The atomic-scale unit, entity

IOP PUBLISHING

METROLOGIA

Metrologia 44 (2007) 402–406

doi:10.1088/0026-1394/44/5/017

The atomic-scale unit, entity: key to a direct and easily understood definition of the SI base unit for amount of substance B P Leonard The University of Akron, Akron, OH 44325, USA

Received 18 June 2007 Published 27 September 2007 Online at stacks.iop.org/Met/44/402 Abstract The atomic-scale unit, entity (ent), is defined as the number-specific amount of substance, n/N , the amount of substance of a single entity. This unit is an invariant physical quantity (the reciprocal of the Avogadro constant) that serves as the basis for redefining the SI base unit for amount of substance in a direct and easily understood manner. It is argued here that the kilomole should be the base unit in order to avoid factors of 10−3 or 103 appearing in relationships involving both mass and amount of substance expressed in base units. Since, in a compatible formulation, the amount-specific number of entities, N/n (=NA ), is equal to Mu /Da, exactly, where Mu = kg kmol−1 = g mol−1 = Da ent −1 , exactly, then NA = (kg/Da) kmol−1 = (g/Da) mol−1 = 1 ent−1 , exactly. The kilomole can thus be defined very simply as: kmol = N ∗ ent, exactly, where N ∗ , the exact kilomole-to-entity amount ratio, is identical to the kilogram-to-dalton mass ratio: N ∗ ≡ kmol/ent ≡ kg/Da. The Avogadro constant, NA = N ∗ kmol−1 , does not appear explicitly in the defining equation, its reciprocal having been replaced by one entity. Like the dalton, the entity would be categorized as a unit in use with SI.

1. Introduction It appears likely that, in the near future, SI base units for mass, electric current and temperature will be redefined in terms of physical invariants. Consideration is also being given to redefining the base unit for amount of substance [1, 2]. The current definition is equivalent to setting the numerical value of the Avogadro constant, NA , expressed in number of entities per mole—i.e. the Avogadro number, AN —equal to the inexact gram-to-dalton mass ratio. Recently proposed redefinitions of the base unit for amount of substance would fix the dimensionless Avogadro number at an exact value [2, 3]. In this paper, we discuss some of the shortcomings of the current definition, then look at some general relationships that imply a much more direct definition. This leads to a proposal for a straightforward and easily comprehended new definition based on an invariant atomic-scale unit of amount of substance: the entity (symbol ent)—the amount of substance of exactly one elementary entity. Since one of the shortcomings of the current definition is that the base unit is too small by a factor of a thousand [4–6], the proposal here is that the new SI base unit should be the kilomole, defined very simply as an exact 0026-1394/07/050402+05$30.00

number of entities, in a manner assuring continuity with the current definition and compatibility with a redefined kilogram and dalton.

2. Shortcomings of the current definition The current definition of the base unit for amount of substance is restated here for reference. 1. The mole is the amount of substance of a system that contains as many elementary entities as there are atoms in 12 grams of carbon 12; its symbol is mol. 2. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles or specified groups of particles. (In this definition, it is understood that unbound atoms of carbon 12 at rest and in their ground state are referred to.) This is equivalent to specifying that the amount-specific mass of carbon 12 is exactly twelve grams per mole. Some of the shortcomings of this definition follow. (i) The indirect nature of the definition makes it somewhat difficult to comprehend.

