Thermal conductivity
The methods of the kinetic theory that we have been using above can be used also to compute the thermal conductivity of a gas. If the gas at the top of a container is hotter than the gas at the bottom, heat will flow from the top to the bottom.  (We think of the top being hotter because otherwise convection currents would be set up and the problem would no longer be one of heat conduction.) The transfer of heat from the hotter gas to the colder gas is by the diffusion of the “hot” molecules—those with more energy—downward and the diffusion of the “cold” molecules upward. To compute the flow of thermal energy we can ask about the energy carried downward across an element of area by the downward-moving molecules, and about the energy carried upward across the surface by the upward-moving molecules. The difference will give us the net downward flow of energy. The thermal conductivity $\kappa$ is defined as the ratio of the rate at which thermal energy is carried across a unit surface area, to the temperature gradient: \begin{equation} \label{Eq:I:43:41} \frac{1}{A}\,\ddt{Q}{t} = -\kappa\,\ddt{T}{z}. \end{equation} Since the details of the calculations are quite similar to those we have done above in considering molecular diffusion, we shall leave it as an exercise for the reader to show that \begin{equation} \label{Eq:I:43:42} \kappa = \frac{knlv}{\gamma - 1}, \end{equation} where $kT/(\gamma - 1)$ is the average energy of a molecule at the temperature $T$. If we use our relation $nl\sigma_c = 1$, the heat conductivity can be written as \begin{equation} \label{Eq:I:43:43} \kappa = \frac{1}{\gamma - 1}\,\frac{kv}{\sigma_c}. \end{equation} We have a rather surprising result. We know that the average velocity of gas molecules depends on the temperature but not on the density. We expect $\sigma_c$ to depend only on the size of the molecules. So our simple result says that the thermal conductivity $\kappa$ (and therefore the rate of flow of heat in any particular circumstance) is independent of the density of the gas! The change in the number of “carriers” of energy with a change in density is just compensated by the larger distance the “carriers” can go between collisions. One may ask: “Is the heat flow independent of the gas density in the limit as the density goes to zero? When there is no gas at all?” Certainly not! The formula (43.43) was derived, as were all the others in this chapter, under the assumption that the mean free path between collisions is much smaller than any of the dimensions of the container. Whenever the gas density is so low that a molecule has a fair chance of crossing from one wall of its container to the other without having a collision, none of the calculations of this chapter apply. We must in such cases go back to kinetic theory and calculate again the details of what will occur.