Generated ERQ

✓ passed E.3 Radioactive decay × A.3 Work, energy and power 13 marks HL 3 passes 117.57s $0.8204

``` ## ERQ · 13 marks · Topics: E.3 Radioactive decay + A.3 Work, energy and power · Archetype: modeling_and_assumptions **Integration:** primary=E.3 Radioactive decay, secondary=A.3 Work, energy and power (strength: supporting) **Stem.** A radiotherapy unit at a regional hospital uses a sealed cobalt-60 source to treat deep-seated tumours. Cobalt-60 decays by beta emission to an excited nickel-60 nucleus, which then emits two gamma photons of average total energy 1.25 MeV per decay. The half-life of Co-60 is 5.27 years. When commissioned, the source had an activity of A₀ = 8.5 × 10¹⁴ Bq. The medical physicist models the source as a constant-power gamma emitter for the purpose of planning a patient's 7-day treatment course, during which the patient receives a continuous fractionated exposure. All emitted gamma energy is assumed to be absorbed by tissue inside the treatment field. ### Part (a) State [2 marks] · AO1 · Topic: E.3 State the relationship between the activity A of a radioactive sample and the number N of undecayed nuclei present, and define the decay constant λ. ### Part (b)(i) Show that [3 marks] · AO2 · Topic: E.3 Using the relationship dN/dt = −λN, show that the decay constant of Co-60 is λ ≈ 4.17 × 10⁻⁹ s⁻¹. ### Part (b)(ii) Calculate [2 marks] · AO2 · Topic: E.3 Calculate the activity of the source after a 7-day treatment course, taking the initial activity as 8.5 × 10¹⁴ Bq. ### Part (c) Determine [3 marks] · AO2 · Topic: E.3 + A.3 Each decay deposits, on average, 1.25 MeV of gamma energy in the patient. Treating the activity as effectively constant over the 7 days, determine the total energy, in joules, delivered to the patient and hence the mean power deposited. ### Part (d) Evaluate [3 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR · Topic: E.3 + A.3 Evaluate the assumption, made in part (c), that the treatment power remains constant. Refer to the clinical implications of activity decay for dose uniformity over both the 7-day treatment window and the multi-year operational lifetime of the source. --- ## Mark Scheme ### Part (a) [2 marks] - M1: Activity A = λN (rate of decay is proportional to number of undecayed nuclei present) [no ECF] - M2: λ is the probability per unit time that a given nucleus will decay (accept: constant of proportionality between A and N, with units s⁻¹) [no ECF] ### Part (b)(i) [3 marks] — Show that - M1: Identifies λ = ln 2 / T₁/₂ from integration of dN/dt = −λN OR uses N = N₀ e^(−λt) with half-life condition [no ECF] - M2: Converts T₁/₂ = 5.27 yr × 3.156 × 10⁷ s/yr = 1.663 × 10⁸ s [no ECF] - M3: λ = 0.693 / 1.663 × 10⁸ = 4.17 × 10⁻⁹ s⁻¹ (accept 4.16–4.18 × 10⁻⁹ s⁻¹) [no ECF] *Show-that target given to 3 sig figs; subsequent parts MUST use λ = 4.17 × 10⁻⁹ s⁻¹.* ### Part (b)(ii) [2 marks] — Calculate - M1: t = 7 × 86 400 = 6.05 × 10⁵ s; λt = 4.17 × 10⁻⁹ × 6.05 × 10⁵ = 2.52 × 10⁻³ [ECF from (b)(i) λ value] - M2: A = 8.5 × 10¹⁴ × e^(−2.52 × 10⁻³) = 8.48 × 10¹⁴ Bq (accept 8.47–8.49 × 10¹⁴ Bq; accept 8.5 × 10¹⁴ Bq if student recognises decay is negligible) [ECF from M1] ### Part (c) [3 marks] — Determine - M1: Total number of decays N_d = A × t = 8.5 × 10¹⁴ × 6.05 × 10⁵ = 5.14 × 10²⁰ decays [ECF from (b)(ii) for A] - M2: Energy per decay = 1.25 × 10⁶ × 1.60 × 10⁻¹⁹ = 2.00 × 10⁻¹³ J; total energy E = 5.14 × 10²⁰ × 2.00 × 10⁻¹³ = 1.03 × 10⁸ J [ECF from M1] - M3: Mean power P = E/t = 1.03 × 10⁸ / 6.05 × 10⁵ = 170 W (accept 1.7 × 10² W) [ECF from M2] ### Part (d) [3 marks] — Evaluate (position + supporting + limiting per §4.4.1) - M1 (Position): The constant-power assumption is justified for the 7-day treatment window but fails over the operational lifetime of the source. - M2 (Supporting consideration): Since λt ≈ 2.5 × 10⁻³ ≪ 1 [ECF from (b)(ii) λt value], activity falls by only ~0.25% over one week — negligible compared to typical clinical dose tolerances (2–5%) — so the power delivered on day 7 is essentially identical to day 1, validating the model for one treatment course. - M3 (Limiting consideration): However, A(t) = A₀ e^(−λt) is exponential, so over months and years the power falls substantially (≈12% per year, 50% after one half-life of 5.27 years); the