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(ii) It is not clear from this indirect definition exactly what an amount of substance is. (iii) A measurement of the number of atoms in 12 g (i.e. 0.012 kg) of carbon 12 has two sources of uncertainty [7]: first, the uncertainty in the mass of the carbon-12 atom measured in terms of the kilogram; and second, the uncertainty in the mass of the kilogram itself—currently defined as the mass of the international prototype of the kilogram (IPK), a physical artefact, the mass of which is known to be drifting due to contamination and handling. This number is the Avogadro number, AN , the numerical value of the Avogadro constant, NA , expressed in number of entities per mole. It is equal to the (inexact) gram-todalton mass ratio, g/Da = 12 × 10−3 m(IPK)/ma (12 C). Thus the definition of the mole (and the value of AN ) is coupled to the IPK and the experimental measurement of the mass of the carbon-12 atom. (iv) The current definition cannot be written as an equation for the mole in terms of some physical invariant. One can, of course, write mol = AN /NA , but, since NA is defined as the number of entities in a sample, N, divided by the corresponding amount of substance, n, this means that, in a sample of 1 mol, NA = AN mol−1 , thereby forming a circular definition. (iv) As the base unit, the mole is too small by a factor of a thousand [4–6]. This stems from the choice of 12 grams— rather than 12 kilograms (the kilogram being the base unit of mass)—as the reference mass of carbon 12. This results in factors of 10−3 or 103 arising in relationships involving mass and amount of substance expressed in base units. Most proposed redefinitions of the mole have focussed on shortcoming (iii) resolved by decoupling the definition from that of the mass unit (and the carbon-12 absolute atomic mass measurement uncertainty) simply by fixing the value of AN at an exact value. A redefinition such as ‘the mole is the amount of substance of a system that contains exactly 6.022 1415 × 1023 elementary entities’—where the numerical value of AN would be chosen to assure continuity with the current definition at the time of transition—although addressing shortcoming (iii), still suffers from shortcomings (i), (ii), (iv) and (v). Shortcomings (i) and (ii) can be resolved by stating that the mole is (i.e. consists of) a certain exact number of elementary entities. This implies that amount of substance, in general, is a collection of a number of specified elementary entities and the definition is then much more easily understood. Shortcoming (iv) would be resolved by formally adopting the entity as an invariant atomic-scale unit in use with SI. A unit of one entity (1 ent), the number-specific amount of substance, n/N, equal to the reciprocal of the Avogadro constant, 1/NA , is the amount of substance of a single entity [8]. The base unit is then defined as being equal to an exact multiple of a unit of one entity, choosing the numerical factor so as to assure seamless continuity with the current definition at the time of transition. Finally, shortcoming (v) is easily resolved by specifying the kilomole to be the base unit for amount of substance, paralleling the use of the kilogram as the base unit for mass. Ideally, a prefix-free name and symbol would be adopted, while keeping the mole, for convenience, as a unit in use with SI, defined as exactly one-thousandth of the new base unit. Metrologia, 44 (2007) 402–406

Perhaps this could be articulated with the adoption of a prefixfree name and symbol for the kilogram, while keeping gram and tonne, for convenience, as units in use with SI.

3. General relationships Consider a sample of N identical atomic-scale entities (i.e. particular molecules, atoms or ions, electrons or other subatomic particles or specified groups of particles, including formula units). The total amount of substance, n, of the sample consists of the N entities—i.e. the collection of the entire number of specified physical particles in the sample. Note that n is neither a dimensionless number nor a mass; it represents a distinct physical quantity consisting of the aggregate of the specified entities—the amount of substance—requiring its own unit. If ma is the effective absolute mass of an individual entity, the total mass of the sample is the sum of the N identical masses: (1) m = Nma . It should be clear that N, n and m are extensive variables— a change in the size of the sample represented by the dimensionless number, N , implies a proportional change in the values of n and m, as well. As is well known, sizeindependent intensive variables can be constructed by forming the quotient of two extensive variables for a particular sample. Referring to the quantities in the numerator and denominator of the quotient, the resulting ‘specific’ quantity is called, in general, the ‘denominator-specific numerator.’ (An exception is sometimes made when the denominator has the dimensions of mass, in which case the intensive quantity is simply called the ‘specific numerator.’) For example, the amount-specific mass, M, is an intensive quantity defined by the quotient: total mass in the sample divided by the corresponding total amount of substance, M = m/n.

(2)

This is usually called the ‘molar mass,’ but for reasons that will become clear, this terminology cannot be used here. The generic term ‘amount-specific mass’ conforms to the general rule of Cohen et al: ‘neither the name of the physical quantity nor the symbol used to denote it implies a particular choice of unit’ [9, p 3]. Similarly, the amount-specific number of entities is an intensive quantity known as the Avogadro constant, NA = N/n.

