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In the SI unit system you have stuff like mass and length which are pretty fundamental. But the mole seems to not be as fundamental as the others. I think we can express it in terms of the other constants. And even if we can, are there any other constants in the SI units system that can also be expressed by other units.? Or is the SI unit system just for practical purposes?

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    $\begingroup$ Why is the number of carbon-12 atoms needed to mass 12 grams not a pretty fundamental unit? $\endgroup$
    – Jon Custer
    Jan 29 at 19:56
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    $\begingroup$ Well, can it be expressed in terms of mass and length? 12 grams is mass and the mass of the carbon atoms is also mass, so...you know. That can be expressed in terms of kilograms. $\endgroup$ Jan 29 at 20:01
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    $\begingroup$ More importantly, @Euler-Masceroni, is there a reason you didn't put the "h" in "Mascheroni"? $\endgroup$
    – Albert
    Jan 29 at 21:09

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The SI is not comprised of the minimal possible number of fundamental dimensional constants. In fact, it is chosen to be fairly redundant, which tends to be somewhat useful for experimental practice. While you pointed out the mole, at least five of the SI base units are not fundamental if you choose to neglect relativity, and only one base unit is necessary if you consider relativity. This is discussed, for example, in arXiv: 2311.09249 [gr-qc]. In a non-relativistic spacetime, all measurements can be carried out by using only clocks and rulers. In a relativistic spacetime, you only need clocks.

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Chemists measure quantities of material in moles because the progress and outcomes of a chemical reaction may depend on having a well-understood stoichiometry among the constituents. If you want to transform some sodium hydroxide (NaOH) and hydrochloric acid (HCl) into salt and water, you'll have reactants left over unless you have equal numbers of sodium and chlorine atoms. But 40 grams of NaOH has the same number of molecules as 35 grams of HCl. Matching the quantity is different from matching the mass.

In previous iterations of the SI, the mole was related to the definition of of the kilogram: twelve grams of carbon-12 contained one mole of atoms. The new definition preserves this old relationship within experimental precision, so old papers still reproducibly describe their experiments. The new SI has decoupled the macroscopic kilogram from the microscopic atomic mass unit. In practice, atomic-scale masses are measured in energy units. While I haven't done the arithmetic to show the connection in terms of realizable experiments, I think you can probably consider the mole as part of the connection between the macroscopic kilogram and joule versus their microscopic counterparts, the a.m.u. and the election-volt.

A mole is like a dozen, only bigger. (A student once asked, "so ... like a baker's dozen?") It's defined in SI because it's used in laboratories and we all have to agree on how big it is.

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In fact, pretty much every unit of measure can be expressed as other units if we wish. Temperature can easily be expressed as average energy. In the original CGS system of electrical units, the unit of charge was equivalent to $\rm cm^{3/2}⋅g^{1/2}⋅s^{−1}$.

If you look up Geometrized Units, everything including mass, energy, etc is reduced to units of length, which is convenient for some physics studies.

The SI system has the 7 base units it does primarily to facilitate trade and commerce, because those units are most useful to be standardized for everyday life, as well as most scientific experiments.

The mole is central to chemistry and controlling chemical reactions, so it is important for the world scientific community to all agree very precisely on "quantity of substance," or basically number of atoms, in any given sample.

