Albert J. Bernatowicz University of Hawaii
Honolulu, Hawaii 96822
Dalton's Rule of Simplicity
There must he something about our psychological makeup that makes infinite regress less than a favorite answer when we question the very nature of things. We have proposed seventh heavens, primum mobiles, or empyrean paradises more often than infinite universes. The earth has rested on Atlas, elephants, and tortoises hut never on an endless succession of bearers. Hutton's "no sign of a heginningno prospect of an end" has been less appealing than creation, uncaused causes, 4004 B.c., or4.5 X lo9years. There has been far more frequent acceptance of spontaneous generation or specific creation than of theories of evolutionary origin of life and matter. So also, in to the notion of indefinitely divisible matter man has time and again postulated a definite end to the regress-an "atom" of some sort. It was not onlythe atomism of Leucippus-Democritus-Lucretius, beloved of the sophomore eager to expose John Dalton's lack of priority. Thinkers as modern and respected as Boyle, Descartes, Gassendi, and Newton are among those holding particulate views of matter before Dalton, views that were sufficiently worked out to have explanatory value even today. What, then, did the Quaker contribute to atomic theory? Dalton's contributions have been described variously: quantitative chemistry, the conception of a different species of atom for each element, the idea of atomic weights, the notion that molecules consist of only a few atoms. We need not he concerned because there is variety of opinion, and because it happens that Dalton had been anticipated in some of these. For, it is scarcely an exaggeration to say that, beyond these purported contributions, the veritable sine p a non of his theory is a set of rules for establishing molecular formulas. Without that set of rules, no contribution, nothing. Any who may wish more details of the overreaching importance of these rules will find explication in Nash and in Greenaway (I), not to mention the implications to he drawn when capsule versions of the theory highlight its Daltonian aspects (2). That particular garden hardly needs further tilling. But there is a generally unmentioned deficiency in the set of rules which I shall pursue here, namely, that except for unique-i.e., monotypic-compounds, Dalton's mles, which at first glance seem so clear and unequivocal, do not stipulate operations. The Rule of Simplicity
To John Dalton, the atoms of one element differed from those of any other in size and weight, and relatively small combinations of such differing atoms constituted the ultimate particles of compounds. From such a conception it follows directly that an element's
weight in a sample is the product of one atom's weight multiplied by the number of atoms present. But a truism of that order would merit scant attention had not Dalton recognized its converse: since the amounts of the elements making up a compound can he obtained by analysis, they are a clue to the weights of the atoms -a clue but not a direct indication. There is the other unknown to he taken into account-the number of atoms present. For this too a truism emerges from the theory: the number of atoms of each element is the count per molecule times the number of molecules in the specimen. Now, even though the New System does not tell us how many molecules might be in a sample, as any hapless freshman can attest bitterly, "it is obvious that" one can cancel out this factor by working simultaneously with two elements of some one compound and getting relative atomic weights instead of absolute figures. (One wonders how many chemists would see such an "obvious" attack on the problem if required to stay within the limitations of the early nineteenth century scientist or of a twentieth century freshman.) Obvious or not to lesser minds, this approach was self-evident to Dalton. Inferring atomic weights from empirical data then reduced to a question of the number of atoms of each element in a molecule of compound. This, be it remembered, before any association of valencies with the elements. How to decide the question? Dalton did it with what is often called his "rule of. simplicity" (3) If there me two bodies, A and B, which are disposed to combine, the following is the order in which the combinations may take place, beginning with the most simple: namely, 1 atom of A 1atom of B = 1 atom [=molecule] of C , hinary. 1 atom of A 2 atoms of B = 1 atom of D, ternary. 2 atoms of A 1 atom of B = 1 atom of E, ternary. 1 atom of A 3 atoms of B = 1 atom of F, quaternary. 1 atom of B = 1 atom of G, quaternary. 3 atoms of A
+
+ + + +
