The Joule-Thomson Effect and The Models We Know

Here's a brain teaser that seems easy but is actually very hard:

We've got a cylinder split in half by a diaphragm.

• On the left side is some Helium gas at 0\degreeC and 1 atmosphere of pressure
• On the right side is a perfect vacuum

We're going to snap our fingers and make the diaphragm instantly disappear:

The gas is now free to occupy twice as much volume, and it will quickly diffuse out:

What happens to the temperature?

Go ahead and actually write down a guess.

Table of Contents

• The High School Answer
• The Thermodynamics 101 Answer
• The Physics 101 Answer
• The Chemistry 101 Answer
• Recap
• The Actual Answer
• What happens with other gasses?
• What happens at higher pressures?
• Conclusion
• Contact me

The High School Answer

In high school physics or chemistry you learn the ideal gas law which governs things like temperature, pressure, and volume for ideal gasses:

In this case we know that V doubles and we know that n and R are constant, but P and T are completely unconstrained. We've got one equation with two unknowns.

This cannot be solved without some other constraint, it must be a trick question! If you told me the final pressure I could answer it.

High school me would have chalked this up as an unknowable thing, outside the reach of equations.

The Thermodynamics 101 Answer

In Thermo class you learn the concept of fluid work.

Gas in a cylinder exerts some pressure on a piston head. The integral of pressure over that area gives a force.

If the piston moves, the integral of that force over distance tells us how much work the fluid did against the piston.

The energy to do the work comes from the internal energy of the gas, which consequently cools down.

That's all well and good but in our case the diaphragm just disappears instantly. There is no piston to perform work against, no area to integrate over.

With no cylinder head to extract work from the gas, its internal energy must stay constant so the temperature must stay constant!

If I were fresh out of Thermodynamics 101, I would say that the temperature remains unchanged, and I'd point at these equations to prove it.

The Physics 101 Answer

Unlike the point-mass particles of Thermo 101, actual atoms do not bounce off of each other instantly. Actual atoms squish, more like kickballs than pool balls.

In any atom, the nucleus carries almost all the mass and it is surrounded by an electron cloud which compresses like a spring whenever it encounters another electron cloud.

In a perfect head-on collision the kinetic energy of both atoms briefly drops to zero because it has been entirely converted into the potential energy of the compressed electron clouds. A moment later the potential energy is converted back to kinetic energy as the particles retreat.

Now, at any given instant a large number of collisions are actively taking place in our cylinder full of Helium so some finite amount of kinetic energy is locked up as potential energy. If we double the cylinder volume we're going to halve our gas density which means one fourth as many collisions per second.

Fewer collisions means that at any given instant, less of the internal energy is locked up as potential energy and is therefore available as kinetic energy. The average kinetic energy of our gas particles will go up.

If the average kinetic energy goes up, the temperature must go up!

The Chemistry 101 Answer

In college chemistry at some point you learn about the London Dispersion Force. This is a type of electrical attraction that occurs between atoms and molecules—anything that has a big electron cloud around it.

Neutral atoms like our Helium gas have exactly as many electrons as they have protons so on the whole, they are uncharged. But sometimes just by chance more electrons than usual will gather on one side of an atom, creating a partial negative charge.

The \delta symbol is a lowercase Greek delta, and in this case it is pronounced partial.

The extra negativity on that side will repel the electrons of a nearby atom, causing that atom to also gain a partial charge which will be aligned with the first atom's.

This creates a kind of weak bond, we might call it a dispersion bond, between the two partially-charged atoms so that they'd prefer to stay next to each other rather than fly apart. It's a nearly instantaneous effect that is real and measurable.

This bonding tendency means that if two gas particles collide with just the right initial conditions, they will actually stick together!

A container of gas is therefore a swirling soup: some atoms traveling alone, some traveling together in dispersion-bonded pairs.

The pairs don't last long though. As soon as a third atom smashes into the bonded pair, some of its kinetic energy will be absorbed and the bond will break.

The result is three free particles which together have less kinetic energy than the bonded pair and incident particle together had.

If chance collisions are what form these bonded pairs, and chance collisions break them, then the collision rate will be a determining factor in how many of them exist at any given instant.

Back to our cylinder full of Helium: if we double the volume then we halve the density so the collision rate falls by 4x. That means the bonds will form at \frac{1}{4} the rate, but also that they will be broken at \frac{1}{4} the rate! So on the whole, there should be no change to the number of bonded pairs.

But wait! If a bonded pair ever collides with the surface of the cylinder, that will also break the bond:

Halving the gas density means there are fewer atoms in the way and any bonded pair has an easier time making its way to the cylinder surface to break, so we should expect more breakages per second.

So the cylinder surface can help break the bonds but it cannot help create them!

That means we should expect far fewer bonded pairs after removing the diaphragm. Breaking each bond robs kinetic energy from the atoms involved, so the average kinetic energy afterwards must be lower.

So the temperature of the gas must go down!

Recap

We've got 4 answers that all sound correct:

• High School Answer: This question requires more information
• Thermo Answer: The temperature must stay the same
• Physics Answer: The temperature must go up
• Chemistry Answer: The temperature must go down

So which is it? Do you want to change your guess?

