# What advantages does the mole have over counting large numbers using SI prefixes?

I'm trying to understand why chemists use the mole unit instead of just counting and using SI prfixes to simplify the big numbers with units such as zetta- or yotta-molecules (yotta=10^24).

Here's what I've found so far, but it doesn't seem to be enough to make moles particularly important:

Reasons for moles:

• You can easily approximate the number of atoms from the mass of a sample because the mass number of an isotope approximately equals the mass of 1 mole of atoms in grams.

• It's a historical convention that would be too expensive or uncomfortable to change now.

• You can use millimole and micromole instead of having to learn several more obscure SI prefixes like exa and peta. EDIT this reason is added from Jan's answer

• Shorter descriptions, eg "mole of carbon" vs "yotta atom of carbon" (from matt_black's answer).

Reasons against moles:

• For accurate calculations (beyond about 3 significant figures), the relationship between mass number and mass in the first reason above breaks down. So this is a risky thing to use.

• It's a additional concept and set of facts that chemists have to spend effort learning but which describes human convention rather than nature.

• We need extra conversion factors such as 1/mol which people often neglect and Faraday's constant used in Q = n(e-) x F  which wouldn't exist without moles.

Non-reasons for moles:

• You can easily calculate the number of atoms from the mass of a sample because the atomic mass of an element equals the mass of 1 mole of atoms in grams. This relationship only exists because of the special units commonly used for atomic mass. If periodic tables listed atomic mass in grams (or perhaps yoctograms), then we could do the same calculations just as easily without moles.

• In practice, we can't measure the number of molecules so we have to measure mass or volume instead and should therefore count them using a unit that's defined in terms of mass. I don't think the precise details of how a unit is defined matter for practical purposes. If you have 1 litre of an ideal gas, you still have to do calculations to find out many moles it contains just as you would to find how many molecules it contains. There is even a proposal  for SI to redefine the mole to be independent of the mass of any substance, indicating that keeping the definition isn't very important.

• Chemists would make mistakes with those big numbers. They'd use "yotta" in the same way they use "mole", not doing calculations with the actual number it represents and not being at risk of other types of error. (from matt_black's answer) However it would be more complex replacing millimole with zetta because you'd have to remember that a zetta-atom is a milli-yotta-atom.

Here's a similar question but it's mixed in with the idea of measuring number rather than mass or volume - Why do people still use the mole (unit) in chemistry?

This other similar question mainly addresses the uncommonness of quantities as big as 1 mole in other areas of life - The mole is used extensively in chemistry, why not elsewhere?

• Your point about the accuracy doesn't make a lot of sense. Many elements have a fractional mass (Chlorine 35.5, Boron 10.8, ...), so for any practical use (particle physicist don't use moles), your "integer mass number" approach breaks immediatly. – Karl Aug 27 '15 at 14:26
• Yep, that's what I mean by the 1st reason against. It's a bit like saying "10N is the weight of 1kg". Enough to get a quick feel for something without looking up tables or getting out a calculator. However, so far, this seems to be the only substantial reason for using moles! – user1318499 Aug 27 '15 at 15:08
• Chemists don't memorise the number of neutrons (it's of no use chemically), they know the ordinal number (=protons) and the average molar mass. – Karl Aug 27 '15 at 15:26
• The unit "moles" is just extremely convenient. One simple additional convention in at least a thousand more complex concepts your learn until you get your degree (or hundreds of concepts for a good high school diploma). – Karl Aug 27 '15 at 15:32
• Karl - what makes it more convenient than are simpler unit like in my question? If chemists don't memorize mass numbers then even this most important reason disappears, along with any value in Jan's suggestion of an imperial mole. – user1318499 Aug 27 '15 at 23:49

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, 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:

: 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 $\ce{^{12}C}$.

• Exactly. High cohesion, low coupling. – ℵ_ϵ Aug 27 '15 at 14:14

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.

• I think these are non-reasons. Without moles, we would have no need for the gas constant R, we'd just use Boltzmann's constant instead. The question already identifies Faraday's constant as redundant for this reason. Without moles, we would just use e instead of N_A * e. You've highlighted that there's this whole parallel system of constants and formulas that exist just to accommodate the mole. n=P.N_A / (R.T) can also be written as n=P/(k_b.T). – user1318499 Aug 31 '15 at 1:14

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.

• You second paragraph about "count of things" applies to SI prefixes too. A "zetta protein" would be a number of protein molecules, just like a "mole of protein" is. You 3rd paragraph about "use much longer descriptions" could be a disadvantage. Is "a mole of hydrogen" referring to atoms or molecules? You need context to know the meaning. Your 5th paragraph about "The actual size" applies equally well to SI prefixes so I think it should be deleted for clarify. – user1318499 Aug 13 '17 at 1:31

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.

