# Nuclear Binding Energy and Mass Defect

## A nucleus weighs less than the sum of its protons and neutrons, a quantity known as the mass defect, which releases energy.

#### Key Points

• Nuclear binding energy is the energy required to split a nucleus of an atom into its component parts. The component parts are neutrons and protons, which are collectively called nucleons.

• Nuclear binding energy is used to determine whether fission or fusion will be a favorable process. For elements lighter than Iron-56, fusion releases energy, whereas for elements heavier, fission releases energy (depending on the product of the reaction).

• The mass defect of a nucleus represents the mass of the energy of binding of the nucleus, and is the difference between the mass of a nucleus and the sum of the masses of the nucleons of which it is composed.

#### Terms

• One of the subatomic particles of the atomic nucleus, i.e. a proton or a neutron.

• The nuclear force, a residual force responsible for the interactions between nucleons, deriving from the color force

• the difference between the unbound system calculated mass and experimentally measured mass of nucleus

#### Examples

• Calculate the average binding energy per mole of a U-235 isotope. Show your answer in kJ/mole. U235 has 92 protons and 143 neutrons. Md=(92(1.00728 amu)+143(1.00867 amu))-235.04393 amuMd=1.86564 amu1.86564 amu * 1kg/6.02214*1026amu=3.09797kg E=3.09797*10-27kg*(299792458m/s)E=2.78432*10-10JEnergy per mole is: 2.78432*10-10J/atom * 6.02*1023 atoms/mole.E=1.67616*1014 J/mole, or 1.67616*1011 kJ/mole.

#### Figures

1. ##### Nuclear binding energy curve

This graph shows the nuclear binding energy (in MeV) per nucleon as a function of the number of nucleons in the nucleus. Notice that Iron-56 has the most binding energy per nucleon, making it the most stable nucleus.

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.

## Mass Defect

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:

$E=mc^2$

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):

$M_d=(m_n+m_p)-m_o$

## 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): (Figure 1)

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.

#### Key Term Glossary

atom
the smallest possible amount of matter that still retains its identity as a chemical element, now known to consist of a nucleus surrounded by electrons
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atomic mass
the quantity of matter of a particle, sub-atomic particle, or molecule
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charge
An electric charge.
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element
Any one of the simplest chemical substances that cannot be decomposed in a chemical reaction or by any chemical means, and are made up of atoms all having the same number of protons.
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energy
a quantity that denotes the ability to do work and is measured in a unit dimensioned in mass × distance²/time² (ML²/T²) or the equivalent
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fissile
Capable of undergoing nuclear fission.
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fission
The process of splitting the nucleus of an atom into smaller particles; nuclear fission
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fusion
A nuclear reaction in which nuclei combine to form more massive nuclei with the concomitant release of energy and often neutrons.
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gamma ray
electromagnetic radiation of high frequency and therefore high energy per photon
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isotope
Any of two or more forms of an element where the atoms have the same number of protons but a different number of neutrons within their nuclei. As a consequence, atoms for the same isotope will have the same atomic number but a different mass number (atomic weight).
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Isotope
Isotopes are variants of a particular chemical element. While all isotopes of a given element share the same number of protons, each isotope differs from the others in its number of neutrons.
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joule
in the International System of Units, the derived unit of energy, work and heat; the work required to exert a force of one newton for a distance of one metre; also equal to the energy of one watt of power for a duration of one second; symbol: J
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kinetic
of or relating to motion
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kinetic energy
The energy possessed by an object because of its motion, equal to one half the mass of the body times the square of its velocity.
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kinetics
The branch of chemistry that is concerned with the rates of chemical reactions.
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mass
The quantity of matter that a body contains, irrespective of its bulk or volume. It is one of four fundamental properties of matter. It is measured in kilograms in the SI system of measurement.
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mass defect
the difference between the unbound system calculated mass and experimentally measured mass of nucleus
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mole
In the International System of Units, the base unit of the amount of substance; the amount of substance of a system that contains as many elementary entities as there are atoms in 0.012 kg of carbon-12. Symbol: mol. The number of atoms in a mole is known as Avogadro’s number.
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neutron
A subatomic particle forming part of the nucleus of an atom and having no charge; it is a combination of an up quark and two down quarks
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nuclear binding energy
the energy required to split a nucleus of an atom into its component parts
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nuclear force
The force that acts between nucleons and binds protons and neutrons into atomic nuclei; the residual strong force
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nuclear power
Power, especially electrical power, obtained using nuclear fission or nuclear fusion.
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nuclear weapon
A weapon that derives its energy from the nuclear reactions of either fission or fusion.
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nucleon
One of the subatomic particles of the atomic nucleus, i.e. a proton or a neutron.
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nucleus
The massive, positively charged central part of an atom, made up of protons and neutrons.
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photon
The quantum of light and other electromagnetic energy, regarded as a discrete particle having zero rest mass, no electric charge, and an indefinitely long lifetime. It is a gauge boson.
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product
a chemical substance formed as a result of a chemical reaction
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proton
A positively charged subatomic particle forming part of the nucleus of an atom and determining the atomic number of an element; the nucleus of the most common isotope of hydrogen; composed of two up quarks and a down quark