Half-lives and the random nature of radioactive decay

SPEC 8463 • MODULE 4 • PAPER 1

Half-Lives & Random Decay

Individual nuclei decay at random, but large samples follow a predictable halving pattern. Learn to translate fluently between elapsed time, number of half-lives, and fraction remaining — the single highest-frequency exam target in this module.

Why This Lesson Matters

Half-life is the most-tested concept in Module 4. Questions are usually multi-step and reward careful tracking of fractions remaining → number of half-lives → elapsed time. Higher tier adds the net decline ratio.

⭐

High Priority

4.4.2.3

Spec Reference

HT+

Net decline (HT)

By the end of this lesson you can…

Define half-life using spec-aligned wording

Explain how half-life relates to the random nature of decay

Calculate activity (or undecayed nuclei) after n half-lives

Read a half-life off an activity-vs-time graph

Translate between fraction remaining and number of half-lives

HT Calculate net decline as a ratio after n half-lives

What Is Half-Life?

Half-life

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

Two equivalent definitions

Population view: time for half the undecayed nuclei in the sample to decay

Measurement view: time for the measured activity / count rate to drop by half

Both definitions describe the same underlying process — and the mark scheme credits either.

Spec-rejected phrasing

"Time for radioactivity to halve" is not credited. The mark scheme requires you to specify what's halving — either "number of undecayed nuclei", "activity", or "count rate".

Random — But Predictable in Bulk

Radioactive decay is fundamentally random. So how can half-life be a fixed value? The answer is in the scale.

Individual nuclei

You cannot predict when any single nucleus will decay, or which nuclei will decay next. Each decay event is a random quantum process.

Large samples

For a sample of millions or billions of nuclei, the fraction that decay in a given time is highly predictable. Large numbers smooth out the randomness.

"Random nature" — 1-mark answer

It is impossible to predict when an individual nucleus will decay, only the average behaviour of a large sample.

Half-Lives Span a Vast Range

Half-life is a fixed property of each isotope — but the values range from microseconds to billions of years.

Polonium-214

≈ 0.0002 seconds

Sub-millisecond — produced in uranium decay chains

Radon-222

≈ 3.8 days

Naturally occurring gas — main source of background radiation in homes

Carbon-14

≈ 5,730 years

Used for radiocarbon dating organic material

Uranium-238

≈ 4.5 × 10⁹ years

Used for geological dating; older than the Earth

Range = over 20 orders of magnitude

From 0.0002 s to 4,500,000,000 years — same underlying physics, vastly different timescales. Half-life is a property of the isotope, not the sample size. Doubling your sample doesn't change the half-life.

After n Half-Lives — Fraction Remaining

This is the most-used pattern in half-life problems. Memorise it.

0 half-lives1100%

1 half-life½50%

2 half-lives¼25%

3 half-lives⅛12.5%

4 half-lives¹⁄₁₆6.25%

The formula

After n half-lives, the fraction remaining = (½)ⁿ = 1 / 2ⁿ.

Conversely, if you're told the fraction remaining, count how many halvings you need to reach it — that's the number of half-lives elapsed.

The Decay Curve

Plotting activity (or number of undecayed nuclei) against time gives a smooth exponential decay curve — halving each half-life forever.

Key features

The curve halves each half-life, never reaching zero. After many half-lives the activity is tiny but technically non-zero. The same shape applies whether the y-axis is activity, count rate, or number of undecayed nuclei.

Worked Examples — Same Math

The spec gives two equivalent definitions of half-life — one in terms of activity, one in terms of the number of undecayed nuclei. The same calculation works for either quantity.

