Is ρV a reliable measure of atom probability in small volumes?

[aaaa202]

Suppose you have a metal with some given density ρ. Now we all know that density is a kind averaged quantity, which only makes sense for large enough volumes to contain fluctuations across the microscopic world.
Now in an exercise, I am asked what is the probability to find an atom in a very small volume V- so small that the product ρV is very small. Is ρV then a measure of the probability of finding an atom in the volume V or is it complicated by the fact that density is an averaged quantity?

 

[Matterwave]

Usually when we look at small volumes in thermodynamics, we are still looking at "big" volumes in comparison to the size of atoms and molecules. So, even though the volume element is "small" in comparison to the total volume, it is big enough that averages still make sense, so that our "micro system" is still internally in equilibrium.

 

[aaaa202]

But in this context I am interested in a volume smaller than this. Does it make sense to talk of a probability?
For instance suppose my atom density is one mole per cubic centimeter. Is the probability to find one atom in 1cubic cm / 1 mole then 1 or how can it be translated into a probability?

 

[Matterwave]

You can use statistical mechanics to do this problem. You basically get a phase space, and a distribution on that phase space to obtain probabilities. However, you should note that the fundamental assumption of statistical mechanics is that all possibilities (microstates) in equilibrium are equally probable. This is just an assumption! (Which works remarkably well for almost all physical processes that we care about). If you want an answer that is NOT based off of this assumption, you might have to try to actually solve the multi-particle wavefunctions from quantum mechanics, but this is typically very (impossibly) difficult due to the extremely large number of particles involved.

 

[Mordred]

Matterwave is correct in that your looking for statistical mechanics of solids, this is a lengthy subject that requires a good understanding of ideal gas laws. the Fermi-Dirac statistics are applicable to metals, you can possibly use Einstein solids or Boltzmann statistics.

I'm not positive which form will best apply in your example, but it should give you something to go on. However this dissertation may help

http://etd.library.vanderbilt.edu/available/etd-07302007-130108/unrestricted/01dissertationYing.pdf

still studying it myself, My interest in statistics are more geared to Cosmology applications. Hope this helps