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The binding energy BE = (Z·m_p + N·m_n − M)c² is the energy released in assembling a nucleus from free nucleons. Binding energy per nucleon BE/A is maximum (~8.8 MeV) near iron, declining for lighter and heavier nuclei. The stability curve plots N versus Z for stable nuclei: light nuclei have N ≈ Z, while heavy nuclei have N > Z (more neutrons) to reduce proton-proton repulsion. Nuclei far from this curve are unstable and undergo radioactive decay.
Calculate binding energies for a few nuclei using atomic mass tables. Plot binding energy per nucleon versus mass number to visualize the curve.
Your prerequisite — mass-energy equivalence, E = mc² — tells you that mass and energy are interchangeable. When protons and neutrons bind together to form a nucleus, the resulting nucleus is *lighter* than the sum of its free parts. This "missing" mass, called the mass defect, has been converted into the binding energy that holds the nucleus together: BE = (Z·m_p + N·m_n − M_nucleus)·c². The binding energy is the energy you would need to supply to completely disassemble the nucleus into isolated protons and neutrons. A larger binding energy means a more tightly bound, more stable nucleus.
Dividing by the number of nucleons A gives the binding energy per nucleon, which is the most useful comparative quantity. When you plot BE/A against mass number A, you find a characteristic curve: it rises steeply from hydrogen (essentially zero), peaks near iron (A ≈ 56, BE/A ≈ 8.8 MeV), and then gradually declines for heavier nuclei out to uranium and beyond. Iron and nickel sit at the bottom of the energy valley — they are the most tightly bound nuclei in nature, the "ashes" of stellar nucleosynthesis that no further nuclear reaction can squeeze energy out of.
The shape of this curve directly explains nuclear fission and fusion. Fusion — combining light nuclei — releases energy because the product lies higher on the curve (more bound per nucleon) than the reactants: hydrogen fusing to helium gains roughly 7 MeV per nucleon. This is the energy source of stars. Fission — splitting heavy nuclei — also releases energy because the fragments land higher on the curve than the original heavy nucleus: uranium splitting into two medium-weight fragments releases about 0.9 MeV per nucleon. Both processes move nuclei toward the peak at iron. Once you reach iron, neither fusion nor fission releases energy.
The stability curve (N vs. Z for stable nuclei) tells a complementary story about which combinations of protons and neutrons are stable. For light nuclei, N ≈ Z: roughly equal numbers of each are needed. For heavier nuclei, the line curves above the N = Z diagonal — stable heavy nuclei have progressively more neutrons than protons. The reason: protons repel each other electrically, and for large Z this repulsion becomes substantial. Adding extra neutrons dilutes the proton density and supplies additional strong-force glue without adding to the electromagnetic repulsion. Nuclei that fall far from the stability curve are radioactive, decaying toward it by beta decay, alpha emission, or other processes.