Atomic Weight of Sodium, history
There is a very close association between the atomic weight of sodium and the atomic weights of potassium, silver, chlorine, bromine, and iodine, each element having been an important factor in the experimental investigation of both its own atomic weight and the atomic weights of the other five. The method of determining the ratio of the atomic weights of these elements to that of oxygen was originated by Berzelius, and was developed by Marignac and Stas. It involves three stages:
 The determination of the molecular weights of the chlorides, bromides, and iodides of sodium, potassium, and silver by analysis of the salts RXO_{3} (R=metal, X=halogen), and induction from the ratios RX:3O.
 The determination of the three ratios Ag: NaX, of the three Ag: RX, and of the three Ag: AgX, values for the atomic weight of silver being obtained from these ratios in conjunction with the previously determined molecular weights of the metallic halides NaX, KX, and AgX 
(RX/3O)×(Ag/RX) = Ag/3O  The determination of the six ratios AgX: RX, and the calculation of additional values for the atomic weight of silver from the expression
(RX/3O)×(AgX/RX)×(Ag/AgX)=Ag/3O
From the results obtained in (2) and (3) for the atomic weight of silver, a mean value is derived. Employing this figure, the molecular weights of the silver halides are calculated from the ratios Ag: AgX, compared with those obtained directly in (1), and mean values derived. Subtraction of the atomic weight of silver from these mean values gives the atomic weights of the halogens. Having thus ascertained the atomic weights of silver and the halogens, the molecular weights of the alkalimetal halides are calculated from the ratios Ag: RX, and also from AgX: RX, the values compared with those obtained directly in (1), and mean values derived. Subtraction of the atomic weights of the halogens then gives the atomic weights of the alkalimetals. In practice all the ratios indicated were not determined, and the experimental work of the earlier investigators lacked some of the precision characteristic of more modern researches. The necessary calculations are complicated by the fact that some ratios appear to have been determined with more accuracy than others. If this discrepancy is due to a greater degree of concordance having been attained in successive experiments upon the ratio, suitable allowance may be made for it if the " probable errors " of all the series of experiments are calculated. If n represents the number of observations or experiments in a series, and S the sum of the squares of the variations of the individual results from the arithmetic mean, the probable error e is given by the formula
e=0.6745×S/(n(n1))
Assuming two experimenters to have measured the same ratio, their results being R_{x} and R_{2}, with probable errors of a and b respectively; then, instead of adopting the arithmetic mean (R_{1}+R_{2})/2, it is better to employ the " weighted mean," the results being " weighted " inversely as the squares of their probable errors. The weighted mean is then
Thus, if b is three times as large as a, in taking the mean the importance attached to is ninefold that laid on R_{2}.
In this manner the initial ratios and their probable errors can be determined. In the calculations made from them, each result obtained by such arithmetical processes as multiplication and division has its probable error computed by known methods. Thus, when all the values for the atomic weight of silver have been calculated, each has its probable error attached; and the weighted mean of the values is readily ascertained, and also its probable error. These data are then utilized in computing the atomic weights of the halogens and of the other elements. The method of calculation described was employed by Clarke. Like all methods of computation, it can make no allowance for chemical errors in the determination of the ratios, such as the presence of traces of impurities in the substances employed. To allow for these errors, it is necessary to examine critically the details of each determination, and form a judgement of its chemical merit. Since such a procedure cannot be made quantitative, critics must necessarily vary in their conclusions. The problem of inducing the most probable set of values for the fundamental atomic weights from the available experimental data is obviously difficult. References to the divergent methods adopted by various computers are appended.
The earlier determinations of the ratios involving the atomic weight of sodium commence with the work of Penny in 1839. Four analyses of sodium chlorate gave as the mean result
NaClO_{3}:3O=100:45.0705 ±0.0029.
Several early series of analyses of sodium chloride are available, in which the silver necessary to combine with the chlorine in a known weight of sodium chloride was determined.
In Stas's second research allowance was made for the solubility of silver chloride in water, and for the presence of a trace of silica in the salt. Clarke has calculated the weighted mean of these four series of experiments to be
Ag: NaCl=100: 54.2071 ±0.00018.
