Introduction

This discussion is in response to Paul De Bièvre’s Column, ‘Clarity about the base quantity “amount of substance” is required before (re)definition of the associated base unit is meaningful’ [1]. The widespread confusion permeating this subject has been well documented [2, 3]. In the following, I argue that this confusion stems from the use of the mysterious Avogadro constant in relating amount of substance to the number of entities in a sample of a pure substance. If X signifies the kind of entities in a sample, the number of entities, N(X), is easily comprehended even though its (integer) value is not known precisely. In fact, N(X) is usually estimated from a knowledge of the total mass, m(X), and the sample-average entity mass, m av(X) = M r(X) Da, where M r(X) is the catalogued average relative entity mass, taking account of the possible presence of various isotopes. The relationship is:

$$ N({\text{X}}) = m({\text{X}})/m_{\text{av}} ({\text{X}}) $$
(1)

However, when it comes to amount of substance, n(X), the formal relationship to N(X), according to the SI (Le Système International d’Unités) [4] is:

$$ n({\text{X}}) = N({\text{X}})\left( {1/N_{\text{A}} } \right) $$
(2)

where N A is called the Avogadro constant, said to be a fundamental constant, independent of the substance. Although the number of entities, N(X), is well understood, this defining relationship—an aggregate of N(X) ‘reciprocal Avogadro constants’—is completely incomprehensible unless some physical meaning can be attached to 1/N A (or to N A itself). Stating that N A is a fundamental constant independent of the substance tells us nothing about the physical meaning of amount of substance, n(X), other than it is proportional to the number of entities, N(X).

By contrast, chemists think of the macroscopic unit for amount of substance, the mole, as an aggregate of a particular number of entities—the particular number being the Avogadro number, \( {\mathcal N}_{\rm{Avo}} \), which has nothing to do with the (inappropriately named) Avogadro constant appearing in Eq. (2). In the next section, I will show that \( {\mathcal N}_{\rm{Avo}} \) is independent of amount of substance, the mole and the Avogadro constant and that it must be equal to the macroscopic-to-atomic-scale mass-unit ratio, g/Da. Chemists have traditionally preferred the gram (rather than the kilogram) as the macroscopic mass unit; since the (current) atomic-scale mass unit, the dalton, is defined as Da = m a(12C)/12, the Avogadro number is \( {\mathcal N}_{\rm{Avo}} \) = g/Da = (0.012 kg)/m a(12C)—the ‘number of carbon-12 atoms in twelve grams of carbon 12’, approximately 6.022 141×1023.

The chemists’ conception of the mole can thus be written down as:

$$ {\text{mol}} = {\mathcal N}_{\rm{Avo}} \;{\text{ent}} = \left( {{\text{g}}/{\text{Da}}} \right)\;{\text{ent}} $$
(3)

where ‘ent’ is the symbol for a single entity, i.e. the smallest possible amount of any substance (while retaining that substance’s chemical properties). Two things need to be emphasized about Eq. (3).

  • Since g/Da (the Avogadro number) is dimensionless, ent must have the same dimension as mol—i.e. amount of substance, N, in both cases. In particular, one entity cannot be the (dimensionless) number 1—if it were, the mole would simply be the (dimensionless) Avogadro number, i.e. an Avogadro number of ones rather than an Avogadro number of entities.

  • There is no dependence on X, i.e. the mole and the entity are both independent of the substance. In this respect, one entity forms an appropriate atomic-scale unit for amount of substance—something I have been proposing for nearly a decade [5–10]. Thus, Eq. (3) links the macroscopic and atomic-scale units for this physical quantity.

If one accepts the chemists’ view of the mole as an aggregate of a particular (Avogadro) number of entities, it is logical to conceptualize amount of substance itself, n(X), as an aggregate of a general number of entities, N(X) ent, in which case the macroscopic unit is obtained by setting N(X) = \( {\mathcal N}_{\rm{Avo}} \) = g/Da—and the atomic-scale unit is obtained by setting N(X) = 1. Thus, we have:

  1. 1.

    Amount of substance is an aggregate of N(X) entities:

    $$ {\text{amount}}\;{\text{of}}\;{\text{substance}},n\left( {\text{X}} \right) = N\left( {\text{X}} \right)\;{\text{ent}} $$
    (4)
  2. 2.

    The macroscopic (base) unit is defined by setting N(X) = \( {\mathcal N}_{\rm{Avo}} \) = g/Da:

    $$ {\text{base}}\;{\text{unit}},{\text{mol}} = {\mathcal N}_{\rm{Avo}} \;{\text{ent}} = \left( {{\text{g}}/{\text{Da}}} \right)\;{\text{ent}} $$
    (5)
  3. 3.

