Nuclear Binding Energy: The Liquid-Drop Model
1 · Predict
The semi-empirical mass formula (liquid-drop model) estimates a nucleus's total binding energy from its proton count Z and mass number A. Holding Z fixed at 26 (iron) and changing only A, does the model predict binding energy increasing steadily with every added nucleon?
- No — the binding energy depends on a balance of competing terms (volume, surface, Coulomb, asymmetry) that shift differently as A grows, so it doesn't simply increase with every added nucleon.
- Yes — more nucleons always means more total binding energy.
- Binding energy always decreases as A increases.
2 · Set Up
- Open the iron-binding preset and press Reset. This preset models iron with Z = 26 protons.
- Enable the binding-energy readout.
- Set the mass number A for each trial (holding Z = 26 fixed) and record the total binding energy.
3 · Collect Data
| Mass number A | Total binding energy E_B (MeV) |
|---|---|
| 54 | |
| 56 | |
| 58 |
Plot binding energy E_B (y-axis) against mass number A (x-axis) for your three trials.
4 · Analyze
- For one trial, compute E_B = a_vA − a_sA^(2/3) − a_cZ(Z−1)/A^(1/3) − a_a(A−2Z)²/A^(1/3) using a_v = 15.75, a_s = 17.8, a_c = 0.711, a_a = 23.7 (MeV), Z = 26. Compare to the table.
- Notice the binding energy drops as A moves away from 2Z = 52 (equal protons and neutrons). Explain why the asymmetry term −a_a(A−2Z)²/A^(1/3) penalizes nuclei with an imbalanced proton-to-neutron ratio.
5 · Extend
- This same liquid-drop model explains why very heavy nuclei (large A) release energy through fission (splitting) while very light nuclei release energy through fusion (combining) — both processes move nuclei toward the peak binding-energy-per-nucleon region near iron. Explain why iron sits near this peak.
- This is a simplified version of the semi-empirical mass formula (it omits the pairing term for even/odd nucleon counts), so its predictions won't exactly match a real nuclear mass table. Why might physicists still find a simplified model like this useful for teaching, even knowing it's not perfectly accurate?
The Physics Behind This Experiment
Semi-Empirical Mass Formula (Liquid-Drop Model)
A nucleus's binding energy is modeled as a competition between a volume term (favoring more nucleons), a surface term (penalizing nucleons at the 'surface' of the nuclear drop), a Coulomb term (penalizing proton-proton repulsion), and an asymmetry term (penalizing unequal proton/neutron counts).