Describe alpha, beta and gamma decay and apply the concept of half-life to radioactive decay

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.

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.