Chemical Education Today
Commentary
SI for Chemists: Persistent Problems, Solid Solutions by Robert D. Freeman
The International System of Units, or SI (1), is a remarkable achievement. It provides seven base units, a scheme for defining other units in terms of those base units, and a set of prefixes that may be applied to the base and defined units to create conveniently-sized units for almost any earthly need. SI, sometimes called the modern metric system, is in routine use throughout the world—except in the United States. However, a well-known aphorism applies, “No system designed and implemented by human beings is perfect.” For chemists, there is significant imperfection in two areas: the unit(s) for mass, and the name and units for “amount of substance”, both of which lie at the heart of chemical logic. The problem with units for mass is simple, straightforward, and easy to correct and is discussed in the next section. The problem with amount of substance, which is the subject of the remainder of this paper, has existed since the term appeared in the official introduction (1971–2) of the base unit “mole”. Numerous papers have been published about this problem. Twenty years ago Dierks (2) reviewed the problem and cited some 100 papers, “mainly (from) German and English language journals”, as representative of about 300 post-1953 papers which he examined. He concluded (correctly) that the situation was confusing and (incorrectly) that the proper quantity for mole is probably “number”. Recently, Mills (3) has commented on the problem and made clear that the situation has not changed significantly. Finally, it is appropriate to note that in the official American edition (4) of the English-language translation of the 6th edition of ref 1, at the end of subsection II.1.1(f ), which defines the unit of amount of substance (mole), there appears this remarkable sentence, “Note that this definition specifies at the same time the nature of the quantity whose unit is mole.” That sentence, which is apparently missing from ref 1, would seem to indicate a clear recognition that amount of substance is a poorly understood and ill-defined physical quantity. An analysis of these problems follows, and some unconventional, but sound, solutions are proposed. Changes in internationally recommended names and units involve several international organizations. For those not familiar with BIPM, CGPM, ISO, IUPAC, IUPAP, and others, these acronyms are defined and the organizations briefly summarized in Appendix 1; relevant URLs are included. The Problem with Units for Mass It is not possible to state the mass of an atom in terms of the number that appears on the periodic table (for example 207.2 for Pb) and in SI units; one is forced either (i) to use “a non-SI unit accepted for use with SI” (Table 7 in ref 1), the “unified atomic mass unit” (u) defined as 1 u = m(12C)12, or (ii) to resort to the cumbersome, awkward “relative atomic mass” nomenclature. It is strange— 16
paradoxical—that this situation should exist at the present stage of development of chemistry, physics, and the SI. The unified atomic mass unit is, of course, the appropriate unit, but the name is obviously too long for convenient use, it is awkward, and it is almost impossible to use with the standard SI prefixes. Given all the other short, oneword names for SI units, the presence of this name almost demands the question, “Why?”, and almost shouts for correction. The name “dalton” (whose symbol is Da) solves the problem and properly honors an influential 19th century chemist who has not been so honored otherwise. Kilodalton, kDa, is useful to, and already used by, biochemists and polymer chemists; millidalton, mDa, and microdalton, µDa, are useful to mass spectroscopists and nuclear chemists. It is instructive to compare these names—millidalton, kilodalton, et cetera—with the following example from the NIST guide for the use of SI (5), “Examples of the use of prefix symbols with eV and u are…and 15 nu (15 nanounified atomic mass units).” CGPM (see Appendix 1) has been adamant that there will be only one SI unit for each physical quantity. In general, this is an eminently sound policy, and for the other base units, the policy has caused no major problem: both global and atomic distances can be measured in terms of the meter; very long and very short time intervals can be measured in seconds; similarly for temperatures and electric currents. But for mass there has been, and still is, the problem that masses of atoms can be compared with greater precision and accuracy than the mass of an atom can be compared with the prototype kilogram. Consequently, it has not been feasible to determine the mass, in kg, of any atom with sufficient precision to define 1 Da = m(12C)12 = 1.xyz…… × 10᎑27 kg. As a result, chemists have operated for a century, more or less, with two necessary mass units: (i) the kilogram, now a base unit of SI, and (ii) the dalton or u, relegated to the status “non-SI unit accepted for use with SI”. If the dalton and the kilogram were commensurable, that is, 1 Da were equal to 1 × 10᎑27 kg, these problems would not exist. But, of course, the kilogram was defined before 1800, before the existence of atoms was established. As the early chemists unraveled the periodic table, their only reasonable choice for a standard of mass was an internal one: first an atom of H, and then later, for a century, an atom of O. During that period, there was no reasonable option for relating the mass of O to the kilogram with sufficient accuracy and precision to justify making a change in either the macroscopic or the microscopic standard of mass. In recent years, metrologists in several national labs have devoted much time to an effort to replace the present definition of the kilogram—the only base unit still defined in terms of an irreproducible artifact rather than in terms of a precisely defined, highly reproducible, natural phenomenon—and simultaneously to decrease the uncertainty in the