The corresponding ratio for sodium bromide was also determined by Stas to be
Ag: NaBr=100: 95.4405.
Two series of determinations of the ratio
AgCl: NaCl =100: x
should be mentioned. The first was carried out by Berzelius in 1811, and gave x=40.885; the second was made incidentally by Ramsay and Miss Aston in their work on the atomic weight of boron, and gave x=40.867. Both these results are much too high.
The values obtained by Stas for the atomic weights of silver, chlorine, and bromine were 107.930, 35.457, and 79.952, and were employed for many years. By their aid the atomic weight of sodium can be calculated from the results already cited. From Penny's experiments (NaClO_{3}: 3O) it follows that NaCl =58.500. From the mean ratio Ag: NaCl it can be calculated that NaCl =58.506. The average result is NaCl =58.503. From the ratio Ag: NaBr it follows that NaBr = 103.009. Hence, two values for the atomic weight of sodium can be induced
Na =NaCl  Cl =58.503 35.457 =23.046; and Na =NaBr Br=103.009 79.952 =23.057.
The mean value Na=23.05 derived from the early work was adopted for many years. It was proved to be too high by Richards and Wells, whose accurate analyses of sodium chloride, computed with Ag=107.93, give Na=23.008.
Richards and his coadjutors have proved one of the fundamental errors in the work of Stas, for years regarded as a model of accuracy, to have been the employment of excessive quantities of substances. His object was to avoid errors in weighing, but his method necessitated a concentration of the solutions such as induced occlusion of extraneous material in his precipitates. This experimental defect and other sources of error have been avoided by the American school, with Richards as its leader, and a new era in the field of atomicweight determination has been initiated.
In their analyses of sodium chloride in 1905, Richards and Wells purified their materials with extreme care, an example being their application of the centrifuge to the removal of the motherliquor from sodium chloride after crystallization. Their work involved the determination of the two ratios AgCl: NaCl and Ag: NaCl. For the experimental details of this classical investigation the original paper should be consulted. In two series of ten experiments each the results were:
AgCl: NaCl=100: 40.7797, whence Na =22.995; and Ag: NaCl=100: 54.185, whence Na =22.998.
The calculation is based upon the value 107.880 for the atomic weight of silver, and 36.457 for that of chlorine, these being the modern numbers obtained by Richards and his coadjutors with O=16 as primary standard.
Two analyses of sodium bromide have been made by Richards and Wells. Their mean is
AgBr: NaBr=100: 54.8010, whence Na =22.998.
In the calculation, the modern value for the atomic weight of bromine, 79.916, has been employed. The result calculated with Stas's values for silver and bromine is Na =23.008, a number much lower than that indicated by the work of Stas himself. This anomaly led Richards and Wells to their research on sodium chloride, and thus caused a revision of the atomic weights of sodium and chlorine.
In 1911 Goldbaum employed a new method in the investigation of the atomic weight of sodium. Aqueous solutions of weighed quantities of sodium chloride and sodium bromide were electrolyzed, using a mercury cathode and a silver anode. The increase in the weight of the anode gave the amount of halogen present. Assuming Cl =35.457 and Br =79.916, the results are:
11 experiments. Cl: NaCl=100: 164.858, whence Na =22.997.
8 experiments. Br: NaBr=100: 128.776, whence Na =22.997.
In the work of Richards and Hoover in 1915, pure sodium carbonate was exactly neutralized with a solution of hydrobromic acid standardized against pure silver. They found that
Na_{2}CO_{3}: 2Ag=29.4350: 59.9168.
Hence, if Ag=107.880, the molecular weight of sodium carbonate is 105.995; so that 2Na+C =57.995. A review of the atomic weight of carbon indicates that the value is probably between 12.000 and 12.005. It follows that the atomic weight of sodium must lie between Na =22.995 and Na =22.998.
The modern work of Richards and Wells, Goldbaum, and Richards and Hoover indicates a value between 22.995 and 22.998 for the atomic weight of sodium. In the account of his work in 1915, Richards gave the preference to the lower number. In this series of textbooks, the value Na=22.996 has been adopted for the computation of other atomic weights. The current table of the International Committee on Atomic Weights gives
Na =23.00. 


