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Home , Amount of substance, Avogadro constant, Molar mass constant

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A critical review of the proposed definitions of fundamental chemical quantities and their impact on chemical For Peer communitiesReview Only

Journal: Pure and Applied

Manuscript ID PAC-REP-16-08-08.R2

Manuscript Type: Report

Date Submitted by the Author: 31-Jan-2017

Complete List of Authors: Marquardt, Roberto; Université Louis Pasteur, Laboratoire de Chimie Quantique Theorique Meija, Juris; National Research Council Canada, Science and Standards Mester, Zoltan; National Research Council of Canada, Institute for National Measurement Standards Towns, Marcy; Purdue University System, Department of Chemistry Weir, Ron; Royal Military College of Canada, Dep. of Chemistry & Chemical Engineering Davis, Richard; Bureau International des Poids et Mesures (BIPM) Stohner, Jürgen; Zurich University for Applied Sciences, Chemistry and Biotechnology; ETH Zurich, Laboratory for

Keywords: Redefinition of the , , proposed new SI

Author-Supplied Keywords:

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4 1 A critical review of the proposed definitions of fundamental 5 6 2 chemical quantities and their impact on chemical 7 8 9 3 communities 10 11 4 (IUPAC Technical Report) 12 5 Revised Version 31 January 2017 13 14 6 doi.org/10.1515/pac-2016-0808 15 16 For Peer Review Only 17 18 19 20 21 (a) (b) (c) (d) 22 7 Roberto Marquardt , Juris Meija , Zoltan Mester ,MarcyTowns , 23 (e) (f) (g) ∗ 24 8 Ron Weir , Richard Davis ,andJ¨urgen Stohner 25 26 9 27 10 (a) Prof. Dr. Roberto Marquardt 28 11 Past-President of Division I 29 30 12 Laboratoire de Chimie Quantique 31 13 Institut de Chimie 32 14 Universit´eLouisPasteur 33 15 1, Rue Blaise Pascal 34 16 F-67008 Strasbourg 35 17 France 36 18 email: [email protected] 37 38 19 (b) Dr. Juris Meija 39 20 Chair Commission II.1 CIAAW, Titular Member ICTNS 40 21 National Research Council Canada 41 22 Measurement Science and Standards 42 23 1200 Montreal Road M-12, B-12 43 24 Ottawa, ON K1A 0R6 44 25 45 Canada 46 26 email: [email protected] 47 27 (c) Dr. Zoltan Mester 48 28 Vice-President Division V 49 29 National Research Council Canada 50 30 51 Measurement Science and Standards 52 31 1200 Montreal Road M-12, G-7 53 32 Ottawa, ON K1A 0R6 54 33 Canada 55 34 email: [email protected] 56 57 58 59 60 i

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1 2 3 4 35 (d) Prof. Dr. Marcy Towns 5 36 Committee on Chemical Education (CCE) 6 37 Purdue University 7 8 38 Department of Chemistry 9 39 Office WTHR 107E 10 40 560 Oval Drive 11 41 West Lafayette, Indiana 47907 12 42 USA 13 43 email: [email protected] 14 15 44 (e) Prof. Dr. Ron D. Weir 16 45 Chair ICTNSFor Peer Review Only 17 46 Royal Military College of Canada 18 47 Dep. of Chemistry & Chemical Engineering 19 48 P.O. Box 17000, Stn. Forces 20 49 Kingston, ON K7K 7B4 21 50 Canada 22 23 51 email: [email protected] 24 52 (f) Dr. Richard Davis 25 53 Bureau International des Poids et Mesures (BIPM) 26 54 Pavillon de Breteuil 27 55 F-92312 S`evres Cedex 28 56 29 France 30 57 email: [email protected]  31 58 (g) Prof. Dr. J¨urgen Stohner FRSC  32 59 Secretary ICTNS, Chair Commission I.1 Green Book, Chair Mole Project 33 60 ZHAW Zurich University of Applied Sciences 34 61 ICBT Institute for Chemistry & Biotechnology 35 36 62 Einsiedlerstrasse 31 37 63 CH-8820 W¨adenswil 38 64 Switzerland 39 65 email: [email protected] 40 66 and 41 67 Guest scientist at 42 68 ETHZ Swiss Federal Institute of Technology 43 69 44 Laboratory for Physical Chemistry 45 70 ETH H¨onggerberg, Wolfgang-Pauli-Strasse 10 46 71 CH-8092 Z¨urich 47 72 Switzerland 48 73 email: [email protected] 49 50 51 52 53 54 55 56 57 58 59 60 ii

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1 2 3 4 74 Prologue 5 6 7 75 In the proposed new SI, the kilogram will be redefined in terms of the and 8 76 the mole will be redefined in terms of the . These redefinitions will have 9 77 some consequences for in chemistry. The goal of the Mole Project (IUPAC 10 11 78 Project Number 2013-048-1-100) was to compile published work related to the definition 12 79 of the quantity ‘’, its unit the ‘mole’, and the consequence of these 13 14 80 definitions on the unit of the quantity , the kilogram. The published work has been 15 81 reviewed critically with the aim of assembling all possible aspects in order to enable IU- 16 For Peer Review Only 82 PAC to judge the adequateness of the existing definitions or new proposals. Compilation 17 18 83 and critical review relies on the broadest spectrum of interested IUPAC members. 19 84 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 iii

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1 2 3 4 85 1 Introduction 5 6 86 ”It is as easy to count atomies as to resolve the propositions of a lover.” 7 8 87 William Shakespeare, As You Like It 9 10 88 The International System of Units (SI) is defined and published in the SI Brochure pro- 11 12 89 duced by the International Committee for Weights and Measures (CIPM) at the request of 13 90 the General Conference on Weights and Measures (CGPM). The Brochure is currently in 14 th 91 its 8 Edition since 1970. The main purpose of this document is ”to define and promote 15 16 92 the SI, whichFor has been Peer used around theReview world as the preferred Only language of science and th 17 93 technology since its adoption in 1960 through a Resolution of the 11 General Conference 18 19 94 on Weights and Measures” [1]. 20 21 22 Table 1: Comparison between the current SI [1] and the proposed new SI [2] for the 23 24 definitions of the kilogram and the mole. This 2011 text of the proposed new SI is not 25 necessarily the final wording that will be recommended by the CIPM to the CGPM prior 26 27 to the next CGPM meeting in 2018. The numbers given in the last column (proposed new 28 SI) have been modified according to the CODATA 2014 recommendation [3]. Final values 29 will be computed by CODATA prior to introduction of the new SI. 30 31 Base Quantity, Unit Name Current SI Proposed new SI 32 [1] [2] 33 mass, The kilogram is the unit of The kilogram, symbol kg, is 34 the kilogram mass; it is equal to the mass the SI unit of mass; its 35 of the international magnitude is set by fixing the 36 prototype of the kilogram. numerical value of the Planck 37 38 constant to be exactly −34 39 6.626 070 040 × 10 when it 40 is expressed in the SI unit for 41 action J s = kg m2 s−1. 42 amount of substance, 1. The mole is the amount of The mole, symbol mol, is the 43 44 the mole substance of a system which SI unit of amount of 45 contains as many elementary substance of a specified 46 entities as there are in elementary entity, which may 47 0.012 kilogram of carbon 12; be an , , , 48 49 its symbol is ”mol”. , any other particle 50 2. When the mole is used, or a specified group of such 51 the elementary entities must particles; its magnitude is set 52 be specified and may be by fixing the numerical value 53 54 atoms, , , of the Avogadro constant to 55 , other particles, or be exactly 56 specified groups of such 6.022 140 857 × 1023 when it 57 particles. is expressed in the SI unit 58 mol−1. 59 60 1

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4 95 5 6 7 96 Resolutions and Recommendations of the CGPM and the CIPM related to the SI are 8 97 listed in the SI Brochure [1].

9 98 10 11 99 Among its principal goals, the proposed new SI is intended to redefine the kilogram. The 12 100 international prototype kilogram (IPK) is an artifact stored at the BIPM near Paris since 13 101 14 1889. The mass mIPK of the IPK is 1 kg by definition; the artifact is a platinum-iridium 15 102 cylinder with a height and diameter of approximately 39 mm. Six additional official copies 16 For Peer Review Only 103 exist. The mass of the IPK and of prototypes in national laboratories have been compared 17 18 104 about every 40 years. It turned out that the prototype kilogram and the six official copies 19 105 showed some divergence in mass over and relative changes are observed on the order 20 −8 106 of 5 × 10 per century with respect to their first calibration, although this trend was not 21 22 107 observed between the two most recent measurement campaigns [4].

23 108 24 25 109 The proposed new SI will redefine the kilogram in terms of physical constants. It is planned 26 110 to redefine the kilogram by fixing the numerical value of Planck constant h. Table 1 com- 27 111 pares the current and the future definitions for the SI units kilogram and mole proposed 28 29 112 in 2011 [2]. The kilogram influences three other base units in the current SI, 30 113 namely the , the mole and the . At the same time when the kilogram is 31 32 114 redefined, it is also intended to redefine three other base units: the ampere, the and 33 115 the mole. The new definition of the kilogram by fixing the numerical value of the Planck 34 116 constant and the new definiton of the mole by fixing the the numerical value of the Avo- 35 36 117 gadro constant NA have implications for chemistry and provoked criticism. Whether or 37 118 not those implications are of any practical importance will be analysed in the present work. 38 39 119 40 120 The quantities mu ( constant), Mu ( constant), Ar (relative atomic 41 121 mass or ”atomic weight”), ma (atomic mass), M (molar mass), and NA (Avogadro con- 42 43 122 stant) are particularly important in chemistry, as chemists often determine amount of 44 123 substance by weighing. These quantities are related for one particular entity X as follows: 45 46 47 M(X) = Ar(X) Mu (1) 48 49 50 M(X) = NA ma(X) (2) 51 52 53 54 ma(X) = Ar(X) mu (3) 55 56 124 Combining Eqs. (1) to (3) results in 57 58 Mu = NA mu (4) 59 60 2

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1 2 3 4 125 The amount of substance X, n(X), is given as 5 6 7 n(X) = N(X)/NA (5) 8 9 126 where N(X) denotes the number of entities X. Eqs. (2), (4) and (5) are of special impor- 10 127 tance for chemistry because they connect the mass of an individual entity (microscopic 11 12 128 mass) with the mass (usually macroscopic) of an arbitrary number of the same entities, 13 129 similar to 14 15 16 For Peer ReviewF = NA e Only (6) 17 18 130 and 19 R = NA k (7) 20 21 131 which connect the F to the e, and the Boltzmann 22 132 constant k to the universal constant R. The Avogadro constant and the Planck 23 24 133 constant are related by 2 25 α Ar(e) Mu c0 = NA h (8) 26 2 R∞ 27 134 to the R∞, the fine-structure constant α, the of 28 29 135 the electron Ar(e), the in vacuum c0 and the Mu. 30 31 136 In order to prepare a critical assessment of facts and opinons about the two changes in 32 33 137 the SI that are most important for chemists, the new definition of the kilogram and that 34 138 of the mole, three IUPAC divisions, Divisions I, II and V, the Committee for Chemical 35 139 Education (CCE) and the Interdivisional Committee for Terminology, Nomenclature and 36 37 140 Symbols (ICTNS), have jointly launched a project in 2013, the outcome of which is the 38 141 present Technical Report. An initial meeting was held in Paris by J. Meija, Z. Mester, 39 40 142 and J. Stohner in April 2014. In July 2014, the first meeting involving all task group 41 143 members was held in Zurich to distribute tasks and to start reviewing relevant literature. 42 144 The second meeting, where this Technical Report was started, was held in Ottawa end of 43 44 145 January 2015. The minutes of all meetings are published on the IUPAC website [5,6] and 45 146 attached to this paper as an electronic supplementary information. 46 47 147 48 148 The outline of this document is as follows. After the introduction, Section 2 summarizes 49 149 briefly the history of the determination of the Avogadro constant and the evolution of its 50 51 150 numerical value over time. A short paragraph is devoted to the history of the process 52 151 by which the positioning within IUPAC in reference to the proposed new SI was reached 53 152 54 before this task group was created. Section 3 describes the new SI and its special relation 55 153 to the chemistry community. Section 4 briefly describes our strategy to assess the com- 56 154 munity’s attitude towards the new SI, mainly by conducting a written survey among the 57 58 155 National Adhering Organizations (NAOs) to IUPAC. Section 5 reviews articles in favor 59 156 or against the proposed new SI and summarizes their major points. We have covered the 60 3

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1 2 3 4 157 published literature which appeared prior to the Ottawa meeting in January 2015 and a 5 6 158 few more papers that appeared since then. Section 6 presents the results and feedback 7 159 from the survey of the NAOs. Section 7 concludes the critical assessment.

8 160 9 10 161 Based on its critical work, the Task Group quite naturally developed its own consolidated 11 162 position with respect to the mole and the name of the quantity ’amount of substance’. 12 163 13 This position is formulated in an Epilogue, and might be of help to IUPAC to review the 14 164 position of the Union with respect to the proposed new SI. 15 16 For Peer Review Only 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 4

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1 2 3 4 165 2History 5 6 166 7 2.1 The mole - A historical flashback 8 167 9 The amount of substance plays a major role in chemistry. This was recognized by the 10 168 scientific community in 1971 through the acceptance of the amount of substance (chemi- 11 169 cal amount), n, as one of the seven base quantities in the SI with its corresponding base 12 13 170 unit mole (symbol: mol) which ultimately led to the introduction of a new constant, the 14 171 Avogadro constant NA. The base unit mole was introduced to resolve the confusion be- 15 172 tween g-mol (‘g-Molek¨ul’, ‘g-Atom’, ‘g-equivalent’, ‘g-formula’, etc.), which has been used 16 For Peer Review Only 17 173 by chemists to refer to a mass equivalent unit, and g, which is a mass unit [7–10], and 18 174 to introduce quantity calculus into chemistry [11–18] (see also [19] and refs. [12,13] cited 19 20 175 therein). We henceforth also carefully distinguish between the amount 21 176 of substance, and the mole as its unit in the SI. Too often, however, this distinction is not 22 177 made in practice and ’mole’ is used to replace ’amount of substance’, which is inappropri- 23 24 178 ate.

