IB DP Physics: Activity, half-life and the radioactive decay law (DP Physics) – Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – the IB holds copyright in its own papers. Use these alongside the official past papers available through your school or the IB store.

These questions cover the second part of sub-topic E.3, Radioactive decay, in the IB Physics guide (first assessment 2025): activity, half-life, background, real-life uses and, for HL only, the decay law. Questions 1 to 8 apply to SL and HL. Questions 9 to 12 are HL only. Learn the content in the study guide and the revision notes. Also: the course hub, the printable checklist and the free 10-minute diagnostics.

Data where needed: half-life of carbon-14 is 5730 years; half-life of sodium-24 is 15 h; the activity per gram of carbon in living material is taken as constant over time.

Questions

1. (a) Define the activity of a radioactive source and state its SI unit. [2] (b) State one reason why the count rate registered by a detector is smaller than the activity of the source. [1]

2. A sample has an activity of 2.4 × 10⁵ Bq and a half-life of 40 minutes. (a) Calculate its activity after 2.0 hours. [2] (b) Calculate the time for the activity to fall to 3750 Bq. [2]

3. A Geiger-Müller tube with a sealed source nearby registers 58 counts per minute. With the source removed, it registers 120 counts in 10 minutes. (a) Calculate the corrected count rate due to the source. [2] (b) State one natural source of background radiation, and explain why the background is measured over a long time. [2]

4. The table shows the corrected count rate from a source with a short half-life.

Time / min	0	2	4	6	8	10
Corrected count rate / counts per second	160	101	63	40	25	16

Determine the half-life of the source, using at least two different intervals, and explain why using two intervals is better than one. [5]

5. A hospital images a patient’s kidneys, and a factory monitors a steel plate about 2 cm thick. (a) For the kidneys, isotope X emits gamma rays, half-life 5.0 h. Isotope Y emits alpha particles, half-life 40 days. Choose one, justifying with radiation type and half-life. [3] (b) For the steel plate, state whether a beta or a gamma source is more suitable, and describe how the count rate changes if the plate gets thicker. [2]

6. Charcoal from an ancient hearth gives an activity of 0.060 Bq per gram of carbon. Living material gives 0.24 Bq per gram of carbon. Calculate the age of the charcoal and state one assumption you made. [4]

7. Water containing a tracer flows in a buried pipe. A detector is moved along the ground above the pipe and records the counts per minute below. The background is 30 counts per minute.

Distance along pipe / m	0	5	10	15	20	25
Counts per minute	31	33	30	118	35	32

(a) Deduce where the leak is and explain how you decided. [2] (b) State the radiation type and half-life suited to the tracer, with a reason for each. [2]

8. Describe how to determine the half-life of a nuclide with a half-life of a few minutes, using a Geiger-Müller tube and counter. Include how you allow for background and analyse the results. [6]

9. (HL only) Sodium-24 is used as a tracer. A sample has activity 5.0 × 10⁸ Bq. (a) Calculate the decay constant, in h⁻¹ and in s⁻¹. [2] (b) Calculate the number of undecayed nuclei in the sample. [2] (c) Calculate the activity after 40 h. [2]

10. (HL only) A nuclide has decay constant λ = 1.5 × 10⁻⁴ s⁻¹. (a) Calculate λΔt for Δt = 10 s, and the exact probability that one nucleus decays in that time. [2] (b) Comment on how well λΔt approximates the probability here. [1] (c) Show that λΔt is not a valid probability for Δt = 1.0 × 10⁴ s, and find the exact one. [2]

11. (HL only) A source has an activity of 6.4 × 10⁶ Bq at t = 0 and 4.0 × 10⁶ Bq at t = 6.0 h. (a) Calculate the decay constant. [2] (b) Calculate the half-life. [1] (c) Calculate the activity at t = 20 h. [2] (d) Calculate the time at which the activity is 5.0% of its initial value. [2]

12. (HL only) An artefact has 37% of the activity per gram of carbon found in living material. (a) Calculate its age. [3] (b) A rock is 1.0 × 10⁶ years old. Show that carbon-14 dating is unsuitable by finding the fraction of carbon-14 that would remain. [2] (c) State the property that an isotope needs for dating a rock of this age. [1]

Answers

1. (a) Activity is the number of nuclei that decay per unit time [1]. The unit is the becquerel, Bq (one decay per second) [1]. (b) Any one of: the source emits in all directions; some radiation is absorbed on the way; the detector misses some particles [1]. [3] Examiner insight: Define activity as decays per unit time, not “radiation given out”, and give the unit.

2. (a) 2.0 h = 120 min, so n = 120/40 = 3 [1]. A = 2.4 × 10⁵ / 2³ = 3.0 × 10⁴ Bq [1]. (b) 2.4 × 10⁵ / 3750 = 64 = 2⁶, so n = 6 [1]. t = 6 × 40 = 240 min = 4.0 h [1]. [4] Examiner insight: Convert the time unit first, and state the number of half-lives so your method is visible.

3. (a) Background = 120 / 10 = 12 counts per minute [1]. Corrected rate = 58 − 12 = 46 counts per minute [1]. (b) One natural source, for example radon gas from rocks and soil, cosmic rays, or building materials [1]. Decay is random, so counts fluctuate, and a long count gives a more reliable average rate [1]. [4] Examiner insight: Convert a background total into a rate before subtracting, and write the subtraction out.

