Roberl M. Hawthorne, Jr. Purdue University
North Central Campus Westv~lle,Indiana 46391
Avogadro's Number: Early Values by Loschmidt and Others
M o s t students of general chemistry will recognize the Avogadro number, N, as the number of molecules in a mole, or in 22.4 liters of a gas under "standard conditions" (T = O°C, P = 1 atm); or perhaps just as the number 6.02 X loz3. Some may be familiar with the related Loschmidt number-the number of molecules in a cubic centimeter of an ideal gas under standard conditions, with a currently accepted value of 2.70 X 1019. Few if any can say where these mysterious numbers come from, or most particularly, how they were first derived in the nineteenth century, without benefit of modern instrumentation. Nor is i t easy for the curious t o find out. Josef Loschmidt (1821-95) is sometimes given (deserved) credit for the first determination of the number, but in most texts with no more than a passing nod. Other nineteenth century figures and their determinations are rarely mentioned. The standard histories of science are of little help. Dampier (I), for example, has only this t o say: "The actual number of molecules in one cubic centimetre of a gas at 0°C and atmospheric pressure was first calculated from the kinetic theory by Loschmidt in 1865 as 2.7 X 1019." This is neither satisfying nor accurate, as the N which can in fact be calculated from Loschmidt's figures is 1.83 X Partington's four-volume series on chemical history does not discuss Loschmidt's method, and indeed has little t o say about any method prior to Rutherford and Geiger's turn-of-the century work with the alpha-decay of radium. If one turns t o treatises on kinetic molecular theory, a curious phenomenon emerges. I n the first decade of this century there is an almost complete discontinuity in attitudes and approaches t o the determination of N. In 1909 Millikan determined the charge of the electron, which by comparison with the known charge of the Faraday enabled the first truly direct evaluation of N. Texts written prior t o this event generally give some kind of review of pre-Millikan methods; those after ignore them almost completely. The obvious reason for dropping the discussion of nineteenth century methods is simply that they were indirect and therefore not likely t o give great accuracy. Behind this reason, however, there lies a considerable conceptual difference between nineteenth and twentieth century determinations. This points t o a possible second reason, rooted in nineteenth versus twentieth century habits of thinking, which may explain why the earlier methods were dropped with such extraordinary alacrity. Presented in part a t the Third Great Lakes Regional Meeting,
American Chemical Society, June, 1969.
Nineteenth versus Twentieth Century Determinations of N
Twentieth century determinations of N have the common characteristic that they all compare some bulk property of matter with a measured property of the individual atom or molecule. Dividing the first of these values by the second yields a figure for the number of molecules in the bulk. Thus, the Millikan method above divides the charge of an entire mole of electrons (the Faraday) by the charge of one electron, and comes out with N. The commonest recent method, the X-ray crystal-density (XRCD) determination, essentially compares the volume of a unit cell with the volume of an entire crystal, the quotient being the Avogadro number N. All the twenty-odd methods which lie chronolo$cally between these carry out some comparison of bulk versus individual molecular property I,-,' A
Determinations of individual molecular properties were of course not available to nineteenth century investigators. Hence they could compare bulk properties only, and values for N were arrived a t by chains of speculation and deductive reasoning of a type quite absent in recent work. Herein lies the conceptual difference between the two centuries' approaches. It is temptingly easy t o ascribe this difference to instrumentation alone. The sixty years since RfilliBan's work have certainly seen a vast refinement of equipment and technique, and a concomitant increase in precision of determinations of N. But if one compares the tone of the nineteenth versus the twentieth century papers on the subject, a marked difference in temperament shows up as well. The latter papersindeed, most of the current research literature-rarely go more than one step beyond their data in drauing a conclusion. Chains of deduction, while not absent in some areas, are not found in determinations of N. And rare indeed is the physicist or chemist \rho dares speculate in print on the philosophical implications of his work, in this or any other area. The nineteenth century, by contrast, was quite at home with deduction and speculation, in and out of the research literature. Boltzmann, whose work in kinetic molecular theory touched on the matter at hand, dared t o cross philosophical lances with no less a person than Schopenhauer (3). Ostwald, who rejected strict materiahsm and favored energy as the ultimate reality, was the darling of spiritualist groups until he recanted and came back t o the kinetic molecular theory. Brief though it is, Loschmidt's paper considered below contains speculations on the nature of the ether and on the distinction between living and non-living matter, both arising from his conclusions about the size of molecules. Volume 47, Number 1 1 , November 1970
