Mole and Chemical Amount A Discussion of the Fundamental Measurements of Chemistry George Gorin Oklahoma State University, Stillwater, OK 74078 Teachers and students continue to report difficulties with the measurement unit called "mole". Although some teachers have claimed that this unit is extraordinarily difficult to understand, they have not made clear what the alleged difficulty might be. Is the mole really more difficult to comprehend that other '%asic"units, such as second, kilogram, and ampere? The aim of this article is to demonstrate that mole and the corresponding quantity are not exceptional, although they are different in one respect. The familiar quantities called volume and mass measure a piven "amount" of matter in terms of units that can be seek or otherwise sensed; in the International System of Units (SI), the pertinent units are called, respectively, cubic meter and kilogram. The sensations provide a qualitative understanding of what is being measured, and then a more precise measurement can be obtained with the help of some appropriate .. . instrument. In the case of mole, this context is lacking, because the unit measures the relative number of atoms, or of particles derived from atoms, in comparison with those present in a standard. and it is not ~ossibleto enumerate atoms exoerimentally. Consequently, mole is always encountered in the context of calculations. One measures mass, or, in some
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cases, volume, and then one proceeds to "calculate moles", by means of some appropriate conversion factor. Arational discussion of this subject has been hindered by the fact that, up to now, we have not had a generally accepted designation for the quantity that is measured in moles. In the SI definition for the mole1 the quantity is called "amount of substance", but this phrase is ambiguous, on the one hand because amount has diverse meanings and, on the other hand, because mole may be used to measure entities, such as Na* ions, which cannot, by themselves, make up a "substance". In this article, we shall henceforth use the term chemical amount for the "molequantity", instead of amount-of-substance, and, in a subsequent section, this choice will be justified. Chemical amount is not merely a calculational device. I t is determined by experiment, just as mass andvolume are. But. in the usual case. the reauired ex~erimentshave been don; by others. Once we know the f o r k l a of a given com-
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fo ole is the amount of substance of a svstem which contains as ~~manv elementaw entities as there are atoms in 0.012ko of - carban- - ~12. ~ h e the n md e s Lseo,me e emenrary en1 ties must be spec11 ed ana may be atoms mo ecL es, tons e ectrons, other pan c es, or specified groups of such particles'' (Ref.1). ~
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pound and the pertinent atomic weights, all that remains to be done is calculations. The Relationship between Atomic Weight and Molar Mass Atomic weight is a unitless quantity. On the other hand, molar mass, MA,is defined following equation:' (molarmass) =MA= d n = (mass)l(chemicalamount) (1)
and, consequently, its numerical magnitude depends on the units used. The SI definition of mole' can be paraphrased as follows: Male is the magnitude of chemical amount that consists of the same number of constituent partides as exactly 12 g afearbon-12. In that substance, the particles are atoms, but mole also can be used to measure any amount of ions, molecules, or combinations of these entities, which can be represented by a chemical formula.
It follows from this definition that the molar mass of carbon-12 is 12 g!mol. Dividing this molar mass by the factor (1g!mol) gives us the number 12, which is the atomic weight; in order to convert atomic weight into molar mass it is only necessary to reverse the process. This relationship has been pointed out explicitly by Gorin (2) and by Nelson (31, and it is, of course, true in general, i.e., any measurement of a quantity can be converted into a number by dividing into it a unit magnitude of that quantity. Students might reasonably ask: But what is the point of going through these operations? The reason is not readily apparent, because most chemists have been conditioned to using a specific SI unit of molar mass, the gram per mole, (glmol), to the exclusion of others. However, there is no such limitation, as is demonstrated by the following example -for HzO, all of the following statements are correct: MA=18.01 glmol= 0.01801kg/mal= 1.801x
lo4 pgimmol
and so on; the units can be changed a t will, by using the appropriate conversion factors. The calculation of chemical amount from eq 1 is completely straightforward. For example, in an amount of Hz0 that has a mass of 100 g: (chemical amount) = n = mlMA= 100 gl(18.01glmal) = 5.55 mol The Relationship between Chemical Amount and Volume I t is a well-known experimental fact that volume is not conserved in chemical reactions. For example, in the reaction of hydrogen with oxygen the volume of water vapor formed is only 2!3 a s large as that of the original substances, and a much greater wntraction takes place as the vapor condenses to liquid. These results are explained by Avogadro's Hypothesis and the Kinetic Molecular Theory (see below). According to this theory, equal volumes of gaseous substances contain very nearly the same number of constituent particles, a t a given temperature and pressure. Thus, the reaction of hydrogen and oxygen may be represented by the equation:
2H2+ 0, + 2H,O i.e., 2 volumes of hydrogen, containing some unspecified number of Hz molecules, react with 1 volume of oxygen, and hence half that number of Oz molecules, to form a number of H90 molecules that also occupv .. two volumes in the gas phase. The kinetic molecular theory is based on the assumption that the volume of the component particles is negligible, 'The letter " M is overworked as a symbol, and the subscript "A" is used in this arlicle to help distinguish molar mass from other quantities.
