Nuclear Binding Energy

The minimum energy required to separate an atomic nucleus completely into its  constituents protons and neutrons, or, to disassemble  the nucleus of an atom into its component parts  equivalently. The energy that would be liberated by combining individual protons and neutrons into a single nucleus is considered as nuclear binding energy. It is always a positive number because we need to supply energy in moving these nucleons, attracted to each other by strong nuclear force, away from each other.

Theory of Relativity and Binding Energy 

Albert Einstein came up with his revolutionary theory, “the theory of relativity” during the 20th century. The theory brilliantly explained that mass and energy are interchangeable, the mass can be converted into energy and vise-versa. This gave a new dimension to physics, and helped to resolve plenty of unsolved problems and formed a forum for a lot of new theories. One among them is the existence of Nuclear Binding energy nuclear mass . 

Mass Defect and Binding Energy 

The structure of an atom comprises a nucleus at the centre and electrons revolving around it in an orbital. The Nuclei itself consists of Protons and Neutrons, which together are called nucleons. Thus, we can say  that the mass of the nucleus will be the same as the sum of individual masses of neutrons and protons. But it is not true. Experimentally, it is found that the mass of an atomic nucleus is less than the sum of the individual masses of the free constituent protons and neutrons. According to Einstein's equation E=mc2. Now this 'missing mass' is known as the mass defect , and it represents the energy that was released when the nucleus was formed.

This difference in the mass is called mass defect given by,

Δm = Zmp+(A−Z)mn−mnuc

Where,

Zmp is the total mass of the protons.

(A-Z) mn is the total mass of the neutrons.

mnuc is the mass of the nucleus.

Now, Einstein’s theory of relativity states that the mass-energy is equivalent to each given by the famous equation E= mc2 . Hence, the total energy of the nucleus is less than the sum of the energies of individual protons and neutrons(nucleons). Which implies that when nucleus disintegrates into constituent, nucleons some energy is released in the form of heat energy. Thus the reaction is  Exothermic in nature .And the energy emitted here is mathematically expressed using,

E= (Δm)c2

To break the nucleus, a certain amount of energy is put into the system. The amount of energy required to achieve this is termed as nuclear binding energy. For example nucleus hydrogen is composed of one proton and one neutron and it can be separated completely by supplying 2.23 million electron volts (MeV) of energy. Conversely, when a slowly moving neutron and proton combine to form a hydrogen-2 nucleus, 2.23 MeV are liberated in the form of gamma radiation. Thus, the total mass of the bound particles is less than the sum of the masses of the separate particles by an amount of the binding energy.

Calculation of Nuclear Binding Energy 

Nuclear binding energy can be calculated following steps:

i) Obtain the mass defect (which is the difference between  the mass of a nucleus and the sum of the masses of the neutron and proton of which it is composed)

ii) Once the mass defect is known, the nuclear binding energy can be obtained by converting that mass to energy using the

Formula Eb= (Δm)c2 ,

where mass is unit of kgs 

iii) Now once the energy obtained is known, it is to be converted into per-nucleon and per- mole quantities.

Example: For finding the mass defect and Nuclear Binding Energy of a copper-63 nucleus if the actual mass of a copper-63 nucleus is given as 62.91367 amu.

Copper atoms have 29 protons and copper-63 also has (63 - 29) 34 neutrons.

Since the mass of a proton is 1.00728 amu and a neutron is 1.00867 amu.

The combined mass is calculated:

29 protons (1.00728 amu/proton) + 34 neutrons(1.00867 amu/neutron)

or 63.50590 amu

Thus, mass defect.

Δm = (63.50590 - 62.91367) amu =  0.59223 amu

Now ,using ΔE = Δmc2, where c = 2.9979 x 108 m/s.

E = (9.8346 x 10-28 kg/nucleus)(2.9979 x 108 m/s)2 = 8.8387 x 10-11 J/nucleus