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
FAQs on Nuclear Binding Energy
1. What exactly is nuclear binding energy?
Nuclear binding energy is the minimum energy required to disassemble the nucleus of an atom into its separate constituent parts: protons and neutrons (collectively known as nucleons). Conversely, it is also the energy that is released when nucleons bind together to form a nucleus. This energy is a direct consequence of the mass defect, as explained by Einstein's equation, E = mc².
2. What is mass defect in the context of a nucleus?
The mass defect (Δm) is the difference between the total mass of an atom's individual protons and neutrons and the actual, measured mass of its nucleus. The nucleus is always lighter than the sum of its parts. This 'missing' mass is not lost but has been converted into the nuclear binding energy that holds the nucleus together.
3. How is the nuclear binding energy of a nucleus calculated from its mass defect?
The nuclear binding energy is calculated using Albert Einstein's famous mass-energy equivalence principle, E = Δmc². Here:
- E is the nuclear binding energy.
- Δm is the mass defect (the difference in mass).
- c is the speed of light in a vacuum (approximately 3 x 10⁸ m/s).
By calculating the mass defect and multiplying it by the square of the speed of light, you find the energy equivalent that holds the nucleus intact.
4. What are the common units used to measure nuclear binding energy?
In nuclear physics, energy is typically measured in Mega-electron Volts (MeV). For comparing the stability of different nuclei, it is more useful to use the binding energy per nucleon, which is expressed in MeV per nucleon. This unit provides a measure of the average energy holding each nucleon within the nucleus.
5. What is the relationship between nuclear binding energy and the stability of a nucleus?
The stability of a nucleus is directly proportional to its binding energy per nucleon. A higher binding energy per nucleon signifies that more energy is required to break the nucleus apart, making it more stable. Nuclei with the highest binding energy per nucleon (like Iron-56) are the most stable elements in the universe.
6. How does the binding energy per nucleon curve help identify stable and unstable elements?
The binding energy per nucleon curve is a graph that plots the binding energy per nucleon against the mass number (A) for various elements. The key insights from the curve are:
- Peak of the Curve: Nuclei with a mass number around 56 (like Iron) are at the peak of the curve. This represents the highest stability.
- Light Nuclei (Left side): Light nuclei can increase their stability and release energy by fusing together (nuclear fusion) to move up the curve.
- Heavy Nuclei (Right side): Very heavy nuclei (like Uranium) are less stable. They can release energy by splitting into lighter, more stable nuclei (nuclear fission), also moving up the curve from the right.
7. What is the key difference between the strong nuclear force and nuclear binding energy?
The strong nuclear force is the fundamental cause, while binding energy is the measurable effect. The strong nuclear force is the powerful, short-range attractive force that holds protons and neutrons together in the nucleus, overcoming the electrostatic repulsion between protons. Nuclear binding energy is the energy equivalent of the mass defect that results from this force; it's the amount of energy you would need to supply to overcome the strong force and break the nucleus apart.
8. Why is energy released in both nuclear fission and fusion, even though they are opposite processes?
Energy is released in any nuclear reaction where the total mass of the products is less than the total mass of the reactants. This occurs because in both fission and fusion, the resulting nuclei have a higher binding energy per nucleon, making them more stable.
- In fusion, light nuclei combine, and the new, heavier nucleus is more tightly bound.
- In fission, a heavy nucleus splits, and the resulting smaller nuclei are more tightly bound.
In both cases, the system moves to a more stable, lower-mass state, and the mass difference is released as energy.
9. If a nucleus has a very high binding energy, does this mean it is in a high-energy, unstable state?
No, this is a common misconception. A high binding energy signifies a very low-energy, stable state. Think of it as a deep energy well. A large amount of energy was *released* when the nucleus formed, so you must supply that same large amount of energy to break it apart. Therefore, a high binding energy means the nucleus is very stable and difficult to disrupt.
