Nuclear Chemistry: Decay Kinetics, Stability and Applications

Decay kinetics

Radioactive decay is a first-order process, with rate proportional to the number of undecayed nuclei:

The band of stability

Plotting neutron number against proton number, stable nuclei fall in a narrow band. Light stable nuclei have roughly equal numbers of each; heavier ones need progressively more neutrons to dilute proton–proton repulsion.

A nucleus outside the band decays in whichever way moves it toward the band, and that is the logic behind predicting decay mode:

Predicting the mode from the neutron-to-proton ratio is the most common question type, and it is answered by comparing with a stable isotope of the same element.

Binding energy

The mass of a nucleus is slightly less than the sum of its constituent nucleons. That mass defect corresponds to the binding energy through E = mc², and dividing by the number of nucleons gives binding energy per nucleon — the proper measure of stability.

The curve of binding energy per nucleon rises steeply for light nuclei, peaks around iron, and falls slowly for heavy ones. That shape explains both nuclear processes at once:

• Fusion releases energy for light nuclei, because combining them moves toward the peak.
• Fission releases energy for heavy nuclei, because splitting them also moves toward the peak.

Iron sits at the maximum, which is why neither process releases energy there. Being asked why fusion works for light elements and fission for heavy ones is answered entirely by this curve.

Magic numbers

Nuclei with certain numbers of protons or neutrons — 2, 8, 20, 28, 50, 82 and 126 — are unusually stable, analogous to closed electron shells. Nuclei magic in both are exceptionally stable. This shell structure explains why certain isotopes are far more abundant than their neighbours.

Applications

Radiochemical dating

Measuring the ratio of a radioactive isotope to its decay product, and knowing the half-life, gives the elapsed time. Carbon dating works for organic material within a range set by the carbon-14 half-life; longer-lived isotopes date rocks over geological timescales.

The assumption that matters is that the initial ratio is known and that the sample has remained closed. Questions often probe exactly those assumptions rather than the arithmetic.

Tracers

A radioactive isotope behaves chemically like the stable one but can be detected in tiny quantities. This is used to follow reaction mechanisms — labelling one atom and finding where it ends up is how many mechanisms were established.

Frequently asked questions

Why is decay first order?

Because each nucleus decays independently with a fixed probability per unit time. The number decaying is therefore proportional to the number present, which is the definition of first order.

Why does temperature not affect decay rate?

Because decay occurs within the nucleus, and thermal energies are far too small to influence nuclear processes. This is a sharp contrast with chemical reactions.

What is the difference between positron emission and electron capture?

Both convert a proton to a neutron. Positron emission ejects a positron; electron capture absorbs an inner-shell electron. They compete, and electron capture is favoured for heavier elements where inner electrons are closer to the nucleus.

Why does gamma emission not change the element?

Because it releases only energy, with no change in proton or neutron number. It usually follows another decay that leaves the nucleus excited.