15.7 Fission, Fusion, and Nuclear Decay

Protons in a nucleus repel each other strongly, yet nuclei hold together. The strong force, an attraction between nucleons (protons and neutrons), is much stronger than the electric force at nuclear distances but has a very short range, only a few nucleon widths (a few times 10⁻¹⁵ m). That's why it dominates inside the nucleus and is negligible outside it.

• Nucleon number: the total of protons plus neutrons (the top numbers, A) is the same before and after.
• Charge: the total charge (the bottom numbers, Z, for nuclei) is the same before and after.
• Energy, counting mass as a form of energy.
• Momentum: the products' momenta add up to the starting momentum, so a nucleus at rest that splits sends its pieces off in opposite directions.

Einstein's E = mc² says mass and energy are interchangeable. In a nuclear reaction, the total mass of the products is often slightly less than the total mass of the reactants. That missing mass Δm has turned into energy, $E = (\Delta m)c^2$, which appears as kinetic energy of the products or as gamma-ray photons.

Nuclear masses are given in unified atomic mass units (u). The equation sheet gives 1 u = 1.66 × 10⁻²⁷ kg = 931 MeV/c², so each 1 u of missing mass releases 931 MeV.

Because c² is enormous, a tiny mass change releases a huge energy. Nuclear reactions release millions of times more energy per atom than chemical reactions.

Fusion joins two or more light nuclei into a heavier one, often with a neutron or other particle given off. The Sun runs on fusion of hydrogen into helium. It needs huge temperatures so the positive nuclei move fast enough to get close despite their repulsion.

Fission splits a heavy nucleus into two or more lighter nuclei plus a few neutrons. Some heavy nuclei split spontaneously; others need an energy input, such as absorbing a neutron, as uranium-235 does in a nuclear reactor. The freed neutrons can trigger more fissions.

Both release energy for the same reason: the products are more tightly bound than the reactants, so they have less total mass.

In radioactive decay, an unstable nucleus changes on its own. Either it turns into a different nucleus (or several), or it settles from an excited state into a lower-energy state of the same nucleus, as in gamma decay (15.8). College Board spelled out that second case in its fall 2026 update to the course description, so if you're asked whether gamma decay counts as radioactive decay, the answer is yes. You can't predict when a particular nucleus will decay, any more than you can predict one coin flip. But a sample of billions of nuclei decays very predictably.

The half-life $t_{1/2}$ is the time for half of the radioactive nuclei in a sample to decay. After one half-life ½ remain, after two ¼, after three ⅛, and so on. Some isotopes have half-lives far shorter than a second and others have half-lives of billions of years. You won't need to memorize any.

The decay constant λ (a different λ from wavelength) is related by $\lambda = \frac{\ln 2}{t_{1/2}}$. The number left after time t is $N = N_0 e^{-\lambda t}$, which is the same as $N = N_0\left(\frac{1}{2}\right)^{t/t_{1/2}}$.

A graph of N against t is a decreasing exponential curve. If you know how much was there at the start, you can use the fraction remaining to find a sample's age; that's how radiocarbon dating works.