Nuclear Binding Energy: Curve, Mass Defect, Examples

Nuclear Binding Energy can be defined as the energy required that disassembles a nucleus into separate neutrons and protons. Albert Einstein proposed the theory of relativity in the 20th century. The theory states that the mass and energy are inconvertible which means the mass can be converted into energy and vice-versa. This additional dimension to physics was helpful in the resolution of many unsolved problems and provided a platform for many new theories. The existence of Nuclear Binding energy is one of them.

Keyterms: Nuclear Binding Energy, Energy, Nucleus, Neutrons, Protons, Binding Energy, Speed of light, Light

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Definition: Nuclear Binding Energy

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A common person rarely considers how much energy it will take to break the log into two but for a physicist, energy plays an extremely important role.The binding energy (B.E.) of the nucleus is the amount of energy necessary to completely break down the nuclei into individual protons and neutrons. The B.E. of the nucleus can be determined using the rest mass.

Nuclear Binding Energy

Mass defect defines the missing mass and Einstein’s equation E = mc² offers a relation of energy to the missing mass.

Where,

• E = Energy
• m = Mass
• c = Speed of light

The well-known Einstein relationship joins both the terms m and c. The bound system has a lower mass than its separate constituents. When the nucleons are more closely bonded together, the nucleus’ mass is reduced.

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Nuclear Binding Energy: Process

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Assume that the nuclide is being pulled apart. The work done to overcome the nuclear forces that hold the nuclei together is the energy put into the system. The energy applied into the system is defined as the work done to control the nuclear forces that hold the nuclei.

Nuclear Binding Energy

The energy applied into pulling apart the nucleus, its individual neutrons and protons causes the system’s mass to grow. Binding energy (BE) is equal to the work needed for disassembling the nucleus.

In comparison, the mass of the bound system is lesser than the total sum of its parts, which is particularly obvious in the nuclei, where the energies and forces are huge.

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Mass Defect

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According to the definition, the binding energy is equal to the input energy. The pieces are at rest when separated. When compared to their glued together state, the total rest mass increases as the energy placed into them increases.

Nuclear Binding Energy: Mass Defect

Thus, the increased mass is,

Δm = BE/c²

Mass Defect or Mass Difference Derivation

The mass difference or mass defect is the sum of the total mass of the nucleus constituents; the mass of the protons and neutrons is more than the mass of the nucleus.

A nuclide AX contains N neutrons & Z protons therefore the difference in mass will be:

\(\bigtriangleup\)m = (Zm_p+ Nm_n) − m_tot

Thus, BE = (Δm) c²= [(Zm_p+ Nm_n) − m_tot]c²

Where,

m_tot = Mass of nuclide AX,

m_p = Mass of the proton

& m_n = Mass of the neutron

We conventionally use the masses of neutral atoms.

To get atomic mass into the last equation, add Z electrons to mtot. This will further give you m(AX), the atomic mass of nuclide.

Then, add Z electrons to Z protons. This will give you Zm (1H), or Z times a mass of hydrogen atom.

Therefore, the binding energy of a nuclide will be,

AX is BE = {[Zm(¹H)+ Nm_n]− m(AX)}c².

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Binding Energy Curve

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The binding energy curve graph can be plotted based on per nucleon versus atomic mass and can be represented as follows:

Nuclear Binding Energy Curve

Due to the presence of only one proton in the nucleus the binding energy of hydrogen is zero and therefore there will be no electrostatic repulsion. It is clearly visible from the graph that the binding energy drastically rises till helium and then falls a bit before climbing back up to the iron56. After reaching iron-56, binding energy gently drops towards U-238.

Nuclear Binding Energy Curve

Anything present on the left-side of Fe-56 can be created in the process of nuclear fusion. In this process, the atoms are compressed under extreme temperature and pressure to generate heavy elements.

Similarly, anything present on the right side of Fe-56 can be created in the process in which heavy atoms are broken into lighter elements.

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Main Features of Binding Energy Curve

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The binding energy per nucleon is the average energy per nucleon required to separate a nucleus into its constituent nucleons. The main features of the binding energy curve are:

1.) For nuclei with a middle mass number (30bn) is constant and practically independent of the atomic number. The curve has a maximum value of around 8.75 MeV for A = 56 and a value of 7.6 MeV for A = 238.

2.) Both light nuclei (A<30) and heavy nuclei (A>170) have lower binding energy per nucleon.

Nuclear Binding Energy Curve

Based on the above two observations, the following conclusion can be drawn:

• The force should be attractive and strong enough to provide binding energy of a few MeV per nucleon.
• As the nuclear force is short-ranged the binding energy’s range of constancy is from 30 < A < 170.

For instance: A heavier nucleus with A = 240 has a lower binding energy per nucleon than a lighter nucleus with A = 120. Therefore, if A = 240 nucleus breaks into two A = 120 nuclei, the nucleons bound more firmly, implying that energy is released in the process.

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Things to Remember

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• Nuclear binding energy is the amount of energy needed to split an atom’s nucleus into its components.
• Nuclear binding energy is used to find out whether fusion or fission will be a favorable process.
• The binding energy (B.E.) of the nucleus is the amount of energy necessary to completely break down the nuclei into individual protons and neutrons.
• When the nucleons are more closely bonded together, the nucleus’ mass is reduced.

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