The nucleus and radioactivity (SACE Stage 2 Physics)

Syllabus dot point

“Describe alpha, beta and gamma decay, balance nuclear equations, and apply the concept of half-life.”

9 min readUpdated 2 Jun 2026

Quick answer

Unstable nuclei decay by emitting radiation. Alpha decay emits a helium nucleus ( $^4_2\text{He}$ ), beta-minus decay emits an electron as a neutron becomes a proton, and gamma decay emits a high-energy photon as the nucleus loses excess energy. Nuclear equations must balance both mass number (top) and atomic number (bottom). Half-life is the time for half the nuclei in a sample to decay.

Jump to a section

What this dot point is asking
Nuclear structure
The three types of decay
Balancing nuclear equations
Half-life
Penetration and uses
How SACE assesses this
Exam-style questions

What this dot point is asking

You need to describe the three decay types, write balanced nuclear equations, and apply the concept of half-life.

Using half-life

A sample has an activity of 800 Bq and a half-life of 6.0 hours

Find the activity after 24 hours, and the fraction of the original nuclei remaining.

Number of half-lives: $n = \frac{24}{6.0} = 4\ \text{half-lives}.$
Fraction remaining: $\left(\tfrac{1}{2}\right)^4 = \frac{1}{16}.$
Activity after 24 h: $800 \times \frac{1}{16} = 50\ \text{Bq}.$

So after 24 hours, $1/16$ of the original nuclei remain and the activity has fallen to 50 Bq.

Nuclear structure

A nucleus contains protons and neutrons (nucleons). The atomic number $Z$ is the number of protons, and the mass number $A$ is the total number of nucleons. A nuclide is written $^A_Z X$ . Isotopes of an element have the same $Z$ but different $A$ . A nucleus is unstable (radioactive) when the balance of protons and neutrons, or its total energy, makes it unable to hold together indefinitely.

The three types of decay

Alpha, beta and gamma decay

Alpha ( $\alpha$ ): emission of a helium nucleus $^4_2\text{He}$ . $A$ falls by 4, $Z$ falls by 2. Highly ionising, weakly penetrating (stopped by paper).
Beta-minus ( $\beta^-$ ): a neutron becomes a proton, emitting an electron $^{\,0}_{-1}e$ (and an antineutrino). $A$ unchanged, $Z$ rises by 1. Moderately penetrating (stopped by aluminium).
Gamma ( $\gamma$ ): a high-energy photon emitted as an excited nucleus loses energy. $A$ and $Z$ unchanged. Very penetrating (reduced by thick lead or concrete).

Balancing nuclear equations

In every nuclear equation, the totals of mass number and atomic number must be the same on both sides.

For alpha decay of uranium-238:

^{238}_{\,92}\text{U} \rightarrow\ ^{234}_{\,90}\text{Th} +\ ^4_2\text{He}.

Check:

238 = 234 + 4

and

92 = 90 + 2

. Balanced.

For beta-minus decay of carbon-14:

^{14}_{\,6}\text{C} \rightarrow\ ^{14}_{\,7}\text{N} +\ ^{\,0}_{-1}e.

Check:

14 = 14 + 0

and

6 = 7 + (-1)

. Balanced - the neutron count drops by one and the proton count rises by one.

Half-life

Radioactive decay is random for any single nucleus, but for a large sample it follows a predictable pattern described by the half-life.

Half-life

The half-life $t_{1/2}$ is the time for half the radioactive nuclei in a sample to decay (and equivalently for the activity to halve). After $n$ half-lives, the fraction remaining is $\left(\tfrac{1}{2}\right)^n$ .

The activity (decays per second, in becquerel) halves over each half-life, giving an exponential decrease.

Penetration and uses

The differing penetration of the three radiations underpins their uses and hazards: alpha is dangerous if ingested but stopped by skin; beta is used in thickness gauges; gamma is used in medical imaging and sterilisation. Half-life governs how long a radioactive source remains active and is the basis of radioactive dating (e.g. carbon-14 dating of once-living material).

How SACE assesses this

SACE Stage 2 questions on this dot point split into two styles. The first asks you to complete or write a balanced nuclear equation, where the reliable method is to balance the mass number (top) and atomic number (bottom) separately on each side; remember that $\beta^-$ decay raises $Z$ by one while leaving $A$ unchanged, and $\alpha$ decay lowers $A$ by four and $Z$ by two. The second style is a half-life calculation: find the number of half-lives $n = \dfrac{t}{t_{1/2}}$ , then multiply the activity or number of nuclei by $\left(\tfrac{1}{2}\right)^n$ . The common slip is treating the decay as linear rather than exponential, so the sample never reaches zero; three half-lives leave one-eighth, not nothing. Track whether the question asks for activity (Bq) or number of nuclei, since both halve each half-life.

Common errors

Not balancing both numbers: Both the mass number (top) and atomic number (bottom) must balance on each side of a nuclear equation.
Getting beta's effect on $Z$ wrong: In $\beta^-$ decay the atomic number increases by 1 (a neutron turns into a proton); the mass number stays the same.
Misusing half-life: After each half-life the amount halves; it does not reach zero. Three half-lives leave $1/8$ , not nothing.
Confusing activity and number of nuclei: Both halve each half-life; just make sure to track the quantity the question asks for, in the right units (Bq for activity).

Exam-style questions

Questions in the style of SACE Board exam questions on this dot point, each with a worked answer. They are written by ExamExplained unless tagged "Past paper"; the year shows the paper a question is modelled on.

SACE 2024-style3 marks

A radioactive source has an activity of

800\ \text{Bq}

and a half-life of

6.0\ \text{hours}

. (a) Calculate the activity after

24\ \text{hours}

. (b) State the fraction of the original radioactive nuclei remaining at that time.

Show worked answer →

(a) Number of half-lives: $n = \dfrac{24}{6.0} = 4$ . The activity halves each half-life, so

A = 800\times\left(\tfrac{1}{2}\right)^4 = 800\times\dfrac{1}{16} = 50\ \text{Bq}.

(2 marks)

(b) The fraction of original nuclei remaining is $\left(\tfrac{1}{2}\right)^4 = \dfrac{1}{16}$ . (1 mark) Markers reward computing the number of half-lives and applying the halving correctly; the sample does not reach zero.

SACE 2022-style3 marks

(a) Complete the alpha decay equation:

^{238}_{\,92}\text{U} \rightarrow\ ^{A}_{Z}\text{Th} +\ ^4_2\text{He}

, stating

A

and

Z

. (b) Carbon-14 decays by beta-minus emission to nitrogen. Write the balanced nuclear equation and state what happens to the proton and neutron numbers.

Show worked answer →

(a) Mass number and atomic number must balance. $A = 238 - 4 = 234$ and $Z = 92 - 2 = 90$ , so the product is $^{234}_{\,90}\text{Th}$ . (1 mark)

(b) $^{14}_{\,6}\text{C} \rightarrow\ ^{14}_{\,7}\text{N} +\ ^{\,0}_{-1}e$ . Check: $14 = 14 + 0$ and $6 = 7 + (-1)$ . A neutron becomes a proton, so the proton number rises by one and the neutron number falls by one, with the mass number unchanged. (2 marks) Markers reward the balanced equation and the proton/neutron change.

Practise this

Exam-style questions 2 on this page
Flashcards Physics deck

Sources & how we know this

Stage 2 Physics subject outline (2025 onwards) — SACE Board
Stage 2 Physics: external assessment — SACE Board