(f) binding energy of nuclei using ΔE = Δm c² and masses of nuclei

The idea

Binding energy is the energy needed to separate a nucleus into its individual protons and neutrons, or released when the nucleus forms. The mass defect Δm is the difference between the total mass of the separate nucleons and the actual nuclear mass; this missing mass appears as binding energy via ΔE = Δm c². Use c = 3.00 × 10⁸ m s⁻¹, so 1 u = 1.66 × 10⁻²⁷ kg converts to about 1.49 × 10⁻¹⁰ J, or roughly 931 MeV. Work in kg and J, or in u and MeV, but never mix units. Binding energy per nucleon peaks near iron-56, explaining why fusion of light nuclei and fission of heavy nuclei both release energy.

What earns the marks

• State that binding energy is the energy required to separate a nucleus into its constituent protons and neutrons.
• Define mass defect Δm as the difference between the sum of the masses of the separate nucleons and the mass of the nucleus.
• Apply ΔE = Δm c² with c = 3.00 × 10⁸ m s⁻¹, keeping mass in kg and energy in J.
• Convert between u and kg using 1 u = 1.66 × 10⁻²⁷ kg, and recognise 1 u c² ≈ 931 MeV.
• Relate binding energy per nucleon to nuclear stability, with a maximum near iron-56.

Common mistakes and how to put them right

• Using the mass of the nucleus minus the sum of nucleon masses, giving a negative Δm.

  Put it right by taking the separate nucleons minus the nucleus so Δm is positive.

• Forgetting to square c, so ΔE is far too small.

  Put it right by computing c² = 9.00 × 10¹⁶ m² s⁻² before multiplying by Δm.

• Mixing units, for example mass in u with c in m s⁻¹.

  Put it right by converting mass to kg first or by using the u-to-MeV conversion consistently.

Exam tips

• Write the equation ΔE = Δm c² before substituting so the method is visible.
• Check the sign of Δm: it must be positive for a bound nucleus.
• If the answer is requested in MeV, divide the energy in J by 1.60 × 10⁻¹³ J MeV⁻¹.