A Sorites Paradox in the Conventional Definition of Amount of Substance

A sorites paradox in the conventional definition of amount of substance

Nigel Wheatley Consultant chemist, E-08786 Capellades, Spain

Abstract. The conventional definition of amount of substance n is as a quantity proportional to number of entities N. This implies that n is discrete for small N while n is considered to be continuous at the macroscopic scale, leading to a sorites paradox. A practical criterion is proposed for distinguishing between amount of substance and number of entities, and the implications for the conventional definition of amount of substance are discussed.

1. Introduction The conventional definition of amount of substance n is as a quantity proportional to the number of elementary entities N, with the same constant of proportionality (1/NA, where NA is the Avogadro constant) for all substances [1]. This has led many authors to treat amount of substance as a quantity of dimension one and the mole as a “counting unit”, a simple shorthand for some large integer “Avogadro’s number”. This approach confuses amount of substance with number of entities. The distinction between the two quantities is important, because amount of substance is not measured by counting entities. It is invariably demonstrated to students of basic chemistry that it is impossible to count the atoms or molecules in a macroscopic sample (there are too many of them), and such students are understandably confused when textbooks or instructors claim that the mole, the unit of amount of substance, is just a way of achieving such an impossible count. This confusion can be compared to the sorites paradox, first attributed to Eubulides of Miletus (believed to have lived in the fourth century BC). The paradox takes its name from the ancient Greek σωρείτης (sōreitēs) meaning “heaped up”, and can be stated as follows:1 A single grain of sand is certainly not a heap. Nor is the addition of a single grain of sand enough to transform a non-heap into a heap: when we have a collection of grains of sand that is not a heap, then adding but one single grain will not create a heap. And yet we know that at some point we will have a heap. In terms of amount of substance, if we have one entity and we add another entity, we get two entities and we count a number of entities. And yet we know that at some point we will have a “heap of entities” and we will measure an amount of substance. This is paradoxical, because amount of substance is a continuous quantity of dimension N while number of entities is a discrete quantity of dimension 1. Some authors have attempted to resolve this paradox by saying that amount of substance is (or should be) a quantity of dimension 1 [2,3]: several reasons why this approach is unsatisfactory have been discussed elsewhere [4]. This paper explains why amount of substance, as that term is generally understood, must be a continuous quantity. A practical criterion is proposed for distinguishing 1 This version of the sorites paradox is taken from the Wikipedia article “Eubulides”, version of 10 December 2010, available at http://en.wikipedia.org/w/index.php?title=Eubulides&oldid=399978056. Different, but generally equivalent wordings are found in other sources. Sorites paradox in the definition of amount of substance between amount of substance and number of entities, that is to resolve this case of the sorites paradox, and the implications for the conventional definition of amount of substance are discussed.

2. Discrete and continuous quantities A discrete quantity is a physical quantity that can only take on certain numerical values. These values may be restricted to the set of integers, although this is not an absolute criterion for discreteness. Examples of discrete quantities include quantum numbers, degeneracy of a state, charge number. The normal definition of continuity in mathematics is known as the (ε,δ)-definition, usually attributed to Bolzano [5]: for any number ε > 0, however small, there exists some number δ > 0 such that for all x in the domain of ƒ with c − δ < x < c + δ, the value of ƒ(x) satisfies (1).2 ƒ(c) − ε < ƒ(x) < ƒ(c) + ε (1) It is impossible to formally satisfy this criterion for a physical quantity, as it is impossible to prove that the quantity is not discrete at some arbitrarily microscopic scale. Nevertheless, many physical quantities are conventionally considered to be continuous, such as mass, length, time etc. For the purposes of this paper, a physical quantity is considered to be continuous if it can be differentiated in conventional quantity equations. This is not mathematically rigorous, as a function must be (ε,δ)- continuous to be differentiable, and a more rigorous criterion is discussed below. On this definition, amount of substance is conventionally considered to be a continuous quantity because it appears as a differential in quantity equations, for example amount of substance in (2), the definition of chemical potential µ [6].

µ = (∂G/∂n)T,p (2) Number of elementary entities, on the other hand, is a discrete quantity, as it can only take integer values. It can be argued, and has been argued [2,3], that there is no real difference between amount of substance and number of entities. Problems with differentiation can be resolved by taking a separate physical quantity which is both continuous and proportional to amount of substance: for example, chemical potential can also be defined by (3), and this is closer to Gibbs’ original definition [7].3

µ = M(∂G/∂m)T,p (3) The many problems caused by ignoring the separate quantity dimension of amount of substance are discussed elsewhere [4]. For the purposes of this discussion, it is sufficient to note that the mass of a given chemical substance is just as discrete at the microscopic scale as the number of entities. The sorites paradox is not resolved by rewriting quantity equations in forms such as (3), because there is no indication of the scale at which the quantity may be considered to be continuous.