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realization of the kilogram. Several of these efforts are directed toward decreasing the uncertainty in the Avogadro constant and others are based on electrical phenomena. A report on the status of these efforts is readily available; the paper by Taylor (pp 181–194) is especially pertinent (6). As meritorious as these efforts are, they are irrelevant to the need for the dalton. Chemists (and most scientists) will continue to express the masses of atoms and molecules in terms of the masses given on the periodic table (PT), that is in daltons, not in yoctograms (1, 5, 7). It is possible, of course, to interpret those masses given on the PT not as real masses in Da, but as relative masses, that is, as numbers with no units. Relative mass, relative atomic mass, et cetera are cumbersome circumlocutions that are confusing to many (especially students), and that lead to further obfuscation such as the so-called “standard molar mass, M ⫺ ° ”(7). Further, the use of relative mass implies the existence of an “absolute mass”. Insofar as I can determine, there is no such thing as an absolute mass. The magnitude of any mass is relative to some chosen standard. There is no inherent difference between choosing one kilogram as the mass of a particular chunk of metal and choosing one dalton as one-twelfth the mass of an atom of carbon-12. After those are chosen, the mass of a liter of water may properly be given as 1.0 kg, and the mass of a molecule of water, as 18.0 Da. In neither case is “relative” needed for clarity; there is no ambiguity if the definitions of kg and Da are known. The only justification for the relative mass nomenclature is CGPM’s insistence that in SI there can be only one unit of mass and consequent refusal to accept Da (or u) as a legitimate unit within SI. The simplest and best solution to the problem of expressing the masses of atoms and molecules in SI units is for CGPM to acknowledge the difficulty in correlating macroscopic masses with atomic and molecular masses and to accept the dalton as a fully legitimate unit within SI. CGPM and BIPM have already taken a small step in this direction by recognizing the use of Da, rather than u, by biochemists (8). At some point it may become possible to redefine the dalton in terms of a newly defined kilogram, without loss of precision in the value of the dalton with respect to m(12C), but that would not negate the usefulness of and need for the dalton as a unit fully acceptable within SI. The only solution that would eliminate the need by chemists for Da is to redefine the kilogram such that m(12C) = 12 × 10᎑27 “new-kilogram”. That would result in 1 new-kg = 1.66 present-kg and create commercial chaos worldwide. Acceptance of Da within SI seems a much better solution. The Problem with Mole The confusion engendered by the present definition of the mole was discussed briefly in the Introduction. Resolution of this confusion requires going back to the beginning. There will be a need for the terms “amount of substance” and “quantity of matter”, abbreviated AoS and QoM, which I take as having the same meaning in general usage. I shall
use QoM for this general meaning and restrict AoS to the usage given in the present SI definition of the mole (1, 7). The use of volume and mass (or weight), or the corresponding words in other languages, to describe a QoM has a very long history and, in fact, may well predate history. Volume was recognized as a property of three-dimensional space by the Greek geometricians and, presumably, by earlier builders and craftsmen. Beginning with Newton, mass was identified as a property of matter and related to inertia and gravity. It is important to note that in these developments, the name that had been used for millennia to describe a QoM (volume, mass) also became the name for what we now call a “physical quantity”. There was no requirement that the same name be used, and it may well have happened without conscious thought about the transfer of names to a new concept. In any case, we have inherited a scheme in which the name mass may refer to a physical quantity (a property of nature), or to a particular object. We seem to handle this dual meaning without undue confusion, and there is no suggestion here that this arrangement be changed. Nevertheless, it is useful to recognize the duality in meaning because it bears on the AoS problem. In the late 19th century, a third way of describing a QoM appeared on the chemical scene: gram-molecular-weight; later, gram-mole; still later shortened to mole. As evidence mounted for the reality of atoms and molecules, it became clear