# How to correctly write “7 apples” according to the international system of units (SI)

According to international system of units (SI), we can write "7 kg of apples" to refer to the mass of these apples.

However, if we want refer to the amount of apples, that is, the number of entities, the unit should be the mole. So, what's the correct way to denote "7 apples" in accordance with the SI convention?

Additionally, is it correct (according to SI) to say "7 atoms of hydrogen"? Or must we use mole?

• Please have a look at the definition of the mole in the IUPAC Gold Book. – Klaus-Dieter Warzecha Nov 19 '16 at 11:20
• @KlausWarzecha: done, but not solved the doubt. – pasaba por aqui Nov 19 '16 at 11:42
• Could you edit to reflect why the answers didn't clear your confusion? I disagree with the close voters that this is unclear, but you need to be more specific if you believe we didn't sufficiently answer you. – M.A.R. Nov 19 '16 at 17:45
• @M.A.R: Answers are very acurate, in particular the one from Loong. I´m waitibg only one expansion or new answer that includes the concept of mole to accept and close the issue. – pasaba por aqui Nov 19 '16 at 19:24
• @Alchimista: according wiki and others, kilo is k, lowercase – pasaba por aqui Oct 25 '17 at 13:47

In accordance with the International System of Units (SI) [Brochure in English, 8th edition, 2006; updated in 2014] and the corresponding International System of Quantities (ISQ) [ISO/IEC 80000 Quantities and units (14 parts)], you can define a suitable new quantity, for example with the quantity name “number of apples” and the quantity symbol “$N_\text{apples}$”.

The number of apples $N_\text{apples}$ is a quantity of dimension one (for historical reasons, a quantity of dimension one is often called dimensionless):

$$\dim N_\text{apples} = 1$$

A quantity of dimension one acquires the unit one, (symbol: $1$); i.e. the coherent SI unit for the number of apples is the unit one.

Generally, the unit one is an SI derived unit; for example, the derived SI unit for friction factor is newton per newton equal to one, (symbol: $N/N = 1$). However, the unit one for counting numbers, e.g. number of protons in an atom or number of apples, is considered as a base quantity because it cannot be expressed in terms of any other base quantities. Hence, in this case, the unit one is usually considered as a base unit, although the CGPM has not yet adopted it as an SI base unit.

The name and symbol of the measurement unit one are generally not indicated. Therefore, you may write: “The number of apples is $N_\text{apples}=7$.”

The unit one or its symbol $1$ may not be combined with SI prefixes. For example, if you have 2000 apples, you must not write “$N_\text{apples}=2\ \mathrm k$” for $N_\text{apples}=2000$. (And by the way, when you see something like “10K reputation” mentioned on any stackexchange site, you are looking at at least three nonconformities at the same time.)

Any attachment to a unit symbol as a means of giving information about the special nature of the quantity or context of measurement under consideration is not permitted. Expressions for units shall contain nothing else than unit symbols and mathematical symbols. Therefore, write

• “the maximum electric potential difference is $U_\text{max}=1000\ \mathrm V$”, not “$U=1000\ \mathrm V_\text{max}$”
• “the gauge pressure is $p_\mathrm e=0.5\ \text{bar}$”, not “$p=0.5\ \text{bar(g)}$”
• “the electric power is $P_\text{el}=1300\ \mathrm{MW}$”, not “$P=1300\ \mathrm{MW_{el}}$”
• “the water content is $170\ \mathrm{g/l}$”, not “$170\ \mathrm{g\ \ce{H2O}/l}$”

and also

• “the number of apples is $N_\text{apples}=7$”, not “$N=7\ \text{apples}$”
• The OP wanted SI units so this seems correct, but rationally any paper will be a mixture of "plain" language and scientific language. The phrase “the number of apples is $\rm{N}_\text{apples} = 7$” as compared to “7 apples” seems to be overly complex, and far less effective for good communication. – MaxW Nov 19 '16 at 16:34
• @Loong: Excellent answer. Could you add any reference to the "mole"? It seems related to the subject. If my understanding of the answer is correct, when finally "unit one" was accepted as base unit, we will have two different base units for same concept, the "unit one" and the mole? – pasaba por aqui Nov 19 '16 at 16:38
• @pasaba_por_aqui - This has nothing to do with the unit moles. You use moles to count atoms and molecules, not apples. – MaxW Nov 19 '16 at 16:51
• Since the number of particles/molecules in a mole of one chemical is the same as the number in a mole of a different chemical, regardless of the volume, mass, etc, using moles as units makes for simple ratios in equations. – barbecue Nov 19 '16 at 21:01
• @pasabaporaqui If you have 12 apples, you could say that you have 19.92646848(26) yoctomoles of apples, but I rather think "12 apples" is clearer. – zwol Nov 20 '16 at 21:07

# Mole is just a scale factor

I find this description very intuitive:

A mole is the amount of pure substance containing the same number of chemical units as there are atoms in exactly 12 grams of carbon-12 (i.e., 6.023 X 1023)

I think this should clear out the main part of your confusion. To go into the side questions:

## It is definitely acceptable to talk about elementary particles without using mole

You will not often say there are XXX patricles in this jar, because you would need to use huge numbers. But this is not wrong.

