Why is the mole/“amount of substance” a dimensional quantity?

According to the BIPM and Wikipedia, "amount of substance" (as measured in moles) is one of the base quantities in our system of weights and measures. Why?

I get why the mole is useful as a unit. In fact, my question isn't really about the mole at all; I just mention it because what little information I could find generally talked about moles, not about "amount of substance". Nor am I asking about why it's chosen as a base quantity and not a derived quantity. I get that any particular choice of bases is more or less arbitrary.

I don't understand why it's a dimensional quantity at all. It is, after all, just a count of things; every student is taught to think of it as "like 'a dozen', only more sciencey". Can't we just call it a dimensionless number?

No, says SI; molar mass doesn't just have dimensions of $\mathsf{M}$, it has dimensions of $\mathsf{M}\cdot\mathsf{N}^{-1}$; and Avogadro's number isn't just a number, it's got units of "per mole" (or dimensions of $\mathsf{N}^{-1}$).

Contrast this with an "actual" dimensionless quantity, plane angle (and its unit the radian). Now, you might say that it's dimensionless because radians are defined as arc length over radius, and so plane angle is just $\mathsf{L}\cdot\mathsf{L}^{-1}$; cancel out and you have no dimensions. That strikes me as arbitrary. We could just as easily argue that arc length is "really" a quantity of $\mathsf{a}\cdot \mathsf{L}$ (where $\mathsf{a}$ is plane angle), because it's the measurement of a quantity that subtends $\mathsf{a}$ at distance $\mathsf{L}$.

But this isn't needed; plane angle isn't even a derived quantity, it's a non-quantity. Plane angle is accepted as dimensionless. Why isn't amount of substance?

As I said, I've found very little on this question. From the Wikipedia article on the mole, I found a PDF of an interesting IUPAC article on atomic weight. It acknowledges the argument (as does the Wikipedia article), but dismisses it out of hand by saying (essentially) "of course counting things is a way of measuring things, so of course we need a unit of measurement for it".

Other than that, Wikipedia (as far as I can tell) touches on eliminating the mole only in the context of eliminating other units (as for example in natural systems of units). The Unified Code for Units of Measure blithely cuts moles from the base units as being "just a count of things", but doesn't go into why SI says it is necessary.

Is there any official rationale for the inclusion of "amount of substance" as a dimension? Failing that, can anyone provide, or point me to, some good reasons why it's so special?

EDIT: Thank you all for your input. The more I've thought about it, the more I've come to feel that there's no reason why "count of stuff" shouldn't be a dimension (it's clearly different from, say, a dimensionless number included as a scale factor), and that my unease with the idea comes from simple habit: in any case not involving moles, it tends to get left out. Really, I'm now more wondering why angles are considered dimensionless...

Reading before coming here:

• – Count Iblis Apr 6 '15 at 19:25
• I always thought if would be funny, just so people would stop thinking that mol is the only unit of amount of substance, to make another unit. It would be doz defined as the number of nucleons in an C-12 atom. – PyRulez Apr 7 '15 at 0:39
• In physics books I've read, angles were not considered dimensionless, but had a dimension of "radian" (or "steradian" for space angles), and the occasional unit that involved angles (usually space angles, but there were a few with plane angles) had a dimension of X per radian or whatever. [EDIT: the big example, which I forgot the first time over, is frequency vs. angular velocity - the dimension is s^-1 for the former, and s^-1 radian for the latter.] – January First-of-May Mar 14 '16 at 21:20
• @JanuaryFirst-of-May: That sounds like units rather than dimensions. An angle can be measured in various units (radians, angles, etc.), but according to the International System of Quantities, it has no dimension. (Contrast something like length, which can have units of metres, miles, parsecs, fathoms... but always has the "length" dimension, identified as $\mathsf{L}$.) – Tim Pederick Mar 16 '16 at 13:33

So, here's the thing. The chemistry that underlies molar mass ratios dates back at least to 1805. We've known that if you divide by a certain "relative mass" number you can get whole-number ratios for atoms in a pile of stuff, for that long. It took us about 60 more years to get a handle on how large atoms were with the estimations of Loschmidt, who worked out that atoms are much smaller than the wavelengths of visible light -- too small to ever "see". This gave a rough count of how many atoms there were in a confined space, too -- but we weren't able to connect these two different quantities (atomic relative masses, count of atoms) together to figure out the mass of a single atom until some work done by Einstein on diffusion in Brownian motion (1905) and some concrete numbers could finally be rolled in with Millikan's oil-drop experiment (1910).

So due to history and convenience, the chemists are basically at the level of saying, "okay, we have N grams of this stuff, our mass spectrometer says that it's M grams per mole, so we've got N/M moles, that includes N/M moles of nitrogen and 15 N/M moles of hydrogen due to the known atomic composition, ..." and so on. You never have to add the uncertainty in Avogadro's number to these calculations; the "size" of a mole isn't important. It's only important when you start to want to know things that are "beyond" historical chemistry approaches, like counting actual numbers of atoms.

