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The binding energy per nucleon BE/A increases from light nuclei, peaks near ⁵⁶Fe (~8.8 MeV per nucleon), and decreases for heavy nuclei. Nuclei near the peak are most stable. Light nuclei prefer equal numbers of protons and neutrons (N ≈ Z), while heavy stable nuclei have more neutrons than protons (due to Coulomb repulsion). Nuclei far from the stability valley are radioactive.
Plot the valley of beta stability (N vs Z for stable isotopes) and compare with the binding energy curve. Explain why fusion of light nuclei and fission of heavy nuclei both release energy.
The most abundant element (iron) is not necessarily the most abundant in the universe (iron-56 is most tightly bound, but helium is more abundant due to primordial nucleosynthesis). Instability is not sudden; nuclei gradually decay as they move away from the stability line.
You already know from mass defect and binding energy that assembling a nucleus releases energy — the total mass of the nucleus is less than the sum of its parts, and the "missing" mass appears as binding energy via E = mc². The binding energy per nucleon, BE/A, is the average energy that would be needed to remove a single nucleon from the nucleus. It is a measure of how tightly bound each nucleon is on average, and it varies dramatically across the periodic table.
The binding energy curve — a plot of BE/A versus mass number A — has a characteristic shape: it rises steeply from hydrogen (essentially zero, since ¹H is just a proton), passes through a hump in the light elements, continues rising more gradually, peaks near ⁵⁶Fe at about 8.8 MeV per nucleon, and then gently falls for heavier nuclei. The peak represents the most stable nuclei: iron-56 holds its nucleons together most tightly per particle. This is the nuclear "valley floor" — nuclei on either side are less stable and will release energy by moving toward iron.
The shape of the curve reflects two competing forces. The strong nuclear force is short-range and attractive, acting between any pair of neighboring nucleons. Adding more nucleons increases binding, but only up to the range of the force — beyond about A ~ 60, new nucleons don't "see" all the other nucleons. Meanwhile, the Coulomb repulsion between protons is long-range: every proton repels every other proton throughout the nucleus regardless of size. For large A, the Coulomb penalty grows faster than the strong-force gain, which is why BE/A decreases for heavy nuclei and why heavy stable nuclei need more neutrons than protons (neutrons contribute strong force but no Coulomb repulsion).
The practical consequences of the curve's shape explain both fusion and fission as energy sources. Fusion of light nuclei (H → He or He → C) moves up the left slope of the curve toward higher BE/A, releasing the difference in binding energy per nucleon times the number of nucleons involved. Fission of heavy nuclei (uranium → barium + krypton, roughly) moves down the right slope toward higher BE/A, again releasing energy. Both processes move nuclei *toward* iron — both release energy for the same underlying reason: the products are more tightly bound per nucleon than the reactants. Iron-56 is the thermodynamic endpoint of all nuclear burning; a star made entirely of iron-56 could release no further nuclear energy.