25 179 26 27 180 Two concepts exist in relation to amount of subtance. A ‘number of entities’ concept can 28 181 be understood by considering an as a limiting case to real . It follows from 29 182 the state equation 30 31 pV = nRT = n(E) kNA T = N(E) kT (9) 32 33 183 that the amount of any ideal gaseous entity E is the same under identical conditions of 34 184 p,volumeV , and thermodynamic T . Thus, the amount of substance 35 36 185 n(E) and consequently the number of entities N(E) are independent of the precise nature 37 186 of the entity E. Equation (9) is the essential summary of the laws of Gay-Lussac and 38 187 von Humboldt that led Avogadro to formulate his hypothesis in 1811 (see ref. [10] cited 39 40 188 in [20]), while debates concerning continuum models of continued to exist until 41 189 about 1900 and beyond [20]. 42 190 43 44 191 Amount of substance is proportional to the number of entities, the proportionality factor is 45 192 the reciprocal of the Avogadro constant. Whether this constant is of fundamental nature 46 47 193 or not has also been subject of debates. Perrin gave the demonstration that the numerical 48 194 value of the Avogadro constant is finite and regarded this as a fundamental advance in 49 195 proffering the existence of molecules [21]. In this context, it is interesting to point to a 50 51 196 debate within the CIPM over whether the mole should be adopted as a base unit. Most 52 197 of the arguments, in favor and against the adoption of the mole as a base unit, that we 53 54 198 read today were already raised in 1970. The position that won the day within the CIPM 55 199 was that IUPAP and IUPAC had both asked for the mole to become a base unit of the SI 56 200 and there was no compelling reason to refuse this request [22]. 57 58 201 59 60 5

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1 2 3 4 202 Table 2 collects numerical values of the Avogadro constant over time from experiments 5 6 203 in kinetic gas theory, diffusion, crystallography, electrolysis, and black-body radiation. 7 204 The table is far from being complete and more information can be found in references 8 205 [7, 20, 23–27]. 9 10 11 Table 2: Change of the numerical value of the quantity which is presently known to be the 12 Avogadro constant over time. Many values have been taken from a collection presented 13 14 in [20, 23]. 15 16 For Peer Review Only 17 Name Year Numerical value Notes 18 22 (∗) 19 J.C. Magnenus ∼ 1646 ≈ 2 × 10 diffusion of incense 20 burnt in a church [20, 25] 21 Loschmidt ∼ 1865 5.8 × 1023 (∗) mean free path in gases [23] 22 R¨ontgen, Rayleigh ∼ 1890 (6 to 7) × 1023 [20] 23 Ostwald ∼ 1899 6.3 × 1023 (∗) [23] 24 . × 23 25 Planck 1900 6 175 10 black-body radiation [20] Einstein, 1905/6 6.17 × 1023 [20] 26 23 27 Smoluchowski 1908 6.0 × 10 23 28 1911 6.56 × 10 [23] 29 Perrin 1909 6.5 × 1023 [20, 21] 23

30 Rutherford 1909 6.16 × 10 counting -particles [23] 31 Millikan 1917 6.064 × 1023 Faraday’s law [23] 32 DuNouy ∼ 1924 6.003 × 1023 [23] 33 Kappler 1931 6.059 × 1023 [23] 34 Birge 1941 6.023 38 × 1023 lattice/XRCD (∗∗) [23] 35 23 36 De Bi`evre 2001 6.022 133 9 × 10 [28] 23 (∗∗) 37 Andreas et al. 2011 6.022 140 78(18) × 10 crystal lattice/XRCD [29] 38 CODATA 2014 6.022 140 857(74) × 1023 recommended [3] 39 40 (∗) Recalculated, numbers originally given in terms of molecules per cm3 or per piece of 41 incense. (∗∗) XRCD: X-ray crystal . 42 43 44 206 The current recommended numerical value for the Avogadro constant is given by CODATA 23 −1 −8 45 207 as 6.022 140 857 × 10 mol with a relative standard uncertainty of 1.2 × 10 [3]. 46 47 208 A second concept can be tied to the stoichiometric equation, which is also central to chem- 48 49 209 istry. In 1792, the German chemist Richter elaborated a new alternative of quantitative 50 210 treatment of matter which he coined ‘’ meaning ‘measure of elements’ [30]. 51 52 211 Since then, chemists have measured matter not only in terms of and mass, but 53 212 also in terms of the amount of substance. 54 213 The stoichiometric equation 55 56 C2H4O=CH4 + CO (10) 57 58 214 can be interpreted in two ways: For a chemist working in the laboratory, the equation 59 215 means that one mole of oxirane (C2H4O) decomposes under appropriate conditions to form 60 6

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1 2 3 4 216 one mole of methane (CH4) and one mole of carbon monoxide (CO). Note that here, the 5 6 217 quantity ‘amount of substance’ is not conserved, since one mole of oxirane decomposes into 7 218 two moles of products, namely one mole of methane and one mole of carbon monoxide. 8 219 However, are always conserved, if one neglects the chemically irrelevant relativistic 9 10 220 variation of mass related to the reaction enthalpy. 11 12 221 Another example using the concept of stoichiometry is the electrolytic decomposition of 13 222 . 9 g water (H2O) decomposes into 1 g (H2)and8goxygen(O2)when 14 223 approximately 96485 of electricity has been consumed during the electrolysis, 15 16 224 therefore, theFor relation ofPeer the mass of hydrogen,Reviewm(H2), and ,Onlym(O2), found in this 17 225 experiment is 1:8. This finding follows from Faraday’s second law of electrolysis. 18 19 226 20 227 Amount of substance allows a quantitive formulation of the law of multiple proportions 21 228 due to and Avogadro: The variation Δn(X) of the amount of substance of X in the 22 23 229 course of a chemical reaction is related to the variation Δn(Y) of the amount of substance 24 230 of Y in the same chemical reaction by a simple ratio Δn(X)/Δn(Y) = νX/νY,whereνX and 25 26 231 νY are the stoichiometric coefficients of X and Y, respectively, in the stoichiometric equa- 27 232 tion of that reaction. This relation follows from the defining equation of the extent of reac- 28 233 tion, ξ [11]. The same, simple ratio does not hold for the variations, Δm(X) and Δm(Y), 29 30 234 of the masses of the involved substances. However, every molecule can be attributed a 31 235 characteristic ”chemical mass quantity” which we today call molar mass (see Eq. (2)): 32 33 236 M(X) = m(X)/n(X) and M(Y) = m(Y)/n(Y); note that, while the symbol ma(X) in 34 237 Eq. (2) refers to the mass of one particular entity X, the symbol m(X) refers here to the 35 238 mass of the amount n(X) of X. The ratio of the variations of masses is expressed in terms 36 37 239 of molar masses and stoichiometric coefficients: Δm(X)/Δm(Y) = νX/νY × M(X)/M (Y). 38 240 As for the total mass, the molar mass is conserved in a chemical reaction within the ac- 39  241 curacy of current measurements of mass: νXM(X) = 0 (the ’reaction mass’ is zero). 40 X 41 242 For this reason we may say that, in Eq. (10), 44.053 g oxirane decomposes to yield 16.043 42 243 g methane and 28.010 g carbon monoxide, with , because the stoi- 43 244 − 44 chiometric coefficients there are νC2H4O = 1, νCH4 = νCO = 1, and because these masses 45 245 are molar masses or mass equivalents; but if we decomposed 1 g of oxirane, we would 46 246 not get 0.5 g methane and 0.5 g carbon monoxide, despite the fact that masses would be 47 48 247 conserved, because the latter masses are not correct molar masses or mass equivalents. We 49 248 also note that the molar quantities, such as molar mass and are intensive. 50 51 249 They are independent of the number of specified entities in a sample. This number may 52 250 be as small as one or may be too large to be counted directly. 53 251 54 55 252 Working in the laboratory and doing synthesis or electrolysis, however, there is no need to 56 253 introduce amount of substance as a new quantity; it is sufficient to know the laws of Avo- 57 58 254 gadro, Dalton and Faraday and the molar mass, as the appropriate mass equivalent, for 59 255 every atom and molecule. The determination of molar masses for any 60 7

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1 2 3 4 256 has been made possible by the use of the relative atomic mass scale. Consequently, with 5 6 257 the use of relative atomic masses (”atomic weights”), chemists are able to convert mass 7 258 measurements into a quantity that is proportional to the number of entities. The distinc- 8 23 259 tion between the chemical amount (1 mol) and the number of entities (about 6 × 10 ), 9 10 260 however, still seems to be unclear and might lead to confusion. In principle, both are 11 261 referring to the same property of matter – the numerosity. 12 262 13 14 263 Finally, the aforementioned example of the oxirane decomposition reaction also allows us 15 264 to interpret amount of substance within its first conceptual frame, namely in chemical 16 For Peer Review Only 17 265 kinetics. Eq. (10) enables us to say that one molecule of oxirane decomposes into one 18 266 molecule of methane and one molecule of carbon monoxide. This interpretation uses a 19 267 microscopic point of view, namely that one possible reaction channel for the decomposition 20 21 268 of oxirane is represented as an unimolecular reaction leading to a rate of reaction which 22 269 is only proportional to the amount (or ) of oxirane and 23 24 270 interpreting 25 C2H4O → CH4 + CO (11) 26 27 271 as a so-called elementary reaction. It is usually not permitted to replace the equal sign in 28 272 the stoichiometric equation by an arrow when entering the field of chemical kinetics [11]. 29 30 273 This reaction has been investigated experimentally to determine the Arrhenius parameters 31 274 of a first order kinetics (see [31, 32] and refs. therein) where the rate of reaction depends 32 33 275 only linearly on the amount concentration of oxirane, as indicated in Eq. (11). 34 276 35 277 We thus have a macroscopic and a microscopic (particulate) interpretation of the same 36 37 278 stoichiometric equation with very different implications. The introduction of the base 38 279 quantity amount of substance reflects both views: within the first concept, it reflects the 39 40 280 microscopic view, within the second concept it reflects the macroscopic view. As men- 41 281 tioned before, the adoption of the mole as a base unit of the SI was made at the joint 42 282 request of IUPAC, IUPAP and ISO [1] and was agreed to by the CIPM after intense 43 44 283 reflection [22]. 45 46 284 2.2 History of IUPAC position concerning the proposed new SI prior to 47 48 285 the present work 49 50 286 On 9 July 2009, Prof. Mills (the IUPAC Representative to and President of the CCU – 51 287 Consultative Committee on Units), sent a letter to IUPAC ICTNS Chair Prof. Lorimer 52 53 288 informing him of CCU’s intentions to propose the redefinition of several SI base units 54 289 including the mole. He stated that ”I would like to hear opinions from members of your 55 290 committee. I certainly wish ICTNS to be aware of this proposal, and I would really like to 56 57 291 come away with some expression of support for the proposal or perhaps of opposition, if 58 292 that should be the case ... The CCU strongly supports making the change to fix the value 59 60 8

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1 2 3 th 4 293 of NA.” At the ICTNS meeting during the 45 IUPAC General Assembly in Glasgow, the 5 6 294 implementation of the new SI and its consequences were part of the Agenda. Among the 7 295 attendees were: J. Dymond (Div. I), J. Reedijk (Div. II), A. Rauter (Div. III), R. Jones 8 296 (Div. IV), B. Hibbert (Div. V), P. Fedotov (Div. VI), M. Nordberg (Div. VII), J. Nyitrai 9 10 297 (Div. VIII), R. Weir (ICTNS), R. Marquardt (ICTNS), A. Fajgelj, P. De Bi`evre, I. Mills, 11 298 T. Coplen, A.J. Thor (ISO TC-12), and F. Pavese. Documents had been circulated in 12 299 13 advance of the meeting. On 3 August 2009, Prof. Mills presented definitions considered 14 300 by CCU for the kilogram, ampere, kelvin, and mole to fix the numerical values of h, e, k, 15 301 and NA, respectively, and a new constant-explicit format for the formal definitions of the 16 For Peer Review Only 17 302 base units of the SI. This presentation was followed by a presentation of Prof. De Bi`evre. 18 303 Prof. Lorimer put forward a motion which was adopted by vote following an extended 19 304 discussion. This motion adopted the ‘ICTNS resolution’ concerning the proposal by the 20 21 305 CCU to redefine the mole.