4. Reading the smooth curve, the rate is 80 at about 3.0 min, so 160 to 80 takes about 3.0 min [1]. The rate is 40 at 6.0 min, so 80 to 40 also takes about 3.0 min [1]. From 4 min (63) to 10 min (16) the rate falls by a factor of about 4, which is two half-lives in 6 min, so T½ = 3.0 min again [1]. The mean half-life is 3.0 min [1]. Count rates fluctuate randomly, so one interval could be unrepresentative; averaging several reduces the effect [1]. [5] Examiner insight: Start the intervals from different values; repeating one interval does not show the half-life is constant.

5. (a) Isotope X [1]. Gamma rays pass out of the body to the detector, whereas alpha particles would be absorbed in the tissue [1]. The 5.0 h half-life means the dose falls quickly after the scan, whereas 40 days would leave the patient radioactive much longer [1]. (b) Gamma [1], since beta radiation of ordinary energy would be stopped by a plate this thick. A thicker plate absorbs more radiation, so the count rate decreases [1]. [5] Examiner insight: Give a point on the radiation and a point on the half-life when the question names both.

6. Ratio = 0.24 / 0.060 = 4 [1]. 4 = 2², so two half-lives have passed [1]. Age = 2 × 5730 = 11 460 years, so 1.15 × 10⁴ years [1]. Assumption: the activity per gram of carbon in living material was the same when the wood was alive [1]. [4] Examiner insight: State the assumption as carbon-14 content in the past; “the measurements are accurate” is not one.

7. (a) Corrected rates are 1, 3, 0, 88, 5, 2 counts per minute [1]. The leak is at 15 m, where the corrected rate is far above the rest [1]. (b) Gamma, because it passes through soil and the pipe wall to the surface detector [1]. A short half-life, so the tracer does not remain in the ground [1]. [4] Examiner insight: Show the background subtraction. The other positions fit random counting, so do not claim a second leak.

8. Measure the background rate with the source far away, over a long time [1]. Fix the source and tube so the geometry stays the same [1]. Record counts in equal intervals, for example every 30 s, for several half-lives [1]. Subtract the background rate to get corrected rates [1]. Plot corrected rate against time and draw a smooth curve [1]. Read the time to halve from several different starting values and take the mean [1]. [6] Examiner insight: Include the background measurement and repeated halving times. Number the steps.

9. (a) λ = ln2 / T½ = 0.693 / 15 = 0.0462 h⁻¹ [1]. In seconds: 0.0462 / 3600 = 1.28 × 10⁻⁵ s⁻¹ [1]. (b) N = A / λ [1] = 5.0 × 10⁸ / 1.28 × 10⁻⁵ = 3.9 × 10¹³ [1]. (c) A = A₀ e^(−λt) = 5.0 × 10⁸ × e^(−0.0462 × 40) [1] = 7.9 × 10⁷ Bq [1]. [6] Examiner insight: Use λ in s⁻¹ when A is in Bq and N is needed. Check part (c): 40 h is 2.7 half-lives, so about 16% remains.

10. (a) λΔt = 1.5 × 10⁻⁴ × 10 = 1.5 × 10⁻³ [1]. Exact probability = 1 − e^(−0.0015) = 1.499 × 10⁻³ [1]. (b) The two agree to better than 0.1%, because λΔt is much less than 1 [1]. (c) λΔt = 1.5 × 10⁻⁴ × 1.0 × 10⁴ = 1.5, which is greater than 1, and a probability cannot exceed 1 [1]. Exact probability = 1 − e^(−1.5) = 0.78 [1]. [5] Examiner insight: Explain the approximation by the size of λΔt; a numerical comparison alone does not.

11. (a) λ = ln(A₁/A₂) / Δt [1] = ln(6.4/4.0) / 6.0 = 0.078 h⁻¹ [1]. (b) T½ = ln2 / λ = 0.693 / 0.0783 = 8.8 h [1]. (c) A = 6.4 × 10⁶ × e^(−0.0783 × 20) [1] = 1.3 × 10⁶ Bq [1]. (d) 0.050 = e^(−λt), so t = ln(20) / λ [1] = 3.00 / 0.0783 = 38 h [1]. [7] Examiner insight: Here 20 h is 2.3 half-lives, so the whole-number method fails. Keep extra figures in λ until the end.

12. (a) λ = ln2 / 5730 = 1.21 × 10⁻⁴ year⁻¹ [1]. t = ln(1/0.37) / λ [1] = 0.994 / 1.21 × 10⁻⁴ = 8.2 × 10³ years [1]. (b) λt = 1.21 × 10⁻⁴ × 1.0 × 10⁶ = 121, so the fraction is e^(−121) [1], about 10⁻⁵³, which is effectively none, so nothing measurable remains [1]. (c) Its half-life must be comparable to the age of the rock (millions of years or longer) [1]. [6] Examiner insight: Write the fraction in (b) as a power of ten, and give the age in years.

Where marks are usually lost

• Halving the measured count rate instead of the corrected count rate.
• Writing the number of half-lives as t × T½ instead of t / T½.
• Justifying an isotope with only the radiation type, or only the half-life.
• Describing the half-life experiment with no background measurement.
• HL: mixing hours and seconds when using λ and A.
• HL: treating λΔt as a probability when it is not small.

Next steps

• Review the revision notes and the study guide.
• Return to the course hub and printable checklist.
• Take the free 10-minute diagnostics, or book a free trial class.

Official syllabus

IB Diploma Programme Physics guide, first assessment 2025, published by the International Baccalaureate Organization. Sub-topic E.3, Radioactive decay.