A gas container has its pressure doubled at constant temperature (doubling the density). According to kinetic theory, what happens to the gas viscosity?
AIt doubles — there are twice as many molecules available to transfer momentum across the velocity gradient
BIt is approximately unchanged — the mean free path halves (canceling the effect of more carriers), so η is density-independent
CIt decreases by half — the shorter mean free path means momentum is deposited locally and doesn't travel far
DIt increases by √2 — viscosity scales with the square root of density
Viscosity is density-independent in an ideal gas: η ~ mv̄/σ, where the number density n has cancelled. Doubling density doubles the number of molecules crossing unit area per second, but it also halves the mean free path λ ~ 1/(nσ), so each molecule deposits its carried momentum only half as far. The two effects exactly cancel. This is the famous counterintuitive result first predicted by Maxwell and confirmed experimentally: denser air is not more viscous. The density-independence breaks down only at high densities where multiple-body collisions become significant.
Question 2 Multiple Choice
How does the viscosity of an ideal gas change as temperature increases?
AIt decreases — hotter molecules collide more frequently, disrupting organized flow more effectively
BIt is unchanged — viscosity depends only on molecular mass and size, not temperature
CIt increases — higher temperature means higher mean thermal speed v̄ ∝ √T, so molecules carry more momentum per crossing and η ∝ √T (or somewhat higher for real gases)
DIt decreases then increases, showing a minimum at intermediate temperatures
Gas viscosity increases with temperature. The mean thermal speed v̄ ∝ √T, and since η ~ mv̄/σ, viscosity scales as √T for ideal hard-sphere gases (modified to roughly T^{0.6–0.8} for real gases with realistic potentials). This is opposite to liquid viscosity, which decreases with temperature. In liquids, viscosity arises from intermolecular attraction holding layers together — higher temperature weakens this. In gases, viscosity arises from momentum-carrying molecular flights — higher temperature means faster molecules carrying more momentum. The underlying mechanism is completely different, so the temperature dependence runs in opposite directions.
Question 3 True / False
A denser gas (at fixed temperature) will have higher viscosity than a less dense sample of the same gas, because more molecules are available to transfer momentum between layers.
TTrue
FFalse
Answer: False
This is the central counterintuitive result of kinetic theory: gas viscosity is independent of density (over the ideal gas range). More molecules do cross the layer per second — but each one also has a shorter mean free path and deposits its momentum closer to where it came from. The increased number of carriers is exactly offset by the reduced distance each carrier travels. Maxwell predicted this in 1860, was reportedly skeptical of his own result, and was vindicated by experiment. The density-independence is a direct consequence of the mean free path λ ~ 1/(nσ) canceling the density factor in the carrier flux.
Question 4 True / False
Gas viscosity arises because faster-moving molecules from a high-velocity layer carry their excess momentum into a slower-moving adjacent layer during collisions.
TTrue
FFalse
Answer: True
This is the correct microscopic picture of viscosity as momentum transport. A fluid with a velocity gradient (fast layers adjacent to slow layers) has molecules thermally wandering across layer boundaries. A molecule from a fast-moving layer carries momentum proportional to its excess velocity; when it collides with a molecule in a slower layer, this momentum is partially transferred, accelerating the slower layer and decelerating the fast layer — which is the macroscopic effect we call viscous drag. Viscosity quantifies the proportionality between the velocity gradient and the resulting momentum flux.
Question 5 Short Answer
Explain qualitatively why the viscosity of an ideal gas is independent of its density, starting from the microscopic picture of momentum transfer.
Think about your answer, then reveal below.
Model answer: Viscosity requires molecules to carry momentum from fast-moving layers to slow-moving ones. The momentum transported per unit time per unit area is proportional to two things: the number of molecules crossing the layer boundary (proportional to density n) and the distance each molecule travels before depositing its momentum (the mean free path λ ~ 1/(nσ), inversely proportional to n). When density increases, more molecules cross the boundary — but each one collides sooner and deposits its momentum locally. The product n × λ ~ n × 1/(nσ) = 1/σ is independent of n. The two effects exactly cancel, leaving viscosity dependent only on molecular mass, size, and thermal speed (hence temperature).
Maxwell famously derived this result and was so surprised he reportedly tested it experimentally himself. The intuition is that what counts for viscosity is not how many molecules are present, but how effective each layer is at communicating its momentum to adjacent layers. Doubling the density doubles both the communication channel capacity and the resistance to long-range transport, leaving the net transport efficiency unchanged. This has practical implications: gas viscosity is relatively easy to predict from first principles using molecular parameters alone, with no empirical density-dependence to fit.