A typical example where you would not use the mole scale factors because it would not be convenient:

A typical H2O molecule is made up of two hydrogen atoms and one oxygen atom.

## It is allowed, but inconvenient to talk about bigger things while using mole

If one were to choose apple, or stars as the relevant elementary entity, one could correctly say:

There are XXX mole apples globally.

However, as XXX would be inconveniently small, (and the expression only understood by a very limited audience) there is no reason reason to use mole.

The existing answer already specifies how to define and use a unit according to SI standards. For apples this may not be that neccesary (as 1 apple is a simple enough unit), but if you were to use it in formulas, or if you needed to define something more complicated (like apples that are more yellow than red and more red than green) please follow the given advice.

• The first statement says "mole is just a scale factor". According to my information, the scale factors in SI are the prefixes, but mole is not a prefix, is a base unit. Moreover, all prefixes are powers of ten. – pasaba por aqui Nov 19 '16 at 21:25
• @pasabaporaqui You're correct that SI does not define it as a scale factor in the same way as the prefixes, but you can find the definition in the link in Loong's answer, SI's definition does effectively make it a scale factor, and I agree with this answer, it's easiest to think of it as such. – hvd Nov 20 '16 at 11:57
• @hvd: It is true that define mole as an scale factor made easier to understand and tech it. However, SI defines it as base unit of a physical property called "amount of substance". That means it is a property of the physical world, like it is the mass, the size, ... . Avogadro constant is the way to convert from/to this unit to unit one. For these reasons, I disagree with the phrase "mole is just a scale factor". – pasaba por aqui Nov 20 '16 at 15:38
• @pasabaporaqui Since you're going full pedant, the mole is a base unit for "amount of substance", which is pedantically distinct from the count number of things. Certainly you can convert a given number of carbon atoms into moles of carbon, but to do so you need a non-SI-defined conversion factor (Avogadro's number), just like you can convert a volume of water into grams using a conversion factor (the density). -- While the count/moles conversion factor is the same for most everything (whereas the density differs widely), to the pedant the distinction still stands. – R.M. Nov 20 '16 at 16:37
• @R.M.: do you mean SI is pedantic? I'm sure they have good reasons to define "mole" as a base unit instead of use unit one. – pasaba por aqui Nov 20 '16 at 17:06

The mole is defined for "elementary entities" (atoms, ions, molecules, clusters, ...), not for macroscopic objects. Therefore, 7 apples are 7 apples.

• Thus, we need define a new SI base unit for microscopic objects (the mole) but not for macroscopic? It is not consistent. If the objective was allow large amount, Mega, Tera, preffixes was enough. – pasaba por aqui Nov 19 '16 at 11:36
• I'm not really sure what confuses you about this, but the amount of things just doesn't have an unit. It's just an amount. We got 1 thing, 5 things, 5 million things or 10^152638 things. There's also no need for a prefix. Besides that it would also be terrible unpractical to use the mole or a similar unit for the amount of things. Just imaging calling Dominos and ordering 3.3210786e-24 large peperroni pizzas... – DSVA Nov 19 '16 at 11:42
• If the amount of things doesn't needs a unit, why the amount of microscopic things (elementary entities) needs it? And yes, your example about pizzas is very accurate to show another paradox of this unit. – pasaba por aqui Nov 19 '16 at 11:45
• It doesn't need it, which is one of the things which is criticised about it. It's just extremly convenient to use it. – DSVA Nov 19 '16 at 11:55
• Obligatory XKCD reference... – barbecue Nov 19 '16 at 20:53

7 apples is just 7 apples. Apples are objects, they don't need units, nor does it make sense to give them units.

7 atoms is just 7 atoms.

The Avogadro Constant is defined as the same number of atoms as are found in 12g of carbon 12. That is, 6.022 x 10^23 mol^-1.

Note the units of mol^-1. The number of dimensionless objects per mol.

Hydrogen atoms are all the same. Carbon atoms are all the same. Apples are not like atoms or molecules because they are all slighty different. So in order to talk about a mol of apples, we need to introduce the concept of a standard Apple with a defined chemical formula. I won't propose what that should be, but let's say a standard Apple weighs 100g.

