Generated ERQ

✓ passed E.3 Radioactive decay × B.1 Thermal energy transfers 12 marks HL 3 passes 126.5s $0.8609

``` ## ERQ · 12 marks · Topics: E.3 Radioactive decay + B.1 Thermal energy transfers · Archetype: data_response **Integration:** primary=E.3 Radioactive decay, secondary=B.1 Thermal energy transfers (strength: supporting) **Stem.** A sealed stainless-steel capsule contains 8.0 × 10¹⁸ nuclei of cobalt-60 (half-life T₁/₂ = 5.27 yr), used as a low-power heat source in a laboratory calibration rig. Each β⁻ decay (followed by two γ photons) deposits, on average, 3.15 MeV of energy in the capsule walls and surrounding argon gas. The capsule sits inside a vacuum-jacketed enclosure of external surface area A_s = 0.045 m² and effective surface heat-transfer coefficient h = 1.8 W m⁻² K⁻¹ to a laboratory at 295 K. Students log the internal gas temperature T_gas and the outer-wall temperature T_wall monthly for 5 years; selected data are shown in the table below. | t / yr | Activity A / 10¹⁰ Bq | T_gas / K | T_wall / K | |---|---|---|---| | 0.0 | 3.34 | 295 | 295 | | 0.5 | 3.13 | 338 | 332 | | 1.0 | 2.93 | 346 | 340 | | 2.0 | 2.57 | 349 | 343 | | 3.0 | 2.26 | 346 | 340 | | 5.0 | 1.74 | 342 | 337 | ### Part (a) State and Define [2 marks] · AO1 · Topic: E.3 State the exponential law of radioactive decay in terms of the decay constant λ, and define the half-life of cobalt-60. ### Part (b)(i) Calculate [3 marks] · AO2 · Topic: E.3 Using T₁/₂ = 5.27 yr, calculate the decay constant λ (in s⁻¹) for cobalt-60, and verify that the initial activity A₀ of the sample is consistent with the t = 0 value in the table. ### Part (b)(ii) Determine [3 marks] · AO2 · Topic: E.3 + B.1 Determine the total thermal energy (in joules) released inside the capsule during the first half-life of the source. ### Part (c) Explain [2 marks] · AO3 · Topic: E.3 + B.1 The temperature data show that, after an initial rise, the gas temperature stabilises near ~348 K rather than continuing to climb as decay energy accumulates. Explain, using the concept of thermal equilibrium with the surroundings, why a steady temperature is reached, and identify the dominant mode of heat transfer from the outer wall to the laboratory. ### Part (d) Evaluate [2 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR A student models the capsule as a constant-power heat source over the full 5-year measurement window. Evaluate the validity of this constant-power assumption using the data in the table. --- ## Mark Scheme ### Part (a) [2 marks] - M1: Statement of decay law N = N₀ e^(−λt) OR equivalently A = A₀ e^(−λt) with symbols defined [no ECF] - M2: Half-life defined as the time for the number of (cobalt-60) nuclei / activity to fall to half its initial value [no ECF] ### Part (b)(i) [3 marks] — Calculate - M1: Convert T₁/₂ to seconds: 5.27 × 365.25 × 24 × 3600 = 1.66 × 10⁸ s [no ECF] - M2: λ = ln 2 / T₁/₂ = 0.693 / 1.66 × 10⁸ = 4.17 × 10⁻⁹ s⁻¹ [ECF from M1] - M3: A₀ = λN₀ = (4.17 × 10⁻⁹)(8.0 × 10¹⁸) = 3.34 × 10¹⁰ Bq, matches table value [ECF from M2] ### Part (b)(ii) [3 marks] — Determine - M1: Mean activity over first half-life calculated as Ā = (A₀ − A₀/2)/ln 2 = A₀/(2 ln 2) = 2.41 × 10¹⁰ Bq, OR equivalently total decays = N₀/2 = 4.0 × 10¹⁸ [ECF from (b)(i)] - M2: Convert energy per decay to SI: E_dec = 3.15 × 10⁶ × 1.60 × 10⁻¹⁹ = 5.04 × 10⁻¹³ J [no ECF] - M3: Total thermal energy = (number of decays) × E_dec = 4.0 × 10¹⁸ × 5.04 × 10⁻¹³ ≈ 2.0 × 10⁶ J [ECF from M1, M2] ### Part (c) [2 marks] — Explain (causal chain per §4.4.1) - M1: Heat is lost from the outer wall to the surroundings at a rate that increases with the wall–lab