The Mole and Avogadro's Number
Roberl M. Hawthorne, Jr.
Purdue University North Central Campus Westville, lndiona 46391
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A forced fusion o f ideas for teaching purposes
In the last two or three decades there has been a growing tendency among teachers of chemistry to abandon the definition of the mole as a mass of material which happens to react with 16 g of oxygen, and to teach instead that the mole is Avogadro's numher, N, of molecules. There has even been a move to do away with the pedantic distinctions of gram-atoms, gram-formula weights, gramions, and so on; and to describe the mole as N units of any sort which can he clearly defined and is of approximately atomic or molecular size. The advantages of defining the mole as N particles are well-known to anyone who has taught from this viewpoint. Stoichiometry is immediately removed from the realm of apparent whimsy ( W h y should 27 g of aluminum react with 3 X 35.5 g of chlorine, but with only 1% X 16 g of oxygen?) and placed on a sound footing of equal numben of reacting units, or simple multiples thereof. The equivalent is easily explained as a mole, N, of charges (nonredox), or a mole, N, of electrons given off or taken up (redox). All of electrochemistry is clarified when the faraday is defined not as the magic number 96,494 (coulombs), but as a mole of electrons. And so on and so on. Enormous areas of quantitative chemistry are organized and clarified for the student by the concept of the mole as a standard number of reacting units. Because of the breadth and simplicity of this idea, those of us who teach it this waylnay be somewhat startled to find that things were not always so. Specifically, the mole and the Avogadro number N have been connected clearly, consistently, and definitionally for only about the last thirtv. vears. " . and almost exclusivelv in textbooks of general chemistry. Individual classroom practice mav carrv the connection further back. but the written record shows only three decades. An even more astonishine corollarv of this observation is that there are textin the 1960's and 70's which do not make book: and anv connection between N and the mole.. . - narticular . even some few which do not mention or use N a t all. These are the conclusions of the present study which are documented below. History of Avogadro's Number and the Mole
That the connection between N a n d the mole came late is less surprising if one examines the history of the two conceuts. The mole. bv that name.. amears to have been .. introduced by ~ s t w a l das late as the beginning of this centurv (1). The relation of the mole to a standard number ofbarticles seems to have begun about this time also. By 1922 we find Trautz insisting (2)l
relation to 16 e of oxveen. " - . he is closer here to a modem definition of tge mole than anyone else would be for a t least another decade. He has recoenized. in this and other passages, the importance of the mole as a standard number. He almost defines the relation between N and the faraday, referring to the electron as an "Elektrizitatsatom" (3), but stops short of making the numerical connection. This treatment is advanced for the time. Prior to Ostwald's definition and Trautz's usage--and indeed long after, in many places-the mole was defined only in relation to 16 g of oxygen and treated only as a combining weight. This is nothing more than an extension and refinement of Cannizzaro's views of 185940 (41, which clarified the diatomic nature of the oxygen molecule, etc., and gave accurate molecular (combining) weights, but showed no interest whatever in absolute numbers of particles. It is a chemist's approach, the quotidian concern of men who must go into the laboratory and actually weigh out quantities of reactants. It is fair to say that most of the chemists and textbook writers who choose this auuroacb to ex.. plain molar quantities are not seriously interested in how many particles they are working with, provided only that there are none left over after reaction. If the mole is a chemist's convenience. Avo~adro'sN is the brainchild almost exclusively of physicis&. A recent review in this Journal (5) has discussed the nineteenth-century derivations of N, from Loschmidt's (1865) onward. The names associated with the determinations are those of physicists-Kelvin, Boltzmann, Rayleigh; and in all cases the N determined is the Loschmidt N (number of molecules in 1 cm3 of a gas under standard conditions-a physical, not a chemical constant), not the Avogadro N which refers to one mole.2 This situation persists into the uresent centurv. Physicists' names continue to appear, and the methods are mostly physical: relating the number of aloha-disinteprations in radium samples to the a c c u m u ~ a t i oof~ the resulting helium gas; measuring the charge of the individual electron and comparing it with Presented before the Division of C h e m d Edurar~onat the l W r h Yat~onal.\Irering. Arncr~uanChemical Society, Chicaw IIlinois, September 14-18,1970. l"From our establishment of the concept of molecular weight it now follows that in I mole of any giuen material there must always be contained the same number of molecules. This numher is called the Loschmidt (formerly the Auogadro) number, and has the value 6.18 X
Aus unserer Festsetzung des Begriffs Molekulargewicht geht jetzt hervor, daas in I Mol eines beliebigen Stoffs st& gleiehviele Molekeln enthalten sein miissen. Diese Zahl heisst die Loschmidt sche (fraherAuogadro sche) Zahl, und . . . hat den Zahlwert: 6,18.102s Molekeln in 1Mol Even though Trautz goes on to define atomic weights in 282
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IPSmolecules in 1 mole."
