Generated ERQ

✓ passed E.3 Radioactive decay × B.1 Thermal energy transfers 12 marks SL 3 passes 112.72s $0.6377

## ERQ · 12 marks · Topics: E.3 Radioactive decay + B.1 Thermal energy transfers **Stem.** A sealed capsule used in a radioisotope thermoelectric generator (RTG) contains 38 g of strontium-90 (⁹⁰Sr, molar mass 90 g mol⁻¹). Strontium-90 undergoes beta-minus decay with a decay constant of 7.63 × 10⁻¹⁰ s⁻¹, releasing on average 1.13 MeV of thermal energy per decay (combined energy from the parent decay and the prompt daughter decay of ⁹⁰Y, deposited locally). The capsule is a sphere of external surface area 1.20 × 10⁻² m² and surface emissivity 0.85, suspended in a vacuum chamber whose walls are held at an ambient temperature of 295 K. The Stefan–Boltzmann constant is σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴, the Avogadro constant is 6.02 × 10²³ mol⁻¹, and 1 eV = 1.60 × 10⁻¹⁹ J. Engineers wish to predict the steady-state surface temperature of the capsule from first principles. ### Part (a) State [2 marks] · AO1 · Topic: E.3 State what is meant by the **decay constant** of a radioactive nuclide, and state its SI unit. ### Part (b)(i) Calculate [3 marks] · AO2/AO3 · Topic: E.3 Calculate the initial activity, in Bq, of the ⁹⁰Sr in the capsule. ### Part (b)(ii) Determine [1 mark] · AO2 · Topic: E.3 Determine the thermal power, in W, generated by the decaying ⁹⁰Sr at the moment the capsule is sealed. ### Part (c) Show that [4 marks] · AO2/AO3 · Topic: E.3 + B.1 By equating the thermal power released by the radioactive decay to the **net** power radiated from the capsule surface (the chamber walls also radiate onto the capsule), show that the steady-state surface temperature of the capsule is approximately 480 K. ### Part (d) Suggest [2 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR The temperature measured experimentally on a real RTG capsule of this design is found to be **lower** than the value calculated in (c). Suggest **two** reasons, referring to assumptions made in the model, why this is the case. --- ## Mark Scheme ### Part (a) [2 marks] - M1: probability per unit time that a given nucleus will decay (accept: constant of proportionality between activity and number of nuclei, A = λN) [ECF: no] - M2: SI unit is s⁻¹ (accept "per second"; do **not** accept Bq for the constant) [ECF: no] ### Part (b)(i) [3 marks] - M1: N₀ = (m/M)·N_A = (38/90)(6.02 × 10²³) → 2.54 × 10²³ nuclei [ECF: no] - M2: uses A₀ = λN₀ with values substituted: (7.63 × 10⁻¹⁰)(2.54 × 10²³) [ECF: yes — from M1] - M3: A₀ ≈ 1.94 × 10¹⁴ Bq (accept 1.9–2.0 × 10¹⁴ Bq, 2–3 s.f., correct unit) [ECF: yes] ### Part (b)(ii) [1 mark] - M1: P = A₀ × E_decay = (1.94 × 10¹⁴)(1.13 × 10⁶ × 1.60 × 10⁻¹⁹) ≈ 35 W (accept 34–36 W; unit required) [ECF: yes from (b)(i)] ### Part (c) [4 marks] - M1: states/uses correct net radiative balance: P = εσA(T⁴ − T_amb⁴) (must include the T_amb⁴ term — this is the integration with B.1) [ECF: no] - M2: equates P from radioactive decay (part b(ii)) to εσA(T⁴ − T_amb⁴), showing the coupling of decay power to thermal radiation [ECF: yes — uses P from (b)(ii)] - M3: correct rearrangement and substitution: T⁴ = P/(εσA) + T_amb⁴ = 35/[(0.85)(5.67 × 10⁻⁸)(1.20 × 10⁻²)] + (295)⁴ = 6.05 × 10¹⁰ + 7.58 × 10⁹ ≈ 6.81 × 10¹⁰ K⁴ [ECF: yes] - M4: T ≈ 482 K, sufficient working shown to justify the "≈ 480 K" target (must show value before rounding) [ECF: yes] ### Part (d) [2 marks] Award 1 mark each for any TWO distinct, physically reasoned points (must reference a model assumption AND the direction of error): - Not all 1.13 MeV per decay is deposited locally — antineutrinos carry energy away undetected, so actual P_thermal < 35 W → lower T. [ECF: no] - Real chamber is not a perfect vacuum / capsule contacts supports → additional conduction or convection losses not included in pure-radiation model → lower T. [ECF: no] - Emissivity of 0.85 may be an underestimate after surface oxidation/ageing → more efficient radiation → lower T. [ECF: no] - Chamber walls may not behave as a perfect blackbody enclosure at 295 K (geometric view factor < 1 was assumed = 1), but candidate must argue this *lowers* T (e.g. cooler effective surroundings via re-radiation losses to external environment). [ECF: no] ### Marker notes - (b)(i): Alternative — candidates who compute moles separately (0.422 mol) then nuclei then activity earn full marks if all steps shown. - (c): Candidates who use P = εσAT⁴ (i.e. omit the T_amb⁴ term) cap at M3 only — they obtain T ≈ 496 K, which does **not** match the "show that ≈ 480 K" target; this failure to integrate B.1's net-radiation form is the discriminator. Do **not** award M1 or M4 in this case. - (c): Accept reverse working — substituting T = 480 K to verify P ≈ 32 W matches (b)(ii) within rounding — provided the radiative-balance equation including T_amb⁴ is explicitly written. - (d): Do **not** accept vague answers ("experimental error", "heat loss") without identifying the specific assumption violated. Reject "decay constant changes with temperature" (unphysical for β-decay). The neutrino-energy-loss point is the strongest E.3-anchored response and should be credited generously when articulated.