3. Conventional definition of amount of substance The conventional definition of amount of substance was drawn up in the late 1960s in preparation for the introduction of the mole as an SI base unit, and approved by the International Union of Pure and Applied Chemistry (IUPAC) in 1969 [8]. It reads: The amount of substance is proportional to the number of specified elementary entities of that substance; the proportionality factor is the same for all substances and is the reciprocal of the Avogadro constant. This definition is also reproduced in the SI brochure [1]. In mathematical terms: n ∝ N (4)

n = N/NA (5)

2 In the criterion, c is the point at which the continuity of ƒ(x) is being tested. 3 Gibbs originally defined chemical potential as an extensive quantity, i.e. without multiplying by the molar mass in (3). The utility of being able to define intensive molar quantities is just one of the arguments in favour of including a dimension of amount of substance in a practical system of quantities [4].

2 Sorites paradox in the definition of amount of substance

Number of entities N is a discrete quantity; division of a discrete quantity by a physical constant gives another discrete quantity (although not necessarily one whose values are confined to integers). Using (5) as a definition of amount of substance leads to n being defined as a discrete quantity, a simple scaled version of N, not the continuous quantity that has always been used by chemists. That is not to say that (4) and (5) are incorrect as quantity equations, merely that they are insufficient as a definition of amount of substance as a continuous quantity.

4. Paradox of the particle in a box The ideal gas equation (9) may be seen as a combination of Boyle’s law (6), Charles’ law (7) and Avogadro’s law (8), with the molar gas constant R serving as the combined proportionality factor. It may also be defined from kinetic theory. p ∝ 1/V (6) V ∝ T (7) n ∝ V (8) pV = nRT (9)

The molar gas constant is also the product of the Avogadro constant and the Boltzmann constant kB, so (5) is often substituted into (9) to give (10):

pV = NkBT (10) This implies that one can set N = 1 in (10), the well-known “particle-in-a-box” situation used to illustrate the quantization of energy. However, if energy, equal to pV, is quantized, then temperature must be a discrete quantity and not the continuous quantity that it is usually assumed to be. There are several ways out of this paradox. One can accept that the quantization of temperature is no more paradoxical than any other quantum effect:4 the quantization of the translational energy of gas molecules has effects at the macroscopic scale, such as in the kinetic isotope effect in oxidative addition of H2/D2 to organometallic complexes [9]. Or one can observe that the usual derivation of (9) from kinetic theory assumes a large number of particles, and so may not be appropriate for a single particle. The very beauty of kinetic theory and expressions such as (9) is that one does not need to know the number of particles or their individual masses, merely the amounts of different particles and their relative masses. However, both these approaches still lead to a sorites paradox: what it the scale at which we can consider temperature to be a continuous quantity or kinetic theory to be valid?

5. Metrological criterion for “large N” Continuous descriptions of amount of substance such as (2) and (9) are applicable because the number of entities is so large that the numerical value of n is practically (if not formally) a continuous function. As the (ε,δ)-criterion of continuity is too rigorous for physical quantities, a different, metrological criterion is required. The following is proposed: • Consider a physical quantity that depends on the amount of substance: it will obviously also depend on the number of entities, from (4), and its numerical value can be expressed as ƒ(x) where x is the numerical value of N. • Consider a measurement of that quantity for N = x and for N = x+1. These measurements will be associated with measurement uncertainties. • If the difference between the two measurement results is significant with respect to the uncertainties in those measurement result, the quantity is considered to be discrete at N = x; if not, the quantity is treated as continuous for that measurement procedure at that scale. In other words, if there is a significant difference in measurement result by adding a single entity, the measurement is a count of number of entities; if there is no significant difference in measurement

4 The present author prefers to see temperature as a property of an ensemble of particles, and use the term “thermal energy” for the property of a single particle. Nevertheless, the use of the term “temperature” for a single particle is widespread in the literature.

