Physics • Year 12 • Module 8 • Lesson 15
Radioactive Decay
Build HSC Band 5–6 extended-response technique on multi-step decay calculations, half-life determination, carbon dating, and evaluation of radiation safety arguments.
1. Multi-step calculation — strontium-90 in nuclear waste (Band 5–6)
8 marks Band 5–6
Scenario. Strontium-90 (90Sr) is a beta-minus emitter produced in nuclear fission with a half-life of 28.8 years. It is of environmental concern because it behaves chemically like calcium and can be incorporated into bone. A nuclear facility monitors a storage container initially containing 2.40 × 1015 nuclei of 90Sr.
(a) Calculate the decay constant λ of 90Sr in s−1. (2 marks)
(b) Calculate the initial activity of the sample in becquerels. (1 mark)
(c) Calculate the number of 90Sr nuclei remaining after 144 years. How many half-lives is this? (2 marks)
(d) Calculate the activity of the sample after 144 years. (1 mark)
(e) The facility requires storage until the activity drops below 1.0 × 106 Bq. Estimate the number of half-lives required (you may use trial and error or logarithms). (2 marks)
2. Data + scenario — evaluating carbon-14 dating (Band 5–6)
8 marks Band 5–6
Scenario. An archaeologist discovers a piece of charcoal from an ancient fire. The measured activity of the charcoal sample is 0.036 Bq per gram of carbon. The activity of freshly formed organic carbon (living organisms) is 0.226 Bq per gram. The half-life of 14C is 5,730 years. The table below summarises the relevant data.
| Parameter | Value | Source / formula |
|---|---|---|
| Initial activity A0 (living carbon) | 0.226 Bq/g | Atmospheric 14C equilibrium |
| Measured activity A (sample) | 0.036 Bq/g | Measured by detector |
| Half-life t1/2 | 5,730 years | Established by independent measurement |
| Decay constant λ | 1.21 × 10−4 yr−1 | λ = ln2/t1/2 |
| Age t | To be calculated | A = A0e−λt |
Q2. Analyse and evaluate the dating method and result. In your response you must:
- Use the data in the table to calculate the age of the charcoal. Show your working.
- Identify and explain the key assumption made in carbon-14 dating and assess whether it is valid for this sample.
- Explain why carbon-14 dating cannot be used to date a rock sample (e.g. granite).
- Identify one source of uncertainty in this measurement and explain how it would affect the calculated age.
- State the age range over which carbon-14 dating is considered reliable, and explain the physical reason for this limit.
Q1(a) — Decay constant (2 marks)
t1/2 = 28.8 yr × 365.25 d/yr × 24 h/d × 3600 s/h = 9.084 × 108 s [1]. λ = ln(2)/9.084 × 108 = 7.63 × 10−10 s−1 [1].
Q1(b) — Initial activity (1 mark)
A = λN0 = 7.63 × 10−10 × 2.40 × 1015 = 1.83 × 106 Bq [1].
Q1(c) — Nuclei after 144 years (2 marks)
n = 144/28.8 = 5.00 half-lives [1]. N = 2.40 × 1015 × (1/2)5 = 2.40 × 1015/32 = 7.50 × 1013 nuclei [1].
Q1(d) — Activity after 144 years (1 mark)
A = λN = 7.63 × 10−10 × 7.50 × 1013 = 5.72 × 104 Bq [1]. (Or A = A0/32 = 1.83 × 106/32 = 5.72 × 104 Bq.)
Q1(e) — Number of half-lives to reach <106 Bq (2 marks)
A(n) = A0(1/2)n < 1.0 × 106. 1.83 × 106 × (1/2)n < 1.0 × 106 → (1/2)n < 0.546 → n > 0.87. [1] So just under 1 additional half-life (~28.8 yr more ≈ 57.6 total years from now) is needed. Using logarithms: n = −log(0.546)/log(2) = 0.87 half-lives; time = 0.87 × 28.8 ≈ 25 years [1]. (Accept any correct method; must state the final time in years.)
Q2 — Sample Band 6 response (8 marks)
Age calculation: Using A = A0e−λt: t = ln(A0/A)/λ = ln(0.226/0.036)/(1.21 × 10−4) = ln(6.28)/(1.21 × 10−4) = 1.837/(1.21 × 10−4) ≈ 15,200 years [1 — correct substitution and calculation shown].
Key assumption: The fundamental assumption is that the ratio of 14C to 12C in the atmosphere has remained constant over time, so A0 (the initial activity per gram in living organisms) is the same today as when the organism died. This is approximately valid for the past ~50,000 years (based on calibration against tree rings and coral records), but it requires correction factors for periods before reliable calibration data exist [1 — identifies assumption; 1 — assesses validity].
Cannot date rocks: Carbon-14 dating requires the organism to have been part of the biological carbon cycle, continuously exchanging carbon with the atmosphere. Rocks such as granite are not organic and contain essentially no 14C. Even if trace carbon is present, it was never in equilibrium with atmospheric 14C, so there is no meaningful A0 to compare against [1].
Source of uncertainty: Background radiation from cosmic rays and other sources contributes to the detector count rate. If not subtracted, it inflates the measured activity A, making the sample appear younger than it actually is (A appears larger, so the ratio A/A0 is closer to 1 and t is smaller) [1]. Accept also: contamination by newer carbon; variability in past atmospheric 14C levels.
Age range and physical reason: Carbon-14 dating is reliable for materials between about 200 and 50,000 years old (~9 half-lives). Below ~200 years, atmospheric variation and contamination are significant compared to the small age-related change in activity. Beyond ~50,000 years, less than 0.1% of the original 14C remains; the activity is too low to measure accurately above background [1].
Marking criteria (8 marks): 1 = correct use of formula t = ln(A0/A)/λ (or equivalent); 1 = identifies constant atmospheric 14C ratio as the key assumption; 1 = assesses validity of the assumption (approximately valid, requires calibration); 1 = explains why rocks cannot be dated (no biological carbon exchange / no meaningful A0); 1 = identifies a specific source of uncertainty with direction of effect on calculated age; 1 = states reliable age range (~50,000 yr upper limit) with physical reason (too little 14C remains); 1 = response uses correct terminology (decay constant, activity, half-life, exponential); 1 = logical, evidence-based structure throughout.