    The (proposed) atomic-scale unit is defined by setting N(X) = 1:

    $$ {\text{atomic}}{-}{\text{scale}}\;{\text{unit}} = 1\;{\text{ent}} $$
    (6)

We will see in the next section that, stemming from the substance-mass equation alone, setting N(X) = g/Da corresponds to a substance mass equal to M r(X) g—the mass in grams numerically equal to the relative entity mass (‘atomic, molecular or formula weight’)—which is the original mole concept [11].

The Avogadro number is independent of amount of substance, the mole and the Avogadro constant

There is a great deal of confusion about the Avogadro number and the Avogadro constant—primarily because of a poor choice of name and symbol for the latter when it was introduced along with the SI definition of the mole in 1971 [4]. In reality, they are totally independent. What is now called the Avogadro number (to distinguish it from the Avogadro constant) is the (dimensionless) number of entities corresponding to a substance mass that, when expressed in grams, has a numerical value equal to the relative entity mass of the substance—i.e. the ‘atomic, molecular or formula weight in grams’, M r(X) g. Although this concept is over one hundred years old [12], the Avogadro number does not have a generally accepted symbol; here, I use \( {\mathcal N}_{\rm{Avo}} \).

By contrast, the (so-called) Avogadro constant, N A, is the (dimensional) constant linking amount of substance, n(X), with the number of entities, N(X), as in Eq. (2). In fact, the Avogadro constant is defined by rearranging Eq. (2) as:

$$ N_{\text{A}} = N\left( {\text{X}} \right)/n\left( {\text{X}} \right) $$
(7)

i.e. the Avogadro constant is any number of entities divided by the corresponding amount of substance. The problem (at the heart of the confusion permeating this subject) is that although the number of entities, N(X), is easily comprehended, without a good concept of amount of substance—appearing in the denominator of Eq. (7)—it is impossible to give physical meaning to the Avogadro constant, N A, from this basic definition. And without a good concept of N A—appearing in the denominator of Eq. (2)—it is impossible to give physical meaning to amount of substance, n(X). Since there is no external reference to any physically meaningful quantity, one is thus caught in an endless loop of circular logic, without resolution. To make matters worse, since N(X) is dimensionless, N A has the incomprehensible dimension of ‘reciprocal amount of substance’, 1/ N, as seen from Eq. (7). Confusingly, the conventional symbol—using ‘N’ with a descriptive subscript—strongly suggests that the Avogadro constant is a (dimensionless) number rather than a dimensional constant. Finally, what is now (since 1971) called the Avogadro number was often previously known as the Avogadro constant (or Avogadro’s constant or Avogadro’s number) [11]. Many textbooks and online tutorials do not make any distinction between these two extremely different concepts. No wonder there is well-documented widespread confusion.

In order to clarify this confusing situation, let us first identify the correct origin of the Avogadro number. First, we rewrite Eq. (1) for the substance mass, expressing it in terms of the atomic-scale mass unit, the dalton:

$$ m\left( {\text{X}} \right) \, = N\left( {\text{X}} \right)m_{\text{av}} \left( {\text{X}} \right) = N\left( {\text{X}} \right)M_{\text{r}} \left( {\text{X}} \right)\;{\text{Da}} $$
(8)

In terms of the (chemists’ preferred) macroscopic mass unit, the gram, this becomes:

$$ m\left( {\text{X}} \right) = \left[ {N\left( {\text{X}} \right)/\left( {{\text{g}}/{\text{Da}}} \right)} \right]M_{\text{r}} \left( {\text{X}} \right)\;{\text{g}} $$
(9)

From this, we see immediately that for m(X) to be equal to M r(X) g, there must be a particular number of entities. By definition, this is called the Avogadro number, \( {\mathcal N}_{\rm{Avo}} \), which, from Eq. (9), is clearly given by:

$$ {\mathcal N}_{\rm{Avo}} = {\text{g}}/{\text{Da}} $$
(10)

the gram-to-dalton mass-unit ratio. Note that this has nothing to do with amount of substance, the mole or the Avogadro constant—it stems entirely from the substance-mass equation written in terms of grams rather than daltons, observing the mathematical identity g ≡ (g/Da) Da, i.e. one gram is an aggregate of an Avogadro number of daltons.

Since \( {\mathcal N}_{\rm{Avo}} \) is also independent of the substance, we have the well-known result that a sample of any substance with a mass equal to its ‘atomic, molecular or formula weight in grams’, M r(X) g, always contains the same number of entities—the Avogadro number, \( {\mathcal N}_{\rm{Avo}} \) = g/Da [12].

