Deriving Avogadro's Number without using "mol"

[Dale]

The learner’s relationship to the BIPM is irrelevant. It is the BIPM’s relationship to the SI that is important: the BIPM defines the SI.

The mol is, in fact, a SI unit because the BIPM says it is

 

[Mike S.]

I never spoke of whether it "is" a SI unit, but whether we should treat it as one. Letting those people tell us what to do can lead to terrible nonsense... would we inflict on any chemistry student a unit finely tuned according to a hypothetical eye of The Genetically Perfect Human? Nope, that stuff stays right out of the chemistry textbook.

 

[Dale]

I never spoke of whether it "is" a SI unit, but whether we should treat it as one.

We don’t have a say in the matter. We have no option whether or not to treat it as a SI unit.

Of course, you don’t have to use SI units if you don’t want to. You can make your Mike units and you can define them anyway you like.

 

[Mayhem]

I don't see how this is particularly important as normal measurements do not even approach the sig figs of Avogadro's number. Most lab scales I have worked with can only measure down to 0,001 mg sensitivity (IIRC; it's been a while).

 

[TeethWhitener]

@Mike S. 's point is that if we treat a mole as a unit, we must also treat a dozen as a unit. It's true in a trivial sense, but I agree with him that it's a nightmare pedagogically, since most students don't tend to think of units as groups of numbers, but rather as labels of physical properties.

In fact, when I used to tutor chemistry, I introduced the idea of a mole by using the example of a dozen, since it's the name of a number that most entry-level students are familiar with. In that sense, a mole is simply Avogadro's number, whether it's a mole of water molecules, electrons, or planets.

 

[DrDu]

The SI system is system of units geared towards engineering. In this respect, the mole makes sense, as, other than with the dozen, a chemist (and most physicists either) have no means to count the atoms or molecules inside the amount of substance they are working with, neither would they have any reason to do so.
Furthermore, the macroscopic concept of "substance" is emergent. A litre of water behaves differently than an imagined collection of isolated water molecules. It makes sense to speak of one mole of liquid water, but not to speak of two atoms of liquid water.
Personally, I have a rather relaxed relation as far as units are concerned. Is it useful to distinguish between Hertz and Becquerel, although both are formally 1/s? Is it useful to distinguish between Gray and Sievert although both are J/kg? Should we really go on to use metres after Einstein?

 

[DrDu]

Here another attempt of an answer:
I think the main question is how people did know let say how many grams of e.g. silver correspond to 1 mole without being able to count the number of atoms.
The point is that chemists observed already in the 18th century, the law of constant proportions, e.g. that the ratio of the masses of hydrogen and oxygen in a compound like water is a constant.
See the article on the law of definite proportions:
https://en.wikipedia.org/wiki/Law_of_definite_proportions
Comparing the ratios of thousands of compounds, they worked out the smallest consistent set of coefficients, which led to the assignment of molar masses "gram atom" and of stochiometric molecular formulas like H2O for water. This process predated the atomic hypothesis and led at the same time to its formulation and acceptance. However, the atomic hypothesis was still not completely accepted even at the end of the 19th century (specifically by positivists like Ernst Mach). Once the atomic hypothesis was accepted, the masses of the elements containing the same amount of atoms where known precisely although the absolute number of atoms was not well known (order of magnitude estimates were first derived in 1865 by Josef Loschmidt).
Specifically, it was known that e.g. the charge of electrons, necessary to deposit 1 mole of Silver (which was known to weigh 107,9 g), is 96485 Coulomb, the Faraday constant, and that therefore 1 mole of electrons corresponds to 96485 Coulomb.