The following general rules may be adoptedas guides in all our investigations respecting chemical synthesis. 1st. When only one combination of two bodies can be obtained, it must be presumed to be a binary one, unless some cause appear to the contrary. 2nd. When two combinrttions are observed, they must be presumed to be a, binary and a ternary. 3rd. When three combinations are obtained, we may expect one to be a. binary, and the other two ternary. 4th. When four combinations are observed, we should expect one binary, two ternary, and one quaternary, etc. 5th. A binary oompound should always be specifically heavier than the mere mixture of its two ingredients. 6th. A ternary compound should be specifically heavier than the mixture of a binary and s. simple, which would, if combined, constitute i t ; etc. 7th. The above rules and observations equally apply, when two bodies, such a s C and D, D and E, etc. are combined. Volume 47, Number 8, August 1970
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Newton's Influence
How the voice of Newton echoes here! The famous expatiation of the first rule of reasoning (La. . .Nature does nothing in vain. . .Nature is pleased with simplicity.. .") pervades the whole, all the more noticeably when we learn of Dalton's belief in "the regularity and simplicity generally observable in the laws of nature," and of his declaration that "the law of chemical synthesis is observed to be simple. . . ."(4). Newton's third rule of reasoning ("Those properties of bodies that are both unchanging and common to all bodies within reach of our experiments are to be considered the universal properties of all bodies whatsoever.") is reflected in Dalton's seventh, which extends his preceding guides to cover aU instances. And, Newtonian in thoroughness, the first half of the rule is a statement of possibilities and definitions that leaves little to the uncertainties of the public's intelligence. The same reluctance to leave something for the reader to fiU in seems to continue into the numbered half, but here the initial impression needs reassessment. What a t first appears to be unnecessarily detailed, even redundant, suddenly is seen to lack an indispensable ingredient-we are not told how to implement the rules. Implementing the Simplicity Rule
There is no problem of application with respect to the first rule. Although it has a certain notoriety in some books as an error which led Dalton to assign the formula HO to water and NH to ammonia (I am using modern symbols, not his), the historical circumstance that more satisfactory formulas for water and ammonia were soon found is beside the point. What is the point is that he provides a basis for assigning formulas in this limited category, monotypic compounds. The rule of simplicity turns out to be not so simple when we leave the monotypic compounds. Faced with more than one combination of the same pair of elements, we are to assume minimal formulas. But which formula to which compound? Or, restating the question, how does one recognise the simplest compound? Viewed operationally, the rule is silent. The difficulty is readily apparent when we get specific. Consider the two copper chlorides, one white, the other green. Analyses disclose that the white powder is about 64.2% copper and 35.8% chlorine; the green crystals run more nearly 47.3% copper and 52.7% chlorine.' Stated as percentages, these analytical results are not visibly useful for the problem a t hand. We must, as Dalton did, convert to figures illustrating the law of multiple proportions, which emerged from the rule of simplicity. Doing so, we learn that the white powder contains 1.79 g of copper per gram of chlorine. In the green crystals the Cu:Cl ratio is 0.895 to 1. Patently, there is twice as much copper per gram of chlorine in the white as against the green compound. The formulas CulCl and CuCl accordingly may seem plausible. But the very same percentage composition can be converted to ratios with the copper instead of the chlorine held constant: per gram of copper the white powder has 0.56 g of chlorine and the green has 1.12 g. The green crystals plainly contain twice as much chlorine per gram
' Water of hydration is omitted to facilitate comparison. 578 / Journal o f Chemical Education
of copper as the white. The content of the two statements is the same, hut the difference in focus may now prompt a tyro to favor CuCl and CuCL. With our modern understanding these may be more acceptable formulas, yet on the basis of the information given they are no more defensible than the first pair. This point-that even if only diatomic and triatomic molecules are admissible, from weight ratios alone we cannot tell which is which-may seem elementary enough today, but it was not recognized by all of Dalton's contemporaries, nor is it clear that Dalton himself was alert to it a t all times. He makes no mention of it in Part I of the first volume of his "New System," which contains the rule of simplicity. In the second part, 2'/2 years later, the point is implicit in the discussion of carbon-oxygen compounds (6) . . . the oxygen in the former is just double what it is in the latter for a given weight of carhone [sic]. Hence, we infer that one is a binary, and the other a ternary compound; but it must be enquired which of the two is the binary, before we can proceed according to system.
