Physics · Simulation and virtual lab Free
Radioactive decay: count rate, half-life and the dice model
Count clicks from a radioactive source every 10 seconds, plot the decay curve and find the half-life, then model random decay with dice.
- You will learn to
- Measure and correct count rates from a radioactive source.
- Use a ln graph to find the decay constant and the half-life.
- Explain why decay is random, using a dice model.
- You will measure
- Counts in each 10 s window; corrected count; ln(count); then λ and t½.
- Time needed
- About 30–40 min
- Equipment
- Nothing extra. Real version: GM tube and counter, sealed source (teacher only), or 100 dice / bottle tops in a tray.
- Price
- Free: no payment, no account.
Virtual lab
Follow the steps with the simulation above (go back to the simulation ↑). Each time you tap Record, a row is added to the table and a point to the graph.
Aim
To measure the count rate from a radioactive source over time, correct it for background radiation, and determine the decay constant and half-life from a graph; and to model random decay with dice.
What to do
- Choose Background only and count for 60 s. Note the background count per 10 s.
- Choose protactinium-234 and start the clock. The counts in each 10 s window go into the table automatically.
- Use clock speed × 5 to save time. Stop after about 300 s.
- Look at the graph of ln(count) against time: it is a straight line. The gradient gives −λ, and t½ = ln 2 ÷ λ.
- Repeat with the unknown source X. Which has the shorter half-life?
- Switch to the dice model: throw 600 dice and remove every six each throw. Compare the graph with the real source.
Results
Questions
What is meant by the half-life of a radioactive source?
The time taken for half of the radioactive nuclei in a sample to decay, or for the activity (count rate) to fall to half.
Why must the background count be subtracted?
The GM tube also counts radiation from rocks, cosmic rays and the air; only the corrected count comes from the source.
Why do the counts in each 10 s window go up and down?
Radioactive decay is random: we cannot predict exactly when a nucleus will decay, so counts fluctuate about the average.
A source has a count rate of 80 per second. Its half-life is 25 s. What is the count rate after 75 s?
75 s is 3 half-lives: 80 → 40 → 20 → 10 per second.
How is the decay constant λ related to the half-life?
λ = ln 2 ÷ t½ ≈ 0.693 ÷ t½.
Why does throwing dice model radioactive decay?
Each die has the same chance (1/6) of showing a six on each throw, just as each nucleus has the same chance of decaying in each second; we cannot say which die will show a six.
For teachers: expected results and how the model works
Each nucleus decays at random with the chosen decay constant (Poisson counts in each 10 s window). Sources: protactinium-234m, half-life 70 s (real value 1.17 min), starting at about 60 counts per second; radon-220 (thoron), 56 s, 45 per second; unknown source X, 25 s, 80 per second. Background is 0.5 counts per second (5 per 10 s). The graph fit uses ln of the counts and ignores windows close to background, so learners get t½ within about 10 % after 300 s. Dice: 600 dice, p = 1/6 gives a "half-life" of ln 2 ÷ −ln(5/6) ≈ 3.8 throws; p = 1/3 gives about 1.7 throws. The clock stops after 400 s.
Exam practice Exam set
National-exam style questions on this simulation (16 marks). Try each one on paper first, then open the marking scheme.
The corrected count rate from a source falls from 640 counts per minute to 40 counts per minute in 60 minutes. Calculate (a) the number of half-lives, (b) the half-life, (c) the decay constant. (5 marks)
Marking scheme: (a) 640 → 320 → 160 → 80 → 40: 4 half-lives (2). (b) 60 ÷ 4 = 15 min (1). (c) λ = 0.693 ÷ 15 = 0.046 min⁻¹ (2).
Explain why a graph of ln(corrected count rate) against time is a straight line, and how the half-life is found from it. (4 marks)
Marking scheme: A = A₀e^(−λt) (1); ln A = ln A₀ − λt, the form y = c + mx (1); gradient = −λ (1); t½ = ln 2 ÷ λ (1).
Iodine-131 (half-life 8 days) is used in hospitals. What fraction of a sample remains after 24 days? Why is a short half-life useful for a medical tracer? (4 marks)
Marking scheme: 24 ÷ 8 = 3 half-lives (1); fraction = (½)³ = 1/8 (1); the source becomes weak quickly so the patient receives a small dose (1), yet lasts long enough to take the images (1).
Thorium-234 decays to protactinium-234 by beta emission. Write the equation. (3 marks)
Marking scheme: ²³⁴₉₀Th → ²³⁴₉₁Pa + ⁰₋₁e (correct mass numbers 1, correct atomic numbers 1, electron shown 1).
Donat Sciences and MathsPhysics · Virtual lab worksheet
School: ........................................ Name: ................................ Class: ........ Date: ............
Radioactive decay: count rate, half-life and the dice model
Aim: To measure the count rate from a radioactive source over time, correct it for background radiation, and determine the decay constant and half-life from a graph; and to model random decay with dice.
Apparatus (real laboratory): GM tube and counter, sealed source (teacher only), or 100 dice / bottle tops in a tray.
