Binding energy per nucleon and its variation with mass number
Binding energy per nucleon peaks near 8.75 MeV at A = 56 and is lower for light and heavy nuclei, so fission and fusion both release energy.
What the curve tells us
The total binding energy grows with A, so we compare nuclei using the binding energy per nucleon, Ebn = Eb/A. A higher Ebn means each nucleon is held more tightly, so the nucleus is more stable. The graph of Ebn against A is one of the most important graphs in nuclear physics. It explains why heavy nuclei split and light nuclei join to release energy.
How to draw the binding energy curve
- Draw the x-axis for mass number A, from 0 to 240, and the y-axis for Ebn (MeV), from 0 to 10.
- Start low at ²H (about 1.1 MeV) and rise steeply for light nuclei.
- Mark small peaks above the curve for ⁴He (7.07 MeV), ¹²C and ¹⁶O.
- Reach a broad maximum of about 8.75 MeV at A = 56 (Fe).
- Fall gently to about 7.6 MeV at A = 238 (U).
- Write fusion with an arrow climbing the left side and fission with an arrow climbing the right side towards the peak.
Reading the three regions of the curve
| Region | Ebn | What it means |
|---|---|---|
| Light nuclei (A < 30) | Low, rises steeply | Joining two light nuclei makes a more tightly bound nucleus, so fusion releases energy |
| Middle nuclei (30 < A < 170) | About 8 MeV, nearly flat | The nuclear force is short range and saturates, so each nucleon adds about the same binding |
| Heavy nuclei (A > 170) | Falls to 7.6 MeV | Splitting into two middle nuclei raises Ebn, so fission releases energy |
Why the middle is flat and the heavy end falls
A nucleon feels the nuclear force only from its nearest neighbours, within a few femtometres. Once a nucleus is big enough, adding more nucleons does not give any one nucleon more neighbours. So its binding stays about the same: this is saturation, and it makes the curve flat. In very heavy nuclei, the many protons repel each other over the whole nucleus. This long-range repulsion lowers Ebn, so the curve falls.
Worked example: energy from splitting a heavy nucleus
A heavy nucleus with A = 236 has Ebn = 7.6 MeV. It splits into a nucleus with A = 140 (Ebn = 8.4 MeV), one with A = 94 (Ebn = 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 curve, so it cannot release energy by fission or fusion. That is why giant stars build an iron core before they collapse, and why the iron in spinach (palak) and in your blood was made inside stars long ago.
2 marks: “Draw the curve of binding energy per nucleon against mass number. Mark the regions where fission and fusion release energy.” Steep rise for light nuclei, peak about 8.75 MeV at A = 56, gentle fall to 7.6 MeV at A = 238. Fusion on the left (A < 30), fission on the right (A > 170).
3 marks: “Using the binding energy curve, explain why (a) heavy nuclei release energy on fission, (b) light nuclei release energy on fusion, (c) Ebn is nearly constant for 30 < A < 170.” (a) Products have higher Ebn than the heavy parent. (b) The fused nucleus has higher Ebn than the light nuclei. (c) The nuclear force is short range and saturates.
2 marks: “Nuclei X and Y have binding energies per nucleon of 7.5 MeV and 8.6 MeV. Which is more stable, and why?” Y, because a higher binding energy per nucleon means each nucleon is more tightly bound.
Iron is the summit: climb up from the left by fusion, climb up from the right by fission.