GCSE · Physics · AQA · Spec 8463

Half-life and the random nature of decay

One unstable nucleus: nobody can say whether it will decay in the next second or thousands of years from now. Billions of them: you can predict how many remain tomorrow.

Pick any point on the curve. How long until it halves?

time (minutes): 5. count-rate (counts per second): 141.42

A made-up sample to practise on. Stop the point anywhere and note the count-rate. Now slide right until the count-rate is half that. How much time passed? Start somewhere else and try again — then once more.

Exam line: To find a half-life from a graph: choose a count-rate, find where it has fallen to half that value, and read the time between the two points. Do it twice from different starts to check.

Watch out: The half-life is the time BETWEEN your two points. It only equals the time you read at the second point if you started at time zero.

Random, yet predictable

Reason it through

If nobody can predict when a nucleus will decay, how can a sample have a fixed half-life?

Link 1 of 3

First link · your turn

Pick one unstable nucleus. Can you say when it will decay?

Locked — reveal the link above first

What do you really think?

A sample with a 3-hour half-life

A sample of a radioactive isotope has a half-life of 3 hours.

Which of these is closest to what you think happens to it?

Half of it decays in the first 3 hours, and the rest has decayed by 6 hours.Each nucleus lasts about 3 hours, then decays.Once half has gone, the rest takes longer than 3 hours to halve, because fewer are left.About a quarter is left after 6 hours, and no one can say which nuclei will decay.

How sure are you?

I'm sureI think soI'm guessing

Half of it decays in the first 3 hours, and the rest has decayed by 6 hours.

This is a common misconception

This is the trap almost everyone falls into first. The second half-life halves what is LEFT, not what you started with. After 3 hours half remains; after 6 hours a quarter; after 9 hours an eighth. It keeps halving — it never simply runs out after two goes.

Each nucleus lasts about 3 hours, then decays.

This is a common misconception

A half-life isn't a lifespan. Decay is random, so some nuclei decay in the first second and others last far longer than 3 hours. The 3 hours describes the whole sample: in that time about half of the nuclei decay — you just can't say which half.

Once half has gone, the rest takes longer than 3 hours to halve, because fewer are left.

This is a common misconception

Half right. With fewer nuclei left, fewer decay each second, so the count-rate drops. But the time to halve doesn't change: on the curve you just used, 800 to 400 took exactly as long as 200 to 100. A smaller sample halves just as quickly.

About a quarter is left after 6 hours, and no one can say which nuclei will decay.

This matches the evidence

Yes. Each half-life halves what is left — a half, then a quarter, then an eighth — and decay is random, so the half-life tells you how many nuclei decay, never which ones.

Each half-life halves what is left, and decay is random: the half-life tells you how many decay, never which ones.

Two ways to measure decay

Activity, count-rate — or both?

For each statement, decide whether it is true of activity, of count-rate, of both, or of neither.

A Activity
B Count-rate

Measured in becquerel (Bq)
The rate at which a source of unstable nuclei decays
The number of decays recorded each second by a detector
Falls to half its starting value in one half-life
Gets smaller as fewer undecayed nuclei remain
Stays the same as the sample gets older

Higher

Net decline, written as a ratio

Problem

A source has an activity of 640 Bq. Calculate the net decline in its activity after 3 half-lives, and express it as a ratio of the starting activity.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Every half-life halves what is left — from any starting point.

What you need to know

Radioactive decay is random: you can't predict which nucleus will decay next, or when.
Half-life is the time for the number of nuclei of the isotope in a sample to halve — or for the count-rate (or activity) from the sample to fall to half its initial level.
Each half-life halves what is left: after 1, 2 and 3 half-lives, a half, a quarter and an eighth remain.
Activity is the rate at which a source of unstable nuclei decays, measured in becquerel (Bq). Count-rate is the number of decays recorded each second by a detector.
You can determine a half-life from a graph or from data by finding how long it takes for a value to halve.
Net decline (Higher only): after a given number of half-lives, start minus what is left, written as a ratio of the start.

The big picture

Radioactive decay is random: you can't predict which nucleus will decay next, or when. But in a large sample the same fraction of nuclei decays in each equal stretch of time, so the number of undecayed nuclei — and the count-rate and activity — halve in a fixed time called the half-life. Each half-life halves what is left, so a sample never simply runs out after two half-lives.

Key points

1One nucleus is unpredictable; in a huge sample, the same fraction decays in each equal time.

2That is why the sample halves in a fixed time — the half-life — whatever amount you start with.

3From a graph: pick any value, find where it has halved, read the time between the two points.

4From data: count the halvings, then divide the time taken by that number.

5Higher only: the net decline is start minus what is left, as a ratio of the start — 7 : 8 after three half-lives.

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Worked example

Problem

A detector records a count-rate of 960 counts per second from a sample. 12 hours later it records 120 counts per second. Find the half-life of the isotope.

⚠ Watch out

Treating decay as a steady countdown. After two half-lives a quarter of the nuclei are still undecayed, not none — each half-life halves what is left.

🧠

Memory hook

Half, then half of that, then half of THAT: 1 → ½ → ¼ → ⅛. Random for one, reliable for a crowd.

✓

Check yourself

A count-rate falls from 400 to 50 counts per second in 30 minutes. What's the half-life, and what fraction is left? (Three halvings in 30 minutes: 10 minutes. One eighth is left.)

Flashcards

(13)

What does it mean to say radioactive decay is random?

You can't predict which nucleus will decay next, or when any particular nucleus will decay.

Half-life: give both ways of defining it.

The time for the number of nuclei of the isotope in a sample to halve, or the time for the count-rate (or activity) from the sample to fall to half its initial level.

How can a random process give a fixed half-life?

In a large sample every nucleus has the same chance of decaying in a given time, so the same fraction decays in each equal time — and the number left halves in a fixed time.

A sample is left for two half-lives. Has it all decayed?

No — a quarter is still undecayed. Each half-life halves what remains.

What fraction of a sample is left after 3 half-lives?

One eighth (½ × ½ × ½).

What is activity?

The rate at which a source of unstable nuclei decays.

What unit is activity measured in?

The becquerel (Bq).

What is count-rate?

The number of decays recorded each second by a detector, such as a Geiger–Muller tube.

How do you read a half-life from a decay curve?

Pick any count-rate, find where it has fallen to half that value, and read the time between the two points.

Does it take longer to halve a small sample than a big one of the same isotope?

No. Fewer nuclei decay each second in the smaller sample, but the time to halve is the same.

Can the half-life tell you when one particular nucleus will decay?

No. It describes the whole sample; which nuclei decay is random.

What is the net decline after a number of half-lives?

How much the value has fallen: start minus what is left, usually written as a ratio of the start.

Why does a decay curve get flatter as time goes on?

Fewer undecayed nuclei are left, so fewer decay each second — but each halving still takes one half-life.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Half-life and the random nature of decay

Pick any point on the curve. How long until it halves?

A sample with a 3-hour half-life

Activity, count-rate — or both?

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