Kinetic estimates of gas transport coefficients ($D\sim \overline{v}\lambda /3,\eta \sim nm\overline{v}\lambda /3,\kappa \sim n{c}_v\overline{v}\lambda /3$)

In a dilute gas whose mean free path $\lambda$ is short compared with macroscopic gradient lengths, particles carry properties from their previous collisions over distances of order $\lambda$. A first-gradient isotropic estimate gives a transported flux $-n\overline{v}\lambda \nabla a/3$ for a property $a$ carried per particle. Applying this reasoning to species abundance, directed momentum $m{u}_x$ and thermal energy with per-particle heat capacity $c_v$ gives diffusivity $D\sim \overline{v}\lambda /3$, dynamic viscosity $\eta \sim nm\overline{v}\lambda /3$ and thermal conductivity $\kappa \sim n{c}_v\overline{v}\lambda /3$. These numerical prefactors are estimates, not exact collision-integral results.

For a constant hard-sphere diameter $d$, $\lambda \propto 1/(n{d}^2)$ and $\overline{v}\propto \sqrt{T/m}$. Consequently $\eta \propto \sqrt{mT}/d^2$, $\kappa \propto c_v\sqrt{T/m}/d^2$ and $D\propto \sqrt{T/m}/(n{d}^2)$. At fixed pressure the last scales as $T^{3/2}/p$. The near pressure-independence of viscosity and thermal conductivity applies only while gas-gas collisions, rather than apparatus walls, determine the mean free path. Temperature-dependent cross-sections and internal molecular excitations change the simple power laws.