The Avogadro number is not a ‘fundamental constant of Nature’—it is simply the ratio of the (chemists’ preferred) macroscopic mass unit, the gram, to the (carbon-12-based) atomic mass unit, the dalton—i.e. a ratio of two human-devised mass units. The International Avogadro Coordination project began as an effort to ‘determine the Avogadro constant, N A’, with a relative uncertainty equal to or less than 2 × 10−8 using an isotopically enriched silicon-28 crystal [13]. In fact, the project actually measures the Avogadro number, \( {\mathcal N}_{\rm{Avo}} \) = g/Da—which is equivalent to measuring the mass of the carbon-12 atom in terms of the kilogram, since g/Da = 0.012/[m a(12C)/kg]. In 2011, the project’s best estimate of g/Da was 6.022 140 82 × 1023, with a relative uncertainty of 3.0 × 10−8 [14]. More recently [15], the value was measured to be 6.022 140 857 × 1023, with a relative uncertainty of 1.2 × 10−8, thereby achieving the desired precision.

We can rewrite the relationship between the number of entities, N(X), and the substance mass, m(X), in dimensionless form in two ways:

  1. (a)

    Atomic scale

$$ N\left( {\text{X}} \right)/1 = m\left( {\text{X}} \right)/\left[ {M_{\text{r}} \left( {\text{X}} \right)\;{\text{Da}}} \right] $$
(11)

noting that m(X) = M r(X) Da = m av(X) when N(X) = 1, and

  1. (b)

    Macroscopic scale

$$ N\left( {\text{X}} \right)/ {\mathcal N}_{\rm{Avo}} = m\left( {\text{X}} \right)/\left[ {M_{\text{r}} \left( {\text{X}} \right)\;{\text{g}}} \right] $$
(12)

noting that m(X) = M r(X) g when N(X) = \( {\mathcal N}_{\rm{Avo}} \) = g/Da.

Amount of substance defined and easily comprehended

Rather than trying to define amount of substance in terms of the reciprocal of a physically incomprehensible ‘fundamental constant’ (with dimension 1/ N), we have seen that the physical quantity can (and should) be defined as an easily understood aggregate of N(X) entities—where one entity is the smallest possible amount of any substance (or collection of entities), while retaining its chemical properties. One entity is easily conceptualized and is the appropriate (substance-independent) fundamental constant for defining amount of substance, n(X), directly, in terms of the corresponding number of entities, N(X):

$$ n\left( {\text{X}} \right) = N\left( {\text{X}} \right)\;{\text{ent}} $$
(13)

where the space before ent indicates that it is being used as an atomic-scale unit for amount of substance.

We can now combine Eq. (13) in dimensionless form with Eqs. (11) and (12) to give:

  1. (a)

    Atomic scale

$$ N\left( {\text{X}} \right)/1 = m\left( {\text{X}} \right)/\left[ {M_{\text{r}} \left( {\text{X}} \right)\;{\text{Da}}} \right] = n\left( {\text{X}} \right)/{\text{ent}} $$
(14)

noting that n(X) = 1 ent when N(X) = 1, and

  1. (b)

    Macroscopic scale

$$ N\left( {\text{X}} \right)/ {\mathcal N}_{\rm{Avo}} = m\left( {\text{X}} \right)/\left[ {M_{\text{r}} \left( {\text{X}} \right)\;{\text{g}}} \right] = n\left( {\text{X}} \right)/{\text{mol}} $$
(15)

noting that n(X) = 1 mol when N(X) = \( {\mathcal N}_{\rm{Avo}} \) = g/Da.

Comparing Eqs. (13) and (2), we see that the (seemingly incomprehensible) reciprocal Avogadro constant, 1/N A, is actually just one entity (exactly). Therefore, it should be stressed that, rather than taking the Avogadro constant, N A, as the fundamental constant (for reciprocal amount of substance), we should take one entity, ent, as the fundamental constant for amount of substance itself (as well as the appropriate atomic-scale unit). The macroscopic unit, mol = \( {\mathcal N}_{\rm{Avo}} \) ent = (g/Da) ent, would remain as the conventional base unit—although, logically, the base unit should be the kilomole (suitably renamed without a prefix), corresponding to the base unit for mass, the kilogram (which should also be renamed without a prefix) rather than the gram [16].

Finally, we note that Eq. (3) can be rearranged as:

$$ {\text{Da}}\;{\text{ent}}^{ - 1} \equiv {\text{g}}\;{\text{mol}}^{ - 1} \equiv {\text{kg}}\;{\text{kmol}}^{ - 1} $$
(16)

showing that the atomic-scale unit for amount-specific mass (commonly called ‘molar’ mass), dalton per entity, is identical to the macroscopic units for this quantity. This identity will be violated by adoption of the ‘New SI’ definition of the mole—formally in terms of the reciprocal Avogadro constant (which, of course, is actually one entity):

$$ {\text{mol}} = N^{*} \left( {1/N_{\text{A}} } \right) $$
(17)

