This problem concerns estimating the thermal conductivity of an ideal gas via the kinetic theory of gases. By Fourier's law of conduction, the heat flux density, or the power delivered per unit perpendicular area, across an area is proportional to the temperature gradient.
$$
\frac{d q}{d t}=-k \frac{d T}{d z}
$$
where the z-direction has been set as the direction along which temperature varies. $q$ is the heat flow per unit area - implying that $\frac{d q}{d t}$ is the power per unit area. The negative sign in the above equation implies that heat flows from regions of higher temperature to regions of lower temperature. Finally, $k$ is the thermal conductivity which we aim to determine in this problem.
Now, consider the following set-up. Two large plates parallel to the xy-plane are located at certain z-coordinates. They are maintained at different temperatures such that a steady, position-dependent temperature $T(z)$, that is strictly decreasing with increasing $z$, is set up in the region between them. An ideal gas with $f$ degrees of freedom fills this region.
(a) Argue qualitatively why there will be power delivered across a plane of a constant $z$-coordinate based on the varying temperature $T(z)$.
(b) It is known that the gas molecules have a mean free path $\lambda$. Now, consider a class of gas molecules with a certain velocity that makes an angle $\theta$ with the z-direction. If the gas molecules cut across a plane of z-coordinate $z$ at a particular instance, what is the average kinetic energy carried by them?
(c) Using the previous result, determine the heat flux density and thermal conductivity $k$ across a plane of z-coordinate $z$, in terms of the degrees of freedom of the gas molecules $f$, the number density $\eta$ (assumed to be uniform throughout), $\lambda$ and the average speed $\langle v\rangle$ of the gas molecules at a plane of z-coordinate $z$. Assume that $\lambda$ is small such that second order and above terms in $\lambda$ are negligible.