22 306 23 24 307 The recommendation of the CCU of the BIPM is supported by the IUPAC, with the 25 308 following suggestions: 26 27 309 1. The greatest effort should be made to change the name of the ISQ base quantity 28 29 310 amount of substance at the same time that a new definition of the mole is approved. 30 12 31 311 2. A note should accompany the new definition to explain that the molar mass of C 32 312 will be an experimental quantity, with a relative measurement uncertainty of about 33 −9 1 34 313 1.4×10 . 35 36 314 On 26 August 2009, these decisions were communicated to the IUPAC Bureau by Prof. 37 38 315 Lorimer. IUPAC Secretary General, Prof. Black, presented this resolution to the IUPAC st 39 316 Executive Committee at its 141 meeting on 3 to 4 October 2009 (Chile). A motion was 40 317 made and seconded that the motion proposed by ICTNS be approved. The motion was 41 42 318 approved unanimously by the IUPAC Executive Committee. This motion was communi- 43 319 cated to BIPM as the official IUPAC position on that matter. The said resolution of the 44 th 45 320 Executive Committee was ratified by the IUPAC Council during its 46 meeting on 3 to 46 321 4 August 2011 (Puerto Rico). 47 322 48 49 323 Whereas the first draft motion put forward by ICTNS Chair Prof. Lorimer stated that: 50 324 ”The name preferred by IUPAC for the SI base unit is ‘chemical amount’”, the final motion 51 52 325 adopted by vote states that: ”The greatest effort should be made to change the name of 53 326 the ISQ base quantity ‘amount of substance’ at the same time that a new definition of the 54 327 mole is approved”, summarized in ref. [33]. 55 56 57 58 59 1Now this value is about 0.45 × 10−9. 60 9

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1 2 3 4 328 3ProposednewSI 5 6 th 329 Resolution 1 of the 24 meeting of the CGPM (2011) [2] introduces the new SI, which will 7 8 330 link the definitions of the kilogram, mole, ampere and kelvin to exact numerical values for 9 331 the Planck constant h, Avogadro constant NA,elementarychargee and Boltzmann con- 10 th 11 332 stant k. Referring to [2], Resolution 1 of the 25 meeting (2014) [34] notes that progress th 12 333 has been made and encourages continued effort to allow the 26 CGPM (2018) to adopt 13 334 a resolution to replace the current SI with the revised SI. References [2] and [34] will be 14 15 335 referred to as the ‘CGPM Resolutions’ in this report. 16 For Peer Review Only 336 17 18 337 The definitions of the and the second are already linked to exactly defined numeri- 19 338 cal values for the speed of light in vacuum c0 and the clock frequency of the caesium-133 20 339 atom. One effect of both CGPM Resolutions will be the creation of a revised SI with new 21 22 340 definitions of the kilogram, mole, ampere and kelvin based on invariant quantities [34]. By 23 341 contrast, since 1889 the kilogram has been defined as equal to the mass of an artifact known 24 25 342 as the international prototype of the kilogram (IPK). Unfortunately the mass, mIPK,of 26 343 the IPK cannot be assumed to be invariant, with the consequence that the magnitude of 27 344 the unit of mass, the kilogram, tracks any possible changes to the quantity mIPK. In addi- 28 29 345 tion, the IPK has only been used during four measurement campaigns, the first ending in 30 346 1889, the second in 1946, the third in 1991 and the fourth just ended. The self-consistency 31 32 347 between the mass of the IPK (always taken to be exactly 1 kg) and similar objects stored 348 33 with it is roughly 75 Ñg over this period of 125 years; the constancy of mIPK with respect 34 349 to true fundamental constants over this period is unknown. Since 1971 the mole has been 35 −1 36 350 defined by assigning the exact value 12 g mol to the molar mass of unbound carbon-12 12 12 −1 37 351 at rest and zero kelvin temperature [1], M( C), with M( C) = 12 g mol (exactly). 38 39 352 Thus the present definition of the mole is linked to the mass of the IPK but mIPK is not 40 353 a perfectly stable quantity.

41 354 42 43 355 The proposed new SI solves the problem of the IPK by linking the kilogram to a fixed 2 −1 44 356 numerical value of the Planck constant, h, whose SI unit is kg m s (or simply J s 45 2 −2 357 because is a special name for kg m s ) and measurements traceable to the SI 46 47 358 metre and second, which are already defined in terms of fixed numerical values assigned to 48 359 two physical constants. The CGPM further proposestoredefinethemolebygivingafixed 49 50 360 numerical value to the Avogadro constant, NA. The choices advocated by the CGPM are 51 361 not the only possibilities for eliminating the artifact definition of the kilogram and defining 52 362 the mole. Here we compare in a systematic way the possible approaches available for the 53 54 363 redefinition of the mole in order to make these choices and their consequences transparent. 55 56 57 58 59 60 10

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1 2 3 4 364 3.1 Choices within constraints 5 6 365 With regard to the definition of the kilogram and the mole in the new SI, we recognize 7 8 366 the following constraints: 9 10 367 1. The kilogram can no longer be defined by the mass of a reference artifact. 11 12 368 2. The number of physical constants with fixed values must not over-constrain the SI; 13 369 there cannot be two independent constants, or groups of constants, that define the 14 15 370 same unit. 16 For Peer Review Only 17 371 3. Existing relations among quantities cannot be affected by redefining the units in 18 372 which the quantities are measured. One of the quantity relations that must remain 19 12 20 373 true under any choice of unit definitions is (see Eq. (2), with X replaced by Cand 12 21 374 m(X) = ma( C)) 22 12 12 ma( C) = M( C)/NA (12) 23 24 12 375 where ma( C) is the mass of a neutral atom of carbon-12. The carbon atom is un- 25 26 376 bound, at rest, and in its ground state thereby ensuring that there are no relativistic 27 377 corrections to the mass. Eq. (12) is equivalent to Eq.(4). 28 29 378 Another quantity relation that must remain true under any choice of unit definitions 30 379 is that given by Eq. (8), which we rewrite here as 31 32 2 h NA h c0 α 33 = = Ar(e) (13) 34 mu Mu 2 R∞ 35 36 380 Note that we cannot fix the values of both h and mu, or fix the values of the three 37 381 quantities NA, h and Mu. Each of these two choices would assign an exact value 38 39 382 to the right-hand side of Eq. (13), which is not a logical constraint on the Rydberg 2 40 383 constant. The problem is also revealed by noting that (muc0/h), which is a frequency, 41 384 would have a fixed numerical value thereby redefining the SI unit of time. 42 43 385 Nevertheless, the relative uncertainty of the experimentally-determined quantity 44 2 45 386 α Ar(e)/R∞ is important because it tells us the relative uncertainty of, say, mu 46 387 if the value of h is fixed; or the relative uncertainty of Mu if both NA and h have 47 388 fixed values as in the CGPM Resolutions; or the relative uncertainty of h if mu, NA 48 49 389 and Mu all have fixed values (see Eq. (2)). Eq. (13) also tells us that the principal 2 50 390 contribution to the uncertainty of Mu in the revised SI will be that of α , uncertain- 51 52 391 ties of the remaining experimental variables being negligible by comparison. From 53 392 the latest values of the fundamental constants recommended by CODATA [3], we 54 393 infer that the relative standard uncertainty of the right-hand side of Eq. (13) is 55 −10 56 394 4.5 × 10 in the proposed new SI. 57 58 59 60 11

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1 2 3 4 5 Table 3: This table presents the relative standard uncertainties ur that result from different 6 ways of defining the SI units of importance to chemistry. The resulting ur are listed for 7 2 −1 the four constants h: Planck constant (SI unit: kg m s ), Mu: molar mass constant (SI 8 −1 −1 9 unit: kg mol ), NA: Avogadro constant (SI unit: mol ), mu: atomic mass constant (SI 10 unit: kg). The uncertainty of the mass mIPK is zero in the present SI and entered as Y to 11 12 indicate a number that will be specified before the proposed new SI is implemented, but 13 which can be expected to be 12 × 10−9 or smaller. 14 Notes 15 ur(mIPK) ur(h) ur(Mu) ur(NA) ur(mu) 16 012For× 10−9 Peer012 Review× 10−9 12 × 10Only−9 1 (present SI) 17 Y0 0.45 × 10−9 00.45 × 10−9 2 (proposed new SI) 18 Y0 0 0.45 × 10−9 0.45 × 10−9 3 (third choice) 19 × −9 20 Y0.45 10 0004(fourthchoice) 21 395 (1) At present, the mass of the IPK defines the kilogram and the molar mass constant is defined 22 −1 396 to be 0.001 kg mol (exactly). The values of other quantities are measured experimentally or 23 24 397 inferred from experiments by making use of well-known equations: all have identical uncertainties 25 398 to the precision shown. 26 399 (2) The CGPM has resolved to define the kilogram through a fixed numerical value of h, combined 27 400 with the present definitions for the second and the metre and to define the mole by giving a 28 29 401 fixed value to the numerical value of the Avogadro constant in the SI. Consequently the molar 30 402 mass constant and the atomic mass constant would have equal relative uncertainties inferred from 31 403 CODATA 2014 [3], although a last CODATA adjustment will be made a few months before the 32 404 redefinitions are adopted. 33 405 (3) The CGPM could have chosen to redefine the kilogram as in Note 2, leaving the definition of 34 35 406 the mole unchanged, i.e. fix h without changing the definition of the mole. This would nevertheless 36 407 have reduced the uncertainties of the Avogadro and atomic mass constants as shown in the table. 37 408 This is a viable option, however we now have the opportunity to define the mole in a more direct 38 409 and universal way, similar to the proposed redefinitions of other units. 39 40 410 (4) For chemical measurements a system that fixes the molar mass constant, the Avogadro constant 41 411 and the atomic mass constant would be very suitable. Note that fixing the values of any two of 42 412 these constants defines the third through the quantity relation Mu = NA mu. However, in this 43 413 case mu defines the kilogram and the value of the Planck constant acquires a finite uncertainty as 44 414 45 other constants shown in Note 2 and Note 3; but this uncertainty is not considered to be negligible 46 415 by electrical metrologists working at national institutes [35,36]. 47 48 416 Table 3 summarizes the relative standard uncertainties ur on the five quantities most rel- 49 417 evant to chemistry (mass of the artifact international prototype of the kilogram, Planck 50 51 418 constant, molar mass constant, Avogadro constant, and atomic mass constant) according 52 419 to different scenarios (Notes 1 to 4). A zero means that the numerical value of the cor- 53 420 54 responding quantity is exactly defined, or ”fixed”. Non-zero values for uncertainties are 55 421 taken from the CODATA 2014 recommendations [3] and are expected to be reduced some- 56 422 what in a future CODATA adjustment preceding the launch of the new SI. In particular, 57 −9 58 423 the relative standard uncertainty of 12×10 shown in the top row of Table 3 has met the −9 59 424 relative uncertainty of 20 × 10 recommended by the community of mass metrologists as 60 12

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1 2 3 4 425 prerequisite for the redefinition of the kilogram [37]. 5 6 426 7 427 We may note that the community of mass metrologists, as represented by the Consulta- 8 428 tive Committee for Mass and Related Quantities (CCM), will realize the new definition 9 10 429 of the kilogram in either of two equivalent ways: (1) through a device whose operation 11 430 resembles that of an analytical balance (the so-called Kibble Watt balance); (2) through 12 431 13 a determination of the number of atoms in a pure, monocrystal of silicon-28 whose mass 14 432 is nominally 1 kg. The latter technique has an additional component of relative standard 15 −9 433 uncertainty at present equal to 0.45×10 [3], the same uncertainty shown in Table 3, but 16 For Peer Review Only 17 434 this is considered by the research teams to be negligible compared to the total uncertainty 18 435 budgets of either of the two ways to realize the new definition of the kilogram (see [38,39], 19 436 however, the uncertainties given therein refer to older values, now superseded [3]). 20 21 437 22 438 A short description on how to define the kilogram based on the fixed numerical value of the 23 24 439 Planck constant can be found in the Appendix. We describe briefly the crystal density or 25 440 atom counting approach as well as the Kibble balance (Watt balance), both are currently 26 441 exploited to link the Planck constant h to the kilogram. 27 28 29 442 3.2 Consequences for chemistry of the CGPM Resolutions 30 31 443 Consequences for chemistry of the CGPM Resolutions can be evaluated by recalling the 32 33 444 following facts: (a) Relations among quantities (such as Eqs. (1) to (9), (12) and (13)) do 34 445 not depend on any choice of units. As a corollary, changing the definitions of existing units 35 446 does not (and cannot) introduce any new constants; (b) Ratios of two quantities of the 36 37 447 same kind are independent of unit systems and thus immune to any changes of the unit 38 448 systems, and (c) the unified atomic mass unit, also called the dalton [11], is independent 39 40 449 of the SI and therefore unaffected by any changes to the SI. The value of the dalton in SI 41 450 units will have an uncertainty an order of magnitude smaller in the proposed new SI than 42 451 at present. 43 44 452 45 453 We now consider the following quantities (Item 1. to Item 8.), important for metrology in 46 454 47 chemistry (see IUPAC Green Book [11]), and discuss how they might be affected by the 48 455 new SI: 49 50 456 1. Ar(X): relative atomic mass of the element X (for historical reasons called ‘atomic 51 457 weight’ of X) 52 53 54 458 2. Mr(X): relative molar mass of entity X (for historical reasons called ‘molecular 55 459 weight’ of entity X) 56 57 460 3. mu : atomic mass constant; mu is equal to the unified atomic mass unit, mu =1u. 58 59 461 It is used to report masses of nuclides and its value in the SI unit kg is not exact, see 60 13

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1 2 3 4 462 Table 3 Note 2. This unit is a non-SI unit accepted for use with the SI and defined 5 6 463 as one twelfth of the mass of an unbound carbon-12 atom at rest and in its ground 7 464 state. The name dalton, symbol Da, is used as an alternative name for the unified 8 465 atomic mass unit [11] 9 10 466 4. Mu : molar mass constant 11 12 13 467 5. NA : Avogadro constant 14 15 468 6. ma(X) : atomic mass of entity X, with 16 For Peer Review Only 17 ma(X) = Ar(X) mu (14) 18 19 20 469 7. M(B): molar mass of entity B (see Eq. (1), with X replaced by B) 21 22 470 8. n(B) : amount of substance B (chemical amount), with 23 24 25 n(B) = m(B)/M (B) (15) 26 27 471 where m(B) is the mass of a sample of entities B determined by analytical weighing. 28 472 Alternatively, n(B) can be determined from the number N(B) of entities in the 29 30 473 sample, according to Eq. (5), when X is replaced by B. 31 32 474 At the top of the list, Ar(X) and Mr(X) are relative quantities. Their values and un- 33 475 certainties are unaffected by the CGPM Resolution (or any other choices in Table 3). 34 35 476 Specifically, tabulated values in the ‘atomic weights’ of the elements produced by IUPAC 36 477 CIAAW [40] are unaffected by any choice of rows in Table 3. 37 38 478 39 479 From the principle of continuity that is followed when redefining SI units, the present 40 480 values of the quantities will be the same as before but their uncertainties will often be 41 42 481 different. From Table 3, we see that the relative uncertainties of NA and mu will be much 43 482 reduced from present estimates; in fact the uncertainty of NA will be zero in the new SI. 44 45 483 Therefore SI atomic masses (in kilogram) will be known to a reduced uncertainty, as can 46 484 be inferred from Item 6. and Table 3. Specifically, the Atomic Mass Evaluations published 47 485 at regular intervals by IUPAP, in which masses of the nuclides are reported in the unified 48 49 486 atomic mass unit, are unaffected by any choices shown in the rows of Table 3. 50 487 51 52 488 However, we also see in Table 3 that following the revision of the SI the molar mass −1 53 489 constant Mu will no longer be exactly 1 g mol , but will acquire a relative standard un- 54 9 490 certainty of less than 1 part in 10 as inferred from the CODATA 2014 recommendations 55 56 491 of the fundamental constants. To assess the importance of this new uncertainty compo- 57 492 nent, it is useful to look at its effect on the estimate of molar mass (Item 7. above). Since 58 493 59 M(B) = Ar(B) Mu and Ar(B) is unaffected by the CGPM Resolutions, all molar masses 60 14