temperature difference (e.g. P_loss = hA_s ΔT), therefore as T_wall rises the loss rate grows until it equals the decay heating rate - M2: At equilibrium, P_loss = P_decay, so T_gas stabilises; the dominant mode across the vacuum jacket / air gap to the laboratory is (free/natural) convection from the outer wall surface [accept radiation as co-dominant if justified] ### Part (d) [2 marks] — Evaluate (position + supporting + limiting per §4.4.1) - M1: Position that the constant-power assumption is **invalid** over the full 5-year window, supported by data: activity falls from 3.34 → 1.74 × 10¹⁰ Bq (≈48% decrease), so decay power P = ĀE_dec falls by the same fraction; the observed peak in T_gas near t ≈ 2 yr followed by decline to 342 K is direct evidence of falling source power - M2: Limiting/qualifying consideration that over **short intervals** (e.g. ≤ 1 yr) activity changes by only ~12%, so a piecewise- or instantaneous-constant-power treatment remains acceptable for short-term thermal modelling [accept: assumption acceptable only if t ≪ T₁/₂] ### Marker notes - Alternative method accepted for (b)(ii): integrate P(t) = λN₀ E_dec e^(−λt) from 0 to T₁/₂ giving E = ½ N₀ E_dec = 2.02 × 10⁶ J — award M1 for setting up integral, M2 for SI energy conversion, M3 for correct evaluation. - (b)(i) accept A₀ in range 3.3–3.4 × 10¹⁰ Bq. - (c) accept radiation (Stefan–Boltzmann) as identified mechanism provided the temperature dependence (∝ T⁴) is referenced; conduction NOT accepted given vacuum jacket. - (d) award M1 for any quantitative use of two table activity values OR for citing the T_gas peak-and-decline trend; award M2 for any valid restriction of the assumption's domain of validity (short Δt, or t ≪ T₁/₂, or ≤ one half-life with stated tolerance). - ECF: candidates using λ or A₀ from (b)(i) in (b)(ii) receive full ECF credit. ```

Critic verdicts

Iteration 0 — failed

ao_ladder fail

The cognitive ladder is broken: part (b)(ii) is labeled 'Show that' (AO2) but falls between two AO2 parts (b)(i) and (b)(ii), leaving no AO3 tier until part (c). The structure should progress AO1 → AO2 → AO3, but instead has AO1 → AO2 → AO2 → AO3 → AO3, violating the strict three-tier hierarchy required for a 12-mark HL ERQ.

FIX: Restructure as: (a) State/Define [2 marks, AO1]; (b) Calculate/Show that [5–6 marks, AO2—combine current b(i) and b(ii)]; (c) Explain [2 marks, AO3—current part]; (d) Evaluate [2–3 marks, AO3—current part]. Alternatively, convert part (c) to an intermediate AO2 task (e.g. 'Determine the rate of thermal energy loss at t = 3 yr from the data') and promote the current (d) to be the sole AO3 Evaluate question, then add a new final AO3 Discuss part. The present layout has three AO-tiers but out of order.
command_structure fail

Part (c) fails the AO3 Explain command-term structure: the mark scheme provides only one explicit causal link (plateau → thermal equilibrium), but §4.4.1 requires at least one 'X therefore Y' chain. While M1 does establish a causal relationship and M2 identifies the mechanism, the marking lacks a second causal step explicitly connecting the physical principle (Newton's law of cooling / heat-transfer coefficient) to why convection dominates over radiation in still air. Additionally, part (b)(ii) contains a critical internal inconsistency: the stem states the target is '1.33 × 10⁶ J', but the mark scheme's M3 and the notes section confirm the correct answer is '2.0 × 10⁶ J' given N₀ = 8.0 × 10¹⁸—this typographical error in the stem creates ambiguity for examiners and violates the principle that the published target must match the scheme's physics.

