Half-Lives and Radioactive Decay | Physics

📖 In-Depth Theory

Radioactive Decay Is Random

Radioactive decay is a RANDOM PROCESS:

It is impossible to predict exactly WHEN any individual nucleus will decay.

It is impossible to predict WHICH nucleus in a sample will decay next.

Decay is spontaneous — NOT triggered by temperature, pressure or chemical state.

However, for a LARGE SAMPLE:

We can predict the PROPORTION that will decay in a given time.

Statistical behaviour becomes predictable even though individual decays are random.

This is similar to flipping a large number of coins — we cannot predict any individual flip, but we can confidently predict about 50% will be heads.

ACTIVITY decreases over time as the number of unstable nuclei falls.

Half-Life

The HALF-LIFE of a radioactive isotope is the time for:

The number of UNDECAYED NUCLEI to halve, OR

The ACTIVITY (or count rate) of the source to halve.

Half-life is CONSTANT for a given isotope — it doesn't change.

EXAMPLES:

Carbon-14: half-life ~5730 years (used in carbon dating)

Iodine-131: half-life ~8 days (medical uses — short enough to leave the body)

Uranium-238: half-life ~4.5 billion years

Radon-222: half-life ~3.8 days

CALCULATING REMAINING ACTIVITY/NUCLEI:

After 1 half-life: ½ remains

After 2 half-lives: ¼ remains

After 3 half-lives: ⅛ remains

After n half-lives: (½)ⁿ remains

EXAMPLE:

Source starts at 800 Bq. Half-life = 2 hours. What is the activity after 6 hours?

6 hours ÷ 2 hours = 3 half-lives

800 → 400 → 200 → 100 Bq

Uses of Half-Life and Decay Curves

DECAY CURVE:

Graph of activity (or count rate) against time.

Curve starts high and decreases exponentially.

To find half-life from graph: find initial activity, halve it, read off time → then verify the next halving takes the same time.

PRACTICAL SELECTION of isotopes:

MEDICAL TRACERS: short half-life needed — activity falls quickly so patient receives minimal long-term dose. Technetium-99m: 6 hours.

CANCER TREATMENT: short enough to deliver dose in treatment window, then decay away.

CARBON DATING: 14C half-life ~5730 years — compares ¹⁴C/¹²C ratio of living things vs sample.

NUCLEAR WASTE: long half-life isotopes are the biggest storage problem — some remain dangerous for thousands of years.

BACKGROUND RADIATION:

All measurements of radioactive sources include BACKGROUND RADIATION — radiation from natural sources (rocks, cosmic rays, radon gas, food).

Background must be measured and SUBTRACTED from readings.

⚠️ Common Mistake

After each half-life, the activity halves AGAIN from its current value — not from the original. After 3 half-lives starting at 1000 Bq: 500 → 250 → 125 Bq. Also: background radiation must be subtracted before half-life calculations.

📐 Key Equations

After n half-lives: fraction remaining = (½)ⁿ

📌 Key Note

Half-life: time for activity (or nuclei count) to halve. Constant for a given isotope. Random decay — can't predict individual nucleus. After n half-lives: (½)ⁿ remains. Decay curve: exponential fall. Background radiation must be subtracted. Medical tracers need short half-lives; carbon dating uses 5730-year ¹⁴C half-life.

🎯 Matching Activity — Half-Life Calculations

Match each scenario to the correct remaining activity. — drag the symbols on the right to match the component names on the left.

400 Bq

Drop here

125 Bq

Drop here

Longer half-life needed

Drop here

Shorter half-life needed

Drop here

Carbon dating — need isotope with half-life comparable to age of sample (thousands of years)

Initial activity 1600 Bq, after 2 half-lives: 1600 → 800 → 400

Initial activity 1000 Bq, after 3 half-lives: 1000 → 500 → 250 → 125

Medical tracer — activity must fall quickly to reduce patient radiation dose

⚽ FIFA Worked Examples

Half-Life Calculation

A source has initial activity 960 Bq. Its half-life is 3 hours. What is the activity after 12 hours?

F

Number of half-lives = total time ÷ half-life; remaining = initial × (½)ⁿ

I

n = 12 ÷ 3 = 4 half-lives

F

960 × (½)⁴ = 960 × 1/16 = 960 ÷ 16

A

Activity = 60 Bq

⭐ Higher Tier Only

Use the concept of half-life to calculate the number of undecayed nuclei or activity remaining after a given number of half-lives. Explain why half-life cannot be changed by physical or chemical means — it is a fundamental nuclear property. Compare the suitability of different isotopes for specific uses based on their half-lives and radiation type.

🎯 Test Yourself

Question 1 of 2

1. A radioactive source has a half-life of 4 hours and initial activity 640 Bq. What is the activity after 12 hours?

2. Why is the half-life of a radioactive isotope described as a 'constant'?

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