31.6 Binding Energy

Nuclear Stability and Binding Energy

Concept of binding energy

Binding energy is the total energy you'd need to completely pull a nucleus apart into separate protons and neutrons. It directly measures how strongly the strong nuclear force holds those nucleons together.

The key idea is mass defect: a nucleus always weighs less than the sum of its individual protons and neutrons. That "missing" mass hasn't vanished. It was converted into the energy that binds the nucleus together. Einstein's famous equation connects the two:

$E_b = (\Delta m)c^2$

• $E_b$ is the binding energy
• $\Delta m$ is the mass defect (the missing mass)
• $c$ is the speed of light

Binding energy per nucleon is what really tells you about stability. You get it by dividing the total binding energy by the number of nucleons (protons + neutrons) in the nucleus.

• Higher binding energy per nucleon = more stable nucleus. Iron-56 sits near the top, making it one of the most stable nuclei in nature.
• Lower binding energy per nucleon = less stable nucleus, more likely to undergo radioactive decay. Uranium-235 is a good example.

Calculation of binding energy

Here's how to calculate binding energy per nucleon, step by step:

1. Find the mass defect ($\Delta m$). Add up the individual masses of all the protons and neutrons, then subtract the actual measured mass of the nucleus.
2. Calculate total binding energy. Plug the mass defect into $E_b = (\Delta m)c^2$. In nuclear physics, it's convenient to use the conversion factor $1 \text{ u} = 931.5 \text{ MeV/c}^2$, so the equation becomes $E_b = (\Delta m)(931.5 \text{ MeV/u})$.
3. Divide by the number of nucleons (A). This gives you the binding energy per nucleon: $E_b / A$.

Example: Carbon-12 ($^{12}C$)

• Carbon-12 has 6 protons and 6 neutrons.
• Mass of 6 protons: $6 \times 1.007276 \text{ u} = 6.043656 \text{ u}$
• Mass of 6 neutrons: $6 \times 1.008665 \text{ u} = 6.051990 \text{ u}$
• Sum of individual masses: $6.043656 + 6.051990 = 12.095646 \text{ u}$
• Actual nuclear mass of $^{12}C$: $11.996709 \text{ u}$
• Mass defect: $\Delta m = 12.095646 - 11.996709 = 0.098937 \text{ u}$
• Total binding energy: $E_b = (0.098937)(931.5) = 92.16 \text{ MeV}$
• Binding energy per nucleon: $E_b / A = 92.16 / 12 = 7.68 \text{ MeV/nucleon}$

Note: masses in these calculations are expressed in atomic mass units (u). Be careful about whether a problem gives you the nuclear mass or the atomic mass (which includes electrons). Your textbook will usually specify which to use.

Binding energy across the periodic table

If you plot binding energy per nucleon against the number of nucleons (A), you get a curve with a distinctive shape:

• For light nuclei, binding energy per nucleon rises steeply as A increases.
• It peaks around $A = 56$ (iron-56), at roughly 8.8 MeV per nucleon.
• For nuclei heavier than iron-56, binding energy per nucleon gradually decreases.

This curve is the reason both fusion and fission can release energy. Nuclei naturally "want" to move toward higher binding energy per nucleon, because that's the more stable state.

• Nuclear fusion combines lighter nuclei into heavier ones. Because lighter nuclei sit on the rising part of the curve, the product has more binding energy per nucleon than the reactants. The difference is released as energy. This is what powers stars: hydrogen fuses into helium, releasing enormous energy.
• Nuclear fission splits heavy nuclei into lighter fragments. Heavy nuclei like uranium-235 sit on the declining part of the curve, so the fragments end up with higher binding energy per nucleon. Again, the difference is released as energy. This is the process used in nuclear power plants and nuclear weapons.

Both reactions move the system toward the iron-56 peak, which is why iron is sometimes called the "most stable nucleus."

Nuclear stability and models

The nuclear stability curve (also called the band of stability) plots the number of neutrons versus the number of protons for all known stable nuclei. Stable light nuclei tend to have roughly equal numbers of protons and neutrons, but heavier stable nuclei need progressively more neutrons than protons to remain stable. Nuclei that fall outside this band are radioactive.

The nuclear shell model treats nucleons as occupying discrete energy levels inside the nucleus, similar to how electrons fill energy levels in atoms. Certain "closed shell" configurations are especially stable, corresponding to magic numbers of protons or neutrons (2, 8, 20, 28, 50, 82, and 126). Nuclei with magic numbers of both protons and neutrons (like helium-4 or oxygen-16) are exceptionally stable.