(3)

And the reciprocal of this, the number-specific amount of substance, is clearly the amount of substance of a single entity [8], ent = n/N = 1/NA , (4) where the symbol ent denotes an amount-of-substance unit of one entity: the amount of substance of exactly one elementary entity. Equation (4) is the definition of the atomic-scale unit of amount of substance, entity. We should note that, since N is dimensionless, the numerical value of NA will depend on the unit used for n; in particular, NA = 1 ent−1 . From equations (1), (2) and (3), we see that NA = M/ma

(5) 403

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and this is sometimes taken to be the definition of the Avogadro constant [5], although equation (3), identifying NA as the amount-specific number of entities, is clearly more fundamental. Equation (4) can be rearranged as n = N/NA = N ent.

(6)

This can be read as ‘the total amount of substance of the sample, n, equals the total number of entities in the sample, N, times the amount of substance of a single entity, ent.’ This suggests that a unit for amount of substance, nunit , could be defined directly in terms of number of entities, simply by specifying the corresponding sample size, N = Nunit : nunit = Nunit ent.

(7)

This follows from the current (indirect) definition together with the adoption of the number-specific amount of substance, entity: ent = n/N, as the atomic-scale unit of amount of substance. The choice of nunit and Nunit is to be made so as to assure continuity with the current definition. For example, using equation (7), we could redefine the mole in a straightforward manner: mol = AN ent,

(8)

in which case, using equation (6), n = N/NA = N/AN mol.

(9)

ubiquitous. This occurs because the mole is based on twelve grams of carbon 12 rather than twelve kilograms. As a base unit, the mole is too small by a factor of a thousand. This has been pointed out by a number of authors. For example, Taylor [4, 5] has considered, and Flowers and Petley [6] have proposed ‘increasing the size of the mole by a factor of 1000’. However, increasing the size while keeping the same name would lead to untold confusion—trying to distinguish between ‘old moles’ and ‘new moles.’ Clearly, the simplest solution is to designate the kilomole as the base unit, paralleling the situation with the mass unit. The drawback to this strategy, of course, is that this would introduce a second SI-prefixed base unit. However, given the likelihood of major changes to the SI occurring soon, with redefinitions of base units for mass, electric current, temperature and amount of substance, the logical solution would be to define a new base unit for amount of substance (as an exact multiple of the atomic-scale unit, entity) equal to the kilomole at the time of transition, but with an appropriate prefix-free name and corresponding symbol. The mole (defined as exactly one-thousandth of the new base unit) would be retained, for convenience, as a unit in use with SI. And this might be the best opportunity to choose an appropriate prefixfree name and symbol for the kilogram, redefined in terms of a physical invariant, keeping gram and tonne, for convenience, as units in use with SI.

5. Summary

From equation (8), a units conversion factor can be defined by CF(ent/mol) = AN ent mol−1 ≡ 1.

(10)

In these equations, the Avogadro number, AN , could be taken to be the current (inexact) value of the gram-to-dalton mass ratio, AN = g/Da, or it could be fixed at an exact value: AN = 10−3 (kg/Da)t=T , exactly, referring to the kilogram-todalton mass ratio at the time of transition, T —and presumably involving a redefined kilogram, based on a physical invariant, satisfying kg = m(IPK)t=T , exactly. However, as explained in the next section, it is better to use a redefined kilomole as the base unit.

4. The kilomole as the base unit In the well-known formula, M = Ar Mu , relating the amountspecific mass, M, to the relative atomic-scale mass, Ar , the amount-specific mass unit is conventionally taken to be Mu = 1 g mol−1 (or 1 kg kmol−1 in some engineering applications). Thus, given a catalogue of relative masses, Ar (X) = ma (X)/Da, we can write, conveniently: M(X) = Ar (X) g mol−1 = Ar (X) kg kmol−1 for any given substance, X. However, in base units: M(X) = 10−3 Ar (X) kg mol−1 .

