Mass defect, made precise
The binding energy B of a nucleus with Z protons and N neutrons is the mass it lost, times c^2. In practice tables list atomic masses (nucleus plus its electrons), so we use the mass of hydrogen m_\mathrm{H} instead of the bare proton, and the Z electron masses cancel on both sides (to the accuracy we need).
Binding energy from a mass table. A positive B means the nucleus is bound.
Worked example — helium-4. With m_\mathrm{H} = 1.007825\,\mathrm{u}, m_n = 1.008665\,\mathrm{u} and M(^4\mathrm{He}) = 4.002602\,\mathrm{u}: the ingredients sum to 2(1.007825)+2(1.008665) = 4.032980\,\mathrm{u}, so the mass defect is 0.030378\,\mathrm{u}. Multiply by 931.494\ \mathrm{MeV/u} and you get a binding energy of about 28.3 MeV, or 7.07 MeV per nucleon — already most of the way up the curve, which is why the alpha particle is so extraordinarily stable.
Reading the curve carefully
The curve is not perfectly smooth. Sharp spikes stand at helium-4, carbon-12 and oxygen-16 — nuclei that are whole numbers of tightly bound alpha particles. The deuteron (^2\mathrm{H}) sits far below everything, bound by a mere 2.22 MeV (1.1 MeV per nucleon) — it barely holds together. These bumps are the first hint that a smooth liquid-drop picture is not the whole story; quantum shell structure (guide 3) hides underneath.
The liquid-drop model
Because the nuclear force saturates and nuclear matter has constant density, a nucleus behaves in bulk remarkably like a tiny drop of incompressible, charged liquid. The liquid-drop model runs with that analogy: model the total binding energy as a sum of classical-looking energy terms, then fit their coefficients to the measured masses of hundreds of nuclei.
The five terms of the SEMF
The result is the Bethe–Weizsäcker semi-empirical mass formula. Each term is a physical effect you can reason out by hand:
Volume, surface, Coulomb, asymmetry and pairing — five terms capturing the bulk of nuclear masses.
Volume a_V A: each nucleon bonds to a fixed number of neighbours (saturation), so binding grows with the number of nucleons — the leading term. Surface -a_S A^{2/3}: nucleons on the skin have fewer neighbours, a correction proportional to surface area, which is why small nuclei pay a bigger penalty. Coulomb -a_C Z(Z-1)/A^{1/3}: every proton pair repels, reducing binding, and this term grows fast with Z. Asymmetry -a_A (A-2Z)^2/A: a quantum effect (the Pauli principle) that favours equal N and Z. Pairing \pm\delta: nucleons like to pair up, so even–even nuclei are extra bound and odd–odd extra loose.
What the formula buys you
Fix the mass number A and let Z vary — these are isobars. The formula makes the nuclear mass a downward parabola in Z (the Coulomb term pushes Z down, the asymmetry term pushes it up toward N=Z). The minimum of that parabola is the most stable charge for that A, and it explains why nuclei off the valley floor beta-decay in the direction that walks them toward it — the physics of guide 4.
The same formula estimates energy released in fission and fusion (guide 5), locates the neutron and proton drip lines where binding fails entirely, and predicts the general trend of binding energy per nucleon. Its blind spot is exactly the shell spikes: it is smooth by construction and cannot see the magic numbers. That is the cue for the shell model.