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder. Part (a) is AO1 (Define), parts (b)(i)–(b)(ii) and (c) progress through AO2/AO3 (Calculate/Determine/Show that), and part (d) is AO3 (Suggest). Command terms match cognitive levels and build systematically toward the final discriminator.
cross_theme pass

Part (c) genuinely integrates both E.3 (radioactive decay → thermal power P from part b) and B.1 (Stefan–Boltzmann thermal radiation law). The decay power is the input driving the thermal equilibrium calculation, and the Stefan–Boltzmann equation is essential to solve for temperature. Neither concept is name-checked or filler; both are structurally required.
discriminator pass

Part (d) explicitly asks students to 'Suggest two reasons, based on the assumptions of the model, why this is the case' — a direct critique of the model's assumptions. The mark scheme confirms this is a model-critique question, requiring students to identify failed assumptions (emissivity, energy absorption, decay history, conduction, residual gas) rather than perform calculations. This is genuine critical evaluation of the physics model.
rule_of_one fail

Part (d) mark count (2 marks) does not match the number of distinct award criteria in the mark scheme. The scheme lists 5 separate valid reasons as bullet points, but only 2 marks are allocated. The instruction 'Award [1] for each distinct valid reason, max [2]' clarifies the scoring but does not resolve the dimensional mismatch: there are 5 distinct operations/criteria listed, yet only 2 marks assigned. This violates the contract that mark count should match award-bullet count (within ±1).

FIX: Either: (A) reduce part (d) to list exactly 2 exemplar reasons with [1] mark each, or (B) increase part (d) to [5 marks] and award [1] for each of the 5 distinct reasons listed (with 'max [2]' removed). Option (A) is recommended for an SL question. Alternatively, restructure as '[2 marks] Suggest **two** reasons...' with only 2 distinct reasons pre-written in the scheme, removing the 5-bullet list.
topic pass

The ERQ genuinely integrates both topics. Part (c) explicitly requires E.3 concepts (activity, decay energy) to calculate thermal power, then applies B.1 (Stefan–Boltzmann radiation law) to find equilibrium temperature. Neither topic is superficial; both are essential to solving the scenario. Part (d) reinforces assumptions bridging both domains.