3 Sorites paradox in the definition of amount of substance result on adding a single entity, it is a measurement of amount of substance. In terms of the sorites paradox, the criterion for what is a heap depends on the uncertainty in our measurements of the potential heap. In the terms of the (ε,δ)-definition of continuity, this criterion identifies a lower limit to the domain of ƒ, above which the smallest possible increment in x, the numerical value of N, (ε = 1) produces no perceptible change in ƒ(x). This is simply Cauchy’s criterion of continuity [10], where a function ƒ(x) is continuous if an infinitesimal change in x produces an infinitesimal change in ƒ(x). It is assumed here that there is a value ε > 1 that does produce a perceptible change in ƒ(x), and that the true, discrete ƒ can be modelled by an (ε,δ)-continuous function through interpolation throughout the domain of interest: these assumptions are fulfilled in practical measurements [11]. Different measurement procedures will have different abilities to resolve individual entities in a sample in a meaningful way. However some generalization can be made for the large class of techniques for which the response is directly proportional to the number of entities. If it is assumed that the measurement uncertainty σ is the same for ƒ(N) and ƒ(N+1), the criterion can be written as [ƒ(N+1) – ƒ(N)]/ƒ(N) = [k(N+1) – kN]/kN = 1/N > tσ/ƒ(N) (11) where k is an arbitrary proportionality factor and t is the multiplier of the measurement uncertainty for any given level of confidence. Taking t = 2 as an example, the measurement method must produce relative uncertainties of less than 0.025 to “count” a number of entities higher than 20. Beyond that limit, the measurement result would more correctly be expressed as an amount of substance (33 yoctomoles in this example). The vast majority of chemical measurements are in the domain where amount of substance can be considered to be a continuous quantity. However, some measurements truly are made on individual atoms or molecules, such as the “atom-at-a-time” chemistry of the transfermium elements [12], and detection limits of the order of tens or hundreds of yoctomoles have already been reported in capillary electrophoresis and HPLC with fluorescence detection [13,14] and in analytical mass spectrometry [15]. A discussion of some of the statistical “shot noise” problems has also been published [16].

6. Discussion Amount of substance is unusual among the physical quantities discussed in the SI brochure, as it is the only one for which a definition is quoted. This is no doubt because the mole was the last of the SI base units to be included and, before its inclusion, there was no acceptable name for the corresponding physical quantity.5 With a new name entering metrological terminology, there must have been a laudable desire to be able to answer the deceptively simple question “what is amount of substance?” Nevertheless, (5) is not the only definition of n, nor is it sufficient to define a continuous quantity unless limited to large N. Amount of substance is also the mass of substance divided by the molar mass, and (12) is the principle behind most measurements of n [17]. n = m/M (12) Mass is conventionally considered to be a continuous quantity; molar mass should probably be treated as a discrete quantity, given the finite number of chemical elements and their possible combinations, but the quotient of a continuous quantity by a discrete quantity is still a continuous quantity. Hence (12) correctly defines amount of substance as a continuous quantity. There are a large number of other quantity equations involving molar quantities, which are completely analogous to (12): any of these are acceptable definitions of n. This paper refers to the “conventional” definition of amount of substance precisely because it is not the only possible definition. Indeed, it could be argued that there is no “correct definition” of a physical quantity, merely an insight into its nature through the various quantity equations into which it enters. These quantity equations may be more or less applicable depending on the scale of the

5 The term “number of moles” is unacceptable for formal use, as it links the quantity to a particular unit of measurement. See also Appendix.

4 Sorites paradox in the definition of amount of substance observation, as has been known for more than a century. This is the case with equations involving amount of substance, which must be reinterpreted as involving discrete quantities if used at the molecular scale. Several authors have commented on the confusion and difficulty, especially for students, surrounding the current definition of the mole [3,18–23]. This is perhaps understandable, when the current definition of the mole and the conventional definition of amount of substance are centred around a number of elementary entities, presumably to be determined. This disguises the fact that the numerical value of the Avogadro constant is irrelevant to the measurement of amount of substance at the macroscopic scale, so long as {NA} is sufficiently large that n can be considered to be continuous. Precise measurements of amount of substance, using (12), are used to calculate precise values of the Avogadro constant [24] and not vice versa. It should be pointed out that amount of substance is not unique in being treated as continuous at the macroscopic scale and discrete at the atomic or molecular scale. Electric charge Q is treated as continuous in quantity equations such as (13), where dW is the infinitesimal work involved in changing the charge on a capacitor of capacitance C, whose plates hold the charges +Q and –Q, by the infinitesimal amount dQ, dW = (Q/C) dQ (13) while charge number q is known to be discrete at the atomic or molecular scale. The present author is unaware of any serious confusion as to the nature of electric charge, even though Q and q are quantities of different kinds [25] used to describe the same phenomenon.

