Understanding Half-Life Revision Notes | GCSE AQA Higher Physics

1. The Radioactive Mystery

Imagine you have a sample containing a million radioactive atoms. After exactly one hour, half of them have decayed. After another hour, half of what's left decays again. This predictable pattern emerges from something completely unpredictable — we can never tell when any individual atom will decay.

This is the central puzzle of radioactivity. Individual atoms decay randomly, like rolling dice where we never know the next result. Yet when we have millions of atoms, the overall pattern becomes as reliable as clockwork. Understanding this paradox is the key to mastering half-life calculations.

Half-life is one of the most important concepts in nuclear physics. It helps us date ancient artefacts, understand nuclear power, and predict how long radioactive waste remains dangerous. The mathematics might look complex, but there's a simple pattern once you see it.

2. What Is Half-Life?

Think of half-life like a countdown timer, but not the kind you're used to. Instead of counting down to zero, it counts down by halves. If you start with 1000 radioactive nuclei, after one half-life you'll have 500 left. After another half-life, you'll have 250. Then 125, then 62.5, and so on.

The crucial point is that the time for each halving stays exactly the same. Whether you start with a million atoms or just a hundred, one half-life always reduces the number by exactly 50%. This consistency makes half-life incredibly useful for calculations and predictions.

We can measure half-life in two equivalent ways. We can count the actual number of radioactive nuclei remaining, or we can measure the count rate — how many decays happen per second. Both methods give the same half-life value because they're measuring the same underlying process.

3. The Random Nature of Radioactive Decay

Here's where radioactivity gets fascinating. Each individual nucleus has absolutely no 'memory' of how long it's been radioactive. A nucleus that formed a billion years ago has exactly the same chance of decaying in the next second as one that formed yesterday. This is what we mean by random decay.

It's like having a coin that you flip every second. Each flip is independent — previous results don't affect future ones. You can't predict when you'll get heads, but if you flip a million coins, you know roughly half will show heads. Radioactive decay works the same way.

This randomness explains why half-life is statistical, not absolute. We can't say 'this nucleus will decay in exactly 2.3 seconds.' Instead, we say 'in a large sample, half the nuclei will decay within one half-life period.' The larger the sample, the more predictable the overall behaviour becomes.

**Trap**: Many students think radioactive decay is like a timer counting down to zero. Actually, it's more like a lottery where each nucleus has the same chance of 'winning' (decaying) every second, regardless of how long it's been waiting.

4. Calculating Half-Life from Data

Exam questions often give you data and ask you to find the half-life. The key is recognising the halving pattern. Look for the time when the count rate or number of nuclei drops to exactly half its starting value.

Sometimes the data won't show a perfect halving at convenient times. You might need to read values from a graph or interpolate between data points. The half-life is always the time difference between any starting point and when that value halves.

Worked example: Finding Half-Life from Count Rate Data

A radioactive sample has these count rates: t=0: 400 counts/s, t=2 hours: 200 counts/s, t=4 hours: 100 counts/s. Find the half-life.

1. Initial count rate = 400 counts/s — This is our reference point for calculating halving
2. Half of 400 = 200 counts/s This occurs at t = 2 hours — Look for exactly half the initial value
3. Half-life = 2 - 0 = 2 hours — Time taken for first halving
4. 200 → 100 counts/s also takes 2 hours (from t=2h to t=4h) — Confirms our answer is correct

Answer: Half-life = 2 hours

Notice how we verified our answer using the second halving. This is crucial for exam success — always check that subsequent halvings take the same time. If they don't, you've made an error somewhere.

5. Half-Life Calculations and Patterns

Once you know the half-life, you can predict the sample's behaviour at any future time. The key insight is that after n half-lives, you're left with (1/2)ⁿ of the original amount. This gives us a powerful calculation tool.

For example, after 1 half-life you have 1/2 remaining. After 2 half-lives: 1/4 (0.25). After 3 half-lives: 1/8 (0.125). The pattern is clear — each additional half-life divides the remaining amount by 2 again.

Worked example: Predicting Future Activity

A sample has initial activity 800 Bq and half-life 4 days. What will the activity be after 12 days?

1. Number of half-lives = 12 ÷ 4 = 3 — Total time divided by half-life period
2. Fraction remaining = (1/2)³ = 1/8 — After n half-lives, fraction = (1/2)ⁿ
3. Final activity = 800 × (1/8) = 100 Bq — Multiply initial activity by fraction remaining
4. Day 0: 800 Bq → Day 4: 400 Bq → Day 8: 200 Bq → Day 12: 100 Bq — Verification confirms our answer

Answer: Activity after 12 days = 100 Bq

**Trap**: Students often confuse 'after 3 half-lives' with 'at 3 half-lives.' If something has a 2-hour half-life, then 'after 3 half-lives' means at t = 6 hours, not t = 3 hours.

6. Exam Technique and Common Mistakes

Half-life questions appear frequently in GCSE physics papers, often worth 4-6 marks. The examiners have favourite tricks, but once you know them, these questions become straightforward mark-winners.

The most common mistake is misreading graphs. Students often pick the wrong axis or misinterpret the scale. Always check whether you're reading count rate, activity, or number of nuclei — and make sure you're reading the correct units on the time axis.

Another frequent error is arithmetic mistakes with the halving calculations. When working through multiple half-lives, write out each step clearly. Don't try to do 1600 ÷ 8 in your head — show 1600 → 800 → 400 → 200 step by step.

In extended response questions, examiners want you to link half-life to the random nature of decay. Explain that we can't predict when individual nuclei decay, but statistical patterns emerge in large samples. This randomness is why half-life is a probability-based measurement, not an absolute countdown.