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1 2 3 9 4 494 will acquire an additional uncertainty component of less than 1 part in 10 . 5 6 495 7 496 Typically, the molar mass is used to estimate the chemical amount n(B) of entity B in a 8 497 macroscopic sample of mass m(B), where m(B) is determined by weighing on an analytical 9 10 498 balance: n(B) = m(B)/M (B) (see Item 8. above). The uncertainty of this calculation 11 499 is dominated by estimates of chemical purity and the accuracy of the analytical weigh- 12 500 13 ing. For most chemicals, variability of isotopic abundances must also be considered for 9 14 501 high-accuracy work. An additional relative uncertainty of 1 part in 10 can be neglected 15 502 [41–43]. The experimental uncertainty of Mu in the proposed new SI was discussed fol- 16 For Peer Review Only 17 503 lowing Eq. (13), where it is seen to be small and unlikely to increase. 18 504 19 505 The ’zero reaction mass’ related to molar mass conservation in a chemical reaction would 20 9 21 506 hence have an additional relative uncertainty of 1 part in 10 . Recalling the reaction of 22 507 Eq. (10), to detect such an additional uncertainty, exactly 44.053 g of oxirane should be 23 24 508 decomposed to yield 16.043 g of methane and 28.010 g of carbon monoxide. Aside from the 25 509 fact that molar masses from standard atomic weights do not have the necessary precision, 26 510 the exact measurement of such masses by weighing in a conventional chemical experiment 27 28 511 seems to be currently unrealistic.

29 512 30 9 31 513 A more fundamental way to think about a relative uncertainty of order 1 part in 10 is to 32 514 consider that it is at this level of accuracy where the assumption of mass conservation in 33 515 chemical reactions begins to break down. For example, exactly 12 g of unbound carbon-12 34 516 35 atoms is reduced to  0  36 × − ΔfH0 (C, graphite) 12 g 1 12 2 (16) 37 M( C) c0 38 0 39 517 when the same number of atoms have formed a graphite crystal [15]. Here Δf H0 (C, graphite) 40 518 is the standard enthalpy of formation of graphite at zero kelvin temperature. The molar 41 2 12 519 mass equivalent of the energy is this enthalpy divided by c . Further division by M( C) 42 0 43 520 gives the mass equivalent relative to the unbound mass of the carbon atoms, approximately −9 44 521 0.66 × 10 . The breakdown of mass conservation in chemical reactions, insignificant for 45 46 522 normal stoichiometry, is one reason not to insist that a chemist’s preference for line 4 47 523 of Table 3 be imposed on the entire community of scientists. While lines 2 and 3 are 48 524 both logically possible, the definition given in line 3 would still require the clarification 49 50 525 appended in 1980 to the present definition of the mole [1,44]. The clarification was needed 51 526 due to the breakdown of conservation of mass, as illustrated in Eq. (16). The definition 52 53 527 ofthemoleinline2isindependentofmass. 54 528 55 529 We conclude that the proposed new SI (line 2 in Table 3) is to be preferred over both the 56 57 530 third and fourth choices in Table 3. 58 59 60 15

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1 2 3 4 531 4 Methodology 5 6 532 To assess and critically review the proposed definitions of fundamental chemical quantities 7 8 533 and their impact on chemical communities, a set of about 100 published articles including 9 534 whole theses and memoranda was identified and classified in the categories “pro”, “contra”, 10 11 535 or “neutral” in reference to the proposed new SI and, in particular, in reference to the 12 536 proposed new definitions of the mole and the kilogram. In these categories, papers have 13 537 been further classified in respect to their relation to general science, metrology or education 14 15 538 issues. Specifically, we identified the following themes: 16 For Peer Review Only 17 539 1. Educational aspects in relation to the mole 18 19 540 2. The Avogadro constant 20 21 22 541 3. Compatibility aspects between molar mass and relative atomic mass (‘atomic weight’) 23 24 542 4. The “entity” concept 25 26 543 5. Further alternative definitions of the mole 27 28 544 6. On the circularity of the current definition of the mole 29 30 545 7. Name of the quantity “amount of substance” 31 32 546 8. Metrological aspects 33 34 35 547 An additional assessment was obtained via the answers obtained from NAO bodies to a 36 548 questionnaire. 37 38 549 39 550 The most fundamental chemical quantity is the amount of substance. Consequently, the 40 551 major part of articles inspected dealt with this quantity and the concept to which it 41 42 552 implicitly refers. Another chemically relevant quantity is mass, as it is related to the 43 553 traditional, and currently accepted definition of the mole. All articles were carefully read 44 45 554 and analyzed with regard to the following questions pertaining the amount of substance: 46 47 555 • Is the quantity of any necessity ? 48 49 556 • What are arguments in favor or against its current definition or name? 50 51 557 • What are arguments in favor or against its proposed new definition? 52 53 558 • Are there alternative new definitions for the quantity or its name? 54 55 559 • mise 56 What are arguments in favor or against the current definition of its unit and its 57 560 en pratique (realization)? 58 59 60 16

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1 2 3 4 561 The papers were distributed among members of the Task Group who individually read 5 6 562 them and collected answers to the aforementioned questions in a table. The result of 7 563 this work is summarized in the following Section 5. Results from the questionnaire are 8 564 summarized and analyzed statistically in Section 6. 9 10 11 12 13 14 15 16 For Peer Review Only 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 17

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1 2 3 4 565 5 Review of selected papers and summary of key issues 5 6 566 raised by the scientific community 7 8 567 Many of the articles analyzed in this work contained similar approaches and contents. We 9 10 568 have focussed on reviewing unique contents and conclusions from selected papers, and this 11 569 review is reported in the present section, where we first address general points raised by 12 570 the scientific community about the proposed new SI and the definition of the mole, while 13 14 571 later giving separate space to specific themes as outlined in the previous section. Where 15 572 appropriate, a critical assessment of the discussion is amended. 16 For Peer Review Only 17 573 18 574 Early in the process of establishing a new SI with relevance to the chemical community, 19 575 Mills et al. [45] presented the argument for the redefinition of the kilogram by fixing either 20 21 576 h or NA. These authors also argued that there are significant advantages moving from the 22 577 current mass definition to a constant based formulation while drawbacks were limited. In 23 24 578 a new definition of the kilogram that fixes either h or NA, the uncertainties of values of 25 579 many fundamental constants (when expressed in SI units) would be reduced significantly 26 580 and significant advantage to the measurement of electrical quantities would also be gained 27 28 581 (see also [35] and Table 3, p. 12). As the single negative effect, they contended that 29 582 the mass of the international prototype mIPK would no longer be known exactly (in kg) 30 31 583 but would have to be determined by experiment. Metrological aspects will be covered in 32 584 Subsection 5.8 below. 33 585 34 35 586 In a subsequent paper [46], Mills et al. provide a comprehensive assessment of the situ- 36 587 ation surrounding the redefinition of the mole, treating it the same way as the kilogram, 37 38 588 ampere and kelvin in terms of need and mandate for redefinition. They specifically ad- 39 589 vocate defining the kilogram by fixing the value of the Planck constant instead of the 40 590 conceptually simpler route of fixing the Avogadro constant and the atomic mass constant, 41 42 591 or the dalton. One reason advanced by these authors is the apparent advantage for precise 43 592 electrical measurements [35], leaving NA ‘free’ to be used for the definition of the mole. A 44 45 593 second reason, for these authors, is that from a more fundamental physical point of view h 46 594 plays a more important role than NA. This is at variance with the view initially presented 47 595 in ref. [45]. The redefinition of the mole in relation to fixed NA (together with h, e,andk) 48 49 596 will have the effect that other constants become exact, for example the Faraday constant 50 597 F , the universal R, the Stefan σ and the molar volume 51 598 52 of an ideal gas Vm at standard pressure and specified temperature. All these constants are 53 599 of major practical importance in chemistry and physics.

54 600 55 56 601 Various comments and concerns related to the proposals for the new SI are assembled in 57 602 a website www.metrologybytes.net [17] for discussion. Details of the proposed revision 58 603 of the SI have been summarized [47]. The proposed explicit-constant formulations of the 59 60 18

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1 2 3 4 604 definitions is said to provide a simple and unambiguous approach applicable both for base 5 6 605 and derived units. Here the rationale for the redefinition of the mole is presented specifi- 7 606 cally as an approach to ”eliminate the present poor understanding of amount of substance, 8 607 which is independent of mass, and its unit mole, which is a unit to count the number of 9 10 608 entities” [47]. Concerns have been raised over the proliferation of correlation among the 11 609 base units in the proposed new SI and it was criticized that units are defined in terms 12 610 13 of a web of interrelated fundamental constants of nature [48,49], and the spatio-temporal 14 611 constancy of such a system could only be tested against other similar systems which would 15 612 not be possible in the case of the proposed new SI, named a ’Zanzibar system’ in ref. [50]. 16 For Peer Review Only 17 613 In ref. [51] potentially serious flaws of the proposed new SI have been listed, ranging from 18 614 ‘wrong definitions of the mole’, the fact that the redefinition of the kilogram relies on 19 615 special relativity and quantum mechanics and that the redefinitions of the SI units are 20 21 616 inconsistent, circular (see also [52]) or even wrong [51].

22 617 23 24 618 In reference to the proliferation of correlation argument, we may expect that correlations 25 619 will remain approximately the same in number as those in the present SI. But this is really 26 620 a non-issue. The proposed new SI assigns fixed numerical values to seven constants when 27 28 621 each is expressed in its SI unit [34]. The Planck and Avogadro constants are among the 29 622 seven. Any other quantity of interest can be measured in terms of one or more of these 30 31 623 defining constants. 32 624 33 625 In reference to the consistency argument, one must distinguish the physical quantity from 34 35 626 a unit used to measure it [53]. In this context, the ’ultimate test of the consistency of 36 627 physics is made by the measurement of dimensionless constants’ [54]. However, all base 37 38 628 quantities are dimensional and ’it is in our best interest to choose some dimensional funda- 39 629 mental constants as the basis of our SI by fixing their numerical values because to test the 40 630 consistency of physics precisely we need units which are intrinsically stable and universally 41 42 631 accessible’ [54].

43 632 44 45 633 If a dimensional is assigned a fixed numerical value, a commonly ex- 46 634 pressed concern is about the consequences that might arise, should the fixed value some- 47 635 how be ‘wrong’. For example, suppose the present value of the Planck constant has some 48 49 636 unsuspected, small error with relation to the mass of the international prototype of the 50 637 kilogram. A consequence might be that a more refined realization of the kilogram af- 51 52 638 ter redefinition might manifest itself as a corresponding change in the calibration of the 53 639 highest-accuracy mass standards. In order to minimize serious consequences from such 54 640 a hypothetical situation, CCM made its recommendations cited above [37]. A pertinent 55 56 641 example that the change in definitions of the mole and kilogram will have no consequences 57 642 in testing physical laws is provided by an experimental test of Einstein’s mass-energy re- 58 643 lation, as discussed in Appendix B of [55]. 59 60 19

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1 2 3

4 644 5 6 645 Every revision to the SI brings advantages to some and disadvantages to others. Criti- 7 646 cism extended to the time-frame and it was argued that the redefinition should come into 8 647 effect only when sufficiently low uncertainties could be achieved for the various constants 9 10 648 needed for the redefinitions [48, 49, 56, 57]. One might question what exactly ’sufficiently 11 649 low’ uncertainties are. Can these even be defined in natural philosophy? As mentioned 12 650 13 above, it is the intrinsic stability of a system of units that counts. From a practical point 14 651 of view, values of fundamental constants are determined from a least squares analysis such 15 652 as ’to discern the best values of fundamental constants based on all available data in order 16 For Peer Review Only 17 653 to avoid discontinuities of the size of a unit after its redefinition’ [54]. 18 654 19 655 The validity of an ‘urgent’ revision of the SI is questioned [57], arguing that the funda- 20 21 656 mental constants h and α might not be constant over time. Furthermore, the numerical 22 657 treatment applied by CODATA to fix the ‘reference’ values of the fundamental constants 23 24 658 at the time of the revision raised some concerns [48, 49]. Various experiments are now or 25 659 will become available in the near future with higher accuracy than current ones, for exam- 26 + 660 ple an optical clock based on Al is more accurate than the Cs-clock [58]; these advances 27 28 661 should be taken into account [57]. In contrast, other authors see all technical requirements 29 662 set by CCU close to being met such that no delay is indicated [59] and the CODATA 2014 30 31 663 recommended values and uncertainties [3] give added support to the latter position. The 32 664 SI being a practical system, the definition of the second will undoubtedly be updated 33 665 to accommodate optical clocks when they become true clocks rather than spectacularly 34 35 666 precise frequency standards, as they are today. 36 37 38 667 5.1 Educational papers concerned about the mole 39 40 668 One of the anchoring concepts in physics and chemistry is that ”matter consists of atoms 41 669 that have internal structures that dictate their chemical and physical behavior” [60]. The 42 670 macroscopic quantities that are used by students and faculty within the practice of chem- 43 44 671 istry contain so many atoms, molecules, ions or elementary particles that they are counted 45 672 in bunches. The unit of such a bunch may be called the mole. Currently, the mole is de- 46 673 47 fined as being that amount of substance which contains as many elementary entities as 48 674 there are atoms in 0.012 kg of carbon-12 [61]. The number of entities in one mole of 49 675 substance is related to the Avogadro constant that is in essence a scaling factor between 50 51 676 macroscopic and microscopic, particulate quantities [60, 62, 63]. 52 53 677 Evidence from peer-reviewed research on teacher and student understanding of the mole 54 678 concept strongly suggests that the current definition is not well understood nor is it well 55 679 communicated in textbooks [62, 64–77]. The classroom contexts of the research with stu- 56 57 680 dents and teachers span the globe. Studies from Lybeck’s lab in Sweden of 30 upper 58 681 secondary school students and 28 teachers of chemistry indicate that few students and 59 60 20

