Using a kinetic model for gases, is it possible to derive a relationship between pressure and thermal conductivity?
From what I have read, I thought pressure was independent of thermal conductivity?
$$\kappa = \frac{n\langle v\rangle\lambda c_V }{3N_A} \tag{1}$$
And $$\lambda =\frac{kT}{\sqrt 2\pi d^2 p} \tag{2}$$ and then just plug this into eq. $(1)$? .
It is always a good first step to think about the limiting cases. The simple limiting case for the thermal conduction of a gas in dependence of the pressure is the thermal conduction of a vacuum (which is the limit $p \to 0$). This tells you that thermal conductivity depends strongly on pressure (at least for some pressures).
You can derive a heat conductivity formula for a gas akin to the Drude formula for the electric conductivity of a classical electron gas (source: Hyperphysics on heat conductivity):
$$ \kappa = \frac{n\langle v \rangle \lambda c_V}{3N_A} $$
Where $\langle v \rangle$ is mean particle speed (that is $\langle \sqrt{ \vec v \cdot \vec v}\rangle$, $n$ is the particle density, $c_V$ is the molar heat capacity at constant volume and $\lambda$ is the mean-free path.
At very low pressures (high vacuum) the dependence on pressure is linear, as the mean free path is limited by the system size and not by collisions in the gas, but the density $n$ is proportional to the pressure (due to $pV = NRT$, which means $n = p/RT$).
For higher pressures the collision probability rises and dominates the mean free path, then $\lambda$ has a dependence approximately like $\lambda \propto 1/p$, which means that the heat conductivity saturates and only slightly rises with a further increase of pressure.
Note, that for gases in an external field (gravity) the dominating heat transport process is usually convection, not conduction. Convection depends even stronger on pressure as can be shown by a crude approximation. We assume a warm object in a large pool of fluid. A first approximation is the the viscosity of a dense gas is independent of the pressure. Convection is driven by density differences, assuming laminar flow, the flow rate is proportional to the driving pressure difference: $$ I \propto \Delta p. $$ The pressure difference that drives the convection is due to the difference in hydrostatic pressure exerted by the heated column above the heat source and the non-heated column. As the hydrostatic pressure is given by $p = \rho h g$ and the density is given by $\rho(p) = \frac{p \rho(p_0)}{p_0}$, we get that $\Delta p \propto \Delta \rho \propto p$. The transferred heat is proportional to the temperature difference of the object and the fluid, the particle number flux and the heat capacity per particle. Combining the above results we get $$\dot Q \propto p^2.$$
Early work in the upper atmosphere showed that electrical conductivity of air depends upon the pressure; the quantitative relationship is known as Paschen's Law, https://en.wikipedia.org/wiki/Paschen%27s_law.
But what about thermal conductivity? As pressure is reduced, so is the density of the gas particles, which changes the mean free path statistics. Thus continuing to reduce the pressure will eventually result in changes in the thermal conductivity: http://www.electronics-cooling.com/2002/11/the-thermal-conductivity-of-air-at-reduced-pressures-and-length-scales/.
