# How many particles are necessary to observe a well-defined state of matter?

The three ordinary states of matter $-$ solid, liquid, gas $-$ can usually be distinguished by a function of the strength of interparticular forces and distance. A shorter distance and a stronger force results in a solid whereas the opposite will yield a gaseous state. In other words, the states of matter are describing macrostates.

With clarity in mind, let us define a practical limit to observing different phases.

Definition. A practical limit of particles is such that a phase transition is concievably observable by experiment. This is the smallest number of individual particles that when probed give us the traditional phase diagrams (omitting different solid phases).

• What is the practical limit for most purposes? I am probably looking for a 'number' of the form $10^a\ldots10^{a + \approx2.5}$ where $a$ is a positive integer.

So while an individual particle under standard conditions can be described as gaseous, it does not fit the definition.

An alternative wording (equivalent as far as I can tell) to this question is given by the Clapeyron equation. Apostrophes mark different phases.

$$\frac{\mathrm{d}P}{\mathrm{d}T} = \frac{\Delta H}{T\left(V'' - V'\right)}$$

• In which range of the number of particles do the entalphy of transition and difference in volumes of phases become measurable? This does not definitively mean that $\Delta H$ has to be a constant w.r.t. $T$ but rather a function of temperature $\Delta H = \Delta H\left(T\right)$ that has predictive capability.
• Surprisingly few compared to a mole. The 'noise' in $10^6$ is $10^3$ so a million molecules would give reliable answers, ignoring any practical difficulties involved. There are also many 'single molecule' experiments, pulling apart individual proteins using AFM for example, or fluorophores measured singly or in a single protein. – porphyrin Feb 24 '17 at 14:45
• I learned it as the thermodynamic limit. In our lab, accurate phase transitions are simulated with 10^3 particles pretty easily via periodic boundary conditions – khaverim Feb 26 '17 at 8:37
• @porphyrin, @ khaverim So currently $PL = 10^3\ldots 10^6$ and the $a$ to beat is about $3$. If you convert your comments into answers, please provide a summary of the experimental techniques applied, including references if possible. – Linear Christmas Feb 26 '17 at 14:42
• ^ @khaverim (feel free to mark comment as obsolete once read) – Linear Christmas Feb 26 '17 at 14:43
• This an interesting question, but it would seem that a different definition is needed for a solid, a liquid and a gas. For a gas you'd have to have enough so that the Maxwell-Boltzmann equation holds. For a liquid it would that you'd need enough molecules to pull the "drop" into a sphere. For a solid I'm not sure. But solids have bulk, small particles, and nonoparticles as well as individual atoms. – MaxW Feb 27 '17 at 17:46

It depends on the material but in general $10^2 - 10^4$ atoms/molecules is a representative range. I learned this concept as the thermodynamic limit; i.e. the number of particles in a system at which increasing particle number does not change density significantly. For obvious reasons it's also called the macroscopic limit. In Statistical Mechanics we bridge the gap between small "countable" systems and macroscopic "mole-quantity" systems with the thermodynamic limit.
Our lab develops and applies models for small molecules like $\ce{H2, CO2}$, noble gases, hydrocarbons etc. for use in computer simulation. The usual method is to parametrize a model such that phase transitions (i.e. density changes as a function of temperature or pressure) are accurately mapped, and intermolecular potentials are comparable to highly accurate QM calculations (CCS(D/T/Q) methods).
One can reproduce accurate phase diagrams of noble gases using just 40-100 atoms by putting the system in a fluctuating periodic box and changing particle positions until an energy minimum is reached (the computational method is called Monte Carlo) under given $T$ and $p$ -- this is called the isothermal-isobaric ensemble, or $NPT$. You can achieve identical results using the $\mu VT$ (grand-canonical ensemble) wherein chemical potential, volume, and temperature are fixed, but $N$ is allowed to change. For professional model development it is more common to use $10^3 - 10^4$ particles for a good fit.
Below is a snapshot of output from a Monte Carlo program obtaining the density of $\ce{He}$ at STP using 100 atoms (the reported value is 0.1786g/L). You can see that $\ce{He}$ in this model is a near-ideal gas with compressibility factor $Z \approx 1$ : I think the answer is more appropriately expressed via computational (Statistical-Mechanical) methods because the concept of the thermodynamic limit arose in and is used by SM. In general, the central-limit theorem is where the concept came from, which states that fluctuations to the mean of a quantity (density here, $N/V$) are expressed as $\frac{1}{\sqrt{N}}$, thus for $N=10,000$, the mean density can expectedly vary by $0.01$. For $N=N_A$, the variance would be certainly negligible.