It tells us that the entropy for an extreme mixture is the same as the
entropy for a pure, monatomic gas ... plus an additive term that is extensive,
independent of temperature, pressure, volume, et cetera, and possibly very large.
In particular, if we perform a Gibbs-type mixing experiment involving extreme
mixtures, starting with N1 particles on one side and N2 particles on the other
side, there will be no entropy of mixing. The entropy of the deal will be
simply additive, namely (N1 + N2) k ln(M).
It is remarkable how this result fits in with other things we know about
Gibbs-type mixing experiments:
* Helium on one side and helium on the other → no entropy of mixing,
because the two samples are the same.
* Helium on one side and neon on the other → considerable entropy of
mixing, because the two samples are different.
* Snow on one side and some different snow on the other side → no
entropy of mixing, even though the two samples are as different
as they possibly could be.
The total entropy (including the entropy of the deal) behaves wildly
differently from the conditional entropy (not including the entropy of
the deal).
There's a lot more that could be said about this, but I'll leave it
here for now.