Generated ERQ

✓ passed E.2 Quantum physics (HL) × C.2 Wave model 12 marks HL 3 passes 143.78s $0.8766

## ERQ · 12 marks · Topics: E.2 Quantum physics + C.2 Wave model · Archetype: modeling_and_assumptions **Integration:** primary=E.2 Quantum physics, secondary=C.2 Wave model (strength: supporting) **Stem.** A group of students investigates the photoelectric effect using a clean potassium photocathode mounted inside an evacuated phototube. Monochromatic light of variable frequency is directed onto the cathode, and the stopping potential V_s is measured by adjusting a reverse bias until the photocurrent just falls to zero. Two of the recorded data points are: at f₁ = 8.2 × 10¹⁴ Hz, V_s = 2.10 V, and at f₂ = 6.8 × 10¹⁴ Hz, V_s = 1.20 V. The students intend to use these data to determine Planck's constant and the work function of the cathode, and then to compare the photon model with the predictions of classical wave theory. Assume the electronic charge is e = 1.60 × 10⁻¹⁹ C. ### Part (a) Define and State [2 marks] · AO1 · Topic: E.2 Define the work function of a metal, and state the condition on the incident photon that must be satisfied for photoelectric emission to occur. ### Part (b)(i) Calculate [3 marks] · AO2 · Topic: E.2 Using the two data points given in the stem, calculate a value for Planck's constant. ### Part (b)(ii) Determine [2 marks] · AO2 · Topic: E.2 Using your value of h from (b)(i) and one of the data points, determine the work function of the potassium cathode. Express your answer in eV. ### Part (c) Explain [3 marks] · AO3 · Topic: E.2 + C.2 Classical wave theory predicts that the kinetic energy of emitted photoelectrons should increase with the intensity of the incident light, and that emission should occur at any frequency provided the light is sufficiently intense. Explain how the experimental observations in this investigation contradict the classical wave model, and how the photon model resolves the contradiction. ### Part (d) Suggest [2 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR · Topic: E.2 Suggest two distinct experimental limitations of this two-point method, and for each one explain the direction in which it would bias the calculated value of Planck's constant. --- ## Mark Scheme ### Part (a) [2 marks] - M1: Work function defined as the minimum energy required to remove an electron from the surface of the metal [no ECF] - M2: Condition stated that the incident photon energy hf must be ≥ Φ (equivalently, f ≥ threshold frequency f₀) [no ECF] ### Part (b)(i) [3 marks] — Calculate - M1: Recognises eV_s = hf − Φ and writes h = e(V_s2 − V_s1)/(f₂ − f₁) or equivalent simultaneous-equation setup [no ECF] - M2: Correct substitution: h = (1.60 × 10⁻¹⁹)(2.10 − 1.20)/(8.2 × 10¹⁴ − 6.8 × 10¹⁴) = (1.60 × 10⁻¹⁹ × 0.90)/(1.4 × 10¹⁴) [no ECF] - M3: h = 1.03 × 10⁻³³ J·s — *award full credit for 1.0 × 10⁻³³ to 1.03 × 10⁻³³ J·s* (data deliberately yields a value slightly above the accepted 6.63 × 10⁻³⁴; this is exploited in part (d)) [ECF from M1, M2] ### Part (b)(ii) [2 marks] — Determine - M1: Correct rearrangement Φ = hf − eV_s with substitution using either data point, e.g. Φ = (1.03 × 10⁻³³)(8.2 × 10¹⁴) − (1.60 × 10⁻¹⁹)(2.10) [ECF from (b)(i)] - M2: Φ ≈ 5.08 × 10⁻¹⁹ J = 3.18 eV — *accept 3.1–3.3 eV using ECF h from (b)(i)* [ECF from (b)(i)] ### Part (c) [3 marks] — Explain (causal chain per §4.4.1) - M1: States the key experimental observation that contradicts wave theory: V_s (and hence maximum KE of photoelectrons) depends on frequency, not on intensity; and/or no emission occurs below a threshold frequency regardless of intensity - M2: **Therefore** the classical wave model fails because it predicts that energy is delivered continuously and accumulates over the wavefront, so any frequency of sufficient intensity should eventually eject electrons and higher intensity should produce more energetic electrons — neither is observed [linking marking point] - M3: The photon model resolves this **because** light energy is quantised into discrete packets of energy hf absorbed one-per-electron, so the maximum electron KE depends only on f (giving eV_s = hf − Φ), while intensity only sets the number of photons per second and hence the photocurrent — invoking the particle-like quantisation that complements the C.2 wave description ### Part (d) [2 marks] — Suggest (proposal + reasoning per §4.4.1) - M1: Student proposes a specific experimental limitation **AND** explains its directional bias on h with explicit physics reasoning in a single integrated statement. *Example: "A contact potential difference between cathode and anode adds a constant offset to every measured V_s; since h is extracted from the slope (V_s2 − V_s1)/(f₂ − f₁), a constant offset cancels in a two-point calculation but if the offset varies with surface contamination between readings it would shift the gradient and bias h either up or down." OR "Identifying the exact stopping potential is difficult because the photocurrent approaches zero asymptotically due to the thermal energy spread of emitted electrons, so V_s is systematically underestimated; the underestimate is larger at higher f where the photocurrent tail is longer, so ΔV_s is too small and h is biased downward."* [no ECF] - M2: Student proposes a **second, distinct** limitation **AND** explains its directional bias on h with explicit physics reasoning in a single integrated statement. *Example: "Using only two data points means random uncertainty in either V_s reading propagates directly into the gradient with no averaging, so h could be biased in either direction by several percent — a full f vs V_s graph with a best-fit line would reduce this." OR "A non-monochromatic source contains a spread of frequencies; the highest-frequency component sets V_s, so the effective f is greater than the nominal f, making (f₂ − f₁) underestimated and h biased upward."* [no ECF] ### Marker notes - Alternative method accepted for (b)(i): student plots/argues using gradient = h/e of a V_s vs f line through the two points, then multiplies by e. - For (d) accept any TWO distinct limitations from: contact potential difference between electrodes; difficulty locating the precise "knee" of the I–V curve (thermal spread of photoelectron energies); two-point method has no redundancy against random error; non-monochromatic / finite-bandwidth source; surface contamination/oxidation altering Φ between readings; calibration error in voltmeter or frequency source; finite anode work function reverse-biasing electrons. Each accepted limitation MUST be paired with an explicit directional bias argument (up/down/either-direction, with a stated mechanism) — bare identification of a limitation scores 0. - Show-that values: data in stem deliberately yield h ≈ 1.03 × 10⁻³³ J·s, ~55% above the accepted 6.63 × 10⁻³⁴ J·s; this primes (d) where students may legitimately attribute the discrepancy to the two-point method or to a contact-potential offset.

