Editing Kinetic theory of gases (section)

=== Diffusion coefficient and diffusion flux ===
{{See also|Fick's laws of diffusion}}Following a similar logic as above, one can derive the kinetic model for [[mass diffusivity]]<ref name="Sears1975" /> of a dilute gas:

Consider a [[Steady state|steady]] diffusion between two regions of the same gas with perfectly flat and parallel boundaries separated by a layer of the same gas. Both regions have uniform [[Number density|number densities]], but the upper region has a higher number density than the lower region. In the steady state, the number density at any point is constant (that is, independent of time). However, the number density <math>n</math> in the layer increases uniformly with distance <math>y</math> above the lower plate. The non-equilibrium molecular flow is superimposed on a [[Maxwell–Boltzmann distribution|Maxwell–Boltzmann equilibrium distribution]] of molecular motions.

Let <math> n_0 </math> be the number density of the gas at an imaginary horizontal surface inside the layer. The number of molecules arriving at an area <math>dA</math> on one side of the gas layer, with speed <math>v</math> at angle <math>\theta</math> from the normal, in time interval <math>dt</math> is
<math display="block"> nv\cos(\theta) \, dA \, dt \times \left(\frac{m}{2 \pi k_\mathrm{B}T}\right)^{3 / 2} e^{- \frac{mv^2}{2k_\text{B}T}} (v^2\sin(\theta) \, dv\, d\theta \, d\phi)</math>

These molecules made their last collision at a distance <math>\ell\cos \theta</math> above and below the gas layer, where the local number density is
<math display="block"> n^{\pm} = \left( n_0 \pm \ell \cos \theta \, \frac{dn}{dy} \right) </math>

Again, plus sign applies to molecules from above, and minus sign below. Note that the number density gradient <math>dn/dy</math> can be considered to be constant over a distance of mean free path.

Integrating over all appropriate velocities within the constraint <math>v > 0</math>, <math display="inline">0 < \theta < \frac{\pi}{2} </math>, <math>0 < \phi < 2\pi</math> yields the molecular transfer per unit time per unit area (also known as [[diffusion flux]]):
<math display="block"> J_y^{\pm} = - \frac{1}{4} \bar v \cdot \left( n_0 \pm \frac{2}{3} \ell \, \frac{dn}{dy} \right) </math>

Note that the molecular transfer from above is in the <math>-y</math> direction, and therefore the overall minus sign in the equation. The net diffusion flux across the imaginary surface is thus
<math display="block"> J = J_y^{+} - J_y^{-} = -\frac {1}{3} \bar{v} \ell \frac{dn}{dy} </math>

Combining the above kinetic equation with [[Fick's laws of diffusion#Fick's first law|Fick's first law of diffusion]]
<math display="block"> J = - D \frac{dn}{dy} </math>
gives the equation for mass diffusivity, which is usually denoted <math> D_0 </math> when it is a dilute gas:
<math display="block"> D_0 = \frac{1}{3} \bar{v} \ell </math>

The corresponding expression obtained from [[Revised Enskog theory|Revised Enskog Theory]] may be written as
<math display="block"> D = \alpha_D D_0 </math>
where <math> \alpha_D </math> is a factor that tends to unity in the limit of infinite dilution, which accounts for excluded volume and the variation [[chemical potential]]s with density.