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1 2 3 4 682 only about three out of 28 teachers use the correct SI definition of the mole [67, 68, 78]. 5 6 683 The majority of students and teachers use the notion that one mole equals ‘Avogadro 7 684 number’ of entities, like one dozen equals the number 12. One should note that, despite 8 685 the fact the we all seem to understand what is meant by ‘Avogadro number’, namely the 9 10 686 numerical value of the Avogadro constant (symbol: {NA}), the ‘Avogadro number’ is not 11 687 defined in the SI Brochure [1]. The definition of the ‘Avogadro number’ is implicit in Eq. 12 688 13 (5) which, indeed, is an equation cited in [1]. 14 689 15 690 The outcome space developed from Lybeck’s interviews indicates that students and teach- 16 For Peer Review Only 17 691 ers are well versed in the quantity calculus needed to move back and forth between the 18 692 macroscopic and microscopic world. Given that the learning outcome emphasized is the 19 693 ability to solve quantitative problems, then it is not surprising that the relationship one 20 23 21 694 mole equals 6.022 × 10 entities serves as a proxy for the definition of the mole (see, for 22 695 instance ref. [79]). In his thesis [78], Str¨omdahl analyzes their work in much detail, while 23 24 696 calling our attention to another philosophical problem: as a vast majority of students and 25 697 teachers understand one mole as the unit measure of number of entities, they adapt a 26 698 concept that belongs to discontinuum physics, whereas the conventional definition relies 27 28 699 on continuum physics.

29 700 30 31 701 In a study of 48 first semester college students in the United States, Staver and Lumpe 32 702 found that students frequently failed to define the mole in terms of carbon-12 and pref- 33 703 erentially defined it as a mass or as ‘Avogadro number’ [69, 80]. Teachers in Spain also 34 35 704 exhibited confusion with the concept of the mole and its quantity amount of substance [70]. 36 705 As in previous studies the mole was most frequently defined as a mass or a number of en- 37 38 706 tities. Similar findings emerged in a study of two chemistry teachers in South Africa [72]. 39 707 In a 2002 review Furi´o et al. concluded that great confusion exists among educators and 40 708 textbooks about the quantity ”amount of substance” and its unit the mole [62]. Moreover 41 42 709 students reflect this confusion by seldom using ”amount of substance” and frequently con- 43 710 ceptualizing and using definitions of the mole as a mass or a number (‘Avogadro number’). 44 45 711 Other researchers have analyzed textbooks used to help students learn chemistry. Furi´oet 46 712 al. found in a study of 87 texts used in secondary and university level in Spain published 47 48 713 between 1976 and 1996 that the quantity ”amount of substance” was not introduced 49 714 explicitly in 95 % of the analyzed texts [70]. Frequently the mole concept was connected 50 51 715 to mass, i.e. molar mass, and to the number of entities. Padilla and Furio-Mas reviewed 52 716 30 university level general chemistry textbooks and discovered that 28 out of 30 did not 53 717 use the current definition of the mole appropriately [71]. Twenty-eight out of 30 describe 54 55 718 a mole as being equivalent to the ‘Avodgadro number’ of entities. Pekdaˇg and Azizoˇglu 56 719 semantically analyzed 15 chemistry texts with three from the United States, five from 57 58 720 France, and seven from Turkey; all were used to teach chemistry at the secondary and 59 721 university level [73]. This study found similar omissions and mistakes comparable to 60 21

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1 2 3 4 722 previous findings where the ”amount of substance” was not used, and the definition of 5 6 723 the mole was equivalent to or conflated with molar mass, ‘Avogadro number’, or molar 7 724 volume. The vast majority of documented teaching practices and textbook descriptions 8 725 do not support the current SI definition of the unit mole and the quantity amount of 9 10 726 substance. This difficulty is consistent with the observed conceptual understanding of 11 727 both students and teachers about the mole and the amount of substance. As a result, it 12 728 13 is unsurprising that a majority of teachers and students define the mole as either a mass 14 729 or as ‘Avogadro number’. 15 16 730 On a hopefulFor note some Peer researchers have Review described pedagogical Only methods to help teach the 17 731 concept amount of substance and the mole. Pekdaˇg and Azizoˇglu describe semantic models 18 19 732 to help students understand these concepts at the particulate (microscopic), macroscopic, 20 733 and symbolic levels [73]. The authors suggest that the models could be used as frameworks 21 734 to guide the development of activities. Fang, Hang, and Clarke [65, 81, 82] developed 22 23 735 concept maps to guide the development of activities and to identify relationships between 24 736 the number aspect and mass aspect of the current SI definition of the mole that they argue 25 26 737 need to be made explicit to students. What is lacking in the field is further research to 27 738 determine the efficacy of these new models in facilitating student learning. 28 29 739 Mills et al. wrote in 2006 ”...it is important that the basis of our measurement system be 30 740 taught in schools and universities, it is preferable, as far as modern science permits, that 31 32 741 the definitions of base units be comprehensible to students in all disciplines, a requirement 33 742 that becomes increasingly difficult to achieve as science advances” [46]. The proposed 34 743 re-definition means that the mole is defined as the amount of substance that contains 35 23 36 744 6.022 140 857 × 10 specified entities (using the CODATA 2014 value [3], see also Table 37 745 1). Thus, the new definition would achieve Mills’ goal that the definition of base units 38 39 746 be comprehensible to students and teachers. In fact, research indicates operationally this 40 747 is the definition that many teachers, students, and textbooks already use. The goal of a 41 748 more comprehensible definition is quite likely best achieved by more closely mirroring the 42 43 749 practical definition already in use by most students and teachers. This is in contrast to 44 750 the view expressed in ref. [83] where it is argued that there is no need for a redefinition 45 46 751 because the current definition of the mole is well understood and established for almost 47 752 50 years. 48 49 50 753 5.2 The Avogadro constant 51 52 754 In many articles we read that the key concepts used by chemists are not well understood. 53 755 54 For example, in the article by Wheatley [84], a widespread divergence of opinion about 55 756 the nature of the Avogadro constant is shown: some believe it is a number whereas others 56 −1 757 believe it is a quantity with a dimension of N . Peculiar and hard-to-understand to many 57 58 758 is also the observation that the Avogadro constant cannot be understood without the mole 59 759 and yet the proposed definition of the mole rests entirely on the Avogadro constant. 60 22

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1 2 3 4 760 Arguments have also been raised that the Avogadro constant is not a fundamental con- 5 6 761 stant, rather a conversion factor [85]. This view is also shared by Andres et al. [83]. 7 762 These authors dispute that lack of comprehensibility of the current definition of the mole 8 763 is an argument for a redefinition, since concepts are well understood for them. The use of 9 10 764 quantum metrology is also not considered a valid argument, nor is the dependence of the 11 765 present definition of the mole on the kilogram an issue, since other dependencies exist in 12 766 13 the present SI and will increase in the proposed new SI. 14 767 The Avogadro constant is an important tool used by chemists. Following Perrin [21], it 15 16 768 guarantees theFor access toPeer the microscopic Review reality by tracing Only the microscopic entity, and 17 769 even proves the existence of atoms. It is of little value to argue whether a certain concept 18 19 770 is ‘fundamental’ or not as any such discussions will abound with subjective opinions. Ul- 20 771 timately, one cannot forget that all decisions regarding creation and classification of the 21 772 measurement units are conventions and practical considerations play an important role. 22 23 773 Likewise, the CIPM ultimately decided that a philosophical discussion on the nature of 24 774 the mole was not relevant, the overriding consideration being whether introduction of the 25 26 775 mole would serve a useful purpose [22]. 27 776 28 777 Leonard argues [86] that one does not need to involve the Avogadro constant in any 29 30 778 SI definition and one could simply define the mole in terms of two units of mass: the 31 779 kilogram and the atomic unit of mass (dalton, see Note 4 in Table 3). In particular, 32 33 780 one has to recognize the ‘Avogadro number’ which is equal to the -to-dalton mass 34 781 ratio [87] and then stipulate that one mole is the amount of substance which contains 35 782 the ‘Avogadro number’ of entities. The proposal by Leonard to explicitly recognize the 36 37 783 ‘Avogadro number’ as the ratio of the gram and the dalton is appealing. A definition of the 38 784 ‘Avogadro number’ as being a scaling factor of individual entities was also supported by 39 785 the IUPAC Analytical Chemistry Division in 2012 [88]. Starting with Eq. (1) to (5) and 40 41 786 adding the present definition of the mole [1], it follows that the ‘Avogadro number’ is the 42 787 gram divided by the value of the atomic mass unit, mu, as determined experimentally in 43 44 788 gram (of course with an experimental uncertainty). If we change the definition of the mole, 45 789 as intended in the proposed new SI, so that the ‘Avogadro number’ is exactly defined, the 46 790 relation g/mu is no longer exact but the uncertainty it acquires is negligible. It is easily 47 48 791 shown that this follows from Table 3, Line 2 and Eq. (1) to (5), which are unchanged. We 49 792 also note that the experimental uncertainty of mu in the proposed new SI will immediately 50 51 793 become an order of magnitude smaller than at present. 52 53 794 5.3 Compatibility between molar mass and relative atomic mass (‘atomic 54 55 795 weight’) 56 57 796 Maintenance of compatibility between the molar masses and the relative atomic masses 58 797 (historically called ‘atomic weights’) is concerning to some. In short, some authors main- 59 60 23

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1 2 3 4 798 tain that chemists appreciate the identity between the numerical values of ‘atomic weights’ 5 6 799 and molar masses [85,89–91]. For example, the molar mass of mercury is M(Hg) = 200.592 −1 7 800 gmol and the atomic weight of mercury is Ar(Hg) = 200.592. In the present system of 8 801 units, mole is defined as the amount of (unbound) carbon-12 atoms in 0.012 kg and dalton 9 10 802 is defined as the 1/12 mass of carbon-12 atom. It follows therefore that the molar mass −1 11 803 of carbon-12 is 12 g mol exactly. In the proposed new SI, however, mole is no longer 12 804 13 defined in terms of carbon-12 and consequently there is no stipulation that the molar mass −1 14 805 of carbon-12 remains 12 g mol exactly. This has the consequence that molar masses 15 806 are not numerically identical to ‘atomic weights’ when they are expressed in the SI units 16 For Peer Review Only −1 17 807 of g mol , with a difference between the two numerical values currently being less than 9 18 808 1partin10. Consequently, many contend that the expression relating molar mass and 19 809 ‘atomic weights’, M(X) = Ar(X) Mu (Eq. (1)) will cease to be used if Mu is no longer 1 g 20 −1 21 810 mol exactly. This argument can be reformulated by introducing a ”correction factor”: 22 23 24 M(X) = (1 + κ)Ar(X) Mu (17) 25 −1 26 811 which allows Mu to remain 1 g mol exactly [46]. κ willbezerowithintoanuncertainty 9 27 812 of less than 1 part in 10 . However, it is also argued that the factor κ is annoying and 28 813 should be avoided by incorporating it into Mu [86,91,92]. Equation (17) was a suggestion 29 30 814 proposed in a 2006 article [46]. The suggestion was subsequently rejected by the CGPM [2] 12 31 815 which proposed that, in the new SI, ‘the molar mass of carbon-12 M( C) will be 0.012 32 −1 33 816 kg mol but with a relative uncertainty equal to that of the recommended value of NA h 34 817 just before redefinition and that subsequently its value will be determined experimentally.’ 35 818 We also find Eq. (17) to be misleading, and even illogical, because Eq. (1) must remain 36 37 819 valid in the proposed new SI. See Section 5 of [41] for a simple explanation of how the 38 820 uncertainty of Mu can be discussed in the same way as the uncertainty of any other ex- 39 40 821 perimental quantity if the proposed new SI is adopted. 41 822 42 823 This Task Group dismisses the concerns in regards to the ”incompatibility” between mo- 43 44 824 lar masses and ‘atomic weights’ and has made several observations in this regard. Firstly, 45 825 important requirements that must be met by a new SI is that the relative atomic mass of 46 12 826 47 carbon-12 keeps its value, Ar( C) = 12. Eq. (1) to Eq. (4), which are essential for chem- 48 827 istry, must be unaltered in the proposed new SI. Consequently, the relative uncertainty of 49 −9 828 the molar mass constant is non-zero, ur(Mu) ≈ 0.45 × 10 (see Table 3). 50 51 829 9 52 830 Secondly, a discrepancy in the molar mass at the level of 1 part in 10 is not in the realm of 53 831 concern for chemists. The uncertainty in real samples is dominated by impurities [93–95]. 54 55 832 To date, the most precise chemical measurement is the measurement of the molar mass 56 833 of the highly-enriched silicon-28 in the 1 kg single crystal used to determine the value of 57 58 834 NA. This project has been aided by remarkable financial support from the International 59 835 Avogadro Consortium and, in 2015, the lowest reported relative standard uncertainty 60 24

