Nuclear fission and fusionEdexcel A-Level Physics: Revision notes

The binding energy per nucleon, $E_B/A$, measures how tightly the average nucleon is held. Plotted against nucleon number $A$:

• The curve rises steeply for light nuclei. Helium-4 is unusually stable compared with hydrogen-2.
• It reaches a maximum of about 8.8 MeV at $A \approx 56$ (iron-56), the most stable region.
• It then falls slowly for heavier nuclei, to about 7.6 MeV for uranium-235.

A nuclear reaction releases energy if the products have a higher binding energy per nucleon than the reactants, because the total binding energy increases and the difference is released, mostly as kinetic energy of the products. Moving towards the maximum from either side releases energy.

Fission is the splitting of a heavy nucleus into two medium-mass nuclei (fragments). In a reactor, a slow neutron is absorbed by uranium-235, which becomes unstable and splits, releasing more neutrons:

${}^{1}_{0}\text{n} + {}^{235}_{92}\text{U} \rightarrow {}^{141}_{56}\text{Ba} + {}^{92}_{36}\text{Kr} + 3\,{}^{1}_{0}\text{n}$

The fragments have a higher binding energy per nucleon than uranium-235 (about 8.5 MeV compared with 7.6 MeV), so energy is released: about 170 to 200 MeV per fission.

The nucleon number and the charge are conserved in the equation: 235 + 1 = 141 + 92 + 3, and 92 = 56 + 36.

Fusion is the joining of two light nuclei to form a heavier nucleus. For example:

${}^{2}_{1}\text{H} + {}^{3}_{1}\text{H} \rightarrow {}^{4}_{2}\text{He} + {}^{1}_{0}\text{n}$

The product has a much higher binding energy per nucleon than hydrogen-2 or hydrogen-3 (helium-4 has about 7.1 MeV), so energy is released. Fusion powers the Sun and other stars.

Per nucleon, fusion of light nuclei releases much more energy than fission: the rise up the curve is about 6 MeV per nucleon for hydrogen-2 to helium-4, but under 1 MeV per nucleon for fission of uranium-235.

Nuclei are positively charged, so they repel each other strongly. They must get within about $10^{-15}$ m, where the short-range strong force can bind them. That needs two conditions:

• Very high temperature (of the order of $10^{7}$ to $10^{8}$ K): the nuclei move fast enough to overcome the electrostatic repulsion. At this temperature the fuel is a plasma of nuclei and free electrons.
• Very high density: the nuclei are close together, so collisions are frequent enough for a useful rate of fusion.

In the Sun, gravity supplies both: the core is at about $1.5 \times 10^{7}$ K with a density of about $1.5 \times 10^{5}$ kg m⁻³. Reactors on Earth cannot use gravity, so they need a much higher temperature, about $10^{8}$ K, and the plasma must be kept away from the walls. This is why net energy gain on Earth is difficult.

To find the energy released in a reaction from binding energies per nucleon:

1. Find the total binding energy of each nucleus: $E_B = A \times (E_B/A)$.
2. Add up the total binding energy of the products and of the reactants.
3. Energy released = total binding energy of products − total binding energy of reactants.

Worked example: uranium-235 ($7.59$ MeV per nucleon) splits into barium-141 ($8.33$) and krypton-92 ($8.51$). Reactant: $235 \times 7.59 = 1784$ MeV. Products: $141 \times 8.33 + 92 \times 8.51 = 1957$ MeV. Energy released $= 1957 - 1784 = 174$ MeV.

The free neutrons have no binding energy, so they do not appear in the totals.