sketch the variation of binding energy per nucleon with nucleon number

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Mass‑Defect, Nuclear Binding Energy and the Binding‑Energy‑per‑Nucleon Curve

Learning objectives

  • Define mass‑defect (Δm), binding energy (Eb) and binding‑energy‑per‑nucleon (Eb/A).
  • State and apply Einstein’s relation \(E=Δmc^{2}\) (including the conversion factor 1 u = 931.5 MeV c⁻²).
  • Write nuclides in the standard notation \({}^{A}_{Z}\!X\) and use it for simple decay equations.
  • Sketch the variation of \(E_{b}/A\) with nucleon number \(A\) and explain the shape of the curve.
  • Relate the curve to the energetics of nuclear fusion and fission.

Key definitions and notation

  • Mass‑defect (Δm) \[ Δm = \bigl[Z\,m_{p}+N\,m_{n}\bigr] - m_{\text{nucleus}} \] where \(Z\) = number of protons, \(N=A-Z\) = number of neutrons, \(A\) = mass number. The tabulated atomic mass already includes the electron masses, so the same value can be used directly for the nucleus.
  • Binding energy (Eb)** \[ E_{b}=Δm\,c^{2}=Δm\times 931.5\;\text{MeV} \] (1 u = 931.5 MeV c⁻² is the energy equivalent of one atomic‑mass unit.)
  • Binding‑energy‑per‑nucleon (Eb/A) – the average energy that holds each nucleon in the nucleus; a convenient measure of nuclear stability.
  • Nuclide notation – \({}^{A}_{Z}\!X\) (e.g. \({}^{56}_{26}\!Fe\)). A simple decay example: \[ {}^{238}_{92}\!U \;\rightarrow\; {}^{234}_{90}\!Th + {}^{4}_{2}\!He \] (α‑decay).

Calculating Δm and Eb

For a nuclide \({}^{A}_{Z}\!X\) :

\[ \Delta m = \bigl[ Z\,m_{p}+ (A-Z)\,m_{n}\bigr] - m_{\text{atom}} \] \[ E_{b}= \Delta m \times 931.5\;\text{MeV} \qquad\qquad \frac{E_{b}}{A}= \frac{E_{b}}{A} \]

with \(m_{p}=1.007276\;\text{u}\) and \(m_{n}=1.008665\;\text{u}\).

Selected data (mass defect, binding energy and binding energy per nucleon)

Nuclide \({}^{A}_{Z}\!X\) A Δm (u) Eb (MeV) Eb/A (MeV)
\({}^{2}_{1}\!H\) (deuterium) 2 0.0022 2.05 1.02
\({}^{4}_{2}\!He\) (α‑particle) 4 0.0304 28.3 7.07
\({}^{12}_{6}\!C\) 12 0.0985 92.2 7.68
\({}^{16}_{8}\!O\) 16 0.1365 127.6 7.98
\({}^{56}_{26}\!Fe\) 56 0.4920 492.3 8.79
\({}^{62}_{28}\!Ni\) 62 0.5436 534.4 8.79 (≈8.794)
\({}^{238}_{92}\!U\) 238 1.7840 1664.0 7.00

Variation of binding‑energy‑per‑nucleon with nucleon number

The curve of \(E_{b}/A\) against \(A\) has a characteristic “mountain‑range” shape. The sketch below includes the key points required by the Cambridge 9702 syllabus.

Nucleon number \(A\) Binding energy per nucleon (MeV) 20 40 60 80 100 120 140 160 6 7 8 9 \(A\approx20\) Peak (Fe‑56, Ni‑62) \(A\approx200\)

Features of the curve

  1. Rapid rise (A ≲ 20) – Adding a nucleon creates many new short‑range attractive pairs, while a large fraction of nucleons are still on the surface.
  2. Broad maximum (A ≈ 56–62) – The most tightly bound nuclei are \({}^{56}_{26}\!Fe\) and \({}^{62}_{28}\!Ni\) (≈ 8.79 MeV per nucleon). Here the gain from extra nuclear attraction is almost cancelled by the growing Coulomb repulsion.
  3. Gradual decline (A > 62) – Electrostatic repulsion between the increasing number of protons outweighs the additional nuclear force, so \(E_{b}/A\) falls to ≈ 7 MeV for the actinides.

Physical interpretation (why the curve has this shape)

  • Short‑range nuclear force – Each nucleon feels a strong attractive force only from its nearest neighbours. In very light nuclei many nucleons are on the surface and have fewer neighbours, giving a low \(E_{b}/A\).
  • Surface effect – The surface‑to‑volume ratio decreases as \(A\) grows, so the proportion of “unbound” surface nucleons falls, raising the average binding.
  • Electrostatic (Coulomb) repulsion – The repulsive energy grows roughly as \(Z^{2}\) and becomes dominant for heavy nuclei, pulling the curve down.
  • Stability – Nuclei near the peak are the most stable. Nuclei left of the peak can release energy by fusing (moving toward higher \(E_{b}/A\)); nuclei right of the peak can release energy by fissioning into fragments that lie nearer the peak.

Link to nuclear energy production

Process Typical reactants Typical products (near the peak) Energy released (≈ per nucleon)
Fusion \({}^{2}_{1}\!H\), \({}^{3}_{1}\!H\), \({}^{4}_{2}\!He\) \({}^{12}_{6}\!C\), \({}^{16}_{8}\!O\) (and heavier nuclei up to Fe‑group) ~7 MeV per nucleon added
Fission \({}^{235}_{92}\!U\), \({}^{239}_{94}\!Pu\) Two fragments with \(A\approx 90–150\) (e.g. \({}^{144}_{56}\!Ba\) + \({}^{89}_{36}\!Kr\)) ~0.9 MeV per nucleon gained

Exam‑style tip

  • When asked to “sketch the variation of \(E_{b}/A\) with \(A\)”, remember the three labelled points:
    • Rapid rise up to \(A\approx20\)
    • Maximum at \(A\approx56\)–\(62\) (Fe‑56, Ni‑62)
    • Slow decline for \(A>62\) (≈ 7 MeV for uranium).
  • To compare fusion and fission, state which side of the curve the reactants lie on and which side the products lie on; the movement towards the peak indicates energy release.
  • Use the conversion \(1\;\text{u}=931.5\;\text{MeV}\) whenever you turn a mass defect into an energy value.

Summary

The binding‑energy‑per‑nucleon curve summarises the competition between the short‑range attractive nuclear force and the long‑range Coulomb repulsion. Its maximum near \({}^{62}_{28}\!Ni\) explains why both fusion (for light nuclei) and fission (for heavy nuclei) can release energy: the products occupy a higher point on the curve, i.e. they have a larger \(E_{b}/A\) and therefore a lower total mass‑energy.

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