Binding Energy: Definition, Equation, How To Calculate

It takes energy to keep the protons, neutrons and electrons together in an atom, to keep atoms together in a molecule or even to keep molecules together. This kind of energy is generally called binding energy.

You have likely heard of Einstein's famous equation E = mc². But what does it mean? This equation relates an object's resting mass to an equivalent energy value. Since ​c​ (the speed of light) is so large, a small amount of resting mass corresponds to a large amount of energy.

In the case of nuclear binding energy, this energy is so strong, it is observed to be part of the object's resting mass. By subtracting the masses of the atom's constituent parts from the total resting mass, you can calculate the binding energy.

We define the nuclear binding energy as the potential energy required to disassemble the nucleus of an atom into its component parts. This is the energy difference between the nuclear mass-energy and the total mass-energy of its individual protons and neutrons. This difference is also called the mass defect.

This energy is created by the force holding the mass of the nucleus together, called the strong nuclear force. When the atomic nucleus breaks apart, this extra potential energy is released in the form of heat or light.

For example, helium-4 (⁴He) is an isotope of helium that has two protons and two neutrons in its nucleus. (Isotopes of elements are labeled by the total number of protons and neutrons in their nucleus, also called their mass number.) It has an atomic mass of 4.002603 amu, where ​amu​ is atomic mass units. (1 amu is 1.66054 × 10^-27kg and is also sometimes written as "1 u".)

If we broke its nucleus into its constituent protons and neutrons, we would have two protons and two neutrons. But we must also keep in mind that there were two electrons. So, to calculate the binding energy, we need to determine the difference in mass between the parts and the whole, which means adding together the mass of two ¹H atoms (hydrogen atoms that consist of 1 proton and 1 electron and have a mass of 1.007825 u), plus the mass of two neutrons (which have a mass of 1.008665 u each), and then subtracting the mass of the ⁴He atom from the sum of the masses, which gives the result 0.030377u. This is the mass defect of helium-4.

1 u = 931.5 MeV/c². Use this to convert your result to MeV/c²: 0.030377 × 931.5 = 28.3MeV/c²

So the binding energy of ⁴He is:

\(BE=\Delta m c^2=28.3\text{ MeV}\)

This can also be converted to joules.

The binding energy per nucleon is the binding energy divided by the total number of neutrons and protons. This gives us information about the stability of a given nucleus. A high binding energy per nucleon would indicate a very stable nucleus – it is not likely to release energy if it decays into something else. A lower binding energy per nucleon might indicate an unstable nucleus. The binding energy per nucleon is sometimes called the BE/A or the BEN.

The binding energy per nucleon for ⁴He is 28.3/4 = 7.075 MeV/c², which is fairly stable. The most stable nucleus, based on binding energy per nucleon, is nickel-62, followed by iron-58 and iron-56. Nickel and iron are also the most common elements in the cores of planets.

If you look at a binding energy curve, you will notice that it peaks at around the number of nucleons: 60, which corresponds to iron.

Recall that with nuclear reactions such as nuclear fusion, two nuclei fuse together to form a heavier nucleus, and with nuclear fission, a nucleus breaks apart into lighter nuclei. The binding energy per nucleus peaks at iron; lighter nuclei than iron are more likely to release energy in a fusion process, and heavier nuclei than iron are more likely to release energy in a fission process.

The reason this curve has a peak is because of the balance between coulombic forces and the strong force. The coulomb force is what makes like charges repel – it makes the protons in atomic nuclei want to escape each other since they are all positively charged. The strong force pulls nucleons (protons and neutrons) together. As the nucleus gets bigger in bigger elements, the strong force is only felt over short distances, between neighboring nucleons. The coulombic force can be felt throughout the nucleus, even as it gets bigger. This makes it easier for larger nuclei to split apart and lowers their binding energy.

The element with the lowest binding energy per nucleon is uranium-238; the most common element used in nuclear fission is uranium-235. Imagine how big the mass of a nucleus of 235 nucleons is compared to that of helium-4! Nuclear fission is the process that occurs in nuclear reactor power plants, which provide electricity from nuclear energy all over the world.

While nuclear fission in a power plant occurs by bombarding a heavy element with high energy particles, fission also happens naturally to heavy elements like uranium and thorium in the Earth's crust, in low concentrations. Natural fission is a form of radioactive decay.

Fusion is the process that goes on in stars to create their heat and light and energy. Stars begin with a large amount of hydrogen; by the end of their fusion process, they end up with a large amount of iron. Iron, the peak of the binding energy per nucleon curve, is the last element in a star's fusion process. Fusing iron, unlike the elements before it, would result in a loss in energy rather than a gain. Once the star runs out of elements it has the energy to fuse, it collapses, its outer layers exploding in a supernova.

Electron binding energy, also called ionization potential, is the amount of energy necessary to remove an electron from an atom. This binding energy is, as a general rule, approximately 1 million times smaller than that of nuclear binding energy.

For electrons, the binding energy is also related to the interplay between two forces. With nuclear binding energy, it is between the coulomb force and the strong nuclear force; with electron binding energy, it is the coulomb force attracting the electrons to the protons balanced with the coulomb force repelling the electrons from each other.

For this reason, the electron binding energy tends to decrease as atoms get larger. Electrons fill the lower orbitals of an atom first, meaning an electron in an outer orbital is somewhat "shielded" from the positive charge at the nucleus by the inner electrons.