Example A — Activity

A radioactive sample has an initial activity of 800 Bq. Its half-life is 4.0 hours. Calculate the activity after 12 hours. 3 marks

1

Number of half-lives elapsed: n = 12 ÷ 4.0 = 3

2

Halve three times: 800 → 400 → 200 → 100 Bq

✓

Or using the formula: activity = 800 / 2³ = 100 Bq

Example B — Number of undecayed nuclei

A sample initially contains 6.4 × 10⁸ undecayed nuclei of an isotope with half-life 10 minutes. How many nuclei remain undecayed after 40 minutes? 3 marks

1

Number of half-lives elapsed: n = 40 ÷ 10 = 4

2

Apply the formula: nuclei remaining = 6.4 × 10⁸ / 2⁴ = 6.4 × 10⁸ / 16

✓

4.0 × 10⁷ undecayed nuclei

Identical method to Example A — just a different quantity on the y-axis.

Reading Half-Life From a Graph

Given a decay curve, finding the half-life is a three-step method.

The 3-step method

Halve the starting activity. If A₀ = 800 Bq, half is 400 Bq.

Draw across from that value on the y-axis to the curve, then drop down to the time axis.

Read off the time. That's the half-life.

Inverse Problem — Time From Fraction

A radioactive isotope has a half-life of 30 minutes. The number of undecayed nuclei in a sample has fallen to one eighth of its starting value.

How much time has passed? 2 marks

1

Translate the fraction into a number of half-lives: ⅛ = (½)³ → 3 half-lives have elapsed

2

Multiply by the half-life: Time = 3 × 30 minutes = 90 minutes (1 h 30 min)

✓

Answer: 90 minutes

Remember the fraction → half-life pattern: ½ = 1, ¼ = 2, ⅛ = 3, ¹⁄₁₆ = 4. Translate first, then multiply.

Net Decline as a Ratio HT only

The Higher Tier extension. The net decline is the ratio of the initial activity to the final activity — expressing how much the activity has dropped overall.

Net decline (HT)

After n half-lives, the ratio:initial activity : final activity = 2ⁿ : 1Equivalently, the fraction that has decayed = 1 − (½)ⁿ.

Worked HT example — forward

After 4 half-lives:

Fraction remaining = (½)⁴ = ¹⁄₁₆ (6.25%)

Fraction decayed = 1 − ¹⁄₁₆ = ¹⁵⁄₁₆ (93.75%)

Net decline ratio = 2⁴ : 1 = 16 : 1

Worked HT example — inverse

If the activity of a source has declined by a ratio of 64 : 1, how many half-lives have elapsed?2ⁿ : 1 = 64 : 1 → 2ⁿ = 64 → n = 6 half-livesMemorise the powers of 2: 2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵=32, 2⁶=64, 2⁷=128. Mark schemes regularly use these specific ratios.

Direction matters

The ratio is initial : final = 2ⁿ : 1, NOT 1 : 2ⁿ. Initial is bigger than final (the source is decaying), so the larger number sits on the "initial" side.

Half-Life Is Constant

For a given isotope, the half-life is fixed. It does not depend on:

Doesn't depend on…

Starting amount of the isotope

Temperature of the sample

Pressure on the sample

Chemical form (e.g. C in CO₂ vs C in a diamond)

Whether the sample has been disturbed

Why?

Because decay is a nuclear process, governed by the nucleus alone. Chemistry, temperature, and pressure affect electrons and bonds — not the nucleus.

Why this matters for dating

Carbon-14 has the same half-life today as it did 50,000 years ago. That's why radiocarbon dating works — and why it would fail if half-life depended on environment. A 5,730-year-old fossil in a hot desert ages exactly the same as one in a cold cave.

Modelling Decay With Coins

A useful classroom analogy: imagine 200 coins, each representing one radioactive nucleus.

The rules

Toss every coin once. Remove every coin that lands tails.

The removed coins represent nuclei that have decayed; the kept coins are still undecayed.

Each toss represents one fixed time interval — the half-life of this "isotope".

Repeat with the remaining coins, then again, and again…

Random — but predictable

You can't predict which individual coin will land tails. But you can predict that roughly half of them will. With 200 coins, the result is reliably close to 100. With 2 coins, randomness dominates.

