Editing Kinetic theory of gases (section)

=== Thermal conductivity and heat flux ===
{{See also|Thermal conductivity}}
Following a similar logic as above, one can derive the kinetic model for [[thermal conductivity]]<ref name="Sears1975" /> of a dilute gas:

Consider two parallel plates separated by a gas layer. Both plates have uniform temperatures, and are so massive compared to the gas layer that they can be treated as [[thermal reservoir]]s. The upper plate has a higher temperature than the lower plate. The molecules in the gas layer have a molecular kinetic energy <math>\varepsilon</math> which increases uniformly with distance <math>y</math> above the lower plate. The non-equilibrium energy flow is superimposed on a [[Maxwell-Boltzmann distribution|Maxwell-Boltzmann equilibrium distribution]] of molecular motions.

Let <math> \varepsilon_0 </math> be the molecular kinetic energy of the gas at an imaginary horizontal surface inside the gas 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, and each will contribute a molecular kinetic energy of
<math display="block"> \varepsilon^{\pm} = \left( \varepsilon_0 \pm m c_v \ell \cos \theta \, \frac{dT}{dy} \right), </math>
where <math>c_v</math> is the [[specific heat capacity]]. Again, plus sign applies to molecules from above, and minus sign below. Note that the temperature gradient <math>dT/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 energy transfer per unit time per unit area (also known as [[heat flux]]):
<math display="block"> q_y^{\pm} = -\frac{1}{4} \bar v n \cdot \left( \varepsilon_0 \pm \frac {2}{3} m c_v \ell \frac{dT}{dy} \right) </math>

Note that the energy transfer from above is in the <math>-y</math> direction, and therefore the overall minus sign in the equation. The net heat flux across the imaginary surface is thus
<math display="block"> q = q_y^{+} - q_y^{-} = -\frac{1}{3} \bar{v} n m c_v \ell \,\frac{dT}{dy} </math>

Combining the above kinetic equation with [[Fourier's law]]
<math display="block"> q = -\kappa \, \frac{dT}{dy} </math>
gives the equation for thermal conductivity, which is usually denoted <math> \kappa_0 </math> when it is a dilute gas:
<math display="block"> \kappa_0 = \frac{1}{3} \bar{v} n m c_v \ell </math>

Similarly to viscosity, [[Revised Enskog theory]] yields an expression for thermal conductivity that reduces to the above expression in the limit of infinite dilution, and which can be written as
<math display="block"> \kappa = \alpha_\kappa \kappa_0 + \kappa_c </math>
where <math> \alpha_\kappa </math> is a term that tends to unity in the limit of infinite dilution, accounting for excluded volume, and <math> \kappa_c </math> is a term accounting for the transfer of energy across a non-zero distance between particles during a collision.