7. Conclusion There is no physical, historical or philosophical reason why amount of substance should not be treated in the same way as electric charge, with one macroscopic quantity, conventionally assumed to be continuous, and a separate discrete microscopic quantity of dimension one. The separation between the two quantities is an artefact of the system of quantities, but it is possible to frame objective criteria to decide whether a measurement result should be expressed as the macroscopic quantity or the microscopic quantity and hence resolve the sorites paradox. The author makes the following proposals: 1. The term “amount of substance” (or any successor term: see Appendix) and its unit the mole should be reserved for the continuous macroscopic quantity that has been measured by chemists for more than 200 years [4]. The term “number of entities” should be reserved for the discrete microscopic quantity. 2. The “conventional definition” of amount of substance should be removed from future editions of the SI brochure and the IUPAC Green Book, as it is neither sufficient to define a continuous quantity nor exclusive of other definitions. This move would bring the treatment of amount of substance into line with the treatment of other base quantities in the International System of Quantities. The discussion of the nature of amount of substance should be left to textbooks and, if necessary, scholarly articles, as with other physical quantities. 3. The Mise en pratique of the mole [17] should be modified to make it clear that the mole should only be used to express a measurement of continuous amount of substance, and that the correct unit for expressing a measurement of discrete number of entities is simply 1. Such a clarification would be similar to that in the Mise en pratique of the metre [26] stating that the metre is a unit of proper length in the context of general relativity. 4. In the context of the probable redefinition of the kilogram and the ampere in the near future, consideration should be given to clarifying the current definition of the mole [1] without changing its current basis. In particular, it should be clarified whether the defining physical

constant for the mole is the molar mass of carbon 12 or the molar mass constant Mu. The second sentence of the current definition, which attempts to define the term “elementary entity” by example, should be moved to the Mise en pratique, while the 1980 clarification from the CIPM (“unbound, at rest and in the ground state”) should be incorporated into the

5 Sorites paradox in the definition of amount of substance

main definition. Three possible revised wordings are given in [4]. As is discussed elsewhere [4,27], there is no reason to change the basis of the definition of the mole to a fixed numerical value of the Avogadro constant: indeed, the present author believes that such a move would only aggravate the confusion surrounding the quantity amount of substance.

Acknowledgements The author thanks his former students at the Lycée Louis Blaringhem, Béthune, France, and the Collège Saint-Exupéry, Hellemmes, France, for their (often vivid) descriptions of the confusion and difficulty in learning about amount of substance.

Appendix 1. Name of the macroscopic physical quantity The name “amount of substance”, proposed by Guggenheim [28] apparently as a translation of the German Stoffmenge, has not been without confusion and controversy. Many alternatives have been proposed, including chemical amount, emplethy, quant, numerosity, numerity, numerousness, cardinality, ontcount, posos, tosos, psammity, metromoriance, chemiance [18,29], although only the first has received any currency. It should be noted that several of the proposals imply, either directly or through their etymology, that this quantity is a pure number or count of entities: such proposals are unacceptable to the thesis of this paper that amount of substance is a continuous quantity whose very utility (apart from being differentiable) is that it allows us to ignore the numerical value of the number of entities at any macroscopic scale. As the IUPAC ICTNS has returned to the subject and requested a new name for the physical quantity as part of any redefinition of the mole [23], the present author humbly proposes “stoichion” (adjectival form: “stoichioic”; French: stoïchion, stoïchioïque), from the ancient Greek στοιχεῖον (stoicheion, element), the root of “stoichiometry” [30,31]. The author is of the firm belief that this proposal will gain the same degree of acceptance as the other alternatives. The author also notes (and endorses) the view of Andres et al. [27] that the definitions of units, and a fortiori the names of the physical quantities measured by those units, are cultural goods that should not be modified without a strong positive reason. While the profound confusion about the nature of amount of substance is evident in the recent discussion about a possible redefinition of the mole, and indeed has long been attested [18,19,32], it is not clear (and has certainly not been proven) that this confusion is caused by the current name for the quantity. The long debate over the terms “atomic weight” and “relative atomic mass”, eminently summarized by de Bièvre and Peiser [2], illustrates the difficulty of achieving consensus within the scientific community without clear evidence for the benefit of any change.

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