physicist must therefore progressively lengthen exposure times or replace the source to maintain dose uniformity across patients treated throughout the source's lifetime. ### Marker notes - Alternative method accepted for (c): computing instantaneous power P = A × E_γ = 8.5 × 10¹⁴ × 2.00 × 10⁻¹³ = 170 W directly, then E = P × t = 1.03 × 10⁸ J — full marks if both quantities stated. - (d) accept any of: ignores attenuation/scattering in tissue, neglects beta-particle energy (absorbed in source capsule, so model is acceptable here), assumes 100% absorption (in reality only fraction deposited in target volume), neglects geometric inverse-square divergence, ignores source self-absorption increasing with age. - Show-that target in (b)(i): λ = 4.17 × 10⁻⁹ s⁻¹ given to 3 sig figs; student-derived 4.16–4.18 × 10⁻⁹ s⁻¹ acceptable. - Sig-fig penalty applied globally per §4.2, not per part. ```

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder. Part (a) is AO1 with State (AO1 command). Parts (b)(i)–(c) are AO2 with Show that/Calculate/Determine (AO2 commands). Part (d) is AO3 with Evaluate (AO3 command). The progression from knowledge → application → higher-order reasoning is intact and appropriate for 13-mark HL integration.
command_structure fail

Part (d) mark scheme violates §4.4.1 AO3 'Evaluate' structure. The three bullets (M1, M2, M3) do NOT form a coherent position + supporting consideration + limiting consideration triplet as required. M1 presents a position and supporting argument together; M2 attempts to introduce a 'limiting' perspective but confuses it with mechanism and then reverts to a supporting point; M3 is a separate clinical implication rather than a discrete limiting consideration paired with the position. The structure is muddled and does not match the mandatory Evaluate framework.

FIX: Restructure Part (d) mark scheme to explicitly separate three discrete, sequential marking points: M1 (position): 'The assumption is justified because activity change over 7 days is ~0.25%, far smaller than clinical dose tolerances (2–5%).' M2 (supporting consideration): 'The fractional decay λt ≈ 0.00252 ≪ 1 validates the constant-power model for the treatment-week timescale; patients on day 1 and day 7 experience negligible dose difference (~0.25%).' M3 (limiting consideration): 'However, over the source lifetime (multiple half-lives), exponential decay is cumulative and beam power falls significantly; thus the assumption fails on monthly/yearly timescales and facility procedures must lengthen exposure times or replace the source to maintain prescribed dose uniformity.' This separates the position and one strong supporting fact (M1–M2) from an explicit, contrasting limitation (M3) showing where the model breaks down.
cross_theme pass

The question successfully integrates both E.3 (radioactive decay) and A.3 (work, energy and power). Part (c) genuinely requires BOTH concepts: students must use the decay constant and activity equations (E.3) to find mean activity, then apply power = energy per unit time and E = Pt (A.3) to calculate total energy deposited. Part (d) evaluates the constant-power assumption by quantifying decay over 7 days and discussing clinical implications for dose uniformity—both topics remain active in the reasoning.
discriminator pass

Part (d) is explicitly framed as 'Evaluate the assumption that the beam power delivered by the source is constant' and requires students to critique the model's validity by examining the fractional decay over 7 days, discussing limitations (cumulative decay over source lifetime), and explaining clinical implications for dose uniformity. This is a genuine model-critique task, not a calculation.
rule_of_one fail

Part (c) is marked as [3 marks] but lists only 3 award bullets (M1, M2, M3). However, M1 conflates two conceptually distinct operations: (i) calculating mean activity AND (ii) computing energy per second from activity and photon energy. These should be separate criteria. The mark scheme lists energy-per-second computation as part of M1, but it is logically a transition step before multiplication by time. This creates ambiguity: is M1 mean activity alone (1 op), or mean activity + power calculation (2 ops)? The structure violates the dimensionality contract.