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Bearing in mind, then, this nineteenth century penchant for deduction, as well as the fact that nineteenth century calculations for N were of necessity indirect and deductive, we can ascribe a double importance to Loschmidt's work. On the one hand we can note its deserved historical priority; on the other, and more interestingly, it can serve as a paradigm for all deductive nineteenth century methods of determining N. Loschrnidt's Calculation
Here then are Loschmidt's arguments as they appeared in his 1865 paper (4), titled "Zur Grosse der Lnftmolekule." Loschmidt begins with the mean-free-path expression derived by Maxwell and modified by Clausius (6) I = 4/aNrlsa in which N = molecules/cmxin a gas a t O°C, 1atm pressure (Loschmidt number), 1 = mean free path, and s = diameter of the molecule. Rearranging, he obtains
In this expression, since gas volume is unity (i.e., 1cma), 1/N may he regarded as the number of cm3 containing one molecule, and is called the "molecular gas volume." On the right of the equation, rlsz/4 is the volume of the cylinder swept out by a molecule between collisions, and is called the "molecular free-path volume." The rearranged Maxwell-Clansius expression now states that the molecular gas volume is exactly 51/3 times the molecular free-path volume. This relates measurable macroscopic quantities to molecular dimensions, but some of these are measurable in theory only, in 1865. So Loschmidt rearranges once more
(This rearrangement is implicit, and not actually expressed by Loschmidt.) Here, as3/6 is the actual volume of one molecule, and NasS/6 is the actual volume of N molecules, completely condensed, with no interstitial space. Since N molecules in the gas phase occupy exactly 1cma,we may regard Nssa/6 divided by unity as a "condensation coefficient," expressing the ratio of the actual volume of condensed gas molecules to the volume they take up in the gas phase. This value is represented by the symbol 6. We now have which says that the molecular diameter is exactly eight times the mean free path, multiplied by the condensation coeficient. Loschmidt has finally got to quantities which can be measured or estimated. The condensation coefficient can he calculated to a fair first approximation by dividing a substance's liquid volume by its gas volume at standard temperature and pressure. Differences in packing geometry indicate that this will give values too high by factors from 1.17 to 1.91, with even greater deviations for non-spherical molecules; but the order of magnitude will be correct. Loschmidt cites a few known condensation coefficients, then observes unhappily that "(f)or air, which is not condensable, the condensation ratio unfortunately 752 / Journal of Chemical Education
cannot be determined directly; and air is exactly the one substance for which the mean free path is known with any certainty." An approximation is available, however. Since 1841, H. Kopp had 8een engaged in a study of the specific volumes (molar volumes, in present-day terminology) of various liquids. He had developed empirical values for the specific volumes of various atoms which, added up, gave those of compounds-rather in the fashion in which one can add equivalent weights of elements to obtain that of a compound. Using Kopp's methods, with modifications, Loschmidt calculates a density (from specific volume) for air as if it were a compound containing 77% nitrogen, 23% oxygen. He makes corrections for packing and temperature, compares the resulting figure with the density of gaseous air, and ultimately obtains a condensation coefficient of 0.000866 for air. (This section of the paper contains a great deal of close reasoning relating to Kopp's method and LoSchmidt's modification of it; to packing considerations; and to the legitimacy of calling air a compound. It is supported by some five pages of observed and calculated specific volumes at the end of the paper, apparently included to holster certainty in the all-im~ortantvalue for air.) Maxwell, in the paper cited above (6),had given a value for the mean free path of air molecules, derived from his expression for the internal friction of gases, with experimental values supplied by Stokes. Loschmidt prefers more recent calculations by Bessel and 0. E. Meyer, however, and uses an average value of 0.000140 mm. This gives s
=
8 X 0.000866 X 0.000140 = 0.000000969m m
"or in round numbers: 1 millionth of a miilimeter for the diameter of an air molecule." This is 10 A, which compares reasonably well with the currently accepted values of about 3 A for the diameters of O2 and N2 molecules. The Mystery of Loschmidl's N
