The Choice of the Most Probable Value for 
 an Atomic Weight: The Atomic 
 Weight of Hydrogen. 
 
 By William A* Noyes. 
 
5<K£. 
 
 
 NV\ 
 
 [Reprinted from the Journal of the American Chemical Society, 
 Vol. XXX, No. i, January, 1908.] 
 
 THE CHOICE OF THE MOST PROBABLE VALUE FOR AN ATOMIC 
 WEIGHT: THE ATOMIC WEIGHT OF HYDROGEN . 1 
 
 By William A. Noyes. 
 Received December 6, 1907. 
 
 A large amount of material has been accumulated from which the 
 atomic wieghts of the more important elements can be calculated. A 
 very superficial examination of this material reveals the fact that the 
 experimental results on which our knowledge of these constants is based, 
 vary very greatly in their value and that many of the older determina- 
 tions have been rendered practically worthless by recent work, which 
 has been more careful and accurate. 
 
 As some of these new determinations affect the values for elements 
 of such fundamental importance that a recalculation of the whole table 
 of atomic weights will be necessary in the near future, it seems desirable 
 to formulate some general principles to aid in the elimination of results 
 which have little or no value and in the combination of the results which 
 remain. Such principles, if they meet with general acceptance, will 
 be of value, riot only for the purpose stated but also as setting a certain 
 standard which must be attained by future workers in this field, if their 
 work is to be of permanent value. 
 
 The most important general principle which has been proposed for 
 the combination of the results of different observers, is the one based on 
 the mathematical discussion of accidental errors of observation. In ac- 
 cordance with the theory of probabilities, these results, if subject only 
 to accidental errors, should be weighted in inverse proportion to their 
 
 1 Presented in abstract at the N. Y. Meeting of the American Chemical Society, 
 Dec. 28, 1906. 
 
the atomic weight of hydrogen. 
 
 5 
 
 probable errors. 1 A very serious objection to this method of treatment 
 lies in the fact that every determination of this kind is subject to con- 
 stant errors, and that the amount of these errors is not proportional to 
 their “probable errors.” 2 Thus Stas obtained 132.8445 ±0.0008 parts 
 of silver chloride from 100 parts of silver, while Richards and Wells 3 
 have obtained 132.8670 ±0.0005 parts. The most probable value 
 calculated by the mathematical rule would be 132.8607. If this value 
 is the true one, the real error of the value obtained by Richards and 
 Wells is 12 times its probable error, while the real error of Stas is 20 times 
 the probable error. And, whatever the true value may be, the real error 
 of one of the results, at least, is many times its “probable error.” An 
 examination of other cases shows that the relations here found are typi- 
 cal, and it seems evident that the question of constant errors requires 
 some other treatment than the simple mathematical one. The proper 
 treatment, which is an experimental one, has been clearly illustrated 
 in the case which we have under consideration. Richards and Wells 
 studied their method very carefully with especial reference to the elimina- 
 tion of constant errors and to secure evidence as to the amount of those 
 errors which could not be wholly excluded. They also pointed out 
 certain errors in the work of Stas and determined, approximately the 
 magnitude of some of these. It is evident for this reason that very much 
 greater weight attaches to the value found by Richards and Wells than 
 to that found by Stas, and it is proposed as a general principle that when 
 a later observer has pointed out sources of error which are considerable 
 in comparison with the “probable errors” and where the later observer 
 has succeeded in avoiding these sources of error, the earlier work must 
 be looked upon as having only confirmatory value and the result of the 
 later work should be accepted without modification. It has been ob- 
 jected to this that the later work is also subject to constant errors which 
 may be in the opposite direction from those of the earlier determina- 
 tion and that if we give a certain weight to the earlier work we may 
 eliminate these errors. But we certainly are not justified in using a value 
 that contains a known error in one direction merely for the chance that 
 we may compensate an unknown error. 
 
 of the atomic weights are nearer to the numbers nov 
 than were the values then given by Professor Clarke. 
 
 ^ TV»1 C Tmirtiol h* r- . . 
 
