IUPAC Recommendations for Concentration Units (ppb, ppm) In a recent interesting post, (Is 1 ppb equal to 1 μg/kg?)
it was pointed out that IUPAC advises to abandon parts per million, parts per billion quantities and instead it suggests to employ micromole/ mole for ppm and nanomole/mole for ppb. An example is given on page 98 of the Green Book
"The volume fraction of helium is 20 ppm".

May be I am missing a point or I see a problem here with the moles in the denominator of these suggested replacement definitions.

a) If we analyze Zr in a rock and find 5 mg Zr in a 20 gram rock, how would one apply the mole definition because one cannot define the moles of a rock. We have to stick to the classical definition:

ppm =10^6 * (mass of analyte / mass of sample).

b) How can one express ppm of Ar in air, and how should we define the moles of air? If we wish to generalize this case, what if our solvent system consists of several components.

The ambiguity I feel in micromole/mole definition is the mole in the denominator. Are we referring to the moles of the solvent or  moles of solvent plus solute. 

If this replacement is analogous to molal concentration, where the denominator is clearly definied to be kg of solvent not the kg of solvent plus kg of solute, then IUPAC should clearly define it.
There are not this kind of problems with the IUPAC recommendation. The only point in it is to specify the parts of  what we refer to, nothing more.
In such a sense it seems clearer than expression as molality or molarity.

Take an amount of atmosphere at given conditions. The concentration of Ar or whatever component can surely be expressed in terms of mole number in a million or whatever number of moles that amount contains.

As for the rock analysis case in the question, note that IUPAC umol/mol is an example, it doesn't mean it should be preferred to mg/kg (or ul/l etc.). The measured amount remains the most convenient both practically and mathematically.

I add that following recommendations would - perhaps - also help clear another practical con of unspecified ppm and percentage concentration values.  Often people forget that 1 l --> 1 kg of solution is acceptable only with very dilute solutions and when the solvent is water and go around telling and even publishing % concentration in various solvent so that the reader must assume a wrong procedure in order to follow up. Explicitly saying mg/kg instead of ppm should evidence that when mg starts to be hundred, than one has to switch concentration units or being more careful when reporting.