Radioactive Half-Life
You cannot predict when a single atom will decay — but with billions of atoms, the pattern is astonishingly regular. That regularity is a clock that can measure tens of thousands of years.
Q1 · If you start with 100 radioactive atoms and half of them decay every minute, how many are left after 1 minute? After 2 minutes? After 3 minutes?
Q2 · Will a radioactive sample ever reach exactly zero atoms by halving repeatedly? Why or why not?
● Know
- That half-life is the time for half the atoms in a sample to decay
- That each radioisotope has its own fixed half-life
- That a decay curve falls steeply at first and then flattens
● Understand
- Why radioactive decay is exponential, halving each half-life rather than falling by a fixed amount
- That you cannot predict when a single atom decays, only the behaviour of a large sample
- How a known half-life lets scientists date objects (radiometric dating)
● Can do
- Calculate how much of a sample remains after a whole number of half-lives
- Find the number of half-lives that have passed from "before and after" amounts
- Read and sketch a decay curve
The half-life of a radioisotope is the time taken for half of the radioactive atoms in a sample to decay. It is a fixed property of each isotope — carbon-14's half-life is about 5,730 years, while some isotopes have half-lives of seconds and others of billions of years.
The key feature is that the sample halves each half-life, no matter how much you start with:
- After 1 half-life: $\tfrac{1}{2}$ of the original remains.
- After 2 half-lives: $\tfrac{1}{2} \times \tfrac{1}{2} = \tfrac{1}{4}$ remains.
- After 3 half-lives: $\tfrac{1}{8}$ remains, and so on.
This is exponential decay — each step multiplies by $\tfrac{1}{2}$, rather than subtracting a fixed amount. Because you keep taking half, the amount gets very small but, in theory, never reaches exactly zero.
Decay is also random: you cannot predict when one particular nucleus will decay. But across billions of atoms, the half-life is remarkably reliable — which is why it makes such a good clock.
A sample contains 80 g of a radioisotope with a half-life of 10 years. How much is left after 30 years? 30 years is 3 half-lives. Halve three times: $80 \to 40 \to 20 \to 10$. So 10 g remains after 30 years.
Dating Australia's deep history: Carbon-14's 5,730-year half-life lets archaeologists date charcoal and bone up to about 50,000 years old. This technique helped establish that Aboriginal and Torres Strait Islander Peoples have lived in Australia for at least 65,000 years (using carbon-14 alongside other dating methods), making theirs one of the oldest continuous cultures on Earth.
Half-life does not mean the sample is all gone after two half-lives. After 2 half-lives, $\tfrac{1}{4}$ still remains, not zero. Each half-life removes half of what is left, not half of the original each time.
Half-life problems come in three common shapes. The trick is always to count how many half-lives have passed.
Type 1 — How much is left? Find the number of half-lives ($n$ = total time ÷ half-life), then halve the starting amount $n$ times. The fraction left is $\left(\tfrac{1}{2}\right)^{n}$.
Type 2 — How long has passed? Work out how many times the amount has halved, then multiply that number by the half-life. For example, going from 100% to 12.5% is 3 halvings ($100 \to 50 \to 25 \to 12.5$), so 3 half-lives have passed.
Type 3 — Finding the half-life. If you know the time and the number of halvings, then half-life = total time ÷ number of half-lives.
Always sanity-check: after each half-life the amount should be exactly half the previous value, and the percentages should follow the sequence 100, 50, 25, 12.5, 6.25 …
Finding time: Iodine-131 has a half-life of 8 days. A hospital sample drops from 100% to 25% of its original activity. That is 2 halvings ($100 \to 50 \to 25$), so the time is $2 \times 8 = 16$ days.
Finding half-life: A sample falls to one-eighth ($\tfrac{1}{8}$) of its starting amount in 24 hours. One-eighth is 3 halvings, so the half-life is $24 \div 3 = 8$ hours.