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1 2 3 9 4 836 of the molar mass of silicon-28 reached 20 parts in 10 [96]. Virtually all chemistry 5 6 837 measurements operate at precision levels several orders of magnitude worse. In addition, 7 838 there are only three elements whose standard ‘atomic weights’ are currently known to a 8 9 839 precision below 1 part in 10 : sodium, fluorine, and phosphorus [40]. Hence, the fact that 9 9 10 840 the molar mass of carbon-12 will now have an uncertainty of less than 1 part in 10 can 11 841 have an impact on the molar masses of only a handful of substances such as F2,NaF, 12 842 13 or PF3. Last but not least, one cannot forget that the current definition of the mole 14 843 stipulates that it applies to unbound atoms. Chemists do not work with unbound atoms 15 2 844 and binding energy of atoms leads to the loss of mass (Δm = E/c0). Consequently, the 16 For Peer Review Only 17 845 mass of chemical substances does not equal the atomic mass the number of entities. 18 846 The molar mass of a crystalline substance X is given by 19 20 0 2 21 M(X, cr) = Ar(X) Mu − Δf H0 (X)/c0 (18) 22 0 23 847 where Δf H0 (X) is the molar cohesive energy of the crystal at zero kelvin temperature [15] 0 −1 24 848 (see Eq. (16)). For a graphite crystal, ΔfH0 (C, graphite) = 711 kJ mol which cor- 25 26 849 responds to a difference between the molar mass of bound and unbound carbon-12, 10 27 850 M(C, graphite) − M(C), of roughly 7 parts in 10 . 28 851 29 30 852 In the current SI, the relative uncertainty of Mu is zero, see Eq. (1). Therefore the error 31 853 in the calculation of M(X, cr) is zero only when mass conservation is assumed in chemical 32 10 33 854 reactions. However, conservation of mass is violated at parts in 10 . Thus, one can have 34 855 identical number of atoms in a gaseous and a solid sample but different masses due to the 35 856 cohesive energy. This underlines the necessary distinction between the concept of mass 36 37 857 and the concept of amount of substance [15].

38 858 39 40 859 In summary, currently we are already faced with the numerical incompatibility between 41 860 molar masses and ‘atomic weights’ due to binding energy without anyone having an issue 42 861 with this. Hence, arguing that the ”kappa”, κ (see Eq. (17)), should be avoided at all 43 44 862 costs [86, 91, 92] is unreasonable. In the proposed new SI it will be correct to state that −1 −9 45 863 Mu =1gmol ; ur(Mu)=0.45 × 10 . The present uncertainty (CODATA 2014 [3]) 46 864 47 will in time become an upper limit when more accurate determinations become available. 48 865 Whereas the current definition of the mole specifies that the carbon-12 atoms referred to 49 866 in the definition are unbound, at rest and in their ground state, no such specification is 50 51 867 needed in the definition of the mole in the proposed new SI. 52 53 54 868 5.4 Introduction of ‘ent’ as the amount of a single entity 55 56 869 The current International System of Quantities (ISQ) distinguishes the number of entities 57 870 B, N(B), and the amount of substance B, n(B), via the Avogadro constant (NA)as 58 59 871 given in Eq. (5). Leonard [86, 92, 97] and De Bi`evre [16, 17] have argued repeatedly that 60 25

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1 2 3 −1 4 872 it would be better to use the reciprocal of NA,ent=NA , an amount consisting of a 5 6 873 single representative entity, as the particulate unit for amount. It is argued that ”major 7 874 simplifications in comprehension” results since the following can then be written: 8 9 n(B) = N(B) · ent (19a) 10 11 12 875 and 13 mol = {NA}·ent (19b) 14 15 876 Mole thus can be regarded as the ‘Avogadro number’ of entities instead of the number of 16 For Peer Review Only 877 the reciprocal Avogadro constant, as it is viewed from the current definition by some [98]. 17 18 878 This requires introducing the unit entity, the amount of single entity (symbol: ent), orig- 19 879 inally proposed in ref. [99], which would be categorized as a unit in use with SI, just like 20 21 880 the dalton. Amount of substance would be understood as a number of entities. In other 22 881 articles, entity was even introduced as the unit of a dimensionless quantity, since atoms, 23 882 molecules, etc. are discrete objects [100, 101]. 24 25 883 26 884 From a different point of view, the use of ‘number of entities’ would spoil the achievement 27 885 gained with the current definition of the mole to resolve the confusion between g-mol and 28 29 886 kg-mol [102]; the term ‘amount of substance’ is used to describe a quantity with three 30 887 conceptually different forms of appearance: chemical mass unit, number of entities and 31 32 888 amount of substance. 33 889 34 890 One could argue that N = n/ent becomes less comprehensible than N = NA n. Introduc- 35 36 891 ing the unit ”entity” would help to clarify that the mole corresponds to the exact number 37 892 of elementary entities; that is also achieved with the proposed new SI, however. Despite 38 39 893 claims to the contrary [97], we note that stoichiometric calculations do not become “more 40 894 easily comprehended” since in practice stoichiometric calculations do not require the con- 41 895 cept of the mole (see Section 2.1). 42 43 896 44 897 Furthermore, the scope of a revision of the SI must avoid step changes and unnecessary new 45 46 898 names. The revision must be to the benefit of all users. Introducing ‘numerosity’ [103], 47 899 ‘avo’ [63], ‘ent’ [86, 99], etc. would provoke endless discussions and finding a consensus 48 900 might be impossible. Alternative names and definitions should only be introduced after 49 50 901 broad consultations. The proposed new SI introduces moderate changes, most of which 51 902 might not even be experienced by the vast majority of users [104]. 52 53 54 903 5.5 Further alternative definitions of the mole 55 56 904 Leonard proposes to redefine the dalton as an exact multiple of the kilogram [97]. Emer- 57 905 son conflates the quantities ‘number of entities’ and ‘amount of substance’ and offers an 58 59 906 alternative version of the current definition: ”a mole is a number of elementary entities 60 26

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1 2 3 4 907 equal to number of atoms in 0.012 kg of carbon 12”. Emerson also contends that the 5 6 908 distinction between n and N is an ”unfortunate result of involving units in definitions of 7 909 quantities” (for the document, see www.metrologybytes.net/opEds2014.php).

8 910 9 10 911 Amount of substance is proportional to a number of entities [1] and understandably many 11 912 often wonder: ”Why is it then not simply a number of entities?” One can indeed concede 12 913 13 that the quantity ‘number of entities’ is logically sufficient and there is no need to intro- 14 914 duce a new base quantity ‘amount of substance’ with its own dimension which has led 15 915 to the definition of the mole and has introduced a new constant: the Avogadro constant. 16 For Peer Review Only 17 916 The decision to consider the amount of substance as a quantity with distinct dimension 18 917 was made in 1971 by the CGPM also in order to facilitate the use of quantity calculus 19 918 by chemists [19] (and refs. [12,13] cited therein), however the proposal was brought to 20 21 919 the CCU jointly by IUPAC, IUPAP, and ISO. Arguments in favor of this decision were 22 920 recalled in the historical retrospective in Section 2.1 above. 23 24 921 25 922 The proposed new SI introduces a change in the definition of mole from a mass basis to a 26 923 number basis but the practical realization of the mole in a routine laboratory work will still 27 28 924 be performed on a mass basis. Consequently, Wheatley finds that the unchanged method 29 925 to realize the mole demonstrates no practical reason to redefine this unit [84]. He argues 30 31 926 that new definitions should only be prompted by better ability to realize the units, which 32 927 is certainly not the case for the mole. The revised definition of the mole does, however, 33 928 divorce the concept of amount from the concept of mass. The routine realization of the 34 35 929 mole is a procedure based on mass equivalence and has normally a lower precision which 36 930 is sufficient, however, for many chemical applications, as outlined in Section 5.3 above, 37 38 931 and also in Section 5.8 below. Furthermore, as the definition of the mass unit kilogram 39 932 will change, keeping the old definition cannot be a solution for the problem raised by the 40 933 author, either. In the New SI this relationship still holds true, however, with a small 41 42 934 uncertainty of no practical disadvantage to chemists. Experiments have been proposed to 43 935 simulate a true realization of the kilogram with very simple means, i.e. with a ’LEGO’ toy 44 45 936 watt balance [105], or with a determination of the atomic mass constant from the crys- 46 937 tallographic structure of an 20 g aluminium cube, which leads to a value for the Planck 47 938 constant [106]. 48 49 939 50 940 Hill et al. [107] compare an ‘electronic kilogram’ (watt balance) and an ‘atomic kilo- 51 52 941 gram’ (based on fixed NA). The watt balance experiment would lead to more precise 53 942 electrical measurements but at the expense of introducing a quantum-mechanical cur- 54 943 rent standard. Fixing h would imply that the Planck constant plays a more central role 55 56 944 than NA; chemists might dispute this and the authors suggest to define the kilogram by 3 57 945 the mass of (84 446 889) × 1000/12 unbound carbon-12 atoms at rest in their electronic 58 946 ground-state, realized by estimating the number of atoms of silicon-28 in a sphere. The 59 60 27

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1 2 3 3 4 947 mole would then simply be the amount that contains exactly (84 446 889) specified ele- 5 6 948 mentary entities; the authors believe that this concept is also easily understandable for 7 949 students. Similar ideas are presented by Khruschov [108] and Si [109]. Both suggest car- 8 950 bon modifications, namely multi-layered graphene [108] or carbon onion structures [109], 9 10 951 that is carbon shells around C60, whereas the size of those shells seems to be precisely 11 952 known. Fox and Hill [110] argue that a perfect cubic structure (fcc) of carbon-12 would 12 953 13 need only one to be described, in contrast to rectangular and parallelpiped or a 14 954 spherical structure. Neither of these proposals have any scientific merit as they rely on 15 955 imagined and yet unrealized molecular structures. Other proposals have been put forward 16 For Peer Review Only 79 17 956 to remedy this apparent pedagogical problem. We have the binary mole (2 , accurate 18 957 to two digits), the empirical mole (23! × (23 + 2/23), accurate to four digits), and the 19 3 958 cubic mole (84 446 889 , accurate to seven digits). Given that the latter is a simple and 20 21 959 accurate representation of the current best estimate of {NA}, one could indeed concede 3 22 960 that stipulating {NA} = 84 446 885 (accurate to eight digits) is perhaps a better way to 23 24 961 define the mole [110]. 25 962 26 963 It was criticized that the current definition of the mole lists the nature of the elementary en- 27 28 964 tities. The term ‘elementary’ should be replaced by ‘of the same kind’ [111]. Furthermore, 29 965 the definition of the mole should not have the same structure as the other four revised 30 31 966 base units in the proposed new SI, since the mole has a different status in physics [111]. 32 33 34 967 5.6 On the circularity of the current definition of the mole 35 36 968 Many have asserted that the current definition of the mole is a seemingly circular definition 37 969 [16]. When rewritten in an explicit constant version, it takes the following form: 38 39 40 970 The mole is the unit of amount of substance. It is defined by taking the fixed 41 971 numerical value of the molar mass of carbon-12 to be 0.012 when expressed in 42 972 the unit kilogram per mole, where the kilogram has already been defined. 43 44 45 973 This definition of the mole appears to be circular because the definition of molar mass 46 974 (mass of one mole of substance) is assumed apriori. The circularity is illusory and is only 47 48 975 due to giving the name ‘molar mass’ to the quantity M = m/n, an unfortunate example 49 976 of conflating names of quantities and units [11]. If this quantity had been called ‘mass-to- 50 977 amount ratio’, the complaints regarding circularity would likely disappear. 51 52 978 53 979 In addition, the definition of the Avogadro constant can be derived from the definition of 54 55 980 the mole without any circularities. There are three ISQ equations that are pertinent to 56 981 understanding any definition of the mole: 57 58 n(X) = N(X)/NA 59 60 28

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1 2 3 4 982 This equation is identical with Eq. (5). 5 6 m(X) = N(X) ma(X) (20a) 7 8 983 9 M(X) = m(X)/n(X) (20b) 10 11 984 Using only the information that the mole is the unit of amount of substance, the first 12 985 equation tells us that n(X) = 1 mol when N(X) = {NA}. By combining the three equa- 13 12 12 14 986 tions, the definition of the mole as stated above also tells us that if X is C, then n( C) 12 12 −1 15 987 =1molwhenN( C) = {NA} = 0.012 kg/ma( C) = 0.001 kg Da . Thus pedagogues 16 For Peer Review Only 988 who teach that the numerical value of the Avogadro constant is the ratio of the gram to 17 18 989 the dalton expressed in gram are being consistent with the present definition of the mole. 19 20 990 5.7 Name of the quantity 21 22 991 In 2009, ICTNS indicated that the name ”amount of substance has been a source of much 23 24 992 confusion and that the greatest effort should be made to change the name ... at the same 25 993 time that a new definition of the mole is approved” [33]. Likewise, the quantity name 26 27 994 ‘amount of substance’ has been criticized as ”not well chosen” [112] and ”is practically 28 995 unknown to most teachers” [70]. Leonard argues that the name for the base quantity 29 996 should be ‘chemical amount’, by analogy with ‘’ [113]. In fact, ‘chemical 30 31 997 amount’ had appeared as the alternative name for ‘amount of substance’ in the IUPAC 32 998 Green Book since 1993 [114]. It has also been said that the name of the quantity ‘amount 33 34 999 of substance’ is too long and should be replaced, for instance by ’numerosity’ [103], by 35 1000 ’ment’ [18], or by ’enplethy’ [115]. 36 1001 37 38 1002 As mentioned in Section 5.4, the scope of a revision of the SI must avoid step changes and 39 1003 unnecessary new names. This Task Group therefore concluded that the name ‘chemical 40 41 1004 amount’ should be preferred. When there is relevant chemical context, one can also talk 42 1005 about the chemical amount of photons or chemical amount of electrons. 43 44 45 1006 5.8 Metrological aspects 46 47 1007 More and more laboratories are currently being equipped with the experimental tools 48 49 1008 of either of the two available experimental protocols to realize the kilogram [38, 39, 116]. 50 1009 Concern was raised that only a few countries in the world will be capable of affording the 51 1010 realization of the kg in the proposed new SI [49]. In the meantime, the groups from NRC 52 53 1011 and NIST have reported excellent uncertainties using Kibble watt balances, 1.5 parts and 8 54 1012 3.4 parts in 10 respectively. NIST expects a further reduction in their uncertainty by 55 56 1013 July 2017. An additional five groups anticipate substantial progress in the near term with 57 1014 their own Kibble watt balances. Several other laboratories are in earlier stages of devel- 58 1015 opment [116]. PTB (Germany) intends to supply primary silicon spheres to laboratories 59 60 29