FIX: Part (c): Restructure M1 and M2 to form an explicit two-step causal chain: M1: 'Plateau at 349 K shows rate of thermal energy generation (P ∝ A) has decreased to equal the rate of heat loss to surroundings [reference data ΔT ≈ 0 at t = 3–4 yr].' M2: 'Therefore P = hA_s(T_wall − T_amb) is satisfied; the dominant loss mechanism is natural convection because the Nusselt number (and thus h) for natural convection in still air far exceeds radiation contribution at T ≈ 349 K, so convection dominates the energy balance.' Part (b)(ii): Correct the stem target from '1.33 × 10⁶ J' to '2.0 × 10⁶ J' and remove the marker-notes disclaimer—derive cleanly without post-hoc justification for a typographical error.
cross_theme pass

The ERQ genuinely integrates both E.3 (Radioactive decay) and B.1 (Thermal energy transfers) throughout the solution. Part (b)(ii) requires students to calculate total energy deposited using decay law and convert MeV to joules—both topics are essential. Part (c) explicitly demands explanation of thermal equilibrium by equating heat generation (from decay activity) to heat loss (convection/radiation), forcing students to deploy both concepts causally. Part (d) further reinforces this by requiring refinement of the thermal model itself. Neither topic is name-checked only; both are operationally required.
discriminator pass

Part (d) successfully asks students to evaluate the validity of a simplifying assumption (constant heat generation rate) and explicitly requires a suggested refinement to the thermal model. This is a genuine model-critique question that satisfies the dimension's contract.
rule_of_one fail

Part (b)(ii) is marked as a 3-mark 'Show that' question but the mark scheme lists 3 award bullets (M1, M2, M3). However, the critical issue is that the stem gives a target of 1.33 × 10⁶ J while the mark scheme notes state the correct answer should be ~2.0 × 10⁶ J. This internal inconsistency violates the contract: a 'Show that' question must have a single, unambiguous target value. The author has created confusion by stating two different targets, making it impossible for markers to apply the scheme consistently.

FIX: Correct the stem target value in Part (b)(ii) from '1.33 × 10⁶ J' to '2.0 × 10⁶ J' (or equivalently, revise N₀ in the stem to 5.33 × 10¹⁸ nuclei to yield 1.33 × 10⁶ J as stated). Delete the marker note acknowledging the discrepancy; the question itself must be internally consistent before it reaches examiners. Alternatively, if 1.33 × 10⁶ J is intentional, recompute the stem data (N₀, T₁/₂, energy/decay) to make it correct.
topic pass

The ERQ authentically integrates both topics: E.3 (radioactive decay) is the primary topic, tested via decay constant, activity decay law, and half-life calculations in parts (a)–(b)(i); B.1 (thermal energy transfers) is the secondary topic, tested via energy deposition from decay, heat loss mechanisms, and thermal equilibrium in parts (b)(ii)–(d). The two topics are genuinely interdependent: the exponential decay of activity directly drives the thermal energy generation rate, and the plateau in temperature arises from the balance between this decaying power and convective/radiative heat loss—neither topic can be solved in isolation.

Iteration 1 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder correctly: (a) AO1 with State/Define; (b)(i)–(c) AO2 with Show that/Calculate/Determine; (d) AO3 with Evaluate. The progression from recall → application → critical judgment is sound and properly sequenced.
command_structure fail

Part (d) mark scheme violates §4.4.1 AO3 structure for 'Evaluate'. The scheme presents M1 as 'position + supporting evidence' and M2 as 'limiting consideration', but this is incomplete: a proper Evaluate requires (per spec) position + supporting consideration + limiting consideration as a unified structure, not split across two marks. The current scheme awards marks for partial reasoning rather than for a single integrated evaluative judgement. Additionally, M2 conflates two distinct limiting considerations (short-interval acceptability AND evidential support from declining T_wall) without clear marking boundaries.