With all that said, you'll be heart-warmed to know that there is a unit revision being considered by the SI organization, and one of the proposals is to fix the number of atoms in 1 mole. But of course they will still use as a guideline that "1 mole of carbon-12 has exactly 12 grams of mass"; it will just transition from what is now "exactly" to what will be "almost exactly."

• This is a good answer and I suspect this is exactly what happens. The real question, however, is why we're still stuck on this 100+ years since it's really no longer a thing to doubt the discrete structure of matter. Even worse, we seem to be wasting the great opportunity afforded by the 'new SI' to knock the mole down from its pedestal. – Emilio Pisanty Apr 6 '15 at 21:23
• SI has to be consistent with prior work in the field. Ideally, we'd all use millionths of a day (that is, an atomic unit corresponding to one millionth of one day circa 2100-2200 CE) as our units of time, something closer to 1 inch as our unit of length, and 10^15 electrons (not protons!) as our unit of charge. We'd then standardize a unit of force based on the Lorentz force of two charges at a certain distance which gives the Maxwell equations a convenient form, and from there we'd derive mass. But we're still stuck with an awful system of time, a length system which doesn't fix $c$, etc. – CR Drost Apr 6 '15 at 21:36
• I don't see why this is an issue. It's perfectly possible to retain the mole as a useful concept (and, indeed, to modernize it as proposed) while also recognizing that it is not really a base unit. – Emilio Pisanty Apr 6 '15 at 21:42
• @ChrisDrost I was under the impression that the meter was defined as some fraction of a light second, which would fix $c$. Or do you mean something else entirely? – k_g Apr 6 '15 at 23:33
• @ChrisDrost The romans had a system in which in desperate times, they could make someone a dictator for a time to fix a problem. I think we need to bring that back, and you dictator (although it should be noted that having positive on the inside and negative on the outside isn't too nonsensical. Its just inconvenient.) – PyRulez Apr 7 '15 at 0:45

Tim, your reservations about the quantity called "amount of substance" is completely justified and many authors argue the same as you do. Let me expand some:

"matter" or "substance" can be quantified in at least three different ways:

• by its mass
• by its volume
• by its numerosity

Some matching examples:

• the "matter" we call bread is usually bought by mass (e.g. a kg bread)
• the "matter" we call milk is usually bought by volume (e.g. a liter milk)
• the "matter" we call eggs is usually bought by numerosity (e.g. a dozen eggs)

In physics we commonly believe that all quantities are valid for any scale. In chemistry, many people think differently. For example, many chemists use the quantity "mass" in the macro scale, but use a different quantity called "relative mass" in the micro scale (atomic or molecular scale). "Relative mass" is considered a dimensionless quantity, and it is a quantity which does not really make sense in the context of modern metrology.

Something similar is the case for numerosity. Many chemists think numerosity is a quantity for the micro scale only, whereas in the macro scale you need to use a different quantity called "amount of substance". Again, many authors think this is not consistent with the rules of modern metrology. They consider the mole as a unit of the quantity "numerosity" analogous to "dozen".

Why do many chemists have these strange opinions? Some of them for historical reasons. There was a time when there was a total disconnect between macro scale and micro scale. The mole existed before there was a consensus that there are atoms and molecules. Without this consensus, it was not possible to consider the mols a numerosity. However, some chemists try even today to rationalize it. They claim that the number of entities in a mole is so large that it is not possible to actually count a mole and that therefore a different quantity is required. Personally I believe this is nonsense. I even believe that with progressing technology we will be able one day to count the entities in a macro scale amount.

Finally, you argue that numerosity should not be considered dimensionless. Here too, I am with you. I even think that numerosity should somehow include in the unit or the dimension what is being counted. For example, 5 apples and 5 oranges are clearly different and they should not be considered the same dimension. This is also the reason why you cannot add the two quantities (you never can add quantities of different dimension). This would mean "apples" would be a quantity dimension and, at the same time, a short version of the unit "entities of apples". Another unit would be "dozzen of apples" or "kapples" (kilo apples).

References:

• That's remarkably well put. – Emilio Pisanty Jul 25 '17 at 21:06

There is a lot of wiggle room in dimensional analysis ("factor-label method"), not just "the right way" and "the wrong way". If I want to define a unit called "dozen" with the universal constant

Steve's constant = 12 dozen-1,

there's nothing wrong with this. It doesn't change anything except superficially:

(5 dozen)2Ã—(12 dozen-1)2 --- vs --- (5Ã—12)2

Which is better? The left one is slightly wordier. But maybe the left one is easier to follow, because I'm ordering dolls by the dozen from a catalog and the label "dozen" is easier to parse than the number 12. The left version might or might not reduce the chance of stupid mistakes like forgetting to square the 12 (e.g. it depends on whether I'm checking the calculations with a computer algebra system).

