Big picture
Rutherford showed that an atom has a tiny nucleus. This chapter looks inside it. A nucleus contains protons and neutrons packed so tightly that its density is about 10¹⁷ kg/m³. They are held together by the nuclear force, far stronger than the electric repulsion between protons.
The mass of a nucleus is slightly less than the total mass of its separate protons and neutrons. That missing mass, the mass defect, appears as binding energy through E = mc². The binding energy per nucleon explains why splitting heavy nuclei (fission) and joining light nuclei (fusion) both release enormous energy.
Learning goals
Describe the composition of the nucleus and define atomic number, mass number, isotopes, isobars and isotones.
Use the atomic mass unit and its energy equivalent.
Calculate nuclear radius and explain why nuclear density is constant.
State the properties of the nuclear force.
Calculate mass defect and binding energy.
Explain the binding energy per nucleon curve.
Explain nuclear fission and fusion and calculate the energy released.
Core ideas
1. Composition of the nucleus
A nucleus contains protons (positive charge) and neutrons (no charge), together called nucleons. The neutron was discovered by Chadwick in 1932.
Atomic number Z = number of protons.
Mass number A = number of protons + neutrons.
Number of neutrons N = A − Z.
A nuclide is written as ᴬ_Z X; for example, ²³⁵₉₂U has 92 protons and 143 neutrons.
Term | Same | Different | Example |
|---|---|---|---|
Isotopes | Z | A (and N) | ¹H, ²H, ³H; ³⁵Cl and ³⁷Cl |
Isobars | A | Z | ³H and ³He; ⁴⁰Ar and ⁴⁰Ca |
Isotones | N | Z and A | ¹⁹⁸Hg and ¹⁹⁷Au (both N = 118) |
2. Atomic mass unit
1 u (atomic mass unit) = 1/12 of the mass of a carbon-12 atom = 1.66 × 10⁻²⁷ kg. Its energy equivalent is 1 u = 931.5 MeV/c².
Masses: proton 1.007276 u; neutron 1.008665 u; hydrogen atom 1.007825 u.
3. Size and density of the nucleus
Experiments show the nuclear radius depends on mass number:
R = R₀ A^(1/3), with R₀ ≈ 1.2 fm (1 fm = 10⁻¹⁵ m)
Example. For ²⁷Al, R = 1.2 × 27^(1/3) = 1.2 × 3 = 3.6 fm.
Since volume ∝ R³ ∝ A, and mass ∝ A, the density of all nuclei is about the same, around 2.3 × 10¹⁷ kg/m³, independent of A. This is about 10¹⁴ times the density of ordinary matter.
Ratio example. The ratio of the radii of ¹²⁵Te and ²⁷Al is (125/27)^(1/3) = 5/3.
4. Nuclear force
The force that holds nucleons together has these properties:
Strongest of all fundamental forces: much stronger than the Coulomb repulsion between protons.
Very short range: effective only up to a few femtometres (about 2–3 fm), and negligible beyond.
Charge independent: the force between proton–proton, neutron–neutron and proton–neutron is about the same.
Attractive at distances greater than about 0.8 fm, and strongly repulsive at smaller distances, which keeps nucleons from collapsing together.
Saturating: each nucleon interacts only with its nearest neighbours.
5. Mass–energy equivalence, mass defect and binding energy
Einstein's relation E = mc² shows mass and energy are equivalent. Converting 1 g of mass completely gives E = (10⁻³)(3 × 10⁸)² = 9 × 10¹³ J.
Mass defect (Δm) = (total mass of the separate nucleons) − (mass of the nucleus):
Δm = [Z mp + (A − Z) mn] − M_nucleus
(Using atomic masses, replace mp by the mass of a hydrogen atom, so the electron masses cancel.)
Binding energy (Eb) = Δm c² = Δm (in u) × 931.5 MeV. It is the energy needed to break the nucleus into separate nucleons, or the energy released when the nucleus forms.
Worked example: helium-4. Mass of ⁴He atom = 4.002603 u; mass of ¹H atom = 1.007825 u; mass of neutron = 1.008665 u.
