Understanding the Mole

Like many chemical terms, the mole has a long and complex history, disguised by the simplicity of its modern definition.

mole [abridged]

One mole contains exactly 6.02214076 \times 10^{23} elementary entities.

IUPAC, 20/05/19

Initially a unit of mass, like the gram-equivalent, the mole was the mass of substance with a specific potential to react with other substances.

The mole morphed into the unit for an ‘amount of substance’, based on the number of particles in a mass of reference substance (hydrogen, oxygen, oxygen-16, and eventually carbon-12).

The redefinition of the mole that came into force on the 20th of May 2019 divorced the mole from any physical measurement entirely. Prior to that, the mole was described in reference to carbon-12:

mole [abridged]

One mole contains the same number of elementary entities as there are atoms in 12 grams of carbon-12.

IUPAC, 1967-2019

While the new definition is based on the results of the superb Avogadro Project, it bears no memory of that; aptly put by Bob Bucat, the number 6.02214076 \times 10^{23} “happens to be our best estimate of the number of carbon-12 atoms in 0.012 kilogram of carbon-12”. However the number was determined, the new definition simply contains the number, without reference to its origin.

Over time, the mole has become more precisely defined, and Avogadro’s number has been more accurately measured. Now, Avogadro’s number is exact, and the mole has the simplest definition it has ever had…

So why is it still so misunderstood?

Amount of Substance

‘Amount of substance’ is the name, like ‘mass’, given to a property of matter. It is a measure of the amount of entities (e.g., particles) in a substance. It is, however, a terrible name. Rather than being brief, memorable, and identifiable (like ‘mass’ is), it is a combination of common words with many meanings. ‘Amount of substance’ does not have a strong enough connection to countable, discrete entities, or to the requisite nature of the substance. The alternative suggested by IUPAC, ‘chemical amount’, is marginally better, mainly because including the word ‘chemical’ helps to confine the scope of its application. Neither term, however, experiences the respect in education that the property deserves.

The truth is that if students were taught ‘amount of substance’ in advance of the mole, it would be a lot easier for them to gain a tangible understanding of the value and meaning of the mole.

Common Misconceptions

It is quite common to find the mole sloppily defined, even in reputable textbooks; ‘amount of substance’, ‘mole’, and ‘1 mol’ are often conflated.

The mole is often treated like a named amount like ‘dozen’. However, the mole is not simply a number, it is a unit. The unit ‘kilogram’ is not the same as ‘1 kilogram’. Likewise ‘mole’ is not the same as ‘1 mole’, and ‘mole’ is definitely not the same as 6.02214076 \times 10^{23}.

Mass is a lot easier to get a feel for, so it’s substantially easier to distinguish between ‘mass’, ‘kilogram’, and ‘1 kilogram’ than it is between ‘amount of substance’, ‘mole’, and ‘1 mol’.

\begin{array}{c c c} \textbf{property} & \textbf{unit} & \textbf{quantity} \\ \hline \text{mass} & \text{kilogram} & \text{1 kg} \\ \text{amount of substance} & \text{mole} & \text{1 mol} \end{array}

In fact, ‘amount of substance’ is rarely even introduced, to the detriment of learning. Imagine trying to teach F = ma without mass… “Force is equal to the number of kilograms times the acceleration” sounds ridiculous.

However, this is exactly what happens in chemistry with ‘amount of substance’ and the mole. The formula m = nM is often read as “mass equals number of moles times molar mass”.

The full definition of the mole highlights the proper distinctions that should be made in and out of the classroom:

The mole, symbol mol, is the SI unit of amount of substance. One mole contains exactly 6.02214076 \times 10^{23} elementary entities. This number is the fixed numerical value of the Avogadro constant, NA, when expressed in mol−1, and is called the Avogadro number.

The amount of substance, symbol n, of a system is a measure of the number of specified elementary entities. An elementary entity may be an atom, a molecule, an ion, an electron, any other particle or specified group of particles.

IUPAC, 08/01/18

The mole is the unit for amount of substance; it is a way of describing the number of entities in a substance in a way that is convenient. It is a powerful tool that humanity chose in the same way we chose the kilogram, the second, or the meter–they frame features of the universe in terms of what humans experience in their day-to-day lives.

Do you think ‘amount of substance’ is worth teaching? Would you rather the mole was redefined as just a number? Let me know in the comments.

What is an atom?


the smallest unit of any chemical element, consisting of a positive nucleus surrounded by negative electrons.