And here Loschmidt's calculations stop. He goes on to compare his value for s with other small dimensions known a t the time-wavelengths of light, thickness of Faraday's gold foils, distance between lines in a grating-and he indulges in the speculations mentioned above. As noted above also, it is possible to plug this value for s into the original mean free path expression and obtain N = 1.83 X lor8. But Loschmidt stops short of doing so. Here lies a minor mystery of the chemical literature. I n the same year, under the same title and under Loschmidt's name, Sehlomileh's Zeitschrift ftir Mathematik und Physik published what is obviously a condensation of the above article (6'1, a regular practice of that journal with papers they considered particularly important. Although there is no attribution to any other writer, the style is quite different from Loschmidt's (7); and more important, the condensation abounds in major errors (e.g., mean free path is given as 0.000170 mm; s is given as 1 (sic) = 1.17 millionths of a millimeter). Yet in this shoddy condensation a value for N appears for the first time. The figure given is 866 billion (866 X
1012) molecules/mmJ, which is 8.66 X 10'' molecules/ cma. This value is not derivable from any set of data, correct or incorrect, in either paper. In the absence of further evidence from the archives of the Vienna Academy of Sciences, or of Schliimilch's Zeitschrift, its authorship must remain a century-old mystery. This then is the first published value for the Loschmidt number-or the Avogadro number, if recalculated for one mole, which was never done in the nineteenth century. It should be noted that Loschmidt set out to calculate not N, buts, the molecular diameter, and that the value for the number of molecules in unit gas volume is a kind of by-product of this quest. This is characteristic of many of the nineteenth century methods for N: the value actually determined is that for the molecular diameter, and the value for N exists only by implication, through the mean free path expression or other kinetic equations linking N and s. Kelvin's Methods (1870)
In 1870 Lord Kelvin (8) (then William Thomson) discussed four available methods of determining molecular diameters, in an article in Nature in the first year of its publication. One of these was substantially similar to Loschmidt's method, although Kelvin was unaware of Loschmidt's work a t the time. The other three show the same deductive ingenuity that characterizes the kinetic-theory approach. The first derives from Cauchy's earlier light experiments, in which Cauchy himself had concluded that "the 'sphere of sensible molecular action' in transparent liquids and solids (is) comparable with the wave length of light." (Light wave lengths were well established in 1870, having been determined by interference methods in the early decades of the century.) By further reasoning Kelvin concludes that the inhomogeneities of matter-that is, center-to-center distances of molecules-are on the order of 0.5 A in size. This is low, as indeed are all of Kelvin's values in this paper; but it is within an order of magnitude. The second method arises from work done by Kelvin himself. A decade earlier he had found that plates of dissimilar metals-e.g., a copper and a zinc plate-held parallel and close to each other, and connected by a fine wire for electrical contact, experience an attraction identical with that of plates deliberately charged by a battery. This charge attractionis directly proportional to surface area, so that if we begin with two plates and a known attraction, then slice each in half and interleave the resulting thinner plates (copper, zinc, copper, zinc), maintaining the electrical contact, we should find twice the attraction. If we go on repeating this process, the attraction will increase by powers of two with each slicing and interleaving. But the total energy of attraction cannot reasonably be greater than the simple energy of solution of these metals, so that when we approach this total energy our slices must he of approximately molecular thickness. Treating this argument quantitatively, Kelvin arrives a t molecular dimensions of about 0.3 A. The third method is a limiting-thickness argument somewhat similar to the foregoing. Beginning with the known contractile force of water (soap-bubble) films, Kelvin calculates the work necessary to draw films out against this force to various areal dimensions, implying
various (decreasing) thicknesses. He reasons that this work cannot be greater than the energy of vaporization for that amount of water, or no liquid film would remain. So when the work calculated is about equal to the heat of vaporization, the film must be exceedingly close to being a vapor-that is, it must be monomolecular. From this Kelvin concludes a molecular diamet,er of about 0.5 A. Summing up the results of these methods, including the Loschmidt-like determination, Kelvin observes, "The four lines of argument which I have now indicated, lead all to substantially t,he same estimate of t,he dimensions of molecular structure. Jointly they establish. . . that. . .the mean distance between the cent,res of contiguous molecules is less than the hundred-millionth, and greater than the two thousand-millionth of a centimetre" (i.e., between 0.05 and 1 A). The values for N which can be calculated from these diameters are necessarily high; indeed, in giving the kinetic argument Kelvin cites as an upper limit a value of 6 X loz1for the Loschmidt number. This is high by more than two orders of magnitude. Nonetheless, the agreement among values derived in quite different ways indicates that the range of magnitudes is probably about right. Determinations by Kelvin and Others (1884-1904)
