# The Cool Joule-Thomson Effect

Most of us notice the refreshing jet of deodorant when it hits our sweaty armpits from a pressurised gas container. Some might have noticed that a soft-drink is somewhat colder right after opening the bottle (or the can) even if it did not come directly out of the refrigerator. canned compressed air also comes out cool. Few of us lose time to think further about this, but the general perception is that gaseous stuff cools down when it is released from a pressurised container, or, in other words, when it expands. Two famous scientists, James Prescott Joule (1818 - 1889) and William Thomson (1824 - 1907) - which later became Lord Kelvin - took a closer look at expanding gases, and found an explanation for the phenomenon which was later named after them: The Joule-Thomson effect.

### Theory

#### Expansion

We will take a look at expanding gases and see whether the theory predicts some form of cooling down or heating up of a gas. In the following passages we divide the universe into two parts, the container containing the gas which expands and gets compressed and all the rest. Two extreme cases of containers can be thought of, we shall have a look at both. First, a non-insulated container, that is, any heat the gas produces or consumes comes from the rest of the universe and second, an insulated container, the expanding and shrinking gas cannot drain any heat from the exterior. (Note that none of both cases exist in the real world, the truth is somewhere in between)...WIPWIP...
. E.g. by opening the valve of a compressed gas can. And second, the compression and expansion that results from pushing or pulling a piston. Both kinds of expansion have significant differences.

#### Ideal Gases Don't Do It

Joule demonstrated that the aforementioned 'cooling down under expansion' does not work just by letting a gas out of a pressurized container. The explanation was that this is so, because an ideal gas1 must always have a constant internal energy. Therefore, it cannot absorb or give away internal energy to the surroundings when it is expanding freely, which would be perceived as cooling or heating, respectively.2 The picture changes if one allows the gas to perform work, e.g. by letting it push a piston up or down, in this case the internal energy is converted into work, and the temperature decreases.

A common interpretation of the Joule-Thomson effect uses the ideal gas state equations, which is one of the simpler equations in thermodynamics - horribly familiar from school exams:

p·V=nRT

In words: The product of pressure (p) and volume (V) is directly proportional to the temperature (T), where n and R are certain constants. According to this law, if one increases the volume of a gas (in a container, eg. a gas in a cylinder with a piston) and holds the temperature constant the pressure will decrease. If the pressure is (somehow) kept constant, then the temperature must decrease. The common mistake is to assume that in a real experiment (still using an ideal gas) neither the pressure nor the temperature is held totally constant, so that something in between those extremes happen. Kind of like this: The volume increases, 90% of this increase is equalized by a decrease in pressure and the remaining 10% cool the surroundings down. This is a wrong interpretation. The temperature only changes if the internal energy of the gas is allowed to change. Which is not the case for an ideal gas expanding against a pressure that is much lower. It is the case if one uses the stored ener

#### Real Gases Do It

To lower the temperature the molecules would have to draw some energy from the surroundings. Ideal gases might do that, but at the same time they cool the surroundings down they become hot. In the end, when equilibrium is reached the same amount they have absorbed is also released to the environment. In the end the temperature will not have changed. So, unless the gas particles find a way to keep the energy - and ideal gases don't do that by definition - the cooling by expansion will not work. Fortunately, for everyone in desperate need of an air conditioner, this is not what gas particles do in real life.

The atoms or molecules that make up a gas are not like snooker balls en miniature. They do interact with each other. There are many possible types of interaction. For example, the gas particles sticking to each other, or pushing each other away when they get too close. Gas particles are more like rubber balls covered with marmelade. Altogether there are two big classes for the so-called intermolecular interactions: attractive (sticking marmelade) and repulsive (pushing rubber) interactions.

Seen from the outside of the gas-container the attractive interactions will lead to a more stable gas, one will have to put in energy to stop the attractive interactions, the gas itself has a lower internal energy. Conversely, to stop the repulsive interactions one will have to take energy away from the molecules, the gas itself has a high internal energy.

