How many molecules does it take to have a phase?

Question

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?

Answer 1

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.

EDIT: It turns out that 21 seems to be the number for water drops.

"How many water molecules are needed to solvate one?" Chem. Sci. 2021

The minimal structure eventually responsible for proper solvation is made of a total of 21 water molecules and includes two complete solvation shells, of which the whole first one is tetrahedrally coordinated to the central molecule.

Answer 2

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.

Answer 3

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.

Answer 4

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.).