Mass defect and binding energy

A nucleus always weighs less than the sum of its separate protons and neutrons. Take ¹⁶₈O, with 8 protons and 8 neutrons: 8 × (1.00727 + 1.00866) u = 8 × 2.01593 u = 16.12744 u.

Mass spectroscopy gives the ¹⁶O atom 15.99493 u. Taking off 8 electrons (8 × 0.00055 u) leaves the nucleus at 15.99053 u, which is 0.13691 u less than its parts.

This shortfall is the mass defect: ΔM = [Zmp + (A − Z)mn] − M (Eq. 13.7).

Because mass is energy, the bound nucleus has less energy than its free nucleons. To pull the nucleus apart into free protons and neutrons, the energy ΔMc² must be supplied: this is the binding energy, Eb = ΔMc² (Eq. 13.8). The same energy is released when the nucleons come together.

Example 13.3: 1 u = 1.6605 × 10⁻²⁷ kg, and multiplying by c² = (2.9979 × 10⁸)² gives 1.4924 × 10⁻¹⁰ J = 931.5 MeV. So 1 u = 931.5 MeV/c².

For ¹⁶O, ΔM = 0.13691 u = 0.13691 × 931.5 MeV/c² = 127.5 MeV/c², so 127.5 MeV is needed to separate it into its nucleons. (NCERT writes this energy with the unit MeV/c²; as an energy it is 127.5 MeV.)

A nucleus cannot actually be torn apart like this, but Eb is still a handy measure of how firmly it holds together. A better measure is the binding energy per nucleon, Ebn = Eb/A (Eq. 13.9): the average energy per nucleon needed to take the nucleus apart. For ¹⁶O it is 127.5/16 ≈ 8.0 MeV.