to a less number. If only two or three determinations have been made, it will be necessary to make additional determinations until the total number is four; if more than four determinations have been made, the test is applicable at once. To apply the test, omit the doubtful observation and compute the arithmetical mean and the average deviation of the other observations. Compute also the difference between the doubtful observation and the arithmetical mean. If this difference is four times or more the average deviation of the other observations, reject the observation, since it can be shown from the theory of probability that the chances are 993 in 1000 that such an observation is the result of a mistake.11 58. Examples. 1. The Committee on Uniformity in Technical Analysis, J. A. C. S. 26, 1648 (1904), show in their report that a sample of oxidized ore from New Jersey, containing franklinite, willemite and zinc spinels and having a zinc content of 18.16% was analyzed by forty-two chemists with results varying from 12.20 to 39.22%. Twenty-three of the chemists were or had been in zinc works, three in other works where zinc is frequently determined, eleven were commercial chemists most of whom make a specialty of zinc, and five were professors or instructors in colleges. There were eight methods used. Analyst 33 found 12.20, 12.73 and 13.74% Zn. What was his average deviation and what his constant error? Ans. Average deviation 44 parts per 1000 Since each analyst must have been satisfied with the concordance of his own results, it is evident from the above report that the concordance of a series of determinations made under similar conditions is in itself no criterion of the absence of a constant error, and often this constant error may be very large in amount. 2. In determining the amount of phosphorus in a sample of disodium phosphate, having the formula Na2HPO4·2H2O the following percentages, calculated as PO4, were obtained, 52.58, 52.86, 52.52, 52.83, 52.47, 52.71, 52.43, 52.79, 52.32, 52.64, 52.39. What is the average deviation and what is the constant error? Should any of the determinations be rejected? Ans. Average deviation 3.1 parts per 1000 11 J. W. Mellor, Higher Mathematics for Students of Chemistry and Physics. Longmans, Green & Co., New York, 1909, p. 533. 3. Richards and Hoover in their article: "The molecular weight of sodium sulphate and the atomic weight of sulphur," J. A. C. S. 37, 112 (1915), give the following figures for the ratio between sodium carbonate and sodium sulphate: What is the average deviation? Should any of the results be rejected? 4. In standardizing an approx. 0.1 molar silver nitrate solution for use in Mohr's method for determining chloride, the following results were obtained by a student What was the average deviation? Should any of the observations be rejected? Ans. Average deviation 1.2 parts 1000 No observation should be rejected 5. In the Eschka method for sulphur in coal, 1 g. of coal, 1 g. of MgO and 0.5 g. of Na2CO3 were used. The weight of BaSO4 obtained was 0.3236 g. A blank analysis on a mixture of 10 g. of the MgO and 5 g. of the Na2CO3 yielded 0.0921 g. of BaSO4. What was the percentage of sulphur in the coal? Ans. 4.30% CHAPTER V WEIGHING1 59. General Considerations. - All quantitative processes resolve themselves, in effect, into two determinations of masses: that of the sample being analyzed and that of the constituent being sought, because it is the ratio of these masses which is the object of every quantitative analysis. Now the determination of mass rests upon the fundamental principle that an object whose mass is M is attracted by the earth with a force F which is proportional to the product of the mass into the intensity of gravity g which is acting upon the mass, namely, F = Mg 1 (1) If we have two bodies of respective masses M1 and M2, then the corresponding gravitational forces F1 and F2 would be From this it follows that if the intensity of gravity is the same in both cases, as it always is, when the two bodies are in the immediate locality of each other,2 the following relationship is true or in other words the ratio of the respective gravitational forces 1 For a lot of very valuable detail, which it is felt is beyond the scope of this book, the reader is referred to the Circular of the Bureau of Standards, No. 3, Design and Test of Standards of Mass, Washington, 1918, 3rd ed. 2 The intensity of gravity varies with the latitude and altitude of a locality. The following values are quoted from Smithsonian Tables, p. 106 of reference cited in § 13: is equal to the ratio of the corresponding masses. For the special case where this ratio is unity, we have as a consequence that or in other words when the gravitational forces are equal the masses are equal. Relationship (5) is the ideal one which we seek to employ in quantitative analysis in conjunction with the equal-arm balance, and if gravity were the only force with which we had to deal, then the measurement of mass would consist simply in using the equal-arm balance and equilibrating the object whose mass M is desired, with a set of reference masses arranged in suitable multiples or sub-multiples, say mı, m2... Mn ... mr, so that we could select them in convenient manner to establish equilibrium. Suppose then, under the restriction that gravity is the only force which is acting, that equilibrium has been established by using the reference masses m1 + m2 + ... m„. Let F' be the gravitational force on M, and F" the gravitational force on the reference masses; also let l' represent the length of one arm and l" the length of the other arm. By the principle of the lever arm and the fact that the system is in equilibrium we have F'VF' = Since 'l" it follows that F' = (6) F" and since F' = F" we have m; all this on the by virtue of (5) that M = m1 + m2 + assumption that gravity is the only force acting on our system. In point of practice, however, since our balance is used in air and the object and reference masses are all surrounded by air, the system is acted upon by another force in addition to that of gravity; this is the force due to the buoyant effect of the air. The force due to gravity acts downward on an object and is proportional to the mass of the object but the force due to the buoyant effect of the air acts upward on an object and is proportional to the volume of the object.3 We, therefore, have when our equalarm balance is in equilibrium in air that 3 By Archimedes' principle any object completely surrounded by a fluid is buoyed up by a force equal to the weight of the volume of fluid displaced. where F', l' and F", " have the same significance as in (6); f' represents the upward force on the object due to the buoyant effect of the air, and f" represents the upward force on the reference masses due to the buoyant effect of the air. Since as before l′ = l'', we may cancel out this factor from (7) and obtain From (8) we can see that F'' can be equal to F" and consequently M equal to mi + m2 + . . . m, under just one condition, namely, that f' =f", which is the same thing as saying that the volume of the object must be equal to the joint volume of the reference masses.4 Equation (8) may be put in another form which will show more clearly the magnitude of the error which is introduced by the buoyant effect of the air. Let d' represent the density of the object, d" the density of the reference masses and s the density of air, all in g./cm.3 Then the volume of the object will be m1 + m + while that of the reference masses will be Whence making the following substitutions in (8), namely, M d' d" F' = Mg; f' F" In practice the following values of d', d", and s, apply: 4 Since when f'f", it also means that the mass of the object is equal to the sum of the reference masses, we can also state that when f' = ƒ" it is the same thing as saying that the density of the object is equal to the density of the reference masses. |