cmb said:
From what is said above, I am willing to be persuaded.
But the remaining hold up I have is that the definition says; "The amount of substance, symbol n, of a system is a measure of the number of specified elementary entities. An elementary entity may be an atom, a molecule, an ion, an electron, any other particle or specified group of particles."
"any other particle or specified group of particles"
So if I have "a mole of electrons", it is physically a half of "a mole of pairs of electrons".
I find this 'unphysical', so can you persuade me that the SI definition does NOT allow me to pick and choose a specified grouping of particles, as the definition seems to say?
It's not "unphysical". To the contrary, it's very physical ;-))).
If I interpret it right, the mole is introduced into physics for specifying many-body systems in terms of macroscopic quantities.
Now macroscopic quantities can be described as a coarse-grained view on the macro-system in terms of some averaging over many microscopic degrees of freedom, which we are unable to resolve, because we simply can not describe all the ##\sim 10^{24}## degrees of freedom of a macroscopic system.
Now the question is, how to determine the "macroscopically relevant" degrees of freedom and the underlying "relevant microscopic" degrees of freedom over which I'm averaging to get the effective description of the macroscopic observables.
Let's consider only the most simple case of thermal equilibrium. The most simple way to describe macroscopic matter in terms of the socalled grand-canonical ensemble. Let's take a somewhat "exotic" example, where the arguments become quite drastically clear: the case of strongly interacting matter under extreme conditions, which is investigated using ultrarelativistic heavy ions in various accelerators (SPS/LHC@CERN, RHIC@BNL, GSI/FAIR@Darmstadt,...) and via observations of neutron stars in (multi-messenger) astronomy (em. waves over a large scale of wavelengths, gravitational waves with LIGO/VIRGO).
As we know after some decades of research, what happens in an ultrarelativistic heavy-ion collision is that a very dense and hot fireball of collectively moving strongly interacting matter exists. You need a lot of different signals measured to achieve this (momentum-distributions of hadrons, relative abundancies of different hadron species, the distribution of hard probes like jets, open heavy flavor, quarkonia, and dileptons and photons).
It turns out that the hadronic spectra in the low-(transverse-)momentum region can be described by the assumption that after some short "formation" time of ##\lesssim 1 \text{fm}/c## a blob of quark-gluon plasma is formed, which is close to local thermal equilibrium, and its further evolution can be well described by (viscous) relativistic fluid dynamics.
Now I've already made a point by calling this "early stage" of the fireball evolution a "quark-gluon plasma". I already made an assumption about the "relevant microscopic degrees of freedom". The reasoning is that the coupling constant of QCD, the underlying fundamental theory describing the strong interaction among quarks and gluons, becomes small for high-energy collisions, and in a very hot and dense medium the particles have large energies and momenta and rattle around with high-energy collisions. That's why in a first naive attempt to understand what's going on, one had assumed that almost massless quarks and massless gluons are the relevant microscopic weakly interacting degrees of freedom in this hot and dense medium and thus a description as a nearly perfect relativistic gas of massless quarks and gluons might be a good, though rough, description.
Now, as lattice-QCD calculations at finite temperature has revealed that's not quite true, but there's still substantial coupling between the quarks and gluons at the temperatures of ##\sim 500 \; \text{MeV}## reached in the early phases of the fireballs of matter created in heavy-ion collisions, and that there's a quite sharp cross-over transition at temperatures of around ##T_{\text{pc}} \simeq 150 \; \text{MeV}##.
The interpretation is that above this "pseudo-critical" temperature the relevant microscopic degrees of freedom are rather quark- and gluon-like quite massive quasi-particles, i.e., something similar as the constitutent (valence) quarks inside hadrons but not anymore sharply bound into usual hadrons. At the cross-over transition one has a strong decrease in pressure, energy density (divided by ##T^3## or ##T^4##, respectively), which shows a strong decrease in "relevant degrees of freedom". This is thus interpreted that at this point something like hadrons is formed, but also these hadrons show some "medium modifications", i.e., they are also quasi-particles with some mass and width.
Now it's clear that the macroscopic quantity "amount of substance" when defined via the "number of microscopic constituents" depends on the state of this substance, which determines which microscopic degrees of freedom are relevant to describe the thermodynamics of the (equilibrated) medium. While in the early hot stages of the fireball evolution in the medium created in heavy-ion collisions the relevant microcopic degrees of freedom are the quark- and gluon-like quasiparticles, in the later colder stages of the fireball evolution the relevant macroscopic degrees of freedom are hadron-like quasiparticles (with medium-modified properties of mass and width). So in the QGP-phase to get the thermodynamics right you have to consider quark- and gluon-like degrees of freedom and thus you'd define "amount of substance" in terms of the corresponding "particle numbers/densities" counting these QGP-degrees of freedom, while in the later hadronic phase you'll rather count hadrons and hadron resonances. This keeps track of the drastic changes at the corresponding cross over.
The same holds for the treatment of most systems. E.g., take the air around us. It consists mostly of nitrogen and oxygen, but of course not in atomar but molecular form. Here the air is described best as a (nearly) ideal gas consising of ##\mathrm{O}_2## and ##\mathrm{N}_2## molecules, which thus make up the relevant microscopic degrees of freedom. You can even describe them as rigid rotators since the vibrational molecular modes are not yet excited at usual room temperature. This of course changes at higher temperatures, and at a certain point of very hot densities you dissociate the molecules to atoms and finally the atoms into a plasma of atomic nuclei and electrons. Always you change your description from one kind of relevant degrees of freedom to another. You can even think further, going back to the very early stages of the big bang: There at some point the atomic nuclei resolve into protons and neutrons and even earlier you rather had a QGP!
So "amount of substance" in the sense of the mole of the SI is defined via the number of "relevant microscopic degrees of freedom", which of course depends on the (thermal) state used to describe it.
Of course the transition between different regions of effective relevant degrees of freedom is particularly interesting and investigated in terms of the phase diagram of the medium. The change between different degrees of relevant effective degrees of freedom indicate transitions like the cross over transition between QGP and hadronic matter, which is expected to become a true first-order transition at higher net-baryon densities as achieved in heavy-ion collisions at lower beam energies, with the first-order transition line in the phase diagram ending in a critical point, where the phase transition becomes 2nd order.