Why the macroscopic approach towards studying matter in classical thermodynamics only applicable when number of constituents are large?

Question

In classical thermodynamics, we study matter macroscopically. That is we are not interested in the individual behavior of its constituents but rather in their bulk behavior. We describe matter using macroscopic quantities like pressure, temperature, volume, mass, etc.

I read in a book, we can apply this macroscopic approach only when the matter contains a large number of constituents. If the number of constituents is small we need to go with statistical thermodynamics.

It wasn't stated explicitly as to why we can use the macroscopic approach with only a large number of constituents but not with a small number. What exactly goes wrong with a small number of particles that we have to resort to statistical thermodynamics?

Answer 1

From the formulation of the question, I assume that the answer should take the point of view of Classical Thermodynamics without mixing it with Statistical Mechanics.

Classical Thermodynamics hinges on a certain number of hypotheses on the systems it can be applied.

One of them is the extensiveness of the energy, which requires that the energy of a system made by putting together two equal subsystems is exactly twice the energy of each subsystem.

Such a property cannot be taken for granted automatically. In particular, if the subsystems are too small, the sum of the energies of the separated subsystems is not the same as the energy of the compound system: the difference being a surface term whose weight with respect to the bulk contributions vanishes as $V^{-\frac13}$ for large volumes $V$.

A second independent hypothesis is that all thermodynamic quantities have well-definite and constant values at thermodynamic equilibrium. It turns out that if a system is too small, thermodynamic quantities may fluctuate around their equilibrium value. Large systems ensure vanishing relative fluctuations. If the system is so small that fluctuations are not negligible, the probability of such fluctuations becomes a central quantity for theory, and we need Statistical Thermodynamics.

Answer 2

The question is so fundamental that cannot be answered in a simple way. But here is a take.

Thermodynamics is intimately connected to probabilities. To know anything about a stochastic process, say the flipping of a coin, we need a lot of observations (or a lot of coins thrown simultaneously). If we toss a die only twice we cannot tell if it is fair or not. But if we toss it 1000 times we can answer this question with more confidence.

In thermodynamics we are interested in predicting various average properties, for example, the mean energy of a system that is in thermal contact with its surroundings, or the mean mean number of molecular collisions with the walls of the container and so on. These statistical averages require a large number of particles.