Physics · Molecular physics and thermodynamics · Year 1

Transport phenomena

Why does the scent of perfume spread slowly through a room, and how does a thermos keep tea hot?

The situation

You open a bottle of perfume in the corner of a room. Its molecules fly at hundreds of metres per second — yet the smell does not reach the far side of the room instantly, but quite a while later. Tea poured into a thermos in the morning stays hot until evening: there is almost no air between the walls of the thermos. And raisins put in water swell up within a few hours. How far do molecules fly without colliding, and how does this determine smell, heat, friction and the movement of water?

Topic

Transport phenomena, mean free path. Vacuum in a thermos flask and the spread of perfume in a room

Key ideas

Mean free path
Diffusion and Fick's law
Vacuum, Dewar flask

Formulas

λ = \frac{k T}{\sqrt{2} π d^{2} p}, ν = \frac{⟨ v ⟩}{λ}, ⟨ v ⟩ = \sqrt{\frac{8 k T}{π m}}

Mean free path ( $π d^{2}$ is the effective collision cross-section) and collision frequency.

J = - D \frac{d n}{d x}, \frac{\partial n}{\partial t} = D \nabla^{2} n, ⟨ r^{2} ⟩ = 6 D t

Fick's laws; the spreading of a cloud of particles is proportional to the square root of time.

D \approx \frac{1}{3} ⟨ v ⟩ λ, η \approx \frac{1}{3} ρ ⟨ v ⟩ λ, ϰ \approx \frac{1}{3} ρ ⟨ v ⟩ λ c_{V}

Transport coefficients: $ρ \propto p$ , $λ \propto 1 / p$ — viscosity and thermal conductivity do not depend on pressure.

Kn = \frac{λ}{L}, q_{m o l} \propto p ⟨ v ⟩ Δ T (Kn ≫ 1)

The Knudsen number: if $λ$ is larger than the size of the vessel, the gas is rarefied and the heat flow is proportional to pressure.

π = c R T, h = \frac{π}{ρ g}, \ln \frac{p_{r}}{p_{\infty}} = \frac{2 σ V_{m}}{r R T}

Van 't Hoff osmotic pressure and the column height; the Kelvin formula — the saturated vapour pressure over a curved surface.

Theory

Free path and collisions

If a molecule has diameter $d$ , it hits every neighbour that falls within a circle of radius $d$ : the effective collision cross-section is $π d^{2}$ . In air under normal conditions $λ \approx 68$ nm, and a molecule collides about $7 \cdot 10^{9}$ times per second. Relaxation time $τ \sim 1 / ν$ : after a few collisions the speeds settle into the Maxwell distribution. In the “Magnifier” the discs start with the same speed — watch how quickly the distribution reaches the Maxwell shape, and the measured $λ$ (in 2D, $λ = 1 / (2 \sqrt{2} n d)$ ).

Diffusion, Fick's law and convection

Where there is a difference in concentration, random motion carries the substance from where there is more to where there is less — Fick's law. The radius of the cloud grows like $\sqrt{6 D t}$ : in air $D \sim 10^{- 5}$ m²/s, so reaching 1 m by diffusion alone takes hours. What brings a smell quickly is convection: the air currents in a room (about 0.1 m/s). Near a constant source there is steady-state diffusion ( $n$ does not depend on time), around an opened bottle it is non-steady. In a mixture the two gases undergo mutual diffusion; a temperature difference can also create a concentration difference (thermal diffusion). The probability density of the particles changes with drift and diffusion according to the Fokker–Planck equation — the experiment models exactly this with random steps.

Internal friction, thermal conductivity, the kinetic equation

Molecules carry momentum (viscosity $η$ ), energy ( $ϰ$ ) and mass ( $D$ ) between layers; all three coefficients are linked through $\frac{1}{3} ⟨ v ⟩ λ$ : $η = ρ D$ , $ϰ = η c_{V}$ . Maxwell's unexpected conclusion: the viscosity of a gas is independent of pressure (the lower the density, the longer $λ$ becomes) — check it in the vacuum mode. A precise theory (Chapman–Enskog) corrects the numerical coefficients: for air $η \approx 1.8 \cdot 10^{- 5}$ Pa·s. These coefficients enter the equations of hydrodynamics and heat conduction. They are obtained from the Boltzmann kinetic equation in the relaxation-time approximation: the collision term is $I [f] \approx - (f - f_{0}) / τ$ . H-theorem: $H = \sum f \ln f$ decreases — entropy grows, even though every single collision is reversible (the magnifier graph).

Rarefied gas, vacuum and how to measure it

When the pressure falls and $λ$ becomes larger than the vessel ( $Kn > 1$ ), molecules hit the walls rather than each other — molecular (Knudsen) flow. Now viscosity and thermal conductivity are proportional to pressure. Producing a vacuum: a rotary pump down to about 1 Pa (rough vacuum), then a turbomolecular or diffusion pump — $10^{- 5}$ Pa and lower; the limit is set by gases released from the walls. Measuring: a deformation gauge (diaphragm, Bourdon) for $10^{5}$ – $10^{2}$ Pa, a thermal gauge (Pirani, thermocouple) for $10^{3}$ – $10^{- 1}$ Pa, where the thermal conductivity of the gas depends on $p$ , and an ionisation gauge below that.

Dewar flask (thermos)

Heat escapes in three ways: conduction, convection and radiation. If the air is pumped out of the gap between the walls ( $Kn ≫ 1$ ), the heat flow through the gas falls in proportion to pressure; convection disappears too. A silvered coating ( $ε \approx 0.03$ ) cuts radiation by tens of times. The remaining loss is mostly through the stopper. Experiment: change the pressure in the gap — from $10^{5}$ to $10^{2}$ Pa the heat flow hardly changes (continuum), and only when $λ$ reaches the width of the gap ( $\sim 1$ Pa) does it drop sharply.

Solutions, osmosis and transport in liquids

A semipermeable membrane lets water molecules through but not sugar molecules: water passes to the solution side and raises the column until the hydrostatic pressure equals the osmotic pressure $π = c R T$ (van 't Hoff's law for a dilute solution). Raisins, plant cells and the kidney work on this principle; if we apply to the solution a pressure greater than $π$ — reverse osmosis (desalination of seawater). The saturated vapour pressure over a curved surface is higher (the Kelvin formula): for a water droplet of radius 10 nm, by about 11 %. In liquids diffusion is $\sim 10^{4}$ times slower than in gases (for sugar $D \approx 5 \cdot 10^{- 10}$ m²/s), and viscosity decreases as temperature rises — the opposite of gases.

Did you know?

The vessel with vacuum walls was invented in 1892 by the Scottish physicist James Dewar to store liquefied gases; today's thermoses are the household version of this vessel.
In air at sea level each molecule collides about $7 \cdot 10^{9}$ times per second, and between two collisions it travels on average only about 70 nm.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Phase transitions — Grade 10
Thermal expansion — Grade 10
Refrigerator and heat pump — Grade 10
Real gases — Grade 10
Maxwell–Boltzmann distribution — Grade 10
Quantum statistics — Year 4
Phase transition and fluctuations — Year 4