and that they exert no attractive force upon one another. This is only an approximation, so that, as the volume and temperature are decreased, the pressure actually deviates more and more from what the theory predicts. However, in the case of small molecules the deviations are slight. At 373 K and 1.01 bar (= 1.00 atm) the actual volumes of hydrogen, oxygen, and water vapor are in the ratio of 2:1:2 within a few percent. The kinetic molecular theory states that, for the socalled "ideal gas", the chemical amount, n, is related to the volume, V, the pressure, p, and the thermodynamic temperature, I: by the equation: n = pVIRT
whereR is a constant, 0.08314 (L.bar/K.mol). So, if we set n = 1mol, T = 298 K, and p = 1bar (= 0.987 atm), we get that the molar volume: V, = nRTlp = 24.8 Wmol this value is changed to 22.7 L a t 273 K, or 22.4 L a t that temperature and 1.00 atm (4). In other words, all gaseous substances have approximately the same volume, 25 L, if the latter is reduced to 298 K and 1.0 bar by the ideal gas equation. For chemical reactions involving only liquids and solids, the volume change cannot be predicted, but it is relatively small. Choosing the Reference Substance In principle, any element can be used as reference and its atomic weight can be assigned any numerical value. But to do so in the absence of a consensus would result in utter confusion. The first tables of atomic weights were compiled by Dalton in 1800-1810. and the reference he used was hvdroeen. to which he iissi&ed nn atomic weight of 1. On the bags of lofic alone, this choice cannot be faulted. Hut 1)alton's valugs were based on imprecise analytical data, and he made the wrong mess about the formula for water, which he assumed tohe "HO. As a consequence, his tables were soon abandoned in favor of those compiled by Berzelius (5). Berzelius introduced the idea of using oxygen as the reference, which is a better choice, from the chemical point of view, because oxygen forms binary compounds with most of the other elements. He got the crucial formula for water right, HzO, and hence the correct ratio for the atomic weights of 0 and H; however, he assigned to them values quite different from those used today, 100 and 6.24, respectively. Moreover, he believed that hydrogen and oxygen were made up of atoms, and that Avogadro's Hypothesis did not apply to gaseous compounds. Consequently, he did not have an adequate basis for establishing a unique and self-consistent set of atomic weights. This development did not take place until after the famous Karlsruhe conference of 1860. There, chemists debated the distinction that had to be made between the concepts of atom and molecule, and, thanks to the effective advocacy of Cannizzaro, in the next few years most chemists came to accept the Kinetic Molecular Theory as one of the axioms on which chemical theorv is based. The merit was not entirely Cannizzaro's. At about the same time, the work of Kroning, Clausius, and Maxwell was instrumental in forging the crucial link between Avogadro's Hypothesis and Newton's mechanical ohiloso~hv.One of the results was the establishment of a set of atomic weights which are close to those in use today. However, two additional refmements had to be added. The first was conseauent uoon the discoverv that oxveen. ... . as well as most other t h n e n t s , are mlxtures ofisotoprs. In all such cases, thc atomic welght is merely the we~ghtrd Volume 71
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averaee .. of the weights .. of the constituent isotooes. . . and its value depends on the isotopic composition. In order to have an exact definition of atomic weinht. - . the standard should be a single nuclide. At first, chemists and mass spectroscopists could not agree on which nuclide to choose. Eventually, however, the resoective International Unions of Pure and Aoolied chemistry and Physics agreed to resolve the probfem by chaneine the reference substance to carbon-12. Soon afterwards, & 1971, mole was admitted into the SI, and, as we have seen, the definition is based on this new standard. In comparison with that standard, the molar mass of oxw-e n atoms is 15.9994 dmol. The ratio 0.0006116, or 0.04 . parts per thousand, is less than the uncertainty of chemical analysis, and, consequently, the change did not affect previously determined values of physiwchemical properties. However, the new standard provides an exact basis for describing the relative masses of individual nuclides, obtained by mass spectrometry. These data are now being measured with a precision of nine or more significant figures (6). Naming the "Mole Quantity" Why have chemists been so dilatory in agreeing on a name for the "mole quantity"? The unit "mole" was introduced into chemistry around 1900 by Ostwald, and he originally defined this unit in terms of gram. Gram is a unit of mass; but what is the mole a unit of? Ostwald did not say;3 however, several years later, he did make it clear that the concept of mole should be linked to the ideal gas.' At any rate, in the period between 1900 and 1970, most chemists were quite content to use the unit mole, and to refer to the corresnondine auantitv " bv " the nhrase "the number of moles". Concern about fmding a better designation did