Radioactivity, Nuclear Equations and Half-Life
How the Atomic Model Has Developed
Our understanding of atomic structure has been revised several times as new experimental evidence emerged. The earliest scientific model, proposed by Dalton in the early 1800s, described atoms as tiny, indivisible solid spheres — a solid sphere model that could not explain electrical phenomena.
In 1897, J.J. Thomson discovered the electron and proposed the plum pudding model. The atom was described as a ball of positive charge with negative electrons scattered throughout it. This model was overturned by Ernest Rutherford’s gold foil experiment.
In 1909, Rutherford, Geiger and Marsden fired alpha particles at a thin gold foil. Most passed straight through (atom is mostly empty space); some deflected at large angles; a very few bounced straight back (dense positive nucleus exists). This led to the nuclear model of the atom.
Niels Bohr (1913) refined the nuclear model by proposing that electrons orbit the nucleus only in specific, fixed energy levels or shells. This explained why atoms emit and absorb specific frequencies of electromagnetic radiation. James Chadwick completed the picture in 1932 by discovering the neutron, confirming that nuclei contain both protons and neutrons.
The structure of an atom is described using two numbers in the standard notation \(^A_Z\text{X}\): the mass (nucleon) number \(A\) gives the total number of protons plus neutrons; the atomic (proton) number \(Z\) gives the number of protons. Electrons orbit the nucleus at set distances, and atoms are electrically neutral because the number of electrons equals the number of protons.
Radioactive Decay
Some nuclei are unstable and emit radiation to become more stable. This is radioactive decay. It is a random process — it is impossible to predict when a specific nucleus will decay, but for a large sample of nuclei the statistical behaviour is predictable.
Radioactive decay can be detected using a Geiger-Müller tube (GM tube) or photographic film. Any measurement of radioactivity must subtract background radiation — the low-level ionising radiation always present from natural and artificial sources.
Four types of ionising radiation are emitted from unstable nuclei: alpha (\(\alpha\)), beta-minus (\(\beta^-\)), beta-plus (\(\beta^+\)) and gamma (\(\gamma\)). Also, neutron emission is possible. All four are ionising. Alpha is most ionising but least penetrating; gamma is least ionising but most penetrating.
Alpha, Beta and Gamma Radiation
Alpha (\(\alpha\)) particle: two protons and two neutrons — equivalent to a helium-4 nucleus (\(^4_2\text{He}\)). Alpha particles are strongly ionising but have very low penetrating power: they are stopped by a few centimetres of air or a sheet of paper. In alpha decay, the parent nucleus loses 2 protons and 2 neutrons: mass number decreases by 4, proton number decreases by 2.
Beta-minus (\(\beta^-\)) decay: a neutron transforms into a proton, emitting a high-speed electron. The proton number increases by 1; the mass number stays the same. Stopped by a few millimetres of aluminium. Moderately ionising.
Beta-plus (\(\beta^+\)) / positron emission: a proton transforms into a neutron, emitting a positron (the antimatter equivalent of an electron). The proton number decreases by 1; the mass number stays the same. Used in PET scanning (positron emission tomography).
Gamma (\(\gamma\)) radiation: high-frequency electromagnetic radiation emitted from the nucleus after alpha or beta decay, as the nucleus loses excess energy. No change to proton or mass number. Only significantly reduced by several centimetres of lead or thick concrete. Weakly ionising but highly penetrating.
Balanced Nuclear Equations
In a nuclear equation, both the total mass number and the total proton number must be conserved (balanced) on both sides of the arrow. The general rule: \(\sum A\) and \(\sum Z\) are each equal before and after the decay.
Alpha decay of radium-226:
$$^{226}_{88}\text{Ra} \rightarrow ^{222}_{86}\text{Rn} + ^{4}_{2}\text{He}$$
Check: top (226 = 222 + 4 ✓); bottom (88 = 86 + 2 ✓).
Beta-minus decay of carbon-14:
$$^{14}_{6}\text{C} \rightarrow ^{14}_{7}\text{N} + ^{0}_{-1}\text{e}$$
Check: top (14 = 14 + 0 ✓); bottom (6 = 7 + (−1) ✓).
Beta-plus decay: proton number decreases by 1, mass number unchanged, positron emitted (\(^0_{+1}\text{e}\)).
In any nuclear equation: mass numbers (top) must balance on both sides; proton numbers (bottom) must balance on both sides. Alpha: −4 mass, −2 proton. Beta-minus: 0 mass, +1 proton. Beta-plus: 0 mass, −1 proton. Gamma: no change to either number.
Half-Life
The half-life of a radioactive isotope is the time taken for half the undecayed nuclei to decay, or for the activity to fall to half its original value. Because decay is random, it cannot be predicted when a specific nucleus will decay — but half-life allows us to predict the statistical behaviour of a very large number of nuclei.
After one half-life: \(\frac{1}{2}\) of the original nuclei remain. After two half-lives: \(\frac{1}{4}\) remain. After three half-lives: \(\frac{1}{8}\) remain. The activity decreases by the same ratio each half-life.
A radioactive sample has an activity of 1,600 Bq. After 40 minutes the activity is 200 Bq. Calculate the half-life.
Number of halvings: \(1600 \rightarrow 800 \rightarrow 400 \rightarrow 200\) = three half-lives.
Three half-lives = 40 minutes, so one half-life = \(40 \div 3 \approx 13.3\) minutes.
Common error: do not confuse ‘fraction remaining’ with ‘fraction decayed’. After 3 half-lives, \(\frac{1}{8}\) remains and \(\frac{7}{8}\) has decayed.
Background Radiation
Background radiation comes from natural sources (radon gas from the ground, cosmic rays, food and drink containing trace radioactive isotopes, building materials) and artificial sources (medical procedures such as X-rays, emissions from the nuclear industry). It must always be subtracted from measured count rates to obtain the count rate from a source alone.