The Joule-Thomson effect, which is the heating or cooling of a gas as it’s depressurized across a porous plug, is caused by intermolecular interactions of attraction and repulsion. But how exactly? I have not been able to find any publication that has answered this question at the micro-level of moving and interacting atoms. I addressed this in a previous post (here) and provide an update below with my build-up to my as-yet unvalidated hypothesis. But first things first. Let’s review the fundamentals.
Intermolecular interactions
Intermolecular interactions of attraction and repulsion occur between all atoms and result from the fact that atoms consist of negatively charged electrons and positively charged protons that are separated by a finite distance, thus setting up a magnet, or, more accurately, a spinning magnet as the electrons are in continual high-speed motion around the protons.
As shared in Chapter 3 of Block by Block: The Historical and Theoretical Foundations of Thermodynamics, consider two hydrogen atoms, each containing one electron and one proton, sitting far apart from each other. While they each contain both positive and negative charges, they appear neutral to each other since the distance between the internal charges is small relative to the distance between the two atoms. The attraction and repulsion forces between the two charges of one and the two charges of the other cancel each other out.
However, as the hydrogen atoms approach each other (illustration on right) and the distance between them shrinks towards the scale of the atoms themselves, the electromagnetic forces between the two no longer cancel out. The charges from one start to ‘see’ the charges from the other and so begin influencing each other’s motions.
When the electron (E1) of the first hydrogen atom moves between the two protons (P1 and P2), it repels the electron (E2) of the second, forcing it to move away and behind P2. E1 is attracted to P2 resulting in net attraction between the two atoms. Since E1 is closer to P2 than P1 is, the attractive force between E1 and P2 is greater than the repulsive force between P1 and P2. One could argue though, that E1 will eventually circle around behind P1, thus reversing this situation. But one can’t ignore E2 in the process. When E1 departs, E2 comes back out front again and creates a new net attractive force with P1. In this way, the two electrons develop a synchronized motion, resulting in a net-net attractive force between the atoms that switches back-and-forth and back-and-forth at very high frequency. This attractive force, called the van der Waals force or London dispersion, is a natural response between any two atoms. The strength of attraction varies, but it’s always there.
In essence, each atomic hydrogen atom is a microscopic magnet containing a positive (proton) and a negative (electron) pole. The opposite poles from the two magnets attract each other. In the real atoms, even though the poles are spinning at very high speeds, the atoms sense their presence and so respond to each other, aligning their motions such that the opposite poles, although moving, can still line up in a synchronized way and create a net attractive force.
Between all atoms, not just atomic hydrogen, van der Waals attraction is always present. If atoms approach each other slowly enough, these attractive forces lead to weak van der Waals bonding (not covalent bonding) and can result in such physical phenomena as condensation. When additional attractive forces are involved, such as exist in polar molecules where certain nuclei attract electrons more than others, an extreme case being polar water molecules in which the oxygen nuclei have much stronger attraction of the electrons than the hydrogen nuclei, the weak bonding tendency (again, not covalent bonding) increases. In these situations, however, if the approach speed is too high, the atoms and molecules collide and bounce off each other like billiard balls. With increasing speeds, the bounce is stronger and prevents weak-bonding attachment from occurring, much like two ‘sticky’ objects will bounce off of each other if you throw one at the other with high enough speed. Replace the word ‘speed’ here with ‘temperature’ and this explains why changes in temperature cause transitions from solid to liquid to vapor.
The cause of the repulsion is more complicated than I presented in the above illustration. With the help of Professor Chérif F. Matta, I worked this issue further in one of my posts (here).
Joule-Thomson Effect
The Joule-Thomson effect, first discovered in the 1850s by the collaborative efforts James Prescott Joule and William Thomson (Lord Kelvin), describes the temperature change of a gas when it depressurizes, without performing work, across a restriction device such as a porous plug, as J-T originally used. Their detailed experimental work showed that most gases cool upon expansion, with the effect depending on the gas’s initial temperature and pressure, while some gases, like hydrogen, heat. Their work enabled significant progress in understanding gas behavior and advancing refrigeration and liquefied gas technologies.
[As a side note, Thomson’s theoretical analysis of the flow across the restriction, including his focus on the change in the product of pressure times volume, i.e., Δ(PV), led to the eventual creation of a new property of matter, later named enthalpy (H). As pointed out by Gibbs in 1875-78 (The Scientific Papers of J. Willard Gibbs. Volume One Thermodynamics. Woodbridge, Conn: Ox Bow Press, 1993, p. 92), the value of H, defined as the sum of internal energy (U) and the product of pressure and volume (PV), does not change during a throttling process. In other words, the flow is isenthalpic. I’m not sure if Gibbs was the first to explicitly state this—Thomson did not state this—but he was certainly one of the earliest to do so.]