By 1884, when Kelvin delivered the series of addresses a t John Hopkins which came to be called the "Baltimore Lectures," he was aware of Loschmidt's work and accorded him due priority. I n addition he offered a new, somewhat related kinetic argument-interestingly enough, with important contributions by Loschmidt. This method arises from the kinetic treatment of molecular diffusion in gases, as well as from the related topic of gas viscosities. Kelvin gives the following expression, adapted from h'laxwell, for the molecular diffusivity of a gas
where D = the diffusion coefficient for the particular gas, i.e., the number of molecules crossing unit area in unit time as they diffuse away from the bulk of the gas, into either an empty space or a space containing another gas; V = the rms average velocity of the molecules; N = the Loschmidt number; and s = the diameter of the molecules. Since, according to Maxwell, diffusivity is proportional to gas viscosity
where p = gas viscosity, and p = gas density, it is possible to rewrite this expression, with some rearrangement, as
This gives two directions for laboratory determinations of values: molecular diffusivitiesand gas viscosities. In 1870 Loschmidt (9) had reported a series of carefully determined values for the interdiffusivity of a number of pairs of gases. From these Kelvin deduces diffusivities for individual gases, calculates them over Volume 47, Number 1 1 , November 1970
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into viscosities according to 1\4axwell's expression, and compares them with viscosities determined directly by Obermayer (10) in 1876. This gives two more or less independent experimental values for Ns2. Unfortunately the rigor of the argument breaks down here, for Kelvin does not possess a reliable value for either N or s. He therefore falls back on a Loschmidtlike argument, comparing liquid and gas densities and introducing a packing factor, q, whose value is unity for the perfect "cubic" array (hexagonal close-packed, in present-day terminology), deviating positively or negatively for other types of packing. Kelvin's final oonclusionis that N = 1.21 X 1020 X q" for argon, for which the best values are available. Since for this monatomic gas q is probably > 1, then N 2 1.21 X loz0. By the time Kelvin gathered t,ogether the "Baltimore Lectures" in book form (If), in 1904, he had three more arguments to add, all due in some degree to Rayleigh. The first of these was Rayleigh's (18) rediscovery in 1890 of an investigation of surface tension and capillary attraction in 1805 by Thomas Young (IS). I n his original paper Young concludes that "the extent of the cohesive force must be limited to about the 250 millionth of an inch" (about 1 A), and further, that "the diameter or di~tance~of the particles of water" is between 0.025 and 0.125 A. Since Young was not clearly thinking in a molecular or atomistic framework-much less a kinetic one-there is no N available; and his calculation of what may be s, the molecular diameter, stands alone as an artifact of almost archeological curiosity. I n 1890 also both Rayleigh and Rontgen, work~ng independently, obtained est,imat,esof molecular dimensions by determining limiting thicknesses of olive oil films on water. Using either bits of camphor or drops of ether as tell-tales, they found that the surface tension of water is not affected by oil films of less t,han 6-10 A thickness. The conclusion is that this is the dhickness of a monomolecular film, i.e., if thicknesses less than this are attempted they are no longer coherent, as evidenced by their lack of effect on the surface tension of the snpporting water. This yields a surprisingly accurate value for molecular sizes, but no N is calculat,ed from it. Rayleigh's third argument, as reported by Kelvin, derives from his 1871 treatment of the light-scattering which makes the sky blue. Rayleigh found that the ratio of scattered to incident light on very small particles (i.e., of molecular size) is directly proportional to N and inversely proportional t o the fourth power of wavelength. Thus the lower wavelengths at the blue end of the visible spectrum are highly scattered, illuminat,ing the sky above us, while the longer red waves come through virtually unchecked, contributing nothing t o the color of the sky. The mathematical expression for this phenomenon, as noted above, contains the Loschmidt N (at standard temperature and pressure), but is independent of the size of the molecular particles, provided that they are quite small in relation to wavelength. This led Rayleigh in 1899 to attempt an evaluation of N b y measuring scattering at various wavelengths. Kelvin's account of this method draws on other sources as well, treating such problems as suspended particulate matter, evaluations a t different altitudes, etc. He arrives at a value of N 2 2.47 X lo'#, which seems impressively close to the currently 754
