HSC Exam Practice

Sample response. The half-life (t_1/2) is the time required for half of the radioactive nuclei in a sample to decay. The decay constant (λ) is the probability of a single nucleus decaying per unit time (units: s⁻¹). They are related by: t_1/2 = ln(2)/λ = 0.693/λ.

Marking notes. 1 mark for defining half-life correctly; 1 mark for defining decay constant correctly; 1 mark for stating t_1/2 = ln(2)/λ.

Sample response. Alpha decay: emits a helium-4 nucleus ($^4_2$He); A decreases by 4, Z decreases by 2. Beta-minus decay: emits an electron (e$^-$) and an antineutrino; A unchanged, Z increases by 1. Gamma decay: emits a high-energy photon; A unchanged, Z unchanged (only energy state changes).

Marking notes. 1 mark per correctly described decay type (particle emitted + changes to A and Z). Must include antineutrino for beta-minus to receive full mark for that type.

Sample response. $^{238}_{92}$U → $^{234}_{90}$Th + $^{4}_{2}$He.

Marking notes. 1 mark for correct daughter nuclide ($^{234}_{90}$Th with correct A and Z); 1 mark for correct alpha particle notation ($^{4}_{2}$He). No antineutrino needed for alpha decay.

Sample response. At the level of an individual nucleus, decay is a quantum mechanical process: the decay constant λ gives only the probability of decay per unit time. There is no way to predict when any particular nucleus will decay — it could decay in the next instant or persist for a very long time. This unpredictability is a fundamental feature of quantum mechanics, not a lack of information. However, when a very large number of nuclei are present, the average rate of decay at any instant is proportional to the number present (dN/dt = −λN). The law of large numbers ensures that the statistical average behaviour is highly predictable, giving the precise exponential decay law N = N₀e^−λt.

Marking notes. 1 mark for identifying individual decay as probabilistic/unpredictable (quantum mechanical); 1 mark for explaining that the average rate is proportional to N (dN/dt = −λN); 1 mark for connecting this to the exponential law N = N₀e^−λt.

Sample response. Alpha particles are massive (mass number 4) and carry charge +2, which means they interact very strongly with the electrons of surrounding atoms, stripping them off and causing intense ionisation. However, this frequent interaction dissipates their energy rapidly over a very short path, so they are stopped by a few centimetres of air or the outer layer of skin. The large mass and charge create many interactions per unit length (high ionisation) but also mean they lose their energy quickly (low penetration).

Marking notes. 1 mark for linking high ionisation to large charge and mass of the alpha particle causing frequent interactions with electrons; 1 mark for explaining that this rapid energy loss per unit length results in low penetrating power (stopped quickly).

Sample response. Carbon-14 dating uses the radioactive decay of ¹⁴C (half-life 5,730 years). While an organism is alive, it continuously exchanges carbon with the atmosphere, maintaining a constant ratio of ¹⁴C to ¹²C. After death, no new carbon is taken in, so the ¹⁴C in the organism decays exponentially. By measuring the remaining ¹⁴C activity and comparing it with the known initial activity, the age can be calculated using t = ln(A₀/A)/λ. Key assumption: the ¹⁴C/¹²C ratio in the atmosphere has been constant over time. Valid age range: approximately 200 to 50,000 years (beyond this, too little ¹⁴C remains to measure accurately above background).

Marking notes. 1 mark for describing the principle (living organisms maintain ¹⁴C equilibrium; after death it decays; age from remaining activity); 1 mark for stating the key assumption (constant atmospheric ¹⁴C/¹²C ratio); 1 mark for stating the valid age range (~50,000 yr upper limit) with a reason.

(a) Decay constant [2 marks]. t_1/2 = 8.02 d × 24 h × 3600 s = 8.02 × 86400 = 6.929 × 10⁵ s [1]. λ = ln(2)/6.929 × 10⁵ = 1.00 × 10⁻⁶ s⁻¹ [1].

(b) Initial activity [1 mark]. A = λN₀ = 1.00 × 10⁻⁶ × 3.20 × 10¹⁴ = 3.20 × 10⁸ Bq [1].

(c) Nuclei after 32.08 days [2 marks]. n = 32.08/8.02 = 4.00 half-lives [1]. N = 3.20 × 10¹⁴ × (1/2)⁴ = 3.20 × 10¹⁴/16 = 2.00 × 10¹³ nuclei [1].

(d) Beta-minus decay equation [2 marks]. $^{131}_{53}$I → $^{131}_{54}$Xe + e$^-$ + $\bar{\nu}_e$. [1 mark for correct daughter nuclide $^{131}_{54}$Xe with correct A and Z; 1 mark for including both electron (e$^-$) and antineutrino ($\bar{\nu}_e$)].

Sample Band 6 response. Radioactive isotopes are used in medicine both for diagnosis (medical imaging) and therapy (cancer treatment). Their suitability is determined by the type of decay, half-life, and initial activity. For diagnostic imaging, gamma-emitting isotopes are preferred. Gamma radiation is highly penetrating and can exit the body to be detected by external cameras, allowing images of organ function to be produced. Alpha and beta radiation, being less penetrating, would be absorbed within the body and could not be detected externally. A short half-life (hours to days) is desirable for diagnostic use: a short half-life ensures the activity decreases rapidly after the procedure, minimising the patient’s long-term radiation dose. Technetium-99m (t_1/2 = 6 h) exemplifies these properties. For therapeutic applications, beta-minus emitters are often preferred because they deposit energy locally in targeted tissue (e.g. thyroid cancer treatment with iodine-131, t_1/2 = 8 days), killing cancer cells without the long-range collateral damage of gamma radiation. The exponential decay law plays a critical safety role: since activity decreases as A = A₀e^−λt, a short half-life guarantees that most of the radiation dose is delivered quickly in the targeted period, after which the isotope becomes effectively inert. A dose that would be hazardous if retained indefinitely becomes negligible after several half-lives. The risks of using radioactive isotopes in medicine include unintended radiation dose to healthy tissue and the risk of DNA damage or induced cancer. The risk is managed by selecting isotopes with short half-lives (so dose is limited), using isotopes that are quickly cleared by the body (biological half-life), and calibrating doses precisely using the activity formula A = λN. In summary, the benefits of radioactive isotopes in medicine — functional imaging and targeted cancer therapy — substantially outweigh the risks when isotopes are selected on the basis of appropriate decay type, short half-life, and controlled initial activity, informed by the exponential decay law.

Marking criteria (7 marks). 1 = correctly identifies gamma emitters as preferred for diagnostic imaging and explains why (penetrating, detectable externally). 1 = correctly identifies beta emitters (or short-range radiation) as appropriate for targeted therapy and explains why (local energy deposition). 1 = explains the role of half-life in diagnostic applications (short t_1/2 minimises ongoing patient dose). 1 = explains the role of half-life in therapy (delivers dose then decays quickly). 1 = demonstrates correct use of the exponential decay law in the context of dose management or safety (A decreases with time). 1 = identifies at least one specific risk of medical use (radiation damage, dose to healthy tissue, etc.) with explanation. 1 = response is balanced, reaches an explicit evaluative judgement, and uses precise terminology (activity, half-life, decay constant, ionising, penetrating, exponential) throughout.