So why would the temperature of a gas change during the throttling process. Let’s first accept that the value of Δ(PV) will influence the temperature change. If work is done on the system, (PV)in > (PV)out, then temperature will increase. Now let’s set this aside.
What are the changes that occur with the internal energy, U? That is the key question. The internal energy of a gas is the sum of the kinetic energy and potential energy of all the molecules comprising the gas. U = KE + PE. When a gas depressurizes via throttling to a low pressure, assume for simplicity sake, PE goes to zero, resulting in an ideal gas in which U is a sole function of temperature. U = KE = f(T) at low pressure. So now we zero in even further, what is the potential energy of the gas at high pressure?
Potential energy is negative for attraction. The closer the molecules are, the lower the potential energy goes and, since energy is conserved, the higher the kinetic energy goes. So let’s play this out. Assume PE = O for the depressurized gas and further than PE < O for the high-pressure gas. This means that KEout < KEin and thus cooling occurs, i.e., temperature decreases. So strong attraction interactions at high pressure leads to cooling during J-T expansion.
But what does “strong attraction interactions” mean? When one molecule approaches another, PE decreases and KE increases. The speed of the molecules near each other increases above the speed when the molecules are far apart. One can envision why increasing pressure would contribute to this. But how about increasing temperature? As temperature increases, molecular speed increases, and one could argue that the net effect of attraction decreases. The molecules spend less time under the influence of the attractive forces. They travel past each other at higher speeds and so experience lower deviations from those speeds. This is what I need to quantify.
So if this thinking is correct, one should observe a decrease in the cooling effect with increasing temperature… AND… MORE ACCURATELY… with increasing speed. Why do I say this? Because it’s speed that actually matters and speed is inversely proportional to square root of molecular weight. Hydrogen (mol wt 2) moves much faster than nitrogen (mol wt 28): 1928 vs. 515 m/s.
My Hypothesis
So my hypothesis as to why hydrogen heats up during J-T expansion? The sum of the PE values of all the H2 molecules at high pressure is closer to zero than it would be for heavier (slower) molecules. Thus, the heat up during the expansion is due solely to the negative value of Δ(PV): (PV)2 – (PV)1 < O. Furthermore, the repulsion interaction plays no role. Note: some literature sources I have read attribute the cause of heating during J-T expansion of hydrogen to strong repulsion between H2 molecules. I don’t believe this to be true.
Regarding the PV effects, per spreadsheet image below, I looked at two states for four gases, one at high pressure (200 atm), one at low pressure (1 atm). I then iterated on temperature using the NIST database until I matched enthalpy (isenthalpic expansion). [One need not model the process itself as it’s really the comparison between two states — high pressure and low pressure at constant enthalpy — that matters.] The main reason for doing this was to obtain the volume at low pressure, which enabled me to quantify the change in PV to account for work done on the system during JT expansion. For each of the four gases I looked at, there’s a decrease in PV, which implies that work is done on the system. I hypothesize that this accounts for the “heating” effect (negative JT coefficient) seen across the board for gases at high temperature (see Fig 1 asymptote below; obtained from Wiki) since intermolecular attraction is no longer significant – – – because all molecules are moving fast.
My hypothesis remains that the faster the molecules move for a given temperature (lower molecular weight), the lower the impact of the attraction forces, meaning less cooling effect. Conversely, the slower moving molecules (higher molecular weight) are more impacted by attraction and show significant cooling upon J-T expansion. Heating only occurs when potential energy at high pressure becomes insignificant.
I need a molecular dynamics simulation model!
I now need to test my hypothesis and the only way I think I can do this is with a molecular dynamics simulation model. I’m not sure a direct equation (written out with pen & paper) would give me what I want but perhaps it could. I feel the MDS model would do it better if it were able to really look at speed distributions as a function of molecular separation. I’m actively seeking someone to assist me in this. If you know someone, please point them my way. I have not yet seen anyone explain the J-T effect with the above hypothesis, or really with any micro-to-macro hypothesis, so a good, solid publication is sitting there in the future for anyone who is interested. [Having said this, if you have read a physical explanation of the J-T effect, please let me know]
Note that for my hypothesis, the repulsion term plays no role.
Note also that the virial equation can quantify the J-T coefficient but the challenge I have is understanding physically what the virial equation is actually telling me. Please correct me if I’m wrong, but unless I’m mistaken, it doesn’t explain what’s going on based on moving and interaction molecules.
Regardless of the above, whatever the correct hypothesis, which may very well be different than mine!, will need to explain:
- why some gases heat while others cool
- why an inversion temperature exists
- why ideal gases have a J-T coefficient of zero (which I don’t believe they do)
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