that chemists were dealing with a new (or newly-recognized) property of nature: “the microgranularity of matter”. However, unlike the situation with mass and volume, there was and is no commonly used word that expresses the concept appropriate to the corresponding new physical quantity—a quantity of matter described by specifying the number of identical, individual units that comprise it. A name for this new physical quantity, other than the inappropriate “number of moles”, was not in (general) use until CGPM produced “amount of substance” in 1971. The situation described in this and the two previous paragraphs is summarized in Table 1. The choice of “amount of substance” by CGPM was most unfortunate. It involved the use of old words in a new meaning without adequate guidance as to precisely what was meant and without any indication that the new physical quantity “AoS” refers to a “new”, or previously undescribed (by CGPM) property of nature, the microgranularity of matter (1). The only way to resolve the AoS–mole problem in a clear, clean manner is to start anew. First, there must be clear indication that we are dealing with a property of nature— the microgranularity of matter—that is not recognized elsewhere in SI units, particularly in the base units. Second, the name given to the physical quantity related to microgranularity must be new—unencumbered by established meaning(s) (Table 2). My suggestion for this new name is “posos”, from the Greek, meaning both/either, how much/ many?, or “tosos” meaning so much/so many. The definition of posos or tosos is “the QoM and energy in a specified number of ‘microgranules’, that is, in a specified number of
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Commentary Table 1. Descriptions of a Quantity of Matter (QoM): Historical Historical usage: Describe a QoM by giving this
“Recently” given this name as a physical quantity
Related property of nature
Volume
3-D space
Mass
Inertia, gravity
Amount of Substance
Granularity of matter
Volume Mass (weight) “Number of g-mol-wts”a (later, “number of moles”) a
gram-molecular-weights
specified elementary entities (atoms, molecules, ions, electrons, other particles, or specified groups of such particles); the entities are understood to be identical, or effectively so.” The meaning of tosos (so much/so many) seems closer to CGPM’s meaning of AoS. However, we need not only a new name but also a new label for the corresponding dimension to be used along with the traditional mass (M), length (L), time (T), electric current (I), et cetera. To avoid confusion, posos (P) seems a better choice than tosos. Posos may not be the best name for this physical quantity, but it does satisfy these essential criteria: (i) short, (ii) easy to pronounce, (iii) amenable to endings (e.g., posar, posarity), (iv) previously unused in any related contexts, and therefore (v) precisely definable without confusion from older meaning(s). In the forty years since Guggenheim (9) first suggested “amount of substance”, there have been numerous suggestions of names to replace that phrase; these are discussed briefly in Appendix 2. From the current emphasis on working with or interpreting processes in terms of individual atoms and molecules, it seems logical that if we are to describe a QoM in terms of the number of microgranules therein, the base unit should be one microgranule—one specified elementary entity. It seems equally logical that the name for this base unit should be “monon”: one monon is the posos in one specified elementary entity. The symbol for monon could be either mn or mnn; below, mn is used. Choosing monon as the base unit for posos would displace mole, a unit the chemist must have and one that must be completely legal within SI. What should we do with mole? For many years, CGPM had a category of SI units
called “Supplementary Units”; there were two: radian and steradian. More recently (1995), CGPM has decided to include these units in the table of “SI Derived Units with Special Names and Symbols”, and the “Supplementary Units” table no longer exists (10). However, that name seems to describe exactly what dalton and mole are: units that do not fit into the carefully (re)constructed SI scheme but which are seriously needed by a large group of scientists and which must be usable routinely with other SI units, without resultant quibbling by SI purists. Therefore, I suggest resurrecting the Table of Supplementary Units, to contain the definitions of the units dalton and mole. What Do These Proposed Changes Accomplish? 1. At present we can write “The mass of a fluorine atom is 19.0 u”, but we can not write “The mass of F is 19.0 u/atom”; atom is not a unit. With dalton and monon available, we can write “the mass of F is 19.0 Da/mn”, and “the mass of F2 is 38.0 Da/mn”; the statements clearly imply that in the first case monon refers to an atom of F, and in the second, to a molecule of F2. 2. With the units Da/mn available, there is no need for the cumbersome nomenclature involving relative masses. One simply gives the mass of the entity in question as xxx Da/mn or xxx Da, as appropriate. 3. The confusing and awkward name “amount of substance” has been replaced by a previously unused name that is precisely defined, is short and is easy to use in terms such as posos (or posar) concentration.
