Nuclear Binding Energy and the Mass Defect

A neutron has a slightly larger mass than the proton. These are often given in terms of an atomic mass unit, where one atomic mass unit (u) is defined as 1/12th the mass of a carbon-12 atom.

Particle Mass (kg) Mass (u) Mass (Mev/c2)
1 atomic mass unit 1.660540 x 10-27 kg 1.000 u 931.5 MeV/c2
neutron 1.674929 x 10-27 kg 1.008664 u 939.57 MeV/c2
proton 1.672623 x 10-27 kg 1.007276 u 938.28 MeV/c2
electron 9.109390 x 10-31 kg 0.00054858 u 0.511 MeV/c2

Einstein's famous equation relates energy and mass:

E = mc2

You can use that to prove that a mass of 1 u is equivalent to an energy of 931.5 MeV.

Something should strike you as strange about the table above. The carbon-12 atom has a mass of 12.000 u, and yet it contains 12 objects (6 protons and 6 neutrons) that each have a mass greater than 1.000 u, not to mention a small contribution from the 6 electrons.

This is true for all nuclei, that the mass of the nucleus is a little less than the mass of the individual neutrons, protons, and electrons. This missing mass is known as the mass defect, and represents the binding energy of the nucleus.

The binding energy is the energy you would need to put in to split the nucleus into individual protons and neutrons. To find the binding energy, add the masses of the individual protons, neutrons, and electrons, subtract the mass of the atom, and convert that mass difference to energy. For carbon-12 this gives:

Mass defect = Dm = 6 * 1.008664 u + 6 * 1.007276 u + 6 * 0.00054858 u - 12.000 u = 0.098931 u

The binding energy in the carbon-12 atom is therefore 0.098931 u * 931.5 MeV/u = 92.15 MeV.

In a typical nucleus the binding energy is measured in MeV, considerably larger than the few eV associated with the binding energy of electrons in the atom. Nuclear reactions involve changes in the nuclear binding energy, which is why nuclear reactions give you much more energy than chemical reactions; those involve changes in electron binding energies.