(11)

For example, for carbon 12, M(12 C) = 12 g mol−1 = 12 kg kmol−1 ; but in base units, M(12 C) = 0.012 kg mol−1 . Similarly, for beryllium 9, M(9 Be) ≈ 9 g mol−1 = 9 kg kmol−1 ; but in base units, this becomes 0.009 kg mol−1 . And in theoretical analyses involving both mass and amount of substance expressed in base units, factors of 10−3 or 103 are 404

This is a proposal to consider adopting the entity (ent), the number-specific amount of substance, ent = n/N = 1/NA , as the atomic-scale unit for amount of substance—categorized as a unit in use with SI. In this way, the base unit for amount of substance can be defined in a direct and easily understood manner simply as an exact multiple of one entity. A consistent definition—avoiding factors of 10−3 or 103 appearing in relationships involving both mass and amount of substance expressed in base units—would take the kilomole to be the base unit. Thus, the definition of the new base unit would be: ‘one kilomole is exactly N ∗ entities.’ As an equation, this appears as: kmol = N ∗ ent, exactly, (12) where N ∗ , the kilomole-to-entity amount-of-substance ratio is identical to the kilogram-to-dalton mass ratio, N ∗ ≡ kmol/ent ≡ kg/Da, thereby defining the dalton exactly in terms of the kilogram: Da = 1/N ∗ kg, exactly.

(13)

kg = N ∗ Da, exactly,

(14)

Rewriting this as

shows the parallel between this and the kilomole definition, equation (12). This guarantees that the amount-specific mass unit can be expressed in base units, ‘convenience’ units or atomic-scale units, all of which are identical: Mu ≡ kg kmol−1 ≡ g mol−1 ≡ Da ent−1 .

(15)

In particular, the numerical value of amount-specific mass, M(X) = Ar (X)Mu (exactly), is the same in each of these Metrologia, 44 (2007) 402–406

The atomic-scale unit, entity

units; and this is identical to the numerical value of ma (X) expressed in dalton—i.e. M(X)/Mu ≡ ma (X)/Da ≡ Ar (X). The Avogadro constant, NA = N/n, is given by ∗

−1

NA = N kmol , exactly.

N ∗ = (kg/Da)t=T = [(12 kg)/ma (12 C)]t=T , exactly. (17)

Using currently available information on the mass of the carbon-12 atom measured in terms of the kilogram [10], we would have N ∗ = 6.022 1415 × 1026 , exactly. Thus, using this value, the proposed definition of the (revised) base unit for amount of substance appears as: ‘The kilomole is the base unit for amount of substance. One kilomole is equal to exactly 6.022 1415 × 1026 entities: kmol = 6.022 1415 × 1026 ent, exactly, where one entity is the unit of amount of substance consisting of exactly one specified atomic-scale entity, which may be a molecule, atom, ion, electron or other sub-atomic particle or a specified group of particles, including formula units.’

Appendix A. The mole as a number It is possible (although not recommended here) to define the mole as a number, rather than a collection of entities. In other words, ‘mole’ would be defined to be a special name for the dimensionless Avogadro number, AN , equal to the gram-todalton mass ratio: (A.1)

But the Avogadro constant is still defined in the usual way as the amount-specific number of entities, N/n—i.e. the number of entities per unit of amount of substance, which gives, for one mole: NA = AN mol−1 , or, equivalently, mol = AN /NA = AN , implying that the Avogadro constant is acting as a units conversion factor; and, as with all conversion factors, it must be identically equal to unity: NA = AN mol−1 ≡ 1,

(A.2)

which should be compared with equation (10). Note that equation (A.1) implies that the amount of substance of a single entity is now defined by ent (= 1/NA ) ≡ 1. For a sample consisting of N entities, the amount of substance, n, is given simply as the total number of entities: n = N . Dividing the right-hand side of this by the conversion factor, NA (≡ 1), gives n = N/NA = N/AN mol,

(A.3)

which is identical in form to equation (9). The total mass of the sample is, as usual, m = Nma = (N/AN )(ma /Da)(AN Da). And, from equation (A.1), this becomes m = (N/AN mol)Ar Mu , where Ar is the relative atomic-scale mass, Ar = ma /Da, and Mu = g mol−1 , giving Metrologia, 44 (2007) 402–406

m = nAr Mu = nAr g mol−1 ,

(A.4)

(16)

The numerical value of N ∗ would be fixed to assure seamless continuity at the time of transition:

mol = AN = g/Da.

the mass in units of gram. Using equation (A.3), the total mass can be written more generally as

which will be recognized as a well-known formula. And from the definition of amount-specific mass, M = m/n, we have M = Ar g mol−1 (= Ar kg kmol−1 ), as usual. Also, since from equation (A.1), g mol−1 ≡ Da, we see that M = Ar Da = ma . Note that equation (A.4) can be written (m/g) = (n/mol)Ar .