 This Journal, 27, 524. 
 
 (,.19 2.9 5 
 
6 
 
 WILLIAM a. NOYES. 
 
 The principle outlined above has been recently proposed, independ- 
 ently, by Professar Guye, 1 in his discussion of the selection of the most 
 probable value for the density of a gas. A second principle proposed by 
 Professor Guye is that when the values obtained by two observers agree 
 while that obtained by a third observer is discordant, the values which 
 are in agreement should be given much greater weight. As an exten- 
 sion of this principle, a value of an individual worker which differs ma- 
 terially from the values obtained by several others, should be rejected 
 entirely. 
 
 After eliminating the results which are excluded by the application 
 of the foregoing principles, it is proposed to arrange those which remain 
 in the order of their probable errors. Any result with a probable error 
 more than five times that of the smallest probable error may be excluded, 
 as such a result will have only one twenty- fifth the weight, according 
 to the theory of probabilities. In practical effect, this is the same as using 
 the mathematical rule which Professor Clarke has so long employed 
 in weighting the results of different workers. As at least five or six ob- 
 servations are necessary to give a probable error which has any sig- 
 nificance, results based on a smaller number of determinations may be 
 excluded unless other evidence warrants the belief that the work is of an 
 unusual degree of accuracy. 
 
 The values for any given ratio which remain after the elimination 
 of results which have little value, may well be combined by weighting 
 them inversely as the squares of their probable errors. 
 
 For further use, the ratios which are selected in this manner should be 
 weighted, not by the probable error calculated by the mathematical 
 rule but by the deviation of the results of different observers from the 
 value selected. If the results of only one observer remain after elimina- 
 ting untrustworthy values (as in the case of the ratio of silver to silver 
 chloride), this result should be weighted in accordance with the average 
 deviation of the results of this observer from his mean. This will, I 
 think, give a much fairer basis than the “probable error” for weighting 
 the value in such cases. Thus the “mean error” of the value of Rich- 
 ards and Wells given above is 0.0018, while the “ probable error” is 0.0005. 
 When we consider the certainty that some sources of constant error 
 will always remain, I think every one will agree that the real error is 
 much more likely to correspond to the former than to the latter value. 
 
 After selecting the most trustworthy experimental ratios as suggested, 
 we have still to combine them for the calculation of atomic weights. 
 This may usually be done in a variety of ways. In choosing among 
 these, the same general principles as before should be applied. For 
 a given atomic weight, only those ratios should be used for which the un- 
 1 Arch. sci. phys. nat., 24, 44. 
 
THE ATOMIC WEIGHT OF HYDROGEN. 
 
 n 
 
 certainties of the values will affect the atomic weight chosen less than 
 five times as much as any other combination of ratios which might be 
 used. ' In most cases this will probably lead to the selection of ratios 
 which furnish a direct comparison with oxygen, silver or one of the halo- 
 gens rather than of those in which the comparison is more indirect. Den- 
 sities of gases corrected to the condition of an ideal gas by the method 
 of D. Berthelot 1 may be considered as direct comparisons with oxygen, 
 and molecular and atomic weights calculated from these densities should 
 be included with those determined by chemical methods. 
 
 The Atomic Weight of Hydrogen. 
 
 The following is a summary of the determinations which have been made 
 of the atomic weight of hydrogen by the chemical method : 
 
 
 
 No. of 
 
 
 Prob. 
 
 Real 
 
 Real error 
 
 
 Date. 
 
 expts. 
 
 Value. 
 
 error. 
 
 error. 
 