Mass of 2 hydrogen atoms + 2 neutrons = 2(1.007825) + 2(1.008665) = 4.032980 u
Mass defect = 4.032980 - 4.002603 = 0.030377 u
Binding energy = 0.030377 × 931.5 = 28.3 MeV
Binding energy per nucleon = 28.3 / 4 = 7.07 MeV6. Binding energy per nucleon curve
The binding energy per nucleon (Ebn = Eb/A) measures how tightly a nucleus is bound. Plotted against A:
It rises sharply for light nuclei, with peaks for especially stable nuclei like ⁴He, ¹²C and ¹⁶O.
It is roughly constant at about 8 MeV for mass numbers 30 to 170, because of the saturation of the nuclear force.
It reaches a maximum of about 8.75 MeV near A = 56 (iron), the most stable nucleus.
It decreases slowly for heavy nuclei (about 7.6 MeV for uranium), because the Coulomb repulsion between many protons grows.
What the curve explains:
Fission: a heavy nucleus (A ≈ 240) splitting into two medium nuclei (A ≈ 120) increases Ebn from about 7.6 to about 8.5 MeV, so energy is released.
Fusion: two light nuclei joining to form a heavier one increases Ebn, so energy is released.
Stability: nuclei with the highest Ebn (around iron) are the most stable.
7. Nuclear fission
Fission is the splitting of a heavy nucleus into two medium-mass nuclei, usually after absorbing a slow neutron.
²³⁵U + n → ¹⁴¹Ba + ⁹²Kr + 3n + about 200 MeV
About 200 MeV is released per fission, mostly as kinetic energy of the fragments.
Each fission releases 2–3 neutrons, which can cause further fissions: a chain reaction.
In a nuclear reactor, the chain reaction is controlled: a moderator (heavy water, graphite) slows neutrons down; control rods (cadmium, boron) absorb extra neutrons; a coolant carries heat to produce steam and electricity.
An uncontrolled chain reaction is the basis of an atomic bomb.
8. Nuclear fusion
Fusion is the joining of light nuclei to form a heavier nucleus.
²H + ³H → ⁴He + n + 17.6 MeV
Nuclei repel each other strongly, so fusion needs very high temperatures (about 10⁷ K or more) to give them enough kinetic energy; hence it is also called a thermonuclear reaction.
The energy of the Sun and stars comes from fusion: in the proton–proton cycle, four hydrogen nuclei fuse into one helium nucleus, releasing about 26.7 MeV.
Per unit mass of fuel, fusion releases more energy than fission, and produces less radioactive waste, but controlled fusion on Earth is still under development.
9. Radioactivity (for reference)
Unstable nuclei decay spontaneously by emitting alpha particles (helium nuclei), beta particles (electrons or positrons) or gamma rays (high-energy photons). The number of undecayed nuclei falls as N = N₀ e^(−λt), and the half-life is T½ = 0.693/λ. After n half-lives, the fraction remaining is (½)ⁿ. (Check whether your current syllabus includes this topic.)
A reliable problem-solving routine
For mass defect, add the masses of separate nucleons (hydrogen atoms + neutrons) and subtract the atomic mass.
Convert to energy with 1 u = 931.5 MeV; divide by A for binding energy per nucleon.
For radius, use R = 1.2 A^(1/3) fm; for ratios, use (A₁/A₂)^(1/3).
For energy released in a reaction, find the mass lost (reactants − products) and multiply by 931.5 MeV.
Explain fission and fusion using the rise in binding energy per nucleon.
Connections that matter
Atoms introduced the nucleus through Rutherford scattering.
Electromagnetic waves: gamma rays come from nuclear transitions.
Chemistry: structure of the atom uses isotopes and mass numbers.
Common mistakes
Thinking the nucleus weighs more than its parts. It weighs less; the difference is the mass defect.
Saying nuclear density increases with A. It is nearly constant.
Calling the nuclear force long-range. It acts only over a few femtometres.
Saying fission and fusion release energy because mass is created. They release energy because mass is lost and binding energy per nucleon increases.
Mixing up isobars and isotopes.
Revision checklist
I can define isotopes, isobars and isotones with examples.
I can calculate nuclear radius and explain constant density.
I can state the properties of the nuclear force.
I can calculate mass defect, binding energy and binding energy per nucleon.
I can explain the binding energy curve and use it to explain fission and fusion.
I can describe a chain reaction, a nuclear reactor and fusion in stars.