FIX: Restructure Part (d) to award marks on a clear 2-mark basis matching §4.4.1: M1 (1 mark) should require a stated position (e.g. 'the constant-power assumption is invalid over 5 years') plus at least one supporting consideration from the data (e.g. 'activity drops ~48% from 3.34 to 1.74 × 10¹⁰ Bq, so power falls by ~48%'). M2 (1 mark) should then require a limiting consideration (e.g. 'however, over short intervals [< 1 year] activity change is <15%, so the assumption remains acceptable for instantaneous analysis' OR 'the temperature peak at t ≈ 2 yr and subsequent decline to 337 K directly evidence falling source power, confirming non-constancy'). Ensure the two marks are logically independent so a candidate can score M1 without M2, yet both together form a complete evaluative argument per the spec.
cross_theme pass

Both E.3 (radioactive decay) and B.1 (thermal energy transfers) are genuinely integrated throughout. E.3 drives the activity decay model and power calculation; B.1 is essential—not filler—in parts (b)(ii), (c), and (d), where heat flow, temperature equilibrium, and thermal losses are central to solving the question. The question requires simultaneous use of both concepts to model the canister's temperature evolution.
discriminator pass

Part (d) explicitly asks students to evaluate a model assumption (constant thermal power) against experimental evidence from the data table. The question requires critique—students must judge whether the assumption holds, cite specific observations (activity drop, temperature profile), and discuss the scope of validity. This is a clear model-critique question that meets the dimension's contract.
rule_of_one fail

Part (c) is marked as 2 marks but the mark scheme lists 2 distinct operations (M1: equate power loss to decay heating; M2: calculate hA_s), which matches. However, part (d) is marked as 2 marks but the scheme lists only 2 broad criteria (M1: position + supporting evidence; M2: limiting consideration), which technically match in count. The critical issue is part (b)(ii): it is marked as 3 marks ('Calculate') but the scheme has only 2 substantive calculation steps (M1: mean activity; M2: energy conversion and power multiplication; M3: final answer). While M3 is listed, it is purely a numerical result from M2 without an independent operation. The scheme should restructure to show 3 distinct conceptual/operational steps (e.g., mean activity → energy per decay in joules → power in watts), not treat final arithmetic as a separate mark.

FIX: Restructure part (b)(ii) mark scheme to make 3 independent award criteria explicit: (M1) calculate mean activity over half-life interval; (M2) convert decay energy to SI units (joules); (M3) multiply mean activity by energy per decay to obtain thermal power with units. Alternatively, reduce the part to 2 marks if only two distinct operations are intended. Verify all other parts align: (a)=2 marks, 2 criteria ✓; (b)(i)=3 marks, 3 criteria ✓; (c)=2 marks, 2 criteria ✓; (d)=2 marks, 2 criteria ✓.
topic pass

The ERQ robustly integrates both topics. E.3 (radioactive decay) is the primary driver: parts (a), (b)(i), and (c)–(d) all require decay-rate calculations, half-life definitions, and activity decay laws. B.1 (thermal energy transfers) is authentically embedded: (b)(ii) and (c) require conversion of radioactive energy release into thermal power and heat-loss rates using Newton's law of cooling (hA_s coefficient), and (d) explicitly evaluates the constant-power assumption against thermal equilibrium dynamics (dT/dt = 0). Neither topic is nominal; both are operationally necessary to complete the question.

