Binding Energy

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What is Binding Energy?

It is very rare for a common person to look at a piece of wood and think that how much energy he will have to exert to split the log into two, but for a physicist, a question related to energy plays a very important role. The energy that is required to disassemble the nuclei into separate protons and neutrons completely is known as the binding energy (B.E.) of the nucleus. From the rest mass, we can determine the BE of the nucleus. 

Mass defect is the missing mass, and Einstein's equation E = mc2 provides a relation of energy to this missing mass, where,

  • E denotes energy

  • m denotes mass

  • c denotes the speed of light

The two are linked through Einstein's famous relationship. Compared to its separate constitutes, the bound system has a smaller mass. The mass of the nucleus will be small when the nucleons are bound together more tightly.


Binding Energy Explained

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Considering the above figure, imagine the nuclide is being pulled apart. The energy put into the system is the work done to overpower the nuclear forces, which is holding the nuclei.

The energy put into pulling apart the nucleus to its constituent protons and neutrons leads to an increase in the mass of the system. Binding energy (BE) is equal to the work required to disassemble the nucleus. 

Comparatively, a bound system has less mass than the total sum of its parts, noticeable in the nuclei, where the energies and forces are huge. 


Mass Defect

By definition, the binding energy is equal to the input energy. When the pieces are separated, they are in rest; the total rest mass is increased as the energy put into them increases when compared to their glued together state as the nucleus.

That mass increase is thus Δm = BE/c2.

Mass defect is known as the mass difference, which means the sum of the total mass of the nucleus constituents; the protons and neutrons are more than that of the mass of the nucleus.

A nuclide AX has N neutrons and Z protons so that the difference in mass is derived as

∆m = (Zmp + Nmn) − mtot.

Thus, BE = (∆m) c2= [(Zmp + Nmn) − mtot]c2

Where,

mtot is the mass of nuclide AX, 

mp is the mass of the proton, and 

mn is the mass of the neutron. 

Conventionally, we deal with the masses of neutral atoms.

First, add Z electrons to mtot to get atomic mass into the last equation, which will give m(AX), the atomic mass of nuclide.

Then Z electrons are added to Z protons, 

This gives Zm (1H), or Z times a mass of hydrogen atom. 

Thus, binding energy of a nuclide AX is BE = {[Zm(1H)+ Nmn]− m(AX)}c2.


Binding Energy Curve

Based on per nucleon versus atomic mass, we can plot the binding energy curve graph and can represent it as follows.

There will be no electrostatic repulsion as the binding energy of hydrogen is zero, as there is only one proton in the nucleus. As it can be viewed in the graph, it shoots up drastically till helium and drops a bit and then climbs again towards the iron-56. This is the most stable nucleus that exists in terms of nuclear. After iron-56, binding energy drops gently towards uranium⁻²³⁸.

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In the chart, anything that is on the left side of Fe-56 can be created in the nuclear fusion process. In this process, the atoms are compressed together at extreme pressure and temperature to form heavy elements. This is how the core of a star is created.

Similarly, anything on the right side of Fe-56 can be created in the nuclear fission process in which heavy atoms are broken into lighter elements. This process occurs in nuclear bombs and nuclear reactors. A massive amount of energy is generated in both fission and fusion reactions.


Main Features of Binding Energy Curve

The average energy per nucleon needed to split a nucleus into its individual nucleons is known as the binding energy per nucleon.

The below graph depicts the binding energy curve, which plots binding energy per nucleon versus the mass number for large nuclei.

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The main features of his curve are listed below.

  1. The binding energy per nucleon (Ebn) is essentially constant and is practically autonomous of the atomic number for nuclei middle mass number (30 < A < 170). The maximum value of the curve is about 8.75 MeV for A = 56 and has a value of the 7.6 MeV for A = 238.

  2. Binding energy per nucleon is lesser for both light nuclei (A <3 0) and heavy nuclei (A > 170).


The following conclusion can be drawn based on the above two observations:

  1. To produce binding energy of a few MeV per nucleon, the force is attractive and adequately strong.

  2. The range of constancy of binding energy is from 30 < A < 170, which is the result of the fact that nuclear force is short-ranged.

  3. Comparatively, a heavy nucleus A = 240 will have a lower binding energy per nucleon to that of A = 120 nucleus; hence if a nucleus of A = 240 breaks into two A = 120 nuclei in which nucleons get more tightly bound as it implies that energy is released in the process.

FAQs (Frequently Asked Questions)

Q1. What is the Binding Energy Formula?

Ans: The amount of energy required to separate a nucleus into individual protons and neutrons is known as binding energy (BE).

In terms of atomic masses, 

BE = {[Zm (1H) + Nmn] – m (AX)}c2 

Where,

M (1H) is a mass of hydrogen atom, 

M (AX) is an atomic mass of nuclide, and

mn is the mass of a neutron.

Q2. Is Binding Energy Positive or Negative?

Ans: The nucleus would separate and escape to space if the binding energy is positive or zero; hence it is negative. 

It can be seen in such a way that the protons and neutrons in a nucleus are held together by strong nuclear force and have potential energy (and negligible kinetic energy).

Q3. What Causes Binding Energy?

Ans: The electromagnetic interaction of the electron with the nucleus and other electrons of the atom, mediated by photons, is through which electron binding energy is derived.

The energy which is required to disassemble the atom into free electrons and nuclei is known as the atomic binding energy.

Q4. Why Does an Increase in Binding Energy Release Energy?

Ans: Nuclear binding energy is given out when protons and neutrons react together making bonds. As the number of nucleons increases, the binding energy gives up a collectively large amount of energy.

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