Chemistry - What advantages does the mole have over counting large numbers using SI prefixes?
Solution 1:
Of course, it’s convention and has been so for a good century. And there is no real reason why this convention (and not a different one) happened in the first place — it is entirely conceiveable to define an ‘imperial mole’ so that the atomic mass of an element is equivalent to the same mass in ounces. 1 imperial mole of carbon atoms would then be twelve ounces of carbon atoms.
So there are basically two questions in this:
Why replace a large number with a unit just meaning ‘this certain large number’?
Why is the presently defined mole a good choice for this unit?
The thing about large numbers is that they are large. Almost everybody who grew up in a metric country can name at least three SI-prefixes: kilo, centi and milli. Thanks to IT, many people now also know mega and giga (and maybe tera), even if they don’t use metric units at home. But tera only gets you to $10^{12}$. We need some $10^{21}$ for moles.
I often work with milli- or micromoles of substances in my research. In plain numbers, that’s $10^{20}$ or $10^{17}$ — I don’t know those prefixes and thus would have to learn an entirely new subset. With the mole, everything one uses in the lab will nicely fall into something between nano and kilo.
It also helps to have a single unit there. Molar mass is expressed in grams per mole, concentrations in moles per litre and many more. If there were no unit, it would be simple grams per 1 or 1 per litre — precisely the reason why some people prefer to use rad or some other way to show radians rather than just writing the number. If the unit is there, you’re unlikely to forget it, you know if your calculations are good and more. If there was no name for this unit, it would have to be invented.
So if the mole didn’t exist, it should be invented for simplicity.
The good thing about the size definition of the mole is, as noted above, that it brings everything into one general range. Whether it’s mass, volume, concentration or amount, every unit is going to be prefixed by only a small subset of the SI-prefixes: kilo, milli, micro, maybe nano. Thankfully, those are the ones that are most used in everyday life, too (excluding nano and maybe micro).
It doesn’t really matter where one ends up. If the mole had originally been defined imperially,[1] that would be fine, I wouldn’t have memorised $12.01\,\frac{\mathrm{g}}{\mathrm{mol}}$ for carbon but $340.48\,\frac{\mathrm{g}}{\mathrm{mol}}$. That would create significantly larger molar masses, but that shouldn’t be a problem it should only mean that nano is more prevalent.
The thing about the definition of the unit is that it doesn’t bother $99.9\,\%$ of the scientists working with the unit. Seconds are defined (by SI) according to a number of transitions occuring in some weird isotope that I wouldn’t even know how to measure. I would explain a second by saying it’s the 86400th fraction of a day if someone asked me. Same meaning, different exactness. If the moles are soon defined by mere counting rather than weighing atoms then so be it. Nothing will change for me in practice. (Maybe the fourth digit of a molar mass but I don’t really count those.) So as long as a new definition doesn’t break anything, we’re good to carry on.
Notes:
[1]: With imperially, I meant to assume the following definition using imperial units:
One mole is the number of atoms in exactly $12.00~\mathrm{oz}$ of atoms of the carbon isotope $\ce{^{12}C}$.
Solution 2:
I think the biggest reason it's still used is because there's no escaping our need of Avogadro's constant, $N_\mathrm A=6.022\times10^{23}\ \mathrm{mol^{-1}}$. Even if we were to use some metric prefixes meaning $10^{24}$, etc. we would still need Avogadro's constant because it is used in the definition of other physical constants.
For instance, Boltzmann's constant, $$k_\mathrm B=\frac{R}{N_\mathrm A}$$
And, Faraday's constant, $$F=N_\mathrm A\cdot e$$ where $-e$ is the elementary charge of an electron.
Or, the number of particles in an ideal gas, $$N=\frac{p\cdot V\cdot N_\mathrm A}{R\cdot T}$$
All of those constants have to stay the same for many of the formulations of Chemistry to be correct, so we would have stumbled across this magic number one way or another.
While it might be true (and I don't know this) that the mole was not defined with any of that in mind, it's really quite convenient if you ask me to have a mole be both a unit and a physical constant which everyone knows.
It's particularly useful because you don't have to memorize what the value of $k_\mathrm B$ is, you just remember the equation and figure it out using a calculator. After all, every knows Avogadro's number and the ideal gas constant.
Solution 3:
The thing about the mole is that it simplifies chemistry terminology in a way that can't be avoided if you want to talk about chemical reactions.
A mole is a count of the things involved in a reaction, not necessarily a count of the atoms involved. So a mole of oxygen gas contains two moles of oxygen atoms. A mole of a protein contains hundreds of moles of amino acid resides and, well, an awful lot of moles of atoms. When thinking about this we don't usually have to worry about the number of units of the thing we are discussing.
If we didn't have the idea of the mole, we would have have to use much longer descriptions every time we talked about chemical entities or chemical reactions to be clear what it was we were actually counting. A mole of carbon atoms would have be described as 602 Zetta carbon atoms: a mole of carbon is shorter. Everything would have its own name. And we'd have to use a lot of exa and zetta SI prefixes which could get awkward in calculations.
The value of the mole doesn't matter that much most of the time when thinking about reactions: only the ratio. That it is an very large number is mostly unimportant and the calculations chemists mostly do use molar ratios where they only need to calculate using atomic or molecular mass for the components in the reaction. Including the actual size of the mole in these calculations would introduce very large and unnecessary numbers that would end up cancelling out in the small number of calculations where nobody made an error and would give completely incorrect answers in all the others where people got confused by all the extra prefixes or digits.
Tha actual size of the mole is a little arbitrary, but has the advantage it is the sort of scale chemistry is often done on. A mole of water is ~18g or about 18mL of water. You can do a reaction with that scale on your tabletop. And you don't need to count or even remember Avogadro's constant to do it or any of the calculations associated with it. 6.02*1023 molecules of water is a much less convenient unit. I could happily talk about fully electrolysing a mole of water to produce a mole of hydrogen and half a mole of oxygen with no mention of Avogadro, no big-number SI units.
PS The mole is truly inconveniently large for some objects not on the atomic scale. A mole of moles (the furry burrowing mammal) would make up a sphere about the same size as the moon.
Solution 4:
Moles are simply a built in conversion factor that keeps us from either having to use amus as our macroscopic unit of mass, or grams or kgs as our microscopic unit of mass.
So the basic advantage is that if you want to find the gram mass of $10^{24}$ oxygen molecules in standard units you would have to use Avogadro's number to do it.
Grams were already entrenched in our units of force, energy, pressure, volume and even length (a meter is equivalent the cube root of the volume of $10^6$ grams of water at what 298K?). You can derive a meter, joule, Newton, second or K from a gram of water in the Earth's gravitational field. Moles are not a fundamental concept in chemistry, they are a practical concept, but the ability to move between non-conforming unit bases is fundamental as long as we live in a universe where not every scale of matter is a precise power of ten up from the last one.
In chemistry, by the way, the amu is effectively/practically close to being the "quantum" unit of mass. We either use the fundamental quantum unit of mass in chemistry when describing molecules and dispense with our units of force, energy, power, pressure, volume, and length, or we ignore the chemical quantum of mass, or we find a way to move from one to another.