Cambridge Dictionary

smallest particle still characterizing a chemical element. It consists of a nucleus of a positive charge (Z is the proton number and e the elementary charge) carrying almost all its mass (more than 99.9%) and Z electrons determining its size.

IUPAC Gold Book

There is something terribly wrong with these definitions.

According to IUPAC, an atom must be neutral: protons and electrons have equal and opposite charges, so Z protons and Z electrons have a net charge of 0.

The Cambridge definition differs, permitting any nucleus with at least one electron to be called an atom. Similar definitions appear at dictionary.com, lexico.com, and macmillandictionary.com. The Cambridge definition is representative of the latitude of common definitions of ‘atom’, and I use it for no other reason.

\begin{array}{r | c c c c} \textbf{number of electrons} & 0 & 0 < x < Z & Z & > Z \\ \textbf{state} & \text{positive nucleus} & \text{positive} & \text{neutral} & \text{negative} \\ \textbf{example} & \text{He}^{2+} & \text{He}^+ & \text{He} &  \text{He}^- \\ \\ \textbf{IUPAC} & & & \text{'atom'} & \\ \textbf{Cambridge} & & \text{'atom'} & \text{'atom'} & \text{'atom'} \end{array}

Some might dismiss the distinction. After all, chemists know the difference? If the layman’s definition isn’t perfect, it’s not going to cause any problems, right?

Alas not. The truth is even chemists’ use of ‘atom’ is confused…

The Neutral Atom

In the US and the UK, understanding an atom in terms of the numbers of protons, neutrons, and electrons is an expectation typically placed on teenagers. ‘Atom’, ‘proton’, ‘neutron’, and ‘electron’ will be taught much earlier but not in a quantitative setting.

Atoms will be introduced as neutral, and questions will rely on that understanding…

An atom contains 6 protons. How many electrons does it have?

‘Correct’ answer: 6

So far, no problem. It’s perfectly reasonable to ask a student to remember and apply a technical definition that differs from the popular one.

However, the use of ‘atom’ rapidly departs from that rigor.

The Non-neutral Atom

Ionic Compounds

This is sodium chloride:


Composed of sodium (\text{Na}^+) and chloride (\text{Cl}^-) ions, sodium chloride–also known as table salt–is often introduced as the prototypical ionic solid.

The synthesis of sodium chloride from sodium and chloride is a relatively common subject for demonstrating balancing a chemical reaction equation:

\begin{aligned} \text{Na} + \text{Cl}_2 &\longrightarrow \text{NaCl} \\ \textbf{2} \text{Na} + \text{Cl}_2 &\longrightarrow \textbf{2}\text{NaCl} \end{aligned}

However, students are repeatedly told to balance atoms, not ions, as if ‘atom’ can mean ‘atom-like’. It is disruptive to learning to so swiftly break the rules that have just been drummed in.

Relative Formula Masses

To calculate the relative formula mass of \text{NaCl}, students are taught to find the relative atomic masses of sodium and chlorine from the periodic table:

\begin{array} {|c|} 11 \\ \text{\Huge Na} \\ \text{sodium} \\ 22.990 \end{array}\begin{array} {|c|} 17 \\ \text{\Huge Cl} \\ \text{chlorine} \\ 35.45 \end{array}

\begin{aligned} RFM(\text{NaCl}) &= 22.990 + 35.45 \\ &= 58.44 \end{aligned}

This explanation bypasses the fact that sodium chloride is ionic. On the surface, this is not a problem because an electron has a relative formula mass of approximately 0.0005.

\text{mass of an electron} = 9.10938 \times 10^{-31} \text{ kg}

\begin{aligned} \frac{1}{12}^{th} \text{ mass of a carbon-12 atom} &= \frac{0.001 \text{ kg}}{\text{Avogadro's number}} \\ &= \frac{0.001 \text{ kg}}{6.02214 \times 10^{23}} \\ &= 1.66054 \times 10^{-27} \text{ kg} \end{aligned}

\begin{aligned} RFM(\text{e}^-) &= \frac{\text{mass of an electron}}{\frac{1}{12}^{th} \text{ mass of a carbon-12 atom}} \\ &= 5.48580 \times 10^{-4} \\ &\approx 0.0005 \end{aligned}

However, this means that the relative formula masses of sodium and the sodium ion actually differ by 0.0005.