Something similar comes up in software engineering. In some languages (like Haskell), you can create multiple types that are inherently the same but semantically different, and only allow explicit (not implicit) conversions between them. For example, "row index" vs "column index" of a matrix are both nonnegative integers, but it is a common mistake to switch them by accident. So, maybe you want them to be two different types, so that the compiler will not let you switch them by accident. OK that's the advantage; but the disadvantage is that the code becomes wordier as you need to write and frequently use functions like row_index_from_int() and column_index_from_row_index() etc.

So treating "mole" (or "dozen" etc.) as an algebraic unit rather than a number is just like that. To the extent that it makes your calculations easier to read and less prone to error, it's a good idea, but doesn't have any deeper meaning beyond that.

Weighing matter isn't the only way to account for how much mass you have, counting the number of fundamental particles g hat comprise it is equally legitimate. Moles , slugs, grams are all units that account for the dimensional quality of mass. Counting moles for that matter is a more precise unit of measure.

• You haven't explained why a count of stuff would be considered a dimension. – curiousdannii Apr 7 '15 at 8:10
• @curiousdannii I'm not saying 'count of stuff' is a dimension. What I'm saying is Mass is a dimension. And counting the number of particles is just another 'unit' of measure to account for the mass. Mass is the dimension. Particle counts, moles is just a unit in the measure of mass. – docscience Apr 7 '15 at 16:50
• but this question is asking why the count itself would be considered a dimension. Mass has an SI unit, so why are moles one too? – curiousdannii Apr 7 '15 at 23:32
• @curiousdannii I'm saying that it should not be considered a dimension. MASS (M) should be the considered dimension. In your last comment you said "Mass has an SI unit". I agree SI units are fine. But I also believe moles are just as good. Dimensions are NOT units. Units provide a name to a type of dimension. Dimensions as used in dimensionless analysis are clearly recognizable properties of a system like Mass and time. Units are names we give dimensions within a dimension type. Am I still missing the point of this question? – docscience Apr 7 '15 at 23:41
• @curiousdannii I suppose I should have added - while it's possible to covert units from one unit to another - you generaly cannot convert dimensions. – docscience Apr 7 '15 at 23:44

In the SI system of units, mole has a dimension. But there is a unit system where the mole is a dimensionless number. See https://www.ncbi.nlm.nih.gov/pmc/articles/PMC61354/.

The number of atoms or molecules is not dimensionless, it is the way we "count" the matter. Here are some examples to illustrate the fundamental importance of the amount of substance:

• While $E = m c^2$, you will not extract the same total energy if your fission process is done with 1 atom or with 1 mole of $^{235}U$.

• Consider the other basic law $PV=nRT$: the amount of substance will determine the volume of your gaz at fixed pressure and temperature.

• The amount of substance is also important in interface physics in processes were the number of molecules may change. For example, evaporation imply a variation of the amount of substance and you must take it into account to describe accurately the evaporation rate.

• To rephrase your points: (1) you will not extract the same energy with 1 atom or $10^{23}$, (2) at fixed pressure and temperature the number of atoms will determine the volume of the gas, and (3) if a pond evaporates you must take into account the number of molecules in it. In no way do these processes describe anything that could not (at least in principle) be described by counting things. – Emilio Pisanty Apr 6 '15 at 21:45
• It appears that (at least in a certain approximation) the world is composed of individual particules and, yes, in many cases we need to count them one by one. It is especially true in chemistry were you consider atoms bounding and breaking. Stoichiometry - or the relatives "amount of substance" between reactives - is of prior importance. But even in (ab initio) quantum mechanics you need to postulate the number of atoms and electrons... you don't get them from first principle. In high energy physics, you may extract particules from vacuum but you need many moles of photon in a small volume! – jvtrudel Apr 6 '15 at 22:04
• It is now 106 years since Perrin conclusively demonstrated the discrete nature of matter. Single ions can be loaded, imaged, and manipulated in ion traps; single molecules can be imaged and manipulated in condensed phases. There's really no excuse for claiming the discrete nature of matter to be an "approximation". In any case, this answer fails to address the points raised by the OP. – Emilio Pisanty Apr 6 '15 at 22:31
• What does OP mean? – jvtrudel Apr 6 '15 at 22:34
• And why my answer failed to address the point? "Nor am I asking about why it's chosen as a base quantity and not a derived quantity. I get that any particular choice of bases is more or less arbitrary. I don't understand why it's a dimensional quantity at all. " Its clearly because the nature is discrete and it was my point. Whatever you trust in Perrin or in the Queen ;-) (inside joke if you catch it) – jvtrudel Apr 6 '15 at 22:38