not arise until after mole had been admitted into the SI, when it became evident that the chemical community was not eager to embrace the term amount-of-substance. Early on, Kell proposed the name "psammity" (91,and many other suggestions followed; the most recent is "numerousness" (10). This proliferation of proposals did not help to solve the real problem, which is to get a consensus on using a single designation, or two a t most. Of the quantities that correspond to the SI base units, only three have distinctive proper names: length, mass, and time. Three others are denoted by a noun plus an adiective: electric current. thermodvnamic temoerature. and iuminous intensity. The a d j e c t i ~ ~an i sessential part df the name:. e.a.. understood to - . current, by itself, is not ~enerallv be the kind of that'is measured inamperes. Chemical amount has the same lineuistic form: the modifying adjective intimates that the amount in q;estion is "
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3"...themolecular weight of a substance, expressed in grams, shall henceforth be called mole [. . . das in Grammen augedruckte [. . .I Molekulargewlcht eines Stoffessol1 fortan ein Mol heissen]" Ref. 7). 4'7hatamount of any gas that occupies a volume of 22414 mL in normal conditions is called one mole [Eine solche Menge irgendeines Gases, welche das Volum von 2241 2 ccm im Normalzustand einnimt nennt man ein Moly (Ref.8). 'A high-precision,but nevertheless only approximate,value of the number of particles in a mole has been determined:6.0221367~loz3; the rounded value 6 x loz3 should be adequate for most purposes. Metrologists prefer to call the product of this number times the unit (moi-') the "AvogadroConstant'(cf.Ref. 1, p 81).
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not a mass or a volume, but an attrihute of matter that is utilized by chemists for their peculiar purposes. Recently, the IUPAC Commission on Phvsicochemical Quantities. ~~-~ Symbols, and Units has recognized chemical amount as admissible svnonvm for amount-of-substance. and this may be help& in the future; but it is too eahy to tell whether this term will eain eeneral acceotance (11). In any case, it should-be ccl;?arfrom thehforegoi'ngdiscussion that the true meanine of the "mole auantitv" is fixed by the ideal gas equation.- ole is the base SI init of the quantity represented by n in that equation. Althoueh havi;lga diitinitive name for this quantity is a convenikce for discourse, it makes no fundamental difference what the quantity is called.
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Why the SI Mole is not a Number Some educators have argued that students would understand the meaning of mole more easily if one explained to them that mole is "like a number". This simile may be enlightening, but only if it is carefully qualified; if it is not, it may well add to the students'confusion. What do we mean by "like a number", in the present context? In his little book on quantities and units, McGlashan reminds us that "anv ~hvsicochemicalauantitv can be defined by a careful specification of the procidure that is used in measurine two instances of lthatl auantitv" (12). Teachers should be careful to point out that'the of measuring chemical amount has nothing whatsoever to do with enumerating the particles in either the sample or the standard, and that the magnitude of that number makes no difference in stoichiometric calculations; all that matters is that the number should be the same in one mole of every s u b ~ t a n c e . ~ If one is not counting particles, what is it that one does in order to ascertain that the mass of one mole of oxygen is 16.00 g? The pertinent experiments are those that show that 16.00 g of oxygen combines with 12.01 g of carbon to form a well-characterized compound, and that, to form the other oxide. the mass of oxvwn reauired is twice as laree. Moreover, the densities of"&ese G o gaseous compounds are, respectively, (28132) = 0.875 and (441321 = 1.38 times as large as that of oxygen. That is why we assign the formula CO to the former oxide, and COz to the latter. In other words, chemical amount is determined from the stoichiometrv of chemical reactions. The relative number of atoms inGolved in the reaction may be deduced from such measurements, but one cannot proceed in the opposite sense, because, as was stated a t the outset, there is no direct experimental procedure by which atoms can be enumerated.
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Literature Cited .~..~~. 1. Mills,I.; Cvltak. T.;KallayN.:Homann,K.; Kuehiteu, K,Quonfitks, UnitsfsondSymbols in Physicol Chamistry; Odord: Blackwell, 1988,p M. 2. Gotin,G. J. Chem. Educ 1984,61,1045. 3. Nelson, P.G . J. Chem. Educ 1890.67.628 4 . Freeman, R. D.; Gorin, 0.J Chem. Edur 1988,65,1044. 5 . C f .Idhe, A. J. 'lhDeuplopmml ofModern Chamktri: New York: Harper. 1964.Ch. 4.6. 6. Ref. 1. pp9CL98. 7. Ostwald, W.GrundliNm dor onorgonischn Chemie, L e i p ~ i gEngelman", 1900,p ~
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8. Osrwald, W.D~undrissderal@emewn Chemie 5th ed.: Dresden: Steinhprr, 1917, n M 7
9. Kell, G . S. Nature 1977,267,663. 10. Roeha-Filho. 8.C. J. Chem.Educ. 1990.67,139. 11. Cvita5.T. Chem. Intemav. 1992,14.1W. ~~
