Definition · Decay curves · Radioactive isotopes in medicine and dating
Learning Objectives
🔬 Define half-life as the time for half the radioactive nuclei in a sample to decay
📈 Read and interpret radioactive decay curves to determine half-life
🧮 Use half-life to calculate the number of nuclei or activity remaining after multiple half-lives
🏥 Describe how radioactive isotopes are used in medical diagnosis and treatment
🌍 Explain how carbon-14 and uranium-lead dating are used to estimate the age of objects and rocks
⚖️ Evaluate the choice of isotope based on its half-life and type of radiation emitted
What is Half-life?
Half-life (t½) is the time taken for the number of radioactive nuclei (or the activity) of a sample to decrease to half its original value.
Radioactive decay is a random and spontaneous process — you cannot predict which nucleus will decay next, or when. However, because samples contain enormous numbers of atoms, statistical patterns emerge and we can reliably measure the average rate of decay.
Every radioactive isotope has its own unique half-life. Some half-lives are fractions of a second; others are billions of years. No physical or chemical process (heating, cooling, applying pressure, forming compounds) can change the half-life of an isotope.
After each half-life, the activity (decays per second, measured in Becquerels, Bq) and the number of undecayed nuclei both halve.
Activity (A) = number of decays per second | Unit: Becquerel (Bq)
After n half-lives, the fraction of the original nuclei remaining is:
Fraction remaining = (½)ⁿ where n = number of half-lives elapsed
For example, after 3 half-lives: (½)³ = 1/8 of the original activity or number of nuclei remains.
Half-lives elapsed (n)
Fraction remaining
% remaining
0
1
100%
1
1/2
50%
2
1/4
25%
3
1/8
12.5%
4
1/16
6.25%
5
1/32
3.125%
Radioactive Decay Curves
A decay curve is a graph of activity (or number of undecayed nuclei) on the y-axis against time on the x-axis. The curve always has a characteristic shape: it falls steeply at first, then more and more gradually, approaching zero asymptotically — this is called an exponential decay.
How to find half-life from a decay curve:
Choose a convenient starting activity value on the y-axis (e.g. 800 Bq).
Halve that value (400 Bq) and draw a horizontal line to the curve.
Drop a vertical line down to the x-axis and read off the time.
Repeat from a different starting point and average your readings for accuracy.
The half-life is the same regardless of where on the curve you start measuring — this is a defining feature of exponential decay.
Background radiation correction: In real experiments, the Geiger counter always detects some background radiation (from cosmic rays, rocks, building materials, etc.). You must subtract the background count rate from all your measured count rates before plotting the decay curve. Failure to do so leads to an overestimate of activity and an incorrect half-life reading from the curve.
The corrected curve will tend to zero, making the half-life easier to read accurately.
Radioactive Isotopes in Medicine
Radioactive isotopes are widely used in hospitals for both diagnosis (imaging) and treatment (therapy). The choice of isotope depends critically on:
The type of radiation it emits (alpha, beta, or gamma)
Its half-life (long enough to be useful, short enough to minimise patient dose)
Whether the element is chemically taken up by the target organ
Technetium-99m (t½ ≈ 6 hours) is the most widely used medical tracer. It emits gamma radiation (detected outside the body by a gamma camera), has a short half-life (minimising radiation dose), and can be chemically attached to molecules that target specific organs.
Diagnosis (tracers): A gamma-emitting tracer is injected into the patient. Gamma rays pass out of the body and are detected to produce an image of organ function. Short half-lives (hours to days) are preferred so the patient is not irradiated for long after the scan.
Treatment (radiotherapy): Cancerous tumours can be treated using radiation that destroys cells. Iodine-131 (t½ ≈ 8 days) is used to treat thyroid cancer because the thyroid gland naturally absorbs iodine. Beta emitters are used internally because beta radiation has a short range in tissue, limiting damage to the target area only.
External beam radiotherapy uses gamma sources (e.g. cobalt-60) directed at tumours from outside the body. The beam is rotated around the patient so only the tumour receives a high dose from all angles.
For medical tracers: use a gamma emitter (detectable outside the body) with a short half-life (minimises patient dose after procedure).
Radioactive Dating
The predictable nature of radioactive decay makes certain isotopes ideal "clocks" for determining the age of materials.
Carbon-14 dating (radiocarbon dating): Used to date organic (once-living) materials up to about 50,000 years old.
Carbon-14 (¹⁴C, t½ ≈ 5,700 years) is continuously produced in the upper atmosphere when cosmic rays convert nitrogen into carbon-14. This radioactive carbon becomes part of CO₂ and enters the food chain. While an organism is alive, it constantly exchanges carbon with its environment, so the proportion of ¹⁴C in its tissues stays roughly constant. When the organism dies, it stops taking in carbon and the ¹⁴C decays without being replaced. By measuring the ratio of ¹⁴C to stable ¹²C in a sample, scientists can calculate how long ago the organism died.
