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Mass-energy and binding energy

Weigh a nucleus and its parts separately and the books refuse to balance: the whole is lighter than the sum. Einstein's equation prices the missing mass in energy, and one curve built from it, peaking at iron, explains where every joule of nuclear power comes from.

Year 13AQA 3.8.1.6CIE 11.1, 23.1OCR A 6.4.4

Builds on Antimatter and photons and Nuclear radius and density.

IN THIS TOPIC

  • Convert between mass and energy using E = mc² and the 931.5 MeV value of the atomic mass unit.
  • Calculate a mass difference and a binding energy from nuclear masses.
  • Sketch the binding energy per nucleon curve and identify the fusion and fission release regions on it.

WHAT YOU PROBABLY THINK

Mass is always conserved.

Einstein's ledger

The equation held back in the particles unit arrives here with its full meaning:

E = mc2ON YOUR DATA SHEET

Mass and energy are one currency, and the exchange rate c2 applies to every energy change, however ordinary. Boil a kettle and the hot water is heavier by a few nanograms; a wound clock spring outweighs a slack one. The changes are immeasurably small because c2 is enormous. Only in the nucleus, where the energies are millions of electronvolts, does the missing mass become large enough to weigh.

Nuclear masses are quoted in the atomic mass unit, u, defined as one twelfth of a carbon-12 atom and printed in the booklet as 1.661 × 10-27 kg. Run that mass through E = mc2 and out comes the conversion the booklet also prints, the one worth memorising in spirit: 1 u is equivalent to 931.5 MeV. Mass differences in u multiply straight into energies in MeV, no joules required.

The missing mass

Now weigh a helium-4 nucleus against its ingredients, using the booklet's proton and neutron masses. Two protons and two neutrons apart come to 4.0319 u; the assembled nucleus is 4.0015 u. The difference, 0.0304 u, is the mass difference, and it did not vanish. It left as 0.0304 × 931.5 = 28.3 MeV of energy when the nucleus formed.

The mass ledger for helium-4: two protons and two neutrons weigh more apart than the nucleus they form, and the difference is the binding energy2p + 2n aparthelium-4 nucleus4.0319 u4.0015 u0.0304 u= 28.3 MeVthe missing mass left as binding energy
FIG. 1The helium-4 ledger: the parts outweigh the whole, and the 0.0304 u gap is worth 28.3 MeV.

That energy is the binding energy: released when the nucleus is assembled, and exactly the energy you must supply to pull it completely apart again into separate nucleons. A bound nucleus sits in an energy well, and the depth of the well is written in its missing mass. So the claim above collapses: in any energy change, mass changes with it, and nuclear reactions are simply the ones where the change shows.

The curve that runs the universe

To compare nuclei fairly, divide each binding energy by its nucleon number. The result, binding energy per nucleon, measures how tightly the average nucleon is held, and plotting it against A gives the most consequential graph in nuclear physics.

Average binding energy per nucleon against nucleon number: a steep climb, a peak at iron-56, and a gentle slope down to uranium, with fusion and fission both moving nuclei towards the peakfusionfissioniron-568.8 MeV per nucleon at the tophelium-4moving towards the peak releases energyAbinding energy per nucleon / MeV
FIG. 2Binding energy per nucleon against nucleon number: a steep climb, the iron-56 peak near 8.8 MeV, and a gentle slope down to uranium.

The curve climbs steeply through the light nuclei, with helium-4 spiking above its neighbours, peaks near iron-56 at about 8.8 MeV per nucleon, then declines gently to uranium at 7.6. Any change that moves nucleons towards the peak leaves them more tightly bound, so the difference is released. Light nuclei climb from the left by fusion; heavy nuclei move in from the right by fission. Iron itself has nowhere better to go, which is why it is the ash of the stars. The exam asks you to point at these two release regions on the plot, so practise marking them.

THE EXAM BIT

  • Keep the currency straight: mass differences in u multiply by 931.5 to give MeV. Only convert to kilograms and joules if the question demands SI, and say which route you took.
  • Check whether you were given nuclear or atomic masses. Atomic masses include the electrons; in most AQA questions the electron masses cancel or the nuclear mass is supplied, but read the data line.
  • Define binding energy as the energy needed to separate a nucleus into its individual nucleons, or the energy released on assembly. Energy holding the nucleus together scores nothing.
  • Comparisons between nuclei always use binding energy per nucleon, never the total; a huge nucleus can have a large total yet sit low on the curve.
  • Sketching the curve: steep rise, helium-4 spike, broad peak at A around 56 labelled near 8.8 MeV, gentle fall to uranium. Mark fusion left of the peak and fission right of it, both pointing towards iron.

CHECK YOURSELF

The nuclear mass of iron-56 is 55.92066 u. Using mp = 1.00728 u and mn = 1.00867 u, find the mass difference, the binding energy, and the binding energy per nucleon.

Show a hint

Iron-56 has 26 protons and 30 neutrons; parts minus whole first.

Show the answer

Separate parts: 26 × 1.00728 + 30 × 1.00867 = 56.44938 u, so Δm = 56.44938 − 55.92066 = 0.52872 u.

Binding energy = 0.52872 × 931.5 = 492 MeV.

Per nucleon: 492 / 56 = 8.8 MeV: the very top of the curve, which is why iron is where both fusion and fission run out of road.

A bound nucleus weighs less than its parts: the gap is the binding energy.

Divide by A and iron-56 tops the curve near 8.8 MeV per nucleon.

No animated video for this topic yet; these notes stand alone. InkPhysics on YouTube.