Generated ERQ

✓ passed A.2 Forces and momentum × E.3 Radioactive decay 12 marks HL 2 passes 87.61s $0.4558

## ERQ · 12 marks · Topics: A.2 Forces and momentum + E.3 Radioactive decay **Stem.** A thin foil contains 8.0 × 10¹² atoms of radon-220 (²²⁰Rn, half-life 55.6 s) deposited at time t = 0. Each radon nucleus decays at rest by alpha emission to polonium-216 (²¹⁶Po, half-life 0.145 s). The Q-value of the ²²⁰Rn → ²¹⁶Po + α reaction is 6.40 MeV, released entirely as kinetic energy of the products. The ²¹⁶Po daughter then itself alpha-decays to ²¹²Pb with Q-value 6.91 MeV. A silicon detector subtends a solid angle equivalent to 0.50 % of 4π around the foil and registers any alpha particle that strikes it, provided the alpha's kinetic energy exceeds a threshold of 6.00 MeV. Air absorption and detector inefficiencies are negligible. Assume the foil is thin enough that recoiling Po nuclei do not escape. ### Part (a) State [1 mark] · AO1 · Topic: A.2 State the principle of conservation of linear momentum. ### Part (b)(i) Show that [3 marks] · AO2/AO3 · Topic: A.2 Show that the kinetic energy of the alpha particle emitted by a stationary ²²⁰Rn nucleus is approximately 6.28 MeV. ### Part (b)(ii) Determine [2 marks] · AO2/AO3 · Topic: A.2 Determine the kinetic energy of the alpha particle emitted in the subsequent ²¹⁶Po → ²¹²Pb decay, assuming the ²¹⁶Po is at rest at the moment of its decay. ### Part (c) Calculate [4 marks] · AO3 · Topic: A.2 + E.3 Both alpha energies from (b)(i) and (b)(ii) lie above the 6.00 MeV detector threshold, so each decay in the foil that emits an alpha toward the detector is counted. Using the fact that ²¹⁶Po has a half-life (0.145 s) much shorter than the 120 s counting window so that ²¹⁶Po decays effectively track ²²⁰Rn decays, calculate the total number of alpha counts registered by the detector between t = 0 and t = 120 s. ### Part (d) Suggest [2 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR The experimentally measured count is found to be lower than the value calculated in (c). Suggest two physically distinct reasons for this discrepancy: one arising from the momentum/recoil dynamics of the decay (A.2), and one arising from the nuclear decay chain assumptions (E.3). --- ## Mark Scheme ### Part (a) [1 mark] - Mark 1: The total linear momentum of an isolated system (no external/net force) remains constant / is conserved [ECF: no] ### Part (b)(i) [3 marks] - Mark 1: Apply momentum conservation: 0 = m_α v_α − m_Po v_Po, hence p_α = p_Po [ECF: no] - Mark 2: Use KE = p²/(2m) ⇒ KE_α / KE_Po = m_Po / m_α, so KE_α = Q × m_Po/(m_α + m_Po) = 6.40 × (216/220) MeV [ECF: yes] - Mark 3: KE_α ≈ 6.28 MeV (accept 6.28–6.29 MeV, must show ≥ 3 sig fig) [ECF: yes] ### Part (b)(ii) [2 marks] - Mark 1: KE_α = Q × m_Pb/(m_α + m_Pb) = 6.91 × (212/216) MeV [ECF: yes from (b)(i) method] - Mark 2: KE_α ≈ 6.78 MeV (accept 6.77–6.79 MeV) [ECF: yes] ### Part (c) [4 marks] - Mark 1: Decay constant λ(Rn) = ln 2 / 55.6 = 1.247 × 10⁻² s⁻¹ [ECF: no] - Mark 2: Number of ²²⁰Rn decays in 120 s: N_decay = N₀(1 − e^(−λt)) = 8.0 × 10¹² × (1 − e^(−1.496)) = 8.0 × 10¹² × 0.7760 ≈ 6.21 × 10¹² [ECF: yes] - Mark 3: Because ²¹⁶Po reaches secular-equilibrium-like tracking on a 0.145 s timescale (≪ 120 s), each Rn decay yields effectively 2 alphas in the counting window; alphas reaching detector = 2 × N_decay × (solid-angle fraction) = 2 × 6.21 × 10¹² × 5.0 × 10⁻³ [ECF: yes] - Mark 4: Total counts ≈ 6.2 × 10¹⁰ alphas (accept 6.1–6.3 × 10¹⁰, 2 sig fig) [ECF: yes] ### Part (d) [2 marks] - Mark 1 (A.2 / momentum-recoil reason — award ONE of): - Recoiling ²¹⁶Po nucleus carries momentum ≈ p_α and may travel a finite distance before its own decay; if it escapes the thin foil or is no longer aligned with the detector solid angle, the second alpha is lost. - Self-absorption: the recoiling daughter buries itself in the foil/backing, attenuating the second alpha below threshold. [ECF: no] - Mark 2 (E.3 / decay-chain reason — award ONE of, must be physically distinct from Mark 1): - The assumption that every Rn decay is matched by a Po decay within 120 s is imperfect at early times t ≲ 5 × 0.145 s, so the factor of 2 over-counts slightly. - A fraction of ²¹⁶Po may undergo a competing decay mode (or branching) rather than alpha emission to ²¹²Pb. - The 6.78 MeV alpha is close to the 6.00 MeV threshold and detector energy straggling/resolution may push some events below threshold. [ECF: no] ### Marker notes - (b)(i) alternative: use exact masses (m_α = 4.0026 u, m_Po = 216.0019 u) — accept 6.28 MeV. - (c) alternative: candidates who model the two isotopes via full Bateman equations and obtain ~6.2 × 10¹⁰ score full marks; the secular-equilibrium shortcut is acceptable because t ≫ 5 × t₁/₂(Po). - (c) common error: forgetting the factor of 2 for the daughter alpha gives ~3.1 × 10¹⁰ — award Marks 1, 2 only (max 2/4). - (d) discriminator: do NOT award both marks for two variants of the same mechanism (e.g. "absorption in foil" + "absorption in air" = 1 mark only). The two marks must come from the two distinct categories explicitly requested in the question (recoil/momentum AND decay-chain).

