## ERQ · 12 marks · Topics: B.2 Greenhouse effect + C.2 Wave model
**Stem.** A research team models Earth's atmosphere as a single thin layer in radiative equilibrium. The Sun delivers an average solar intensity of S = 340 W m⁻² to the top of the atmosphere. Of this, a fraction α = 0.30 is reflected back to space (the planetary albedo), while the remaining shortwave radiation passes through the atmosphere unimpeded and is absorbed by Earth's surface. The surface, treated as a perfect blackbody at temperature T_s, emits infrared radiation that is fully absorbed by the atmospheric layer. The layer itself radiates as a blackbody at temperature T_a, emitting equally upward to space and downward to the surface. The team then examines the spectrum of the outgoing infrared and observes that CO₂ absorbs strongly in a narrow band centred on wavelength λ = 15 μm, where the absorption arises from resonance between the electromagnetic wave and a vibrational mode of the CO₂ molecule. Use σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴ and c = 3.00 × 10⁸ m s⁻¹.
### Part (a) State [2 marks] · AO1 · Topic: B.2
State what is meant by (i) the albedo of a planet and (ii) radiative equilibrium of Earth's atmosphere.
### Part (b)(i) Show that [3 marks] · AO2/AO3 · Topic: B.2
Using the single-layer model, show that the surface temperature satisfies σT_s⁴ = 2 × S(1 − α)/2 × ... and hence calculate T_s.
### Part (b)(ii) Determine [3 marks] · AO2/AO3 · Topic: B.2
Determine the temperature T_a of the atmospheric layer in this model.
### Part (c) Explain [2 marks] · AO3 · Topic: B.2 + C.2
The CO₂ absorption band is centred on λ = 15 μm. Explain, in terms of the wave model, why CO₂ absorbs strongly at this specific wavelength but is essentially transparent to visible light.
### Part (d) Suggest [2 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR
The measured global mean surface temperature of Earth is approximately 288 K, which differs from the value calculated in (b)(i). Suggest two reasons why the single-layer model gives a different value from the observed temperature.
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## Mark Scheme
### Part (a) [2 marks]
- Mark 1: Albedo = fraction (or ratio) of incident solar (shortwave) radiation reflected back to space by the planet [ECF: no]
- Mark 2: Radiative equilibrium = (average) power absorbed by Earth from the Sun equals the (average) power radiated by Earth to space / energy in = energy out [ECF: no]
### Part (b)(i) [3 marks]
- Mark 1: Energy balance at surface: σT_s⁴ = S(1 − α) + σT_a⁴, AND atmosphere balance giving σT_a⁴ = σT_s⁴/2, leading to σT_s⁴ = 2 · S(1 − α)/... → σT_s⁴ = 2S(1−α) [accept: σT_s⁴ = S(1−α)/(... ) provided correct factor of 2 shown] [ECF: no]
- Mark 2: Correct substitution: T_s⁴ = 2 × 340 × (1 − 0.30)/(5.67 × 10⁻⁸) = 8.40 × 10⁹ K⁴ [ECF: yes from M1]
- Mark 3: T_s = 303 K (accept 302–304 K) to 3 s.f. with units [ECF: yes]
### Part (b)(ii) [3 marks]
- Mark 1: Top-of-atmosphere balance: S(1 − α) = σT_a⁴ (recognises atmosphere alone radiates to space) [ECF: no]
- Mark 2: Substitution: T_a⁴ = (340 × 0.70)/(5.67 × 10⁻⁸) = 4.20 × 10⁹ K⁴ [ECF: yes]
- Mark 3: T_a = 255 K (accept 254–256 K) with units, to 3 s.f. [ECF: yes; alternatively T_a = T_s/2^(1/4) using (b)(i) value]
### Part (c) [2 marks]
- Mark 1: The CO₂ molecule has a natural (vibrational) frequency; the EM wave at λ = 15 μm has frequency f = c/λ ≈ 2.0 × 10¹³ Hz that matches this natural frequency, producing resonance and strong absorption [ECF: no]
- Mark 2: Visible light has a much higher frequency (much shorter wavelength, ~10¹⁴–10¹⁵ Hz) that does not match any vibrational/resonant frequency of CO₂, so the wave is not absorbed and is transmitted [ECF: no]
### Part (d) [2 marks]
Award 1 mark for each distinct valid reason, max 2:
- The model treats the atmosphere as a single layer that fully absorbs all surface IR; in reality absorption is wavelength-selective (atmospheric "window" lets some IR escape directly), changing the energy balance
- The model ignores convection, latent heat transport and conduction, which redistribute energy from the surface and lower the surface temperature
- Albedo is taken as constant; clouds, ice and surface variations make it spatially/temporally variable
- The atmosphere is not isothermal — temperature decreases with altitude (lapse rate), so a single T_a is unrealistic
- The surface is treated as a perfect blackbody; real emissivity < 1 over some IR ranges
- Latitudinal variation of incoming solar radiation is averaged out
### Marker notes
- (b)(i): Accept derivations that explicitly write both atmospheric balance (2σT_a⁴ = σT_s⁴ ⇒ T_a⁴ = T_s⁴/2) and surface balance. Bald answer T_s = 303 K with no derivation: max 1/3.
- (b)(ii): Alternative route: T_a = T_s × (1/2)^(1/4) = 303 × 0.841 = 255 K — full ECF from (b)(i).
- (c): Do not credit "CO₂ is a greenhouse gas" alone — must reference frequency matching / resonance and mismatch with visible frequencies.
- (d): Do not credit "human activity / more CO₂" — the question is about model assumptions, not real-world forcing. Reject vague "experimental error".