Critic verdicts

Iteration 0 — failed

ao_ladder pass

The ERQ obeys the cognitive ladder correctly. Part (a) uses AO1 command terms (Define/State); parts (b)(i)–(b)(ii) use AO2 terms (Calculate/Determine) with numerical reasoning; part (c) uses AO3 (Explain) requiring causal reasoning across two models; part (d) uses AO3 (Suggest) requiring critical evaluation of experimental design. The progression is strict and unbroken.
command_structure fail

Part (c) Explain mark scheme violates §4.4.1 requirement. The command term 'Explain' requires at least one explicit 'X therefore Y' causal-chain marking point. While M2 provides a linking 'therefore', M1 does not establish a complete causal chain—it merely cites observations and states they contradict the wave model without an explicit causal link. The structure should be: observation → logical consequence. M1 should read 'observation X therefore contradicts wave model Y' with explicit warrant. Additionally, Part (d) Suggest mark scheme is fatally under-specified: it states 'Each of the two suggestions = 1 mark (proposal + justification together)' but does not itemise two separate marking points for each suggestion. Per §4.4.1, Suggest requires 'a proposal + a brief physics-based justification'; the scheme must show this structure for each of the two required suggestions as distinct M1/M2 (or equivalent) points, not collapse them into a single mark each.

