# How many molecules does it take to have a phase?

A single molecule can't be solid, liquid or gas. It's just a molecule. A mole of something can be any of the three. So, how many molecules does it take for phases to be meaningful?

I realize that this may depend on the molecules in question. The minimum number that can form a solid might depend on the configuration that molecules adopt when solid. The smallest crystal of a lattice for example. How about a gas though? Can two molecules be called a gas if they exist under conditions where a larger quantity of the same moiety would be in a gaseous state?

In short, can a minimum number of molecules necessary to adopt a particular phase be defined?

• Could you (theoretically) say that when the average kinetic energy of one particle is at a certain point, it is at a certain phase? – Noel Aug 12 '16 at 20:35
• @Noel No, because it is possible to have two phases (e.g. ice in water) or three phases (triple point) at equilibrium, and there is no difference in average kinetic energy (phases at same temperature). Also, phase changes can occur with pressure changes while the average kinetic energy remains constant (e.g. isothermal expansion). – Karsten Theis Feb 3 at 19:55

This is actually an active area of research for water clusters. In principle, for $\ce{(H2O)_{n}}$ there should be a "melting" phase transition, much like for ice to liquid water.

So, in principle, if you had an accurate enough theoretical method, you could do molecular dynamics and see when the melting point of the cluster matches bulk water.

Defining a line between liquid and gas seems tricky to me, but I don't do statistical mechanics, and perhaps there are molecular dynamics techniques for defining a boiling point.

Defining the solid-liquid phase transition is much easier, because the radial distribution function of the system can be monitored to see the transition (i.e., the density changes).

At the moment, I don't think the best dynamics methods get the melting transitions accurately yet, but I think n~20 is about right. I'll revise this when I get a better answer.

# 無

How many grains of sand does it take to make a pile?

The question is nonsensical, because a phase of matter is an emergent property of a collection of atoms or molecules, just like being a pile is an emergent property of a bunch of sand.

Add to that the notion that "molecule" is kind of vague in any case, and you've got a world-class mess on your hands.

Sure, it's quite clear what the unit of C2H5OH is, and that it is a molecule, but for NaCl, what constitutes the unit? What constitutes the "molecule"? Is there one?

What about polymers, or diamonds? Is a diamond with its multitudes of covalent bonds a single molecule? Is it many? Is (C3H6)n n molecules, or one?

I'm afraid the question leads you down a philosophical rabbit hole that goes far beyond the science of chemistry.

• I know, and it was kind of tongue-in cheek. I was indeed thinking of the sorites paradox. Still, I was hoping that there may be a more formal definition used in the field or that, at least for gases, it may even be definable to a given number of molecules. – terdon Sep 24 '14 at 14:04
• @terdon Typically, phases and whatnot is more about physics than it is about chemistry, meaning that the exact number of bits of stuff and microscopic properties tends to be meaningless, unless you're studying the microscopic properties themselves, in which case you concern yourself with a tiny number of species, so phase becomes irrelevant. – Williham Totland Sep 24 '14 at 14:12
• Sorry, this is a very silly objection the the question. Some chemicals don't have clear definitions of a "molecule" (e.g. ionic solids) but this is irrelevant to the very large class of chemicals where the molecule is well defined. For this (large) class, the question is perfectly well sensible. – matt_black Sep 24 '14 at 20:03
• @matt_black: Phase is still a question of physics, not of chemistry, and when it comes to how many molecules you need to have a phase, that's not even science, that's philosophy. The question has no answer, because it's asking for the bulk properties of individual items, which, again, makes no sense. – Williham Totland Sep 24 '14 at 20:06
• @WillihamTotland Disagree. It's chemistry (perhaps physical chemistry but still chemistry). And it is not just not philosophy but can be studied and produce real answers (like how many gold atoms do you need before a cluster behaves just like bulk gold metal?) – matt_black Sep 24 '14 at 20:19

360... I'll explain.

As Geoff Hutchison pointed out, one place to look for this type of thing is in the molecular dynamics literature. His answer is certainly correct, but I'll expand on it based on some papers I've looked at in the past.

That number 360 really only works as an upper bound for water when trying to simulate the properties of liquid water or ice.