Age ≈ (t½ ÷ 0.693) × ln(N₀ ÷ N) [Higher tier context — usually solved graphically at GCSE]
In practice at GCSE, you simply use the half-life to work out how many half-lives have elapsed and multiply by the half-life value.
Uranium-lead dating: Used to date very old rocks (millions to billions of years). Uranium-238 (t½ ≈ 4.5 billion years) decays through a series of steps to stable lead-206. The ratio of U-238 to Pb-206 in a rock sample indicates how long the rock has existed in solid form.
Different isotopes are suited to different timescales:
Isotope
Half-life
Used to date
Carbon-14
5,700 years
Organic material up to ~50,000 years
Potassium-40
1.25 billion years
Rocks and minerals
Uranium-238
4.5 billion years
Very old rocks, age of Earth
For accurate dating, the half-life of the chosen isotope must be comparable in magnitude to the age of the object being dated.
Key Quantities and Units
Quantity
Symbol
Unit
Notes
Activity
A
Becquerel (Bq)
1 Bq = 1 decay per second
Half-life
t½
seconds (s), minutes, hours, years
Depends on isotope
Number of nuclei
N
No unit (a count)
Proportional to activity
Count rate
—
counts per second (cps) or counts per minute (cpm)
Measured by Geiger counter
Activity and number of undecayed nuclei both follow the same pattern of decay — they halve every half-life. You can use either quantity when reading a decay curve.
📝 Example 1 — Reading half-life from a decay curve
A radioactive sample has an initial activity of 960 Bq. After 20 minutes the activity is 480 Bq, after 40 minutes it is 240 Bq, and after 60 minutes it is 120 Bq. Determine the half-life of the sample.
1Identify what happens each time interval: 960 → 480 (halved) in 20 min; 480 → 240 (halved) in 20 min; 240 → 120 (halved) in 20 min.
2Each 20-minute interval causes the activity to halve. This is the definition of one half-life.
3The half-life is the same for each interval, confirming exponential decay.
Half-life t½ = 20 minutes
📝 Example 2 — Calculating remaining activity after multiple half-lives
A radioactive source has an initial activity of 6400 Bq and a half-life of 5 days. Calculate the activity after 25 days.
1Find the number of half-lives elapsed: n = total time ÷ half-life = 25 days ÷ 5 days = 5 half-lives
2Apply the halving formula: fraction remaining = (½)⁵ = 1/32
4Check using a step-by-step table:
Day 0: 6400 Bq → Day 5: 3200 Bq → Day 10: 1600 Bq → Day 15: 800 Bq → Day 20: 400 Bq → Day 25: 200 Bq ✓
Activity after 25 days = 200 Bq
📝 Example 3 — Finding half-life from a graph (background corrected)
A Geiger counter records a count rate of 520 counts/min from a sample, and the background count rate is 20 counts/min. After 30 minutes, the total count rate is 270 counts/min. After 60 minutes it is 145 counts/min. Calculate the half-life of the sample.
1Subtract background from all readings to get the corrected count rates.
At t = 0 min: 520 − 20 = 500 counts/min
At t = 30 min: 270 − 20 = 250 counts/min
At t = 60 min: 145 − 20 = 125 counts/min
2Check: 500 → 250 in 30 minutes (halved ✓). 250 → 125 in 30 minutes (halved ✓).
3Both intervals are 30 minutes for one halving, confirming consistent exponential decay.
📝 Example 4 — Choosing an isotope for a medical procedure
A doctor needs a radioactive tracer to image a patient's kidney function. The scan takes about 2 hours. Explain which of the following would be most appropriate: (A) Iodine-131, t½ = 8 days, beta and gamma emitter; (B) Technetium-99m, t½ = 6 hours, gamma only emitter; (C) Uranium-238, t½ = 4.5 billion years, alpha emitter.
1Type of radiation: The tracer must emit gamma radiation to be detected outside the body. Alpha particles are stopped by a few cm of air and would not leave the body. This rules out Uranium-238 (C).
2Half-life relative to scan duration: The half-life should be long enough for the scan (2 hours) to be completed, but short enough that the patient is not significantly irradiated for days afterwards.
3Evaluate Iodine-131 (A): t½ = 8 days — far too long; the patient would remain radioactive for weeks. It also emits beta radiation which would irradiate internal tissues without providing useful diagnostic information.
4Evaluate Technetium-99m (B): t½ = 6 hours — just right. After the 2-hour scan, the activity falls to about (½)^(2/6) ≈ 79% initially, but within 24–48 hours the activity is negligible. It emits only gamma radiation, ideal for external detection.
Best choice: Technetium-99m (B) — gamma emitter, half-life of 6 hours is ideal for a 2-hour scan with minimal long-term patient dose.
Question 1. What is the correct definition of half-life?
