SOME REMARKS ON THE DETERMINATION OF FORMULAS KENNETHD. DODDS, BELLEWE HIOXSCROOL, BBLLEVW, PENNSYLVANU
The chemical reasons behind the rules for calculating formules are obscure to many bezinners primipally became of the impression that the validity of such calculations i s somehow dependent both upon the statement of the analyticel data in terms of percentage composition, and upon the means by which the final numbers (whenfractional) which represent the numbers of atoms of each element zn the compound are reduced to whole numbers. It i s the purpose of these remarks to elucidate the chemical aspects of the arithmetical operations by which formulas are determined, and to show that they can be deduced without regard to percentage composition, and that there i s a choice of ways for reducingfractional results to whole numbers of atoms.
. . . . . . The first operation in calculating formulas, by present methods, requires the use of the percentage composition of the compound. This is the case both with the traditional method, in which the percentages of the constituent elements are divided by their atomic weights; and with Stone's excellent method ( I ) , by which a provisional molecular weight is found as the quotient of the atomic weight of any element in the compound by its percentage. In case the composition is not given in percentage, the textbooks state that the known data must be reduced to that basis before proceeding with the calculation. The beginner is usually unfamiliar with the reasons for expressing analytical results in terms of percentage, and therefore gets the impression that chemical formulas are linked in some obscure way with the purely arithmetical phases of percentage composition. The unnecessary rigidness of the stipulation regarding percentage is indicated by two considerations: first, that theweights of the elements in a sample of a compound are proportional to their numbers and to their atomic weights; and second, that the expression of analytical data in terms of percentage cannot have any peculiar virtue, since elements are present in any weight of sample in relatively the same proportions as in one hundred units of weight. For these reasons i t would seem that neither the weight of sample analyzed nor the system of weights employed have any indispensable bearing on the computation of formulas. To transfer this discussion to a concrete problem, let us attempt t o find the formula of a compound directly from the following analytical data: Calcium Carbon Oxygen
0.2581 gram 0.1548 gram 0.4129 gram
Designating numbers of atoms by 7, Weight Cs
or &a
X atomic weight Ca. 2244
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DETERMINATION OF FORMULAS
Hence
The above quotients can be reduced to whole numbers of atoms either through division of each by their highest common factor*-the traditional method--or by multiplication of each by the least quantity that will yield whole numbered productsStoue's method (1). It is easily seen that this choice of operations is due to the equivalence of division and multiplication when the divisors and multipliers are reciprocals. In our example, it is evident that
Also, although the multiplier is not so readily apparent,
Identical results are obtained by following the general outline of Stone's method (1). Thus on the basis of carbon, the molecular weixht of the 12 compound is represented by 0.1548 Or 77'51 Then Weight Ca = 77.51 X 0.2581 or20 77.51 X O.IM8 or 12 Weight C 0;
Weieht 0
a
77.51 X 0.4129 or 32
. . = 2 X 2(0r + 0.5) = 4 * Although many textbooks direct that such quotients be divided by the smallest of them, this pracedure often leads to further fractions and thus necessitates an additional operation. This complication does not arise when the divisor is the H. C. F. The smallest quotient i s of course frequently also the H. C. F.
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JOURNAL OF CHEMICAL EDUCATION
NOVEXBER. 1931
While the foregoing solutions demonstrate that formulas can readily be computed from weights, the author does not advocate that this method supersedes those which proceed from a percentage basis, for aformula would not rest very securely on the results of a single analysis. On the contrary, the purpose of this discussion is to demonstrate the fundamental relation of these methods to analytical data, and thereby to provide a means of counteracting the all-too-frequent tendency of pupils-and students likewise--to miss the chemical significance of the arithmetical operations involved in the calculation of formulas.
Literature Cited (I) STONE, "A Question of Procedure," J. Cwmd. Eonc., 5, 465-8 (Apr., 1928).
