[Bill S (OP)]

A frequently found example of the process of increasing entropy involves a closed box containing a quantity of O₂ molecules. At the start, they are bunched together. With time, they spread out to fill the space. It is argued that there are many more ways of achieving the second configuration than the first, therefore entropy has increased.   

Intuitively, that seems fine; but let’s look more closely.  Are all the O₂ molecules identical?  If they are, what difference might that make?

Consider a box containing three identical molecules.  Call them A, B and C. We will have positions x, y and z.
This would seem to give us three possible combinations,

1)  A - x,  B - y, C – z   
2)  A – y, B – z, C - x
3)  A – z, B – x, C – y

but, if the molecules are indistinguishable, how, for example, does configuration 1 differ from 3?

If one cannot distinguish between A, B and C; aren’t all three configurations identical?

[jeffreyH]

What is the kinetic energy of each molecule? How are they oriented? Are they rotating?

[Colin2B]

Let’s say you have identical triplets Alice, Belinda and Chris. Try again. You might not think they are different arrangements, but they are. You are really looking for permutations not combinations.

The thing about wanting to determine equivalence in probability depends on how you want to count. For example 3 dice roll with each one showing 4 on the upface. If you are only interested in the sum of faces = 12 then order doesnt matter.

[alancalverd]

Common confusion between similarity and identity. A = B or A ≡ B does not mean A is B.

Bosons, on the other hand, are more fun.

[chiralSPO]

It makes no difference whether the molecules are the same or not (10,000 distinct gas molecules will still disperse).

Imagine the coordinates of each molecule in 3D space. We can represent the location of any molecule using a vector [x,y,z]. For N identical molecules, there are N vectors of the form [x_n,y_n,z_n] (1<n<N). If the molecules are in a sealed box 10 cm on a side, then –5<x_n<5, –5<y_n<5, and –5<z_n<5 for all N molecules.

If each of these coordinates is randomly distributed, one can calculate the probability of all of the molecules being in the same 1 cm³. As N increases, this becomes astronomically small. I think it would be 0.001^N–1

1000 equivalent cm³ units
10% chance each coord is right for a molecule, and 3 coords, so 0.1³
N–1 molecules to be in the same place as the first

∴ 0.001^N–1 *100%

# molecules__%chance of being in the same cm³
1,   100%
2,   0.1%
3,   0.00001%
4,   0.00000001%
etc.

[chiralSPO]

To answer the other part of the question. Oxygen molecules can be distinguished from one another using some techniques.

Oxygen molecules (O₂) can contain multiple isotopes of oxygen, for instance ¹⁶O¹⁷O vs ¹⁶O¹⁶O. Under normal conditions the oxygen isotopes cannot transmute into each other, and the oxygen atoms will not exchange between molecules (as H and D atoms do in liquid H₂O/HDO/D₂O mixtures). Thus these are permanently distinguishable molecules.

There are also techniques that one can use to "tag" an oxygen molecule for a short time (and can therefore theoretically see where an individual molecules ends up moving to). For instance O₂ typically exists in one of two indistinguishable triplet states (↑↑ or ↓↓), but given light of a specific frequency can change to one of two indistinguishable singlet states (↑↓ or ↓↑). Singlet oxygen is less stable and will either convert back to triplet oxygen (if left alone) or will react with the first oxidizable species it finds. Or one can put the sample in a strong magnetic field, such that the two triplet states become distinguishable (given a B field of ↑, the ↑↑ and ↓↓ will have different energies--in general this type of technique would be involved in EPR (ESR) spectroscopy).

[Bill S (OP)]

Thanks for those great responses, folks.  Lots to think about - little time for thinking.