## Abstract

The base quantity ‘amount of substance’ is poorly understood and the name and symbol usually avoided. This is because of its formal interpretation as the number of entities multiplied by the *reciprocal* of the mysterious Avogadro *constant*, *N*
_{A}. If X signifies the kind of entities involved, the number of entities in a sample, *N*(X), is easily comprehended, and if *m*
_{av}(X) is the sample-average entity mass, the total mass, *m*(X) = *N*(X)*m*
_{av}(X)—an aggregate of *N*(X) average entity masses—is also conceptually straightforward. However, the corresponding amount of substance, *n*(X) = *N*(X)(1/*N*
_{A})—an aggregate of *N*(X) ‘reciprocal Avogadro constants’—is incomprehensible unless some physical meaning can be attached to 1/*N*
_{A}. By contrast, the base unit, mole, is thought of by chemists as an aggregate of a *particular* number of entities: mol = \( {\mathcal N}_{\rm{Avo}} \) ent, where \( {\mathcal N}_{\rm{Avo}} \) is the Avogadro *number* (equal to g/Da) and ent represents one entity. It makes sense, therefore, to interpret amount of substance as an aggregate of a *general* number of entities: *n*(X) = *N*(X) ent—an easily grasped concept. A ‘reciprocal Avogadro *constant*’ is thus seen to actually be exactly one entity. One mole then corresponds to setting *N*(X) = \( {\mathcal N}_{\rm{Avo}} \), for which the total mass is the relative entity mass in grams—conforming to the original mole concept.

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## References

De Bièvre P (2015) Clarity about the base quantity ‘amount of substance is required before (re)definition of the associated base unit mole is meaningful. Accred Qual Assur 20:441–443

Furio C, Azcona R, Guisasola J (2002) The learning and Teaching of the concepts of amount of substance and mole: a review of the literature. Chem Educ Res Pract 3:277–292

Fang S-C, Hart C, Clarke D (2014) Unpacking the meaning of the mole concept for secondary school teachers and students. J Chem Educ 91:351–356

BIPM, SI Brochure [8th edition, 2006; updated 2014] www.bipm.org/en/si/si_brochure/

Leonard BP (2007) The atomic-scale unit, entity: key to a direct and easily understood definition of the SI base unit for amount of substance. Metrologia 44:402–406

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

Leonard BP (2010) Why the invariant atomic-scale unit, entity, is essential for understanding stoichiometry without ‘Avogadro anxiety’. Accred Qual Assur 16:133–141

Leonard BP (2011) Alternative interpretations of the mole and the ideal gas equation. Accred Qual Assur 16:577–581

Leonard BP (2012) Why the dalton should be redefined exactly in terms of the kilogram. Metrologia 49:487–491

Leonard BP (2014) The mole is an Avogadro number of entities, the macroscopic unit for chemical amount. Accred Qual Assur 19:213–220

Milton JT (2011) A new definition for the mole based on the Avogadro constant: a journey from physics to chemistry. Philos Trans R Soc A 369:3993–4003

Perrin JB (1909) Mouvement Brownien et réalité moléculaire. Ann Chim Phys 18:5–114 [trans: Soddy F (1910) Brownian movement and molecular reality, Taylor and Francis (London)]. http://web.lemoyne.edu/~giunta/perrin.html

International Avogadro Project (2015) www.bipm.org/en/bipm/mass/avogadro/

Massa E, Nicolau A (2011) International determination of the Avogadro constant. Metrologia 48 (Foreword)

Mohr PJ, Newell DB, Taylor BN (2015) CODATA recommended values of the fundamental physical constants 2014. arXiv:1507.07956v1

Leonard BP (2011) The avo (Av), gali (G), entity (ent) and exact dalton. Accred Qual Assur 16:173–174

Mills IM, Mohr PJ, Quinn TJ, Taylor BN, Williams ER (2006) Redefinition of the kilogram, ampere, kelvin and mole: a proposed approach to implementing CIPM recommendation 1 (CI-2005). Metrologia 43:227–246

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## Appendix: Resolution of the ‘(1 + κ)’ problem

### Appendix: Resolution of the ‘(1 + *κ*)’ problem

With the redefinition of the SI base units scheduled for 2018, the relationship between the number of entities, *N*(X), and the substance *mass* expressed in (redefined) grams will *not* change from that shown in Eq. (12). In dimensionless form, it remains as:

where \( {\mathcal N}_{\rm{Avo}} \) is the (inexactly known) Avogadro *number*, \( {\mathcal N}_{\rm{Avo}} \) = g/*m*
_{u}, where *m*
_{u} = *m*
_{a}(^{12}C)/12. However, the relationship between *N*(X) and the amount of substance, *n*(X), expressed in (redefined) moles will change from that shown in Eq. (15). Instead, it becomes:

where \( N^{*} \) is a fixed exact integer chosen to be as close as possible to \( {\mathcal N}_{\rm{Avo}} \) at the time of adoption of the new units. If the new unit definitions were to be adopted today, \( N^{*} \) would be set equal to exactly 6.022 140 8568 × 10^{23} (the value is adjusted within the uncertainty in \( {\mathcal N}_{\rm{Avo}} \) to be an integer multiple of 12). Equations (19) and (20) give:

where (1 + *κ*) is the ‘molar mass correction factor’, (1 + *κ*) = \( N^{*} \)/\( {\mathcal N}_{\rm{Avo}} \) = \( N^{*} \)
*m*
_{u}/g, i.e.:

In relating substance mass to amount of substance directly, the architects of the redefined units [17] recommend grouping (1 + *κ*) with g mol^{−1}:

where *M*
_{u} is the (inexactly known) ‘molar mass constant’, *M*
_{u} = (1 + *κ*) g mol^{−1}.

However, chemists will continue to work with *fixed exact units*, grams and moles, not the inexactly known *M*
_{u}. So it makes much more sense to group the (1 + *κ*) factor with the term *m*
_{av}(X)/*m*
_{u} in equation (23), giving:

where the denominator is:

an atomic-scale mass *exactly* related to the (redefined) kilogram.

I have previously recommended that the dalton should be redefined this way, for use in cataloguing (sample-average) entity masses used in stoichiometry—that have relatively low precision due to uncertainties in the isotopic composition and the (usually ignored) mass equivalent of binding energy—while retaining the unified atomic mass unit, u = *m*
_{u} = *m*
_{a}(^{12}C)/12, for cataloguing individual nuclidic masses (to very high precision) [9]. If we redefine the dalton as:

Eq. (24) becomes:

where *M*
_{r}(X) is the (sample-average) relative entity mass, *m*
_{av}(X)/Da, referred to the exact dalton. This has the added advantage of *fixing* the value of the Avogadro *number*:

so that Eq. (15) can be written:

*with no explicit or implicit uncertainty factors*. The exact dalton also means that the redefined mole can be written as:

an aggregate of an (exact) Avogadro *number* of entities, as it is thought of by chemists. It also means that we retain the exact relationships between the dalton per entity and the corresponding macroscopic units:

so that the amount-specific mass of a substance, *M*(X) = *m*(X)/*n*(X), can be written directly as:

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Leonard, B.P. Why is ‘amount of substance’ so poorly understood? The mysterious Avogadro constant is the culprit!.
*Accred Qual Assur* **21**, 231–236 (2016). https://doi.org/10.1007/s00769-016-1201-4

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DOI: https://doi.org/10.1007/s00769-016-1201-4

### Keywords

- Amount of substance
- Avogadro constant
- Mole
- Entity