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder correctly. Part (a) is AO1 (State); parts (b)(i)–(b)(ii) and (c) are AO2/AO3 with Calculate/Show that/Determine command terms; part (d) is AO3 with Suggest. Progression from recall → procedural calculation → analysis/application is sound.
cross_theme fail

Part (c) is labeled 'A.2 + E.3' but uses only A.2 (conservation of momentum and kinematics applied to the decay geometry and activity decay law). E.3 (radioactive decay) appears only as the physical context for the activity calculation—the exponential decay formula N = (A₀/λ)(1 − e^(−λt)) is applied mechanically but the E.3 concept is not genuinely blended with A.2 problem-solving. Part (d) is purely E.3 assumptions and does not integrate A.2. The two topics operate in separate domains rather than both being essential to solving a unified problem.

FIX: Restructure part (c) to require explicit application of both A.2 and E.3 together. For example: 'The detector measures a momentum-dependent count rate: only alpha particles with recoil momentum within a certain range trigger the silicon sensor due to its energy threshold. Given the spread of recoil momenta from the isotropic decay (derived from A.2 momentum conservation), and the exponential decay of activity (E.3), calculate the actual number detected in 120 s.' This forces both topics to be integral to the solution, not sequential.
discriminator pass

Part (d) explicitly asks students to 'suggest two reasons for this discrepancy' between calculated and measured alpha counts. This directly requires critique of the model assumptions underlying part (c) — specifically, the assumptions of isotropic emission, perfect detector efficiency, ideal vacuum, point-source geometry, and particle energy retention. The mark scheme confirms this is a model-critique question by listing physical limitations (air absorption, detector efficiency, self-absorption, source size) rather than further calculations.
rule_of_one fail

Part (c) is allocated 3 marks but the mark scheme lists 3 distinct award criteria (geometric fraction, decay calculation, final count). However, the decay calculation in Mark 2 is exceptionally complex and involves two sub-steps (finding λ and then applying the exponential decay formula with numerical evaluation), which compressed into a single criterion creates ambiguity about whether this single mark adequately weights the two distinct operations. More critically, Part (d) is allocated 2 marks with a scheme stating "any TWO of the following, 1 mark each" — this is acceptable in isolation, but the overall structure allows candidates to pass by selecting from a pool of 5+ possible reasons. This violates the strict IB principle that each mark should correspond to a clearly defined, non-redundant operation. The scheme itself acknowledges the risk of "restatements of the same physical effect" but provides no structured differentiation to prevent double-counting. This is a design flaw that makes the marking non-uniform across candidates.