where \( N^{*} \) is an exactly specified integer, which would not be equal to the (inexactly known) Avogadro number, \( {\mathcal N}_{\rm{Avo}} \) = g/Da, if the carbon-12-based dalton is retained, as seems likely [9]. This will require the introduction of a ‘molar-mass correction factor’, (1 + κ) [17], modifying the right-hand side of Eq. (15) to the form:

$$ N\left( {\text{X}} \right)/ {\mathcal N}_{\rm{Avo}} = m\left( {\text{X}} \right)/\left[ {M_{\text{r}} \left( {\text{X}} \right)\;{\text{g}}} \right] = (1 \, + \kappa )\left[ {n\left( {\text{X}} \right)/{\text{mol}}} \right] $$
(18)

where \( {\mathcal N}_{\rm{Avo}} \) = g/Da and (1 + κ) = \( N^{*} \)/\( {\mathcal N}_{\rm{Avo}} \) = \( N^{*}\)/(g/Da) = (1000 \( N^{*} \)/12)m a(12C)/kg—a factor very close to 1, but with a relative uncertainty equal to that of the carbon-12 atomic mass relative to the (redefined) kilogram, and a nominal value that will change slightly as new experimental values of m a(12C)/kg are found. Clearly, this will add yet another level of confusion to an already confusing subject. A simple resolution of the ‘(1 + κ)’ problem is discussed in ‘Appendix’.

Summary

The reason that the physical quantity amount of substance has, from its introduction in 1971, been universally misunderstood is because of a very bad choice for the fundamental constant—the Avogadro constant, with the dimension of ‘reciprocal amount of substance’—relating amount of substance to the corresponding number of entities. Even though the number of entities, N(X), is easily comprehended, unless one has a concept of the physical meaning of N A, (or 1/N A), there is no way to grasp the meaning of amount of substance, interpreted as n(X) = N(X)(1/N A). And there is no way of understanding the physical meaning of the Avogadro constant from its fundamental definition as N A = N(X)/n(X) without a good concept of n(X). Although N A is often (incorrectly) ‘defined’ as N A = 6.022… × 1023 mol−1 (i.e. an Avogadro number per mole), this is no help because in order to understand the physical meaning of a mole, one would, once again, have to have a good concept of the meaning of amount of substance. We are thus again left trapped in a rudderless loop of circular logic without reference to any physically meaningful independent concept.

However, stemming from the chemists’ concept of the mole as an aggregate of a particular (Avogadro) number of entities, it clearly makes sense to conceptualize amount of substance as an aggregate of a general number of entities: n(X) = N(X) ent. This corresponds to the obvious fact that when the number of entities in a sample is one, the corresponding amount of the substance is one entity. The macroscopic base unit then corresponds to setting N(X) = \( {\mathcal N}_{\rm{Avo}} \), giving the mole definition as: mol = \( {\mathcal N}_{\rm{Avo}} \) ent = (g/Da) ent.

One entity—the smallest amount of any substance—is an easily comprehended fundamental constant for amount of substance itself (i.e. with dimension N) and is the appropriate atomic-scale unit for this quantity. The Avogadro constant, correctly defined as N A = N(X)/n(X), is seen to be given by N A = N(X)/[N(X) ent] = 1 ent−1, exactly, i.e. the Avogadro constant is exactly one per entity. And, from Eq. (5), we see that 1 per entity is equal to \( {\mathcal N}_{\rm{Avo}} \) per mole. In other words, N A ≡ 1 ent−1, is the definition of the Avogadro constant in terms of the real fundamental constant, one entity, whereas N A = \( {\mathcal N}_{\rm{Avo}} \) mol−1 is its expression in terms of the conventional base unit.

Without external reference to a physically meaningful quantity, the Avogadro constant is a very mysterious and incomprehensible fundamental constant—and this is at the core of the well-known confusion surrounding amount of substance and the mole. But once it is realized that its reciprocal is simply one entity (exactly)—an easily comprehended fundamental constant of Nature—everything suddenly falls into place.

Unfortunately, with the adoption of the set of so-called New SI definitions of the base units in 2018 [11]—in which the mole will be redefined as ‘mol = \( N^{*} \)(1/N A)’, an aggregate of an exactly specified (integer) number of reciprocal Avogadro constants—an uncertainty factor, (1 + κ) = \( N^{*} \)/(g/Da), will arise when relating substance mass to the corresponding amount of substance. Although the uncertainty is so small that it will not affect practical calculations, its existence adds yet another layer of confusion to the subject. As shown in ‘Appendix’, this could be avoided by redefining the dalton exactly in terms of the (redefined) kilogram, Da = 1/(1000 \( N^{*} \)) kg—as the appropriate atomic-scale mass unit used in stoichiometry—while retaining the (inexactly known) carbon-12-based unified atomic mass unit, u = m a(12C)/12, for cataloguing individual nuclidic masses in terms of u to very high precision [9].