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1 2 3 4 1016 who may wish to buy them and less expensive secondary standards made of natural single- 5 6 1017 crystal silicon. They will also provide ‘service after sale’ to keep the primary standards 7 1018 primary and to recalibrate the secondary standards. It seems, however, even a situation 8 1019 envisioned in [49] was considered to be acceptable by the CCM experts in mass metrol- 9 10 1020 ogy [37]. 11 1021 12 1022 13 A comprehensive treatise on the subject of redefinition providing desirable qualities for the 14 1023 definition of a unit has been put forward [102,117], with qualities for the definition of base 15 1024 units as follows: definitions should be simple and freely available to everyone; standards 16 For Peer Review Only 17 1025 should be related to stable ‘invariants of nature’; realization of the definitions must be 18 1026 reproduceable and of high precision (best measurements), if precision increases, revising 19 1027 the definitions should be made possible [117]. 20 21 1028 22 1029 Traceability within the SI ensures that measurement results are comparable toconsistent 23 24 1030 with results from different measurement methods. As implied in [102, 118] a primary 25 1031 (direct) method of measurement guarantees highest metrological quality. It is accepted 26 1032 without reference to a standard measuring the same quantity. 27 28 1033 29 1034 What is appealing to some metrologists is that in the proposed new SI, the uncertain- 30 31 1035 ties for the atomic and the molar mass constant become identical once NA is fixed, see 32 1036 Eq. (4) [94, 95, 119]. 33 1037 34 35 1038 Chemical measurements do depend on a realization of the mole through a primary (direct) 36 1039 method of measurement [102, 118]. The mole, however, does not depend on a particular 37 38 1040 method of realization. The use of a primary (direct) method of measurement is important 39 1041 to realize the unit at a high standard, but various secondary methods of realization can be 40 1042 devised. Weighing a sample of material with known purity (and therefore known relative 41 42 1043 molar mass), for instance, is used to determine the amount of substance 43 44 m(X) n(X) = (21) 45 Ar(X) Mu 46 47 1044 and this would be unchanged in the proposed new SI [94]. In Eq. (21), m(X) is the mass 48 1045 of a sample of entity X corrected for impurities; the molar mass constant Mu would have 49 50 1046 an uncertainty in the proposed new SI. As discussed in Section 5.3 above, the change to 51 1047 the experimental accuracy is entirely negligible as explained in Section 5 of [41]. This 52 1048 argument should alleviate the anxiety expressed in [89]. 53 54 1049 55 1050 If one could instead determine the number of entities N(X) of X using a primary method, 56 57 1051 the amount of substance would be given by Eq. (5), n(X) = N(X)/NA, and the amount −1 58 1052 of substance that corresponds to a single entity X (N(X) = 1) would be n(X) = NA 59 60 30

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1 2 3 4 1053 exactly, where NA is the Avogadro constant [119]. 5 6 1054 7 1055 The silicon XRCD experiment that can be used to realize the kilogram in the proposed 8 28 1056 new SI [39] also realizes the value of n( Si) to very high accuracy with no weighing 9 10 1057 involved [41]. Weighing the silicon sphere will give the same result at potentially the same 11 1058 accuracy using Eq. (21). The uncertainty of Mu in the proposed new SI is completely 12 1059 13 negligible even in this extreme case. 14 15 16 For Peer Review Only 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 31

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1 2 3 4 1060 6 Questionnaire 5 6 1061 The letter to NAOs was sent out in June 2014 to the official representatives of IUPAC 7 8 1062 National Adhering Organizations (NAOs) using email addresses as they appear on iu- 9 1063 pac.org. The announcement of this activity was published in the official IUPAC news 10 11 1064 magazine in July 2014 [120] and in September 2014 [121]. In addition, an announcement 12 1065 of this consultation also appeared in August 2014 in the journal Accreditation and Quality 13 1066 Assurance [122], an announcement of this action was posted on the IUPAC NAO forum 14 15 1067 of the IUPAC Discussion Board (at forum.iupac.org) by the IUPAC Secretariat (10 July 16 For Peer Review Only 1068 2014). 17 18 1069 19 1070 In this letter, all the NAOs were asked to answer in written form to the below listed 20 1071 questions and return their document no later than 2014-10-01. 21 22 1072 23 1073 QUESTIONNAIRE 24 25 1074 IUPAC NAOs are hereby asked the following: 26 27 1075 1. Are you (as NAO representing your members) satisfied with the current definition 28 1076 of the mole? 29 30 1077 (a) YES or NO? 31 32 1078 (b) If NO, please specify in a few sentences why you opted for NO. 33 34 1079 (c) If NO, please provide some suggestion on what to change. 35 36 1080 2. Are you (as NAO representing your members) satisfied with the new definition of 37 1081 the mole as proposed by the 24th General Conference of Weights and Measures? 38 39 1082 (a) YES or NO? 40 41 1083 (b) If NO, please specify in a few sentences why you opted for NO. 42 43 1084 (c) If NO, please provide some suggestion on what to change. 44 45 1085 3. Are you (as NAO representing your members) satisfied with the current definition 46 47 1086 of the quantity amount of substance? 48 49 1087 (a) YES or NO? 50 1088 (b) If NO, please specify in a few sentences why you opted for NO. 51 52 1089 (c) If NO, please provide some suggestion on what to change. 53 54 1090 4. Are you (as NAO representing your members) satisfied with the current name of the 55 56 1091 quantity amount of substance? 57 58 1092 (a) YES or NO? 59 60 32

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1 2 3 4 1093 (b) If NO, please specify in a few sentences why you opted for NO. 5 6 1094 (c) If NO, please provide a suggestion for a new name. 7 8 1095 The responses from the following twenty countries were received : Australia, Belgium, 9 1096 Brazil, Canada, Egypt, France, Great Britain, Greece, Hungary, Ireland, , Nepal, 10 11 1097 Netherlands, Norway, Portugal, Slovakia, Slovenia, United Kingdom, United States of 12 1098 America, and Uruguay. 13 14 1099 15 1100 Summaries of all answers are listed in the following Table 4. Some NAOs answered very 16 For Peer Review Only 1101 short, others wrote helpful comments. All answers are available at full length as PDF on 17 18 the IUPAC webpage for download [6]. Inspection of the table shows that the replies from 19 20 21 Table 4: Answers received to the questionnaire which have been mailed to all the IUPAC 22 NAOs and their respective summarized answers to the four questions. 23 Country 24 1971 unit 2014 unit quantity quantity 25 definition definition definition name 26 Q1 Q2 Q3 Q4 27 28 Australia YES YES YES NO 29 Belgium NO NO NO NO 30 Brazil NO NO NO NO 31 Canada YES NO YES YES 32 Egypt YES − YES YES 33 France YES NO YES YES 34 Great Britain NO YES YES YES 35 Greece NO YES NO YES 36 Hungary NO YES NO NO 37 38 Ireland NO YES NO YES 39 Italy NO NO NO NO 40 Nepal YES YES YES YES 41 Netherlands NO NO − NO 42 Norway − YES −− 43 Portugal NO YES YES NO 44 Slovakia NO YES YES YES 45 Slovenia −−−− 46 Uruguay YES YES YES YES 47 48 USA NO NO NO NO 49 Total YES 6 10 9 9 50 51 Total NO 11 7 7 8 52

53 1102 54 1103 the NAOs were diverse rather than tending to any unified opinion either on the proposed 55 56 1104 definition of the mole or in regard to the current definition or the name of the quantity 57 1105 ‘amount of substance’. 58 59 1106 60 33

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1 2 3 4 1107 Some IUPAC members and international organizations felt that the questionnaire was be- 5 6 1108 ing used as a vote on the matter, which was not its aim. The purpose of the questionnaire 7 1109 was simply, however, ”to collect opinions and comments”, if any, by the NAOs of IUPAC 8 1110 and this was clearly stated in the cover letter. 9 10 1111 11 1112 The total count at the bottom of this table shows that a majority is not satisfied with the 12 1113 13 current definition of the unit mole. A slight majority is in favour of the new definition 14 1114 as being proposed in the new SI [2]. As far as the definition of the quantity amount of 15 1115 substance and its name are concerned, there seems to be no clear position. From the 16 For Peer Review Only 17 1116 answers to four questions raised, there was no emerging consensus on the definition of the 18 1117 mole. Opinion on all questions was, in fact, equally divided. 19 1118 20 21 1119 Despite the diversity of ’yes’ and ’no’ answers, the detailed replies show unambiguously 22 1120 that there is a need for a better formulation of what is ‘amount of substance’. Many 23 24 1121 replies also raised numerous technical issues the treatment of which are beyond the scope 25 1122 of this project. The full detail of the answers will be made public at the aforementioned 26 1123 web site of the minutes of the Task Group meeting in Ottawa, in January 2015 [6]. 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 34

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1 2 3 4 1124 7 Conclusions 5 6 1125 7 7.1 Conclusions on the definition of the mole 8 1126 9 The contents of this manuscript are the result of a Task Group work commanded by three 10 1127 divisions of the International Union of Pure and Applied Chemistry, IUPAC (Divisions I, 11 1128 II and V), as well as by IUPAC’s Interdivisional Committee on Terminology, Nomencla- 12 13 1129 ture and Symbols, ICTNS, and IUPAC’s Commission for Chemical Education, CCE. The 14 1130 aim of this work was the preparation of a text containing a summary of facts and opinions 15 1131 about two imminent changes in the International System of units, SI: the new definition 16 For Peer Review Only 17 1132 of the kilogram and that of the mole [2]. Further to this, a critical assessment of these 18 1133 facts and opinions should be made, where appropriate and convenient. 19 20 1134 21 1135 More than 100 published contributions related to this matter were read, carefully ana- 22 1136 lyzed and discussed. Additionally, a poll survey was launched among national adhering 23 24 1137 organizations (NAO) of IUPAC. The Task Group concluded that that the proposed new 25 1138 definitions have been studied sufficiently well to be successfully implemented. The opin- 26 27 1139 ions expressed by members from educational and metrological scientific communities, as 28 1140 well as by scientists practicing chemistry hold that the new definitions are needed and 29 1141 are even highly desirable. A careful analysis has shown that a remaining reticence among 30 31 1142 some chemists with respect to the new definition of the mole is not justified, either in 32 1143 concern over accuracy issues, or in reference to routine work in a chemical laboratory. A 33 34 1144 majority of opinions from the published material analyzed in this work were in accord 35 1145 with the results from the questionnaire study. 36 1146 37 38 1147 Based on its own critical work, the Task Group naturally developed its own consolidated 39 1148 position with respect to the mole. This position is formulated as an epilogue, and appended 40 41 1149 to this manuscript. It might be of help to IUPAC to review the position of the Union with 42 1150 respect to the proposed new SI. 43 44 45 1151 7.2 Conclusions on the name of the quantity 46 47 1152 In the educational literature there is confusion among students, teachers, and textbook 48 49 1153 authors regarding the term ‘amount of substance’. Also the poll evaluation carried out 50 1154 during this work proves that many IUPAC NAOs are not satisfied with the current formal 51 1155 definition of the quantity ‘amount of substance’. Pursuant to the 2009 recommendation of 52 53 1156 ICTNS to the IUPAC Bureau [33], this Task Group discussed a suitable alternative name 54 1157 for the amount of substance. A discussion arose regarding many proposed names for the 55 56 1158 quantity. Based on the papers reviewed in this work, it seems that the name ‘chemical 57 1159 amount’ would be the best choice and that the concept would also be useful when no 58 1160 chemical substances are involved. 59 60 35

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1 2 3

4 1161 5 6 1162 As some authors have reflected, the name ‘chemical amount’ has an analogy to ‘electrical 7 1163 current’. Many have criticized the name amount of substance largely on the basis that 8 1164 it is a three-word name. However, this is a widespread misunderstanding in regards to 9 10 1165 the use of the name amount of substance. The IUPAC Green Book (3rd edition) points 11 1166 out that ”the words of substance may be replaced by the specification of the entity” (see 12 1167 13 Section 2.10 [11]). Hence, in practice we speak of amount of oxygen and not amount-of- 14 1168 substance of oxygen. In that sense, ‘substance’ in ‘amount of substance’ is a placeholder 15 1169 name and ‘chemical amount’ might avoid this misunderstanding. In addition, the adjective 16 For Peer Review Only 17 1170 ‘chemical’ can be omitted when sufficient context permits much like the name ‘electrical 18 1171 current’ is often shortened to ‘current’ in the scientific literature. 19 20 Examples 21 chemical amount of dioxygen is 5 mol, n(O2)=5mol 22 1172 chemical amount of (III) oxide is 2 mol, n(Fe2O3)=2mol 23 24 amount of dioxygen is 5 mol, n(O2)=5mol 25 amount of iron(III) oxide is 2 mol, n(Fe2O3)=2mol 26 27 1173 In fact, ‘chemical amount’ has appeared as an alternative name for ‘amount of substance’ 28 29 1174 in the IUPAC Green Book since 1993 [114]. This Task Group concluded that the name 30 1175 ‘chemical amount’ might be the preferred name among all names suggested so far. When 31 32 1176 there is relevant chemical context, one can also talk about the chemical amount of photons 33 1177 or the chemical amount of electrons. However, amount of photons and amount of electrons 34 1178 are sufficiently well understandable expressions. We finally note that the task of formulat- 35 36 1179 ing a recommendation for a new name for the quantity amount of substance was outside 37 1180 the scope of the present IUPAC Technical Report. Our work might nevertheless serve as 38 39 1181 a basis for a future thorough discussion about this specific question within IUPAC. 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 36

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1 2 3 4 1182 8Acknowledgements 5 6 1183 This work has benefited from fruitful discussions with many bodies and members of IUPAC 7 8 1184 and outside the Union. In particular, we thank Professor Ian Mills, Dr. Franco Pavese 9 1185 and Dr. Barry Wood for numerous exchanges on this matter. We also thank the National 10 11 1186 Research Council of Canada for the hospitality during our Ottawa meeting. We gratefully 12 1187 remember the late Professor Paul De Bi`evre, who inspired many of us with his enthusiasm 13 1188 and critical view on this matter. 14 15 16 For Peer Review Only 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 37

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1 2 3 4 1189 Epilogue 5 6 1190 During several intense discussions conducted by this Task Group in the context of the 7 8 1191 present work, the following text emerged as an honest union of converging opinions. It 9 1192 contains a proposal, by this Task Group, of a new wording for the definition of the mole 10 11 1193 and in that it reflects the 2009 ICTNS final motion that the quantity amount of substance 12 1194 should also be renamed. 13 1195 14 15 16 1196 The chemicalFor amount, Peern, is a measure Review of the number ofOnly specified elementary 17 1197 entities. An elementary entity may be an atom, a molecule, an ion, an electron, 18 19 1198 any other particle or specified group of particles. The mole, symbol mol, is the 23 20 1199 SI unit of chemical amount. One mole contains exactly 6.022 140 857 × 10 21 1200 elementary entities. This number of elementary entities is called the Avogadro 22 23 1201 number.

