2.6 Transport from kinetic theory

A gas out of uniformity — with a velocity gradient, a temperature gradient, or a concentration gradient — relaxes back toward uniformity by carrying the non-uniform quantity from place to place. The carrier is the molecule, and the distance it carries a quantity before giving it up is the mean free path of 2.5. This single picture yields the gas’s viscosity, thermal conductivity, and diffusion coefficient, and explains a result that startled the nineteenth century.

The common mechanism

A molecule crossing a plane carries whatever it last equilibrated to, one mean free path back. If some property per molecule $\phi$ varies with height $y$ , a molecule crossing from below carries $\phi(y-\ell)$ and one crossing from above carries $\phi(y+\ell)$ ; the net flux of $\phi$ across the plane is proportional to the difference, hence to the gradient $d\phi/dy$ . Working the bookkeeping through gives every transport coefficient the same form: a carrier flux $n\langle v\rangle$ times a mean free path $\ell$ times the amount of the property carried.

▶ Viscosity η ≈ ⅓ ρ ⟨v⟩ ℓ, independent of pressure Derivation

Let the gas have a mean flow $u_x(y)$ varying with height $y$ . A molecule crossing the plane at $y$ from below last collided at $y-\ell$ and carries the extra $x$ -momentum $m\,[u_x(y-\ell) - u_x(y)] \approx -m\ell\, du_x/dy$ . About $\tfrac16 n\langle v\rangle$ molecules cross unit area per unit time in each of the six axis directions, so the net $x$ -momentum flux — the shear stress — is

\tau \;\sim\; \tfrac13\, n\langle v\rangle \cdot m\ell\,\frac{du_x}{dy} \;=\; \tfrac13\, \rho\,\langle v\rangle\,\ell\,\frac{du_x}{dy},

with $\rho = nm$ . Comparing with the definition of viscosity $\tau = \eta\, du_x/dy$ ,

\eta \;\approx\; \tfrac13\,\rho\,\langle v\rangle\,\ell.

Now substitute $\rho = nm$ , $\langle v\rangle \propto \sqrt{T/m}$ , and $\ell = 1/(\sqrt2\,n\pi d^2)$ : the number density $n$ cancels, leaving

\eta \;\sim\; \frac{m\langle v\rangle}{\pi d^2} \;\propto\; \frac{\sqrt{m k_B T}}{d^2}.

The viscosity of a gas is independent of its pressure (at fixed temperature) and rises as $\sqrt{T}$ . ✓

Maxwell derived the pressure-independence in 1860 and, disbelieving it, tested it himself: a pendulum damped by air loses energy at the same rate whether the air is at full pressure or half. The microscopic reason is a cancellation — halving the density doubles the mean free path, so each molecule carries momentum twice as far but there are half as many carriers.

The unity of transport

The same argument with a different carried quantity gives the other transport coefficients, all of the form $\tfrac13\,(\text{carrier density})\times\langle v\rangle\times\ell\times(\text{amount per molecule})$ :

Diffusion carries the molecules themselves: $D \approx \tfrac13\langle v\rangle\,\ell$ .
Thermal conduction carries kinetic energy: $\kappa \approx \tfrac13\, n\, c\,\langle v\rangle\,\ell$ , with $c$ the heat capacity per molecule.
Viscosity carries momentum: $\eta \approx \tfrac13\,\rho\,\langle v\rangle\,\ell$ , as above.

Their ratios are pure numbers of order unity — $\eta/(\rho D) \sim 1$ , $\kappa/(\eta c) \sim 1$ — because all three are the same molecular shuttle measuring out the same mean free path. Momentum, heat, and matter diffuse through a gas at comparable rates for one reason: they ride the same molecules. The continuum laws built on these coefficients — the Navier–Stokes equation, Fourier’s law, Fick’s law — are developed in the viscosity and diffusion chapter.

Brownian motion: kinetic theory made visible

A particle large enough to see through a microscope but small enough to feel the molecular bombardment receives an unbalanced kick at every instant and wanders. Each collision delivers a small random impulse uncorrelated with the last, so the particle performs a random walk and its mean-square displacement grows linearly in time, $\langle x^2\rangle = 2 D t$ in one dimension. The same molecular collisions that drive the wander also resist the particle when it is pushed, and the two are tied by the temperature through the Einstein relation $D = k_B T/\gamma$ , with $\gamma$ the drag coefficient. Brownian motion is the molecular agitation of kinetic theory, scaled up until a single object makes it visible; the random-walk mathematics is in Math Foundations 11.3, and the drag and the Einstein relation in the viscosity and diffusion chapter.