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    $\begingroup$ Beware that temperature is only proportional to average energy when that energy is partitioned among degrees of freedom with a particular mathematical form. That form is satisfied for translational, rotational, and (in the SHM approximation) vibrational motion, which covers many common situations. But temperature is fundamentally about the relationship between internal energy and internal entropy; the proportionality is a frequent result, not an axiom. $\endgroup$
    – rob
    Jan 30 at 6:02
  • $\begingroup$ Acknowledged. But in any case, the Boltzmann constant could be chosen as dimensionless, and we would then measure temperature with energy units. In that case perhaps the physical interpretation would be "quantity of energy corresponding to a unit of increase in (dimensionless) entropy." $\endgroup$
    – RC_23
    Jan 30 at 17:44
  • $\begingroup$ Isn't it true that if energy states do not follow a Boltzmann distribution, then "Temperature" of the system is not well defined anyway? Or at least requires some generalized definition to be made? $\endgroup$
    – RC_23
    Jan 30 at 17:46
  • $\begingroup$ No, you get a different $\partial S/\partial U$ for non-quadratic degrees of freedom, even if the energy obeys equipartition. There are also cases where $\partial S/\partial U$, and therefore $T$, are negative; the most famous example is in spin systems after a population inversion. It's pretty inside-baseball for the present question about the mole, but it's important to remember in general that "temperature $\propto$ energy" is only correct under certain (very common) assumptions. $\endgroup$
    – rob
    Jan 30 at 18:10
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Mole is base SI unit, so it's fundamental, as per Avogadro law, which states that all gasses of same temperature and pressure bounded by same volume - has same number of molecules. Or in terms of ideal gas law : $$ n(moles) =\frac {PV} {RT} = \text {const} $$ for all possible gasses in the universe.

It's very important fact when you need to count particles in chemical reaction carefully. For example you construct graphene layers for a quantum processor and know that each layer must be composed exactly from 1 mole of atoms for it to operate effectively.

The only question can arise why number of moles definition was chosen to be : $$n = \frac {N} {N_A} = \frac {N} {6.02214076×10^{23} } $$

Why exactly such number of particles in a mole was chosen? It's simply due to historical reasons. At the time of definition it was chosen to be such number of particles which has 12 grams of ${} ^{12}C$ isotope material.

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Honestly, I'm surprised that this is the one that you are objecting to. I personally have a problem with the candle (luminous intensity), since it is not really tied to something physical but rather to the "average" human eye. I don't object to such a tool existing for light fixture design and so on, I just object to it being a "fundamental unit."

With regards to the mole, it is the unit with which we count discrete objects. We can obviously count by ones, but if we are doing statistical mechanics or chemical reaction kinetics or whatever, that is not always convenient. We could have just picked something rounder like "heptillion" or something similar, but for experimental convenience we use the Avagodro number definition.

Regarding your comment, the "12 grams of carbon 12" definition does not reduce the mole to just a mass any more than the old definition of the meter being "the distance light travels in X seconds" (where seconds was defined by atomic transitions) reduced the meter to just a time. It means that the (old) definition of mole depended upon the mass standard. However, what mole actually defined was the number of individual elements involved in creating that mass. That number of elements then is the same for everything. You can have a mole of hydrogen, and it's mass will be different.

I will point out one thing. I have often thought of the mole as "sort of not a unit" in the sense that you are just counting things. You could have a mole of apples, for example. You can't have a meter of apples, or a second of apples. In this way, the mole is more like a grouping. Like a said above, we could have used "heptillion" or something like that instead. This would be the same thing as using Avogadro's number. Heptillion makes more sense in base 10 counting, Avogadro's number makes more sense as something experimentally repeatable (due to the 12 grams of carbon 12 definition).

It is worth noting that if we count by ones, we get the Boltzman constant (in things like the ideal gas law) rather than the universal gas constant. The universal gas constant is really just what happens when you count by Avogadro's number, rather than by ones. It is really the per particle energy that contains physical information about the nature of reality. The Avogadro number (and the mole) are just convenience factors for human scales.

It is also worth noting that we would understand what is meant in saying that the mass of the earth is about a mole of kilograms. This is what I mean in saying that the mole is "sort of not a unit."

People are free to disagree.

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Just like the meter is a unit for the fundamental quantity of lenght, kilogram is the unit for the fundamental quantity of mass and so on, the mole is the unit for the fundamental quantity of counting (the number of things).

It didn't have to be a mole that was chosen as the SI unit; we could also have chosen a dozen, a pair, a singleton or other counts. Just like the kilogram also didn't have to be the SI unit for mass, but could have been the gram or the pound or the like. When we identity a fundamental quantity or property of nature we just need a unit in order to measure it, and it doesn't matter which.

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