry term penalising any imbalance between neutron and proton number. The liquid-drop picture behind it reproduces most nuclidic masses to within about one percent but systematically misses shell effects at the magic numbers 2, 8, 20, 28, 50, 82 and 126, where real nuclei are more tightly bound than the formula predicts.

Precise atomic masses became measurable when Francis Aston built his mass spectrograph at the Cavendish Laboratory in 1919. Aston resolved the isotopes of neon, chlorine and dozens of other elements, and formulated the whole number rule: isotope masses are close to integers, so the fractional atomic weights on the periodic table reflect mixtures rather than fractional nucleons. He then noticed the systematic deviations from exact integers, quantified them as the packing fraction, and recognised that they encoded nuclear binding. Aston received the Nobel Prize in Chemistry in 1922, and Arthur Eddington had already suggested in 1920 that the mass difference between four hydrogen atoms and one helium atom could power the stars. The modelling followed. Carl Friedrich von Weizsacker published the semi-empirical mass formula in 1935, building on George Gamow liquid-drop analogy, and Hans Bethe and Robert Bacher elaborated it in their 1936 review, hence the name Bethe-Weizsacker formula. Its asymmetry and Coulomb terms explained the observed valley of stability, and Lise Meitner and Otto Frisch drew on the same liquid-drop reasoning in 1939 when they interpreted nuclear fission. The mass unit itself was contested for decades: chemists referenced oxygen as a natural mixture while physicists used oxygen-16, giving two slightly different scales. The dispute was settled between 1959 and 1961 when IUPAP and IUPAC agreed to define the unified atomic mass unit as one twelfth the mass of a carbon-12 atom, the convention still in force.