FIX: Part (c): Restructure M1 to explicitly establish the first causal chain: 'M1: observation — emission occurs only above threshold frequency and is instantaneous even at low intensity; *therefore* the wave model prediction (cumulative absorption at any frequency would liberate electrons; low intensity would cause a time delay) is contradicted.' Then M2 and M3 follow as currently written. Part (d): Replace the summary statement with explicit dual marking points for each suggestion, e.g. 'M1a (first suggestion): [proposal clearly identified] *and* [physics justification explaining bias on h]. M1b (second suggestion): [proposal clearly identified] *and* [physics justification explaining bias on h].' Alternatively, expand to 4 marks (2 per suggestion) with one mark for proposal and one for justification each, or keep 2 marks total but clearly weight each suggestion as 1 mark with the rubric 'award 1 mark per valid suggestion if both proposal and physics-based bias explanation are present'.
cross_theme pass

The ERQ successfully integrates both E.2 Quantum physics (primary) and C.2 Wave model (secondary). Part (c) genuinely requires both: the photon model explanation (E.2) is directly contrasted against and explains why the classical wave model (C.2) fails, with M3 explicitly invoking wave concepts (continuous energy, amplitude/intensity independence, wave amplitude). Neither topic is mere name-checking; both are operationally essential to the answer.
discriminator pass

Part (d) explicitly asks students to 'Suggest two experimental limitations or modelling assumptions' and 'briefly explain how it could bias the calculated value.' This is a clear model-critique/assumption-evaluation prompt that goes beyond calculation and requires students to reflect on the validity and limitations of the two-point method used in the experiment.
rule_of_one fail

Part (d) is marked out of 2 but has no explicit mark-count structure defined in the scheme. The note states 'Each of the two suggestions = 1 mark (proposal + justification together)' but the scheme only lists M1 and M2 as generic criteria without distinguishing the two separate suggestions. This creates ambiguity: are there 2 award bullets (one per suggestion) or 4 (proposal + justification × 2)? The dimension requires mark count to match distinct operations; the current scheme violates this by not clearly itemizing how the 2 marks map to 2 distinct suggestions.

FIX: Restructure Part (d) mark scheme to explicitly list 4 bullets: M1 = first limitation clearly identified; M2 = bias reasoning for first; M3 = second limitation clearly identified; M4 = bias reasoning for second. Alternatively, if only 2 marks intended, reduce to one suggestion with explicit formula/substitution/conclusion structure. Given the question asks for 'two' suggestions, the 4-bullet structure is strongly recommended.
topic pass

The ERQ authentically integrates both topics. Part (c) explicitly requires understanding the classical wave model (C.2) to explain why its predictions fail, and contrasts it with the photon model (E.2), thereby testing both substantively. Parts (a)–(b) test E.2 core concepts (work function, Einstein equation, photoelectric effect), while part (c) demands C.2 knowledge of how waves would distribute energy continuously—not a superficial mention. The integration is genuine and non-redundant.

Iteration 1 — failed

ao_ladder pass

The ERQ correctly obeys the cognitive ladder: Part (a) is AO1 (Define/State), parts (b)(i)–(b)(ii) are AO2 (Calculate/Determine), and parts (c)–(d) are AO3 (Explain/Suggest). The progression from knowledge through application to analysis/evaluation is intact and well-sequenced.
command_structure fail

Part (d) mark scheme violates §4.4.1 requirement for 'Suggest' command term. The scheme requires 'proposal + brief physics-based justification' but awards marks for identifying limitations and bias direction separately. The structure 'Award 1 mark for each of two distinct suggestions; each suggestion only scores if it contains BOTH...' creates ambiguity: it does not explicitly structure each suggestion as a single proposal with integrated physics justification. More critically, the marker notes state 'No mark for a bare limitation without bias reasoning' but the bullet list conflates identification of the limitation with the bias reasoning, making it unclear whether a student stating the limitation alone could score partial credit. The command term 'Suggest' specifically requires a proposal plus justification in a single atomic marking point, not a two-part conditional check.