ZThe distinction between the Avogadro and the Loschmidt N used in this paper reflects the predominant American usage. I am indebted to Professor E. Broda of the Institut fiir Physikalische Chemie, University of Vienna, for painting out that in Germanspeaking countries the number of molecules in 1 mole is usually (though not invariably)called the Loschmidt number.
the charge of a whole mole of electrons, the (chemically derived) faraday; measuring all variables except N in Einstein's equations for viscosities of suspensions of particles, etc. The Avogadro N and the gram-molecule (later mole) began to be associated a t roughly the turn of the century, and the research literature of the neriod 1900-1920 abounds in determinations and re-determinations of the constant (6). But let us he clear about the seauence of ideas in these' papers, for it is central to the presknt study. The men working at this time have a prior definition of the mole (gram-atom, gram-molecule, etc.) as a weight of material which reacts with 16 e of oxveen:. the Avoeadro " numher is appended to this definition almost as an afterthought: "Oh, yes. and this is the number of atoms in it . . . . " 6 i v i a l as i; sounds, the inversion of this sequence to define the mole as N particles is all-important from the pedagogical point of view, as discussed above. This redefinition was not to he clearly articulated for nearly half a century after the first associations of N and the mole.
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Nand the Mole Associated: Sources and Methods Evidence in support of this assertion has been gathered by a survey of texts and treatises from 1891 to the present; by review of the chemical education literature of the last fifty years; and by spot-checking the popular scientific literature, hooks, and periodicals for the last two decades or so. This last source can be eliminated from consideration immediately, as the popular literature appears seldom to have considered N or the mole separately, and never together. The hooks examined were the holdings in general and introductory inorganic chemistry of five major libraries: the chemistry library of Harvard University; the chemistry library of the University of Massachusetts, Amherst; the Chemists' Club library in New York; the Kemper Library of the Illinois Institute of Technology, Chicago; and the John Crerar Library, Chicago (holdings examined in part). A few texts from my own lihrary were added, as well as half-a-dozen or so treatises in general, inorganic, or physical chemistry from the libraries mentioned above. Altogether 104 books were examined. This semi-random selection process seems to have provided a reasonably thorough sampling: by the end of the survey, books began to turn up which had already been examined at another lihrary. T o determine the use of Avogadro's numher N and how well it was related to molar concepts, each book was evaluated in its treatment of five specific points
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Figure 1. Total number of c k m i s t r y texts examined, by dlcades of publication dates: showing portion of total which give numerical value for Avogedro's number N: and smaller porfion which use N pedagogically.
1) Presentation of
2) 3) 4) 5)
a numerical value for N A (or NL, the Loschmidt number, particularly in older texts) Use of numerical N in explaining atomic and molecular weights (AW/MW, in Figure 2) Relation of N to the gram-molecular volume in gases (GMV, in Figure 2) Use of numerical N in explaining stoichiometry, including solution concentrations (STOICH) Use of numerical N in ex~lainine - Faradav relationshios and equivalents (F)
decade-group is a (somewhat subjective) average of the four preceding hars. This average is carried over to Figure 1 as the double-shaded "Use N" portion of the graph. I t should be noted that the numbers of hooks shown in Figure 2 will neither sum nor average exactly, as the population represented in any given bar does not necessarily include the same individuals as that in the bar next to it. The table shows the national origins o f t h e hpoks examined, again by deqades. Although American texts seem to have led the way in using N for teaching purposes, no reallv firm conclusions can he drawn here. hecause the sampling was done, after all, in American libraries. In addition to the texts renresented in Firmres 1 and 2 a numher of treatises, as described above, were examined and eliminated from further consideration. These (except for the earliest, dated 1899) give values for N, sometimes with a little review of methods, hut do not use it pedagogically a t all. Two other eliminations should be mentioned also. The first of these is intermediate-level texts such as analytical or organic texts, which are dismissed because as a group they are not concerned in any direct way with N a n d the mole. The second elimination is made with more regret. Recollections and reminiscences of colleagues indicate that N and the mole were often more clearly connected in the classrwm than in texts (sometimes even by the au-
The results of these evaluations are presented in graphic form in Figures 1 and 2. Figure 1 is a summary of findings by decades, showing (in the bar for each decade) the total numher of hooks examined; the number within that total which give a numerical value for N ; and the smaller number, still within the total, which use N thoroughly throughout the text. Figure 2 shows where this last number comes from, in the four decades in which N is used thoroughly. The hars in each decade-group indicate the numher of texts which use N in explaining each of the points listed above; the "Overall" bar at the end of each
Figure 2. Number of chemistry texts, by decades, which use N in explaining various paints of chemistry. See text for explanation.
National Origin of Texts and Treatises Examined, by Decades Number of
Total Decade 1891-1900 1901-1910 1911-1920 1921-1930
Books
Exammed 3 6 5 15
American 0 3 4 9
English
German
Other
1 0
1 3
Russ. 1
11 13 1951-1960 1961-1970
14 28
12 15
0 6
1 4
Totals
104
67
12
14
Russ. Russ. Fren. Russ.
