Mass–energy equivalence
Einstein showed that mass and energy are equivalent: E = mc². A small mass corresponds to a huge energy: converting 1 g of mass completely gives 10⁻³ × (3 × 10⁸)² = 9 × 10¹³ J. In nuclear physics it is convenient to use 1 u ≡ 931.5 MeV.
Mass defect
The mass of a nucleus is always less than the total mass of its separate protons and neutrons. The difference is the mass defect: Δm = [Z m_p + (A – Z) m_n] – M_nucleus When using atomic masses, use the mass of a hydrogen atom in place of m_p so that electron masses cancel.
Binding energy
The binding energy E_b = Δm c² is the energy released when a nucleus forms from its nucleons, or the energy needed to separate it completely into free nucleons. Example: deuterium. m(¹H) = 1.007825 u, m_n = 1.008665 u, m(²H) = 2.014102 u. Δm = 0.002388 u, so E_b ≈ 0.002388 × 931.5 ≈ 2.22 MeV. Example: helium-4. Δm = 2(1.007825) + 2(1.008665) – 4.002603 = 0.030377 u, so E_b ≈ 28.3 MeV and the binding energy per nucleon ≈ 7.07 MeV.
Binding energy per nucleon
E_bn = E_b/A measures how tightly a nucleus is held together. Its graph against A shows:
- It is small for light nuclei, rises quickly and is roughly constant (about 8 MeV) for mass numbers from about 30 to 170.
- It reaches a maximum of about 8.75 MeV near A = 56 (iron), the most stable region.
- It falls slowly for heavy nuclei (about 7.6 MeV for uranium).
Consequences: when a heavy nucleus splits into two medium nuclei (fission), or when light nuclei join (fusion), the products are more tightly bound and energy is released. The near-constant value in the middle reflects the short range and saturation of nuclear forces.
Key points
- E = mc²; 1 u = 931.5 MeV.
- Mass defect = mass of nucleons – mass of nucleus.
- Binding energy = Δm × 931.5 MeV (Δm in u).
- Peak binding energy per nucleon near A = 56.
Exam tips
- Keep six decimal places in mass-defect calculations; the answer depends on small differences.