Then came Volume 11, the opening section of which (printed 1817, issued 1827) contains the assertion, "The same metal combines with one, two, or perhaps more atoms of oxygen.. .." Having disallowed the alternative, one oxygen atom combining with two or more atoms of a metal, Dalton cannot come up with today's Hg20 or CuzO. His oxides of mercury, copper, and some other metals could only be stated as monoxides and dioxides. Finally, during the ensuing decade he learned of authors who were using the same one-sided approach to multiple proportions problems. In the appendix he instructs them appropriately, pointing out that the pair of weight ratios 10:7 and 10: 14 is the same as 20:14 and 10: 14, therefore two sets of formulas are equally probable (6). (He does not, by the way, reexamine his own handling of the metallic oxides.) How Others Have Treated the Matter
Curiously enough, the dilemma-which is binary, which ternary?-is not always mentioned in the many histories of chemistry and general education texts that present Dalton's rule of simplicity. Curious on the one hand because the claims on behalf of critical thinking among enthusiasts of general education hardly suggest they would ignore the fatal flaw in an historically crucial rule; curious on the other hand because the dilemma looms so obviously in today's light that it is hard to believe writers are oblivious to it. This is not to say the rule has escaped criticism, but the criticisms usually refer to its arbitrariness, rarely to the stumbling block that it is not operational. Among the many who must have recognized the difficulties of implementation, a t least three provide ostensible ~Iarificationsof the rule, or interpretations of "most simple." Moore says ". . .the compound which was best known received the simplest formula. . . "(7). This merely shifts the question so it becomes a matter of how to determine which is best known. Moreover, there is the awkwardness that Dalton himself, comparing carbonic oxide and carbonic acid, notes it is carbonic acid "which has been longest known, and the proportion of its elements more generally investigated" (81, and then goes on to use chemical characteristics, not familiarity, to assign the simpler formula to car-
bonic oxide. According to Ihde (9) it is the "more common" of two compounds which is binary. Even as Moore's version, this also seems eminently plausibleuntil we try to formulate general criteria for commonness. A third neo-Daltonism appears in the recent book by Easley and Tatsnoka (lo), where the "most abundant" compound is said to have received the simplest formula from Dalton. I elicited some sputters from a chemist when I requested of him a general rule for establishing "most abundant." How Dalton Applies His Rule
But what of Dalton? How does he work with the set of rules? No simple answer is possible, for "it all depends." Not a fundamentalist in reading his own text, he is, for example, quite able to ignore the injunction "must be presumed" in the second rule when he takes up two compounds of oxygen with phosphorus. Having made one out to be a binary compound, he does not presume that the other must be ternary. Instead, he explores the possibility that it may be quaternary (one phosphorus atom with three of oxygen) and then from evidence rather than rules concludes it is ternary after all (11). Similarly, he reports on three oxygensulfur compounds and decides they comprise a binary, a ternary, and a quaternary instead of the binary and two ternaries expected from rule three (18). When he does not bypass the simplicity rule, his use of it is tentative and pragmatic, designedly heuristic. Consider his handling of water. When an author of a systematic chemistry treatise describes five substances as hydrogen-oxygen compounds (IS), there is a reasonable presumption that he intends this as his best professional opinion; any lack of certainty he admits to is merely honest caution, not strong enough to change this opinion. Accordingly, for water a strict reading of his rules would have guided Dalton away from the first one, which covers monotypic compounds. All the same, he invokes the first rule, his excuse being that water alone is "certainly known" to consist of oxygen and hydrogen (14). Having gotten a foothold, now he is able to continue with the other compounds in that family. Not only in that family, but perhaps with everything else on his agenda. A recent biographer of Dalton (16), after presenting reasons why particle weights for water and its constituents were of singular importance to Dalton and to the chemistry of the time, writes I cannot show conclusively, but if I may invoke a rule of greatest simplicity in historical explanation, in emulation of my subject, I suggest that it was to water that Dalton prohably first applied his own rule of simplicity, and calculated %n atomic weight.
Pragmatic for heuristic reasons, then, but also tentativd, for a doubt remains in Dalton's mind (16) After all, it must be allowed to be possible that wster may be a. ternary comp,ound. In this case, if two atoms of hydrogen unite to one of oxygen, then an atom of oxygen must weigh 14 times as much as one of hydrogen. . .