Question 2. A sample has an initial activity of 800 Bq and a half-life of 4 years. What is the activity after 12 years?
Question 3. Why must background radiation be subtracted before plotting a decay curve?
Question 4 (Short answer). A radioactive source has an activity of 3200 Bq. After 3 half-lives, what is the activity in Bq?
Question 5. Which isotope would be most suitable as a medical tracer for a 4-hour scan?
Challenge 1 (Higher). A radioactive sample has an initial count rate of 1200 counts/min. The background count rate is 50 counts/min. After 60 minutes the total count rate is 625 counts/min. After a further 60 minutes it is 337.5 counts/min.
(a) Calculate the corrected count rates at t = 0, t = 60 min, and t = 120 min.
(b) Determine the half-life of the sample.
(c) Calculate the corrected count rate after a total of 4 hours.
(c) After 4 hours = 240 minutes:
n = 240 ÷ 60 = 4 half-lives
Count rate = 1150 × (1/2)⁴ = 1150 ÷ 16 = 71.875 ≈ 71.9 counts/min
(Add background back if asked for total: 71.9 + 50 ≈ 122 counts/min, but corrected = 71.9 counts/min)
Challenge 2. An archaeologist discovers a piece of charcoal from an ancient campfire. She measures that the charcoal has only 1/8 of the carbon-14 activity of a living tree of similar mass. Carbon-14 has a half-life of 5,700 years.
(a) How many half-lives have elapsed since the tree was cut down?
(b) Estimate the age of the campfire.
(c) Suggest why carbon-14 dating would be unreliable for dating a rock thought to be 500 million years old.
(a) Number of half-lives:
Activity is 1/8 = (1/2)³ of original ∴ n = 3 half-lives
(b) Age of campfire:
Age = n × t½ = 3 × 5,700 = 17,100 years
(c) Unreliability for old rocks:
Carbon-14's half-life is 5,700 years. After 500 million years, approximately 500,000,000 ÷ 5,700 ≈ 87,700 half-lives would have elapsed. The fraction of C-14 remaining would be (1/2)^87700 — essentially zero. There would be no detectable C-14 left to measure. A much longer half-life isotope (e.g. uranium-238, t½ = 4.5 billion years) should be used instead.
Challenge 3 (Extended writing). Iodine-131 (t½ = 8 days, beta and gamma emitter) is used to treat thyroid cancer. Explain why iodine-131 is a suitable choice for this treatment, and why it would be unsuitable as a diagnostic tracer for a different organ such as the liver. Include reference to the type of radiation, the half-life, and the chemical behaviour of iodine in your answer.
Suitability for thyroid treatment:
• The thyroid gland naturally absorbs iodine to produce hormones, so iodine-131 is chemically targeted to the thyroid without needing special delivery methods.
• It emits beta radiation, which has a short range in tissue (a few mm), meaning it destroys cancerous thyroid cells while causing minimal damage to surrounding organs.
• The half-life of 8 days is appropriate — long enough to deliver a sustained therapeutic dose to the tumour, but short enough that activity decreases significantly within weeks, reducing long-term radiation risk.
Unsuitability as a liver tracer:
• The liver does not naturally absorb iodine, so iodine-131 would not be selectively taken up by the liver — the image would not show liver function.
• For diagnosis, a gamma-only emitter is needed so radiation can pass out of the body to be detected by a gamma camera. Iodine-131 emits beta radiation which is absorbed within the body and contributes unnecessary dose without aiding the image.
• A half-life of 8 days is too long for a diagnostic procedure — the patient would remain radioactive for weeks after a scan, receiving unnecessary radiation dose. A short half-life tracer like technetium-99m (t½ = 6 hours) would be far more suitable.
Challenge 4 (Higher calculation). A hospital has a sample of technetium-99m with an activity of 400 MBq (400 × 10⁶ Bq) at 8:00 am. The half-life of technetium-99m is 6 hours. A patient needs to receive a dose of at least 25 MBq for a scan.
(a) Calculate the activity of the sample at 8:00 pm (noon that same day is 12 hours after 8am — 8pm is 12 hours later).
(b) What is the latest time the scan can be performed using this sample if the minimum activity required is 25 MBq?
(a) Activity at 8:00 pm (12 hours after 8:00 am):
n = 12 ÷ 6 = 2 half-lives
Activity = 400 × (1/2)² = 400 ÷ 4 = 100 MBq
(b) Latest time for scan (minimum 25 MBq):
Find n such that 400 × (1/2)ⁿ ≥ 25:
(1/2)ⁿ ≥ 25/400 = 1/16
(1/2)⁴ = 1/16 → n = 4 half-lives
Time elapsed = 4 × 6 hours = 24 hours
Latest time = 8:00 am + 24 hours = 8:00 am the following day
(At exactly n = 4, activity = exactly 25 MBq, so the scan must be performed by 8:00 am the next morning.)