The Actual Answer

If you perform this experiment, the Helium will get much warmer.

But if you start the experiment with much colder Helium at around -222\degree C, the Helium will stay the same temperature.

And if you start even colder, at say -243\degree C, the Helium will get colder.

Every gas has a characteristic temperature, its Joule-Thomson Inversion Temperature, at which point this type of expansion causes no temperature change. Whenever a gas starts out hotter than than its inversion temperature, it will cascade to even higher temperatures. If it starts lower than its inversion temperature, it gets even colder.

The Thermodynamics answer above is correct in that the total internal energy of the gas does stay constant. The hair we need to split is that:

While

The Physics explanation above is correct and fewer collisions really does free up more kinetic energy. The Chemistry explanation above is also correct and lower density really does mean fewer bonded pairs which absorbs some kinetic energy. Which effect dominates depends on the initial temperature of the gas!

What happens with other gasses?

Larger electron clouds are easier to polarize, so they form stronger dispersion bonds. Consequently, big atoms like Krypton or Xenon have very high inversion temperatures because their capacity for dispersion bonding is so strong.

Almost all gasses have inversion temperatures higher than room temperature so almost all gasses will cool if you start at room temp. Hydrogen, Helium, and Neon are the only gasses which heat up.

In Hydrogen fuel stations, high-pressure Hydrogen stored at the station gets transferred to lower-pressure tanks inside the cars. This causes significant heating, so most stations will refrigerate their high-pressure tanks. Starting at a lower temperature attenuates the heating effect.

The capacity for strong dispersion bonding is why atoms with similar chemistry exist in different phases at room temperature. Chlorine is a gas at room temperature and pressure, but Bromine is a liquid and Iodine is a solid, despite them all being Halogens!

More complex molecular gasses like CO2, Methane, or Butane all exhibit strong dispersion bonding and so all have high inversion temperatures.

Speaking of CO2, there is a fun science experiment you can run if you have a CO2-filled fire extinguisher. If you cover the nozzle with a pillow case and empty the extinguisher, you will end up with lots of powdery dry ice in the pillow case.

This happens because the CO2 inside the cylinder is well under its inversion temperature of 695° C. It is stored at very high pressure (around 55 atmospheres) so allowing it to expand into the atmosphere is almost like allowing it to expand into a vacuum. The cooling effect is so strong that it can pull the CO2 below its own freezing point.

It is possible to expand a fluid so much that it becomes a solid.

This not just a fun novelty, it is an important step in the Linde Process which is widely used for liquifying gasses.

What happens at higher pressures?

Everything I've said so far is true if your gas starts at 1 atmosphere of pressure. But the story changes at higher pressures!

As the starting pressure gets higher, the number of collisions per second goes up. More collisions means the squishy kickball effect that causes heating becomes more important and the dispersion pair effect which causes cooling becomes less important.

So generally, as the initial pressure increases the heating effect dominates, so the inversion temperature gets lower.

Said again: The inversion temperature of a gas is not a constant. It depends on the initial pressure.

But here's where things get weird. At sufficiently high pressure there are just so many collisions that the heating effect dominates no matter the initial temperature! Beyond some maximum pressure, there is no inversion!

To understand what's going on in this regime we have to remember that at pressures this high most gasses are no longer gasses, they are supercritical fluids:

A supercritical fluid is a phase of matter where the properties of the fluid smoothly transition between liquid-like and gas-like.

At high pressures and low temperatures the supercritical fluid starts to get very dense and liquid-like. In that regime, there are again so many collisions ocurring that the heating effect begins to dominate over the cooling effect:

Here we have finally arrived at the full picture. At most achievable pressures, every fluid actually has two inversion temperatures! Between these temperatures, the fluid will cool. Beyond them, the fluid will heat up. But starting from ultra-high pressures, all fluids heat up.

The easy way to remember this is that large numbers of collisions happen in two regimes: very high temperature and very high density. In both regimes, heating will dominate.

Conclusion

It is tempting to teach students that the Ideal Gas Law as some sort of Truth™, some sort of inescapable Law of the Universe. But it's just a convenient, algebra-only model that sorta-kinda works under a narrow range of circumstances.

With the inclusion of Calculus we can start to do basic Thermodynamics, but this too is a dramatic oversimplification of the real world. It's just a model.

Over and over again we learn more and produce richer models. With Physics and Chemistry under our belts we can invent concepts like Inversion Temperature but even this is just a model. It doesn't always work. Its predictions aren't always right.

In every field of science we have to start small, teaching simple models which are known to be wrong but which lay a useful foundation. If you've ever taken Psych 101, Econ 101, Biology 101, Electricity and Magnetism, Classical Mechanics, or really any introductory course, please keep in mind that you've only learned the equivalent of the Ideal Gas Law in each of those fields.

The role of an engineer is to know many models and when to apply which ones.

The role of a scientist is to create new models that can explain phenomena that our current models miss.

Whatever your role is, remember not to mistake a model for reality.

Contact me

You can find me on twitter or send me an email at mattferraro.dev@gmail.com.