Why this is a good analogy

Each coin's outcome is random

The probability of "decay" is fixed per coin per toss

The fraction surviving halves on average each round

Real radioactive decay behaves identically — same math

Examiner Alerts

Alert 1 — Wording for the definition

"Time for radioactivity to halve" is not credited — the mark scheme needs you to specify what's halving. Use: "number of undecayed nuclei", or "activity", or "count rate" — never "radioactivity".

Alert 2 — Linear subtraction error

Decay is exponential, not linear. Starting at 800 Bq with half-life 4 h, students sometimes wrongly subtract a fixed value each half-life (e.g. 800 → 600 → 400 → 200). The correct method is always halving: 800 → 400 → 200 → 100.

Alert 3 — Fraction ↔ half-lives

The recurring exam target. If you're told the fraction remaining, immediately translate it to a number of half-lives: ½ = 1 · ¼ = 2 · ⅛ = 3 · ¹⁄₁₆ = 4. Many students lose marks here by treating ¼ as "two half-lives' worth of subtracted activity" instead of two halvings.

Alert 4 — HT ratio direction

HT The net decline ratio is initial : final = 2ⁿ : 1. Writing it the wrong way round (1 : 2ⁿ) reverses the meaning and is marked wrong.

Common Errors — and the Fixes

Students often write…Mark-scheme versionWhy it matters

"Half-life = time for the radioactivity to halve"Time for the number of undecayed nuclei (or activity / count rate ) to halve"Radioactivity" is rejected — must name what's halving

"After 2 half-lives, ½ + ½ = 0 left"After 2 half-lives, (½)² = ¼ remainsFractions multiply, not add

Subtracts a fixed amount each half-life (linear)Halves the activity each half-life (exponential)Wrong physics — recurring error

"Half-life depends on the amount of sample"Half-life is a fixed property of the isotope , independent of sample sizeCommon misconception

Divides initial activity by n (number of half-lives) instead of 2ⁿActivity = initial ÷ 2ⁿWrong formula

"The random nature means we can't predict anything"Can't predict individual nuclei; large samples are predictableMisses the bulk-statistics point

Stretch Question

A researcher wants to date an organic artefact that is about 23,000 years old. They have access to two isotopes:

Carbon-14 — half-life ≈ 5,730 years

Uranium-238 — half-life ≈ 4.5 × 10⁹ years

Explain which isotope is better suited for dating this artefact, and how many half-lives have elapsed. 4 marks

1

Carbon-14 is the right choice. Its half-life is on a similar timescale to the artefact's age.

2

Number of half-lives: 23,000 ÷ 5,730 ≈ 4 half-lives.

3

After 4 half-lives, the fraction of C-14 remaining = (½)⁴ = ¹⁄₁₆ — measurable but reduced significantly. This gives a strong signal to date the artefact accurately.

4

Uranium-238 would be useless here. With a half-life of 4.5 billion years, the fraction decayed in 23,000 years is too small to measure — the difference between "original" and "23,000 years old" would be lost in measurement uncertainty.

General rule: choose an isotope whose half-life is roughly the same order of magnitude as the timescale you're measuring.

Checkpoint — What You Can Now Do

Tick each one off before moving on

✓ Define half-life using mark-scheme wording (number of undecayed nuclei / activity / count rate — never "radioactivity")

✓ Explain the random nature of decay — individual nuclei random, large samples predictable

✓ Use the fraction-remaining pattern (½ → ¼ → ⅛ → ¹⁄₁₆) to translate between fraction, number of half-lives, and elapsed time

✓ Calculate activity after a known number of half-lives using activity = initial ÷ 2ⁿ

✓ Read a half-life off an activity-vs-time graph using the half-initial method

✓ State that half-life is a fixed property of the isotope, independent of sample size, temperature, or chemistry

✓ HT Calculate net decline as a ratio (initial : final = 2ⁿ : 1)

Next Lesson

Lesson 4.05 — Radioactive Contamination and Background Radiation. The distinction between contamination (radioactive material on/in something) and irradiation (exposure to radiation, where the object itself doesn't become radioactive). Plus background radiation sources.