Generated ERQ

✓ passed A.2 Forces and momentum × E.3 Radioactive decay 12 marks HL 3 passes 127.54s $0.8199

``` ## ERQ · 12 marks · Topics: A.2 Forces and momentum + E.3 Radioactive decay · Archetype: theory_application **Integration:** primary=A.2 Forces and momentum, secondary=E.3 Radioactive decay (strength: co_equal) **Stem.** A laboratory source contains a small mass of ²²⁶Ra, which alpha-decays to ²²²Rn with Q-value 4.87 MeV and half-life 1.60 × 10³ years. A silicon surface-barrier detector counts the emitted alpha particles, while a separate recoil-ion detector records the daughter nuclei. At the start of an experiment the measured activity of the source is 3.70 × 10⁵ Bq. Atomic masses: m(²²⁶Ra) = 226.025 u, m(²²²Rn) = 222.018 u, m(⁴He) = 4.0026 u; 1 u = 1.66 × 10⁻²⁷ kg = 931.5 MeV c⁻². The parent nuclei may be assumed stationary in the laboratory frame. ### Part (a) State [2 marks] · AO1 · Topic: A.2 State the law of conservation of linear momentum and state the condition under which it applies. ### Part (b)(i) Show that [3 marks] · AO2 · Topic: E.3 + A.2 Using the decay equation ²²⁶Ra → ²²²Rn + ⁴He and the atomic masses above, show that the Q-value of the decay is approximately 4.87 MeV. ### Part (b)(ii) Calculate [3 marks] · AO2 · Topic: A.2 Assuming the alpha particle carries kinetic energy equal to 4.78 MeV, calculate the recoil speed of the ²²²Rn nucleus immediately after the decay. Use non-relativistic mechanics. ### Part (c) Determine [3 marks] · AO3 · Topic: E.3 + A.2 The detector resolves the alpha energy to ±15 keV. The decay actually has two alpha branches: α₁ (to the ²²²Rn ground state, Q₁ = 4.87 MeV) and α₂ (to an excited state, Q₂ = 4.68 MeV). After 4.80 × 10³ years of operation, determine the activity of the source AND the number of α₁ events detected during a 1.00 hour counting interval, given that 94.5% of decays follow the α₁ branch and the detector subtends a solid-angle fraction of 1.20 × 10⁻³ of 4π sr. ### Part (d) Suggest [1 mark] · AO3 · ASSUMPTIONS DISCRIMINATOR Suggest one physical reason why the recoil ²²²Rn nucleus may NOT, in practice, be detected with the kinetic energy predicted in part (b)(ii), even when the alpha energy is measured exactly. --- ## Mark Scheme ### Part (a) [2 marks] — State - M1: The total linear momentum of a system is constant / remains the same before and after an interaction [no ECF] - M2: Provided no (net) external force acts on the system / system is isolated [no ECF] ### Part (b)(i) [3 marks] — Show that - M1: Mass defect Δm = 226.025 − (222.018 + 4.0026) = 4.4 × 10⁻³ u (accept 0.0044 u) [no ECF] - M2: Q = Δm × 931.5 MeV — substitution shown [no ECF] - M3: Q ≈ 4.87 MeV (student value 4.84–4.90 MeV acceptable; target given to 3 sf) [no ECF] ### Part (b)(ii) [3 marks] — Calculate - M1: Apply conservation of momentum: p(Rn) = p(α), and use E_k(α) = p²/(2m_α) to obtain p(α) = √(2 m_α E_k(α)); substitution with m_α = 4.0026 × 1.66 × 10⁻²⁷ kg and E_k = 4.78 × 1.60 × 10⁻¹³ J [no ECF] - M2: p(α) ≈ 1.01 × 10⁻¹⁹ kg m s⁻¹ (accept 1.00–1.02 × 10⁻¹⁹) [ECF from M1] - M3: v(Rn) = p / m(Rn) ≈ 1.01 × 10⁻¹⁹ / (222.018 × 1.66 × 10⁻²⁷) ≈ 2.74 × 10⁵ m s⁻¹ [ECF from M2] ### Part (c) [3 marks] — Determine (E.3 + A.2 co-equal) - M1: Decay constant λ = ln2 / T₁/₂ = ln2 / (1.60 × 10³ × 3.156 × 10⁷) = 1.37 × 10⁻¹¹ s⁻¹; OR equivalently use t/T₁/₂ = 3.00 [no ECF] - M2: A = A₀ e^(−λt) = 3.70 × 10⁵ × 2^(−3.00) = 4.63 × 10⁴ Bq (accept 4.6 × 10⁴ Bq) [no ECF] - M3: N(α₁ detected) = A × 0.945 × (1.20 × 10⁻³) × 3600 ≈ 1.89 × 10⁵ counts (accept 1.8–1.9 × 10⁵) [ECF from M2] ### Part (d) [1 mark] — Suggest (proposal + physics justification per §4.4.1) - M1: ONE valid proposal with physics-based reason, e.g.: the recoil nucleus loses kinetic energy by ionising/scattering within the source material (self-absorption) before reaching the recoil detector, because its range in solid matter is only tens of nm at ~100 keV; OR the parent ²²⁶Ra is bound in a crystal lattice, so momentum is partly transferred to neighbouring atoms rather than to a free ²²²Rn nucleus; OR a fraction of decays populate the excited ²²²Rn* state (α₂ branch), so the recoil energy is shared with a subsequent γ photon and is lower than the value calculated for the ground-state branch. ### Marker notes - (b)(i) Show-that target: 4.87 MeV given to 3 sf; student-derived 4.84–4.90 MeV acceptable per §4.5 - (b)(ii) Alternative method: use m_α v_α = m_Rn v_Rn with v_α from ½ m_α v_α² = E_k directly → v_Rn = (m_α/m_Rn) v_α; full marks if executed correctly - (c) Accept use of (½)³ = 0.125 in place of exponential since t = 3 T₁/₂ exactly - (d) Accept any ONE physically valid limitation with mechanism stated; bare "relativistic effects" without quantitative or causal reasoning is NOT sufficient (γ ≈ 1.00004 at v_α ~ 1.5 × 10⁷ m s⁻¹, so relativity is negligible here and should be rejected as the primary limitation) - ECF chain: (b)(ii) M2→M3; (c) M2→M3; (b)(i) is no-ECF (Show-that) ```

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) are AO2 (Calculate/Show that), part (c) is AO3 (Explain with causal chain), and part (d) is AO3 (Suggest with justified reasoning). The progression from knowledge → application → analysis is strict and properly sequenced.
command_structure fail

Part (c) fails the §4.4.1 Explain structure. M1 states a fact (E_k ∝ 1/m) but is not a causal-chain marking point; M2 provides the 'therefore' link but lacks explicit warrant connecting to the physical principle of kinetic energy formula. The mark scheme must show a single integrated causal chain: 'because p is equal AND E_k = p²/(2m), therefore kinetic energy is inversely proportional to mass, therefore the light alpha carries much more energy than the heavy radon.' Currently the structure is fragmented into fact + conclusion rather than explicit X→Y causality.

FIX: Rewrite Part (c) M1–M2 as a single causal chain: M1: 'Since both particles have equal momentum magnitude (from momentum conservation), and kinetic energy E_k = p²/(2m), kinetic energy is inversely proportional to mass.' M2: 'Therefore, the radon nucleus (≈55× heavier) carries ≈1/55 of the kinetic energy, so the alpha particle takes almost all the energy released.' This makes the warrant explicit: equal momentum + inverse mass dependence in the E_k formula = explains the energy distribution.
cross_theme pass

Both A.2 (Forces and momentum) and E.3 (Radioactive decay) are genuinely integrated throughout. A.2 drives the quantitative core (momentum conservation, kinetic energy from momentum, mass-energy inversion in part c), while E.3 provides the physical context (alpha decay process, Q-value, daughter nucleus recoil). Neither topic is name-checked only; both are actively used in solving each part (b–d).
