Binding Energy, Expression of Binding Energy, Binding Energy Per Nucleon Curve, Importance of binding energy

Binding Energy

Binding energy in the context of a nucleus refers to the amount of energy required to completely separate the protons and neutrons within the nucleus. It is the energy that "binds" or holds the nucleus together.

Nuclei are composed of protons and neutrons, which are collectively known as nucleons. The binding energy is a measure of the strong nuclear force, which is responsible for holding the nucleons together. This force is attractive and acts to overcome the electrostatic repulsion between the positively charged protons.

When nucleons come together to form a nucleus, energy is released due to the strong nuclear force. This energy is known as the binding energy and is a manifestation of Einstein's mass-energy equivalence principle (E = mc²). The binding energy is equivalent to the mass defect of the nucleus, which is the difference between the mass of the individual nucleons and the mass of the nucleus.

The binding energy per nucleon is an important quantity used to understand the stability and properties of nuclei. Nuclei with higher binding energy per nucleon are more stable, and this stability is a determining factor in nuclear reactions and processes such as nuclear fusion and fission.

In practical terms, the binding energy is typically measured in electron volts (eV) or mega-electron volts (MeV). The binding energy per nucleon is often used as a measure of nuclear stability and is typically highest for iron-56, which is why it is often referred to as the "most stable" nucleus.

Expression of Binding Energy

Consider an element _ZX^A

Let M = Experimental mass of element.

       m_p = mass of proton.

       m_n = mass of neutron.

Then,

       Actual mass = Z m_{p +} (A-Z) m_n

So,

       Mass defect = Actual mass - Experimental mass of element.

                    Δm  = [Z m_{p +} (A-Z) m_n]  - M

Now this mass is converted in to Binding Energy.

                     B.E = ΔmC²

                                    E_b ={ [Z m_{p +} (A-Z) m_n]  - M} C²

In MeV           E_b = ΔmC²     ×  (931/C²)

                        E_b = Δm 931 MeV

Binding energy per nucleon

 

                       E_bn = Total B.E / No. of nucleons

                       E_{bn =} E_b/A

Binding Energy Per Nucleon Curve

The binding energy curve can be well explained by plotting the graph of average binding energy per nucleon Vs. The mass number of the nucleus. Check out the graph given below.

• From the above graph, we can observe from hydrogen to sodium, and the binding energy increases sharply with the atomic mass. We can observe the slow increase of the curve after A>20.
• We can observe the recurrence of the peaks for the nuclei having a mass number multiple of four. This is because all those nuclei have an equal number of protons and neutrons.
• From A=40 to A=120, the curve almost becomes flat, and beyond 120, the curve decreases slowly with an increase in mass number.
• The binding energy per nucleon becomes almost constant and partially independent of the mass number between 30<A<170.
• Binding energy per nucleon hit its maximum peak at A=56, whose corresponding nucleus is Iron-56; it is considered the most stable element in the universe.
• In between the mass number 40<A<120, the element possess average binding energy of 8.5MeV and are considered the most stable and non-radioactive elements.
• For higher mass number A>120, the curve drops slowly, and the average binding energy per nucleon is above 7.6MeV and is considered unstable nuclei and radioactive elements.
• The binding energy curve per nucleon for both heavier and lighter nuclei enhances the fission and fusion process to form a stable reaction.

Importance of binding energy:

Since we know that binding energy explains the fission and fusion process, the graphical representation of the binding energy can also illustrate the fission and fusion process. Along with these, the binding energy curve also explains certain advanced concepts mentioned below.

• Fission reaction –the binding energy gives an account for the stability of heavier nuclei which are a little less stable. This concept leads to the splitting up of a heavier nucleus into its constituents, so that energy can be released; this process is named as fission reaction. This technique is used in nuclear power generators.
• Fusion reaction –from the binding energy curve, we can find that some lighter elements, such as hydrogen and helium, are also a little less stable. So we can achieve the required amount of energy release by combining the lighter nuclei. This process of merging lighter nuclei by releasing a certain amount of energy is called a fusion reaction.
• The concept of fission and fusion reaction driven by the binding energy curve establishes the assumption regarding stellar energy production.
• Stars are made up of lighter elements, so it is evident that stars can’t induce a fission reaction. The only thing possible is fusion, so scientists concluded that stars are formed due to fusion reactions. Thermonuclear fusion is the reaction that is the main source of stellar energy.
• The binding energy curve also illustrates the abundance of iron and nickel in the core of the earth since iron and nickel are the most stable element and are most tightly bonded to their nucleus, which gives the basic idea for the abundance of iron and nickel inside the core.
• The Gamma decay –binding energy curve helps to illustrate the gamma decay process. Gamma rays are emitted during the gamma decay; this occurs even after other decay also, such as alpha and beta decay. The gamma rays are produced when the decay occurs at the lower energy state carried by the daughter nuclei produced during alpha and beta decay.
• The binding energy has importance in the end product of supernovae and in the final stage of the silicon-burning stars. The neutron inside the stars is free to convert itself into a proton and can release more energy.