One of my chemical engineering professors at MIT told us in one of our classes, that if we want to understand what’s happening in the physical world, we need to picture ourselves as a molecule and ask, what do you see? I bring this philosophy to address this question. I don’t have the final answer to this question but wanted to share my thinking.

Picture an ideal gas, one that follows the equation PV = nRT. This equation can be derived from first principles as shown in a separate post here. Of note in the derivation, the connection between pressure and the kinetic energy of the moving molecules (1/2 mv²) is not as direct as one might think. Based on Newton’s concept of force as the rate of change of momentum, the derivation quantifies the pressure of a gas as the change in momentum of single molecule times the rate of collision of molecules against wall. Since both of these terms are directly proportional to velocity, pressure is thus proportional to velocity². Additionally, since momentum is proportional to mass, one ends up (with some more derivations) with pressure being proportional to 1/2 mv². Connecting this result with the experimentally determined PV = nRT, the hypothesis is made that temperature is equal to the average kinetic energy of the moving molecules. This discussion is important as I move forward in this discussion.

So again, picture the ideal gas, and then lower temperature. What happens to the molecules?

The intermolecular interactions of attraction and repulsion always exist. They never turn off. The question is, how significant is their impact? At high temperatures, i.e., high speeds, their impact is low and gases demonstrate ideal behavior (PV = nRT). In fact, a key assumption in the Kinetic Theory of Gases is that the interactions are insignificant. Molecules still collide at high temperature, as manifested by the mean free path theory, and these collisions are caused by the repulsive interaction. So it’s really the attractive interaction that becomes insignificant. When molecules pass-by each other at high speed, the attraction interaction is present but the deflection of motion resulting from the interaction is minimal.

So now slow the relative speeds down. Deflection becomes more significant. The molecules spend more time interacting with each other and more time means more deflection as the molecules are drawn toward each other. The amount of deflection is related to the magnitude of the force and also to the amount of time the force acts.

The role of the attractive interaction

The attractive interaction is the source of force acting between the molecules and thus the cause of acceleration between them. This results in an increase in speed. Molecules that are near each other move faster than when they are far apart. The average molecular speed of molecules in an ideal gas state, largely free of intermolecular interactions, is LOWER than the same average of the same molecules being impacted by intermolecular attraction. Thus, if one could place gas molecules into two buckets, bucket #1 for the “ideal” molecules (no interactions) and bucket #2 for the “non-ideal bucket (attraction interaction significant), the distribution of molecules in bucket #1 should be shifted to slower speeds than the distribution of molecules in bucket #2. I hypothesize that this explains the physical underpinning of the Joule-Thomson effect as discussed here.

So as temperature decreases, the significance of the attraction interaction increases and molecules experience greater total acceleration when they are close to each other. Ultimately, one could imagine that at a certain point the total acceleration pulls them together, causes them to collide, and causes them to “stick” together. This is the first step of condensation; the formation of dimers and eventually clusters of molecules stuck together. I discuss more about what parameters govern “sticking” below. But for now, let’s place this on the table for consideration.

So why do gases deviate from ideal behavior?

To answer this question, each term in the ideal gas equation must be considered. What is temperature (T) really measuring? Is it simply the average kinetic energy of the molecules in the system? And what about pressure (P)? At either low temperature or high pressure, does the calculation of pressure based on moving and interacting molecules deviate from the calculation based on the ideal gas assumptions? And let’s not forget n. When does the decrease in n due to the formation of dimers and higher start taking place and how does this affect the compressibility factor (Z = PV/nRT)? I need to better understand how T, P, and n all vary with conditions to cause gases to deviate from ideal gas behavior (Z = 1.0). Bottom line, I need to explain the graph:

What parameters govern sticking?

For sticking, the incoming molecule must enter the attractive well (potential energy well – E_well) and lose enough kinetic energy to become trapped (bound). If KE > E_well, then there will be a bounce. If KE < E_well, then there will be temporary orbit/stick, but sticking requires dissipation to prevent escape. So is a 3rd body needed? or a transfer of KE of translation to the KE of rotation/vibration? How about surface phonons (lattice vibrations). Without dissipation, even low KE collisions could bounce elastically.

Considering the role of mass is important. Mass is clearly involved with kinetic energy, but what about the potential energy well (Lennard-Jones). Is mass involved here? Recall the the escape velocity of a body on Earth is independent of mass.

An interesting point is this: The critical physical requirement for sticking is that KE must be low enough for the collision duration (τ≈σ/v_rel) to allow sufficient energy transfer (ΔE>KE – E_well) before rebound. This threshold is not sharp but probabilistic, influenced by impact parameter (offset from head-on collision) and orientation.

In summary, sticking physically requires relative velocity low enough that KE < E_well.

What do I need to learn? Exactly why duration of collision is important. In essence, collision duration is the physical bridge between kinetic energy and dissipation kinetics—if too brief, you get bounce; if extended, you get stick.

And then I have to return to the original question: What physically happens to cause a gas to deviate from ideal behavior?

Notes from literature: The process of condensation between molecules (e.g., in gas-phase homogeneous nucleation or on surfaces during heterogeneous nucleation) hinges on whether colliding molecules can dissipate enough relative kinetic energy to overcome the barrier for sticking, driven by intermolecular forces. While velocity and mass are intertwined in kinetic energy, their roles differ in how they influence collision dynamics, energy distribution, and dissipation efficiency. In practice, for condensation at fixed (T), heavier molecules condense more easily due to combined lower v_rel and better dissipation. Velocity, and specifically, the relative velocity v_rel between colliding molecules, is the dominant direct factor in determining whether a collision results in sticking or bouncing. It affects both the energy barrier and the dynamics of dissipation.

Thoughts

• Is a third body always need to enable condensation?
• How about chemical reaction?
• What is the criteria governing the rebound effect? Is there a certain ratio of incoming speed relative to strength of attraction involved?

References

Consider these references:

Brann, M. R., Hansknecht, S. P., Ma, X., & Sibener, S. J. (2021). Differential condensation of methane isotopologues leading to isotopic enrichment under non-equilibrium gas–surface collision conditions. The Journal of Physical Chemistry A, 125(42), 9405–9413. https://doi.org/10.1021/acs.jpca.1c07826

Ergin, G., & Takahama, S. (2016). Carbon density is an indicator of mass accommodation coefficient of water on organic-coated water surface. The Journal of Physical Chemistry A, 120(18), 2885–2893. https://doi.org/10.1021/acs.jpca.6b01748

Julin, J., Shiraiwa, M., Miles, R. E. H., Reid, J. P., Pöschl, U., & Riipinen, I. (2013). Mass accommodation of water: Bridging the gap between molecular dynamics simulations and kinetic condensation models. The Journal of Physical Chemistry A, 117(2), 410–420. https://doi.org/10.1021/jp310594e

Ohashi, K., Kobayashi, K., Ida, K., Kurata, Y., Nagano, Y., & Tsuruta, T. (2020). Evaporation coefficient and condensation coefficient of vapor under high gas pressure conditions. Scientific Reports, 10(1), 8143. https://doi.org/10.1038/s41598-020-64905-5

Sun, J., & Wang, H. S. (2016). On the early and developed stages of surface condensation: Competition mechanism between interfacial and condensate bulk thermal resistances. Scientific Reports, 6, 35003. https://doi.org/10.1038/srep35003

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