Table 2. Descriptions of a Quantity of Matter (QoM): Modern Property of nature
Related physical quantity
Dimension
Describe a QoM by giving its
Volume
L3
IDa and volume
Inertia, gravity
Mass
M
IDa and mass
Granularity of matter
Posos
P
Posos onlyb
3-D space
a
ID = identification; composition is not required
b
Specification of a posos requires both the number and the type of monons; complete description may require the state of aggregation (19)
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Chemical Education Today
4. Posos is clearly identified as a dimension (P) to be used with mass, length, time, et cetera. 5. With monon available, the SI unit for the Avogadro constant becomes mn/mol, instead of the confusing mol᎑1 , which seems to imply that 1 mole = 6.0p23 (11) and seems to support the idea that mole is a number. Elimination of this confusion also provides a simple solution to Rocha-Filho’s concern (12) about distinguishing between Avogadro’s constant and Avogadro’s number: Avogadro’s constant, Av (not NA, which implies a number), is 6.0p23 mn/mol; Avogadro’s number, NA, is the numerical value of Avogadro’s constant, 6.0p23. In most usage, Avogadro’s number probably should be replaced by Avogadro’s constant. 6. Regardless of changes that may befall the definitions of mole and dalton in the future, the base unit monon should be secure for all time, as should be the case for a base unit.
Recommendations for CGPM, ISO/TC12, IUPAC, and Others 1. In ref 1, Table 1. SI Base Units: replace amount of substance and mole (mol) with posos and monon (mn). 2. Define posos: the quantity of matter in a specified number of specified elementary entities. 3. Define monon: one monon is the posos of one specified elementary entity. 4. In supplementary comments, make it clear that, unlike (most) other SI units, posos is based on the microgranularity of matter, that it refers to a quantity of matter described by the number of specified elementary entities in it, and that its dimension is neither M (a mass) nor 1 (a number) but P (posos). 5. In a new “Table 2. Supplementary Units”, Define NA: NA is the number of 12C atoms in 0.012 kg (12 g) of 12C.
would be somewhat analogous to the redefinition of the meter in terms of the speed of light.
Recommendations for the ACS and its Division of Chemical Education It is possible, but unlikely, that CIPM, CGPM, and ISO/ TC12 will be persuaded by the logic of the above recommendations and adopt them rather quickly. Realistically, most of the members of these organizations are not chemists and may well not appreciate the problems that the discussed deficiencies in SI create for chemists (and physicists, etc.). If these recommendations or modifications are to be adopted, almost certainly they will need to be promoted by interested, knowledgeable, and respected organizations. The current Committee on Nomenclature, Terminology, and Symbols of the American Chemical Society and a new Committee on Units and Quantities in the Division of Chemical Education should take the lead in recruiting the support of such organizations as the American Association of Physics Teachers, American Physics Society, American Institute of Physics, IUPAC, and IUPAP for these recommendations. An Alternative Solution I am reluctant to pose this alternative, which I consider much less logical than the solution(s) described above, yet given the importance placed by many chemists on the mole being a base unit, I feel it necessary to propose a solution that retains that base unit and preserves most of the other features proposed above. The alternative is this: replace AoS with posos, as above; leave the base unit of posos as the mole; in the proposed new table of “SI Supplementary Units”, define dalton (Da) and monon (mn). Acknowledgments I thank Paul Epstein (OSU, Foreign Languages) for consultations and suggestions about the names posos and tosos, and George Gorin (OSU, Chemistry) for many years of discussion, sometimes heated, about the topics herein.
