Mass defect, binding energy and the binding energy curve
A nucleus weighs less than its free nucleons. The mass defect, as energy Δm c², is the binding energy. Per nucleon it peaks near iron, about 8.8 MeV.
Mass defect and binding energy
Einstein showed that mass and energy are equivalent: E = mc². Careful measurement shows that the mass of a nucleus is less than the total mass of its free protons and neutrons. This difference is the mass defect, Δm. When nucleons join to form a nucleus, energy is released, and it carries away the mass Δm. To break the nucleus back into free nucleons, the same energy must be supplied. This is the binding energy, BE = Δm c². Dividing by A gives the binding energy per nucleon, B = BE/A. A larger B means a more stable nucleus.
How to find the binding energy, step by step
- 1Count Z (protons) and N = A − Z (neutrons).
- 2Find the total mass of the free nucleons: Zmₚ + (A − Z)mₙ.
- 3Subtract the mass of the nucleus: Δm = Zmₚ + (A − Z)mₙ − M.
- 4Turn mass into energy: BE (in MeV) = Δm (in u) × 931.
- 5★Divide by A to get the binding energy per nucleon, B = BE/A.
★ key step
Worked example: binding energy of helium-4
The mass of a ⁴₂He nucleus is 4.00151 u. Find its mass defect, binding energy and binding energy per nucleon. (mₚ = 1.00727 u, mₙ = 1.00865 u, 1 u = 931 MeV)
- Z = 2, N = 2.
- Mass of 2 protons = 2 × 1.00727 = 2.01454 u; mass of 2 neutrons = 2 × 1.00865 = 2.01730 u.
- Total mass of the free nucleons = 4.03184 u.
- Δm = 4.03184 − 4.00151 = 0.03033 u.
- BE = 0.03033 × 931 MeV = 28.2 MeV.
- Per nucleon: 28.2 ÷ 4 = 7.06 MeV.
Δm = 0.03033 u; BE ≈ 28.2 MeV; about 7.06 MeV per nucleon
How to draw the binding energy curve
- Draw the x-axis for mass number A (0 to 240) and the y-axis for binding energy per nucleon B (MeV) (0 to 9).
- Start low at ²H (about 1.1 MeV) and rise steeply for light nuclei.
- Mark small peaks for ⁴He, ¹²C and ¹⁶O, which are more stable than their neighbours.
- Reach a broad maximum of about 8.8 MeV near A = 56 (iron).
- Fall gently to about 7.6 MeV at A = 238 (uranium).
- Write fusion with an arrow climbing the left side and fission with an arrow climbing the right side towards the peak.
Reading the curve
| Region | B | What it means |
|---|---|---|
| Light nuclei (A below about 20) | Low, rising steeply | Less stable. Joining two light nuclei gives a more tightly bound nucleus, so fusion releases energy |
| Middle nuclei (around A = 56) | Highest, about 8 to 8.8 MeV | The most stable nuclei, such as iron |
| Heavy nuclei (A above about 200) | Falls to about 7.6 MeV | Splitting into two middle nuclei raises B, so fission releases energy |
Worked example: energy from splitting a heavy nucleus
A heavy nucleus with A = 236 has B = 7.6 MeV. It splits into a nucleus with A = 140 (B = 8.4 MeV), one with A = 94 (B = 8.6 MeV) and 2 free neutrons. Estimate the energy released.
- Binding energy before = 236 × 7.6 MeV = 1793.6 MeV.
- Binding energy after = 140 × 8.4 + 94 × 8.6 = 1176 + 808.4 = 1984.4 MeV (free neutrons have no binding energy).
- Energy released = 1984.4 − 1793.6 = 190.8 MeV.
About 190 MeV, close to the 200 MeV usually quoted per fission
Iron sits at the top of the binding energy curve. That is why the Sun gets its energy by joining light nuclei, while nuclear power stations get theirs by splitting heavy uranium nuclei.
3 marks: “Define mass defect and binding energy. The mass of a deuteron (²₁H nucleus) is 2.01355 u. Find its binding energy. (mₚ = 1.00727 u, mₙ = 1.00865 u, 1 u = 931 MeV)” Mass defect: total mass of the free nucleons minus the mass of the nucleus; BE = Δm c². Δm = 1.00727 + 1.00865 − 2.01355 = 0.00237 u. BE = 0.00237 × 931 ≈ 2.2 MeV (about 1.1 MeV per nucleon).
5 marks: “Draw the curve of binding energy per nucleon against mass number. Use it to explain why both fission and fusion release energy.” Curve: low for light nuclei, small peaks at ⁴He, ¹²C and ¹⁶O, maximum about 8.8 MeV near A = 56, falling to 7.6 MeV at A = 238. Heavy nuclei split into middle ones with higher B (fission); light nuclei join into heavier ones with higher B (fusion). The gain in total binding energy is released.
1 marks: “What is the energy equivalent of 1 u?” About 931 MeV.
Lost mass is locked energy: Δm (in u) × 931 = BE (in MeV), and iron is the summit of the curve.