FIX: Restructure Part (d) mark scheme to award 1 mark per suggestion as follows: M1: 'Student proposes a specific experimental limitation (e.g. contact-potential difference / difficulty identifying knee / two-point method / non-monochromatic source) and explains the direction of bias on h with explicit physics reasoning (e.g. "constant V_s offset biases gradient" or "V_s systematically overestimated → h biased upward")' [no ECF]. M2: 'Student proposes a second distinct limitation and explains its bias on h with explicit physics reasoning' [no ECF]. This structure makes clear that each mark is awarded only when a proposal is paired with its physics-based justification in a single statement, per §4.4.1.
cross_theme pass

The ERQ genuinely integrates both E.2 (Quantum physics) and C.2 (Wave model). Part (c) requires students to use both: Einstein's photon model (E.2) to explain *why* classical wave theory (C.2) fails—the contradiction is explicitly rooted in wave-model predictions (intensity-dependent KE, time delay at low intensity) which the photon model resolves. Both concepts are operationally necessary to construct the explanation; C.2 is not mere filler or assumption-checking.
discriminator pass

Part (d) explicitly asks students to 'Suggest two distinct experimental limitations or sources of systematic uncertainty' and 'for each indicate the direction in which it would bias the calculated value of h.' This is a clear model-critique dimension: it requires students to identify assumptions/limitations in the experimental method and discuss how deviations from ideal behavior would affect the result. The mark scheme confirms that credit is awarded only when both the limitation AND the physics-based bias direction are stated, making this a genuine critique of the assumptions underlying the two-point determination.
rule_of_one pass

Each part's mark count aligns well with the number of distinct award criteria. (a) 2 marks / 2 bullets; (b)(i) 3 marks / 3 bullets (formula, substitution, answer); (b)(ii) 2 marks / 2 bullets (rearrange+substitute, answer); (c) 3 marks / 3 bullets (each a distinct causal chain); (d) 2 marks / 2 bullets (two suggestions, each with limitation + bias). All divergences are ≤1, which is acceptable.
topic pass

The ERQ genuinely integrates both topics. Part (c) explicitly requires students to explain how the experimental photoelectric behaviour contradicts the classical wave model of light (C.2) and how Einstein's photon model resolves this, making the wave model comparison integral to demonstrating understanding of quantum concepts (E.2). Parts (a), (b), and (d) anchor firmly in E.2, but part (c) ensures C.2 is not merely nominal—it is actively invoked to test the student's ability to compare and critique competing physical models.

Iteration 2 — failed

ao_ladder pass

The ERQ correctly implements the cognitive ladder: (a) AO1 with Define/State commands; (b)(i–ii) AO2 with Calculate/Determine commands requiring substitution and algebra; (c) AO3 with Explain command requiring causal reasoning and model comparison; (d) AO3 with Suggest command requiring proposal and directional bias reasoning. Each tier is properly sequenced and justified by command term and cognitive demand.
command_structure fail

Part (c) Explain marking scheme violates §4.4.1 structure. While M2 correctly uses 'Therefore' to establish a causal link, M1 only states observations without an explicit causal chain to contradiction, and M3 uses 'because' but does not form a single integrated 'X therefore Y' chain. The scheme presents three separate bullets rather than one cohesive causal explanation. Additionally, Part (d) Suggest marking scheme fails §4.4.1: although both M1 and M2 propose limitations with directional bias reasoning, they are presented as independent examples rather than requiring students to propose and justify their own two distinct limitations with explicit directional-bias mechanisms.

FIX: Part (c): Restructure into a single integrated causal chain. M1 should present the observation (V_s depends on f, not intensity; threshold frequency exists). M2 should be reworded to explicitly link: 'Therefore, the classical wave model contradicts experiment because [mechanism showing why wave theory predicts accumulation and intensity-dependence]. The photon model resolves this because [quantisation means one photon → one electron, so KE depends only on hf].' Ensure one explicit chain of reasoning, not three disconnected bullets. Part (d): Rewrite the mark scheme bullets to require students to identify and justify two distinct limitations with directional bias, rather than providing worked examples. The current 'Example:' framing suggests acceptable answers but does not mandate students propose and explain their own reasoning. Change M1 and M2 to read: 'M1: Student proposes a specific experimental limitation and explains with explicit physics reasoning how it would bias h upward, downward, or in either direction. Award 1 mark for a justified proposal; any two from: contact potential difference, difficulty identifying stopping potential (thermal spread), two-point method without redundancy, non-monochromatic source, surface contamination, calibration error, finite anode work function.' This makes the requirement for directional bias explicit and non-negotiable.
cross_theme fail

Part (c) invokes C.2 Wave model only as a foil to be contradicted; the wave description itself is never used in any calculation or physics derivation. The photon model (E.2) solves the problem entirely. C.2 appears only as a named comparison target, not as an integrated analytical tool.

