Cambridge International AS & A Level · 15 · structured
Derive the molecular energy bridge
From moles to molecules to mean translational energy · Original GioPhysics question with a detailed, mark-by-mark answer guide.
- Demand
- demanding
- Marks
- 8
- Topics
- 2
- Answer
- Complete
State the relation connecting the Boltzmann constant k, molar gas constant R and Avogadro constant N_A, then justify it using pV = nRT, pV = NkT and N = nN_A. Next use pV = (1/3)Nm⟨c²⟩ to deduce average translational kinetic energy per molecule. For 0.200 mol at 300 K, calculate total translational energy using R = 8.31 J mol⁻¹ K⁻¹. Is this necessarily the full internal energy of a molecular gas?
- (a)
State from memory the relation connecting k, R and N_A, then justify it using the supplied gas equations.
2 marks - (b)
Deduce mean translational kinetic energy from the pressure equations.
3 marks - (c)
Calculate total translational energy and distinguish it from full molecular-gas internal energy.
3 marks
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(a)
State from memory the relation connecting k, R and N_A, then justify it using the supplied gas equations.
Identify what the question is testing, organise the response into distinct mark-earning points, and make every conclusion traceable to a physical principle or to the evidence supplied.
- 1
Recall k=R/N_A without it being printed in the question (1).
- 2
Use nRT=nN_AkT and cancel nT to justify the relation (1).
(b)
Deduce mean translational kinetic energy from the pressure equations.
List the given quantities with units, identify the required quantity, write the governing relationship before substituting, and keep extra digits until the final line so rounding does not distort the result.
- 1
(b) Equate pV expressions; cancel N; identify (1/2)m⟨c²⟩ = (3/2)kT.
(c)
Calculate total translational energy and distinguish it from full molecular-gas internal energy.
List the given quantities with units, identify the required quantity, write the governing relationship before substituting, and keep extra digits until the final line so rounding does not distort the result.
- 1
Use total translational energy = (3/2)nRT (1).
- 2
Substitute 1.5 × 0.200 × 8.31 × 300 = 747.9 J ≈ 748 J (1).
- 3
Molecular rotational and vibrational modes may also contribute: this need not be the full internal energy (1).
Complete marking guidance
Use these worked steps alongside the part-by-part answers. They explain the method and mark allocation; they do not add extra marks.
- (a) Recall k=R/N_A without it being printed in the question (1).
- (a) Use nRT=nN_AkT and cancel nT to justify the relation (1).
- (b) Equate pV expressions; cancel N; identify (1/2)m⟨c²⟩ = (3/2)kT.
- (c) Total translational energy = (3/2)nRT = 1.5 × 0.200 × 8.31 × 300 = 747.9 J ≈ 748 J (2). Molecular rotational and vibrational modes may also contribute, so this need not be the full internal energy (1).
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