# Binding energy

Binding energy is the energy required to disassemble a whole system into separate parts. A bound system typically has a lower potential energy than the sum of its constituent parts; this is what keeps the system together. Often this means that energy is released upon the creation of a bound state. This definition corresponds to a positive binding energy.

## General idea

In general, binding energy represents the mechanical work that must be done against the forces which hold an object together, disassembling the object into component parts separated by sufficient distance that further separation requires negligible additional work.

In bound systems, if the binding energy is removed from the system, it must be subtracted from the mass of the unbound system, simply because this energy has mass. Thus, if energy is removed (or emitted) from the system at the time it is bound, the loss of energy from the system will also result in the loss of the mass of the energy from the system.[1] System mass is not conserved in this process because the system is "open" (i.e., is not an isolated system to mass or energy input or loss) during the binding process.

There are several types of binding energy, each operating over a different distance and energy scale. The smaller the scale of a bound system, the higher its associated binding energy.

## Types of binding energy

In astrophysics, the gravitational binding energy of a celestial body is the energy required to expand the material to infinity. Solely for the purpose of comparison with the other types of binding energy, if a body with the mass and radius of the Earth were made purely of carbon-12, then the gravitational binding energy of that body would be about 4.66 eV per atom. If a carbon-12 body had the mass and radius of the Sun, its gravitational binding energy would be about 14.24 keV per atom.

At the molecular level, bond energy and bond-dissociation energy are measures of the binding energy between the atoms in a chemical bond. It is the energy required to disassemble a molecule into its constituent atoms. This energy appears as chemical energy, such as that released in chemical explosions, the burning of chemical fuel and biological processes. Bond energies and bond-dissociation energies are typically in the range of few eV per bond. For example, the bond-dissociation energy of a carbon-carbon bond is about 3.6 eV.

At the atomic level, the atomic binding energy of the atom derives from electromagnetic interaction, mediated by photons. It is the energy required to disassemble an atom into free electrons and a nucleus.[2] Electron binding energy is a measure of the energy required to free electrons from their atomic orbits. This is more commonly known as ionization energy.[3] Among the chemical elements, the range of ionization energies is from 3.8939 eV for the first electron in an atom of caesium to 11.567617 keV for the 29th electron in an atom of copper.

At the nuclear level, nuclear binding energy is the energy required to disassemble a nucleus into the free, unbound neutrons and protons it is composed of. It is the energy equivalent of the mass excess, the difference between the mass number of a nucleus and its true measured mass.[4][5] Nuclear binding energy derives from the nuclear force or residual strong force, which is mediated by three types of mesons. The average nuclear binding energy per nucleon ranges from 2.22452 MeV for hydrogen-2 to 8.7945 MeV for nickel-62.

At a yet more fundamental level, quantum chromodynamics binding energy is the energy which binds the various quarks together inside a hadron. This energy derives from the strong interaction, which is mediated by gluons. The chromodynamic binding energy inside a nucleon, for example, amounts to approximately 99% of the nucleon's mass. The chromodynamic binding energy of a proton is about 928.9 MeV, while that of a neutron is about 927.7 MeV.

## Mass-energy relation

Classically, a bound system is at a lower energy level than its unbound constituents. Its mass must be less than the total mass of its unbound constituents. For systems with low binding energies, this "lost" mass after binding may be fractionally small. For systems with high binding energies, however, the missing mass may be an easily measurable fraction. This missing mass may be lost during the process of binding as energy in the form of heat or light, with the removed energy corresponding to removed mass through Einstein's equation E = mc2. Note that in the process of binding, the constituents of the system might enter higher energy states of the nucleus/atom/molecule, but these types of energy also have mass. It is necessary that they be removed from the system before its mass may decrease. Once the system cools to normal temperatures and returns to ground states in terms of energy levels, there is less mass remaining in the system than there was when it first combined and was at high energy. In that case, the removed heat represents exactly the mass "deficit" and the heat itself retains the mass which was lost (from the point of view of the initial system). This mass appears in any other system which absorbs the heat and gains thermal energy.[6]

As an illustration, consider two objects attracting each other in space through their gravitational field. The attraction force accelerates the objects and they gain some speed toward each other converting the potential (gravity) energy into kinetic (movement) energy. When either the particles 1) pass through each other without interaction or 2) elastically repel during the collision, the gained kinetic energy (related to speed), starts to revert into potential form driving the collided particles apart. The decelerating particles will return to the initial distance and beyond into infinity or stop and repeat the collision (oscillation takes place). This shows that the system, which loses no energy, does not combine (bind) into a solid object, parts of which oscillate at short distances. Therefore, in order to bind the particles, the kinetic energy gained due to the attraction must be dissipated (by resistive force). Complex objects in collision ordinarily undergo inelastic collision, transforming some kinetic energy into internal energy (heat content, which is atomic movement), which is further radiated in the form of photons—the light and heat. Once the energy to escape the gravity is dissipated in the collision, the parts will oscillate at closer, possibly atomic, distance, thus looking like one solid object. This lost energy, necessary to overcome the potential barrier in order to separate the objects, is the binding energy. If this binding energy were retained in the system as heat, its mass would not decrease. However, binding energy lost from the system (as heat radiation) would itself have mass. It directly represents the "mass deficit" of the cold, bound system.

Closely analogous considerations apply in chemical and nuclear considerations. Exothermic chemical reactions in closed systems do not change mass, but become less massive once the heat of reaction is removed, though this mass change is much too small to measure with standard equipment. In nuclear reactions, however, the fraction of mass that may be removed as light or heat, i.e., binding energy, is often a much larger fraction of the system mass. It may thus be measured directly as a mass difference between rest masses of reactants and (cooled) products. This is because nuclear forces are comparatively stronger than the Coulombic forces associated with the interactions between electrons and protons, that generate heat in chemistry.

### Mass change

Mass change (decrease) in bound systems, particularly atomic nuclei, has also been termed mass defect, mass deficit, or mass packing fraction.

The difference between the unbound system calculated mass and experimentally measured mass of nucleus (mass change) is denoted as Δm. It can be calculated as follows:

Mass change = (unbound system calculated mass) − (measured mass of system)
i.e., (sum of masses of protons and neutrons) − (measured mass of nucleus)

After nuclear reactions that result in an excited nucleus, the energy that must be radiated or otherwise removed as binding energy for a single nucleus to produce the unexcited state may be in any of several forms. This may be electromagnetic waves, such as gamma radiation; the kinetic energy of an ejected particle, such as an electron, in internal conversion decay; or partly as the rest mass of one or more emitted particles, such as the particles of beta decay. No mass deficit can appear, in theory, until this radiation or this energy has been emitted and is no longer part of the system.

When nucleons bind together to form a nucleus, they must lose a small amount of mass, i.e., there is a change in mass, in order to stay bound. This mass change must be released as various types of photon or other particle energy as above, according to the relation E = mc2. Thus, after binding energy has been removed, binding energy = mass change × c2. This energy is a measure of the forces that hold the nucleons together. It represents energy that must be supplied again from the environment, if the nucleus were to be broken up into individual nucleons.

The energy given off during either nuclear fusion or nuclear fission is the difference of the binding energies of the "fuel", i.e., the initial nuclide(s), from that of the fission or fusion products. In practice, this energy may also be calculated from the substantial mass differences between the fuel and products, which uses previous measurement of the atomic masses of known nuclides, which always have the same mass for each species. This mass difference appears once evolved heat and radiation have been removed, which is a given requirement for measuring the (rest) masses of the (non-excited) nuclides involved in such calculations.