(A.5)

In words: ‘number of grams’ equals ‘number of moles’ times relative atomic-scale mass. Although this shows how to define amount of substance as a number, it is conceptually challenging to keep track of the difference between a number with a unit name and a ‘pure’ number, such as the number of entities or the ratio of two quantities with the same units, for example. In addition, as explained in section 3 above, this is not what is meant by ‘amount of substance.’ And the concept of NA (= AN mol−1 ) as a conversion factor, identically equal to 1, might appear to some to be a rather radical departure from the traditional definition. For these reasons, it seems that redefinitions using the atomic-scale amount-of-substance unit, entity, should be much more readily comprehended. In [3], Becker et al proposed redefining the mole as a number. However, they did not identify the Avogadro constant as a conversion factor. Instead, they proposed that ‘the Avogadro constant be converted into the dimensionless Avogadro number.’ This appears to mean replacing the usual formula, n = N/NA , stemming from the basic definition of the Avogadro constant as the amount-specific number of entities, by (A.6) ‘n’ = N/{NA }, using the notation {NA } as the numerical value of the Avogadro constant expressed in number per mole—i.e. the Avogadro number (equal to AN , as defined above). The quotation marks around ‘n’ in equation (A.6) are to distinguish Becker et al’s definition from the usual notation using n as the symbol for amount of substance. This is an important distinction because although the authors used the phrase ‘amount of substance, n,’ they defined their ‘n’ as ‘the number ratio between the number of entities in a sample and the number of entities in one mole {NA }’—as in equation (A.6). Comparing this with equation (A.3), we see that Becker et al’s ‘n’ is actually the dimensionless numerical value of the amount of substance expressed in mole, rather than the amount of substance, itself. In other words, using these authors’ definition of ‘n’ as the number ratio, amount of substance = ‘n’ mol. Or, stated yet another way: ‘n’ is the ‘number of moles.’ This leads to nonstandard definitions of M and Mu . In the expression for total mass of a sample of N entities, according to Becker et al’s analysis, using ‘n’ as N/{NA }, m = ‘n’Ar Mu = (N/{NA })Ar Mu = (N/{NA })M.

(A.7)

With m in gram and ‘n’ and Ar both dimensionless, this means that M must be in units of gram (rather than gram per mole) and 405

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that Mu = 1 g (rather than 1 g mol−1 ). Note that equation (A.7) can be written in a form equivalent to equation (A.5): (m/g) = ‘n’Ar

[4]

(A.8)

—i.e. ‘number of grams’ equals ‘number of moles’ times relative atomic-scale mass.

[5]

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[7]

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[8] [9] [10]

the kilogram, the mole and of other units Metrologia 44 1–14 Taylor B N 1991 The possible role of the fundamental constants in replacing the kilogram IEEE Trans. Instrum. Meas. 40 86–91 Taylor B N 1994 Determining the Avogadro constant from electrical measurements Metrologia 31 181–94 Flowers J L and Petley B W 2005 The kilogram redefinition—an interim solution Metrologia 42 L31–4 Leonard B P 2007 On the role of the Avogadro constant in redefining SI units for mass and amount of substance Metrologia 44 82–6 Morikawa T and Newbold B T 2004 ‘Amount of substance’ and a single elementary entity Chemistry 13 431–5 Cohen E R et al 2007 Quantities, Units and Symbols in Physical Chemistry 3rd edn (London: RSC Publishing) Mohr P J and Taylor B N 2005 CODATA recommended values of the fundamental constants 2002 Rev. Mod. Phys. 77 1–107

Metrologia, 44 (2007) 402–406