/
Journal of Chemical Education
accepted value, except that Kelvin regarded it as an extreme lower limit. One final nineteenth century method for N should be mentioned. The "b" correction factor in the van der Waals gas equation
is commonly interpreted as correcting for the finite volume occupied by real molecules which are in fact not dimensionless points. As such it has the form b = '/~rNsa
It is possible, by use of the expression for gas viscosity, to eliminate N and solve for s (molecular diameter) in terms of b and other measurable quantities. The N which is in turn available from this value of s has been determined, for example, by Perrin (14) as about 2.8 X 1019for mercury vapor. Summary
Of the methods of nineteenth century investigators in determining N , the Avogadro or Loschmidt number, the most prominent. are the derivations from equations of the kinetic molecular theory. As the latter is one of the triumphs of nineteenth century physics, it is not surprising that it should have shed light in many other areas as well. There are also a number of estimates from limiting-thickness arguments, however; eit,her for liquid films or for interleaved metal plat,es. Most of the remaining values arise from optical considerat,ions -what size particles will give this degree of diffraction, scattering, etc.? The only example lying outside these categories is the derivation from t,he van der Waals equation, immediately above. I t is interesting, though probably not significant, that the value derived is t,he most accurate of the lot. It is also interesting, and a great deal more significant, that these are physical methods, not chemical. Chemists in the nineteenth century were for the most part indifferent to the number of ultimate particles they worked with. They were happy with atomic and molecular weights as combining ratios only, and few of them seemed impelled to go beyond this empiricism to ask how many molecules these combining weights might represent. That they left to t,he physicists. The Avogadro number was not to be firmly tied up with that uniquely chemical concept, the mole, until well into the twentieth century. The methods reviewed here, then, are all physical, all indirect, and many are highly ingenious in their deductions. But by the turn of the century deduction was already yielding to direct determination. Rutherford and Geiger would soon be counting scintillations from individual alpha particles given off by radium samples. Perrin would be observing suspended gamboge particles set into Brownian motion by the impingement of individual molecules. And finally Millikan would find the charge of the individual electron and usher in an era of truly direct determination. The cast of characters changes, too. The nineteenth century giants-Kelvin, Rayleigh, Boltzmann-disappear from the scene and are replaced by a new breed of experimenters. Precision and accuracy are ever greater, and today we know N within a few parts per
hundred thousand. This is immensely important and useful-but somehow today's precision seems less exciting than yesterday's inspired guesswork. Literature Cited (1) D ~ m m n Srrr , W ~ r c r ~ C.,"A aa Hiatorv of Soienoe," 4th ed.. Cambridge University Preaa. 1949, p. 229. (2) For recent partial revieva of determinations of N, ace for example L o n r ~ A,, , Qiorn. Pi& Sac. Ilol. Fia.. 7 (I), 11-23 (1966): Aoomn. N. M. H., Nuooo Cimcdo, 6 (10). Suppl.,.221-3 (1957): or Su*aob& A,. AND KALNAJB, J., i b i d . , P P 214-20. (3) BOLTEXANN. LODWIQ."tiber eine These Sohopenhauers," i n "Popullre Sahriften," Barth. Leiprig. 1905, p ~385-402. . (4) L o s c x u r o ~J.. , Sitrungsber. der k . Ahad. Wiss., Wien, 52. 11. Abth.. 395413 (1865).
M ~ x w e mJ. , C.. Phil. Map., XIX (Series 4). 19-32 (1860); Cwaaros, . R.. ibid.. ~ p434-6. L o s c n ~ mJ., ~ , Schlbmileh's 2.fur Math. und Phya.. 10. 511-12 (1865). FANTEL. H A N H.. ~ personal eommunioatian. I am indebted to Mr. Fantel for confirmation of this point. T m o x s o ~W., , Nature, 1, 551-3 (1870). L o s o x r m ~J., , Sitrunoder. der k. Ahod. Wisa., W i m , 61.11. Abth., 216, 367-80. 652 (1870): 62, 11. Abth., 46%78 (1870): W n ~ ~ s o ~ A,, xo. ibid., 62, 11. Abth., 569, 575-89 (1870): B E N ~ A R J... ibid. D. 676. 687-98. OeenuAmn. Silnmgsbsr. dark. Akod. Wiss., Wien. 7 3 , 11. Abth.. 433
,.-,",. ,70-.2,
KELVIN. WI~,I*MTHOMSON LORD."Baltimore Lectures." Cambridge University Press. London. 1004, Lecture XYII. pp. 279423. R ~ r ~ e rJ. oW ~ ,. , Phil. Mag., 30, 474 (1890). Y o a m T., "On the Caheaion of Fluids," Phil. Tvans. (1805): "Collected Works." Yol. 1, p. 461 fi. PERRIN.J.. atom^,',', Constable, 1923. Ch. 11: Vmoo. S. E., "Loeohmidt's Number. Science Pvogresn. 27, $108, 634-49 (1933).
