Mass defect and binding energy

What is mass defect?

A nucleus's mass is smaller than the sum of the masses of the free protons and neutrons that make it up. This missing piece of mass is the mass defect $\Delta m$. It vanished because energy was released when the nucleons were bound together — and energy has mass.

Invariant — binding energy:

Binding energy is the energy you must supply to pull a nucleus apart into separate nucleons — and it equals exactly the mass defect times $c^2$. Dividing by the number of nucleons gives $E_{\text{bind}}/A$ — a measure of how tightly a single nucleon is held.

Why is Fe-56 the most stable?

The $E_{\text{bind}}/A$ curve climbs steeply for light nuclei, reaches a peak at A ≈ 56 (iron, ~8.8 MeV per nucleon), then slowly falls. The top = the largest binding energy per nucleon = the most tightly bound, most stable nucleus. All other nuclei "want" to move toward this peak.

Where does nuclear energy come from?

From the binding-energy difference. Fusion of light nuclei (the Sun) and fission of heavy ones (a reactor) — both shift nuclei toward Fe-56, into a region of higher $E_{\text{bind}}/A$. The surplus binding energy is released as the reaction's energy.

Why you need this for the exam:

Nuclear problems almost always come down to $E = \Delta m\,c^2$: compute the mass defect (sum of nucleon masses minus the nuclear mass), multiply by $c^2$. And the shape of the curve, with its maximum at iron, explains why the Sun shines and a reactor works.

Remember: a nucleus's mass < the sum of nucleon masses, and that difference is the binding energy: $E_{\text{bind}} = \Delta m\,c^2$. The peak of the curve at Fe-56 is the most stable nucleus in the Universe.