FIX: Reorganise Part (c) into 4 marks with 4 distinct bullets: M1 = Calculate mean activity (or justify using A₀); M2 = Convert photon energy to joules; M3 = Calculate power (activity × energy per photon); M4 = Calculate total energy (power × time). Alternatively, reduce to 2 marks if the part is intended to be formula + substitution + answer only, but then combine mean activity and power into one bullet and omit explicit conversion step. Current 3-mark structure with 3 bullets that each conflate multiple operations is ambiguous.
topic pass

The ERQ successfully integrates both topics. E.3 (Radioactive decay) is primary and tested throughout (a)–(d): decay constant, activity decay law, and quantitative use of λ and half-life. A.3 (Work, energy and power) is explicitly required in part (c), where students must convert photon energy to joules, calculate power (energy per unit time), and determine total energy deposited over the time interval—all core A.3 concepts. Part (d) evaluates the constant-power assumption, which bridges both topics conceptually. Neither topic is superficial; both are genuinely necessary to solve the problem.

Iteration 1 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder correctly. Part (a) is AO1 (State/Define), parts (b) and (c) are AO2 (Show that/Calculate/Determine), and part (d) is AO3 (Evaluate). The command terms, cognitive levels, and mark distribution align with IB Physics guidelines for a 13-mark HL extended-response question.
command_structure fail

Part (d) violates §4.4.1 AO3 structure for Evaluate. The mark scheme provides position + supporting consideration + limiting consideration, but M1 conflates position and supporting consideration into a single bullet, and M2 is labeled 'limiting consideration' without explicit warrant linking to physical principles. Additionally, M1 cites '[ECF from (b)(i)/(b)(ii)]' but should clarify which specific numerical result is being recycled per §4.6.

FIX: Restructure Part (d) as follows: M1 (Position): The constant-power assumption is valid for the 7-day treatment window. M2 (Supporting consideration): Since λt ≈ 2.5 × 10⁻³ ≪ 1 [ECF from (b)(ii)], the activity decays by only ~0.25% over one week, which is negligible compared to clinical dose tolerances (2–5%). Therefore patients treated on day 1 and day 7 receive essentially the same dose rate, justifying the constant-power model for short timescales. M3 (Limiting consideration): However, radioactive decay is exponential: A(t) = A₀ e^(−λt). Over the operational lifetime (months to years, multiple half-lives), the beam power falls substantially (~12% per year, reaching 50% after 5.27 years). This violates the constant-power assumption; the physicist must progressively increase exposure times or replace the source to maintain dose uniformity across patient cohorts treated over the source's lifetime.
cross_theme pass

Both theme letters differ (E vs A), and both physics concepts are genuinely integrated. Part (c) uses A.3 (power as energy per unit time, P = E/t) to calculate total energy delivered, which is central to the solution. Part (d) then uses E.3 (exponential decay, λt) to evaluate the constant-power assumption's validity—demonstrating why A.3's power model works short-term but fails long-term due to E.3 decay physics. The secondary topic is woven throughout the problem-solving chain, not mere window-dressing.
discriminator pass

Part (d) explicitly asks students to evaluate a key assumption (constant source power over 7 days) and to discuss its validity at different timescales. The mark scheme requires both justification for the short-term assumption AND a critical limitation for extended operational lifetime, which is model critique in the required spirit.
rule_of_one pass

All parts have mark counts that align well with their award criteria. Part (a) has 2 marks and 2 bullets; (b)(i) has 3 marks and 3 bullets; (b)(ii) has 2 marks and 2 bullets; (c) has 4 marks and 4 bullets; (d) has 2 marks and 2 substantive bullets. No divergence exceeds 1.
topic pass

The ERQ authentically integrates both topics. E.3 (radioactive decay) is primary throughout: activity definition, decay constant derivation, exponential decay law, and half-life all appear centrally in parts (a)–(d). A.3 (work, energy and power) is genuinely embedded in part (c), where the energy per photon (1.25 MeV) must be converted to joules, the activity is used to calculate delivered power P = A₀E, and energy is obtained from E = Pt. Part (d) evaluates the constant-power assumption, which directly requires understanding how power relates to the time-dependent activity. Neither topic is nominal; both are essential to solving the problem.