Iteration 1 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder correctly. Part (a) is AO1 (Define), part (b) progresses through AO2 (Calculate/Determine), part (c) is AO2/AO3 (Calculate with justification of equilibrium condition), and part (d) is AO3 (Suggest with reasoning). The command terms and cognitive levels increase appropriately.
cross_theme fail

The question fails the dimension requirement. While both topics are present (E.3 and B.1 differ in first character), the Stefan–Boltzmann thermal radiation law is only invoked in part (c) as a standalone calculation tool; it is not genuinely integrated with radioactive decay physics. Parts (a)–(b) are purely E.3, part (c) uses B.1 mechanically without requiring decay concepts within the thermal balance, and part (d) relegates the blending to assumptions/limitations rather than core problem-solving. The two physics concepts operate sequentially, not synergistically.

FIX: Restructure to require active coupling of both concepts in the main solution. For example: (c) could ask students to derive the equilibrium condition by equating the power from radioactive decay (E.3) to radiated power (B.1) in a single step, making clear that neither concept can be omitted. Alternatively, add a part that asks how the surface temperature changes as the sample decays over time, forcing students to combine the time-dependent activity A(t) = A₀e^(−λt) with the Stefan–Boltzmann equation in a unified thermal model.
discriminator pass

Part (d) explicitly asks students to critique the model by suggesting reasons for discrepancy between calculated and measured temperature. It requires discussion of model assumptions (radiation-only, complete energy absorption) and their breakdown in practice, fulfilling the dimension's requirement for model critique rather than further calculation.
rule_of_one fail

Part (b)(ii) is marked as 2 marks but contains only 1 distinct operation (power = activity × energy per decay), violating the dimension rule that mark count must match award-criterion count within ±1. The substitution and numerical answer should be separated into distinct criteria, or the mark allocation must be reduced to 1.

FIX: Either: (1) Split part (b)(ii) into 3 marks with separate criteria for formula identification, substitution with unit conversion (MeV → J), and final answer with unit; OR (2) Reduce part (b)(ii) to 1 mark and reassign the freed mark(s) elsewhere to maintain the 12-mark total. The current 2-mark allocation with only 1 formula-based operation is under-specified.
topic pass

The ERQ authentically tests both topics. Part (b)(i–ii) and (d) require E.3 concepts (activity, decay constant, energy per decay). Parts (c) and (d) critically depend on B.1 (Stefan–Boltzmann radiation law, thermal equilibrium, heat-loss mechanisms). The thermal power from decay is physically coupled to radiative energy balance; neither topic is superficial or nominally invoked.

Iteration 2 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder: Part (a) is AO1 with State (defining decay constant); Parts (b)(i–ii) and (c) progress through AO2/AO3 with Calculate/Determine/Show that (requiring decay activity, power calculation, and algebraic proof of the target temperature); Part (d) is AO3 with Suggest (requiring reasoned, assumption-aware evaluation of model discrepancies). The integration of E.3 and B.1 is well-structured, with the net-radiation formula P = εσA(T⁴ − T_amb⁴) explicitly demanded in (c) as the discriminator. All parts respect the cognitive hierarchy.
cross_theme pass

The ERQ genuinely blends E.3 (Radioactive decay) and B.1 (Thermal energy transfers). Part (c) is the critical integration point: students must use the activity and decay power from E.3, then equate it to the net radiative power εσA(T⁴ − T_amb⁴) from B.1. The Stefan–Boltzmann net-radiation formula with the ambient-temperature term is explicitly required in the mark scheme (M1), and omitting it prevents achieving the target answer of 480 K. Both physics concepts are functionally necessary to solve the problem, not merely name-checked.
discriminator pass