• I have a suggestion that your 4th paragraph isn't helpful and perhaps could be deleted. I think the "effectively quantum" property is covered in the first bullet point of the question. Karl's comments on the question say that this property isn't useful. And using macroscopic units for atoms is already the main question. – user1318499 Aug 13 '17 at 1:20

I am a chemist and I am still trying to understand why SI prefixes are not used by chemists.

I am familiar with all the SI prefixes, but even if I was not familiar, I would prefer to learn a pair of universal prefixes rather than deal with the mole and all the issues, inconsistencies, and headaches it brings to us:

1. I have to learn SI prefixes the same. I.e. the mole is redundant.
2. I have to learn two mole concepts: the chemical mole and the thermodynamic mole .
3. I have to learn three definitions of mole in the SI. The original definition in the SI (1971), the amended definition (1980), and the new definition proposed for the future SI . I have to learn the differences and check what one is being used in a given reference. For instance, the definition of mole in the current SI gives a uncertainty to the Avogadro constant, the proposal for the new SI sets a exact value for this constant and moves the uncertainty to $M(^{12}C)$.
4. Then I have to deal with "correction factors" to compensate for the incompatibility between the redefinitions of mole and kilogram .
5. I have to learn an abstruse name for the physical quantity that supposedly is measured in moles, "amount of substance", like if a mol of electrons is a substance. And then I have to deal with the proposals to change that physical quantity to "chemical amount", "stoichiometric amount", or "enplethy" .
6. I have to deal with the ontological confusion between counting and measuring .
7. I have to deal with redundant equations such as $pV=nRT$ and $pV=Nk_BT$.

REFERENCES

 Failures of the global measurement system. Part 1: the case of chemistry. Gary Price. Accred Qual Assur (2010) 15:421–427.

 What is a Mole? Old Concepts and New. Jack Lorimer. In CHEMISTRY International January-February 2010.

 Comments on recent proposals for redefining the mole and kilogram. B. P. Leonard. Metrologia 47 (2010) L5–L8.

 The Mole is Not an Ordinary Measurement Unit. Ingvar Johansson Accred Qual Assur (2011) 16:467–470.

EDIT

I want to add the words that a synthetic chemist wrote in a letter to C&EN:

In theory, we can dispense with the mole (C&EN, Aug. 4, page 32). The mole, molarity, and molality were adopted as convenient ways to express relative amounts and concentrations of substances when we didn’t know the actual mass of atoms and molecules—when the very existence of atoms and molecules was in dispute.

Now we know, which has enabled us to come up with the number $6.022 × 10^{23}$, give or take, and requires us to explain to students—who question why anyone would specify such a ridiculous number— that it wasn’t chosen but is experimentally derived. Now that we know the actual mass of atoms, ions, and molecules, we can describe the actual number of entities directly—for example, an acid as $1.5 \; \mathrm{YH^{+}}$ per liter ($\mathrm{Y} = \mathrm{yotta} = 10^{24}$), which is about $2.5 \; \mathrm{mol \, H^{+}/L}$. We could rid ourselves of an unnecessary and confusing-to-beginners concept, but of course it will never happen.

I agree with his viewpoint, except maybe with his "it will never happen". Time will say.

• I think it’s safe to assume that over $90~\%$ of the users here are chemists. As a chemist, I fail to grasp your points one to seven, although not grasping the difficulty of two to four could be due to me being an organic chemist, not a physical one. In order: 1) In my life I have not yet experienced an SI prefix larger than tera ($10^{12}$). I need thousands or millions of ‘tera’ to reach the range of a mole, so I would be learning new prefixes just for that. 2) Nobody in organic or inorganic chemistry even knows these exist. – Jan Dec 10 '16 at 2:40
• 3) learning the definitions is not required, you just need to understand how to apply them. Whether the molar mass of carbon or the Avogadro constant have an uncertainty is fully irrelevant because their uncertainties are way below any other lab uncertainties. 4) The correction factors average out to $1$ for practically all lab work. 5) I don’t find amount abstruse at all. And I can safely ignore weird renamings much like I can mostly ignore the unit psi if it pops up. 6) … seriously …? 7) Technically, all equations are redundant. In this case, you just forgot the transformation – Jan Dec 10 '16 at 2:43
• factor $k_\mathrm{B} N_\mathrm{A} = R$ which links the two. In the same vein, there are numerous ‘redundant’ equations for energy. – Jan Dec 10 '16 at 2:45
• In a world where many chemists still rely on kcal/mol, I'm afraid your vision of abandoning the mole will be difficult to pursue. – Martin - マーチン Dec 12 '16 at 12:10
• Why oh why don't we just abandon kcal and just use the kJ? :( – LordStryker Dec 12 '16 at 17:16