Explain nuclear binding energy and the stability of nuclei using the binding-energy curve

What this dot point is asking

WACE wants you to explain what holds the nucleus together, define binding energy and binding energy per nucleon, and use the binding-energy curve to explain why fission and fusion release energy. This ties mass-energy equivalence to nuclear stability.

The strong nuclear force

Protons in a nucleus repel each other electrostatically, yet nuclei hold together. The strong nuclear force, attractive and far stronger than the electrostatic force but acting only over very short ranges (about the size of a nucleus), binds protons and neutrons (nucleons) together. Neutrons add this binding without adding repulsion, which is why heavier stable nuclei need more neutrons than protons.

Mass defect and binding energy

The mass of a nucleus is less than the total mass of its separate nucleons. This mass defect $\Delta m$ corresponds to the binding energy, the energy that would be needed to pull the nucleus apart into free nucleons:

Equivalently, this energy was released when the nucleus formed. A larger binding energy means a more tightly bound nucleus.

Binding energy per nucleon

Total binding energy grows with nucleus size, so to compare stability fairly we use the binding energy per nucleon,

where $A$ is the number of nucleons. This is the better stability measure: the higher it is, the more energy holds each nucleon and the more stable the nucleus.

The binding-energy curve

Plotting binding energy per nucleon against mass number gives a curve that rises steeply for light nuclei, peaks around iron and nickel (mass number near $56$), then declines slowly for heavy nuclei. Iron-region nuclei are the most stable. Any reaction that moves nuclei toward this peak releases energy, because the products are more tightly bound than the reactants.

Linking to fission and fusion

Heavy nuclei such as uranium lie to the right of the peak, so splitting them (fission) moves the fragments up toward the peak and releases energy. Light nuclei such as hydrogen isotopes lie to the left, so joining them (fusion) also moves toward the peak and releases energy. Both processes increase binding energy per nucleon, which is the unifying reason they are energy sources.

Calculating a mass defect from nucleon masses

The most common quantitative task is to build the mass defect from scratch. Count the protons ($Z$, the proton number) and neutrons ($A-Z$) in the nucleus, multiply by their individual masses, and add to get the total mass of the separate nucleons. Subtract the measured mass of the bound nucleus to get the mass defect $\Delta m$. Then convert with either $E_b=\Delta mc^2$ (if $\Delta m$ is in kilograms) or, more conveniently, by multiplying $\Delta m$ in atomic mass units by $931.5\ \text{MeV per u}$. Keep all five decimal places of the nucleon masses through the subtraction, because the mass defect is a small difference of large numbers and rounding early destroys the answer.

Comparing nuclei

To judge which of two nuclei is more stable, compare binding energy per nucleon, not total binding energy. A large nucleus can have a big total binding energy yet a lower per-nucleon value, making it less stable than a small, tightly bound nucleus.