If only he had followed the doubt and not the certainty!
Not every family of two-element compounds is handled in the "New System" the way the supposed series .containing hydrogen and oxygen is solved. In the well-publicized attack on the nitrogen oxides, and on those of carbon ( l 7 ) , specific gravities are Dalton's criteria of simplicity. For what are now known as ethylene and methane he uses chemical e v i d e n c e offering no apologies when his conclusions are the reverse of what specific gravities would have led him to (18). Obviously, he is not wedded to specific gravities as a measure of simplicity. Indeed, in the appendix to the final volume he concedes that the nitrogen oxides can be arranged in either of two ways and it is here that he gives his full opinion of how the number of atoms is to be determined, simplicity rules notwithstanding (19) It is necessary not only to consider the combinations of A with B, hut also those of A with C, D, E, etc.; as well as those of B with C, D, etc., before we can have good reason to be satisfied with our determinations as to the number of atoms which enter into the various compounds.
That was nineteen years after the rule of simplicity, but it seems evident he had been using this "all things considered" attitude all along. Summary
In sum, it appears that Dalton's famous rule is less an obiter dictum than an adaptable recommendation to use the briefest formulas as far as possible. If more than one combination of the same elements exists, the rule is of no help in deciding which is the simplest, and modern translations of "most simple" into "best known," 6' most common," or "most abundant" merely substitute one operationally undefined term for another. Dalton himself brought his full knowledge of chemistry to bear in deciphering molecular composition, ignoring the rule when there was empirical evidence on which to base conclusions, and using the rule as an expedient in other cases, apparently quite aware that a t any time "some cause [may] appear to the contrary." Literature Cited (1) N ~ s n L. , K.."The Atomic-Molecular Theow," Harvard Univeraitv Press. Cambridee. 1950. no. 2 4 4 1 : G n a m * w * ~ .F.. "John Dalton and the ~ t ~ ~ : ~ ~ ~ ~~ t ~h ~ i966, ~o ~ ~,P. 133. i u h , "ON. "A Hietory of Chemistry," (lat (2) For examplea, see M E Y B ~E. English ed.). Maomillan & Co.. London. 1891, p. 179: Mum. M. M. P.. "A History of Chemioal Theories and Laws." (lat ed.). John Wlley & Sons, New Y o r k . 1907, p. 87; PART~N~TON. J. R.,"A History of Chemistry," Maemillan & Co.. London, 1962, Vol. 3, p. 802. (3) DALTON.J., "A N e v System of Chemical Philosophy," Manohester. 180&1810, Vol. I. pp. 213-14. (Page referenoea are to the facsimile remint by William Dawson & Sons Ltd., London. 1953, not to the reprint by Philosophical Library. New Y o r k . 1964.) Val. I. PP.131. 223. (4) DALTON. V d . I , p. 368. (5) DALTON. 1827, Vol. 11. P. 350. (6) DALTON, (7) MOORE. F. J., "A History of Cheml&w." (1st ed.), McGrsw-Hill B w k Co.. New Y o r k . 1918, p. 76. The same statement is also found on p. 121 of the 3rd edition, 1939, revised by W. T. HA% Val. I. P. 368. (8) DALTON, (a) IRDE. A. J.. "The Development of Modern Chemistry," Harper & Row. New Y o r k . 1964, p. 109. J. A,, Jn., A N D T*~8001*. M. M., "scientific Thought: Cases (10) E*s~er. from Classical Physics." Allyn & Bacon. Boston. 1968. P. 126. (11) DALTON, VOI. I. PP. 414-15. (12) DALTON. VOI. I, PP. 383 R. and figs. 4 9 5 0 , PI. 5. Vd. I, chap. V, seot. 1. (13) DALTON, (14) DALTON. Vol. I. D. 275. (15) G n ~ a n ~ w * p. r . 145. (10) DALTON. Vol. I, p. 276. VOI. I, p. 317 and p. 369 (17) DALTON. (18) DALTON. Vol. I. PP. 437-47. (191 DALTON, Vd. 11, P. 350.
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