Iteration 2 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder strictly. Part (a) uses State (AO1), parts (b)(i)–(c) use Show that/Calculate/Determine (AO2), and part (d) uses Evaluate (AO3). Command terms and cognitive demand increase monotonically with no inversions or missing tiers.
command_structure fail

Part (d) mark scheme violates §4.4.1 AO3 structure for 'Evaluate'. M1 states a position but M2 and M3 are not explicitly structured as 'supporting consideration' + 'limiting consideration' with clear physics-based warrant. M2 conflates numerical justification (λt ≪ 1) with clinical context in a single bullet; M3 lacks explicit linkage between exponential decay mechanism and operational consequence. The marking points do not cleanly separate position → support → limit as required.

FIX: Restructure part (d) M1–M3 to explicitly separate: (M1) Position: constant-power assumption valid short-term, invalid long-term. (M2) Supporting: λt ≈ 2.5 × 10⁻³ ≪ 1 therefore activity decay negligible over 7 days (~0.25%), within clinical tolerance, validating assumption for single course. (M3) Limiting: exponential decay A = A₀e^(−λt) causes substantial power loss over operational lifetime (~12% yr⁻¹), requiring progressive lengthening of exposure times to maintain dose uniformity across multi-year patient cohort, thus assumption fails operationally. Include explicit 'therefore' causal chains in M2 and M3 linking physics (exponential decay rate, half-life value) to clinical impact (dose uniformity).
cross_theme pass

Both E.3 (radioactive decay) and A.3 (work, energy and power) are genuinely integrated throughout the question. Parts (b) and (c) use decay constant and activity decay (E.3) to calculate total energy and mean power (A.3). Part (d) critically evaluates the constant-power assumption by comparing the exponential decay model (E.3) against clinical power uniformity requirements (A.3), requiring synthesis of both concepts.
discriminator pass

Part (d) explicitly asks the student to 'Evaluate the assumption... that the treatment power remains constant' and requires discussion of both the 7-day window (where the assumption holds) and the multi-year operational lifetime (where it fails). The mark scheme structure (position + supporting + limiting) and model notes confirm this is a genuine model-critique question, not a calculation repeat.
rule_of_one pass

All parts satisfy the mark-count-to-award-criteria contract. Part (a) has 2 marks and 2 bullets; (b)(i) has 3 marks and 3 bullets (formula derivation, unit conversion, numerical result); (b)(ii) has 2 marks and 2 bullets (time calculation with λt, exponential decay application); (c) has 3 marks and 3 bullets (decay count, energy conversion, power division); (d) has 3 marks and 3 bullets (position, supporting, limiting). No divergence exceeds 1.
topic pass

The ERQ authentically integrates both topics. E.3 (radioactive decay) is primary and pervasive: parts (a)–(b) establish decay kinetics, part (c) uses activity decay to calculate total decays, and part (d) evaluates the exponential decay model. A.3 (work, energy and power) is genuinely secondary but structurally necessary: part (c) requires conversion of decay energy (MeV) to joules and explicit calculation of power (energy/time), and part (d) critically examines constancy of power delivery over timescales where decay becomes significant. Neither topic is superficial; both are required for the physical scenario to work.

Planner output (stage 1)

Archetype: real-world device
Scenario: A radiotherapy facility uses a Co-60 gamma source to treat tumours. The source has initial activity A₀ and decays over time. Clinicians must balance therapeutic dose delivery (energy deposited in tissue) against safe exposure limits. Students analyse the decay kinetics, calculate energy released, and evaluate whether a simplified constant-power model holds over a treatment week.
Cross-pollination part: c
Discriminator part: d

Part	Marks	AO	Command
a	2	AO1	State the relationship between activity and the number of undecayed nuclei, and define the decay constant.
b(i)	3	AO2	Show that the activity of the Co-60 source after time t is given by A(t) = A₀ e^(−λt), given that the half-life is 5.27 years.
b(ii)	2	AO2	Calculate the activity remaining after 7 days if the initial activity is 8.5 × 10¹⁴ Bq.
c	3	AO2\|AO3	Each decay of Co-60 releases a mean gamma-ray energy of 1.25 MeV. Determine the total energy delivered to the patient during the 7-day treatment period, assuming all emitted gammas are absorbed.
d	3	AO3	Evaluate the assumption that the treatment power (energy per unit time) remains constant over the 7 days. Comment on the clinical implications of activity decay for dose uniformity.

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