Therefore we can say

1 Apple = 100g of Apples

10 Apples = 1000g of Apples

1 Apple = 100g = 1/(6.022 x 10^23 mol-1) = 1.66 x 10^-22 mol of Apples

Note this also holds for atoms or molecules, should you wish to express a quantity of entities in mol.

1 mol of Apples = 1 mol x 6.023 x 10^23 mol-1 = 6.02 x 10^23 Apples. Which, given that an Apple weighs 100g, weighs 6.022 x 10^25 grams.

Hopefully this illustrates why talking about moles of apples is a strange and rather inconvenient thing to do. If we are discussing hydrogen atoms, which weigh 1.66 x 10^-22 g each, talking about vast quantities of them in terms of moles makes a lot more sense.

• Statement "1 Apple = 100g = 1/(6.022 x 10^23 mol-1) = 1.66 x 10^-22 mol of Apples" equals magnitudes that are dimensionality different (grams equal to moles). Is that correct? – pasaba por aqui Nov 19 '16 at 21:28
• @pasabaporaqui indeed, moles and grams are not dimensionally equivalent. What I said is 1 Apple = 100g and 1 Apple = 1.66 x 10^-22 mol. These statements allow us to derive a third relation 100g = 1.66 x 10^-22 mol (of Apples.) From this we can find the molar mass of Apples 6.022 x 10^25 g mol^-1 which relates the two. – Level River St Nov 19 '16 at 22:54

The unit mole is qualitatively different from the dimensionless unit $1$ because a mole represents an imprecise range of numbers rather than an exact number. It is imprecise since a mole is defined in terms of the number of carbon-12 atoms that together have a mass of $12 \cdot 10^{-3}$ kg, and that number is unfortunately not really a constant since the kg unit is defined by a particular physical metal object that has a mass that varies slightly over time.

So specifying the number of apples as fractions of moles instead of whole numbers adds an unnecessary extra element of uncertainty that is not present if you just specify the number of apples as integer multiples of the unit $1$, assuming your apple-counting device is reliable.

The uncertainty inherent in the definition of the mole could be removed by changing the definition of the kg unit in a way that fixes Avogadro's constant to be an exact value. But that hasn't happened yet.

A mole isn't really a unit, it is a quantity and (although someone beat me to it in a comment) you can have a mole of moles (at least in theory). Although SI defines it as a base unit this does imply a specific context.

The point here is that the SI system is a practical standard not an axiom of scientific philosophy and no sane person would expect Avogadro's number to be a base unit for macro scale quantities of some arbitrary item.

Indeed it is a ratio of carbon atoms per gram so it is not independently defined as it depends on the definition of the kg.

Moles come about from the fact that in chemistry you are often interested in the specific number of molecules which are taking part in an reaction but at the same time you need to be able to relate this to a measurable quantity as it is not normally convenient to weight out individual molecules. For example in combustion one molecule of methane reacts with two of oxygen to form one molecule carbon dioxide and two of water. Molar mass allows us to translate the ration of molecules to mass as long as we know the molar masses of all the elements involved which we do because the value of a mole is chosen to easily relate atomic mass to kg (or more usually grams). E.g. as carbon has an atomic mass of 12 (ish) we know that 1 mole of carbon atoms has a mass of 12 g

Units tell you the specific property which is being described. As far as the pure units are concerned 7 kg is 7 kg whether it is apples, oranges or plutonium.

7 kg describes the quantity of mass, of course apples have many other properties which may or may not be described by SI units so 'apples' needs to be stated to tell the greengrocer what it is that we want 7 kg of. If you are less picky you could say you want 7 kg of edible biomass.

You can also have entirely generic quantities of a unit, physics textbooks can talk entirely legitimately of a mass of 10 kg travelling at 10 metres per second.

This also brings up the concept of dimensional analysis in mathematics which is the principle that for an equality to be valid it must have the same base units on both sides of the equation.

Here it is also useful to introduce the concept of base units these are essentially properties of matter/space which cannot be defined in any terms other than themselves and are the philosophical core of the SI system.

In summary SI units can describe both property and scale, quantity is just a number. Since as far as I am aware there is no SI unit for 'applyness' 7 apples is just a quantity of an ad-hoc unit and not (nor can or should it be) an SI quantity.

Also the Mole is only really useful when you are talking about molecules and atoms which have well defined molecular/atomic masses.

• Well it is a dimensionless unit, which for the propose of explanation in this context is a quantity, although I do take you point. the crucial thing is that a mole does not define any fundamental property and is not defined by any SI base units. – Chris Johns Nov 20 '16 at 17:03