The Nucleus: Quick Revision

A brief note on The Nucleus. Read it before a test. Try each question at the end before you open its answer.

The nucleus and its size

• Nucleons: protons and neutrons. The atomic number $Z$ counts protons, the neutron number is $N$, and the mass number is $A = Z + N$.
• Isotopes: same $Z$. Isobars: same $A$. Isotones: same $N$.
• Atomic mass unit: $1\ \text{u} = \dfrac{1}{12}$ of a carbon-12 atom $= 1.660539 \times 10^{-27}$ kg.
• Radius: $R = R_0A^{1/3}$, with $R_0 = 1.2$ fm and $1\ \text{fm} = 10^{-15}$ m.
• Density: about $2.3 \times 10^{17}\ \text{kg/m}^3$ for every nucleus, because the mass and the volume are both proportional to $A$.

Mass defect and binding energy

• Mass defect: $\Delta m = Zm_H + Nm_n - M$, with atomic masses in u.
• Mass and energy: $E = mc^2$, and $1\ \text{u}$ is equivalent to $931.494$ MeV. NCERT rounds this to $931.5$ MeV.
• Binding energy: $B = \Delta m \times 931.494\ \text{MeV/u}$. It is the energy needed to pull the nucleus apart.
• Per nucleon: $B/A$. Helium-4 has $7.07$ MeV, and iron-56 has $8.79$ MeV.

The binding energy curve

• $B/A$ rises steeply for light nuclei, peaks near $A = 56$ to $62$ at about $8.8$ MeV, and falls to $7.6$ MeV at uranium.
• Fusion of light nuclei and fission of heavy nuclei both climb the curve, so both release energy: $E = A\,(b_\text{after} - b_\text{before})$.
• Nuclear force: strong, attractive, short range, charge independent, and saturating.
• Heavy nuclei have more neutrons than protons, because the proton repulsion has a long range.

A worked example

Nitrogen-14 has an atomic mass of $14.003074$ u, with $7$ protons and $7$ neutrons. Find its binding energy per nucleon.

1. The parts. $7 \times 1.007825\ \text{u} + 7 \times 1.008665\ \text{u} = 14.115430$ u.
2. The mass defect. $14.115430\ \text{u} - 14.003074\ \text{u} = 0.112356$ u.
3. The binding energy. $0.112356\ \text{u} \times 931.494\ \text{MeV/u} = 104.66$ MeV.
4. Per nucleon. $\dfrac{104.66\ \text{MeV}}{14} = 7.476$ MeV.

Practise with new numbers

Check yourself

1. A nucleus has $Z = 82$ and $A = 208$. How many neutrons does it have?

2. A mass defect of $0.050$ u is found for a nucleus of $40$ nucleons. What is the binding energy per nucleon?

3. Why does fission of a heavy nucleus release energy?

Sources

• NCERT Physics Part II (Class 12, rationalised edition), Chapter 13 Nuclei.
• OpenStax University Physics Volume 3, Chapter 10 Nuclear Physics.
• AME2020 atomic mass evaluation, for the atomic masses.