Questions: Thermal Conductivity from Kinetic Theory

Question 1

An engineer expects that compressing a gas to twice its pressure will roughly double its thermal conductivity, since twice as many molecules are available to carry heat. Is this reasoning correct?

• A. Yes — more molecules per unit volume means more heat carriers and proportionally higher κ
• B. No — higher pressure increases molecular density but decreases mean free path proportionally, so the two effects cancel and κ is pressure-independent
• C. No — κ decreases with pressure because more frequent collisions rapidly dissipate any thermal gradient
• D. Yes — this is the basis of using high-pressure gases as coolants in industrial applications

Answer: (B)

This is one of kinetic theory's most counterintuitive predictions, confirmed experimentally. κ ~ (1/3) ρ c_v v̄ λ. When pressure doubles, ρ doubles — but λ (the mean free path) halves, because molecules collide twice as often at twice the density. The product ρλ remains constant, so κ is unchanged. The two effects cancel exactly. This pressure-independence was one of Maxwell's surprising early successes with kinetic theory and is well-verified experimentally over many decades of pressure.

Question 2

For a monatomic ideal gas, kinetic theory predicts that thermal conductivity κ scales with temperature as:

• A. κ ∝ T (linear in temperature)
• B. κ ∝ T⁻¹ (decreasing with temperature)
• C. κ ∝ √T (square root of temperature)
• D. κ is temperature-independent

Answer: (C)

The mean thermal speed v̄ ∝ √(k_B T / m), and since κ ~ ρ c_v v̄ λ and the product ρλ is pressure-independent, κ grows as v̄ ∝ √T. The Chapman-Enskog result confirms κ = (5/2)(k_B / σ_eff) √(k_B T / πm) ∝ √T. This means hot gases conduct heat better than cold gases — a non-obvious result that distinguishes gases from most solids and liquids, where conductivity typically decreases with temperature.

Question 3

Kinetic theory predicts that the thermal conductivity of an ideal gas is independent of pressure, because increasing pressure simultaneously increases molecular density and decreases mean free path, with the two effects exactly canceling.

• T. True
• F. False

Answer: True

The simple mean-free-path estimate κ ~ (1/3) ρ c_v v̄ λ contains the density ρ ~ nM and the mean free path λ ~ 1/(nσ), where n is number density and σ is cross-section. The product ρλ ~ M/σ is independent of n and hence of pressure. This is a genuinely surprising prediction — naively, compressing a gas should make it conduct heat better — and it holds well experimentally over many orders of magnitude of pressure (breaking down only at very low pressures where λ approaches the container size, or very high pressures where molecules interact continuously).

Question 4

Internal degrees of freedom in polyatomic molecules reduce thermal conductivity because energy stored in rotation cannot be transported across a temperature gradient.

• T. True
• F. False

Answer: False

Internal degrees of freedom provide additional channels for energy transport, which INCREASES κ relative to the monatomic case. A rotating molecule can absorb translational kinetic energy in a collision, carry that rotational energy across a temperature gradient, and transfer it to translational motion on the other side. This is an additional transport pathway. The Eucken correction accounts for this by adding the contribution of internal modes to the monatomic result. More channels for energy transport means higher conductivity, not lower.

Question 5

Why do kinetic theory derivations show that thermal conductivity and viscosity arise from the same transport mechanism, and how are they related?

Model answer: Both κ (thermal conductivity) and η (viscosity) arise from molecules transporting a conserved quantity down a gradient via random thermal motion and mean-free-path transport. In thermal conductivity, molecules carry energy down a temperature gradient; in viscosity, they carry momentum down a velocity gradient. Because the same mean free path and thermal speed govern both processes, κ and η are proportional: the Eucken relation gives κ = (1/4)(9γ − 5) c_v η. Both are independent of pressure for the same reason (ρλ = constant), and both increase as √T.

This connection is one of kinetic theory's unifying insights: transport coefficients that look macroscopically unrelated (heat flow vs. fluid friction) turn out to be two faces of the same microscopic mechanism — molecules randomly walking across gradients and mixing their properties. The Chapman-Enskog expansion reveals this unity systematically by treating them as different moments of the same distribution-function perturbation.