Iteration 2 — passed

ao_ladder pass

The ERQ correctly implements the cognitive ladder: (a) is pure AO1 (State/Define); (b)(i)–(b)(ii) are solidly AO2 (Calculate/Determine with numerical reasoning); (c)–(d) are AO3 (Explain causal chains, Evaluate with position + evidence + limitations). The progression is strict and unbroken.
command_structure pass

The mark scheme meets the §4.4.1 AO3 command-term structure requirements. Part (c) Explain provides an explicit causal chain ('heat loss rate increases with ΔT, therefore equilibrium is reached when P_loss = P_decay'). Part (d) Evaluate supplies position (constant-power assumption is invalid) + supporting consideration (48% activity decrease with peak-and-decline T_gas trend) + limiting consideration (acceptable over short intervals where activity change ≤ 12%). All method marks avoid formula-statement-only errors, and ECF annotations are correctly placed in (b)(i)→(b)(ii) and (b)(i)→(b)(ii) chains.
cross_theme pass

The question genuinely integrates both E (Radioactive decay) and B (Thermal energy transfers). E.3 drives the heat source (decay constant, activity, number of decays), while B.1 is actively deployed in parts (b)(ii)–(c) to calculate total thermal energy released and explain steady-state equilibrium via heat-loss rate balancing. Both concepts are essential to the solution; neither is mere name-checking or filler.
discriminator pass

Part (d) explicitly asks students to 'Evaluate the validity of this constant-power assumption' by comparing it against experimental data. This is a clear model-critique task: students must identify that the assumption breaks down over 5 years (activity falls ~48%), support this with quantitative evidence from the table, and discuss limitations/valid domains. The mark scheme confirms M1 awards for quantitative critique and M2 for qualifying/restricting the model's applicability.
rule_of_one pass

Each marked part's award-bullet count matches its mark allocation: (a) 2 marks / 2 bullets; (b)(i) 3 marks / 3 bullets; (b)(ii) 3 marks / 3 bullets; (c) 2 marks / 2 bullets; (d) 2 marks / 2 bullets. All Calculate and Determine parts include formula/method, substitution, and numerical answer-with-units as required. The scheme is internally consistent and meets the dimensional contract.
topic pass

The ERQ robustly integrates both topics. E.3 (radioactive decay) is primary: parts (a), (b)(i) test decay law and half-life; part (b)(ii) requires calculating decays from activity decay. B.1 (thermal energy transfers) is genuinely secondary but essential: part (b)(ii) converts decay count to thermal energy deposited; part (c) explicitly demands thermal equilibrium analysis (heat loss rate balancing decay power) and identification of heat-transfer mode; part (d) uses temperature data to evaluate the constant-power assumption. Neither topic is nominal—both concepts are operationally required to solve the problem.

Planner output (stage 1)

Archetype: real-world device
Scenario: A sealed container of radioactive cobalt-60 (half-life 5.27 years) undergoes beta decay, releasing energy into its surroundings. The container is thermally insulated but imperfectly; heat generated by decay gradually warms the enclosed gas. Students measure the activity of the sample over time and record temperature changes, then must reconcile the decay dynamics with thermal energy dissipation from the system.
Cross-pollination part: b(ii)
Discriminator part: d

Part	Marks	AO	Command
a	2	AO1	State the equation for radioactive decay (decay constant form) and define the half-life of cobalt-60.
b(i)	3	AO2	Calculate the decay constant λ for cobalt-60 and determine the initial activity A₀ given that the sample contains 8.0 × 10¹⁸ nuclei at t = 0.
b(ii)	3	AO2	Derive an expression for the total thermal energy released into the container over one half-life, given that each beta decay releases 3.15 MeV on average. Calculate this energy numerically.
c	2	AO3	The measured temperature rise inside the container is significantly less than predicted by the energy released from decay. Explain why thermal energy transfer to the surroundings must be considered, and identify the primary mechanism of heat loss in this scenario.
d	2	AO3	Evaluate whether the assumption that the thermal energy released is constant over time is valid. Suggest how the rate of thermal energy transfer would need to be modified to achieve equilibrium between energy generation and loss.

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