Nuclear Energy & Binding Energy

Mass-energy equivalence, mass defect, binding energy per nucleon, nuclear fission, fusion, and the operation and safety of nuclear reactors.

Spec Points Covered
  • State and apply $E = mc^{2}$ to convert between mass and energyThe capacity to do work. Measured in joules (J)..
  • Calculate mass defectThe difference between the total mass of the individual nucleons and the actual mass of the nucleus. This mass is converted to binding energyThe capacity to do work. Measured in joules (J).. and binding energyThe capacity to do work. Measured in joules (J).The energy required to completely separate a nucleus into its individual protons and neutrons. Equal to the mass defectThe difference between the total mass of the individual nucleons and the actual mass of the nucleus. This mass is converted to binding energy. multiplied by c². for a given nucleus.
  • Interpret the binding energy per nucleonThe binding energyThe energy required to completely separate a nucleus into its individual protons and neutrons. Equal to the mass defectThe difference between the total mass of the individual nucleons and the actual mass of the nucleus. This mass is converted to binding energy. multiplied by c². of a nucleus divided by its nucleon number (mass number). Higher values indicate greater nuclear stability. graph to explain why fission and fusion release energy.
  • Describe induced fission, chain reactions, critical mass and the role of thermal neutrons.
  • Explain the roles of the moderator, control rods and coolant in a nuclear reactor.
  • Discuss the safety aspects and waste management of nuclear powerThe rate of energy transfer. Measured in watts (W)..
Notes
01 Mass and energy are equivalent: E = mc² Mass-energy equivalence 3.8.1.6 → 02 The atomic mass unit (u) is the standard unit for nuclear masses Unified atomic mass unit (u) 3.8.1.6 → 03 Mass defect is the difference between a nucleus and its separated nucleons Mass defect 3.8.1.6 → 04 Binding energy is the energy needed to completely separate a nucleus Binding energy 3.8.1.6 → 05 The binding energy per nucleon graph shows iron-56 as the most stable nucleus Binding energy per nucleon 3.8.1.6 → 06 Fusion combines light nuclei — requires extreme temperatures to overcome repulsion Nuclear fusion 3.8.1.7 → 07 Fission splits heavy nuclei into smaller, more stable fragments Nuclear fission 3.8.1.7 → 08 Induced fission requires absorption of a thermal (slow) neutron Induced fission 3.8.1.7 → 09 Chain reactions sustain fission — critical mass keeps them controlled Chain reaction 3.8.1.7 → 10 A nuclear reactor uses a moderator, control rods and coolant 3.8.1.7 → 11 Nuclear fuel is enriched uranium: U-238 mixed with fissile U-235 3.8.1.8 → 12 Nuclear waste is classified by activity level and stored accordingly 3.8.1.8 → 13 Nuclear power: high energy density, no greenhouse gases, but radioactive waste 3.8.1.8 → 14 Calculating energy released: binding energy after minus binding energy before 3.8.1.6 →
Σ Key Equations Full Reference →
On Data Sheet
Not on Data Sheet
Mass-energy equivalence
$$E = mc^2$$
  • Where:
    • $E$ = energy (J)
    • $m$ = mass (kg)
    • $c$ = speed of light = 3.0 × 10⁸ m s⁻¹
Applies to all nuclear reactions. A small mass defect corresponds to a large energy release.
Mass defect
$$\Delta m = Zm_p + (A - Z)m_n - m_{\text{total}}$$
  • Where:
    • $Δm$ = mass defect (kg or u)
    • $Z$ = proton number
    • $A$ = nucleon number
    • $mₚ$ = proton mass
    • $mₙ$ = neutron mass
    • $m_total$ = measured nuclear mass
Total mass of separated nucleons minus measured mass of nucleus.
Average kinetic energy of a thermal neutron
$$E = \frac{3}{2}kT$$
  • Where:
    • $E$ = kinetic energy (J)
    • $k$ = Boltzmann constant = 1.38 × 10⁻²³ J K⁻¹
    • $T$ = temperature (K)
At T = 300 K, E ≈ 0.04 eV. This is the energy needed for neutrons to induce fission in U-235.
Binding energy from mass defect
$$E = \Delta m \, c^2$$
  • Where:
    • $E$ = binding energy (J)
    • $Δm$ = mass defect (kg)
    • $c$ = speed of light (m s⁻¹)
Convert Δm from u to kg first (1 u = 1.661 × 10⁻²⁷ kg), or use 1 u = 931.5 MeV directly.
Number of nuclei from mass
$$N = \frac{m \times N_A}{M}$$
  • Where:
    • $N$ = number of nuclei
    • $m$ = mass of sample (g)
    • $N_A$ = Avogadro constant = 6.02 × 10²³ mol⁻¹
    • $M$ = molar mass (g mol⁻¹)
Used to find the number of atoms in a sample for activity and energy calculations.
Q Retrieval Practice All 14 Questions →
Q1. State the equation for mass-energy equivalence and define each term.
  • E = mc².
  • E = energy (J), m = mass (kg), c = speed of light (3.0 × 10⁸ m s⁻¹).
Q2. Define the atomic mass unit (u) and state its value in kg and MeV.
One-twelfth of the mass of a carbon-12 atom. 1 u = 1.661 × 10⁻²⁷ kg = 931.5 MeV.
Q3. Define mass defect and state the equation to calculate it.
The difference between the total mass of separated nucleons and the measured mass of the nucleus. Δm = Zmₚ + (A−Z)mₙ − m_total.
Q4. Define binding energyThe energy required to completely separate a nucleus into its individual protons and neutrons. Equal to the mass defect multiplied by c²..
  • The energy required to completely separate a nucleus into its individual protons and neutrons.
  • Equivalently, the energy released when the nucleus is formed from isolated nucleons.
Q5. Why does iron-56 sit at the peak of the binding energy per nucleonThe binding energy of a nucleus divided by its nucleon number (mass number). Higher values indicate greater nuclear stability. graph?
  • Iron-56 has the highest binding energy per nucleonThe binding energy of a nucleus divided by its nucleon number (mass number). Higher values indicate greater nuclear stability. (~8.8 MeV), making it the most stable nucleus.
  • Nuclei lighter than iron can gain stability by fusion; nuclei heavier than iron gain stability by fission.