Why a Helium Nucleus Weighs Less Than the Parts It Is Made Of

A nucleus always has slightly less mass than its separate protons and neutrons. That missing mass is the energy holding the nucleus together, and rearranging nuclei to bind them more tightly releases part of it — in a reactor, or in the Sun.

This lesson covers nuclear classification with mass defect and binding energy, the binding energy curve, and energy released in fission and fusion.

**Isotopes share the same atomic number Z, isobars the same mass number A, and isotones the same neutron number N; the mass defect is the mass lost when nucleons combine, and the binding energy is that mass times $c^2$, with 1 u equivalent to 931.5 MeV.

Classifying nuclei:

- Isotopes — same Z: carbon-12 and carbon-14
- Isobars — same A: argon-40 and calcium-40
- Isotones — same N: carbon-14 and oxygen-16, both with 8 neutrons

Mass defect.** $\Delta m=[Z{m}_H+(A-Z)m_n]-M$, using atomic masses so that electron masses cancel.

Worked example — helium-4. With $m_H=1.007825$ u, $m_n=1.008665$ u and $M=4.002603$ u:

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

$$E_b=0.030377\times 931.5=28.3\text{ MeV}\frac{E_b}{A}=\frac{28.3}{4}=7.07\text{ MeV per nucleon}$$

An everyday example. The heavy water used in many Indian nuclear reactors contains deuterium, an isotope of hydrogen with one proton and one neutron.

The substance. Mass is not conserved in nuclear changes, but mass-energy is — the missing 0.030 u did not vanish; it left as energy when the nucleus formed.

Binding energy per nucleon rises steeply for light nuclei, peaks near mass number 56, and falls slowly for heavy nuclei, so energy is released when heavy nuclei split or light nuclei join to move closer to the peak.

Reading the curve:

- Light nuclei — few neighbours per nucleon, so binding energy per nucleon is low and fusion releases energy
- Middle nuclei near iron — the most tightly bound, and so the most stable
- Heavy nuclei — growing proton repulsion lowers binding energy per nucleon, so fission releases energy

Why the curve flattens. The nuclear force is strong but short-range, so each nucleon attracts only its nearest neighbours. Once a nucleus is large enough, adding nucleons no longer adds neighbours.

Worked example — estimating fission energy. Uranium-235 plus a neutron has about 7.6 MeV per nucleon. If it splits into two middle-sized nuclei with about 8.5 MeV per nucleon, for all 236 nucleons:

$$E\approx 236\times (8.5-7.6)=212\text{ MeV}$$

That is roughly 200 MeV from a single fission.

An everyday example. India's nuclear power stations generate electricity by moving uranium nuclei from the heavy end of this curve towards its peak.

The substance. A larger total binding energy does not mean a more stable nucleus — uranium has far more total binding energy than iron, but less per nucleon.

**The energy released, or Q-value, equals the mass lost in the reaction multiplied by $c^2$: $Q=(\sum m_{\text{reactants}}-\sum m_{\text{products}})\times 931.5$ MeV per u.

The method. Add the masses on each side, subtract, and convert. A positive Q means energy is released.

Worked example 1 — deuterium-tritium fusion.** Deuterium and tritium fuse into helium-4 and a neutron, with masses 2.014102 u, 3.016049 u, 4.002603 u and 1.008665 u:

$$\Delta m=(2.014102+3.016049)-(4.002603+1.008665)=0.018883\text{ u}$$

$$Q=0.018883\times 931.5=17.6\text{ MeV}$$

Worked example 2 — mass into energy. Converting just 1.0 g of mass entirely into energy gives

$$E=m{c}^2=1.0\times 10^{-3}\times (3.0\times 10^8{)}^2=9.0\times 10^{13}\text{ J}$$

Comparing per nucleon. Fusion gives $17.6/5=3.5$ MeV per nucleon, while fission gives about $200/236=0.85$ MeV per nucleon.

An everyday example. The sunlight ripening mangoes in summer comes from fusion of hydrogen into helium in the Sun's core.

The substance. Fission releases more energy per reaction, but fusion releases more per nucleon, and so more per kilogram of fuel.