Nuclear binding energy is the energy required to split a nucleus of an atom into its component parts, the nucleons (protons and neutrons). The binding energy of nuclei is always a positive number, since all nuclei require net energy to separate them into individual protons and neutrons.
Nuclear binding energy accounts for a noticeable difference between the actual mass of an atom's nucleus and its expected mass based on the sum of masses of its non-bound components.
Recall that energy (E) and mass (m) are related by the equation:
where c is the speed of light. In the case of nuclei, binding energy is so great that it accounts for a significant amount of mass.
The actual mass is always less than the sum of the individual masses of the constituent protons and neutrons because energy is removed when when the nucleus is formed. This energy has mass, which is removed from the total mass of the original particles. This mass, known as the mass defect, is missing in the resulting nucleus and represents the energy released when the nucleus is formed. Mass defect (Md) can be calculated as the difference between observed atomic mass (mo) and that expected from the combined masses of its protons (mp, each proton having a mass of 1.00728 amu) and neutrons (mn, 1.00867 amu):
Nuclear Binding Energy
Once mass defect is known, nuclear binding energy can be calculated by converting that mass to energy by the aforementioned E=mc2. Mass must be in units of kg.
Once this energy (which is a quantity of joules for one nucleus) is known, it can be scaled into per-nucleon and per-mole quantities. To convert to joules/mole, simply multiply by Avogadro's number. To convert to joules per nucleon, simply divide by the number of nucleons.
Nuclear binding energy can also apply to situations when the nucleus splits into fragments composed of more than one nucleon, and in this case the binding energies for the fragments (as compared to the whole) may be either positive or negative, depending on where the parent nucleus and the daughter fragments fall on the nuclear binding energy curve (see below). If new binding energy is available when light nuclei fuse, or when heavy nuclei split, either of these processes result releases the binding energy. This energy—available as nuclear energy—can be used produce nuclear power or build nuclear weapons. When a large nucleus splits into pieces, excess energy is emitted as photons (gamma rays) and as kinetic energy of a number of different ejected particles.
Nuclear binding energy is also used to determine whether fission or fusion will be a favorable process. For elements lighter than Iron-56, fusion will release energy (if the fusion product is still lighter than or equal to Iron-56, for fusion creating heavier elements the situation becomes more nuanced) because the nuclear binding energy increases with increasing mass, whereas elements heaver than Iron-56 will generally release energy upon fission, as the lighter elements produced contain greater nuclear binding energy. The nuclear binding energy can be seen in the following graph (notice the peak at Iron-56):
The rationale for this peak in binding energy is the interplay between the coulombic repulsion of the protons in the nucleus (like charges repel each other), and the strong nuclear force (strong force), which hold nucleons (protons and neutrons) together at short distances. As the size of the nucleus increases, the strong nuclear force is only felt between nucleons close together, while the coulombic repulsion continues to be felt throughout the nucleus, leading the instability and hence the radioactivity and fissile nature of the heavier elements.