Mass defect and binding energy
Binding energy — The energy that would be needed to separate a nucleus completely into its individual nucleons — equivalently, the energy released when it formed.
Weigh a nucleus carefully and it comes out lighter than the protons and neutrons it is made of. Helium-4 is about 0.7% lighter than two protons plus two neutrons weighed separately.
That missing mass is the mass defect, and it is not an error. When the nucleons came together, energy was released, and by E = mc² that released energy came out of the mass. The nucleus is lighter by exactly the amount of energy it gave up.
The same energy is what now holds it together: to pull the nucleus apart you would have to put that energy back. It is therefore called the binding energy.
Because c² is so enormous, a tiny mass difference corresponds to a huge energy — which is why nuclear processes release millions of times more energy per atom than chemical ones. Burning a carbon atom releases a few electronvolts; fissioning a uranium nucleus releases about 200 million.
- Δm
- mass defectkg or u
- Z
- protons
- N
- neutrons
- c
- 3.0 × 10⁸m s⁻¹
Binding energy per nucleon
Total binding energy is not the useful comparison, because a big nucleus has more of everything. Divide by the number of nucleons and you get binding energy per nucleon — a fair measure of how tightly bound each particle is, and therefore how stable the nucleus is.
Plot it against nucleon number and the curve rises steeply from hydrogen, peaks around iron-56 at about 8.8 MeV per nucleon, then falls slowly towards uranium.
Iron sits at the top because it is the most stable nucleus there is. Everything else is somewhere down one side of that peak, and that single fact governs which nuclear reactions release energy.
Move up the curve and energy is released. For light nuclei that means joining them together — fusion. For heavy nuclei it means splitting them apart — fission. Both routes end nearer iron, and both release the difference.
Drag the marker. To the left of iron, joining nuclei climbs the curve — fusion releases energy. To the right, splitting them climbs it — fission releases energy. Iron itself yields nothing either way, which is why stellar fusion stops there.
Why stars die at iron
A star fuses lighter elements and releases energy each time, climbing the curve. Once its core is iron there is nowhere left to go — fusing iron would absorb energy, not release it. The outward pressure fails, and a massive star collapses and explodes as a supernova.
Fission and the chain reaction
In fission, a heavy nucleus such as uranium-235 absorbs a slow neutron, becomes unstable, and splits into two lighter nuclei — plus, crucially, two or three more neutrons.
Those spare neutrons can go on to split further nuclei, which release more neutrons again. That is a chain reaction. Left uncontrolled it grows exponentially, which is a bomb; controlled, it is a power station.
A reactor controls it with three components. Fuel rods hold the uranium. A moderator — usually graphite or water — slows the fast neutrons down, because slow neutrons are far more readily absorbed and are what sustain the reaction. Control rods of boron or cadmium absorb neutrons, and are pushed in or drawn out to hold the reaction exactly steady, at one neutron from each fission going on to cause the next.
The energy appears as kinetic energy of the fragments, which heats a coolant, which raises steam, which drives a turbine. The nuclear part is only the boiler; the rest is an ordinary power station.
The difficulty is the waste. The fragments are themselves radioactive, some with half-lives of thousands of years, and must be stored securely for far longer than any institution has ever lasted.
In a fission reaction the total mass decreases by 0.215 u. Calculate the energy released in MeV and in joules. Take 1 u = 931.5 MeV and 1 eV = 1.6 × 10⁻¹⁹ J.
- Uses
E = Δm × 931.5 MeV.The conversion avoids going through kilograms and joules. E = 0.215 × 931.5.E = 200 MeV.A typical fission yield — a good check that the arithmetic is right.- Converts:
200 × 10⁶ × 1.6 × 10⁻¹⁹. E = 3.2 × 10⁻¹¹ J.Tiny per nucleus, but a kilogram of uranium holds about 10²⁴ of them.
200 MeV, or 3.2 × 10⁻¹¹ J
Fusion
In fusion, light nuclei join to form a heavier one. In the Sun, hydrogen becomes helium, and about four million tonnes of mass become energy every second.
Fusion releases more energy per kilogram than fission, its fuel is effectively unlimited — hydrogen isotopes from seawater — and its products are not long-lived radioactive waste. On every measure it is the better option.
The obstacle is getting the nuclei close enough. Both are positive and repel fiercely, and the strong nuclear force only takes over at extremely short range. Overcoming that repulsion needs temperatures of millions of kelvin, at which matter is a plasma that no container can touch.
The Sun manages it with gravity, which confines and compresses its core. On Earth the same conditions must be produced with magnetic fields or lasers, and holding a plasma stable long enough to get more energy out than went in has taken seventy years and is not finished.
That is why every power station running today uses fission, and why fusion remains the thing that would change everything if it worked.
| Fission | Fusion | |
|---|---|---|
| What happens | a heavy nucleus splits | light nuclei join |
| Fuel | uranium, plutonium | hydrogen isotopes |
| Conditions | slow neutrons, room temperature | millions of kelvin |
| Waste | long-lived radioactive fragments | helium, essentially harmless |
| Energy per kg | large | larger still |
| In use today? | yes, worldwide | not yet |
Key points
- A nucleus is lighter than its separate nucleons; that mass defect is its binding energy.
E = Δmc², and1 u = 931.5 MeV.- Binding energy per nucleon peaks at iron-56 — the most stable nucleus.
- Fusion releases energy below iron; fission releases it above.
- A reactor needs a moderator to slow neutrons and control rods to absorb them.