Mass-Energy and Nuclear Binding Energy: Concept and Examples

The theory of Special Relativity propounded by Albert Einstein denotes the fact the mass is nothing but another form of energy. Hence, it is possible to convert mass-energy into another form of energy. The theory promoted the deeper understanding of the nuclear masses and the interaction nuclei have with each other. The phenomenon of nuclear fission and nuclear fusion can also be better understood with the help of gaining more insight into the binding energy of nuclei with each other.

Key Terms: Mass energy, Conservation of energy, Nuclear Binding energy, Mass Defect, Binding energy per nucleon.

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Mass Energy – Main Points

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• Einstein’s famous mass-energy equivalence, i.e, E = mc², defines the relationship between mass of a particle and energy. Here, “E” is the energy that is equivalent to the product of “m” that is mass and the speed of light squared. The speed of light is almost equal to 3 X 10⁸ meters per second.
• Equivalently, the mass of a particle also equals to its Energy divided by the speed of light squared, i.e, m = E / c².
• As per the formula, a little amount of rest mass is capable of generating enormous amount of energy, independent of its matter.
• As per the law of Conservation of Energy, energy is incapable of being created or destroyed. It can only be transferred or transformed into various forms. Taking the example of a dynamic stick explosion, the chemical energy gets transformed into kinetic energy. 
• In classical physics, the conservation of energy and the conservation of mass were two unequal phenomenon, however, E=mc² infers that mass-energy is conserved as a whole, which goes on to prove that any object with mass can be converted to pure energy.

Nuclei Class 12 Important Notes PDF

Nuclei Class 12 Important Notes

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Nuclear Binding Energy

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The energy required to split the nucleus of an atom further into its constituents, that is, protons and neutrons, is called Nuclear Binding Energy.

In order to prove the above, let us take an example of Oxygen particle. The Oxygen nucleus consists of 8 protons and 8 neutrons. It’s got 8 electrons as well. Now, we can calculate the total mass of the particle as follows:

8 neutrons (mass) = 8 x 1.00866 u

8 protons (mass) = 8 x 1.00727 u

8 electrons (mass) = 8 x 0.00055 u

Since nucleus is made up of protons and neutrons, the approximate mass of Oxygen nucleus is 8 x (1.00866 + 1.00727) = 8 x 2.01593 u = 16.12744 u

As we know the atomic mass of a particle is the sum of protons, neutrons and electrons, with electrons having almost negligible weight. So, the standard atomic mass of an oxygen particle is determined to be 15.99493 u. We can get the mass of oxygen nucleus by subtracting the mass of 8 electrons from 15.99493 u ;

Oxygen nucleus mass = 15.99493 u – (8 x 0.00055u) which turns out to be 15.99053 u.

So, there is a difference of 0.13691u (16.12744 u – 15.99053 u)

In other words, we can say that the mass of oxygen nucleus is less than the total mass of combined constituents to the number of 0.13691u.

The term “Mass Defect” is given to this difference between the mass of the nucleus and the mass of its constituents and is reflected by the following equation ;

Δ M = [Zm_p + (A – Z)m_n] – M

We can say that, in order to break down the 8 protons and 8 neutrons from the oxygen nucleus, the extra energy (Δ Mc²) must be provided.

Einstein’s mass-energy equivalence relation equation (E = mc²) helps us derive the relation between this energy (E_b) to the mass defect (ΔM). Therefore,

E_b = Δ Mc²

In a nutshell, we can say that Energy E_b is released when the protons and neutrons of a particle are bound together to form a nucleus of a certain mass and charge. This is known as the Binding Energy. In order to separate the neutrons and protons from the aforementioned particle, the amount of energy required will be E_b

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Binding Energy Per Nucleon

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The binding energy per nucleon denoted as E_bn is another measure of binding between protons and neutrons. It is the ratio of the binding energy of a nucleus to the number of nucleons in the nucleus:

E_bn = E_b/A

Where, E_b is the binding energy of the nucleus and A is the number of nucleons in it. We can say that the binding energy per nucleon is the average energy per nucleon required for a nucleus to be separated into its individual nucleons. Following is a plot of the binding energy per nucleon versus the mass number for a large number of nuclei :

Binding Energy Per Nucleon

The following observations can be made from the diagram above:

• 8.75 MeV per nucleon is the maximum binding energy for mass number (A) = 56.
• 7.6 MeV per nucleon is the minimum binding energy for mass number (A) = 238.
• For 30 < A < 170, E_bn is nearly constant.
• For light nuclei (A < 30) and heavy nuclei (A > 170), the value of E_bn is low.

Following conclusion can be made based on the observations above :

• With the force being attractive in nature and strong, it produces a binding energy of a few MeV per nucleon.
• The reason that E_bn is nearly constant in the range 30 < A < 170 is because of nuclear force being short-ranged. 
• If a nucleon has ‘p’ neighbors within the range of the nuclear force, then its binding energy is proportional to ‘p’.
• In a large nucleus, as most of the nucleons reside inside it, the change in binding energy would be small.
• A very heavy nucleus having A=240 has low binding energy. Hence, in case a nucleus A=240 disintegrates into two A=120 nuclei, the nucleons are bound tightly. The phenomenon of Nuclear Fission uses this concept.
• The nucleons get tightly bound post fusion if we consider two very light nuclei with A <10. Imagining these two nuclei joining to form a heavier nucleus, the binding energy per nucleon of the fused and heavier nucleus is more than the E_bn of the lighter nuclei. This phenomenon forms the basis of the Sun’s working.

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Things to Remember

• The theory of Special Relativity propounded by Albert Einstein denotes the fact the mass is nothing but another form of energy.
• Einstein’s famous mass-energy equivalence, i.e, E = mc², defines the relationship between mass of a particle and energy.
• The energy required to split the nucleus of an atom further into its constituents, that is, protons and neutrons, is called Nuclear Binding Energy.
• The binding energy per nucleon denoted as E_bn is another measure of binding between protons and neutrons. It is the ratio of the binding energy of a nucleus to the number of nucleons in the nucleus: E_bn = E_b/A
• With the force being attractive in nature and strong, it produces a binding energy of a few MeV per nucleon.
• The reason that E_bn is nearly constant in the range 30 < A < 170 is because of nuclear force being short-ranged. 

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