nuclear binding energy

Take a nucleus apart, one proton and one neutron at a time, until nothing is left but free, well-separated nucleons at rest. That job costs energy, and the total bill is the nuclear binding energy. Turn the picture around: it is exactly the energy released when the free nucleons snap together into the nucleus. A bound nucleus sits in an energy well, and the binding energy is how deep that well is. The deeper the well, the more tightly the nucleus is glued and the harder it is to break.

The defining relation is a direct application of mass-energy equivalence. A nucleus of Z protons and N neutrons has a rest mass M(Z,N) that is measurably less than the sum of its separated parts, so B = [Z m_p + N m_n - M(Z,N)] c^2, where m_p and m_n are the free proton and neutron masses. The missing mass, Delta m = Z m_p + N m_n - M, is the mass defect, and B = Delta m c^2. Because the binding energy is positive for any stable nucleus, the assembled nucleus is genuinely lighter than its ingredients; the released energy carried away the difference in mass.

This one number governs nuclear energetics. A reaction releases energy whenever the products are more tightly bound (larger total B) than the reactants, and the plot of binding energy per nucleon versus mass number tells you at a glance which way energy flows. Fusing light nuclei climbs the curve toward the iron peak and releases energy; splitting a very heavy nucleus does the same from the other side. Typical nuclear binding energies are millions of electron-volts per nucleon, roughly a million times the electron-volt scale of chemical bonds, which is why nuclear processes are so energetic.