1 1 1 1
Swed. 1 11
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th& themselves). It is tempting to pursue this line of inquiry, but to do so systematically would mean constructing an oral history quite beyond the scope of this paper. N a n d the Mole Associated: Discussion and Conclusions
Figures 1 and 2 show an association of Avogadro's number N and the mole (as defined by the use of N in explaining mole-centered concepts) which has grown up over the last thirty-five years or so. Note that this generalization refers to the pedagogical use of N, not to its mere mention. The latter has remained roughly constant since the first reference found, in 1913 (7), with an average of about 70% of all texts presenting a numerical value for N. Use of the number, on the other hand, has increased irregularly from its first instance in 1938 (8), with an average of about 30% of texts thoroughly connecting N and the mole over the last two decades. We might note in passing that that first usage is by Joel Hildebrand-which should come as no surprise to anyone. The most frequent and consistent use of Avogadro's N, as shown clearly in Figure 2, has been in explaining the faraday and the electrochemical equivalent. This is also the earliest use, with the faraday as N electrons stated almost explicitly in some books in the teens and 20's, before it actually appears in Figure 2. It is interesting to speculate that the ease of this association of N and the faraday may arise from the fact that it is not hindered by the semantic blocks contained in the terms gram-atom, gram-ion, etc. At any rate, the connection grew up long before N and the mole were similarly associated. This chronology of the increasing connection of N and the mole in texts is roughly paralleled in the literature of chemical education, as found in this Journal; hut the record is much scantier here. In 1933 Brown (9) expressed the remarkable view that "Avogadro's number . . . is not primarily the number of molecules in a mol, but . . . the ratio of a gram to an atomic weight unit." By 1961 no less an authority than Guggenheim defines the mole as "the amount of substance containing the same number of molecules (or atoms or radicals or ions or electrons as the case mav be) as there are atoms in 12 erams of 12C" (11: italics;n o;iginal). Between these twostatements is " ' e jlittle showine the develo~mentof Gueaenheim's clear-cut association: In 1941 ~ G d r i e t hand Copley (11) put forward a definition of the "gram-electron" (i.e., N electrons), a step
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in the right direction. In 1942 a letter from Coolidge (12) a t Duke University reported that he was teaching the mole as a number-to medical students! A brief flurry of articles and letters (13) followed Guggenheim's definition in 1961. Other than this the record is empty of explicit connections. In summary, then, the textbook and chemical education literature show a growth in the use of N as a definition for the mole, beginning some time in the 1930's; with a mild quantum jump in the use of N for thorough discussion of mole-associated concepts around 1950. At present about 415 of the general chemistry texts examined give a value for N,and about 113 use it thoroughly. I t is to be hoped that this trend continues. At the same time we note that 213 of the recent texts are using N less than completely in explaining the mole; and more important, that about 115 of them fail even to mention the number. This is truly surprising. Even granting the right of a teacher or writer to choose his material and present it in his own fashion, we may still observe that this is an exceedingly peculiar thing to leave out. Let me add in closing that I regard this study as openended and unfinished even now. I will welcome further information filling the gaps in my data; and I would he most interested to learn of the priority of some other author over those cited above. Finally, I wish to express my thanks to the librarians at the various institutions whose collections I examined, for their assistance. Literature Cited (11 Ttautz. Mar, "Lehrbueh der Chemie," Vol. I. "Stoffe.'' de Gmyter. Berlin and Iripzig. 1922; p. 56. (2) Ref. ( I ) , 0 . 57. I31 Ref.(IJ. p.273. 14) Cannizzara, Stanidso. '"Sunfa di un e o m dl Fllonofla ehimica," Nuovo Cimanto. 7. 321 (Iffi8): English translation: Alomhie Club Reprints. No. 18. Edinburgh. 80,"
151 Hevfhorne.RobertM. J r . J C H E M EDUC.,47.751-5 119701. (6) See, for example. Reinganum. M.. Ann. Physik. 28, 142 (1939); Bsuer, E.. and Moulin. M., Radium, 7. 372 l19111; Einstein. A,. 2. Ekklmchem., 13. 41; 14, 235, and Ann. Physik, I.591 l19111; Penin. J.. A n n chim. phys.. 18. 5 (1910): C o m Z ?end., 147. 530: 119, 477, 549 (19101: Millikan, R. A,. P h w . Reuieru, 2 121,io9 119131: among many othen. (7) Noyes, William A,. "ATexthmkofChemlstcy." Holf.NewYork, 1913. (81 Hildebrand. Joel H.. "Prineiplas of Chemistv." 13d d l , Macmillan & Co.. Now York, 1 9 3 8 . p ~ 2L.5ML. . I91 Broan,F. E.. J . CHEM.EDUC., 10,308l1933). (10) Guggenhoim.E.A.,J . CHEM. EDUC.,38.86II8611. (11) Audrieth. L.F..endCoploy.M.J.. J . CHEM. EDUC.. I8.373l194l). 112) Coolidge,ThomasB.. J . CHEMEDUC.. 11.143 (19421. I131 Lo., Shiu, J. CHEM. EDUC., 38, 549 (19611: Copley, G. N.. J . C H E M EDUC., 38. 551 (1961: Cohen, Invin. J . CHEM. EDUC.. 38. 554. 555 119611: Foy. John R.. J.CHEM.EDUC., 38,554(19611.