Part (d) explicitly asks students to 'Suggest two reasons, referring to assumptions made in the model, why this is the case' (why measured T is lower than calculated). This is a genuine model-critique task: students must identify violated assumptions AND connect them to the directional error. The mark scheme confirms each point must reference an assumption AND justify the direction of the discrepancy.
rule_of_one fail

Part (c) is marked out of 4, but the mark scheme lists only 4 award criteria (M1–M4), which nominally matches. However, the structural problem is that M1 is a "states/uses" criterion (definition of the equation form) while M2–M4 are calculation steps. The scheme conflates definitional and computational work: M1 awards for writing the net-radiation law (not a calculation operation), then M2–M4 perform the calculation proper. For a "Show that" question worth 4 marks in AO2/AO3, the marking should allocate marks transparently to distinct *operations* (substitute, rearrange, compute intermediate, final answer). As written, M1 does not represent an operation—it is a conceptual statement—making the total operation count effectively 3 (M2, M3, M4), not 4. This violates the dimension's requirement that mark count must align with distinct operation count within ±1.

FIX: Restructure part (c) to 3 marks and rewrite as: (1) M1: State or use the net radiative-balance equation P = εσA(T⁴ − T_amb⁴) [conceptual], (2) M2: Equate and rearrange to T⁴ = P/(εσA) + T_amb⁴, substituting values correctly [manipulation + substitution], (3) M3: Compute T and verify it equals ≈480 K [final answer with units and rounding justification]. Alternatively, keep 4 marks but restructure: M1 = write equation, M2 = substitute P value from (b)(ii), M3 = substitute all thermal and geometric constants, M4 = compute T to 480 K. The current scheme's implicit operation count (3 calculations after stating the law) does not match the 4-mark allocation when the first mark is purely definitional.
topic pass

The ERQ robustly tests both E.3 (radioactive decay) and B.1 (thermal energy transfers). Parts (a)–(b) establish decay constant, activity, and thermal power output from ⁹⁰Sr decay (E.3). Part (c) is the critical integration point: it requires equating the decay power to Stefan–Boltzmann net radiation P = εσA(T⁴ − T_amb⁴), which is the defining concept of B.1 thermal transfer via radiation. The mark scheme explicitly confirms that omitting T_amb⁴ (pure-radiation-only model) fails to reach the ≈480 K target and caps at M3, demonstrating that B.1's net-radiation form is genuinely necessary. Part (d) discriminates further by inviting discussion of model assumptions (neutrino losses, vacuum quality, emissivity aging), anchoring back to both topics. Neither topic is superficial or nominally mentioned.

Planner output (stage 1)

Archetype: real-world device
Scenario: A nuclear waste container emits heat via radioactive decay of a long-lived isotope. The container's surface temperature rises above ambient due to the thermal energy released by decaying nuclei. Students must relate the decay rate and energy per decay to the observed temperature increase, then evaluate assumptions about heat dissipation and insulation that affect the container's thermal equilibrium.
Cross-pollination part: c
Discriminator part: d

Part	Marks	AO	Command
a	2	AO1\|AO2	Define the term 'activity' and state its SI unit
b(i)	3	AO2\|AO3	Calculate the initial activity of the sample given the number of nuclei and the decay constant
b(ii)	2	AO2	Determine the total thermal power generated by the decaying isotope per unit time using the activity and energy per decay
c	3	AO2\|AO3	Calculate the equilibrium surface temperature of the container given the power output, ambient temperature, and thermal properties (emissivity, surface area, Stefan–Boltzmann constant)
d	2	AO3	Suggest how heat loss mechanisms (convection, conduction through the container support, and radiation) might differ from the simplified model, and evaluate whether the calculated temperature is likely an overestimate or underestimate

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