 Prob. error' 
 
 Dulong and Berzelius 
 
 1821 
 
 3 
 
 I .00667 
 
 356 
 
 108 
 
 0-3 
 
 Leduc 
 
 1892 
 
 2 
 
 I .00749 
 
 83 
 
 26 
 
 0-3 
 
 Erdmann and Marchand 
 
 1842 
 
 8 
 
 I .00160 
 
 71 
 
 615 
 
 8.7 
 
 Thomsen 
 
 1870 
 
 8 
 
 I.00570 
 
 7 1 
 
 205 
 
 2.9 
 
 Rayleigh 
 
 1889 
 
 5 
 
 I .00692 
 
 56 
 
 85 
 
 i -5 
 
 Dumas 
 
 1842 
 
 19 
 
 1 00250 
 
 44 
 
 525 
 
 12 .0 
 
 Keiser 
 
 1898 
 
 8 
 
 I.OO753 
 
 3 i 
 
 22 
 
 0.7 
 
 Dittmar and Henderson. . . . 
 
 1890 
 
 24 
 
 I . 00840 
 
 29 
 
 65 
 
 2 . 2 
 
 Noyes (recalculated) 
 
 1890 
 
 24 
 
 1 .00765 
 
 17 
 
 IO 
 
 0.6 
 
 Thomsen 
 
 1895 
 
 21 
 
 I .00826 
 
 14 
 
 51 
 
 3-6 
 
 Cooke and Richards 
 
 1887 
 
 16 
 
 I .00826 
 
 13 
 
 5 i 
 
 3-9 
 
 Noyes (original) 
 
 1890 
 
 24 
 
 I .00654 
 
 1 1 
 
 121 
 
 1 1 .0 
 
 Keiser 
 
 1888 
 
 10 
 
 I . 00306 
 
 7 
 
 469 
 
 67.0 
 
 Noyes 
 
 1907 
 
 48 
 
 I .00787 
 
 2 
 
 12 
 
 6.0 
 
 Morley 
 
 1895 
 
 23 
 
 I .00762 
 
 2 
 
 12 
 
 6.0 
 
 The probable errors of the 
 
 table 
 
 are 
 
 calculated 
 
 from those assigned 
 
 by Professor Clarke . 2 For the results of Erdmann and Marchand and 
 Leduc, the values are arbitrary. For convenience these errors are given 
 in units corresponding to the last significant figure of the values for the 
 atomic weights. 
 
 On applying the principles which have been outlined, we find that 
 the results of Dulong and Berzelius, Erdmann and Marchand, and of 
 Dumas, are excluded because the later work of Dittmar and Hender- 
 son by the same method, demonstrates that serious constant errors were 
 involved in the earlier work. Leduc’s value is to be rejected because 
 the number of experiments was too small. Reiser’ s earlier value is to 
 be rejected because it is not in accord with any of the later work and be- 
 cause he has himself given us a later and better value. My own original 
 value must be rejected because it was subject to a constant error and 
 
 1 Compt. rend., 144, 76. 
 
 2 “Constants of Nature.” Part V., p. 24 (1897). 
 
8 
 
 WILLIAM a. NOYES. 
 
 the recalculated result may be considered as superseded by my later 
 and more careful work. Because the probable errors of all of the other 
 determinations are more than five times as great as those of Morley and 
 myself, they would be excluded by the third principle proposed. The 
 final value, if calculated from these two results, is 1.00775, 
 
 It is interesting to notice the relation between the real errors of the 
 various values (assuming this value as true) and the probable errors. 
 Only in those cases where we now know that there were serious constant er- 
 rors, is the real error more than six times the probable error. 
 
 Morley calculates a value corresponding to 1.00762 from his deter- 
 minations of the densities of the gases and their combining volumes. 
 This value has not been considered here, partly because the probable 
 error of the density of hydrogen is about 3 in 100,000, instead of 2 for 
 the chemical method, but chiefly because of the uncertainty of the ratio 
 of the combining volumes. 1 
 
 If a value is calculated by Professor Clarke’s method, weighting each 
 result in inverse proportion to its probable error, only Keiser’s older 
 value and my own original value would affect the value which I have se- 
 lected by more than about one part in 100,000. Keiser’s older value would, 
 however, reduce it by about 40 parts and my own original value by about 
 4 parts in 100,000. 
 
 1 Morley: “Smithsonian Contribution to Knowledge,” No. 980, p. no (1895). 
 
 University of Illinois. 
 
 Urbana, III.