Radioactivity and half-life: WACE Year 12 Physics Unit 4

What this dot point is asking

WACE wants you to describe the three decay modes, write balanced nuclear equations, and use half-life to calculate how much of a sample remains. The key statistical idea is that decay is random per nucleus but predictable in bulk.

The three decay modes

Alpha decay emits an alpha particle, a helium-4 nucleus ($^4_2\text{He}$), reducing the mass number by $4$ and the atomic number by $2$. Beta-minus decay emits an electron when a neutron becomes a proton, raising the atomic number by $1$ with the mass number unchanged. Gamma decay emits a high-energy photon as an excited nucleus settles to a lower energy state, changing neither the mass number nor the atomic number. Alpha is the least penetrating and gamma the most.

Balancing nuclear equations

In any nuclear equation the total mass number (top) and the total atomic number (bottom) must be the same on both sides. For example, alpha decay of radium-226:

The mass numbers ($226=222+4$) and atomic numbers ($88=86+2$) both balance, which lets you identify an unknown product.

Randomness and half-life

You cannot predict when a given nucleus will decay; the process is random and spontaneous. Yet across a large sample a fixed fraction decays each second, so the number remaining falls exponentially. The half-life $t_{1/2}$ is the time for half the nuclei (and half the activity) to decay, and it is a fixed property of each isotope, unaffected by temperature, pressure or chemical state.

Calculating with half-lives

After $n$ whole half-lives, the fraction remaining is

Activity (decays per second) falls in exactly the same proportion, since it is proportional to the number of undecayed nuclei. This produces the familiar exponential decay curve, which never quite reaches zero but halves each half-life.

Penetrating power and the decay particles

WACE expects you to rank the three radiations by penetrating and ionising power. Alpha particles are large and highly charged, so they ionise strongly but are stopped by a sheet of paper or a few centimetres of air. Beta particles are lighter and faster, ionise less, and are stopped by a few millimetres of aluminium. Gamma rays are uncharged photons, ionise weakly, and are the most penetrating, needing thick lead or concrete to attenuate significantly. There is an inverse relationship between ionising power and penetration: the most strongly ionising radiation (alpha) is the least penetrating, because it deposits its energy quickly. This determines the hazard each presents depending on whether the source is outside or inside the body.

Activity, decay constant and dating

Activity is the number of decays per second, measured in becquerels, and it is proportional to the number of undecayed nuclei present, so it falls exponentially with exactly the same half-life as the nuclei themselves. This is why a measured activity can be used in place of a count of nuclei in half-life problems. The exponential fall underpins radioactive dating: in carbon-14 dating, a once-living sample stops taking in carbon-14 at death, so the ratio of carbon-14 to stable carbon then halves every $5730$ years. Measuring how much carbon-14 remains (or its current activity) and comparing with the living value tells you how many half-lives have passed and hence the age. The same principle, with longer-lived isotopes such as uranium, dates rocks over billions of years, which is a frequent application question.

Working with whole and partial half-lives

When the elapsed time is a whole number of half-lives, halving repeatedly is quickest. For non-whole numbers, use $N=N_0(\tfrac{1}{2})^{t/t_{1/2}}$ directly. Always check your nuclear equations balance before trusting an identified daughter nucleus.