In molecular dynamics, one treats all the molecules using only classical, Newtonian interactions, but one must still decide what degrees of freedom to give your molecules. For instance, one common model of water is called the TIP4P/2005 model. The details here aren't important besides that they have to tell us how well their model simulates liquid water when they introduce this model.

In that paper which is free online, they simulate liquid water using 360 of their TIP4P/2005 water monomers (with a slight twist I'll explain below) and achieve quite good success in matching the experimental properties of bulk liquid water. For instance, they match the enthalpy of vaporization over a wide range of temperatures, and find a density maximum above the freezing point of water which has often been a very difficult property to simulate.

Now, the twist I was talking about. In molecular dynamics, one must place molecule(s) in a box and run the simulation. This presents a problem, however, in that molecules are never just confined to some arbitrary box. Thus, a common technique is to mirror this box when doing the simulation to make the surface area where there are edges much smaller than the total volume where there are molecules. This means that in this paper I linked to, there only 360 unique monomers, but the properties come from a larger effective number of waters.

Geoff Hutchison mentioned the number 20, which for water I would think is a bit low (though he certainly knows more chemistry than I do), as there are gas phase water clusters composed of 21 (and more) waters. See Cui, J., Liu, H., & Jordan, K. D. (2006). Theoretical Characterization of the (H2O) 21 Cluster: Application of an n-body Decomposition Procedure. The Journal of Physical Chemistry B, 110(38), 18872-18878. for a study of one of these larger clusters (this was actually done by other researchers at Pitt).

The idea, however, is that as these clusters get larger and larger, their properties approach the properties of bulk water. For instance, one could potentially make the argument that locally some of the water molecules in a larger cluster like $\ce{(H_2O)21}$ behave as if they were in bulk water.

Another way of answering the question is through spectroscopic means. Then the question becomes, what is the smallest number of molecules one can associate together and reproduce the absorption spectrum for the bulk phase? With something like water, if one only wants to reproduce the big features in an infrared absorption spectrum, I would put this number at around 60 because this means you're getting 180 unique vibrational frequencies contributing to the spectrum, which, if you blur your eyes, might look about right.

Finally, because I like this question, I must say I disagree with the other rather cynical answer. I think anytime you fall in these rabbit holes of questions which seem unanswerable, it is because the question isn't yet defined well enough.

• Also my apologies for answering an old question. I just like this one a lot :) – jheindel Aug 12 '16 at 21:58
• There's never anything wrong about answering a question, no matter how old it is. That's one of the great things about SE. However, I'm not sure about your 360 rule. For one thing, that 360 were enough doesn't imply that 359 wouldn't be. For another, water is rather special, I'm not sure that the rules found for water would apply to other molecules. Very good point about the spectroscopic properties though. That may well be the right way of approaching this. – terdon Aug 13 '16 at 11:51
• Ya that's why I was saying it's an upper bound. I think I remembered to write that... I wasn't sure if I expected that it would take more or less molecules to accurately simulate something with fewer intermolecular forces like propane or something similar. And I only threw in the whole 360 thing just cause the paper I cited used that many molecules so I could at least be certain that should work... It's a very difficult question. – jheindel Aug 13 '16 at 18:09

As far as computational chemistry is concerned, you could use periodic boundary conditions to simulate solids and liquids. For crystalline solids, all you need is the contents of the unit cell, which might be as little as one particle. While these system would capture some aspects of a bulk phase, they will lack e.g. the broad distribution of kinetic energies with a defined average, so there will be limitations.

As far as experimental chemistry is concerned, many quantities and concepts that we understand clearly in a bulk phase change. For example, if I split a bulk solution in half, each half contains the same solute concentrations. That is not the case with a phase containing only a handful of solute molecules. If a gas phase has only two molecules, it is difficult to define pressure and temperature. For solids and liquids, surface effects become much more important as a large fraction of molecules will be at the surface.

Even in something as large as a bacterial cell (volume around $$10^{-15}$$ liter with about $$3\times 10^{10}$$ water molecules), you are no longer dealing with a familiar bulk situation for molecules at low concentrations (e.g. only one or two molecules of DNA, only a handful of the rarer enzymes etc.).