What to do (with the simulation on www.donatstem.co.rw)
- Choose Background only and count for 60 s. Note the background count per 10 s.
- Choose protactinium-234 and start the clock. The counts in each 10 s window go into the table automatically.
- Use clock speed × 5 to save time. Stop after about 300 s.
- Look at the graph of ln(count) against time: it is a straight line. The gradient gives −λ, and t½ = ln 2 ÷ λ.
- Repeat with the unknown source X. Which has the shorter half-life?
- Switch to the dice model: throw 600 dice and remove every six each throw. Compare the graph with the real source.
Results
| # | Time (s) | Count in 10 s | Corrected count | ln(corrected count) |
|---|
| 1 | | | | |
| 2 | | | | |
| 3 | | | | |
| 4 | | | | |
| 5 | | | | |
| 6 | | | | |
| 7 | | | | |
| 8 | | | | |
| 9 | | | | |
| 10 | | | | |
| 11 | | | | |
| 12 | | | | |
Graph
Plot ln(corrected count) against time (s). Draw the best straight line and use its gradient to find λ and the half-life.
Questions
- What is meant by the half-life of a radioactive source? [2]
- Why must the background count be subtracted? [2]
- Why do the counts in each 10 s window go up and down? [2]
- A source has a count rate of 80 per second. Its half-life is 25 s. What is the count rate after 75 s? [2]
- How is the decay constant λ related to the half-life? [2]
- Why does throwing dice model radioactive decay? [2]
- The corrected count rate from a source falls from 640 counts per minute to 40 counts per minute in 60 minutes. Calculate (a) the number of half-lives, (b) the half-life, (c) the decay constant. [5]
- Explain why a graph of ln(corrected count rate) against time is a straight line, and how the half-life is found from it. [4]
- Iodine-131 (half-life 8 days) is used in hospitals. What fraction of a sample remains after 24 days? Why is a short half-life useful for a medical tracer? [4]
- Thorium-234 decays to protactinium-234 by beta emission. Write the equation. [3]
Donat Sciences and MathsPhysics · Virtual lab worksheet
School: ........................................ Name: ................................ Class: ........ Date: ............
Marking scheme: Radioactive decay: count rate, half-life and the dice model
Expected results
Each nucleus decays at random with the chosen decay constant (Poisson counts in each 10 s window). Sources: protactinium-234m, half-life 70 s (real value 1.17 min), starting at about 60 counts per second; radon-220 (thoron), 56 s, 45 per second; unknown source X, 25 s, 80 per second. Background is 0.5 counts per second (5 per 10 s). The graph fit uses ln of the counts and ignores windows close to background, so learners get t½ within about 10 % after 300 s. Dice: 600 dice, p = 1/6 gives a "half-life" of ln 2 ÷ −ln(5/6) ≈ 3.8 throws; p = 1/3 gives about 1.7 throws. The clock stops after 400 s.
Answers
- What is meant by the half-life of a radioactive source? [2]
The time taken for half of the radioactive nuclei in a sample to decay, or for the activity (count rate) to fall to half.
- Why must the background count be subtracted? [2]
The GM tube also counts radiation from rocks, cosmic rays and the air; only the corrected count comes from the source.
- Why do the counts in each 10 s window go up and down? [2]
Radioactive decay is random: we cannot predict exactly when a nucleus will decay, so counts fluctuate about the average.
- A source has a count rate of 80 per second. Its half-life is 25 s. What is the count rate after 75 s? [2]
75 s is 3 half-lives: 80 → 40 → 20 → 10 per second.
- How is the decay constant λ related to the half-life? [2]
λ = ln 2 ÷ t½ ≈ 0.693 ÷ t½.
- Why does throwing dice model radioactive decay? [2]
Each die has the same chance (1/6) of showing a six on each throw, just as each nucleus has the same chance of decaying in each second; we cannot say which die will show a six.
- The corrected count rate from a source falls from 640 counts per minute to 40 counts per minute in 60 minutes. Calculate (a) the number of half-lives, (b) the half-life, (c) the decay constant. [5]
(a) 640 → 320 → 160 → 80 → 40: 4 half-lives (2). (b) 60 ÷ 4 = 15 min (1). (c) λ = 0.693 ÷ 15 = 0.046 min⁻¹ (2).
- Explain why a graph of ln(corrected count rate) against time is a straight line, and how the half-life is found from it. [4]
A = A₀e^(−λt) (1); ln A = ln A₀ − λt, the form y = c + mx (1); gradient = −λ (1); t½ = ln 2 ÷ λ (1).
- Iodine-131 (half-life 8 days) is used in hospitals. What fraction of a sample remains after 24 days? Why is a short half-life useful for a medical tracer? [4]
24 ÷ 8 = 3 half-lives (1); fraction = (½)³ = 1/8 (1); the source becomes weak quickly so the patient receives a small dose (1), yet lasts long enough to take the images (1).
- Thorium-234 decays to protactinium-234 by beta emission. Write the equation. [3]
²³⁴₉₀Th → ²³⁴₉₁Pa + ⁰₋₁e (correct mass numbers 1, correct atomic numbers 1, electron shown 1).