discriminator pass

Part (d) explicitly asks students to 'Suggest one physical reason for this discrepancy and justify, with reference to the physics, whether the non-relativistic treatment is appropriate.' This is a clear model-critique dimension: it requires identification of a limitation (experimental/theoretical), and evaluation of an assumption (non-relativistic approximation). The mark scheme confirms M1 expects proposal of a physical limitation and M2 requires justification of the approximation's validity.
rule_of_one pass

Each part's mark count aligns well with the number of distinct award criteria. Part (a): 2 marks, 2 bullets (definition + law). Part (b)(i): 3 marks, 3 bullets (unit conversion, momentum calculation, velocity calculation). Part (b)(ii): 3 marks, 3 bullets (formula choice, substitution/show-that, total energy). Part (c): 2 marks, 2 bullets (inverse mass relationship + mass ratio linking). Part (d): 2 marks, 2 bullets (physical suggestion + relativity justification). All divergences are ≤1 and proportional to the complexity of each operation.
topic pass

The ERQ robustly tests both A.2 (Forces and momentum) and E.3 (Radioactive decay). A.2 is primary and carries the conceptual load: conservation of momentum (part a), momentum calculations (b)(i), and the inverse mass–kinetic energy relationship (c). E.3 is secondary but genuinely integral: the decay scenario, Q-value concept, and mass-energy considerations (d) require understanding radioactive decay physics. Neither topic is superficial; both are essential 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)(i)–(b)(ii) are AO2 (Calculate/Show that); parts (c)–(d) are AO3 (Explain/Suggest with evaluation). The progression from knowledge → application → synthesis is properly ordered and marked appropriately.
command_structure fail

Part (d) mark scheme violates §4.4.1 AO3 Suggest structure. The command term 'Suggest' requires 'a proposal + a brief physics-based justification,' but M2 reads as an 'Evaluate' structure (position + supporting + limiting consideration) rather than justification. Additionally, M1 offers multiple alternatives ('e.g. ... OR ...') without clarity on which constitutes the single required proposal; the mark scheme should specify one proposal per bullet or restructure to award 1 mark per distinct proposal with matching justification.

FIX: Restructure Part (d) to enforce the Suggest command term per §4.4.1: M1 should present ONE specific proposal (e.g., 'The alpha particle's speed (~1.5 × 10⁷ m s⁻¹ ≈ 0.05c) approaches the relativistic regime, requiring relativistic momentum p = γm_α v_α instead of the Newtonian value'); M2 should give the brief physics-based justification of why this matters (e.g., 'A relativistic Lorentz factor γ ≈ 1.0013 would increase the alpha momentum slightly, requiring marginally larger recoil momentum and thus slightly larger recoil kinetic energy, an effect of order 0.1%'). Alternatively, split the alternative suggestions into separate optional proposals (e.g., 'accept either: relativistic correction to alpha (M1a + M2a) OR thermal motion of parent nucleus (M1b + M2b)'), ensuring each pairing has proposal + justification. The current M2 structure conflates suggestion justification with comparative evaluation against the 2% tolerance, which is extraneous to the Suggest rubric.
cross_theme fail

While the question contextualizes the problem within radioactive decay (E.3), the actual physics required to solve parts (a)–(d) relies almost entirely on A.2 (momentum conservation, kinetic energy, and the mass-ratio argument). E.3 provides only the decay equation and masses; no genuine radioactive decay physics (half-life, activity, decay constants, branching ratios, or decay modes) is used in the solution. Part (d) mentions relativistic effects and thermal motion but does not require E.3 concepts. The two topics do not genuinely blend; E.3 is decorative framing rather than an integral physics input.

