Know that isotopes of an element may be radioactive due to an excess of neutrons in the nucleus and/or the nucleus being too heavy

Published by Patrick Mutisya · 14 days ago

IGCSE Physics 0625 – 5.2.3 Radioactive Decay

5.2.3 Radioactive Decay

Learning Objective

Understand that isotopes of an element may be radioactive because they contain an excess of neutrons in the nucleus and/or because the nucleus is too heavy.

Key Concepts

  • Isotopes – atoms of the same element (same number of protons, \$Z\$) but different numbers of neutrons (\$N\$).

  • Stability of a nucleus depends on the ratio \$N/Z\$ and on the total number of nucleons (\$A = Z+N\$).

  • When the nucleus is unstable it can undergo radioactive decay to reach a more stable configuration.

Why Some Isotopes Are Radioactive

Two main reasons make a nucleus unstable:

  1. Excess neutrons: For light nuclei (low \$Z\$) a roughly 1:1 neutron‑to‑proton ratio is stable. If \$N\$ is too large, the strong nuclear force cannot hold the nucleus together, leading to beta decay.

  2. Too heavy a nucleus: For heavy nuclei (high \$Z\$) the electrostatic repulsion between protons becomes very large. Even with a higher \$N/Z\$ ratio, the nucleus may still be too massive, favouring alpha decay.

Common Types of Radioactive Decay

Decay TypeParticle EmittedChange in NucleusTypical Occurrence
Alpha (\$\alpha\$) decayHelium nucleus \$^{4}_{2}\mathrm{He}\$\$^{A}{Z}\mathrm{X} \;\rightarrow\; ^{A-4}{Z-2}\mathrm{Y} + ^{4}_{2}\mathrm{He}\$Heavy nuclei (e.g., \$^{238}_{92}\mathrm{U}\$)
Beta minus (\$\beta^{-}\$) decayElectron \$e^{-}\$ (and antineutrino \$\bar{\nu}\$)\$^{A}{Z}\mathrm{X} \;\rightarrow\; ^{A}{Z+1}\mathrm{Y} + e^{-} + \bar{\nu}\$Neutron‑rich nuclei (e.g., \$^{14}_{6}\mathrm{C}\$)
Beta plus (\$\beta^{+}\$) decayPositron \$e^{+}\$ (and neutrino \$\nu\$)\$^{A}{Z}\mathrm{X} \;\rightarrow\; ^{A}{Z-1}\mathrm{Y} + e^{+} + \nu\$Proton‑rich nuclei (rare in nature)
Gamma (\$\gamma\$) emissionHigh‑energy photonExcited nucleus \$^{*A}{Z}\mathrm{X} \;\rightarrow\; ^{A}{Z}\mathrm{X} + \gamma\$Often follows \$\alpha\$ or \$\beta\$ decay

Neutron‑to‑Proton Ratio and Stability

The stable \$N/Z\$ ratio increases with atomic number. Approximate ranges are:

  • Light elements (\$Z \le 20\$): \$N/Z \approx 1.0\$

  • Medium elements (\$20 < Z \le 40\$): \$N/Z \approx 1.2\$–\$1.3\$

  • Heavy elements (\$Z > 40\$): \$N/Z \approx 1.5\$–\$1.6\$

If an isotope’s actual \$N/Z\$ lies outside the stable range, it will tend to undergo a decay process that moves the ratio toward the stable region.

Examples of Radioactive Isotopes

IsotopeAtomic Number (\$Z\$)Neutron Number (\$N\$)Decay ModeReason for Instability
\$^{14}_{6}\mathrm{C}\$68\$\beta^{-}\$Excess neutrons (ratio \$N/Z = 1.33\$ > stable for \$Z=6\$)
\$^{238}_{92}\mathrm{U}\$92146\$\alpha\$Very heavy nucleus; strong proton repulsion
\$^{90}_{38}\mathrm{Sr}\$3852\$\beta^{-}\$Neutron‑rich relative to stable Sr isotopes
\$^{210}_{82}\mathrm{Pb}\$82128\$\alpha\$Heavy nucleus, close to the limit of stability

Half‑Life Concept

The half‑life (\$t_{1/2}\$) is the time required for half of a sample of radioactive nuclei to decay. It is related to the decay constant (\$\lambda\$) by:

\$ t_{1/2} = \frac{\ln 2}{\lambda} \$

Half‑lives can range from fractions of a second to billions of years, reflecting the wide range of nuclear stability.

Summary Checklist

  • Isotopes have the same \$Z\$ but different \$N\$.

  • Excess neutrons → \$\beta^{-}\$ decay (neutron → proton).

  • Very heavy nuclei → \$\alpha\$ decay (helium nucleus emitted).

  • After \$\alpha\$ or \$\beta\$ decay, the daughter nucleus may be left in an excited state and emit \$\gamma\$ radiation.

  • Stability is linked to the \$N/Z\$ ratio and overall nuclear mass.

Suggested diagram: A chart showing the valley of stability on a plot of neutron number (N) versus proton number (Z), with arrows indicating typical \$\alpha\$ and \$\beta^{-}\$ decay paths toward the valley.