Why technetium-99m's 6-hour half-life is perfect for scans: The medical tracer technetium-99m, produced from Australian-made molybdenum, has a half-life of about 6 hours. That is long enough to complete a scan but short enough that it decays away quickly, minimising the radiation dose to the patient. Half-life calculations let doctors time the dose precisely.
Do not subtract a fixed amount each half-life. A sample of 100 g does not go 100 → 75 → 50 → 25 (subtracting 25 each time). It halves: 100 → 50 → 25 → 12.5. Exponential decay multiplies; it does not subtract.
Quick-fire true or false on half-life.
Half-life is the time for half the atoms in a sample to decay.
After two half-lives, none of the original sample remains.
After three half-lives, one-eighth of the sample remains.
Each radioisotope has its own fixed half-life.
A decay curve is a straight line.
You cannot predict when a single nucleus will decay.
Carbon-14 dating relies on a known half-life.
Radioactive decay subtracts a fixed mass each half-life.
A decay curve plots the amount of radioactive material (or its activity) against time. It always has the same shape: it falls steeply at first, then more and more gently, flattening out as it approaches (but never quite reaches) zero. You can read a half-life straight off the curve — find the time it takes for the amount to drop from any value to half that value.
This is the basis of radiometric dating. Living things constantly take in carbon-14 from the environment, so while alive they have a fixed proportion of it. When they die, they stop taking it in and the carbon-14 they already contain decays with its 5,730-year half-life. By measuring how much carbon-14 is left, scientists count how many half-lives have passed and so estimate the age.
For very old rocks (millions to billions of years), carbon-14 has decayed away long ago, so geologists use isotopes with much longer half-lives, such as uranium-238 (about 4.5 billion years) decaying to lead. Choosing an isotope with a suitable half-life is itself a key Working Scientifically decision.
A piece of ancient charcoal has only $\tfrac{1}{4}$ of the carbon-14 a living tree would have. One-quarter is 2 halvings, so 2 half-lives have passed: $2 \times 5,730 = 11,460$ years. The charcoal is about 11,500 years old.
Dating Australia's rocks: Some of the oldest material ever found on Earth comes from the Jack Hills in Western Australia — tiny zircon crystals dated to about 4.4 billion years old using uranium–lead radiometric dating. These crystals are direct evidence of conditions on the very early Earth, and they were dated using exactly the half-life ideas in this lesson.
You cannot use carbon-14 to date a dinosaur fossil or a rock millions of years old. After about 50,000 years (roughly 9 half-lives) so little carbon-14 is left that it can't be measured reliably. For very old samples, an isotope with a much longer half-life must be chosen.
Connect the key ideas about half-life. Click two connected ideas to explain the link.
Wrong: "After two half-lives the sample is completely gone." No — after 2 half-lives one-quarter remains. Each half-life removes half of what is left, so the amount approaches zero but never reaches it.
Right: After each half-life, half of the remaining sample decays: 100% → 50% → 25% → 12.5% → … The amount keeps halving and only approaches zero.
Wrong: "Heating a sample or adding a catalyst will speed up its radioactive decay." No — half-life is a fixed property of the nucleus and is not changed by temperature, pressure or chemical reactions.
Right: A radioisotope's half-life is constant and cannot be sped up or slowed down by ordinary conditions, unlike the rate of a chemical reaction.
Wrong: "You can predict exactly when a particular atom will decay." No — individual decay is random. Only the average behaviour of a very large number of atoms is predictable, which is what gives a reliable half-life.
Right: Decay of any single nucleus is random and unpredictable, but the half-life describes the reliable average behaviour of the whole sample.
A Clock for the Oldest Continent
Half-life turns radioactivity into a clock, and Australia has used it to reveal some of the deepest history on Earth. Carbon dating of charcoal and shell middens has documented at least 65,000 years of Aboriginal and Torres Strait Islander presence. Uranium–lead dating of zircon crystals from the Jack Hills in Western Australia has revealed material 4.4 billion years old — close to the age of the Earth itself.