FIX: Redesign the question to require students to use both A.2 and E.3 fundamentally. For example: (i) use activity or count rate from E.3 to constrain the initial mass of radium, then (ii) apply momentum conservation to find recoil energy, (iii) relate the recoil energy to stopping power or range, and (iv) predict detection efficiency using both decay statistics (E.3) and momentum transfer (A.2). Alternatively, ask students to design a detector system where momentum resolution (A.2) must match the energy resolution needed to distinguish decay branches with different Q-values (E.3). The current variant should be split into two separate A.2-only and E.3-only questions.
discriminator pass

Part (d) explicitly asks students to 'Suggest one physical limitation of this model and evaluate how accounting for it would alter the calculated recoil energy.' This is a genuine model-critique task: students must identify an assumption (non-relativistic treatment or parent at rest) and discuss how relaxing it changes the physics. The mark scheme confirms this dual structure (M1: proposal of limitation, M2: evaluation of impact).
rule_of_one fail

Part (b)(ii) is marked as [3 marks] but contains 4 distinct award criteria (M1, M2, M3, and an implicit verification step in M3). This diverges by more than 1 from the stated mark count, violating the dimension's contract that a 3-mark Calculate/Show part must have exactly 3 award bullets.

FIX: Either: (1) combine M2 and M3 into a single criterion ('Calculate E_k(Rn) and verify total energy equals Q-value to within tolerance'), reducing to 2 substantive marks plus setup = 3 total; or (2) award 4 marks to part (b)(ii) to match the 4 distinct operations (formula, substitution, answer conversion, verification against tolerance). Option (1) is preferred for a Show-that question.
topic pass

The ERQ authentically integrates both topics. A.2 (Forces and momentum) is primary and essential: parts (a)–(d) all require conservation of momentum and the momentum–kinetic energy relationship (p²/2m). E.3 (Radioactive decay) is secondary but substantive: the alpha decay of ²²⁶Ra is the physical scenario generating the momentum and energy that must be analyzed. Part (c) and (d) explicitly demand understanding of why the decay products have the mass and momentum characteristics they do. Neither topic is nominal; both are operationally invoked.

Iteration 2 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder correctly. Part (a) is AO1 (State); parts (b)(i)–(b)(ii) are AO2 (Show that, Calculate); parts (c)–(d) are AO3 (Determine, Suggest). The progression from recall → application → analysis/evaluation is properly ordered, with command terms matching cognitive demands at each tier.
command_structure fail

Part (d) fails the AO3 Suggest command-term structure. Per §4.4.1, Suggest requires 'a proposal + a brief physics-based justification'. The mark scheme M1 provides acceptable examples, but crucially the marking criterion itself does not clearly separate or label the proposal from the justification as distinct marking components. More critically, the mark scheme awards only 1 mark total for part (d), yet the stem asks to 'Suggest one physical reason' — the single mark should be split into at least two sub-bullets: one for the proposal itself and one for the physics warrant. As written, M1 conflates proposal and justification into one undifferentiated bullet, violating the explicit REQUIRED structure that Suggest answers must exhibit proposal AND justification as separable marking points.

FIX: Restructure part (d) mark scheme as: M1 (0.5 marks): Proposal stated (e.g., 'recoil nucleus loses energy by ionisation/scattering in source material' OR 'momentum shared with lattice atoms' OR 'excited-state branch shares energy with γ photon'). M2 (0.5 marks): Physics-based justification provided (e.g., 'range ~tens of nm at ~100 keV means self-absorption occurs' OR 'crystal binding prevents free recoil' OR 'α₂ branch reduces recoil energy'). Alternatively, if 1 mark total is fixed, explicitly label the bullet as 'M1: Proposal of one physical mechanism with explicit causal link to kinetic-energy loss' to signal that both elements are required within that single mark.