FIX: For Part (c): retain 3 marks but explicitly separate Mark 2 into two sub-operations: (Mark 2a) calculate decay constant λ = ln(2)/t₁/₂; (Mark 2b) evaluate total decays using N(t) = (A₀/λ)(1 − e^(−λt)). Alternatively, compress to 2 marks by combining geometric and decay steps. For Part (d): convert from a pool-selection format to a structured 2-part response: (Mark 1) one reason related to transport/attenuation (air absorption, range); (Mark 2) one reason related to detector or source properties (efficiency, self-absorption, finite size). Explicitly forbid awarding both marks for variations of the same mechanism.
topic fail

Part (c) nominally invokes E.3 (radioactive decay) only in its decay-rate calculation via activity and half-life, but the core physics requirement—determining the number of particles detected—is entirely geometric and kinematic (A.2: isotropic emission, solid angle, linear motion). The E.3 content is superficial formula-substitution; no decay mechanism, nuclear instability, or properties of the decay product are substantively tested. Part (d) does not remedy this, as it addresses experimental sources of error, not conceptual integration of both topics.

FIX: Restructure (c) and/or (d) to require deeper E.3 reasoning. For example: (c) could ask students to account for the decay of the polonium-216 daughter (half-life ~0.15 s) as a source of competing/secondary alpha particles and how this affects the detector response model; or (d) could explicitly require calculation of what fraction of Po-216 decays during the 120 s window and how their alpha emissions contaminate the count. Alternatively, replace (c)–(d) with a question that requires simultaneous application of momentum conservation (A.2) to the recoiling Po nucleus *and* its subsequent beta decay or gamma emission (E.3) to determine the final energy/momentum state. This would test genuine integration, not sequential application of independent topics.

Iteration 1 — passed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder across all four parts. Part (a) is AO1 (State); parts (b)(i)–(b)(ii) are AO2/AO3 (Show that/Determine); part (c) is AO3 (Calculate, requiring synthesis of decay kinetics and geometry); part (d) is AO3 (Suggest, requiring critical evaluation and discrimination between distinct physical mechanisms). The progression from recall → application → analysis is maintained throughout, and the final part explicitly demands evaluation and synthesis across two distinct topic domains.
cross_theme pass

The ERQ genuinely blends both A.2 (Forces and momentum) and E.3 (Radioactive decay). Parts (b)(i)–(c) require momentum conservation and recoil kinematics to calculate alpha energies and track decay chains; part (d) explicitly demands one discriminator from A.2 (recoil/momentum dynamics) and one from E.3 (decay-chain assumptions), ensuring both concepts are substantively used throughout, not merely name-checked.
discriminator pass

Part (d) explicitly asks students to 'Suggest two physically distinct reasons for this discrepancy: one arising from the momentum/recoil dynamics of the decay (A.2), and one arising from the nuclear decay chain assumptions (E.3).' This is a clear model-critique question that requires evaluation of assumptions and limitations in both the mechanical (recoil/momentum) and nuclear physics (decay-chain tracking) models used in parts (b)–(c). The mark scheme correctly enforces distinctness and topic separation.
rule_of_one pass

Each part's mark count aligns with the number of distinct award criteria in the scheme. Part (a): 1 mark, 1 bullet. Part (b)(i): 3 marks, 3 bullets (momentum setup, KE formula/substitution, numerical answer). Part (b)(ii): 2 marks, 2 bullets (formula application, answer). Part (c): 4 marks, 4 bullets (decay constant, decay count, two-alpha accounting, final count). Part (d): 2 marks, 2 bullets (one recoil reason, one chain reason). All divergences are ≤1, meeting the strict IB standard.
topic pass

The ERQ effectively integrates both topics. Parts (a)–(c) rigorously apply momentum conservation and kinematics (A.2) to compute alpha energies and recoil dynamics, while parts (c)–(d) require understanding of radioactive decay kinetics, half-lives, decay chains, and secular equilibrium (E.3). Part (d) explicitly demands one discrepancy explanation from A.2 (recoil/momentum) and one from E.3 (decay-chain assumptions), ensuring neither topic is superficial.

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