Thermal diffusion

Take two bulbs, as we did in the previous section, joined by the same narrow tube, but fill them with an even, 50–50 mixture of helium and argon. Hold one bulb at 400 K and the other at 300 K. Nothing else. No membrane, no rotor, no chemistry. Wait for the system to settle, then sample each bulb. The hot bulb now holds about 51.5 percent helium, the cold bulb about 48.9 percent.[1] The gas has partly unmixed due to nothing but a temperature difference: lighter atoms toward the heat, heavier atoms toward the cold, at least for helium and argon. The effect is small, under three percentage points, but it is real.

This is thermal diffusion.[2] In liquids it carries the name Ludwig–Soret effect, after Carl Ludwig, who saw it in salt solutions in 1856, and Charles Soret, who studied it in 1879. In gases it was predicted on paper before anyone saw it, by David Enskog in 1911 and independently by Sydney Chapman in 1916, and confirmed in 1917 by Chapman and F. W. Dootson, using nearly the arrangement above.[3]

What is happening at the molecular level

At first glance, an ideal gas should be easy to figure out, since all we have are molecules that move and collide. And yet thermal diffusion resists a direct cause and effect. Its explanation is a chain of cause, effect, cause, effect. Let’s see why.

Picture a thin slice of gas somewhere along the tube, with the hotter end on one side and the cooler end on the other. Nothing is flowing, so atoms cross the slice equally in each direction. But the atoms arriving from the hot side are moving faster, on average, than those arriving from the cold side. That is true of helium and argon alike, and both gain the same fraction, about 15 percent, between 300 K and 400 K. What differs is the absolute gain. The average speed of helium rises by about 195 m/s, while argon, much heavier and slower, gains only about 62 m/s.

Now let the atoms collide with one another. To keep the bookkeeping simple, the table below follows four collisions in which a moving atom strikes an atom of the other kind that is standing still. We can leave out collisions between atoms of the same kind. In such a collision, whatever momentum one atom gains the other loses, so the total momentum of that species is unchanged, and only collisions between unlike atoms can push one species relative to the other.

A helium atom arrives from the hot side and strikes a resting argon atom. A helium atom arrives from the cold side and strikes a resting argon atom. An argon atom arrives from the hot side and strikes a resting helium atom, and an argon atom arrives from the cold side and does the same. Each moving atom arrives at the average speed of the end of the tube it came from, head-on, and each collision is perfectly elastic, meaning that both total momentum and total kinetic energy of the pair are the same after the collision as before. Because equal numbers of atoms cross in each direction, each of the four can be counted once.

Four head-on collisions using end-of-tube speeds

Moving atom	Arrives from	Speed (m/s)	Atom at rest	Change in helium velocity (m/s)	Change in argon velocity (m/s)
He	hot end	1455	Ar	+2644	−265
He	cold end	1260	Ar	−2290	+229
Ar	hot end	460	He	−837	+84
Ar	cold end	399	He	+725	−73
Net				+242	−24

In every collision the moving atom strikes an atom of the other kind that starts at rest. Speeds are the mean speeds at 400 K and 300 K. Collisions are head-on and perfectly elastic. Positive velocity changes are toward the hot end.

Looking at the bottom row, the net values are far larger than the imbalance in a real gas, because the table uses speeds from the two ends of the tube. Real atoms arrive from only about one mean free path away, where the temperature differs by a tiny fraction of a degree. But each atom collides billions of times a second, and the tiny imbalances add up to a slow drift of helium toward the hot end and argon toward the cold end. As the drift builds a difference in composition, ordinary diffusion pushes back. When the two balance, the bulbs stop changing, and steady state is reached.

The table illustrates the effect and does not derive it. It uses head-on collisions with a resting partner, which real gases do not, and the result it shows is not general. What happens in a real gas depends on how the molecules push and pull on one another and on how that changes with temperature, because faster collisions bring molecules closer together and so change their effective size. Size matters in its own right: the effect grows as the masses and diameters of the two kinds of molecule become more unequal. For hard spheres, the lighter atoms go toward the heat, and helium and argon follow that pattern. But for the molecule Maxwell favored, which repels its neighbors with a force falling as the fifth power of distance, the same speed gains give no separation at all, and for softer molecules the effect can reverse.[4] These are real complications, and a simple picture cannot untangle them. The full theory, due to Enskog and Chapman, solves the kinetic equation for the whole gas. Readers who want to understand where the direction, the size, and the occasional absence of the effect come from should turn to the literature. Furry’s 1948 article gives the clearest qualitative rules, Chapman and Cowling’s treatise gives the full theory, and Grew and Ibbs’s book, cited above, is devoted to the subject.[4]

From curiosity to column

On its own, the separation effect is too small to use. Sampling two bulbs gives a few percentage points. In 1938 Klaus Clusius and Gerhard Dickel, as discussed in [2], found a way to multiply the effect. They replaced the two bulbs with a tall, narrow, vertical gap between a hot surface and a cold one, such as a heated wire running down the center of a cooled tube. Across the gap, thermal diffusion does what it did in the bulbs: the lighter atoms drift toward the hot surface and the heavier atoms toward the cold. Then gravity takes over. Gas next to the hot surface is less dense and rises, and gas next to the cold surface is denser and sinks, so a slow loop of convection forms in the gap. The loop transports the helium-enriched gas at the hot wall upward and the argon-enriched gas at the cold wall downward. Each circuit adds a little more to the separation, and over a tall column the small sideways effect accumulates into a large difference between top and bottom. Instead of sampling a bulb, you draw gas from the ends of the column.

Like the two bulbs, the column has no membrane, no rotor, and no chemistry: only a hot surface, a cold surface, and gravity. Thermal diffusion columns were tried at large scale for uranium at Oak Ridge during the Manhattan Project, using liquid uranium hexafluoride, before other methods replaced them. [5]

References

[1] Calculated for two bulbs of equal volume at equal pressure, from the thermal diffusion ratio of helium–argon measured by B. E. Atkins, R. E. Bastick and T. L. Ibbs, Proceedings of the Royal Society A 172, 142–158 (1939), Table II: k_T = 0.0931 for a 50 percent mixture. The calculation is mine.

[2] Grew, K. E. and Ibbs, T. L. (1952). Thermal Diffusion in Gases. Cambridge Monographs on Physics. Cambridge: Cambridge University Press. xi + 143 pp.

[3] Chapman and Dootson used two glass bulbs of about 100 cm³ joined by a short tube with a stopcock, a nearly equal mixture of hydrogen and carbon dioxide at atmospheric pressure, one bulb at 230 °C and the other at 10 °C, held four hours. The hot bulb held 44.9 percent hydrogen and the cold bulb 41.3 percent. S. Chapman and F. W. Dootson, “A note on thermal diffusion,” Philosophical Magazine (6) 33, 248–253 (1917). The dates for Enskog and Chapman follow Atkins, Bastick and Ibbs (1939).

[4] Chapman and Dootson (1917), p. 248; W. H. Furry, American Journal of Physics 16, 63 (1948); S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases (Cambridge University Press).

[5] U.S. Department of Energy, Office of Scientific and Technical Information, “Manhattan Project: Processes > Uranium Isotope Separation > Thermal Diffusion,”; U.S. National Park Service, “S-50 Liquid Thermal Diffusion Plant”; and U.S. National Park Service, “Oak Ridge Wayside: S-50 Thermal Diffusion Plant”.

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