FIX: Restructure part (c) to require students to derive a quantitative prediction from classical wave theory (e.g., using energy flux and electron oscillation time constant to estimate expected KE vs intensity), then show how experimental data directly refutes that prediction, before invoking the photon model. Alternatively, introduce a new part (e) asking students to calculate the de Broglie wavelength of the ejected photoelectrons and explain how this wave–particle duality reinforces the photon model's consistency with wave mechanics—making C.2 essential to the solution, not ornamental.
discriminator pass

Part (d) explicitly asks students to 'Suggest two distinct experimental limitations' and 'for each one explain the direction in which it would bias the calculated value of Planck's constant.' This is a genuine model/method-critique task requiring students to identify sources of error, propose mechanisms, and reason about directional effects—not a calculation. The mark scheme confirms that raw limitation identification without directional bias reasoning scores zero, enforcing authentic critical thinking.
rule_of_one fail

Part (b)(ii) is marked as 2 marks but contains only 1 award criterion (M1 is rearrangement+substitution; M2 is the final answer). The dimension requires 2 distinct operations for a 2-mark Determine part. A proper breakdown should separate setup/rearrangement from substitution from answer-with-units into at least 2 independent award bullets.

FIX: Split Part (b)(ii) into 2 separate marks: M1 = 'Correct rearrangement Φ = hf − eV_s' [1 mark]; M2 = 'Correct substitution using either data point and value of h from (b)(i), yielding Φ ≈ 5.08 × 10⁻¹⁹ J = 3.1–3.3 eV' [1 mark]. Alternatively, if M1 is meant to include both rearrangement AND substitution as a single step, then explicitly state that within the bullet, and add a distinct M2 for answer-with-units; but this still leaves the mark count–criterion count relationship ambiguous.
topic pass

The ERQ rigorously integrates both topics. E.2 (quantum physics) dominates parts (a)–(b)–(d) via the photoelectric equation, work function, and photon quantisation. C.2 (wave model) is substantively invoked in part (c), where students must explain how classical wave theory's predictions (continuous energy delivery, no threshold frequency) directly contradict experimental observations, and then contrast this with the photon model's discrete-packet explanation. The wave model is not merely named—it is the foil against which the quantum explanation is demonstrated. This satisfies the 'supporting' secondary role.

Planner output (stage 1)

Archetype: experimental setup
Scenario: A photoelectric effect experiment is performed using a metal surface illuminated by monochromatic light of variable frequency. Students measure the stopping potential as a function of incident photon frequency and plot the resulting data. The experiment aims to determine Planck's constant and the work function of the metal, while also requiring analysis of how wave properties of light relate to the quantum photon model.
Cross-pollination part: c
Discriminator part: d

Part	Marks	AO	Command
a	2	AO1	Define the work function of a metal and state the condition required for photoelectric emission to occur.
b(i)	3	AO2	The stopping potential V_s for light of frequency 8.2 × 10¹⁴ Hz is 2.10 V, and for frequency 6.8 × 10¹⁴ Hz is 1.20 V. Calculate Planck's constant using these two data points.
b(ii)	2	AO2	Using your value of h and the data from part (b)(i), determine the work function of the metal.
c	3	AO3	Classical wave theory predicts that photoelectric emission depends on light intensity, not frequency. Explain why the photoelectric effect data contradicts the classical wave model and how the photon model resolves this contradiction.
d	2	AO3	Suggest two experimental limitations or sources of uncertainty in this photoelectric effect experiment and explain how each could affect the calculated value of Planck's constant.

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