# Can Kinetic Theory obtain fluids behavior?

My understanding is that it can't. But when presenting it here I received comments that contradict this. So, as suggested, I am asking the question to be able to receive concrete approaches on this point from other users.

While I have put thought into it, and researched to some extent, I recognize I, by no means, am expert on this. So I will present my line of reasoning and arguments, and the question here is the one on the title.

My view of Kinetic Theory of Matter is that it visualizes macroscopic behavior of systems with sufficiently large number $$N$$ of constituents, as a result of the microscopic dynamics. That is, knowing the interactions between the $$N$$ particles to some extent, you can obtain macroscopic characteristics of the big system.

This theory is able to explain completely the macroscopic behavior of the ideal gas, which is to say real gases in a large portion of their phase space. It does an excellent job for the case of thermodynamic equilibrium, but is also used in describing macroscopic diffusion of some magnitude in gases, and is the choice when describing diffusion at molecular level.

But when we talk about liquids, there are some problems and I am only address here a system in thermodynamic equilibrium.

We can imagine a liquid made of the same molecules of the corresponding gas, interacting with the same potentials, but they just are closer and hence more restrained in movement because now they spend more time close to each other than in the gas.

Here is the first problem with that picture: if it was correct, we would never observe a gaseous and a liquid phase as different as they are. The gas would slowly turn into a liquid as energy was removed from it: the compressibility, the density, etc. would change continuously. But we know this is not true, because we have discontinuities in these magnitudes in phase transitions.

Another problem with this picture is the large incompressibility of liquids, while at the same time being so malleable. To give an idea, water at room temperature has a compressibility of about $$450 MPa^{-1}$$ (link)! This means that you need a pressure more than 4 times the atmospheric in order to compress it by $$1\%$$! And a simple calculation with the Lennard-Jones potential (used often in liquid water simulations) tells you that a decreasing the intermolecular distance in $$1\%$$ from the equillibrium distance ($$r_m$$) leads to an increase in the potential are less than $$0.4\%$$.

So this means that particles in the liquid overreact in the macro scale with respect to what we would expect if the average distance would scale linearly with macroscopic scale compression. But there seems to be non-linearity in the scaling to micro world, and this is not accounted for in KTM.

Furthermore, a liquid keeps his volume as opposed to how gasses fill their containers. This picture is more consistent with a microscopic system where the $$N$$ particles move freely, like in the ideal gas, but contained by some potential well of the size of the macroscopic system. However the KTM would describe it as particles interacting with those at some range, and is not been proven how these kinds of interactions could generate this type of big potential well, although admittedly it seems to be the case.

Also when a gas and a liquid in thermodynamic equilibrium with a clear boundary between them, how can this be explained? If molecules in the liquid are just closer to each other, since both are at the same temperature, the average kinetic energies should be the same in both phases, so what makes some of them remain at interacting distances while those in the gas, with no larger kinetic energies, move freely? Why do we see phenomena like vapor saturation, where inside the same container, the liquid and the gas are completely equilibrated, while the microscopic picture there is nothing preventing the liquid molecules from expanding, and the whole system becoming one with an average interparticle larger than what was in the liquid, and shorter than what was in the gas?

So these are my main concerns regarding KTM as applied to liquids, and I think they are not resolved. I saw other questions in this site like these (q1, q2, q3) and they seem to have not received attention from the KTM connoisseurs, or there is no valid answer to provide.

I will much appreciate to be proven wrong, but if not then it just means there is room for theory :). Also arguments on my case which I missed are also very welcome.

• It depends by what you mean by "kinetic theory". It's not a term that is, as far as I can tell, widely used for the more general thermodynamic/statistical mechanics case that includes interaction energy and non-trivial properties of atoms/molecules. Statistical mechanics can, at least in theory, handle the phenomena you are asking about, but in practice it's hard to make predictions from first principles. – CuriousOne Sep 16 '15 at 23:07

First principles calculations of liquid-gas equilibrium phenomena are widely available. The numerical technique is molecular dynamics. You mentioned the Lennard-Jones potential. As an example, here is a paper that calculates liquid-gas interface properties for a lennard-jones potential.

More information on molecular dynamics can be found here and here and here.

To address the issues you raise more specifically:

... if it was correct, we would never observe a gaseous and a liquid phase as different as they are. The gas would slowly turn into a liquid as energy was removed from it: the compressibility, the density, etc. would change continuously.

No, theory does predict separate phases. That there are regions of distinct densities/phases is a natural outcome of the theory.

So this means that particles in the liquid overreact in the macro scale with respect to what we would expect if the average distance would scale linearly with macroscopic scale compression.

Your approach assumes that, in liquid phase, all inter-particles distances are at the minimum of the potential. That is not geometrically possible.

• Just FMI, is the term "kinetic theory" still used for the case with an interacting potential? – CuriousOne Sep 16 '15 at 23:52
• Professionally, I've only heard "kinetic theory" used in the context of gases. Google tells me, though, that there are those who use that term more generally in the way that the OP was using it. – John1024 Sep 17 '15 at 0:36
• Thanks for this answer, I will get back to it after studying the references. However, in a fast reading of your first link, I found they do not exactly obtain the surface tension of liquid: "Particles close to the liquid–gas interface especially feel an additional attractive force from the liquid slab outside of their truncation spheres. This additional force has to be taken into consideration when solving the equations of motion in the simulation of inhomogeneous fluids". Which explains the broad boundaries they get in density profiles. I will go deeper for a thorough understanding. – rmhleo Sep 17 '15 at 8:09