As a consequence of these interactions the internal energy cannot be independent from the volume. One example: Under normal circumstances, the attractive interactions are by far the most dominant. Hence, a gas packed into a small volume at a high pressure (the deodorant gas) will be subject to a lot of attractive interactions, or, in other words have a relatively low internal energy. When it is released into a big volume at a lower pressure (the air between the nozzle and the armpit, for example) the particles will be subject to less of these interactions, or have a higher internal energy. This transition from the low internal energy to the high internal energy consumes energy which is drained from the environment, this in its turn is perceived as a drop in temperature.

#### Details

The stickier the molecule the bigger the effect. CFC, for example, infamous for the destructive effect on the ozone layer, is a very sticky gas molecule. The effect is not so pronounced with the likes of nitrogen and oxygen - which are the gases used by Joule. Under the conditions of his experiment air behaves like an ideal gas. Some few gases, like Helium or Hydrogen even show the inverse behaviour. They heat up under expansion. The repulsive forces in helium and hydrogen outweigh the sticking forces, leading to the opposite effect as described for the other gases.

#### Inversion Temperatures

The aforementioned repulsive forces in their turn depend on the temperature. Above a certain temperature the repulsive forces outweigh the attractive forces. For most gases the inversion temperature lies somewhere around and above 350°C (e.g. oxygen: 477°C, nitrogen: 352°C). That is why one can liquefy gases by compressing and expanding them, they get a little colder after every expansion step. Exceptions are helium (Ti= -239°C), hydrogen (Ti= -69°C) and neon (Ti= -3°C), which have to be cooled down below their inversion temperatures before starting the liquefaction steps.

### The Joule-Thomson Effect in Quotidian Life

Apart from the deodorant and the soft-drink bottles the Joule-Thomson effect is used to cool down and liquefy gases. It is also the priciple behind the mechanisms of air-conditioners and the refrigerators. In all cases the expansion of a compressed gas is used to drain energy from a recipient (which will therefore become colder). This process is optimized by using as much gas as possible over the highest possible difference in volume - and by using particularly sticky gas molecules. After the expansion step the gas is re-compressed in a different chamber, which will heat up due to the inverse process. In the end, any compressor will produce more heat than 'coldness', the trick is just to keep the chambers well separated. The most effective gases, with high stickiness (or cohesion forces) are the chloro-fluorocarbons (also known as CFC or freon gases), which harm the ozone layer. The usage of the Joule-Thomson effect to cool things down will not lose its importance in the near future, even though there are a few alternative ways to get things cold (e.g. Peltier elements).

Joule started studying science in 1834 when he went to Cambridge to learn his stuff with science top-banana John Dalton. Later on, Joule established the equivalence between amounts of heat and mechanical work, studied the effectivity of electrical motors, invented the technique of arc-welding and the displacement pump. He had a lot of money from the family's prospering brewery, but he ended up flat broke after he spent all his money on his private scientific experiments. James had a big, impressive beard and the unit of energy, the Joule, is named in his honour.

William Thomson (1824 - 1907) and since 1892 also Baron Kelvin of Largs (Scotland), went to university in Glasgow aged 10. In 1852 he worked with Joule on the Joule-Thomson effect. He became obscenely rich from patents and consulting activities related to submarine cable technology. Some say he was a lousy lecturer and the official unit of absolute temperature, the Kelvin, is named in his honour.

### Summary

The Joule-Thomson effect is the name of the commonly observed phenomenon, where a gas will cool down when it expands. It is a misconception that this effect results from the ideal gas equation (pV=nRT). An ideal gas does not cool down or heat up under expansion. The explanation for the effect lies in the non-ideal intermolecular interactions, also termed cohesion, present in real gases. When a real gas expands the interactions end, and this generally costs energy. This energy is drained from the environment, which in the end has a lower temperature. The Joule-Thomson effect is used to produce 'coldness' in most air-conditioners and refrigerators. It is also the key step in the liquefaction of gases.

1An ideal gas is composed of particles that do not interact with each other, except for fully elastic collisions - like miniaturised snooker balls. An ideal gas is a virtual concept.2To calculate the variation of temperature with the change of volume under constant internal energy one could write the differential: (dT/dV)U = 1/cV(P-T(dp/dT)V). Using the equation for an ideal gas: (dp/dT)V = nR / V. Substituting that into the differential yields: (dT/dV)U=0. No change of temperature with volume.