cross_theme pass

Both A.2 (Forces and momentum) and E.3 (Radioactive decay) are genuinely integrated throughout the question. Conservation of momentum (A.2) is explicitly required in part (a), then actively used to solve the recoil kinematics in part (b)(ii). Radioactive decay concepts (E.3)—Q-value calculation, activity decay law, branching ratios, half-life—are central to parts (b)(i) and (c). Part (d) tests understanding of how recoil energy predictions from momentum conservation break down in realistic nuclear decay scenarios, blending both themes. Neither topic is merely name-checked; both are necessary for full solution.
discriminator pass

Part (d) explicitly asks students to 'Suggest one physical reason why the recoil ²²²Rn nucleus may NOT, in practice, be detected with the kinetic energy predicted in part (b)(ii)' — a classic model-critique prompt that requires identifying a limitation or assumption breaking. The mark scheme confirms this is a single-mark justification of a physical mechanism (self-absorption in solid matter, lattice binding, or excited-state branch), not a further calculation.
rule_of_one fail

Part (c) is marked as 3 marks but contains 4 distinct award criteria (M1: decay constant, M2: activity after time, M3: number of detected α₁ events, plus implicit final answer). Additionally, M3 depends on M2 via ECF but performs a separate composite calculation (multiplication of activity by branch ratio, solid angle fraction, and time). This exceeds the 3-mark contract by mixing decay-constant calculation, decay-law application, and event-counting logic without clear separation. The mark count–to–criterion count mismatch violates the dimension requirement.

FIX: Restructure Part (c) as two separate parts: (c)(i) [2 marks] Calculate activity after 4.80 × 10³ years using decay law (M1: λ calculation; M2: A final value). (c)(ii) [2 marks] Determine number of α₁ events detected in 1.00 hour (M1: branch ratio and solid-angle correction factor; M2: N final value with ECF from c(i)). Alternatively, if Part (c) must remain 3 marks total, explicitly combine the decay-constant step with activity into a single M1 ("Determine λ and calculate A after 4.80 × 10³ years"), leaving M2 and M3 for α₁ event counting and final answer respectively.
topic pass

The ERQ genuinely integrates both A.2 (Forces and momentum) and E.3 (Radioactive decay) as co-equal topics. Part (a) tests conservation of momentum directly; part (b)(i) applies the Q-value equation (decay physics, E.3) and uses atomic masses; part (b)(ii) uses conservation of momentum to distribute recoil energy between products (A.2 applied to a decay scenario); part (c) applies radioactive decay law λ and activity decay (E.3) while requiring momentum-based energy sharing logic; part (d) asks students to identify physical limitations affecting the momentum-predicted recoil energy. Neither topic is superficial—both are essential to solving the problem.

Planner output (stage 1)

Archetype: real-world device
Scenario: A particle detector used in nuclear physics employs a recoil system to measure the momentum of alpha particles emitted during radioactive decay. When an alpha particle is ejected from a decaying nucleus, the detector records both the particle's kinetic energy and the recoil momentum of the residual nucleus. Students are given experimental data from a radium-226 decay event and must apply conservation of momentum and energy principles to verify the decay process and evaluate the detector's accuracy under idealized assumptions.
Cross-pollination part: b(i), b(ii), d
Discriminator part: d

Part	Marks	AO	Command
a	2	AO1	State the law of conservation of momentum and define the term 'momentum'.
b(i)	3	AO2	Calculate the recoil velocity of the daughter nucleus when an alpha particle with kinetic energy 4.78 MeV is emitted from a stationary radium-226 nucleus. [Assume non-relativistic mechanics and use provided atomic masses.]
b(ii)	3	AO2	Determine the kinetic energy of the recoil nucleus and verify that total kinetic energy released in the decay equals the Q-value (6.40 MeV for ²²⁶Ra → ²²²Rn + ⁴He).
c	2	AO3	Explain why the recoil nucleus's kinetic energy is much smaller than the alpha particle's kinetic energy, despite both having equal magnitudes of momentum.
d	2	AO3	Suggest one physical limitation of the model used in parts (b) and (c), and evaluate how relativistic effects or nuclear binding energy loss might affect the momentum and energy calculations in real decay events.

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