Define dalton: 1 dalton (Da) = m(12C)12.
Appendix 1. Relevant International Organizations
Define mole: 1 mole (mol) is the posos of N A specified monons.
BIPM, CIPM, CGPM: These exist as a result of the 1875 Convention du Mètre (in the United States, usually written as: Treaty of the Meter) and subsequent amendments. Some fifty signatory states have delegated to these organizations responsibility for defining and maintaining standards for units. The General Conference for Weights and Measures (CGPM) is the diplomatic body, with delegates from all member states, which meets every four years and acts on proposals presented and recommended to it by CIPM, the International Committee for Weights and Measures. CIPM is the scientific body that meets annually, has one member from each of 18 states, and oversees the work of BIPM, the International Bureau for Weights and Measures. BIPM was established in 1875 by the original Treaty and is located in a Paris suburb; it has its own building(s) with offices and laboratories. For more, see http://www.bipm.fr (accessed Oct 2002).
Append these comments: • From these definitions, 1 mole contains NA monons, and 1 g is equivalent to NA Da. • The Avogadro constant, Av, may be written as NA mnmol or as NA Dag. • The value of Avogadro’s number, NA, is (at present) determined experimentally. If and when NA is determined with sufficient accuracy and precision, it may be convenient to define NA as that high-precision value, thereby fixing the number of monons in a mole and defining the dalton and m(12C) in terms of the (perhaps newly-defined) kilogram; these changes
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Commentary ISO: ISO is not an acronym, but a name to be used worldwide, regardless of the acronym in the local language, for example in English, IOS for International Organization for Standardization. ISO is a network of the national standards institutes of some 130 countries, with a coordinating headquarters in Geneva, Switzerland. A member body of ISO is the national body most representative of standardization in its country. The member body in the United States is ANSI, American National Standards Institute. The work of ISO is done primarily by Technical Committees; TC 12 is the Technical Committee responsible for recommendations concerning “Quantities, Units, Symbols, …”, which are published in ISO 31/0 through ISO 31/13. For more, see http://www.iso.ch (accessed Oct 2002). CODATA: Committee on Data for Science and Technology, of the International Council of Scientific Unions (ICSU). Members of ICSU are IUPAC, IUPAP, et cetera. The review and adjustment of values of the fundamental physical constants has been done under CODATA auspices for many years (13, 14). For more, see http://www.codata.org (accessed Oct 2002); for values of the constants, see http://physics.nist.gov/cuu/ (accessed Oct 2002). IUPAC: International Union of Pure and Applied Chemistry deals with matters of interest to all fields of chemistry. It is perhaps best known for the IUPAC Nomenclature Rules—for both organic and inorganic chemistry, the so-called Blue Book and Red Book. The Green Book (7) provides recommendations for quantities and units for chemists. The primary members of IUPAC are the National Adhering Organizations, one from each member country; for the United States the NAO is the National Academy of Science, not the American Chemical Society. An individual may be an affiliate member. For more, see http://www.iupac.org (accessed Oct 2002). IUPAP: International Union of Pure and Applied Physics is an organization for physics and physicists, similar to IUPAC. For more, see http://www.iupap.org (accessed Oct 2002). ICSU: International Council of Scientific Unions is an attempt to coordinate the efforts of many unions such as IUPAC and IUPAP. For more, see CODATA. Authority of these organizations: BIPM and CGPM have statutory authority by virtue of the Treaty of the Meter and subsequent amendments. All of the others are independent, voluntary organizations. There is no hierarchy among them. None has authority superior to another; they have no legal or enforcement authority. Some European scientists have argued (e.g., at IUPAC meetings) that ISO has a higher authority than IUPAC; ISO’s Web site indicates otherwise. In some countries, certain standards have been given the force of law, but that is a result of action by that country’s government and has no standing outside that country. Further, while it obviously would be desirable for all of these organizations to agree on specific standards (e.g., names for units), there is no such requirement, except, of course, all are expected to endorse CGPM’s definitions of the SI base units and coherent derived units.