RFM(\text{Na}) = 22.990

\begin{aligned} RFM(\text{Na}^+) &= 22.990 - 0.0005 \\ &= 22.9895 \end{aligned}

Again, a meaningless difference, particularly given that the missing mass from the missing electron in \text{Na}^+ is compensated for by the extra electron in \text{Cl}^-.

\begin{aligned} RFM(\text{NaCl}) &= RFM(\text{Na}^+) + RFM(\text{Cl}^-) \\ &= RFM(\text{Na}) - RFM(\text{e}^-) + RFM(\text{e}^-) + RFM(\text{Cl}) \\ &= RFM(\text{Na}) + RFM(\text{Cl}) \end{aligned}

The issue is that this is one more case where ions are treated like atoms, compounding the confusion in students over what an atom actually is.

Resolving the Conflict

This is the current landscape:

\begin{array}{r|c c} \textbf{entity} & \text{Na} & \text{Na}^+ \\ \textbf{IUPAC} & \text{atom (neutral)} & \text{ion (charged)} \\ \textbf{usage} & \text{atom} & \text{ion/atom} \end{array}

The word ‘atom’ is applied to entities that do not fit the IUPAC definition.

There are three solutions I propose…

Solution 1 – A new word for atom or ion

A new word that means “a nucleus with at least one bound electron” would empower chemists to refer to atoms, ions, or mono-nuclear entities that could be either.

My expertise in etymology and lexicography is limited, so my tentative suggestion is: “monon” (mon-on).


smallest particle still characterizing a chemical element. It consists of a nucleus containing Z protons with or without neutrons, and one or more bound electrons.

Nathan March

Solution 2 – Adopting the common usage of ‘atom’

Instead of using ‘monon’, ‘atom’ could be used for any mononuclear entity with electrons. The phrase ‘neutral atom’ could then apply when neutrality is required, or another word could be coined.

Solution 3 – Using ‘atom’ rigorously

The last option would be to simply use ‘atom’ according to its definition, and never for an ‘ion’. The constituents of ionic compounds would always be referred to as ions, and balancing of chemical equations would be by balancing nuclei and charge.


\begin{array}{r | c c c} \textbf{number of electrons} & 0 < x < Z & Z & > Z \\ \textbf{state} & \text{positive} & \text{neutral} & \text{negative} \\ \textbf{example} & \text{He}^+ & \text{He} & \text{He}^- \\ \\ \textbf{IUPAC} & \text{'ion'}& \text{'atom'} & \text{'ion'} \\ \textbf{common usage} & \text{'atom'/'ion'} & \text{'atom'} & \text{'atom'/'ion'} \\ \\ \textbf{solution 1} & \text{'monon'/'ion'} & \text{'neutral monon'/'atom'} & \text{'monon'/'ion'} \\ \textbf{solution 2} & \text{'atom'/'ion'} & \text{'neutral atom'} & \text{'atom'/'ion'} \\ \textbf{solution 3} & \text{'ion'} & \text{'atom'} & \text{'ion'}\end{array}

The Advantages of the Monon

Solution 3: Applying ‘atom’ according to the IUPAC definition would apply considerable pressure to educators to update their materials and be explicit about the charge of a mononuclear entity in all cases. It is unlikely to occur as it would highlight past materials as ‘incorrect’ and create a discrepancy between past and future materials.

Solution 2: Adopting the common usage of ‘atom’ might work and, in fact, ‘neutral atom’ is a phrase that occurs embarrassingly often (science-direct.com, khanacademy.org, nature.com), despite it being technically a tautology–no doubt adopted because of the inevitable clash between the common usage and the technical definition.

Solution 1: ‘monon’ solves the conflict by the path of least resistance, letting the ‘atom’ recede over time to its technical definition, and providing no pressure to revise past works.

Another advantage of ‘monon’ is that it can be extended to multiples of itself using the common numerical prefixes.

\begin{array}{r | l} \textbf{monon} & \text{single atom or mononuclear ion} \\ \textbf{duon} & \text{two monons} \\ \textbf{trion} & \text{three monons} \\ \textbf{tetron} & \text{four monons} \\ \textbf{...} & \\ \textbf{polyon} & \text{many monons}\end{array}

This allows new families of chemicals to be talked about with ease.

\begin{array}{r | l} \textbf{family} & \textbf{examples} \\ \text{monons} & \text{H}^+, \text{He}, \text{Cl}^- \\ \text{duons} & \text{H}_2, \text{HBr}, \text{NaCl} \\ \text{trions} & \text{MgCl}_2, \text{O}_3, {\text{N}_3}^- \\ \text{tetrons} & \text{AlCl}_3, \text{BH}_3, {\text{NO}_3}^- \end{array}

Would you like to adopt this or do you have a better suggestion? Let me know in the comments.