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VKC Notes · Gaseous state · Civil Services P-I · IFoS P-I · Geo-Scientist P-II

4.2 Kinetic Theory of Gases: Maxwell Distribution, Collisions, Mean Free Path, Effusion and Transport

★★ 2021–2025: 12 parts, 95 marks (Civil Services 8 parts / 65 m · IFoS 2 / 15 m · Geo-Scientist 2 / 15 m). Civil Services sets one 5–10-mark numerical on speeds, wall collisions, effusion or viscosity every year.

Syllabus lines. Civil Services: “Maxwell’s distribution of speeds, intermolecular collisions, collisions on the wall and effusion; thermal conductivity and viscosity of ideal gases.” IFoS: the same without transport properties. Geo-Scientist asks the kinetic-theory postulates under real gases. Real gases, van der Waals and critical constants have their own notes.

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CSE 2024 Q7(a) = asked there; tap to open the solved answerSN = line kept in the short notes

4.2.1 The kinetic model of a perfect gas ★★

  1. A gas is a large number of molecules of mass m in continuous, random, straight-line motion.
  2. Their own volume is negligible compared with the volume of the container (point masses).
  3. They do not attract or repel each other; they interact only in perfectly elastic collisions.
  4. Pressure is the momentum given to the walls per unit area per second; the mean kinetic energy is proportional to T: ½m⟨c²⟩ = (3/2)kBT.SN
(4.2.1)
  • Which postulates fail for real gases: (2) at high pressure molecules have a finite volume, free volume V − nb, repulsion gives Z > 1; (3) at low temperature attraction lowers the wall pressure by an²/V², Z < 1, and gases liquefy. Correcting both gives the van der Waals equation Geo 2022 Q1(a) 5m.SN
  • A gas that stays a gas at constant V obeys p ∝ T. Sealed sample at 1.00 atm, 300 K → 0.37 atm at 150 K: pure Ne (Tc 44 K) would give 0.50 atm; pure C₂H₆ would condense to its vapour pressure 0.10 atm; so a mixture: p(Ne) = 0.27 atm at 150 K, i.e. Ne 0.54 and C₂H₆ 0.46 mole fraction CSE 2025 Q1(h) 5m.

Solved PYQs CSE 2023 Q1(d) 10m Geo 2024 Q12(a) 10m

  1. crms = 719 m s⁻¹ for H₂O: T = Mc²/3R = 0.018 × 719²/(3 × 8.314) = 373 K; at 473 K, crms = √(3 × 8.314 × 473/0.018) = 810 m s⁻¹ (∝ √T: 719 × √(473/373)).SN
  2. 4 dm³ of O₂ pumped down to 0.10 Pa at 298 K: (i) the gas left still fills 4 dm³; (ii) n = pV/RT = 0.10 × 4 × 10⁻³/(8.314 × 298) = 1.61 × 10⁻⁷ mol = 9.7 × 10¹⁶ molecules; (iii) + 2 g N₂ (0.0714 mol): p = nRT/V = 4.42 × 10⁴ Pa (the 0.10 Pa of O₂ is negligible).

4.2.2 Maxwell distribution of speeds ★★★

(4.2.2)
(4.2.3)
040080012001600c / m s−1f(c)300 K1000 KN20100020003000c / m s−1f(c)Ar: cmp 352 m s−1He: cmp 1113 m s−1same T = 298 K
Fig. 4.2.1 Computed Maxwell curves: N₂ at 300 K and 1000 K (left); Ar and He at 298 K (right).
Draw it in 60 s: left: a hump starting at the origin (v2 rise, exponential tail); hotter curve lower, broader, peak to the right; three dashed lines cmp < c̄ < crms. Right: tall narrow Ar hump, low wide He hump.
  • Shape: the v² factor makes f(0) = 0; the exponential cuts off high speeds. Raising T or lowering M moves the peak to higher speed, lowers it and broadens it; the area stays 1. Only T/M matters.SN
  • N₂ at 300 K: cmp 422, c̄ 476, crms 517 m s⁻¹.
  • Ar and He at the same T: Ar (cmp 352 m s⁻¹) is the tall narrow curve, He (1113 m s⁻¹) the low broad one. Ar effuses more slowly (rate ∝ 1/√M); F₂ (M 38.0) would follow almost the Ar curve (M 39.95) CSE 2022 Q3(c) 10m.SN
  • Equal speeds need equal T/M: O₂ matches H₂ at 20 K when T = 20 × 32/2 = 320 K; this holds for c̄, crms and cmp alike. Strictly, the mean velocity (a vector) of a gas at rest is zero IFoS 2024 Q8(a) 10m.SN

4.2.3 Intermolecular collisions and mean free path ★★

dtube swept in time Δt: volume σ c̄rel Δtσ = πd2missed: centre outside the tubehit: centre inside the tube
Fig. 4.2.2 Collision cylinder.
Draw it in 60 s: a circle of radius d (red) and a long tube of radius d; three molecule centres inside the tube (hit), two outside (missed); label σ = πd2.
(4.2.4)
  • σ = πd² (collision cross-section), 𝒩 = N/V = p/kBT. The √2 converts the mean speed into the mean relative speed of two like molecules; the ½ in ZAA stops each collision being counted twice.SN
  • N₂ at 1 atm, 298 K (σ = 0.43 nm²): c̄ = 475 m s⁻¹, λ = 67 nm (about 200 molecular diameters), z = 7.1 × 10⁹ s⁻¹.
  • λ ∝ T/p (∝ 1/𝒩); at fixed density λ does not depend on T; z ∝ p/√T at fixed pressure.SN

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