Appendix 2. Other Names Suggested to Replace AoS As noted above, AoS was suggested by Guggenheim (9), apparently as a direct translation of the German Stoffmenge. Since Rocha-Filho (15) has provided, in this Journal, references to many of the suggested replacements, those references are not repeated here. Many of the proposed names imply that mole is a number—
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number, cardinality, numerosity, numerousness, numerity, and ontcount—all are unacceptable on the grounds that mole is not a number. Kell (16) has suggested “psammity”. My primary objection is that the “silent p” would cause difficulties in spelling and pronunciation. Gorin (17) has suggested three names—“metromoriance”, “chemiance”, and “chemical amount”, in that order chronologically. The first is derived from Greek and means “to measure small particles”; the meaning is excellent, but the word is much too long for routine, everyday use. Chemiance suffers from the attachment to chemi- of the suffix -ance, none of the several meanings (18) of which combines coherently with (the meaning of ) chemi-. Chemical amount is the only one of the many suggested replacements that has received any recognition from the standards organizations (7). My view is that chemical amount is an improvement over AoS, but only an incremental one. Most of the criticisms of AoS apply equally well to chemical amount.
Literature Cited 1. BIPM (see Appendix 1), The International System of Units, 7th ed., 1998. 2. Dierks, W. Teaching the Mole, Eur. J. Sci. Educ. 1981, 3 (2), 145–158. 3. Mills, I. M. Metrologia 1997, 34, 101–109. 4. Taylor, B. N., U.S.A. Editor The International System of Units (SI), NIST Special Pub. 330. U.S. GPO: Washington, 1991. 5. Taylor, B. N., Guide for the Use of the International System of Units (SI), NIST Special Pub. 811. U.S. GPO: Washington, 1995. (Downloadable in Adobe Acrobat PDF format [423 kB] from: http://physics.nist.gov/Document/sp811.pdf.) 6. Metrologia 1994, 31 (3), the entire issue. 7. IUPAC (see Appendix 1), Quantities, Units, and Symbols in Physical Chemistry, 2nd ed., prepared for publication by Mills, I.; Cvitas, T.; et al. Blackwell, Oxford, 1993; p 41, footnote (6). 8. Ref. (1), p 106, Table 7, footnote (c). 9. Guggenheim, E. A. J. Chem. Educ. 1961, 38, 86–87. 10. Ref. (1), pp 127–128. 11. 6.0p23 = 6.0 × 1023; 1.6n27 = 1.6 × 10᎑27. Freeman, R. D. J. Chem. Educ. 1978, 55, 103. Peckham, G. D. J.Chem. Educ. 1997, 74, 64. 12. Rocha-Filho, R. C. J. Chem. Educ. 1992, 69, 36. 13. Cohen, E. R.; Taylor, B. N. The 1986 Adjustment of the Fundamental Physical Constants, CODATA Bull. 1986, 63, 1–49. 14. Mohr, P. J.; Taylor, B. N. J. Phys. Chem. Ref. Data 1999, 28, 1713–1852, and Rev. Mod. Phys. 2000, 72, 351–495. 15. Rocha-Filho, R. C. J. Chem. Educ. 1990, 67, 139–140. 16. Kell, G. S. Nature, 1977, 267, 665. 17. Gorin, G. J. Chem. Educ. 1982, 59, 508. 18. Webster’s New World Dictionary, 2nd College ed.; William Collins & World Publishing Co., Inc.: Cleveland and New York, 1974. 19. Ref. (7), pp 46–47.
Robert D. Freeman is retired from the Department of Chemistry, Oklahoma State University, Stillwater, OK 74078, and affiliated with ENODY Unlimited, 116 South Kings Street, Stillwater, OK 74074;
[email protected].
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