Every atomic nucleus contains protons and neutrons, with the sole exception of the most common form of hydrogen, which has only a single proton.
Within a nucleus that has two or more protons, a powerful repulsion exists between these positively charged particles (the Coulomb force). This electrostatic repulsion should cause the nucleus to break apart.
However, matter around us is stable, meaning the nuclei are intact. This stability is due to the binding energy of the nucleus, which overcomes the repulsive force.
The Origin of Binding Energy
Where does this crucial binding energy come from?
It is derived from a small amount of mass loss during the formation of the nucleus, described by Einstein’s famous mass-energy equivalence relation: E = ∆m c².
The term ∆m is called the mass defect. To understand this, consider the total mass of the individual, unbound nucleons (protons and neutrons).
If a nucleus contains p protons and n neutrons, and the mass of a single proton is P and a single neutron is N, the expected total mass would be: m=nN+pP.
However, when scientists experimentally measure the actual mass of the assembled nucleus, m’, they find that it is less than the calculated mass; m'<m.
This difference between the actual mass and experimentally found mass is known as mass defect ∆m. Which is ∆m=m-m’.
Mass Defect and Conservation of Mass:
This “missing” mass, ∆m, even though it might look like it violates the law of conservation of mass, it does not violate the fundamental laws of conservation of mass and energy.
Instead, the mass has been converted into energy. The law is upheld because the missing mass is precisely equivalent to the energy required to hold the nucleus together, the binding energy (E_b).
The magnitude of this binding energy is calculated using Einstein’s relation E=∆mc².
Conclusion
This binding energy is the dominant source of energy in the nucleus (neglecting thermal/vibrational energy).
Generally, the total binding energy tends to increase with the total number of nucleons (the mass number).
However, the key factor for stability is the binding energy per nucleon.
Very heavy nuclei, like Uranium-235 (U²³⁵) and Plutonium-239 (Pu²³⁹), have a lower binding energy per nucleon compared to nuclei near the middle of the periodic table.
These heavy nuclei can fission (split) into smaller “daughter” nuclei.
This splitting releases a large amount of energy because the resulting, smaller nuclei have a higher binding energy per nucleon than the original heavy nucleus, allowing them to achieve a more stable state.