24 1202 25 26 1203 Further specifications: 27 1204 28 29 30 1205 (1) The chemical amount of a substance B, n(B), is proportional to the num- 31 1206 ber of entities of B, N(B), with 32 33 −1 34 n(B) = NA N(B) 35 36 1207 The proportionality factor is a universal physical constant that is indepen- 37 1208 dent of the nature of the substance. Its reciprocal, NA, is the Avogadro 38 39 1209 constant which is the same for all substances. 40 14 1210 (2) The stipulated Avogadro number {NA} = 6 022 140 857 × 10 , will be 41 42 1211 the numerical value of the Avogadro constant, NA =6.022 140 857 × 23 −1 −1 43 1212 10 mol . The Avogadro constant has the SI unit mol , because the 44 45 1213 chemical amount n is a base quantity with the SI unit mol and because 46 1214 the number of entities, being a number, is a quantity of dimension one 47 1215 (i.e. dimensionless). 48 49 1216 (3) The chemical amount of B, n(B), is also proportional to the mass of B, 50 1217 m(B), with 51 −1 52 n(B) = M(B) m(B) 53 54 1218 The proportionality factor is the reciprocal of the molar mass, M(B), 55 1219 which is a characteristic constant of the substance B. 56 57 1220 (4) The previous definition of the mole implied that the Avogadro number is 58 1221 the ratio of the gram to the dalton, with the value of the dalton expressed 59 60 38

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1 2 3 4 1222 in gram. The historical continuity of the present definition preserves this 5 6 1223 relation to within an uncertainty negligible for practical purposes. 7 8 1224 A few remarks:

9 1225 10 11 1226 The molar mass of any atom or molecule B, M(B), may still be obtained from 12 13 1227 its relative atomic mass (”atomic weight”), Ar(B), from the equation 14 12 15 M(B) = Ar(B) M( C)/12 = Ar(B) Mu 16 For Peer Review Only 12 17 1228 In this equation Mu is the molar mass constant, equal to M( C)/12. Because 18 12 −1 19 1229 the molar mass of unbound carbon-12, M( C), is no longer 12 g mol exactly, −1 20 1230 the molar mass constant, Mu, is no longer 1 g mol exactly. Its uncertainty, 21 1231 however, is of no practical relevance in chemistry, being smaller than 1 part in 22 9 −1 23 1232 10 . Within this uncertainty, the value of Mu remains1gmol .Themolar 24 1233 mass of any atom or molecule B is also related to the mass of the elementary 25 26 1234 entity, m(B), by the equation 27 28 M(B) = NA m(B) = NA Ar(B) mu 29 12 30 1235 and mu is the atomic mass constant equal to ma( C)/12. Mu and mu are 31 32 1236 related with the Avogadro constant through the equation 33 34 Mu = NA mu. 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 39

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1 2 3 4 1237 Appendix – Definition of the kilogram based on the Planck 5 6 1238 constant 7 8 1239 The kilogram, symbol kg, is the SI unit of mass; its magnitude is set by fixing the numerical 9 10 1240 value of the Planck constant. The proposed new SI sets the numerical value of the Planck −34 11 1241 constant, symbol {h},toexactly6.626069Y×10 when expressed in the SI unit for action 12 2 −1 1242 Js=kg m s thereby defining its magnitude (the final missing digits abbreviated by 13 14 1243 Y are still being determined). Thus, in the proposed new SI the same experiments that 15 1244 are currently able to determine the value of the Planck constant will become the primary 16 For Peer Review Only 17 1245 means to realize the new definition of the kilogram based on the fixed numerical value for 18 1246 h. Two kinds of such experiments are currently persued and we give here a short summary 19 1247 of the main ideas linking the Planck constant h to the kilogram. 20 21 22 1248 Crystal density approach or atom counting 23 24 1249 The h-based definition of the kilogram can be realized by what is known as the crystal 25 26 1250 density approach (also known as the atom counting) which relies on the fact that the 27 1251 density of a perfect macroscopic crystal with mass m and volume V is the same as the 28 1252 ratio of mass to volume of its atomic-scale unit cell: 29 30 m ma(X) 31 = k (22) V Va 32 33 1253 The crystal is composed of the X whose atomic mass is ma(X). The 34 35 1254 volume Va of each unit cell contains exactly k atoms (for example, if X is silicon, then 36 1255 k =8the unit cell of silicon is a face-centred, diamond-cubic structure which is the 37 1256 building-block of defect-free, ultra-pure silicon monocrystals). The number of atoms in 38 39 1257 the crystal equals N = m/ma(X). Following Eq. (22), N can be determined from 40 V 41 N = k (23) 42 Va 43 44 1258 where k is known exactly whereas V and Va can be determined to high accuracy by optical 45 1259 interferometry and X-ray interferometry, respectively. Thus, 46 47 48 m = Nma(X) = NAr(X) mu (24) 49 12 50 1260 mu is the atomic mass constant, 12 mu = ma( C), which has the SI unit kg, and Ar(X) 51 1261 is the relative atomic mass (‘atomic weight’) of X, which is of dimension one (i.e. di- 52 53 1262 mensionless),asisN. Multiplying the right-hand side of Eq. (24) by 1 = h/h results 54 1263 in 55 ma(X) mu m = hN = hNAr(X) (25) 56 h h 57 −2 1264 The quantity mu/h is a physical constant with the SI unit m s. It has been measured 58 59 1265 to high accuracy in two different types of atomic-physics experiments [42,43]. In fact, the 60 40

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1 2 3 8 4 1266 relative uncertainty of mu/h is far smaller than the relative uncertainty (parts in 10 )to 5 6 1267 which N can presently be determined. 7 1268 8 1269 Thus, if one can produce a pure crystalline substance of suitable size and shape, it is 9 10 1270 possible to calculate its mass in kilogram traceable to a fixed value of the Planck constant 11 1271 h [106]. The crystal can then be used as a primary mass standard to disseminate the SI 12 1272 13 unit of mass to secondary standards. 14 15 1273 Kibble balance (Watt balance) 16 For Peer Review Only 17 1274 The h-based definition of the kilogram can be realized by a second kind of experiments that 18 19 1275 relies on comparing electrical power to mechanical power (see for example [123]), which is 20 1276 popularized in a form of a LEGO watt-balance [105] or a loudspeaker-based balance [124]. 21 1277 22 23 1278 Consider a horizontal wire carrying an electric current I.IfalengthL of the wire is in 24 1279 a horizontal magnetic field of flux density B oriented perpendicular to the flow of the 25 26 1280 current, the Lorentz force F on the wire will be vertical and is given by 27 28 F = BLI (26) 29 30 1281 In an appropriate geometry, of which there are many, F can be made to compensate the 31 1282 weight G = mg of an object of mass m,whereg is the acceleration of at the 32 1283 position of the object. At equilibrium, 33 34 35 mg = BLI (27) 36 37 1284 This is the principle behind the operation of a modern analytical balance, but the balance 38 1285 response, I(BL/g), is displayed as a unit of mass simply by calibrating the balance with 39 1286 a standard whose mass is traceable to the international prototype kilogram (IPK). 40 41 1287 The Kibble balance eliminates the term BL by a second measurement: The same wire is 42 1288 made to move vertically with velocity v, which causes a tension U to be induced between 43 44 1289 the ends of the wire. Under these conditions, 45 46 U = BLv (28) 47 48 1290 Eliminating BL from Eq. (27), 49 mgv = UI (29) 50 51 1291 where the left side of Eq. (29) is a virtual mechanical power and the right side is a 52 1292 virtual electrical power; hence the earlier name Watt balance, after the SI unit of power. 53 54 1293 The power in both cases is ”virtual” because it is derived by combining results from two 55 1294 different operations, neither one of which involving power generation or loss. 56  1295 By measuring I in Eq. (29) from Ohm’s law, U = RI, 57 58 UU m = (30) 59 gvR 60 41

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1 2 3 4 1296 To determine the mass m to the highest accuracy and at the same to introduce h,allfive 5 6 1297 quantities on the right side of Eq. (30) must be measured as accurately as possible. By 7 1298 eliminating the current I, the electrical quantities can be measured in terms of quantum 8 1299 standards commonly used to obtain maximum precision in resistance [125] and tension 9 10 1300 (voltage) [126] metrology. This also introduces h to the right side of Eq. (30), which is 11 1301 the goal. The quantum standards are based on the quantum Hall effect, first reported in 12 1302 13 1980 by von Klitzing et al. [127] and the AC Josephson effect, first predicted and then 14 1303 observed in 1962/1963 [128, 129]. Both have been perfected over the ensuing years. The 15 1304 AC Josephson effect provides ideal frequency-to-tension conversion, described by 16 For Peer Review Only 17 h U = nf (31) 18 n 2e 19 20 1305 where e is the elementary charge, h is the Planck constant and f is the frequency of 21 1306 22 microwaves which irradiate the two weakly coupled superconductors that form the circuit 23 1307 element. The unit of h/e is V/Hz. The characteristic curve of observed tension across 24 1308 the junction as a function of applied current passing through the circuit reveals quantized 25 26 1309 ”steps” in the tension. The height of each step is hf/2e and the stable tension of the n-th 27 1310 step is given by Eq. (31). 28 1311 The quantum-Hall effect was first observed in semiconductors, where a test sample has a 29 30 1312 width that is much greater than its thickness but much less than its length. At cryogenic 31 1313 temperature, and in the presence of a strong magnetic field perpendicular to the length- 32 33 1314 width surface, a constant current flowing along the length of the sample will create a 34 1315 tension, the ”Hall voltage”, perpendicular to both the current and the magnetic field, i.e. 35 1316 across the width of the sample. The ratio of the Hall tension (‘voltage’) to the current is 36 37 1317 the Hall resistance RH. Constant values of RH are observed as a function of increasing 38 1318 magnetic flux density. These ”plateaus” are described by 39 40 1 h RH = (32) 41 i e2 42 43 1319 where i is the integer defining the i-th plateau. The value of i decreases as the applied 44 2 1320 magnetic flux density increases. The quantity h/e is a quantum of resistance, equal to 45 46 1321 about 26 kΩ, which is the resistance of the highest possible plateau (i =1). 47 1322 Finally, apart from some manageable details discussed in [123], Eq. (30) becomes 48   49 nn i ff 50 m = h (33) 4 gv 51 52 1323 where m will be determined from the fixed value of h, a collection of exact integers, two 53 54 1324 frequencies, a velocity and the acceleration of gravity. Note that e does not appear in 55 1325 Eq. (33), which means that a value of e is superfluous to this method. Exact values of 56 2 57 1326 both e and h are, of course, needed for electrical metrology based on h/2e and h/e . 58 1327 Simplified presentations can be found in [105, 124], with a deeper analysis in [123]. 59 60 42

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1 2 3 4 1328 Acronyms 5 6 1329 This paragraph contains a list of acronmys used in the preceeding text together with their 7 8 1330 english or (when applicable) french translation. 9 10 Acronym English/French 11 BIPM International Bureau of Weights and Measures 12 / Bureau International des Poids et Mesures 13 CCU Consultative Committee on Units 14 15 /Comit´e Consultatif des Unit´es 16 CGPMFor General Conference Peer on WeightsReview and Measures Only 17 /Conf´erence G´en´erale des Poids et Mesures 18 CIPM International Committee for Weights and Measures 19 /Comit´e International des Poids et Mesures 20 CODATA Committee on Data for Science and Technology 1331 21 ICTNS Interdivisional Committee on Terminology, Nomenclature and Symbols 22 IPK international prototype kilogram 23 24 ISO International Organization for Standardization 25 ISQ International System of Quantities 26 IUPAC International Union of Pure and Applied Chemistry 27 IUPAP International Union of Pure and Applied Physics 28 NAO National Adhering Organizations 29 PDF portable document format 30 SI International System of Units 31 /Syst`eme International d’Unit´es 32 XRCD X-ray crystal density 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 43

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