# What is the physical basis behind thermal diffusion in a Clusius-Dickel column?

A Clusius-Dickel column allows separation of two species by the application of a thermal gradient. In its simplest form, two bulbs are maintained at two different temperatures $T_1$ and $T_2$ and joined by a thin, insulated tube. If two species $A$ and $B$ are present, at steady state the Maxwell-Stefan equation reduce to:

$$\frac{dx_A}{dz}+\frac{k_T}{T}\frac{dT}{dz}=0$$

Where $x_A$ is the mole fraction of $A$, $k_T$ is the thermal diffusion ratio and $z$ the coordinate along the tube.

For ideal gas, I expect $k_T=0$ since the number of moles in each bulb obeys the ideal gas law and is therfore equal to $n_i=p_iV/RT$, independent on the nature of the gas. If one assumes real gases, what is the physical phenomenon responsible for $k_T$ ?

Edit: to frame the question differently, what is the mechanism responsible for gaseous thermal diffusion, a process once considered for the separation of isotopes of uranium?

More atoms from the cold side try to enter the hot side than visa-versa, because there are more collisions/cm^2/s on the cold side. The pressure is the same because a cold atom exerts less force when it collides with the walls than a hot atom.

So how much pressure builds up as atoms rush over to the hot side? If the pipe that connects the hot and cold bulbs is tiny, we can assume that the molecules that enter the pipe on one side will usually make it through. Setting the flow rates equal, the final pressure ratio that is ~ the square root of the temperature ratio.

However, larger pipes leave us with much smaller pressure gradients. Now the molecules can't simply zip through, they collide with each-other on the way. This slows down diffusion. When a gradient builds up, "bulk flow" takes over and aggressively relieves any pressure differences.

But what happens when we have a gas made up with both large and small molecules? The molecules with a larger surface area are more affected by bulk flow, since they collide more, so they will concentrate on the cold side. Mass isn't as important as it sounds. Quadrupling the larger mass species roughly cuts in half the opposing effects of diffusion (they move at half the speed) and collisions (they have twice the momentum). The main effect of high mass would be to slow down the rate at which equilibrium is achieved.

Most of the time, we observe large particles diffusing away from heat. However, there are exotic situations involving liquids in which particles diffuse toward heat; liquids are complex!

• Could you expand on why larger molecules tend to concentrate on the cold side? Sep 10, 2013 at 21:44
• The main point is that the (suseptabality to pressure gradients)/(suseptability to diffusion) ratio determines whether it is on the hot side (diffusion wins) or the cold side (pressure wins). Surface area increases this ratio, but mass keeps it approximatly constant. Thus, it is higher for large molecules, so they concentrate on the colder side where pressure is lower. Sep 11, 2013 at 23:56
• Thanks. Could you point me to a reference that works out the details? Sep 12, 2013 at 16:05