What Is Nuclear Binding Energy and Mass Defect?

Nuclear binding energy is the energy required to separate a nucleus completely into its individual protons and neutrons. Mass defect is the difference between the total mass of those separate nucleons and the smaller mass of the bound nucleus.

How Are Mass Defect and Binding Energy Related?

When nucleons combine, the attractive strong nuclear force creates a bound system and energy is released. Because energy has left the system, the nucleus has less mass than its separated nucleons, according to mass-energy equivalence:

$$E = \Delta m c^2$$

Here, $\Delta m$ is the mass defect and $c$ is the speed of light. The same amount of energy must be supplied to separate the nucleus again, so the nuclear binding energy is positive even though the nucleus has lost mass during formation.

For a nucleus containing $Z$ protons and $N$ neutrons:

$$\Delta m = Zm_p + Nm_n - m_{\text{nucleus}}$$

Using atomic masses is also valid if electron masses cancel consistently. If atomic masses are supplied, use hydrogen-atom masses rather than bare proton masses so that the electron masses cancel on both sides.

Worked Example: Helium-4

Using atomic mass values:

$$\Delta m = 2(1.007825\,\text{u}) + 2(1.008665\,\text{u}) - 4.002603\,\text{u}$$ $$\Delta m = 0.030377\,\text{u}$$

Since $1\,\text{u} = 931.5\,\text{MeV}\,c^{-2}$:

$$E = 0.030377 \times 931.5 = 28.3\,\text{MeV}$$

The binding energy per nucleon is therefore:

$$\frac{28.3}{4} = 7.07\,\text{MeV per nucleon}$$

A larger binding energy per nucleon generally indicates a more tightly bound nucleus. This quantity allows nuclei with different nucleon numbers to be compared. The binding-energy-per-nucleon curve rises rapidly for light nuclei, reaches a maximum near iron, and then declines slowly. Consequently, fusion of light nuclei and fission of very heavy nuclei can release energy because the products move toward a higher binding energy per nucleon.

A common misconception is that mass defect means mass has simply disappeared. It has instead been converted into energy and transferred from the nuclear system.

IB Exam Technique

This is SL and HL content in Topic E.3, Radioactive decay. In a calculation, show the nucleon masses, subtract the nuclear mass, convert units consistently, and then apply $E=\Delta mc^2$. Do not confuse total binding energy with binding energy per nucleon.