Binding Energy: Mass Defect, Binding Energy per Nucleon and Mass Discrepancy in Helium Atom

Binding energy is equivalent to mass defect or mass discrepancy. What is meant by mass defect or discrepancy will be treated in details under the sub-title, "Mass defect and binding energy".

It is established that the Nucleus of the atom of elements contain two basic particles: Proton and Neutrons. The mass of the protons and neutrons are known, and as such can be used to calculate the total mass of each of them in a defined atom of an element. And the masses of each should be the same in both combined and uncombined state. But reality is that the total mass of the individual sub-atomic particles measured from the standard mass of proton and neutron is different from the actual mass of the atom. That is the mass of the atom is less than the calculated mass from the measured standard masses of free proton and neutrons. This mass difference is termed Mass defect or mass discrepancy.

The difference in the Ideal mass of an atom and the real mass is accounted for by Albert Einstein's theory of relativity which states that mass and energy are interconvertible. According to the theory, the loss in mass is directly proportional to the release of energy. This is expressed by the Einstein's Equation:

E = mc²

Where, E = Energy in Joules (J), m = is loss in mass in Kilogram (Kg) and c is the velocity of light (299,792,458m/s or approximately, 3.0 × 10⁸ms^-1) in ms^-1.

Note: It is worthwhile to state that the value of the velocity of light was recently contested by experimental results from the Large Hadron Collider at the CERN.

The implication of this is that certain amount of energy is evolved during the process of binding the protons and neutron to form the atom. Hence, we can use the explanation of the mass defect to define the binding energy as, the energy evolved during the formation of the nucleus from free protons and neutrons.

The Helium atom is made up of two protons and two electrons. The actual mass of the Helium atom (mass of proton and neutron) is 4.0026 amu. Where amu implies "atomic mass unit".

But the standard masses of protons and neutrons are:

Mass of Proton - 1.0078 amu.

Mass of neutron - 1.0087 amu, respectively.

But the Helium atom is made up of two Neutrons and two protons, hence, the ideal mass of the Helium atom can be calculated from the standard masses of the proton and neutron respectively as,

Mass of two Helium Protons - 2 × 1.0078 amu = 2.0156 amu

Mass of two Helium Neutrons - 2 × 1.0087 amu = 2.0174 amu

Total ideal mass of the Helium atom = Mass of the two Helium Protons + the mass of the two Helium Neutrons = 2.0156 amu + 2.0174 amu = 4.0330 amu.

This is different from the real mass of the Helium atom: 4.0026 amu.

The difference in mass can be calculated by subtracting the real mass from the Ideal mass, i.e.

4.0330 amu - 4.0026 amu = 0.0304 amu.

This called the mass defect or mass discrepancy of the Helium atom.

Binding energy per Nucleon is relevant in the determination of the stability of the nucleus of elements. But before describing how it is used, we must state that, the particles in a nucleus (proton and neutrons) are collectively known as Nucleons. Hence, binding energy per nucleon is obtained when we divide the total binding energy by the number of nucleons it contains.

The binding energy per nucleon is plotted against the atomic mass number, A to obtain the stability of the nucleus of elements of the periodic table. Here, the binding energy rises sharply from the lightest element until it reaches the vicinity of atomic mass 56 (the Iron nuclei). This is the maximum, and the binding energy per nucleon of heavier nucleus begin to decline with increase in mass.

The binding energy per nucleon helps to determine elements suitable for nuclear fission and nuclear fusion. Elements whose atomic numbers are close to 56 are the most stable and would require large amount of energy for disintegration. Those at the two extremes are unstable and as such, larger nuclei can be split into lighter ones by nuclear fission, and lighter ones can be combined to form larger nuclei by nuclear fusion.