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