The same idea keeps Australians safe and healthy. Short half-life isotopes made at Lucas Heights are timed so they do their diagnostic job and then decay away quickly. Whether dating the past or treating patients, the constant, reliable half-life is the key — exactly what this lesson is about.
✍ Copy Into Your Books
▾Half-life basics
- Half-life = time for half the atoms to decay
- Fixed for each isotope; not changed by heat or chemistry
- Decay is exponential (×½ each half-life)
- Fraction left = (½)ⁿ where n = number of half-lives
The pattern
- 0 half-lives: 100%
- 1: 50% 2: 25% 3: 12.5% 4: 6.25%
- Decay curve: steep then flattens, never quite 0
Dating
- n half-lives passed → age = n × half-life
- Carbon-14 (half-life ~5,730 yr) dates once-living things
- Uranium-238 (~4.5 billion yr) dates ancient rocks
Half-Life Calculations
Curves and Dating
At the start, the hook asked how a tiny amount of carbon-14 can reveal the age of something to within a few hundred years.
Now explain, using the words half-life and halving, how scientists turn the amount of carbon-14 left in a sample into an age. Then revisit your Q1 answer — were your numbers (50, 25, 12.5) correct?
Q1. A 160 g sample of a radioisotope has a half-life of 8 days. Calculate the mass remaining after 24 days, showing your halving steps. (3 marks)
Q2. Explain why radioactive decay produces a curve that gets less steep over time rather than a straight-line decrease. Use the idea that the sample halves each half-life. (4 marks)
Q3. A scientist wants to date a wooden tool thought to be about 10,000 years old. Carbon-14 has a half-life of 5,730 years. Evaluate whether carbon-14 is a suitable choice for this task and justify your answer. (3 marks)
Revisit Your Thinking
Go back to your Think First answers. Has your understanding changed?
- Were your numbers for 1, 2 and 3 minutes (50, 25, 12.5) correct?
- Can you now explain why a sample never quite reaches zero?
Model answers (click to reveal)
Answers
▾MCQ 1
B — Half-life is the time taken for half of the radioactive atoms in a sample to decay.
MCQ 2
C — Three half-lives means halving three times: 800 → 400 → 200 → 100.
MCQ 3
A — 100% to 25% is 2 halvings (100 → 50 → 25), so 2 half-lives. With a 6-hour half-life that is 2 × 6 = 12 hours.
MCQ 4
D — Individual decay is random and unpredictable, but a large sample shows a reliable, fixed half-life. Half-life is unaffected by temperature or chemistry, and the decay curve is not straight.
MCQ 5
B — After billions of years essentially no carbon-14 would be left to measure, because its half-life (~5,730 years) is far too short. An isotope with a much longer half-life, such as uranium-238, is used for ancient rocks.
Short Answer 1
Model answer: 24 days ÷ 8 days = 3 half-lives. Halve the mass three times: 160 → 80 → 40 → 20. So 20 g remains after 24 days.
Short Answer 2
Model answer: Each half-life, half of the atoms still present decay. Because there are fewer and fewer atoms left as time goes on, the actual number decaying in each equal time interval gets smaller and smaller. Early on, lots of atoms decay quickly (the curve is steep); later, only a few decay (the curve is shallow). This halving of what remains produces a curve that falls steeply at first and then flattens, approaching but never reaching zero — quite different from a straight line, which would mean a fixed amount decaying each interval.
Short Answer 3
Model answer: Carbon-14 is a suitable choice. 10,000 years is a little under 2 half-lives of carbon-14 (since 2 × 5,730 = 11,460 years), so a measurable amount of carbon-14 would still remain — roughly a quarter to a third of the original. Because the object is once-living wood and is well within the ~50,000-year range of carbon-14 dating, the isotope's half-life is well matched to the age being measured, making it a valid and reliable choice.